Solubility-diffusion model

The ideas of Overton are reflected in the classical solubility-diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst-Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the... 

Meanwhile, computational methods have reached a level of sophistication that makes them an important complement to experimental work. These methods take into account the inhomogeneities of the bilayer, and present molecular details contrary to the continuum models like the classical solubility-diffusion model. The first solutes for which permeation through (polymeric) membranes was described using MD simulations were small molecules like methane and helium [128]. Soon after this, the passage of biologically more interesting molecules like water and protons [129,130] and sodium and chloride ions [131] over lipid membranes was considered. We will come back to this later in this section. 

If the pore-mechanism applies, the rate of permeation should be related to the probability at which pores of sufficient size and depth appear in the bilayer. The correlation is given by the semi-empirical model of Hamilton and Kaler [150], which predicts a much stronger dependence on the thickness d of the membrane than the solubility-diffusion model (proportional to exp(-d) instead of the 1 Id dependence given in equation (14)). This has been confirmed for potassium by experiments with bilayers composed of lipids with different hydrocarbon chain lengths [148], The sensitivity to the solute size, however, is in the model of Hamilton and Kaler much less pronounced than in the solubility-diffusion model. 

The rate of transmembrane diffusion of ions and molecules across a membrane is usually described in terms of a permeability constant (P), defined so that the unitary flux of molecules per unit time [J) across the membrane is 7 = P(co - f,), where co and Ci are the concentrations of the permeant species on opposite sides of membrane correspondingly, P has units of cm s. Two theoretical models have been proposed to account for solute permeation of bilayer membranes. The most generally accepted description for polar nonelectrolytes is the solubility-diffusion model [24]. This model treats the membrane as a thin slab of hydrophobic matter embedded in an aqueous environment. To cross the membrane, the permeating particle dissolves in the hydrophobic region of the membrane, diffuses to the opposite interface, and leaves the membrane by redissolving in the second aqueous phase. If the membrane thickness and the diffusion and partition coefficients of the permeating species are known, the permeability coefficient can be calculated. In some cases, the permeabilities of small molecules (water, urea) and ions (proton, potassium ion) calculated from the solubility-diffusion model are much smaller than experimentally observed values. This has led to an alternative model wherein permeation occurs through transient hydrophilic defects, or pores , formed by thermal fluctuations of surfactant monomers in the membrane [25]. 

Computer simulations of both equilibrium and dynamic properties of small solutes indicate that the solubility-diffusion model is not an accurate approximation to the behavior of small, neutral solutes in membranes. This conclusion is supported experimentally [57]. Clearly, packing and ordering effects, as well as electrostatic solute-solvent interactions need to be included. One extreme example are changes in membrane permeability near the gel-liquid crystalline phase transition temperature [56]. Another example is unassisted ion transport across membranes, discussed in the following section. 

When a polymer film is exposed to a gas or vapour at one side and to vacuum or low pressure at the other, the mechanism generally accepted for the penetrant transport is an activated solution-diffusion model. The gas dissolved in the film surface diffuses through the film by a series of activated steps and evaporates at the lower pressure side. It is clear that both solubility and diffusivity are involved and that the polymer molecular and morphological features will affect the penetrant transport behaviour. Some of the chemical and morphological modification that have been observed for some epoxy-water systems to induce changes of the solubility and diffusivity will be briefly reviewed. 

A rather important aspeet that should be eonsidered is that interfaeial quenching of dyes does not neeessarily imply an eleetron-transfer step. Indeed, many photoehemieal reactions involving anthracene oeeur via energy transfer rather than ET [128]. A way to discern between both kinds of meehanisms is via monitoring the accumulation of photoproducts at the interfaee. Eor instance, heterogeneous quenehing of water-soluble porphyrins by TCNQ at the water-toluene interfaee showed a elear accumulation of the radical TCNQ under illumination [129]. This system was also analyzed within the framework of the exeited-state diffusion model where time-resolved absorption of the porphyrin triplet state provided a quenehing rate eonstant of the order of 92M ems. ... 

Mooney et al. [70] investigated the effect of pH on the solubility and dissolution of ionizable drugs based on a film model with total component material balances for reactive species, proposed by Olander. McNamara and Amidon [71] developed a convective diffusion model that included the effects of ionization at the solid-liquid surface and irreversible reaction of the dissolved species in the hydrodynamic boundary layer. Jinno et al. [72], and Kasim et al. [73] investigated the combined effects of pH and surfactants on the dissolution of the ionizable, poorly water-soluble BCS Class II weak acid NSAIDs piroxicam and ketoprofen, respectively. 

The permeation of small molecules in amorphous polymers is typically following the solution diffusion model, that is, the permeability P of a feed component i can be envisioned as the product of the respective solubility S and constant of diffusion Dj. Both parameters can be obtained experimentally and in principle also by atomistic simulations. 

Experimental parameters or variables involved included solubility, diffusivity, suspension concentration, release opening radius, declination angle, and number of cells. Experimental results indicated that this device behaved in accordance with the theoretical model. 

The use of models and particularly those of a sophisticated nature is, however, seriously restricted by the limitations of the parameters involved in the model equations. It is actually the determination of certain parameters which becomes the crucial point in designing. The major uncertainties originate from two sources. Firstly, the process data, i.e. estimation of phase equilibria (solubilities), diffusivities and especially kinetic rate data,involves inaccuracies. The second major source of large uncertainties is the reliability of the nonadjustable hydrodynamic quantities. 

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