J . Phys. Chem. 1987, 91, 605-611
605
correlation deserves further investigation.
the fractal dimension reflects also its distribution (it is related to the specific surface area). There is not a priori relationship between roughness and fractal dimension. This is illustrated in Figure 10, for a self-similar fractal. Figufe 10a shows two surfaces with the same roughness, but different self-similar fractal dimension. Also, surfaces with the same self-similar fractal dimension may have different roughness, as seen in Figure lob. However, we observe that in our system, which is self-affine, there is a smooth variation of the fractal dimension with roughness (or vice versa) (Figure 11, a and b), but it is different for the interface and the apparent interface. While the change of the box dimension with roughness has a power law form for the interface, for the apparent interface, the dependence is linear. We believe that this
Conclusions In this work we propose a computer model for simulation of dissolution precipitation phenomena, based on randomization of the microscopic processes. We show that, for experimentally measurable quantities, the model obeys the deterministic description of the system. We illustrate the application of the model and show the type of information that can be obtained from its analysis.
Acknowledgment. This work was supported by the Office of Naval Research.
Micromechanics of Surfactant Microstructures P. Neogi,*+ Myungsoo Kim,+ and S. E. Fribergt Department of Chemical Engineering and Department of Chemistry, University of Missouri-Rolla, Rolla. Missouri 65401 (Received: May 20, 1986; In Final Form: September 1 1 , 1986)
Force balance and stability criteria have been used to distinguish among various shapes of surfactant aggregates. The model aggregate is assumed to be liquidlike in the interior with a bounding interfacial region. Inclusion of higher moments of surface excess stress leads to the conclusion that not all spheres are stable, that stable short cylinders of -3-3.5 length to diameter ratio can exist (cylindrical micelles) and even grow to infinite lengths (cylinders in hexagonal liquid crystals), and that there are regions where only lamellar structures are possible. A model relating surface packing to surface tension and the higher order effects shows that cylindrical structures can compete with spherical ones, except at high packing densities where only lamellar forms can exist and except when the interfacial energies are ultralow, the latter case being established separately. A model for the isotropic surfactant phase allows one to obtain a persistence length in terms of the surface properties.
Introduction The simple picture of micelles as spheres is neither general nor adequate in many cases, and as reviewed by Anacker' have led to numerous difficulties in interpreting experimental data on micelles. X-ray methods have shown the existence of cylindrical micelles,2 but not much is known of their shapes at the ends. Indirect evidence to the existence of other shapes comes from systems with aggregation numbers in hundreds, where it is impossible to form a sphere without a vacuum at the center: data show however that the densities of the hydrophobic tails inside the micelles are very close to that of pure hydrocarbons. This has led to speculations about the actual shapes of micelles as prolate or oblate spheroid^.^ Incentive to postulate other than spherical shapes comes from another source. Besides micelles, lamellar and cylindrical forms of aggregates are known which are infinite bimolecular sheets stacked in lamellar form or infinite cylinders stacked in hexagonal or cubic arrays. Micelles are often in equilibrium with such structures and to provide a full description of the two-phase coexistence, intermediate shapes need to be known. These intermediate shapes would serve to bridge the gap between simple spherical micelles and the liquid crystals, which are infinite at least in one of the dimensions. Some of these structures may well be thermodynamically unstable falling within the spinodal r e g i ~ n . ~ The use of oblate and prolate spheroids as the bridging structures appear to be attractive, but in recent years doubts have been expressed regarding their fea~ibility~-~ and other structures, some as exotic as toroids, have been suggested.' Given a competition among various structures, one feasible way of picking the "right" one under a given set of conditions is to analyze their free energies of formation and choose the one with the lowest. However, Mukherjee5s6has pointed out that with tiny Department of Chemical Engineering. *Department of Chemistry.
