Modeling Retrograde Vitrification in the PolystyreneâToluene System

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Modelling Retrograde Vitrification in Polystyrene-Toluene System Giuseppe Scherillo, Valerio Loianno, Davide Pierleoni, Rosario Esposito, Antonio Brasiello, Matteo Minelli, Ferruccio Doghieri, and Giuseppe Mensitieri J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b01766 • Publication Date (Web): 02 Mar 2018 Downloaded from http://pubs.acs.org on March 4, 2018

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The Journal of Physical Chemistry

Modelling Retrograde Vitrification in Polystyrene-Toluene System Giuseppe Scherillo1, Valerio Loianno1, Davide Pierleoni2, Rosario Esposito1, Antonio Brasiello3, Matteo Minelli2, Ferruccio Doghieri2, Giuseppe Mensitieri1* 1

Department of Chemical, Materials and Production Engineering (DICMAPI), University of Naples Federico II, Naples, Italy 2 Department of Civil, Chemical, Environmental and Materials Engineering (DICAM), Alma Mater Studiorum University of Bologna, Italy 3 Department of Industrial Engineering (DIIn), Università degli Studi di Salerno, Fisciano (Salerno), Italy *

Corresponding author. E-mail: [email protected]

Abstract Atactic polystyrene, as reported in a recent contribution by our group, displays a marked change in glass transition when exposed to toluene vapor due to plasticization associated to vapor sorption within the polymer. The dependence of glass transition temperature of the polymerpenetrant mixture on the pressure of toluene vapor is characterized by the so called ‘retrograde vitrification’ phenomenon, in that, at a constant pressure, a rubber to glass transition occurs by increasing the temperature. In this contribution, we have used a theoretical approach, based on nonrandom lattice fluid thermodynamic model for the polymer-toluene mixture, to predict the state of this system, i.e. rubbery or glassy, as a function of fluid pressure and system temperature. The experimentally detectable glass transition is assumed to be a kinetically affected evidence of an underlying II order thermodynamic transition of the polymer mixture. Based on this hypothesis, the Gibbs – Di Marzio criterion, stating that equilibrium configurational entropy is zeroed at the glass transition, has been applied to locate the transition. The working set of equations consists in the expression of configurational entropy obtained from the adopted lattice fluid model equated to zero, coupled with the equation expressing the phase equilibrium between the polymer phase and the pure toluene vapor phase in contact and with the equations of state for the two phases. Theoretical predictions are in good qualitative and quantitative agreement with the experimental results previously obtained gravimetrically performing ‘dynamic’ sorption experiments, which represent a neat example of occurrence of so-called ‘Type IV’ glass transition temperature vs. pressure behavior. The peculiar retrograde vitrification phenomenon and the glass transition temperature vs. pressure envelope determined experimentally are well described by the proposed theoretical approach.

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Introduction Glass transition temperature, Tg, of a polymer is affected by contact with a fluid that can be absorbed within the material. This phenomenon is ruled by the mechanical action of fluid pressure, P, and by the thermodynamic affinity between fluid and polymer. In fact, a fluid can have counteracting effects. It may act both as a pressure-generating medium, thus promoting, in principle, an increase of Tg with increasing pressure, and as an effective plasticizer due to increase in free volume and possible establishment of polymer-penetrant interactions occurring in association with penetrant sorption, thus promoting a decrease of Tg. In particular, the plasticizing action depends in a complex fashion upon the combined effects of pressure, P, and temperature, T, on fluid sorption within the polymer. Such an issue has been theoretically addressed by Condo et al.,1 who predicted the relevant features of polymer-penetrant systems in terms of Tg value vs. fluid pressure. The adopted modelling approach, which is rooted on a consistent framework combining Equation of State of random mixtures based on a lattice fluid theory to describe sorption equilibrium and the Gibbs-Di Marzio criterion2-4 for glass transition (i.e. the mixture equilibrium configurational entropy is zero at the glass transition), evidenced four fundamental types of behavior (i.e. types I, II, III and IV) for the Tg vs. penetrant pressure plot. Worth of mention in the present context is the so-called retrograde vitrification phenomenon, consisting in a rubber to glass transition, at constant P, for the polymer-penetrant mixture, occurring at increasing temperature. In brief, this phenomenon consists in the fact that, in isobaric conditions, the decrease in segmental motion, taking place as the temperature is reduced, is overcome by the increase in the diluent concentration due to increase in solubility. Of interest here is the so-called type IV behavior,1 schematically represented in Figure 1, which displays the mentioned retrograde vitrification behavior. Several experimental approaches have been proposed in the literature to investigate the effects on the Tg of a polymer system promoted by sorption of low molecular weight compounds, including methods based on in situ measurement of creep compliance,5,6 on gravimetric sorption measurement by identifying a discontinuity in the sorption isotherm from the glassy to rubbery-types of behavior,7,8 on stepwise temperature- and pressure-scanning thermal analysis,9 on a minimum foaming temperature approach,10 on in situ spectroscopic ellipsometry11 and on detection of sharp increase in mutual diffusion coefficient in the polymer-penetrant system.12 Experimental evidence of retrograde vitrification has been actually provided only for a very limited number of systems, including poly(methyl methacrylate) (PMMA) and poly(ethyl methacrylate) (PEMA) in contact with pressurized CO2,5,

