Behavior near the wall: density-pressure relation

is the error function. The subscript h in (51) stands for ‘half space’. The width of the depletion zone ofE^hnear the wall is∝√

L, i.e. of the order of the root mean square end-to-end distance or radius of gyration, see (37).

From (47) the bulk normalized monomer density profile is given by

M = 1 + 4 [−2 i²erfc(y/2) + i²erfc(y)], (55) where i²erfc is the twofold iterated error function (Abramowitz and Stegun 1972). The monomer density profileMhas the parabolic form

M →z²/L (56)

Expressions for the ‘magnetic analogs’, i.e. the Laplace transforms G(t;r,r⁰) =

0.4.2 Behavior near the wall: density-pressure relation and boundary operator expansion

Here we consider both ideal polymers and real polymers in a good solvent. For a dilute, monodisperse solution of free polymers in the half space with a planar boundary wall, the bulk-normalized density profilesEfor ends orMfor monomers have the scaling form

P(z) =Y(z/R^g), (59)

with universal scaling functionsY =YeorYmforP =EorM. Herez, the distance from the wall, is large on the microscopic scale, and forzÀ R^g,Y →1. For

microscopic distances¿z¿ R^g, (60)

EandMhave a power law behavior inz. The power law exponents are positive, in accordance with the depletion phenomenon, and are known as ‘surface exponents’. In the case ofEthe exponent is new (Eisenriegler 1993) and not a simple combination of the bulk exponentsν andγ. In the case ofMthe exponent is1/ν.

The reason for the second power law is that the monomer density near the wall is related (de Gennes 1979) to the force that the polymers exert onto the wall. The force per area is given by

f

A =kBT n≡p0, (61)

withnthe chain density in the bulk, since it equals the chain osmotic pressure in the bulk, which by the ideal gas law is kBT n. This suggests (de Gennes 1979) that the monomer densityN nM(z)in the region (60) must also be independent ofN, implying via (1) that

M ≡Ym(z/R^g)→Bg(z/R^g)^1/ν (62) forz¿ R^g, withBga universal amplitude.

As in Eq. (15) it is advantageous to introduce a modified monomer density m(r) =

N

X

P=1

(R^1/νg /N)

N

X

j=1

δ(r−r_P,j) (63)

for a system ofN chains. This quantity is less dependent on microscopic details than the ordinary monomer density(N/R^1/νg )m. For free chains in the half space (fc,h)

{m(r)}(fc,h) = R^1/νg nM(z)→Bgz^1/νp(fc,h)/(kBT), (64) where the curly brackets denote a chain ensemble average, andp(fc,h) =p0 ≡kBT nis the polymer pressure on the wall, according to (61).

The relation between density and pressure close to a planar boundary wall

{m(r_k, z)}(ensemble) → Bgz^1/νp(ensemble)(r_k)/(kBT) (65) not only applies to dilute free chains in the half space as given in (64), with the pressure independent of the positionr_kin the wall, but also in other situations. Of particular interest are (i) a single chain with one end (or the two ends) fixed in the half space, (ii) a single chain trapped between two parallel plates, (iii) a dilute or semidilute polymer solution in the half space , and (iv) a dilute or semidilute polymer solution in the half space containing a mesoscopic obstacle (particle). While densities and pressures are quite different in these different cases, their ratioBgz^1/ν is the same. The density-pressure relation, with the same factorBgz^1/ν, even applies if the boundary is not planar but has a nonvanishing mesoscopic radius R of curvature. One example, the surface of a spherical particle of radius R in a polymer solution, is discussed in Sec. V. The mesoscopic distancezofrfrom the point in the boundary has to be small not only compared toR^g(or the Edwards’ correlation lengthξ), but also compared toR. The amplitudeBgis independent of microscopic details to the same extent as the exponentν. In particular it depends on the spatial dimensiondand is different for ideal chains and chains with excluded volume interaction. For ideal chains

Bg=B_g^(ideal)=d/3 (66)

follows from Eqs. (56) and (64).

