Hydrodynamic theory, direct molecular

The main message of these lectures is the need to amend the classical hydrodynamic theory by direct inclusion of intermolecular interactions. This is necessary not only in the theory of contact line motion outlined here, but in all mesoscopic hydrodynamic problems, e.g. in fluid mechanics of microdevices, which attracts lately a lot of attention. The specific feature of the contact line problem is the connection between microscopic and macroscopic. The motion in the precursor film can and should be treated more precisely, on the statistical level with due account for fluctuations or directly through molecular dynamics simulations. A challenging problem is matching the microscopic theory with classical hydrodynamics applicable in macroscopic domains away from the immediate vicinity of the contact line. 

In order to apply the Smoluchowski equation (Equations (1.3), (2.1), (3.29)), we need values for the least distance of approach (rAn) and the diffusion coefficient (Dab)- The value of tab can be estimated from molecular volumes (Section 2.5.1.2). The diffusion coefficient can be determined by various methods, but experimental values are available only for a minority of the myriad possible situations. A common practice is to use the Stokes-Einstein relation (Section 1.2.3), which rests on the assumption that solute molecules in motion behave like macroscopic particles to which classical hydrodynamic theory can be applied. We shall first outline (a) the relation between the diffusion coefficient D and the mechanics of motion of particles in fluids, leading to the Stokes-Einstein equation relating D to solute size and solvent viscosity and (b) the direct experimental determination of D. We shall then (c) compare the results and note the reservations that are required in relying on the Stokes-Einstein estimates of D in various cases. 

These results make it clear that the forms of t]0 — rjs and Je° are completely independent of model details. Only the numerical coefficient of Je° contains information on the properties of the model, and even then the result depends on both molecular asymmetry and flexibility. Furthermore, polydispersity effects are the same in all such free-draining models. The forms from the Rouse theory cany over directly, so that t]0 - t]s, translated to macroscopic terms, is proportional to Mw and Je° is proportional to the factor A/2M2+, /A/w. Unfortunately, no such general analysis has been made for models with intramolecular hydrodynamic interaction, and of course these results apply in principle only to cases where intermolecular interactions are negligible. 

One of these proposed a non-equilibrium process in which the separation was controlled by differing rates of diffusion for different molecular masses [23]. Other workers have proposed a separation by flow mechanism [24, 25] in which the larger molecules are excluded from the surface of the gel particles and remain in the centre of the solvent channels and are thus eluted first. The original theory did not invoke a porous structure for the gel, but this was modified later. The mechanism bears resemblance to that proposed for hydrodynamic chromatography (see Chapter 10). A further model suggested that the pore size distribution of the gel was directly responsible for its ability to separate molecules by size, and that there is a one-to-one correspondence with size of pores and size of molecules [26]. All these theories have been critically reviewed in the book by Yau et al [6]. 

Modern synthetic methods allow preparation of highly monodisperse spherical particles that at least approach closely the behavior of hard-spheres, in that interactions other than volume exclusion have only small influences on the thermodynamic properties of the system. These particles provide simple model systems for comparison with theories of colloidal dynamics. Because the hard-sphere potential energy is 0 or 00, the thermodynamic and static structural properties of a hard-sphere system are determined by the volume fraction of the spheres but are not affected by the temperature. Solutions of hard spheres are not simple hard-sphere systems. At very small separations, the molecular granularity of the solvent modifies the direct and hydrodynamic interactions between suspended particles. 

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