Langmuir 1994,10, 1084-1092
1084
Bending Moduli and Spontaneous Curvature. 2. Bilayers and Monolayers of Pure and Mixed Ionic Surfactants P. A. Barneveld, D. E. Hesselink, F. A. M. Leermakers; J. Lyklema, and J. M. H. M. Scheutjens Laboratory for Physical and Colloid Chemistry, Agricultural University Wageningen, Dreijenplein 6, 6703 BC Wageningen, The Netherlands Received September 14, 1993. I n Final Form: January 3,1994' Bending elasticity moduli of equilibrium bilayers and monolayers of surfactants are calculated using a previously developed self-consistentfield lattice model. The model is extended by incorporating ionic interactions at curved interfaces, so that ionic surfactants can be treated as well. The interfaces are formed by self-assemblingof the surfactants. It is found that the size of the counterionsis an important parameter in determining the bending moduli of charged interfaces. Screening the electric double layer by salt has two opposing effects on the rigidity of monolayers and bilayers of ionic surfactants. The f i t is that suppression of the double layer as such would make the layers less rigid. However,this trend is outweighed by the simultaneously occurring thickness growth. The s u m effect is therefore that the surfactant layers are more rigid in higher salt concentrations. In the case that salt ions decrease the solvent quality, as salting-out ions do, the rigidity of the layer passes through a maximum in high salt concentrations (ca. 1kmol/m% The spontaneous curvature of a water-dodecane-sodium dodecyl sulfate system depends on the ionic strength in solution. We predict that the preferential curvature changes sign twice when the ionic strength increases. This can be explainedby a continuousincrease in packingdensity of the surfactant layer upon an increase in ionic strength. There are at least two factors contributing to the observed behavior: (1)The shape of the surfactant becomes increasinglyimportant when packing density increases. (2) The difference in the ability of the solvents (dodecaneand water) to penetrate into the surfactant layer becomes, especially at high packing densities, a curvature determining factor.
Introduction Each fluid interface is the seat of an elastic free energy of bending A,, for which Helfrich wrote'
In this equation, A, is the area of the interface and J and K are the mean and Gaussian curvatures, respectively. They are defined in terms of the principal curvatures c1 and c2 as c1 + c2 and c1c2, respectively. Similarly, k, is called the mean bending and k, the Gaussian bending is observed when modulus. The spontaneous curvature Jsp the elastic free energy of bending is a t ita minimum. In Helfrich's approach the contribution of a spontaneous Gaussian curvature, Kap,is neglected. The parameters k,, h,, and Jsp play important roles in the physics of curved surfactant layers, which occur, for example, in vesicles and microemulsions. Bending elasticity parameters can also be important in planar layers because of their role in thermally induced undulations of the interface.2 Examples of such systems are surfactant monolayers in a Langmuir trough and macroscopic free liquid films for which the thickness may be influenced by undulations.3 Hence it is desirable to know the bending parameters for various systems. Experimentally, bending elasticity parameters are hard to determine. Nevertheless, several techniques have been applied to a variety of systems. Without being complete, we mention an ESR study on microemulsions by di Meglio
* To whom correspondence should be addressed.
Abstract published in Advance ACS Abstracts, March 1,1994.
(1) Helfrich, W. 2.Naturforsch. 1973,Bc, 693. (2)Helfrich, W. 2.Naturforsch. 1978,33a, 305.
(3)Barneveld, P.A.;Scheutjens, J. M. H. M.; Lyklema, J. ColZoids Surf. 1991,52,107-121.
et al.,4 an ellipsometry study on surfactant monolayers by Meunier? and an X-ray study on lamellar phases by Safhya et al.8 Song and Waugh have studied mixed phosphatidylcholinephosphatidylserine bilayers and found that the surface charge has, at variance with current theoretical predictions, no effect on the intrinsic rigidity of the membranes.' Statistical thermodynamic approaches have been used to predict the rigidity of interfaces. The bending elasticity parameters can be related to the curvature dependence of the interfacial t e n s i ~ n . ~Szleifer *~ et al. investigated the influence of the length of short uncharged end-grafted chain molecules, also in the presence of shorter chains (alcohols), using the statistical thermodynamic model of Ben-Sha~l.~Milner and Witten have developed an analytical self-consistent field lattice model valid for high densities of end-grafted long chains.1° Both groups disregarded the head groups of the surfactants. Barneveld et al. used the self-consistent field lattice approach of Scheutjens and Fleer to study curvature elasticity parameters of full equilibrium bilayers and monolayers of poly(oxyethy1ene) ~urfactants.~ Systems containing ionic surfactants, however, involve electrical double layer phenomena. Winterhalter and Helfrich" and Lekkerkerkerl2JS were the first to inves(4) di Meglio, J. M.;Dvolaitzky, M.; Ober, R.; Taupin, C.J. Phys.Lett. 1983,44,L-229. (5) Meunier, J. J. Phys. Lett. 198S, 46,L-1005. (6)Safinya, C. R.;Roux, D.; Smith, G.S.;Sinha, S. K.; Dimon, P.; Clark, N.A.;Bellocq, A. M. Phys. Rev. Lett. 1986,57,2718. (7) Song, J.; Waugh, R. E. J. Biomech. Eng. 1990,112,236-240. ( 8 ) Kozlov, M. M.: Leikin, S. L.: Markin, V. S. J. Chem. SOC.,Faraday Trans. 2 1989,85,277-292. (9)Barneveld, P.A.;Scheutjens, J. Langmuir - - J. M. H. M.:. Lyklema. 1992,8,3122-3130. . (10)Milner, S.T.;Witten, T.A. J. Phys. (Paris) 1988,49, 1961. (11)Winterhalter,. N.: . Helfrich, W. J. Phrs. Chem. 1988,92,68666867. (12)Lekkerkerker, H. N.W. Physica A 1989,159,31!3-328. (13)Lekkerkerker, H.N. W. Physica A 1990,167,384-394.
