Polymers theta conditions

Fig. XI-5. Adsorption isotherm from Ref. 61 for polystyrene on chrome in cyclohexane at the polymer theta condition. The polymer molecular weights x 10 are (-0) 11, (O) 67, (( )) 242, (( )) 762, and (O) 1340. Note that all the isotherms have a high-affinity form except for the two lowest molecular weights.

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

A. Milchev, W. Paul, K. Binder. Off-lattice Monte Carlo simulation of dilute and concentrated polymer solutions under theta conditions. J Chem Phys 99 4786-4798, 1993. 

Theta temperature is one of the most important thermodynamic parameters of polymer solutions. At theta temperature, the long-range interactions vanish, segmental interactions become more effective and the polymer chains assume their unperturbed dimensions. It can be determined by light scattering and osmotic pressure measurements. These techniques are based on the fact that the second virial coefficient, A2, becomes zero at the theta conditions. 

An illustrative example is the work of Clark et al, on the conformation of poly(vinyl pyrrolidone) (PVP) adsorbed on silica 0). These authors determined bound fractions from magnetic resonance experiments. In one instance they added acetone to an aqueous solution of PVP in order to achieve theta conditions for this polymer. They expected to observe an increase in the bound fraction on the basis of solvency effects as predicted by all modern polymer adsorption theory (2-6), but found exactly the opposite effect. Their explanation was plausible, namely that acetone, with ability to adsorb strongly on silica due to its carbonyl group, would be able to partially displace the polymer by competing for the available surface sites. 

The stability of these dispersions has been investigated. A strong dependence of critical flocculation conditions (temperature or volume fraction of added non-solvent) on particle concentration was found. Moreover, there seems to be little or no correlation between the critical flocculation conditions and the corresponding theta-conditions for the stabilising polymer chains, as proposed by Napper. Although a detailed explanation is difficult to give a tentative explanation for this unexpected behaviour is suggested in terms of the weak flocculation theory of Vincent et al. 

It should also be noted that ternary and higher order polymer-polymer interactions persist in the theta condition. In fact, the three-parameter theoretical treatment of flexible chains in the theta state shows that in real polymers with finite units, the theta point corresponds to the cancellation of effective binary interactions which include both two body and fundamentally repulsive three body terms [26]. This causes a shift of the theta point and an increase of the chain mean size, with respect to Eq. (2). However, the power-law dependence, Eq. (3), is still valid. The RG calculations in the theta (tricritical) state [26] show that size effect deviations from this law are only manifested in linear chains through logarithmic corrections, in agreement with the previous arguments sketched by de Gennes [16]. The presence of these corrections in the macroscopic properties of experimental samples of linear chains is very difficult to detect. 

A theoretical expression for the concentration dependence of the polymer diffusion coefficient is derived. The final result is shown to describe experimental results for polystyrene at theta conditions within experimental errors without adjustable parameters. The basic theoretical expression is applied to theta solvents and good solvents and to polymer gels and polyelectrolytes. 

We have shown that the microscopic expression for the polymer diffusion coefficient. Equation 2, is the starting point for a discussion of diffusion in a wide range of polymer systems. For the example worked out, polymer diffusion at theta conditions, the resulting expresssion describes the experimental data without adjustable parameters. It should be possible to derive expressions for diffusion... 

It is however possible to find conditions, called unperturbed or theta conditions (because for each polymer-solvent pair they correspond to a well-defined temperature called d temperature) in which a tends to 1 and the mean-square distance reduces to Q. In 6 conditions well-separated chain segments experience neither attraction nor repulsion. In other words, there are no long-range interactions and the conformational statistics of the macromolecule may be derived from the energy of interaction between neighboring monomer units. For a high molecular weight chain in unperturbed conditions there is a simple relationship between the mean-square end-to-end distance < > and the mean-... 

The deterioration of the solvent qnality, that is, the weakening of the attractive interactions between the polymer segments and solvent molecules, brings about the reduction in the coil size down to the state when the interaction between polymer segments and solvent molecules is the same as the mutual interaction between the polymer segments. This situation is called the theta state. Under theta conditions, the Flory-Huggins parameter % assumes a value of 0.5, the virial coefficient A2 is 0, and exponent a in the viscosity law is 0.5. Further deterioration of solvent quality leads to the collapse of coiled structure of macromolecules, to their aggregation and eventually to their precipitation, the phase separation. 

The unconventional applications of SEC usually produce estimated values of various characteristics, which are valuable for further analyses. These embrace assessment of theta conditions for given polymer (mixed solvent-eluent composition and temperature Section 16.2.2), second virial coefficients A2 [109], coefficients of preferential solvation of macromolecules in mixed solvents (eluents) [40], as well as estimation of pore size distribution within porous bodies (inverse SEC) [136-140] and rates of diffusion of macromolecules within porous bodies. Some semiquantitative information on polymer samples can be obtained from the SEC results indirectly, for example, the assessment of the polymer stereoregularity from the stability of macromolecular aggregates (PVC [140]), of the segment lengths in polymer crystallites after their controlled partial degradation [141], and of the enthalpic interactions between unlike polymers in solution (in eluent) [142], as well as between polymer and column packing [123,143]. 

