Langmuir 1991, 7, 590-595
590
The Bending Modulus of Ionic Lamellar Phases A. Fogden and B. W. Ninham’ Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, ACT 2601, Australia Received J u n e 13, 1990. I n Final Form: September 20, 1990 The electrostatic double layer free energy, bending rigidity, and elastic modulus of Gaussian curvature are calculated for undulating ionic lamellar phases. A range of situations, appropriate to exploration of the Helfrich mechanism for the stabilization of swollen lamellar phases, is covered. 1. Introduction Application of the concept of curvature elasticity to thermal fluctuations of amphiphilic membranes has allowed some justification for, and prediction of, the stability of a variety of structures in self-assembling systems. The common basis of such theories is the phenomenological formula of Helfrich’ for the bending energy per unit area of membrane
Here H and K denote the mean and Gaussian curvatures, respectively, defined in terms of the local principal curvatures c1 and c2 by
H = (1/2)(c1 + c2), K = clc2 (2) Ho and KOare the corresponding preferred spontaneous values, k , is the bending rigidity, and kc is the elastic modulus of Gaussian curvature. For spontaneously curved surfaces a recent study2raised fundamental objections to this formula on the basis of simple stability considerations. It is apparent that the predicted bending energy is not minimized at H = HOand K = KO (unless both quantities are zero). This circumstance is in conflict with the identification of these parameters as preferred curvatures characterizing the rest state of the membrane. The consequences of this observation are of broad relevance since the majority of phases formed in surfactant aggregation are driven by spontaneous membrane curvatures dictated by the preferred geometry of the surfactant molecule set by intramolecular forces. However, for lamellar phases comprised of planar bilayers, objections to the Helfrich approach do not arise since the spontaneous curvatures are zero. The Helfrich form
+
(3) gbend = 2kcH2 k$ is then appropriate, provided the length scales of interaction of the membrane (with itself and with neighboring lamellae) are very much smaller than the average principal radii of curvature and the length scale of their variation on the surface. The splay contribution to this bending energy of outof-plane fluctuations of the membrane has been long established, while the saddle splay term is of a more subtle nature and is commonly discarded in analyses of lamellar systems. In particular, the mechanism for the strong, longranged steric repulsion between membranes postulated (1)Helfrich, W. 2.Naturforsch., C: Biochem., Biophys., B i d , Virol. 1973, 28c, 693.
( 2 ) Fogden, A.; Hyde, S. T.; Lundberg, G. J. Chem. SOC.,Faraday
Trans, in^ press.
by Helfrich3 has been attributed to the splay term. This repulsion has been invoked to account for the observed stability of swollen lamellar phase^.^ Interest in the swelling properties of such systems has motivated the advent of sophisticated techniques for measurement of the splay modulus hc.596 Interpretation of such experiments depends on theoretical estimates of the moduli, which it is our purpose to obtain. For lamellar phases comprising bilayers of ionic surfactants (and oil) swollen with aqueous electrolyte, the validity of the Helfrich form, eq 3, can be assessed. These systems are characterized by the Poisson-Boltzmann equation so that direct calculation of the electrostatic energy of the perturbed system is possible. Within the range of validity of the Poisson-Boltzmann theory, the resulting predictions of the electrostatic contribution to the modulus k , may be compared with values inferred from X-ray scattering and other technique^.^!^ 2. Preliminary Considerations To facilitate calculations of the splay term, consider an isolated bilayer defined by parallel planar interfaces undergoing small, two-dimensional, sinusoidal undulations of amplitude a and wavelength 2 a l k for which eq 3 predicts a surface-averaged bending energy per unit area g = (1/4)kck4a2 (4) A previous study7 analyzed a single such surface separating a 1-1 electrolyte and a medium in which the field is assumed to vanish. With the boundary condition
relating the normal derivative of the electrostatic potential to the uniform surface charge density u, the free energy of the double layer is given by
If the constituent monolayers are free from electrostatic connection and oscillate independently, then the energy of the bilayer in this decoupled limit is twice this value. Specifically, with the potential given by the nonlinear Gouy-Chapman theory (3) Helfrich, W. 2.Naturforsch., A: Phys., Phys. Chem.,Kosmophys. 1978,33a, 305. (4) Lipowsky, R.; Leibler, S. Phys. Reu. Lett. 1986, 56, 2541. (5) Bassereau, P.; Marignan, J.; Porte, G. J. Phys. 1987, 48, 673.
(6) Ricetti, P.; Kekicheff, P.; Parker, A. J.; Ninham, B. W. Nature (London) 1990,346, 252. (7) Fogden, A.; Mitchell, D. J.; Ninham, B. W. Langmuir 1990,6,159.
