SPHERE-WALL INTERACTION

dzE^film(z)] = p0D(1−U). (134) The free energy difference in (134) increases for increasingD, and the force per unit area

∂DF/A = ∂DδF/Apushing the plate towards the wall is given by (Asakura and Oosawa 1954)

p0{1−U−D∂U/∂D} = p0−nDU f , (135) where the forcef of a single chain between plates is given by (124). The number of chains per unit area between the large plate and the wall equalsnR∞

0 dzE^film=nDU, since it can be expressed by the number of chain-ends via the bulk-normalized end densityEand the number densitynof chains in the bulk. Thus the force per unit area in the last expression of (135) is the difference of the ideal gas pressurep0 of the chains outside and the disjoining force per unit areanDU fof the chains in between the wall and the plate.

For ideal chains, substituting (126) into (134) leads to the expression δF/A=−pL 4 for the free energy of interaction between plate and wall per unit area. Here the integration path in the complexτ plane encircles the cut that extends fromτ =−∞toτ = 0 counter-clockwise, and the scaling variableθ =D/L^1/2 equalsD/R^gind = 3. Eq. (136) implies the limiting behavior

δF/A=−p0{2(2(L/π)^1/2) −D +... , 16(L/π)^1/2θ⁻²e^−(θ/2)2} (137) forD ¿L^1/2, D ÀL^1/2. This is consistent with the general results that forD/R^g →0 the force per unit area on the plate equals the bulk pressurep0and thatδF/Aequals minus twice the free energy per unit area due to a single boundary wall. For an ideal chain this is 2p0(L/π)^1/2, see (84) or the second term on the right hand side of (97).

0.7 SPHERE-WALL INTERACTION

Here we consider a spherical particle S in a polymer solution in the half space h bounded by a planar wall w, as shown in Fig. 6.

0.7.1 Derjaguin approximation for a large sphere

For a large spherical particle withRÀ R^g, D, one may apply the Derjaguin approximation , which replaces the sphere by a local superposition of immersed plates with local distance

D˜ =D+ r_k²

2R (138)

from the wall and leads to the form δF =

Z

d^d−1r_k[δFpkw/A]_D→D˜ (139)

of the free energy of interaction between the sphere and the wall. HereδFpkw/Ais the free energy of interaction per unit area between the wall and a plate considered in Sec. VI B. For ideal chains withδF_pkw/Agiven by Eq. (136), one finds

δF = −2p0RR²gV(D/R^g) (140)

ind= 3, where V(θ) = 4π

Z dτ

2πie^ττ⁻²ln[1 + exp(−θτ^1/2)], (141) implying

V → {4πln2−4√

πθ+ (π/2)θ²+... ,32√

πθ⁻³exp(−(θ/2)²)} (142) forD¿ R^g, DÀ R^g.

The supposedly exact Derjaguin expressions (140), (141), and (142) should be compared with the Asakura-Oosawa type (Asakura and Oosawa 1954; Asakura and Oosawa 1958) pre-dictionδFphs=−pv, wherevis the volume of the overlap between a layer of widthR˜around the sphere and a layer of width2 ˜Rwith center on the wall. Choosing the effective sphere ra-diusR˜ of the polymer as in (101) and definingV^phsas in (140), one finds, forR À R^g, D, that

V^phs = (2√ 2−p

π/2θ)², (143)

providedD <2 ˜R ≡4R^g/√π, andV^phs = 0forD ≥2 ˜R. While the linear and quadratic terms inθare identical to those in (142),V^phs(0) = 8is about 10% smaller thanV(0) = 8.71.

A comparison for arbitraryD/R^gis shown in Fig. 7.

0.7.2 Small particle expansion for a small sphere

Immersing a small spherical particle withR¿D,R^gin a polymer solution in the half space changes the polymer free energy perkBTby

F

kBT = −{W−1}^(fc,hs) = AgR^d−(1/ν)R^1/νg nM^h(zS), (144) whereM^his the bulk-normalized monomer density in the half space without the sphere, and zS =D+ (R/2)is the distance of the center of the sphere from the boundary wall. As in the derivation of (116), we have used here the small sphere expansion (109) and the form on the left hand side of (64) of the modified monomer density{m}^(fc,hs) in the half space.