0022-3654/87/2091-0605$01.50/0
differences in the characteristic energies a large change in the shapes of the aggregates formed can result. One other method of choosing a particular form appears to be first proposed by Tanford3 in the first edition of his book. The basic idea is that in a given shape with a certain aggregation number it should be possible to pack surfactant molecules such that the interior, made up of tails, pack with a reasonable density and the head groups at the surface occupy reasonable areas and only under those conditions is the postulated shape acceptable. This idea was later taken up by M ~ k h e r j e e . ~Israelachvilli .~ et al.' have correlated the surface-tail packing ratio to the formation of spheres, cylinders, and lamellae. The numerical values in the correlation do not appear to be unique; nevertheless we shall use their numerical criterion for comparison. A different method, which emphasizes force balances, is used here. It is well-known that in interfacial phenomena the force methods and the thermodynamic methods produce identical results, differing only by the levels of difficulty under which these are reached. To provide a full (or a reasonable) thermodynamic discussion regarding all possible structures appears to be very difficult at this point in time. At an interface, Murphy: who has generalized the work of BufP and Buff and Saltzburg,Io has shown (1) Anacker, E. W. in CarionicSurfactants; Marcel Dekker: New York, 1970; p 203. (2) Reiss-Husson, F.; Luzzati, V. J. Phys. Chem. 1964, 68, 3504. (3) Tanford, C. The Hydrophobic Effecfs, 2nd ed.; Wiley-Interscience: New York, 1980. (4) Montroll, E. W.; Lebowitz, J. L. Studies in Statistical Mechanics; North-Holland: New York, 1979; Vol. 7. (5) Mukherjee, P. J. Pharm. Sci. 1974, 63, 972. (6) Mukherjee, P. In Micellization, Solubilization, and Microemulsions; Mittal, K. L., Ed.; Plenum: New York, 1977; Vol. 7, p 171. (7) Israelachvilli, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Sac., Faraday Trans. 2 1916, 72, 1525. ( 8 ) Murphy, C . L. Ph.D. Thesis, Department of Chemical Engineering, University of Minnesota, Minneapolis, 1966. (9) Buff, F. P. J. Chem. Phys. 1956, 25, 146.
0 1987 American Chemical Society
606
The Journal of Physical Chemistry, Vol. 91, No. 3, 1987
that the thermodynamic description is obtained by minimizing the free energy locally, where for thick interfaces, for interfaces with ultralow surface tensions, and for highly curved interfaces the interfacial tension y (the zeroth moment of excess interfacial stress) is insufficient; higher moments, that is, the first moment or the bending modulus 7f and the second moment or the torsional modulus 2, are needed. This brings about some changes; for instance, Gibbs adsorption equation becomes -dy = ss d T +
kr, dp, + 7f d(2H) - X d K
(1)
i= 1
where ss is the surface excess entropy per unit area, T is the absolute temperature, r i and p i are the surface excess and the chemical potential of the i-th species. 2H is mean curvature, the sum of the two principal curvatures c1 and c2 2H =
C]
+ ~2
Neogi et al. ternatively, one may analyze the global system consisting of two phases, one continuous and the other dispersed. Identical results are obtained. (We state this without proof, which follows the derivation of BufP and Murphys almost identically, with the exception that summations over i-mer aggregates of numbers j have to be introduced.) It is noteworthy that global energy minimization also yields the result that the chemical potentials of individual species must be the same everywhere-the starting point of the conventional theories. To utilize this, one has had to postulate a shape; we emphasize here that the shapes can be obtained through the force balances. We have used simple models for surfactant aggregates in order to apply these balances, which does not exclude the possibility of using more sophisticated models in the future. Formalism
According to Murphys
and K is the Gaussian curvature K =
CIC~
Both are invariants. The question arises as to whether the formalism obtained by Murphys for a conventional fluid-liquid interface can be applied to the surface of a surfactant aggregate. That the hydrocarbon tails inside are fluidlike has been mentioned earlier.3 Consequently one may assume to the first approximation that the stresses inside can be described by an isotropic pressure as in a regular fluid. Further, in principle, a surface is described both by its scalar area and by an orientation given by unit outward normal n which varies from point to point. The surface free energy here will depend on the scalar surface area and on the two scalar invariants of n,
under a small but arbitrary perturbation u = qn + 711to the surface. Here n is the outward unit normal to the surface and T~~ is a vector tangential to the surface at the point where n is evaluated. V, is the surface gradient operator
V, = ( I - nn).V
$1
= 2H7
+ F ( 4 P - 2 K ) - 2 H K X + ($1
$2 $3
K = det (-V,n)
+ 7fVS(2H) - X V , K = 0 = -V,% + V , X * ( 2 H l s- b) = 0
= V,?
( 1 1) DeGennes, P. G. In Liquid Crystals and Plastic Crystals; Gray, G . W . , Winsor, P. A., Eds.; Ellis Horwood: Chichester, U.K., 1974; Vol. 1. (12) Miller, C. A.; Neogi, P. AIChE J . 1980, 26, 212.
