Figures

Figure 1: Unbalanced pressure due to polymer depletion pushes the particle towards the wall.

-2 R 2 R e

Figure 2: Colloidal particle interacting with a rigid polymer sphere.

Figure 3: Small colloidal particle interacting with a rigid (left) and with a flexible (right) polymer.

n R 3 g large particle radius small particle radius semi-dilute solution

dilute solution R _g /R 0

R g = ξ

R g = R ξ = R

Figure 4: Various limits of a single spherical particle in a solution of nonadsorbing polymers in a good solvent. The sphere becomes a planar wall for vanishingR^g/R(i.e. for points on the vertical axis), and becomes a ”small” sphere, with radius much smaller than the characteristic mesoscopic polymer lengths (such as the radius of gyrationR^gof a polymer chain in the dilute solution or the mesh-sizeξin the semidilute solution ), asR^g/Rbecomes large with the inter-chain overlapnR³gkept fixed. The following limits are shown: planar wall in a dilute solution (lower left corner), planar wall in a semidilute solution (upper left corner), small sphere in a dilute solution (lower right corner), and small sphere in a semidilute solution (upper right corner). The dashed lines are crossover regions.

0 2 4 6 8 10 12 14

0 0.2 0.4 0.6 0.8 1

F/kBT

R/R_g

Figure 5: Size-ratio dependence of the free energy cost F for an inter-chain overlap of (4π/3)R³gn = 1.16. The ‘renormalized mean-field approximation’ (full line) is compared with Monte Carlo results (dots). The dashed line shows the asymptotic behavior for small spheres.

r z

s w

D R

r

h

Figure 6: A spherical particle s immersed in a polymer solution near a planar wall w. The monomer and end densities of polymer chains (not shown) depend on the positionr= (r_k, z) in the half space h.

Y(ϑ,∞) Y^phs(ϑ,∞)

ϑ=D/(√ 2R^g) δF/(2R2 gRp)

2 1.5

1 0.5

0 -2

-4

-6

-8

Figure 7: Scaled interaction free energy δF/(2R²gRp) = Y(D/(√

2R^g), R/(√

2R^g))of a sphere and a wall for large particle to polymer size ratioR/R^g. The lower curveY(ϑ,∞) =

−V(D/R^g)is the exact Derjaguin result (141), and the upper curveY^phs(ϑ,∞) =−V^phsis the Asakura-Oosawa approximation (143).

0.5

Figure 8: Bulk-normalized density of chain endsE(r_k, z)forR/(√

2R^g) = 1/4andD/R= 1(a), 3 (b), 5 (c). The deviationE^h− E from the density without the sphere, and thus the number−∆N of chains removed on inserting the sphere, decreases as the sphere approaches the wall.

0 0.5 1 1.5

−8

−6

−4

−2 0

0 0.5 1.0 1.5

ϑ

0 2 4 6 8 10

ρ = 0 ρ = 0.2 ρ = 0.5 ρ = ρ

_c

≈ 0.7 ρ = 1 ρ = ∞

(a)

(b)

Figure 9: (a) Scaling functionY(ϑ, ρ)for the interaction free energy vsϑ = D/(√ 2R^g) for various fixed values ofρ =R/(√

2R^g), ranging fromρ= ∞(lowest curve, Derjaguin approximation (140), (141)) toρ= 0(uppermost curve, small radius expansion (145)). The squares, dots, triangles, and diamonds show numerical data. Forρ < ρc ≈ 0.7there is an inflection point in theϑ-dependence ofY, which is absent forρ > ρc. (b) Scaling function f(ϑ, ρ) of the polymer-induced force. The points of inflection of Y in (a) correspond to maxima off. Forρ > ρcthe functionf(ϑ, ρ)has a maximum atϑ= 0, corresponding to a sphere which touches the wall.

0 1 2 3 4 r

_||

/ R

0 0.2 0.4 0.6 0.8 1.0

[ M (r) / M

h

(z) ]

(as)

ϑ = 2 ϑ = 3/2 ϑ = 1 ϑ = 1/2 ρ = 1

Figure 10: Depletion of the normalized monomer density[M(r)/M^h(z)]^(as)near the wall due to the sphere as a function ofr_k/R for ρ = R/(√

2R^g) = 1and various values of ϑ = D/(√

2R^g) (compare Fig. 6). Forρ = 1 > ρc the depletion is more pronounced, and hence the force increases, as the sphere moves towards the wall, i.e. asϑis decreased (compare Fig. 9).

simple anisotropic shapes

ellipsoids: two spherical pieces:

prolate

sphere

oblate

cigar dumbbell

pancake lens

Figure 11: Simple shapes of anisotropic colloidal particles with a symmetry axis of revolution and reflection symmetry about the center.

4 R

2 √ 2 R

Figure 12: Dumbbell of two tangentially touching spheres of radiusR (full lines) and its circumscribing ellipsoid (lines of dashes).

