2346
J . Phys. Chem. 1988, 92, 2346-2352
On the Structure of Aggregates of Adsorbed Surfactants: The Surface Charge Density at the Hemimicelle/Admicelle Transition Mark A. Yeskie and Jeffrey H. Harwell* The Institute of Applied Surfactant Science and School of Chemical Engineering and Materials Science, The University of Oklahoma, Norman, Oklahoma 73019 (Received: September 8, 1987)
Surfactants adsorbed from aqueous solution onto mineral oxide surfaces are known to form micellelike aggregates at the interface. It has long been the view that at low coverages these surfactant aggregates were principally in the form of monolayers, formed on patches of the surface. Monomers in these aggregates are viewed as being oriented in such a manner that the hydrophilic groups of the surfactants are next to the surface, with the surfactant tail groups forming a hydrophobic film in contact with the aqueous solution. Aggregates of this structure are commonly referred to as hemimicelles. It has recently been propssed that bilayered aggregates, termed admicelles, were more consistent with the known role of the hydrophobic effect in the aggregate formation. This paper presents the results of a study that examines whether the hydrophobic interactions involved in the aggregation of adsorbed surfactants are large enough to induce admicelle formation at a lower chemical potential than that at which a hemimicelle would form. It is shown that while there are conditions under which a second layer of surfactant will always form simultaneously with the first layer of surfactant (admicelle formation), there are also wide ranges of conditions under which hemimicelles will form; Le., under certain conditions surfactants will aggregate to form a first layer without the simultaneous formation of a second layer. This is shown to be because the electrostatic contributions to the free energy of transferring a monomer from a hemimicelle to an admicelle are potentially much larger in magnitude than the largest possible hydrophobic contributions to the free energy of transfer. It is also concluded that when admicelles do form, there is little if any interpenetration of the hydrocarbon tails of the second layer of surfactant monomers between the hydrocarbon tails of the first layer, because the electrostatic repulsion between the two layers of head groups is larger than the largest possible gain in hydrophobic bonding that could result from the interpenetration.
Introduction Numerous aspects of anionic surfactant adsorption on oppositely charged mineral oxide surfaces are widely agreed upon by workers in the field. Among the most important of these aspects is the conclusion that at very low coverages surfactant aggregates form locally at the mineral oxide/aqueous solution interface.'-'0 These local aggregates have been referred to as hemimicelles since the term was introduced by Gaudin and Fuerstenau'J in a study of the role of surfactant adsorption in mineral flotation phenomena. The aqueous-phase surfactant composition at which these aggregates first begin to form on the surface was later termed the hemimicelle concentration, often abbreviated HMC.3,4 Both terms clearly communicate that these aggregates are very micellelike in many aspects of their behavior. The prefix hemi also correctly conveys, however, one unmicellelike aspect of the structures, as they have long been viewed. As illustrated in Figure 1, the structure termed a hemimicelle has been viewed as being a local surfactant monolayer, formed so that the ionic head group interacts strongly with the charged mineral oxide surface, thus forming a hydrophobic patch on the mineral surface. The principal rationale for this structure has been the phenomenon of mineral flotation. Suspended particles of minerals with hydrophilic surfaces do not readily adhere to the surface of (1) Gaudin, A. M.;Fuerstenau, D. W. Trans. Am. Inst. Min., Metall. Pet. Eng. 1955,202,66. (2) Gaudin, A. M.; Fuerstenau, D. W. Trans. Am. Inst. Min., Metall. Pet. Eng. 1955,202,958. ( 3 ) Somasundaran, P.; Healy, T. W.; Fuerstenau, D. W. J. Phys. Chem. 1964,68,3562. (4) Somasundaran, P.; Fuerstenau, D. W. J. Phys. Chem. 1966,70, 90. ( 5 ) Gouion, G.; Cases, J. M.;Mutaftschiev, B. J. Colloid Interface Sri. i9i6;56, 587. (6) Scamehorn, J. F.; Schechter, R. S.; Wade, W. H. J. Colloid Interface Sci. 1982,85, 463. (7) Harwell, J. H.; Hoskins, J. C.; Schechter, R. S.; Wade, W. H. Langmuir 1985,I , 251. (8) Bisio, P. D.; Cartledge, J. G.; Keesom, W. H.; Radke, C. J. J . Colloid Interface Sci. 1980,78, 225. (9) Chander, S.; Fuerstenau, D. W.; Stigter, D. In Adsorption From Solution; Ottewill, R. H., Rochester, C. H., Smith, A. L., Eds.; Academic: London, 1985; p 197. (10) Chandar, P.; Somasundaran, P.; Turro, N. .I. J . Colloid Interface Sci. 1987,117, 31.
