Free-volume hole relaxation in molecularly ... - APS Link Manager

PHYSICAL REVIEW E 89, 022603 (2014)

Free-volume hole relaxation in molecularly oriented glassy polymers Zhiyong Xia,1,* Morgana Trexler,1 Fei Wu,2 Yan-Ching Jean,2 and J. David Van Horn2 1 The Johns Hopkins University, Applied Physics Laboratory, Laurel, Maryland 20723, USA University of Missouri–Kansas City, Department of Chemistry, Kansas City, Missouri 64110, USA (Received 24 September 2013; revised manuscript received 19 December 2013; published 24 February 2014) 2

The free-volume hole relaxation in polycarbonate and poly(methyl methacrylate) with different levels of molecular orientation was studied by positron annihilation lifetime spectroscopy at variable pressures. The molecular orientation was achieved through a simple shear process performed at different temperatures and extrusion rates. It has been demonstrated that the β relaxation is largely responsible for the free-volume hole anisotropy after simple shear orientation. Upon the removal of mechanical force, the deformation of the free volume is mostly reversible at temperatures much lower than the glass transition. No strong correlation between macroscopic deformation and the free-volume hole deformation was found regardless of molecular orientation. DOI: 10.1103/PhysRevE.89.022603

PACS number(s): 36.20.−r, 81.05.Qk, 78.70.Bj

I. INTRODUCTION

It is well known that polymers are not in their equilibrium state at temperatures below the glass transition (Tg ). During the formation of glassy polymers, the supercooled liquid with high internal energy is typically frozen into a nonequilibrium state. The mismatch between the polymer chain relaxation and supercooling rate leads to the entrapment of unoccupied volume in the polymers, i.e., the formation of free-volume holes [1]. Free volume in polymers is generally thermodynamically unstable and tends to decrease as polymers age. At temperatures above α relaxation, long-range micro-Brownian motion becomes possible, and the free volume is more sensitive to the change in temperature. At temperatures below the main chain α relaxation or Tg , only local conformational changes are allowed, and the fraction of free volume (the ratio of free volume to total volume, which includes both the actual molecular volume and free volume) is relatively small and remains rather constant with the change in temperature (Fig. 1) [2]. Free volume in polymers exists as a distribution of holes or cavities that range from a few to tens of angstroms in size. It is worth noting that, since Doolittle proposed the free-volume concept in the early 1950s [3], this physical parameter remained mostly a concept in nature in the early days. However, with the development of positron annihilation lifetime spectroscopy (PALS), direct detection of free-volume holes in polymers has become possible and has attracted much attention in modern polymer physics [4–9]. In the past, much work has been performed on the free-volume hole information in isotropic polymers [10–15] with only limited reports on oriented polymers (mostly oriented thin films) [16,17]. Due to the growing interest in oriented bulk polymers for structural and ballistic applications [18,19], the understanding of freevolume hole is particularly useful in addressing microcrack initiation and propagation in polymers under stress. In this study, the free-volume hole information in two bulk glassy polymers, polycarbonate (PC) and poly(methyl methacrylate) (PMMA) (Fig. 2) with controlled molecular orientation, is studied through PALS bulk measurement and pressure dependence analysis. PC and PMMA were selected because

*

[email protected]

1539-3755/2014/89(2)/022603(12)

of their differences in β relaxation. That is, PMMA has much stronger β-relaxation (mainly from the pendant group) relative to its α-relaxation, as opposed to PC where the β-relaxation is mostly lower than the main chain α-relaxation [20]. Mechanically, PMMA has a lower chain entanglement density and is thus typically referred to as a brittle material. PC, on the other hand, has a higher chain entanglement density and is typically referred to as a ductile polymer [21,22]. The main objective of this study is to establish the correlation between molecular orientation and the relaxation of free-volume holes both under stress and after the stress is removed. Learning from this will help the understanding of microcrack initiation and growth in oriented polymer for both structural and high strain rate applications. II. EXPERIMENT A. Materials

PC (Makrolon) with a number average molecular weight (Mn) and weight average molecular weight (Mw) of 20 and 73 kg/mol, respectively, was obtained from Bayer Material Science. The PC has a notch Izod impact strength (ASTM D256) of 954 J/m. Cell-cast grade PMMA (Plexiglas G) with a Mn and Mw of 1500 and 2200 kg/mol, respectively, was obtained from Arkema and has a much lower notch Izod impact strength (ASTM D256) of 16 J/m. Both PC and PMMA were obtained in sheet form with a nominal thickness of about 12.7 mm. Prior to simple shear orientation, both PC and PMMA samples were cut into 152 mm squares and annealed at 140 °C and 100 °C, respectively, for 60 min to release the manufacturing induced residual stress. B. Polymer orientation

In this work, controlled molecular orientation in PC and PMMA was achieved through the previously developed equal channel angular extrusion (ECAE) [23,24]. ECAE is a solid state polymer orientation technique that was developed originally to introduce large plastic deformation in bulk metallic materials [24]. Application of ECAE to orient bulk polymer parts has received lots of attention recently [24–29]. During the ECAE extrusion process, a billet is first inserted into the vertical channel of the die fixture (Fig. 3). With the increase

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©2014 American Physical Society

XIA, TREXLER, WU, JEAN, AND VAN HORN

Increase in volume

Equilibrium cooling trace


Unoccupied free volume

Chain occupied volume

Tg

Increase in temperature

FIG. 1. (Color online) Schematic plot showing the change of free volume as a function of temperature.

