5654
J. Phys. Chem. 1993,97, 5654-5660
Catanionic Amphiphilic Layers. A Monte Carlo Simulation Study of Surface Forces U. Nilsson, B. Jiinsson,' and H.Wennerstram Division of Physical Chemistry 1, Chemical Center, P.O. Box 124, S-22100 Lund, Sweden Received: July 13, 1992; In Final Form: March 11, 1993
Monte Carlo simulations have been performed to study the interaction between catanionic amphiphilic surfaces in an aqueous solution. Experimentally these systems show a phase behavior similar to phospholipid systems, where a lamellar phase is in equilibrium with almost pure water, and the swelling of the lamellar phase cannot be explained by the DLVO theory. It has recently been proposed that the short-range repulsive force between bilayers has its origin in the confinements of the protrusional motions of the amphiphilic molecules as a second bilayer approach. A model has been developed which allows for the amphiphiles to move in a direction perpendicular to the hydrocarbon-water interface and for the correlations between these motions. The result shows that the protrusional degree of freedom extends the range of the repulsive force to about 1 nm; it is decaying exponentially with a decay length consistent with the protrusion force model of Israelachvili and Wennerstriim. The structure of the surface and its dependence on the interactions between the headgroups of the amphiphiles were also investigated. The results indicate rough surfaces with varying headgroup protrusions, the degree of protrusion being dependent on the electrostatic interactions within the surface and of the size of the polar headgroups. When the surface consists of negatively charged amphiphiles, such as for example an SDS-water system, a more diffuse surface results compared to the catanionic lamellar system. Increasing the size of the headgroups also gives rise to larger protrusions. The nonhomogeneity in relative permittivity across the hydrocarbon-water interface has also been taken into account within a simple model, and it is found that the introduction of a dielectric discontinuity results in less diffuse surfaces. The simulated forces are used to calculate the phase equilibria in catanionic systems, and we find a good quantitative agreement.
1. Introduction
A range of amphiphilic molecules forms lamellar phases that swell in a solvent. For ionic systems the swelling in water and even in oil is caused by electrostatic double layer For nonionic surfactants with an oligo(ethy1ene oxide) headgroup the swelling at intermediate water content is caused by an overlap between headgroups4.5 while the swelling to high dilution6 is caused by undulation forces.' For liquid bilayers the swelling in water have traditionally been interpreted as due to a hydration force89 whose molecular origin is connected to a structure in the aqueous solvent.10 This view has been challenged5J1-13 on the basis that configurational excitations of the lipid are essential in a molecular description of the force. Israelachvili and WennerstriSmI3concluded that such excitations can involve both the configurations of the headgroup or the protrusionsof whole molecules, and in general a combination of these effects. In an effort to delineate the importance of the different contributions, we have performed a series of Monte Carlo simulations on simplified model systems. In two previous reports14J5we have developed simplifiedmodels that mimic the zwitterionic headgroups of phospholipids and demonstrated that their configurational freedom contributes to the force. In this paper we take another lamellar system of current interest as the basis for our model simulations, namely, the socalled catanionic surfactants.16J7 They are characterized by the bilayer, which consists of a mixture of a cationic surfactant and an anionic one (Figure 1). If they occur in a nonstochiometricratio, the bilayer acquires a charge and there is a double-layer repulsion18leadingto extensive swelling. It has been demonstrated that one in this case can get equilibriumvesicle formation at high dilution.19 However, when the stochiometric catanionic salt is exposed to water, a liquidcrystalline lamellar phase is formed. This swells to incorporate 20-3076 water by weight as shown in the phasediagramsof Figure 2. The swelling of the lamellar phase shows clear phenomenological similaritiesto the zwitterionic lipid systems and there are good reasons to believe that similar mechanisms are controlling 0022-3654/93/2097-5654$04.00/0
Figure 1. Schematic representation of a lamellar phase consisting of the catanionic amphiphile dodecyltrimethylammoniumdodecyl sulfate.
