Predictions of Volume Relaxation in Glass Forming Materials Using a

Article pubs.acs.org/Macromolecules

Predictions of Volume Relaxation in Glass Forming Materials Using a Stochastic Constitutive Model Grigori A. Medvedev and James M. Caruthers* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907, United States ABSTRACT: In a previous publication (Macromolecules2012, 45, 7237), a stochastic model of the glassy state that acknowledges dynamic heterogeneity was shown to describe the classic Kovacs poly(vinyl acetate) volume relaxation data set with the exception of the expansion gap and the pressure dependence of the glass transition temperature. This initial model included two assumptions−the relaxation time of the meso-domain was controlled by the specific volume and the fluctuations were isotropic. Recently, a fully tensorial stochastic constitutive model (SCM) was developed where (i) the relaxation time was a function of both the configurational entropy and the stress tensor and (ii) nonisotropic fluctuations were allowed. In the present work, the SCM is shown to predict the Kovacs data, including the expansion gap as well as the pressure dependence of the glass transition temperature.

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fluctuations, the local mobility in a meso-domain also fluctuates. Thus, a stochastic description of the response of a glassy material is a fundamentally different approach than the traditional approach used by KAHR and related models, where the average relaxation time depends upon the average value of the variable controlling the relaxation time. If the function that relates the relaxation time to the local structural variables is very strong, the difference in the values of these variables for different meso-domains results in a significant difference in the relaxation rates, i.e., dynamic heterogeneity. The experimentally measured macroscopic response is an average over an ensemble of meso-domains. Complete characterization of the state of the glassy material requires knowing the distributions of all structural variables from which the distribution of the relaxation times can be determined. In the stochastic approach, the shapes of these distributions and the associated relaxation time spectrum evolve in a complicated fashion during the externally applied thermal and deformation history; i.e., the predicted response is thermo-rheologically complex. There are two key issues in the explicit formulation of a stochastic based model of the relaxation behavior of glassy polymers. First, what are the independent, deterministic variables and what quantities fluctuate? And second, what is the local mobility, i.e., the log a model? In the previous stochastic model for volume relaxation (which we designate as the “volume stochastic model” or VSM6), the local specific volume was fluctuating whereas temperature and pressure were the externally controlled macroscopic values and the local log a was a function of all three variables. Thus, the temperature and pressure histories were the same for each of the meso-domains and the specific volume history for a given meso-domain was one realization of the defining stochastic differential equation

INTRODUCTION The classic series of experiments by Kovacs1 on volume relaxation following temperature jumps for poly(vinyl acetate), PVAc, in the Θg29 region reveals the richness and complexity of the viscoelastic behavior of glassy materials, including such features as nonlinearity, asymmetry of approach to equilibrium, and “memory” for a two-step thermal history. For years a variety of theoretical models have been unable to predict all the features of the Kovacs data.2−5 In a previous publication6 we demonstrated that the phenomenological Kovacs−Aklonis− Hutchinson−Ramos (KAHR) model,7 which is widely considered as having come the closest to describing the full Kovacs data, fails in a dramatic fashion to describe the “short anneal” experiments of Kovacsalthough this should intuitively be one of the easier experiments to predict, since the amount of relaxation is small. This failure occurs even if the spectrum of the relaxation times can have any shape as well as if the fit to the rest if the Kovacs volume relaxation data set is sacrificed. We argued that the KAHR model failure is representative of the entire class of thermo-rheologically simple models that employ a “material clock” t* and implicitly assume that the shape of the relaxation spectrum is invariant to changes in temperature, volume or other state variables. In our previous communication6 we proposed that one must explicitly acknowledge dynamic heterogeneity, which is a key signature of the glass region,8 and then developed a stochastic model for volume relaxation that was able to describe the Kovacs PVAc volume relaxation data, including the short time annealing experiments. The primary assumption in the stochastic approach to describing glasses is that the behavior of a meso-domain is governed by local values of the variable(s) upon which the local relaxation time depends.6,9 Since the meso-domains are of the order of several nanometers, there are inherent fluctuations in the various thermodynamic quantities associated with a given meso-domain; hence, a stochastic description of the mesodomain response is required. Because of these local © 2015 American Chemical Society

Received: September 9, 2014 Revised: January 12, 2015 Published: January 27, 2015 788

DOI: 10.1021/ma501870k Macromolecules 2015, 48, 788−800

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Macromolecules (SDE). In a more recent publication,9 a full tensorial stochastic constitutive model (SCM) was developed, where both the local entropy and local stress tensor were fluctuating and the temperature and strain tensor were deterministic and hence the same for all meso-domains. Using the VSM, it was possible to predict for the first time the entire Kovacs data set, including the “short anneal” experiment. One of the parameters in the VSM is the size of the meso-domain, which was determined to be 2.8 nm via optimization of the VSM to the Kovacs data set. The size associated with dynamic heterogeneity has been determined experimentally via multidimensional NMR10 and light scattering11 to be in range of 2−4 nm, confirming that incorporating fluctuations as in the VSM is physically reasonable. In addition, the VSM provided insight into the behavior of individual mesodomain; specifically, the volume trajectory of an individual meso-domain had large fluctuations consistent with the molecular picture of small systems, where the macroscopic volume relaxation response only emerged when a large number of realizations were averaged. Also, the trajectory of the mobility of an individual meso-domain as characterized by the material time t* exhibited a start−stop like behavior characteristic of a Levy flight.12 However, the VSM had three important unresolved issues, as described below: (1) Expansion Gap. In the Kovacs volume expansion experiment a material equilibrated at Θi is rapidly brought to temperature Θf (Θf > Θi) and the subsequent volume relaxation is monitored, where the effective relaxation time, i.e. τeff, is obtained via differentiation of the specific volume response with respect to time. Consider a series of up-jump experiments with different initial temperatures, but with the same final temperature. As the relaxation approaches equilibrium all of these upjump experiments have the same temperature and specific volume, where it would be expected that τeff would be the same regardless of Θi; however, Kovacs observed1 that the dependence of τeff on Θi persisted seemingly all the way to equilibrium. Specifically, when the instantaneous τeff is plotted vs the instantaneous specific volume, an “expansion gap” occurs between the different Θi experiments as the equilibrium is approached i.e. as δ → 0, where δ = ((V − V∞)/V∞), V is the current specific volume, and V∞(Θ) is the equilibrium specific volume. McKenna and co-workers have carefully analyzed the original Kovacs data and have definitively concluded that the expansion gap is statistically well supported.13 To date no volume relaxation model, including the VSM, has been able to capture the expansion gap. However, the failures to predict the Kovacs τeff vs volume plot by the traditional models and by the VSM were qualitatively different. The traditional models of KAHR type are thermo-rheologically simple, where the final approach to equilibrium is controlled by the long relaxation time tail of the spectrum regardless of the history experienced by the material. Specifically, early in the response multiple relaxation times are operative and the response can be different for different initial conditions; in contrast, at the end of the response only the longest relaxation time contributes and the response is independent of the initial conditions; i.e., the τeff curves for different initial temperatures merge. Thus, the traditional thermo-rheologically simple models have no

