Materials Chemistry and Physics 66 (2000) 126–131 Multicomponent diffusion in solutions where crystals grow A. Vergara, L. Paduano, V. Vitagliano, R. Sartorio ∗ Dipartimento di Chimica dell’Università “Federico II” di Napoli, via Mezzocannone 4, 80134 Napoli, Italy Abstract The correct study of diffusive time evolution of concentration boundaries in n-component systems requires the use of all the (n−1) 2 diffusion coefficients defined by Fick’s law. However, to simplify the analysis, the so-called pseudo-binary approximation is very often used. This can lead to very misleading results. On the other hand, the possibility to predict the diffusional behaviour of n-component systems from the properties of corresponding binaries and from the knowledge of the solute–solute cross-interactions should be a very important goal. If no “chemical” solute–solute interactions are present in solution, the diffusion coefficients depend only on the “hydrodynamic” or volumetric solute–solute interactions. This contribution, which is mostly reflected in the cross-diffusion coefficient values, is always present and assumes an important role in solutions containing macromolecular solutes. It is then very important in modelling the diffusion phenomena in systems where a protein can crystallise in the presence of polymeric solutes as precipitating agents. The present paper is devoted to the study of the hydrodynamic effects on the diffusion coefficients of poly(ethyleneglycol) samples, which is one of the widely precipitating agents used in the protein precipitation. A predictive model to evaluate the diffusion coefficients in the presence of the only hydrodynamic effect was applied with good success to the systems presented and to a literature system NaCl–lysozyme–water. © 2000 Published by Elsevier Science S.A. Keywords: Multicomponent diffusion; Polyethyleneglycol; Hydrodynamic interactions; Protein crystallisation 1. Introduction The correct description of the diffusive phenomena in an n-component system (solute 1, solute 2, ... , solute n−1, solvent 0) is based on the generalised Fick equation [1]: J i =− n−1  j =1 D ij ∇C j , i = 1,... ,n − 1 (1) with (n−1) 2 diffusion coefficients. Here the D ii are the main-terms diffusion coefficients relating the flux of a com- ponent to its own concentration gradient, and the D ij (i=j) are the cross-diffusion coefficients relating the flux of each component to the concentration gradient of the others. Nevertheless in studying the time evolution of real bound- aries, the so-called pseudo-binary approximation is widely used. In this approach the cross-diffusion coefficients are supposed to be zero and the main terms are assumed to be equal to the corresponding binary diffusion coefficients. The use of the pseudo-binary description for a multicomponent system is in principle incorrect and leads to serious errors for at least three reasons: (1) the cross-diffusion coefficients are usually not zero and in many cases they can be of the same order of magnitude or even larger than the main terms ∗ Corresponding author. [2], (2) the presence of another solute can substantially af- fect the diffusion of component i under its concentration gradient and then the D ii can be very different from the corresponding D i values [3], (3) in some real boundaries, very large concentration gradients can set up and then, even in the presence of small cross-terms, the contributions of cross-flows are not negligible [4]. Regarding points (1) and (2), the cases of large cross-terms and significant difference between the D ii and D i are essen- tially three: (i) strong electrostatic solute–solute interactions (electrolyte mixtures [5]), (ii) chemical equilibrium between the solutes (enzyme–substrate, cyclodextrin–host [6]), (iii) the presence of a macromolecule in solution (polymers [7] and proteins [8], etc.). Regarding point (3), large concentra- tion gradients are present in asymmetric membrane prepa- ration by casting method [9], in the release of the guest to a lipidic membrane by a cyclodextrin–host complex [10], in protein growth in the presence of some precipitating solute [11], and in any case when phase transition phenomena oc- cur. In spite of the recommendations of Gosting [12], al- ready in the 1950s, and recently of Lin et al. [13] about the necessity to include cross-terms in diffusive models, the pseudo-binary approximation has been widely used in cytoplasmatic systems [14], and to model crystal growth of proteins [15] and virus [16]. 0254-0584/00/$ – see front matter © 2000 Published by Elsevier Science S.A. PII:S0254-0584(00)00308-4