the interbilayer force in the two systems. Despite the similarities, there are two distinct differencesbetween the two types of systems. In the catanionics there is no covalent link between the anion and cation centers, and thus the surface is not necessarily locally neutral and the in-plane electrical polarizability of the surface is high. Second, the typical headgroups of the surfactants in the catanionic system have few internal degrees of freedom. We can thus anticipate that the protrusions of whole molecules are relatively more important in these cases relative to the zwitterionic lipids. In this paper we will try to establish whether the IsraelachviliWennerstriim protrusion force theory can explain the swelling of the catanionic bilayer systems. These systems are suitable for a theoretical study of the hydration interaction, due to their molecular simplicity. We have developed a model where the correlations between the protrusional motion of the amphiphiles can fully be taken into account. The interbilayer force has been calculated with the Monte Carlo simulation technique. We have also investigated, within our model, how the thickness of the surface layer varies when the interactions between the headgroups of the amphiphiles are changed. For the purpose of
0 1993 American Chemical Society
Catanionic Amphiphilic Layers
The Journal of Physical Chemistry, Vol. 97, No. 21, 1993 5655 separation is defined as the average distance between the centers of the headgroups located on opposite surfaces. The solvent is assumed to behave as an ideal dielectricum: it responds linearly and locally to an electric field, with relative dielectric permittivity chosen as el = 80. By treating water as a continuum, the contribution to the net force coming from the water structure cannot, of course, be taken into account. The uncharged and charged lamellar phases are modeled analogously. In the uncharged system all headgroups have zero charge. In the charged system the surface has a net negative charge, and it consists of an either anionic headgroups or a mixture of anionic and neutral headgroups. The counterions are modeled as hard spheres of radii r, and they are moving in the solvent between the surfaces. The surface is prevented from falling apart due to the hydrophobic effect. The cost of creating a hydrocarbon-water interface is taken into account by introducing a potential proportional to the area of the hydrocarbon chain exposed to the solvent. The proportionality constant is chosen to y = 18 mJ/ m2, a value obtained semiempirically for charged surfactant systems.2-20A similar value has also been obtained from the solubility of alkanes in water.21 The protrusion motion of the amphiphilic molecules will, of course, be strongly dependent on the choice of this parameter. A single amphiphilic molecule at a smooth hydrocarbon-water interface will in this formulation of the hydrophobic effect move in a symmetric V-shaped, rather than a parabolic, potential. The increase in free energy of the system will consequently be the same whether the amphiphile moves a distance z out into the water region or a distance z into the hydrocarbon region. The area of the created hydrocarbonwater interface is in both cases the same. In the simulation the interfacial free energy can be calculated as a sum over nearestneighbor pair interactions. The centers of the headgroups interact electrostatically with a Coloumb potential and the relative permittivity of the particles is the same as the surrounding solvent. There is also a hardsphere interaction between the particles and a hard-wall interaction between the particles and the hydrocarbon chains on both surfaces. This is a well-defined model system, and its properties can be exactly calculated, within numerical accuracy, in a Monte Carlo simulation. Most of the results to be presented will be from simulations of this system. However, the description of the electrostatic interactions as though the charged headgroups are located in a homogeneous dielectricmedium is simplistic. The polar water and the nonpolar hydrocarbon have very different dielectric properties. One way to take this into account would be to assign to each hydrocarbon chain a lower relative dielectric permittivity and thus introducing dielectric discontinuitiesin the system. The charged headgroups will then induce polarization charges at the interfaces between the hydrocarbon and the solvent,but it is difficult to solve exactly for these polarization. In a correct treatment the electrostatic pair interaction depends on thedetailed configurationsof the two surfaces. In a simulation this is difficult to handle, and the effect of the dielectric inhomogeneity on the interaction between two ionic headgroups was therefore neglected. It is not easy to predict how this approximation affects the protrusional motion of the amphiphiles. However, it is our belief that the most important effect of the dielectric discontinuitieson the protrusions can be considered on a one-particle level. The electrostatic self-energy for an ionic amphiphile in a lamellar phase is strongly dependent on the locations and shapes of the hydrocarbon-water interfaces. This contribution to the systems free energy was taken into account in the simulations in an approximate way. The surfaces were considered to be far apart and the roughness of the surfaces was treated in a meanfield manner, i.e., when calculating the self-energy for an amphiphile, a smooth hydrocarbon-water interface was assumed
: II 80
- 80 Y
Y
602
1
Inmeliar
NE 20
water + crystals
water + c r y N l s
-0
0
60
70
80 90
100
% W/W Dodeeylsmmonium dodernaoste
- 20
60
70
80 90
100
% WIW DcdefylMmethyInmmonlum
dodecylsulpbte
Figure 2. Experimental phase diagrams for the binary catanionic systems dodecylammonium dodecanoate and dodecyltrimethylammoniumdodecyl sulfate. The maximum swelling of the lamellar phases corresponds to water layer thicknesses of 6.0 and 11 A, respectively (from ref 16).
