Illustration of the extrapolation length 1/c. The initial tangent of G(z, s) extrapolated to the z axis hits this axis at z = −1/c. For depletion (a) the intercept is negative and c is positive, for adsorption (c) the intercept is positive and c is negative. At the critical point (b) G = 1 for any z, so 1/c = ∞ and c = 0.
The end-point distribution G of an ideal chain (a) and its density profile ρ (b) as a function of ζ = z/2R, for four positive values of C = cR: C√60 = 1, 4, 10, 100. Solid curves are the exact Eq. (11) (a) or Eq. (A1) (b). Dashed curves are the approximation according to Eq. (13) (a) or Eq. (14) (b). Symbols are numerical SCF data for a 6-choice lattice, χ = 0, and φb = 10−7, for different chain lengths: N = 4000 (circles), 100 (squares), 40 (crosses), 10 (plusses) in (a), and 4000 (circles), 1000 (diamonds), 100 (squares), 40 (crosses) in (b). The c values were converted to the lattice parameter χs as discussed in the text. The vertical dashed line in (a) is ζ = δe/2R = , that in (b) ζ = δd/2R = .
Theoretical and numerical results for the inverse extrapolation length c as a function of Δχs for depletion. The solid curve is Eq. (22), the dotted line is c = 5Δχs. Symbols are numerical SCF data for five solvencies: χ = 0 (circles), 0.3 (diamonds), 0.4 (squares), 0.45 (plusses), and 0.5 (crosses).
The proximal length p as a function of the extrapolation length 1/c for depletion, for χ = 0 and N = 1000. Solid curves are Eq. (26c) for φb = 0, 0.001, 0.004, 0.01, and 0.05. The dashed line is the strong repulsion limit p = 1/c (Eq. (27a)). The dotted curves are the logarithmic form (Eq. (27b)) for weak repulsion.
Numerical (symbols) and analytical (curves) depletion profiles ρ(ζ) for χ = 0, N = 1000, and five concentrations: φb = 10−7 (circles), 0.001 (squares), 0.004 (diamonds), 0.01 (crosses), and 0.02 (plusses). The interaction strength is c = 1, C = 100/√60 (a); c = 0.1, C = 10/√60 (b); c = 0.04, C = 4/√60 (c); and c = 0.01, C = 1/√60 (d). These are the same C values as in Fig. 2: the circles in each diagram (10−7) are thus the same as in Fig. 2(b). The SCF results are for χs values computed from Eq. (22). The analytical curves are ρ = g 2 with g from Eq. (26).
Train density φ0 as a function of Δχs in the adsorption regime, for N = 1000, φb = 10−6, and five solvencies. Symbols are the SCF data for χ = 0 (crosses), 0.3 (plusses), 0.4 (squares), 0.45 (diamonds), and 0.5 (circles). Curves are the analytical results of Eq. (34).
Theoretical and numerical results for the inverse extrapolation length c as a function of Δχs for adsorption. Symbols are the numerical SCF data for N = 1000, φb = 10−6, and five solvencies: χ = 0 (circles), 0.3 (diamonds), 0.4 (squares), 0.45 (plusses), and 0.5 (crosses). The solid curves are theoretical GSA from Eq. (35). The dotted (steepest) line is c = 5Δχs, the dashed curve (which in this range is close to c = 5Δχs) is Eq. (32) for ideal chains.
Adsorption profiles φ(z) for N = 5000 and φb = 0.001 (a) and φb = 0.05 (b), for four solvencies: χ = 0 (circles), 0.3 (squares), 0.4 (diamonds), and 0.5 (crosses). Curves are analytical according to φ = g 2 with g from Eq. (31) (a) or according to φ = φb g 2 with g from Eq. (29) (b); in both cases φ0 was set to 0.6, which fixes the parameters p and d. Symbols are SCF, with (a) “loops only” and (b) the overall profile (see text). The χs parameter needed for the SCF data was computed from Eq. (34) with φ0 = 0.6.
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