Authors: J.N. Zemel
Affilation: University of Pennsylvania and Scitefair International Inc, United States
Pages: 469 - 47
Keywords: Surface recombination velocity model, fluid flow, microchannels.
A number of years ago, Eyring introduced the free volume concept to explain why the viscosity of liquids decreased with increasing temperature while at the same time the density for most liquids also decreased (water at 0?C being an outstanding and fortunate exception)(1,2). The coefficient of expansion of solids has been reasonably accounted for in terms of the increase in phonon density as the temperature increases. The obvious difference between liquids and solids is the structural order in solids and the general absence of a corresponding long range order in most simple liquids. Eyring's concept of a "free volume" creates space within the liquid as it expands. There is a corresponding concept in solids, namely vacancies. The presence of vacancies in solids has been used extensively to understand their mechanical, electrical and optical properties. The diffusion of vacancies to and from an interface with their subsequent destruction or creation has been often used to explain the annealing of solids. Vacancies or "holes" also arise in electron transport in semiconductors. Both vacancies and holes arise naturally by thermal activation and exist at thermal equilibrium in solids. However, it is possible to generate excess vacancies and holes over and above their equilibrium value. Given enough time, the holes will recombine with the excess electrons and the excess vacancies will recombine with either with excess interstitial molecules or will diffuse to the surface where they disappear. In the former case, the process is accelerated due to the charge state of holes and electrons while in the latter case, bulk vacancies are generally removed by thermal processes (even though in some instances, stress fields can accelerate the process). What is intriguing about the electronhole situation is that there is excess current flow so long as the excess carriers exist. The removal of the electron-hole pairs is due to recombination process in the bulk and at the surface. All recombination processes obeys the reaction e+p <=> O which is identical to the generation recombination process for vacancies, V+SO where e represents the electron, p represents the hole, V is the vacancy and S represents ether an nterstitial molecule or a surface molecule. In this paper, we examine this analogy a bit further, following Eyring's original reasoning, by associating the motion of a liquid as a vacancy process. However, the generally accepted view that the flow of liquids is incompressible would imply that vacancy generation cannot occur because of continuity requirements. Our view is that the actual change in density due to excess vacancy generation is sufficiently small so that continuity still applies. By focusing on the vacancy motion rather than the molecular motion, some simplicity is introduced into the problem. It is the presence of excess vacancies due to shearing of the liquid that mediate the flow. What we are interested in is to see if the same approach used in electron-hole recombination can be applied to the flow process, especially in very small channels. The model proposed is greatly simplified. First, we assume thatthe motion of the liquid is associated with the density of excess vacancies generated by the shearing process in the medium. Second, we assume that the flow is laminar in order to have a simple shear distribution. The justification for the laminar assumption is that we will apply this model to data taken by Urbanek et al. in microchannels that have hydraulic diameters, d, in the 3-10 ,um range (3). The flow studied had suffciently small Reynold numbers so that laminar flow was assured. Third, we assume that a no-slip boundary condition applies to this problem. In a sense, this assumption is somewhat contradictory since the shear is then greatest at the walls, just where the flow is going to zero. However, there already exists a large density of vacancies due to therrnal equilibrium in the liquid so that the additional shear induced vacancies are a perturbation on the overall concentration. By increasing the temperature, the equilibrium concentration of vacancies should increase but we would anticipate that the diffusion coeffcient of the excess vacancies will also increase. This will allow more vacancies to survive, thereby reducing the overall viscosity as observed. If there is a recombination process at the walls of the channel, the excess density of vacancies should decrease more rapidly not only at the surface, but also in the bulk of the liquid as a result of diffusion. The concentration of the excess vacancies would involve not only the generation rate due to fluid motion but also the relative rate of recombination at the walls to the the bulk. This is very similar to the surface recombination of electrons and holes in semiconductors. We will present the model and discuss its application to available data.
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