Figure 3. Schematic representation of one of the monolayers in a catanionic system. A0 is the cross-sectional area of a hydrocarbon chain andris theradiusoftheheadgroup. Thehydrocarbonchainsareassumed to fill the bilayer.
comparison simulations were also performed for an uncharged system, where all the amphiphiles have zero charge, and for a charged system, where the amphiphilic surface has a net charge and oppositely charged counterions are moving in the aqueous medium. 2. Model System
A realistic molecular description of a system consisting of one or more bilayers and water is hard to achieve in a Monte Carlo simulation. This is mainly due to uncertainties in the intermolecular potentials and to the limitations of computer power. A simplified model must therefore be developed, and this can be done by averaging over some molecular degrees of freedom and explicitly treat only the degrees of freedom one believes are important for the investigated properties. The catanionic system was modeled as two opposing monolayers, and in Figure 3 one of the monolayers is shown schematically. In the model the motional degrees of freedom of the hydrocarbon chains have been neglected in all but one direction, Le., perpendicular to the surface. They are treated as hard rectangular parallelepipeds, and they have been placed in a square lattice. Their cross-sectionalarea is A0 = 25 A2 and they are sufficiently long to always have contact with the four nearest neighbors. That means that an amphiphilic molecule cannot leave the surface during the simulation. Only two opposing halves of bilayers have been explicitely treated. The headgroups are modeled as hard spheres with a radius r and a charge of magnitude e (the surface has a zero net charge). Each hard sphere is confined to a hydrocarbon chain, on which cross-sectional surface it is free to move. The intersurface
Nilsson et al.
5656 The Journal of Physical Chemistry, Vol. 97, No. 21, 1993
I
E,
E,
-5
0
5
10
z,, (A, Figure 5. Electrostatic self-energy for an amphiphile modeled according to Figure 4a and with the same parameters as in Figure 4b. The entrance into the hydrocarbon region is treated within two different models. The cylindrical hole created is filled with either water (data from Figure 4b) or hydrocarbon. The electrostatic self-energy is assumed to be zero for an infinitely long amphiphile in a bulk water solution.
E v
-50 -40 .30 -20
-10
0
1
n
\
-5
0
5
10
z c ,(A,
15
20
Figure 4. (a, top) Model system used for calculating the electrostatic self-energy for an amphiphilic molecule at a smooth hydrocarbon-water interface. The relative dielectric permittivity for the water and hydrocarbon regions are € 1 = 80 and c2 = 2, respectively. (b, bottom) Electrostatic self-energy calculated from the model shown in Figure 4a. The radius of the half-sphere modeling the headgroup is r = 2 A, and its charge is -e. The radius of the cylinder is r, = 2.82 A, corresponding to a cross-sectional surface of 25 A2. The electrostatic self-energy is set to zero for an infinitely long cylinder in water, Le., zc is infinite (see Figure 4a).
