The Free-Volume Theory

The free-volume concept has been mentioned in previous sections, but it is instructive now to consider this idea more closely and to draw together the various points alluded to earlier. Free volume, Vf, is defined as the imoccupied space in a sample, arising from the inefficient packing of disordered chains in the amorphous regions of a polymer sample. The presence of these empty spaces can be inferred from the fact that when a polystyrene glass is dissolved in benzene, there is a contraction in the total volume. This and similar other observations indicate that the polymer can occupy less volume when surrounded by benzene molecules, and that there must have been unused space in the glassy matrix to allow this increase in packing effidraicy to occur. 

On that basis, the observed specific volume of a sample, V, will be composed of the volume acmally occupied by the polymer molecules, Vq, and the free volume in the system, i.e.. 

Each term will, of course, be tranpraature depmdmt The free volume is a measure of the space available for the polymer to undeigo rotation and translation, and when the polymer is in the liquid or rubberlike state the amount of free volume will increase with temperature as the molecular motion increases. If the temperature is decreased, this free volume will contract and eventually reach a critical value where there is insufficient free space to allow large-scale segmental motion to take place. The temperature at which this critical value is reached is the glass transition temperature. Below Tg, the free volume will remain essentially constant as the temperature decreases further because the chains have now been immobilized and frozen in position. In contrast, the occupied volume will alter because of the changing amplimde of thermal vibrations in the chains and, to the first approximation, will be a linear function of temperature, irrespective of whether the polymer is in the liquid or glassy state. 

The precise definition of the average amount of free volume present in a totally amorphous polymer remains unclear, but it must also depend to some extent on the thermal history of the sample. A number of suggestions have been made. 

Simha and Boyer (S-B) observed that a general empirical relationship exists between the and the difference in expansion coefficients of the liquid and glass states. From the examination of a wide range of polymers, they concluded that 

The free volume theory originated some years later than the lubricity and the gel theories, when the evolution of different properties of polymers as a function of temperature, specific volume, thermal expansion coefficients, or viscosity was attempted to be explained.The relationships between these properties and some variables corresponding to polymer stracture, such as molecular weight or terminal groups content, the presence of another monomer and, of course, the presence of plasticizers, was also explained. For plasticized polymers the theory attempted to explain the diminution of the glass transition temperature with the plasticizer content. This theory is a contribution of different authors, but it was postulated by Fox and Floiy. The theory is still being used to explain some properties of plasticized polymers, i.e., viscoelastic properties.  

Glass transition temperatures are characterized by a change from a hard, nonciystal-line, glass-like material to a rubbery sohd. The viscosity at glass transition temperature for all polymers was found to be around lO Pa.s, independent of the chemical stracture of the polymer. In this way friction between molecules (or viscosity) was related to the volume between them, and so to the glass transition temperature, Tg. Consequently, it was accepted that the glass transitions happen at that physical state where all materials exhibit the same fractional free volume . 

From these findings, the free volume theory shows that between atoms and molecules there is nothing but free volume. The free volume was first defined by Fox and Flory at temperatures above the transition temperature as the specific volume above the transition temperature minus the solid specific volume extrapolated to the same temperature above the transition temperature. This early definition implies that the free volume at 

The free volume defined in this way remains somehow vague because of the difficulties in measuring the specific volmne at absolnte zero temperature. Various free volume fractions are discussed in the literature. For example Kanig divided the free volume into two fractions a continuous part that results from oscillations and increases slightly as the temperature is raised, and a discontinuous part, called holes, which increases greatly with the temperature. 

Some well established models that correlate free volume fraction to the glass transition temperature for polymers support the free volume concept. The fractional free volume of a liquid, f, is defined as f=Vi/vq, where Vj is the free volume and Vg is the occupied volume. The empirical Boyer-Sumha mle relates the ftee volume fraction, f to the cubic 

There are three main groups of theories of the glass transition [28-30] (a) the free volume theory, (b) the kinetic theory, and (c) the thermodynamic theory. Although these three theories may at first appear to be different, they really examine three aspects of the same phenomenon and can be unified successfully, though only in a qualitative way. 