(5) (6)
Here (4) is the modified Laplace equation, and the curvature tensor b = -V,n (7) According to ( 5 ) there is no Marangoni effect at equilibrium, and (6)-which would not exist in the classical case of 7f = X = 0-says that the net surface torque is zero. To obtain the higher order perturbation, we follow Tyuptsov13 and obtain from ( 2 )
dS) where for equilibrium structures (4)-(6) can be used, and one has
Following the same argument as before, we note that in the variations 6($2) and a($,) only the components in the tangential directions contribute to the integrals. Since, for equilibrium surfaces, 4J2 and 4J3 do not vary on the surface (eq 5 and 6), the last two integrals are zero. The variation 6($,) can be written as8
6($,) = [ ( 4 P - 2K)O + V,2111(Y - X K ) + % [ 4 H ( 4 P - 2K)7 - 4HK7 2bV,V,q] + X 2 H [ 2 H K q (2HI, b):VsV,~l+ ~ I I - V (9) ~JI
+
+
where the last term denoting variation on the surface is zero. Substituting eq 9 into 8, one has 6% = -J([(4HZ
- 2K)q
+ V,2q] x
+ % [ 4 H ( 4 p - 2K)q - 4HKq + 2b:V,Vsq] + X 2 H [ 2 H K q + (2HZs - b):V,V,q] + 7 ~ V ( p -pz)Jq l dS
(7- X K ) (10) Buff, F. p.: Saltzburg, H. J . Chem. Phys. 1957, 26, 23.
- p 2 ) = 0 (4)
where p1 and p 2 are the inside and outside pressures
2H = tr (-V,n)
where tr is the trace and det is the determinant. Thus the total interfacial energies can be expressed in the way given in eq 1 by choosing the moduli to correspond to those of MurphyG8The above rationale is similar to that given in computing the energies of liquid crystals. I I It can also be shown that the minimization of the local free energy is equivalent to the minimization of a “surface mechanical energy”. At equilibrium, the fact that 6~ = 0, that is, the energy must be an extremum, has been used by them to find the equations for stress balance at the interface. In forthcoming sections we have extended Murphy’s treatment to obtain h2e which for stable equilibrium must be 20; that is, the energy must be a minimum in a stable system. Murphy’s equilibrium results have been used to test shapes of finite microstructures which are bodies of revolution (axisymmetric), and their stability has been investigated by using the results obtained here. The fact that microstructure interiors should have constant densities means that these bodies under infinitesimal perturbations retain constant volumes, since a significant amount of energy is needed to eject or incorporate a surfactant molecule into the structure which is the hydrophobic interaction energy. Since the interior is fluidlike, we also assume that it is possible to ascribe to it a pressure pI 1 p2, the outside pressure, most systems of interest being convex bodies (though not exactly so in some cases). The constraints which thus arise have also been derived and used. Finally, a model has been used to estimate the nature of packing of the head groups; such a constraint should exist as noted by a number of investigator^.^-'*'^ If the global energy is to be minimized then one must have a prior knowledge of interfacial energies obtained through local (interfacial region) minimization and hence accompanied by the conditions of force balances at all points on the interfaces. Al-
(3)
where I is the identity tensor, I , = I - nn is the surface identity tensor and V is the three-dimensional gradient operator. At equilibrium, he = 0 and hence
(13) Tyuptsov, A. D. Fluid Dyn. 1966, 1(2), 51.
The Journal of Physical Chemistry, Vol. 91, No. 3, 1987 607
Micromechanics of Surfactant Microstructures
7
As the fluids are also at equilibrium V p , = V p 2 = 0. Finally as I 0 for stability, the stability criterion becomes
t
J M 4 P - 2K)V + 7,211 x (7 - X K ) % [ 4 H ( 4 P - 2K)v - 4HK7 2b:VsVst]]+ X2H[2HKt] (2HIs - b):V,V,q])qd S I0 (10)
+
+
+
Equation 10 has been derived for % = X = 0 by TyuptsovI3 and Huh.I4 Now V = S y d V = ' f 3 s Y V - rdV, which reduces by Green's theorem to V=
y3*
dS
where R describes the surface. Further, 6V = 0 = Jv6(dV) = S y V & dV, which under Green's theorem becomes
RJ Figure 1. Schemati? diagEam of stability condition for spheres. The plane is 9 ?t 31 = P / 2 , and the shaded region is stable to all perturbations.