in extrusion force, the billet will be sheared as it being forced through the right angle of the die fixture. As a result, controlled molecular orientation will be introduced in the billet. Since the vertical channel and horizontal channel of the die fixture have the same dimension, the orientation is thus a simple shear process and is achieved without compromising sample dimensions. The latter cannot be readily achieved by any conventional polymer orientation techniques, such as drawing, stretching, and rolling [24]. Since the billet retains its dimensions after one ECAE treatment, the billet can be reinserted into the die and extruded again. Depending on how the billet is arranged (180° or 0°) with respect to the extrusion force direction (Fig. 3) in subsequent extrusions, different modes of molecular orientation can be achieved [23,26]. Microscopically, due to the shear stress field associated with ECAE, spherical structures are usually elongated along the maximum principle stress direction (α) and form ellipsoidal shapes (Fig. 3) [30,31]. This effect has been demonstrated in the study of semicrystalline polymers in the past [25,30–34]. The amount of the shear strain (γ ) in the oriented sample can be quantified by measuring the tilt angle (θ ) of the initial squared grid lines on the side of the billet.

FIG. 3. (Color online) Schematic showing the change of grid lines on the side of the billet during ECAE process. The deformation of an initial sphere into an ellipsoid as in semicrystalline polymers is also illustrated.

In this work, PC and PMMA have been treated by ECAE under different conditions to achieve different levels of shear strain using a custom-built servohydraulic-driven four-post ECAE extrusion system with a load frame capacity of 150 000 kg. To maintain a steady flow and to prevent microcracking of the billet, PC was extruded at 140 °C at an extrusion rate of 0.38 mm/s, whereas PMMA was extruded under three conditions: 100 °C at 0.38 mm/s, 105 °C at 0.38 mm/s, and 105 °C at 0.76 mm/s. To quantify the shear strain induced by each condition, grid lines with the mesh size of 6 mm × 6 mm were inscribed on one side of each sample. Permatex 80208 antiseize grease was used to facilitate the extrusion process. After ECAE treatment, all samples were machined to remove the residual antiseize grease followed by sandpaper polishing (Buehler 800 grit). The polished plaques were then used for preparing test samples in this work. All samples were prepared in a consistent manner. C. Specific gravity analysis and compressive strength determination

PC

Specific gravity of all samples before and after ECAE processing was analyzed according to ASTM D792. Four repeats were performed per sample. Uniaxial compressive yield strength of PC and PMMA in the direction perpendicular to the plate’s biggest surface (152 mm × 152 mm) was determined according to ASTM D695 at 0.02 mm/s using a screw-driven MTS 30G. The yield strength was determined from the 1% offset of compressive strain. D. Glass transition temperature analysis

PMMA

FIG. 2. Molecular structures of PC and PMMA.

Glass transition temperature (Tg ) in samples before and after the simple shear treatment was analyzed by using the Mettler-Toledo Differential Scanning Calorimeter-DSC1 under N2 (99.99%) purge at 50 ml/min. Prior to the experiment, the instrument was calibrated for both melting point and 022603-2

FREE-VOLUME HOLE RELAXATION IN MOLECULARLY . . .


melting enthalpy by using indium and zinc standards. During the measurement, test samples of approximately 20 mg were encapsulated in aluminum pans. All samples were heated from −40 °C to 270 °C at a heating rate of 20 °C/min. One needs to exercise caution in analyzing Tg in mechanically deformed samples. As pointed out by McKenna [35] and Lacks and Osborne [36], due to the complex interaction between mechanically induced deformation and the thermal relaxation nature around Tg , heat capacity change around Tg can be complicated and sometimes difficult for drawing the baselines and obtaining the midpoint Tg value. In our case, baselines in the ECAE treated samples are still reasonably well defined as compared to the control. As a result, the Tg was obtained by finding the midpoint of the baseline shift using the method proposed by Moscato and Seyler for oriented polymers [37]. However, in order to differentiate these values from the “true” Tg , we refer to the Tg in this paper as “apparent Tg ” for technical convenience. Further understanding of the simple deformation on the Tg is beyond the scope of this study and will be reported elsewhere.

wave function will be localized in the attractive potential of defects, i.e., the lifetime of positronium increases in a vacancy. Thus, the o-Ps lifetime correlates directly with the dimensions where Ps is localized, i.e., larger defects and free-volume holes which contain lower mean electron density will lead to longer Ps lifetime [39,40]. The high sensitivity of PALS in measuring free-volume properties can be attributed to the unique localization of the positron and the positronium atom in open spaces such as holes, free volume, and voids. In this work, PALS bulk measurements were performed by using a conventional fast-fast coincidence method at 25 °C. The time resolution of the spectrometer was around 300 ps, measured using a 60 Co source in the 22 Na energy window settings [41]. All samples were conditioned under dry Ar atmosphere for 24 h before the PALS measurements. A 22 NaCl positron source (sealed between Kapton disks) was sandwiched between two identical test samples (40 mm × 40 mm × 20 mm). Positron interaction with the source coating (when used) is taken into account in the data analysis. All PAL spectra were resolved into three components by using the PATFIT program [42,43]. The spectra were then fitted to three exponential lifetime components, which are assigned as follows: the shortest lifetime, τ1 0.125 ns, is attributed to p-Ps (the singlet Ps state) annihilation; the intermediate lifetime (τ2 0.4 ns) is due to bulk positron annihilation, and the component with the longest lifetime (τ3 1.8–2.1 ns) and its relative intensity (I3 ) is due to pick-off annihilation of o-Ps. Since the variations of τ1 and τ2 are less than that of τ3 , the observed lifetime of o-Ps (the triplet state of Ps) was found to be directly correlated to the size of free-volume holes, whereas the corresponding intensity (I ) is a measure of free volume fraction (fv ). Bulk phenomena and radius distribution at ambient conditions were also obtained. On the basis of the spherical model, fv can be expressed as a product of average free volume (Vf ) (as determined from τ3 ) and the corresponding relative intensity (I3 ) via Eq. (2) below:

