Zimm Model in the Theta Solvent

2 Zimm Model in the Theta Solvent For a polymer chain in the theta solvent, the Gaussian chain model conveniently gives ns an analytical expression of Eq. 3.163. With Eq. 3.56, 

Equation 3.164 is not correct for n = m. It diverges as n and m approach each other. Upon integration with respect to n and m, however, the singularity is removed. We do not treat // separately here. There is no need for that. The sum of the hydrodynamic interactions from other beads far exceeds the friction a given bead would receive in the absence of the hydrodynamic interactions. We will discuss this problem later when we derive the center-of-mass diffusion coefficient. 

For g we use the same definition as Eq. 3.133 with I given by Eq. 3.130. It does not mean that h is the same in the Rouse model and the Zimm model. We just use the same formula in the Zimm model to express the random force in the normal coordinates. As Eq. 3.125 is an inverse transform of Eq. 3.118, the following gives an inverse transform of Eq. 3.133 (Problem 3.21)  

all of the equations appear to be coupled with each other. Fortunately, they can be decoupled. It can be shown (Problem 3.22) that, when N 1, % is approximated by 

Formally, the normal modes of the Zimm model follow the same differential equations as those of the Rouse model (Eq. 3.131). 

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With these modified and Iq, q, follows the same equation as in the Zimm model in the theta solvent. 

Figure 3.55. Initial decay rate Finit of i(f), reduced by its low-fc asymptote Dok, is plotted as a function of kRu. The data obtained for polystyrene in various solvents are on a theoretical curve (solid line) obtained for the Zimm model in the theta solvent. (From Ref. 40.)...

The expression for [r (t)] ) in the long time is the same as that of the Rouse model, because is common between the two models. Thus, Eq. 3.240 holds as it does for the Zimm model in the theta solvent. Note, however, that Dq is different between the two models. 

S.4.6.4 Zimm Model in the Good Solvent The Zimm model we used for the theta chains needs a small modification when we apply it to the chains in the good solvent. We must give up the numerical coefficients but can stiU obtain the exponents that agree with experimental results. 

In the Rouse model, there is no excluded volume effect. In the Zimm model for the theta solvent, the chain conformation is the same as that of the Rouse model. For these two models, U is given by a sum of the elastic energy of the springs (Eq. 1.51), and therefore... 

Initial Slope in the Zimm Model, Theta Solvent Equation 3.211 is valid also in the Zimm model for the theta solvent because the equation depends on the chain conformation only. However, Eqs. 3.212 and 3.213 are different. We evaluate the sum in Eq. 3.211 for small k and large k separately. For small k. 

Problem 3.22 Show that, in the Zimm model for the theta solvent. 

In dilute solutions in a theta solvent, flexible polymer chains are in an isolated, ideal random coil state. For description of the dynamics in such solutions, Zimm model subdivides the chain into subchains (cf Figure 2), represents the friction and elasticity of the subchain by a bead (friction center of the subchain) and the entropic springs conneaing the beads, respectively, and analyzes the motion of this bead-spring chain in the presence of hydrodynamic and thermal... 

Universal Calibration Curve in SEC In the Zimm model (theta and good solvents), the intrinsic viscosity is essentially the ratio of the volume of the polymer chain, R/, to the mass of each polymer chain, M/Np. The solvent viscosity and the temperature do not show up explicitly in the final expression. Thus we can define hydrodynamic volume VMby... 

The viscosity factor of 2.86 x 10 mol" is 3-18% larger than experimental values for typical flexible polymers " in theta solvents and about 12% larger than the value obtained by Zimm" from Monte Carlo calculations for Gaussian chains without the preaveraging approximation of the HI tensor. Hence, the viscosity theories ofYamakawa and co-workers ° ° ° based on the cylinder and touched-bead models involve appreciable errors near the coil limit and data analysis based on them may lead to tmderestimation of either 2" or ljM.1 or both for flexible chains. 

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