Exact and Monte Carlo study of adsorption of a self-interacting polymer chain for a family of three-dimensional fractals

Abstract

We study the problem of adsorption of self-interacting linear polymers situated in fractal containers that belong to the three-dimensional (3d) Sierpinski gasket (SG) family of fractals. Each member of the 3d SG fractal family has a fractal impenetrable 2d adsorbing surface (which is, in fact, 2d SG fractal) and can be labelled by an integer b (2 ≤ b ≤ ∞). By applying the exact and Monte Carlo renormalization group (MCRG) method, we calculate the critical exponents ν (associated with the mean squared end-to-end distance of polymers) and φ (associated with the number of adsorbed monomers), for a sequence of fractals with 2 ≤ b ≤ 4 (exactly) and 2 ≤ b ≤ 40 (Monte Carlo). We find that both ν and φ monotonically decrease with increasing b (that is, with increasing of the container fractal dimension d f ), and the interesting fact that both functions, ν(b) and φ(b), cross the estimated Euclidean values. Besides, we establish the phase diagrams, for fractals with b = 3 and b = 4, which reveal existence of six different phases that merge together at a multi-critical point, whose nature depends on the value of the monomer energy in the layer adjacent to the adsorbing surface.

Statistical mechanics of polymer chains grafted to adsorbing boundaries of fractal lattices embedded in three-dimensional space

“…We have been able to complete the exact form of required RG transformations and carry out a comprehensive analysis for the first three members (b = 2, 3 and 4) of the 3d SG family of fractals. The RG equations (2.8)-(2.14) were found in previous studies [5,6,[19][20][21][22], and in this work we have determined the additional RG transformation (2.15) by applying an exact enumeration method. For b = 2 fractal these are given in the Appendix, while for b = 3 and 4 fractals they can be obtained upon request to the authors (to be more precise, for the critical exponents calculation only transformations defined with M S were needed, and consequently only that transformations are quoted).…”

“…In this model interactions between monomers and adsorbing wall are described by parameters w = e −εw/k B T , and t = e −εt/k B T , where ε w is the energy of a monomer lying on the adsorbing surface, and ε t is the energy of a monomer in the layer adjacent to the surface. Depending of the values of w and t the polymer chain can exists in one of three possible states: adsorbed, desorbed and surface attached [5,6]. For the first three members (b = 2, 3 and 4) of the 3d SG fractal family we have performed the exact RG analysis to evaluate the surface critical exponents γ 11 , γ 1 ,and γ s that govern the numbers of all possible polymer configurations with both, one, and no ends grafted on the adsorbing boundary, respectively.…”

“…2 , B (r) 1 ) tend to (A * , B * , 0, 0, 0), when r → ∞, which indicates that polymer, stays away from the attractive surface [5,6]. This state is referred to as The mean squared end-to-end distance R 2 N of the polymer chain scales with its length N as N 2ν , where the critical exponent ν is equal to…”

“…The numbers a 1 , a 2 , and b 1 do not depend on r [5,6]. Analogously, we can construct additional recursion relations for surface one-leg generating functions…”

“…When w is increased beyond w c (t), and for x = x c (w, t) [5,6] RG parameters flow towards the new fixed point that describes the adsorbed polymer chain…”