Abstract
We reconsider the depletion interaction of an ideal polymer chain, characterized by the gyration radius RG and bond length a , and an impenetrable spherical colloid particle of radius R . Forbidding the polymer-colloid penetration explicitly (by the use of Mayer functions) without any other requirement we derive and solve analytically an integral equation for the chain partition function of a long ideal polymer chain for the spherical geometry. We find that the correction to the solution of the Dirichlet problem depends on the ratios R/R G and R/a . The correction vanishes for the continuous chain model (i.e. in the limit R/R G → 0 and R/a → ∞ but stays finite (even for an infinite chain) for the discrete chain model. The correction can become substantial in the case of nano-colloids (the so-called protein limit).
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In fact, the diagram expansion of the integral (R1) contains some fictitious loop diagrams corresponding to cycled polymers with all possible degrees of polymerization. To exclude these diagrams, one should make use of the famous de Gennes $n\rightarrow 0$ trick PGGbook,deGennes72, i.e. consider the field $\phi(\mathbf{r})$ as an $n$-component vector field and set $n=0$ in the final results. However, for the Gaussian integral (R1) the fictitious loop diagrams appearing in the numerator and denominator of (R1) precisely cancel each other. It is worth noticing that it is this property of the Gaussian integral which enables one to disregard the loop diagrams within the mean-field (sadle-point) and random phase (Gaussian vicinity of the saddle-point) approximations. So, in this paper we also disregard the loop diagrams and, thus, consider the field $\phi(\mathbf{r})$ as a scalar one
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Erukhimovich, I., Johner, A. & Joanny, J.F. Depletion of ideal polymer chains near a spherical colloid particle beyond the Dirichlet boundary conditions. Eur. Phys. J. E 31, 115–124 (2010). https://doi.org/10.1140/epje/i2010-10568-4
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DOI: https://doi.org/10.1140/epje/i2010-10568-4