Transport processes friction

Most descriptions of the dynamics of molecular or particle motion in solution require a knowledge of the frictional properties of the system. This is especially true for polymer solutions, colloidal suspensions, molecular transport processes, and biomolecular conformational changes. Particle friction also plays an important role in the calculation of diffusion-influenced reaction rates, which will be discussed later. Solvent multiparticle collision dynamics, in conjunction with molecular dynamics of solute particles, provides a means to study such systems. In this section we show how the frictional properties and hydrodynamic interactions among solute or colloidal particles can be studied using hybrid MPC-MD schemes. 

The analogy for transport processes is readily interpreted from Stokes theory if we consider the generalization that forces or fluxes of a property are proportional to a diffusion coefficient, the surface area of the body, and a gradient in property being transported. In the case of momentum, the transfer rate is related to the frictional and pressure forces on the body. The diffusion coefficient in this case is the kinematic viscosity of the gas (vg = p-g/pg, where pg is the gas density). The momentum gradient is Pjg Uoo/B. 

In Chapter 3 we found that all relative transport processes, whether induced by external fields or diffusion, proceed at a rate inversely proportional to the friction coefficient /. Since virtually all separation methods require a certain level of completion of transport, or a certain number of transport steps, the time scale of the separation is linked to the time scale of the required transport both ultimately hinge on the magnitude of /. This conclusion is valid whether one is using methods such as chromatography where the transport processes must maintain the distribution of components between phases at a point near equilibrium, or electrophoresis where transport proceeds only fractionally to equilibrium. 

In addition to mass and energy, other quantities can also experience transfer. Flowing layers with different flow rates in a convection stream can influence one another. The slower flowing layer acts as a brake on the faster layer, while at the same time the faster layer acts to accelerate the slower one. The cause of this behavior is the inner friction of the liquid appearing as a viscosity difference, which is a consequence of the attractive forces between the molecules. Viscosity can be explained as the transport of momentum. The viscosity of different media can be very different and thus plays an important role in transport processes. 

It turns out that Eq. (5-56) can also be applied to turbulent flow over a flat plate and in a modified way to turbulent flow in a tube. It does not apply to laminar tube flow. In general, a more rigorous treatment of the governing equations is necessary when embarking on new applications of the heat-trans-fer-fluid-friction analogy, and the results do not always take the simple form of Eq. (5-56). The interested reader may consult the references at the end of the chapter for more information on this important subject. At this point, the simple analogy developed above has served to amplify ouf understanding of the physical processes in convection and to reinforce the notion that heat-transfer and viscous-transport processes are related at both the microscopic and macroscopic levels. 

Since both these transport processes depend on the frictional coefficient of the molecule in the same way, this is a very reasonable assumption, and explains why under conditions of severe retardation zones remain extremely sharp. By application of Pick s laws, Richards and Lecanidou determine the evolution of the zone half-width with time, and derive the important result, confirmed by experiment, that the zone-width, provided the ratio of diffusion coefficient to retardation factor is of the right order, depends to only a slight extent on the width of the starting zone applied at one end of the gel. For typical conditions there is no detectable loss in resolution for an initial zone width of up to 2 mm, and even for a 1 cm column of RNA solution applied to the gel there is only a 25% increase in band width. 

The friction coefficient is a quantity fundamental to most particle transport processes. The Stokes law form. / = 3jt fidp, holds for a rigid sphere that moves through a fluid at constant velocity with a Reynolds number. dpUjv, much less than unity. Here U is the velocity, and V is the kinematic viscosity. The particle must be many diameters away from any surfaces and much larger than the mean free path of the gas molecules, ip, which is about 0.065 jum at 25°C. 

In using Eqs. 19 and 34 as starting points for the development of the theory of transport processes, several tasks remain. The first and perhaps simplest is to develop expressions for the transport coefficients in terms of the single and pair densities /(1> and /< >. Next the continuity equations must be solved to give explicit expressions for the densities when the system is perturbed from its equilibrium state by the transport process under consideration. Finally, it is required to obtain the frictional coefficient f in terms of the intermolecular potential energy function according to Eq. 33. 

A precise determination of the frictional coefficient C in terms of the intermolecular potential and the radial distribution function at present constitutes the principal unresolved problem of the Brownian motion approach to liquid transport processes. It has been suggested by Kirkwood that an analysis of the molecular basis of self-diffusion might be a fruitful approach. The diffusion constant so calculated would be related to the frictional coefficient through the Einstein equation, Eq. 46. 

We will consider an instructive prototype transport process. This process is the gravity (or ultracentrifuge) induced sedimentation of a collection of identical noninteracting macroscopic particles each of mass M. Assume that the particles are suspended in a long tube filled with a viscous fluid and that the tube is aligned parallel to the Z-axis. The particles thus descend in the negative Z-direction. Further assume that the fluid exerts only a simple frictional force - z t) on each particle. Then the equation of motion for a typical particle is... 

