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Density profiles around nanoparticles and distant perturbations
2.M. E. Fisher and P. G. de Gennes, C. R. Acad. Sci. Paris B 287, 207 (1978).
3.P. G. de Gennes, C. R. Acad. Sci. Paris II 292, 701 (1981).
4.P. G. de Gennes, C. R. Acad. Sci. Paris B 288, 359 (1979).
8.E. Eisenriegler and M. Stapper, Phys. Rev. B 50, 10009 (1994);
8.In these two references the amplitude of Eq. (3.3) is denoted by and , respectively, and is denoted by .
9.H. W. Diehl, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1986), Vol. 10;
9.For earlier literature on boundary critical phenomena, see K. Binder, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1983), Vol. 8.
10.For the special case of the leading isotropic density correction for a nanoparticle with “ordinary” boundary condition, the fusion approach has been used before in E. Eisenriegler, J. Phys.: Condens. Matter 12, A227 (2000). Equation (2) of this reference corresponds to Eq. (2.7) in the absence of the distant perturbation P.
12.Due to universality the concentration profiles induced by boundaries and the corresponding interactions in a near critical binary liquid mixture follow (Refs. 1, 2, 9, and 11) from density profiles and thermodynamic Casimir forces in the Ising model with boundary magnetic fields.
13.E. Eisenriegler, in Lecture Notes in Physics, edited by H. Meyer-Ortmanns and A. Kluemper (Springer, New York, 1998), Vol. 508, pp. 1–24;
13.E. Eisenriegler, in Soft Matter, edited by G. Gompper and M. Schick (Wiley, New York, 2005), Vol. 2.
14.J. L. Cardy, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1986), Vol. 11, p. 55.
16.The small sphere expansion applies if the sphere radius is much smaller than the correlation length and the distances from the sphere center to other surfaces and operators.
19.Neglecting the anomalous dimension in , results consistent with the predictions of the small sphere expansion (2.8) were obtained in Ref. 3 for the order-parameter density profile at a large distance from a single sphere and the interaction free energy of two distant spheres at the critical point of demixing.
22.In the Gaussian model, , , , and follow from and , and from Ref. 8.
23. equals the correction to the stress tensor on the surface of the sphere and , the corresponding correction on the planar wall . The two quantities equal the corrections to the pressure/ on and , respectively, due to the presence of .
24.For and , the profile correction tends to the sum of and const , where const is the constant multiplying on the right-hand side of the second Eq. (3.14). While the leading behavior for with fixed is given by the second one of the two terms above, the limit with fixed is given by the first term in accordance with Eq. (3.15).
26.In the Ising model, , , , , , and .
27.In the Ising model , , and attain their Gaussian values 1, 2, and , respectively, while , , , equals , and equals , compare Refs. 8, 18, and 25.
28.P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University, Ithaca, 1979).
32.Generally, for a polymer perturbation of finite extent, the densities have a nontrivial limit for , which is given by the critical ratios in Eqs. (3.1) and (3.2) in Ref. 30, while for an infinitely extended planar wall , tends to zero.
33.The amplitudes depend on the internal lengths of the small object and, like a multipole expansion, the expansion only includes operators that are consistent with all symmetries of the object. The usual operator product expansions and the small particle expansions for spherical (Refs. 17 and 18) or nonspherical (Ref. 25) particles are other expansions with these characteristics.
34.Interesting examples are the density profiles of or induced by two circular disks in the critical Ising model.
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