to be placed in a position corresponding to the mean value of the locations of its four nearest neighbors. In the numerical calculations (see appendix A) of the selfenergy for an amphiphilic molecule at a smooth hydrocarbonwater interface the model shown in Figure 4a was used. To simplify the calculations, the hydrocarbon chain was modeled as a cylinder with the same cross-sectional area as in the parallelepipedic description above. The relative permittivity of the cylinder was chosen as €2 = 2. The headgroup was modeled as a half-sphere, and it was placed on top of the cylinder. In Figure 4a the situation is depicted where the amphiphile protrudes into the water region. When the amphiphile enters the hydrocarbon region, a cylindricalhole is being created and in our model this hole was assumed to be filled with water. Figure 4b gives the free energy cost of pushing an amphiphile into the hydrocarbon region when the headgroup has a unit charge, a radius of r = 2 A, and a cross-sectional area of the hydrocarbon chain of 25 Az.Observethat for the system as a whole the increase in free energy will be the same whether the amphiphile moves a distance z out into the water region or a distance z into the hydrocarbon region. The reason for this is that when an amphiphile moves, for example, out into the water region the self-energy of the amphiphile will decrease but at the same time the self-energyof its neighborswill increasesince the hydrocarbon chain of the outgoing amphiphile gives a more hydrophobic surrounding. Our mean-field treatment of the position of the neighbors then gives this symmetric behavior of the system's electrostatic self-energy. In the simulation the electrostatic self-energy, as well as the interfacial free energy, can be calculated as a sum over nearestneighbor pair interactions. The self-energy pair interaction, for
the above discussed case, is proportional to the sum of the selfenergy from Figure 4b and its mirror image in the axis zc = 0. The result is a symmetrical function with the minimum at zc = 0, indicating that the self-energy of the system has a minimum when the surface is smooth. Thus in our description of the dielectric inhomogeneity, it has been assumed that the induced polarization charges will affect only their sources and no other charges in the system and furthermore they are present only on one surface. For a catanionic lamellar system and when the distance between the surfaces is large, the approximations made will hopefully not be too severe and the essential feature of the dielectric discontinuities will be maintained. When an amphiphile moves into the hydrocarbon region, one may ask whether water also enters, as is assumed in our model, or if the hydrocarbons enclose the charged headgroup. In Figure 5 it can be seen that if the headgroups are enclosed by hydrocarbon, the electrostatic self-energy drastically increases. When water enters, a hydrocarbon-water interface is created, but with an interfacial tension of y = 18 mJ/m2 and a cross-sectional surface of A0 = 25 A2,this additional cost is only about 0.9 kT/A. These considerations support our model.
3. Simulation Procedure The Monte Carlo, MC, simulations were performed using the Metropolis algorithmzz and the minimum image convention. Corrections for the long-range electrostatic interaction from charges outside the MC box was made using a self-consistent mean-field approximation as described previously.l5,*3 The hydrocarbon chains were allowed to move only normal to the surface, while the headgroups change their lateral positions in two different ways. Locally they were allowed to move on the square lattice site of 0.25 nmz allotted to the hydrocarbon chain. In addition there were MC steps where two randomly selected oppositely charged groups could change place. This allows for totally disordered headgroup distributions, and it drastically increases the electrostatic polarizability of the surface. One of our main objectives is to calculate the force between two surfaces, and it is then essential to design the model system so that this goal can be accomplished. We have thus chosen to use an isothermal-(anisotropic) isobaric ensemble. In addition to the direct interaction terms there is a p,,,Vcontribution to the energy. Having a fixed lateral density, the volume changes only with displacementsin the perpendicular (z)direction. By choosing the external pressurep,,,, the force per unit area is predetermined and the simulation is continued until we obtain a stable average separation between the opposing monolayers. The simulation is then repeated for another value of pext.
Catanionic Amphiphilic Layers
The Journal of Physical Chemistry, Vol. 97, No. 21, 1993 5657 i
n
qa,, = +e, -e *?
5
.e
,
0.004 0.003
L
4
0.002
C
0.001
-1 15
10
5
20
25
separation distance, ( A ) Figure 6. Schematic representation of the interaction energy between two surfaces as a function of the separation distance and its change when an external pressure is applied. 0.006 h
3
4
0.005
v
0.004
0
-30
-10
-20 (
0
z - z,
10
20
(A)
Figure 8. Number density distributions for three different surfacecharge densities. The surface is successively being charged up. The surface is either electrical neutral consisting of an equimolar amount of positive and negative ions or negatively charged consisting of either an equimolar amount of neutral and negative ions or only negative ions. In the latter cases when the surface is negativelycharged, free counterions are moving in the solvent. The radii of the headgroups and the counterions are 2 A. 0.008 I
1
0.003 0.002
0.001 0
-10
-5
0
(z
- ZCM)(A)
5
10
Figure 7. Number density distribution for a single surface when the headgroups of the amphiphiles have zero charge and a radius of r = 2 A. The cross-sectional area of the hydrocarbon chain is 25 A* and the interfacial tension is 18 mJ/m*.
