Apr 15, 2016 - 7.3.2 Geometry and fractional conversion of a composite sphere. 155. 7.3.3 Relative .... particle size distribution (PSD) of cement, and curing temperature. Expansion is ...... CmS with medium particle size cured at 250C. 3 5C.
Expansion and Shrinkage of Early Age Cementitious Materials Under Saturated Conditions: The Role of Colloidal Eigenstresses A TsINSTITUTE MASSkCHU Of TECHNOLOgY
by
Muhannad Abuhaikal
JUN 0 7 2016
B.S., Birzeit University (2007) S.M., Massachusetts Institute of Technology (2011)
LIBRARIES
Submitted to the Department of Civil and Environmental Engineering in partial fulfillment of the requirements for the degree of Ph.D. in Civil Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2016 Massachusetts Institute of Technology 2016. All rights reserved.
I Signature redacted
Author ...... D
.
Certified by.
rtment of Cvil and Environmental Engineering April 15, 2016
Signature redacted
21,
Franz-Josef Ulh4"f Professor of Civil and Environmental Enginee i' Thesip Supervisor Accepted by ..
Signature redacted (
Donald and Martha Harleman Professor of Ci vil and
frieidi M. Nepf nvironmental Engineering
2
Expansion and Shrinkage of Early Age Cementitious Materials Under Saturated Conditions: The Role of Colloidal Eigenstresses by Muhannad Abuhaikal Submitted to the Department of Civil and Environmental Engineering
on April 15, 2016, in partial fulfillment of the requirements for the degree of
Ph.D. in Civil Engineering
Abstract Mixing water with anhydrous cement powder and other additives results in a viscid cement slurry and triggers a set of complex exothermic reactions. As the cement slurry transitions from a suspension to a gel and ultimately to a stone-like porous solid, the material develops mechanical properties. This transition, however, is also accompanied by bulk volume changes, which -if restrained- lead to premature cracking of materials and structures. The main objective of this study is to relate bulk volume changes as measured at a macroscopic scale to their finer colloidal origin under controlled temperature and pressure conditions. To achieve this goal, an original set of macroscopic scale experiments is designed and a multiscale microporomechanics model is employed to rationalize the experimental results. While bulk volume changes have been classically attributed to capillary pressure, surface tension, and disjoining pressure that all relate to changes in relative humidity, we herein argue that they are a consequence of eigenstresses that develop in the solid phase of the hydrating matter due to attractive and repulsive colloidal forces at mesoscale. To prove our hypothesis, we experimentally investigate volume changes under saturated and drained conditions that eliminate any volume changes associated with humidity and (effective) pressure changes. Under these conditions, we observe first a volume expansion followed at later stages of hydration by volume shrinkage that cannot be explained by classical theories. By analyzing both expansion and shrinkage within the framework of incremental micro-poro-mechanics, we suggest that the expansion is caused by the relative volume change between the reactive solids and hydration products in the hydration reaction. After the solid percolates, this volume change is restrained by the percolated solid phase. This induces a compressive eigenstress in the solid phase that entails a swelling of the material under overall stress-free conditions. In return, as the material further densifies, the attractive forces between charged C-S-H grains prevail causing the whole system to shrink. These attractive (tensile) forces compete with
3
the compressive solid eigenstress development, reversing the expansion into shrinkage. By carrying out tests at different temperatures, we provide strong experimental evidence that this tensile eigenstress development is an out-of-equilibrium phenomenon that occurs close to jamming. Furthermore, the tensile eigenstresses calculated from our shrinkage measurements agree qualitatively with those from meso-scale coarsegrained simulations of C-S-H precipitation originating from the electrostatic coupling between charged C-S-H particles mediated by the electrolyte pore solution. Thesis Supervisor: Franz-Josef Ulm Title: Professor of Civil and Environmental Engineering
4
Acknowledgments I would like to express my gratitude to my advisor Prof. Franz-Josef Ulm for his help and guidance throughout my masters and PhD studies and especially, I would like to thank him for his patience.
I would also like to thank Dr. John Germaine
for the great help in designing the main experimental setup in this study, Benjamin Druecke, James Haug, Ivan Prestini, Tom Bell, Konrad Krakowiak, and Miranda Amarante for the great help in the lab. I am also thankful to Enrico Masoero, Jeffrey Thomas, Roland Pellenq, Elizabeth Dussan, and Henri Van Damme for the insightful discussions. My thoughts go to my friends and colleagues who all made this a more enjoyable experience and special thanks to Donna Hudson. I gratefully acknowledge that this research was made possible by the funding from Schlumberger as part of the collaboration on the X-Cem project with SchlumbergerDoll Research Center. The greatest thanks go to my parents and my siblings.
5
6
Contents I
Overview
12
1
Introduction
13
2
II 3
1.1
Industrial context . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.2
Research question . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.3
M ethodology
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.4
Outline of thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17 19
Literature Review 2.1
Measurements of bulk volume changes
. . . . . . . . . . . . . . . . .
19
2.2
Driving forces and models of bulk volume changes . . . . . . . . . . .
21
2.2.1
Capillary pressure . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2.2
Disjoining/Joining pressure
. . . . . . . . . . . . . . . . . . .
30
2.2.3
Surface energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.2.4
Crystallization pressure . . . . . . . . . . . . . . . . . . . . . .
32
2.3
Mitigation strategies of bulk volume changes . . . . . . . . . . . . . .
33
2.4
Colloidal forces in C-S-H gel . . . . . . . . . . . . . . . . . . . . . . .
35
2.5
Sum mary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
38
Materials and Methods
39
Materials 3.1
Light-burn magnesium oxide . . . . . . . . . . . . . . . . . . . . . . .
39
3.2
C 2 S and C 3 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
7
40
3.4
Sum mary
46
.
Class G cem ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental Setup
4.3
. . . . . . .. . . . . . . . . . . . .
. . . .
49
4.1.2
Syringe pumps
. . . .
50
4.1.3
Bladder accumulator
. . . . . . . . . . . . . . . . . .
. . . .
52
4.1.4
Flow meters . . . . . . . . . . . . . . . . . . . . . . .
. . . .
52
4.1.5
Check valve . . . . . . . . . . . . . . . . . . . . . . .
. . . .
53
4.1.6
F ilter . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
54
4.1.7
Water bath
. . . . . . . . . . . . . . . . . . . . . . .
. . . .
56
Configurations . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
56
.
Pressure cell.. . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
. . . . . . . . . . . . . . . . . . . . .
(PVC)
4.2.2
Rubber Diaphragm Configuration (RDC) . . . . . . .
. . . .
58
4.2.3
Liquid Barrier Configuration (LBC) . . . . . . . . . .
. . . .
58
. . . .
60
.
Permeability and Volume of the reaction Configuration
Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
Microporomechanics framework . . . . . . . . . . . . . . . . . . . .
5.1.2
Useful relations and tools
.
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
.
Poroviscoelasticity
The Permeability and Volume of the reaction Configuration (PVC)
.
5.1.1
Stress and displacement boundary conditions . . . . . . . . .
5.2.2
Volume of the reaction Vrxn (chemical shrinkage)
5.2.3
PVC test with bottom surface connected to the pressure trans-
.
5.2.1
. . . . . .
.
5.2
.
ducer only . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
.
The Rubber Diaphragm Configuration (RDC) . . . . . . . . . . . . Boundary conditions . . . . . . . . . . . . . . . . . . . . . .
5.3.2
Field equations . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
5.3.1
The Liquid Barrier Configuration (LBC)
8
. . . . . . . . . . . . . . .
.
5.3
58
4.2.1
5 Analytical Framework and Boundary Value Problems 5.1
49
4.1.1
.
4.2
General Layout
.
4.1
47
.
4
3.3
5.5
III 6
.125
5.4.1
Boundary conditions . . . . . . . . . . . . . . . . . . . . . . .
96
5.4.2
Wetting, interfacial tension and viscous forces . . . . . . . . .
97
5.4.3
State of stress and strain in the liquid barrier test . . . . . . .
100
Sum mary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results
110
Results
111
6.1
Magnesium Oxide.. .........
6.2
M onoclinic C3 S . . . . ..
6.3
f-C 2 S
6.4
Class G cement ...
IV
106
. ... .. .. ... . . 111
.. ......
. . . . . . . . . . . ..
. . . ..
..
.
119
.. .
128
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ..
.. .
......
. . . ..........
6.4.1
Hydration kinetics
. . . . . . . . . . . . . . . . . . . . . . . .
128
6.4.2
Bulk volume changes . . . . . . . . . . . . . . . . . . . . . . .
129
140
Discussion
7 Expansion During the Hydration of Magnesium Oxide
141
7.1
Hydration kinetics of MgO . . . . . . . . . . . . . . . . . . . . . . . .
142
7.2
Microstructure of MgO and Mg(OH) 2 . . . . . . . . . . . . . . . . . .
143
7.3
Volume fractions and relative volume changes
. . . . . . . . . . . . .
152
7.4
7.3.1
Evolution of volume fractions during the hydration of MgO
153
7.3.2
Geometry and fractional conversion of a composite sphere
155
7.3.3
Relative volume change in the porous particles . . . . . . . . .
157
Bulk volume changes and multiscale microporomechanics for the hydration of M gO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159
7.4.1
Multiscale Thought Model of MgO hydration
. . . . . . . . .
159
7.4.2
LEVEL I: The composite sphere . . . . . . . . . . . . . . . . .
160
7.4.3
LEVEL II: The porous MgO-Mg(OH)
. . . . . . . .
167
7.4.4
LEVEL III: The continuum scale . . . . . . . . . . . . . . . .
172
9
2
particle
7.6
. . . . 176
.
.
Hydrostatic state of stress (RD test)
. .
. . . . 176
7.5.2
Oedometric state of stress (LB test) . . .
. . . . 178
.
.
7.5.1
Sum m ary
. . . . 180
. . . . . . . . . . . . . . . . . . . . .
181
8.1
Analysis of kinetics of C3 S hydration . . . . . . . . . . . . . . . . .
182
8.2
Microstructure of C 3 S
. . . . . . . . . . . . . . . . . . . . . . . . .
185
8.2.1
SE M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
187
8.2.2
Sorption isotherms . . . . . . . . . . . . . . . . . . . . . . .
187
8.2.3
Volume fractions
191
.
. .
. . . . . . . . . . . . . . . . . . . . . . . .
Eigenstress development during C 3 S hydration through a multiscale thought-m odel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
201
8.3.1
LEVEL I (The C-S-H gel) . . . . . . . . . . . . . . . . . . .
202
8.3.2
LEVEL II (C-S-H gel with capillary porosity)
. . . . . . . .
205
8.3.3
LEVEL III. Reinforcement by Rigid (but Slippery) Inclusions
206
8.3.4
Eigenstress development . . . . . . . . . . . . . . . . . . . .
210
Speculation about the origin of Eigenstresses . . . . . . . . . . . . .
215
8.4.1
Powers and Brownyard Model . . . . . . . . . . . . . . . . .
216
8.4.2
Densification of C-S-H . . . . . . . . . . . . . . . . . . . . .
219
8.4.3
Meso-scale simulations . . . . . . . . . . . . . . . . . . . . .
223
. . . . .
.
Expansion of
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
227 229
O-C 2 S
9.1
Kinetics of C 2 S hydration
U.2
I!VoLUtU"- I E
9.3
Eigenstress development during the hydration of C 2 S
9.4
Sum m ary
.
. . . . . . . . . . . . . . . . .
. . . . 229
.
. . . . 233
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 233
VoULUcklln
ofvlmefatin
AUIng-L
the h1ydratlin
Uf C-20 .
.
9
Sum m ary
.
8.5
.
8.4
.
.
8.3
.
Eigenstress Development During C3 S Hydration
.
8
Comparison with experimental measurements
.
7.5
10 Development of Eigenstresses During the Hydration of Class-G Cement
235 10
10.1 Evolution of volume fractions during the hydration of Class-G cement
236
10.2 Bulk volume changes and multiscale microporomechanics model for the hydration of Class-G cement . . . . . . . . . . . . . . . . . . . . . . .
247
10.2.1 Bulk volume changes in the Rubber Diaphragm Configuration
251
10.2.2 Bulk volume changes in the Liquid Barrier Configuration . . .
254
10.3 Sum m ary
V
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions and Perspectives
263
264
11 Conclusions and Perspectives
265
11.1 Summary of main findings . . . . . . . . . . . . . . . . . . . . . . . .
265
11.2 Research contribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
268
11.2.1 New experimental apparatus . . . . . . . . . . . . . . . . . . .
268
11.2.2 A multiscale microporomechanics model
. . . . . . . . . . . .
268
11.2.3 Driving forces of bulk volume changes in CBM . . . . . . . . .
269
11.3 Industrial benefits and impact . . . . . . . . . . . . . . . . . . . . . .
269
11.4 Limitations and Perspectives . . . . . . . . . . . . . . . . . . . . . . .
270
A Evolution of volume fractions during hydration reactions
291
A.1
Hydration kinetics and mechanisms . . . . . . . . . . . . . . . . . . .
291
A.2
Evolution of volume fractions
297
. . . . . . . . . . . . . . . . . . . . . .
A.2.1
Minimum water content and initial porosity for complete reaction304
A.2.2
Porosity of the individual solid phases
. . . . . . . . . . . . . 305
B Elementary concepts and tools for micro-poro-mechanics modelling309 B.1
Representative volume and scale separability . . . . . . . . . . . . . .
309
B.2
Stress and strain field averages . . . . . . . . . . . . . . . . . . . . . .
310
B.3
Statistical indentation
. . . . . . . . . . . . . . . . . . . . . . . . . .
311
11
Part I Overview
12
Chapter 1 Introduction 1.1
Industrial context
The main objectives of the primary cementing of an oil well is to support the casing string and provide zonal isolation by filling and sealing the annular gap between the casing string and the borehole [136]. Lack of understanding of the state of stress in the cement sheath and how the sheath may fail can lead to loss of zonal isolation. Among all the possible reasons that can lead to failure of cement sheath, bulk volume changes that cement-based materials (CBM) experience during hydration in a constrained environment is the focus of this study.
When cement slurry develops its mechanical properties, it also undergoes volume changes during the early age of hydration. If constrained, these volume changes will lead to the development of stresses in the hydrating cement. To predict failure due to loading during the service life of the well, one needs to understand initial stresses endured by these volume changes.
Prediction of stress and strain development in the cement sheath at early ages is critical for performance evaluation.
Yet, such a prediction calls for a profound
understanding of the chemical and physical mechanisms at early ages in the cement sheath.
13
This study will involve the development of an experimental program designed for the measurement of volume changes, and eigenstresses of hydrating cement pastes and slurries under controlled pressure and temperature conditions. Appropriate mechanics and kinetics models will also be developed for the interpretation of the experimental data. The models will eventually be used for the development of an engineering tool necessary for the prediction of the cement sheath behavior.
1.2
Research question
Can the driving forces found in cement literature explain the bulk volume changes observed in saturated CBM? If not, then what are the driving forces of bulk volume changes in CBM during hydration under saturated conditions? More specifically, can the growth and densification of C-S-H explain the observed bulk volume changes in saturated CBM?
Bulk volume changes of Cement-Based Materials (CBM) under autogenous (sealed) conditions and drying (through evaporation) conditions have been investigated extensively in the cement literature. Driving forces of such bulk volume changes have been attributed to capillary tension, disjoining pressure, crystallization pressure, and surface energy of the solids as detailed in later sections.
Yet, these driving forces
cannot explain the bulk volume changes observed under saturated conditions during the hydration of the CBM. In this study, we investigate the bulk volume changes of saturated CBM during hydration in an attempt to understand and quantify the driving forces of bulk volume changes.
The driving forces generally accepted in the literature for autogenous and drying shrinkage cannot explain the bulk volume changes of saturated CBM. For this purpose, alternative driving forces thought to control the bulk volume changes are considered in this study. The first driving force is related to changes in colloidal forces
14
leading to contraction of the Low Density C-S-H (LD-CSH) whether it was due to the progression of the chemical reaction or due to densification of the LD-CSH that originally existed in an out-of-equilibrium state. This kind of force has never been explored before but the recently published work of Ulm et al [195] shows that colloidal particles precipitating inside the porous C-S-H gel, can lead to volume changes. The second driving force is due to the formation of High Density C-S-H (HD-CSH) which is believed to form through a topochemical reaction leading to expansion of the material. Expansion can also be a consequence of the volume difference between the dissolving solids and the hydration products where the volume of the solids can double at full hydration. The formation of the LD-CSH and the subsequent densification depends primarily on the reaction mechanism, rate of the reaction, presence of nucleation sites (seeds), particle size distribution (PSD) of cement, and curing temperature.
Expansion is
attributed to the volume difference between the reactive solids (anhydrous cement) and the hydration products and leads to the observed expansion.
1.3
Methodology
This section details the approach to study the origin of bulk volume changes including a general description of the experimental program employed to pinpoint the driving forces of bulk volume changes. This section will also describe how to use the microporomechanics model to relate the macroscopic measurements to their microscopic origin. Advances in experimentation and multiscale microporomechanics models made it possible to establish the relationship between the macroscopic behavior of a heterogeneous material and its finer-scale origins (see, for example, [20] [153] [61]). With such models, mechanisms and intrinsic properties that dominate the behavior of the material can be incorporated at their corresponding scale, and their contribution to the macroscopic behavior can be obtained. Investigating the origin of volume changes by means of multiscale microporomechanics modeling of macroscopic scale experiments
15
will be the focus of this study. With focus on CBM, a number of reactive porous materials will also be investigated, including the hydration of magnesium oxide and pure cement phases. Relevant measurements in this study will be obtained at macroscopic scale and a multiscale microporomechanics model will be developed in an attempt to relate the macroscopic observations to their finer scale origin. Multiscale micromechanics in conjunction with understanding of the microstructure properties of the porous media are necessary to quantify the mechanical properties and bulk volume changes at the macroscopic scale. Bulk volume changes of CBM are possibly controlled by several mechanisms that operate in various parts of the microstructure and at multiple scales. The origin of these volume changes will be introduced in a multiscale micromechanics model as eigenstresses at the appropriate scale based on the understanding of the driving forces. A model for volume changes is developed by combining the contribution of the different mechanisms in the multiscale micromechanics model.
The primary experimental setup employed in this study is a new apparatus designed to measure bulk volume changes during the hydration of cement-based materials under controlled curing pressure and temperature. The apparatus is also capable of monitoring the hydration kinetics by measuring the volume of the reaction (chemical shrinkage) of saturated samples. Other standardized tests will also be employed to test the proposed mechanisms. These tests include: 1. Thermogravimetric Analysis (TGA) to monitor the progression of the hydration reaction and the chemical composition of the material. 2. Scanning Electron Microscope (SEM) to investigate the microstructure of the material. 3. Electron Probe MicroAnalysis (EPMA) to investigate the chemical composition at the microscale. 4. Sorption isotherms to investigate the microstructure. 16
5. Isothermal calorimetry to monitor the reaction kinetics.
6. Indentations to investigate the microstructure. The methodology followed to answer the research question can be summarized as follows: 1. Bulk volume changes are measured at macroscopic scale using a new experimental setup designed to measure the bulk volume changes under saturated conditions while simultaneously measuring the volume of the reaction. 2. Systematic testing of a set of materials ranging in complexity from magnesium oxide, C 3 S and C 2 S, and the Class-G cement are investigated.
The tested
parameters also included the particle size distribution, curing temperature, and the w/c ratio. 3. The microstructure of the material and the reaction kinetics are investigated using a set of standardized tests. 4. Based on the understanding of the microstructure and the evolution of material properties, a multiscale microporomechanics model is developed to interpret the experimental data.
1.4
Outline of thesis
The goal of this thesis is to study the possible contribution of the densification of CS-H and the evolution of solid volume fractions to the observed bulk volume changes. To achieve this goal, we start in chapter 2 by a review of the literature relevant to the research question including driving forces and models of bulk volume changes, mitigation strategies of bulk volume changes, and the available methods for the measurement of the bulk volume changes. The main variables manipulated in this research are presented in chapter 3 in form of a set of materials ranging in complexity from the simplest Magnesium Oxide, then introducing the more complex C 3 S and C 25
17
followed by the most complex of all, Class-G cement. Chapter 4 introduces the main experimental setup used to study bulk volume changes for the different materials and test parameters, including curing pressure and temperature, effective stress, w/c ratio, and the stress and displacement boundary conditions. Chapter 5 details the framework for the analysis of the measured bulk volume changes and the different test configuration obtained in the main experimental setup. Chapter 6 presents the obtained results from the different tests on the studied materials. Chapters 7-10 discuss the results obtained on the studied materials along with the multiscale microporomechanics model developed for each material for the analysis and interpretation of the experimental measurements. In chapter 11, we draw the conclusions by presenting a summary of the main findings as well as a highlight of the research contribution and the possible industrial benefits, and gives suggestions for future research.
18
Chapter 2
Literature Review In this chapter, we review the measurements and interpretations of bulk volume changes in general while focusing on the relevant conditions. This review is necessary to highlight the contribution of this study and set the ground for the experimental design and data interpretation.
2.1
Measurements of bulk volume changes
Bulk volume changes have been the subject of extensive research due to their critical role in durability and performance of CBM, especially in high performance concrete. The main body of the literature focuses on autogenous shrinkage, drying shrinkage and, to a much lesser extent, the bulk volume changes under saturated conditions. Of special interest in the study of bulk volume changes are saturated and autogenous (sealed) curing conditions. Autogenous volume changes are the bulk volume changes (expansion or shrinkage) of cementitious materials during the hydration of cement paste.
Autogenous volume changes are measured under isothermal conditions for
sealed (no mass exchange) and free of stress samples. Autogenous shrinkage of CBM have been measured both by volumetric and linear methods. Although designed to measure the same quantity, volumetric and linear measurements are fundamentally different due to experimental considerations related to boundary conditions. In volumetric measurements, cement slurry is placed inside a tight rubber mem19
brane and immersed in water; the change in slurry volume is then measured by dilatometric or gravimetric measurements [7,115,117,132,172]. One of the main challenges in the volumetric measurement of autogenous shrinkage is the sedimentation and the associated bleeding water. Investigation of the bleed water on the measured autogenous shrinkage has been the subject of investigation of many researchers (see, for example, [24,107,130]). While bleeding in volumetric measurements resulted in an overestimation of autogenous shrinkage at early ages [107], it could have been the reason expansion was observed in linear measurements [130]. The linear measurement is carried out by placing the cement slurry in a prismatic mold with displacement sensors on one or both ends to measure the deformations [6,7,9,24,59,84,95,108,117,126,127]. In both volumetric and linear measurements the focus of the measurement is on the bulk volume changes without pore pressure measurement (and effective stress) or monitoring the hydration kinetics. Bulk volume changes have been measured for samples with different mix designs, compositions, and curing conditions. In general, shrinkage is observed and is typically attributed to the self-desiccation resulting from the negative volume of the reaction (chemical shrinkage). In some cases, an expansion was measured at early ages during the hydration of CBM under autogenous conditions [7]. Another interesting observation is the impact of particle size distribution on the measured autogenous shrinkage. Bentz et al [17,18] studied the influence of particle size distribution on the bulk volume changes under autogenous conditions. By comparing the bulk volume changes of cements with different particle sizes at the same water-to-cement ratio, they observed significant variation in the bulk volume changes and the associated eigenstresses. These variations were then attributed to the pore size distribution and the subsequent decay in relative humidity which reduces the associated capillary stresses in the hydrating CBM. Among the main parameters that control the properties of the hydrating and hardened CBM is the initial water-to-cement mass ratio (w/c), which can also be viewed as the initial porosity. Another way to view the initial w/c is by relating it directly to the average spacing between cement particles [12]. The impact of w/c on 20
w
1400 1200
-1400-
C 1200-
.