E. Dynamic mechanical analysis

Molecular chain entanglement density (Xc ) in PC and PMMA before simple shear orientation was measured by dynamic mechanical analysis (DMA) using the TA Instruments RSA-G2 solid analyzer. Both samples were tested under the single cantilever mode with 50 g force precompression to ensure proper sample-fixture contact during the measurement. All samples were heated from −130 °C to 200 °C at 3 °C/min at 1 Hz under N2 purge. Xc was quantified by measuring the rubbery plateau modulus (E) according to [38]: Xc = E/3RT ,

(1)

where R is the universal gas constant (8.31 J/mol K) and T is the corresponding temperature (K). Stress relaxation at 25 °C for both PC and PMMA was performed using the same instrument under 1% and 2% strain levels, respectively. F. PALS analysis

Two types of PALS analyses were performed in this work: bulk measurements and pressure dependence analyses.

fv = AVf I3 ,

(2)

where A is an empirical constant that has been determined to be 0.018 ± 0.002 nm−3 for most glassy polymers [44]. In an infinitely deep spherical potential well, a semiempirical correlation between the measured o-Ps lifetime (τ3 ) and hole radius (R) is shown below [45–47]:

1. Bulk measurements

In PALS analysis, a positronium atom (Ps) with a Bohr ˚ is a bound state consisting of an diameter of 1.06 A electron and a positron. Positronium is typically formed in two states, ortho-positronium (o-Ps) with parallel spins and para-positronium (p-Ps) with antiparallel spins. Ps is generated by using positrons from the β + decay of 22 NaCl: 22 Na → 22 Ne + β + + ν + γ (1.27 MeV). The 1.27-MeV γ radiation appears almost simultaneously with positrons and is thus used as a start signal for tracking the decay of positronium trapped in the free volume in polymers. Since the Ps atom is relatively small compared to the expected hole sizes in most polymers (ranging from a few to tens of angstroms), upon radiation of polymers, o-Ps is preferentially trapped in the “empty space” or free volume in polymers. The positron

R 2π R −1 1 1 1− sin + , τ3 = 2 R0 2π R0

(3)

where Ro = R + R, and R is the o-Ps penetration depth into the wall of the hole. The value of R in molecular solids ˚ such as plastic crystals has been determined to be 1.66 A [48]. In Eq. (3), τ3 and R are expressed in nanoseconds and angstroms, respectively. We would like to point out that Eqs. (2) and (3) are simplifications of the actual free-volume morphology and are based on the assumption that free volume has a spherical shape. In the past, much work has been performed to elucidate the morphology of free volume in several polymer systems using

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both classical Voronoi tessellation and molecular dynamics simulation [49–51]. Different free-volume shapes have been proposed including cube, bar, cylinder, ellipsoid and spheres; under certain conditions, some shapes correlate to PALS data better over other geometries [47,52,53]. For example, Raˇcko et al. [53] found that an ellipsoidal shape matches o-Ps well when computed using the number average method, but the matching index drops substantially when the volume average was used. On the other hand, the work by Arrizi et al. [54], who based their analysis on the Delaunay tetrahedral tessellation, showed that the unoccupied cluster volume in PC is generally spherical in shape. While there is some correlation of PALS to free-volume shape, it appears to be sample dependent. As a result, in this paper we have adopted the generally accepted spherical free-volume shape model [53,55].

2. Pressure-dependent analyses

An anvil system with a 25 000 kg hydraulic press (Carver Co.) was used to generate quasi-isotropic pressure up to 138 MPa by using the Bridgman method [56,57]. The outer diameter of the anvil is 25 mm with a working diameter of 14 mm and a taper angle of 10°. Sample polymer pieces and gaskets were machined to match each other. All polymer samples were machined into disk form (diameter = 4.0 mm; thickness = 1.0 mm) and housed in a pyrophyllite ceramic gasket to meet the quasi-isotropic condition [58]. A drop of 22 NaCl solution was directly deposited on one side of one sample disk and dried; the second disk was then placed on the source side of the first disk to complete the sample sandwich. For the ease of handling, the sample sandwich was made by placing the disks sequentially into the surrounding pressure ring. The anvil and gasket sample assembly were placed in the press and pressure was applied to the predetermined levels. In the pressure setup, the start and stop detectors were placed at an approximately 135° angle and as close to the assembly as possible. Counting rates between 50 and 200 cps were obtained, relating to 22 Na sources of approximately 100 μCi.

III. RESULTS AND DISCUSSION A. ECAE induced simple shear orientation

As mentioned earlier, ECAE induced shear strain (γ ) can be quantified by measuring the shear angle (θ ) of the deformed

FIG. 4. (Color online) Micrograph showing the deformation of the grid lines on PMMA after simple shear. a and b are the lengths of the major and minor axes of the ellipsoid after ECAE. In this example, the PMMA flow direction (as shown in Fig. 3) is from left to right.

grid line initially inscribed on the side of the billet (Fig. 4), and is computed through Eq. (4): γ = cot θ.

