Expansion factor end distance

Thus, in the Douglas-Freed scheme, it follows that if eq 3.49 holds, the end distance expansion factor for 3-dimensional chains in poor solvents obeys the two-parameter theory. This consequence may be utilized to determine the function /r(w). With eq 3.36 substituted for this function, the right-hand side of eq 3.50 is expanded in powers of z and the result is compared with the familiar z expansion for in the two-parameter theory, i.e., or = l4-(4/3)z —2.075z - -... (see Section 1.3 of Chapter 2). Then we find... 

Fig. 3-2. RG predictions of the end distance expansion factor OR. Curve a, in poor solvents curve b, in very good solvents (self-avoiding chains) curve c, a smooth coimec-tion of a and b. 

With eq 1.4 the end distance expansion factor and the radius expansion factor as can be calculated analytically (in the continuous chain limit). The results are as follows ... 

The average end distance expansion factor aR(n) for subchains containing n beads and located at different positions is defined by... 

If the average end distance expansion factor for a subchain with fixed n and i is denoted by aR(n i), we have... 

Yamakawa [34] was the first to calculate by perturbation the end distance expansion factor or on the basis of eq 2.12, and, recently, Cherayil et al. [35] made a similar calculation with the result which may be written... 

For a touched-bead chain whose backbone behaves like a wormlike chain Yamakawa and Stockmayer [51] calculated perturbatively the end distance expansion factor ttR and derived... 

To scrutinize the screening effect we look at a subchain (ij) between beads i and j of the test chain. As it becomes longer, the subchain is surrounded with more beads of other chains and hence undergoes a stronger screening effect. Therefore, if we denote the end-distance expansion factor for the subchain (ij) by we can expect that aR ij) decreases monotonically to unity with... 

Thus, the char2M ter of the concentration dependence of such quantities as the mean square end-to-end distance expansion factor (Equation 39), the correlation length (Equation 40), and the osmotic pressure (Equation 43) coincides with the results of familiar scaling laws (see Table 4.1). 

In a good solvent, the end-to-end distance is greater than the 1q value owing to the coil expansion resulting from solvent imbibed into the domain of the polymer. The effect is quantitatively expressed in terms of an expansion factor a defined by the relationship... 

Equation (10) directs attention to a number of important characteristics of the molecular expansion factor a. In the first place, it is predicted to increase slowly with molecular weight (assuming t/ i(1 — 0/T) >0) and without limit even when the molecular weight becomes very large. Thus, the root-mean-square end-to-end distance of the molecule should increase more rapidly than in proportion to the square root of the molecular weight. This follows from the theory of random chain configuration according to which the unperturbed root-mean-square end- o-end distance is proportional to (Chap. X), whereas /r = ay/rl. 

Recalling that i/ = (R )/N ) is the expansion factor for the mean-square end-to-end distance and the radius of gyration Rg is - /(R2)76 within the uniform expansion approximation, we have... 

Ratio of a dimensional characteristic of a macromolecule in a given solvent at a given temperature to the same dimensional characteristic in the theta state at the same temperature. The most frequently used expansion factors are expansion factor of the mean-square end-to-end distance, Ur = (/o) expansion factor of the radius oj gyration, as = (/0) relative viscosity, = ([ /]/[ /]o), where [ ] and [ /]o are the intrinsic viscosity in a given solvent and in the theta state at the same temperature, respectively. 

Figure 5. (a) Expansion factor of mean-square end-to-end distance aj and fbj radius of gyration as function of z = xBJN for open (Op.) and periodic (Per.) chain. Other results shown here are Domb and Barrett (DB) [67], Douglas and Freed (DF) [66], Muthukumar and Nickel (MN) [64], and Flory modified (Fm) [17,69]. (From ref. 68, by permission of the publishers, Butterworth Co. (Publishers) Ltd. .)... 

Problem 3.24 The intrinsic viscosity of polystyrene of molecular weight 3.2x10 in toluene at 30°C was determined to be 0.846 dlVg. In a theta solvent (cyclohexane at 34°C) the same polymer had an intrinsic viscosity of 0.464 dL/g. Calculate (a) unperturbed end-to-end distance of the polymer molecule, (b) end-to-end distance of the polymer in toluene solution at 30°C, and (c) volume expansion factor in toluene solution. (3> = 2.5x10 mol )... 

What is the excluded volume effect Show the dependence of the expansion factor a on the molecular weight of the polymer. How does the mean square end-to-end distance (r ) vary with the molecular weight for theta and better solvents ... 

The parameter a is close to but not equal to the linear expansion factor of the excluded volume effect with respect to the end-to-end distance aM or that with respect to the radius of gyration as (46). 

After Eqnation (3 9), the expansion factor is given by the ratio of the respective end-to-end distances. Since it is assnmed that the segments are randomly arranged in both the real and the perfect chain, the problem might be formnlated in terms of corresponding Ganssian distribntions. Application of Eqnation (lb) yields the elastic contribntion. 

The distance between the chain ends is often expressed in terms of unperturbed dimensions (So orRo) and an expansion factor (a) that is the result of interaction between the solvent and the polymer... 

The unperturbed dimensions refers to molecular size exclusive of solvent effects. It arises from intramolecular polar and steric interactions and free rotation. The expansion factor is the result of solvent and polymer molecule interaction. For linear polymers, the square of the radius of gyration is related to the mean-square end-to-end distance by the following relationship ... 

When the chains are extended, their conformations may be considered as being determined by equilibrium between the forces of expansion due to excluded volume and the forces of contraction due to chain segments expanding into less probable conformations. Based on random flight statistics, the chains are extended linearly by a factor a over their dimensions. The acmal root-mean-square end-to-end distance is equal to The change in the elastic part of free energy is... 

The expansion factor is unity for an unperturbed coil. An expansion factor based on the end-to-end chain distance can similarly be defined. 

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