Adam-Gibbs based model to describe the single component dynamics in miscible polymer blends under hydrostatic pressure

We present in this work a new model to describe the component segmental dynamics in miscible polymers blends as a function of pressure, temperature, and composition. The model is based on a combination of the Adam-Gibbs AG theory and the concept of the chain connectivity. In this paper we have extended our previous approach D. Cangialosi et al. J. Chem. Phys. 123, 144908 2005 to include the effects of pressure in the component dynamics of miscible polymer blends. The resulting model has been tested on poly vinyl methyl ether PVME /polystyrene PS blends at different concentrations and in the temperature range where the system is in equilibrium. The results show an excellent agreement between the experimental and calculated relaxation times using only one fitting parameter. Once this parameter is known the model allows calculating the size of the relevant length scale where the segmental relaxation of the dielectrically active component takes place, i.e., the so called cooperative rearrangement region CRR in the AG framework. Thus the size of the CRR for PVME in the blends with PS has been determined as well as its dependence with pressure, temperature, and concentration. © 2007 American Institute of Physics. DOI: 10.1063/1.2780157


I. INTRODUCTION
The dynamics of polymer blends has attracted the interest of many researchers during the last years because the peculiar features displayed by these systems.Among them one of the most interesting is the presence of two distinct time scales for the component segmental dynamics due to the heterogeneous dynamics at the length scale of the segmental relaxation.2][3][4][5][6] Some of these models are based on the influence of concentration fluctuations on the component dynamics. 1,2owever, despite the general agreement about the ability of concentration fluctuations to affect the distribution of relaxation times, several authors have argued about their role in determining the mean relaxation time. 3,7It is now generally accepted that concentration fluctuations alone cannot account quantitatively the presence of the two distinct dynamics and the effect of chain connectivity has to be considered. 3Lodge and co-workers 3,6 have used the idea that in miscible polymer blends the local concentration of one component will, on average, be richer in that component compared to the bulk composition.This is a direct consequence of the chain connectivity.Based on these arguments and by defining the selfand effective concentration, they calculated the effective T g of each component in terms of the macroscopic T g of the neat polymers through the Fox equation.The approach of Lodge and McLeish has been later used by Leroy et al. 4 who described the component segmental dynamics taking into account both the effect of concentration fluctuations and chain connectivity.
More recently, a different approach to describe the component segmental dynamics in miscible polymer blends has been proposed by us 8,9 starting from the theory of Adam and Gibbs ͑AG͒, 10 which relates the dynamic and thermodynamics behaviors of glass formers and has allowed us to describe the molecular dynamics close ͑and above͒ T g . 11,12In those works we provided a new approach to describe the component segmental dynamics of athermal miscible polymer blends ͑at atmospheric pressure͒ combining the concept of chain connectivity, expressed in terms of the selfconcentration, and the AG model.This approach has been successfully applied to different polymer blends and polymer-solvent mixtures. 8,9,13,14On the other hand, several models have been proposed in the last years to describe the pressure-temperature dependence of the relaxation time of neat glass forming materials. 15,16In particular, an extension of the AG theory has been recently proposed by Casalini et al. 17 These authors basically proposed to write the configurational entropy in terms of the excess thermal heat capacity and of the excess thermal expansion to give account of both thermal and pressure effects, respectively.][19][20][21] The aim of this work is to present a model based on the AG theory to describe the component segmental dynamics of miscible polymer blends, at any concentration, as a function of both pressure and temperature.The idea is to combine the approach for describing the component dynamics in miscible polymer blends at atmospheric pressure 8 with the pressure extension of the AG theory. 17To test the resulting model we have analyzed the segmental dynamics of poly͑vinyl meth-ylether͒ ͑PVME͒ on the well known PVME/polysterere ͑PS͒ system, at different concentrations.6][27][28] The previous works on PVME/PS blends allow us to test the accuracy of the new model in describing PVME dynamics.Furthermore, using this new approach we have also analyzed the pressure dependence of the relevant length scale for the segmental dynamics.