According to the hole theory of liquids, first developed by Eyring [31], molecular motion in liquids depends on the presence of holes or voids, i.e., places where there are vacancies. When a molecule moves into a hole, a 

Simha and Boyer [34] thereafter postulated that the free volume at 

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The Free Volume Theory. This extends the lubricity and gel theories and also allows a quantitative assessment of the plasticization... 

In the literature there is only one serious attempt to develop a detailed mechanistic model of free radical polymerization at high conversions (l. > ) This model after Cardenas and 0 Driscoll is discussed in some detail pointing out its important limitations. The present authors then describe the development of a semi-empirical model based on the free volume theory and show that this model adequately accounts for chain entanglements and glassy-state transition in bulk and solution polymerization of methyl methacrylate over wide ranges of temperature and solvent concentration. 

A useful model should account for a reduction of kt and kp with increase in polymer molecular weight and concentration and decrease in solvent concentration at polymerization temperatures both below and above the Tg of the polymer produced. For a mechanistic model this would involve many complex steps and a large number of adjustable parameters. It appears that the only realistic solution is to develop a semi-empirical model. In this context the free-volume theory appears to be a good starting point. 

Vrentas, JS Duda, JL, Diffusion in Polymer-Solvent Systems. I. Reexamination of the Free-Volume Theory, Journal of Polymer Science Polymer Physics Edition 15, 403, 1977. Vrentas, JS Duda, JL, Diffusion in Polymer-Solvent Systems. II. A Predictive Theory for the Dependence of Diffusion Coefficients on Temperature, Concentration, and Molecnlar Weight, Journal of Polymer Science Polymer Physics Edition 15, 417, 1977. 

In addition to temperature and concentration, diffusion in polymers can be influenced by the penetrant size, polymer molecular weight, and polymer morphology factors such as crystallinity and cross-linking density. These factors render the prediction of the penetrant diffusion coefficient a rather complex task. However, in simpler systems such as non-cross-linked amorphous polymers, theories have been developed to predict the mutual diffusion coefficient with various degrees of success [12-19], Among these, the most notable are the free volume theories [12,17], In the following subsection, these free volume based theories are introduced to illustrate the principles involved. 

JS Vrentas, JL Duda. Diffusion in polymer-solvent systems. I. Reexamination of the free volume theory. J Polym Sci, Polym Phys Ed 15 403-416, 1977. 

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

Figure 14 The free volume theory of Yasuda and coworkers holds for the diffusion of acetaminophen in swollen 10 X 4 poly(lV-isopropyl acrylamide) gel. (Adapted from Ref. 176.)...

Yasuda s free volume theory [57] has been proposed to explain the mechanism of permeation of solutes through hydrated homogeneous polymer membranes. The free volume theory relates the permeability coefficients in water-swollen homogeneous membranes to the degree of hydration and molecular size of the permeant by the following mathematical expression ... 

Equation 39 has the structure proposed for the rate constants on the basis of the free volume theory (1,5,9). From this, it would be expected that the models developed from the free volume theory would be very successful in predicting both, the rate behaviour and the molecular properties at high conversions. The reason why these models have been only partially successful stems from the... 

The rate parameters follow similar conversion trajectories. Therefore, the rate constants and the initiator efficiency can be modelled with the same equation. An equation of the form of equation 39 is suggested. The theoretical Justification for the form of equation 39 stems from the free volume theory. 

It would be an advantage to have a detailed understanding of the glass transition in order to get an idea of the structural and dynamic features that are important for photophysical deactivation pathways or solid-state photochemical reactions in molecular glasses. Unfortunately, the formation of a glass is one of the least understood problems in solid-state science. At least three different theories have been developed for a description of the glass transition that we can sketch only briefly in this context the free volume theory, a thermodynamic approach, and the mode coupling theory. 

In the free-volume theory of hquids, the molar Helmholtz function is dehned by the equation... 

In the free volume theory, translational diffusion of a lipid molecule in the bilayer occurs only when a free volume larger than a certain critical size appears in the vicinity of the lipid molecule. The free volume theory implies that the smaller the overall volume, the lower the probability for a molecule to associate with a free volume of a critical size. The molecules diffuse slower if the probability for a molecule to associate with a free volume of critical size is small. With increasing pressure, the overall volume decreases and the lateral diffusion is thus reduced. The activation volume for diffusion in the LC phase was calculated using the expression ... 