0022-3654/88/2092-2346$01.50/0
bubbles sparged into an aqueous slurry. Addition of small concentrations of surface-active compounds, most often at concentrations well below the critical micelle concentration, results in adherence of the particles to the bubble surface. The mineral particles can then be separated from the suspension by foam flotation. It is also well established that the surfactant concentration at which the mineral exhibits a dramatic improvement in flotation response coincides with the concentration at which aggregates begin to form on the surface.'-4 This is suggestive of the formation of a hydrophobic layer on parts of the mineral surface and, hence, of the hemimicelle structure. This view of hemimicelles has been reinforced by contact-angle studies of the air/water/mineral interface: the onset of aggregate formation has been found to coincide with a decrease in contact angle. This has been viewed as a confirmation of both the flotation mechanism and the hemimicelle structure. To our knowledge, it was not until 1982 that the structure of aggregates of adsorbed surfactant began to be systematically reexamined. At that time Scamehorn et aL6 published experimental evidence interpreted by them to indicate that monoisomeric petroleum sulfonates would form complete bilayers on y-alumina under proper conditions of added electrolyte and pH. This should not be interpreted as suggesting that earlier workers had not recognized that bilayered structures could form. At least the beginning of bilayer formation had been recognized as early as Gaudin and Fuerstenau, who proposed it as a mechanism for the decrease in flotation efficiency with increasing surfactant adsorption, which is often seen in flotation studies. Scamehorn et al. also presented a theoretical model of the formation of the surfactant aggregate on the surface. The results of their modeling were important in the development of what we will call the admicelle hypothesis, namely, that local bilayered structures (Figure 1) could form on patches of the oxide surface without a hemimicelle having been formed at lower surfactant concentrations. In their model, the Fowler isotherm was modified to allow bilayer formation to occur either subsequent to hemimicelle formation or simultaneous with hemimicelle formation. This model was then applied to a surface made up of a distribution of patches of different adsorption energies, to simulate an actual surfactant adsorption isotherm. By assumption of a Gaussian distribution of standard-state free energies of aggregate formation for the patches, which were pictured as making up the particle 0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 8, 1988 2347
The Hemimicelle/Admicelle Transition Hemimicelle
Admrelle
c Headgrouor
gregate or between the surfactant head groups and the charged mineral oxide surface. These interactions are known to be dominant in the formation of anionic micelles. It is possible that the Scamehorn model predicted substantially simultaneous first- and second-layer formation because the surfactant/mineral surface interaction was too small. This required that the lateral interaction parameter, which results in prediction of a two-dimensional phase transition, had to be made too large.
Theory Figure 1. Two proposed structures for aggregatesof adsorbed surfactant
molecules. surface, isotherm shapes resulted that were like those typically observed for aqueous/mineral oxide/ionic surfactant systems. Scamehorn et al. also determined that a value of the surfactant/surfatant interaction parameter necessary to result in a local phase transition from low coverage to local aggregate formation led to a nearly simultaneous formation of both a first and a second layer (- 1.8 layers were found to form simultaneously). This led them to hypothesize that, even at low coverages, the system they were studying did not result in substantial formation of local monolayers (hemimicelles); rather, the adsorbed surfactant layer was built up primarily by the patchwise formation of essentially bilayered surfactant aggregates. Subsequently, Harwell et al.' proposed calling these local aggregates admicelles (Figure l ) , to distinguish them from the monolayered aggregates that had been pictured by earlier workers. The surfactant isotherm proposed by Harwell et al. simply assumed that admicelles were the only surfactant aggregates to form a t the solid/solution interface; their isotherm did not allow for the possibility of hemimicelle formation. While the results of Scamehorn et al. were important in leading Harwell et al. to propose the existence of admicelles at low adsorption densities, the principal reason for their advancement of this model was that it removed the one major difference between the formation of micelles in the bulk solution and the formation of aggregates at the solid/solution interface, namely, that surfactant aggregate formation on the surface resulted in formation of a hydrophobic layer in contact with the aqueous solution. Though the model of Harwell et al. did not explain the contact-angle and particleflotation behavior observed by earlier workers, the complexity of the interactions at the air/water/mineral interface in the presence of adsorbed surfactant' 1-13 is sufficient to raise questions about the conclusiveness of these observations with regard to the structure of the surfactant aggregate in the absence of the an air/water interface. The purpose of the study presented in this paper is to examine the relative importance of hydrophobic interactions compared to electrostatic interactions in determining whether a local monolayer of surfactant (a hemimicelle) will form on a surface patch rather than a local bilayer of surfactant (an admicelle). It is hoped that this will