Flammable solid A solid (other than an explosive) that ignites readily and continues to bum. It is liable to cause fires under ordinary conditions or during transportation through friction or retained heat from manufacturing or processing. It bums so vigorously and persistently as to create a serious transportation hazard. Included in this class are spontaneously combustible and water-reactive materials. An example is white phosphoms. 

In this chapter, we will re-examine these processes, but from the approach developed by Maxwell and Stefan. This approach basically involves the concept of force and friction between molecules of different types. It is from this frictional concept that the diffusion coefficient naturally arises as we shall see. We first present the diffusion of a homogeneous mixture to give the reader a good grasp of the Maxwell-Stefan approach, then later account for diffusion in a porous medium where the Knudsen diffusion as well as the viscous flow play a part in the transport process. Readers should refer to Jackson (1977) and Taylor and Krishna (1994) for more exposure to this Maxwell-Stefan approach. 

In using such equations, the transport process is considered as being macroscopic and the membrane as a black box. The factor membrane structure can be considered as an interphase in which a permeating molecule or particle experience a friction or resistance. 

Particles are the bed material employed in fluidized bed reactors and can be reactants (e.g., coal and limestone), products (e.g., polyethylene), catalysts, or inert. The choice of particle size, in general, affects the hydrodynamics, transport processes, and hence the extent of reactor conversion. Particles experience particle particle collisions, friction between particles and walls or internals, and cyclones. In some cases, the catalyst material is inherently susceptible to attrition, and special preparation to enhance the attrition resistance is required. For example, the vanadium phosphate metal oxide (VPO) catalysts developed for butane oxidation... 

The dominant forces that determine deviations from ideal behaviour of transport processes in electrolytes are the relaxation and electrophoretic forces [16]. The first of these forces was discussed by Debye [6, 17]. When the equilibrium ionic distribution is perturbed by some external force in an ionic solution, electrostatic forces appear, which will tend to restore the equilibrium distribution of the ions. There is also a hydrodynamic effect. It was first discussed by Onsager [2, 3]. Different ions in a solution will respond differently to external forces, and will thus tend to have different drift velocities The hydrodynamic (friction) forces, mediated by the solvent, will tend to equalize these velocities. The electrophoretic ( hydrodynamic) correction can be evaluated by means ofNavier-Stokes equation [18, 19]. Calculating the relaxation effect requires the evaluation of the electrostatic drag of the ions by their surroundings. The time lag of this effect is known as the Debye relaxation time. 

As stressed in section 4, induced spectra in liquids arise from fluctuations in the local density around a particle. The interest in the study of induced spectra per se is that these fluctuations are the elementary events involved in such transport processes as viscous flow, diffusion or vibrational dephasing. Consider the diffusion coefficient, it is related to a friction coefficient by... 

At room temperature, the prereduced catalyst can still react slowly with the oxygen in air, but this can only be obviously perceived after several months. In the transportation process, the mutual friction and collision among the particles leads to the local shell-ofT of the oxide film, but a new oxide film will be formed by the reoxidation in air. 

Biological macromolecules in solutions can be distinctly characterized from their transport behaviour in solution phase. The study of transport processes yields physical parameters like the diflusion coefficient, sedimentation coefficient, intrinsic viscosity, friction constant etc. of the dissolved solute molecule. These coefficients are dependent on two parameters. First, is the size and shape of the solute particle Second, is the type of the solvent medium and its environment (pH, temperature, pressure, ionic strength etc.). The solvent medium can force the diffusing particles to assume a special shape and/or to get distributed in a special fashion in space through solvent-solute interactions. At the same time a pair of solute molecules will also influence each other s behaviour and/or their physical shape and size. This process may or may not be mediated by the solvent. To account for all these mechanisms, we need to discuss the solute-solvent, solvent-solvent and solute-solute interactions. Interestingly enough, much of this information is contained in the transport coefficients of a solute and physical parameters describing a solvent. 

It is worthwhile to emphasize the constraints (4) which indeed underlies the application of Einstein s generalized friction model. The conditions (4) to (6), which have been thoroughly discussed [7, 9] for unidimensional transport processes in multi-component systems of uncharged and charged particles, are mostly fulfilled experimentally for the transport processes we shall discuss here (for a more precise account concerning this point the reader is referred to the pertaining literature [10, 11]). 

Actually, if we devise a transport process in a three component polyelectrolyte system at infinite dilution in which the salt is maintained steadily at constant chemical potential throughout the measuring cell, the dissipative process occurring in the system can be completely described by the frictional properties of the polyion. This point becomes apparent if we focus attention on the significance of the diffusion Equation (27). In the limiting case of excess salt and the condition Vju = 0, no driving force acts on the small ions the polyion in its own concentration gradient acquires with... 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

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