Adsorption kinetics under the influence of barriers at the subsurface layer

Abstract

At the initial stage of surfactant adsorption (when the layer is relatively diluted), the kinetics may be dominated by factors related to the transfer of molecules through the subsurface region and onto the interface. We consider two independent physical effects: (1) diffusion through a subsurface layer with nanometer thickness, where structuring or molecular interactions can impose substantial changes on the transfer rate, as compared with the bulk diffusion and (2) hindrance to the act of adsorption itself, when the molecules hit the interface from a place directly adjacent to it. These two effects are taken into account by formulating a model which includes the balance of fluxes in the subsurface layer. This model allows one to find analytical solution for the adsorption as a function of time. Application of the theory is illustrated by analyzing experimental data for two proteins which adsorb on air/water interface. Attention is paid to the particular case when the resistance to adsorption is relatively small but is still significant as compared with the bulk diffusion. Then, the theoretical fit of the adsorption vs. time can be implemented in a specific linear scale. The overall resistance of the interfacial zone comprises additive contributions from the hindrance to the act of adsorption and from the (retarded) diffusion through the subsurface layer. They are incorporated into one physical parameter (or characteristic time), which influences the kinetics.

Appendix

Several inverse Laplace transformations, used for the derivations in the text, are listed below. With the auxiliary function

$$ F(p) \equiv {e^p}{\text{erfc}}\left( {\sqrt {p} } \right) $$

(38)

where ${\text{erfc}}\;(y) \equiv 1 - \frac{2}{{\sqrt {\pi } }}\int\limits_0^y {{e^{{ - {z^2}}}}} {\text{d}}z$ is the complementary error function, one writes:

$$ {L^{{ - 1}}}\left[ {\frac{1}{{\sqrt {s} + \alpha }}} \right] = \frac{1}{{\sqrt {{\pi \;t}} }} - \alpha F\left( {{\alpha^2}t} \right) $$

(39)

$$ {L^{{ - 1}}}\left[ {\frac{1}{{\sqrt {s} \left( {\sqrt {s} + \alpha } \right)}}} \right] = F\left( {{\alpha^2}t} \right) $$

(40)

$$ {L^{{ - 1}}}\left[ {\frac{1}{{s\left( {\sqrt {s} + \alpha } \right)}}} \right] = \frac{1}{\alpha }\left[ {1 - F\left( {{\alpha^2}t} \right)} \right] $$

(41)

$$ {L^{{ - 1}}}\left[ {\frac{1}{{s\sqrt {s} \left( {\sqrt {s} + \alpha } \right)}}} \right] = \frac{2}{{\sqrt {\pi } }}\frac{{\sqrt {t} }}{\alpha } - \frac{1}{{{\alpha^2}}}\left[ {1 - F\left( {{\alpha^2}t} \right)} \right] $$

(42)

Here, L ⁻¹ denotes the inverse Laplace transform, α is an arbitrary constant (may be complex), t is time, and s is the Laplace variable.

Besides, from Eq. (34) it follows that F(0) = 1; F(∞) = 0.