hope of predicting the expansion gap.14 In contrast to the KAHR model and its variations, the VSM is thermorheologically complex, where the current relaxation spectrum depends on the thermal and pressure history. It follows that at least in principle the VSM has the ability to predict the expansion gap effect. We previously demonstrated that with an appropriate choice of material parameters the VSM can describe the expansion gap;6 however, these parameter values resulted in a poor fit to the rest of the Kovacs data. We speculated that this difficulty of VSM was caused by either a deficiency of the particular version of the meso-domain relaxation time function τ = τ(Θ,p,V) or the crudeness of the mean-field approximation employed to capture interaction between neighboring meso-domains. (2) Pressure Dependence of the Glass Formation Line. When a glass is formed via isobaric cooling, Θg exhibits significant pressure dependence. The glass formation line must be the iso-mobility line, where τ(Θ,p,V) is a constant and the slope of the glass formation line is given by (dV/ dΘ)τ=Const. In case of the VSM this slope calculated at the p = 1 atm point was the model parameter s that was determined by optimization to the isobaric Kovacs data set, where the pressure dependence of Θg was not included in the data set used in the parameter optimization. The (dV/dΘ)τ=Const slope needed in the VSM to describe the Kovacs data was qualitatively different from the slope of the experimentally measured glass formation line for PVAc. We interpreted this discrepancy as a clear indication that despite the success of the VSM in predicting the Kovacs volume relaxation data, the specific volume was not the correct variable that controls the relaxation rate of a meso-domain. (3) Isotropic Fluctuations. Implicit in the VSM is the assumption that the fluctuations experienced by a meso-domain are always isotropic, since volume is the fluctuating variable. The macroscopic boundary conditions in the Kovacs experiments are, of course, isotropic, which makes density/specific volume an appropriate variable to use in a continuous constitutive model. However, local boundary conditions experienced by a given meso-domain are hydrostatic only on average, where limiting the fluctuations experienced by a nanometer size domain to only isotropic ones appears unjustifiably restrictive. In addition to above shortcomings, by its structure the VSM can only describe deformations that are macroscopically isotropic. Recently the VSM was extended to include arbitrary, threedimensional deformations, where the fluctuations were not required to be isotropic. The resulting SCM9 assumes that the relaxation time of a meso-domain is a function of the strain and stress tensors, i.e. τ = τ(Θ,H,S,T) where H is the Hencky strain tensor, T is the stress tensor that is conjugant to the Hencky strain tensor and S is the entropy which is conjugant to the temperature Θ. As stated above, the entropy and stress are local (i.e., fluctuating) and temperature and strain are macroscopic (i.e., deterministic) and common to all meso-domains. Thus, in the SCM there are seven independently fluctuating variables− the six components of the symmetric mesoscopic stress tensor and the entropy. It should be noted that a straightforward generalization of VSM would be to have the strain tensor as the 789


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Macromolecules fluctuating variable; however, since the VSM was unable to capture the pressure dependence of Θg (see point 2 above) the choice of a fluctuating stress was deemed more promising. Because the fluctuating variable is stress instead of strain, the SCM is physically quite different than the VSM even in isotropic deformations. The SCM is able to qualitatively predict features of uniaxial constant strain rate experiments that have not to date been captured by the traditional constitutive models−models that do not explicitly incorporate mesoscopic fluctuations. In particular, the SCM was able to predict (i) the postyield softening and its dependence on the aging time,9 (ii) the dependence of the rate of stress relaxation at various points along the stress−strain curve on the strain rate of the loading,15 and (iii) the dependence of the stress overshoot in the so-called “stress memory” experiment on the strain rate of the loading and unloading steps.16 Considering the ability of the SCM to capture these challenging features of the nonlinear relaxation behavior of glass forming materials, we will now turn attention back to the Kovacs volume relaxation data to determine if the SCM can still predict the Kovacs volume relaxation and perhaps even address some of the limitations of the VSM detailed above. The paper is organized as follows: in the Stochastic Constitutive Model section we briefly recap the structure of the SCM, where the focus will be on the modifications made to the form of the log a function. In the Results section it will be shown that SCM is able to describe the Kovacs data as well as the VSM, but where the SCM is now able to also predict the expansion gap and capture the proper pressure dependence of Θg. The features of the SCM responsible for these improved predictions will be shown. In the Discussion, we put into perspective the predictive capabilities of the SCM for describing the time-dependent thermo-mechanical behavior of glassy polymers. Finally, a comment is necessary concerning our approach toward the analysis of constitutive models for describing the thermo-mechanical behavior of polymer glasses. We believe the one must test the qualitative predictions of a model against as diverse a set of data as possible. If a potential model exhibits qualitative failure for a particular type of experiment there is no need for quantitative fitting of other experiments where said model may do a much better job. So one of the goals of the current paper is to demonstrate that the recently developed SCM does not experience qualitative failure (in fact the predictions will be shown to be very good), when put to a stringent test of describing the volume relaxation data of Kovacs.

a=

τ τ0

(1)

where τ is the relaxation time and τ0 is the relaxation time in reference state. In the previous paper on the SCM9 the relaxation time for a meso-domain was assumed to depend on (i) the macroscopic temperature, (ii) the volumetric component of the strain, and (iii) the fluctuating entropy and the fluctuating symmetric stress tensor as represented by the dimensionless, seven-component vectorx̂(see Appendix A for the definition of x̂). In that paper we argued that an expression for the shift factor based on the ‘configurational internal energy’ is a reasonable first guess (see also Caruthers et al.17); specifically, ⎛ ⎞ ecR log a(Θ, V , x̂) = c1⎜ ∞ − 1⎟ ⎝ ec (Θ, V ) + eĉ (x̂) ⎠

(2)

where the nonfluctuating/equilibrium (i.e., e∞ c (Θ, V)) and the fluctuating (i.e., êc(x̂)) contributions in the denominator are assumed to be additive. The nonfluctuating term in eq 2 is constructed so that the Williams−Landel−Ferry (WLF) form18 of log a is recovered, when (i) the fluctuations are formally set to zero, i.e. x̂ = 0, (ii) the specific volume is at equilibrium above Θg, and (iii) the deformation is at atmospheric pressure. Specifically9 ⎧ V − VR ec∞(Θ, V ) = 2ΘR ⎨c 2(ΔAα∞ + ΔC) + ΔA VR ⎩ ⎫ + ΔC(Θ − ΘR )⎬ ⎭

(3)

where the various material constants are defined in Appendix A and can be determined from equilibrium and linear viscoelastic data. In this paper we assume that êc(x̂)is constructed from a linear combination of fluctuations in the (i) configurational entropy, i.e.x̂0, (ii) hydrostatic stress, i.e. x̂1, and (iii) the deviatoric part of the stress as given by the second stress invariant I2 ; specifically, eĉ (x̂) = nΘR σSx0̂ − mVRσK x1̂ +

1 VR I2(x̂) 4 ΔG

(4)

where coefficients n and m are log a model parameters. Previously the second term on the right-hand-side of eq 4 was not included.9 It should be noted that the expression for êc(x̂) given in eq 4 (i) is not linear with respect to the fluctuating variable x̂ since I2 (x̂) is quadratic, (ii) does not contain terms that are quadratic with respect to x̂1, and (iii) is not in the form suggested by the internal energy, because the internal energy does not have a linear pressure term, i.e. the x̂1 term. Thus, the effect of fluctuations on the local mobility as defined by eqs 2 and 4 is empirical, where a rational understanding of why this particular empirical form is successful for a variety of thermaldeformation histories remains a very significant research challenge. Expanding upon a postulate by Robertson,2a the configurational internal energy responsible for controlling the mobility of a specific meso-domain is assumed to be a combination of the configuration internal energies of that meso-domain and the surrounding meso-domains as given via a mean field approximation. Thus, the fluctuating term êc(x̂) in the mesoscopic log a expression given by eq 2 is replaced with

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STOCHASTIC CONSTITUTIVE MODEL The SCM predicts the response to a general three-dimensional, thermal-deformation history, where in Appendix A the formulation of the SCM for the specific case of a macroscopically isotropic thermal deformation is developed. One should remember that even though the macroscopic deformation of interest in this paper is isotropic, there is no constraint that the mesoscopic fluctuating stress be isotropic. The key nonlinear feature in the SCM is the mesoscopic shift factor “a”. A brief summary of the development of the log a model is provided below, where a more complete development is given in Appendix B. The shift factor “a” is a generalization of the traditional shift factor aΘ used in the time−temperature superposition procedure when analyzing linear viscoelastic data and is defined 790


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Macromolecules Table 1. Material Properties of PVAc ΘR (K) 303.9 a

a

α∞ × 104 (K−1)

VR (cm3/g) 0.8430

a

7.175

αg × 104 (K−1)

a

2.842

K∞ (GPa)

a

1.96

a

Kg (GPa) 3.23

Cp0∞ (J/kg/K) 1632.0

b

Cp1∞ (J/kg/K2) 0.72

b

Cp0g (J/kg/K) 263.5

b

Cp1g (J/kg/K2) 3.45b

b

Reference 20. Reference 21.

