Exact results and the effect of monomer–monomer bond type for a grafted ideal chain

We determine the equilibrium chain end probability density P_N(x) for a one-dimensional ideal chain of N monomers grafted to a planar surface. This distribution is also the distribution function of a three-dimensional chain where the tangential dimensions y, z are integrated out of the distribution. With a small modification of the analysis of Erukhimovich, Johner, and Joanny for free chains near a wall, we are able to obtain exact results for P_N(x) for any monomer number N though with the restriction that the monomer–monomer bonds lengths are exponentially distributed. In particular, we obtain exact values for P_N and its derivative d P_N/ d x at the surface for any N and the full profile P_N(x) for selected values of N. To determine the effect of the bond distribution, we find P_N(x) numerically for Gaussian and uniformly distributed one-dimensional bonds and compare with the exact results for exponential bonds. We suggest several ways to quantify the effect of bond type based both on P_N(x) near the surface and in the scaling region x∼Na. We then extract the large N limit of P_N(x) and show that it is similar to the chain end probability for continuous chains but shifted toward smaller x. We show the amount of shift is a measure of the magnitude of the correction to the continuous chain P_N(x). We derive the value of the shift for exponential bonds and show that the value is different for other bond types. We argue the shift can be interpreted as an effective surface behind the actual surface.