help clarify conditions under which one or the other structure will be preferred by the system. Specifically, we hope to provide at least a preliminary answer to the following question: Are the hydrophobic contributions to the free energy of surface aggregate formation large enough to imply that second-layer formation should always be expected to accompany first-layer formation? A positive answer to this question would mean that the admicelle structure is the more correct visualization of the structure of the aggregated adsorbed surfactant. We are aware of no other theoretical attempts to examine the simultaneous formation of the first and second layers of these aggregates. While the modified Fowler isotherm of Scamehorn et al. is capable of reproducing surfactant adsorption isotherms, its major weakness is that it does not explicitly include electrostatic interactions either between surfactant monomers within the ag(1 1) Ralston, J. Ado. Colloid Interface Sci. 1983, 19, 1. (12) Somasundaran, P.; Chandar, P.; Chari, K. Colloids Surf.1983, 8,
Following the procedure of Ben Naim,I4 Harwell et al.' presented the following expression for formation of a surfactant aggregate a t the interface:
This expression was obtained by equating expressions for the chemical potential of the surfactant in the bulk and in the aggregate. In this expression, 4 is the mean electrical potential experienced by the surfactant monomer in the aggregate relative to the bulk potential (in units of kT); mw/kT (also dimensionless) is the free-energy change on removing the surfactant hydrophobic moiety from the aqueous phase to the aggregate of adsorbed surfactants, with m representing the number of CH2/CH3groups removed from contact with the aqueous phase per molecule of surfactant in the aggregate; w is the net free-energy change per CH2/CH3 group, k is the Boltzmann constant, and Tis the absolute temperature. The surfactant adsorption density in each layer of the aggregate is represented by r M (mol&ules/nm2), while p is the bulk surfactant monomer concentration (molecules/nm3) and A is the de Broglie thermal wavelength of the surfactant (nm). The term 47r2 arises in the analysis because of the loss of orientational degrees of freedom when the molecule is in the aggregate.' For a given value of the surface charge density on a local patch, us,let pA be the concentration of surfactant monomers at which an admicelle will form on surface patches with charge density us. Let pH be the concentration at which a hemimicelle will form on patches at the same surface charge density. Using eq 1, we can subtract In pH from In p A to obtain an expression relating the hydrophobic and electrostatic contributions to admicelle and hemimicelle formation:
If In (/JA/PH) < 0, then the concentration of surfactant monomers in the bulk solution necessary for an admicelle to form would be lower than the concentration of monomers necessary for a hemimicelle to form; Le., an admicelle will form at a lower chemical potential than a hemimicelle. Therefore, if A(mw/kZ')- A4 < 0, then a surfactant molecule in an admicelle will be at a lower chemical potential than a surfactant molecule in a hemimicelle. Equivalently, if A ( m w / k T ) < A+, then an admicelle will form first. Note that A+ can be throught of as the mean electrical work (in units of kT) of transporting a surfactant ion from a hemimicelle to an admicelle, while Amw/kT is the free-energy change per molecule due to hydrophobic bonding of the first layer of the admicelle to the second layer of the admicelle. Thus, by examining the effect of factors like surface charge density, added electrolyte concentration, counterion binding to the admicelle, etc., on A4 = A ( m w / k T ) , we can examine the conditions under which a hemimicelle or an admicelle would be the structure with the lowest free energy per molecule. In order to estimate the electrostatic contribution to the freeenergy change of admicelle and hemimicelle formation, we assume the model structures in Figures 2 and 3, respectively. Because of our present inability to quantify the effect of surfactant adsorption density on the hydrophobic contribution to the free energy of aggregate formation, we must make the ad hoc assumption that the density of surfactant molecules in the hemimicelle is the same as the density of surfactant molecules in both the top and bottom
121.
(13) Blake, P.; Ralston, J. Colloids Surf. 1985, 16, 41.
(14) Ben Naim, A. Hydrophobic Interactions; Plenum: New York, 1980.
2348
The Journal of Physical Chemistry, Vol. 92, No. 8, 1988
Yeskie and Harwell
/ TODl a y e r
headgroups plus counterions
-a
=T
V2Yr
-)
trP
Lower l a y e r headgroups plus counterions
+
O x - d e sdrface
+
1
- 0
V2VB
/
=B
,r
/
s/
TI/
p
p I
= e ( r ~ a-, r~
~ )
(7) Here r N a , T is the adsorption density of bound (specifically adsorbed) counterions in the top layer (ion/nm2), and r N a , B is the density of counterions in the bottom layer. Assuming that the Poisson-Boltzmann equation is valid for the diffuse layer, we obtain (TB
,;;;;;;; ,;;;;;;;;;:
Figure 2. Schematic of proposed model for the electrical potential in the admicelle.
where p o is the total electrolyte concentration in the bulk (molecules/nm3) and cD is the effective dielectric constant across the diffuse layer. For the hemimicelle structure shown in Figure 3, similar expressions can be obtained to approximate the ensemble average electrical potential: 4H = $ r ~+ ( e / k T ) [ ( @ / z c+~Lo/ec)us
( 3 P / 8 e B + LO/tC)a’BI (9)
TABLE I: Typical Mineral Oxide Surface Charge Densities suspended solid
added electrolyte
surface charge range, e/nm2
pH range
a-FeOOH TiO, y-A1203
KNO,
-0.6 to +0.6
10-5
NaN03 NaCl
-0.4 to +0.4 -0.6 to +0.8
9-4 11-4
pzc“ U’B
7.4 5.9 8.6
“Point of zero charge.