(4)

The maximum principal molecular orientation direction (α) is correlated to θ via Eq. (5) [25]: 1 2 α = arctan . (5) 2 cot θ Experimentally, θ can be affected by (1) extrusion temperature, (2) extrusion rate, (3) polymer type, (4) polymer chain relaxation time, and (5) frictional force between the sample surface and the die inner surface. In this work, the frictional force is held constant by applying antiseize grease on the surface of the plate sample. Effects of extrusion temperature and extrusion rate on the shear angle, apparent glass transition temperature, compressive strength, and specific gravity of the extruded billet are shown in Table I. As can be seen in Table I, after only one ECAE pass, greater than 100% shear strain can be induced in the specimen without compromising sample geometry. In PMMA, at a fixed extrusion rate of 0.38 mm/s, the shear strain increased by 15% (from 128% to 143%) with a decrease of 5 °C in extrusion temperature (from 105 °C to 100 °C). At the same shearing temperature of 105 °C, doubling the extrusion rate (from 0.38 to 0.76 mm/s) led to 13% decrease in shear strain. It is worth noting the simple shear extrusion was carried out at a temperature 15 °C lower than the Tg of PMMA to facilitate polymer flow through the die fixture. At temperatures above Tg , the polymer chain relaxation increases leading to reduced extrusion induced shear strain. Due to the simple shear nature, the specific gravity of the samples before and after ECAE remains almost unchanged (last column in Table I). The uniaxial compressive yield stress in simple sheared samples

TABLE I. Effects of simple shear conditions on shear strain (γ ), apparent glass transition temperature (Tg ), compressive yield strength, and specific gravity of PC and PMMA.

Sample

Extrusion Extrusion Shear Shear Principal orientation Apparent glass transition Compressive Specific yield strength gravity rate temperature angle (θ ) strain (γ ) direction (α) temperature (Tg ) (mm/s) (°C) (deg) (%) (= cot θ) (deg) (°C) (MPa) (g/cm3 )

PC control Oriented PC

— 0.38

— 140

— 40

— 119

— 29.6

151 153

69 68

1.190 1.189

PMMA control Oriented PMMA Oriented PMMA Oriented PMMA

— 0.76 0.38 0.38

— 105 105 100

— 41 38 35

— 115 128 143

— 30 29 27

119 121 120 127

113 105 107 102

1.185 1.185 1.184 1.185

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10 1

10 1

PC control PMMA control

10 0

Normalized counts

Tan

10 0

10-1

10-1

10-2 10-3

10-2

10-4

-5

10-3

-150

-100

-50

0

50

100

150

0

5

200

Temperature (°C) FIG. 5. (Color online) Comparison of different relaxation peaks in PC and PMMA as illustrated by dynamic mechanical analysis (DMA) performed at 1 Hz with a heating rate of 3 °C/min.

was found to be slightly smaller than the controls for both PC and PMMA. Apparent Tg has been found to increase in all sheared samples regardless of sample type due to the mechanically stored energy. Compared to PMMA, the shear strain in PC is slightly higher at the same extrusion rate of 0.38 mm/s and a similar extrusion temperature gradient (Tg − Textrusion = 11 ◦ C). However, PC exhibits a higher molecular entanglement density (541 mol/m3 ) compared to that of PMMA (217 mol/m3 ) at 170 °C as derived from the rubbery plateau moduli. As such, the ductility after one simple shear pass is so low that additional extrusion would require a much higher load, and in most cases lead to surface cracking in the material. As a result, PC processed after only one ECAE pass will be used in this work. For comparison purposes, PMMA processed with one ECAE pass is also presented in this work. Figure 5 shows the tanδ curve of both PC and PMMA control samples analyzed by DMA at 1 Hz with a heating rate of 3 °C/min. Both polymers show multiple relaxation peaks. Going from high temperature to low temperature, these peaks are generally referred to as α, β, and γ relaxations [20]. Each of these relaxations corresponds to a different mode of molecular motion in glassy polymers. For example, α relaxation comes from large scale cooperative motion of the polymer main chains, while β relaxation is generally due to the thermal history effect and often appears as a shoulder on the low temperature side of the α peak. The γ peak is

10

15

Lifeme (ns) FIG. 6. (Color online) A representative example of the positron lifetime spectra in PC control sample. The line indicates the component of the spectra related to the lifetime and intensity of the o-Ps in the material defects.

a result of molecular excitation and its location with respect to room temperature has been used as an indication of impact strength [59–61]. In this work, it is clear that the two prominent relaxation peaks in PC are α peak (151 °C) and γ peak (–100 °C), whereas the β peak only appears as a shoulder (80 °C) next to the α peak. PMMA, on the other hand, shows a broadened β peak (25 °C) but its relative size with respect to the α peak (131 °C) is bigger than that in PC. B. Positron annihilation lifetime analysis 1. Bulk analysis

As mentioned earlier, PALS is unique in the measurement of free-volume properties because of the localization of the positron (e+ ) and positronium (Ps) in holes and free volume and voids in polymers. Two key parameters that can be obtained from the PALS analyses are positronium lifetime (τ3 ) and the associated relative intensity (I3 ) [7–17]. Figure 6 is a representative PALS curve showing the normalized counts versus lifetime for PC control used in this study. The initial portion of the curve is from p-Ps and bulk annihilations, and the lower right part of the curve corresponds to annihilation related to o-Ps trapped in the material defects. Experimental parameters including τ3 , I3 are derived from these spectra and are shown in Table II. Table II also shows the effects of shear strain on positron lifetime, intensity, hole size, hole shape, and the fractional free volume under atmospheric conditions.

TABLE II. PALS results under 0.1 MPa pressure on PC and PMMA at different levels of shear strain (γ ). The terms are the corresponding errors of each measurement. Sample PC PMMA

γ (%)

τ3 (ns)

τ3 (ns)

I3 (%)

I3 (%)

0 119 0 115 128 143

2.068 1.964 1.869 1.856 1.880 1.834

0.010 0.020 0.019 0.014 0.015 0.016

29.46 24.73 27.56 28.08 27.83 28.81

0.19 0.47 0.50 0.31 0.37 0.42

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ellipsoid sphere τ3

τ3

1.00 0.95 1.00 0.99 1.00 0.98

εhole

εmacro

0.00 0.19 0.00 0.12 0.00 0.14

0.00 0.87 0.00 0.91 0.89 0.87


PHYSICAL REVIEW E 89, 022603 (2014) 6.0 PC control 5.0

sphere τ3

= (1 + 0.4ε − 4.16ε2 + 2.76ε3 ).

