Collapse of a polymer chain: A Born–Green–Yvon integral equation study

Abstract

The collapse transition of a single polymer chain in two and three dimensions: A Monte Carlo study A Born-Green-Yvon ͑BGY͒ type integral equation is developed for the intramolecular distribution functions of an isolated flexible polymer chain. The polymer is modeled as a linear array of n identical spherical interaction sites connected by universal joints of bond length . In particular we study chains composed of up to nϭ400 square-well spheres with hard-core diameters and well diameters ͑1рр2͒. Intramolecular distribution functions and the resulting average configurational and energetic properties are computed over a wide range of temperatures. In the high temperature ͑good solvent͒ limit this model is identical to the tangent hard-sphere chain. With decreasing temperature ͑worsening solvent͒ the square-well chain undergoes a collapse transition identified by a sudden reduction in chain dimensions and a peak in the single chain specific heat. Extensive comparison is made between the BGY results and Monte Carlo results for square-well chains with ϭ1.5. The BGY theory is extremely accurate for square-well 4-mers at all temperatures. For longer chains the theory yields reasonably accurate results for reduced temperatures greater than T*Ϸ1 ͑expanded and theta states͒ and qualitatively correct behavior for T*Ͻ1 ͑collapsed state͒. Very accurate values for the theta temperatures for square-well chains with 1.25рр2.0 are also obtained.

An integral-equation theory for a self-interacting polymer adsorbed at an interface

“…Thus, with increasing attraction, adsorption increases. As shown in reference [23], for a square-well chain in the bulk, the scaling law (radius of gyration: < ) shows that the well depth βζ corresponds to the theta chain. However, due to interactions between segments, the density distribution from a random-flight model is different from that of a square-well chain with .…”

“…(15). References [23] and [32] show that the BGY equation overestimates the collapse. The stronger the collapse, the stronger the depletion layer.…”

“…a square-well chain or a Lennard-Jones chain), the integral-equation theory of liquids is more suitable [23][24][25][26][27][28]. An advantage of integral-equation theory is that its results can be compared with Monte Carlo simulations.…”

“…(6) is not useful unless we know the intra-molecular correlation function. Although the second-order distribution function in the bulk is easily obtained [23,32], it is difficult to obtain it for the inhomogeneous system. To proceed, we must make some approximations.…”

“…The normalized correlation function w can be found in references [23,32]. The undetermined normalization constant of the non-normalized correlation function is cancelled with the normalization constant C in Eq.…”