1 z−1 eĉ (x̂) + ⟨eĉ (x̂)⟩ z z

resourcesan effort that was not part of the current research. Thus, the parameters in the SCM given in Table 2 were determined based on a number of by-hand iterations, where the parameter evolution was guided by expert knowledge. On the basis of this experience, the SCM model is highly nonlinear, where relatively precise combinations of parameters are needed for the predictions to be in even qualitative agreement with multiple thermal histories; thus, this initial “human-driven” optimization is needed to provide reasonable initial conditions for any future “computer-driven” optimizer. As a result, the set of parameter values used in this communication can only be considered as preliminary. It is fully expected that a set that is optimized further will produce a better fit; however, we believe that the fit already achieved provides convincing evidence that the SCM is capable of describing the Kovacs data. The first prediction of the SCM is the constant rate (1 K/ min) cooling volume response, where the glass formation line as a function of pressure is predicted as shown in Figure 1. The

(5)

where z is the coordination number and < > indicates an ensemble average.

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RESULTS The ΔK, ΔG, ΔA, and ΔC parameters in the SCM are directly determined from the elastic behavior of the glass and rubber/ melt as given in eqs A.10−A.13. The glass and rubbery material property data for PVAc are given in Table 1 and the associated SCM model parameters are given in Table 2. The c1 and c2 Table 2. SCM Model Parameters independently determined parameters

WLF parameters ΔG (GPa)

ΔK (GPa)

ΔA (J/kg/ K)

207 1.3a Optimized Parameters

1.27

τ0 (s) this work ref 6 ref 9

5.0 × 10 2.58 × 104 1 × 104 4

c1

c2 (K)

16.8

42.56

mesoscopic log a model parameters

basic SCM parameters

a

a

L (nm)

z

ω

n

m

2.7 2.84 9.0

7.0 6.9 1.0

0.75 − 1.0

0.3 − 1.0

0.185 − 0.0

Reference 22.

WLF parameters for PVAc are also given in Table 2. The values of c1 and c2 are chosen to be the same as in the previous paper6 to facilitate comparison with the VSM results. It should be noted that these values are in agreement with the ones given originally by Ferry,19a although the most recent edition of Ferry19b provides somewhat different numbers for PVAc. These changes in values for c1 and c2 will affect the optimized parameters in the SCM, but will not qualitatively affect the predictions shown in this paper. The remaining SCM parameters for PVAc were determined from fitting the Kovacs volume relaxation data. In the VSM, it was possible to convert the defining stochastic differential equations into a master equation form that allowed a numerically efficient solution procedure that enabled optimization of the VSM model parameters with the Kovacs PVAc data set.6 In contrast, the SCM cannot be transformed into a useful master equation, thereby requiring direct solution of the set of stochastic differential equations which is significantly more numerically intense. Specifically, as shown previously with the VSM model6 it takes the average of order 104 realizations to obtain a reasonably smooth macroscopic response for a just one thermal history, where one must include at least 22 of the thermal histories in the Kovacs data set to effectively evaluate the predictions of just one parameter set in the SCM. The development of a full optimization code for the SCM would require massive computer resources and the development of a significant code to efficiently manage such

Figure 1. V vs Θ (on cooling) at various pressures: circles, McKinney−Goldstein data for PVAc;20 upper curve, 0.1 MPa; lower curve, 80 MPa; dashed blue line, experimental formation line; solid black line, SCM predictions.

slight discrepancy between the experimental and predicted glassy asymptotes at 80 MPa in Figure 1 is a consequence of linearization, i.e., the glassy and equilibrium coefficients of thermal expansion at 80 MPa are assumed to be the same as their counterparts at 0.1 MPa, and thus can be straightforwardly improved. The successful prediction of the formation line by the SCM is due in large part to the nonfluctuating contribution to the log a given by eq 3. Using the material parameters for PVAc from Tables 1 and 2 we obtain for the slope of the formation line the value of −23 × 10−4 (cm3/g·K), which is relatively close (and has the correct sign) to the experimental value of −16 × 10−4 (cm3/g·K).23 In comparison, the VSM predicted slope of the glass formation line was +1 × 10−4 (cm3/ g·K),6 which has the wrong sign. The effect of the fluctuating term in the log a given by eq 4 on the formation line is nonnegligible due to nonlinearity of the expression in eq 2, where the further improvement from −23 × 10−4 toward −16 × 10−4 (cm3/g·K) is the direct consequence of including fluctuations. Specifically, since the e∞ c (Θ, V) value is different at different 791


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Figure 2. Volume relaxation following down temperature jumps: symbols, data; solid line, SCM prediction. (a) From 40 to 25 °C: − blue, 27.5 °C; green, 30 °C; red, 32.5 °C; teal, 35 °C; magenta, 37.5 °C; yellow. (b) From 40 °C, teal; 37.5 °C, red, 35 °C, green; and 32.5 °C, blue; all to a final temperature of 30 °C.

Figure 3. (a) Volume relaxation following up-jump from 37.5 °C, teal; 35 °C, red; 32.5 °C, green; and 30 °C, blue, to 40 °C; (b) “Asymmetry” experiment−volume relaxation: up-jump from 30 to 35 °C, blue; down-jump from 40 to 35 °C, green. Key: symbols, data; solid line, SCM prediction.

Figure 4. Volume relaxation following histories consisting of multiple temperature jumps. (a) “Memory” experiment: down-jump from 40 to 10 °C, anneal for 160 h, up-jump to 30 °C, blue; down-jump from 40 to 15 °C, anneal for 140 h, up-jump to 30 °C, green; down-jump from 40 to 25 °C, anneal for 90 h, up-jump to 30 °C, red. (b) “Anneal” experiment: down-jump from 40 to 25 °C, anneal for ta, up-jump to 40 °C; ta = 0.3 h, magenta; ta =4 h, red; ta =28 h, teal; ta =160 h, green; ta =1500 h, blue. Key: symbols, data; solid line, SCM prediction.

pressures, the relative effect of the fluctuating term êc(x̂) is also slightly different, although the magnitude of the fluctuations is approximately the same. The prediction of the pressure dependence of the glass transition per se is not difficult to achieve, e.g. by employing an appropriately modified (pressure

dependent) log a of the type used in the KAHR models.4 However, to do it while simultaneously describing the rest of the Kovacs volume relaxation data presents a bigger challenge− a challenge that the previous version of the stochastic model (i.e., VSM) failed to overcome.6 792