A. Adam-Gibbs theory
The AG theory, which is based on the assumption of cooperative rearranging regions, gives an expression that relates the increase of structural relaxation time, , to the reduction of configurational entropy, S c , by 10 = 0 expͩ C TS c ͪ, where 0 is the value of at very high temperature and C is a material constant.This quantity is here assumed independent of temperature and pressure, although this assumption has been recently questioned. 29Since S c is not experimentally accessible, it is usually identified with the excess entropy ͑S c ϰ S ex = S melt − S crystal ͒.The physical meaning of this assumption is still under debate and no general agreement has been achieved, [30][31][32] however, some findings seem to suggest a general validity of the proportionality between S ex and S c which has been successfully used during the last years.Thus S c , at atmospheric pressure, can be estimated as where ⌬C P = C P melt -C P crystal is the excess heat capacity at atmospheric pressure and T K is the Kauzmann temperature, where the excess entropy would vanish.If an inverse temperature dependence is assumed for ⌬C P ͑⌬C P = K / T͒, the empirical Vogel-Fulcher-Tammann ͑VFT͒ equation 33 ͑T͒ = 0 exp͑DT 0 / ͑T − T 0 ͒͒ is recovered ͑with T 0 = T K ͒.However, this temperature dependence for ⌬C P is not always valid and a linear dependence of ⌬C P with temperature is often a better approximation. 34If a linear dependence of the type ⌬C P = b − m T T is assumed for the temperature dependence of the excess heat capacity, then by integrating Eq. ͑2͒ and replacing in Eq. ͑1͒ we obtain the following temperature dependence for the segmental relaxation time at atmospheric pressure: which in the experimentally accessible temperature range provides as good description as the VFT equation with the same number of parameters.It is worth to mention that this equation was first reported by Scherer 35 in 1984.By using Eq.͑3͒ to fit the experimental data of neat polymers at atmospheric pressure, the three unknown parameters log͑ 0 ͓s͔͒, C / g T , and T K are obtained.

B. Pressure extension of the AG theory for neat components
Among the several models developed to describe the pressure-temperature dependence of the relaxation time of glass forming materials, 15,16 a pressure extension of the AG theory has been recently proposed by Casalini et al. 17 In their work they have proposed an extension of Eq. ͑2͒ for the configurational entropy ͑S c ͒, by adding a term related to the pressure change.Thus, the temperature and pressure dependence of the S c is given by

͑4͒
where ⌬͑‫ץ‬V / ‫ץ‬T͒ P = ͑‫ץ‬V / ‫ץ‬T͒ P melt − ͑‫ץ‬V / ‫ץ‬T͒ P crystal is the difference of the thermal expansivity of the melt and the crystal; if g P does not depend on temperature, then the thermal and volumetric contributions to the configurational entropy are fully decoupled.The second term in Eq. ͑4͒ gives the pressure dependence of S c ͑and therefore of the heat capacity͒.We have shown in previous work 20 that this dependence is rather weak for PVME; these results are in agreement with those obtained by Takahara et al. 36,37 on other glass formers.
It is worth noting here that S ex and ⌬͑‫ץ‬V / ‫ץ‬T͒ P are defined as the difference between melt and crystal corresponding quantities.However, it is sometimes impossible for polymers to get the thermodynamic properties of the crystalline state.Thus, it is usually assumed that the heat capacity and thermal expansion for the crystal state are similar to those found for the glass. 38 .Additionally, the pressure dependence of ⌬͑‫ץ‬V / ‫ץ‬T͒ P has been empirically described according to 20 ⌬͑‫ץ‬V/‫ץ‬T͒ P = ⌬͑‫ץ‬V/‫ץ‬T͒ P=0 − A͓1 − exp͑− P/P 0 ͔͒.͑5͒ Thus, by replacing Eq. ͑5͒ in Eq. ͑4͒ and integrating, the following expression for the pressure-temperature dependence of the excess entropy is obtained: Finally, by replacing Eq. ͑6͒ in Eq. ͑1͒ we obtain the pressure-temperature dependence of the segmental relaxation time for neat polymers as follows:

͑7͒
Note that most parameters of Eq. ͑7͒ can be obtained from independent experiments: by differential scanning calorimeter ͑DSC͒ ͑b and m T ͒, by pressure-volume-temperature ͑PVT͒ ͑⌬͑‫ץ‬V / ‫ץ‬T͒ P=0 , A and P 0 ͒, and by the temperature dependence of the relaxation times at atmospheric pressure ͑log͑ 0 ͓s͔͒, CЈ = C / g T , and T K ͒ through Eq. ͑3͒.In this way the only unknown parameter in Eq. ͑7͒ is the ratio g P / g T .This is the single free parameter which has to be obtained from the measurements of the relaxation time at higher pressures. [19][20][21]