layers of the admicelle. This assumption is consistent, however, with measurements, by fluorescence probing, of constant micropolarity and microviscosity all along the aggregate region of the surfactant adsorption isotherm.I0 The mean electrical work of transporting the surfactant from the bulk solution to the admicelle is the average of the work of transporting it to the top layer and the work of transporting it to the bottom layer:
4~ =
( ~ A , T+ + A , B ) / ~
(3)
where is the mean electrical potential in the admicelle (dimensionless), 4A.Tis the mean electrical potential in the top layer of the admicelle, and ‘$A,B is the mean electrical potential in the bottom layer of the admicelle. Solving the equations in Figure 2, with appropriate boundary conditions, and assuming that the mean electrical potential in the top layer of the admicelle is approximated by the electrical potential in the middle of the top surfactant head-group layer, and the electrical potential in the bottom of the admicelle by the electrical potential in the middle of the bottom head-group layer, we obtain 4A,T
4A.B
= 4 D + ( e / k T ) [ ( @ / 2 t T ) u+ s (@/2EB)uB + (3@/8ET)uTl (4) =
‘$D
+ ( e / k T ) [ ( 3 @ / 2 € B+ L/cC)us + (11@/8eB+ L/cC)uB + ( @ / 2 t T ) u T 1
In these expressions e is the elementary electrical quantum, @ (nanometers) is the thickness of the charge layer consisting of the surfactant head groups and bound counterions, L (nanometers) is the thickness of the hydrocarbon core of the admicelle (Lois the length of the hydrocarbon chain of the surfactant, and L = 2L0 for the admicelle in Figure 2 ) , tT is the dielectric constant of the top head-group layer, usis the charge density of the solid surface below the admicelle after the admicelle has formed, uT is the net charge on the top layer of the admicelle, which consists of the head groups of the second-layer surfactant monomers plus their specifically adsorbed counterions (e/nm2), uB is analogously defined for the bottom layer, and ‘$D (units of kT) is the electrical potential drop across the diffuse layer. The charge densities in the two head-group regions of the admicelle are given by the following expressions:
=
e(rNa,T
- rM)
(6)
= W ’ N ~ -, BFM)
( 1 1)
All terms are defined in a manner analogous to those for the admicelle expressions. Assuming that eB i= cT (the possible effect of this assumption on the conclusions is examined in the discussion, and the assumption is shown to be acceptable), we can solve for A 4 = @* - 4H:
A4 = ($D - 4 ’ ~ )+ ( e / k T ) [ ( 8 / 2 Q + L / 2 t c - L o / d g s + e(7P/16tT)(rNa,T - rM) + e(15@/16€T+ L/2tC)(rNa,B rM) - e(3@/8cT+ LO/eC)(r’hua,B- r M ) l ( 1 2 ) Since we are comparing a surfactant in an admicelle to a surfactant in a hemimicelle on the same surface patch, we have assumed that u, has the same value in eq 4, 5 , and 9 in arriving at eq 12. There is certainly some error in this assumption. Except for 100%binding in the second layer of the admicelle, the electrical potential of the admicelle should be more negative than that of the hemimicelle; this will both cause a higher counterion binding on the lower layer of the admicelle and also induce a higher positive charge on the alumina surface below the admicelle through a local increase in the hydrogen ion concentration. There is some experimental evidence that this effect does actually occur to a measurable extent.lg Our assumptions thus bias the results in favor of the hemimicelle structure. Using eq 12, we can examine the electrostatic component of the work of transferring a surfactant ion from the hemimicelle to the admicelle as a function of the variables us,p o , r N a , T , r N a , B , and rrNa,B. The term A(mw/kT) in eq 2 represents the hydrophobic contribution to the work of transferring the surfactant ion from the hemimicelle to the admicelle. Though a theoretical expression analogous to eq 12 is not currently possible for this term, an empirical estimate can be made. Tanford,lS citing Hermann16 and Reynolds et al.,I7 has suggested that the hydrophobic contribution to the free energy of transfer of the hydrocarbon moiety from bulk water to a surfactant micelle is proportional to the “area” of hydrocarbon/water contact. Using Tanford’s figure of 25 cal/(mol A2) for the hydrophobic contribution to the freeenergy change and the estimate of Scamehorn et aL6 of a density of molecules in the adsorbed surfactant bilayer of 0.44 nm2/ molecule (2.29 molecules/nm2), we obtain A(mw/kT) = -2. This (15) Tanford, C. The Hydrophobic Effect: Formarion of Micelles and Biological Membranes; Wiley: New York, 1980. (16) Hermann, P. B. J. Phys. Chem. 1972, 76, 2754.
(17) Reynolds, J. A.; Gilbert, D.B.; Tanford, C. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 2925. (18) James, R. 0.; Parks, G. A. Surf. Colloid Sri. 1982, 12, 119. (19) Bitting, D.; Harwell, J . H. Langmuir 1987, 3, 500
The Journal of Physical Chemistry, Vol. 92, No. 8, 1988 2349
The Hemimicelle/Admicelle Transition
I
I
Hydrocarbon core -31
,'
Y Lower layer headgroups plus -3 counterions
I
/
TABLE 11: Parameter Values Used in the Base Case parameter
value
rM,molecules/nm2
2.29 2
360 VZY+ : V2VB
0
.-og '6
p
T p I
I/
1.4
2.2c0 35c0 25
,:;;;;;; ,;;;;;;;;;:
Figure 3. Schematic of proposed model for the electrical potential in the
hemimicelle. value should probably be considered as a reasonable lower bound
on the hydrophobic contribution, as it assumes that there is no penetration of water between the surfactant tails in the hemimicelle. A maximum possible hydrophobic contribution can then be estimated by assuming that the hydrophobic contribution to the work of transferring a surfactant ion from the admicelle to the hemimicelle is the same as the hydrophobic contribution to micelle formation. With sodium dodecyl sulfate (SDS) as a model compound, 1.5 A as the distance of closest approach of a water molecule to the hydrocarbon chain, and 1.5 nm as the length of the hydrocarbon chain, this estimate yields A(mw/kT) = -6. The average of the maximum and minimum estimates of the hydrophobic contribution, -4kT/molecule, will be used as a reference in evaluating the significance of the hydrophobic contribution to the work of transferring the surfactant from the hemimicelle to the admicelle. There is some evidence in the literature that the hydrophobic contribution to surfactant aggregation at the mineral/solution interface may be somewhat higher than that for micelle f ~ r m a t i o n . ~ ,Even ' if the figures derived from Tanford are as much as 50% in error because of this, it will be shown that the conclusions of this study are still valid. Once again, however, this assumption biases the results in favor of hemimicelle formation. In summary, if for a given set of values for the parameters us, po, rNa,T, rNa,B, and r'Na,B, we find A$ > -4 from eq 12, it seems reasonable to assume that admicelle formation is favored; if A$ < -4, then hemimicelle formation is favored. The effects on the conclusions of using the maximum or minimum values of A$ are examined in the next section.