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Macromolecules Kovacs PVAc data and SCM predictions of the single step temperature down-jump experiments from the same initial temperature to a variety of final temperatures are shown in Figure 2a and down-jumps from a variety of initial temperatures to the same final temperature are shown in Figure 2b. The predictions of the single step temperature up-jump are shown in Figure 3a. The SCM predictions for a temperature up-jump and down-jump both from a material initially at equilibrium to the same final temperature are shown in Figure 3bthis is the so-called asymmetry experiment. The predictions of the SCM are compared to Kovacs data for the multiple step histories for both the memory experiment (Figure 4a) and the annealing experiments (i.e., Figure 4b) where particular attention should be on the short time (i.e., 0.3 h) anneal experiment which could not be predicted by the KAHR class of models.6 Overall the SCM predictions shown in Figures 2 through 4 are good except perhaps for (i) the up-jump in Figure 3b, where the predicted response should be a little later and (ii) the intermediate anneal experiments in Figure 4b, where the predicted up-jump response should be a little earlier. In summary, the SCM is able to adequately describe the Kovacs volume relaxation data set, including the short time anneal experiments, where full parameter optimization (see the discussion at the beginning of this section) should improve these results. A comment is in order regarding the amount of scatter in the predicted curves appearing in Figures 2−4. As mentioned previously, the specific volume in the SCM is a macroscopic quantity and as such does not fluctuate. However, the value of the specific volume is obtained as a result of solving the boundary conditions via eq C.1, where the procedure involves ensemble averaging of the local stresses that do fluctuate. Thus, if the number of the ensemble members were equal to the exceedingly large number of nanometer size domains in a macroscopic specimen then the averaging would have produced a smooth volume response. However, in the actual simulation the number of realizations/members of ensemble was much smaller due to obvious limitations on the computational resources; thus, the scatter in the specific volume predictions is seen. In our computations the number of realizations depended on the time spanned by a given history and to a lesser degree the temperature (and hence the rate of relaxation) where the shorter histories as in Figure 3a allowed for a larger number of realizations and consequently a smoother curves. Also, it is significant that for a given trajectory (and hence the entire ensemble) the correlations decay with time; thus, when the response is plotted on a logarithmic scale the scatter in the predictions appear to increase toward longer times. The “Expansion Gap” paradox as reported by Kovacs in his seminal paper1 has long been a subject of debate.13 It refers to the observation that when the so-called effective relaxation time calculated as 1 1 dδ ≡ τeff δ dt

Figure 5. Kovacs τeff versus volume departure δ-plot. The color scheme is the same as in Figure 3a. Key: solid lines, data; symbols, numerical differentiation of SCM predictions.

10−4. Application of eq 6 requires numerical differentiation of data, where McKenna and co-workers13 have clearly shown that the expansion gap is statistically justified because of the high quality of the Kovacs data. Although the δ versus time curves predicted by the SCM in Figure 3a appear relatively smooth, the corresponding τeff versus δ curves predicted by the SCM as shown in Figure 5 exhibit considerable scatter especially when δ approaches zero. While it is not possible to claim that the SCM captures the expansion isotherms perfectly, the existence of the “expansion gap” is clearly evident in Figure 5, and its magnitude is close to that observed experimentally. To the best of our knowledge, this is a first prediction of the expansion gap by any model in the literature. The explanation of the expansion gap effect as produced by the SCM is illustrated in Figure 6, where the instantaneous spectrum of the relaxation times at 40 °C predicted by the SCM is shown. The spectrum is defined as a fraction of mesodomains whose relaxation time falls into the interval from τ to τ + dτ, where the area under the curve is normalized to unity. The relaxation spectrum in Figure 6a spans approximately 5 decades, which is consistent with the width of the PVAc viscoelastic bulk modulus as measured by McKinney− Belcher.23 For elucidating the cause of the expansion gap we compare the equilibrium spectrum (i.e., black curve) with the two spectra resulting from the temperature up-jump and subsequent relaxation for up-jumps from 35 °C (i.e., red curve) and from 30 °C (i.e., blue curve). The times when the spectra are obtained are 330s for the up-jump from 35 °C and 2100 s for the up-jump from 30 °C, where the volume departure from the equilibrium in both cases is δ = 5 × 10−4. It is evident from Figure 6a that both the “red spectrum” and the “blue spectrum” have not yet reached their equilibrium shape, where the blue spectrum is farther from equilibrium. The expansion gap results from the difference between the red and the blue spectra despite the volume departure from equilibrium for these cases being the same. The difference between the spectra originating from the jumps from the different initial temperatures is more apparent in Figure 6b, where equilibrium spectrum has been subtracted from the two nonequilibrium spectra. As compared to the equilibrium spectrum a spectrum that is a result of an upjump in temperature has a dearth of short relaxation times and an excess of long relaxation times (or more precisely, the fraction of the meso-domains with the short relaxation times is too small compared to the equilibrium distribution, while the fraction of the meso-domains with the long relaxation times is

(6)

is plotted versus the volume departure from equilibrium, i.e. δ = [V − V∞ (Θ)]/V∞ (Θ)], τeff does not appear to converge to a single value, which would surely be expected as δ → 0 for upjumps with different initial temperatures but the same final temperature. The experimental τeff data for PVAc (see the solid curves in Figure 5) were determined from the PVAc volume relaxation data shown in Figure 3a, where the expansion gap in the experimental data is clearly evident for δ at least as small as 793


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Figure 6. Explanation of the expansion gap effect in the SCM. (a) Spectrum of relaxation times at 40 °C: at equilibrium, black; after up-jump from 35 to 40 °C at δ = 5 × 10−4, red; after up-jump from 30 to 40 °C at δ = 5 × 10−4, blue. (b) Difference between “red” and “black” spectra, red; difference between “blue” and “black” spectra, blue.

predictions can be reduced by substantially increasing the number of simulation; however, increasing the number of simulation cannot significantly reduce the scatter as δ approaches zero in the expansion gap data shown in Figure 5.

too large). This deviation from equilibrium is different for different up-jumps, resulting in the τeff expansion gap. The shapes of the spectra shown in Figure 6a are all quite similar, which is why thermo-rheologically simple volume relaxation models like KAHR do an acceptable job on most of the Kovacs volume relaxation data. However, there is a clear change in the shape of the relaxation spectra as shown in Figure 6b, which is the origin of the expansion gap and the reason why the SCM that naturally incorporates the thermo-rheological complexity is able to capture the τeff expansion gap. The amount of scatter in the SCM predictions for a particular thermal history is determined by (i) the number of realizations available for averaging and (ii) the magnitude of volume departure from equilibrium. The signal-to-noise ratio significantly worsens as the signal approaches zero. The magnitude of the noise can be reduced by increasing the number of simulations used in computing the macroscopic volume. For the short anneal time (ta = 0.1 h), up-jump curve 5 × 105 realizations were averaged to obtain the response shown in Figure 4b; however, for of the longer annealing time experiments and the memory experiments only 3 × 104 realizations were employed because of the increased computational demand for a single realization. Examining the predictions in Figure 3a, one observes that at long times δ does not reach 0, which is due to accumulation of error when integrating the SDEs corresponding to the temperature ramp portion of the thermal history. In contrast, the annealing experiments in Figure 4b do not exhibit the same problem, which is due to a cancelation of positive and negative errors during the two-step thermal history. Using a more sophisticated numerical scheme than a simple Eulerian method for solving the SDEs could potentially reduce the accumulation of error; however, it will result in a significantly higher computational burden. The expansion gap shown in Figure 5 exhibits the most significant error which is due to (i) the inherently noisy operation of differentiating data and (ii) dividing the differentiated data by δ which is approaching zero. Substantially increasing the number of simulation could reduce the error in τeff for δs that are away from zero; however, significant reduction of the τeff error as δ approaches zero (which is where the expansion gap appears) is not possible due to the amplification of the noise from data differentiation by an increasing small denominator in eq 6. In summary, the volume relaxation predictions shown in Figures 2−4 are relatively smooth, where the fluctuations in the specific volume