C. Adam-Gibbs theory in miscible polymer blends
We will summarize here the basic hypothesis of the model we have recently proposed to describe the segmental component dynamics in miscible polymer blends at atmospheric pressure.For a full detailed discussion, see Refs. 8 and 9.
Starting from the AG equation ͓Eq.͑1͔͒ we assume that for a given component in a miscible polymer blend we can write the temperature dependence of the segmental relaxation time as where A/blend is the segmental relaxation time of component A in the blend and 0 is the same as defined in Eq. ͑1͒ for the same component; CЈ A/blend ͑T͒ and S ex A/blend ͑T͒ refer to regions centered around a segment of polymer A and are calculated as where CЈ A/blend will depend on temperature if eff A does depend ͑see below͒.In these equations, which are only valid for athermal mixtures, we have used the effective concentration ͑ eff ͒ to take into account the connectivity effects ͑see Ref. 8͒.At length scales of the order of few monomeric units, eff can deviate significantly from the macroscopic concentration.eff can be calculated as where is the macroscopic concentration of the component under consideration and s is the self-concentration of the same component.The self-concentration is defined as the volume fraction within a sphere of radius r c , centered in a given monomer, occupied by the polymer chain of such a monomer.For radius ͑r c ͒ of the relevant volume smaller than the Kuhn length, s can be related with the excess entropy as where l P is the packing length and ␣ is a proportionality constant between r c and excess entropy 8 according to Similar equations can be written for component B. This set of equations ͓Eqs.͑8͒-͑12͔͒ can be exactly solved being ␣ the only unknown parameter; however, we have shown in a later work 9 that the parameter ␣ can be determined for each polymer from an independent experiment.This model has shown to be very accurate to describe the single component dynamics in miscible polymer blends at atmospheric pressure. 8,9,13,14

A. Component dynamics in miscible polymer blends under hydrostatic pressure
The natural extension to describe the component segmental dynamics in miscible polymer blends under hydrostatic pressure is to combine the previous two extensions of the AG theory in a new model.By mimicking Eq. ͑8͒ we can write

͑13͒
where now A/blend ͑T , P͒ is the segmental relaxation time of component A in the blend as a function of pressure and temperature and 0 is the same as defined in Eq. ͑1͒ also for component A; as previously, CЈ A/blend ͑T , P͒ and S ex A/blend ͑T , P͒ refer to regions centered around a segment of polymer A and are calculated as where S ex A ͑T , P͒ and S ex B ͑T , P͒ are the excess entropy as a function of the pressure and temperature for the neat homopolymers as defined in Eq. ͑6͒.As in the case of Eq. ͑9͒, these expressions are only valid for athermal mixtures.eff is the effective concentration as defined in Eq. ͑10͒ with the self-concentration defined as where all the parameters are defined as before, being now ␣ pressure dependent.This system of equations can be exactly solved once the pressure-temperature dynamics of both neat components are known, being ␣͑P͒ the only fitting parameter for a given component.

B. Implementation of the model
In order to apply the here proposed model to a given polymer blend, it is first necessary to perform a full characterization of the pressure-temperature dynamics of both components of the blend.According to the previous section ͓Eqs.͑6͒ and ͑7͔͒ we have to perform DSC and PVT measurements on each of the neat components to calculated b and m T and ⌬͑‫ץ‬V / ‫ץ‬T͒ P=0 , A, and P 0 , respectively.Then, from the temperature dependence of the relaxation times at atmospheric pressure log͑ 0 ͓s͔͒, C / g T , and T K can be obtained through Eq. ͑3͒.Finally, the only unknown parameter in Eq. ͑7͒, i.e., the ratio g P / g T , can be estimated by minimizing the mean square deviation between the experimental data and the relaxation times given by Eq. ͑7͒.This procedure gives the full temperature-pressure dependence of the segmental relaxation time for each component of the blend ͑see Table I͒.
According to Eqs. ͑11͒-͑15͒ we need to know the Kuhn and packing length of both components to calculate their self-concentration.In the case of PS and PVME here studied these data are available in the literature and are also listed in Table I.The remaining two parameters are the macroscopic composition of the blend ͑͒, which is known for each blend, and ␣͑P͒ which is the only unknown parameter in our model and has to be estimated from the fitting of the experimental relaxation times at different temperatures and pressures.

