Theta condition

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. 

FIGURE 19.1 Theta condition effects on pMMA elution. Upper trace acetonitrile 65°C Lower trace THF 35°C... 

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. 

The adsorption of block and random copolymers of styrene and methyl methacrylate on to silica from their solutions in carbon tetrachloride/n-heptane, and the resulting dispersion stability, has been investigated. Theta-conditions for the homopolymers and analogous critical non-solvent volume fractions for random copolymers were determined by cloud-point titration. The adsorption of block copolymers varied steadily with the non-solvent content, whilst that of the random copolymers became progressively more dependent on solvent quality only as theta-conditions and phase separation were approached. 

Colloid stability conferred by random copolymers decreased as solvent quality worsened and became increasingly solvent dependent around theta-conditions. However, dispersions maintain some stability at the theta-point but destabilize close to the appropriate phase separation condition. 

Our conclusions are that a) theta conditions are not necessary to obtain experimental mK values unperturbed by excluded volume interactions, b) the RIS and BAA approximations are applicable and c) mK is a very sensitive to the details of the RIS model, the tacticity and the composition of a copolymer. 

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]. 

K, a = Mark-Houwink parameters Kg = K value at theta condition... 

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

Theta conditions are identified experimentally as the situation in which the second virial coefficient of the osmotic pressure is zero. 

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... 

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