6, 9,10,13

thin Polystyrene (PS) film-CO2 2


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systems,11 the poly(lactic acid) (PLA)-CO2 system12 and, more recently, by our group, the PStoluene system.14

Figure 1. Schematic illustration of glass transition temperature – pressure map displaying the type IV behavior in a mixture of a polymer with a low molecular weight compound. In this latter contribution, Pierleoni et al. obtained the Tg vs P map from data of toluene vapor ‘dynamic’ sorption experiments in PS conducted by gravimetric measurements performed at a controlled rate of variation of temperature and/or pressure of the toluene vapor. In particular, in isothermal tests, the toluene mass uptake in PS was recorded as a function of pressure, performing the experiments at prescribed rates of change of pressure. In isobaric tests, absorbed mass of toluene in PS was recorded as a function of temperature, in experiments at prescribed rates of change of temperature. Finally, sorption tests were performed at a constant toluene activity by concurrently changing both temperature and pressure; in this case, absorbed mass data were plotted as a function of both temperature and pressure. In analyzing experimental results, the glass transition was assumed to be located in correspondence of the discontinuity in the slope evident in the curves reporting the mass of toluene sorbed in PS as a function of temperature and/or pressure. From such data, a glass transition temperature – fluid pressure envelope was constructed, which revealed that the PS-toluene system is characterized by a type IV behavior, displaying a ‘retrograde vitrification’ phenomenon. The experimentally accessible rubber-to-glass transition was considered as an evidence of an underlying II order thermodynamic transition. The unknown T, P coordinates of the actual thermodynamic transition are different from the coordinates of the experimentally detectable rubber-to-glass transition, since kinetic effects, related to the intrinsic mobility of the 3



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macromolecules and ruled by the rate of change of temperature and/or pressure, determine the occurrence of ‘freezing’ of the polymeric system before the manifestation of the ‘true’ underlying thermodynamic transition. Figure 2 reports schematically the type IV behavior along with the patterns of the three different types of ‘dynamic’ sorption tests performed by Pierleoni et al..14

Figure 2. Schematic illustration of the patterns of the three different types of ‘dynamic’ sorption tests performed by Pierleoni et al..14 In the present contribution, we provide a theoretical interpretation of the PS-toluene data, following an approach similar to that proposed by Condo et al..1 In detail, differently from Condo et al.,1 we have used the Non Random Hydrogen Bonding, NRHB, compressible lattice fluid model15,16 to describe the thermodynamics of the PS-toluene mixture, both to interpret sorption isotherms of toluene in PS and to calculate the entropy of the mixture. In view of the chemical structures of the two components, the specific interaction contributions present in NRHB model have been omitted in the present analysis. It is worth noting that accounting for non-randomness provided a better interpretation of sorption isotherms in comparison to the one attainable by using simpler random lattice fluid models, such as the one adopted in ref. [1]. In fact, non-randomness of lattice contacts always arises due to difference of mean field interaction energies between the different species (i.e. components and holes). The prediction of the location of glass transition as a function of pressure of toluene vapor has been then calculated, according to the Gibbs-Di Marzio criterion, imposing that the configurational entropy of the mixture, in a rubbery state at sorption equilibrium with external toluene vapor phase, 4


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is zero. It is worth noting that, since the glass transition envelope complies also with the conditions for phase equilibrium, uniform values of P and T have been assumed to hold within the two-phase systems considered, as is the case in which the effects of external force fields are neglected.