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DISCUSSION Previously a stochastic approach for describing volume relaxation was developed that explicitly acknowledges dynamic heterogeneity in glassy polymers, which was able to predict many features of the classic Kovacs PVAc volume relaxation experiments.6 The volume stochastic model (VSM) was able to predict the full set of Kovacs data, including the short time anneal experiments that had escaped prediction by other nonstochastic volume relaxation models; however, the VSM was not able to predict the expansion gap and the pressure dependence of the glass formation line. In addition, the VSM is conceptually unsatisfactory because the meso-domain fluctuations are required to be isotropic which cannot be physically justified. In this paper, we have examined the predictions of a recently developed stochastic constitutive model (SCM)9 that is fully tensorial as well as allowing anisotropic fluctuations. An important feature of the SCM is that the rate of relaxation for a meso-domain is postulated to be controlled by the fluctuating configurational entropy and the invariants of the fluctuating symmetric stress tensor as well as the macroscopic temperature and strain in contrast to the fluctuating volume and the macroscopic pressure and temperature used in the log a term in the VSM. In this communication we have shown that the SCM (i) can describe the Kovacs PVAc volume relaxation including the short annealing time data, which is a problem for the KAHR class of volume relaxation models, (ii) predicts the volume vs temperature isobars at the elevated pressures, i.e., the pressure dependence of the glass formation line, and (iii) predicts the expansion gap in the τeff vs specific volume deviation plot. Prediction of the expansion gap is of particular significance as it has been a longstanding challenge using the traditional thermo-rheologically simple models. It is shown that the ability of the SCM to predict the τ-effective behavior is due to its inherent thermo-rheological complexity An important feature of the SCM is that it makes predictions of the macroscopic response of a glassy material for any temperature-deformation history. In a series of publications we have already demonstrated that the SCM qualitatively predicts (i) the postyield softening,9 (ii) the dependence of the rate of stress relaxation on the loading rate (in particular how the 794


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It should be noted that in fitting the Kovacs volume relaxation data reported in this communication no attempt has been made at optimizing the term in the log a corresponding to the anisotropic stress fluctuations, i.e., the third term in the right-hand-side of eq 4. Since this term is unaffected by the isotropic temperature and pressure histories considered here, in the first approximation the role of this term is just the rescaling of the reference time τ0 as compared to the case when it is not present. More specifically, since the components of the vector x̂ are independent normally distributed random variables with the mean of zero and the variance of unity, the average value of the third term in eq 4 is

relative order of the strain dependence of the stress relaxation response changes direction with strain rate−a very unexpected and counterintuitive result),15 and (iii) the stress memory following the uniaxial constant strain rate deformation experiments and most importantly its dependence on the strain rate.16 These uniaxial deformation results cannot be predicted using the traditional, nonstochastic constitutive models, except in the case of postyield softening, where some constitutive models employ an additional, ad-hoc relaxation function.24 These predictive results along with predictions shown in this communication are beginning to provide evidence that the explicit incorporation of fluctuations that acknowledges the dynamic heterogeneity in the glassy state may be a necessary ingredient of constitutive models for describing glassy materials. The model parameters in both the SCM9 and the VSM6 are compared in Table 2 along with the SCM model parameters that were employed previously to describe the postyield softening. The τ0, L, and z as determined via fitting of the Kovacs volume relaxation data are essentially the same for the SCM and VSM models; however, they are somewhat different than the parameters appearing in ref 9, where the simplified version of the stochastic model that does not employ the meanfield assumption (i.e., z was set to 1 and L to 9 nm) was used. The reason for this choice in ref 9 was to demonstrate that the prediction of the postyield softening was not due to the meanfield assumption. One of the main differences between VSM and the SCM is in the form of log a, i.e., equation B.1−B.2 vs eqs 2 and 4, where the only VSM log a model parameter is the slope s of the glass formation line in V−Θ space. Comparing the ω, n, and m parameters between ref 9 and this communication, one sees that log a now has explicit dependence upon the hydrostatic stress via m = 0.185 vs 0 as well as optimization of the entropic contribution where n = 0.3 vs 1.0. It is satisfying that even though the details of the fluctuations are quite different in the VSM and SCM models, fitting the two models to the Kovacs PVAc data set produces very similar parameters for τ0, L and z, which are the parameters that control the magnitude of the fluctuations. The optimized model parameters are given in Table 2, where the SCM has six model parameters vs five in the VSM (two of which, i.e., z and L, are highly correlated resulting in effectively four parameters) as compared to four parameters in the KAHR model (although if the relaxation spectrum is made more complex than a single stretched exponential form, additional KAHR parameters will appear). The problem with this type of comparison is that the KAHR models fail to fit the entire data setin particular both the short time annealing experiments and the expansion gap. And, the VSM is unable to fit the pressure dependence of the glass formation line as well as the expansion gap. The principle of parsimony (i.e., Occam’s razor) is an excellent principle to choose between competing models; however, parsimony applies only if a model can describe all of the data. If a model can only describe a fraction of the data and qualitatively fails at predicting other parts of the data it is a clear indication that the model is inadequate. With respect to specific volume relaxation in the glass transition region the SCM is the only model to-date that can describe the whole Kovacs data set, even though it has 2 additional model parameters. Finally, the increase in the number of model parameters in the SCM is due to the use of a more involved log a function, where a better, more physically motivated log a function might decrease the number of parameters.

2

VR 2 σG ΔG

(7)

which is effectively combined with the equilibrium term in the denominator of eq 2. A more subtle effect of the presence of the anisotropic stress term is the aforementioned decoupling between the parameters z and L. An important model assumption in the SCM is the specific form of the local log a as given in eqs 2−4. In light of dynamic heterogeneity, the assumption that the local mobility depends upon the local state of the material is much more physically appealing than the approach used in most constitutive models that only include a macroscopic average mobility that depends upon the macroscopic average state of the material. However, at this point one must postulate a specific relationship between local mobility and the local state of the materialthe determination of this relationship is perhaps the outstanding problem in the physics of the glass. In this communication we have assumed that the local log a expression depends upon the fluctuating term êc(x̂) that includes linear, but not quadratic, contributions of x̂0, x̂1 and I2; in addition, as defined in eq 5 the fluctuating component êc(x̂) local log a is assumed to be an average of the local value of êc(x̂) with the ensemble average ⟨êc(x̂)⟩. This particular log a model is a reasonable choice that includes the fluctuating configurational entropy and the fluctuating invariants of the stress (where the third invariant was not needed for the deformations analyzed in this communication), where this log a model was able to capture the volume relaxation data of Kovacs. However, there is a point of caution: the success of this particular log a expression is possible partly due to the fact that for all thermal histories in the Kovacs data set the average nonequilibrium hydrostatic stress ⟨x̂1⟩ is small, where for large anisotropic deformations quadratic terms may have to be included. To summarize, the form of the local log a in eqs 2−4 is not final and the determination of the magnitude of the terms describing quadratic stress contributions to log a can only be done by analyzing thermo-mechanical experiments involving large deformations such as yield under constant strain rate uniaxial loading. Notwithstanding the non-final form for log a, we believe that the prediction of the expansion gap (as well as other features of the Kovacs volume relaxation data set reported in this communication) by the SCM is significant, considering that prediction of the expansion gap has proven challenging over the years. Perhaps more important though is that this prediction results not via the invention of ever more elaborate log a functions, but rather via the introduction of new physics, i.e., the incorporation of mesoscopic fluctuations. In addition, this new physics in the SCM shows promise in describing the nonlinear, thermo-mechanical behavior. A more complete 795


