A mathematical model of electrophoretic separation processes has been developed and adapted for computer simulations. The model is used to predict the characteristic behavior of a variety of electrophoretic techniques from a knowledge of chemical equilibria and physical transport phenomena. The model provides a unifying basis for a rational classification of all electrophoretic processes.
A general model is developed for the electrophoresis of soluble materials. The model describes the evolution of concentration fields for a set of compounds which undergo transport by flow, diffusion, and migration in an electric field and simultaneously participate in rapid dissociation-association reactions. Modes of electrophoresis requiring special treatment can now be studied in a unified context. As an example of its utility, the model is used analytically to study a process known as isotachophoresis. In Part I1 two electrophoretic separation processes are simulated numerically, demonstrating the model's versatility. D. A. SAVILLE Department of Chemical EngineeringPrinceton University Princeton, NJ 08544 A. PALUSINSKI Biophysics Technology Laboratory and Department of Electrical and Computer Engineering University of ArizonaTucson, AZ 85721 SCOPEWhen a solution containing amphoteric compounds is exposed to an electric field, the migration of ions and uncharged species occurs in the presence of rapid dissociation-recombination reactions. If a zone containingseveral species is inserted in a column containinga homogeneous buffer, the various species will migrate at different rates according to their relative electrophoretic mobilities. Conversely, if the ends of the column are made impermeable to the amphoteric species, a stationary pH gradient will eventually be formed and sample constituents will migrate to their equilibrium isoelectric points under the influence ofthe electric field. These phenomena form the basis of several separation methodologies that have been difficult to model mathematically. The purpose of this paper is to present a generally applicable model of electrophoretic processes and apply it to a specific situation. The application is designed to illustrate the interplay between reaction, electromigration, and diffusion in simple configurations where complications due to lateral boundaries, bulk flow, and nonuniform temperature are supressed. The situation studied in detail is isotachophoresis in a one-dimensional column; more diverse applications that require numerical treatment of the equations are described in Part 11. CONCLUSIONS AND SIGNIFICANCEAlthough electrophoretic processes can be described by familiar conservation relations, the structure of the mathematical model differs from that used to describe systems with strong electrolytes due to the presence of rapid dissociation-recombination reactions that tie the concentrations of ionic species to those of the undissociated solutes. It is shown that since the reactions are fast relative to transport by diffusion and electromigration, it is possible to treat the reactions as being in local equilibrium. Similarly, the ratio of an electrical length (the Debye scale) to the physical scale of the process is small, and this leads to the electroneutrality approximation. The model that is developed consists of a set of conservation equations for the total concentration of each amphoteric compound and the current. Appended to this set of partial...
The mathematical model outlined in Part I is recast in a form suitable for numerical computation. The spatial derivatives are replaced by finite-difference expressions, which leads to a set of ordinary differential equations coupled to a set of nonlinear algebraic relations. This system is solved using existing integration techniques. The resulting algorithm simulates the characteristic behavior of the classical modes of electrophoresis, which is shown by examples involving moving boundary electrophoresis and isoelectric focusing. In the first example two different integration schemes are used and their accuracy and stability investigated. The second example illustrates the versatility of the methodology. 0. SCOPEIn the model presented in Part I (Saville and Palusinski), the electrophoresis of amphoteric compounds is described by a set of partial differential equations coupled to a system of algebraic equations. Separation of sample components arises from interactions between the chemical equilibria and the transport processes. These interactions alter the effective mobilities of the various species and induce them to separate under the action of the electric field in nonequilibrium processes such as isotachophoresis. In equilibrium processes such as isoelectric focusing, the action of the field produces a pH gradient and the amphoteric constituents move to positions where they are isoelectric. In either case, the evolution of the process is best followed by numerical methods. The purpose of this paper is twofold: 1. To show how the mathematical model derived in Part I can be expressed in a form suitable for numerical computation. 2.To demonstrate the model depicts the detailed characteristics of electrophoretic processes.The numerical algorithm selected employs a five-point finite-difference expression to approximate the spatial derivatives at a set of mesh points. This converts the set of partial differential equations into a set of ordinary differential equations describing the temporal evolution of the concentration fields at each mesh point. These equations can be integrated using any one of a variety of schemes for solving sets of firstorder ordinary differential equations. Boundary conditions at either end of a separation column are incorporated by adjusting the form of the finite-difference expressions at the boundary points. In a similar fashion, the algorithm can easily be adapted for simulation of novel separation methods such as the use of immobilized ampholytes, molecular sieving, or ion-selective membranes at the boundaries. CONCLUSIONS AND SIGNIFICANCEThe implementation of the algorithm describing electrophoretic transport processes is illustrated by simulating moving boundary electrophoresis and isoelectric focusing with immobilized species. Both examples are intended to illustrate particular electrophoretic processes using rather simple systems. Attention concentrates on the essential characteristics of a particular mode, For the isoelectric focusing example, the central feature is the migration...
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