adsorbed polyelectrolyte circles (the second terms with the double sum) decrease in
magnitude when the number of circles increases (provided that the total charge of the
wrapped polymer is kept constant). Hence, within this model the adsorbed polyelectrolytes cannot overcharge the sphere. This result is similar to that discussed in the
previous section for polyelectrolyte adsorption on an oppositely charged cylinder.
The mechanical bending energy follows from (58) by parametrization in terms
of spherical coordinates. The contour s of a circle is related to its angle via s ¼ a#.
Hence, we obtain for N s semicircles:
E b ¼
l p pN s k B T
2a
:
(93)
6.2.3 Results
The inevitable chain bending of polyelectrolytes with a rather large persistence
length during adsorption on a spherical surface does not favor polyelectrolyte
wrapping and shifts the equilibrium towards undercharged complexes. With the
linear charge density e/l 0 , the neutralization fraction:
^ y ¼
e
s s
N s
4al 0
(94)
decreases as the adsorbed polyelectrolyte becomes stiffer, as displayed in Fig. 18. With
increasing number of circles, the energy corrections due to discreteness decrease in
magnitude, i.e., the uniform energy term proportional to ð1 À ^ yÞ
2
dominates, resulting
in ^ y values close to unity. According to (94), the optimal number of wrapped
polyelectrolyte circles exhibits a similar persistence length dependence as ^ yðl p Þ. For
the parameters of Fig. 18, the maximal value of N s (l p ! 0) is 27. Because N s is an
integer number by definition, a staircase-like behavior for the optimal ^ y is observed;
with increasing l p , the value of N s drops like ^ y from 27 to 0 (94). For a weaker screening
of the polyelectrolyte–sphere electrostatic attraction, more persistent polymers can be
wrapped around the sphere (see Fig. 18). Similarly to Fig. 15 for the polyelectrolyte–
cylinder charge compensation, Fig. 18 illustrates that the polyelectrolyte–sphere
compensation fraction never exceeds unity, even for highly flexible chains and strong
attraction conditions (realized at small k values).
At some critical value of l p , the polyelectrolyte bending energy penalty exceeds
the electrostatic attraction energy and ^ y drops abruptly to zero, which corresponds
to polyelectrolyte unwrapping, in complete analogy to the adsorption–desorption
transition in the weak adsorption case. In the current model of strong adsorption of
polyelectrolyte semicircles, the wrapping–unwrapping transition is of first order, in
44
R.G. Winkler and A.G. Cherstvy
magnitude when the number of circles increases (provided that the total charge of the
wrapped polymer is kept constant). Hence, within this model the adsorbed polyelectrolytes cannot overcharge the sphere. This result is similar to that discussed in the
previous section for polyelectrolyte adsorption on an oppositely charged cylinder.
The mechanical bending energy follows from (58) by parametrization in terms
of spherical coordinates. The contour s of a circle is related to its angle via s ¼ a#.
Hence, we obtain for N s semicircles:
E b ¼
l p pN s k B T
2a
:
(93)
6.2.3 Results
The inevitable chain bending of polyelectrolytes with a rather large persistence
length during adsorption on a spherical surface does not favor polyelectrolyte
wrapping and shifts the equilibrium towards undercharged complexes. With the
linear charge density e/l 0 , the neutralization fraction:
^ y ¼
e
s s
N s
4al 0
(94)
decreases as the adsorbed polyelectrolyte becomes stiffer, as displayed in Fig. 18. With
increasing number of circles, the energy corrections due to discreteness decrease in
magnitude, i.e., the uniform energy term proportional to ð1 À ^ yÞ
2
dominates, resulting
in ^ y values close to unity. According to (94), the optimal number of wrapped
polyelectrolyte circles exhibits a similar persistence length dependence as ^ yðl p Þ. For
the parameters of Fig. 18, the maximal value of N s (l p ! 0) is 27. Because N s is an
integer number by definition, a staircase-like behavior for the optimal ^ y is observed;
with increasing l p , the value of N s drops like ^ y from 27 to 0 (94). For a weaker screening
of the polyelectrolyte–sphere electrostatic attraction, more persistent polymers can be
wrapped around the sphere (see Fig. 18). Similarly to Fig. 15 for the polyelectrolyte–
cylinder charge compensation, Fig. 18 illustrates that the polyelectrolyte–sphere
compensation fraction never exceeds unity, even for highly flexible chains and strong
attraction conditions (realized at small k values).
At some critical value of l p , the polyelectrolyte bending energy penalty exceeds
the electrostatic attraction energy and ^ y drops abruptly to zero, which corresponds
to polyelectrolyte unwrapping, in complete analogy to the adsorption–desorption
transition in the weak adsorption case. In the current model of strong adsorption of
polyelectrolyte semicircles, the wrapping–unwrapping transition is of first order, in
44
R.G. Winkler and A.G. Cherstvy