1.0 0.0

(6)

In Eq. (6), ε = 1 − (b/a)2 , where a and b are the length of the semimajor and semiminor axes of an ellipsoid, respectively. In the calculation of the eccentricity factor, the following procedure was used. For the control polymer, the o-Ps lifetime sphere ) is directly translated into a spherical hole radius, (τ3 whereas for the sheared samples, the o-Ps lifetime for sheared ellipsoid ) is compared to that of the control polymer, sample (τ3 and ε is computed through Eq. (6). In Eq. (6), for a circular structure, the eccentricity factor ε would equal to zero, and for a linear streak, ε would equal 1. Two eccentricity factors are involved in the evaluation of the oriented PC and PMMA: macroscopic eccentricity factor (εmacro ) and microscopic free-volume hole eccentricity factor (εhole ). εhole is calculated on the basis of the measured τ3 by using Eq. (6), and the results are shown in Table II. εmacro was calculated by using the experimentally measured shear angle (θ ) and maximum tensile direction (α), and the results are also reported in Table II. It is surprising to note the difference in εmacro and εhole . Samples after the simple shear deformation still exhibit a large macroscopic structural anisotropy, as shown by the large εmacro values. However, as compared to macroscopic anisotropy, the free-volume holes only exhibit low levels of anisotropy (εhole ) after ECAE. The difference between εmacro and εhole can be accounted for by the β relaxation under room temperature in both polymers for the following two reasons: The relaxation of glassy polymers under room temperature is dominated by β relaxation which also has a low activation energy [61]; The relative magnitude of β relaxation (relative to α relaxation) in PC is much smaller than that in PMMA (Fig. 5). The fact that εhole of simple sheared PC is higher than that of PMMA (0.19 versus 0.12 and 0.14 as shown in Table II) suggest that the β relaxation at room temperature in PMMA is mostly responsible for the free-volume hole shape relaxation. We would also like to point out that even the εhole difference between PC and PMMA is largely due to β relaxation. Contributions from α relaxation, which can be accelerated by mechanical deformation [36], can also be possible even though this requires more cooperative molecular motion [36]. Quantification of the contribution from α and β relaxations requires in-depth characterization and will be reported elsewhere. The discrepancy between εmacro and εhole

3.0

2.0

ellipsoid

τ3

PMMA control

4.0

fv (%)

As discussed in Secs. II B and III A, because of the stress field associated with the simple shear process, elongated ellipsoids are generally the preferred morphology in polymeric materials [30–34]. Further, due to the constant values of specific gravity before and after the simple shear process (Table I), it is thus reasonable to attribute the reduction in o-Ps lifetime largely to the change in free-volume hole shape from spheres into ellipsoids. As a result, the correlation between spherical holes and ellipsoidal holes developed by Jean and Shi [17] is used to understand the anisotropy of the free-volume holes in both oriented PC and PMMA [Eq.(6)]. Equation (6) relates positron lifetime (τ3 ) with the eccentricity factor (ε) through the following:

0

25

50

75

100

125

150

Pressure (MPa) FIG. 7. Comparison of the fractional free volume (fv ) for both PC and PMMA controls as a function of PALS test pressure.

has also been noted by Jean and Shi in the study of oriented polyaryl-ether-ether-ketone [17]. 2. Pressure analysis

To probe the stability of the free-volume holes before and after the simple shear orientation, PALS was measured for all samples while under quasi-isotropic pressure ranging from 1 atm (0.1 MPa) to 138 MPa at 25 °C. In the PALS pressure test, the test sample was loaded into the hydraulic press under a 0.1 MPa condition to obtain the PALS results under atmospheric condition. Subsequently, the sample was gradually pressed to the desired pressure and held in place for acquiring the PALS data. After being tested at the highest pressure (138 MPa), the pressure on the test specimen was released and the PALS measurement at 0.1 MPa was repeated. Due to the nature of the isotropic deformation and the fact that the Poisson’s ratios for both PMMA (0.35) and PC (0.38) are lower than 0.5 (constant volume), the microscopic deformation of the free-volume holes during the PALS pressure test is likely dominated by the shrinking and collapsing of the free-volume holes even though some level of shape change is also possible. As a result, in this analysis, the decrease in o-Ps lifetime is attributed mainly to the decrease in free volume rather than to the change in free-volume hole shape. Figure 7 shows fv in unoriented PC control and unoriented PMMA control at different quasi-isotropic pressures. The two polymers show similar pressure dependence, although PC appears to be slightly more compressible (slightly steeper initial slope in fv curve) than PMMA under pressures less than 50 MPa. At higher pressure range, the two materials show similar compressibility. As the pressure increases beyond 75 MPa, both PC and PMMA show similar hole shape change, and level off after 120 MPa. The changes in fv with respect to the change in quasiisotropic pressure in PC with and without simple shear orientation are shown in Table III. Values of τ3 are found to decrease distinctly with the increase in PALS test pressure, whereas the corresponding I3 decreases slightly in a roughly

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TABLE III. PALS results under different pressures for PC with shear strain (γ ) of 0% and 119%. The terms are the corresponding errors of each measurement.

