Critical adsorption profiles around a sphere and a cylinder in a fluid at criticality: Local functional theory

We study universal critical adsorption on a solid sphere and a solid cylinder in a fluid at bulk criticality, where preferential adsorption occurs. We use a local functional theory proposed by Fisher et al. [M. E. Fisher and P. G. de Gennes, C. R. Acad. Sci. Paris Ser. B 287, 207 (1978); M. E. Fisher and H. Au-Yang, Physica A 101, 255 (1980)]. We calculate the mean order parameter profile $\psi \left(r\right)$, where $r$ is the distance from the sphere center and the cylinder axis, respectively. The resultant differential equation for $\psi \left(r\right)$ is solved exactly around a sphere and numerically around a cylinder. A strong adsorption regime is realized except for very small surface field $h_1$, where the surface order parameter $\psi \left(a\right)$ is determined by $h_1$ and is independent of the radius $a$. If $r$ considerably exceeds $a, \psi \left(r\right)$ decays as $r^{-(1+\eta )}$ for a sphere and $r^{-(1+\eta )/2}$ for a cylinder in three dimensions, where $\eta$ is the critical exponent in the order parameter correlation at bulk criticality.

Figure 1

(a) $\mathrm{\Psi }\left(x\right)$ and (b) $x\mathrm{\Psi }\left(x\right)$ as functions of $x=r/a$ around a solid sphere with radius $a$ in a fluid at criticality. They are written according to the exact solution (20) including the sphere interior ($x<1$). Three curves are obtained for (from top to bottom) $x_0=0.9$, 0.5, and 0.1, for which $(H_1,{\mathrm{\Psi }}_1,\alpha )=(15.1,2.86,1.64), (1.43,1.08,1.22)$, and $(0.423,0.418,0.548)$, respectively. Here, $\mathrm{\Psi }\left(x\right)$ diverges as $x\to x_0$ and decays as $\alpha /x$ with $\alpha =3^{1/4}{x}_0^{1/2}$ for $x>2$.

Figure 2

(a) $H_1$ and ${\mathrm{\Psi }}_1$ vs $x_0$ from Eq. (21), where $0<x_0<1$. (b) $H_1$ and ${\mathrm{\Psi }}_1$ vs $\alpha =3^{1/4}{x}_0^{1/2}(<3^{1/4})$. (c) ${\mathrm{\Psi }}_1$ vs $H_1$, which behaves as in Eqs. (22) and (23).

Figure 3

(a) $\mathrm{\Psi }\left(x\right)$ and (b) $\sqrt{2x}\mathrm{\Psi }\left(x\right)$ as functions of $x=r/a$ around a solid cylinder with radius $a$ in a fluid at criticality. They are numerically obtained from Eqs. (33) and (19) for $H_1=10.2$, 0.691, and 0.230, for which ${\mathrm{\Psi }}_1=1.61$, 0.941, and 0.579, respectively. They are written in the cylinder exterior and interior. For $H_1>2^{-3/2}=0.354, \mathrm{\Psi }\left(x\right)$ diverges as $x\to x_0$ as in Eq. (40), where $x_0$ is 0.871 for $H_1=10.2$ and 0.248 for $H_1=0.691$. For $H_1<2^{-3/2}, \mathrm{\Psi }\left(x\right)$ diverges as $x\to 0$ as in Eq. (41). In (b), $\sqrt{2x}\mathrm{\Psi }\left(x\right)$ tends to 1 at large $x$ as in Eq. (38) for any $H_1$.

Figure 4

$2^{1/2}{\mathrm{\Phi }}_1$ vs $2^{3/2}{H}_1$ for a cylinder for (a) $H_1<2^{-3/2}$ and (b) $H_1>2^{-3/2}$. These curves are calculated from Eq. (37) using two special solutions in these cases. The curves behave as in Eqs. (42) and (43).