Tensile Properties of Latex Paint Films with TiO2 ...

Tensile Properties of Latex Paint Films with TiO2 Pigment Eric W.S. Hagan1,2, Maria N. Charalambides1, Christina R. T. Young3, Thomas J.S. Learner4, Stephen Hackney2 1

Department of Mechanical Engineering, Imperial College London, UK Conservation Department, Tate, London, UK 3 Department of Conservation & Technology, Courtauld Institute of Art, London, UK 4 Getty Conservation Institute, Los Angeles, CA 2

Abstract. The tensile properties of latex paint films containing TiO2 pigment were studied with respect to temperature, strain-rate and moisture content. The purpose of performing these experiments was to assist museums in defining safe conditions for modern paintings held in collections. The glass transition temperature of latex paint binders is in close proximity to ambient temperature, resulting in high strain-rate dependence in typical exposure environments. Time dependence of modulus and failure strain is discussed in the context of time-temperature superposition, which was used to extend the experimental time scale. Nonlinear viscoelastic material models are also presented, which incorporate a Prony series with the Ogden or Neo-Hookean hyperelastic function for different TiO2 concentrations. Keywords: Latex paint, Viscoelastic modeling, Finite strain, Hyperelastic, Prony series, Time-temperature superposition, Moisture content

1. Introduction Latex paint is an extremely common medium for modern artists and it is typically applied to a canvas substrate that is restrained on a stretcher. Even when other materials are used, the surface preparation layer most often contains a latex binder with a small amount of TiO2 for opacity, and larger quantities of filler—such as calcium carbonate, kaolin, or a variety of other white inorganic materials. The frequent appearance of latex paint in collections makes it important to determine its mechanical properties, and it is necessary to consider a wide range of conditions. A typical painting consists of interacting layers that may be subjected to various types of load: vibration in transport, impact, and environmental fluctuations. The aims of this study were to illustrate the large-strain tensile properties exhibited by latex paints between the glassy and rubbery regimes, and to determine a set of constitutive constants that would facilitate modeling with finite element (FE) software. Finite element analysis has been performed for oil paints on canvas by Mecklenburg and Tumosa (1991) using a linear-elastic model to predict the stresses that develop during changes in temperature and relative humidity. The use of such a model is justified for oil paints due to the large degree of cross-linking in the binder. In the current study on latex paints, there is a strong influence of strain-rate on the mechanical properties, and the stress-strain curve is highly nonlinear. This behavior is expected for a high molecular weight polymer with little cross-linking, at a temperature slightly above its glass-transition. The approach taken for the analysis was to measure the tensile properties of free films with different amounts of TiO2 pigment, and apply the principle of time-temperature

superposition (TTS) (Williams, Landel et al. 1955) to extend the time-scale of the experiments. This technique has been applied to other art-related materials at small strains. Ogawa (1998) investigated the effects of humidity on the properties of urushi and showed a decrease in the equilibrium modulus with increased moisture content. Obataya et al (2001) further studied the time-dependent behaviour of urushi with respect to its vibrational response on musical instrument soundboards. From industrial research, the work of Zosel (1980) is particularly of interest as it compares the mechanical properties of many coating types and includes the effects of pigmentation. Early research into the stress-strain properties of rubber is extremely useful for interpreting tensile data of latex films at large deformations. Smith (1962) suggests representing stress as a separable function of time and strain for elastomeric materials and describes ways for validating this theory. For example, the curves at different strains on an isometric plot must lie parallel for the theory to hold true. It is also indicated in literature that the ultimate properties (i.e. failure stress and strain) of SBR rubber follow the same time-temperature relationship as that determined for the modulus (Smith and Stedry 1960). The application of TTS at large strains forms a basis for the analysis in this study.