+ +
Surfaces of Revolution The equations under consideration cannot be solved in their generality, nor are we concerned with the general solutions here. Here we are interested in the axisymmetric figures generated by the revolution of a curve about the axis which are also symmetric about the equator. In that case the change in the position vector R describing the surface can be written in cylindrical coordinates ( r , z , 0) and its unit vectors (e,, e,, e& as15
dR = d r e,
+ r d0 es + dh e,
J
t
(13)
Y
where z = h ( r ) describes the surface. The tangent vectors are
d
where h' = dh/dr and the unit normal is
Figure 2. Schemaiic diagram of stability condition for cylinders. The plane is 9 ?t = P,and the shaded region is stable to all perturbations.
+
The details of the calculations for the various surface quantities are given in the Appendix. We use eq 4-6 and 10 (eq 12 is satisfied with a particular representation for t] as shown in the Appendix) and the results of the Appendix to obtain solutions for spheres, cylinders,and lamellae as given below. w e use dimensionless forms of He= hlL,, R = r/Lo, ;i. = y/yo, % = % / y J o , X = X f y J ; , and P = (p, -p2)Lo/y0, where Lo is an appropriate length scale-thickness of a lamella, radius of a sphere, or a cyGnder and yois a representative interfacial tension. 9, %, and X are taken to be positive. ( a ) Spheres. The equilibrium condition is
4 + k + k= P / 2
,?.52R
(16)
1 . When the bending modulus dominates, the sphere cannot retain its shape. ( b ) Cylinders. The equilibrium condition is
y+k=P where 9, k,and for stability is
k cannot vary on the surface.
411 - ( 5 2
+ m2)1 + k ( 2 m 2 - 2 ) - A$
(18) The condition 5
o
(19)
For k = ,?I = 0, the most dangerous case is given by m = 0, where 5 1 1 or X 5 27r for stability, In this case if the length to radius ratio in cylinders exceeds 2a, they are unstable, a result first obtained by Huh.14 However when bending and torsion are considered
where 9 and k + k cannot vary on the surface. The condition for stability gives where 5 is a dimensionless wavenumber. For k = k = 0, we have (a2 m2) 2 2. The situation of m = 0 (the axisymmetric perturbation) is seen to be the mostcangerous case. There 5 1 2II2or the dimensionless wavelength X I2Il27r for stability. Since these waves are perpendicular to the surface and run from pole to pole along the surface, the disturbance of the largest dimensionless wavelength a sphere can accommodate is 7r. Consequently in this case the sphere is stable to all small disturbances, a result first obtained by Huh.14 Using similar arguments, it can be shown th$ spheres are stable to all small perturbations for 9 2 k.> 2% and unstable to all small perturbation2 for-? + X < 2%. The situation is shown schematically on a 9 - % - X diagram in Figure
+
(14) Huh, C. Ph.D. Thesis, Department of Chemical Engineering, University of Minnesota, Minneapolis, 1969. (15) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Martinus Nijhoff Boston, 1983; p 474.
for stability.- We consider the different cases. (i) 4 > 2%. For m = 0 only, the system can be unstable to disturbances of some wavelengths. For stability
Thus if the (lengthlradius) ratio of a cylinder is less than the right-hand side in eq 21, then it is stable to all disturbances it can accopmodate. This (length/radius) ratio becomes infinite at 4 = 2%. (ii) 9 C 2%. In this case the system is unstable to all disturbances for m > 1 . The situation has been shown on a y - k - % diagram in Figure 2. (c) Lamellae. At equilibrium, P = 0 and 7 and k cannot vary from point to point. The system is also stable to all small disturbances.
608
The Journal of Physical Chemistry, Vol. 91, No. 3, 1987
TABLE I: Structural Organization ?rli sphere cylinder 0
stable
1/2
unstable
short cylinders, stable infinite cylinders, stable unstable
Models lamella stable
(d)Discussion. We first discuss why results different than those obtained under classical theories, % = X = 0, may be expected. From Murphy8 de = T dss
+ chidm; + y d S - 8%
d(2H)
Neogi et al.