5



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Theoretical background Modeling of sorption thermodynamics in rubbery polymers by NRHB approach Significant improvement of the classic compressible Lattice Fluid theories to describe rubbery polymer-penetrant mixture thermodynamics was obtained by accounting for the non-randomness distribution of all contacts - including non random distribution of free volume hole sites - and for highly specific forces, like hydrogen bonding, between neighboring molecules. A relevant example of this class of approaches is the non-random hydrogen-bonding (NRHB) model,15-19 that has been used in this study. In view of the absence of hydrogen bonding and other specific interactions in the case of the polystyrene-toluene system, the contribution to model equations related to hydrogen bonding has been here omitted. In the case of pure components displaying no hydrogen bonding, the NHRB model has only three characteristic parameters for each species i: one is represented by ∗ ,, , used to calculate the inverse of the close packed density according to equation reported in

reference;16 another is represented by the average mean field interaction energy per molecule ∗ .

The latter parameter is, in turn, made of two contributions, i.e. ∗ and ∗ , which represent,

respectively, the enthalpic and entropic contribution to ∗ (reader is referred to the original contribution on NRHB model16

for details about the calculation of ∗ from these two

contributions). It is worth noting that the model assumes that the volume occupied by a cell of a molecule of species i, ∗, is represented by a universal value equal to 9.75/NAV cm3/molecule, in which NAV represents the Avogadro number. Consistently, in the case of a mixture, the molar volume of lattice cells, indicated as v*, takes the same value, independently of concentration. From this set of parameters, the number of lattice cells ri occupied by one molecule of species i, is calculated.16 ∗ The values of the two mean field energy parameters, as well as of ,, , can be retrieved

from pure component vapor pressure and/or volumetric properties for low molecular weight fluids, while, in the case of polymers, they are commonly retrieved from PVT data in the melt state. Another relevant lattice model parameter for each of the components is represented by the surfaceto-volume ratio, si, that is a geometric characteristic of a molecule of component i. It is not a fitted parameter, but can be calculated according to well established theories, i.e. UNIFAC group contribution method.16 In the case of systems endowed with specific interactions, such as hydrogen bonding, three additional parameters are also needed, namely the energy, the volume and the entropy of formation for each kind of hydrogen bond established in the system. For the PS-toluene system under analysis, no self- and cross-hydrogen bonding are present, neither in the case of pure components nor in the 6


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case of their mixtures. NRHB mean field scaling parameters for pure toluene were taken from ref. [20], while those for pure PS from ref. [16], and their numerical values are reported in Table 1, along with values of si. Table 1. Values of NRHB mean field scaling parameters and of si for toluene20 and PS16

∗ (J/mol)

∗ ∗ (J/mol K) , (cm3/g)

si

Toluene

5097.2

0.0768

1.06205

0.757

Polystyrene

5341.5

4.5361

0.9027

0.667

In the following, we will consider only the specific case of a binary system made of a polymer and a low molecular weight penetrant, which is not endowed with any type of specific interaction. We will indicate, respectively, the quantities referred to the penetrant with subscript ‘1’ and those related to the polymer with subscript ‘2’. For the case of a mixture of non-hydrogen bonding fluids, the corresponding parameters, i.e. ε* and the average number of cells occupied per molecule in the mixture, r, are obtained through mixing rules containing a binary interaction parameter. In detail, in the case of a binary mixture, we have that ε* is defined as: ∗ ∗ = ∗ + 2 + ∗

(1)

where θ1 and θ2 are the so-called surface contact fractions [16], which depend on concentration and ∗ = 1 − ∗ ∗

(2)

The binary interaction parameter, k12, measures the departure of mean field interaction energy from the one provided by geometric mixing rule. The corresponding scaling parameters for the temperature, T*, pressure, P* and density, ρ*, are interrelated as:

∗ =

(3)

∗

∗ = ∗ = ∗ ∗

(4)