2. Materials and Methods The materials investigated were latex paint films with a poly(butyl acrylate-co-methyl methacrylate) binder exhibiting a glass transition (Tg) in the vicinity of 10°C with dynamic scanning calorimetry (DSC). Golden Artist Colors supplied the materials for investigation and their exact composition remains unknown to the authors. The primary interest was to obtain a high quality artist paint formulation with the volume fraction of inorganic material as a control variable. The manufacturer provided a formulation with the highest volume fraction of pigment (0.38 TiO2 when dry) and another without pigment. These two batches were mixed together in different ratios to create a spectrum of pigment concentrations. The amount of TiO2 in the dry films was verified with a rule of mixtures calculation involving three measured densities: TiO2 pigment particles (3.86g/cm3), non-pigmented latex (1.12g/cm3), and the pigmented film. The calculated amounts of TiO2 were also confirmed with thermal gravimetric analysis. Extraction of water-soluble additives from the films showed the non-pigmented latex to have 4% extractable material by weight. The majority of this substance is likely a combination of surfactants that remain after film formation. Depending on their molecular weight and structure, they may be soluble in the binder, or form crystalline inclusions. Similar extractions from the pigmented films showed a 2% by weight increase of extractable material in the binder fraction for a TiO2 volume fraction of 0.38. This increase is attributed to the dispersant used when adding the pigment to the latex. The films were simply treated as two component polymer/pigment composites for the purposes of this investigation, and the influence of additives was not studied further. The composition of latex paint is discussed in detail by Jablonski et al (2003) with respect to concerns in art conservation. For general information on the structure of non-pigmented latex, and the nature of film formation, there is a vast amount of literature available. Steward (2000) and Keddie (1997) each provide reviews of the process. Paints were cast on polyester sheets using parallel tape layers to control thickness and a razor blade to draw down smooth films. The materials were then left in the laboratory for over one year to allow the latex to fully coalesce and material properties to stabilize. One

day prior to testing, the substrate was removed and rectangular tensile specimens were cut to 60mm by 6mm dimensions using a template and razor. Thickness of these free films varied from 0.12 to 0.2mm across all materials. Paper tabs were bonded to the ends of each sample with a cyanoacrylate adhesive to provide a rigid gripping surface with a free gauge length of 40mm. The paint films were then conditioned overnight to the required equilibrium moisture content in small enclosures containing silica gel buffering at 50%RH. Samples were also equilibrated at 5%RH for a smaller set of tests under dry conditions. Tensile tests were performed on a universal testing machine at constant speeds of 50, 5, 0.5 and 0.05mm/min using a 100N load cell. An environmental chamber was constructed inhouse to maintain constant temperature and relative humidity (for T>10°C) during the experiments (Hagan 2008). The results are summarized using secant modulus values, failure strains, and full stress-strain curves. Secant modulus was taken at a strain of 0.005, and true strain is used throughout the discussion.

3. Experimental 3.1 The effect of pigment content Figure 1 gives atomic force microscopy (AFM) images at two magnifications for a nonpigmented latex at an age of two days. The early film structure consists of closely packed latex particles, and the interface material is most likely surfactant. When the temperature is above Tg, or in the presence of a coalescing aid, these particles inter-diffuse to form a more homogeneous film over time. Surfactant will also migrate through the bulk film to reach the attracting surfaces.

Figure 1. AFM images at different magnifications for a two-day-old BA/MMA latex film without pigment.

The first set of tensile tests was performed on the latex paint binder alone to identify the relaxation spectrum before any pigment was added. Figure 2a summarizes the secant modulus measured at different temperatures and strain rates. These data are plotted against log(1/R), where R is the initial strain rate (speed/gauge length) of the test in s-1. It is noted that strain rate decreases with extension in a constant speed test; however, at small strains it is possible to use the defined variable R to generate a master curve of the modulus. An alternative approach would involve plotting these data against log(t0.5%), where t 0.5% is the time at 0.5% strain. Figure 2b shows the result of superposition at a reference temperature, Tref, of 20°C, and the time scale now extends over ten decades for the single curve. These superposed data are plotted against log[1/(RaT)] where a T is the shift factor (see Appendix

Fig. A1) along the time axis. Sufficient overlap was found between curves if 10°C increments were used in combination with R spanning four decades. The range of test conditions was adequate to show the transition from glassy to rubbery over long times, and near equilibrium conditions were achieved. The melting of surfactant crystals made it difficult to determine equilibrium properties. DSC measurements showed the onset of surfactant melting at ~35°C with a peak at 47°C. For this reason, the highest test temperature was 30°C and the data point at the longest time in Figure 2b is from the lowest test speed at this temperature. Tensile tests at higher temperatures confirmed the DSC results by showing a sudden drop in the modulus as the film structure was altered.

