A repetitive sequence of quiescent fluid layers of differing viscosities through which small spherical Brownian particles move is analyzed in order to illustrate in a simple context how the theory of macrotransport processes, a generalization of Taylor dispersion theory, may be employed to rigorously analyze spatially periodic micropatterned chromatographic separation devices for circumstances in which the solute species to be separated are animated by the action of species-specific external forces oriented asymmetrically relative to the body-fixed pattern. In the generic "vector" separation scheme, illustrated by our elementary example, the different species undergoing separation move, on average, in different directions relative to pattern-fixed axes, whence their chromatographic sorting is effected according to their different mean angular trajectories through the device. This scheme differs fundamentally from traditional "scalar" chromatographic separation schemes, wherein all species move on average parallel to the animating force (including circumstances in which they are passively entrained in a unidirectional solvent flow) and hence for which the sorting is effected by the relative speeds of the several species through the chromatographic column. Vector chromatography is quantified by two global "macrotransport coefficients," namely the solute mobility dyadic M̄* (representing the tensor proportionality coefficient between the mean solute velocity vector Ū* and the external force vector F acting upon the solute molecules) and the dispersivity dyadic D̄* (resulting from the deviation of the instantaneous position of the particle from its mean position based upon its mean velocity vector). In the present example these coefficients are studied parametrically as functions of: (i) the orientation of the external force relative to the symmetry axis of the fluid layers; (ii) the local viscosity distribution within a layer; (iii) the vector particle Peclet number (constructed from the vector force, the length of the viscosity period, and the Boltzmann factor kT); and (iv) the thermodynamic interphase solute partition distribution coefficient between the two fluid layers comprising a unit cell.
Bibliographical noteFunding Information:
This work was partially supported by a Graduate Research Fellowship from the National Science Foundation to K.D.D. and a grant to H.B. from the Office of Basic Energy Sciences of the U.S. Department of Energy, as well as a “Grand Challenge Award” from the Mathematical, Information and Computational Sciences Division within the Office of Computational and Technology Research of the Office of Energy Research of the D.O.E.
- Generalized Taylor dispersion