7



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where Mw is the molecular weight and k is the Boltzmann constant. The Equation of State (EoS) expressions derived in the NRHB framework are written in terms of dimensionless reduced variables, i.e. reduced temperature , reduced pressure and reduced density : = ! ∗ , = !∗ , = !∗ = 1/

(5)

The EoS for a mixture takes the following form: + , + ) + #$%1 − − &∑ ( * − $% &1 − + * + $%- . = 0

(6)

where 01 = 2/3 41 − 51 − 41 − 6 and z is the coordination number of the lattice where

molecules are assumed to be assembled, 71 represents the “close packed” volumetric fraction of species i, qi represents the number of lattice contacts per molecule for component i , q represents the average number of lattice contacts per molecule in the mixture and ri is number of lattice cells occupied by one molecule of component i. The molecular shape factor, si, mentioned above, is defined as the ratio of qi and ri. The state variables 819 , represent the multiplicative corrective factors accounting for the non-randomness of contacts among molecular sites of species j and molecular sites of species i within the lattice (i,j = 0,1,2; in particular, index equal to 0 stands for the empty cells of lattice). Their values can be obtained by solving a set of equations, obtained by minimizing Gibbs free energy as a function of number of different kinds of lattice fluid contacts and by imposing material balance expressions for the lattice fluid contacts.

The phase equilibrium condition between a binary polymer-penetrant mixture and the pure penetrant in vapour or liquid phase implies the equality of the chemical potentials of the penetrant in the two coexisting phases:

:; = :
?

@A ,

* +

where

=

%$+,>

C> >

− D ∑FG

BE )E E

+

+ $% + D − 1$%1 − − D & − 1 +

#$%Γ + ,> − 1$%Γ . + D

? =>,IJ

@A

B>

>

A

+

,> >

* $% &1 − +

? =>,IJ

@A

(8)

represents the hydrogen bonding contribution, that is not considered in the case at hand.

R represents the universal gas constant and, finally, K represents a characteristic quantity that accounts for flexibility and symmetry of molecule of kind i, and it is defined in ref. [15, 16]. The expressions of the EoS and of the chemical potential for pure penetrant in the vapor or liquid phase can be obtained, respectively, from equations (6) and (8) by setting ( = 1 and the number of components in the summation equal to 1. Once P and T are fixed, solution of the set of equations made of eq. (7) and of the EoS equations for the mixture and for the pure penetrant in contact with it, provides the composition of the polymer mixture at phase equilibrium with pure component in the vapor or liquid state.

Results and Discussion Prediction of glass transition temperature One of the first models for the prediction of polymer glass transition in the presence of solutes rooted on a thermodynamic framework is the theory of Gibbs and Di Marzio.2-4 According to this approach, polymers as well as mixtures of polymers with low molecular weight compounds are modelled, at equilibrium, using a lattice fluid theory that allows also for the presence of vacant sites. In the case of pure compounds, the rubber-to-glass transition is identified as the occurrence of a true thermodynamic second order transition, for a fixed pressure, at a temperature T2 where the equilibrium configurational entropy of the system, as calculated by means of the lattice fluid theory, becomes equal to zero. This is the ideal glass transition temperature that would be observed experimentally in the case of non-crystallizable materials, if one changed temperature at an infinitely slow rate. Clearly, experimental measurement of T2 is not feasible. In fact, the experimentally measurable Tg is higher, since it is a kinetically affected manifestation of T2, which constitutes a lower bound of the experimentally accessible glass transition temperature. As argued by Panayiotou et al.,21 Tg should be linked to entropy by the same relationship holding between T2 and entropy, thus resulting, at equilibrium, in S=0 at the Tg. The same approach can be adopted to treat the occurrence of glass transition of pure compounds at a fixed temperature as a consequence of a change in pressure. In the case of mixtures, adopting the same arguments, one can again define 9



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the experimentally accessible glass transition as occurring at zero value of equilibrium configurational entropy of the mixture as a whole. The Gibbs-Di Marzio criterion for the identification of the glass transition temperature for pure polymers and polymer-penetrant mixtures, as the point at which the equilibrium configurational entropy becomes zero, can be applied to any thermodynamic model able to provide an expression for entropy. In a series of papers,15, 21-23 Panayiotou and co-workers used the NRHB lattice fluid model to calculate the entropy of pure amorphous polymers and of polymer-penetrant mixtures and, in turn, to evaluate Tg according to the Gibbs – Di Marzio criterion. It is worth noting that in the NRHB model the entropy accounted for is only the configurational entropy. As a consequence, the application of the criterion consists in taking equal to zero the equilibrium total entropy of a polymer-penetrant binary mixture calculated using NRHB model, whose expression is given by the sum of three contributions:

LMNM = LOP + LQR = LOP,S + LOP,TS + LQR

(9)

in which Stot is the total configurational entropy of the system, that is made of a ‘lattice fluid’ contribution, LOP , and an ‘Hydrogen Bonding’ contribution, LQR . In turn, the ‘lattice fluid’

contribution is expressed as the sum of a randomicity contribution, LOP,S (that would be the total

lattice fluid contribution in case of random mixing) and of a contribution accounting for nonrandomicity of the site contacts in the lattice, LOP,TS. Moreover, we have:

LOP,S = LUOP,S + LVTW OP,S

(10)

where LUOP,S is an external ‘configurational’ contribution while LVTW OP,S is an ‘internal’ or ‘molecular flexibility’ contribution to entropy,23 given by the following expressions:

\ + ]0 + 0NZ4Y^⁄4 − ∑1 _1 0NZ_1 + 2⁄3Y − 6 + LUOP,S ⁄S4T = 6 − Y0NZ6 − [

\ + 5⁄4 5⁄46 − [

(11) `

1 LVTW OP,S ⁄S4T = ∑1 & 4 * 0NZa1

(12)

1

from which:

\ + ]0 + 0NZ4Y^⁄4 − ∑1 _1 0NZ_1 + LOP,S⁄S4T = 6 − Y0NZ6 − [ `

2⁄3Y − 6 + 5⁄46 − [ \ + 5⁄4 + ∑1 & 1 * 0NZa1 41

(13)

The non-randomicity contribution to the total configurational entropy is given by:24 10


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2

3

LOP,TS ⁄S4T =

5

5

#−Y − 60b8cc − 46 76 0b866 − 4 3 73 0b833 + 6

3

Y − 6d6 d4 0bec6 8c6 + d3 d4 0bec3 8c3 +

56 46

76 d3 d4 0be63 863 .

(14) We do not report here the expression of SHB, since it is not relevant in the present context, in view of the absence of Hydrogen Bonding in the polymer-penetrant system considered. In equations (11)-(14), xi represents the molar fraction of species i, and c.m

where hi19 = i11 + i99 − 36 − j63 ]i11 i99 ^ 5/4

Finally, d4 = 5/4nY

o6

e19 = f

g

hi19 k jW

3

and i11 = i∗1 2 . Here i,j=0,1,2 and ic∗ = c.

.

To calculate the value of the Tg of the mixture, one needs to know the values of δi, which is the so-called chain flexibility parameter appearing in the random term of entropy.1,2,25,26 This is a characteristic quantity for each pure component i of the mixture (i.e. the polymer and the penetrant, in the case at hand), and it is given by:21

lnr = lns + t D lns − 2 − t D lnt − D 1 − t ln1 − t −

u vw

xA

(15)

In eq. (15), fi represents the equilibrium fraction of flexed bonds, i.e. in a high energy state, in a semiflexible molecule of type i consisting of ri segments or "mers". Note that in the framework of lattice fluid models, ri coincides with the number of cells occupied by a molecule of type i, as previously assumed by Condo et al..1 Following Condo et al.,1 a two-state model for bond energies in both the components molecules (i.e. bonds can have a high energy state or a low energy state) is adopted here. The value of fi is provided by the following expression:1

t =

y oz{ovw ⁄xA

(16)

ny oz{ovw ⁄xA

In equations (15) and (16), Δ} represents the molecular flex energy for bonds of type i, that is the increase in intramolecular energy that accompanies the "flexing" of a bond in a type i chain molecule. It is worth noting that a limitation of the present approach is that it is assumed that the flex energy is neither dependent on temperature nor on composition. This assumption is relevant also for the calculation of chemical potentials, since the theoretical mathematical form obtained in the framework of lattice fluid theory (eq. (8)) would be affected by the dependence of flex energy upon mixing.25 Based on this simplifying assumption, the flex energy of each component could be then straightforwardly calculated by zeroing the equilibrium entropy of a pure component - polymer 11



