6.2 Non-linear mean-field theory for the osmotic brush
6.2.4 Optimal brush height and its limiting behavior
The total free energy of the brush per unit cell is the sum of the electrostatic and elastic free energies obtained in Eqs. (6.16) or (6.17) and (6.25), i.e.
Ftot =FPB+FFJC. (6.29)
This compares with the scaling free energy (6.1), where the interaction and counterionic contributions (per unit area), i.e. Fci+Fint, are now incorporated in the cell-model PB free
energy, FPB.
The total free energy can be viewed as a function of the effective Manning parameter of the system, ξ, which varies according to the brush height L (see Eq. (6.20)). The typical form of Ftot is shown in Figure 6.5a as a function of ξ, where other parameters are fixed as
f = 1, ℓB = 0.1b0,q =qm = 1, rc =b0/2, and D= 1.5b0. Recall that the effective Manning
parameter, ξ, is bounded from below by ξ0, and from above by ξu, which are here ξ0 = 0.1
and ξu = 0.4. The total free energy increases for small and large ξ and has a minimum
at an intermediate Manning parameter, ξ∗, which corresponds to an optimal brush height
L∗=ξ0L0/ξ∗.
The reason for this behavior is that by decreasingξ down toξ0, the chain becomes highly
stretched and its elastic free energy diverges due to its finite extensibility (see Eq. (6.28)). For large ξ → ξu, on the other hand, the available space for counterions decreases (since Rhc→D), and the translational entropy due counterion confinement diverges.
In this section, I will focus on the generic predictions of the present mean-field cell model for the brush height, L∗, as a function of grafting density, ρa. To this end, I will take
counterions as point-like particles (rc = 0) with charge valencies q =qm = 1. More general
Figure 6.5b shows the optimal brush height plotted as a function of ρa for the Bjerrum
length ℓB = 0.1b0, and for three different values of the charge fraction f = 1, 1/2 and 1/3
(solid curves). As seen, both at large and small grafting densities, the brush height increases and eventually tends to its maximum valueL0 =Nmb0. Also a lower bound is predicted for
the equilibrium height of the brush depending on the charge fraction and the Bjerrum length. Forf >1/2 andℓB/b0 ∼0.1, this lower bound is about 50% of the contour length indicating
that the chains remain increasingly stretched over a wide range of grafting densities. The limiting behavior of the brush height with grafting density can be understood both by the asymptotic expansion of the free energy (Appendix F.2) and by simple physical arguments as I will present now.
In the limit of large grafting densitiesρa →ρmaxa (with ρmaxa = 1/b20 being the maximum
grafting density here), one deals with small cell radius, i.e. D/R0→1, whereR0 =b0/2 (recall
from Section 6.2.1 that in the present model ρab20 = R20/D2). In this limit, the dominant
contribution to the PB free energy comes from the confinement entropy of counterions inside the cell, which may be approximated by that of an ideal gas of particles (see Appendix F.2), i.e.
Sci
NmkB ≃
fln[π(D2−R2)L], (6.30) where R is defined in Eq. (6.19). Therefore, osmotic pressure of counterions becomes the major repulsive pressure swelling the rod against stretching pressure of the elasticity. In this limit, the chain has a large extension and its elastic free energy per number of monomers is given by Eq. (6.28), i.e.
Felas
NmkBT ≃ −
ln(1−L/L0). (6.31)
Balancing the longitudinal pressures due to these two opposing contributions using∂/∂L(Felas−
T Sci) = 0 at fixed cell radius D, I obtain
L∗(ρa)
L0 ≃
f+ρab20
1 +f , (6.32)
which is equivalent to Eq. (6.8) obtained within the non-linear scaling theory in Section 6.1. The expression given by Eq. (6.32) is plotted in Figure 6.5b for the charge fractionsf = 1, 1/2, 1/3 (dashed curves) together with the results obtained from minimization of the full free energy in Eq. (6.29) (solid curves). The coincidence is evident at large grafting densities. The linear dependence on the grafting density in Eq. (6.32) is induced by the conserved polymer volume constraint. The weaker dependence and deviations from Eq. (6.32) observed at lower grafting densities (e.g., at about ρab20 ∼ 0.1, which approximately corresponds to
the simulated regime) result from lateral electrostatic effects, which become significant and generate a minimum for the brush at intermediate grafting densities.