PC with γ = 119%

I3 (%)

I3 fv fv (%) (%) (%)

0.1 55 69 83 103 124 0.1

2.068 1.611 1.404 1.280 1.223 1.182 1.943

0.010 0.013 0.014 0.013 0.014 0.016 0.025

29.46 26.68 23.75 24.54 22.70 19.59 23.63

0.19 0.36 0.41 0.45 0.48 0.50 0.58

5.51 3.02 1.99 1.65 1.36 1.08 3.92

0.07 0.09 0.08 0.07 0.07 0.06 0.18

0.1 55 69 83 103 124 138 0.1

1.964 1.393 1.330 1.183 1.076 1.011 0.961 1.988

0.020 0.013 0.018 0.015 0.015 0.017 0.016 0.016

24.73 23.67 19.40 19.21 17.14 16.30 14.85 25.27

0.47 0.41 0.54 0.49 0.48 0.59 0.48 0.30

4.19 1.94 1.43 1.06 0.73 0.58 0.45 4.38

0.15 0.07 0.08 0.06 0.05 0.05 0.04 0.11

linear fashion. This indicates that a microdensification has occurred at the molecular level during the quasi-isotropically pressurized PALS test. Table III also demonstrated that, in PC with γ = 0%, a large fraction of the compressed free volume has been recovered upon the removal of the highest isotropic PALS pressure even though the highest PALS pressure (124 MPa) is much higher than its compressive yield strength (69 MPa) (Table I). On the other hand, in PC with γ = 119%, the free volume has become higher after the removal of the highest isotropic PALS test pressure (138 MPa). Combined with these observations, it is appears that simple shear induced free-volume hole anisotropy even at very low levels (εhole = 0.19) can lead to the growth and coalescence of the free-volume holes upon the removal of PALS test pressure. The latter is likely due to the anisotropy induced stress concentration at the end of the ellipsoids in glassy polymers [51,62]. Figure 8 shows the change in fv with the increase in the quasi-isotropic PALS test pressure for PMMA control and PMMAs with three levels of shear strain. All four samples show similar pressure dependence with respect to fv regardless of the level of shear strain with values for fv converge at 138 MPa level. The largest variation is found to appear in the intermediate pressure range (from 50 to 75 MPa). This is most likely due to the fact that the PALS test pressure induced deformations at these stress levels are lower than the yield stress of PMMA (Table I). To compare the fv change after the pressure analysis, all samples were retested at 0.1 MPa after the removal of the highest PALS test pressure and the results are compared to the initial 0.1 MPa data (Fig. 9). For PMMA [Fig. 9(a)], fv has been found to increase after the removal of the highest test pressure regardless of the shear strain level. This is due to the microcracking of PMMA at 138 MPa, especially since this pressure is higher than the maximum compressive strength of PMMA (113 MPa in Table I). For PC [Fig. 9(b)], a slightly different trend has been observed. In PC control, even the highest PALS pressure of 138 MPa is much higher than the

4.0

fv (%)

τ3 (ns)

3.0 2.0

1.0 0.0 0

25

50

75

100

125

150

Pressure (MPa) FIG. 8. Comparison of fractional free volume (fv ) for PMMA with different levels of shear strain at different quasi-isotropic test pressures. Maximum test variation for fv is 0.15%.

maximum compressive strength of PC (68 MPa in Table I); fv still decreases upon the removal of the PALS test pressure mostly due to the intrinsic plasticity in PC materials [21,63]. 6.0 fv inial at 0.1MPa

fv ﬁnal at 0.1MPa

5.5 5.0

fv (%)

PC with γ = 0%

Pressure τ3 (MPa) (ns)

PMMA control PMMA 143% PMMA 128% PMMA 115%

5.0

4.5 4.0 3.5 3.0 0%

115%

128%

143%

Levels of shear strain in PMMA (a) 6.0 fv inial at 0.1MPa

fv ﬁnal at 0.1MPa

5.5 5.0

fv (%)

Sample

6.0

4.5

4.0 3.5

3.0 0%

115%

Levels of shear strain in PC (b)

FIG. 9. (Color online) fv recovery at 0.1 MPa during the PALS pressure test for (a) PMMA and (b) PC with different levels of shear strain.

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6.0

Radius (Å)

PC with 119% strain 5.0

1.66

PMMA with 115% strain

2.34

2.86

3.28

3.64

PDF

fv (%)

0.6

2.0

4.24

0.1 MPa 55 MPa 69 MPa 83 MPa 103 MPa 124 MPa 0.1 MPa*

4.0 3.0

3.96

0.4

1.0

0.2 0.0 0

25

50

75

100

125

150

0.0

Pressure (MPa)

0.5

1.0

1.5

3.0

3.5

3.64

3.96

4.0

(a) γ = 0 %

Radius (Å)

In comparison, fv in sheared PC (γ = 115%) has been found to increase by 4% (4.19–4.38) upon the removal of 138 MPa pressure suggesting the formation of microcracks in oriented PC. This result is consistent with our observation of the flow behavior of PC during the ECAE process; that is, after just one ECAE pass (γ = 115% strain), the intrinsic ductility of PC has dropped so much that additional straining significantly cracked the material. To compare the material effect, fv values for PC and PMMA with similar levels of shear strain (115%) are plotted in Fig. 10. Both PC and PMMA show similar compressibility upon pressurization and once again with the largest variation around 50 MPa due to the PALS pressure being lower than the yield strength of both materials.