0 7 4 ~ - 1 4 6 3 / 9 ~ ~ 2 4 0 ~ - 0 5 9 0 $JQ O 2 . 50 0 1991 American Chemical Society
Langmuir, Vol. 7, No. 3, 1991 591
The Bending Modulus of Ionic Lamellar Phases
the Helfrich form eq 4 with bending moduluss
where e , is the dielectric constant of water, q is the protonic charge, 0 = l/kBT (where kg is the Boltzmann's constant and T is the absolute temperature), and K is the inverse Debye length. Also p1 = (1 + (k2/K2))1/2and s = 47ra@q/ t,,,~. In the linear limit (small s), eq 7 reduces to
An alternative definitiong of the free energy of double layer systems results from consideration of a hypothetical equilibrium discharging process in which the ionic charge is decremented to zero. Under the assumptions of BenSimon et al., the corresponding formulas, analogous to eq 11, for the insulating and conducting membrane cases are
For the long wavelength regime k 1 + a cos k y are given by the PoissonBoltzmann equation
where no is the bulk electrolyte concentration, while the potential J/ in the intervening hydrocarbon layer -1 + a cos k y C x C 1 + a cos k y satisfies Laplace's equation v2J/ = 0 (A2) In order to derive the electrostatic free energy of the system, these partial differential equations are solved subject to the interfacial coupling conditions (17) and the boundary conditions of vanishing field and potential infinitely far from the membrane. To facilitate this, introduce the dimensionless potentials u+ = Pq++ and u = @qJ/and the scaled coordinates 5 = K X and 7 = KY, where the inverse Debye length K is defined as
(27)
To complete our treatment of the bilayer, it remains to analyze the odd mode of oscillation of the monolayers. With the undulating monolayer surfaces defined by x = f1 f LY cos k y , the system is symmetric about the plane x = 0 and the general electrostatic bending energy formula analogous to that for the even mode in the Appendix is simply obtained on interchange of cosh kl and sinh kl throughout. However for the odd mode the coupling is not solely of electrostatic origin since the bilayer thickness 2(1 a cos k y ) is fluctuating, introducing additional packing constraints. In particular, the physical constraint of local conservation of volume in the hydrocarbon region demands that 1
case and the charge density optimization case for the even mode are not relevant in general for the odd mode. Now the bending energy is given by substitution of eq 28 into the variable surface charge density electrostatic free energy formula, yielding
Then within the Debye-Huckel approximation the governing equations and coupling conditions become V2Uf
= u*
v2u = 0 u*-u=O
A ( u* at
3)+
ak sin
(5);-uf 3)
?si(?)
-
(1+ a2k2sin2
(A44 (A4b) (A54
=
&))'I2
(A5b)
(28)
This completely specifies the surface charge distribution, so previous considerations of the uniform surface charge
are the general, dimensionless surface charge densities.
Fogden and Ninham
594 Langmuir, Vol. 7, No. 3, 1991
We seek a perturbation solution in powers of the scaled amplitude of undulation m
m
aL2) i=O
1 + ipl(a,a;l
- a&l:,)
1
+ -(a;Oa,o + a;al0) + 4
i=O
The periodicity of the boundary conditions admit Fourier cosine series representations for the potentials and surface charge densities
S*(
1
+ a2k2sin2 (f q))"'
=
2's;
3
cos ( n q )
n=O
(A9)
where, as in eq A7, we write
where at0 and ai0 were set to zero in this formula to remove the linear term. To specify the internal energy the parameters a,*, a20*, a,,*, C l n , and d l , ( n I1) must be determined from the coupling condition eq A5. To achieve this, insert eq A7 into eq A5 and expand in powers of KCY up to second order. Then introduce eqs A8-Al2 and equate Fourier series for the coefficients of each power. In particular we find that, subject to the expected consistency condition sin + S t n = 0 ( n 1 1) a,+ = a,- = 2s a,,
+ a10 = s +
- s ( p , - 1))-sinh kl
Dl
i=O
Substitution of eqs A7-A8 into eqs A4 produces the general solutions Ain* = ain*eTP"(ET') Ai, = cin cosh qnt + din sinh q,t
all -a11 +
= -2(s
+ (S:,
- ~ ( p -, I))*)
(A171
Dl
(All) cl:
(-412)
where q, = nk/K and pn = (1 + qn2)'i2. By definition of the planar perturbation, the family son*, Con, don, Son* are zero for n 2 1. Also, from eqs A9-AlO the mean charge densities on the surfaces are
implyingsio* = 2s6i0,where 6 denotes the Kronecker delta. The total electrostatic internal energy & of the system, from which the free energy is derived, is given by
In terms of dimensionless quantities, the mean internal energy per unit monolayer area is thus
+ dl:
= (s:,
- dpl-
l)&,,)l/Dn)'
where
D, = pn sinh nkl
'0 + -qn
cosh nkl
(-418)
W '
Combining eqs A16 and A17 and simplifying, the change in internal energy per unit area on fluctuation of the bilayer is
where
O
2Pl
(-2p,
+ 1) + 2(P,-
2sinh kl 1) --
Dl
"'>
(P1- 1 sinh2 )0,2
The corresponding free energy of fluctuation per unit area is
Substituting eqs A7-A8 and All-A12 into this expression and expanding to quadratic order in the amplitude yield
where the prime denotes replacement of q by Xq and ab) by XuCy) in the function. Inserting the form (A19) then yields
Langmuir, Vol. 7, No. 3, 1991 595
The Bending Modulus of Ionic Lamellar Phases
63 - ‘w _
K
sll 2J1dhe’2
s2J1dXh2e’,
+ ss,:
JldXhe’,
+
+ ~ s ~ n 2 ~ d h h ’ ,(A25) .,) n=2
The integrals, involving the quantities defined in eqs A20A23, are readily evaluated via the change in integration variable from h to P ’ ~ and , it may be verified that
The first term, proportional to the magnitude of the relative area change on fluctuation, represents an electrostatic stretching energy which is alleviated if an
opposing bilayer surface tension term is introduced and the mean free energy optimized with respect to the mean surface charge density. The remaining terms recover the bending energies in the uniform charge density case for s&, = 0, n L 1,and in the minimizing charge density case for stl = s(p1 - I), sin = 0, n L 2. Note also that the quantities defining this quadratic energy expression also specify the monolayer potentials up to first order, which are given by
