Segment density of a block copolymer chain tethered at both ends
The exact solution for the end-to-end distance distribution of unperturbed block copolymer molecules and its inverse-langevin, sine-series, and Gaussian mathematical approximations are derived from the formal integral solution to the problem of random flights. Using the Markov property of random flights, these end-to-end distance distributions combine to give segment densities of perturbed block copolymer molecules, where the perturbation manifests itself as a geometric constraint on specific segments, called tethering. Computations for the case of a singly or doubly tethered diblock copolymer molecule, symmetric with respect to the number of bonds, having short bonds and long bonds, elucidate differences between the above mathematical approximations and three homopolymer approximations, and between the segment types. The Gaussian approximation is best and easiest for several reasons, giving only a few per cent error for chains of more than 40 bonds. Among the homopolymer approximations, the one that preserves the contour length and the mean-square end-to-end distance of the copolymer molecule yields the best agreement, within a few per cent, with the copolymer distribution. For the doubly tethered case, segments that are close, along the contour, to the tethered ends display a maximum probability density about an order of magnitude greater and a full width at half maximum about an order of magnitude smaller than segments that are far, along the contour, from the tethered ends. When the distributions are averaged over all segments of a block, the maxima of the distributions for the two segment types are within a few bond lengths of each other and are located near the tethered end of the segments connected by short bonds. The distribution for the segments connected by short bonds is highly peaked and localized, but the distribution for the segments connected by long bonds has a moderate maximum and a broad plateau of segment density extending to the other tethered end. The different segment types have maxima and full widths at half maximum that differ by about an order of magnitude from each other. The scaling of the distributions with contour length suggests that the reduced distributions are within a few per cent of each other for chains greater than 40 bonds, and the Gaussian approximation is within a few per cent of the exact solution above this length. Tabulation of the reduced distributions in the form of four-dimensional histograms, with only a moderate number of intervals in each dimension, appears promising for use in computer simulations of physical properties in the condensed phase
Fischel, L. B., Newman, J., & Theodorou, D. N. (1997). Segment density of a block copolymer chain tethered at both ends. None, 93(24), 4355-4370.