0
800
M
-o- W/C=0.30
40
W/C=0.35 .E + W/C=0.40. -%-/C=0.45
600-
400shrn 400
.C4) 0) 200
0 1
0
01
-200
shrnka
swelling
-400
0
50
100 150 200 250 300 350 400 Age (days)
+ W/C=0.25 -o- W/C=0.30 -W/C=0.35 -e- W/C=0.40
800
-
CD
W/C=0.25
--
1000
600-
400 -- W/C=0.50 8 2000 -+- W/C=0.60 r_ i W Of CD 2 -2004 -400-
-U-W/C=0.45
-
4"
0
-o- W/C=0.50 +- W/C=0.60 I
10
I
20 30 Age (days)
40
50
b) Zoom: 0 - 50 days
a) 0 - 400 days
Figure 2-1: Linear autogenous deformations vs. age., initialized at Vicat setting time up to 1 year on c'p20 x 160-n1 sealed samples of cement pastes with variable w/c. at, T=20 C. Reproduced from [9].
the bulk volume changes has been investigated extensively ill the literature of cement and, ill general, the lower the w/c the higher is the autogenous shrinkage [9, 18]. A relevant observation related to the effect of w/c on the bulk voluimne changes
can be found in the work of Baroghel-Bounv et al [9]. where cenenlt pastes
with
high w/c experienced expansion up to 20 days followed by shrinkage (see figure 21). Shrinkage in low w/c materials is usually attributed to the smaller porosity and smaller separation distance between cement particles as a result of capillary tension. In the light of the previously discussed densification of C-S-H. one can argue that capillary tension imight be a consequence of the ongoing chemical reaction, but it is the densification of C-S-H that is controlling the overall behavior of the material. Densification of C-S-H and bulk volume changes seem to follow similar trends, yet. only the densification of C-S-H
can explain the expansion anld shrinkage of saturated
naterials as will be shown in the main results of this study.
2.2
Driving forces and models of bulk volume changes
This section reviews the drivilig forces of bulk vohune changes in mature ald hydrating CBM. This section will also help determiie the lmeasllralble quantities required to build the microporomechanics model. The driving forces of bulk volume changes will be listed ill the following subsections
21
with the corresponding models and experimental measurements. It is important to build some understanding of the driving forces of bulk volume changes in order to be able to isolate and elucidate the contribution of each force. In this section, we review the driving forces found in the literature with focus on the controlling parameters of each force. The fact that hydration of CBM is accompanied by bulk volume changes indicates the development of eigenstresses inside the material. These eigenstresses can originate at multiple scales depending on the mechanism that is causing them.
A
number of volume-change-inducing mechanisms can be found in the cement literature, among which, capillary tension predominates as the most accepted mechanism (see for example [94]). In addition to capillary tension, eigenstresses in hydrating or drying cement paste are generally attributed to stresses like changes in surface tension of colloidal particles [209], or changes in disjoining pressure [10]. Eigenstresses can also result from temperature changes, crystallization pressure [206], and possibly the densification and hardening of hydration products [195]. Whether it is one or a combination of these mechanisms that dominate the bulk volume changes in cement, it has been established experimentally that changes in relative humidity is necessary for significant bulk volume changes to occur in drying specimens. Similar to drying shrinkage the driving force of autogenous shrinkage is usually attributed to capillary tension resulting from drop in relative humidity. While in drying shrinkage drop in relative humidity is due to loss of water through evaporation, under autogenous test conditions it is attributed to consumption of water in the chemical reaction [94]. It should be noted here that bulk volume change due to change in relative humidity does not necessarily imply that capillary tension is the driving force due to the fact that drying is associated with various physical and chemical changes in CBM. A bulk volume change of a porous material is a manifestation of the interplay between a number of possible driving forces. Significant bulk volume changes of CBM can be induced experimentally by desaturation, solvent exchange, and temperature variation. There exists ample experimental evidence that variation of degree of saturation (degree of saturation, Sw, is the volume fraction of the pore space occupied by
22
the pore solution) is one of the most influential means to induce bulk volume changes. Irrespective of the driving forces, which will be discussed later on, the change in degree of saturation is associated with certain changes in the pore space and the pore solution.
These changes include the formation of gas bubbles and changes in the
chemical and physical properties of the pore solution. Desaturation is the loss of water from saturated porous solids, whether it was due to the chemical reactions (Self-desiccation [158]) or through evaporation (drying). It is also necessary here to understand the difference between self-desiccation and drying. Drying involves the evaporation of all volatile phases from the pore solution while keeping the amount of non-volatile solute constant. Drying is also associated with the development of stress/strain gradients and the dimensions of the test specimen are critical for the analysis of such phenomenon. That is to say, homogeneous test conditions for bulk volume changes are virtually impossible to achieve in a drying test, nonetheless, reasonably-thin specimens can be employed to create near homogeneous conditions. Self-desiccation in sealed specimens on the other hand, involves the loss of only water to the chemical reaction and it is possible that this kind of test is the only test to provide homogeneous conditions. Pore solution is always saturated with all ionic species that cement can provide (see for example [167]). Calcium ions are released continuously in the pore solution from the dissolution of gypsum, free lime, and all the major cement phases. The concentration of calcium ions is maintained slightly above the portlandite saturation point [182]. Alkali ions (Na+,K+) are mainly generated from the dissolution of alkali sulfates in the pore solution [182] and the major part of the alkalis are introduced into the pore solution at very early ages during the mixing process [180,188]. The very high solubility limit of alkali products ensures that the pore solution is always under-saturated with alkali ions [150]. Precipitation of hydration products during the hydration reaction is driven by the dissolution of cement. The chemical composition of the pore solution during the reaction depends on the composition of cement, and it can change during the reaction. Another way to trigger precipitation of hydration products is the loss of solvent by 23
self-desiccation or drying which can increase the supersaturation with respect to a number of hydration products. The precipitation of hydration products due to loss of solvent can change the chemical composition of the pore solution especially in the presence of alkalis. The change in chemical composition of the pore solution, in general, will have an impact on the activity of water (Raoult's law and relative humidity) [64, 145], the surface tension [140,175], the dielectric permittivity [203], and the viscosity of the electrolyte solution [63,93, 103]. The type of ions in the pore solution will also have an impact on the colloidal forces between charged surfaces by changing the counter ions and consequently changing the electrostatic coupling [154]. pH of the solution is also expected to change if Ca(OH) 2 is to precipitate and consequently changing the surface charge density of the colloids. Chemical composition of the pore solution can also change in saturated material due to the decrease in the porosity, where depending on the w/c ratio, porosity can decrease significantly. The decrease in porosity can also lead to the increase in the concentration of alkali ions in the pore solution and change its chemical composition. As part of this study, an evaporative analysis is conducted to illustrate the impact of alkali ions on the properties of the pore solution due to evaporation. To investigate the impact of evaporation, half a liter of pore solution is extracted from Class-G cement mixed continuously for 24 hours in a sealed container with w/c = 1.0. The pore solution is extracted in a MILLIPORE Stericup with 0.22 pm filter under vacuum. A sample of the pore solution is then collected, and the remainder of the solution is placed in an evaporation dish inside a desiccator under vacuum with silica gel beads. Vacuum is used to prevent interaction with carbon dioxide from the atmosphere and the silica gel is used for evaporation through capillary condensation. Samples of the pore solution are then collected at different evaporation stages. Figure 2-2 shows the color change during evaporation and the development of pH resulting from the presence of alkali ions and the common ion effect. 24
-
13.4 13.3
13.1 13
-
-M G"12.912.8 0
20
40
60
80
100
Water loss (%) (a)
(b) Figure 2-2: Evaporative behavior of pore solution of Class-G cement. (a) a picture of the collected samples of the pore solution showing the color change due to evaporation.
(b) development of pH during evaporation due to the common ion effect. Water loss is the percentage of water evaporated from the original pore solution.
2.2.1
Capillary pressure
Drop in relative humidity due to moisture exchange with the environment or self-
desiccation will lead to the formation of a gas-liquid interface (gas bubbles). which if present in narrow pores, will result in pressure drop due to the formation of menisci
[65]. Drop in pore pressure resulting from the formation of liquid-vapor interface can stress the solid skeleton and cause the material to shrink [94]. This section is a review of the capillary pressure as a possible driving force of bulk volume changes in hydrating and mature cement-based materials.
We start with a general discussion about, the
possible causes that can lead to the formation of liquid/gas interface followed by a discussion of the Kelvin-Laplace equation and an estiiate of the bulk volume changes induced by capillary pressure. Pure liquids can be driven into a imetastable state either by superheating above the boiling point T (boiling) or by stretching below its saturated -vapor pressure p,t (cavitation).
The liquid will eventually return to equilibrium by nucleation of
vapor hu lles [39].
If a vapor Nibble nucleus is formed ii a pure liquid. it tends
to grow or collapse depending on whether the radius of the mewlv formed mucleus is
25
I
larger or smaller than the critical radius r, [76]. The nucleation of the vapor bubble is associated with energy gain proportional to the bubble volume. However, a minimum amount of work is required to form the bubble in a metastable liquid
E(r) =
4
3
7rr3(p - psat) + 47r 2'7
(2.1)
where -y is the liquid-vapor surface tension, p is the pressure in the liquid, and r is the radius of the bubble. This competition results in an energy barrier
Eb=
167733 l~v 3 (p - psat)
(2.2)
2
resulting in a critical radius 27y
_
2 -y
)
(2 .3
(Psat - P)
and a nucleation rate (how long one should wait for the nucleation to occur) [89,152]
F = Fo exp
Fo exp
Eb=
kBT
3kBT
(2.4)
where kB is Boltzmann's constant and T is the absolute temperature. The prefactor FO represents the number of nucleation sites and the frequency for nucleation [89]. Cavitation pressures of tens of MPa's were reported for pure water [88,215], but pore solution in CBM is not a pure liquid.
In addition to boiling and cavitation, if a gas or volatile vapor is dissolved in the liquid, then changing the conditions to create a supersaturated solution can lead to the formation of gas bubbles [116].
Supersaturation can be achieved in three
ways: First, decreasing the solubility of the gas in the liquid by decompressing the solution which has been saturated at a higher pressure or increasing (typically) the temperature of the saturated solution [31]. Second, supersaturated solutions can be prepared by reducing the amount of liquid (solvent) in the saturated solution while keeping the amount of dissolved gas constant (see for example the formation air bubbles in ice [38,118,173] or crystals [207]). Third, gas can be generated chemically
26
or electrochemically in a solution [31, 147]. The saturation concentration ceq of a gas in a liquid is determined by the partial pressure Ph of the gas above the solution and is described by Henry's law:
(2.5)
Ceq = rKhPh
where Kh is Henry's constant. The presence of dissolved gas in water strongly affects the initiation of nucleation in the solution. In general, the higher the gas concentration, the lower the energy barrier for nucleation, the smaller is the critical radius, and the higher the nucleation rate [204]. The gas bubble nucleation rate in the presence of dissolved gas is estimated in the same way as for the pure liquid, i.e. Eq. (2.4) with the difference in the critical radius, where the critical radius becomes [191,204]: 2 'y
rc =-
(2.6)
where c is the concentration of the gas in the liquid, p is the pressure in the liquid, and ,= exp(P-
sat
)
(2.7)
It is evident that the critical radius is smaller than in pure liquids and nucleation will therefore be easier.
Dissolved gases can also act as surfactants at the solution-gas
interface and consequently, reducing the energy barrier [116]. Ionic strength [34,54],
impurities [3, 54, 216], microbubbles [26, 105,119], and solid surfaces [116, 131, 214] can all reduce the energy barrier for nucleation. Cavitation has also been reported in porous media under tension [81,141] and cavitation due to drying was observed experimentally in micro capillaries [29,30]. Air is typically present in CBM as dissolved air in the mixing water. Air bubbles can be entrapped in the slurry during the mixing process [156].
Gas can also be
introduced as part of the reaction products like the release of hydrogen gas from aluminum or zinc reactions in an alkaline environment [40, 134,147, 151] and such technology is used in oilwell cementing to maintain the hydrostatic pressure and
27
counteract the negative volume of the reaction of the cement sheath slurry [86,178]. For a conservative estimation of the critical radius of air bubbles in pure water at 25'C with psat = 3000 Pa and y = 0.072 N/m under 0.5 atmospheres, the critical radius using Eq. (2.3) is approximately 3 microns. Pore sizes of larger than 3 microns are abundant in cement-based materials. In conclusion, formation of vapor bubbles in cement is feasible whether due to drying or self-desiccation.
Formation of gas bubbles in cement is possible due to drying or self-desiccation in sealed samples. Following is a discussion of the capillary pressure and the associated stresses. Capillary pressure is the jump in pressure across the interface between two immiscible fluids in a capillary:
Pc = Pnw - Pw
(2.8)
where pc is the capillary pressure, Pn, is the pressure in the non-wetting (less wetting) phase, pw is the pressure in the wetting phase (more wetting). In capillary tubes, the pressure jump across the interface depends on the interfacial tension between the fluids -y, contact angle 9, and capillary radius r.
For 0 = 0, it is reasonable to
assume that the meniscus is hemispherical with radius of curvature equal to that of the capillary.
Pc =
-
r
cos()
(2.9)
In porous media, the pressure jump across the interface depends on the interfacial tension between the fluids -y, contact angle 9, pore size distribution, and degree of saturation Sw. The pore size distribution represents the volume fraction of each of the pore sizes present in the porous medium and the degree of saturation Sw is the volume fraction of porosity occupied by the liquid phase. Depending on the pore size distribution, the loss of saturation will drive the liquid-gas interface (meniscus) into pores with progressively smaller diameters, which results in drop in relative humidity.
28
Relative humidity can be related to the curvature of the meniscus by Kelvin equation
=_ 2Vm
In (
Po
(2.10)
rRT
where p is the partial vapor pressure of wetting fluid, po is the partial vapor pressure at equilibrium, -y is the surface tension between the two immiscible fluids, Vm is the molar volume of the wetting fluid, T is the temperature, r is the radius of curvature, and R is the universal gas constant. Leading to the Laplace-kelvin equation
PC=
RT
(2.11)
-
-In
In partially saturated porous media, volume changes due to drop in capillary pressure can be attributed to changes in relative humidity, degree of saturation (pore size distribution) and the compressibility of the material.
Detailed discussion and
derivation of the asymptotic shrinkage due to capillary pressure can be found in [192] as summarized below: For an elastic porous solid, the free energy of the solid and the interfaces is a function of the strain, porosity, and degree of saturation,
T = 'I(e,
, SW)
(2.12)
where Sc, is the degree of saturation of fluid a:
Sa =S=
where
#
is the total porosity and
#
(2.13)
is the fraction of porosity occupied by fluid a.
The state equations are obtained in the form:
= S.+ SnwPnw - , #PC
(2.14)
o- -
FmOdwE=
,h
and
P
Eq(.)ed
For undeformable solid, for which 0 = Oo, the state equation and Eq. (2.8) reduce 29
to: 0P
(2.15)
dI(S)
dSw
For an undeformable solid skeleton, the energy T reduces to the interface energy qoU; and use in Eq. (2.15) yields:
Pc = Pnw - Pw=
dU dSU
(2.16)
with U the integral over the capillary pressure-saturation curve. Then, Kelvin's law (2.10) provides the link between relative humidity and capillary pressure. The mean stress in a non-saturated porous solid, reads: oUm = KE - b7r(t)
(2.17)
The asymptotic shrinkage due to drying in a stress-free material (Urn = 0), thus is: b E(t) = -ir(t) K
(2.18)
where b is Biot coefficient, K is the drained bulk modulus, and 7 is defined as: , = S.pw + Sn.pnw - U = S.pw + SnwPnw -]
P(S)dS
(2.19)
Strain due to capillary pressure requires changes in the relative humidity, which can be achieved by drying for example.
To prevent possible contribution of such
pressure in this study, samples are maintained saturated throughout the tests by providing water to the sample under controlled pressure.
2.2.2
Disjoining/Joining pressure
One of the driving forces of bulk volume changes is the disjoining pressure. Disjoining pressure is the hydration pressure built up between surfaces at nanometer separation due to the interaction between the water molecules and the hydrophilic surfaces of 30
the solid [120].
During the hydration reaction of cement a nanoporous C-S-H gel
is formed in addition to Portlandite and Calcium-Alumino-Ferrite phases.
C-S-H
gel has especially a large internal surface, which in addition to the cations adsorbed on these surfaces will give rise to attractive and repulsive colloidal forces. At high relative humidity, water adsorbed on the solid surface is at the origin of disjoining pressure [10,209]. Disjoining pressure is defined as the sum of attractive and repulsive colloidal forces including Van der Waals forces, the double layer repulsion and other colloidal forces [72] and it is activated due to loss of moisture from the gelpore space in C-S-H (changes in relative humidity).
2.2.3
Surface energy
The hydration products of cement have a very high internal surface with the main contribution being from the C-S-H. Down to very low relative humidity, large quantities of electrolytic liquid are adsorbed on the internal surface of the hydration products. Adsorption of water on a solid surface at very low relative humidity changes the solid surface energy which can create considerable hydrostatic stresses at the colloidal scale [209]. In general, adsorption of water on the solid surface will result in relaxation of the surface tension while desorption of water will increase the surface tension and compress the solid [66,94]. The change in free energy of pure adsorbent can be written as [69,162]:
AF
rRT
P2
dp
(2.20)
where n is the number of moles of adsorbate on a fixed mass of adsorbent, S is the solid surface area, R is the gas constant, T is the temperature, and p is the vapor pressure. The length strain i
due to change in surface energy can be written
as [5,66, 157, 208] --l
kAF;
k = Ps E
(2.21)
And -L 1
(ki Pc
2'Pnw - Pw = -- cos(08 ) r
(5.115)
where pt is the pressure at distance z behind the meniscus in the liquid barrier, PA is the pressure in the pore fluid in front of the meniscus, pc is the pressure jump across the interface, Pnw is the pressure on the less wetting side of the meniscus, p, is the pressure on the more wetting side of the meniscus, -y is the interfacial tension between the two fluids, r is the pore radius, and 0, is the static wetting angle at the triple point. The liquid barrier will ingress, however, at a lower pressure or even against the pressure differential if the barrier liquid is more wetting than the pore fluid (Figure 5-13(b)). In this case and since the pressure in the liquid barrier is always maintained equal or higher than the pore pressure (Pt ;> Pb), the liquid barrier will continue to ingress inside the porous solid until it replaces the pore fluid completely. Such fluid should not be used as a liquid barrier. To illustrate the principle of the liquid barrier, we consider a case where two
97
immiscible and perfectly wetting fluids flow through a capillary (Figure 5-14).
In
a capillary tube, the equilibrium contact angle 0, results from equilibrium between several forces as a direct consequence of the molecular interactions among the three materials at the contact line [58] (Figure 5-14(a)). The equilibrium contact angle 0, is different from the dynamic contact angle
0
d
when the interface between liquids is
set in motion (Figure 5-14(b)). Once the interface is set in motion, the rate of displacement of the meniscus will depend on the dynamic contact angle. A pressure differential Ap must be imposed between the pore fluid in front of the meniscus pw and the oil at a distance z behind the meniscus pt in order to produce a flow rate Q based on Hagen-Poiseuille equation
[58] [62]: Ap=
8QpLBx rr
2
-y -cos(Od) r
-
(5.116)
where PLB is the viscosity of the liquid barrier. In the problem at hand, the viscosity of liquid barrier
fluid
[pf.
ALB
is 4 orders of magnitude larger than the viscosity of the pore
Eq. (5.115) can then be modified for porous solids:
AP =
P. - Pt
PLB 1LBz - pc K
(5.117)
where q = -'Vp is the Darcy flux and K is the permeability of the porous solid. Now consider a porous solid specimen with height h supported at the bottom surface uz(z = 0)
=
0 and a free top surface (Figure 5-15). The specimen is initially
saturated with the pore fluid only (Figure 5-15(a)). Flow is allowed only through the top and bottom surfaces where the top surface is in direct contact with the liquid barrier at pressure pt. The bottom surface is in contact with a fluid that has the same properties as the pore fluid at pressure Pb. The vertical surface of the specimen is free to slide vertically (greased). Then consider a liquid barrier that has penetrated to a depth z inside a porous solid with porosity 0 in a sample with cross sectional area A (Figure 5-15(b)).
The
volume of the liquid barrier that crossed the surface of the porous solid (assuming
98
that the liquid barrier completely replaced the pore fluid) is
(5.118)
Ax
VLB =
And by definition, the flow rate QLB (measured in m 3 /s) is
QLB
QLB
dVLB
A
dt
(5.119)
Hence (5.120)
d dt
QLB =
Now using Eq. (5.117) and (5.120) with the pressure gradient given by Vp = Ap/z, and Ap = p.(t) - pt in Figure 5-15(c) where p.(t) is the pressure in the pore fluid at the wetting end of the meniscus. It is also possible to relate pw to the pressure at the bottom of the sample PA using Eq. (5.80). pt is the pressure in the liquid barrier at the top surface of the specimen. Then we have
Assuming Pc, Pt, K,
#
dx
K AP+
dt
p
(5.121)
x
are constants, we have
xdx =
(p. - pt + pe)dt
(5.122)
- Pt + PC)dt)
(5.123)
Since z (t = 0) = 0, then
x(t)
(
]
(Pw
The ingressed volume is then
VLB(t) = OAx(t) =5AQ
J
(pw - pt + pC)dt
(5.124)
In case the pressure at the top and bottom surfaces of the sample are identical in
99
a liquid barrier test, the inequality (Pc > Pt - Pw) should be satisfied for the liquid barrier to function.
5.4.3
State of stress and strain in the liquid barrier test
It is virtually impossible to obtain perfectly homogeneous fields in a CBM sample during hydration except in autogenous conditions due to the presence of a water sink. Homogeneous fields can be assumed if the permeability of the sample is high enough or the dimensions of the sample are small enough to prevent the development of a pressure gradient.
For a reactive porous sample with thickness h where the
bottom surface is exposed to water at pressure PA and the flux is restricted at the top surface, then the following relation must be satisfied to obtain homogeneous pressure distribution (Pb a pt based on Eq. (5.80)):
h2 2
Pb
[l
-
7-
800
800
600
600
400
400
200
200
0
________
I
L-
0 0
50
100 Time (hr)
0
200
150
0.2
0.4
0.6
0.8
(b)
(a)
Figure 6-25: Bulk volume changes of Class G cement samples in the rubber diaphragm test. All samples experience fast expansion up to 40 hours followed by slower expansion and shrinkage at about 100 hours. Data are initiated after consumption of the bleed water or setting of the slurry as observed by pressure variation at the bottom
surface of the sample (see Figure 6-26).