0743-7463/94/2410-10S4$04.50/0 0 1994 American Chemical Society
Bending Moduli and Spontaneous Curvature
tigate the influence of the double layer on the bending parameters k, and K, using Debye-Huckel and PoissonBoltzmann equations, respectively. They considered “solid”particles (the particles have fiied surface composition) of variable radii at fixed surface charge density. The excluded volumes of surfactant and ions were neglected. The results are translated to the prediction of the mechanical properties of membranes and microemulsions as a function of the ionic strength. They predict that charged membranes become more flexible when the ionic strength increases. Other authors arrive, using similar assumptions, at the same conclusion.1k17 As we will show below, these predictions have limited applicability,because they exclusively apply to surfaces that cannot respond to changes in solution. Hence, for surfactant systems this is not a good approximation. Nevertheless, Odijk,l*Hardon et al.,l9 and Pincus et al.,O used these predictions to investigate the behavior of charged multibilayer systems. One of the aims of the present study is to calculate the bending elasticity parameters for a more realistic situation, i.e., in which all molecules, including the small ions, have finite volumes and where self-assembling ionic surfactants spontaneously form monolayers or bilayers that are in equilibrium with the free surfactants in the solution. Consequently, the restriction that the charge of the surfactant layer is fixed, is relaxed. It is now a function of parameters such as the ionic strength of the system. The results are compared with those for uncharged surfactants of the same size and for chargedlayers without surfactant. Recently, Ennis reported a statistical thermodynamical treatment which accounts for the fact that surfactant layers change composition upon changes in solution conditions.21 Although this last work was less rigorous than ours, similar conclusions regarding the effect of the ionic strength on the mechanical properties of charged bilayers were obtained. A second topic of this paper is to predict the mechanical properties of surfactant monolayers as they exist in oil in water (OIW) or water in oil (WIO) emulsions. The important issue here is to compute the preferential curvature of the oil-water interface. In our previous paperg we discussed oil-surfactant-water systems, where the surfactant was a nonionic. In this paper we concentrate on ionic surfactants and investigate the effect of the ionic strength at fixed surfactant concentration. In the following section we briefly review the relations between bending parameters and thermodynamic quantities. We proceed with a section on the self-consistent field lattice model that we use to calculate the thermodynamic quantities and that we extend to charged interfaces with curvature. Following a discussion on the parameters used, the Results and Discussion section presents calculations on charged “solid” interface, surfactant bilayers and monolayers. Bending Elasticity Parameters Aggregation of ionic surfactants introduces an inhomogeneous charge density distribution in and around the aggregate. Since the double layer and surroundings are (14)Mitchell, D. J.; Ninham, B. W. Langmuir 1989,5,1121-1123. (15)Duplantier, B.;Goldstein, R. E.; Romero-Rochin,V.; Pesci, A. I. Phys. Rev. Lett. 1990,65,50&511. (16)Fogden, A.; Ninham, B. W. Langmuir 1991, 7,590-595. (17)Winterhalter,M.; Helfrich,W. J. Phys. Chem. 1992,96,327-330. (18)Odijk, T.Langmuir 1992,8,1690-1691. (19)Harden, J. L.;Marques, C.;Joanny,J.-F.;Andelman,D. Langmuir 1992,8,1170-1176. (20)Pincus, P.;Joanny, J.-F.;Andelman,D.Europhys. Lett. 1990,11, 763-768. (21)Ennis, J. J. Chem. Phys. 1992,97, 663-678.