+ SX dK
(22)
I
Consider a sphere: its mean curvature 2H is negative and Gaussian curvature K is positive. If the sphere is made smaller S decreases and surface tension y term causes a decrease in the energy in e . But 2H decreases (it increases in magnitude but has a negative sign), causing an increase in e . Similarly K increases thus increasing e . consequently the effects of curvature can not only offset the conventional effects, that is, of surface tension alone, but can even reverse such effects particularly if y is small or curvatures are high. These arguments also hold in the other systems studied here but become progressively weaker; in cylinders K = 0 and in lamellae 2H = K = 0. We concentrate on the nature of the results obtained here and for the time ignore lamellar structures. Comparing Figures 1 and 2 , we note that for the same conditions of the tails, that is, the same value of P, systems with lower surface tensions favor spherical geometry and systems with higher surface tensions favor cylindrical geometry as the equilibrium plane for spheres is located below the equilibrium plane for cylinders. In oil-watersurfactant systems where ultralow tensions are observed dispersions are spherical and cylindrical dispersions are not known. If the surface tension is high as in surfactant-water systems, cylinders can form either short ones as in cylindrical micelles or long ones as in cylindrical liquid crystals. We now look at the stability results. These have been given in Table I under the assumption that X can be neglected. It should be emphasized here thst the stability results can be interpreted only through the ratio ? f / qand not through their absolute values. At the top of the table, all shapes are seen to be stable. However, the cylinders formed have low values of stable maximum length to radius ratio, and may not actually form there. The cylinders gain in importance as the ratio 7f/4increases toward 112. Everything else remains the same but the maximum length to radius ratio of the cylinders grow toward infinity; t t a t is, hexagonal cylindrical liquid crystals can form. Above the 7 f / y ratio of 112, only lamellar structures are stable. Previous calculations based on energy considerationsI6 show that if spheres form then bending energies have to be low, in keeping with the observation made in Table I. The peculiarity of the behavior in this region is that only lamellar microstructures can form. This provides an interesting explanation for the lamellar phase seen by Ekwalll’ at low surfactant concentrations. It is noteworthy that such phases are seen only with higher alcohols, where the surface tension is changed due to the dilution of the surface charge densities and the bending modulus is perhaps less affected due to matching chain lengths. In that case our explanation for the formation of these lamellar phases differs significantly from the one by Wennerstrom.I8 In principle, one could proceed now with the thermodynamic calculations for a structure under constraints just derived. This would provide the regions of concentrations where it would exist on a phase diagram and the actual dimensions of the aggregates. We have not carried out these calculations at present and have confined ourselves to the study of force balances alone. (16) Kim, M.; Neogi, P.; Friberg, S . E. Paper presented at AIChE National Meeting, New Orleans, LA, 1986; paper No. 74e. (17) Ekwall, P. In Advances in Liquid Crystals; Brown, G . H., Ed.; Academic: New York, 1975; p 1. (18) Wennerstrom, H. J . Colloid Interface Sci. 1979, 68, 589.
To proceed further w_e need some information on “equations of state” models for 4,7f, and X.In general these are expected to be functions of surface packing and curvatures, and for a particular system can even vary from point to point on the surface. The surface models should take into account the molecular aspects. At present statistical physics shows that on highly curved interfaces, dividing surface and interfacial tensions cannot be uniquely defined,19*20 a feature that should not obscure phenomenology but does cause some difficulty in interpreting molecular models. In view of these difficulties we confine ourselves to the simplest model stress distribution. Previously Murphy8 had suggested a model where the excess stresses are located o n 9 at chosen surfaces. This allows the computation of 4, ?f,and X in terms of these excess stresses and physical dimensions in these systems’2J6,21 Hall and Mitchell,22using a “hydrostatics” approach where the interior is also assumed to be fluidlike, derived an equivalent format. We use the two-surface approach, one located above the head groups and the other located immediately below the first methylene group attached to the head group. The stretching-bending modulus of the two surfaces are assumed to be different, that is K A K / ~ for the upper surface and K - A K / for ~ the lower where K A K / ~ 1 0 . Further as the datum, we assume that stretching gives rise to tension only (at the cost of the reference value of the surface tension y*) and it is least when SmLo/uH = 1, where Sm is the surface area per head group and uH is the volume of the hydrophobic tail. Explicitly, the results are as follows. Spheres.
+
*
Y =
A,(
S m - Sm* Sm*
)+
2~ + -AK L*
w=
and for L* =
m
where the parameters are as follows: 6/Lo = fractional thickness (19) Hemingway, S. J.; Henderson, J. R.; Rowlinson, J. S . Faraday Symp. Chem. SOC.1981, 16, 3 3 . ( 2 0 ) Schofiled, P.; Henderson, J. R. Proc. R. SOC.London 1982, 3 7 9 4 321. (21) Miller, C. A. J . Dispersion Sci. Technol. 1985, 6, 159. (22) Hall, D. 0.;Mitchell, D. J. J. Chem. SOC.Faraday Trans. 2 1983, 79, 185.