I Pc (CGC-Full-RD-W38-T25) (CGC-FuII-RD-W38-T25)
-Pc ---
Pc (CGC-FuII-RD-W38-T25)
1030 1020 1010 w w
E
1000
990 980
C
970
C
960 950 940
0
20
60 40 Time (hr)
80
100
Figure 6-26: The evolution of pressure at the bottom surface of the sample during hydration in the rubber diaphragm test, where the pressure drop is dIc to the check valve. Sample CGC-RDC-01 is tested with a 14 kPa check valve while all the other samples are tested with 60 kPa check valve.
135
I
-Ev
Ev (CGC-FuI-RD-W38-T25) (CGC-C1-RD-W31-T25)
---
Ev (CGC-Full-RD-W38-T25)
--
Ev (CGC-C1-RD-W31-T25) Ev (CGC-C2-RD-W36-T25)
2000
-
4000
-
6000
-
-
2000
-
4000
-
6000
8000
-
8000
10000
-
10000
12000
-
Ev (CGC-C2-RD-W36-T25) 12000
-
-0.
0
00
50
0.1
0
250
200
150
100
0.2
0.3
0.4
0.5
0.6
0.7
Time (hr)
(b)
(a)
Figure 6-27: Bulk volume changes of Class G cement samples in the rubber diaphragn test with coarse PSD. CGC-R.DC-05 is the reference sample with full PSD and w/c
0.38.
I -Pc
(CGC-C1-RD-W31-T25)
-Pc
(CGC-Full-RD-W38-T25) Pc (CGC-RD-C2-W36-T25-P1)
1040
1030 a) a)
E
C C
1020 1010 1000 990 980 970
960 950 940
-u0
20
40
60
80
100
Time (hr)
Figure 6-28: The evolution of )ressure at the bottom surface of the sample during hydration in the rubber diaphragm test. where the pressure drop is due to the
check
valve. Sample CGC-RDC-02 is teste(l with a 14 kPa check valve while all the other samples are tested with 60 kPa check valve. It is evident here that sample CGCRDC-02 experienced significant sedimentation and hence the delayed pressure drop. Pressure diro) at the bottom of sample CGC-RDC'-06 is innnediate due to the low w/ c.
0 136
____________
I
-
Ev (CGC-F1-RD-W38-T25)
-
Ev (CGC-F1-RD-W38-T25)
-
Ev (CGC-F2-RD-W49-T25)
-
Ev (CGC-F2-RD-W49-T25) Ev (CGC-Full-RD-W38-T25)
Ev (CGC-Full-RD-W38-T25)
3000 2500 2000 1500 1000 500 0 -500 -1000
0
50
0
250
200
150
100
-
4000 3500 3000 2500 2000 1500 1000 500 0 -500 -1000
4000 3500
0.1
0.2
0.3
E
Time (hr)
(b)
(a)
Figure 6-29: Bulk volume changes of Class G cement samples in the rubber diaphragimi test, with fine PSD. CGC-RDC-05 is the reference sample with full PSD and w/c= 0.38.
-- Pc (CGC-F2-RD-W49-T25) (CGC-F1-RD-W38-T25)
_Pc
Pc (CGC-Full-RD-W38-T25) 1040 1030
Ln E
1020 1010 1000 990 980 970 960 950 940
0
20
40
60
80
100
Time (hr)
Figure 6-30: The evolution of pressure at the bottom surface of the sample during hydration in the rubber diaphragm test. where the pressure drop is due to the check valve. Sample CGC-RDC-03 has a 14 kPa check valve while all the other samples are tested with 60 kPa check valve.
I 137
-
(CGC-LB-P1O-W42-T25-Full)
-Ev
Ev (CGC-LB-P1-W42-T25-Full)
--
Ev (CGC-LB-P1-W42-T25-Full)
-
Ev (CGC-LB-P10-W42-T25-Full)
Ev (CGC-LB-P5-W42-T25-Full)
Ev (CGC-LB-P5-W42-T25-Full) 0-
0-
-2-
-2
~1~
-4-
-
..-
__
____
-4-
-______
E -6
-6-8
-
-10
-10
-
-8
-
--_
-12 -[ 0.2
-12 100
50
0
150
200
250
0.8
0.6
0.4
Time (hr)
(b)
(a)
Figure 6-31: Bulk volume changes of Class G cement samples in the liquid barrier test with full PSD with variable curing pressure.
I -Vrxn
(CGC-LB-P1O-W42-T25-Full)
-Vrxn
(CGC-LB-P1-W42-T25-Full)
-( --
0.7 0.6
-
0.5
X
-
25
/-
20
0.4
-
________
________
____________
_________
-
30
-
-
35
0.8
-
40
(CGC-LB-P1-W42-T25-Full)
( (CGC-LB-P5-W42-T25-FuLII)
Vrxn (CGC-LB-P5-W42-T25-Full) 45
(CGC-LB-P1O-W42-T25-Full)
_
0.3
I I
___________
15
5
0.1
0
0 0
200
100
,1 0
300
50
100
150
Time (hr)
Time (hr)
(a)
(b)
Figure 6-32: Volunie of reaction and degree of hydration of Class in the liquid barrier test with variable curing pressure.
138
i _----. -
________
1~
~1
0.2
10
_____
_____
-a
200
250
G cenent saniples
MUM-w:
(CGC-LB-PO.2-W42-T25-H18.6) -Ev
Ev (CGC-LB-PO.2-W42-T25-H18.6)
Ev (CGC-LB-PO.2-W42-T25-H17.8)
Ev (CGC-LB-PO.2-W42-T25-H17.8)
-
Ev (CGC-LB-PO.2-W42-T25-H47.3)
Ev (CGC-LB-P.2-W42-T25-H47.3)
0
0
-0.5
-0.5
-1
-1
-1.5
-
-
-j
E
-1.5
- - - --
-2
-2
-2.5
-2.5
-3
-3
-3.5
-3.5
-4
-4 0
50
150
100
200
250
0.2
0
0.4
0.6
0.8
Time (hr)
(b)
(a)
Figure 6-33: Bulk volume changes of Class G cement samples in the liquid barrier test with full PSD with variable thicknesses.
-
Vrxn (CGC-LB-PO.2-W42-T25-H18.6)
--
Vrxn (CGC-LB-PO.2-W42-T25-H17.8)
-h
(CGC-LB-P.2-W42-T25-H 18.6) (CGC-LB-PO.2-W42-T25-H17.8)
( (CGC-LB-P.2-W42-T25-H47.3) 0.8
-
40
0.7
-
0.6
-
-
45
-
Vrxn (CGC-LB-PO.2-W42-T25-H47.3)
35 IJ.
30
7/~~_
-
0.5
25
-K--~-~ -~.
10
_____
I
5
-/
-~
~1 ~
0
0.3
0.2 0.1
-~
-
--
200
2
200
2 50
0-
-
0
______
-
-
-
15
-
0.4
20-
-
>
50
100
150
200
0
250
50
1()0
150
Time (hr)
Time (hr)
(a)
(b)
Figure 6-34: Volume of reaction and degree of hydration of Class G cement smles in the liquid barrier test with variable thicknesses.
139
Part IV Discussion
140
Chapter 7 Expansion During the Hydration of Magnesium Oxide The main purpose of including in our investigation the hydration of magnesium oxide (MgO) is to calibrate and assess the performance of the main experimental setup (the pressure vessel). MgO serves as a simple model material with well-known chemical reactions, enthalpy of reaction, volume of reaction, and mechanical properties to study the bulk volume changes of reactive porous solids. The hydration kinetics of MgO also resembles the kinetics of cement-based materials, but through a single chemical reaction with a single hydration product while the complete chemical reaction can be achieved in one week at 25'C. The enthalpy of hydration of MgO is 930 J/g based on thermochemical data tabulated in Ref. [35]. The volume of reaction can be calculated based on the molar volumes of the different constituents as listed in Table (A.2) to
be -109 pL/g The morphology and reactivity of MgO depends on the synthesis procedures and mainly on the calcination temperature. There exists a number of MgO grades classified based on the calcination temperature and consequently their reactivity. This chapter discusses the results obtained for the hydration of light-burn MgO presented in Section 6.1. Hydration of MgO is investigated with multiple techniques including SEM, nitrogen sorption, calorimetry, TGA, and indentations, as well as volume of the reaction and bulk volume change measurements in the pressure vessel.
141
The development of the microstructure during the hydration of MgO is first investigated with SEM, nitrogen sorption, calorimetry, TGA, and micro-indentations. The resulting understanding of the microstructure of the material is then employed to construct and validate a multiscale microporomechonics model. The second part of the discussion focusses on the development and validation of the model followed by interpretation of the bulk volume change measurements to relate the macroscopic measurements to the origin of bulk volume changes.
7.1
Hydration kinetics of MgO
Reactivity of MgO depends on the calcination temperature. The hydration kinetics of MgO resembles the kinetics of cementitious materials with an induction period followed by acceleration and deceleration periods as shown in Figure 6-2 and Figure 6-3. Such behavior can be attributed to a boundary nucleation and growth acceleration followed by a diffusion-limited deceleration. The close resemblance between the hydration kinetics of MgO and cement can help elucidate the kinetics and controlling mechanisms of cement hydration. For example; while the induction period observed during the hydration of cement is sometimes attributed to the formation of a semipermeable membrane [22], it is not clear as to how a semi-permeable membrane of Mg(OH) 2 can lead to the induction during the hydration of MgO. Hydration kinetics is monitored with both volume and enthalpy of the reaction. The tested light-burn MgO is initially partially hydrated as shown by the TGA results and the measured enthalpy of 860 J/gMgo is necessarily an under estimation of the known 930 J/gMgo. The volume of the reaction on the other hand was overestimated when measured in the pressure vessel.
Light-burn MgO consists of aggregates of
approximately 100 nm crystallites forming porous particles ranging in size from submicrometers up to 10's of micrometers. It is possible that part of the MgO porosity is occluded, which can result in overestimation of the measured volume of the reaction.
142
7.2
Microstructure of MgO and Mg(OH) 2
To build an appropriate micro-poro-mechanics model to characterize the material at hand, we need first to establish an understanding of the microstructure of the material. To build this understanding, the material is investigated under the scanning electron microscope at different stages of the chemical reaction and under varying reaction conditions. Sorption isotherms are also collected for these samples to investigate the surface area and pore size distribution. The specific surface area of light burn MgO for both batches was about 25 m 2 /g within the range of values reported for the material in the material datasheet (Table 3.1). Assuming a mono-disperse particle size distribution of MgO and a density of 3.58 g/mL, the average radius is calculated to be 34 nm,which is close to the size observed under the SEM (Figure 7-1). The total pore volume of light-burn MgO is 0.23 cm 3 /g (Figure 7-3e). This pore volume is mainly encapsulated inside the MgO clusters and part of this pore volume is due to the contact between the grains. If this volume is assumed to be encapsulated ,
entirely inside the MgO clusters and taking the density of MgO to be 3.58g/cm 3 one can estimate the porosity of these clusters as 0 =0.23/(1/3.58+0.23)=0.45 a packing density of 0.55.
and
This packing density is very close to the random loose
packing of a mechanically stable system of the fully hydrated Mg(OH)
2
TIRLP=0-
55
[139]. Similarly, the pore volume
is 0.06 cm 3 /g corresponding to a packing density of
0.88. The packing density of 0.88 indicates that sintering of the Mg(OH) 2 has occured during hydration. The surface area of the fully hydrated Mg(OH) 2 was also measured with nitrogen sorption to be about 10 m 2 /g. Taking into account that the molar volume of Mg(OH) 2 is 2.21 time the molar volume of MgO (see Table A.2), one can estimate a specific surface area (SSA) of 28 m 2 /g. (sintering) of Mg(OH)
2
The small measured SSA indicates either merging
particles as observed in the SEM micrographs (see Figure 7-2)
or possibly, complete dissolution of the smaller MgO particles. Hydration of light burned magnesium oxide produces a granular cohesive solid of
143
I Ii (b) 545X
2 G 0 -N
It"
W wo
Irv,
ALM
Tk
Jlii-@ v-r (d) 45.25kX
(c) 4.31kX Figure 7-1:
MNlicrostructure of anhydrous MgO under the SEM. The material in dry
powder form was placed on a conductive adhesive carbon tape and carbon coated for imaging. The basic building block of the material is crystallite with crystal size of around 100 inn forming particles of agglomerates that can go up to 20 nm in size.
144
-tpm
OC-S
100pm
CM 83..POP
SWW
()
(a) 2.66kX(BSE)
j
2pm
E
172X
I&O
CM -S E 2pm
CMSE
(d) 8.78kX
(c) 3.41 kX
I WA. 200
n
200 nm
(f) 84.66kX
(e) 46.81kX
Figure 7-2: SEM micrographs of the microstructure of Mg(OH) 2 at full hydration at (liffereit magnifications. A fractured surface of fully hydrated sample of Mg(OH)2 was exposed and coated with carbon for the investigation. The basic building block of the material is crystallite with crystal size of around 200 min. The original structure of the anhydrous MgO is still observable at low magmification.
145
160
70
140
60
120
50
60
1----o, 0z
40
cjz
20 0
J(MgOR-1t-9)
30 20
10
:5
0
0.2
0.6
0.4
0.8
0 0
1
0.2
7 6
2
5
1.5
-
17 -2
-A
-
SU
-I-i-t
-L
1-
'-I
-o 0.5
1
~0
0 1
01
100
10
0.25
-"-T
0.06
0.2 U
0.15
C.)
(vi{RD&O RDC4 ( d RDC0
10
0.05
0
100
-
RDC-07i -'RDC--:0 IC-09
g 0.04
4----
0.02
C.) 0
0 10
-4(g
S
7
0.1
1
f
-4-(
I
(d)
(c)
C.)
1
0.8
Pore Radius (nm)
Pore Radius (nm)
0
0.6
(b)
(a)
3 2
0.4
Relative Pressure (P/Po)
Relative Pressure (P/Po)
4
(MgO-RD-08)
-
40
.
80
(MgO-RD-07)
-
100
-4
0
100
10
100
Pore Radius (nm)
Pore Radius (nm)
(f)
(e)
Figure 7-3: Nitrogen sorption isotheris for Mgo an(d fully hydrated Mg(OH) 2 . Fig.
.
(a) and (1) are the niitrogen sorJptioli isotheruis for the alllydrous light burn MgO and fully hlydrated Mg(OH) 2 respectively. Figs. (c) and (d) show the dl/dr pore volume and Figs. (e) and (f) show the cumulative pore volume for MgO an Mg(OH) 2 respectively.
146
magnesiuni
hydroxide.
Here we investigate the properties of fully hydrated inagne-
sium oxide by means of indentations.
The application of indentation analysis to
heterogeneous porous materials is made possible by an extensive development of
the grid indentation technique by Constantinides and Ulm [44, 196], Vandainne and
Uhin [201. 202], and Gathier and Uln [80]. Samples of MgO were mixed at high water-to-solid mass ratios to obtain reasonable workability.
Then water is squeezed out under different effective stresses to obtain
materials with different porosities (Table 6.1 shows the 6 tested samples). Six samples
were prepared with two different water-squeezing mechanisms in a rubber diaphragm test and in a confinement cell loaded with a piston. In the rubber diaphragm. samples with known mass and water-to-solid ratio are subjected to controlled effective stress for a specific period of time to squeeze the wa-
ter out of the sample (see Table 6.1). The sample is then kept saturated until the tine of testing (see Figure 7-4). Samples are then obtained for the indentation test. porosity measurements by total evaporable water, SEM and TGA. For indentation tests, the sample is cut down to the appropriate dimensions and left to equilibrate to the ambient relative humidity which also is the relative humidity during the indentation test.
(b)
(a) Figure 7-4:
Sample sqlleezed in the rubber diaphragm: (a) Sample inside the rub-
ber diaphragm. (b) Sample outside the rubber diaphragm. (buckled) edge due to squeezing.
147
Notice the corrugated
In the confinement cell and the piston, samples with known mass and water-tosolid ratio are subjected to controlled effective stress for a specific period of time. Once the desired amount of water is squeezed out, the volume of the sample is held constant while measuring the evolution of stress (see Figure 7-5). The initial drop in stress shown in Figure (7-5) is due to stress relaxation after switching from constant stress to constant displacement and the stress development afterwards is due to the hydration and crystallization pressure of MgO. 3.5
0.5
3
0.4 -
-
6.3 -
0.3
2.5
5.9 --
0.2
2
5.7
0.1
1.5
-
6.1 - -
1
0
5.5. 0
50
100
-
-
6.5
0
50
100
0
50
Time (hr)
Time (hr)
Time (hr)
(a)
(b)
(c)
100
Figure 7-5: Evolution of compressive stress in the samples cured under the piston. Fig. (a) is sample MgO-Pis-01, Fig. (b) is sample MgO-Pis-02, and Fig. (c) is sample MgO-Pis-03 (see Table 6.1). Samples with known mass and water-to-solid ratio are subjected to controlled effective stress for a specific period of time. Once the desired amount of water is squeezed out, the volume of the sample is held constant while measuring the evolution of stress. For the porosity measurement with the total evaporable water, a representative saturated-surface-dry sample is oven dried at 105'C and the porosity was calculated assuming a density of 2.34 for the magnesium hydroxide (see Table 6.1 for the measured porosity). TGA test were also conducted on each sample to guarantee the full hydration which was achieved after 6 days. It is necessary to establish an understanding of the material behavior for the choice of indentation load profile and suitability of the micromechanics model. We start by studying the structure of the material under the scanning electron microscope followed by micro-indentations carried out at different loads and loading times. The SEM images (Fig.7-2) show the granular nature of the material while the BSE image shows the non-homogeneous porosity distribution were black represents pores 148
and bright is dense contamination by polishing pads. The particles seem to have a single size of about 200 nm in diameter while the local porosity has a characteristic length of about 5 micrometers as seen in the SEM images. Based on these characteristics, the activated area in an indentation test should be approximately one order of magnitude larger than the characteristic size of the crystallites, namely, about 1 micrometer. An indentation depth of about 1 micrometer should provide information about the heterogeneous local porosity distribution. An indentation depth of more than 10 microns is required to capture the homogeneous response of the material, but not too large to cause cracking. The loading profile of an indentation test is typically designed in such a way to capture the desired material properties without interference from other undesired properties. In this study, we measure the indentation modulus and indentation hardness to extract the solid modulus in addition to the cohesion and friction angle. For this purpose, we need to eliminate any possible interference by creep properties and/or fracture.
The effect of maximum load and hold time on the measured indentation
modulus and hardness are investigated here to help chose the appropriate values for the loading profile. Figure 7-6 shows the variation of mechanical properties as a function of the indentation load on sample MgO-Pis-03. There seems to be a decrease followed by a slight increase of the mechanical properties by increasing the maximum indentation load. Figure 7-7 shows the variation of the mechanical properties as a function of the hold time on sample MgO-Pis-03. Hold time was varied from 5 to 600 seconds and there was not a recognizable pattern.
The hold time did not have any significant
impact on the measured values of the indentation modulus and hardness. The compiled indentation results are shown in Figure 7-8 for the six tested samples. The model fit is shown in Figure 7-9 (see Section B.3 for the details of the model). The resulting properties from the fit are m, = 73.00
GPa, a = 0.4185
0.40 GPa, c, = 0.2301
0.0024
0.0045, and h, = 3.157 GPa. The value of 73 GPa for the solid
modulus is very close to the reported values in the literature of 71GPa (calculated
149
I 28.5
~1 4 -i
-
-
580
29
-
-
-F
-
29.5
620
-
28
cjz
e-1 0t
27.5
-
560
-
600
-
30
-
640
-
26.5
-
26
-
520
27 -
540
-
-)
25.5
500 0
0
6000
4000
2000
2000
6000
4000
Fn (mN)
Fn (mN)
(b)
(a)
-
28
-
-
560
28.5
27.5
-
580
-
Figure 7-6: Variation of indentation hardness (a) and indentation iodulus (b) as a function of the max indentation load Fn on sainple MgO-Pis-03. Five indents were carried out at each indentation load with 30 seconds loading, 30 seconds hold. and 30 seconds unloading.
I
-o 0
-
--------
26
480
25.5
-
460
26.5
--
1
100 10 Hold time (s)
-
500
-A
27
.-
-
520
-
540
-
T
1
1000
100 10 Hold time (s)
1000
(b)
(a)
Variation of indentation hardness (a) and indentation niodulus (b) as on sainple MgO-Pis-03. Five in(lents were carried out a function of the hold tie at each hold time with 30 seconds loadig. and 30 seconds unloading and 2000 nIN Figure 7-7:
maxiiiniuin indentation load.
150
I
Bow - - -
pw
--
- -- -
-=
40 35 30 25 -
20
----- M model (pa) --4 Gpa}{MgG-RD-G9) M (Gpa) (MgO-RD-08) M (Gpa{MgO-RD-07) M (Gpa) (MgO-Pis-03) M (Gpa) (MgO-Pis-02) M (Gpa) (MgO-Pis-01)
15 /
10
I
/
I I I
5
I.
I
I
I
0-0
200
400
600
800
1000
H (MPa) Figure 7-8: Indentation modulus M and indentation hardness H compiled from the different specinens shown in Table 6.1.
ms=73 cs=0.2301
alpha=0.4185 hs=3.1574 error=0.00056795 3.5
80 70
-/
3
60
-/
2.5
-50
2
(40 1.5 1 0.5
10 W.
7
/
20
/'
'
30
0.6
0.7
0.8
0.9
1
T.
eta
0.6
0.7
O
0.9
1
eta
Figure 7-9: Relationship between the packing density and indentation modulus and hardness from1 the experime1nt and the nodel for the niiero indetations on all dry Sam11ples.
I 151
from the elastic modulus of 67 GPa). The cohesion of 230 MPa seems to be related to the cementing and capillary reinforcement at the contact points between the particles as observed in the SEM images. The elastic properties of a single crystal of brucite were measured by Jiang et al. [102] by means of brillouin scattering. They carried out the experiments at pressure ranging from ambient pressure to 15 GPa in a diamond anvil cell. The resulting elastic constants were C1 = 154.0(14) GPa, C33= 49.7(7) GPa, C 12 = 42.1(17) GPa, C13 = 7.8(25) GPa, C14 = 1.3(10) GPa, C4 4 = 21.3(4) GPa and aggregate moduli K, =36.4(9) GPa, G, = 31.3(2) GPa, E, = 73.0 GPa from the Reuss bound and K
= 43.8(8) GPa, G, = 35.2(3) GPa, E, = 83.3 GPa
from the Voigt-Reuss-Hill average. The elastic modulus of anhydrous polycrystaline MgO is E = 300 GPa and the shear modulus G = 130 GPa [177]. The relationship between the indentation modulus and hardness of a pure porous solid can be obtained by synthesis of that material at different packing densities (porosities). This approach was particularly applicable to Mg(OH)
2
as obtained from
the hydration of light burn MgO due to the small and mono-disperse particle sizes and due to the presence of agglomeration of MgO crystallites. The origin of cohesion in Mg(OH) 2 compacted during hydration seems to be related to the observed reinforcement (Solid capillary bridges) between the particles in the SEM images. Compacted slurries of previously-hydrated magnesium hydroxide did not form a solid cohesive material. Spatial variation of packing density is due to the original clusters of MgO crystallites. This variation allows the use of nanoindentation to extract properties of the material over a wide range of porosities on one sample.
