Logarithmic growth in surface adsorption

A review is made of experimental data on surface adsorption of particles and polymers from water solutions, their analysis and interpretation in terms of general theoretical models of surface adsorption. A characteristic isotherm and kinetics is found and defined as logistic growth. The discussion is focused on literature indicating a possibility of describing logistic growth by using statistical (probabilistic) models based on the mean stay time of molecules on the surface. The statistical approach is further elaborated as follows: ligands arriving at a surface have a binary choice-to bind or to become reflected. Since the number of attempts to bind, n, will be high we can use the true mean of the binomial distribution to describe the reaction and write: S = n a; where S is the number of successful attempts and a is the probability of binding. The probability, a, will depend on the site density and on the sticking probability of the ligand at the binding site. Several experimental studies show that surface reactions have a nonlinear time- and concentration dependence and can be described by a Boltzmann factor of the form, I(1-e-t/tau); where I is the flux of ligands to the surface and tau = stay-time. The exponential form indicates that the reactions are self-dependent, and a statistical model for description of such reactions will be of the form: S(t) = N(o)(1-2-alpha t) (2-beta t); where N(o) is the number of molecules present in the system, alpha relates to the probability of positive cooperativity or t-dependent binding, and beta relates to the probability of desorption.