For very small grafting densitiesρab20≪1 (or equivalentlyD/R0 ≫1), the present model
is applicable for very long chains, since only in this case, counterions will be confined within the brush (see the Discussion). In this limit, the brush height shows different asymptotic behavior depending on whether the optimal Manning parameter, ξ∗, is below or above the Alfrey-Fuoss threshold ΛAF ≃ 1, Eq. (6.24). First I will consider the case with ξ∗ < ΛAF.
For the parameters chosen in Figure 6.5b (ℓB = 0.1b0, qm = q = 1), this condition holds
for ρab20 < 0.1, as can be checked from Eq. (6.24). In this case, the counterion cloud is
6.2 Non-linear mean-field theory for the osmotic brush 97
1e-6 1e-4 1e-2 1
1 (a) (b) (c) 0.4
L/
* ( Nm bo ) ρab02Figure 6.6: Log-log plot of the rescaled optimal height of the brush as a function of grafting density. Solid curves show the results obtained from the minimization of the full free energy, Eq. (6.29), with f = 1, q=qm= 1, and point-like counterions, for a) ℓB = 0.1b0 (ξ0= 0.1), b)ℓB = 0.7b0 (ξ0 = 0.7) and c) ℓB = 1.2b0 (ξ0 = 1.2). The dashed and dot-dashed curves show the asymptotic estimates at small grafting densities for the cases a) (using Eq. (6.34)) and c) (Eq. (6.38)) respectively. The dotted line shows the brush height in the absence of lateral effects forf = 1 (Eq. (6.39)).
screening on the bare electrostatic potential of the rod ψ(r) = 2ξln(r/R). This potential is used to calculate the electrostatic energy per unit cell as
Uelec
NmkBT ≃
f ξlnD
R. (6.33)
The entropic contribution of counterions may still be accounted for using Eq. (6.30). Conse- quently, the electrostatic free energy of the system, Felec≃Uelec−T Sci, is obtained as
Felec
NmkBT ≃
f(ξ−2) lnD
R (6.34)
for very large D/R0. This expression can be derived also by expanding the PB free energy
(6.16) in powers of R/D as shown in Appendix F.2.
Using Eqs. (6.34) and (6.19), the longitudinal electrostatic pressure can be calculated by differentiatingFelec with respect toL=ξ0L0/ξ. Balancing this with the elastic pressure from
Eq. (6.31), one obtains the equilibrium brush height which has to be calculated numerically and is shown in Figure 6.6 (dashed curve) for f = 1 and ℓB = 0.1b0. In the Figure, I also
show the results from minimization of the full free energy, Eq. (6.29) (solid curve a). The plot is made for ρa down to 10−6b0−2. For vanishingly small grafting densities ρab20 → 0,
the entropic contribution becomes negligible compared with the bare electrostatic repulsion between monomers, and the equilibrium brush height behaves asymptotically as
L∗(ρa)
L0 ≃
flnρab20
flnρab20−2ξ0−1
. (6.35)
This function is not shown in Figure 6.6, because it is valid for smaller grafting densities. In the second scenario, i.e. when the optimal Manning parameter becomes larger than the threshold ξ∗ >ΛAF≃1, bare electrostatic interactions are partially screened as a result
of the counterion condensation process (Chapter 3). The PB electrostatic potential (up to some logarithmic corrections) reduces to the bare electrostatic potential of a rod with critical Manning parameter ξM = 1, i.e., ψ(r) = 2 ln(r/R), when D/R0 → ∞. The electrostatic
energy per unit cell can be estimated using this potential as
Uelec NmkBT ≃ f ξ ln D R. (6.36)
To estimate entropic contributions in this case, I shall adopt the counterion-condensation picture [20, 39] that only a fraction of 1/ξ of counterions are unbound and may contribute to the entropic pressure. Thus, the corresponding ideal-gas entropy of counterions,Sci in Eq.