Results and Discussion A systematic analysis of the variables involved is necessary in order to determine from eq 12 the conditions under which one aggregate structure is preferred over the other. The procedure selected here consists of choosing approximate values for the parameters in eq 12 and then solving eq 12 to determine usfor A$ = -4. In any real system, u, will depend on both the solid and the pH of the solution with which the solid is in contact. Typical values of a, estimated from electrophoretic mobilities found in James and Parks'* are shown in Table I. The ranges of the surface charge densities are estimated for a-FeOOH suspended in a KN03solution, TiOz suspended in a N a N 0 3 solution, and 7-A1203suspended in a NaCl solution. Although the surface charge densities vary between -0.6 and +0.8 e/nm*, such estimates are average values for the entire surface, and the range in a, is probably larger because of the heterogeneous nature of the surface, as seen by the adsorbing surfactant aggregate^;^^^ nevertheless, almost all reasonable values of usshould certainly lie between -2 and +2 e/nm2, and values between -0.5 and + O S e/nm2 are probably representative of most patch charge densities. Of course, for a particular patch on a mineral oxide surface, the charge on the patch is a function of the pH. It is not necessary to include this phenomenon explicitly in our study to reach our goals. As mentioned, the primary variables of interest in our investigation are the adsorption densities of the bound counterions ( r N a , T , rNa,B, and r'Na,B) and the surface charge density of the
1.0-
r,,,, /rn
(Fmctlonal counterlon blndlng on the top layer of an aomlcelle)
Figure 4. Effect of fractional counterion binding to the top layer of the admicelle on the surface charge density at which the chemical potential of a surfactant ion would be the same in an admicelle as in a hemimicelle, while the binding on the lower layers are held equal. 4.0
,,,
-I.Oato' '
1
1
1
bll' ' ' lo!;
r,,,, /r,
1
1
I ,
1
"
b!3'
'
"0141
1
' '
1
1
b$'
1
1
' '
1
5
1
'ola' ' ' b17' ' ' o' !;
1 1
I
I
''
1 1
I
I
r s
b!s' ' ' '?o
(Fractlonil counterion blndlng on the ~ D layer D o f an admlcelle)
Figure 5. Effect of counterion binding to the top layer of the admicelle on the surface charge density at which the chemical potential of a surfactant ion would be the same in an admicelle as in a hemimicelle when the counterion binding in the bottom layer of both structures is different.
patch of the mineral oxide surface below the surfactant aggregates (us). We will assume that r N a , T , rNa,+ and r'Na,B all can vary i.e., the fractional counterion binding on the between 0 and rM; adsorbed aggregates varies between 0 and 1. We will use the value of rMobtained from Scamehorn et al. as a reasonable approximation of the density of surfactants in a layer of a surfactant aggregate. The counterion density in the bulk, po, will be assumed to be 0.1 ion/nm3, corresponding to a counterion concentration of 0.15 M; varying the counterion density had negligible effect on the solution to eq 12, as is discussed below. Approximate values for the remaining parameters of the model are given in Table II.' In Figures 4-6, a series of lines is drawn in the r N a , T , u, plane. These lines represent the solution to eq 12 for A$ = -4. Each point on these lines is then a point a t which the work of transporting a surfactant ion from a hemimicelle to an admicelle would be zero. For the same values of the other parameters, at a higher surface charge density we would calculate A(mw/kT) < A$, and the surfactant ion would be at a lower chemical potential in an admicelle; at a less positive surface charge density, we would calculate A(mw/kT) > A$, and the surfactant ion would be at a lower chemical potential in a hemimicelle.
2350 The Journal of Physical Chemistry, Vol. 92, No. 8, 1988
,
20.0
t
c
2.0
D
2.2
E
2.27
2.0 2.2 2.27
t
i
E
\ \
1o.cp
rk,T/r,
IFractionol counterinn a i n a i n g
on
.
t h e t o o l a y e r 01 an aamicoiie)
Figure 6. Effect of counterion binding on the top layer of the surfactant aggregates on the surface charge density at which the chemical potential of the surfactant ion would be the same in an admicelle as in a hemimicelle when the hydrocarbon tails of the two layers of the admicelle completely interpenetrate. Counterion bindings to the lower layers are assumed equal.