The density-pressure relation (65) can be understood from a field-theoretic analysis, which also allows one to calculate the universal and situation-independent amplitudeBg. For exam-ple, consider the case (i)=(A,B ; h) of a chain with ends fixed atrAandrB. Since Eq. (17) also applies to a chain in the presence of a boundary (Eisenriegler 1993; Diehl 1986; Diehl 1997), one finds, on comparing (15) with (63), that

{m(r_k, z)}(A,B; h) = L hΨg(r_k, z)·ϕ⁽¹⁾_ABi^h|^k=0

L hϕ⁽¹⁾_ABi^h|^k=0 , (67) where

Ψg(r_k, z) =R^1/νg (N l²)⁻¹1

2Φ²(r_k, z). (68)

Herehi^hdenotes the field-theoretical half space average with the Dirichlet boundary condition (34). The behavior on approaching the wall follows from the boundary operator expansion (Dietrich and Diehl 1981; Eisenriegler 1997)

Φ²(r_k, z) ∝ z^d−xΦ2T_⊥,⊥(r_k,0), (69) which is a short distance relation analogous to the bulk relation (32) in which one operator approaches another one. The operator

T_⊥,⊥(r_k,0) = 1 2

£(∂zΦ(r_k, z))¤2 z=0≡1

2Φ²_⊥(r_k) (70)

is the diagonal component, perpendicular to the wall, of the stress tensor at the Dirichlet surface. It is the boundary operator of lowest inverse length dimension that is even in Φ and nonvanishing at the Dirichlet boundary, and it has scaling dimensiond. Taking (30) into account, Eq. (69) provides the field-theoretical explanation of thez^1/νbehavior ofmnear the boundary. The scaling dimensiondof the surface operator (70) follows from the role of the stress tensor in generating coordinate transformations. For example, integratingT_⊥⊥over the planar boundary generates a shift away from the surface⁹, so that

Z

d^d−1r_khT_⊥⊥(r_k,0)·ϕ⁽¹⁾_ABi^h= (∂zA+∂zB)hϕ⁽¹⁾_ABi^h (71) ifzA, zB>0.

While the factor of proportionality in (69) is nonuniversal, the corresponding factor in

Ψg(r_k, z)→Bgz^1/ν T_⊥⊥(r_k,0) (72)

is given by the universal numberBg. On using the shift identity, (71) implies (Eisenriegler 1997)

Z

d^d−1r_k{m(r_k, z)}^{(A,B; h)}→Bgz^1/ν(∂zB+∂zA) lnZ_N^(h)(rA,r_B). (73)

9For the Gaussian model the shift identity (71) follows directly from (57) and (70), using Wick’s theorem to show that both sides of (71) equalR

dp(2π)^1−de^{ip(rAk−rBk)}e^−(zA+zB)w, withwfrom (57). For non-Gaussian field theories of theΦ⁴-type, see Diehl et al. (1983) and Appendix 5C in Eisenriegler (1993).

Since the right hand side equals Bgz^1/ν times the modulus of the force onto the wall, (73) is consistent with the density-pressure relation (65). In cases (ii) and (iv) one may argue similarly (Eisenriegler 1997). The estimate

Bg ≈ 0.99 , d= 3 (74)

for polymer chains in good solvent ind= 3follows from a first order expansion (Eisenriegler 1997; Eisenriegler 1993) inε= 4−dand is very close to the ideal chain valueB_g^ideal= 1.

For an ideal chain ind= 3with two ends and one end fixed, respectively, the forces on the wall are given by

f /kBT = R⁻g¹(yA+yB)/[exp(yAyB)−1] (75) and

f /kBT = ∂zAln erfyA (76)

whereyA=zA/R^g,yB=zB/R^g. They change from anR^gindependent power law behav-ior for small distanceszAorzBto an exponential dependence for large distances, e.g. for one end fixed from∝z⁻¹_A atzA¿ R^gto∝ R⁻¹g exp(−z_A²/R²g)atzAÀ R^g.