1.66

1.0

2.34

2.86

3.28

4.24

0.1 MPa 55 MPa 69 MPa 83 MPa 103 MPa 124 MPa 138 MPa 0.1 MPa*

0.8

PDF

0.6 0.4 0.2 0.0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Lifeme (ns) Lifeme (ns)

3. Lifetime distribution analysis

To explore the mechanical differences in each polymer series, it is of interest to study the hole volume probability density function (PDF) for theoretical comparison. Maximum entropy lifetime (MELT) analysis [64] was used to assess the lifetime distribution of PC and PMMA under the conditions related above. We would like to point out that this method only assumes a continuous distribution of positronium lifetimes in a given spectrum without previous assumptions of number of lifetimes, types of defects, or shapes of voids in the sample under evaluation. For convenience, rather than using complicated shape parameters to describe the pore size distribution, in this section, we have simplified the shapes into equivalent spherical pores to demonstrate the lifetime distribution change for samples with different levels of shear strains. As a result, hole volume PDF g(v) was obtained by assuming spherical holes and is expressed as below [7,65]: f (R) . 4π R 2

2.5

Lifeme (ns) (ns) LIfeme

FIG. 10. Dependence of fractional free volume (fv ) on PALS test pressure for PC and PMMA with similar level of shear strain.

g(v) =

2.0

(7)

The fraction of hole volume distribution as determined by o-Ps annihilation in holes with volume between V and V + dV is given by g(V )dV. This approach does not take into account the change in free-volume eccentricity expected in the samples under pressure in this study, but is useful for comparing the

(b) γ = 119 %

FIG. 11. (Color online) Lifetime distribution and estimated radius versus PDF in PC with two strain levels (γ = 0% and 119%) under various pressures using MELT analysis. The repeated experiment under ambient conditions after applied pressure is denoted by an asterisk.

two types of polymers. The estimated radii are included for a general comparison of the trend of free-volume size changes in the samples under pressure, but do not account for the expected shape of the free volume under pressurized conditions. Figure 11 shows the lifetime distributions and estimated free-volume hole radius in PC control (γ = 0%) and in PC with 119% shear strain. In the PC control, the distribution remains broad as the pressure is increased, and finally narrows at the highest pressure. Similarly, in the oriented PC, the lifetime distribution becomes narrower at higher pressures. The narrow distribution shows that some of the holes collapsed upon mechanical loading. Upon unloading, the distribution has become wider in both oriented and unoriented samples. However, the distribution has almost completely recovered upon unloading in the oriented sample, whereas in the unoriented sample, the free volume was only partially recovered upon unloading. On the basis of the size distribution, the oriented sample shows

022603-8



Radius (Å)

Radius (Å) 1.66

2.5

2.34

2.86

3.28

3.64

3.96

4.24

1.66

2.5 0.1 MPa 55 MPa 69 MPa 83 MPa 103 MPa 124 MPa 138 MPa 0.1 MPa*

PDF

1.5

1.0

2.86

3.28

3.64

1.5

1.0

0.5

3.96

4.24


2.0

PDF

2.0

2.34

0.5

0.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.5

1.0

1.5

1.66

2.34

2.5

Lifeme Lifeme(ns) (ns)

Lifeme (ns)

(a) γ = 0 %

(b) γ = 115 %

2.86

3.28

3.0

3.5

4.0

3.64

3.96

4.24

Radius (Å)

Radius (Å) 2.5

2.0

3.64

3.96

4.24


PDF

1.5 1.0

2.34

2.86

3.28


2.0 1.5

PDF

2.0

1.66

2.5

1.0 0.5

0.5

0.0

0.0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.5

4.0

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Lifeme (ns)

Lifeme Lifeme(ns) (ns)

(c) γ = 128 %

(d) γ = 143 %

FIG. 12. (Color online) Lifetime distributions and estimated radius versus PDF in PMMA with different levels of shear strain (γ ) under various pressures using MELT analysis: (a) γ = 0%; (b) γ = 115%; (c) γ = 128%; (d) γ = 143%. The repeated experiment under ambient conditions after applied pressure is denoted by an asterisk.

a wider distribution than the unoriented sample at the initial 0.1 MPa condition. The PDF analyses of PMMA are shown in Fig. 12. The distributions become narrowed after the application of mechanical pressure during PALS measurement for all four samples. As in PC, the distribution remains broad as pressure is applied and then narrows at the higher pressures. This trend may be slightly more rapid for the PMMA than for PC, and the mechanically stressed samples may also approach a narrower distribution more rapidly per simple shear orientation. The latter samples led to the smallest distribution and radius at the highest pressure for those samples with the highest level of shear orientation. This appears to indicate that as pressure is applied, all free-volume voids are being constricted at a similar distribution until they reach the smallest size at the highest pressure.

The biggest difference observed between PMMA and PC was seen during the second experiment under ambient conditions after depressurization. All four PMMA samples regardless of the level of molecular orientation showed two populations of lifetime distribution upon the removal of the pressure, whereas all samples in the initial measurement under the atmospheric condition only showed one distribution. The MELT analysis indicated two well-resolved features in the decompressed sample analysis for PMMA control and PMMA with 143% shear strain. In addition, the free-volume values derived above for these samples appeared to be anomalous, i.e., larger than the initial free volume as shown in Table III. These data suggested that after mechanical compression, these samples generated two distributions of free-volume holes, i.e., the polymer free volume did not fully relax to a single distribution of free volume. To evaluate this, the sample data

022603-9



TABLE IV. Reevaluated lifetime, intensity, and free-volume results (PATFIT), including τ4 and I4 at 0.1 MPa for depressurized PMMA with different levels of shear strain (γ ). γ 0% 143%

τ3 (ns)

τ 3 (ns)

I3 (%)

I3 (%)

R∗ ˚ (A)

R ∗ ˚ (A)

fv+ (%)

fv+ (%)

τ4 (ns)

τ 4 (ns)

I4 (%)

I4 (%)

R ∗∗ ˚ (A)

R ∗∗ ˚ (A)

fv++ (%)

fv++ (%)

2.069 2.084

0.038 0.048

25.45 23.73

1.10 1.48

2.918 2.931

0.028 0.035

4.76 4.51

0.34 0.44

0.977 0.841

0.373 0.214

5.653 7.989

4.441 1.486

1.622 1.371

0.631 0.417

0.18 0.15

0.35 0.17

R ∗ , based on τ3 ; fv+ , based on I3 ; R ∗∗ , based on τ4 ; fv++ , based on I4 . The terms are the corresponding errors of each measurement.

