Multicomponent aqueous ion diffusion  

Thread by Akshay Chibber on 03 Mar 2009 at 15:28:54 
I have an interest in getting diffusion rates for metal ions through cation exchange membranes or other cationic media (such as ion exchange resins) in an aqueous environment. Example: diffusion rate for K+ ions across a 1 cm barrier of sulfonated poly styrene particles, where [K+] High=1M, and [K+] Lo is 0.0.01M (all components are in water, and the resin is in the K+ form).

For the sake of experimental design, what is the best reputable calculation method to acquire insight in the multicomponent diffusion coefficients of ions and water in polymers and multilayer laminate materials? I read on a migration model (using molecular weight, mol / gram) by Hinrichs and Piringer, but I am not convinced of the applicability of the migration model to my problem (and the rigorous scientific character of it).

Akshay Chibber

    Comment by Composite Analytica on 06 Mar 2009 at 17:07:59  | |responses: 0|
    Hi Akshay, furher to your questions:

    1. Diffusion coefficient according to the free volume theory

    Probably still the best theory for calculation of an unknown diffusion coefficient of a liquid or gas in a polymer, is the free volume theory for diffusion. The theory is developed by Cohen and Turnbull (1959) who considered transport in a liquid of hard spheres. Molecules reside, most of the time, in cages bound by their neighbours. Occasionaly a fluctuation in density opens up a hole within a cage large enough to permit considerable displacement of the molecule contained by it. Succesful diffusive transport occurs if another molecule jumps into the hole before the first can return to its original position. In the model of Cohen and Turnbull, diffusion is treated as translation of a molecule across the void within its cage.

    Diffusion occurs not only as a result of an activation in the ordinary sense, but rather as a result of redistribution of the free volume within the liquid or frozen liquid: a polymer.

    Important advantage of the volume theory over other theories is the limited amount of parameters, which can be calculated rather well by using the work from Guggenheim (for the free volume of the liquid) and Positron Lifetime Spectroscopic information (for the free volume in the polymer and resins).

    2. Maxwell - Stefan mass balance for (multicomponent) diffusion

    For studying permeation of one or more gas or liquid components diffusing alone (binary diffuson) or simultaneously (multicomponent) in a polymer or composite material, a mass balance is essential.

    The mass balance is used for interpretation of the diffusion experiment, according to gravimetric methods (ASTM D570, ASTM E96) or gas chromatography measurements (ASTM D1434) and subsequently for application of the diffusive mass transfer in the real life application, such as mass transport through a membrane, containment, pipeline or package.

    Usually Fick's first and second laws are used to balance the driving force (the concentration gradient) with the friction force (the diffusivity).

    This goes well as long as we have [1] binary diffusion + [2] the concentration gradients acting as the only driving force + [3] the solubility obeys Henry Law. If we have to consider i.e. multicomponent diffusion, Fick's laws hardly make any sense.

    The Maxwell - Stefan equation uses the chemical potential gradient as the driving force for diffusing chemicals. The motion with respect to other chemicals (especially the polymer matrix) causes friction. The driving force is equal to the sum of these friction forces, because acceleration effects are negligible in diffusion.

    Composite Analytica