α π/2

α/2 π−α/2

−π/2

α/2−π −α/2 π

−π

0

α < π

α

α/2

π/2 π−α/2

π

−π α/2−π

−π/2

−α/2

0

L D

α π/2

α/2 π−α/2

−π/2

α/2−π

−α/2 π

−π

0

α > π

α

π−α/2

π/2 α/2 π

−π

−α/2

−π/2

α/2−π

0

Figure 13: Conformal mapping of a dumbbell or lens onto a wedge.

(2 + √ 2) R

2 √

1 + 2

^−1/2

R

√ 2 R

Figure 14: Dumbbell of two spheres of radiusRintersecting at an angleα=π/2(full lines) and the circumscribing ellipsoid (lines of dashes).

z

P

prolate ϑ

oblate ϑ

Figure 15: Particles of prolate and oblate ellipsoidal shape near a planar wall. The ellipsoid is oriented parallel to the wall forϑ=π/2in the prolate case and forϑ= 0in the oblate case.

− 5

− 4

− 3

− 2

− 1 0 1

0 1 2 3 4

z

^P

/ R

g

A

_DISK

Figure 16: The amplitudeA ∝ B/(particle size)³which specifies the dependence (191) of the free energy of immersion(δPF)Won the orientation of the small anisotropic particle with respect to the planar wall. The caseA^DISK = B/R³_DISKof a circular disk of radiusRDISK

given in Eq. (194) is shown. On decreasing the distancezP between the center of the disk and the wall,A^DISKpasses through a maximum value of 0.501 atzP/R^g = 1.55, changes sign at zP/R^g = y0,DISK = 0.99, and drops to the value−16/3 for zP/R^g ¿ 1. For zP/R^g> y0,DISKand< y0,DISK, the most favorable orientation of the disk is perpendicular and parallel to the wall withcosϑ= 0and 1, respectively, see Eq. (191) and Fig. 15. For a general prolate or oblate small ellipsoid, the qualitative form ofB/(particle size)³is that of

−A^DISKorA^DISK, respectively.

Abramowitz, M. and Stegun, I., 1972, Handbook of Mathematical Functions. Dover Pub-lications, New York.

Asakura, S. and Oosawa, F., 1954, J. Chem. Phys. 22, 1255.

Asakura, S. and Oosawa, F., 1958, J. Polym. Sci. 33, 183.

Binder, K., 1986, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz, vol. 8, pp. 1–144. Academic Press, London.

Breidenich, M., Netz, R., and Lipowsky, R., 2000, Europhys. Lett. 49, 431.

Bringer, A., Eisenriegler, E., Schlesener, F., and Hanke, A., 1999, Eur. Phys. J. B 11, 101.

Brown, L. and Collins, J., 1980, Ann. Phys. 130, 215.

Burkhardt, T. and Eisenriegler, E., 1995, Phys. Rev. Lett. 74, 3189.

Cardy, J., 1983, J. Phys. A 16, 3617.

Cardy, J., 1986, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz, vol. 11, pp. 55–126. Academic Press, London.

Cardy, J. and Hamber, H., 1980, Phys. Rev. Lett. 45, 499.

Chatterjee, A. and Schweizer, K., 1998, J. Chem. Phys. 109, 10464.

de Gennes, P. G., 1979, Scaling Concepts in Polymer Physics. Cornell University, Ithaca.

des Cloizeaux, J. and Jannink, G., 1990, Polymers in Solution. Clarendon, Oxford.

Diehl, H., Dietrich, S., and Eisenriegler, E., 1983, Phys. Rev. B 27, 2937.

Diehl, H. W., 1986, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz, vol. 10, pp. 75–267. Academic Press, London.

Diehl, H. W., 1997, International J. of Mod. Physics B 11, 3503.

Dietrich, S. and Diehl, H., 1981, Z. Physik B 43, 315.

Eisenriegler, E., 1993, Polymers near Surfaces. World Scientific, Singapore.

Eisenriegler, E., 1997, Phys. Rev. E 55, 3116.

Eisenriegler, E., 1998, in Lecture Notes in Physics, edited by H. Meyer-Ortmanns and A. Kluemper, vol. 508, pp. 1–24. Springer, Berlin.

Eisenriegler, E., 2000, J. Chem. Phys. 113, 5091.

Eisenriegler, E., 2004, J. Chem. Phys. 121, 3299.

Eisenriegler, E., Bringer, A., and Maassen, R., 2003, J. Chem. Phys. 118, 8093.

Eisenriegler, E., Hanke, A., and Dietrich, S., 1996, Phys. Rev. E 54, 1134.

Eisenriegler, E. and Ritschel, U., 1995, Phys. Rev. B 51, 13717.

Flammer, C., 1957, Spheroidal wave functions. Stanford University Press, Stanford.

Hanke, A., Eisenriegler, E., and Dietrich, S., 1999, Phys. Rev. B 59, 6853.