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Macromolecules S∞ = S∞(Θ, H) T∞ = T∞(Θ, H)

unification of the relaxation behavior in glassy polymers using the SCM will be the subject of future research. The description of the thermo-mechanical behavior of a glassy material via the SCM framework is based on solving the stochastic constitutive equations of an individual meso-domain for a specific set of boundary conditions and then averaging multiple realizations of the constitutive equations to obtain the experimentally observed macroscopic response. The language that we have employed in describing the SCM uses the concept of a meso-domain that has nanometer dimensions. However, the mathematical structure of the SCM is a set of stochastic differential equations that apply at a point not over a domain, where the magnitudes of the fluctuations, i.e., the variances given in eqs A.5, are parametrized via the domain size L. It is assumed that the boundary conditions characterized by the global strain tensor and temperature are applied uniformly to each domain, where the macroscopic stress tensor and the entropy are obtained as averages over ensemble of mesodomains. The interaction between meso-domains enters the model in two ways: (i) via the aforementioned requirement that the macroscopic stress equal average of the mesoscopic stresses and (ii) via the “mean field” assumption that the relaxation time for a given meso-domain depends on the local stress and entropy as well as the average stress and entropy. The mean field approximation is a significant model assumption, where one would physically expect that adjacent domains would interact via contact consistent with the standard conservation laws (i.e., mass, momentum, and energy with appropriate boundary conditions). This latter approach would require describing the parameters of the meso-domains as spatially dependent quantities, i.e., as fields, where the formulation of an appropriate nonlinear, stochastic field theory remains a significant challenge in condensed matter physics. The mean field SCM model is admittedly an approximation to the full spatial interaction between meso-domains, but this mean field approach seems to have value based upon the predictions shown in this communication for the Kovacs PVAc volume relaxation data.

■

It is convenient to combine entropy and stress deviations in a single vector of dimensionless variables, x, with the components defined as (i.e., eq 49 in ref 9): x0 =

1 ( −2δT1 + δT2 + δT3), 6 σG

xn =

δTn , σG

n = 4, 5, 6

(A.4c)

The variances of the equilibrium distributions of the fluctuations in entropy and stresses are determined from the standard statistical mechanics25 and are given by σS 2 ≡

Cpg Cpg 1 1 1 3K g σG 2 ≡ 3 2kBΘ 2 VkBCpg σK 2 ≡ 3 2kBΘ 3 CVg CVg L L L

Gg

(A.5)

where Cpg is the constant pressure heat capacity, CVg is the constant volume heat capacity, Kg is the glassy bulk modulus and Gg is the glassy shear modulus. L is the size of the mesodomain which is a model parameter. The stochastic differential equations that define the evolution of the fluctuating variables are given by Eqns. (52a-e), and Eqn. (53) in Ref9. When the macroscopic deformation is isotropic, these general constitutive equations simplify to dx0̂ = dr0̂ − dx1̂ = dr1̂ +

2ΔC 2ΔA dV dΘ − σS σS VR 3

ΔK dV + σK VR

3

(A.6)

2ΔA dΘ σK VR

(A.7)

dxn̂ = drn̂ n = 2, ..., 6

T3 = T33 ,

drn̂ = −

(A.1)

i = 1, ..., 6

1 a(x̂)τ0

2 dWn̂ n (A.9)

Here and in the rest of the paper quantities with the “hat” are stochastic/mesoscopic quantities and the quantities without the “hat” are macroscopic. dŴ denotes a Wiener process.26 The coefficients in eqs A.6 and A.7 are determined from the material parameters for the glass (denoted via a “g” subscript) and equilibrium states (denoted via an “∞” subscript) as given below:

Deviations from equilibrium for the entropy and stress respectively are denoted as δS = S − S∞ δTi = Ti − Ti∞ ,

⎤ dt ⎡ ∂ ln(a(x̂))⎥ + ⎢xn̂ + ∂xn̂ a(x̂)τ0 ⎣ ⎦

= 0, ..., 6

T4 = T12 ,

T6 = T23

(A.8)

where the stochastic terms are given by

In this appendix the general SCM which was derived elsewhere9 will be applied for the specific case of a macroscopically isotropic deformation. Even though the macroscopic stress tensor is isotropic, the mesoscopic stress tensor can exhibit anisotropic fluctuations as well. The deformation is described via the Hencky strain measure H, where tr (H) = ln(V/VR), V is the specific volume, and VR is the specific volume in a reference state. The stress tensor T is conjugate to the Hencky strain tensor and related to the wellknown Cauchy stress TCauchy via T = V/VR TCauchy. For the components of a symmetric tensor we use the notation:

T5 = T13 ,

(A.4a)

1 (δT2 − δT3), 2 σG

x3 =

Stochastic Constitutive Model for Isotropic Deformations

T2 = T22 ,

δS σS

x2 =

APPENDIX A

T1 = T11 ,

(A.3)

(A.2)

ΔG = Gg − G∞

(A.10)

ΔK = K g − K∞

(A.11)

2ΔA = −VR[K g αg − K∞α∞]

(A.12)

2ΔC = VR[K g αg 2 − K∞α∞2] −

where the equilibrium entropy and stress are functions of temperature and strain 796

1 [Cpg − Cp ∞] ΘR

(A.13)


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(Θ, ⟨V̂ ⟩); thus, averaging of the log a trajectories from the solution of the defining SDE will not formally result in a macroscopic log a with the macroscopic volume and temperature dependence given in eqs B.1 and B.2). Nevertheless, numerical experimentation using the VSM that employs Eqns. (B.1 and B.2 shows the emergence of the WLF dependence above Θg, where the relaxation time at a given temperature is evaluated from the volume response to a small temperature perturbation applied at said temperature. Even though eqs B.1 and B.2 are for the fluctuating mesodomain, if they are applied to macroscopic behavior, then the slope of the iso-log a line at the reference point, i.e. the glass transition point at 1 atm, is given by

where α is the coefficient of thermal expansion, Cp is the constant pressure heat capacity, G is the shear modulus and K is the bulk modulus. All quantities in eqs A.10−A.13, which are typically weak functions of temperature and density, are evaluated at the reference point (ΘR,VR). An important caveat is that the entropy S, i.e., x0, is standard thermodynamic entropy whereas it is assumed that log a is a function of the configurational portion of entropy. To avoid introducing new notation and since the coefficient n in eq 4 is unknown, from now on x0 will represent the configurational entropy. The only additional change is that the dynamic equation eq A.6 will now have the form ⎡ 2ΔC 2ΔA dV ⎤ dx0̂ = dr0̂ − ω⎢ dΘ + ⎥ σS VR ⎦ ⎣ σS

(A.14)

dV dΘ

In other words, an additional model parameter ω is introduced since the elastic/glassy response to change in temperature or volume of the configurational entropy is not necessarily the same as of the full entropy (which would correspond to ω = 1). The rest of the dynamic equations given in eqs A.7 and A.8 remain the same. The SCM for macroscopically isotropic deformation involves eqs A.14 and A.7−A.9) where the log a shift function is defined in eqs 2−4. The numerical procedure for solving these equations is developed in Appendix C.

