Theta-point

The essential idea of the Alexander model, a global balance of interaction and stretching energies, can be applied to other situations involving tethered chains besides the good solvent case. In theta or poor solvents, the interaction term must be modified to account for poorer solvent quality. A simple limit is precisely at the theta point [29, 30] where binary interactions effectively vanish (% = 1/2 or v = 0). The leading term in Fim now accounts for three-body interactions ... 

The linearity of L with N is maintained at the theta point. Relative to Eq. 5, the chains have shrunk by a factor of (a/d),/3 but the linear variation indicates that the chains are still distorted at the theta point and characteristic dimensions do not shrink through a series of decreasing power laws as do free chains [29-31]. Experimentally, Auroy [25] has produced evidence for this linearity even in poor solvents. Pincus [32] has recently applied this type of analysis to tethered polyelectrolyte chains, where the electrostatic interactions can produce even stronger stretching effects than those that have been discussed for good solvents. Tethered polyelectrolytes have also been studied by others [33-35],... 

Lai, P. Y. and Binder, K. (1992) Structure and dynamics of polymer brushes near the theta point - a Monte-Carlo simulation. 

An excluded-volume random-coil conformation will be achieved when the solvent quality exceeds the theta point, the temperature or denatu-rant concentration at which the solvent-monomer interactions exactly balance the monomer—monomer interactions that cause the polymer to collapse into a globule under more benign solvent conditions. A number of lines of small-angle scattering—based evidence are consistent with the suggestion that typical chemical or thermal denaturation conditions are good solvents (i.e., are beyond the theta point) and thus that chemically or thermally unfolded proteins adopt a near random-coil conformation. 

The thickness of the adsorbed layer, t, is approximately proportional to at the Theta point but in good solvent the... 

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. 

Figure 3. Plateau adsorption from solvent/non-solvent mixtures, curve 1, PMM I 2, RC 08 3, RC 51 A, RC 86 5, PS III. 8 marks the theta-point. Vertical lines are (Vns) for phase separation at c = 10 2 g cm-3.

In the absence of polymer the sediment volume of silica depends on the non-solvent fraction of the medium as shown in Figure 6. The sediment volume assessment of steric stabilization behavior of the copolymers is illustrated in Figures 7a to 7c. At low styrene contents, both the random and block copolymers show a steady increase in sediment volume as the non-solvent content is raised up to the phase separation value. With polystyrene and random copolymers of high styrene content, the sediment volume stays largely constant with alteration in the non-solvent fraction until the theta-point is approached and then continues to become larger as the limit of solubility is reached. In Figure 7b only the data points of RC 86 are shown, RC 94 giving almost identical values. 

Influence of Addition of Electrolyte and Increase of Temperature Addition of electrolyte or increase of temperature at a given electrolyte concentration to a sterically stabilized dispersion may result in its flocculation at a critical concentration or temperature, which in many cases coincides with the theta point for the stabilizing chain. At the theta point the mixing term in the steric interaction is zero and any yield value measured should correspond to the residual van der Waals attraction. The energy arising from van der Waals attraction may be calculated from the following approximate relationship,... 

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. 

The osmotic second virial coefficient A2 is another interesting solution property, whose value should be zero at the theta point. It can be directly related with the molecular second virial coefficient, expressed as B2=A2M /N2 (in volume units). For an EV chain in a good solvent, the second virial coefficient should be proportional to the chain volume and therefore scales proportionally to the cube of the mean size [ 16]. It can, therefore, be expressed in terms of a dimensionless interpenetration factor that is defined as... 

Figure 2. Concentration dependence of the diffusion coefficient for polystyrene in two solvents at the theta point for various molecular weights. The line is the theoretical curve. (Reproduced from Ref. 12. Copyright 1983 American Chemical Society.)...

The organization of this paper is as follows First, we will present some of the evidences which point to the existence of the chain theta point and to the possibility of the configurational transition, based on the recently conducted Monte Carlo studies by McCrackin and Mazur.2 Next, we will describe the chain as a stochastic process of dependent events. This stochastic process will serve as a basis for the formulation of the chain partition function. The chain partition function will be subsequently expanded in terms of the eigenvalues of the transition matrix. We will also present a general outline showing how the study of the distribution of the eigenvalues of the transition matrix could be employed in conjunction with the Monte Carlo calculations in order to study the thermodynamic... 

Since a branched polymer has more than two chain ends, the determination of both the thickness of the adsorbed polymer and the adsorbance is of particular interest. Kawaguchi and Takahashi106) investigated well-characterized comb-branched polystyrene adsorbed on a chromium plate from cyclohexane solution at the theta point by ellip-sometry and compared the results with those for the corresponding linear polystyrene. 

To determine the conformation of adsorbed sodium polyacrylate, we first assume a loop-train conformation. Then, t at the theta point can be evaluated by Eq. (B-108). Since 1.5N NaBr at 15 °C is the theta solvent for sodium polyacrylate120, this calculation... 

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