Hsu, H.-P. and Grassberger, P., 2003, Eur. Phys. J. B 36, 209.

Hsu, H.-P. and Grassberger, P., 2004, J. Chem. Phys. 120, 2034.

Jansons, K. and Phillips, C., 1990, J. Colloid Interface Sci. 137, 75.

Lipowsky, R., 1995, Europhys. Lett. 30, 197.

Louis, A., Bolhuis, P., Meijer, E., and Hansen, J.-P., 2002, J. Chem. Phys. 116, 10 547.

Maassen, R., Eisenriegler, E., and Bringer, A., 2001, J. Chem. Phys. 115, 5292.

Nelson, D., 1976, Phys. Rev. B 14, 1123.

Nienhuis, B., 1982, Phys. Rev. Lett. 49, 1062.

Obukhov, S. P. and Semenov, A. N., 2004, preprint.

Odijk, T., 2000, Physica A 278, 347.

Ohshima, Y. N., Sakagami, H., Okumoto, K., Tokoyoda, A., Igarishi, T., Shintaku, K. B., Toride, S., Sekino, H., Kabuto, K., and Nishio, I., 1997, Phys. Rev. Lett. 78, 3963.

Rudhardt, D., Bechinger, C., and Leiderer, P., 1998, Phys. Rev. Lett. 81, 1330.

Sch¨afer, L., 1998, Excluded Volume Effects in Polymer Solutions. Springer, Heidelberg.

Snoeks, E., van Blaaderen, A., van Dillen, T., van Kats, C. M., Brongersma, M. L., and Polman, A., 2000, Adv. Mater. 12, 1511.

Taniguchi, T., Kawakatsu, T., and Kawasaki, K., 1992, Slow Dyn. Condens. Matter 256, 503.

Tuinier, R., Vliegenthart, G., and Lekkerkerker, H., 2000, J. Chem. Phys. 113, 10768.

van Blaaderen, A., 2003, Science 301, 470.

van Blaaderen, A., 2004, MRS Bulletin February, 85.

Verma, R., Crocker, J. C., Lubensky, T. C., and Yodh, A. G., 1998, Phys. Rev. Lett. 81, 4004.

Zinn-Justin, J., 1989, Quantum Field Theory and Critical Phenomena. Clarendon, Oxford.

anisotropic particles, 9, 37, 38, 42, 45 Asakura-Oosawa approximation, 9, 25, 33, 45 boundary operator expansion, 9, 20, 44 conformal invariance, 9, 38, 39, 44

critical behavior, 8, 9, 11, 12, 14, 17, 38, 39, 44

density pressure relation, 9, 19–21, 26, 30, 34, 44, 45

Derjaguin approximation, 9, 26, 32, 35, 45 dilute solution, 8, 9, 16, 18, 19, 23, 24, 28, 31,

34, 36, 45, 47, 53

Dirichlet boundary condition, 15, 20, 24, 28 dumbbells, 9, 37, 39, 44, 45

ellipsoids, 9, 37, 41, 45

end density, 16–18, 23, 25, 27, 31, 34, 45 end-to-end distance, 8, 10, 13, 15, 18 excluded volume interaction in a good solvent,

8, 9, 11, 14, 17, 19, 21, 22, 28, 31, 34, 45

field theory, 8–10, 12, 14, 15, 20, 21, 23, 28, 31, 34, 36, 38, 44

free energy cost of immersion, 9, 24, 27, 42, 45

ideal chain, 9, 10, 15–17, 19, 22, 24, 26–28, 30–35, 37, 45, 46, 49

induced surface tension, 22, 45 induced torque, 42, 43

inter-chain overlap, 8, 9, 22, 28, 36, 45 Laplace transform, 10–12, 16, 27, 30 lenses, 39, 44 osmotic pressure, 8, 19, 23, 28

particle to polymer size ratio, 9, 24, 25, 28, 34, 45

planar wall, 9, 17, 21, 22, 24, 29, 45 polymer depletion, 8, 9, 15, 18, 24, 35, 45 polymer-colloid mixtures, 8, 45

polymer-induced interaction, 9, 28, 31, 32, 34, 35, 42, 44, 45

polymer-magnet analogy, 8, 16, 18, 28, 36 radius of gyration, 8, 15, 18

renormalization group, 17, 23

renormalized mean-field approximation, 17, 23, 28, 44

scaling, 8, 13, 18, 23, 32, 34 scaling dimension, 12–14, 20, 27, 38 screening length, 23, 28

semidilute solution, 8, 19, 23, 28, 34, 45, 47, 53

small particle operator expansion, 9, 14, 26, 33, 35, 36, 38, 42, 44

spherical particle, 9, 19, 23, 24, 26, 32, 35, 44, 45

stress tensor, 20, 38

universal, 8, 9, 12, 19, 20, 23, 27, 31, 44

Im Dokument Field theory of polymer-colloid interactions (Seite 45-63)