Langmuir, Vol. 10, No. 4, 1994 1085
both electroneutral, no net charges are transported from or to the surroundings upon formation of the system. Hence, no electrostatic work is performed by the system. The change in the interfacial grand potential (interfacial grand canonical characteristic function), as,as given in ref 9, can also be used to find expressions for the bending elasticity parameters in the case of systems with charged interfaces dOs = S d T - E n : dpi + y dA, + A,Cl d J + A$, I
dK (2)
where Tis the temperature, S the entropy, p the chemical potential, and n the number of molecules. The subscript i denotes any componentin the systemand the superscript s refers to the interfacial part of the quantity. The coefficients AsC1 and A,C2 are referred to as the fiit and second bending moment, respectively.22 The curvature elasticity energy, A,, per surface area, as given in eq 1,is essentially the excess interfacial tension due to bending with respect to the same at spontaneous curvatureJw,i.e.,ACIA8=y(J,K)-y(Jw,Kw). InHelfrich’s approach, eq 1,Ksp = 0. The interfacial tension can be expanded in the same way as in our previous work: where ionic surfactants were not considered. Combining eq 2 and the integrated form of this equation Os= yA,, we arrive at the following GibbsDuhem type relation dpi + Cl d J + C, dK
d r = -8’ d T -
(3)
I
where sa is the interfacial entropy per unit area and ri is the amount adsorbed per unit area. At constant T and (pi),y is a function of J and K and can be expanded, e.g., in a Taylor series around the point where the interfacial tension is at its minimum, i.e., where the interface is in the spontaneousstate characterized by Jsp, and K,,,. Retaining terms up to second order, we obtain
The elasticity parameters,
E,b
a2ylaaab, are given by
Ejj = acl/aJ E, E,
(5)
= acdaK
= E~ = acl/aK = acdaJ
For a cylindrical system (K = 0) of which the interfacial tension as a function of curvature J is known,EJJcan be calculated using eq 4. If for that system EJKcan be neglected, the spontaneous mean curvature JSp can be calculated. The elasticity EJJ coincides with Helfrich’s k,, occurring in eq 1. With the results of a spherical interface (K= 52/4), E m and KBpcan be retrieved. Note that Helfrich’s parameter k,, which equals Cz(0,O) = -(E,Ksp + EJKJ,~) (cf. eqs 1,3, and 4) can be obtained from a planar interface (see ref 22, eq 40). In the next section, a lattice model is presented which allows for the calculation of the interfacial tension of ionic monolayers and bilayers. Essentially, we extend the self(22)Markin, V.S.;Kozlov, M. M., Leikin, 5.L. J. Chem. Soc.,Faraday Tram. 2 1988,84,1149-1162.
Bameveld et al.
1086 Langmuir, Vol. 10, No. 4,1994 from the planar ones,24i.e. the A’s a t z
h+r(z)=
Figure 1. Definition of the lattice layers. The fiist available layer is layer 1at position 1. In situation (a) all layers are available for molecules (20 = l / 2 ) . In situation (b) a solid core of radius R = I is inaccessible for molecules (20 = 4 2 ) . consistent field lattice theory of curved nonionic monolayers and bilayers as described in our previous papere by incorporating electrostatic interactions, using the method developed by Bohmer et al.23but modified for curved surfaces. The equations are solved by a newer and simpler iteration scheme: see Appendix A.
Self-Consistent Field Lattice Theory The model is based on a lattice. The lattice can be either flat, cylindrical, or spherical. Each chain segment, solvent molecule, or (hydrated) ion occupies one lattice site. All sites have equal volume. Inhomogeneities are allowed in only one dimension, which leads to the concept of lattice layers of constant segment volume fraction. For convenience, the layers are numbered from z = 1up to M, where conventionally layer 1is the innermost layer and layer M is in the bulk solution. A lattice layer extends from z l - 112 to zl + 112, where 1 is the thickness of a layer. The center of the lattice, 20, is located at zo = 112 - R, where R is the radius of an optional solid, inaccessible core. This is illustrated in Figure 1. The area and volume of the lattice up to layer z are given by: A,(zl+
i)= planar symmetry cylindrical symmetry
lA,(zl+ 112) L(z) A+,(=)
&(z) = 1- X-,(z)
- X+l(Z)
-
cylindrical symmetry (9) spherical symmetry
Note that AZ~&) = 0 when Iz - 2’1 > 1. Chain Statistics. We consider linear molecules as chains of segments. Each segment has the same size but can be of different chemical nature. For a segment of typex, irrespective to which chain it belongs, the potential energy in layer z relative to bulk solution (phase B) is u,(z). The probability of finding a free (detached) segment x in layer z, e.g., a solvent molecule or small ion, is given by its Boltzmann factor G,(z), defined as
G , ( ~ )= e+Az)/kT
(10)
where k is Boltzmann’s constant. Generally, u,(z) depends on all possible interactions of segment x with ita environment. Here, we take into account the excluded volume effect, specific nearest neighbor interactions, and electrostatic interactions u,(z) = UW+ kT
X,~[(CP,(Z))
- 381 + ev,($(z) -IC“)
Y
(11)
where cp is the volume fraction, x the Flory-Huggins parameter, v the valency of a segment, $the electrostatic potential, and e the elementary charge. The subscripts x and y refer to any segment type A, B, C, ...,in the system and the superscript indicates the bulk solution. The angular brackets, (...), indicate averaging of cp over the neighboring cells. For a site in layer z, this involves three consecutive layers, z - 1, z, and z + 1
(6)
(44z))= A-,(z)cp(z-l) + Ao(z)cp(z) + A+,(z)cc(z+l) (12)
spherical symmetry The excluded volume term in eq 11,u’(z), is a Lagrange multiplier which is chosen such that C, (cx(z) = 1for each
and
V(Zl+
z.