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or penetrant - at T2. However, for all practical purposes, in the case of a polymer (component ‘2’ in the present context), one may readily retrieve the value of flex energy from the experimentally accessible glass transition temperature, as it will be discussed in the following section, according to the procedure proposed in ref. [21]. This value of molecular flex energy, should be better interpreted as an ‘apparent’ flex energy. Conversely, in the case of the low molecular weight component (component ‘1’ in the present context), it is assumed here that Δ } = 0, following again Condo et al..1 This is equivalent to assume the penetrant to be fully flexible, differently from the polymer. It is worth noting that Zi, the bond coordination number (bond conformations) for bond of type i, appearing in eqs. (15) and (16), is, in general, different from the coordination number of the lattice (indicated by z in the present context). For the sake of simplicity, we assume here that polymer and penetrant share the same value of Zi (i.e. we set here Z1= Z 2= Z), as already proposed in ref. [21]. In particular, a value of Z=4 has been chosen here for both polymer and penetrant, as originally suggested by Gibbs and Di Marzio.2

Calculation procedure for Tg The theoretical prediction of glass transition temperature as a function of fluid pressure has been performed by solving the set of equations including the phase equilibrium condition between polymeric mixture phase and toluene vapor, the equations of state for pressure pertinent to each of the two phases, and the equation expressing the zeroing of equilibrium configurational entropy of the mixture. It is worth noting that, as already discussed for the pressure and temperature fields, also the toluene chemical potential field is considered to be uniform in the two-phase system, as is the case for thermodynamic equilibrium conditions in multiphase systems in which external force fields contribution are neglected. Before solving the set of equations, we first determined the two relevant parameters of the model, i.e. the NRHB binary interaction parameter of PS-toluene system and the molecular flex energy of PS. The former was determined by fitting concurrently sorption isotherms of toluene in PS at several temperatures, while the latter was determined by zeroing the expression of equilibrium configurational entropy of the pure polymer at its Tg, as measured by differential scanning calorimetry. A subtle issue has to be mentioned in association to this procedure. In fact, calorimetric analysis has been performed by cooling PS at a rate of 5 °C/min, while the ‘dynamic’ sorption experimental data for the mixture refer to a rate of change of pressure of 0.16 mbar/min and a rate of change of temperature of 2°C/h, respectively for isothermal tests and for isobaric and isoactivity tests. It is impossible to state if the conditions at which the calorimetric and ‘dynamic’ sorption experiments have been actually performed at similar conditions in terms of molecular 12


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mobility. In fact, the kinetic effects that influence the accessible transition are likely very different in the two type of experiments, thus possibly resulting in some inconsistency in assuming a unique value for the ‘apparent’ flex energy for all the experimental condition analyzed. As already discussed, since in this system there is no hydrogen bonding, the terms in NRHB model related to specific interactions have not been considered. In the following section, we provide a short summary of the steps followed in the procedure. First, concurrent fitting of PS-toluene sorption isotherms, limited to conditions at which the polymer-toluene mixture is rubbery, was performed adopting the NRHB approach, by imposing the equivalence of toluene chemical potential in the polymer mixture and in the external vapor phase (eq. (7)) and by simultaneously imposing the EoS of the polymer mixture and of the vapor phase. The experimental isotherm at 40 °C was taken from our previous work,14 while isotherms at the other temperatures were retrieved from reference [27]. Results of the simultaneous fitting of these isotherms with the NRHB model are reported in Figure 3. An excellent interpretation of data was obtained with a single value of the binary interaction parameter (k12 = -0.0015 ± 0.0002). This is a relevant point, since it allows a significant simplification in the implementation of the calculation procedure used to predict the location of glass transition. The subsequent step has been the evaluation of flex energy of pure polymer by zeroing the expression of the equilibrium configurational entropy, SLF, at the Tg for pure PS. In the case at hand no hydrogen bond entropy contribution (SHB) is present. By calorimetric analysis, it has been determined a value of glass transition temperature for PS, at atmospheric pressure, equal to 99.9 °C, from which, a value of the flex energy for the pure polymer equal to 3389.20 J/mol has been calculated.

