Adsorption of polydisperse polymer chains

The adsorption of polydisperse ideal polymer chains is shown to be sensitive to the large N tail of the distribution of chains. If and only if the number of chains decays more slowly than exponentially then there is an adsorption transition like that of monodisperse infinite chains. If the number decays exponentially the adsorption density diverges continuously at a temperature which is a function of the mean chain length. At low coverages, chains with repulsive monomer–monomer interactions show the same qualitative behaviour. PACS: 36.20.-r, 67.70.+n, 36.20.Cw Synthetic polymers are almost inevitably polydisperse. The polymer chains are not all of the same length, and so a polymer is not a single component but a mixture of chains of different lengths. However, almost all of the large number of theoretical studies of polymer adsorption have considered monodisperse polymer chains, where the chains are all of the same length. This is done to simplify matters and is a reasonable approximation when, as is often the case, the width of the distribution of lengths of polymer chain is small. Here we study polymers in which there are polymer chains with a very wide range of lengths, paying particular attention to the longest polymers in the distribution. Our motivation is not just the experimental fact that synthetic polymers are polydisperse yet are almost invariably treated within theory as being monodisperse. Polydisperse polymers may, and we show that they do, exhibit behaviour which is qualitatively different from that of monodisperse polymers. We study a very highly idealised model of polymer adsorption: ideal chains adsorbing onto a wall due to a short-ranged attraction between the wall and the monomers. This problem has been extensively studied for monodisperse chains [2–6], but not as far as the author is aware for polydisperse chains. The adsorption of an ideal chain is a textbook problem. It is highly idealised: almost invariably the density of polymer adsorbed onto a surface is too high for interactions between the polymer segments to be neglected. However, it has served well as a simple first model of adsorption of monodisperse polymer chains [2] and we shall use it as such for polydisperse chains. There has been previous work within the Scheutjens–Fleer theory [7], on the adsorption of polydisperse polymer chains but not, as far as the author is aware, on ideal chains. Roefs et al. [7] did not consider the large N tail of the distribution and so did not find the behaviour we will describe below. Note that in ref. [7] the ratio between the volume of the solution and the surface area of the wall was only of the order of the radius of gyration of the polymers. By contrast here we study a wall in contact with a bulk polymer solution so the ratio between the volume and surface area is infinity. We start by briefly reviewing the behaviour of monodisperse chains, with particular emphasis on the dependence of the adsorption on the length of the chain. The chains are ideal and each consists of a linear chain of N monomers of length a. The chains are at a non-zero number density ρ in the bulk. The bulk polymer solution is in contact with a wall. This wall attracts monomers via a short-ranged attraction; the range is taken to be a for simplicity. The strength of attraction is ǫ. Having discussed monodisperse chains we then generalise the theory to describe an arbitrary polydisperse mixture of chains of differing lengths. We show that not all polydisperse mixtures behave in the same way: there are three qualitatively different behaviours possible. Which behaviour a mixture exhibits depends on the large N tail of the distribution. This is also true of the cloud-point curve of polydisperse polymers, as was shown by Šolc [8]. Recent work by the author on the bulk phase behaviour of polydisperse hard spheres [9] has found a similar sensitivity to the tail of the distribution. Ideal chains do not interact with each other and so finite ideal chains are independent systems with a finite number of degrees of freedom. They therefore cannot exhibit a phase transition. For ideal chains there is only an adsorption transition in the limit that the number of monomersN