6, =
t+
fL( zl+
\./,ck : sh z1
-20)
-1 - zo)2 - z0)a
planar symmetry cylindrical symmetry
(7)
spherical symmetry
respectively. In these equations, h is the length of the cylinder and L ( z )the volume of layerz. The latter quantity is simply related to V and given by
L ( z ) = V(zl+ 1/2) - V(z1- 112)
To find the volume fraction profile &,s) of a particular segment s in chains of type i, we define end segment distribution functions Gi(z,sll). These functions describe the average Boltzmann weight of all conformations of a chain of s segments with segment s in layer z. The functions are evaluated by step-weighted walks along the contour of chains i, starting with the distribution of a (detached) segment 1 and finishing after - 1 steps at segments in layer z. Each step generates the distribution of the end segment of a chain from that of a chain which is one segment shorter, according to the recurrence equation: Gi(z,sll) = Gi(z,sIs)( Gi(z,s-lll))
(13)
(8)
The a priori probability for a chain that is generated to have adjacent segments in layers z and z’is given by Xz~-z(z) and is dictated by the geometry of the lattice. In the case of a curved lattice these probabilities can be calculated
In this equation, the angular brackets represent an average of Gi(Z,s-lll) over three layers, in the same fashion as (c is averaged in eq 12. The weight of segments in layer z, Gi(z,sIs), equals G,(z) when segment s is of type x . The sequence starts with Gi(z,lll). A similar end segment
(23) Bdhmer, M. R.; Evers, 0. A.; Scheutjens, J. M. H. M. Macromolecules 1990,23, 2286-2301.
(24) Leermakers, F. A. M.; Scheutjens, J. M. H. M.J. Chem. Phys. 1989,93, 7417-7426.
Bending Moduli and Spontaneous Curvature
Langmuir, Vol. 10, No. 4, 1994 1087
distribution function, Gi(Z,.+), is calculated by starting the sequence a t the other end of the chain (segment r). Now, cpi(z,s) can be evaluated from v&s) = ci
express the electrostatic potential in layer z in terms of the local charge and the electric potentials in the adjacent layers
Gi(z,sll)Gi(z,slr) Gi(z,s(s)
(14)
which combines the two chain ends at segments in layer z. The division by Gi(z&) corrects for double counting of the statistical weight of segments. The normalization factor Ci can be obtained from the equilibrium volume fraction (48 in the bulk solution. Since in bulk solution all end segment distribution functions are unity, vi’ = C, a@(8) = riCi or Ci = v!/ri
(15)
Alternatively, Ci can be related to the total number of chains ni in the system ni
c:=
where C(zl,zl+l) is the capacitance of the system formed by the plates at zl and zl + 1, while the dielectric permittivity changes at zl + 112 and is given by
c-’(zl,zl+l) =
0(21,21+1/2)
D(z1+1/2,zZ+Z) (20) An(zl+l/2)e(z+l)
+
A,(zl)e(z)
In this equation, the equivalent planar thickness D(zl,zl+l/ 2) of the dielectric of half a lattice layer is defined by the integral A,(zl) J2t1+1/2 dz’/A,(z’), i.e., D(zl,zZ+Z/2) =
planar symmetry
zl+Z/2-z0
(16)
because the number of any segment s of chains i equals ni and the total statistical weight of the end segment r in Equations 13-16 are valid chains i is 14~2L(z)Gi(z,r11). for first-order Markov or freely jointed chain statistics. It is possible to use higher order chain statistics and more We have chosen not to include exact SCF modeling.2S*26 these extensions in this paper, as it cannot be expected that these will change the results qualitatively. Solving the equations requires an iteration procedure. For a given initial guess (ux(z))we calculate (pi(z,s) and hence the volume fraction profile of each moiety in the system from eqs 10 and 13-16. With the volume fraction profiles we check ux(z)using eq 11 and the boundary condition Cxcpz(z)= 1 for all z. Then the values of (ur(z)) are improved and the calculation is repeated until a selfconsistent solution is found. In Appendix A more details on the numerical method are given. Electrostatic Potential Profile. When evaluating eq 11 the electrostatic potential profile should be known. Its evaluation from the volume fraction profiles is here. Each segment of type x has a valency vx and a dielectric permittivity e,. The charge distribution q ( z ) (sum of charges in layer z ) and the permittivity profile e(%) can readily be obtained from the volume fraction profile
I
cylindrical symmetry
( z l - 2,)1/2 21 - zo 112
spherical symmetry
+
\
(21) With proper boundary conditions, e.g., $(O) = $(1), @ = 0 and the electroneutrality conditions C,q(z) = 0, the potential profile is obtained from eq 19. During the iterations of the potential energy profile u(z), the electroneutrality condition is hard to obey. B6hmer et al. have developed an iteration scheme to satisfy electron e ~ t r a l i t y . In ~ ~ the Appendix, we propose a simpler method which does not require that the neutrality condition is continually met during the iterations but neverthelessguarantees that in the final numerical solution the system is electroneutral. Thermodynamic Parameters. Expressions for the chemical potential pj of component i
-
( p i pi*)/kT = In
v;
-+
‘pib + 1 - ri J
“j
and the interfacial tension y
and
respectively. In this model, the charges q ( z ) are assumed to be located on the planes in the center of each layer z, thus forming a multiplate capacitor of the same geometry as the lattice. The contribution of any charged plate in the system to the electrostatic field E ( z ) can be obtained from Gauss’ law j E dA, = q / e . Since E = -V$, this enables us to ~
(25) Leermakers, F. A. M.; Scheutjens, J. M. H. M. J. Chem. Phys. 1988,89,3264-3274. (26) Leermakers, F. A. M.; Scheutjens, J. M. H. M. J. Chem. Phys. 1988,89,6912-6924.
have been derived byEver~,~~Leermakers,~and B6hmer.a In eq 22, the asterisk denotes the reference state of unmixed components. The quantity vxi* is the volume fraction of segment x in pure amorphous component i, which equals the fraction of x segments in the chain. (27) Evers, 0.A.; Scheutjens,J. M. H. M.; Fleer, G. J.Macromolecules 1990,23, 5221.
