A theory for the orderhyphen;disorder transitions found in associative linear colloids has been developed from a onehyphen;dimensional lattice model using the matrix method of statistical mechanics. Our theory bears exactly the same relationship to the helixhyphen;coil theory of singlehyphen;chain helices as the lattice theory for separation of binary liquid mixtures or condensation of a lattice gas bears to the Ising theory of ferromagnetism. In this paper we show that the condensation of a onehyphen;dimensional lattice gas (or the separation of a onehyphen;dimensional immiscible phase) involving only finite (nearest) neighbor interactions, can be arbitrarily sharp as the statistical weights assigned to the various occupied sites tend to certain values, but that the onehyphen;dimensional condensed phase must always exist in segments. These linear segments of condensed phase are, in fact, the colloid particles or micelles. The critical volume fraction for micelle formation is given as the product of two parameters, sgr;2and zgr;, while the sharpness of the transition and the resulting degree of aggregation is dependent on the ratio sgr;2/zgr;. In general, the weighthyphen;average degree of association increase much faster above the critical point than the number average. The distribution of molecular weights of a linear associative colloid has been calculated and a method for the determination of the molecular weight distribution has been suggested.
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