Figure 3. PS-toluene solubility isotherms at different temperatures. Experimental data from Pierleoni et al.14 (40°C), and Krüger et al.27 (70-115°C). Lines represent fitting using NHRB model. 13



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The final step, has been the numerical solution of the following set of equations, by using Newton-Raphson method: a) phase equilibrium between polymer mixture and toluene vapor, consisting in the equivalence of toluene chemical potential in both phases (it is assumed that no polymer is present in the vapor phase); b) Equation of State of the two phases at sorption equilibrium; c) ‘Internal’ equations of NRHB model accounting for minimization of Gibbs energy as function of non random variables. d) Equivalence to zero of the expression for configurational entropy of the mixture, SLF. The solution of this set of equations corresponds to a null value of the equilibrium configurational entropy for a rubbery polymer-penetrant mixture in equilibrium of phase with an external fluid phase at fixed value of pressure. For any fixed pressure selected to construct the Tg vs P envelope, one obtains, by solving the set of equations (a) – (d), the values of equilibrium density and composition of binary mixture, equilibrium density of vapor phase, and of the temperature at which the transition occurs. Following this procedure the Tg vs. P envelope was predicted and compared to the experimental values in Figure 4a, where the vapor pressure of toluene was also reported (in red), to allow for the comparison with partial pressure for toluene at the glass transition. As reported in figure 4a, the model predictions are in good agreement with experimental data, displaying all the relevant features highlighted by the experimental analysis, including the retrograde vitrification phenomenon. The experimental values were determined at a rate of change of pressure equal to 0.16 mbar/min in the case of isothermal tests and at a rate of change of temperature equal to 2°C/h in the case of isobaric and isoactivity tests, as described in detail by Pierleoni et al..14 As discussed in a previous contribution,14 the location of transition depends upon the rate of change of boundary conditions imposed during ‘dynamic’ sorption experiments. Figure 4b reports some unpublished results, previously obtained by our group by performing ‘dynamic’ isothermal tests at different rate of change of pressure at 40°C. It is evident that the transition occurs at decreasing values of pressure as the rate of change of pressure decreases, approaching the values predicted theoretically. In the light of the previous discussion, this result suggests that the cooling rate of calorimetric experiments are consistent, for the purpose of determining the correct value of the of apparent flex energy, with values of rate of change of pressure lower than those actually adopted in the performed experiments. 14


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Figure 4a. Comparison of experimental data for Tg vs P taken from ref.[14] with prediction of theoretical approach.

Figure 4b. Comparison of theoretical prediction with experimental data highlighting the effect of varying the rate of pressure decrease (investgated rates are 0.16, 0.04, 0.01and 0.004 mbar/min) in isothermal experiments performed at 40°C. Comparison offered in Figures 4a and 4b shows that peculiar results for rubber-to-glass transition in PS-toluene system, as obtained through ‘dynamic’ sorption experiments introduced in a recent paper [14], are consistent with the thermodynamic picture for glass transition provided by the Gibbs-Di Marzio approach. Data for pressure (vapor fugacity) vs temperature relation at rubber-toglass transition identified through either isothermal desorption, isobaric sorption/desorption or isoactivity cooling experiments all appear to be closely related to a change in thermodynamic 15



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properties of equilibrium phases underlying the states experimentally accessible in ‘dynamic’ scanning experiments. The latter change, in turn, is satisfactorily captured by a relatively simple thermodynamic model based non-random lattice fluid theory and tuned on independent pure component glass transition and binary equilibrium experimental data. Finaly, we have assessed the sensitivity of theoretical predictions of the Tg vs. P envelopes to the values of glass transition temperature of pure polymer and, in turn, to the corresponding values of ‘apparent’ flex energy. The effect on model calculations of changing the value of Tg of pure PS around the experimental value determined calorimetrically (i.e. 373.05 K) has been evaluated by assuming several values for the glass transition temperature of pure polymer, i.e. 371.15, 372.15, 373.15, 374.15 and 375.15 K. The results of the numerical calculations are illustrated in Figure 5. The values of the ‘apparent’ flex energy of the polymer corresponding to the selected values of Tg are reported in Table 2. As expected the predictions are quantitatively quite sensitive to the assumed value of Tg and of ‘apparent’ flex energy.

