On the nature of the free volume in amorphous polymers

In a previous publication¹ it was shown that the free volume in amorphous polymers cannot consist of empty sites according to the quasilattice model. In this paper it is demonstrated why theories which are based on the assumptions of this model are nevertheless partly successful. It is even possible to calculate all three jumplike alterations in the second derivatives of Gibbs free energy at the glass transition in accordance with the experimental results. As far as we know, this is the first time that, in addition to the steplike changes in the thermal expansion coefficient Δα and the specific heat Δc_p, the analogous change in the compressibility Δ_χ can also be described correctly by a theory. The parameters for accommodation however reveal, that the quasilattice model does not apply. Consequently better-founded equations are derived, which despite their simplicity, give excellent agreement with the experimental data. One conclusion is that if the free volume consists of cavities in the material, their mean size has to be at least ten times smaller than that of a segment of the chainlike polymer molecules.     

SCATTERING FUNCTION OF POLYMER BLENDS

For a system of flexible polymer molecules, the concepts of two concentrations, namely the segmental and the molecular concentrations, have been proposed in this paper. The former is equivalent to the volume fraction. The latter can be defined as the number of the gravity centers of macromolecules in a unit volume. The two concentrations should be correlated with each other by the conformational function of the polymer chain and should be discussed in different thermodynamic equations. On the basis of these concepts it has been proved that the Flory-Huggins entropy of mixing should be the result of the mixing “ideal gases of the gravity centers of macromolecules“. The general correlation between the free energy of mixing and the scattering function (structural factor) of polymer blends has been studied based on the general fluctuation theory. When the Flory-Huggins free energy of mixing is adopted, the de Gennes scattering function of a polymer blend can be derived.     

A theoretical interpretation of free volume at glass transition

We device a relaxed lattice model (RLM) to study the mechanism of glass transition,which unifies the cageeffects from particle-particle interaction and entropy.By analyzing entropy in RLM with considering the influence of interactions on equilibrium,we demonstrate that glass transition is a second-order phase transition.For a perfect onedimensional linked particle system like linear polymer under normal pressure,the free volume at glass transition is rigorously deduced out to be 2.6％,which provides a theoretical basis for the iso-free volume of 2.5％ given by Willian,Landel and Ferry (WLF) equation.Extending to system with dead particles linked with higher dimensions like branched or cross-linked chains under positive or negative pressure,free volume at glass transition is varied,based on which we construct a phase diagram of glass transition in the space of free volume-dead particle-pressure.This demonstrates that free volume is not the single parameter determining glass transition,while either dead particles like cross-linked points or external force fields like pressure can vary free volume at the glass transition.     

The effect of lattice compressibility on the thermodynamics of gas sorption in polymers

A compressible lattice model with holes, the glassy polymer lattice sorption model (GPLSM), was used to model the sorption of carbon dioxide, methane, and ethylene in glassy polycarbonate and carbon dioxide in glassy tetramethyl polycarbonate. For glassy polymers, an incompressible lattice model, such as the Flory–Huggins theory, requires concentration-dependent and physically unrealistic values for the lattice site volumes in order to satisfy lattice incompressibility. Rather than forcing lattice incompressibility, GPLSM was used and reasonable parameter values were obtained. The effect of conditioning on gas sorption in glassy polymers was analyzed quantitatively with GPLSM. The Henry's law constant decreases significantly upon gas conditioning, reflecting changes in the polymer matrix at infinite dilution. Treating the Henry's law constant as a hypothetical vapor pressure at infinite dilution, gas molecules in the conditioned polymer are less “volatile” than those in the unconditioned polymer. Flory–Huggins theory was used to model the sorption of carbon dioxide, methane, and ethylene in silicone rubber. Above the glass transition temperature, the criterion of lattice incompressibility for Flory-Huggins theory was satisfied with physically realistic and constant values for the lattice site volumes. © 1992 John Wiley & Sons, Inc.     

