On the possibility of modeling of polymers glass transition in a wide range of cooling and heating rates

    On the possibility of modeling of polymers glass transition in a wide range of cooling and heating rates T.V. Tropin, J.W.P. Schmelzer, V.L. Aksenov PII: DOI: Reference:

S0167-7322(16)32396-0 doi:10.1016/j.molliq.2016.12.009 MOLLIQ 6690

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

21 September 2016 22 November 2016 2 December 2016

Please cite this article as: T.V. Tropin, J.W.P. Schmelzer, V.L. Aksenov, On the possibility of modeling of polymers glass transition in a wide range of cooling and heating rates, Journal of Molecular Liquids (2016), doi:10.1016/j.molliq.2016.12.009

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ACCEPTED MANUSCRIPT On the possibility of modeling of polymers glass transition in a wide range of cooling and heating rates T.V. Tropin1, J.W.P. Schmelzer2,1, V.L. Aksenov3,1 1

Joint Institute for Nuclear Research, Joliot-Curie Street 6, 141980 Dubna, Russia Institute of Physics, University of Rostock, Wismarsche Str. 43-45, 18057 Rostock, Germany 3 National Research Centre “Kurchatov Institute”, Petersburg Nuclear Physics Insitute, Gatchina, Russia The presented work continues the investigation of the problems connected with modeling of the kinetics of polymers glass transition in a wide range of temperature change rates. In our previous work [1], an attempt to model a big set of heat capacity curves of polystyrene glass transition has been made, and the inability of the common methods to do this within a single set of parameters has been demonstrated. To go a step further, in this work we proceed with the common and several novel methods of modeling. To normalize the models with each other, a fit of the 10 K/min cooling/heating DSC curves of polystyrene is made, and the literature model parameters readjusted. Further, the modeling of the reduced heat capacity curves at the cooling and heating rates in a wide range of q=10-6-106 K/s with a logarithmic step is performed. The comparison of Tg(q) behavior with lately measured data for polystyrene is made. It is shown, that the methods need some modifications to qualitatively describe details of the glass transition kinetics in a wide range of q. Some of the possibilities to advance the models are discussed.

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1. Introduction

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The glass transition, or freezing-in, of various liquids is presently considered to be a very interesting phenomenon [2]. The non-stopping efforts of the scientists to provide a consecutive and complete description of this kinetic effect, as well as the structure of the glass, are not yet successful. The amount of different theoretical methods and approaches in this field can be compared only with the diversity of the types of the systems, where transitions bearing similar features have been discovered [3,4]. To the classical molecular liquids one can add polymers, colloids, granular materials, spin and Coulomb glasses, and even certain systems in computer science. There are no big doubts in the similarity of the features of the occurring transitions, thus from theoretical point of view it is both important to provide the general models of these effects, and also to develop separate theoretical approaches to account for specific features in each case. Polymers of different kinds present an important system for various applications, and also an interesting object of research. The transition of different (mainly atactic) polymers from liquid to the glassy state has been intensively investigated both experimentally and theoretically for almost a decade. Different phenomenological and microscopic theoretical models were proposed [5]. Yet, presently, there exist no method for consistent description of the effect for all the possible thermal histories applied to the sample. The modern calorimetric techniques allow one to probe a wide range of cooling and heating rates, starting from as slow as 10-6 K/s and going up to high values of 106 K/s. Lately [1] an attempt to describe the glass transition of polystyrene measured via temperature dependencies of the heat capacity, Cp(T), by such theoretical methods as Tool-Narayanaswamy-Moynihan (TNM), Adam-Gibbs (AG), or Gutzow-Schmelzer (GS) has been made. As the result, it was shown, that if a single set of model parameters is used, an

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inevitable discrepancy in the heat capacity experimental and modeled curves will be present. As it was shown in [6] (and lately again in [1]) variation of a couple of model parameters with cooling rate improves the situation. In this work we continue the research on capabilities of common theoretical methods for description of the isobaric heat capacity curves of a polymer, measured with the differential scanning calorimetry (DSC) techniques. To advance the previous work, some new methods are applied, and a different approach to the modeling is chosen.

