In this section, we assume that an emulsifier molecule is adsorbed to an interface as soon as it encounters it (i.e., there are no energy barriers that retard adsorption). In an isothermal quiescent liquid, emulsifier molecules move from a bulk liquid to an interface by molecular diffusion, with an initial adsorption rate given by (Tadros and Vincent 1983, Magdassi and Kamyshny 1996):

where D is the translational diffusion coefficient, r is the surface excess concentration, t is the time, and c is the concentration of emulsifier initially in the bulk liquid. The variation of the surface excess concentration with time is obtained by integrating this equation with respect to time:

Thus, a plot of the surface excess concentration versus Vt should be a straight line that passes through the origin. This equation indicates that the accumulation of an emulsifier at an interface occurs more rapidly as the concentration of emulsifier in the bulk liquid increases or as the diffusion coefficient of the emulsifier increases. The diffusion coefficient increases as the size of molecules decreases, and one would therefore expect smaller molecules to adsorb more rapidly than larger ones. Experiments with proteins have shown that Equation 5.11 gives a good description of the early stages of adsorption to clean interfaces (Damodaran 1989, Walstra 1996b). After the initial stages, the adsorption rate decreases because the interface becomes saturated with emulsifier molecules and therefore there are fewer sites available for the emulsifier to adsorb to (Figure 5.11). In practice, the rate may be faster than that given by Equation 5.10 because of convection currents caused by temperature gradients within a liquid. Consequently, considerable care must be taken to

ensure that the temperature within a sample is uniform when measuring diffusion-controlled adsorption processes.

The above equations do not apply during the homogenization of emulsions, because homogenization is a highly dynamic process and mass transport is governed mainly by convection rather than diffusion (Dickinson 1992, Walstra 1996b). Under isotropic turbulent conditions, the initial increase of the surface excess concentration with time is given by (Dukhin et al. 1995):

ld y

where C is a constant which depends on the experimental conditions, and rd and re are the radii of the droplet and emulsifier, respectively. This equation predicts that the adsorption rate increases as the concentration of emulsifier increases, the size of the emulsion droplets increases, or the size of the emulsifier molecules increases relative to the size of the droplets. This equation implies that when an emulsion is homogenized, the emulsifier molecules initially adsorb preferentially to the larger droplets and that larger emulsifier molecules adsorb more rapidly than smaller ones (which is the opposite of diffusion-controlled adsorption). This explains why large casein micelles adsorb faster than individual casein molecules during the homogenization of milk (Mulder and Walstra 1974).

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