Micromechanics of Surfactant Microstructures
The Journal of Physical Chemistry, Vol. 91, No. 3, 1987 609
I
I
-0.25
I
I 2
6
I
I 3
I
I
4
I I
I
-0.25
I 2
I
I
1
I
3
4
Sm Lo/ vH
Sm LOAH Figure 3. h/(T+ k)against SmLo/uHfor different values of AK/Y* in sphere. The values of K / Y * = 100 and 6/Lo = 0.3 have been used. Below the dashed horizontal line, the system is stable and the vertical line denotes the packing criteria for sphere, SmL,/uH > 3.
Figure 4. k/T against SmL,/uH for different values of AK/Y* in cylinder. The values of K / Y * = 100 and 6/L, = 0.3 have been used. Below the dashed horizontal line, the system is stable and the vertical lines denote the packing criteria for cylinders, 2 < SmLa/uH< 3.
of the head groups; SmLo/vH = inverse of Israelachvilli et al.'s7 packing parameter; K/Y*= ratio of the bendingstretching modulus to the surface tension; AK/Y*= ratio of the inhomogeneity in the bending-stretching modulus at the two interfaces to the surface tension. Cylinders. v =
6
S m - Sm* 2Y* + 2K( Sm*
)+
I
I
1
I
Sm b/vH
x,,
Figure 5. The maximum length to radius ratio, in cylinder against SmL,/uH for different values of A K / Y * . The values of K / Y * = 100 and &/Lo= 0.3 have been used. It is noteworthy that the length to diameter ratios of all short cylinders fall in the 3-3.5 range over wide values of parameters.
H=
and for L* =
they reach constant values and at small SmL,/uH values they tend to values of infinity in some cases. Surfactant Phase The model for the surfactant phase is taken here to be a bicontinuous one, where the surfactant layer separates regions of oil and water which are randomly arranged.23 Since the surfactant phase can be in equilibrium with an oil-continuous phase and a water-continuous phase at the same pressures, it would require P = 0. Further, as the oil and water domains are randomly arranged, the interface would have the two radii of curvature in opposite directions. Hence we take 2 H = 0 as a model in these systems. The curvature tensor is thus written as
m
% -_ 5
b = -cIu~uI + C ~ U ' U+~ C
(24) In Figure 3 we have plotted %/(y + ?I) against SmLo,lvHfor K / Y * = 100 and various values of AK/y*, from eq 23. Below the dashed horizontal line the system is stable. Spheres are thus unstable only when the aggregates are closed packed. The vertical line at SmLo/vH = 3 denotes the criterion of Israelachvilli et aL7 which is that spheres can exist only for SmLo/vH > 3. In Figure 4 a similar plot has been drawn for cylinders, where above the horizontal line is the region of instability. The criterion of Israelachvilli et a].' places cylinders between the two vertical lines at SmLo/vH = 2 and 3. In general, we notice in Figure 3 and 4 that unstable spheres and cylinder may occur, if they at all do occur for given values of physical properties, in the region of dense packing with SmLo/vH = 1-2, a region which according to Israelachvilli et aL7 can only be occupied by lamellae. In Figure 5 , we have plotted ,A,, the maximum allowable length to radius ratio in cylinders as a function of SmL,/uH. At large SmL,/vH
+
~ U ~ UC~U'U~ ~
(25)
where ul and u2are the basis vectors in the tangential directions on the surface. Substituting 2 H = 0 and eq 25 into eq 4-6, one has
+ c,2)% = 0 v,y - XV,(CI2 + c22) = 0 ax + Y(-c~u' + c~u') + ax -(c~u' + c~u') = 0 -(c12
-V,%
(26)
(27)
(28) aY aY2 where y' and y 2 are the coordinates in the two principal directions. According to eq 26, either c1 and c2 are both zeros or Tf is zero. The first of these lead to lamellae and the second presents a more interesting feature. The stability condition in the second case becomes
+
[2(c12 + V;v][y + R(c12+ c 1 2 ) ] v I0 (29) for positive values of y and X and VS2a= -a2q;eq 29 yields a2 (23) Scriven, L. E. Nufure 1976, 263, 123.