In Figure 4 us is plotted versus r N a , T / r M (the fractional counterion binding to the top layer of the admicelle) for five different values of rNa,B = r’Na,$ (counterion adsorption densities in the hemimicelle and in the lower layer of the admicelle of 0.5, 1.0, 2.0, 2.2, and 2.27 counterions/nm2). These counterion adsorption densities correspond to fractional counterion binding to the hemimicelle and to the bottom layer of the admicelle of 0.22, 0.44, 0.87, 0.96, and 0.99, respectively. The plot indicates that at lower values of counterion binding in the surfactant layer next to the surface, the chemical potential of the surfactant ions would be lower in the hemimicelle structure for all reasonable values of the surface charge density. For example, for r N a , B / f M = r’Na:B/.rM = 0.22 (line A, 0.5 adsorbed counterion/nm* In the hemmeelle and in the lower layer of the admicelle) and rNa,T//rM = 0.22 (0.5 adsorbed counterion/nm2 in the top layer of the admicelle), the chemical potential would not be lower in the admicelle unit the surface charge density on the patch of the mineral surface below the aggregate exceeded +3.6 e/nm2. Increasing r N a , B and r’Ng,B to 2.2 molecules/nm2 (line D, 0.96 fractional counterion binding in hemimicelle and in the lower layer of the admicelle) would indicate that an admicelle would be favored over a hemimicelle at a surface coverage of only + 1.4 e/nm2. Increasing r N a T also to 2.2 moIecuIes/nm2 (0.96 fractional counterion binding in the top layer of the admicelle) reduces the surface charge necessary to reach the transition point to +O. 17 e/nm2. At fractional bindings of 0.99 in all layers of the aggregates, the transition from hemimicelle to admicelle is found to occur at a surface charge density of only 0.03 e/nm2. The solutions to eq 12 plotted in Figure 4 assume the same counterion binding on the bottom layer of the hemimicelle and of the admicelle. If the counterion binding on the top layer of the admicelle is less than 100%(rNa,T < r M ) , eq 12 also indicates that the electrical potential in the bottom layer of the admicelle will be more negative than the electrical potential in the headgroup region of the hemimicelle; thus, we would expect the counterion binding in the lower level of the admicelle to be higher than the counterion binding in the head-group region of the hemimieelle if the aggregates were on patches of the surface having the same surface charge density. In Figure 5 we have plotted solutions to eq 12 for A+ = -4 with r’Na,B reduced by 10%relative to r N a , B (lines H and I). This exercise results in a significant reduction of the value of the surface charge density necessary for admicelle formation to occur. For example, for a counterion binding of 0.87 in the bottom layer of the admicelle, we find from Figure 4 that at a counterion binding of 0.8 in the upper layer of the admicelle, a surface charge density of +0.72 e/nm2 is necessary for admicelle formation to be favored over hemimicelle formation. With, however, a reduction in the counterion binding in the head-group region of the hemimicelle of lo%, from 0.87 to 0.79, the surface charge density at which the admicelle is
Yeskie and Harwell favored decreases to +0.17 e/nm2 (line H). While +0.72 e/nm2 is near the upper limit of the surface charge densities reported by Parks and James, +O. 17 e/nm2 is a very reasonable figure. For a fractional counterion binding of 0.96 in the bottom layer of both structures (line D, redrawn from Figure 4) and at a fractional counterion binding of 0.8 in the upper layer of the admicelle, a surface charge density of +OS0 e/nm2 is necessary for admicelle formation to be favored over hemimicelle formation. Reducing the counterion binding in the head-group region of the hemimicelle by 10%(line I) results in a reduction of the surface charge density at which the admicelle is favored to -0.10 e/nm2; Le., the chemical potential of the surfactant ion would be lower in the admicelle even at a slightly negative surface charge density. Conversely, when rNa,B (the counterion adsorption density in the lower layer of the admicelle) is reduced by 10%relative to r’Na,B (the counterion adsorption density in the lower layer of the hemimicelle), it can be seen from lines F and G in Figure 5 that this causes a significant increase in the surface charge density at which the chemical potential of the surfactant ion becomes lower in the admicelle. For example, at a fractional counterion binding in the upper layer of the admicelle of 0.80 ( r N a , T = 1.8 counterions/nm2), the surface charge density at which the chemical potential of the surfactant ion is lower in the admicelle increases from +OS0 e/nm2 for bindings of 0.96 (2.2 counterions/nm*, line D) on the lower layers of both structures to +1.3 e/nm2 for a binding of 0.86 (2.0 counterions/nm2) on the lower layer of the admicelle (line G). Thus, a reduction in counterion adsorption density in the admicelle’s lower layer of only 0.2 counterions/nm2 requires an increase in surface charge density of +0.8 e/nm2 for the admicelle to still be the structure in which the surfactant ion would be at the lower chemical potential. In Figure 2 the structure of the admicelle was depicted such that there was no overlapping of the hydrocarbon tails of the surfactants. This configuration has been suggested by Bisio et aL8 from a study of the effects of surfactant adsorption on the pressure drop through the pores of a track-etched membrane. In Figures 4 and 5 all curves corresponded to the case of an admicelle with a hydrocarbon core equal in thickness to twice the length of the hydrocarbon tail of the surfactant. In Figure 6 we have plotted curves correspanding to complete overlapping of the hydrocarbon tails. With all other conditions equal, the chemical potential of the surfactant molecule is now lower in the admicelle only at much higher values of the surface charge density. The effect is even more dramatic than that of increasing the counterion binding in the bottom layers of the aggregates. For example, at a fractional counterion binding in all layers of the admicelle and the hemimicelle of 0.96 (r’Na,B = r N a , B = r N a , T = 2.2 e/nm2), the surface charge density at which the chemical potential of the surfactant ion is lower in the admicelle increases from +0.17 e/nm2 (line D, Figure 