Barneveld et al.
1088 Langmuir, Vol. 10, No. 4,1994
1-0.1 C,
2
“ t
MJ.0)- MO.0)
c,=0.1 M
KT 0.4
= 0.4 M
I
O-M J
-0.3 0
0.02
0.04
0.06
JI
0.08
0
0.1
0.2 f/nm
0.3
Figure 3. Mean bending modulus, k,, of a charged surface in
Figure 2. Contribution of the liquid to the interfacial tension of cylindrical solid rods as a function of the surface curvature J. The slope of each curve equals VakdkT (cf. eq 4). The surface charge density is 0.15C m-2. The diameter of the smallmolecules is 0.3 nm. All pparameters are zero. The ionic strength of the solution is indicated.
various electrolyte solutions as a function of the diameter of the ions, represented by the lattice layer thickness 1. The surface charge density is 0.15 C m-2and all X-parametersare zero. The curvesare extrapolated (dashed parts) tothe Poisson-Boltzmann limit (open circles), where the ions are point charges.
As the volume fractions and the electric potentials are known everywhere in the system, the Laplace pressure pfl - pa can be calculated by applying eq 22 twice, Le., substitute for b first phase a and then j3 and taking the difference. Equation 23 is then used to obtain y, which is needed to calculate the bending elasticity parameters (eq 4). Some caution has to be exercised to avoid artificial influences of the lattice on the r e s ~ l t . ~
In Figure 3,k , is plotted versus the diameter of the ions (i.e., the lattice spacing I) for three different electrolyte concentrations. The limiting values of k, for 1 0 must equal the values, which are indicated by open symbols on the k , axis, as obtained by Lekkerkerker12 using the Poisson-Boltzmann approach. As intuitively expected, the bending elasticity modulus increases strongly with 1. For ions with a diameter of 0.3nm, k, is more than a factor of 2 larger than for point charges. The correct limit of our lattice approach with the Poisson-Boltzmann theory serves as a proof that our lattice model is basically correct. We note that we did not observe any influence of 1 on the ion density profile in a classical flat Gouy-Chapman layer. Here, the finite size effect is thus negligible. The calculation of the elasticity modulus is, as shown in Figure 3,a very sensitive probe for finite size effects. We also mention that we did not include socalled cavity effects in the electrostatic calculations. It is possible to do so and preliminary results% show that for flat interfaces and univalent salt, the effects are very small indeed. Only for divalent ions at high concentration can one retrieve the famous result of the primitive model that attraction is found between two equally charged surfaces. 11. Bilayers. In this section, we show the dependence of the bending parameters k, and h, of ionic surfactant bilayers on the ionic strength of the solution. Effects of chain length and Flory-Huggins interaction parameters of nonionic surfactants have been extensively discussed in our previous paper.g Here, we focus on one ionic surfactant, sodium dodecyl sulfate (SDS),mixed with the nonionic surfactant C12E7. The procedure to obtain the bending parameters from aplotofyllJkTvs Jl(cf.eq4) isthesameasinourprevious workg and is therefore not repeated here. Like in the nonionic case, bilayers of ionic surfactants are symmetrical and, consequently, their spontaneousmean curvatures are zero. The stability of bilayers depends on the values of k, and k,. To obtain thermodynamic stability the total elastic free energy of bending (eq 1)should be positive. If k, is negative, as it usually is, k, has to be positive. For spheres (K= 52/4), the relevant parameter is k, kd2. Negative values for this parameter characterize unstable films, whereas positive values pertain to stable layers. Figure 4 shows the bending moduli k, and fi, of SDS bilayers mixed with 0, 10, or 20% C12E7 in the bilayer. In Figure 4a, the mean bending modulus k , appears toincrease
Parameters We represent a dodecyl sulfate molecule by the segment sequence C12B3, where C stands for either a -CH2- or the terminal -CH3 group and B3 represents the large sulfate head group. The B segments have a valency vs of -Vs, Le., the total valency of a surfactant molecule is -1. For all polar segments we have chosen to take the relative dielectric constant equal to that of water: e, = 80. For the apolar C groups a low value was taken: e, = 2. The nonionic surfactant C12E7 is represented by the segment sequence C12(0CC)70, where 0 stands either for an -0- or the terminal -OH group. The solvent consists of monomers W (water). The following Flory-Huggins interaction parameters are used: xcw = xco = 2.0, xwo -1.6: XCB = 2.0,and XBO = XWB = 0. The ionic strength of the solution is determined by the two ionic monomer types with valency +1 and -1, respectively. Except for their charge, the electrolyte ions are identical to the solvent monomers W; Le., they have the same size and the same chemical interactions (xparameters) with the surfactant moieties as water (W). In all calculations we have used a hexagonal lattice, for which hl(=) = h+l(=) = l / 4 and bo(=) = l/z. Results and Discussion I. Charged Particles. We start with a very simple system consisting of hard cylindrical rods of various radii in a 1/1 electrolyte solution. All three types of molecules (solvent molecules and the two ions) have the volume of one lattice site. In addition, all X-parameters are zero. The cylinders have a fixed surface charge density of 0.15 C m-2 (arbitrarily chosen). In Figure 2 the dimensionless curvature dependent surface tension l(y(J,O) - y(O,O))/ J k T versus J is shown for three different electrolyte concentrations. Here, the diameter of the ions is the same as 1 and equal to 0.3 nm. By applying eq 4, kJkT is easily obtained from the slope of the curves. The results are 0.457, 0.366,and 0.290 in 0.1, 0.2, and 0.4 M electrolyte solutions, respectively ( k , = EJJ).
