Solutions ideal polymer solution

Figure 9.6 (a) Molar entropy of mixing of ideal polymer solutions for r = 10, 100 and 1000 plotted as a function of the mole fraction of polymer compared with the entropy of mixing of two atoms of similar size, r = 1. (b) Activity of the two components for the same conditions. 

V, is the molar volume of polymer or solvent, as appropriate, and the concentration is in mass per unit volume. It can be seen from Equation (2.42) that the interaction term changes with the square of the polymer concentration but more importantly for our discussion is the implications of the value of x- When x = 0.5 we are left with the van t Hoff expression which describes the osmotic pressure of an ideal polymer solution. A sol vent/temperature condition that yields this result is known as the 0-condition. For example, the 0-temperature for poly(styrene) in cyclohexane is 311.5 K. At this temperature, the poly(styrene) molecule is at its closest to a random coil configuration because its conformation is unperturbed by specific solvent effects. If x is greater than 0.5 we have a poor solvent for our polymer and the coil will collapse. At x values less than 0.5 we have the polymer in a good solvent and the conformation will be expanded in order to pack as many solvent molecules around each chain segment as possible. A 0-condition is often used when determining the molecular weight of a polymer by measurement of the concentration dependence of viscosity, for example, but solution polymers are invariably used in better than 0-conditions. 

Figure 1. Curve 1 could represent an ideal polymer solution containing A and B but undergoing no association the nonideal counterpart of this is shown in curve 2. An ideal mixed association between A and B, such as described by Equation 1 might be described by curve 3, whereas, curve 4 could represent a nonideal, mixed association.

For non-ideal polymer solutions where there are interactions between the polymer molecules, Einstein and Debye showed independently that if the solute is uniformly distributed throughout the solution, no light is scattered by the solution because light scattered by one particle will interfere destructively with light scattered by the neighbouring particle. Random Brownian motion causes fluctuations in concentration, the extent of fluctuations is inversely proportional to the osmotic pressure developed by the concentration difference. It is found that... 

Equation (P3.6.3) derived above for ideal solutions is invalid for real solutions and polymer solutions are rarely ideal even at the highest dilution that can be used in practice. It is, however, useful to retain the form of the ideal equation and express the deviation of real solutions in terms of empirical parameters. Thus, for a real solution Eq. (P3.6.3) is expressed in a parallel form as... 

Since N 1, the threshold density c is fairly small (just as it was for an ideal polymer solution see (6.14)). Thus the semi-dilute regime is appropriate for a wide range of concentrations. 

The discussion of real (i.e., non-ideal) polymer solutions will be deferred till section 6 d. We shall first consider the solubility" of macromolecules from the point of view of ideal solutions. This aspect has played an important part in the physical chemistry of high polymers especially in the theory of fractional precipitation. 

Here R is the gas constant, T is the temperature, A2 is the second virial coefficient and A3 is the third virial coefficient. This equation should be compared to the virial expansion for a real gas. The higher order terms (second and third on the right-hand side of the equation) account for in-termolecular interactions. In an ideal polymer solution, virial coefficients of order two and higher are zero. Such a solution should be compared to an ideal gas, in which there are no intermolecular interactions. The... 

Quasi-elastic scattering by dilute, ideal, polymer solutions II effects of hydrodynamic interactions (with E. Dubois-Violette). Phvs. (N.Y.) 2, n 4, 181-198 (1967). 

From oscillation rheometry, separating viscous and elastic effects, a practical fluid relaxation time can be derived, which is indicative of the critical flow rate of non-ideal polymer solutions. 

Between these two extremes, the polymer and solvent can interact just enough so that the segments of the polymer necklace will be randomly distributed. This limit of a random coil of polymer is conventionally chosen as the ideal polymer solution, and a solvent showing these characteristics is called a 0 solvent. Under these conditions, the diffusion of the polymer can be calculated as a correction to the Stokes-Einstein equation ... 

As shown in Section 8.3 of Chapter 8, the osmotic pressure of an ideal polymer solution having solvent mole fiaction xj is —(/ 7 /n,)lnxj, where n, is the molar volume of the solvent. Consequently, the osmotic pressure difference between two solutions having solvent volume fiactions (p and (p is given as follows ... 

Evaluate ASj for ideal solutions and for athermal solutions of polymers having n values of 50, 100, and 500 by solving Eqs. (8.28) and (8.38) at regular intervals of mole fraction. Compare these calculated quantities by preparing a suitable plot of the results. 

In Chap. 1 we referred to these as 0 conditions, and we shall examine the significance of this term presently. Note that 0 conditions for a polymer solution are analogous to the Boyle temperature of a gas Each behaves ideally under its respective conditions. 

At the end of the 1930s, the only generally available method for determining mean MWs of polymers was by chemical analysis of the concentration of chain end-groups this was not very accurate and not applicable to all polymers. The difficulty of applying well tried physical chemical methods to this problem has been well put in a reminiscence of early days in polymer science by Stockmayer and Zimm (1984). The determination of MWs of a solute in dilute solution depends on the ideal, Raoult s Law term (which diminishes as the reciprocal of the MW), but to eliminate the non-ideal terms which can be substantial for polymers and which are independent of MW, one has to go to ever lower concentrations, and eventually one runs out of measurement accuracy . The methods which were introduced in the 1940s and 1950s are analysed in Chapter 11 of Morawetz s book. 

The last quantity that we discuss is the mean repulsive force / exerted on the wall. For a single chain this is defined taking the derivative of the logarithm of the chain partition function with respect to the position of the wall (in the —z direction). In the case of a semi-infinite system exposed to a dilute solution of polymer chains at polymer density one can equate the pressure on the wall to the pressure in the bulk which is simply given by the ideal gas law The conclusion then is that [74]... 

Polymer solutions always exhibit large deviations from Raoult s law, though at extreme dilutions they do approach ideality. Generally however, deviation from ideal behaviour is too great to make Raoult s law of any use for describing the thermodynamic properties of polymer solutions. 

The approach of Rory and Krigbaum was to consider an excess (E) chemical potential that exists arising from the non-ideality of the polymer solution. Then ... 

The term 6 is important it has the same units as temperature and at critical value (0 = T) causes the excess chemical potential to disappear. This point is known as the 6 temperature and at it the polymer solution behaves in a thermodynamically ideal way. 

Thus, as this short section has shown, the fact that polymer solutions are non-ideal in the sense that they do not obey Raoult s law leads to numerous important applications in the world beyond the chemical laboratory. The use of polymers as thickeners, while lacking the apparent glamour of some applications of these materials, is significant commercially and accounts for the consumption of many tonnes of polymer throughout the world each year. 

Gee and Orr have pointed out that the deviations from theory of the heat of dilution and of the entropy of dilution are to some extent mutually compensating. Hence the theoretical expression for the free energy affords a considerably better working approximation than either Eq. (29) for the heat of dilution or Eq. (28) for the configurational entropy of dilution. One must not overlook the fact that, in spite of its shortcomings, the theory as given here is a vast improvement over classical ideal solution theory in applications to polymer solutions. 

The analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

First approaches at modeling the viscoelasticity of polymer solutions on the basis of a molecular theory can be traced back to Rouse [33], who derived the so-called bead-spring model for flexible coiled polymers. It is assumed that the macromolecules can be treated as threads consisting of N beads freely jointed by (N-l) springs. Furthermore, it is considered that the solution is ideally dilute, so that intermolecular interactions can be neglected. 

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