IV. EXPERIMENT
The experimental data used in this paper have been taken from a previous work 28 where the experimental details were given and discussed.Only a brief summary is presented here.
Samples DSC and PVT measurements.Calorimetric measurements were performed on a Q1000 TA Instruments DSC in the modulated mode, with amplitude of 0.5 K, a period of 100 s, and underlying cooling rate of 0.25 K / min.PVT measurements were performed on a Thermo Haake PVT100 equipment on isobaric cooling mode in the pressure range of 20-200 MPa with a cooling rate of 5 K / min.Dielectric measurements at atmospheric pressure.Dielectric measurements were performed using a broadband dielectric spectrometer based on two high precision dielectric analyzers, one for the frequency range 10 −2 -10 7 Hz ͑Alpha analyzer Novocontrol GmbH͒ and the other for the range ͑10 6 -1.8ϫ 10 9 ͒ Hz ͑Agilent 4192B͒, and a Novocontrol Quatro cryosystem for temperature control with a precision better than 0.1 K.
Dielectric measurements under pressure.Dielectric measurements were carried out in a pressure cell ͑0 -300 MPa͒ supplied by Novocontrol GmbH.The measurements were performed by frequency sweeps ͑10 −2 -10 6 Hz͒ at constant temperature, after stabilizing the temperature of the cell for about 2 h, with stability better than ±0.1 K, and constant pressure, with stability better than ±2 MPa.

A. Dynamics of neat polymers
Full DSC and PVT characterization of PVME and PS has been performed in previous works. 8,20,21The corresponding parameters are listed in Table I.Additionally, the pressure-temperature dependence of the segmental relaxation time for both neat polymers has been also analyzed in pre-vious works 20,21 by means of Eq. ͑7͒. Figure 1 shows the temperature dependence of the maximum relaxation times for PVME and PS at several pressures.The solid lines in Fig. 1 represent the best fit of the experimental data with Eq. ͑7͒.
The corresponding fitting parameters are listed in Table I.

B. Dynamics of PVME in polymer blends
Raw data of dielectric loss ͑Љ͒ as a function of the frequency at constant temperature, as well as at constant pressure, for the different blend's composition are shown in Ref. 28.From those spectra the temperature dependence of the maximum relaxation time ͑ max ͒ at different pressures was obtained and is shown in Figs.2-4.In these figures only those experimental data points within the so called equilibrium window, 28 i.e., within the temperature range where the system is in thermodynamic equilibrium, are shown.It is worth to remind here that we are dealing with the dynamics of the low T g component of PVME/PS blends with components having very different T g values.Thus, by decreasing the temperature from the high temperature regime, where the whole system is at equilibrium, the high T g component starts freezing first and the system begins falling out of equilibrium whereas the other component is still moving relatively fast.
In this low temperature range the low T g dynamics is affected by the confinement imposed by the other component.We know from a previous work 8 that the lower limit of the equilibrium ͑T eq ͒, at atmospheric pressure, is approximately 280, 300, and 330 K for 75%, 50%, and 25% of PVME in the blend.Since the here presented model, in its current form, is limited to describe the relaxation dynamics only at equilibrium conditions, the low temperature range, i.e., below the equilibrium window, has not been considered in this work.
Figures 2-4 show the comparison between the experimental data and the calculated relaxation times, using the here proposed model, for PVME in blends with 75%, 50%, and 25% of this component as a function of the temperature at different pressures.The only fitting parameter for each Figures 2-4 show that at low temperatures the calculated relaxation times tend to be systematically slower than the experimental maximum relaxation times.This fact is not an evidence of the limitations of the model but an effect which can be rationalized by the presence of the statistical thermal concentration fluctuations which yield a local heterogeneous distribution of both components in the blend.Since the di-electric technique enhances those regions richer in the dielectrically active component ͑PVME͒, due to its higher contribution to the dielectric signal, the resulting relaxation time for such regions is shorter than the corresponding for regions with average ͑or even lower than͒ macroscopic concentration.Thus, the main contribution to the dielectric loss peak comes from regions richer in PVME giving an effective maximum relaxation time shorter than that predicted by the model, which would correspond to an average concentration.This is especially important at temperatures close to T g where concentration fluctuations originate the dramatic relaxation broadening experimentally observed for PVME in blends with PS, 4 and therefore the relaxation time of the average concentration can be significantly slower than that obtained from the dielectric peak maximum.Contrary, at higher temperatures the concentration fluctuations do not play a major role in determining the shape of the segmental dynamics relaxation since it becomes close to that of the pure polymer.Therefore, in this high temperature range, maximum relaxation times obtained from dielectric spectra are found to be very close to those corresponding to the average concentration.
In order to test this hypothesis we have made an estimation of the relaxation time characteristic of the mean concentration.Based on previous results 4 we have used the distribution of effective concentrations, accounting for the statistical thermal concentration fluctuations, to calculate the dielectric response in the blends with 75% and 50% of PVME and hence found the corresponding PVME relaxation time for the mean concentration.The so obtained values for some representative pressures and temperatures ͑at relatively long times͒ are shown as open symbols in Figs. 2 and 3.These new relaxation times show that the model gives an excellent description of the PVME relaxation time for the mean concentration.This also shows the importance of the statistical thermal concentration fluctuations in the relaxation behavior close to T g .Unfortunately, for the blend with 25% of PVME the distribution function of the effective concentrations is not available and therefore we cannot perform the same estimation.
These results evidence the difficulties in developing a complete and quantitative model of the component segmental dynamics in athermal miscible polymer blends.However, the proposed model gives a very good description of the component dynamics over a broad range of pressures, temperatures, and concentrations with just only one fitting parameter ͑␣͑P͒͒.At atmospheric pressure it has been shown that the parameter ␣ can be determined from an independent experiment and thus the model becomes completely predictive.It is expected that a similar independent determination of the parameter ␣͑P͒ at higher pressures can be done.