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We start with fitting the standard 10 K/min (~0.1667 K/s) cooling/heating rate Cp curves of polystyrene with each method to evaluate the model parameters and compare the fitting quality. Further, for the two selected methods the heat capacity curves are modeled for the case of thermal treatment of the polymer, where the range of cooling/heating rate, q, varies as 10-6-106 K/s with logarithmical step. As a comparison parameter the limiting fictive temperature, corresponding to the thermodynamic glass transition temperature, Tg, is chosen and computed. The Tg values are compared with the lately performed calorimetric measurements by Schawe [7]. In that work, the temperatures of glass transition have been measured for different cooling rates in the range of 0.003-4000 K/s. The author has performed a successful approximation of the measurements by the common Vogel-Fulcher-Tammann-Hesse (VFTH) law and provided the fit parameters. We use these data to compare our modeling results with experiments and to analyze the methods applicability in a wide temperature change range.

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The structure of the work is the following. After this introduction, the short description of experimental data and theoretical methods is provided. Further, we present the modeling results and discuss this data. The conclusions are given in the end.

2. Experimental data

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The experimental data that is used in this work have been measured elsewhere [1,7]. A form of atactic polystyrene, PS 168N (form BASF, ρ=1.047 g/cm3, Mn=95,000 g/mol, Mw=270,000 g/mol) has been used in both works. From Ref. [1] a common DSC measurement of glass transition of the PS sample during cooling at 10 K/min and reheating at same rate has been taken, the experimental data set is visualized in figures in next parts of the article. The glass transition temperatures of the same material Tg were measured in [7] at two different DSC instruments. The Tg definition, and the following measurement method, were taken from [8]. This procedure implies the partial integration of Cp curves, thus the obtained Tg is often called the thermodynamic glass transition temperature [7,8]. This way of Tg measurement has the advantage of producing equal values after both cooling and heating experiments. Also, it is by definition the same as theoretically modeled limiting fictive temperature. A set of correction procedures applied in [7] allowed obtaining a smooth Tg(q) dependence in a wide range of q. It was shown, that this dependence can be very well fitted by the VFTH law:

 q  B log    A  Tg  Tv K s

(1)

ACCEPTED MANUSCRIPT The fit parameters are: Tv=333K, B=556K and Aβ=13.2. From the equation (1) one can obtain the Tg(q) dependence which will be further used in this work.

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3. Methods of modeling

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Each of the methods applied in this work describes the glass transition in a phenomenological way by introducing a single structural parameter. This parameter reflects the departure of system from equilibrium liquid state and is used in the expression for the relaxation time. Often the so-called fictive temperature Tf of the system, introduced by Tool [9], is used. In this work, we also apply the model, developed by Schmelzer and Gutzow for vitrification of the liquids [2,10–12], where the structural parameter, ξ, is the number of unoccupied sites per mole of system. Generally, the applied methods (or expressions) can be subdivided into two groups, first containing the commonly applied approaches and the second one the relatively new ones or those that are expected to be more suitable for the system considered. A detailed description of the methods can be found, for example, in [1,5].

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As of the first group, the modern methods take advantage of the fictive temperature concept to calculate the change of Tf basing on the thermal history of the modeled experiment. Using the approach proposed by Narayanaswamy [13], one can obtain the following expression for calculating the evolution of the fictive temperature numerically:

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   n t      j   T f ,n  T0   Ti 1  exp     (2)      i 1 j i  TNM , j      where T0 is the initial temperature, Δtj are the time steps, corresponding to the selected temperature steps, τTNM,j is the relaxation time, β is a model parameter, reflecting the nonexponentiality of relaxation in the glass transition process in the commonly employed stretched exponential (or Kohlrausch) form. Using Eq. (2) one can further introduce an arbitrary form for the relaxation time τ to model the dependence of systems time of relaxation to equilibrium state using any systems properties and its thermal history. In our work, we shall apply the AdamGibbs and the common VFTH form for the relaxation time:

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 B B x (3) 1  x  R T  T  R T f  T    where τ0 is the preexponential factor for the relaxation time, B is the parameter, related with the activation enthalpy, T∞ is the limiting temperature, x – non-linearity parameter. Here and further R is the universal gas constant.