Figure 2. Secant modulus of the non-pigmented latex versus time: a. test data at different temperatures and strain rates showing 95% confidence intervals calculated from the Student’s t-test; b. modulus data reduced to Tref=20°C using the shift factor, aT.

The effect of adding TiO2 pigment particles to the latex was of significant interest in this study. Figure 3a shows an SEM image of TiO2 powder with particles approximately 0.3µm in diameter. The freeze-fractured surface of a one year-old commercial latex paint containing TiO2 is given in Figure 3b for comparison. This paint film contains particles of similar shape and size, and no latex particles are visible because the film has significantly coalesced over time. A pigment volume fraction of 0.32 was estimated from density measurements on this artist material. The large amount of expensive pigment highlights a fundamental difference between artist paints and their household counterparts (containing greater amounts of filler and extender).

Figure 3. a. TiO2 powder imaged with SEM at 20,000X; b. freeze-fractured paint surface imaged with AFM.

Paints with TiO2 volume fractions of 0.13, 0.25, and 0.38 were formulated with the latex, and master curves of modulus data were created in the manner shown earlier. Figure 4a gives the secant modulus at Tref=20°C for each of the formulations with the curve for nonpigmented latex (Fig. 2b) included for comparison. As expected, the curves show an upward shift with increased volume fraction of TiO2 due to constraint posed on the binder fraction. The strong modulus enhancement in the rubbery region is attributed to the large specific surface area (surface area/volume) of the particles since the mobility of polymer chain segments is limited at the polymer-pigment interface (Zosel 1980). Figure 4b shows failure strains, εf, plotted against failure time, tf, at T ref=20°C using shift factors determined from the secant modulus data (see Appendix Fig. A1). It is apparent that strain to failure decreases with the addition of pigment particles for a given failure time. This is expected due to the rigidity of the TiO2 particles and the constraint they impose on the polymer fraction. An inverse interpretation is that failure time increased with pigment content for a given failure strain. As an example, consider a failure strain of 2% for the latex with 0.38 TiO2. The logarithm of failure time is read approximately from Figure 4b as log(tf/aT)=-2 at log(0.02)=-1.7. The material will fail if it is deformed to 2% strain in any time less than 10-2=0.01s. At longer times, stress sufficiently relaxes so that failure does not occur. Films with less pigment must be deformed to this strain in a much shorter time to reach failure. The results presented in Figure 4b show the potential for assessing the risk of failure for latex films over a broad range of conditions. At 20°C the films will only fracture if they are deformed at extremely high rates, which are unlikely to occur. A higher risk condition would involve a drop in temperature (e.g. during transportation) and the experimentally determined shift factors permit referencing these data at any temperature between –10°C and 30°C.

Figure 4. Tensile properties of BA/MMA latex films with different volume fractions of TiO2: a. secant modulus reduced to 20°C; b. failure strains reduced to 20°C with the same shift factors as the modulus data.

With respect to Figure 4b, it is noted that several factors may influence the results. Rectangular samples are not ideal for measuring strain to failure, due to the concentration of stress at the grip edges. Bonded tabs on the ends of each sample were used to reduce this effect by lowering the compressive force produced by grip tightening. This preparation method had the additional advantage of eliminating slippage within the grips during extension of the specimen. The cast films were also prepared very carefully to minimize foaming at the surface. Air bubbles are easily created by the surfactants if the latex is quickly agitated, and these can lead to small imperfections in the dry film. It is assumed

from the precautions taken during the experiments that Figure 4b provides at least a reasonable estimate of strain to failure. Relative standard error (RSE) in failure strain measurements was below 10% in the rubbery region (long times) and averaged from 715% for films in the glassy region (short times).