This procedure of determining the force is computationally very efficient and adapted to the protrusion force problem, where there is no natural reference point that can be used to fix the separation. However, there are two drawbacks. First the procedure works well only for repulsive forces, i.e., pext> 0. This is illustrated in Figure 6, showing a typical interaction curve between two surfaces and how it is changed when a pcxlVterm is added. When pext< 0, the energy decreases without limit for large separations. Thus mean distances are inaccessible that are larger than for the minimum in the original force curve. Second, a high lateral repulsive pressure can lead to a single surface tilting to effectively increase its cross-sectionalarea. This is a true complication for simulations on highly charged surfaces but not for the neutral catanionic system. It was typically sufficient to have 36 particleson each monolayer to obtain reliable distributions and force-distance curves. The temperature is 300 K. Of course, the width of the distributions depends on the number of amphiphilic molecules building up the surface, but in the results to be shown each surface consists of 36 molecules so interesting comparisons between different runs can be made. The box length was varied to ensure that the longwavelength undulation contribution to the force was negligible. All undulations with a wavelength longer than the size of the system are suppressed due to the periodic boundary conditions. 4.
Results and Discussion
In the Monte Carlo simulations of the model system the correlations between the motions of the amphiphilic molecules gives a very rough surface. In Figure 7 the density profile, sampled relative to the center of mass of the surface, of the headgroups of a single isolated uncharged (all amphiphilic molecules have zero charge) surface is shown.
-10
0
-5 ( 2-z,,
5
10
1 (ii,
Figure 9. Number density distribution for two different values of the radius for the catanionic system.
The figureclearlyshows that the protrusional degree of freedom of the molecules gives a broadening of the surface layer. The width of the distribution depends on the cross-sectional area of the hydrocarbon chain and the interfacial tension, which in all simulations were set equal to A0 = 25 A2 and y = 18 mJ/m2, respectively, as well as the chemical properties of the headgroup. Decreasing the interfacial tension or the cross-sectionalarea of the hydrocarbon chain will give a broader distribution. If the correlations between the protruding amphiphilic molecules are neglected, i.e., consider the motion of a single molecule at a smooth hydrocarbon-water interface, the distribution could be described by the equation
n(z) = const exp(-u/kT) (1) where u is the energy, which symbolically can be written as
The first term on the right-hand side is the interfacial energy (z the distance from the headgroup to the smooth surface), AuCl the electrostatic energy, and Auhs an energy associated with the
hard-sphere description of the headgroup. The y and A0 dependence of the distribution width can be clearly seen in eq 2. The electrostatic part Auel contains several Darts. The role of the direct interaction between charged headgroups is shown in Figure 8. Giving the surface a net charge leads to a broader distribution. The attractive interactionbetween the cat- and anionic groups leads to a smoother surface than without interactions, q = 0. A similar effect is seen in Figure 9. The surface is smoother in a system where the radius of the ions is 1.5 A compared with
Nilsson et al.
The Journal of Physical Chemistry, Vol. 97, No. 21, 1993 0.01
,
,
. .
I
"
~
'
I
"
- - - no diel. disc
'
'
I
"
"
-diel. disc.
0.008
--cr = l . 5 A
-
0.006
\-
-D-
'
r=2A
-0- r = 2 A (diel.
0.004
0.002 0
-10
-5
-
0
5
10
(2 2 C M ) (A, Figure 10. Number density distributions with and without dielectric discontinuity for the catanionic system. The dielectric constant of the hydrocarbon and water regions are 2 and 80, respectively. The radius of the headgroups is r = 2 A.
a system where r = 2 A. The reason for this is that there can be a closer contact between the anions and cations when r = 1.5
A.
Yet another electrostatic effect is shown in Figure 10. With a dielectric nonhomogeneity the self-energy of the ions is minimized for a smooth surface and this has a damping effect on the protrusions. If the protrusions contribute significantly to the force between two monolayers, we expect that the broader surface distributions will result in the strongest repulsive forces. However, note that both the electrostatic and the hard-sphere term contribute to the net force. Figure 11 shows the calculated force-distance curves for the three cases r = 2.0 A, r = 1.5 A, and r = 2.0 A with dielectric discontinuity. In all three cases there is a change in the force of 2 orders of magnitude with a distance change of 6-7 A, which implies a very strong repulsive force. The force decays roughly exponentially with a decaylength of approximately 1.3 A. This value is consistent with the simple theory of ref 29, which yields a decay length of 1.1 A for the parameters used. The forces for r = 1.5 A and for r = 2.0 A with dielectric discontinuity are distinctly different, although the roughness of the surfaces appears very similar as seen in Figures 9 and 10. This demonstrates that the direct electrostatic interaction also influences the force.