 VFTH (T , T f )  exp ln  0 

The next two expressions for τ0 follow the Adam-Gibbs (AG) theory (for details, see [1,5,14,15]). The general AG expression for the relaxation time is obtained as one of the results of a molecular-kinetic theory of glass-forming liquids, basing on the supposition that relaxation involves a cooperative rearrangement of subsystems (cooperatively rearranging regions). As the result, an exponential dependence of relaxation time on macroscopic configurational entropy, Sc(T), is obtained. Considerations of the dependence of Sc(T) on temperature and other parameters result in different expressions for relaxation time. Configurational entropy is

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presently defined as an integral of ∆𝐶𝑝 /𝑇 in the region from Kauzmann temperature T2 to Tf, thus expressing the non-linearity of the process [14]. ∆𝐶𝑝 is the difference between the liquid and glass heat capacities. A hyperbolic form for ∆𝐶𝑝 reproduces the VFTH expression for τ(T). A detailed review of the derivation of further expressions following Adam-Gibbs theory can be found, for example, in [16], some of the results are listed next. The AGL form (L denoting the logarithmic term) is obtained if we suppose ∆𝐶𝑝 to be constant:

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    B  .  AGL (T , T f )   0 exp (4)  T  f   RT ln     T2    The so-called Scherer’s form is obtained, if an expression of the form ∆𝐶𝑝 = 𝐶𝑝0 + 𝐷𝑇 is used as the one applicable near the glass transition range [14]:

(5)

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    B    AGS (T , T f )   0 exp  ,   Tf    RT  ln    C T f  T2        T2    where C is the constant obtained from linear expression for ∆𝐶𝑝 .

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The detailed concepts of the GS method can be found in [2,10]. A relaxation equation for ξ is expressed in the common form:

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d 1      e  (6) dt  , where τ is the characteristic relaxation time and ξe is the equilibrium value of structural order parameter. Here again, the classical expression for the relaxation time would be of the VFT model: B ). (7) R(T  T ) The methods belonging to the second group consist of recently proposed expression for the GS approach [1]:

 (T )   0 exp(

    B A   (T ,  )   0 exp    R(T  T )    RT  ln 1     ln     1   , 

(8)

where an additional entropic term is added to the exponent. This expression has proved to be qualitatively suitable for modeling the heat capacity curves of PS during glass transition. This is mostly due to the non-linearity, introduced in (8). Also, we will apply the expression, proposed by Hutchinson and co-workers in [17]. In this work a new non-linear parameter xs is introduced in the expression for configurational entropy, partitioning it as follows:

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(9) As it is described in [17], the new non-linearity parameter xs is supposed to be completely the property of material. It should not depend on thermal history of the sample, or cooling/heating rate. For obtaining the expression for Sc(T) in (9), the ∆𝐶𝑝 (𝑇) is supposed to be constant in the glass transition range (similar to the AGL form, (4)).

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One last approach, that we partly utilize in the work, was proposed by a group of authors in [18,19] to describe the structural relaxation and ageing of different polymers. The authors suppose that there exists an intermediate state of entropy relaxation (above equilibrium line), at which the system stays during the physical ageing process. The structural parameter within this method is not the fictive temperature, but the configurational entropy of the polymer itself, and the model, introduces one additional parameter δ, reflecting the departure of the “intermediate plateau” from the equilibrium line in terms of heat capacity Cp. In calculations one obtains the evolution of the configurational entropy and enthalpy of the system, which are directly connected with Cp. An attempt to perform the calculations has been also made for the modified free-volume model, proposed in [20], but the minimization of parameters resulted in worse than expected agreement with experiment, probably due to their interconnection and the following failing of the Nelder-Mead procedure.

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4. Results and discussion

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As a first step in the comparison of models applicability we have performed the curve fitting of the 10 K/min cooling/heating experiment on polystyrene for each of the methods described above. For convenience, the reduced heat capacity curves were used. The criterion for the fit quality, σ, was the mean square deviation of model from experiment in the significant part of the curves with the step of 0.01 K, divided by the number of points. A standard Nelder-Mead minimization procedure was applied. Taking account of parameter correlations, care was taken to keep their values within realistic limits. The obtained parameter values and the fit quality, σ, are presented in Table 1. Table 1. Parameter values and fit quality for the modeling of the 10 K/min (0.1667 K/sec) cooling-heating curves of PS168N. The definitions of the models and their parameters are given in section 3 of the paper. Method TNM-VFTH AGL AGS GS-VFT Hutchinson Conf. entropy GS I

Ref. [21] [16] [14] [10] [17] [19] [1]

ln(τ0/s) -15.8 -99.3 -93.3 -8.18 -65.4 -37.4 -21.4

β 0.514 0.549 0.55 0.559 0.556 -

Parameter values T∞, T2, K B/R, K 343.6 632 138.8 38500 290 1610 328.2 500.2 226.5 13200 264.9 1000 282.1 2210

σ*103 Additional x=0.19 C=-0.0025 xs=0.217 δ=0.247 A/R=0.00022

2.0 2.1 2.1 44 2.1 1.8 14

The quality of the obtained fits is comparable in most cases, the GS method showing worse agreement. Interestingly, this disagreement comes mostly from the undershoot appearing before the peak in Cp dependence, while the agreement with the other parts of the curve is better.