-
+
(28) Huinink, H.Ion-Ioninteractiesin de Dubbellang. MSc Theah, Agricultural University Wageningen, The Netherlands, 1992.
Langmuir, Vol. 10, No. 4, 1994 1089
Bending Moduli and Spontaneous Curvature
2
I
3
4
-3 I 0
'
1
2
I
3
c,/M
4
c,/M 0.1
Figure 4. Mean (a)and Gaussian (b) bending elasticity moduli of bilayers of ionic surfactant C12B.9 and 0, 10,or 20 % nonionic surfactant Cl2E7 in the bilayer as a function of salt concentration. For parameters see text.
JJ
0.05
pc'
1
I
Figure 6. Bending elasticity parameters k, (a) and h, (b), and spontaneous curvature (c) of a monolayer of SDS (modeled as C12B3) on a dodecane (Cl2)-water interface as a function of salt concentration. The concentration of SDS in the water phase was fixed to cpb = 2.5 X lW.For parameters see text. Figure 5. The same graphs as in Figure 4, but for salt ions having unfavorable interactions with solvent and surfactant tails, representing salting-out electrolytes. For parameters see text. with increasing electrolyte concentration. This trend is not found when the composition of the film is kept constant.12 The difference is that in our model, a bilayer of finite thickness is built spontaneouslyby self-assembling of surfactants. When the ionic strength of the solution increases, electric repulsion decreases, which allows the membrane to become thicker. As was emphasized in our previous paper? thicker bilayers are harder to bend and, hence, have lar er k, values. The effect of the salt concentration of 7 3,also indicates a more stable bilayer at higher ionic strength; see Figure 4b. It is inferred that incorporating nonionic surfactant in the bilayer destabilizes the bilayer, since both k, and k, decrease. The monotonous increase of the stability of the mixed bilayer with increasing salt concentration has not yet been experimentally confirmed. There is an example29 where bilayers of mixed surfactants are only stable over a distinct range of electrolyteconcentrations,which was in the order of kmol/m3. However, these results cannot be invoked to check our theory, because salting-out electrolytes were used, not satisfying our choice for the X-parameters. To that end we present in Figure 5 the counterpart of Figure 4 for the case that the ions have unfavorable interactions with the solvent. A Flory-Huggins parameter of 1.0 is assumed for the interactions between the electrolyte ions and water (instead of zero), as well as between the ions and the carbon segments (instead of 2.0). These values cause the quality of the solvent to decrease with increasing electrolyte concentration, as is the case with salting-out electrolytes like NaCl and KC1. In this situation, the mean bending modulus k, passes through a maximum (Figure 5a). The effect of increasing bilayer thickness, which increases k,, is now opposed by a decreasing solvent quality, as explained in our previous paper.9 On combination of the results for k, with those for h, (Figure 5b), a stable bilayer could be expected in a salt concentration of around 1 M electrolyte, in agreement with the experiments.B 111. Monolayers. Here, we discuss the bending elasticity parameters of a monolayer SDS on an oil-water interface. The oil is dodecane, (212. In all calculations shown below we fixed the surfactant concentration in the
water phase. The chosen value, (pb = 2.5 X lW,is lower than the critical micelle concentration of the surfactant in the water phase, even at high ionic strength. As monolayers are asymmetrical, a nonzero spontaneous curvature is expected, which is anticipated to be strongly dependent on salt concentration. In practice, phase inversion may occur when the salt concentration is increased. This phenomenon is usually ad hoc interpreted as caused by a changing effective head group area. However, this change is the result rather than the origin. We use the convention that J < 0 for W/O and J > 0 for O/W emulsions. The salt concentration mentioned below is specified, as for the surfactant, for the water phase. Note that the salt concentration in the oil phase is also finite, its value is controlled by the interaction parameters. In Figure 6 we will present the mechanical properties of these monolayers. We conclude the paper by displaying some representative segment density profiles of the monolayers in Figure 7. Figure 6 shows the effect of the concentration of indifferent electrolyte on the bending moduli k, and k, and the spontaneouscurvature JSp.The salt concentration has a dramatic effect on k,. At low ionic strength, k, is strongly negative, indicating that a cylindricalmonolayer is unstable. Adding salt increases k, considerably; this is a consequence of the higher surfactant adsorption when the electrical double layer is screened. In this case, a positive k, is found for salt concentrations larger than 0.4 M. The influence on K, is just the opposite, like in the case of pure nonioni~s.~ The variations of k, and k, on the salt concentration can be explained by the fact that the packing density of surfactant increase continuously upon increasing salt concentration. The increase in packing density is nicely illustrated in Figure 7. Figure 6c presents the dependency of the spontaneous curvature of a SDS monolayer on the ionic strength. The most striking result is that the sign of JIpchanges near a salt concentration of 0.4 M and again at about 1.0 M. In the ionic strength range from 0.4to 1M Jspindicates that water in oil emulsions are preferred, whereas above 1M and below 0.4 M oil in water emulsions are expected. This result is not consistentwith current ideas in the literature.so It is generally expected that the suppression of the double layer and ensuing decrease of the effective head group (30)Mitchell, D.J.; Ninham, B.W.J. Chem. Soc., Faraday Trcms. 2
~~
(29) van de Pae, J. C. Tenside Surf. Det. 1991,28,15&162.