610
The Journal of Physical Chemistry, Vol. 91, No. 3, 1987
+ cZ2), and hence in terms of wavelengths
2 2(c12
x
I21/2?r/(c12
+ c22)1/2
(30)
The above result can be explained in the following way. If the surfactant phase can be subdivided into domains where inside a domain one region has a distinct relation with the other regions but there are no such relations with regions outside the domain, then the maximum scale of these domains is given by the righthand side in eq 30. The latter is thus the persistence length scale introduced by de Gennes and T a u p h Z 4 If in eq 27 X is assumed to be a constant, then one has on integration cI2 c? = 71% on assuming that there are no shear stresses at the surface, or
+
Under ultracentrifuge, Hwan et aLzs observed that the surfactant phase fragmented into a mixture of oil-in-water and water-in-oil microemulsion phases. If we assume that the fragments represent these domains, then IclI lczl c, the curvature of microemulsion droplets. This would also imply that oil and water domains are uncorrelated which is also the current mathematical representation of bicontinuous structures.26 It is very intriguing to note that 2 H = 0 and 7;( = 0 can both be proven by using thermodynamic argumentsz7 In the force balance method used here the condition 2 H = 0 has to be assumed as the starting point in the model.
- -
Discussion It has been possible to show using the force balances and stability that not all spheres are stable and not all infinite cylinders are unstable. In that, we are able to show for the first time that under restricted conditions infinitely long cylinders can both form and remain stable. We have also found that there are regions where only lamellae can form and be stable; the stability of lamellae is unrestricted under our theory. This shows that if aggregates form with the above properties lamellar structures will be seen even if the range of surfactant concentrations are smaller than those observed in lamellar liquid crystals under normal conditions. That lamellar liquid crystals, and only these, act out of character as described above has been noted by Ekwall.17 The above results are general, in the sense that no assumptions regarding the dependence of interfacial tension, bending modulus, and the torsional modulus on surface packing or the curvatures of the surface are being made. Next, with a simple model relating these effects it was found that the ratio between the bending modulus to the interfacial tension was usually small such that behaviors such as infinitely long and stable cylinders or unstable behaviors occurred only over a limited range. As used here the curvatures of a specific structure are constant when related to the fully extended length of the surfactant molecule L,. Consequently, the contribution of the bending of the surfactant films to the interfacial energy is a constant. However, the moduli of tension are strongly influenced by the stretching parameter SmLo/cH,but their ratios quickly reach constant values. For instance ? f / y is seen to be insensitive to the stretching parameter except very near the value which produces the lowest interfacial tension, namely, Sm*LO/oH= 1. Consequently, the calculations depend on the choice of how the moduli depend on the stretching parameter. We have modelled the surfactant phase as a bicontinuous structure with zero mean curvature. The results show that in this case the bending modulus 7f = 0 and a persistence length scale proportional to ( X / y ) l / * can be defined. If two points in the surfactant phase are separated by a distance greater than this scale then these two points can be assumed to be uncorrelated in their structures and assumed to belong to disjointed regions. Equations 4-6, 10, and 12 can be used to obtain all permissible shapes; however, the calculations have to be done numerically and (24) DeGennes, P. G.; Taupin, C. J . Phys. Chem. 1982, 86, 2294. (25) Hwan, R.-N.; Miller, C. A,; Fort, Jr., T. J . Colloid Znterface Sci. 1979, 68, 221. (26) Talmon, Y . ; Prager, S.J . Chem. Phys. 1978, 69, 2984; 1982, 76, IC7C
LJJJ.
(27) Ruckenstein, E. Fluid Phase Equilib. 1985, 20, 189
Neogi et al. have not been carried out. At least some of the calculations need to be performed in the future without which we are unable to utilize the force balance conditions completely. Lastly we point out the reason why the results from geometrical packing considerations, such as those of Israelachvilli et aL7 do not agree with ours except in that both predict lamellar forms at high packing densities. As MukherjeeS shows, the basic idea behind geometrical packing lies in visualizing the surfactant molecules as hard cylinders with a taper. The hard cylinder assumption implies a long-range correlation among methylene groups in a hydrocarbon tail. Here we have investigated the other extreme where only short-range correlation exists.
