Free volumes

At a certain polymer concentration, however, the viscosity of the mixture becomes so great that the polymer segments can no longer move freely. Because of this freezing-in process the specific volume of the amorphous polymer v is larger than the specific volume of the liquid polymer v° would be at the same temperature. Conversely, the density of the liquid polymer is higher than the density of the solid The solid polymer has vacant sites or what is called free volume. These vacant sites should be seen as having diameters of the order of those of atoms. The free-volume fraction/w, p is calculated as 

The same free volume also occurs in the Williams-Landel-Ferry dynamic glass-transition-temperature equation (see Section 10.5.2). The free-volume fraction is independent of the polymer type. It has a value of about 2.5% (Table 5-7). 

In addition to the WLF free volume, a series of other free volumes can be defined and discussed. The vacant volume can be obtained from the specific volume of the amorphous polymer v m measured at the temperature Tand the specific volume v dw calculated from the van der Waals radii. The vacant-volume fraction /vac is 

As is true for simple liquids, the vacant-volume fraction here is 

Finally, a fluctuation volume/nue can be determined from sound-velocity measurements, and this describes the motion of the center of gravity of a molecule as a result of thermal motion (see Table 5-7). 

The principal physical structural parameters that control the modes of deformation and failure and mechanical response of epoxies are (1) macroscopic inhomogenieties such as microvoids or concentrations of unreacted monomer, (2) the glassy-state free volume and (3) the crosslinked network structure characteristics. 

When you heat a crystalline solid the kinetic energy and hence the amplitude of the oscillations of the molecules increases. The material expands. As it does so the attractive force between the molecules decreases (see Chapter 8, page 208). At the melting temperature this force of attraction is not enough to keep the molecules in place and the material melts. Now the motion becomes a complex coupling of vibrational oscillations and translational movement, as holes open up as a result of random displacements of neighbors. Clearly, there has to be enough empty space in the material as a whole for this to occur. 

let us imagine that at very low temperatures in a material that for whatever reason has not or cannot crystallize, the molecules are more or less randomly close packed. The total volume of the system is then that of the hard core of the molecules plus the unoccupied volume between them. As the tem- 

FIGURE10-54 Schematic plots of temperature versus volume (top) and specific heat (bottom) for a glassy polymer. 

FIGURE 10-55 Schematic diagram showing the unoccupied volume and oscillations around a mean position in ordered and random close packing of spheres. 

FIGURE 10-56 Schematic diagram showing the free volume as temperature is increased. 

Using the above ideas, Doolittle (1951) proposed an expression for the viscosity Tj oi Sl liquid given by 

Cohen and Turnbull (1959) clarified the physical significance of the Doolittle equation by demonstrating that the probability P v ) of finding a hole of size v or larger in a field of free volume can be expressed as 

Since the volume per atom v is temperature-dependent and will vary with fluctuations, the free volume will also fluctuate and in a liquid will wander around, not necessarily being associated with specific atoms. In the Cohen and Grest model the probability distribution of v, P(v), and the local free-energy function, /(v), sketched in Figs. 1.7(a) and (b) permit the determination of a specific expression for the average free volume vf and its temperature dependence given by 

The appearance of two phases in IPNs during curing and the existence of the interphase show that the packing density of IPNs is looser as compared with 

For polymer blends it was shown [121] that in the interphase region, the fraction of free volume increases compared with coexisting phases. The higher is the fraction of the interphase, the larger is the contribution of the free volume in the interphase to the total free volume of the system. 

The free volume in IPNs has not been investigated extensively. The first attempt to estimate this value was done [65] using the data on vapor sorption based on the theory developed by Fujita [122]. The sequential IPNs based on styrene-DVB copolymer (network I) and cross-linked PU (network II) were studied. The PU content was up to 0.24 by mass. From the data on the sorption kinetics the interdiffusion coefficients Dy, solvent self-diffusion coefficient D, and relative diffusion coefficient D were found. The value of D was calculated from the relation Dy = I (1 - Vg), where Vg is the volume fraction of a solvent in the IPN. According to Fujita, the change in self-diffusion coefficient in isothermal conditions is described by the equation  

The dependence of g[log(D /D(0, T)] - 1 on og is Hnear, which indicates the applicabihty of the free volume model to the system under investigation. From these data the values of/p(T) were found from known values of/. The data for various ratios of networks were compared with those calculated according to the additivity rule (Fig. 16) [65]. 

As we see, values of A/ change in a nonmonotonous way depending on the network ratio, the fraction of free volume being higher in IPNs at all ratios but W2/W1 = 0.03-0.07. The compHcated dependence of the free volume on composition may be attributed to the formation of the intermediate region of the interphase between two phases in the IPN. A looser interphase contributes to the increasing free volume of the whole system. 

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Otlier expressions for tire diffusion coefficient are based on tire concept of free volume [57], i.e. tire amount of volume in tire sample tliat is not occupied by tire polymer molecules. Computer simulations have also been used to quantify tire mobility of small molecules in polymers [58]. In a first approach, tire partition functions of tire ground... 

