Confined model systems mean values

A molecular simulation of a macroscopic biosensor is still beyond our computational means. We have therefore used only the field theory to model its dynamical behavior. We have found that the eflfect of the presence of adsorbed particles at the confining walls is to slow down the coarsening of domains in the middle of the cell. When the concentration of adsorbed particles reaches a critical value, the correlation length seems to be frozen and the system never orders into a single cell-encompassing domain, as it does in the absence of adsorbed particles. Hence, the time dependent response of the sensor encodes additional information regarding the amount (and perhaps also the distribution) of adsorbed particles. 

Later, in a model where each cylindrical polymer rod was confined to a concentric, cylindrical, electroneutral shell whose volume represents the mean volume available to the macromolecule, the concept was extended to the macroscopic system itself which was considered as an assembly of electroneutral shells at whose periphery the gradient of potential goes to zero and the potential itself has a constant value. Closed analytical expressions which represent exact solutions of the Poisson-Boltzmann equation can be given for the infinite cylinder model. These solutions, moreover, were seen to describe the essence of the problem. The potential field close in to the chain was found to be the determining factor and under most practical circumstances a sizable fraction of the counter-ions was trapped and held closely paired to the chain, in the Bjerrum sense, by the potential. The counter-ions thus behave as though distributed between two phases, a condensed phase near in and a free phase further out. The fraction which is free behaves as though subject to the Debye-Hiickel potential in the ordinary way, the fraction condensed as though bound . 

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