Subtracting the free energy change atzS=∞, i.e. in the bulk, whereM^h= 1, one finds δF = −kBT nAgR^d−(1/ν)R^1/νg [1− M^h(zS)] (145)

for the free energy of interaction (133) between sphere and wall. Eq. (145) not only applies to a solution of ideal chains whereM^his given by (55) but also, with the appropriate form of M^h, to dilute or semidilute polymer solutions in a good solvent .

For illustration and later use we confirm (144) by a field-theoretical calculation in the case of a dilute polymer solution, where Eq. (132) applies. Substituting the formE(rA) = RdrBhϕ⁽¹⁾_ABi/R

drBhϕ⁽¹⁾_ABi^bof the bulk-normalized end-density in (132) and using (119) to expand

hϕ⁽¹⁾_ABi^h− hϕ⁽¹⁾_ABi^S,h → hϕ⁽¹⁾_AB·Ψg(rS)i^hAgR^d−(1/ν) (146) for a small sphere radiusR, one finds

F = p0AgR^d−(1/ν){m(rS)}(fc,h)/n , (147)

since the half space profile is given by (Eisenriegler 1997) {m(r)}^(fc,h) = n

RdrAR

drBLhϕ⁽¹⁾_AB·Ψg(r)i^h

R drBLhϕ⁽¹⁾_ABi^b , (148)

where the limitk = 0is understood. Eq. (147) is in agreement with (144) if the form of {m}^(fc,h)on the left hand side of (64) is taken into account.

For ideal chains ind = 3the above result simplifies, since R

drBLhϕ⁽¹⁾_ABi^b = 1and Ψg = Φ²/2. One can use (49) and (47) to obtainF =pAgRLM^h(zS), which is consistent with (144).

0.7.3 Arbitrary size ratios

Here we consider the sphere-wall interaction mediated by a solution of ideal chains for ar-bitrary size ratioR/R^g. The half space profile M^hin (145) and the function−V in (140) increase monotonically with increasing distance from the surface so that the interaction is at-tractive for both small and large spheres. However, the behavior for small and large spheres is quite different. For a small sphere δF has a point of inflection where the mean force

∂DδF/∂D pushing the particle towards the wall has a maximum. For a large sphere the force decreases monotonically with increasing distance.

Fig. 8 shows numerical results (Bringer et al. 1999) for the end densityE(r)obtained by solving the diffusion equation (36) and from which the immersion free energy for a sphere near a wall can be calculated via Eq. (132). The resulting dependence ofδF/(2R²gRp0)≡ Y(ϑ, ρ) on the scaled sphere-wall distanceϑ=D/(√

2R^g)≡θ/√

2is shown for various size ratios ρ = R/(√

2R^g)in Fig. 9. Also shown is the scaling function f(ϑ, ρ) = ∂ϑY(ϑ, ρ) ≡

∂DδF/(√

2R^gRp)of the mean force∂DδF pushing the particle towards the wall.

Besides expressingF via (132) in terms of the number−∆N of removed chains, which is a global quantity, one may study the polymer-induced interaction between sphere and wall by means of the local density-pressure relation (65). Since∂DδF equals the reduction of the force onto the wall due to the sphere insertion and must be related to the near wall behavior of the monomer densitiesMandM^hin presence and absence of the sphere (Eisenriegler 1997;

Bringer et al. 1999),

∂DδF = p Z

dr_k[1−(M(r)/M^h(z))z→0]. (149)

This result shows that the depletion M(r) < M^h(z)of the monomer density due to the sphere leads to a positive∂DδF, i.e. to an attractive sphere-wall interaction. Some results for the monomer depletion are shown in Fig. 10.

0.8 INTERACTION BETWEEN TWO OR MORE