■

⎛ ∂Θ ⎞ ⎤ ⎛ ∂Θ ⎞ −1⎡ ⎟ ⎟ ⎥ = s ⎢1 + ⎜ = −⎜ ⎝ ∂Θ ⎠V ⎦ ⎝ ∂V ⎠Θ ⎣

Relaxation Time Dependence upon State of the Glass

To complete the characterization of the behavior of a mesodomain the relaxation time τ = a(x̂)τ0 in eq A.9 has to be defined as a function of x̂. As a formal exercise any non-negative function is admissible, but only the one that captures the correct dependence of mobility on the local state variables has a chance of predicting the data. In this Appendix the choice of the loga function used in the current version of the SCM is described.

SCM log a

In case of a fully tensorial SCM9 the relaxation time for a mesodomain was assumed to depend on the macroscopic temperature and the volumetric component of the strain and the fluctuating entropy and stress as given in eqs 2 and 3. The slope of the iso-mobility line at reference point in the absence of fluctuations (i.e., x̂ = 0) is

VSM log a

In the VSM6 the meso-domain shift factor is assumed to be a function of the fluctuating specific volume and the externally controlled temperature and pressure as given by

dV dΘ

⎛ ⎞ c2 log a(Θ, p , V̂ ) = c1⎜ − 1⎟ ⎝ c 2 + Θ − ΘR + Ξ(Θ, p , V̂ ) ⎠

where

(B.4)

Unlike in eq B.3, the expression in the right-hand-side of eq B.4 does not contain any adjustable parameters; moreover, eq B.4 predicts that Vg decreases with increasing Θg in agreement with experimental data. The analysis above assumes that there is one-to-one relationship between the fluctuating mesodomain mobility and the macroscopic mobility, which is obviously not true; however, it is encouraging that at least the slope of the glass formation line is of the proper sign. The glass formation line prediction of the SCM is shown in the Results. The discussion so far did not include fluctuations, where the contribution of the fluctuations to the mesoscopic log a, i.e. the functional form of êc(x̂) needs to be determined. Since êc(x̂) is a scalar, the stress tensor contributions contained in x̂ must be in terms of the invariants. In terms of the dimensionless variables defined by eq A.4, the three invariants of the stress tensor are

∞

(B.2) 18

loga = const

⎛ ∂[log a] ⎞ −1⎛ ∂[log a] ⎞ = −⎜ ⎟ ⎜ ⎟ ⎝ ∂V ⎠Θ ⎝ ∂Θ ⎠V

⎛ ∂e∞ ⎞ −1⎛ ∂e∞ ⎞ V ΔC = −⎜ c ⎟ ⎜ c ⎟ = − R ⎝ ∂V ⎠Θ ⎝ ∂Θ ⎠V ΔA

(B.1)

V ̂ − V (Θ , p ) VRα∞ − s

(B.3)

Thus, the parameter s is the slope of the iso-mobility line. However, as was pointed out previously,6 the value of s from optimization of the Kovacs data has a different sign than the slope obtained experimentally by measuring the glass transition temperature at various pressures. This clearly indicates that something is fundamentally wrong with the functional form of the fluctuating log a in the VSM, but since volume is the only variable in the VSM, there is no alternative.

APPENDIX B

Ξ=

loga = const

⎛ ∂[log a] ⎞ −1⎛ ∂[log a] ⎞ = −⎜ ⎟ ⎜ ⎟ ⎝ ∂V ⎠Θ ⎝ ∂Θ ⎠V

∞

c1 and c2 are the WLF constants, V is the equilibrium volume, and s is a model parameter that describes the slope of the iso-mobility lines in V-Θ space.6 Note that the dependence on pressure in eqs B.1 and B.2 is only via V∞. No additional explicit dependence on pressure is required since its effect on log a is supposed to be accounted for by volume−this is what the assumption of volume being mobility controlling variable means. The logic behind the expression eq B.1 is that at atmospheric pressure and above Θg if one replaces the fluctuating specific volume V̂ with the equilibrium volume V∞ (Θ, p = 1 atm) then the empirically established WLF behavior is recovered. For a nonlinear function ⟨log a (Θ, V̂ )⟩ ≠ log a 797


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given by eq 5. The case of z = 1 corresponds to independent meso-domains and the case of z → ∞ corresponds to the macroscopic average, i.e., when there are no fluctuations. Following the original suggestion by Robertson,2a a similar approach was used in the VSM, where z = 6.9 was determined to be the optimal coordination number to fit the isobaric Kovacs data.6 If êc(x̂) is linear with respect to the fluctuating variables (i.e. there is no I2 term) then eq 5 implies a strong correlation between the coordination parameter z and the size of the meso-domain L, because in that case eq 5 depends on the combination z−1σ (where σ is the variance of the fluctuating variable defined in eq A.5) and hence z−1L−3/2. It follows that simultaneously changing the values of z and L so as to maintain z−1L−3/2 constant largely preserves the value of log a as given by eq 5, which was found to be the case for the VSM.6 However, the êc(x̂)expression given in eq 4 used in the SCM contains a quadratic term; consequently, there will not be a simple correlation between z and L.

3 σK x1

1 (tr(δ T))2 3 + x32 + 2(x4 2 + x52 + x6 2)]

I2 = δ T : δ T − = σG 2[x 2 2

I3 = det(δ T)

(B.5)

As shown previously,9 the elastic energy due to a deviatoric (i.e. volume preserving) deformation is given by e (̂ dev)(x̂) =

1 VR I2(x̂) 4 ΔG

(B.6)

and the energy due to an isotropic deformation is given by e (̂ iso)(x̂) = ΘσSx0̂ +

1 VR 1 VR ΔK (I1(x̂))2 + 18 ΔK 2 PD − 1

2 ⎡ σSx0̂ I (x ̂ ) ⎤ + 1 ⎥ ⎢ ⎣ 2ΔA 3ΔK ⎦

■

(B.7)

where PD =

Solution Procedure

VR ΔK ΔC 2(ΔA)2

In this appendix we will describe how the volumetric response to an arbitrary thermal-pressure history is determined for the SCM. The macroscopic and mesoscopic stress are related by a simple condition9

(B.8)

is the Prigogine−Defay ratio.27 Thermodynamic stability requires that PD > 1 (see details in ref 9). In the previous publication9 we used the following expression for the fluctuating contribution to the denominator in eq 2 eĉ (x̂) = ΘR σSx0̂ +

APPENDIX C

1 VR I2(x̂) 4 ΔG

T = ⟨T̂ ⟩

(C.1)

where ⟨....⟩ indicates an ensemble average. The boundary conditions for an isotropic experiment imply that the average/ macroscopic stresses are the hydrostatic contributions given by

(B.9)

Comparing eqs B.6 and B.7 with B.9, only the first linear term in eq B.7 has been retained and in that term the reference temperature ΘR is substituted for Θ for simplicity. As discussed in ref 9, the logic behind omitting the quadratic terms in eq B.7 is as follows: the expression in eq B.7 is incomplete in that there are potentially other quadratic terms resulting from taking a derivative of the third-order terms in the expansion of the Helmholtz free energy, which has been formally truncated at the second order to arrive at eq B.7; thus, quadratic terms have been excluded altogether instead of including just some of them. An important feature of eq B.9 is that it contains a term linear with respect to x̂0 but does not contain a term linear in I1 (or x̂1). Equation B.9 has proven inadequate to predict the Kovacs data and needs modification; specifically, the term linear in I1 needs to be included. An appropriate combination having the units of energy per unit mass is

Tn = −

V p; VR

Tn = 0;

n = 1, 2, 3

(C.2)

n = 4, 5, 6

(C.3)

where eqs C.3 state that there are no macroscopic shear stresses on the material. The equilibrium stresses are given by the linearized equations Tn∞ = −

