The Uniform Thin Film Approximation UTFA

For many of the applications relevant to this volume, nanoscale particles such as proteins are added to the interface, and the question arises whether the resulting monolayer deposit, which is inherently heterogeneous, can be modelled as a homogeneous, uniform thin film. Were that to be the case the analysis of the raw data would be much easier than otherwise. Careful comparison of results assuming such a film with explicit consideration (using Mie theory) of the actual heterogeneity of such layers has revealed that the UTFA is an excellent approximation up to particle diameters of around 50 nm (Mann et al. 1997). These films are often isotropic, or anisotropic to a negligible degree, which means that they are fully characterized by just two parameters, refractive index nA and geometrical thickness dA4.

In OWLS, these two parameters can be obtained by a straightforward analytical solution of two simultaneous mode equations using the effective refractive indices of two independent modes as the input. A disadvantage of

4 One way in which the assumptions of the UTFA could be invalidated even though minute nanoparticles are being adsorbed as a monolayer is if those particles cluster together to create large islands. Such behaviour is evidenced by a broadening of the TM incoupling peak but not the TE one (J.J. Ramsden and T. Jaworek, unpublished observations), and has a definite kinetic signature as discussed in Sect. 2.6.1.

the other optical methods (viz., SAR, ellipsometry and SPR) is that tedious fitting calculations are required to extract nA and dA. While the general availability of powerful personal computers makes this less of a problem than formerly, the fitting procedure is itself not robust insofar as a large family of possible solutions satisfy the equations.

In many applications, however, only the number v of particles per unit area of interface (or the amount M of protein deposited on a membrane) is required, and this can be straightforwardly obtained from any nA and dA pair yielding a solution5 by v = M/m = dA(nA - nC)/(m dn/dc) (18)

where m is the mass of one adsorbate particle (e.g. a single protein molecule) and dn/dc is the refractive index increment (rii) of the adsorbate solution. The rii simply expresses the slope of a plot of solution refractive index versus the concentration of protein which it contains. For a given protein dissolved in a given solvent, this has been shown to be linear up to concentrations of around 0.5 g/ml, in which range even dense monolayers comfortably fall (de Feijter et al. 1978). Some common and other blood proteins dissolved in common buffers used in biological research all have a value of about 0.18 cm3/g (de Feijter et al. 1978, Ball and Ramsden 1998).

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