7.3
Volume fractions and relative volume changes
This section presents the relationships and calculations of the evolution of volume fractions during the hydration of MgO as well as the relative volume changes due to the evolution of volume fractions. The evolution of volume fractions are first discussed in light of the known hydration reaction and properties of the different components. 152
Then the relative volume changes are discussed for a single MgO grain based on pure geometric considerations and the possible contribution of the reaction mechanisms. This section also presents the calculations of the relative volume changes of the porous MgO particles as a possible interpretation of the observed bulk volume changes.
7.3.1
Evolution of volume fractions during the hydration of MgO
Evolution of the volume fractions of the system components can be related directly to the enthalpy or volume of the reaction given a known chemical reaction with known stoichiometric coefficients. To estimate the evolution of the volume fractions of the different constituents of the system, we consider the following chemical reaction along with material properties in Table A.2:
MgO(S) + H20(1) ,
Mg(OH) 2 (s)
(7.1)
The reaction proceeds with the dissolution of MgO and precipitation of Mg(OH) 2 and the resulting volume fractions of the different components is then:
frxn = (Rmg(OH)
2 ~
RH
-
1)(1
f(Mgo) = (1 - no)(1 -
f Mw = no f(Mg(OH)
2) =
-
(1 - no)RH
(1
-
-
no)
)
no)RMg(OH) 2
(7.2)
.3) (7.4) (7.5)
And the total porosity of the system in the absence of bulk volume changes is
#tot
=
1
-
f(MgO)
-
f(Mg(OH) 2 )
= no + (1 - no)(Rhp + 1)
153
(7.6)
where R,, = 2.21,
f
is the fractional volume of the reaction. j(NgO) is the volume
fraction occupied by the non-porous solid MgO.
f...
is the volume fraction
occupied
by the remaining initial water used for mixing (initial water is treated as a separate phase just like solids),
f(Mg(OH) 2 ) is
the volume fraction occupied by the solid
non-porous Mg(OH) 2 , 0O is the initial Porosity upon mixing. and
(
is the fractional
conversion. A plot of the evolution of the volume fractions is shown in Figure 7-10. For more details on the definition of each term, see Appendix A. As observed under the SEM in conjunction with nitrogen sorption isotherms, the total porosity of the system
4,t
is divided into particle porosity (4,, and the macro porositv
), such
that: (7.7)
Otot = OPar + Om
The distribution of the porosity will be further detailed in the discussion of the mutiscale thought-model for the hydration of MgO.
1t 0.9 0.8 .00.7
fm-
00.6 10.5-
0.4 Mfg(OH)2
0.1 0-0
fJgo 0.2
0.4
0.6
0.8
1
Figurie 7-10: E-volution of volume fraction of the various comnponents during tHe hy. dratin of Mlgo with w/cl( = 1.
154
7.3.2
Geometry and fractional conversion of a composite sphere
For the purpose of this analysis, we consider a single non-porous crystal of reactive solid (the core co) and the growing reaction products are also composed of non-porous solids (the shell sh) as illustrated in Figure 7-11.
/a
(a) S=
0
(b) 0 0 is the incremental volume fraction of the reactive solid (MgO) consumed in the reaction. The radius of the growing sphere is:
b = ao ((Rhp
1
-
+ 1)1/3
(7.11)
The volume fraction of the shell is: Vsh f
V=C
b3
_
-
a3
Rh(1
b3 (Rhp - 1) +1
where Vc, is the volume of the composite sphere. The volume fraction of the core is:
fCO=
V _ * C8e
1 (Rhp- 1) + 1
a3
b
(7.13)
The relative volume change is calculated relative to the original volume of the core as: E* =
O
= (Rhp - 1)
(7.14)
It is important to note here that depending on the reaction kinetics and mechanisms, the ratios can be drastically different. This analysis is valid if the reaction is a surface reaction or topochemical reaction, but it should be modified at larger scales if the reaction is a through-solution reaction. In a through solution reaction, not all the dissolving core will contribute to the volumetric strain because part of the dissolving ions will precipitate in the water-filled pore space. To account for the fact that not the entire dissolving core contributes to the eigenstrain, Rhp should be multiplied by a coefficient c accounting for the reaction mechanism and kinetics, and the relative volume change becomes: * = (cRhp - 1)
156
(7.15)
The value of c depends on the degree of hydration and the thickness of the hydration layer which in turn controls the reaction mechanism. In case of topochemical reaction the entire volume difference between the reactive solid (MgO) and the hydration products (Mg(OH) 2 ) is manifested as bulk volume change with c = 1. However, If 1 to 1 if expansion
the chemical reaction proceeds through solution, c can range from is measured and it can be smaller than
1 Rhp
if shrinkage is possible. The thickness of
the hydration layer assuming radial growth only is:
t = b - a = ao (
7.3.3
/1-)
1)+ 1-
(cR,-
(7.16)
Relative volume change in the porous particles
Another possible way to view the relative volume change in the MgO particles is by considering all the MgO grains in a single particle to maintain the relative positions inside the particle. In this scenario, only the grains on the outer perimeter of the particle contribute to the volume change as illustrated in Figure 7-12. The relative radial deformations in this case can be described in terms of the crystal and particle radii as follows:
E* =
R
- Ro 0
a0 [((Rhp- 1)
b - ao =t
+ 1)1/3 _
o
(1
(7.17)
And the relative volume changes is ,
-3
R3 3
R
-R
RR
-
(o -b -ao)
3
-
3
(7.18)
where b is the radius of the composite sphere and ao is the original radius of the MgO grains while RO is the initial radius of the porous particle. Taking the average radius of the particle to range between 3 and 8 microns from Table 3.1, which is also in line with the particle sizes observed under the SEM in Figure 7-1, and an approximate grain size of MgO of 100 nm, the relative radial deformation can be calculated as shown in Figure 7-13. The value obtained for volumetric strain obtained in this way 157
a-.
a
is
close to the vohne strain ieasured experimentally and is discussed further in the
following sections.
0.30.
RRe
00
..
of
HO~*
3o3~ ***~ (a.0
(b) Figure 7-12: A schematic of the hydration of MgO particles where the volume change is only attributed to the MgO grains on the outer boundary of the particles. (a) is the original anhydrous MgO particle (b) The MgO particle when fully hydrated into Mg(OH) 2 . It is assumed in this model that all the grains maintain the original positions due to the sintering of the grains. The sintering of MgO grains is a consequence of the calcination process while sintering of Mg(OH) 2 is due to the progression of the hydration reaction.
10
10
10
10
10 10
10
102
1? (pim) Figure 7-13: Rate of eigenstrain in the MgO-Mg(OH) 2 particles as function of the radius of the porous particles (1ie line). The dashed red line represents the strain ineasurd experimentally of about 0.015 for w/.s = I and the corresponding initial raldils of the porous Mg() particlc R ~ 3pmi.
158
7.4
Bulk volume changes and multiscale microporomechanics for the hydration of MgO
A single MgO particle consists of a large number of agglomerated MgO grains as observed in the SEM images in Figure 7-1. is about ripar
=
The packing density of these particles
0.55 as shown in Section 7.2. All samples were initially mixed with
high water-to-solid mass ratio (w/s ~_ 1) due to the very high specific surface area (~- 25m2 /g) of the hydrostatically neutral but hydrophilic surface of MgO (periclase). Such high w/s ratio produced coherent, yet very weak porous solid of Mg(OH)
7.4.1
2
Multiscale Thought Model of MgO hydration
During the hydration of MgO and due to the fact that the volume of the solid reaction products is larger than the volume of the solid reactants, the material experiences a bulk volume change.
The bulk volume change in this case is expansive, but the
extent of the expansion and the associated stress development are also related to the microstructure of the material, the reaction mechanisms, and the deformation behavior of the solid. The origin of the stress and the expansion during the hydration of MgO is assumed to be related to the difference in volumes between the reactive solids and the hydration products. To estimate the elastic properties and the state of stress of the porous solid resulting from the hydration of MgO, a 3-level multiscale microporomechanics model is developed here as illustrated in Figure 7-14. Based on the microstructure investigated previously, the multiscale microporomechanics model will consist of a three-level model of the MgO-Mg(OH)
2
microstructure.
The first level (level I) of this model
consists of nano scale crystals of hydrating MgO as a composite sphere where the reacting MgO forms the core and the precipitating Mg(OH)
2
forms the shell (Figure
7-14(a)). The second level (level II) includes multiple grains (crystals) from level I in addition to the interstitial space which will be referred to as the particle porosity (Figure 7-14(b)). The second level (Level II) is analyzed using two methods: the first
159
method is the self-consistent method in which the aggregate of crystals is modeled as larger scale particles. The second method is the Mori-Tanaka scheme in which the continuous porous matrix is composed of the composite spheres. Finally, at the third level (Level III), the aggregates of crystals are assumed to form a continuous porous matrix composed of the porous particles (Fig.
7-14(c)) for which a Mori-Tanaka
scheme is ap)lied.
(a)
(b) Continuum
(b) Particle
Composite Sphere
N~~7C o Core Shell
0
Level (III)
Level (II)
Level (I)
Figure 7-14: Multiscale microporo mechanics model for the hydrating MgO.
7.4.2
LEVEL I: The composite sphere
A relevant model to the hydration of reactive solids like cement-based materials and Magnesium Oxide is the shrinking-core model of a composite sphere. The composite sphere is made up of the anhydrous MgO grains forming the core and the Mg(OH)
2
forming the shell. Here we develop the evolution of the elastic properties of a colm)osite sphere with a reactive core followed by multiscale imicroporomechanics of granular solids. The first level of the nmltiscale though-nodel is composed of anlhydrous MgO and Mg(OH), and the volume fraction occupied by these phases is shown in Figure 7-10
as LEVEL I: J
= flid
=
fAJIO 160
+
fAIf((O)I2
(7.19)
To estimate the elastic properties and the evolution of eigenstresses in a composite sphere, we start by a geometrical description of the composite sphere.
Eigenstrain in a rigid composite sphere
Volume changes of the core and the shell can be considered individually by applying the divergence theorem. A rigid composite sphere is considered here with a shrinking core and swelling shell (see Figure 7-15).
The volume change rate to which the
core is subjected is defined in terms of the velocity vector at the core-shell interface V - n = V, =
dur dt
with n
e, the outward unit normal to the shrinking core surface
(see Figure 7-15): dveo S
V - n dS=
Vco
'9Vco(t)
divV dV VCo0
(7.20)
v"
Similarly, one can apply a similar approach to the shell assuming continuity of the radial velocity at the core-shell interface
V - n dS = -Vo
where
s(t) 8O
= &Ve(t) -
+
V -n dS
(7.21)
WVco(t), with &Ve(t) the outer surface of the composite
sphere oriented by the unit normal n = er, whereas the inner surface of the shell is oriented by a unit normal n = -e, (see Figure 7-15). Normalizing the previous expressions by the total volume of the composite sphere, we obtain:
V
osa
V - n dS =
c~t- c +Dv Ves dt
(7.22)
where D, is the total volume strain rate of the composite sphere. If we then define the average volumetric strain rate of the shell, dh, by:
Vsh (t)dsh =
VhV
161
- n dS
(7.23)
we obtain the consistencv equation:
Dv =
v;--((t ) S
d
CO
+
V1,(t)' ('1J
(7.24)
I Cs
t nr
no
O3Vcob
no aa r
Ves
aVcs
VCS+aVCo =0Vsh
Figure 7-15:
A schematic of the coniposite sphere of shrinking M gO (core) and
swelling M\g(OH) 2 (shell).
Upscaling Following the framework developed in Section 5.1. eigenstress can be incorporated in the nuiltiscale inicromechanics model at the scale of the composite spheres in level I (Figure 7-17). The assumption of the rigid sphere that underlines the geometrical description above does not account for confinement effects the shell exerts onto the core and vise versa. The focus of the inicroporoniechaiiics model is to account for the deforiability of the core and shell in this process. Consider thus a distribution of eigenstresses in incremental form:
(7.25)
d, (z) = k(z)dc(z) + iog, (z)
with the distribution of the bulk modulus and the incremental pre-stress given by:
K(z) k ( z-=SI 11
dcCO
Vz Ce
KC{ (
Vz e VA
Kal, 162
(7.26) Vz
EI
where Ke, = 160 GPa is the bulk modulus of the core (MgO), Keh
=
43 GPa is the
bulk modulus of the shell (Mg(OH) 2 ), Vc0 is the volume of the core, Vh is the volume of the shell, and do& is the incremental eigenstress that develops in the solid phase due to the progression of the hydration reaction. dEc, = dc 0dt and
dEch =
dshdt stand
for the volumetric strain increments related to the volume change rates previously defined for the rigid system. Homogenization of Eq. (7.25) with Eq. (7.26) entails the macroscopic incremental stress state equation:
dE' = KsdE[ + dEPI
(7.27)
where E, is the macroscopic mean stress,dEv, is the incremental macroscopic volumetric strain,
fc,
is the volume fraction of the core and fsh is the volume fraction
of the shell (fc, +
fah
= 1). K,, is the homogenized bulk modulus of the composite
sphere shown in Eq. (7.35) and dEP is the homogenized pre-stress of the composite sphere. They are obtained in the following way:
* From strain localization d&(z) = A(z)dEv, with A' 0 = (A)vco and A"h
(A)Vh
are the volume averages of the fourth order strain distribution tensors, we obtain the homogenized bulk modulus of the composite sphere:
VCO M KeOAvO +
-
V
(t)KshAh
(
)
Kes = (k(z)A(z))VS
+
The strain compatibility condition requires dEl = (dE(z))vcs = fco(de(z))vco
-Vs.
fsh(de(z))Vsh 0
T1-A Figure 7-18: A schematic of the porous MgO-Mg(OH) 2 particle with the mean stress resulting from homogeneous stress boundary conditions. The initial packing density as measured with nitrogen sorption is 0.55.
and in the presence of eigenstresses, we shall apply the following distribution:
E
Kcs
Vz
0
Vz E VP
dEPI
eCs
Vz E V
(7.41)
k (z) = dppar Vz E Vp
where K, is the effective bulk modulus of the composite sphere, V,, is the volume of the composite sphere (solids), Vp is the volume of pores and p is the pore pressure. Homogenization of Eq. (7.40) with Eq. (7.41) entails:
dzfJ + dppar = Ke 8 (1 =
where E
-
+ (1 - b")(dZE' + dppar) 1)dE{'
b
K" dEf + (1 - b)(dEP' + dppar)
(7.42) (7.43)
is the macroscopic mean stress at level II, par is the pore pressure in the
porous particles, (dEm, +dp) is the incremental effective stress, dEf' is the incremental macroscopic volumetric strain at Level II. dE' = (de(z)) = 1(de(z)) " +#(Kde(z))
is
the incremental macroscopic volumetric strain, rj is the solid volume fraction occupied by the composite spheres and
#
is the porosity (7"7
= 1 --
(A)' and (A) "s are ().
the volume averages of the fourth order strain distribution tensors, dc(z) = A(z)dE. 168
The Biot coefficient at level II reads:
I='(A(1-r= A)'c= 1-
K"
(7.44)
A cs
The level II state equation defines changes in stresses and particle porosity at constant hydration degree. and read as: Incremental mean stress in the porous particles (Level II)
KEI do,,,,(z)) =KI'dEf' + (I
where
Ppar
-
b") dZ'P'
b, dppar
-
(7.45)
is the pressure prevailing in the particle porosity.
At, level II, the manifestation of the strain (dEf') is critically dependent on the reaction mechanism and the nicrostructure of the material as introduced in the discussion of Level I. Porosity of the porous MgO-Mg(OH)
2
particles plays a critical
role in the bulk volume changes at this level due to the possible rearrangement of the hydrating grains during the hydration reaction. And porosity also provides additional space for the hydration products to form without contributing to the radial growth of the hydrating grains if a through-solution reaction controls the reaction mechanism. In this model, it is assumed that the relative positions of the hydrating conposite spheres only expands without rearrangement of the particles.
The lack of
rearrangement of the grains requires that the eigenstrain of the porous particles must equal exactly the eigenstrain in the composite sphere if the entire hydration products contribute to the radial strain of the composite sphere. The initial packing density as measured with nitrogen sorption is 0.55 and in the case where all the formed IVg(OH)
2
contributing to the eigenstrain, then this packing density will remain con-
staiit throughout the hydration reaction. Due to the precipitation of Mg(OH) 2 in the water-filled pore space. the packing density of the particle will increase at a rate proportional to the contributing fraction and the non-contributing fraction of Mg(OH) 2 . This conclusion is supported by the decrease of the surface area of the MgO particles and the merging of the hydrated 169
I particles observed under the SEM as well as the small impact of the w/c ratio on the measured Imulk volume changes. The prefactor c in Eq. (7.15) and (7.34) is requiired to correct the eigenstrains in the material to account for this phenomenon.
Ppar
~ MgO Core Mg(OH) 2 Contributing
d -
Mg(OH) 2 Non-Contributing
-s
Fignre 7-19: An ilinstration of the porous MgO-Mg(OH) 2 particle with the part of Mg(OH) 2 contributing to the measured eigenstrain and the part that precipitates
inside the water-filled pore space withont contribution to eigenstrain.
The initial
packing density as measured with nitrogen sorption is 0.55 and in the case where all the fornied Mg(OH) 2 contributing to the eigenstrain. then this packing density will remain constant throughout the hydration reaction. Due to the precipitation of Mg(OH) 2 in the water-filled pore space, the packing density of the particle will increase at a rate pIroportional to the contributing fraction and the non-contributing .
fraction of Mg(OH) 2
Evolution of the poroelastic properties of the porous MgO-Mg(OH)
2
par-
ticles (Level II) Elastic properties of the 1)0r0us particles (Level II) is calculated with both selfconsistent and Mori-Tanaka schemes. It is imperative here to estimate the evohition of the packing density by taking into account the impact of the reaction mechanism. If we assume that all the measured eigenstrain originate at this level. then the packing density of the particle should be corrected accordingly. tion occulpied by the porois parti(les
can
Initially the volume frac-
be estimated directly based on the volune
fractions of the colmponents as follows: a AJ0
(
I~ 170
glOr
(7.46)
where fpar is the volume fraction occupied by the porous particles.
This volume
fraction includes the porosity of the particle; rII = rpa, is the packing density of the particle, that is the volume of solids in the particle relative to the volume occupied by the porous particle. The volumetric strain at level II is defined as:
VP ar
I
-YP-a
r
-
par
fpar
(7.47)
1
fpar
with the volume fraction occupied by the porous particles:
fpar
flgO
-
+ fMg(OH) 2 ripar
1
_ ("
(7.48)
Finally, we obtain the packing density of the porous particles:
'ar=
77par
II
fMgo + f.Mg(oH) CII + If 0 H) 2
=
=
(7.49)
~ )p'ar
The packing density of the fully hydrated particles of Mg(OH) 2 is =pa 0
=II
with
er0
=
0.55 and n4"aax
=
+
2max
r'
0ar g
_
= 0.88:
(7.50)
0.88 based on the results of the nitrogen sorption
isotherms.
Again, the prefactor c in Eq. (7.15) and (7.34) is considered here to estimate the evolution of the packing density of the hydrating MgO-Mg(OH)
2
particles. The
poroelastic properties at this scale are calculated with the self-consistent scheme and read as:
K"GI
=
Kcs
1 - b
=
b"
=
4GII/Ges+ 3 (1 -qrs)
where r' = KcS/Gcs = 2 (1 + vC) /3 (1 - 2vc"), with j
171
=
r(
(7.51)
rII the packing density of
GII C(
2
porous particles. and:
1
5
+1
1144 (1
-
3-r' (2 + r7)
I) -
(7.52)
- rs) - 4807 + 400
02
+ 408rsr/ - 120rs (2
with r = r" the packing density of the MgO-Mg(OH)
2
+ 9
()
(2
+
the MgO-Mg(OH)
porous particles.
The second method employed to estimate the elastic properties at Level II is the Mori-Tanaka scheme. In this scheme, it is assumed that the second level can be modeled as a continuous matrix with the particle porosity. Based on the Mori-Tanaka scheme (mt), the shear modulus is [61]:
GI_
(
t =9
GCS
_
_ + 9rs) _(8
r)
(7.53)
8 + 9rs +6(1 - rII)(2+ rs)
And the effective bulk modulus is:
-m,
,"
4 4n,,(
3(1 - r/I)r8 + 4
7.54 )
Kcs
A comparison of the elastic properties obtained by the two schemes is shown in Figure 7-20.
7.4.4
LEVEL III: The continuum scale
The third level of the multiscale microporomechanics model for the hydration of MgO consists of the first two levels (the composite sphere and the particle porosity), in addition to the macroscale porosity. The continuum scale (Level III) is modeled with two methods similar to level II. To estimate the packing density of Level III, it is assumed that part of the Mg(OH) 2 precipitates inside the porous particle foH)2 and the other part precipitates in the water-filled porosity outside the original boundary of the porous particle
fMg(OH). -The
fraction of Mg(OH) 2 that precipitates inside the particle can be related 172
150
120
100 7-
80
-GeS
60 50
40
-
020.
0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
(a)
0.6
0.8
1
(b)
Figure 7-20: Evolution of the bulk modulus of the porous particles (Level II). where A" is the bulk modulus of the porous sphere estimated based on the self-consistent is the bulk modulus of the porous sphere calculated from the moritanaka scheme. and K,., is the bulk modulus of the composite sphere. (b) Evolution scheme, K'/
of the shear modulus of the composite sphere. where Gc, is the bulk modulus of the composite sphere.
to the packing density of the particle:
H
-
(7.55)
I/fparx
and the part that precipitates outside the boundary of the particle is:
.t'g(O>11)2
=Mg(O1)
2
-
fjg(OH) 2
(7.56)
The fraction of Mg(OH). that precipitates in the macroporosity is assumed to have the same packing density of the porous particles. The packing density of level III is then: f
7/=
7' + J'g(OH) 2 IIIII or +
__
.fl\g(oH-I)
2
+ 'fNg
(757) 7.57
The method employed to estimate the elastic properties of Level III is the MoriTanaka scheme. In this scheme. it is assumed that the third level can be modeled as a continuous matrix with micro porosity. Based
173
oi1 the Mori-Tanaka scheme. the
shear modulus is
[61]: I/7 H (8 + 9r1-)
GJ,[ G"
(7.58)
8 + 9,r, + 6(1 - VIT")(2+ r,)
The effective bulk modulus is:
where r, = K''G'.