(6.30), may be corrected by such a factor and used, together with Eq. (6.36), to obtain the leading term of the electrostatic free energy per unit cell,Felec ≃Uelec−T Sci, as
Felec NmkBT ≃ − f ξ ln D R. (6.37)
This expression may be obtained by a limiting expansion of the PB free energy (6.17)– see Appendix F.2. Calculating the longitudinal electrostatic pressure from Eq. (6.37) and balancing it with the non-linear stretching pressure from Eq. (6.31), I obtain
L∗(ρa)
L0 ≃
1 + 2ξ0
flnρab20
. (6.38)
The asymptotic expression, Eq. (6.38), is shown in Figure 6.6 (dot-dashed curve) along with the result from minimization of the full free energy, Eq. (6.29) (solid curve c) for a system withf = 1 andℓB = 1.2b0 (ξ0 = 1.2) for which the optimal Manning parameter,ξ∗, remains always above the Alfrey-Fuoss threshold (see footnote 2). As seen, the above asymptotic estimate, Eq. (6.38), becomes accurate forρab20 <10−4. Similar behavior is obtained within
the present model for the whole range of Bjerrum lengths; see, e.g., the result forf = 1 and
ℓB = 0.7b0 (ξ0 = 0.7) in Figure 6.6 (solid curve b).
The preceding results on the low-grafting-density behavior of the brush thickness demon- strate the important role of lateral electrostatic effects (especially that of lateral distribu- tion of counterions), which are systematically included in the PB free energy and generate re-stretching of the chains at small grafting densities. In this limit, the constant volume con- straint becomes unimportant. Both for weakly charged (ξ0 < 1) and highly charged chains
(ξ0 > 1), lateral electrostatic contributions produce a repulsive longitudinal force acting on
the chains, which increases logarithmically by decreasing the grafting density (Eqs. (6.34) and (6.37)). In particular, for highly charged chains, the force is independent of the brush height, i.e. −∂Felec/∂L ∼lnD/R (using Eqs. (6.37) and (6.20)), which is a direct consequence of
the electrostatic screening due to condensation of counterions. In any case, the increase of the brush height, which converges to the contour length, is logarithmically weak. If lateral effects are neglected, the brush height remains independent of the grafting density and reads
L∗
L0
= f
1 +f, (6.39)
which is shown by a dotted line in Figure 6.6. This result follows from Eqs. (6.30), (6.31) and neglecting the volume constraint (compare with Eq. (6.7) in the non-linear scaling theory in Section 6.1).
6.2 Non-linear mean-field theory for the osmotic brush 99 0.04 0.06 0.08 0.1 0.12 ρaσ2 8 10 12 14 16 18 20 22
L
*/
σ (i) (ii) 0.04 0.06 0.08 0.1 0.12 ρaσ2 2 4 6 8 10 12 14 16 (i) (ii)L
*/
σa)
b)
Figure 6.7: Brush height as a function of the grafting density for polyelectrolyte chains ofNm= 30 monomers (contour lengthL0≃30σ). Circles show the simulation data and squares are the predictions of the present mean-field cell model for a) charge fractionf = 1 and the Bjerrum lengthℓB =σ, and b) charge fractionf = 1/3 andℓB ≃2σ. The dotted lines (i) and (ii) show the scaling predictions, Eqs. (6.6) and (6.7), with Gaussian and non-linear elasticity respectively.
Finally, I emphasize that the non-monotonic behavior of the brush thickness is not influ- enced by the elasticity model and qualitatively similar features are obtained when a Gaussian chain elasticity is used.3