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An example of comparison of the modeled and experimental curves is presented on Fig. 1. As for the other methods, it should be noted, that the configuration entropy method and the Hutchinson method provide visually better agreement with the heating curve (Fig. 2).

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Experiment GS-VFT TNM-VFTH

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Fig. 1. Comparison of experimentally measured Cp curves of polystyrene during 10 K/min heatingcooling cycle (measurement performed by C. Schick’s group, Rostock University [1]) with modeled curves via common TNM-VFTH approach and GS-VFT method (first group). The blue lines correspond to cooling, the red lines – to heating. Line styles definitions are given in the legend.

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A few observations should be made with respect to these results. First, for each of the methods, the value of non-exponentiality parameter, β, stays in a narrow range ~0.51-0.56. Thus, this is a general feature of the investigated system that is well (and also in a similar way) reflected in each of the models via Kohlrausch law. This parameter affects the shape of the curve and thus cannot be varied significantly beyond these limits, though its possible dependence on cooling rate may be discussed in some cases [1]. Further, one should mention significant deviations of parameters τ0 and reference temperatures (T2, or T∞, depending on the method) for different models applied. A lower value of the reference temperature is a general feature of the AG-based methods. Small τ0 values are also common in these cases.

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For the following discussion, we shall limit the demonstrations for the two methods, namely TNM-VFTH and the Hutchinson model. Belonging to the different method groups, they generally reflect the differences and advantages of approaches. Some of the model specific comments will also be given in the end of the section.

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Experiment Hutchinson model Conf. entropy model

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Fig. 2. Comparison of experimentally measured Cp curves of polystyrene during 10 K/min heatingcooling cycle with modeled curves via Hutchinson and configurational entropy methods.

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Further, the modeling of glass transition on cooling and the following heating of polystyrene was performed within the different frameworks for the range of q from 10-6 K/s up to 106 K/s. The examples for two different methods are presented on Fig. 3. As already was mentioned by different researchers, the distinctions of these curves from the measured ones at different cooling/heating rates increase as the range of these rates widens [1,16]. To visualize this difference we need to compare some curve parameters with experimental data. Here we will compare the glass transition temperature, obtained in [7], with the models. For the theoretical curves the so-called limiting fictive temperature for each different cooling rate is computed. This comparison is presented on Fig. 4. One can see, that the TNM model is in a good agreement at low cooling rates, approximately for q<10 K/s, then the deviation appears and increases with q. This is also in agreement with Fig. 3a, where we can see that at higher cooling rates an unexpectedly early glass transition occurs. At high temperatures this model predicts a too quick decrease of the liquids relaxation time. Yet, one should notice, that the character of dependence of TNM-VFTH modeling is similar to experimental one – thus we suppose that the VFTH expression should still be used in certain form for the modeling. In contrast, the glass transition temperatures from the Hutchinson model are mostly closer to experiment, but the dependence on Fig. 4 is seemingly different from experiment and VFTH.

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Fig. 3. A set of Cp curves of polystyrene glass transition modeled with the TNM-VFTH (a) and Hutchinson (b) models for the parameters from Table 1.

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Fig. 4. Comparison of experimentally obtained glass transition temperatures, Tf', with modeled one for different theoretical approaches. The solid black line represents a VFTH fit of experimental data [7], symbols – different models, as described in the legend. a – the Tf' values are calculated using parameter values obtained after fitting the 10 K/min curves (Table 1). b – a direct fit of the Tf'(q) dependence via different models.

