Universal density-force relations for polymers near a repulsive wall

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 55, Issue 3 SUPPL. B, 1997, Pages 3116-3123

  Eisenriegler, E ^a  

^a FORSCHUNGSZENTRUM JÜLICH   (Germany)

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EID: 4244017793     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/physreve.55.3116     Document Type: Article

References (55)

1. P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University, Ithaca, 1979).
2. J.F. Joanny, L. Leibler, and P.G. de Gennes, J. Polym. Sci. Polym. Phys. Ed. 17, 1073 (1979).
3. If the wall acts on the monomers with a short-range repulsive potential ν(z) the force per area follows [2] from the monomer density on multiplying by -dνldz and on integrating over z. Note that v is a "nonuniversal" quantity that depends on microscopic details characterizing the monomers and the wall. For potentials ν that repel long polymer chains the "universal" quantities such as the density profile exponent and the density-force amplitude are independent of these details and have definite values 1/v and B, respectively. However, for potentials that characterize an attractive wall or a wall at the adsorption threshold [1.4] and for which -dν/dz changes sign on varying z, the argument [1.2] for proportionality between the osmotic pressure and the monomer density close to the wall (just out of the potential range) breaks down and the profile exponent is different from 1/v [1,4].
4. E. Eisenriegler, Polymers Near Surfaces (World Scientific, Singapore, 1993).
5. b the polymer density in the bulk.
6. BT times the derivative with respect to the position of the wall (in the direction perpendicular to its surface and away from the polymer chain) of the logarithm of the polymer chain partition function.
7. In this case 1/v=2.
8. BTA) is the same as the force which the polymer solution exerts onto the particle, pushing it towards the wall.
9. M.E. Fisher and P.C. de Gennes, C.R. Acad. Sci. Ser B 287, 207 (1978).
10. P.C. de Gennes, C.R. Acad. Sci. Ser. B 292, 701 (1981).
11. (a) T.W. Burkhardt and E. Eisenriegler, Phys. Rev. Lett. 74, 3189 (1995)
12. (b) E. Eisenriegler and U. Ritschel, Phys. Rev. B 51, 13717 (1995).
13. x, even in the presence of excluded-volume interaction, compare Sec. 5.6.3 in Ref. [4].
14. 1/v. This fundamental property which is consistent with Refs. [5,6] is not restricted to the present situation, compare the discussion in Sec. V.
15. (as) and an amplitude ratio of the form of the rhs of Eq (11). From Eqs. (4)-(8) one sees that from the density-force ratio (11) one may predict the corresponding ratios in the other situations (i)-(iv), since all these ratios are the same (compare the introductory remarks and Ref. [37]). For the behavior at z- a see the last paragraph of Sec.V and Ref. [47].
16. M. Daoud and P.G. de Gennes, J. Phys. (Paris) 38, 85 (1977).
17. -1/v of the critical temperature of the magnet confined between parallel plates with separation D.
18. x are essentially the ground-state energies [1].
19. R in the field theory.
20. E. Eisenriegler, A. Hanke, and S. Dietrich, Phys. Rev. E 54, 1134 (1996).
21. k introduced in [19], can be related to a half space amplitude [11]. For ideal chains in d = 3, A=2π.
22. In addition to the short-distance expansion and the shift identity discussed below, one needs a "small radius expansion" explained in Refs. [11] and [19]. The "small radius expansion" involves the universal amplitude of [20] and applies also to other problems where a polymer interacts with a small repulsive sphere. Compare P. G. de Gennes, C. R. Acad. Sci. Ser B 288, 359 (1979) and T. Odijk, Macromolecules 29, 1842 (1996).
23. In addition to the short-distance expansion and the shift identity discussed below, one needs a "small radius expansion" explained in Refs. [11] and [19]. The "small radius expansion" involves the universal amplitude of [20] and applies also to other problems where a polymer interacts with a small repulsive sphere. Compare P. G. de Gennes, C. R. Acad. Sci. Ser B 288, 359 (1979) and T. Odijk, Macromolecules 29, 1842 (1996).
24. To be precise we consider a plate of microscopic thickness. However, most of our conclusions are independent of the thickness.
25. x/Â, compare the counterpart of the normalization (18) discussed in front of Eq. (20b) and Ref. [33].
26. L in Eq. (4.16) of Ref. [19] is given by the product of ÂD(U- 1) and the partition function of a chain with one end fixed in the bulk.
27. S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954). The quantities P and f in Eqs. (3.1), (3.2) and Fig. 3 of this reference correspond to our quantities δf and U.
28. B.V. Deriagin, Kolloid Z. 69, 155 (1934).
29. Compare the Appendix.
30. s).
31. J. des Cloizeaux and G. Jannink, Polymers in Solution: Their Modelling and Structure (Clarendon, Oxford, 1990).
32. K. Binder, in Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz (Academic, London, 1983), Vol. 8, pp. 1-144.
33. (a) H.W. Diehl, in Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz (Academic, London, 1986), Vol. 10, pp. 75-267
34. (b) H.W. Diehl, in Proceedings of the Third International Conference "Renormalization Group-96," Dubna, Russia, August 25-September 1, 1996 (World Scientific, Singapore, 1997, in press).
35. 1 is the first of the n components of the order parameter density (or GL field) Φ. The formal limit n→0 [1,4,29] is taken after the calculation of the GL averages.
36. c, which inside & may be converted to a factor L by partial integration.
37. S. Dietrich and H.W. Diehl, Z. Phys. B 43, 315 (1981).
38. J.L. Cardy, Phys. Rev. Lett. 65, 1443 (1990).
39. E. Eisenriegler, M. Krech, and S. Dietrich, Phys. Rev. B 53, 14377 (1996).
40. 1/v always remains the same (as long as the wall at z = 0 is repulsive) and is independent of the variables D introduced in Sec. I. This is a well known feature of short-distance expansions.
41. 4 coupling constant is to be taken at its fixed-point value.
42. H.W. Diehl, S. Dietrich, and E. Eisenriegler, Phys. Rev. B 27, 2937 (1983).
43. B are integrated around (r∥,0).
44. 2 term in the Hamiltonian [31].
45. p is the distance of the particle from the wall, compare, e.g., Sec. 4 D in Ref. [19].
46. E. Eisenriegler, J. Chem. Phys. 79, 1052 (1983).
47. Q/Ω and equals 〈Ψ〉 Here Q labels the polymer chains in a macroscopic volume Ω. Compare, e.g., Eq (A.12) in Ref. [43].
48. This relation is quite general (see, e.g., Ref, [35] and references contained therein) and has been verified for the case we are considering in Appendix B 3 of Ref. [43].
49. T.W. Burkhardt and H.W. Diehl, Phys. Rev. B 50, 3894 (1994).
50. x and D.
51. b are equal functions of z for z≪D.
52. A) which readily follows from the expressions for G and X given in Ref. [19].
53. R. Lipowsky, Europhys. Lett. 30, 197 (1995).
54. A away from the center of the sphere.
55. This complication is, of course, absent for the case of a GL system describing a critical fluid right at the critical point [11], where t = 0. In this case proving the asymptotic validity of the Deriagin approximation for the free energy of interaction is completely straightforward [U].