5 Weak Adsorption: Discussion
5.1 Comparing Geometries
Applying various approaches (see Sects. 3 and 4), we determined the distinct
scaling behavior for the critical surface charge density required to trigger adsorption of flexible polyelectrolyte chains onto a plane, a cylinder and a spherical
surface as function of the Debye screening length 1/k (see Fig. 3). A main
conclusion is that the critical surface charge density scales as s c ~ k
3 for a planar
interface and all k, whereas for cylindrical and spherical surfaces in the largecurvature or low-salt limit we find the relations s c ~ k
2 and s c ~ k
1 , respectively.
Generally, as the radius of surface curvature increases and the parameter ka grows,
a transition takes place from the large curvature limit to the planar limit. For
both the cylinder and sphere cases, this change in scaling laws appears at ka % 1,
i.e., when the radii of surface curvature become comparable to the Debye screening
length (Fig. 10). For adsorption onto a cylindrical surface, the crossover to the
planar limit occurs at somewhat smaller ka values than for the sphere situation.
Comparing the results obtained by the WKB method with the exact solutions for
the planar and spherical surface, we find, within 2% error, quantitative agreement in
the planar case. For a sphere, we find the same asymptotic dependence of s c on k in
the limit ka ( 1 and ka ) 1 for both approaches. Quantitatively, the results deviate
somewhat for ka ( 1. Hence, the WKB method is a very valuable approach for
studying critical adsorption behavior for a wide range of geometries. The main
advantage of the WKB method is a unified approach for the various geometries
based on the same level of approximations. It can be applied at the same level of
complexity to virtually any shape of the polylectrolyte–surface adsorption potential. Recent advances in polyelectrolyte adsorption under confinement [49, 167] and
adsorption onto low-dielectric interfaces [50] have been presented.
For a given ka value, the critical surface charge density required to initiate
polyelectrolyte adsorption onto a sphere is always larger than that for adsorption on
a cylinder. This, in turn, is larger than |s c | required for adsorption onto a plane.
Physically, as one changes from a plane to a rod and to a sphere, the entropic
penalty of polyelectrolyte confinement near the surface is likely to grow. Correspondingly, the necessary surface charge densities js c j to initiate adsorption
increase as well. Another, energy-based explanation is that a spherical surface
offers less contact area [129] and thus larger surface charge densities are necessary
for polyelectrolyte adsorption as compared with cylindrical and planar complexation. In the limit of ka ( 1, the reduction in the number of translational degrees of
freedom for adsorbed polyelectrolytes appears to be coupled to the power of js c j
$ ðkaÞ
gþ1 . Namely, g ¼ 2 for the planar case with possible chain translations
in two dimensions, g ¼ 1 for a cylinder with possible translations along its axis,
and g ¼ 0 for sphere adsorption with no chain translational freedom at all.
Strong and Weak Polyelectrolyte Adsorption onto Oppositely Charged Curved. . .
27
5.1 Comparing Geometries
Applying various approaches (see Sects. 3 and 4), we determined the distinct
scaling behavior for the critical surface charge density required to trigger adsorption of flexible polyelectrolyte chains onto a plane, a cylinder and a spherical
surface as function of the Debye screening length 1/k (see Fig. 3). A main
conclusion is that the critical surface charge density scales as s c ~ k
3 for a planar
interface and all k, whereas for cylindrical and spherical surfaces in the largecurvature or low-salt limit we find the relations s c ~ k
2 and s c ~ k
1 , respectively.
Generally, as the radius of surface curvature increases and the parameter ka grows,
a transition takes place from the large curvature limit to the planar limit. For
both the cylinder and sphere cases, this change in scaling laws appears at ka % 1,
i.e., when the radii of surface curvature become comparable to the Debye screening
length (Fig. 10). For adsorption onto a cylindrical surface, the crossover to the
planar limit occurs at somewhat smaller ka values than for the sphere situation.
Comparing the results obtained by the WKB method with the exact solutions for
the planar and spherical surface, we find, within 2% error, quantitative agreement in
the planar case. For a sphere, we find the same asymptotic dependence of s c on k in
the limit ka ( 1 and ka ) 1 for both approaches. Quantitatively, the results deviate
somewhat for ka ( 1. Hence, the WKB method is a very valuable approach for
studying critical adsorption behavior for a wide range of geometries. The main
advantage of the WKB method is a unified approach for the various geometries
based on the same level of approximations. It can be applied at the same level of
complexity to virtually any shape of the polylectrolyte–surface adsorption potential. Recent advances in polyelectrolyte adsorption under confinement [49, 167] and
adsorption onto low-dielectric interfaces [50] have been presented.
For a given ka value, the critical surface charge density required to initiate
polyelectrolyte adsorption onto a sphere is always larger than that for adsorption on
a cylinder. This, in turn, is larger than |s c | required for adsorption onto a plane.
Physically, as one changes from a plane to a rod and to a sphere, the entropic
penalty of polyelectrolyte confinement near the surface is likely to grow. Correspondingly, the necessary surface charge densities js c j to initiate adsorption
increase as well. Another, energy-based explanation is that a spherical surface
offers less contact area [129] and thus larger surface charge densities are necessary
for polyelectrolyte adsorption as compared with cylindrical and planar complexation. In the limit of ka ( 1, the reduction in the number of translational degrees of
freedom for adsorbed polyelectrolytes appears to be coupled to the power of js c j
$ ðkaÞ
gþ1 . Namely, g ¼ 2 for the planar case with possible chain translations
in two dimensions, g ¼ 1 for a cylinder with possible translations along its axis,
and g ¼ 0 for sphere adsorption with no chain translational freedom at all.
Strong and Weak Polyelectrolyte Adsorption onto Oppositely Charged Curved. . .
27