Stick boundary conditions

A physical value for 7 for each particle can be chosen according to Stokes law (with stick boundary conditions) ... 

Ghoreishy, M. H. R. and Nassehi, V., 1997. Modelling the transient flow of rubber compounds in the dispersive section of an internal mixer with slip-stick boundary conditions. Adv. Poly. Tech. 16, 45-68. 

The friction coefficient of a large B particle with radius ct in a fluid with viscosity r is well known and is given by the Stokes law, Q, = 67tT CT for stick boundary conditions or ( = 4jit ct for slip boundary conditions. For smaller particles, kinetic and mode coupling theories, as well as considerations based on microscopic boundary layers, show that the friction coefficient can be written approximately in terms of microscopic and hydrodynamic contributions as ( 1 = (,(H 1 + (,/( 1. The physical basis of this form can be understood as follows for a B particle with radius ct a hydrodynamic description of the solvent should... 

Perrin [223] extended Debye s theory of rotational relaxation to consider spheroids and ellipsoids. Using Edwards analysis [224] of the torque on such bodies, Perrin found two or three rotational relaxation times, respectively. However, except for bodies very far from spherical, these times are within a factor of two of the Debye rotational times [eqn. (108)] for stick boundary conditions. 

Since then, many studies have been made of the rotational relaxation of bodies of various shapes with slip or stick boundary conditions [225— 228]. These studies all show that, for inertialess particles, the rotational relaxation time is proportional to the coefficient of viscosity. Furthermore, the coefficient of proportionality is not very different from V/kBT [see eqn. (108)]. 

Bauer et al. [237] and Alms et al. [238] have studied a wide range of organic molecules (benzene, mesitylene, methyl iodide, nitrobenzene, etc.) in solution. They have compared the measured long-time rotational relaxation times with both Perrin s ellipsoid rotational times with stick boundary conditions [223] and with those from Hu and Zwanzig s similar calculation based on slip boundary conditions [227]. There is close agreement between experiment and the slip boundary condition model of Hu and Zwanzig. Typical rotational times could be expressed as... 

It is to be noted that in the above discussion although the numerical values of the prefactor is close to 6n, it does not in any way imply the stick boundary condition. The above calculation is based only on microscopic considerations on the other hand, the boundary condition can only be obtained by studying the somewhat macroscopic velocity profile of the solvent. Thus, the main point here is that in the high-density liquid regime, the ratio of the friction to the viscosity attains a constant value independent of the density and the temperature. 

The effect of ion concentrations appears only through the ionic conductivity a, as the parameters p and p for the ionic and dipole depolarization contributions to Ae are either 2/3 or 1 for "slip" or "stick" boundary conditions at the ion surface, and the factor (e - edipole contribution (e - e ) to the sum e of dipole and induced dipole (e ) polarizations of solvent. 

In a series of papers, Felderhof has devised various methods to solve anew one- and two-sphere Stokes flow problems. First, the classical method of reflections (Happel and Brenner, 1965) was modified and employed to examine two-sphere interactions with mixed slip-stick boundary conditions (Felderhof, 1977 Renland et al, 1978). A novel feature of the latter approach is the use of superposition of forces rather than of velocities as such, the mobility matrix (rather than its inverse, the grand resistance matrix) was derived. Calculations based thereon proved easier, and convergence was more rapid explicit results through terms of 0(/T7) were derived, where p is the nondimensional center-to-center distance between spheres. In a related work, Schmitz and Felderhof (1978) solved Stokes equations around a sphere by the so-called Cartesian ansatz method, avoiding the use of spherical coordinates. They also devised a second method (Schmitz and Felderhof, 1982a), in which... 

The orientation relaxation time of I in ethanol was found to be very close to that predicted for slip boundary condition. Important point of our results is that the rotational diffusion in ethanol is faster than that expected for the stick boundary condition. Our results may suggest that the strong hydrogen bonding between the solvent molecules allows the solute to rotate more freely within the solvent cage than in other polar solvents. 

