A simple network thermodynamic method for series-parallel coupled flows: II. The non-linear theory, with applications to coupled solute and volume flow in a series membrane.

Building on the linear network thermodynamic model presented in the first paper in this series (Mikulecky, Wiegand & Shiner, 1977), a method of non-linear analysis is developed based mainly on the methods of Chua (1969). Using coupled salt and volume flow through membranes as an example, it is shown that the Kedem & Katchalsky (1963a,b,c) linear model of series combinations of membranes behaves in a physically unrealistic manner (negative and infinite concentrations occur in the compartment between the membranes). Using the model introduced by Patlak, Goldstein & Hoffman (1963) to obtain characteristics for a non-linear 2-port, the series network is well behaved and self-regulated. In the coordinate system of the linear reciprocal 2-port, the non-linear 2-port is shown to be a non-reciprocal element. Another co-ordinate system, which utilizes unidirectional fluxes (as measured by isotope fluxes, for example) and their conjugate driving forces, shows the non-linear 2-port to be separable in the sense of Li (1962) and Rosen (1968). A set of three pseudo-independent force-flow pairs completely characterizes the system and is compatible with a very simple non-linear network analysis which utilizes experimentally accessible and practical parameters such as the unidirectional solute fluxes, concentrations in the baths, volume flow and the hydrostatic minus effective osmotic pressure. As constitutive relations, the forward and backward permeabilities, which are dependent on volume flow and the filtration coefficient are all that are needed. A reflection coefficient appears in the driving force conjugate to volume flow in that the effective osmotic pressure is δΔπ and thus four independent coefficients, two of which depend on volume flow, are needed to determine the system. From the network analysis, the global properties of the series and/or parallel combinations of non-linear 2-ports are readily obtained. Although applied to a particular example, these non-linear methods are perfectly general. They are essentially the graphical form of the algebraic analysis in the linear theory, and are essentially based on the generalized version of Kirchhoff's laws used in graph and network theory. In the particular case of coupled solute and volume flow through membranes the non-linear element has a natural piecewise linearization which allows the linear theory to be applied using different values for the elements in each region. In an initial analysis, a linear network is drawn using the I-equivalent network of 1-port elements to model the linear 2-port in a series system. From this over-simplication a great deal of the qualitative behavior of the system can be visualized. For example, it is an easy way to see that solute will be accumulated or depleted in the central compartment between the membranes in an asymmetric series system. After this the non-linear analysis is used to further quantitatively describe the system's behavior and such phenomena as rectification of volume flow in the series membrane system. An appendix introduces a general theory for a class of non-linear transport phenomena.