KO
(3 (
K11
(3(1
14(i> -- )!
(7.59) s+ 4)
I '4
-
I
~- --GI
14
-GS
12,
14
~12 Z110 6
8
4
6
0.2
-KI! 0.8 06
0.4
1
0
0.6
04
0.2
0.8
1
(b)
(a)
Figure 7-21: Evolution of the bulk modulus of the continuum (level III). where K"' is the bulk modulus calculated based on the Mori-Tanaka scheme at level III and
the self-consistent scheme at level II. Kf/ is the bulk modulus calculated from the mori-tanaka scheme at both level II and level III. (b) Evolution of the shear modulus at Level III.
9 At this level, we consider a second pore space. the macroscale porosity. in which another pressure prevails, denoted by p and the porous MgO-Mg(OH) 2 particles make up the solid phase. The vohune occupied by the hydration products is designated ,r and the space occupied by the capillary porosity is
ca.
The incremental stress state equation for the stress distribution at level III rcals:
d(IU, (z) = k(z)dc(z) + du')(z)
(7.60)
174
-
--
I
with:
{
do") (z)
dB"=(1
-
{
k(z)
K"
Vz E
Vpar
0
Vz E
VCa
(7.61)
b") dYZP' - b" dppar Vz E Vz
-dp
Vpar
(7.62)
E Vcap
The eigenstress at Level III is obtain by application of Levin's Theorem: dEP"' = A(z)daP(z)) = (1 - b{") dEZ" - b'"dp
(7.63)
Whence, the incremental equations of state:
dEf;
= (doa, (z)) !
= KN',dEt
1 + (1- b4" ) (1 - b") dEP' - (1 - b ") b(1(ipar - bdp dD' 1
(7.64) where:
K",
(7.65)
(1I - b, 11) K'"
=
1K"
0.8 is the C/S ratio and it was calculated from solubility data.
For a C/S ratio of 1.7, the composition of C-S-H would be
C 1. 7SH 2 .5 , where 2.5 is the combined water. Sierra [174] examined the different forms
of the combined water in the C-S-H solid by molecular spectroscopy and thermal analysis and suggested a distribution of water between hydroxylic, interlayer, and surface water. The hydroxylic water is the part of the water that decomposed into hydroxyl groups and became part of the C-S-H solid. The interlayer water is the water confined inside the C-S-H solid layers and the surface water is adsorbed on the surface of the C-S-H solid and both interlayer and surface water are characterized by a density larger than 1 g/mL due to electrostriction of water. Based on this classification of water in the gel and as relevant to calculation of volume fractions and volume of the reaction, we distinguish three conditions of the C-S-H gel based on the degree of saturation (water content). The first condition is the completely saturated C-S-H gel where the gel porosity if filled completely with water. The density of the gel in this condition is the density that is used later on in the multiscale thought-model and it also defines the packing density of the C-S-H gel. The second condition is defined at the degree of saturation where part of the gel water with bulk water properties (free water) is removed and only combined water remains.
The density of the C-S-H gel in this condition is related directly to the 192
specific surface area and surface charge of C-S-H. The importance of this condition is related to the average density of the hydration products and the measured volume of the reaction. The third condition is achieved when free and adsorbed water are removed and only the dry C-S-H solid remains with hydroxylic "water" and interlayer water only. This condition defines the density of the C-S-H solid as used for the calculations of packing density of the gel. To estimate the composition of the C-S-H necessary for the calculation of the molar volumes and water content of the C-S-H gel, we consider here two models for the nanostructure of the C-S-H gel. Glasser et al. [83] suggested a simplified model that was developed for the thermodynamic treatment and dissolution equilibria of C-S-H and is represented by the formula:
CaH 6 -2xSi 2 O 7 - zCa(OH)2 -nH 2 0
(8.3)
The C/S ratio is given by: Ca/Si = X
(8.4)
2
This model is applicable to young C-S-H gels with C/S ratio ranging from 1.0 to 1.4. The value 3+ z - x is related to the water bound as hydroxyl groups; and n refers to the interlayer water and surface water. The requirement of the value of z is related to the silicate chain length as will be seen next. The second model considered here is the generalized nanostructure model developed by Richardson and Groves [165,166]. The model suggests a generalized formulation for the nanostructure of C-S-H based on a modified Tobermorite/Portlandite (T/CH) and Tobermorite/Jennite (T/J) structural viewpoints. Richardson and Groves gave the formulation of the C-S-H structure as:
{Ca 2nHwSi( 3n-1)O(9n- 2 )} - (OH)w+n(y- 2) - Cay2 - mH 20 where w is the number of silanol groups, (3ni 193
-
(8.5)
1) is the silicate chain length, and the
calcium to silicon ratio is given by
n(4 +y)
Ca/Si = n(4 + 1) 2(3n - 1)
(8.6)
For a Ca/Si=1.7 and assuming that the dimeric silicate chains are the predominant form in young C 3 S or OPC paste [83], n = 1, y = 2.8 The density of the nanoscale C-S-H gel particles ("solid" C-S-H) including the hydroxilic water and the inter-layer water is assumed constant and independent of the curing conditions and the degree of saturation of the C-S-H gel. Allen et al. [2] combined measurements from small-angle neutron and X-ray scattering to measure the average composition and density of the C-S-H solid.
They considered C-S-H
particles including all the physically bound water in the internal structure of the particles and excluding the free and adsorbed water. After Allen et al., the density .
of C-S-H is taken as 2.60 g/mL for an average composition of C 1 .7 SH 1 .8
Young and Hansen [212] and Thomas et al. [184] estimated the density of the C-S-H particles including the interlayer water and adsorbed water as measured with pycnometry and neutron scattering as 2.18 g/cm 3 with an average composition of C 1 .7 SH 2 . 1 . It is imperative to mention here that this density and composition include the water adsorbed on the surface of the C-S-H solid and consequently, it depends on the specific surface area of the solid. A compilation of C-S-H density measurements are listed in Table 8.3 for different degrees of saturation and drying conditions. In conclusion, the composition and density of the C-S-H phase used to calculate the enthalpy and volume of the reaction must include, at least, all the combined water (hydroxylic, interlayer, and adsorbed water) due to the contributions to enthalpy and density of the hydration products. The minimum H/S ratio required for the balanced chemical reaction is H/S=2.1. For the calculations of the packing density of the CS-H gel on the other hand, the composition and density must include the water that defines the C-S-H solid, which in general is considered as the solid with the interlayer only. An average density suitable for such calculation is found in the work of Allen
194
3
and a compositions of C 1 .7SH 1 .8
.
et al. [2] with an average density of 2.6 g/cm
Before we embark on calculating the volume fractions of the different components of the C 3 S paste, we start by introducing the necessary definitions of the volume fractions and densities occupied by the C-S-H phase. PCSH: is the density of the C-S-H solid including all the combined water and excluding
the free water. This density depends on the specific surface area, but does not depend on the packing density of the C-S-H gel.
fCSH:
is the volume fraction occupied by the C-S-H solid including the hydroxylic water and interlayer water and excluding the adsorbed (surface) water and the free water. This volume fraction is assumed to be independent of the surface area and the packing density of the gel.
fCPsH:
is the volume fraction occupied by the C-S-H solid including all the combined water and excluding the free water. This volume fraction depends on the specific surface area. But does not depend on the packing density of the C-S-H gel.
Pgel: is the average density of the C-S-H gel including all the combined water and
the free water inside the C-S-H gel (gel water). This density depends on the packing density and specific surface area of the C-S-H phase:
Pgei
fipi = Tge1fCS
=
where Pgei = 1 - 7lgel, and pw
=
+ 'PgelPw
(8.7)
1.0 g/mL is the density of the free gel water.
fgel: is the volume fraction occupied by the C-S-H gel including the combined water
and the free water inside the gel porosity (the saturated gel).
This volume
fraction depends on the packing density and specific surface area of the C-S-H phase: fgei = fCSH "7gel
(8.8)
To estimate the evolution of the volume fractions of the different components of the system, we consider the chemical reaction below along with material properties
195
in Table A.2. The reaction is based on the formulation of Young et al. [212].
- CxSHy + (3-x)CH
C 3 S + (3-x+y)H ,
(8.9)
where x is the C/S ratio and y is the H/S molar ratio. The volume fractions of the different components are then:
(Rhp -
- 1) (1 - no)(
RH
(8.10)
)
frxn =
(8.11)
fmw = no - (1 - no)RH
(8.12)
fc 3 s =
(1 - no)(1 -
fcsH
=
(1
-
no)RcSH
(8-13)
fcH
=
(1
-
no)RCH
(8.14)
fgel
=
(1
-
no)Rei
(8.15)
The total porosity of the system is:
di,=1
f
-
fbp
-
(8.16) =
no - (1 - no)(Rhp
-
1)
while the capillary porosity is:
Ocap =
1
f(cs)
-
fgel -
geI -- Otot
-
Ocap
-
fCH
(8-17)
Finally, the gel porosity is:
where Rh
-
(3=
I
H
VCsH
(8.18)
is the ratio between the molar volumes of the hydration
products and that of C 3 S, RH
-
(3-xy)V, is the ratio between the molar volumes
of water and the reactive solid (C 3 S), RCSH is the ratio between the molar volume of C-S-H and C 3 5. frxn is the fractional volume of the reaction, f(c 3 s) is the volume 196
fraction occupied by the remaining solid non-porous CS, fm" is the volume fraction occupied by the remaining initial water used for mixing (initial water is treated as a separate phase just like solids), fCSH is the volume fraction occupied by C-S-H as defined by
RCSH,
0
is the initial porosity upon mixing, and
is the fractional
conversion (degree of hydration). fhp is the volume fraction occupied by the hydration products CH and C-S-H (see Figure 8-8). For more details on the definition of each term, see Appendix A.
The volume fraction of the porous C-S-H can then be calculated as follows
fgei = fCSH qgel
where
'lgeI
is the packing density of the C-S-H gel.
parameters in this study. fsH
(8.19)
It is one of the most critical
is the volume fraction occupied by the C-S-H solid
(no adsorbed water on the surface).
On the other hand, the density of the C-S-H gel and packing density thereof is not an intrinsic property of the C-S-H gel and it varies spatially and temporally depending mainly on the water-to-cement ratio, curing temperature, and particle size distribution of the cement powder.
The derivation of the evolution of volume fractions during the hydration of ClassG cement are discussed in the Class-G chapter below and relevant models in the literature are also reviewed. The reviewed models include measurements other than the enthalpy and volume of the reaction, such as ThermoGravimetric analysis (TGA) [157], Quantitative X-Ray Diffraction (QXRD) [90], Nuclear Magnetic Resonance
(NMR) [133], QuasiElastic Neutron Scattering (QENS) [73], and Scanning Electron Microscope (SEM) [71]. Other measurements can also be used in a similar fashion without the resort to approximate chemical reactions.
197
Table 8.3: Variation of C-S-H density and molar volume p (g/cm 3 ) (g/mol) 1.85-1.90 227.5 C1 .7 SH 4 (*) 1.90-2.10 227.5 2.12 229 177.0 2.86 C 1 .7SH 1 .2 180.6 C1. 7 SH(1. 3 - 1 .5 ) 2.44-2.50 2.70-2.86 180.6 187.8 2.604 C 1.7SH 1.8 193.2 2.18 C1.7SH2.1 193.2 2.43-2.45 191.6 2.46 C 1 .6 7 SH 2 . 1 (t) 293.3 1.73 C 1 .7 5 SH 7 .5 233.9 1.91 C1.75SH4.2 255.5 1.86 C 1 .7 5 SH 5 .4 253.7 1.96 C 1 .75SH 5 .3 251.9 1.83 C 1 .7 5 SH 5 .2 * tandarrd enthailn of formntion i: AH
" (cm 3 /mol) 121.3 113.7 108 61.9 73.1 65.0 72.1 88.6 79.2 78 170.0 122.5 137.4 129.4 137.8 =
Remarks
Refs
D-dry D-dry
11% RH 11% RH Jennite Saturated/inner Saturated/outer Saturated/inner Saturated/outer Saturated
-1283 kJT/mol from reference
[212] [157] [15] [212] [67,157] [32,99,163] [2] [184, 212] [67] [114] [55] [55] [55] [55] [55]
r2121
.
Formula
t standard enthalpy of formation AHY - -2723 kJ/mol from reference [114]. Both enthalpies are equivalent if the difference in water content of the C-S-H gel has the standard enthalpy of formation of water.
198
Average density of C 3 S paste The total density of the C3 S paste during hydration can be calculated from the volume
fractions as:
Ppaste = (p) =
where
fi and pi
fipi
=
fC3 spC3 s + fCsHPCSH
+ fCHPCH + totPw
(8.20)
are the volume fraction and the density occupied by phase i, respec-
tively. The density of the paste can also be calculated from the initial density and the fractional volume of the reaction as:
Ppaste = (P) = Po + frxnPrxn
where po = (1 - no)pC3 s + nopw is the initial density of the mix and Prxn
(8.21)
= Pw =
1.0
g/mL is the density of the fluid used to keep the sample saturated (water). The density of the tested samples was measured at the end of each test and the w/c is corrected to account for the water squeezed out of the sample during setting up the test and the volume changes at early age before setting and upon complete consumption of the bleed water. The corrected w/c are shown in Table 3.3.
Mass fraction of C-S-H The mass fraction of the C-S-H solid is calculated here as a necessary quantity for surface area calculations. The mass fraction can be calculated at a given degree of hydration and water-to-solid mass ratio as follows:
PCSHfCSH __ PCsHfCSH + nopw + frxnPw no)pcas (1 p) where (p) is the average density of the C3 S paste.
199
I
0.9
0.9
0.8
fH
0.7
0.7
0. 4
0. 4 fCSH
0.303
ICS
'
0.2
-
- ----
---------------------
-
f.
'0.2
fc,3s
kas 0.1
0.1
01
0
0.2
0.6
0.4
0.8
0.2
10
0.4
0.6
0.8
C
(b) 'v/c
(a) w/c=0.34
0.49
Figure 8-8: Evolution of volume fraction of the various components during the bydration of C 3 S with water-to-solid ratio of (a) iv/c = 0.34 and (b) w/c = 0.49. The yellow spot marks the degree of hydration where all the remaining free water is inside the C-S-H gel. f,,. is the volume fraction occupied by the free water remaining from the original mixing water only (not including any water provided from the outside during the hydration).
-
The
fc,, is
the volume fraction occupied by free water remaining
from the initial mixing water in the capillary porosity. fTsi is the volume fraction of the solid C-S-H only. Pgl is tie volume fraction occupied by the gel porosity. The capillary porosity d/c-, is represented by the areas shaded in blue and grey. The grey . area represents part of the water uptake from outside the sample The volume fraction occupied by total porosity of the system is 010t = cap +-,,I. the C-S-H solids and the gel porosity is defined as the C-S-H gel f+l
200
fCS
-
OgPel-
8.3
Eigenstress development during C 3S hydration through a multiscale thought-model
A multiscale thought-model is developed here to interpret the bulk volume changes and to calculate the eigenstress development during the hydration of C3 5 paste. Bulk volume changes of C3 S are investigated for three different particle size distributions, different water-to-solid ratios, and curing temperatures as shown in Figures 6-8 to 6-14. The water-to-solid ratio is chosen to obtain reasonable workability and prevent excessive sedimentation for each particle size distribution. In general, the bulk volume changes of C3 5 pastes include an expansion part during the first 200 hours of hydration of saturated samples followed by shrinkage for the remainder of the tests. The initiation of the bulk volume change measurements shown in Figure 6-14 for different PSDs and in Figure 6-22b for variable curing temperatures is related to the evolution of pressure at the bottom surface of the sample. Initially, pressure at the bottom surface is identical to the pressure delivered through the top port and the flow rate through the bottom port is completely restricted due to the presence of the check valve in the flow network of the RDC test. Once the slurry is set and the bleed water is completely consumed by the sample, pressure at the bottom port starts dropping at a rate proportional to the rate of hydration and a maximum drop equal to the cracking pressure of the check valve, as shown in Figure 8-9. The measured bulk volume changes during the hydration of C 3 S are attributed solely to the development of eigenstresses assuming that the 60kPa effective stress does not contribute to the measured bulk volume changes in the set paste. Eigenstresses develop in cement-based materials due to a combination of possible driving forces as discussed in Section 2.2. A multiscale microporomechanics is developed to interpret the measured bulk volume changes of saturated hydrating C3 S paste. The multiscale microporomechanics model consists of a three-level model of the cement microstructure. In this model, CBM will be modeled as a matrix of C-S-H particles in which all other solid phases and the pore space are embedded. As shown schematically in Figure 8-10, the first level (level I) of the model includes 201
Pc (C3S-M-RD-W42-T25-P1)
Pc (C3S-G-RD-W34-T25-P1) -Pc (C3S-F-RD-W49-T25-P1)
1040 -
-
1020
--
-
-
1060 -
1000 980
0
960
~rn
-~
940 0
10
5
15
20
Time (hr)
Figure 8-9: Evolution of pressure at the bottom surface of the sample during the RDC test with 60 kPa check valve. Pressure at the bottom port remains constant until the slurry is set and the bleed water is consuned by the volume of the reaction. Pressure
at the bottom port starts dropping at a rate proportional to the rate of hydration and a maximum drop equal to the cracking pressure of the check valve.
the C-S-H colloids and the interstitial space also known as gel porosity. The second
level (level II) includes level I in addition to the capillary porosity. Finally. residual (un-reacted) cement grains and other non-reactive solid inclusions embedded inside
the porous matrix define the continuum scale (level III). Driving forces of bulk volume changes are modeled as eigenstresses at the corresponding scales.
The goal of the
nicromechanical modeling is to establish the relationship between the macroscopic observations and their finer-scale origin by considering the intrinsic properties of the
material constituents and the driviNig forces of bulk volume changes.
8.3.1
LEVEL I (The C-S-H gel)
Following the framework developed in Section 5.1. the first level (level I in Figure
8-1() of this model includes the C-S-H colloids and the interstitial space also known as gel porosity.
Here, we consider a mli(roporomnechallics model of the C-S-H phmse.
which is conposed of the C-S-H colloids and the gel porosity. in additiol
202
to the
OU,
o
OC C(CH
Level (I)
Level (II)
Level (III)
Figure 8-10: Three-Level thought model of hydrating C3 S paste. The first, level (level I) of this model includes the C-S-H colloids and the interstitial space also known as gel porosity. The second level (level II) includes level I in addition to the capillary porosity. Finally. residual anhydrous cement grains and other non-reactive solid inclusions are embedded inside the porous matrix define the continuum scale
(level III).
eigenstress due to the densification process.
The volume fraction occupied by the
components at the first level is:
f
(8.23)
= fgei = fCsH 1/gel
where i1gel is the packing density of the C-S-H gel and fJJISH is the volume fraction
occupied by the C-S-H solids as defined by Eq. (9.7). The level I state equations define changes in stresses and gelporosity at constant hydration degree., and read as: Incremental mean stress in the C-S-H gel (Level I)
(1
= do(z)) = KIdE' + (I - b') do-* - b' dp
where p(; is the pressure prevailing in the gel porosity.
(8.24)
The gel porosity change is
defined at constant hydration degree. Applying the self-consistent scheme, the poroelastic properties at this scale read
203
as [193-195]: K' ks
1 - bV 4GI/gs + 3 (1
-
(8.25)
r') rs
where rs = k/g8 = 2 (1 + v) /3 (1 - 2v') and:
-
=
(1 -
-
+
16
1r9 (2 + 171)
(8.26)
V"144 (1 - rs) - 480,q' + 400 (r7)
2
+ 408rsrI - 120r8 (71)2 + 9 (rs) 2 (2 +
with rTI the packing density of the C-S-H gel:
(8.27)
n= (/a)/
/In
with 3 = In
/a
(7rn),and
oo
m=
./r ( o is the hydration degree
threshold at which the solid particles percolate r1o = 1/2). The packing density is derived by fitting meso-scale simulations by Masoero et al. [121] as discussed in the subsequent sections.
This densification is captured by a power relation of the form [195]:
(8.28) where qlim is the C-S-H limit packing density (at complete hydration
jim), which
depends on poly-dispersity of the C-S-H particles [121], (typically, qlim = 0.64 for Lowdensity C-S-H;
ulim
= 0.74 for High-density C-S-H, [47,200]); While a is determined
from the percolation threshold: In a-=
tm
ln
(-)lim
(8.29)
with /o = 0.5 the packing density beyond which the hydrating matter can support a deviatoric loading. 204
ql)2
8.3.2
LEVEL II (C-S-H gel with capillary porosity)
At level II, we consider a second pore space, the capillary porosity, in which another pressure prevails, denoted by p and the C-S-H gel (C-S-H solid and the gel porosity) makes up the solid phase. The volume occupied by the hydration products is designated Vel and the space occupied by the capillary porosity is Vcap. The volume fraction occupied by the second level is:
f=
-
fI +
qcap
fgei
=
(8.30)
+ Ocap
The porosity of the second level is then defined as: II =
~ap -f
(8.31)
'I
The incremental stress state equation for the stress distribution at level II reads:
dum (z) = k(z)dc(z) + duP (z)
(8.32)
with:
k(z)=
d-P (z) =
d PI = (1
KI Vz (2 Vge 0 Vz E Vcap
(8.33)
b') do-* - b' dpG
Vz G
Vgle
-dp
Vz
Vap
8.34)
The eigenstress at Level II is obtained by the application of Levine's Theorem:
=
(1
-
W) dZPI - bIdp
8.35)
(
dEP1 = {A,(z)doP(z))
Whence, the incremental equation of state: d E' = (do,,, (z)) = K'dEII + (1 - bI - bW) do-* - bI dpG - b,'dp
205
(8.36)
wiere:
(1-
=
1)/i'
-
2)
7"Ai41
(8.37)
K' K"
(8.38)
-
K"
K' K'\
-bj)
i j'= = K1I
(1 - b/) (I - b')
(8.39) (8.40)
For identical pressure in the capillary porosity and the gel porosity dp; = dp. we have: Incremental mean stress at the second level (Level II)
d(EI
=
Kdo1
(z)) = KId Ef + (1 - b") du*
-
b" dp
(8.41)
dZP'
where:
b"l = bbl=1b.?)bI +bW =I-
(8.42)
The elastic properties at this level are calculated with a M\Jori-Talnaka scheme [193
K" KI
Gil GI
4 (1 3
-
195]: (8.43)
, IIrhyd + 4
(1 -") (8 + 9.h4d) 6H" (2 + rhlY() + 8 + 91r.Yd
(8.44)
where rhyd = K 1 /G.
8.3.3
LEVEL III. Reinforcement by Rigid (but Slippery) Inclusions reinforcing effect of the residual anhydrous cement grains
The third level
includes the
as a two-phase
composite. One phase is the porous matrix formed of the hydration 206
Iw
products in addition to the gel and capillary porosity (Vm = Vgei
+ Vcap). The sec-
ond phase is the residual anhydrous cement grains and non-reactive solid additives and portlandite. A convenient way to model the second phase is by considering the anhydrous cement grains and the solid additives as rigid inclusions un-bonded to the porous matrix (slippery inclusions). The assumption of discontinuous slippery interface simplifies the strain localization tensor but does not affect the volume integral. Inclusions like the residual cement grains can also undergo eigenstrains due to the progression of the hydration reaction. Such contribution to the overall behavior of the hydrating material will be included in this level of the multiscale microporomechanics model.