1981, 77,601-629.
1090 Langmuir, Vol. 10, No. 4, 1994
Barneveld et al. B
water
10'
salt
cp
cp 1 0.3
45
50
60
55
tail
r
I o -40 ~
65
50
60
70
Z
80
90
100
80
90
100
Z
water
1
0.8
cp
cp
1
Oa6I 0.4
I\
0.2
45
50
55
t
salt
60
65
1
40
50
60
70
z
Figure 7. Segment density rofiles through a cross section of a flat oilaurfactan-water interface at two ionic strength conditions: Parts a and b are for 1M = 1.62 X lo-*) added salt and parts c and d are for 0.01 M ((pb = 1.62 X 10-9 salt. Parts a and c are on a linear-linear scale, whereas parts b and d are plotted log-linear. Other quantities are chosen as in Figure 6. The layer numbers are chosen arbitrarily.
(3
*
area, giving the surfactant molecule a less conical shape, would lead to flatter layers. Indeed the head group area is decreased quite considerably. However, this also leads (at constant bulk concentration of the surfactant) to more adsorption and hence to a thicker monolayer. In the traditional way of thinking, an increase or decrease of packing density of the surfactant upon variations in, e.g., ionic strength is not considered. Even when it is realized that the packing density increases upon an increase in ionic strength, the behavior found in Figure 6c for Jspis rather difficult to understand. As can be seen Jspalways increases upon an increase of ionic strength (except at the phase inversion near 0.4 M salt). We believe that below a salt concentration of 0.4 M the physics is fundamentally different from that above 0.4 M. We therefore discuss the two regimes one after the other. ( 1 ) Below 0.4 M . In this part of the curve we have a rather loose surfactant layer on the interface between the oil and the water (see also parts c and d of Figure 7). The effective head group area is relatively large for low ionic strength because the head group charge is not yet fully screened. With increasing ionic strength the packing density increases and the effective shape of the surfactant becomes less conical. The first effect will induce stronger curvature, whereas the second effect tends to reduce the curvature. Clearly the fiist effect dominates: Jspincreases. Gradually the oil droplets become smaller. Ultimately there is no space available for the oil molecules in the center of the emulsion droplets. The solution to this problem is perhaps surprising: Jspchanges sign. (2) Above 0.4 M. Although at this high ionic strength the Debye radius becomes of the same length scale as the
surfactant head groups, the packing of the surfactants still increases upon an increase of ionicstrength. We might expect that a t high packing densities the penetration of the two solvents (CIZoil and monomeric water) into the surfactant film becomes an issue (compair parts a and c of Figure 7). Monomeric components have (per unit mass) more translational entropy then chain molecules, and thus monomers are more effective solvents. When the space to penetrate in the surfactant layer becomes limited, oil molecules are easier pushed out of the surfactant layer than the monomeric water molecules. The effect of this is that the spontaneous curvature increases. As shown in Figure 6c, this effect can lead to a change in sign of Jsp. We thus propose that for loose layers a competition between varying surfactant shape and packing density causes the increase in Jsp,whereas for dense surfactant layers the preferential curvature is controlled by the ability of the solvents to penetrate the surfactant layer. One can envision several tests for this (novel) explanation. Important in this respect are highly symmetric emulsions, i.e. symmetric both in the surfactant shape (range of interactions and architecture) and with respect to the two solvents used. Probably these symmetry relations can experimentally be realized in polymeric systems. We plan to investigate this theoretically in the near future. We note that the curve in Figure 6c is very similar to that of the spontaneous curvature of nonionic surfactants as a function of the surfactant concentration (Figure 8 in ref 9). In that case the concomitant increase of the monolayer thickness also explains the observed behavior. We conclude this paper by presenting some typical density profiles of oil-surfactant-water interfaces. The
Bending Moduli and Spontaneous Curvature
Langmuir, Vol. 10, No. 4, 1994 1091
Table 1. Selected Properties Corresponding to the Profiles Shown in Finure 7 ~~
salt Gibbs concentration(M) excess ( d L ) 1
0.15256
0.01
0.044778
~~~
surface tension (mN/m)
A$ (mV)
1.69 15.16
25.9 80.5
curves as presented in Figure 6 are a result of many calculations. Both in cylindrical and in spherical coordinate systems the curvature was varied systematically. In Figure 7 we plot, for the same conditions as in Figure 6, some representative density profiles for low and high ionic strength. These profiles are obtained in a flat geometry. In Table 1 we collect a few details for the two interfaces: (1) the Gibbs excess given in the number of molecules per lattice site (the inverse head group area), (2) the surface tension, and (3) the electrostatic potential difference generated over the oil-water interface due to the adsorption of the ionic surfactant. As anticipated the adsorbed amount is a strong function of the ionic strength in solution. There is more than 3 times more surfactants adsorbed at the oil-water interface at high than at low ionic strength. We have plotted in Figure 7 density profiles both on a linear-linear and on a log-linear plot. This enables us to present the profiles of co- and counterions in both phases. The profiles support our discussion of Figure 6. There is a strong penetration of the oil in the tail region of the surfactant, which is considerably less strong at high ionic strength, and the head groups are fully hydrated by water, even at high ionic strength. It is noted that there are many oil-water contacts, despite the presence of the surfactant molecules at the interface, even at high ionic strength. Obviously,the surface tension is lowest for the highest adsorbed amount and the electrostatic potential gradient generated over the oilwater interface is highest for the lowest ionic strengths. This is logical because the screening is best at the highest salt concentration. The diffuse part of the electric double layer in the water phase extends over many layers a t low ionic strengths but dies away at high salinity. This is fully consistent with Gouy-Chapman theory. Note that in our model the ions are able to penetrate the head group region and that they even partition in the oil phase. Also in the oil phase there is a diffuse double layer. Due to the low local dielectric constant the screening length is larger in the oil than in the water phase. Note that far away from the interface the excess charge density is zero, both in the water and in the oil phase. A final remark on the surfactant profile is perhaps useful. We see that the profiles are not extremely sharp. The width of the head group profile is about five layers, fairly independent of the adsorbed amount. The tail profile also shows a pronounced maximum and one can easily imagine that the tail of the surfactant can assume many different conformations.