3.2 Effect of moisture content A smaller set of experiments was performed to illustrate the effect of moisture on the relaxation spectrum, and the importance of environmental control. All of the data discussed thus far were collected after conditioning the paint samples to equilibrium moisture content at 50%RH, 23°C. Figure 5a compares two of the master curves shown earlier with data collected from samples conditioned at 5%RH. Tests were performed at 20°C and 30°C; therefore, only partial curves are given. The loss of moisture results in a significant increase in film stiffness from two possible factors: an increase of the polymer Tg, and an increase of entanglements or pseudo cross-links (Furukawa 2003). The latter makes it impossible to apply a simple time-moisture superposition principle since the equilibrium modulus appears to be altered in the presence of water. Figure 5b gives the equilibrium moisture content, Me, (per unit weight of paint binder) as a function of relative humidity for different volume fractions of pigment. It is evident that the presence of TiO2 particles increases the equilibrium moisture content at a given relative humidity. Water is known to adsorb on the surface of metal oxides (Zhang and Lindan 2003) and it is also attracted to polar functional groups in polymers (Barrie 1968). As the pigment fraction is increased, less polymeric material is present to absorb water; however, the available surface area of the pigment particles increases. The size of the pigment particles is also expected to affect water adsorption at a given volume fraction since the specific surface area (surface area/volume) is high for small particles.

Figure 5. Influence of moisture content: a. secant modulus reduced to 20°C for 0 and 0.38 TiO2 volume fractions at 50%RH and 5%RH; b. equilibrium moisture content in the binder fraction for 0, 0.13 and 0.38 TiO2 fractions at 23°C.

4 Modeling The experimental results presented a clear picture of the material behavior under a wide range of conditions. It is also desirable to establish a set of constitutive constants for input to finite element software. When modeling the stress-strain response of elastomeric materials it is convenient to represent stress as a separable function of time and strain: σ (ε,t) = σ o (ε)g(t)

(1)

Validation of Equation 1 is performed with the aid of isometric curves of experimental data that are plotted as log(σ ) against log(t). The curves at different strains are parallel when € strain and time are separable functions (Smith 1962). Figure 6a shows an isometric plot reduced to 20°C for the non-pigmented latex. It is apparent that the curves are not parallel at short times when ε>0.02, and data in this region are associated with experiments at low temperatures and/or high rates of deformation. Stress-strain curves are given in Figure 6b to illustrate the necking behavior that is observed for the faster test speeds at 0°C. This is specifically shown by a bump in the curve at small strains, after which the strain in the sample becomes non-uniform. An analysis of the necking phenomenon is given elsewhere (Hagan 2008) for these data using Considere’s criterion to establish the onset conditions. The following viscoelastic models are limited to conditions where strain was uniform in the test sample; therefore, data in the necking region were removed from the calibration where appropriate.

Figure 6 a. Isometric curves for the non-pigmented latex reduced to 20°C; b. Stress-strain curves illustrating necking behavior of the latex at 0°C (i.e. bump at low strains for 5 and 50mm/min).

In Equation 1, the normalized function g(t) was defined in the form of a Prony series, where N

g(t) = g∞ + ∑ gie−t τ i

(2)

i=1

and €

N

g∞ + ∑ gi = 1. i=1

€

(3)

Each gi term in Equation 2 represents the relative contribution of the ith Maxwell element to the initial modulus of the material, at the corresponding relaxation time, τ i. A non-zero value for g∞ implies that the stress does not diminish to zero at long times. Many hyperelastic models are available to define the time-independent stress function σo(ε) in Equation 1 (Marckmann and Verron 2006). In this study, the Ogden and Neo-Hookean (ABAQUS 1998) hyperelastic functions were employed for large and moderate strains respectively. In uni-axial tension, the Ogden hyperelastic potential gives σ 0 (λ) =

2µo λ α −1 (−α 2)−1 λ −λ α

[

]

(4)

for true stress, when written in terms of the stretch ratio λ = exp( ε ). The Ogden €α parameter is an empirical variable that alters the form of the curve at large strains, and µo is the initial shear modulus. The Neo-Hookean hyperelastic potential provides a much simpler function under the same deformation mode:  1 σ o ( λ ) = 2C10 λ2 −   λ

(5)

This equation is equivalent to the Mooney-Rivlin form with C01=0 (ABAQUS 1998) and is generally useful for low to moderate strains. For either hyperelastic equation, the initial € Young’s modulus, Eo, is found by differentiating σ o(λ ) with respect to λ, and taking the limit as λ→1. This results in Eo=3µo for Equation 4 and Eo=6C10=3µo for Equation 5. The factor of three difference between shear and Young’s moduli arises from the assumption of incompressibility. Solving for the stress given by an arbitrary strain history, ε (t), involves evaluating the convolution integral: σ (ε,t) =

t

∫ g(t − s) 0

dσ o (ε) ds ds

(6)