玻璃化转变的热力学理论错在哪里？

现行国内高分子物理学教科书上都介绍玻璃化转变的热力学理论,其把玻璃化转变描述成为一个二级相转变.但是现在人们已经普遍接受玻璃化转变的本质是一个动力学过程的观点.我们通过讨论玻璃化转变热力学理论的来历,试图弄清这一理论到底错在哪里.首先,该理论所基于的Kauzmann佯谬可能不是一个真正的佯谬;其次,半柔顺链高分子溶液的经典格子统计理论结果中所谓的熵灾难可能是出于对构型熵的错误理解所致.因此,把以上二者联系起来构成玻璃化转变热力学理论的基本假定就从根本上来说是不可靠的.     

Mesoscale model parameters from molecular cluster calculations

We present an efficient, systematic, and universal method to estimate the interaction parameters used in mesoscale simulation methods such as dissipative particle dynamics and self-consistent field methods from molecular cluster calculations. The method is based on a generalized Flory-Huggins model in which molecules, or fragments thereof, are in contact with their van der Waals surface. We sample the density of states of molecular clusters in the space spanned by the coarse-grained degrees of freedom. From here, we calculate the sum over states and free energy of the cluster at a temperature of interest by histogram reweighting. The method allows to calculate the energy and entropy contributions to the cluster free energy explicitly. For two components, we then obtain the excess free energy of mixing and the Flory-Huggins chi-parameter, and their energetic and entropic contributions. We present two applications of the method: a simple liquid mixture of hexane and nitrobenzene, and a series of polymer blends. In the case of hexane/nitrobenzene, we compare to alternative simulation methods; here we find that the energy of mixing alone is too high to explain the critical point. By including the excess entropy of mixing, however, the predicted phase behavior is in reasonable agreement with experiment. The tendency of calculations based on average energy alone to overestimate the chi-parameter is also apparent in the polymer blend calculations.     

A free-volume hole-filling model for the solubility of liquid molecules in glassy polymers 1: Model derivation

The use of hole-filling models is quite common for the sorption of gases into glassy polymers, but these have yet to be explicitly applied to the case of liquid immersion, where extensive use of Flory-Huggins theory dominates. This paper explores how the models based on the idea of sequential filling of a Gaussian distribution of pre-existing free-volume holes within the structure of the glassy polymer can be modified to allow the prediction of the equilibrium solubility of a liquid penetrant. For liquids, the driving force for sorption is more subtle than for gases, with more emphasis on molecular interactions rather than external pressure. For this reason, terms relating to the molecular interactions of a liquid molecule filling a hole were developed, including the effects of elastic constraint for small holes. Consideration of thermal fluctuations show that configurational entropy provides much of the driving force for sorption. Some comparisons with experimental data show a reasonable agreement, and one which is far better than the Flory-Huggins theory.     

Prototype of Low Thermal Expansion Materials: Fabrication of Mesoporous Silica/Polymer Composites with Densely Filled Polymer inside Mesopore Space

A prototype of novel low thermal expansion materials using mesoporous silica particles is demonstrated. Mesoporous silica/polymer composites with densely filled polymer inside the mesopore space are fabricated by mechanically mixing both organically modified mesoporous silica and epoxy polymer. The mesopores are easily penetrated by polymers as a result of the capillary force during the mechanical composite processing. Furthermore, we propose a new model of polymer mobility restriction using mesoporous silica with a large pore space. The robust inorganic frameworks covering the polymer effectively restrict the polymer mobility against thermal energy. As a result, the degree of total thermal expansion of the composites is drastically decreased. From the mass‐normalized thermal mechanical analysis (TMA) charts of various composites with different amounts of mesoporous silica particles, it is observed that the coefficient of thermal expansion (CTE) values gradually increase with an increase of the polymer amount outside the mesopores. It is proven that the CTE values in the range over the glass‐transition temperatures (T_g) are perfectly proportional to the outside polymer amounts. Importantly, the Y‐intercept of the relation equation obtained by a least‐square method is the CTE value and is almost zero. This means that thermal expansion does not occur if no polymers are outside the mesopores. Through such a quantative discussion, we clarify that only the outside polymer affects the thermal expansion of the composites, that is, the embedded polymers inside the mesopores do not expand at all during the thermal treatment.     