Stress and Strain Localization While the inclusions are rigid, it's worthwhile considering the strain localization in the matrix. To do this, we consider an eigenstress-free situation according to which
d EII = KII'd EI
where K"'I is the homogenized stiffness tensor. Then consider:
dZEII = (dom(z)) = (1 - fi ) (K dEvI) + fcdo-m where
fc, is the volume fraction occupied by the residual anhydrous cement, and do-
is the mean stress in the inclusion:
fi douIn = K"'!dEI"1 - (1 - fci) (KIldE Assuming that dorjc = KintdEn (with Kin' an interface stiffness, and dEv"nc 1M
f du dS ). Then,
feidE In V = KI"' - (1 - fci) Kint KI" dE" it dEI"' v V
207
or, with strain localization factors (dEvflC = AincdEv!II, and dEf'' = AmdEtIII), while
considering strain compatibility, fcjAinc + (1 - fc) Am -nc
fiAi
=
1:
K"'I - KI
Kint (1 - KII/Kint)
Thus, the inclusion stress: Kint inc
Incw
KI'I - K "I
1-
1l
=1 1 - KII/KInt fel I - KIII/ int
K fci1 - KII/Kint d
(lm
"M
and the total stress:
dY:Il = ill m For Kint
-+
d !!=K"
1 + KIIIK int (I - KIIII/Kint
EI d E"
oc, and hence Ain, -+ 0, we naturally recover dEI' = KI'IdE7"', and
the stress localization: do Ic = BdE4I ; B, =
1
KI KIII
(8.45)
Finally, an application of a self-consistent model for rigid inclusions with slippery
K"' KII
G"' Gil
=
(
.'
GI"/K", and:
1
1
24 (2 - 3fjc) + 9(8fc -
The model is restricted to
(8 (3 -
2.fc) - (15 - 24fe) r-im
5)2 (r7)2
f,
"A/vm 180 deGY
3
.
--- - -
)
.1
i
.7
-3
5
.9
-
2
-
.b
--
------
-
-
4
I
PA
Fig. 3-8 (above)--Effect of original w/c on w/V. curves 0
Fig. 3-9 (right)-Effect of wet-curing on w/V curves
.1
.2
.3
.4
.5
6
.7
b
.9
0
p
to show the effect of w/c (left) and the effect of age (right) Figure 8-19: Typical w/l behavior of cement-based materials, from [157]. on the adsorption
8.4.2
Densification of C-S-H
One way to view the densification process of C-S-H gel is possible
)v
considering the
reaction mechanisms and the inner- and outer-C-S-H porducts. C-S-H as formned by the hydration of Alite is typically classified into two types, the outer product C-S-H (Op-CSH) and the inner product C-S-H (Ip-CSH) [179]. Inner product C-S-H (IpCSH) is the C-S-H formed inside the original boundaries of the cement grains (alite and belite particles). Ip-CSH is generally a high density C-S-H (HD-CSH); but IpCSH from small cemnent grains is usually LD-CSH [164]. It is highly likely that the IpCSH is formed through a topochemical reaction in a diffusion-inited reaction [179].
Outer product C-S-H (Op-CSH) is the C-S-H formed outside the boun(arv of the 219
I
5
A Fig. 3-10
Figure 8-20: Schematic representing the model of cement paste iicrostructure as depicted by Powers and Brownyard model from Ref. [157]. The shaded area represents the porous cement gel with inclusions of non-colloidal (microcrystalline) material. (A) is the cement paste with high water-to-solid ratio and the capillary lenses (capillary bridges) represent the capillary water in the model. (B) is the model for the structure of cement paste with low water-solid ratio where all the capillary pores are filled with porous hydration products (cement gel).
4c
structure. Gel particles are repreSimpliied model $ pte sented as needles or plates; C designates capillary cavities. Ca(OH2 Fig. 1.
crystals, unhydrated cement, and minor hydrates are not represented.
Figure 8-21: Schematic represe11tinlg the niodel of cement paste microstructure as (lepicte(d iby Powers from Ref. [159].
I 220
--
original
cement grains: and it usually has a low density and incorporates all kinds of
ions available in the solution in its structure. Another classification of C-S-H
call be found in the work of Tennis and Jen-
nings [183] and Constantinides and Ulin [46] in which C-S-H is classified based on its packing density. High Density C-S-H (HD-CSH) is characterized by high density and constitutes the major part of the Ip-CSH of the large cement particles and its porosity is not accessible by Nitrogen. Low Density C-S-H (LD-CSH) is characterized by the low density and constitutes most of the Op-CSH. Most of the porosity of the LD-CSH is accessible by nitrogen [183]. Densification of C-S-H gel was also studied experimentally for different drying conditions [163] and for saturated cement paste [133]. The desification of C-S-H in the work of Muller et al. [133] seems to be dependent on the initial porosity (waterto-solid ratio) with the ultimate packing density decreasing as the water-to-solid ratio increases. Mikhail et al. [128] investigated the pore structure of fully hydrated cement paste with w/c ranging from 0.35 to 0.70 and found that aln increase in w/c will result in an increase in the average size of the gel pores and consequently, a lower density C-S-H gel (see Figure 8-22).
120 140 120
0
0
0
0
0
0.2
0.4
0.6
0.8
1.0
S
Figure 8-22: Nitrogen sorption isotlierms on hardened portland cement paste. after
[128]. Whether thc densification of the C-S-H gel is a continuous process or simnply due 221
to the formation of inner and outer products, it seems that the initial porosity has a significant impact on the densification. The fact that the densification of the C-S-H gel is dependent upon the initial water-to-solid ratio indicates that the impingement of the hydration products plays an important role in the process. The rate of the hydration reaction during the early stage of hydration is driven by nucleation and growth and nearly independent of the water-to-cement ratio [19]. Once the reaction becomes diffusion-limited, the rate of hydration becomes strongly dependent on the water-to-cement ratio [16].
The influence of the w/c is typically attributed to the
available space for the hydration products to form by linearly relating the rate of hydration to both the volume fraction of water-filled porosity and the volume fraction of residual anhydrous cement [16]. A closer look at the evolution of porosity (Figure 8-22) indicates that the impact of w/c on the reaction rate starts long before the w/c becomes a determining parameter for the diffusion-limited reaction. That is, for a large range of water-to-cement ratios, the rate of the reaction is affected even though there is still enough empty space for the hydration products to form. This leads to the conclusion that the rate determining parameter is not the availability of water-filled pore space but rather the mean diffusion distance traversed by the both the solute and the solvent. The concept of the effect of the water-to-cement ratio on the mean diffusion distance (MDD) is illustrated in Figure (8-23). At higher w/c, the MDD remains low due the high initial separation distance between the cement grains.
At lower w/c
ratios, the impingement of hydration products occurs at earlier age and at a higher rate leading to a drastic increase in the MDD. For higher water-to-cement ratios, the mean separation distance between the cement grains is larger and the impingement of the hydration products occurs at later ages and at a slower rate.
Impingement
of the growing particles increases the MDD drastically depending on the initial w/c ratio. The increase in the MDD leads to an increase in the degree of super saturation with respect to the hydration products through the hydration shell which leads to the precipitation and densification of the C-S-H gel. Berliner et al. [19] found an exponential relationship between the effective diffusion coefficient and the w/c ratio
222
independent of the particle size distribution. r
Supersaturated Bulk solution
Bulk solution Neighbor particle
High supersaturation
Shell Core +
Concent ration Core/Shell Interface
Outer --------
*-P4 Solute *-- Solvent
surface
(a) Reduced accessible sesurface area
acces sible surface area
-Complete -
VNew
(b)
outer hydration products New inner hydration products
(c)
Figure 8-23: A cartoon illustrating the impact of w/c on the mean diffusion distance (MDD). For higher water-to-cement ratios, the mean separation distance between the cement grains is larger and the impingement of the hydration products occurs at later ages and at a slower rate. Impingement of the growing particles increases the MDD drastically depending on the initial w/c ratio. Fig. (a) shows the effect of thickness of the hydration shell on the concentration gradient of the solute ions of a partially hydrated cement grain. Fig. (b) the MDD is exactly the thickness of the hydration shell. Fig. (c) The MDD increased drastically due to the impingement of the growing hydration products. The increase in the MDD leads to an increase in the degree of super saturation with respect to the hydration products through the hydration shell which leads to the precipitation and densification of the C-S-H gel.
I 8.4.3
Meso-scale simulations
Advances in miolecular modeling of cement-based materials provide a powerful and versatile mieans to exploring the structure and properties of C-S-H [96.121,122,148].
223
Among all the possible driving forces of bulk volume changes in cement-based materials, we discuss here the possible contribution of stresses resulting from the precipitation of the C-S-H gel. Precipitation and the subsequent densification of the C-S-H gel seems to induce significant eigenstresses in the material as shown by the simulations of Masoero et al. [121], Ioannidou et al. [96] , and micromechanical modelling by Ulm et al. [195]. Stresses can be estimated in a simulation box using the known expressions for the virial stress (at zero kelvins for simplicity) [135] 1
N
1
=E
SS(x x) -
k1
(8.58)
k=1 j>k
where, k,l are colloids in the REV, VREV is the volume of the REV, x
is the ith
component of position of colloid k, and J' is the JIh component of the force applied on colloid k from colloid 1. The idea is then to relate the measured stress to quantities that can be controlled experimentally and can be measured at a macroscopic scale. This link is based on the understanding of colloidal forces and the controlling parameters of these forces. In general, these forces are dependent upon the properties of the interstitial liquid, like pH, ionic strength I, and dielectric permittivity Er. These forces can also depend on the surface charge density a, of the solid particles, which in turn depends on the properties of the interstitial solution and the constitution of the solid particles. Another important parameter that can control the colloidal forces is the temperature
T. These forces can also depend on the chemical composition of the interstitial solution, which might dictate the type of counter ions available between the interacting surfaces. Relating the stress developed during hydration and the aforementioned parameters can then be incorporated in a multiscale micromechanics model to study the contribution of precipitation and densification forces on the bulk volume changes in cement-based materials [193-195]. Development of such relations can be equally important to study the contribution of other colloidal forces (other than precipitation forces) in hydrating and mature cement-based materials. Colloidal forces in cement224
based materials can be modified by changing the chemical and physical properties of the pores solution including the solvent exchange. From a simulation point of view, this requires the development of interaction potentials that take into account these variables and then use these potentials to estimate stresses.
In the following we discuss and clarify the significance of the tensile stresses developed inside the simulated C-S-H gel as modeled by Masoero et al. [121]. In these simulations, the precipitation of the C-S-H gel is simulated by successively and randomly inserting particles into a cubic periodic simulation cell. At early stages of the simulations and due to the absence of mechanical percolation, all the inserted particles are at equilibrium and the overall stress in the system is zero. Once mechanical percolation if reached, however, insertion of a new particle will induce stresses in the system because the existing particles are not free to rearrange anymore.
Stresses
induced by insertion in a percolated system are generally tensile stresses because the long range interactions are attractive. This can simply be understood as increasing the attractive interactions per unit volume of the material which will result in reduction of the volume (eigenstrain) in an NPT ensemble. Such effect can be seen in the
virial stress Eq. (8.58).
In Eq. 8.58, and due to the attractive potential and keeping
VREV
constant and
inserting a new particle will result in an increase in tensile stress if the resultant of the interactions of the newly inserted particle is attractive.
The magnitude of this
eigenstress will depend on the PSD and the interaction potential between the particles. The interaction potential used in these simulations is a generalized Lennard-Jones potential form:
Uij (rij) = 46(&ij)
I(
(rij
2) (On) rj
(8.59)
The potential is parameterized to describe the physical properties of C-S-H in mature cement paste.
Other potentials are possible. E.g.
Ioannidou et al.'s potential [96]
for early hydration (eqn. 8.60), and can give different results for the volume changes, where the potential is based on generalized Lennard-Jones attraction and Yukawa
225
[154].
repulsion and based on the experimental measurements by Plassard et al.
U(r)
As a
4E
-
-
21
rI )r
+ AC I
(8.60)
consequence of the insertion procedures in Masoero's simulations, the system is in
tension at the termination of insertion because of the strong inter-particle attraction and because the structural rearrangements are carried out under pVT condition. An example of the evolution of such stresses is shown in Figure 8-24.
900 800
-
700 -
*
600
1
POLY 10 POLY 5 POLY 2 MONO
llini
=
(S
0.60 0)
500
1
400
(6 = 0.19)
rm
=
0.68
-
.RP 300 (b)
200 100
hin1 =
0.75
(6 = 0.47)
0 -100 0
0.2
0.8
0.6
0.4 Packing density
(Vor
Figure 8-24: Virial stress in the system resulting from the simulated precipitation process. Tension due to jammning of "out-of-equilibrium" colloidal systems where (lensification (increase in packiniig density) is due to progressioi of the hvdrationm reactions. Eigemnstress as calculated from Virial Stress at the end of Insertion Delay ill 'T enisemble. Delay niot log enough (precipitation too fast) to reach equilibrium from the work of M\asoero et al. [121, 122]. The value of eigenstress and associated volume camiges developed iin the simulated material depenid to some extent on the mmechaiism of precipitation. the deep interaction well, the shape amid polydispersity of the particles. and the plT
ensemble.
The interaction potential sed in Masoero s silmllations was parameterized to simulate the interactionms iii mature
cemeit paste, so its applicability to hydratimg cenmelit 226
at early ages is not obvious. A more relevant approach to simulate the volume changes during the hydration of cement at early ages is using potentials based on the ionic correlation forces as appears in the work of Ioannidou et al. [96]. In Ioannidou's simulations and due to the presence of a repulsive shoulder (based on AFM experiments by Plassard et al. [154]), the net stresses developed in the system were compressive and swelling was measured upon relaxation of the system. A more realistic case would be an NPT ensemble with account for the aqueous solution and transformation of part of the solution to form the particles. Thus far, both the work of Masoero and Ioannidou have worked with implicit solution and neglected the effect of the solution on the pressure.
Hence, only after setting we can
impose a control of the pressure, alternating pVT (Grand Canonical) with NPT (isobaric), to have both precipitation and control over pressure to measure the associated shrinkage.
8.5
Summary
Bulk volume changes during the hydration of C 3 S showed a general behavior of expansion during early age followed by shrinkage at later ages. The expansion part of the bulk volume changes are interpreted in light of the findings during the hydration of MgO; and is related to the difference in volume between the reactive solids and the hydration products. The solid volume change in conjunction of the reaction mechanisms are then employed to model the expansion. To interpret the shrinkage strain on the other hand, a multiscale thought-model is developed for the saturated hydrating
C3 S paste. As shown schematically in Figure 8-10, the first level (level I) of the model includes the C-S-H colloids and the interstitial space also known as gel porosity. The second level (level II) includes level I in addition to the capillary porosity. Finally, residual (un-reacted) cement grains and other non-reactive solid inclusions embedded inside the porous matrix define the continuum scale (level III). Driving forces of bulk volume changes are modeled as eigenstresses at the corresponding scales. It was possible with this model to quantify the development of eigenstresses during the hy-
227
dration reaction and relate it to the main parameters such as the degree of hydration, rate of reaction, packing density of the C-S-H gel, and the solid volume fractions. The resulting eigenstresses are then attributed to the densification of the C-S-H gel resulting from an out-of-equilibrium precipitation of the C-S-H particles.
228
Chapter 9 Expansion of
#-C 2S
The main differences between C 3 S and C 2 S is the quantity of CH formed during the
hydration reaction and the rate of the reaction. The difference in the formed CH is used to assess the contribution of CH to the measured bulk volume changes.
9.1
Kinetics of C 2 S hydration
The hydration kinetics of C 2 S is monitored with both the enthalpy of the reaction in isothermal calorimeter and the volume of the reaction in the pressure vessel (see figure 6-16 and 6-17).
The relationship between the measurements is fairly linear
with a ratio of 7.60, 7.46, and 7.23 J/pL for the fine, medium and coarse materials, respectively. Assuming the enthalpy of hydration of C 2 S to be Hrxn = 260 J/g [182], the corresponding volume of the reaction is 35.00 t 0.88
9.2
/L/gc
s.
2
Evolution of volume fractions during the hydration of C 2 S
To estimate the evolution of the volume fractions of the different components of the system, we consider the chemical reaction below along with material properties in 229
-Hrxn (J) (C2S-F-W45-T40) - Hrxn (J) (C2S-M-W45-T40)
40
-
20
-
10
-
30
-
50
-
60 x =
-
70
-
-
-
Hrxn (J) (C2S-G-W45-T40)
100 90 80
0
10
5
0
15
Vrxn (pL/g)
Figure 9-1: The relationship between the volume and enthalpy of reaction as measured with isothermal calorimetry and the pressure vessel for C 2 S samples with different PSD cured at 40'C with w/c = 0.45. The relationship is fairly linear with a ratio of 7.60, 7.46, and 7.23 J/pL for the fine, medium and coarse materials, respectively. Assuming the enthalpy of hydration of C 2 S to be H,,, = 260 J/g, the corresponding 0.88 pIL/gc 2 s. volume of the reaction is 35.00 Table A.2. the reaction is based on the forntulation of Young et al. [212].
CSHY + (2-x)CH
C2 S + (2-x+y)H ,
where x is the C/S ratio and y is the H/S molar ratio.
(9.1)
For the chemical reaction
considered here x = 1.7 and y = 2.1. The resulting balanced chemical reaction is:
CS+ 2.4 H
-"
C. 7 SH2 .1 + 0.3 CH
(9.2)
The volume fractious of the different components are then:
f(s= (1-no)(l ,I =- -
1)(1
-
-n)
(9.4) (9.5)
(1 - O)R 21
(9.3)
)
fr-n = (Rh, - RH
)RCsH
(9.6)
230
I
fCH
=
no)RCH
(9.7)
fgel
- (1 - no)Rgei
(9.8)
(1
-
And the total porosity of the system is: 1
otot =
fC 2 s
-
fAp
-
no - (1 - no)(Rhp -
=
1)
while the capillary porosity is:
cap =
1
f(C2 s)
-
fgel
bgel = Otot -
#cap
-
-
(9.10)
fCH
Finally, the gel porosity is:
where Rh
-x)V (2
H +VCSH
(9.11)
is the ratio between the molar volumes of the hydration
products and that of C 2 S, RH
=
(2-x+y)VX VU26
is the ratio between the molar volumes
of water and the reactive solid (C 2 S), RCSH is the ratio between the molar volume of C-S-H and C 2S. frxn is the fractional volume of the reaction, fraction occupied by the remaining non-porous solid C 2S,
fmw
f(C
2
s) is the volume
is the volume fraction
occupied by the remaining initial water used for mixing (initial water is treated as a separate phase just like solids), fCSH is the volume fraction occupied by C-S-H as defined by RCSH, no is the initial porosity upon mixing, and
is the fractional
conversion (degree of hydration). fhp is the volume fraction occupied by the hydration products CH and C-S-H (see Figure 9-2).
Figure 9-2 presents the evolution of volume fraction of the various components during the hydration of C 2S with water-to-solid ratio ofw/c = 0.34 and w/c = 0.42. The yellow spot marks the degree of hydration where all the remaining free water is inside the C-S-H gel. fm w is the volume fraction occupied by the free water remaining from the original mixing water only (not including any water provided from the outside during the hydration).
fe,
is the volume fraction occupied by free water 231
tCusH is the volume
remaining from the initial mixing water in the capillary porosity. fraction of the solid C-S-H only. porosity.
1
2I
fl
0.7 Eht +AFm
--------
fCH
~0.6
+ .fEtt +
AFm 11
-C--
0.5 0.4
'-
-
CH
0. 4
fCS
0.3
0.3
0
0.2
--------------
.k'i-
0.1
J''
0.1
------------------
0.4 (a) w/c
0.6
0.8
0.2
10
0.4 (b) w/c
0.30
0.6
0.8
0.42
Figure 10-2: Evolution of volume fraction of the various components during the hydration of CGC with water-to-solid ratio of (a) w/c = 0.30 and (b) w/c = 0.42. The yellow spot marks the degree of hydration where all the remaining free water is inside the C-S-H gel. f 1 1, is the volume fraction occupied by the free water remaining from the original mixing water only (not including any water provided from the outside during the hydration). fe t is the volume fraction occupied by free water remaining from the initial mixing water in the capillary porosity. fCSH is the volume fraction of the solid C-S-H only. 9qd is the volume fraction occupied by the gel porosity. The capillary porosity (/co is represented by the areas shaded in blue. The dark blue area ) The total represents part of the water uptake from outside the sample (Oca, porosity of the system is cb, =- co + +ge. The volne fraction occupied by the =fC[SH i OgelC-S-H solids and the gel porosity is defined as the C-S-H gel
248
resulted in large sedimentation. To interpret the measured bulk volume changes during the hydration of saturated Class-G cement samples, we employ a multiscale microporomechanics model similar to the model developed for the hydration of C 3S in Section 8.3. The main difference between the hydration of C3 S and Class-G cement is the evolution of the volume fractions of the paste components as shown in Section 8.2.3 and Section 10.1.
In
the simpler C 3 S hydration, the reactive solid is C 3 S and the hydration products are CH and C-S-H gel only. In Class-G cement the reactive solids include C 3 S, C 2 S, C3 A, C 4 AF, and CSH 2 and the hydration products include CH, C-S-H gel, Ettringite C6 AS 3H 32 , monosulfate C 4ASH 12 , and other calcium aluminates like C 4 AH 13 and .
C 2 AH8
As shown in Section 10.1, the hydration of gypsum and the formation of ettringite occurs during the early age of hydration, mainly during the first 16 hours at ambient conditions (T = 25'C and p = 1 atm). The depletion of CSH 2 is followed by the hydration of C3 A to form mainly C 4 AH13 and C 2 AH8 during the first 30 hours. The formation of ettringite and AFm phases can lead to the observed expansion during the first 30 hours of Class-G hydration as shown in Section 6.4.2. The behavior of the material after the first 30 hours depends on the w/c and the particle size distribution. To model this behavior, we employ the multiscale microporomechanics model with the following volume fractions: 1. Level I includes the C-S-H gel as composed of the C-S-H solid and the gel porosity. The eigenstress that can lead to contraction in the paste is attributed to the gel formed at this scale: =frCHi+g
=
fCSH
+ge -Tgel
_
1
fCSH ~
Pge(
2. Level II includes the C-S-H gel (Level I) in addition to the capillary porosity similar to the model of the C 3 S:
P
=
f' +
249
Ocap
(10.28)
3. Level III includes the residual anhydrous cement grains in addition to all the hydration products that do not, contribute to the (leveloinent of eigenstresses as well as gel and capillary porosity and non-reactive solids. The evolution of elastic properties of Class-G cement during hydration with the volume fractions discussed in this section are shown in Figures 10-3. 10-4, and 10-5. -- -----
-Level
I
-Level
III
0. 8~ -Lvel
0.5_
-
-
0.4
II
0 .6
0.3
0 .4
0.2
Level
1
-- Level 1.1 -- Level III
0 .2
01 0.6
0.4
0.2
0o
0.8
0
1
0.2
0.6
0.4
0.8
1
(a)
Figure 10-3: Evolution of the shear modulus G/gs during the hydration of CGC with (a) w/c =- 0.30 and (b) w/c = 0.42 as calculated by the mmultiscale nicroporomnechanics model for the three levels in function of hydration degree
0.5--
1.4-
1.2
-Level -Level S-Level
I 11 III
I -- eel II -Level III -Level
0.4
0.3
0.8
0.6
0.2
0.4 0. 1
0
0.2
0.6
0.4
0.