Acknowledgment. This work was performed under DOE grant DE-AC02-83ER 13083. Appendix From the tangent vectors one has arr = ( Q ’ y = 1 hI2 ass = ( Q @ S ) - ~ = r2
+
Following well-known rules,* the dual set is defined as
For a scalar 17
From the unit normal one has an V,n = -ar ar and on using the relationsl5
an + -as a0
where all other derivatives of the unit vectors (e, e,, ea) with respect to (r, z, 0) are zeroes, one has an - h” h ’h” _ (1 + h 9 3 ~ -~(1r + h y / : e z ar
from which we get
b=-Vn= (1
+
h” a’a, h’2)3/2
+ r(l +h’h ’ 2 ) 1 /asas 2
In a similar fashion we get +
[ -23 r3 88
8%
(vii)
-a& as2
From (vii) we get
and
-+
b: V,V,q = 1
h’
a27
- (ix)
J. Phys. Chem. 1987, 91, 611-617 If we assume that t=
t o cos
(mQf,(r)
x
such that fa satisfies 1
(1
Lo m n P
2h’h’’ hr2)
8; = -2.f,
+ ht2+“-(1 +
P
and m = 0, 1, 2, ..., then t satisfies
r
R (xii)
9
Sm
s
and
T
b
v,v,s = -
I
a2h”
(1
7 r3 (1 + h’ h’2)1/2 + h’2)3/2+ -m2
Now, the term
U
(xiii)
un V Y Superscript
*
611
bending stress identity tensor Gaussian curvature torsional stress fully extended chain length of surfactant tail integer unit normal vector pressure pressure difference position vector surface vector surface excess entropy surface area per surfactant head group surface absolute temperature arbitrary perturbation vector volume of surfactant tail volume coordinate in the principal direction reference
Subscript
for such values of 7, because of the cos (me) term here. This meets the requirement (12). From (vi) we also have
2H = tr ( b ) =
(1
h” + h’2)3/2 r ( l
+
K = det (b) =
r(l
h’ h’2)1/2
+
S
0
Greek Letters (Y
(xiv)
h ’h” h’2)2
+
Y
r
6 c
v K
x
Symbols a
b C
e
h 2H
tangent vector curvature tensor defined in eq 7 principal curvature unit basis vector position at surface mean curvature
surface appropriate reference
P T
$1 $2
$3
V
wavenumber surface tension surface excess thickness of surfactant head group mechanical energy small perturbation normal to surface stretching-bending modulus wavelength chemical potential tangential vector scalar defined in eq 4 vector defined in eq 5 vector defined in eq 6 gradient operator
Amlnes as Microemulsion Cosurfactants Klaus R. Wormuth and Eric W. Kaler* DeDartment of Chemical Engineering, BF- 10, rniversity of Washington, Seattle, Washington 981 9 (Rkceived: J&e 3, 1986; In Final Firm: September 15, 1986)
A systematic study of the phase behavior of short straight-chain amine amphiphiles in oil and water mixtures reveals large differences between amine and corresponding alcohol systems. The primary amine head group is more hydrophilic than alcohol, nitrile, carboxylic acid, ketone, and aldehyde head groups. The hydrophilicity of the amine head group is about the same as the hydrophilicity of the triethylene glycol head group, but butylamine is a more effective amphiphile on a mass basis than triethylene glycol monobutyl ether. Addition of acid strongly increases amine hydrophilicity while addition of base decreases amine hydrophilicity. In microemulsions, the effective hydrophilicity of the amine/anionic surfactant combination is lower than expected, indicating that a specific ionic interaction between amine and anionic surfactant is occurring in the surfactant-rich film separating oil and water domains.
Introduction Microemulsions are thermodynamically stable microstructured mixtures containing oil, water, surfactant, usually salt, and often a small amphiphilic molecule called a cosurfactant. The most common cosurfactants are low molecular weight alcohols. The richly varied phase behavior and microstructure of microemulsions have attracted attention because microemulsions mix all ratios of oil and water into transparent phases with ultralow interfacial tensions. Systems forming microemulsions present complex phase 0022-3654/87/2091-0611$01 .50/0
behavior, and according to current theory and experiment, the chemical structure of the cosurfactant plays an important role in such phase behavior.’-s We have studied a novel type of (1) Holt, S. L. J . Dispersion Sci. Technol. 1980, I , 423. (2) Clausse, M.; Perelyasse, J.; Boned, C.; Nicolas-Morgantini, L.; Zradba, A. In Surfactants in Solution; Mittal, K. L., Lindman, B., a s . ; Plenum: New York, 1984; Vol. 3, p 1583. (3) Stilbs, P.; Rapacki, K.; Lindman, B. J . Colloid Interface Sci. 1983,95, 583.
0 1987 American Chemical Society