C. Pressure-temperature dependence of the size of the CRR
As previously mentioned the AG theory allows calculation of the temperature and pressure dependence of the size of the relevant volume for the segmental relaxation ͑CRR͒ by means of Eq. ͑12͒.Thus, once the excess entropy and ␣ parameter are known, Eq. ͑12͒ can be used to estimate the size ͑2r c ͒ of the CRR at any temperature, pressure, and composition.Figures 6-8 show the temperature dependence, at different pressures, of the size of the CRR for PVME in PVME/PS ͑75/ 25͒, ͑50/ 50͒, and ͑25/ 75͒ blends, respectively.The size of the CRR is around 1 nm, in agreement with previous results at atmospheric pressure on the same blend. 8The increasing length upon temperature reduction imposed by the AG theory is clearly observed in these figures although less pronounced for ͑25/ 75͒ blend because the narrower temperature range experimentally available.As regards the effects of pressure, a slight but systematic increment of the temperature sensitivity of the CRR's size with pressure is observed for the three concentrations.This is very clear for the 50/ 50 blend where the effects of pressure are more evident.This fact is probably due to differences in the monomeric volume of both polymers and its effects on local packing which has shown to play a key role in controlling the dynamics of glass formers. 39However, the local packing of polymer blends and its effects on the blend dynamics are beyond the scope of the present study and are worth to be analyzed in future works.

VI. CONCLUSIONS
We have presented a model based on the AG theory to describe the segmental component dynamics in miscible polymer blends as a function of pressure, temperature, and concentration.A very good agreement between calculated and experimental relaxation times has been shown.Based on the knowledge of the pressure-temperature dependence of the dynamics of the neat polymers, the model gives account of the component dynamics in the blend by means of only one fitting parameter ͑␣͒.This parameter is characteristic of a given component and it is expected, as in the case of atmospheric pressure, that it can be determined by means of independent experiments.If so, the here presented model would become completely predictive, once the relevance of the statistical thermal concentration fluctuations on the dielectric relaxation time are considered.Additionally, we were able to evaluate the size of the relevant length scale ͑CRR͒ where the relaxation process takes place as well as its temperature, pressure, and composition dependence.Our results show a decreasing CRR size with increasing dilution, temperature, or pressure with values between 0.8 and 1.2 nm in agreement with previous results.

FIG. 1 .
FIG. 1. Segmental relaxation time of PS and PVME as a function of the temperature at different pressures ͓from bottom to top: P atm = 0.1, 50, 100, 150, 200, and 250 ͑and 300 only for PVME͒ MPA͔.The lines represent the best fit of the experimental data by means of Eq. ͑7͒ ͑see text͒.

FIG. 4 .
FIG. 4. Logarithm of the relaxation time of PVME vs temperature at different pressures ͑from bottom to top: P atm = 0.1, 50, 100, 150, 200, 250, and 300 MPa͒ for PVME/PS ͑25/ 75͒.The filled symbols represent the experimental maximum relaxation time.The solid lines represent the fit of the experimental data with the here proposed model.

FIG. 6 .
FIG.6.Diameter of the relevant volume for segmental relaxation vs temperature, at different pressures, as calculated using the model for PVME in blend PVME/PS ͑75/ 25͒.

TABLE I .
Relevant parameters for the pure components of the here studied blends.Errors are ±1 of the least significant digit unless specified.