The important characteristic is that the size dependence in this regime is R. On the other hand, in the continuum limit the Stokes-Einstein result applies, where for stick boundary conditions... 

Cl(jj) (w 25 ps ) and not two. We note that the initial decay of Cy(t) corresponds to an apparent solvent drag which is much smaller than hydrodynamic estimates that assume stick boundary conditions. For example, treating the methyl group as a sphere of radius a = 2.5 X (the van der Waals radius) to obtain the frictional coefficient, f, f = Sirria (ri is the shear viscosity, 0.01 P) we find 2I/f equal to 0.0003 ps, i.e., about 150 times smaller than that observed. In this sense, the observed drag is nearer to the hydrodynamic slip boundary condition limit the exact slip limit for a sphere corresponds to f = 0 and an infinite relaxation time. The relatively long relaxation time is consistent with the results of experimental studies of the rotational motion of small nonassoclated molecules ( ). 

The shppage at the interface between a thin film of density Amf and the substrate is usually described in terms of an interfacial friction coefficient ( coefficient of shding friction ), x- This coefficient determines the stress acting between the film and the substrate, which move at different velocities. An infinite value of x implies that the non-sHp (sticking) boundary condition is applicable. When the interfacial friction coefficient equals zero, the film is free to slide with no energy dissipation. 

Hu and Zwanzig (1974) have performed hydrodynamic calculations of the rotational friction coefficents of prolate and oblate ellipsoids as a function of the axial ratio using slip boundary conditions. The ratio of the friction calculated with slip to that calculated with stick boundary conditions is shown in Fig. 7.8.3 as a function of the axial ratio p. 

Fig. 7.8.3. The ratio of the friction coefficient calculated using slip-boundary conditions to that calculated using stick-boundary conditions for prolate and oblate ellipsoids versus p, the ratio of the shorter to longer axis. (From Bauer, et al., 1974.)...

Solutions of macromolecules are often sufficiently dilute that Eq. (13.5.21) applies. Moreover for large molecules can be computed from hydrodynamics. For a sphere with stick boundary conditions Cs = 6jirjas. Thus in dilute solutions D° and thereby as, the particle radius, can be determined (see Chapters 5 and 8). Since D° depends on the temperature and the solvent, it is important to report the data in a standardized manner. Usually the measurements are performed at room temperature and are extrapolated to inifinite dilution. Thus for example the notation D%0,a denotes the diffusion coefficient of the solute at 20°C in the solvent H20 extrapolated to infinite dilution. For nonideal solutions... 

The linearity of the plot in Fig. 7 suggests that e acquires a similar hydrodynamic volume (F) in each of these alcohols, but when an effective radius is calculated from V (assuming stick boundary conditions) from the slope, the result is clearly too small, / =1.1 A. In fact, this is a typical result for rotating molecules where extensive slip is actually occurring. Thus a linear relationship between and i is not sufficient evidence to ignore the molecular aspects of the liquid dynamics. 

Figure 7.10 Illustration of Stick Boundary Condition. Solute biological macromolecule shown as single grey sphere moves with an average velocity through water (shown as light blue spheres). Water molecules in immediate hydration layer move at the same average velocity due to tight hydration interactions. Under Slip Boundary Conditions, water molecules do not possess hydration interactions and therefore do not move with the biological macromolecule at all.

In the case of the translational frictional forces opposing translational motion, there are two extremes known as the stick boundary condition (strong intermolecular interactions) and the slip boundary condition (negligible intermolecular interactions) that are characterised by the following two basic equations ((7.2) and (7.3) respectively) ... 

Streamline plots, with the change in shade indicating the final position of points that were initially in a line perpendicular to the weld, (a) and (c) use a stick boundary condition (b) and (d) use a slip model, with a limiting shear stress of 40 MPa (5 ksi). Adapted from Ref 45... 

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