Solution theta

Here m (i = A, B, or AJBW) is the molar chemical potential of reacting species i. Equation 3 is valid for self-associations as well since n or m is zero in that case. Under ideal (theta) solution conditions the activity coefficient ifo of each of the associating species is one, so that... 

Er. Gulari, Es. Gulari, Y. Sunashima and B. Chu, Polymer diffusion in a dilute theta solution, 1, Polystyrene in cyclohexane, Polymer 20 (1979) 347-355. 

Here Lw is the contour length of the chain. In an effort to generalize Eq. (14) for non-theta solutions, Reed et al. [46] have provided an ad hoc treatment by combining theories of Odijk [35], Odijk and Houwaart [36], Skolnick and Fixman [37], and Gupta and Forsman [49]. Here they append an additional contribution f30 to A arising from short-ranged non-electro-static interactions, so that f3 of Eq. (11) is given by... 

Eq. (9.41) may be used to calculate the limiting viscosity number of a theta solution, if the unperturbed dimensions of the macromolecule have been determined by some other method. 

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. 

The intrinsic viscosity in a theta solution is labeled Equation (3-39) can thus be expressed as follows for theta conditions ... 

In a better solution than that provided by a theta solvent the polymer coil will be more expanded. The radius of gyration will exceed the which is characteristic of the bulk amorphous state or a theta solution. If the polymer radius in a good solvent is times its unperturbed /-g, then the ratio of hydrodynamic volumes will be equal to a and its intrinsic viscosity will be related to [ /] by... 

Thus, at some special temperature T = 9, A2 vanishes and the solutions become pseudoideal. Such solutions are also called theta solutions. The second virial coefficient is positive at temperatures higher than 9 and negative at lower temperatures. 

The pseudoideal or theta solution is an important special case of irregular solutions in macromolecular science. The enthalpy of mixing and the excess entropy of mixing exactly compensate each other at a certain temperature with the dilute theta solution. Theta solutions at this theta temperature thus behave like ideal solutions. In contrast to ideal solutions, however, the enthalpy of mixing is not zero and the entropy of mixing differs considerably from the ideal entropy of mixing. Thus, an ideal solution exhibits ideal behavior at all temperatures, the pseudoideal solution only behaves ideally at... 

Consequently, < Mgpc ) = < Mw ) for spheres, where ar, = 0, and < Mgpc ) = < Afz +1 > for rigid rods. An average close to the weight average is obtained for coil-shaped molecules with the usual distributions when the experiments are run in theta solutions. 

Properties of Theta Solutions Solutions in the theta condition have A2 = 0. When A2 = 0, the second-order term in Il/Ilideai = 1 + A2MC + A Mc + (Eq. 2.20) is absent. The nonideality of the solution does not become apparent until the third-order term A Mc becomes sufficiently large. The osmotic pressure is... 

The chain dimension such as and Rp in theta solution increases with the molecular weight M just as the ideal chains do. Figure 2.25 shows an example obtained for polystyrene in cyclohexane at The data for R are plotted as... 

Figure 2.26 compares a plot of the root mean square end-to-end distance Rp for polymer chains on the cubic lattice for the theta chains with a plot for athermal chains. The chain contraction in the theta condition is evident. The data for the theta solution follow a power law of Rp Af / when N 1. 

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