Theta conditions correspond to a solvent so poor that precipitation would occur for a polymer of infinite molecular weight. 

Although a is ordinarily greater than unity, fractional values are also possible. The range of fractional values is more limited, however, since the polymer tends to precipitate rather than squeeze out much more solvent under conditions poorer than theta conditions. Incorporating a into Equation (89) gives... 

Summary The classical treatment of the physicochemical behavior of polymers is presented in such a way that the chapter will meet the requirements of a beginner in the study of polymeric systems in solution. This chapter is an introduction to the classical conformational and thermodynamic analysis of polymeric solutions where the different theories that describe these behaviors of polymers are analyzed. Owing to the importance of the basic knowledge of the solution properties of polymers, the description of the conformational and thermodynamic behavior of polymers is presented in a classical way. The basic concepts like theta condition, excluded volume, good and poor solvents, critical phenomena, concentration regime, cosolvent effect of polymers in binary solvents, preferential adsorption are analyzed in an intelligible way. The thermodynamic theory of association equilibria which is capable to describe quantitatively the preferential adsorption of polymers by polar binary solvents is also analyzed. 

For polymer chains in two dimensions in good solvents, the theoretical predictions point to a v value narrowly centered in 0.75 [61], Monte Carlo simulations predicts a value of 0.753 [66], while by the matrix-transfer method a value of 0.7503 is predicted. In the case of theta condition the situation is not clear, the predictions are less precise. Monte Carlo simulation [66] has suggested ve 0.51 while matrix-transfer data suggest ve 0.55 [67],... 

Our calculation of (ro2) is therefore, for polymer solutions, only valid for theta conditions. We shall consider this case in somewhat more detail. We have seen that for the determination of the viscosity average molar mass, the Mark-Houwink... 

When the concentration of the free polymer is set equal to zero, the situation corresponds to pure steric stabilization. The free energy of interaction due to the interpenetration of the adsorbed polymer chains has a range of 26, where 6 is the thickness of the adsorbed layer. This free energy is proportional to the quantity (0.5 - x), where x is the Flory interaction parameter for the polymer-solvent system. Thus, a repulsive potential is expected between two particles when x < 0.5 and this repulsion is absent when x = 0.5. For this reason, it was suggested [25] that instabilities in sterically stabilized dispersions occur for x > 0.5, hence for theta or worse-than-theta conditions. However, the correlation with the theta point only holds when the molecular weight of the added polymer is sufficiently high... 

Analysis of the steric stabilization in the absence of free polymer in these dispersions indicates that destabilization is possible under better-than-theta conditions. It is shown that smaller particles and thicker layers of adsorbed polymer result in improved stability, i.e. the critical particle concentration, above which instability sets in, decreases with increasing particle size and with decreasing thickness of the adsorbed layer. 

The method outlined above is now well established. Its application, however, is often limited by the difficulty of finding appropriate theta solvents. This limitation becomes especially serious for the investigation of crystalline polymers with high melting points, because for such polymers the theta condition can rarely be attained at ordinary temperatures. It is therefore highly desirable to develop a method for estimating the unperturbed dimensions without the aid of theta-solvent experiments. 

With polymers that have ionizable groups, adsorption of a polymer will alter the charge of the surface altering the electrostatic interaction energy and also provide steric protection for the colloid, because the ionized groups will give better than theta conditions for the poisoner in an aqueous solution. This type of polymer stabilization is called electrosteric stabilization because both the electrostatic and the steric play a role in stabilization. The equations for this total interaction are simply the sum of electrostatic and steric terms as well as the van der Waals attraction. 

Theta conditions are of great theoretical interest because the diameter of the polymer chain random coil in solution is thenequal to the diameter it would have in the amorphous bulk polymer at the same temperature. The solvent neither expands nor contracts the macromolecule, which is said to be in its unperturbed state. The theta solution allows the experimenter to obtain polymer molecules which are unperturbed by solvent but separated from each other far enough not to be entangled. Theta solutions are not normally used for molecular weight measurements, because they are on the verge of precipitation. The excluded volume vanishes under theta conditions, along with the second virial coelTicient. 

Under theta conditions the polymer coil is not expanded (or contracted) by the solvent and is said to be in its unperturbed state. The radius of gyration of such a macromolecule is shown in Section 4.4.1 to be proportional to the square root of the number of bonds in the main polymer chain. That is to say, if M is the polymer molecular weight and A/q is the formula weight of its repeating unit, then... 

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