Detonation, Free Volume Theory of the Liquid State Developed by Eyring et al and by Lennard-Jones-Devonshire. The free volume theory of the liquid state developed by Eyring Hirshfelder (Ref 1) and by Lennard-Jones Devonshire (Ref 2) has provided a useful approximate description of the thermodynamic props of liquids in terms of intermolecular forces... 

Detonation, Free Volume Theory of Multi-component Fluid Mixtures. The free volume theory of the liq state is extended to multi-component fluid mixts by using the method of moments in the treatment of the order-disorder problem. The results of this extension are given in the article by Z.W. 

Polyvinylchloride was the host polymer in a study of the diffusion of dimethyl-phthalate, dibutylphthalate, and dioctylphthalate, performed by Maklakov, Smechko, and Maklakov 60) between room temperature and 110 °C. Azancheev and Maklakov 61) extended this work to include polystyrene as host, and to dependences of diffusion on concentration. They concluded that the macromolecules did constrain and trap the phthalate molecules at high polymer concentration, but without inhibiting the mobility of these diluents at lower polymer concentrations, e.g., in the gel. They used a version of the free volume theory to give a semi-quantitative explanation of the temperature and molecular size dependence of phthalate diffusion. 

It was found that Afi Tg and Aa Tg are not constant and therefore the SB equation has limited applicability. Hie results indicate an increase in Aa Te with increasing Tg. Therefore it is inadmissible to use the product A a Tg as a universal value in any theoretical discussion of the glass-transition phenomenon. At the same time, this conclusion in no way excludes the free-volume theory and the role of free-volume in the transition from the glassy to the liquid or rubberlike state. 

In reality, the data on isothermal contraction for many polymers6 treated according to the free-volume theory show that quantitatively the kinetics of the process does not correspond to the simplified model of a polymer with one average relaxation time. It is therefore necessary to consider the relaxation spectra and relaxation time distribution. Kastner72 made an attempt to link this distribution with the distribution of free-volume. Covacs6 concluded in this connection that, when considering the macroscopic properties of polymers (complex moduli, volume, etc.), the free-volume concept has to be coordinated with changes in molecular mobility and the different types of molecular motion. These processes include the broad distribution of the retardation times, which may be associated with the local distribution of the holes. 

However, in many real systems both in block copolymers and in polymer blends91 the components may mutually influence each other due to interphase interaction90,92. Such interaction may cause the system behavior to derivate from that predicted within the framework of the free-volume theory for a two-phase system. 

We believe the difficulty is that the free-volume theory as applied to the glass-transition does not take account of the essential role of intra- and intermolecular interaction in the system and the flexibility of the polymer chains, all of which factors play an important role in the glass-transition phenomena. 

Fig. 16. The diffusion coefficient of acetaminophen in 10 x 4 PNIPAAm gels falls as the swelling degree (Q) of the gel decreases due to increasing temperature. Below the transition temperature of the gel, the linear relationship between log D and (Q — 1) 1 predicted by the free volume theory of Yasuda et al. [10] is observed. Above the transition temperature, the theory underestimates D by 35 times. Reprinted from the Journal of Controlled Release (1992) 18 1, by permission of the publishers, Elsevier Science Publishers BV [70]...

A modified version of the free-volume theory is used to calculate the viscoelastic scaling factor or the Newtonian viscosity reduction where the fractional free volumes of pure polymer and polymer-SCF mixtures are determined from thermodynamic data and equation-of-state models. The significance of the combined EOS and free-volume theory is that the viscoelastic scaling factor can be predicted accurately without requiring any mixture rheological data. 

A simplified version of the free volume theory, considering that Tg is an iso-free volume point and that the free volume absorbed by one crosslink is independent of the crosslink density, leads to (Fox et al., 1955) ... 

Using the classical hypotheses of the free volume theory, the glass transition temperature for a polymer (p) and solvent (s) solution, with a volume fraction of solvent, v, is given by (Kelley and Bueche, 1960) ... 

According to the free volume theory as the temperature increase the frequency and amplitude of chain jumping (i.e. thermal agitation) increase and the resulting free volumes become larger. Increase in temperature also decreases the interaction between acetic acid and water molecules so it will be easy for both acetic acid and water molecules to... 

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