Rigid ellipsoid

In addition to an array of experimental methods, we also consider a more diverse assortment of polymeric systems than has been true in other chapters. Besides synthetic polymer solutions, we also consider aqueous protein solutions. The former polymers are well represented by the random coil model the latter are approximated by rigid ellipsoids or spheres. For random coils changes in the goodness of the solvent affects coil dimensions. For aqueous proteins the solvent-solute interaction results in various degrees of hydration, which also changes the size of the molecules. Hence the methods we discuss are all potential sources of information about these interactions between polymers and their solvent environments. 

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 ... 

Yet there is no theoretical justification for the use of Buckingham s equation for macromolecules other than rigid ellipsoids of revolution. Probably, it is insuperably difficult to calculate theoretically the factor qjfg in Eq. (E-8) for polymer molecules of arbitrary conformation. Facing this difficult situation, Omura et al. (117) took a tentative step in which pin Eq. (E-l 1) is related to fN by... 

When the overall motion is not isotropic, the diagonal elements of the rotational diffusion tensor are no longer equivalent and rotation about the three principal axes of the diffusion tensor may be described by different diffusion coefficients or correlation times. For anisotropic motion, the correlation time in Eqs. 16 and 25 is an effective correlation time, r ff, containing contributions from the various modes of reorientation. Partitioning of the various components of rff can be achieved through appropriate dynamic models. The simplest case of anisotropic motion is that for a symmetric-top molecule. The r ff of a rigid ellipsoid is expressed in terms of two parameters, Dn and DL these two parameters respectively describe the rotational diffusion about the C3 symmetry axis (major axis) and the two perpendicular axes (minor axes), which are assumed to be equivalent25-44 (Fig. 4) ... 

Riseman (139) or Kuhn and Kuhn (153 , 153"). For the sake of comparison, Fig. 12 also shows the theta-solvent intrinsic viscosities of polystyrene in cyclohexane [experimental valuesofKRiGBAUMandFuoRY (149), small black points theoretical values, broken line] and the theoretical intrinsic viscosities of rigid ellipsoids with axial ratios p = M/500 (chain curve). As a matter of course, the chain curve reduces to the Einstein value of [rf in the range of M below500 [see, for example, Petehlin (16) ]z. 

Fig. IS. Molecular-weight dependence of sedimentation constant (rc) and intrinsic viscosity ( ), for various degrees of draining and coil expansion. Full line is for coiled polymers without draining. Dotted curve is for rigid ellipsoids of revolntion at various axial ratios p. Experimental points a, cellulose nitrate in ethyl acetate 729) b, cellulose nitrate in acetone (181) c, cellulose acetate in acetone (125) d., ethyl cellulose in ethyl acetate 223 ) e, ethyl bydroxyethyl cellulose in water (772)...

Fig. 27. Viscosity-molecular weight relation lor paiy-y-benzyl-L-glutamate. Solid curve, theory lor random coil in DCA, Eqs. (58) and (86). Chain curve, theory lor rigid ellipsoids in DMF. Dashed curve, random coils in theta solvent, K — 58 10". Dotted curve, hypothetical curve lor random coils (or interrupted helices)...

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 ... 

Jeffrey [1923] extended Einstein s analysis to flow around an impermeable, rigid ellipsoid of revolution, and Simha [1940] further incorporated the effect of Brownian motion, deriving an equation of the form... 

Similarly, asymptotic expressions for rigid ellipsoids of revolution were derived that are valid for axial ratios larger than 10 and are given by... 

The theory for rotational diffusion of ellipsoids, and measurements by fluorescence polarization, can be traced to the classic reports by F. Perrin. Since these seminal reports, the theory has been modified to include a description of expected anisotropy decays. Hiis theory has been summarized in several reviews.For a rigid ellipsoid with three unequal axes, it is now agreed that the anisotropy decays with five correlation times. The correlation times depend on the three rotational diffiision coefficients, and the amplitudes depend on the orientation of the absorption and emission transition moments widiin the fluoroi iore and/or ellipsoid. While the the( predicts five correlation times, it is known diat two pairs of correlation times will be very close in magniOide, so that in practice only three correlation times are expect for a nonsf oical molecule. ... 

The orientation of the molecules brought about by the velocity gradient is revealed by a double refraction. A detailed discussion of this double refraction on the basis of rigid ellipsoidal particles can be found in the work of Peterlin and Stuart In recent work the subject was treated from the point of view of randomly coiled flexible structures. The principal features of this treatment may be outlined as follows. 

For the case of rf jri of 20, the droplet began to rotate as it was a rigid ellipsoid in a shear flow. In the elongational flow four-roller device, the same droplet in the same fluid moves, stretched and burst rapidly when rjay/K was 0.14. 

Motion of a fiber in flow is described by Jeffery s model [3]. It is assumed that the fiber is a single rigid ellipsoidal partide suspended in a viscous fluid, the flow is a creeping flow of a Newtonian and incompressible fluid, and Brownian motion and inertia terms of the fiber are neglected. Jeffery s model was used for prediction of fiber orientation in the early period of injection molding CAE. Since it is, however, for dilute suspension, the model is replaced with the Folgar-Tucker model for concentrated suspension. 

In the present paper we present the results of a numerical analysis of the hydrodynamic lubrication of two spinning, rigid ellipsoids. Solutions were initially obtained for a half-Sommerfeld cavitation boundary condition... 

If two rigid ellipsoids have common principal z-axes at their point of contact and principal radii of curvature r ) and... 

Takserman-Krozer, R. and Ziabicki, A. (1963) Behaviour of polymer solutions in a velocity field with parallel gradient. L Orientation of rigid ellipsoids in a dilute solution. J. Polym. Sci., Al, 491-506. 

Theodosopoulou, M. Dahler, J.S. (1974b). The kinetic theory of polyatomic liquids II. The rough sphere, rigid ellipsoid and square-well ellipsoid models. J. Chem. Phys., 60, 4048-4057. 

Following Einstein, Jeffery (1922) investigated the motion of non-spherical particles (rigid ellipsoidal particles) in a shear field of Newtonian liquid, on the basis of the creeping flow equation, and obtained the following expression for the bulk viscosity ... 

For a rigid ellipsoid of revolution, two rotational diffusion constants are required one for rotation about the symmetry axis, Dy, and one for leorientation about any axis perpendicular to the symmetry axis, Dj. These diffusion constants can be calculated knowing the dimensions of the ellipsoid of revolution, and the viscosity of the medium ... 

---