Vitrification of polymer solutions as a function of solvent quality, analyzed via vapor pressures

Vapor pressures (headspace sampling in combination with gas chromatography) and glass transition temperatures [differential scanning calorimetry (DSC)] have been measured for solutions of polystyrene (PS) in either toluene (TL) (10-70 degrees C) or cyclohexane (CH) (32-60 degrees C) from moderately concentrated solutions up to the pure polymer. As long as the mixtures are liquid, the vapor pressure of TL (good solvent) is considerably lower than that of CH (theta solvent) under other identical conditions. These differences vanish upon the vitrification of the solutions. For TL the isothermal liquid-solid transition induced by an increase of polymer concentration takes place within a finite composition interval at constant vapor pressure; with CH this phenomenon is either absent or too insignificant to be detected. For PS solutions in TL the DSC traces look as usual, whereas these curves may become bimodal for solutions in CH. The implications of the vitrification of the polymer solutions for the determination of Flory-Huggins interaction parameters from vapor pressure data are discussed. A comparison of the results for TL/PS with recently published data on the same system demonstrates that the experimental method employed for the determination of vapor pressures plays an important role at high polymer concentrations and low temperatures.     

Visualization of structural rearrangements during annealing of solvent-crazed poly(ethylene terephthalate)

A direct microscopic procedure is used for studying structural rearrangements during the annealing of PET samples after solvent crazing. Even at room temperature, solvent-crazed PET samples experience shrinkage which is provided by processes taking place in crazes. This shrinkage is observed at temperatures up to the glass transition temperature of PET and proceeds via drawing together of crack walls. Once the glass transition temperature is attained during annealing, the spontaneous self-elongation of the polymer sample occurs. The mechanism of this phenomenon is proposed. The low-temperature shrinkage of the polymer sample is related to the entropy contraction of highly dispersed material in crazes that has a lower glass transition temperature than that of the bulk polymer. This shrinkage cannot be complete, owing to crystallization of the oriented polymer in the volume of the crazes. As a result of crystallization, the oriented and crystallized polymer in the crazes coexists with the regions of the unoriented initial PET. As the annealing temperature approaches the glass transition temperature of the bulk PET, its strain-induced crystallization takes place. As a result, the regions of the unoriented polymer between crazes are elongated along the direction of tensile drawing and the sample experiences contraction in the normal direction.     

Sorption and diffusion of water in poly(vinylpyrrolidone)

The effect of molecular mass, thermal prehistory, physical state, and three-dimensional chemical crosslinked structure of a polymer on dissolution and diffusion in the PVP-water system has been studied. The kinetic dependences of sorption that correspond to the Fickian or pseudonormal type have been measured. In a certain concentration range, sorption is accompanied by transition of the system to the rubbery state. In the glassy state, the negative concentration dependence of the diffusion coefficient related to the nonequilibrium state of the polymer sorbent is observed. Sorption isotherms are described by S-shaped curves. It has been shown that the thermal prehistory of the polymer sorbent has the most pronounced effect on its sorption behavior. The effect of molecular mass is insignificant, while three-dimensional chemical crosslinks in PVP manifest themselves only in the region of the rubbery state. In accordance with the double sorption model, the experimental isotherms are represented as the superposition of two isotherms described by the Langmuir and Flory-Huggins equations. For the glassy state of the polymer sorbent, the degree of the nonequilibrium state has been estimated. With due regard for the excess free volume, the detailed thermodynamic analysis of isotherms has been performed; namely, the pair interaction parameters and the free energy of mixing have been calculated. The state of water in the polymer has been examined within the framework of hydrate contributions and clusterization theory.     

