Ellipsoidal rigid particles

Rigid particles other than unsolvated spheres. It is easy to conclude qualitatively that either solvation or ellipticity (or both) produces a friction factor which is larger than that obtained for a nonsolvated sphere of the same mass. This conclusion is illustrated in Fig. 9.10, which shows the swelling of a sphere due to solvation and also the spherical excluded volume that an ellipsoidal particle requires to rotate through all possible orientations. 

Equation (3-33) shows how the inertia term changes the pressure distribution at the surface of a rigid particle. The same general conclusion applies for fluid spheres, so that the normal stress boundary condition, Eq. (3-6), is no longer satisfied. As a result, increasing Re causes a fluid particle to distort towards an oblate ellipsoidal shape (Tl). The onset of deformation of fluid particles is discussed in Chapter 7. 

We may note that, for a nonzero internal viscosity, the system of equations for the moments is found to be open the equations for the second-order moments contain the fourth-order moments, etc. This situation is encountered in the theory of the relaxation of the suspension of rigid particles (Pokrovskii 1978). Incidentally, for 7 —> 00, equation (F.28) becomes identical to the relaxation equation for the orientation of infinitely extended ellipsoids of rotation (Pokrovskii 1978, p. 58). 

The Staudinger index is a function of the hydrodynamic volume, and this itself depends on the mass, shape, and density of the particle. The density of the particle is dependent on the flexibility of the molecular chain and the solvation. The flexibility and solvation of rigid particles such as spheres, ellipsoids, and rods vary only slightly with temperature. Thus, the Staudinger index of these particles is practically temperature independent. 

In the last section we noted that Simha and others have derived theoretical expressions for q pl(p for rigid ellipsoids of revolution. Solving the equation of motion for this case is even more involved than for spherical particles, so we simply present the final result. Several comments are necessary to appreciate these results ... 

This is obvious for the simplest case of nondeformable anisotropic particles. Even if such particles do not change the form, i.e. they are rigid, a new in principle effect in comparison to spherical particles, is their turn upon the flow of dispersion. For suspensions of anisodiametrical particles we can introduce a new characteristic time parameter Dr-1, equal to an inverse value of the coefficient of rotational diffusion and, correspondingly, a dimensionless parameter C = yDr 1. The value of Dr is expressed via the ratio of semiaxes of ellipsoid to the viscosity of a dispersion medium. 

Theory. In the general case where rigid revolution ellipsoidal particles in solution possess both a permanent and an induced dipolar moment colinear with the particle optical axis, the theory derived by Tinoco predicts the following behaviour of the solution birefringence An(t) in the limit of weak electric field (6). 

Particle asymmetry has a marked effect on viscosity and a number of complex expressions relating intrinsic viscosity (usually extrapolated to zero velocity gradient to eliminate the effect of orientation) to axial ratio for rods, ellipsoids, flexible chains, etc., have been proposed. For randomly orientated, rigid, elongated particles, the intrinsic viscosity is approximately proportional to the square of the axial ratio. 

Fig. 3 A comparison of different coarse grain lipid models. The Shelley model " of DMPC, and Marrink and Essex models of DPPC are compared to their atomistic equivalents (for ease of comparison, hydrogen atoms of the atomistic models are not shown). Solid lines represent harmonic bonds connecting CG particles, and the CG particle types for the Shelley and Marrink models are labelled (the labels are the same as those used in the main text). The point charges (represented by + and —) and point dipoles (represented by arrows) are shown for the Essex model (the charges and dipoles are located at the centre of their associated CG particle). The Shelley and Marrink models use LJ particles (represented by spheres), while the Essex model uses a combination of LJ particles (spheres) and Gay-Berne particles (ellipsoids). Finally, the blob model proposed by Chao et al is also shown for comparison. This model represents groups of atoms as rigid non-spherical blobs that use interaction potentials based on multipole expansions.

Precise expressions for W have been obtained for the model of a rigid ellipsoid of revolution (spheroid) rotating about the central axis normal to the symmetry axis of the particle ... 

These phenomena can be interpreted in terms of molecular orientation by the velocity gradient in the flowing liquid, opposed by the rotary Brownian movement which produces disorientation and a tendency toward a purely random distribution. The intensity of this Brownian movement is charaterized by the rotary diffusion constants, 0, discussed in the preceding section. The fundamental treatment of this problem, for very thin rod-shaped particles, was given by Boeder (5) the treatment has been generalized, and extended to rigid ellipsoids of revolution of any axial ratio, by Peterlin and STUARTi 56), [98), (99) and by Snell-MAN and Bj5knstAhl (J9J). The main features of their treatment are as follows 1 ... 

Figure 13.1 GBEMP mappings for (A] water, (B] methanol and (C) benzene, their rigid bodies, which are enclosed by dash lines, consist of CG particles represented by sphere, ellipsoid, and disk respectively. For each CG particle, a point multipole is included at the mass centers for water and benzene and at the oxygen atom for methanol respectively. As for water and benzene, EMP and Gay-Berne interaction sites share the same spot indicated by black filled circle enclosed by red open circle. In the case of methanol, EMP and Gay-Berne interaction sites are located at different positions illustrated by red and black filled circles respectively.

Figure 13.2 GBEMP mappings for amino acid dipeptides each rigid body, being enclosed by a dash line, consists of a Gay-Beme particle (represented by shadowed ellipsoid, or sphere, or disk] with or without electric multipoles. The indices of the rigid bodies, Gay-Berne sites, interacting BMP sites and non-interaction BMP sites (just serve as connecting different rigid bodies], are indicated by Roman numerals and Arabic numbers in black, red and blue, respectively.

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