6 8 1 0 1 2 1 4 < D > , (A) Figure 11. Calculated distance dependence of the force per unit area. The standard deviation is given by the size of the symbols.
0
2
4
6 A for dodecyltrimethylammonium dodecyl sulfate and dodecylammonium dodecanoate systems, respectively. We see that the theoretically estimated swelling limits agree well with the actually observed ones. Furthermore both theory and experiment give that there is a more extensive swelling with bulkier headgroups, a trend also observed for phospholipids. To date we are not aware of any explicit force measurements with catanionic surfactants. However, in the phase diagrams of Figure 2 there is also a phase boundary lamellar liquid crystal to crystal, and this boundary contains information on the forces in the liquid crystal. At temperatures of 43 and 35 OC,respectively, there is in the two systems a eutectic three-phase line dilute solution-lamellar liquid crystal-crystals. Passing this line from a lower to a higher temperature involvesmainly the melting of surfactant chain order as in a gel-to-liquid crystal transition for phospholipids. At the three-phase line the chemical potential of the surfactant is the same in the maximally swollen lamellar phase as in the crystal. At a temperature above this three-phase line more surfactant can be incorporated into the lamellar phase before the chemical potential reaches that of the crystal, Le., the stability range of the lamellar liquid crystal increases. The change in surfactant chemical potential with composition can be calculated from the change in the force/area (water chemical potential) using the Gibbs-Duhem equation:25,26
5. Interpretation of the Catanionic Surfactant-Water Phase Equilibria
In the real system of catanionic surfactants and water, there are additional forces operating that were not accounted for in the simulations. The dispersion force acts between all molecular species in the system. For the hydrocarbon-hydrocarbon interaction across water the Hamaker constant, H,is approximately 6 X 10-2' J and with an expression for the force, F Flarea = -H/(6aD3) this implies an attractive force of 3 X lo5 N/mZ at a formal separation D of 10 A. There is also an attractive chargecharge correlation force,24 which is included in the simulation and which seems to be as large as the dispersion force at the lo-A range judging from the results in ref 15. Due to these two attractive components, we expect the net forcetoturnattractiveintherange (D)= 8-14Afortherepulsive force cases shown in Figure 11. When the net force is zero, a lamellar phase is in equilibrium with pure water and the swelling limit in the phase diagrams of Figure 3 provides an estimate of the distance of zero force. Assuming a bilayer thickness of approximately 2 X 16 A for the surfactants with dodecyl chains we obtain water layer thicknesses at the swelling limit of 10 and
where X = nH20/nsurfis the molar ratio water - surfactant, V H ~ O is the water molar volume, and upis the area per surfactant polar groups. The main effect of a temperature change is a change in the difference in the standard chemical potential, Ape, of the surfactant in liquid-crystalline and solid states. Expanding around the temperature TOof the three-phase line ' 0
where Ah is the measured transition enthalpy at the three-phase line. This has not yet been measured for the catanionics, but for the analogous compound dilauryl phosphatidylethanolamine it is 16kJ/mol. Using thisvalue, the calculated forces,and estimated van der Waals contribution at long range, we calculate thedotted phase boundaries in Figure 12. We find the agreement between theory and experiment quite satisfactory. 6. Conclusions
We have used a novel procedure for calculating the force between two surfaces with liquidlike degrees of freedom. The
The Journal of Physical Chemistry, Vol. 97, No. 21, 1993 5659
Catanionic Amphiphilic Layers
The potential in the center of band i can now be written
I
isomvic solution
N
@(Iti)= C V ( R i )+
+ afrce(Ri)
(AS)
j= I
40
water + crystals
60
70
80 90
water + crystals
100
60 70 % W/W
% WIW Dodecylammonlum dodecanoate
80 90 100
Dodecyltrimethylammonium
where W is the potential from band j and cPfrce the potential from all free charges in the system. The sum is not taken for the value j = i. Equations AS and A3a give
aw(Ri) I
an
dodecylsulphate
Figure 12. Calculated (dotted lines) and experimental phase boundaries of thelamellar phase for twodifferent catanionicsystems. Thecalculated force curve for the catanionic system with r = 1.5 A shown in Figure 11 has been used to construct the phase diagram of the dodecylammonium dodecanoatewater system and the force curve from the r = 2.0A system has been used in the construction of the dodecyltrimethylammonium dodecyl sulfatewater system. The experimental phase diagrams are from ref 16.