The nature of dielectric losses in H-Film

Based upon the high-temperature solid-state dielectric behavior of H-Film and supporting evidence from infrared spectroscopy, heat capacity, thermal expansion, and dynamic mechanical measurements, it is concluded that the actual chemical structure of H-Film is far more complex than previously expected. The very unusual thermal properties of this polymer, such as lack of a glass transition temperature, its extreme insolubility, and high dielectric losses above 250°C with only weakly corresponding mechanical losses, suggest that in addition to the pyromellitimide repeat unit, a significant part of the structure consists of amide groups, probably in the form of crosslinks.     

PVT properties of dodecane/polystyrene systems

The PVT properties of crosslinked polystyrene samples containing various amounts of dodecane were measured. The Tait equation was used to describe the PVT behavior of each system in both the glassy and rubbery regions. The glass transition temperature was determined from the abrupt change of the thermal expansion coefficient. Increase in the dodecane content in the samples resulted in a significant decrease of the difference between the expansion coefficients in the glassy and rubbery regions. Addition of dodecane lowered the glass transition temperature linearly. However, the dependence of the glass transition temperature on pressure was not affected by the presence of dodecane in the polymer samples. Above the glass transition temperature, the volume of the swollen polymer, V_m, could be determined by simple addition of the volumes of the pure components at the appropriate temperature and pressure; the volume change of mixing, δV_m, was independent of temperature and pressure. Below the glass transition temperature, volume additivity of the two components was also applicable after appropriate adjustment of the glass transition temperature of the polymer to that of the dodecane/polymer samples. © 1994 John Wiley & Sons, Inc.     

Inhomogeneous polymer solutions: Theory of the interfacial free energy and the composition profile near the consolute point

The Flory–Huggins formulation of the combinatorial entropy, supplemented with residual free energy, is applied locally to obtain the interfacial free energy and the concentration profile of polymer in the interface between two demixed polymer solution phases. Two choices were investigated for the residual free energy: a “regular solution” formulation and an empirical formulation of Koningsveld for polystyrene in cyclohexane. Asymptotic, analytical solutions of the equations near the critical solution point and solutions obtained by numerical calculations are given as a function of temperature for several molecular weights. At temperatures farther below the critical temperature the equations have no solutions. The reason for this is not entirely clear. The local formulation of the free energy used here is an improved version of a previous one, which gave wrong results for asymmetric systems (polymer in a low molecular weight solvent). This newer version is consistent with our theory of critical opalescence and gives a relation between the interface “thickness” and the correlation range of the concentration fluctuations. The calculated correlation ranges were in good accord with those found experimentally by Debye, Chu, and Woerman. That the newer version of our equations for an interface gives no acceptable solutions at lower temperatures could be caused by a “collapse” of a diffuse to a sharp interface as suggested by Nose.     

Steric and bridging interactions between two plates induced by grafted polyelectrolytes

If colloidal particles are grafted with a polymer, then the grafted chains can provide steric repulsion between them. If some of the grafted polymer chains are also adsorbed to a second particle, then a bridging force is generated as well. For uncharged plates and polymer, the following contributions to the free energy of the system have been taken into account in the calculation of the interaction force: (i) the Flory-Huggins expression for the mixing free energy of the grafted chains with the liquid; (ii) the entropy loss due to the connectivity of the polymeric segments; (iii) the van der Waals interactions between the segments and the plates; and (iv) the free energy of adsorption of the polymer segments of the grafted chains on the other plate. For charged plates, the electrostatic free energy as well as the free energy of the electrolyte are included in the total free energy of the system. By minimizing the free energy with respect to the segment concentration and, when it is the case, with respect to the electrical potential, equations for the segment number density distribution and for the electrical potential are obtained, on the basis of which the interactions between two plates grafted with polymer chains that can be also adsorbed on the other plate were calculated. The interaction thus obtained includes steric and bridging forces.     