There are a number of important concepts which emerge in our discussion of viscosity. Most of these will come up again in subsequent chapters as we discuss other mechanical states of polymers. The important concepts include free volume, relaxation time, spectrum of relaxation times, entanglement, the friction factor, and reptation. Special attention should be paid to these terms as they are introduced. 

If we were required to pack beads in a beaker, we know from experience that by jostling the container we could achieve some compaction or decrease in free volume. In fact, we can picture the flow of a huge array of beads through a pipe by considering the beaker as a volume element in that pipe. By vibration, the beads are jostled downward that is, the holes work their way to the top. 

The concept of free volume is taken up again in Chap. 4. 

For the evaporation process we mentioned above, the thermodynamic probability of the gas phase is given by the number of places a molecule can occupy in the vapor. This, in turn, is proportional to the volume of the gas (subscript g) 12- oc V In the last chapter we discussed the free volume in a liquid. The total free volume in a liquid is a measure of places for molecules to occupy in the liquid. The thermodynamic probability of a liquid (subscript 1) is thus V, oc V, frgg. Based on these ideas, the entropy of the evaporation process can be written as... 

Neither the volume occupied by a mole of gas at the boiling point nor the free volume of a liquid vary too widely from substance to substance. Taking the former to be about 30,000 ml and the latter to be about 3 ml gives... 

There are two ways in which the volume occupied by a sample can influence the Gibbs free energy of the system. One of these involves the average distance of separation between the molecules and therefore influences G through the energetics of molecular interactions. The second volume effect on G arises from the contribution of free-volume considerations. In Chap. 2 we described the molecular texture of the liquid state in terms of a model which allowed for vacancies or holes. The number and size of the holes influence G through entropy considerations. Each of these volume effects varies differently with changing temperature and each behaves differently on opposite sides of Tg. We shall call free volume that volume which makes the second type of contribution to G. 

On the basis of these ideas, the observed volume of a sample can be written as the sum of the volume occupied by the molecules (subscript 0) and the free volume (subscript f). Acknowledging that each of these is a function of temperature, we write... 

Figure 4.15 Geometrical representation of the temperature variation of the actual volume (solid line) and the occupied volume (broken line). The shaded difference indicates the free volume which decreases to a critical value at T .

In discussing Fig. 4.1 we noted that the apparent location of Tg is dependent on the time allowed for the specific volume measurements. Volume contractions occur for a long time below Tg The lower the temperature, the longer it takes to reach an equilibrium volume. It is the equilibrium volume which should be used in the representation summarized by Fig. 4.15. In actual practice, what is often done is to allow a convenient and standardized time between changing the temperature and reading the volume. Instead of directly tackling the rate of collapse of free volume, we shall approach this subject empirically, using a property which we have previously described in terms of free volume, namely, viscosity. 

Next we assume that the state designated by the subscript 2 in Eq. (4.61) corresponds to Tg, we designate the fraction free volume at Tg by fg. Likewise,... 

Things appear to have taken a strange turn We started out discussing the free volume and have ended up with an equation which contains no volume at all More specifically, we set out to examine the rate at which the free volume collapses at Tg. A final development of Eq. (4.63) will produce the desired result. 

These results make sense physically what we must remember is that they follow from an analysis of the vanishing free volume. 

Although it is easy to discuss free volume, it is necessary to come up with a numerical value for this quantity in order to test these concepts. There is... 

It is possible to derive an expression equivalent to Eq. (4.67) starting from entropy rather than free volume concepts. We have emphasized the latter approach, since it is easier to visualize and hence to use for qualitative predictions about Tg. 

The collapse of the free volume below a critical size for molecular motion is... 

Hydrocarbons without bulky side groups are held together by London forces, the weakest of intermolecular attractions. This means that the free volume tends to be large for these compounds, so a relatively large amount of cooUng is necessary before the free volume collapses. Thus Tg is low for these compounds. 

The effect of a bulky substituent like a phenyl group on the hydrocarbon chain apparently decreases chain flexibiUty sufficiently to allow more intimate alignment between molecules, less free volume, and therefore a high value for Tg. 

In the methacrylate homologous series, the effect of side-chain bulkiness is just the opposite. In this case, however, the pendant groups are flexible and offer less of an obstacle to free rotation than the phenyl group in polystyrene. As chain bulk increases, molecules are wedged apart by these substituents, free volume increases, and Tg decreases. 

An extra amount of free volume is associated with chain ends, which are capable of wagging in a way that is not possible in the middle of a chain. Accordingly, as molecular weight decreases, Vj- increases, which, in turn, decreases Tg. The following expression has been found to describe this molecular weight dependence ... 

The effect of branching is to increase the number of chain ends and, therefore, free volume, which decreases Tg. Conversely, crosslinking ties together separate molecules, decreases the number of loose ends, and raises Tg. Copolymers show different effects on T, depending on the microstructure... 

In this section we resume our examination of the equivalency of time and temperature in the determination of the mechanical properties of polymers. In the last chapter we had several occasions to mention this equivalency, but never developed it in detail. In examining this, we shall not only acquire some practical knowledge for the collection and representation of experimental data, but also shall gain additional insight into the free-volume aspect of the glass transition. 

The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

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