0
0
-
0.2 0.2
0.6
0.4
0.8
(1) (a) Figure 10-4: Evolution of tie bulk modulus K/Ai during the hydration of CGC with (a) wc = 0.30 and (1) c = 0.42 as calculated by tlhe multiscale microporomeehanics model for the three levels in funiction of hydratioii degree
0.95
0.95
0.9
0.9-
-
-
-
-
Af_-
-0.85
0.8 50.75
.0.75
0.7
0.7 --
0.65 -Level ---
0
--
Level I
II
--- Level III
Level III
0.2
Level I
0.65r -Level
11 0.6
0.4
0.8
1
0
0.2
0.4
0.6
0.8
1
C
(.
(a)
(b)
Figure 10-5: Evolution of the Biot coefficient b during the hydration of CGC with (a) 'w/c = 0.30 and (b) w/c = 0.42 as calculated by the multiscale inicroporomliechanics model for the three levels in function of hydration degree
10.2.1
Bulk volume changes in the Rubber Diaphragm Configuration
The experimental results of the bulk volume change study on Class-G cement (CGC) were presented in Section 6.4.2.
Depending on the PSD, degree of hydration, and
solid volume fractions, bulk volume change of Class-G cement displays high rate of expansion at early ages due to the hydration of aluminate and ferrite phases to form ettringite and monosulfates as well as calcium aluminate hydrates. The rate of early expansion is observed to change its rate upon the depletion of sulfate while it continues to expand at a comparable rate due to the hydration of C3A and formation of calcium aluninate hydrates. After the first ~30 hours of hydration, the expansion typically continues at a lower rate due to the formation of all hydration Products in a processes reminiscent of the expansion of MgO and C 3 5. Shrinkage of CGC is observed at later ages as shown in Figures 10-6 and 10-7 which indicates that large solid volume fraction is required for the densification of
C-S-H to occur. For all samples shrinkage starts at solid volume fraction larger than 0.8 except for sample (CGC-F2-RD-W45-T25) in Figure 10-6 where shrinkage starts at solid volume fraction of 0.76. The solid volume fraction for Class-G cement is 251
__
defined as follows:
SVTF
=
CH + f-s-Hos
1
(10.29)
-- + fAFm + fAFt
The development of eigenstresses during the hydration of Class-G cement are
then calculated using the multiscale incroporomechanics model. The resulting eigenstresses due to the observed expansion and shrinkage are shown in Figure 10-8 and Figure 10-9 as function of the degree of hydration and the solid volume fraction. Figure 10-8 shows the results of coarse
CGC compared to the reference CGC and
extreme expansion is observed in the sieved coarse sample; probably due to the very low solid volume fraction. Figure 10-9 shows the evolution of eigenstresses in Class-G cement with fine particle sizes. It is evident from both sets of tests that the initial 1/c ratio and the evolution of solid volume fraction play an important role in the observed bulk volume changes and eigenstress development.
---
-- Ev (CGC-F2-RD-W45-T25) -Ev (CGC-F1-RD-W33-T25) Ev (Ref-CGC-Full-RD-W31-T25)
Ev (CGC-F2-RD-W45-T25) Ev (CGC-F1-RD-W33-T25) Ev (Ref-CGC-Full-RD-W31-T25)
4
4
3
3
2
2
1
1
0
0
j
I 4 __
__
____________
__________
______
-1
-1 0
0.2
0.4
0. 6
0.4
0.8
0.5
0.7
0.6
0.8
0.9
SVF
(b)
(a)
Figure 10-6: Bulk volume changes as measured during the hydration of Class-G cenmet with fine particle size distribution and the reference material with full PSD (a) as function of degree of hydration (b) as function of the solid volume fraction assuming a constant packing density of the C-S-H gel at 0.64. Sample nomenclature is shown in Figure 6-1.
I 252
-
Ev (CGC-Cl-RD-W26-T25) Ev (CGC-C2-RD-W25-T25) Ev (Ref-CGC-Full-RD-W31-T25)
Ev (CGC-C1-RD-W26-T25) Ev (CGC-C2-RD-W25-T25) Ev (Ref-CGC-Full-RD-W31-T25) 12
- -- _
10 d
-
-_
10
_
8
-
-
12
8-
E
4 2
0-1 0
0.2
0.4
0.6
0.8
-
2
-
4
-
6
0.5
0.7 SVF
0.6
ceieit
0.9
(b)
(a) Figure 10-7:
0.8
Bulk volume changes as measured during the hydration of Class-G
with Coarse particle size distribution and the reference material with full
PSD (a) as function of degree of hydration (b) as function of the solid volume fraction assuming a constant packing density of the C-S-H gel at 0.64. Sample noInenclature is shown iii Figure 6-1.
I y*
a*
(CGC-C1-RD-W26-T25) (CGC-C2-RD-W25-T25) cr* (CGC-Ref-FuIl-RD-W31-T25)
(CGC-C1-RD-W26-T25)
a* (CGC-C2-RD-W25-T25) a* 0
-c*
(CGC-Ref-FuI-RD-W31-T25) 0.4
0.2
0.6
0.5
0.8
0.9
-500-
-
-1500
-1500
-
-2000
-2000
-
-2500
-2500
-c
-
-
-1000
-
-500
-1000
0.8
-
0
0-
SVF 0.7
0.6
(b)
(a)
Figure 10-8: Development of eigenstresses as calculated from the multiscale microporoiechanics model for different PSD and v/c ratios of class G-Ceient (CGC). (a) in function of degree of hydration and (b) as function of the solid volume fraction. A reference material with (Full) PSD is colnpared to coarse CGC as obtained by sieving (CI) and sedimentation (C2).
253
-I
--
- * (CGC-F2-RD-W45-T25) --- * (CGC-F1-RD-W33-T25) (y* (CGC-Ref-FuIl-RD-W31-T25)
-Y*
* (CGC-F2-RD-W45-T25) (CGC-F1-RD-W33-T25) a* (CGC-Ref-Full-RD-W31-T25)
SVF 0.2
0.4
0.6
0.8
0.4
1
100
100
0.5
0.6
0.7
0.8
0.9
-
0
0-
-100
-100
-200
-200
ci
-4--
-
0
-400
-
-
-400 -500
-
-500
-
-300
-
ci
-300
(b)
(a)
Figure 10-9: Development of eigenstresses as calculated from the nultiscale imicroporomiechanics model for different PSD and w/c ratios of class G-Cement (CGC). (a)
in function of degree of hydration and (b) as function of the solid volume fraction. A reference material with (Full) PSD is conipared to fine CGC as obtained by sieving (Fl) and sedimentation (F2).
10.2.2
Bulk volume changes in the Liquid Barrier Configuration
The second configuration employed to measure the bulk volume changes during the
hydration of CGC is the Liquid Barrier Configuration (LBC). A set of tests are carried out on CGC with w/c = 0.42 at 25'C with different thicknesses and curing pressures as listed in Table 3.5. The liquid barrier test separates. by design. volumie change neasureients from the
water mass absorbed as discussed in Section 5.4. To achieve this separation, LBC test is performed by placing the slurry directly inside the pressure vessel and covering with approximately 10nun of the liquid barrier. The pressure and temperature (25'C) are then applied and bulk volume changes and water uptake are monitored
contiiously
for the duration of the test. The test starts with water access through the top port only aid after a specific period of timie related to the setting of ceient. water is allowed through the bottom port.
Vith both top and bottom ports connected to the
saMe source of pressurized water. the effective stress developed inside the speciiimen 254
is negligible in the axial direction and depends on the volume changes in the radial direction (see Section 5.4). The liquid of choice for this test is the Poly[methyl(3,3,3-trifluoropropyl)siloxane], Dow Corning FS-1265 chemically inert liquid. FS-1265 has a viscosity of 10000 cSt, a specific gravity of 1.3, a surface tension of 28.7 mN/m, and a molecular weight of 14000 g/mol. The values for the surface tension and contact angle of water in cement are usually assumed to be similar to that of water and glass. The contact angle between the siloxane, water, and cement is taken as 104' [104] (page 214). As observed in Section 6.4, it was only possible to measure the bulk volume changes after setting of the cement slurry and complete consumption of the bleed water. The measured volumetric strain of the tested samples with different thicknesses and curing pressures are shown in Figure (10-10). Similar to other bulk volume measurements, the samples display large shrinkage that coincides with the known volume of the reaction until setting and consumption of bleed water is complete.
The results in
Figure 10-10 are initiated at this instance. By comparing the strains measured in the RDC (Figure 10-6) and the LBC (Figure 10-10), a discrepancy is evident. The differences between the RDC and the LBC tests are detailed here in an attempt to understand the discrepancy between the volumetric strain measurements. The first difference is the dimension of the samples. In the RDC all samples have similar size of about 46.7 mm in diameter and 25.4 mm in thickness while the samples in the LBC have variable thickness with constant diameter of 53.9 mm (diameter of the pressure vessel).
The impact of sample thickness may cause
some pressure distribution inside the sample due to the evolution of permeability and the negative volume of the reaction. Shrinkage in thick samples is smaller than that of thin samples prepared under identical conditions. Another difference between LBC and RDC is the boundary conditions. In the RDC test, the sample is exposed to a hydrostatic state of stress and controlled (very small) effective stress. The liquid barrier test on the other hand, is constrained radially with an effective stress in the axial direction that is dependent on the evolution of permeability and the resulting drop in pore pressure due to the negative volume of 255
-
Ev (CGC-LB-PO.2-W42-T25-H47.3)
-
Ev (CGC-LB-PO.2-W42-T25-H17.8)
Ev (CGC-LB-P.2-W42-T25-H18.6)
-
Ev (CGC-LB-P5-W42-T25-H12.4) Ev (CGC-LB-P10-W42-T25-H44.1)
Ev (CGC-LB-P1-W42-T25-H44.1)
0.8
0.6
0.4
0.2
0 0
-1000
--
_
-___
-
-500
5.
-
-1500
-2000
-2500
__-
-_
-3000
-3500
--
-----
-
-
-4000
Figure 10-10: Volumetric strain (iii micro strains) as function of the degree of hydration of Class-G cement paste hydrated in the LBC test. Samples have different thicknesses (H in iii) and variable curing pressure (P in M\IPa) with constant w/c = 0.42 and constant temperature T =25'C.
256
---
(CGC-LB-PO.2-W42-T25-H47.3) 18.6) (CGC-LB-P1-W42-T25-H 44.1)
a* (CGC-LB-P.2-W42-T25-H
* 0*
(CGC-LB-P5-W42-T25-H 12.4) (CGC-LB-P1O-W42-T25-H44.1)
600
1~~~~~~
500
-i
__________
i
________
________________________
-
400
i
__________________________
/ /
0*
a* (CGC-LB-PO.2-W42-T25-H17.8)
0~ *
7 __
____/
____
200 ______
_____---
-
--
I--- --7~-
-~
/
/
____-
__
-
300
100
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 10-11: Development of eigenstresses (r* as function of the degree of hydration of Class-G cement paste hydrated in the LBC test.
257
* (CGC-LB-PO.2-W42-T25-H47.3)
--
c* (CGC-LB-PO.2-W42-T25-H18.6)
600
-*
a*
(CGC-LB-P1-W42-T25-H44.1)
500
(CGC-LB-P5-W42-T25-H12.4) (CGC-LB-P1-W42-T25-H44.1)
I-
I
-
400
(CGC-LB-PO.2-W42-T25-H 17.8)
,*
-
*
--.
*
300
100
-
200
0 0.55
0.65
0.6
0.7
1lgel Figure 10-12: Development of eigenstresses as function of the C-S-H gel packing density of Class-G cement paste hydrated in the LBC test.
the reaction. Radial stresses in the LBC depend on the volumetric strain as detailed in Section 5.4. The ratio between the elastic deformations of the LBC to RDC strains as described in Eq. (5.134) can range from 1/3 to 1/2 for Poisson's ratios of 0 and 0.2 respectively. That is, in all cases the strains measured in the LBC must be smaller in magnitude that that measured in the RDC. Yet, this is not the case as seen in the RDC (Figure 10-6) and the LBC (Figure 10-10). One can speculate other possible causes of this discrepancy include possible formation of water pockets between the barrier liquid and the top surface of the sample and creep of the FS-1265 under pressure. The formation of water pockets was observed in a test carried out in a separate container where such pockets formed by the bleed water due to the difference in density of the liquids (see Figure 10-13).
The
complication resulting from the formation of such pockets is mainly due to the small fraction of sample surface exposed to water which takes longer time to be consumed by the ongoing reaction and gives false shrinkage. To check for possible creep of the FS-1265, a representative sample of FS-1265 was placed inside the pressure vessel and the injected water volume was measured under a pressure of lOMP. Other than the effect of temperature fluctuation, there was no significant volume change in the system. FS-1265 does not creep (see Figure 10-14). Another possible source of discrepancy between the LBC and RDC tests can be speculated to be the ingression of the barrier liquid inside the specimen due to pressure gradient resulting from low permeability and the negative volume of the reaction. Evolution of the pressure for CGC samples with different w/c ratios is shown in Figure 6-24, where a pressure transducer is connected directly to the bottom port (port-3) in Figure 5-1(b). The permeability of identical sample is measured directly with the PVC configuration (Figure 5-1) by applying a pressure differential and measuring the flow rate through the top and bottom ports. To measure the volume ingress of FS-1265 inside the sample, we employ Eq. (5.124) in Section 5.4, reproduced here for convenience:
VLB(t)
= qAx(t) = A
-_ 259
J
(Pw - Pt + PC dt)
(10.30)
Water pocket Liquid barrier
Specimen
(b)
(a)
Figure 10-13: Formation of water pockets between the liquid barrier and the top surface of the specimen due to bleeding and the differences in densities. (a) a schematic illustrating the geometry of the water pocket (b) water pockets as observed in similar conditions to the LBC test. Water pockets can provide water to the specimen at a very low rate resembling a false bulk volume change.
12
10 6 0 0
0
0
20
30
40
50
60
70
Time (hr) Figure 10-14: Volume changes measulre( with only a sample of typical volunie of FS1265 inside the vessel. The measured volume is a result of temperature fluctuatiomis on the syringe pumIp within 0jpL corresponding to alout less than 1% of the measured bulk volume changes.
I
The cross sectional area of the specimen is A = 2.29 x 10-W2 , porosity
#
is esti-
mated from the evolution of volume fractions discussed previously, pw is the pressure in water right underneath the liquid barrier, pt is the pressure on top of the liquid barrier, pc is the capillary pressure,
[1
is the viscosity of the liquid barrier (1 = 13
Pa.s), and K is the permeability of the specimen.
To measure permeability, two samples are investigated, the first sample was cured under water for 6 days (144 hours) under 5 MPa pressure. Two pumps were used to supply water to the top and bottom of the sample, the pressure at the bottom was lowered to 1 MPa, resulting in nearly steady state flow over 22 hours with 4 MPa differential pressure. Figure 10-15 shows flow into the sample through the top surface (red) and flow out of the sample through the bottom surface (green) as measured by the pumps. The difference in the measured flow rates coincides with the volume of the reaction due to continuing hydration reaction.
Using Darcy's law with the
.
average flow rates, the permeability was 3 x 10- 20 m2
Pressure in the pore solution right underneath the liquid barrier can be estimated using Eq. 5.80, reproduced here:
h p dVxn Ms
PW = W2
dt
A8 +
(10.31)
where A, is the area of the specimen, m, is the mass of the anhydrous cement, K is the permeability of the specimen, p is the viscosity of the pore solution, h is the thickness of the specimen, and PA is the curing pressure applied through the liquid barrier and the bottom surface of the specimen.
A sample calculation of the Ingression of the
liquid barrier inside the sample is shown in Figure 10-16, where the resulting strain is in the order of the measured strains in the LBC. The 120pL at 200 hours for a 50 mL sample correspond to a strain of 0.0024 (2400 pE), which is comparable to the strains shown in Figure 10-10.
261
600
250 25.293t - 3602.7
-'=
400
-
-
200
11= 5.0004/- 956.34
150 200
100 E)
-_- Linrar (Vtop) -- LiiI "ar ' )r (VH
0
(pL-) -Vtop 50 ) 9Vbot (pl-) Linear (Vtop
50 0
-200
L4)
Linear (Vbot (pl))
-50 -400
-100
18.403t + 2605.2
-
- -2to
2
200
210
2011+ 416.23
-150
-600
190
170
160
150
140
230
220
Time (hr)
Time (hr)
(a)
(b)
240
.
Figure 10-15: Direct neasureinenit of permeability on Class-G ceient with w/c = 0.42 cured and 25'C curing temperature.(a) cured for 144 hr under 5 MPa curing pressure with differential pressure of 4 MPa and thickness of 51.9 nun. the permeability is 3.4 x 10-20 (b)cured for 193 hr under 200 kPa curing pressure with pressure differential 2 of 0.8 IPa and thickness of 54.2 nn. the permeability was 3.0 x 10-201,
g 140
phi=0.2
-phi=0.3
phi=0.4
---
phi=0.5
-
I-
120
1
100
80
_---_-
60 40 20
0 0
50
1 00 Tim ? (hr)
150
200
e the specimenildle to 50kPa Figure 10-16: Volnme of the liquid barrier ingressed isid( pressure dlifferential and 3 x 10-201m2 permeability. The measured vohunetric strain is ablout 0.001. which is in the order of the llieasured strains in the LBC.
262
10.3
Summary
Bulk volume changes of Class-G cement have been investigated under two different boundary conditions in the rubber diaphragm configuration (RDC) and the liquid barrier configuration (LBC). In the RDC, expansion at early ages was observed followed by shrinkage at later ages. Expansion is a result of the difference in volume between the anhydrous solids and the hydration products.
Eigenstresses are then
calculated based on the multiscale microporomechanics model. Eigenstresses seem to be dependent on the solid volume fraction as is the case during the hydration of C 3 S. Early expansion observed in CGC is attributed to the hydration of C3 A and C 4 AF and the origin of this expansion is similar to the expansion of MgO and C3 S, and not necessarily related to the crystallinity of the hydration products. In the LBC on the other hand, only shrinkage is observed after setting and consumption of the bleed water. Shrinkage is then modeled using the multiscale microporomechanics model and the eigenstresses are calculated. The discrepancy between the two boundary conditions is then discussed in light of the possible differences between the RDC and the LBC, including sample dimensions, elastic stress field, possible formation of water pockets, and the possible ingression of the liquid barrier inside the sample. Thin samples in LBC showed larger volumetric strain than thicker samples indicating that ingression of the liquid barrier might be a reason of the observed shrinkage.
263
Part
V
Conclusions and Perspectives
264
Chapter 11 Conclusions and Perspectives This chapter presents a summary of the main findings of the study related to the driving forces of bulk volume changes of cement-based materials in addition to the main contribution of this thesis. Based on the findings of this study, the limitation of the current study are presented and further research is suggested.
11.1
Summary of main findings
Bulk volume changes of cement-based materials are generally attributed to several driving forces related to changes in relative humidity including capillary tension, disjoining pressure, and surface energy (see Section 2.2 for more details). Such driving forces require a drop in the degree of saturation (relative humidity), which is typically attributed to self-desiccation (see for example [65]) or drying of the material due to evaporation of the pore solution. In this study, samples are maintained saturated and bulk shrinkage and expansion are still observed. It is possible to monitor the hydration kinetics for all the investigated materials (MgO, CA5, C 2 , CGC) with volume of reaction (chemical shrinkage) as well as enthalpy of reaction. All material display a negative enthalpy and volume of reaction as listed in Table 11.1.
The negative volume of reaction entails reduction in the
absolute volume of the hydrating system associated with a significant increase in the solid volume due to the difference in density between the reactive solids and 265
the hydration products. The increase in the solid volume during hydration can be represented by the ratio of the total volume of the products relative to the volume of the reactive solids, which is listed as Rhp in Table 11.1. Hydration of all investigated materials is associated with more than 100% increase in the solid volume fraction. The increase in the solid volume leads to the conclusion that an expansion should be expected in all materials, yet, the experimental observations show otherwise. During the hydration of the simplest investigated material (MgO), expansion is observed during the conversion of MgO to Mg(OH) 2 , representing only a small fraction of the difference in volume between the reactive solid and the hydration product. On the other hand, hydration of C 3 S displayed amore complex behavior starting with expansion followed by shrinkage at later ages.
The main difference between MgO
and C 3 S originates from the nature of cohesive forces in these materials. Cohesion in the MgO-Mg(OH)
2
is attributed to development of ionic-covalent bonds between
the precipitating Mg(OH)
2
grains with low surface charge density in a process that
resembles sintering of ceramics. Cohesion in C 3 S, C 2 S, and Class-G cement, on the other hand, is attributed to the strong electrostatic coupling between the colloidal particles of C-S-H, which forms the binding phase in these materials. The main difference between hydrating MgO and cement-based materials is the presence of the colloidal C-S-H phase that binds all other phases through surface colloidal forces. A possible driving force of such bulk volume changes is the precipitation and densification of the C-S-H gel as originally presented by Ulm et al. [195]. Densification of C-S-H is suggested to be driven mainly by the mean diffusion distance of the solute and the solvent, which is controlled by the thickness of hydration products surrounding the anhydrous cement grains and leads to the observed shrinkage. Thus, the densification of C-S-H depends mainly on the w/c, PSD, and curing temperature. It is found in this study that the main parameter controlling the densification of CS-H is the solid volume fraction which is a function of the w/c ratio and the degree of hydration. Densification of C-S-H and the resulting stresses are further supported by the available meso-scale simulations of the precipitation and densification of C-S-H [96,
266
121,122,148] as discussed in Section 8.4.3 and Section 2.4. In these simulations, the precipitation of the C-S-H gel is simulated by successively and randomly inserting particles into a cubic periodic simulation cell. At early stages of the simulations and due to the absence of mechanical percolation, all the inserted particles are close to equilibrium and the overall stress in the system is close to zero.