4) to 1.71 e/nm2 (line D, Figure 6) as the thickness of the hydrocarbon core is decreased from twice the length of the hydrocarbon tail ( L = 2&) to equal the length of the hydrocarbon tail ( L = &). Only at the highest possible values of the counterion binding on all layers of the aggregates does the surface charge density necessary for admicelle formation decrease into the range of feasible values. For example, at r’Na,B = rNa,B f rNa,T = 2.27 e/nm2, which corresponds to a fractional counterion binding of 0.99 on all layers, the surface charge density necessary for the chemical potential of the surfactant to be lower in the admicelle is only 0.10 e/nm2. It is interesting to note that this indicates that the electrical work necessary to compress the admicelle from the configuration depicted in Figure 1 to a configuration in which the tails of the top layer penetrate between the tails of the bottom layer is larger than can reasonably be obtained from any possible increase in hydrophobic contribution to aggregate formation, which would result from some corresponding decrease in area of hydrocarbon/water contact in the compressed configuration. It has already been noted that the value of A@ used in the calculations plotted in Figures 4-6 is an average of the estimated maximum and minimum values of the hydrophobic contribution to the hemimicelle/admicelle transition. All the calculations shown in these figures were repeated for A,$ = -2 and A,$ = -6 to
The Hemimicelle/Admicelle Transition determine the sensitivity of the calculations to this approximation. This change resulted in a slight shift either to higher values of the surface charge density for A 4 = -2 or to lower values for A# = -6, but by only 0.008 f 0.0004 e/nm2, for the lines in Figures 4 and 5. The respective shifts in Figure 6 were somewhat higher (0.1872 f 0.0002 e/nm2), but the range of values for the surface charge is also much higher for this case. The lines were typically shifted the same amount over the entire range of fractional counterion bindings in the top layer of the admicelle. In short, electrostatics dominate the hemimicelle/admicelle transition, and the conclusions of this study are not highly dependent on an extremely accurate estimate of the hydrophobic contribution to the transition. The effects on the hemimicelle/admicelle transition of two other variables, 0 (the thickness of the head-group regions of the aggregates) and po (the concentration of counterions in the supernatant), were also examined. The effect of changes in po was negligible. Changing po from 0.15 M counterion concentration to 0.001 M added counterion generally resulted in a increase in the surface charge density of much less than 1%. This is because the procedure adopted here allows the electrolyte concentration to affect the chemical potential of the surfactant ion only by compressing the diffuse part of the electrical double layer. Since the potential drop across the diffuse double layer is almost the same in both the admicelle model and the hemimicelle model, po has little effect on the surface charge density at which the hemimicelle/admicelle transition occurs. If the electrolyte concentration were allowed to affect the counterion binding, the results might be somewhat different. In Figures 4-6 the value of 0 was equal to 2.0 nm; this figure was arrived at from considerations of steric constraints on the packing of sulfate groups and hydrated counterions into the head-group region.’ Ignoring these steric constraints and repeating the calculations for p = 1.O nm produced some interesting results: The lines in Figure 4 shifted only slightly, the surface charge density changing by only f0.02 e/nm2. For those lines in Figure 5, however, for which the counterion binding on the bottom layer of the admicelle was not equal to the counterion binding on the hemimicelle, the lines were all shifted by approximately 0.4 e/nm2; lines H and I were lowered by 0.40 and 0.44 e/nm2, respectively, and lines F and G were raised by 0.37 and 0.40 e/nm2. While these results significantly increase the range of bindings and surface charge densities for which the admicelle is the favored structure, they still do not eliminate the possibility of the formation of hemimicelles. The effect of reducing p is, of course, exactly equivalent to that of increasing the dielectric constant in the head-group regions, tT = tB. The most important conclusion to be drawn from this part of the study, then, is that the overall conclusions, relative to the likelihood of the formation of an admicelle rather than a hemimicelle, are not significantly affected by possible errors in estimates of 0, tT, and tB. It does not seem possible for the thickness of the head-group region to be more than trivially different from the values examined here. Only if the dielectric constant of the head-group regions were to turn out to be significantly greater than 6to could the hemimicelle structure be ruled out from this study. When, however, p / q is reduced simultaneously with a reduction of L (the thickness of the hydrocarbon core of the admicelle) to (the length of the hydrocarbon chain of the surfactant), a dramatic change occurs. All the lines in Figure 6 then have a positive slope, and almost all of the solutions correspond to negative values of the surface charge density. If the admicelle is compressed and the dielectric constant of the head-group regions is increased by only a factor of 2 from that in Table 11, then a surfactant monomer in the admicelle structure will have a lower chemical potential than a surfactant monomer in the hemimicelle structure, for almost all possible combinations of binding and surface charge density. It should be pointed out, though, that this conclusion is relatively independent of the hydrophobic contribution to the transition from hemimicelle to admicelle and is still a function of the electrostatics. What has happened in this last case is that the improved shielding of the head groups from one another and
The Journal of Physical Chemistry, Vol. 92, No. 8, 1988 2351 the corresponding improved shielding of the two head-group layers of the admicelle from one another make it so much more favorable to form an admicelle than a hemimicelle that only at negative values of the surface charge will a hemimicelle form instead of an admicelle.