With certain strain histories and hyperelastic potentials, it is possible to solve Equation 6 €analytically. To determine the parameters under other conditions it is necessary to approximate the integral using numerical methods. The technique used in this study is discussed in detail by Goh et al (2004) with reference to the work of Taylor (1970) and Kaliske (1997). A simple modification was made to the solver routine in order to extend the experimental time-scale by means of TTS. For tensile data at temperatures other than the pre-described reference temperature, the relaxation times were modified according to the relationship τ i = aT τ iref ,

(7)

while the g i terms remained unaltered. The shift factor, aT, was the experimentally €determined value from the modulus master curves in Figure 4a (see Appendix Figure 9 for values). It was found that relaxation times spaced equally by one decade produced satisfactory results with the least possible number of terms.

Figure 7 shows experimental stress-strain curves from the non-pigmented latex equilibrated at 50%RH prior to testing, and the Ogden/Prony series model that best fits these data at large strains. Tensile curves are only given at 20°C and 30°C in this case since necking occurred at lower temperatures and/or higher strain rates. The Ogden model performs extremely well at following the sharp rise in stress at high strains when T>Tg, and the ratedependence is accurately captured by the Prony series. Additional model calibrations were performed for the latex containing TiO2 volume fractions up to 0.25, and similar results were obtained at 20°C and 30°C. The parameters for each concentration are given in Appendix Table 1 along with the root-mean-squared error (RMSE) as a measure of ‘goodness of fit’ for comparison. With the existing theory, it was not possible to develop an accurate model for films containing 0.38 TiO2. The use of a damage function is likely required for such a high inorganic concentration, and this will be studied in future work.

Figure 7. Comparison of experimental stress-strain curves of the non-pigmented latex with predictions from the Ogden/Prony series model at large strains: a. 30°C; b. 20°C.

Capturing the stress-strain response at moderate strains may be sufficient for the application to paintings conservation. Figure 8 compares experimental data with the same Ogden/Prony series model (Table 1) at strains up to 0.02 for T≤Tg and 0.05 for T>Tg. The results show a very accurate fit at each test temperature and crosshead speed. Data at 0°C, 10°C, and –10°C (not shown here) were included in this case since the necking region is avoided below strains of approximately 0.02. Within this range of deformation, it is also possible to simplify the model by replacing the Ogden function with the Neo-Hookean form. This eliminates one parameter and the difference in error is negligible as long as the initial shear modulus, µo, is maintained between models.

Figure 8. Comparison of experimental stress-strain curves of the non-pigmented latex with predictions from the Ogden/Prony series model at small strains: a. 30°C; b. 20°C; c. 10°C; d. 0°C.

The results presented in Figures 7 and 8 show excellent potential for predicting the stressstrain behavior of latex paint films at two different strain regimes on the first loading cycle. The parameters determined for these models are easily incorporated into existing FE software packages, and it is noted that accurate modeling may require additional calibration with other deformation modes (i.e. pure shear & equibiaxial modes). Further work is required for modeling of paint films with high volume fractions of inorganic material, and the behavior on multiple loading cycles must be quantified.

5 Conclusions This work highlights the use of TTS for extending the time-scale of tensile data collected from latex paint films, and its applicability up to large strains. Measured shift factors were successfully used to generate secant modulus and failure strain master curves at TiO2 volume fractions ranging from 0 to 0.38. The described method for predicting failure strains as a function of time and temperature may provide a valuable risk assessment tool for conservation. The failure strains were shown to decrease by roughly three orders of magnitude between rubbery and glassy regions. This is important to consider in paintings conservation since the glass transition of latex artist paints is close to ambient temperature. In order to interpret the significance of these data, it is necessary to investigate the stresses that develop in an actual painting. Finite element analysis may provide a useful tool for this purpose. To facilitate future FE modeling, shift factors from TTS were used to calibrate non-linear viscoelastic models that combined Ogden or Neo-Hookean hyperelastic functions with a Prony series. Acceptable agreement was found between these models and experimental data; however, limitations were found at low temperatures (or equivalently high strain rates), where necking was observed in the experiments. In future work, the use of FEA may require calibration with experimental data from additional deformation modes to