Figure 5. Glass transition temperature vs. pressure envelopes predicted theoretically for several values of Tg of pure PS and, in turn, of polymer ‘apparent’ flex energy.

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Table 2. Values of glass transition temperature of PS and of the corresponding values of ‘apparent’ flex energy of the polymer calculated numerically by zeroing the entropy. Assumed value for Tg of pure PS [K] 371.15 372.15 373.15 374.15 375.15

Corresponding values of ‘apparent’ flex energy of pure PS [J/mol] 3367.3 3380.6 3393.9 3407.3 3418.6

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Conclusions A theoretical approach, based on the combination of Gibbs-Di Marzio criterion for glass transition and expressions for configurational entropy and phase equilibrium obtained from NRHB lattice fluid theory, has been adopted to predict the Tg – P envelopes for the PS-toluene system. Binary interaction parameter and polymer flex energy, that are the only unknown parameters of the model, were obtained from experimental data. A temperature independent value for the binary interaction parameter was obtained by concurrent fitting of toluene sorption isotherms in polystyrene at several temperatures, performed with phase equilibrium equations provided by NRHB theory. Conversely, polymer flex energy was obtained from the value of glass transition temperature of pure polystyrene determined calorimetrically, applying the Gibbs-Di Marzio criterion coupled with expression for configurational entropy of pure polymer. The theoretical predictions for the Tg – P envelopes show a remarkable agreement with data providing a reliable framework to interpret the dependence of glass transition of PS on pressure of toluene vapor as well as the peculiar retrograde vitrification phenomenon. The experimental data and the theoretical interpretation, taken together, represent one of a few examples of rather complete and consistent description of type IV behavior for glass transition vs pressure in a polymer-penetrant system.

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[16] Panayiotou, C.; Tsivintzelis, I.; Economou, I.G. Nonrandom hydrogen-bonding model of fluids and their mixtures. 2.Multicomponent mixtures. Ind. Eng. Chem. Res. 2007, 46, 2628-2636. [17] Panayiotou, C. New expressions for non-randomness in equation-of-state models. Fluid Phase Equilib. 2005, 237, 130–139. [18] Panayiotou, C.G. In Handbook of Surface and colloid chemistry; 2rd ed.; Birdi, K. S., Ed.; CRC Press Taylor and Francis group, New York, 2003, pp 5-66. [19] Panayiotou, C.G. In Handbook of Surface and colloid chemistry; 3rd ed.; Birdi, K. S., Ed.; CRC Press Taylor and Francis group, New York, 2009; pp 45-89. [20] Tsivintzelis, I; Spyriouni, T.; Economou, I.G. Modeling of fluid phase equilibria with two thermodynamic theories: Non-random-hydrogen bonding (NRHB) and statistical associating fluid theory (SAFT). Fluid Phase Equilib. 2007, 253, 19-28. [21] Prinos, J.; Panayiotou, C. Glass transition temperature in hydrogen bonded polymer mixtures. Polymer 1995, 36, 1223–1227. [22] Panayiotou, C.; Pantoula, M. Sorption and swelling in glassy polymer/carbon dioxide systems: Part 1. Sorption. J. Supercr. Fluids 2006, 37, 254-262. [23] Tsivintzelis, I.; Angelopoulou, A.G.; Panayiotou, C. Foaming of polymers with supercritical CO2: An experimental and theoretical study. Polymer 2007 48, 5928-5939. [24] Mensiteri, G.; Scherillo, G. In Wiley Encyclopedia of Composites; 2nd ed., Nicolais, L., Borzacchiello, A., Eds.; John Wiley & Sons, Hoboken, NJ, 2012; pp 804-829. [25] Flory, P.J. Statistical thermodynamics of semi-flexible chain molecules. Proc. R. Soc. Lond. A 1956, 234, 60-73. [26] Panayiotou, C.G. Glass transition temperature in polymer mixtures. Polym. J. 1986, 18, 895902. [27] Krüger, K.M.; Sadowski, G. Fickian and non-Fickian sorption kinetics of toluene in glassy polystyrene. Macromolecules 2005, 38, 8408-8417.

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Modeling Retrograde Vitrification in the PolystyreneâToluene System

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Modeling Retrograde Vitrification in the PolystyreneâToluene System