4. Stress relaxation test

Room temperature (25 °C) time domain stress relaxation for both PMMA and PC was analyzed at 1% and 2% strain levels, respectively. The data were analyzed using the stretched exponential Kohlrausch-Williams-Watts equation [66,67]. n t . (8) E(t) = E0 exp − λ E0 and E(t) are the relaxation modulus at time zero and time t, λ is the characteristic relaxation time, n is the polydispersity of relaxation time, and n = 1 for a single relaxation time. The relaxation time distribution increases with the decrease in n. Normalized stress versus relaxation time comparison between PC and PMMA in the linear viscoelastic range is shown in Fig. 13, which clearly shows a faster relaxation time in PMMA than in PC. The n values for PC and PMMA controls were found to be 0.25 and 0.21, respectively. The relaxation time for PC and PMMA based on curve fitting has been found to be e13 and e12 s, respectively, indicating at 25 °C, PMMA has a faster relaxation time than PC. The relaxation time found PC is higher than the values reported for β relaxation of PC at room temperature, which is in the neighborhood of 107 s [61]. This is likely due to the high molecular weight in our case. On the basis of the aforementioned analysis, it is plausible to propose the following. With the application of the simple shear deformation, polymer chains are aligned along the maximum tensile direction causing the free-volume holes

to be deformed into ellipsoids. PMMA showed >100% fv recovery upon the removal of test pressure. Combined with the lifetime distribution analysis which shows two populations of lifetime distribution, it is likely that additional microcracks have been formed during the pressure test, especially since the highest PALS test pressure is beyond the compressive yield stress of PMMA. As shown by molecular dynamic simulation [62], upon the removal of hydrostatic pressure, all the principle strains can be released in glassy polymers. In our case, due to the low molecular entanglement density, it is possible that the strain induced microcracks are formed upon the release of hydrostatic pressure leading to the formation of an additional population of free volumes, i.e., forming a bimodal free-volume distribution. PC, on the other hand, behaves differently; in control PC, fv only recovered partially upon PALS pressure removal, whereas in oriented PC, fv was recovered >100%. However, the lifetime distribution analysis did not show any bimodal lifetime distribution indicating the initial formed free-volume holes have become larger upon the application of 124 MPa PALS pressure. This difference is mainly due to higher entanglement molecular weight and slower chain relaxation time in PC than PMMA as shown by the relaxation time difference between the two. The combined effect from these two features leading to the free-volume recovery is slow enough to allow additional mechanical deformation before forming a population of microscopic cracks in PC. PMMA, on the other hand, has a much faster chain relaxation time and a much lower chain entanglement density which makes the material less mechanically damping

1.10 PMMA at 1% strain PC at 2% strain

1.00 0.90

E(t)/E0

were analyzed again using PATFIT (as above), and an additional lifetime for the second distribution of free-volume voids was considered. The results of this analysis are shown in Table IV. While all the decompressed samples were reevaluated, only PMMA control and PMMA with 143% shear strain gave satisfactory results for two defect lifetimes in samples after the pressure extreme, but with larger attendant errors. The first lifetime (τ3 ) is close to the lifetime of positronium species in the starting polymer sample. The second lifetime (τ4 ) is similar to that of the highly compressed samples and suggests a set of smaller free volumes present after decompression. PMMA samples with low levels of shear orientation did not exhibit this bifurcated free-volume phenomenon. Although this result helps explain the well-resolved lifetime distributions derived from MELT discussed above, it should also be noted that there is a larger associated error for the evaluated lifetimes with lower intensity. This observation of a bimodal lifetime distribution after a macroscopic physical process is important in its own right, and will be the focus of future experiments as outlined in [51].

0.80 0.70 0.60 0.50 1

10

100

1000

Relaxaon me (s) FIG. 13. Normalized modulus versus relaxation time comparison for PC and PMMA without simple shear deformation.

022603-10



and is thus prone to form populations of microcracks upon deformation.

in simple sheared glassy polymer at ambient conditions (far below glass transition) must happen at a relatively fast rate. The removal of stress led to free-volume relaxation in all samples studied. The alignment of the polymer chains and free-volume holes imposes molecular restrictions on the molecular mobility of both PC and PMMA in their glassy states.

IV. SUMMARY

On the basis of the PALS analyses of both PC and PMMA, it is clear that the two polymers in the unstressed state exhibit different levels of fractional free volume; PC has a higher fractional free volume than PMMA, which might contribute to the high fracture toughness of PC compared to PMMA under quasistatic conditions. A large discrepancy was found between the macroscopic eccentricity factor and microscopic hole eccentricity factor in the simple sheared samples. Combining the bulk- and pressure-dependent PALS analyses, it is clear that the relaxation of the free-volume holes

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ACKNOWLEDGMENTS

This work was supported by the internal research and development (IRAD) funding by the Research and Exploratory Development Business Area of the Johns Hopkins University Applied Physics Laboratory (JHU/APL). Writing of this paper was made possible through the Stuart S. Janney fellowship from JHU/APL.

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