provide accurate predictions, and the effect of multiple loading cycles must be considered. The theory used in the present work did not perform well at high inorganic concentrations. Modeling of such materials may require the addition of a damage function to the existing equations. Several areas remain for further analysis since there are many factors that influence the tensile properties of the paint films. From micromechanics theory, it is expected that inorganic pigments with an aspect ratio much different from unity will pose greater restraint on the binder. The TiO2 particles in the present work were approximately spherical and are expected to have a lower stiffening effect than particles such as clay for example. In a similar set of experiments, the effects of adding inorganic particles with different shapes (short fibers, platelets) was investigated, and this information will be published at a later date (Hagan 2008). Other areas requiring further analysis include the influence of surfactants and moisture on the relaxation spectra. The role of moisture was only touched upon in this study to emphasize that it has a strong effect on the mechanical properties, and accurate environmental control is necessary for satisfactory results.

Acknowledgements The authors would like to thank the Deborah Loeb Brice Foundation, Tate, and the Natural Sciences and Engineering Research Council of Canada (NSERC) for funding this project. We would also like to thank Golden Artist Colors for supplying the materials, and staff at the Getty Conservation Institute for their support with thermal analysis and microscopy.

Appendix Figure 9a gives the shift factors, aT, used throughout this work at Tref=20°C. Results are not presented in terms of the well-known WLF equation since it is only applicable to homopolymers above Tg. The fact that the latex is comprised of a random copolymer with a crystalline surfactant phase makes it necessary to determine these shift factors experimentally. It is interesting to note that the a T values are very similar for different volume fractions of pigment. Figure 9b shows the shear modulus weight distribution for the Ogden/Prony series model at T ref=20°C. Recall that relaxation times at different temperatures are calculated from aT according to Equation 7.

Figure 9. a. Shift factor versus temperature for different TiO2 concentrations; b. Initial shear modulus of Maxwell elements (µi=giµo) in the Ogden/Prony series model at their corresponding relaxation times, τi.

Table 1 lists the parameters determined for the Ogden/Prony series model at different TiO2 volume fractions. The root-mean-square error (RMSE) is provided at the bottom of this table as a measure of ‘goodness of fit’, which is defined here as E M 1 n σ j − σ j RMSE =  ∑ n j=1  σ Ej

2  

(A1)

In this equation, σjE and σ jM are the experimental and model stress values respectively for the jth data point up to n points. The RMSE values are given for both strain regimes € investigated in Figures 7 and 8. As an example, at low strains (Fig. 8), the 20 stress strain curves (five temperatures and four speeds) and 14 points per curve gave n=280 points for the error calculation. Relaxation Time Maxwell Element, i

τi , s

1 2.00E-06 2 2.00E-05 3 2.00E-04 4 2.00E-03 5 2.00E-02 6 2.00E-01 7 2.00E+00 8 2.00E+01 9 2.00E+02 10 2.00E+03 11 2.00E+04 Equilibrium, e Goodness of fit, Eqn. A1 (small strain, see Fig. 8) Goodness of fit, Eqn. A1 (large strain, see Fig. 7)

Latex with no pigment (µo=581Mpa, α=-5.7)

Latex with 13% TiO2 (µo=843Mpa, α=-5.52)

Latex with 25% TiO2 (µo=1228Mpa, α=-4.16)

gi

gi

gi

0.09544687 0.17722885 0.23616776 0.22585009 0.1550006 0.07634137 0.02698361 0.00411039 0.00151003 0.00035926 0.00035926 0.00064191

0.12127185 0.17682613 0.20771348 0.19656885 0.14986373 0.09204711 0.04554656 0.00583675 0.00230889 0.00053209 0.00051057 0.00141245

0.12540418 0.18444016 0.21464184 0.19764631 0.14400529 0.08302017 0.03787078 0.00621745 0.00303431 0.00120317 0.00089949 0.00341389

0.061

0.086

0.13

0.15

0.17

0.22

Table 1. Ogden/Prony series model parameters for different TiO2 concentrations and the associated error.

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