Environmental swap energy and role of configurational entropy in transfer of small molecules from water into alkanes

We studied the effect of segmented solvent molecules on the free energy of transfer of small molecules from water into alkanes (hexane, heptane, octane, decane, dodecane, tetradecane, and hexadecane). For these alkanes we measured partition coefficients of benzene, 3-methylindole (3MI), 2,3,4,6-tetrachlorophenol (TeCP), and 2,4,6-tribromophenol (TriBP) at 3, 11, 20, 33 [corrected], and 47 degrees C. For 3MI, TeCP, and TriBP the dependence of free energy of transfer on length of alkane chains was found to be very different from that for benzene. In contrast to benzene, the energy of transfer for 3MI, TeCP, and TriBP was independent of the number of carbons in alkanes. To interpret data, we used the classic Flory-Huggins (FH) theory of concentrated polymer solutions for the alkane phase. For benzene, the measured dependence of energy of transfer on the number of carbons in alkanes agreed well with predictions based on FH model in which the size of alkane segments was obtained from the ratio of molar volumes of alkanes and the solute. We show that for benzene, the energy of transfer can be divided into two components, one called environmental swap energy (ESE), and one representing the contribution of configurational entropy of alkane chains. For 3MI, TeCP, and TriBP the contribution of configurational entropy was not measurable even though the magnitude of the effect predicted from the FH model for short chain alkanes was as much as 20 times greater than experimental uncertainties. From the temperature dependence of ESE we obtained enthalpy and entropy of transfer for benzene, 3MI, TeCP, and TriBP. Experimental results are discussed in terms of a thermodynamic cycle considering creation of cavity, insertion of solute, and activation of solute-medium attractive interactions. Our results suggest that correcting experimental free energy of transfer by Flory-Huggins configurational entropy term is not generally appropriate and cannot be applied indiscriminately.     

Determination of thermal expansion coefficient of a monofilament polyamide fiber using digital image correlation

Coiled polymer actuators are a type of artificial muscles that are a promising development in the field of smart materials. The coefficient of thermal expansion of monofilament polyamide fibers is a crucial parameter for understanding the actuation of coiled fibers. The main purpose of this work is to develop a new methodology for estimating the coefficient of thermal expansion and the transition temperature of monofilament polymer fibers. In the experimental procedure, axial deformations of monofilament polyamide fiber samples were induced by temperature variations using a controlled thermal system. These deformations were determined from images of polyamide samples using the digital image correlation method. Two different approaches based on distinct temperature conditions were conducted. An alternative model with three parameters, including the coefficient of thermal expansion, was introduced to describe the thermal-mechanical behavior of monofilament polyamide fibers. Moreover, polyamide samples were also characterized using four conventional methodologies. Results indicated that the coefficient of thermal expansion changed of a modest negative value to a large negative value and this transition occurred around the glass transition temperature of the polyamide. The thermal expansion curves demonstrate good repeatability and all estimated parameters were in accordance with literature, indicating that the proposed approach can be suitable for the proposed study. This investigation may help in understanding of the intrinsic thermal-mechanical behavior of polymeric monofilaments employed as actuators.     

Thermodynamics and structure of the ordered amorphous phase in polymers

Experimental data are analysed to show that the activation enthalpy and the structural entropy appear to follow empirical relations for structural relaxation of polymeric systems. The relations indicate that melting temperature, glass transition temperature, relaxation time, coefficient of thermal expansion of free volume are interrelated in polymers. The parameters of structural relaxations, measured by mechanical and dielectric spectroscopies, are reviewed for polyethylene, poly(4-methyl-1-pentene) and a liquid-crystalline polynorbornene derivative. The thermodynamic parameters obtained from calorimetric measurements, are reported for zero heating rate extrapolation and they are used in the empirical relation, which combines the Arrhenius and the Vogel-Fulcher formulae.     

Theoretical calculations of glass transition temperatures of plasticized polymers

Based on free volume, an equation has been derived enabling calculation of glass transition temperature (T_g) of plasticized polymers, knowing the plasticizer content, the thermal expansion coefficients and values of T_g of the polymer and plasticizer. Determined and calculated T_g are in satisfactory agreement up to the plasticizer concentration which dissolves the polymer.