Segmental diffusion segment density

Meier solved the diffusion equation with the appropriate boundary conditions. Curiously, as first pointed out by Di Marzio (1965), this corresponds to the placement of adsorbing barriers at jc=0 and x=d, even though the physical surface at x=0 corresponds to an impenetrable reflective surface. Meier obtained rather unwieldy expressions for the segment density distribution functions, which will not be reproduced here. Results were, however, obtained for both low and high surface coverages. 

For the computation of segment density profiles in polymer solutions near interfaces one can use the fact that there is a close analogy between the diffusion of a Brownian particle and the flight of a random walk [20, 21]. A dififusion-like equation can be derived to evaluate the partition function of polymer chains. Given the boundary condition this diffusion equation can be solved. The partition function z h) of one confined chain is given by [1, 22, 23]... 

The transition from dilute solution behaviour (isolated polymer coils) to semi-dilute (interpenetration of coils, uniform polymer segment density) usually occurs over a narrow range of concentration, and a critical concentration c identified. This c will, however, depend to some extent upon the particular experiment performed, e.g., solution viscosity, diffusion (cf., e.g., ref. 99) ... 

The local mobility A in general depends on the environment. In the simplest model (local coupling approximation), A is assumed to be a constant related to an effective diffusion coefficient D . This assumption is valid when the segment density fluctuations are small compared to the average densities o. Another simple modd is to assume that A depends on the segment density as A =Ao< (r,t), where Aq is a constant. The formal theory of time evolution of density variables with general kinetic coeffidents was developed by Kawasaki and Sddmoto, but this is rather difficult to implement in practice. 

The equation above builds on a shear model that assumes that all material in the distance range O-dgcM oscillates with the crystal and thus contributes fully to the tme sensed mass, whereas the material located further away from the surface does not oscillate with the crystal and does not contribute to the sensed mass. This is clearly a simplification, as are the optical models used for evaluating the ellipsometric thickness. It has been shown that the QCM thickness and the ellipsometric thickness are similar for relatively compact and homogeneous layers [28]. We do not expect this to be the case for more diffuse polymer layers since the ellipsometric thickness is directly influenced by the segment density profile [29], whereas the QCM thickness is influenced by the amount of water that oscillates with the crystal, and tlus quantity is at present an unknown function of the segment density profile. 

The average monomer interpenetration depth of segments on the interdiffusing chains, jc(0, is obtained from an integration over the Gaussian segment density profiles of the inter-diffused minor chain population,(2i) such that... 

Mahabadi and O Driscoll also studied the effect of dissolved polymer upon the termination rate coefficient [82, 83]. They derived a theoretical relationship for the dependence of kt upon conversion [82], based on a previously derived segmental diffusion model [100] that allowed for a concentration dependent linear expansion coefficient and polymer-solvent interactions. Also Mahabadi and O Driscoll pointed out that the rate of segmental diffusion would increase if the segment density gradient was increased as a result of coil shrinkage. In a simplified form, the dependence of kt upon polymer concentration, C, could be represented by ... 

Fig. 8.16 (a) Segment density profiles for three thicknesses of adsorbed n-hexadecane films at 350 K, plotted vs. distance for the solid surface, with the original at the center of atoms in the top layer of the solid substrate. An error-function fit for the tail region (marked by arrows) of the thickest film is shown in the inset, (b) Diffusion constants in the directions parallel and perpendicular to the surface plotted vs. distance. Circles and triangles correspond to the thinnest (about 0.1 nm) and thickest (about 4.0 nm) films, respectively. SoUd symbols denote parallel diffusion and empty ones perpendicular diffusion. (Redrawn from Ref. 30.)... 

In the region of c)Rg> 1, we may be able to observe intemal motions of a particle related to /p(rigid sphere, there is no internal mode of motions, so that/p = 1. For rigid particles with a nonspherical shape, /p comes from rotational diffusion, while it also reflects shape fluctuations, such as bending, in semifiexible particles. On the other hand, in maaomolecules of flexible chains, we see the dynamics of segment density fluctuations. 

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