simulationsshow unequivocally that protrusions of molecules from the surface cause very strong repulsions at short range, providing additional support to the existence of protrusion forces in all types of amphiphilic systems. For the catanionic systems protrusions can be expected to be especially large due to the small cross-sectionalarea of the molecule. We have also demonstrated that the calculated forces are consistent with the observed phase equilibria. Wealso find it significant that thenet forceisattractive when the surfactant is in a crystalline state as shown by the fact that there is no swelling of the crystals. This further supports the conclusion that surface excitationsare essentialfor generating a repulsive force.
- eo(e,
an
- 1)-
a@.,(@ an
where ai is the electrostatic potential in medium i and n is the unit normal to the interface, directed from medium 1 to 2. Using the boundary condition for the discontinuity of the normal component of the dielectric displacement aa1(R) El-
an
=E
a 9 , ~ an
(A6)
a@;(Ri)
an
.
E;
- E;
t2
+ €1
L' = 2eo-
. .. +=u p ' 2e0
together with eq A4 yields the system of linear equations
which can be solved numerically. The coupling elements di are calculated according to the following procedure. Any point, R', on band j is defined by
r'= rj
(Al)
an
2c0
j=l
.
The polarization surface charge density, uPlrinduced at a point R on the interface between two perfect dielectric media 1 and 2 can be calculated from26
d@.,(R)
a@F(Ri)
a;l
N
Introducing the quantities
Appendix. Electrostatic Self-Energy in a Cylindrically Symmetrical Dielectric Nonhomogeneous Medium
a,,(R) = E0(E2 - 1)-
E-a ydn( R , )+-+-+an
+ t sin(uj) where -Atj
z'= z,
+ t cos(uj)
(A10)
< t < Atj
and any point R, lying on a line perpendicular to band i and going through its center by
r = ri - u cos(ui)
z = zi
+ u sin(ui)
(A1 1)
The electrostatic potential from band j in R can now be calculated as a surface integral:
(A21
and assuming no free surface charge density where
4rr' (A12b) (r rq2 where k(m) is the complete elliptic integral of the first kind.27 mji =
gives the equation Upol(R)
e2 - el aayR) el an
= 2E0-E,
+
(A41
By dividing the interface between the different dielectric media into a finite number, N, of cylindrical bands, this equation can be used to solve the polarization charge density, which is assumed constant on each band. The band is defined by a set of parameters (ri,zi, Bi, Ari) where ri and zi are its cylindrical coordinates, Bi its azimuth angle, and 2AOi its width.
(z - z12
+ +
Now since the surface is divided into a finite number of bands and thederivativeofthe potential will becalculatedin themidpoint of band i , it will be calculated with an accuracy only of the order of At?. By setting r = 0 and taking the derivative with respect to u of eq A12 and using eq A8a, the following approximate expression is found for the coupling-elements di:
where
Nilsson et al.
5660 The Journal of Physical Chemistry, Vol. 97, No. 21, 1993
References and Notes (1) (2) 482. (3) (4)
An approximate expression for the coupling elements c" can be found by doing a Taylor expansion of the integrand in eq A12 (small values of u and t) and then take its derivativewith respect tou. Withthe helpofeqA8b thefollowingequationwasobtained:
cii = 2 At. -{26,[ cos(u.) ri 2m0
In(
2)
- 1 1 + a,]
(A14)
If the free charges are assumed to be line charges of cylindrical symmetry going through a point (r', 23, then the electrostatic potential in a point (r, z) can easily be shown to be
where q is the total charge. The derivative with respect to u can now simply be calculated. The electrostatic free energy, A,], of the system can now be calculated according to N"
where Nq is the number of free charges in the system and a(&) is the total electrostatic potential coming from all the cylindrical bands and the rest of the free charges.
Parsegian, V. A. Trans. Faraday SOC.1966,62, 848. Jbnsson, B.; Wennerstrbm, H. J. Colloid Interface Sci. 1981, 80,
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