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Basing on the results obtained above a further fit could be proposed: to clarify the quality of the applied models one could fit their results to the VFTH dependence for Tf'(q) of polystyrene. Next, we could analyze the agreement of the separately modeled 10 K/min curve for the obtained parameters with experiment. The procedure of the fit to the Tf'(q) dependence was carried out with constant values of β and certain limitations to other parameters. The result for some of the methods is presented on Fig. 4b. The new parameters values for TNM method are: B/R=977.2 K, ln(τ0/s)=-21.95, T∞=337.2 K, x=0.19, β=0.52; the improved parameters for Hutchinson model are: B/R=1043.1 K, ln(τ0/s)=-23.13, T2=335.83 K, xs=0.604, β=0.559. It is interesting that, except the non-linearity parameters, other parameters of these two different models are very close to each other after minimization. To clarify how well the obtained parameters fit the heat capacity curves the modeling of the 10 K/min experiment has also been performed (Fig. 5). One can see that while the glass transition temperature dependence has been fitted, the quality of approximation of the Cp curves has been lost. In numbers this can be expressed, as σ=0.0035 for TNM-VFTH (σ=0.0085 for heating curve) and σ=0.0067 (σ=0.017) for the Hutchinson method. Thus for both (cooling and heating) curves the quality of the fit has decreased about twice for TNM and ~3 times for Hutchinson method. Separately, for the heating curve the decrease is even larger: ~4-10 times for both methods.

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Fig. 5. Comparison of the dependence of modeled glass transition temperature to VFTH fit of experimental measurements [7] for two different methods.

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Presently, there is no clear explanation of the reason of these discrepancies, as also there are no proposals as how to overcome it. A minor correction to the experimental Cp curves with respect to the influence of thermal lag, as described, for example, in [22,23] could be one of the steps to improve the quality of agreement. The difference in Tg for the sample with the mass of ~4 mg may be a few Kelvin at the given heating rate. In the modeling part, we can suppose that the expression for polystyrene relaxation time should contain the VFTH law in a certain form, perhaps modified to meet the glass transition non-linearity properties. On the other hand, some further modification of the general method or partial expressions should be made to obtain the satisfactory fits of the calorimetric data. The growing disagreement at different cooling rates may be connected with a certain change of relaxation character in the glass transition interval. A further step could also be made by considering the specifics of polymer systems or from microscopic modeling of fast polymer relaxation in liquids.

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With respect to the Gutzow-Schmelzer approach, basing on the irreversible thermodynamics and being a general theory for the liquids, there are two reasons for the high value of σ. First, the modeling requires the definition of the crystals melting temperature Tm, which, for atactic polymers, is a non-existent parameter (no crystallization, and, thus, crystal melting in this case for atactic PS). In this work Tm was chosen as 512K, the melting temperature of isotactic PS, but, in fact, it is indeed a model parameter that should be varied. Its variation scales the Cp curves obtained by the model. Second, the VFT expression should be changed together with basic approach, expressed by equation (6), to account for both non-exponentiality and non-linearity of the polymers glass transition. As it can be seen, the non-linear expression (8) improves the fit quality. Also, non-exponentiality may (and must) also be introduced within the approach, for example, as discussed in Ref. [24].

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5. Conclusions

In the present work we have performed the modeling of a single glass transition experiment by several theoretical methods, applied commonly, or lately, for polymers. The obtained single curves are in good agreement with experiment, yet as one starts to model the glass transition in a wide range, a disagreement even with such characteristic parameters as glass transition temperature is reached. The minimization of model parameters in order to obtain a good agreement for the Tg(q) dependence, on the opposite, results in the disagreement of the 10 K/min heat capacity curve with models. This basic shortcoming of the described models is quite evident from the presented data and should be carefully analyzed in the further research. A method, that would succeed to describe both the kinetics and the characteristic properties of the transition for a certain polymer in a wider range of temperature change rates, would be in advantage in comparison to the others.

6. Acknowledgments The calculations performed during this research were run at the computational servers of the Bogoliubov Laboratory of Theoretical Physics, JINR.

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The present research was supported by the Heisenberg-Landau program of the German Ministry of Science and Technology (BMBF), and by Russian Foundation for Basic Research, project #16-32-50067. The financial support is gratefully acknowledged.

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ACCEPTED MANUSCRIPT J.M. Hutchinson, S. Montserrat, Y. Calventus, P. Cortés, Application of the Adam−Gibbs Equation to the Non-Equilibrium Glassy State, Macromolecules. 33 (2000) 5252–5262.

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G. Liu, D. Zhao, Y. Zuo, Modified Adam–Gibbs models based on free volume concept and their application in the enthalpy relaxation of glassy polystyrene, Journal of NonCrystalline Solids. 417–418 (2015) 52–59.

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ACCEPTED MANUSCRIPT Highlights

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Several common and late theoretical models are used to model the Cp curves of polystyrene glass transition; The obtained parameter values are used to model glass transition in cooling rate range 10-6-106 K/s; It is shown, that with a single set of parameters considerable deviations from experiment occur; The limitations of the approaches and possible modifications are discussed.

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