V − VR V ∞ p (Θ, V ) ≈ −pa + K∞ VR VR

− K∞α∞(Θ − ΘR );

n = 1, 2, 3

(C.4)

where pa indicates atmospheric pressure. The linearization employed in eq C.4 has been done for convenience since the nonlinear terms have a negligible contribution for the volume relaxation experiments of interest in this paper, although these terms may have a small contribution to the description of the glass formation line at higher pressures. Using the eqs A.7−A.9, A.14, 2, 4 and 5, and C.2−C.4, the response of a meso-domain is determined by solution of this set of stochastic differential equations for an ensemble of trajectories. Even though stress is the fluctuating variable, the volume of the meso-domain (which is the same as the macroscopic volume) does change in order to enforce the macroscopic stress boundary condition given in eq C.1 for a specific temperature-pressure history. The time evolution of an ensemble of meso-domains is governed by the SDEs given in eqs A.6−A.9 for each individual meso-domain, subject to algebraic constraint eq C.1 that is imposed on the entire ensemble. For the isotropic deformations with the boundary conditions considered here, the temperature and pressure are externally prescribed and the variable of

1 1 1 (p ̂ − p∞ )V = − tr(δ T̂ )V = − VσK x1̂ ≈ − VRσK x1̂ 3 3 3 (B.10)

In the right-hand-side of eq B.10 the coefficient in front x̂1 is assumed to be constant; i.e., VR is used instead of V for simplicity. Mean Field Approximation

The configurational internal energy responsible for controlling the mobility of a specific meso-domain is assumed to be a combination of the configuration internal energies of that meso-domain and the surrounding meso-domains, whose contribution is assumed to be given via a mean field approximation. Thus, the fluctuating term êc(x̂) in the mesoscopic log a in eq 2 is replaced with the expression 798


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Macromolecules interest, i.e. the specific volume, is obtained from the solution of eq C.1, which involves averaging over the ensemble. The solution is carried out as an iterative procedure where the quantities at the time t + dt are evaluated based on their values at t. Specifically, assume that the external pressure is fixed at the value of p and at time t the following macro- and meso-scopic quantities are known: Θ(t ), V (t ), {xn̂ i(t )}iN=dom 1;

n = 0, ..., 6

4 With dV having been determined, the local entropy and stress can now be updated using eqs A.6−A.8 for each meso-domain. The solution via steps 1−4 results in two distinct asymptotic behaviors for the specific volume response during cooling at a constant rate: (i) an ideal glass where the relaxation time is large i.e., a → ∞, and (ii) an equilibrium melt/rubber where the relaxation is rapid, i.e., a → 0 (or more precisely, examining eq 2, the smallest value the shift factor can attain is a → 10−c1 = 10−16.8 (see Table 2)). In the ideal glass the stochastic contribution ⟨dr̂1⟩ is zero; thus, applying eq C.10, the specific volume follows the glassy asymptote dV = VRαgdΘ. In the equilibrium rubber/melt case at every time step the solution of the SDEs eqs A.6−A.9 for each meso-domain is such that x̂ fluctuates around the value that renders the coefficient of the quantity 1/a to be nearly zero as a → 0 in eq A.9; otherwise, the restoring force overwhelms the random fluctuations which scales as (1/√a) . In case of the component x̂1 this implies

(C.5)

where Θ (t) is the externally imposed temperature at time t, V (t) is the macroscopic volume at time t, x̂in (t), n = 0, ..., 6 are the entropy, and the six components of the symmetric stress tensor (see eq A.4) for the ith domain and Ndom is the number of meso-domains that are part of the ensemble under consideration. Note that, by assumption, the values in eq C.5 are such that the condition given by eq C.1 is satisfied ⟨x1̂ (t )⟩ = [−p + p∞ (Θ(t ), V (t ))]

1 σK 3

(C.6)

0 ≈ x1̂ +

Here we will only show the determination of the hydrostatic component of stress, i.e., x̂1, where the rest of the x̂n(t) variables are treated in an analogous manner. After a time increment dt, the temperature becomes Θ(t + dt) = Θ(t) + dΘ, where dΘ is known, and the values of the rest of the variables need to be obtained from solving the eqs A.6−A.9 under the condition eq C.1 with a constant external pressure p. Equation C.1 implies that the unknown x̂in (t + dt) and V(t + dt) satisfy ⟨x1̂ (t + dt )⟩ =

1 Ndom

⎞2 2.303c1 1 ⎛ ln(a(x̂)) ∂ ln(a(x̂)) = ⎜ mVRσK + 1⎟ ∂x1̂ ⎠ ecR z ⎝ 2.303c1 (C.12)

The condition of rapid relaxation (i.e., a → 0) along with consideration of eq 2 implies that the quantity in parentheses in eq C.12 is less than unity. Moreover, substituting the parameter values we find

∑ x1̂ i(t + dt ) i=1

= [−p + p (Θ(t + dt ), V (t + dt ))]

1 σK 3

(C.7)

2.303c1 1 mVRσK < 10−2 ecR z

By subtracting (C.6) from (C.7), we arrive at ⟨dx1̂ ⟩ = {p∞ [Θ(t + dt ), V (t + dt )] − p∞ [Θ(t ), V (t )]}

1 σK 3

(C.8)

⎞2 ⎛ ln(a(x̂)) ⟨x1̂ ⟩ ≈ − ⎜ + 1⎟ ⎠ ⎝ 2.303c1

where eq C.4 has been used for the equilibrium pressure. On the other hand, by averaging eqs A.7 the same quantity is expressed as ⟨dx1̂ ⟩ = ⟨dr1̂ ⟩ +

2ΔA 3 dΘ σK VR

2.303c1 1 mVRσK = const ecR z (C.14)

which in turn implies that ⟨dx̂1⟩ → 0. Then eq C.8 gives rise to the equilibrium asymptote dV = VRα∞dΘ. Thus, the solution of the defining SDEs gives rise to the expected isobaric specific volume responses in both the glassy and rubbery/melt states.

(C.9)

Finally, equating right-hand-sides of eqs C.9 and C.8 gives rise to the desired formula for the unknown dV: σ dV = − K ⟨dr1̂ ⟩ + αg dΘ VR 3 Kg

(C.13)

In other words, at conditions well into the rubbery state the righ hand side of eq C.12 is (i) small compared to the variance of the stochastic variable x̂1 (which is unity) and (ii) is a very weak function of x̂, i.e., is nearly a constant. Thus, averaging eq C.11 over the entire ensemble, we have

⎡ ⎤ 1 dV = ⎢ − K∞ + K∞α∞dΘ⎥ VR ⎣ ⎦ σK 3

ΔK dV 3 + σK VR

(C.11)

Performing the differentiation with the expression for a (x̂)given in eqs 2 and 4

Ndom

∞

∂ ln(a(x̂)) ∂x1̂

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AUTHOR INFORMATION

Corresponding Author

(C.10)

*(J.M.C.) E-mail: [email protected].

In light of (C.10) the solution algorithm consists of the following steps: 1 The stochastic increments {dr̂in}Ni=1dom; n = 0, ..., 6 in eqs A.9 for each meso-domain are determined via Eulerian integration28 using the known x̂in(t). This involves calls to the random number generator to determine the dWn′s. 2 ⟨dr̂1⟩ is calculated by averaging over the ensemble of domains 3 dV is calculated using eq C.10

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by National Science Foundation Grant Number 1363326-CMMI. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant Number ACI-1053575. 799


Article

Macromolecules

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