Form factor Gaussian chain

Fig. 4.27 SAXS intensity as a function of wavevector for a PS-P1 diblock (Mw = 60 kg mol-1, 17wt% PS) (points) in dibutyl phthalate with a polymer volume fraction

Thus all seems perfect. We have constructed an RG mapping, wliich indeed shows a fixed point. However, the expression (8.32) for / is not satisfactory. It must be independent of A, otherwise dilatation by A2 does not lead to the same result as repeated dilatation by A. Now Eq. (8.32) is only approximate since in Eq. (8,31) we omitted terms O 0 2. This is justified only if 0 is small. We thus need a parameter which allows us to make. If arbitrarily small, irrespective of A. Only e — 4 — d can take this role. In all our results the dimension of the system occurs oidy in the form of explicit factors of d or It thus can be used formally as a continuous parameter. To make our expansion a consistent theory, we have to introduce the formal trick of expanding in powers of e — 4 — d. 3 vanishes for = 0, consistent with the observation (see Chap, fi) that the excluded volume is negligible above d = 4, not changing the Gaussian chain behavior qualitatively. For e > 0 Eq. (8.32) to first order in yields... 

FIG. 15 Schematic representation of a Kratky plot q2P(q) vs. q for a Gaussian chain. P(q) is the chain form factor, q is the scattering vector. The inset shows a Guinier plot In P(q) vs. q2 (see text for details). 

Figure 18.7 Calculated form factors for a Gaussian chain, sphere and a rod. The R s used were for a homogenous sphere of radius a, Rg = 3/5 and for a rod of length L, P = I 1.

However, the equilibrium constant for the rings includes an extra factor of the probability to form a ring. This factor is proportional to (mn) / for a Gaussian chain of the length mn, but again we have to divide it by the symmetry factor or = tm for a ring, because we can close a chain at any one of m bonds to form a ring. 

The form factor of ideal polymer chain can be calculated explicitly making use of the Gaussian distribution of distances between each pair of the monomer units and is described by the Debye function ... 

Several calculations have been given for the form factor of the chains, after a step-strain of given deformation tensor. These are restricted to a uniaxial deformation (extension to other cases is simple) and to a melt (no solvent) this latter condition allows to admit that, similarly to the isotropic case, the chains behave as Gaussian chains. Then the equality... 

Fig. 11. Calculated form factors for isotropic Gaussian chains (dotted dashed line), completely affine deformation (dotted), Rouse,model = 5 10 , 10 (tiny dots), 5 10 , 10 , 5 10 ,...

Form Factors The plot of Si(k) as a function of k at small k gives the radius of gyration for any conformation, but, beyond that range, Si(k) depends on the conformation. For a Gaussian chain, SiCk) follows the Debye function. Equations 2.59 and 2.76 allow us to calculate SiCk) for other conformations. Let us first define a form factor P(k) by... 

Figure 2 5. Form factor P(k) for a spherical molecule, a rodlike molecule, and a Gaussian chain, plotted as a function of kR. ...

Problem 2.21 Verify that the form factor of a two-arm star polymer (% = 2 in Eq. 2.98) reproduces the form factor of a Gaussian chain. 

The same nomenclature, P(Q), is used for the single-chain form factor both in the amorphous state (Eq. (7.6)) and in the crystalline state (Eq. (7.24)), though the chain is not assumed to be Gaussian in the latter. 

Gaussian chain structure factors take on a perfectly self-similar form and no large k crossover to locally rigid behavior, (i) k) oc, occurs. This mathematical feature allows an exact analytic... 

Inserting this into the expression for the correlation function in the scattering function, we get the form factor of a Gaussian polymer chain equal the Debye function, g ... 

Borsali et al. [73] started with the general Eq. (2.24) derived by Hess and Klein and evaluated the shear viscosity by replacing the dynamic structure factor by the mean fidd expression S(q,t) = exp[ — Dq t/S(q)] and assuming Gaussian chain statistics for the calculation of the form factor. Numerical calculations of the resulting integral show the peak position to vary as... 

This form factor for a Gaussian chain is known as the Debye structure factor. The asymptotic limits of this equation are as in Equation 2.11 with v = 1/2. 

This closed-form result for a readily gives its dependence on b (i.e., T (T)), the salt concentration Cs, the monomer concentration p, and the dielectric mismatch parameter 5. The value of a given by Equation 4.65 is substituted into Equation 4.55 to calculate the expansion factor li. A comparison of a and ii thus calculated with the separation approximation and with the numerical computation with full coupling reveals that the separation approximation is very good as long as the chain is not collapsed below the Gaussian chain size. 

The RPA is a mean field approximation that neglects contributions from thermal composition fluctuations and that assumes the chain conformations to be unperturbed Gaussian chains. The last assumption becomes visible from the Debye form factor in the first two terms, which for Vp, = are in accordance with Eq. 7, while the third term involves the FH interaction parameter. 

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