Conclusions In our self-consistentfield lattice model, the contribution of the electric double layer to the mean bending elasticity modulus k, increases significantly with the diameter of the electrolyte ions. Extrapolated to zero diameter, our values agree with those found with Poisson-Boltzmann theory.12 However, screening the electric double layer by an increasing ionic strength does not diminish the rigidity of ionic surfactant mono- and bilayer systems. Instead, k, increases because suppressing the electric double layer promotes a thicker surfactant layer. In bilayers, the
Gaussian elasticity modulus h, becomes larger with increasing ionic strength, whereas in monolayersit becomes smaller. In bilayers, addition of nonionic surfactant decreases both k, and h,. In monolayers, additional electrolyteincreases the spontaneous curvature, promoting the formation of small oil droplets in water. At a certain salt concentration Jspchanges sign and k, becomes positive, leading to small water droplets in oil that become larger changes when the ionic strength increases. Eventually, Jap sign again, so that the water in oil system is inverted into an oil in water system with the big oil droplets becoming smaller when the salt concentration is further increased. The physics is dominated by the continuous increase of the packing density of the surfactants at the oil-water interface, when the ionic strength increases. Two contributing factors are suggested (1) The increase in ionic strength induces a denser layer so that the shape of the surfactant becomes increasingly important. (2) For dense layers (high ionic strength) the asymmetry in solvation of the surfactant layer by the solvents determines the preferential curvature.
Appendix A NumericalMethod. The procedure to f i d a numerical solution consistent with eqs 10, 11, and 13-16 is as follows. We define the potentials u,(z) as iterationvariables. Using eq 11, we can write u&) as u,(z) = u’(z)
+ uxint(z)+ ev,(+(z) - @)
(Al)
where uxht(z) = kTC,x*,[(cp,(z)) - (p,a3, which can be calculated from the iteration variables {ux(z)),using eqs 10 and 13-16 for the calculation of {cp,(z)). The electrical can be deduced from {ux(z)) potential profile ($(z) by combining eq A1 for each charged species x into two independent equations for each layer z, containing the contributions of the positive and negative charges (segments and salt ions), respectively
v)
t
+(z) = u’(z) + U + ~ ~ ( +Z e(+(z) ) - @)v+ = u’(z) + uint(z) + e(+(z) - @)v-
UAZ)
(A21
In this equation, the subscript + (-1 means that only the positive (negative) charges are involved; e.g., u+(z)is the average value of u&) where x represents all types of positively charged segments and salt ions in the system. Equation A2 is a set of two equations per layer with u’(z) and (+(z) as unknowns. Therefore
v)
Note that u+ is positive and v- is negative, so that (u+ - u-) > 0. For a segment x in layer z we define the hard-core potential u’&), which must become independent of x , as (cf. Eq 11)
C(zZ-Z,zl)+(z-l)
ev,
+ q ( 2 ) + C(zl,zl+l)+(z+l)
C(Zl-l,Zl)
+ C(zl,zl+l)
(A41
where eq 19 is used to implement the correct interdependence between the electric potentials that are based
1092 Langmuir, Vol. 10, No. 4,1994
on the current iteration variables. The division by E,+&) is allowed because at equilibrium this quantity is equal to unity. Also at equilibrium u’,(z) is independent of the segment type and thus ut&) = C,U’~(Z)/E~VX.We define functions f x ( z ) ,which are only zero in the self-consistent situation
The point of zero of these simultaneous equations in {ux(z))
Barneueld et al. can be obtained by standard numerical methods. From all needed quantities can be the self-consistentset {ux(z>}
calculated to find yA, from eq 23. The bending elasticity moduli are obtained from the dependence of yA, on the curvatures J and K.
Acknowledgment. This work was made possible by the support of Unilever Research Laboratory Vlaardingen. This help is greatly appreciated. Special thanks are due to A. Jurgens,F. Schepers,and J. van de Pasfor stimulating discussions.