Once mechanical
percolation if reached, however, insertion of a new particle will induce stresses in the system because the existing particles are not free to rearrange anymore. Stresses calculated from the experimental results agree qualitatively with those from meso-scale coarse-grained simulations of C-S-H precipitation originating from the electrostatic coupling between charged C-S-H particles mediated by the electrolyte pore solution. Expansion of the paste, on the other hand, is due to the fact that the volume of the hydration products is larger that the original volume of cement powder. The extent of the expansion depends on the mechanism of the hydration reaction as well as the microstructure of the porous solid that is formed during the hydration reaction. Depending on the reaction mechanism, only a fraction of the increase of the solid volume can contribute to the observed bulk volume changes at the macroscopic scale. Hydration of MgO is associated with an expansion that is linearly related to the degree of hydration, indicating a constant fraction of the formed solid is contributing to the expansion. The hydration of CBM, on the other hand, displayed a non-linear relationship between the expansion and the degree of hydration due to the more complex microstructure.
Table 11.1: A summary of the investigated materials and reaction properties. Material
MgO C3 S C2S CGC
AVn rnRhp
AHrxn
pL/g
J/g
R
109 58 35 60
930 520 260 413
2.21 2.16 2.31 2.13
267
Binder
Cohesive
Phase
Forces
Mg(OH) C-S-H C-S-H C-S-H
2
Ionic-covalent Colloidal Colloidal Colloidal
11.2
Research contribution
The contribution of this study can be classified in three parts relating to the main experimental setup, the multiscale microporomechanics model, and the driving forces of the bulk volume changes.
11.2.1
New experimental apparatus
For purpose of this study, a new experimental apparatus was designed and tested on a number of materials with a range of chemical and physical properties.
The
experimental apparatus was motivated by the lack of proper apparatuses capable of simultaneous measurements of all the necessary poromechanical quantities; especially the control of curing pressure, effective stress, and curing temperature.
The main
advantage of the pressure control lies in the prevention of possible contribution of permeability and self-desiccation due to "partially sealed" conditions. For this reason, tests were carried out mainly under a pressure of 1 MPa to guarantee complete saturation throughout the test and eliminate any possible contribution from hygral forces like capillary pressure.
11.2.2
A multiscale microporomechanics model
A second contribution of this study is the implementation of a multiscale microporomechanics model. This model is supported by extensive experimental investigations to predict the evolution of stress in cement-based materials during hydration under saturated conditions. A multiscale microporomechanics is developed to interpret the measured bulk volume changes of saturated hydrating C 3 S paste. The multiscale microporomechanics model consists of a three-level model of the cement microstructure.
In this model,
CBM is modeled as a matrix of C-S-H particles in which all other solid phases and the pore space are embedded. As shown schematically in Figure 8-10, the first level (level I) of the model includes the C-S-H colloids and the interstitial space also known as gel porosity. The second 268
level (level II) includes level I in addition to the capillary porosity. Finally, residual (un-reacted) cement grains and other non-reactive solid inclusions embedded inside the porous matrix define the continuum scale (level III). Driving forces of bulk volume changes are modeled as eigenstresses at the corresponding scales.
The goal of the
micromechanical modeling is to establish the relationship between the macroscopic observations and their finer-scale origin by considering the intrinsic properties of the material constituents and the driving forces (eigenstresses) of bulk volume changes.
11.2.3
Driving forces of bulk volume changes in CBM
In saturated samples, the densification shrinkage starts due to the decrease of the space available for hydration products to form and the initiation of precipitation of C-S-H colloids inside the existing low density C-S-H. Precipitation of hydration products can only occur on surface covered with pore solution or inside the pore solution only and it cannot occur inside the gas (vapor) phase. Yet, the densification of C-S-H that leads to shrinkage of the paste is dependent on the mean diffusion distance, which in turn depends on the w/c, PSD, and curing temperature.
The
densification of C-S-H resembles very closely the behavior observed for the sealed samples as well as the saturated samples. Expansion on the other hand is attributed to the difference in volume between the reactive solid and the hydration products.
11.3
Industrial benefits and impact
The main objectives of the primary cementing of an oil well is to support the casing string and provide zonal isolation by filling and sealing the annular gap between the casing string and the borehole [136]. Lack of understanding of the state of stress in the cement sheath and how the sheath may fail can lead to loss of zonal isolation. Among all the possible reasons that can lead to failure of cement sheath, bulk volume changes that cement-based materials (CBM) experience during hydration was investigated in this study. When cement slurry develops its mechanical properties, it also undergoes volume
269
changes during the early age of hydration. If constrained, these volume changes will lead to the development of stresses in the hydrating cement. Figure 11-1 shows the development of macroscopic stress in a fully constrained material during hydration of Class-G cement at 25'C with w/c = 0.42. To predict failure due to loading during the service life of the well, one needs to understand initial stresses endured by these volume changes. Prediction of stress and strain development in the cement sheath at early ages is critical for performance evaluation. Yet, such a prediction calls for a profound understanding of the chemical and physical mechanisms at early ages in the cement sheath. The main contribution of this study is a multiscale microporomechanics model. This model is supported by extensive experimental investigations to predict the evolution of stress in cement-based materials during hydration under controlled conditions. Besides predicting the evolution of stresses and strains in hydrating CBM, this model also pins down the main parameters that control the evolution of the eigenstresses and strains.
Once the impact of these parameters on the bulk volume changes is
understood, only then it is possible to control the resulting stresses and performance of the cement sheath or CBM in general. The implications of this model extend to the development of an engineering model to predict the performance of the cement sheath during the service life of the oil well.
11.4
Limitations and Perspectives
Results obtained with the rubber diaphragm configuration (RDC) and the liquid barrier configuration (LBC) provided an apparent Poisson's ratio v = 0 for the hydrating magnesium oxide if elastic deformations are assumed, yet, this is not the case during the hydration of Class-G cement. The discrepancy in bulk volume changes of ClassG cement as measured with RDC and LBC cannot bc explained with the Poisson's ratio like is the case in the MgO. This discrepancy is attributed to several possible differences between the RDC and LBC. Yet, a conclusive analysis was not possible. Bulk volume changes have been studied at a macroscopic scale and it is found that
270
-250
-200
-150
o* (MPa) -100
-50
0 5
0
-10
-15
Figure 11-1: Development of macroscopic (engineering) stress during the hydration of Class-G cement of a fully constrained material in function of the solid eigenstress determined from measurements.
the densification of C-S-H gel can be the driving force for the observed shrinkage in saturated cement-based materials. Further investigation of the densification of the CS-H gel with nanoindentation in conjunction with water vapor and nitrogen sorption isotherms can prove very helpful at understanding the densification and the associated volume changes. It is of particular interest to investigate the water-to-cement ratio on the mean diffusion distance and the resulting densification.
The mean diffusion
distance can be studied by perneability measurements and impact of w/c ratio and seeding on the rate of hydration. At the mneso-scale simulations front, current studies indicate a possible contribution of the simulated densification of C-S-H on the development of eigenstresses during precipitation. The initial promising results of these simulations show the potential of deeper understanding of the driving forces of bulk volume changes, yet a systemiatic study of such contribution is not yet available.
271
272
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[214] ZHANG, X. H., ZHANG, X. D., Lou, S. T., ZHANG, Z. X., SUN, J. L., AND Hu, J. Degassing and Temperature Effects on the Formation of Nanobubbles at the Mica-Water Interface. Langmuir 20, 9 (apr 2004), 3813-3815. [215] ZHENG, Q., DURBEN, D. J., WOLF, G. H., AND ANGELL, C. A. Liquids at large negative pressures: water at the homogeneous nucleation limit. Science (New York, N. Y.) 254, 5033 (nov 1991), 829-32. [216] ZIMA, P., MARSIK, F., AND SEDLAR, M. Cavitation rates in water with dissolved gas and other impurities. Journal of Thermal Science 12, 2 (may 2003), 151-156.
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290
Appendix A Evolution of volume fractions during hydration reactions In this appendix, volumetric relationships are derived for the solids and liquids during the hydration reactions of cement-based materials (CBM). Evolution of the volume fractions are first derived for a single chemical reaction in which a reactant solid reacts with water to form the solid hydration products.
A.1
Hydration kinetics and mechanisms
Mixing cement and water triggers a series of chemical reactions between the anhydrous cement grains and water that lead eventually to the setting and hardening of cementbased materials. The chemical reactions between water and cement are referred to as the hydration reactions and the solids formed during the hydration reactions are referred to as the hydration products [129].
Similar to other chemical reactions,
the hydration reactions of cement are characterized by the enthalpy of the reaction, volume of the reaction, and reaction stoichiometries. This section reviews the concept of the degree of hydration and relate the calorimetry data to the volume of the reaction (chemical shrinkage). Monitoring the kinetics and progression of the hydration reactions is of critical importance for the interpretation of bulk volume change measurements and estima291
tion of the evolution of the mechanical properties and strength of CBM. Monitoring the progression of the reaction is also necessary to study the impact of the different mix variables, constituents, and curing conditions on the hydration kinetics. These variables include the water-to-cement mass ratio (w/c), particle size distribution and specific surface area, sample size, mixing procedures, the presence of seeds and admixtures, the chemical composition of cement, curing pressure, and curing temperature. The quantity that is commonly used to describe the kinetics and progression of the hydration reaction is the degree of hydration
and the rate of hydration 9. The
degree of hydration is usually defined as the percent of mass (or amount in moles) of the reactive part of the solid consumed in the hydration reaction. It is used to describe as ratios how much of cement has reacted and it ranges between zero and
one [144]: m(t)
TO -
m(t)
_
1 -
MO
O
(A. 1)
where m0 is the initial mass and m(t) is the mass of the reactive solid (cement) at time t respectively. For a polyphase material like CBM, the degree of hydration can be defined as the weighted sum of the degrees of hydration of all the phases:
Z
i/
=
where p,
fi,
and
(A.2)
j are the density, volume fraction, and degree of hydration of phase
i, respectively; while the mass fraction mi of phase i is defined as:
m =
fi and (p) =
pifi
(A.3)
The concept of the degree of hydration is closely related to the fractional conversion Xi. It provides an approximate indication of the progression of the chemical reaction. In fact, the degree of hydration is some sort of a mass average of the fractional conversion of multiphase materials, where the fractional conversion is defined
as [77]: Xi(t)
=
n - ni (t) n0
292
(A.4)
where Xi is the fractional conversion of reactant i, n? is the initial amount of reactant i, and ni(t) is the amount of reactant i at time t. The fractional conversion X and the degree of hydration
are identical for a
single solid phase material when the solid is the limiting reactant. On the other hand, there exists a quantity used by chemists to describe a concept similar to that of the hydration degree and it is called the extent of the reaction. The extent of the reaction a, also called the degree of advancement of the reaction, is defined as [123] [124]:
ai
ni - no i0
vi
(A.5)
where ni is the amount of substance (moles), vi is the stoichiometric number (dimensionless) and a has units of moles. The extent of the reaction a will not be used in this study and only the degree of hydration
defined in eqn.A.1 and the conversion
Xi defined in Eq. (A.4) will be used in this study. It is obvious from Eq. (A.1) that the hydration of CBM is treated as an elementary chemical reaction while it is known to be otherwise. The consequences of such assumption depend on the use and interpretation of the degree of hydration or the rate thereof.
For example, determination of the volume fraction of C-S-H formed
during the hydration of CBM requires a direct determination of the conversion of each of the cement minerals. For purposes of this study, monitoring the hydration kinetics is necessary to model the evolution of the microstructure of the material during hydration. Hence, it is necessary to establish the relationship between the methods used to monitor the hydration reactions and the evolution of the volume fractions of the various hydrates. A number of methods are available to monitor the hydration kinetics of CBM. These methods can be divided into direct and indirect methods. In the direct methods, the volume fractions of the different phases are measured directly. These methods include ThermoGravimetric analysis (TGA) [157], Quantitative X-Ray Diffraction (QXRD) [90], Nuclear Magnetic Resonance (NMR) [133], QuasiElastic Neutron Scattering (QENS) [73], and Scanning Electron Microscope (SEM) [71]. In the indirect 293
methods, the volume fractions of the various phases are inferred based on models. This includes the enthalpy of the reaction as measured with conduction calorimetry and the volume of the reaction (chemical shrinkage). Calorimetry (Enthalpy of reaction (AZHrxn)) Enthalpy of reaction is the difference in enthalpy between the reactants and the products of a chemical reaction at specified temperature and pressure. The enthalpy of reaction can be measured experimentally in isothermal calorimetry, or it can be calculated based on the known chemical reactions and the standard enthalpy of formation for the reactants and the products:
AZHrxn = 13 AvpAH5 (products) -
AvAZHf (reactants)
(A.6)
where vp are the stoichiometric coefficients of the products from the known chemical reaction, v, are the stoichiometric coefficients of the reactants from the known chemical reaction, LAH5 is the standard enthalpy of formation for the reactants or the products. Mixing of cement and water triggers an exothermic hydration reaction. makes iskotherm-"Al LtL.LXLA'Q
. JL U11iL.LI..k
c-Alrieryr+-
odut %I"i'LL
U~I"Jii
'"L
L.1Ii1% U.LJy
the
mostq
populaiIr teh
U11%,
L.L.LJOU
F
e
IW
U%'%'JJ..iII'
ique us 4
LLJ-%A
This
tf-
Ynm;ni
U
XIJ~_
tor the hydration kinetics of cement-based materials. Enthalpy of hydration is generally modeled as a linear function of the enthalpies of hydration of individual cement phases assuming that the cement reactions proceed without interaction between the compounds.
Assuming a linear relationship between the phase composition of ce-
ment and its heat evolution, Woods et al. [211] estimated the enthalpy of reaction at complete hydration as: AHrxn = Z miAHi where AHrx,
(A.7)
is the total heat of hydration per unit mass of cement at complete
hydration (typically measured in J/g), AHj is the enthalpy of reaction of phase i per unit mass at complete hydration (J/g), and ni is the mass fraction of phase i, respectively (i = C 3 S, C2 S, C 3A, C 4 AF, Gypsum, MgO, CaO, etc.). This formula 294
has been adopted by several authors and it seems to estimate the heat of reaction with reasonable accuracy for the complete reaction (see for example [171]). To estimate the heat of hydration for incomplete reaction, Eq. (A.7) requires the determination of the degree of hydration of each phase individually. In other words, the total heat of hydration cannot be related to the volume fractions of the hydration products unless the degree of hydration of each phase is known. The degree of hydration is typically defined as the ratio between the cumulative enthalpy of hydration H,,, and the enthalpy of the reaction AHrx =
Hrxn
(A.8)
where the subscript H is used here to distinguish the degree of hydration as defined in Eq. (A.1) and the degree of hydration as measured by isothermal calorimetry.
Volume of reaction (chemical shrinkage AVrx,,) Most of volume change occurs due to decomposition of water to form hydroxyl groups in the hydration products, like Ca(OH) 2 , Mg(OH) 2 , C-S-H, etc. Part of this volume change is due to electrostriction of water in confined space like the interlayer water in C-S-H or adsorbed water on the charged surface of the solids.
Similar to the
enthalpy of reaction, the volume of reaction can also be used to monitor the hydration kinetics. While enthalpy is monitored by means of isothermal calorimetry, the volume of reaction is typically monitored by measuring the changes in the absolute volume of a hydrating cement paste sample. In addition to the enthalpy of reaction (AHrxn), hydration of cement is accompanied by an increase in solid volume (one volume of anhydrous material produces on the average more than two volumes of hydrates) and, simultaneously, by a reduction of absolute volume due to difference in average densities between the reactants (including water) and the products. This reduction in volume is referred to as the volume of reaction (AVrn), also known as chemical shrinkage or Le Chatelier contraction. The volume of reaction (chemical shrinkage) has been directly quantified by 295
covering cement slurry with water (curing water) and measuring the volume change of the entire system [155]. Although the fundamental technique has not changed over time, other researchers have measured the change in the pressure at the bottom of a burette placed on the top of the cement container [213]. Another method consists of measuring the change in the buoyancy force on an immersed specimen in direct contact with water [169]. A standard test method is also available for the measurement
of chemical shrinkage; ASTM COI [36]. The volume of reaction is another property of the chemical reaction similar to the enthalpy of the reaction. Consequently, it can be used to monitor the reaction kinetics. Unlike heat of hydration and due to hardware limitations, the specimen size required for the chemical shrinkage measurement is much larger than that required for the measurement of the enthalpy of reaction. The large size of the specimen introduces new complications to the measurement. Heat capacity and thermal conductivity can interfere with the enthalpy measurement but the small size of the specimen will make this interference negligible. On the other hand, compressibility of the cement paste constituents in conjunction with its permeability will influence the chemical shrinkage measurement and the extent of this influence will depend on the specimen size. Similar to the enthalpy of reaction, we define the volume of reaction for multiple simultaneous chemical reactions as
AVrxn =
miAVi
(A.9)
And the degree of hydration as =
AVrxn A=rxn
(A.1O)
where the subscript V is used here to distinguish the degree of hydration as defined by Eq. (A. 1) and the degree of hydration as measured by the volume of the reaction. The volume of the reaction (Chemical shrinkage) can cause changes in the reaction environment, which in turn can activate a number of bulk volume change mechanisms. Despite this, there is no direct relationship between the chemical shrinkage and bulk volume change and it is only the bulk volume change that can lead to development
296
of stresses in cement. Development of stresses in CBM at early ages is a consequence of a combination of curing conditions, evolution of mechanical properties, types and degree of constrains and the volume changes. The first step in the attempt to estimate the evolution of stresses and straints in hydrating cement-based materials (H-CBM) is to understand the mechanisms of volume changes.
A.2
Evolution of volume fractions
To derive the evolution of volume fractions for a single reaction, we consider a single reactive solid (RS) reacting with water (RW) to form the hydration product (HP).
VrsRS(s) + VrwRW() 7-
vhpHP(s)
where vrs,vrwand vhp are the stoichiometric coefficients of the reactive solid, water consumed in the reaction, and the solid hydration products, respectively. A compilation of possible chemical reactions in cement based materials are shown in Table 10.2 with material properties in Table A.2 and the cement chemist notation in Table A.1 by. We define the fractional conversion of substance i (degree of hydration) by: ni0 - ni(t)
_
m? - mrc(t)
foc -
fTc
frs
(A.11)
where no is the initial amount of reactant i at time t = 0, ni(t) is the amount of reactant i at time t, mrc(t) is the mass of reactant consumed in the reaction at time t, and m% is the initial mass of the reactant. solids consumed in the reaction,
fr,
is the volume fraction of the reactive
fre is the initial volume
fraction of the reactive solids,
and frc is the volume fraction of the residual un-reacted solid. We will assume that RS is the limiting reactant in this reaction and it will be used as the reference to calculate the fractional conversion. In terms of measured quantities in this study, the degree of hydration can be approximated as: Vrxn(t) AVrxn
297
_
Hrxn(t) A Hrxn
(A.12)
where
A~ n.AHe,,
are the volume of the reaction and the enthalpy of the reaction,
respectively. For the hy dration of C3 S- AV
-58pL/gcs
AH,
= -520k J/g.:Ss,
V%'n (t) and H,.x., (t) being the volumne of the reaction and the enthalpy of the reaction as measured at time t. respectively.
Afrxn
extra water = Afl mixing
~ nwil
water
no Constant mass-2
-_Constant mass-]
1-no
residual cement frc
0 0
4ma
1
Figure A-1: Evolution of volume fractions where the red lines represent the constant volume. The entire system here has a constant mass but the absolute volume is decreasing while the volume of the paste is considered constant (red lines). fr is the volume fraction occupied by the anhydrous cement, powder. f;, is the volume fraction occupied by the hydration products. f, is the volume fraction occupied by the remaining water used for mixing, and r is the porosity of the system. The yellow circle represents the degree of hydration when all the initial mixing water is consumied in the reaction. It can be seen in the figure that the total volunie of the system is decreasilng due to the negative volume of the reaction (chemical shrinkage) while the volunie of the solids is increasing. Most importantly,. the total mass of the system is constant (Conservation of M\ass). Note that the extra water added to maintain the systen saturated is exactly Afr,, which is the mininum amount of water required to maintain the system saturated. 1/ is the volume of the paste and it is assumed constant in the derivations of volume fractions. "Constant mass-1" is the mass of the seahed system and the reaction stops at "PWN when all the initial water is consuied ili the reaction (water is the limiting reactant). "Constant nmass-2" is the mass of the paste in additioi to the mnininni amount of curing fluid required to maintain the pore pressure constant.
Now consider the reference volunie to be composed of the voluinme fraction of
I
residual reactive solids frc, hydration products fhp (in the absence of air and nonreactive solids). Then as illustrated in Figure A-1, we have:
(A.13)
#+ frc + fp = 1
where
4
is the total porosity of the system (at all scales). For a saturated material,
the porosity is filled with the mixing water fm w and water provided from the outside,
fmw is the volume fraction of water that remained from the initial water used for mixing. This definition is important because under constant pressure in a saturated system, the volume fraction of water in the system will be fmw + cfrxn, where c < 1 is
a factor depending on the initial w/c. Also, in sealed specimens, if the initial amount of water available for the reaction is small, then water can be the limiting reactant. That is, during the hydration reaction:
f
0, and MSC is the composite shear-to-solid shear moduli ratio:
1 5 Msc = I _ 2 4
-
3 3 - (3 -#) 16
1 1 \/144(1 - y) - 480(1 - #) + 400(1 16
#) 2
+ 4087(1 -
4)
- 120y,(l -
#) 2
+ 972(3 -
0)2
(B.18)
The indentation hardness on the other hand depends on three parameters:
the
strength properties of the granular solid (cohesion c' and friction coefficient as), porosity
= 1 - y, and the geometry of the indenter probe.
H we= iH hs
(B. 19)
where h. is the asymptotic hardness of the frictional-cohesive granular solid that
313
obeys Drucker-Prager mlodel: h' is given by Eq. (B.12) and H1 , is given by Eq. (B.3). Finally. to extract the packing density distribution and the asyiptotic solid properties of a single granular porous solid with heterogeneous distribution of porosity. we apply the statistical analysis of the grid indentation.
Grid indentation test on
a porous solid provides a bivariate data set with N entries of indentation
M-
and indentation hardness Hjj=1
.N
. N.
moduli
Making use of this data set, we can back
calculate the indentation mnodulis 'm, Poisson's ratio ius. indentation hardness he, friction coefficient o,. packing deinsity at percolation 1 /0. as well as a set local packing
by nfninizing the quadratic error between model predictions and
i=J ...N
densities
the experimental values:
2]
1 --
Hi
T/1gS
/)
2
+
t 1
h.H1/(c+ Ili, '1/0) -
IIIin
l ofn1. s io A exS
-
(B.20)
Oli -=1...N A)
An examiple of such
inimiization is show~ln in Figure B-1.
40 35 30 25
--
M -(-Gpa)-(M gO-RD-09) M (Gpa) (M gO-RD-08) M (Gpa) (M gO-RD-07) M (Gpa) (M gO-Pis-03) M (Gpa) (M gO-Pis-02) M (Gpa) (Ng gO-Pis-01)
20 15 I /
10 I
5-
I
I
M model (G pa)
I
I
0 0
200
600
400
80 0
1000
H (MPa) Figure B-1: Indentation noduills NI and indentation laIrdness H diffient M'\Igo specinens shown in Table 6.1.
314
comnpiled fron the