Conclusions It is probably good to begin this section with a reminder that the equations derived in this paper do not address whether or not any surfactant aggregate will in fact form at a given set of conditions. They only address whether a monomer in a hypothetical aggregate at those conditions would be at a lower chemical potential in a hemimicelle or in an admicelle. The most important conclusion that can be drawn from this study is that despite the importance of hydrophobic interactions in the formation of surfactant aggregates at the solid/solution interface, electrostatic interactions govern the transition from the hemimicelle structure to the admicelle structure. High counterion bindings, high surface charge densities, and higher dielectric constants all favor the formation of an admicelle before the formation of a hemimicelle. Low counterion bindings, low surface charge densities, and low dielectric constants all favor formation of a hemimicelle at a lower surfactant concentration than necessary for an admicelle to form. Bindings have been reported that are as high or higher than those necessary for surfactant monomers to be at a lower chemical potential in an admicelle than in a hemimicelle, based on the results of this model. Bindings of 0.85 are reported for SDS micelles, and since the sulfate group density is significantly higher in the adsorbed surfactant aggregate than in the micelle (0.44 versus 0.6 nm2/molecule), one would expect counterion bindings to be at least as high on the adsorbed aggregates. Studies of counterion bindings on aggregated adsorbed surfactants in our own laboratory have recently indicated counterion bindings on admicelles varying from 0.5 to upward of 0.95 and frequently indistinguishable from 0.99.19All of these indicate that it is feasible for the admicelle to form in lieu of the formation of a hemimicelle over a large range of physically accessible conditions. The admicelle structure becomes more favored as one moves away from the point of zero charge of the mineral surface to higher surface charge densities. As one moves in the direction of the point of zero charge, conditions are more favorable to hemimicelle formation. This generalization can be wrong, however, depending on the counterion bindings observed at these conditions. It is especially interesting that the transition point at which an admicelle will form at a lower chemical potential than a hemimicelle falls within the range of feasible values of counterion binding and of surface charge density for most sets of parameter values examined. Only for relatively high values of the dielectric constant in the head-group region (6to or greater) will admicelles almost always be the first structure to form. It might be expected that one of the proposed structures would provide a lower chemical potential for the monomers over almost the entire range of feasible parameters. Instead, the model indicates that both structures are feasible for providing the lowest chemical potential for the monomers over wide ranges of the actual conditions. This is basically because the maximum value of the hydrophobic contribution to the transfer of a monomer from a hemimicelle to an admicelle is small relative to the electrostatic interactions. Were the hydrophobic contribution much larger or the magnitude of the electrostatic interactions much smaller, then admicelles would always form at lower surfactant concentrations than hemimicelles. The fact that the lines for the hemimicelle/admicelle transition pass through the regions of physically possible values for the model parameters also suggests that in at least some systems the actual aggregate structure may be intermediate between the hemimicelle structure and the admicelle structure, just as was found by Scamehorn et a1$ thus, if one could view the structure of an individual aggregate over a period of time, it would appear to flicker between the extremes of the two structures examined in this study and might appear on a time average to have, for example, 1.5 or 1.8 layers as opposed to either 1 or 2 layers.
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J . Phys. Chem. 1988,92, 2352-2356
It should also be pointed out that if the mineral surface on which the surfactant is adsorbing is heterogenous, systems should exist in which portions of the surface are covered with admicelles while other portions of the surface are covered with hemimicelles. At some surfactant concentrations the aggregates formed on the most highly charged patches might be bilayered, while the aggregates formed on the less highly charged patches might be hemimicelles. It is, of course, also possible for the second layer to form subsequent to the formation of the first layer, as has been recognized by many workers. Thus, the assumption of Harwell et al. that only admicelles would form on local patches is shown by this study to be unwarranted-based on our current knowledge of these systems-and the model presented by them should be modified to allow for formation of both structures in order to be generally applicable to surfactant adsorption on mineral oxides.
Finally, it should be reiterated that this work supports the interpretation of Bisio et al. that the tail groups of the upper layer of the bilayered aggregate do not significantly interpenetrate the tail-group region of the bottom layer. This conclusion is based on a consideration of the electrostatic effects relative to the hydrophobic contribution, regardless of any possible steric considerations. Only in the case of dielectric constants in the head-group regions being significantly higher that those in Table I1 does this conclusion not hold.
Acknowledgment. This material is based upon work supported by the National Science Foundation under Grant No. CPE8404676. Financial support from the Mobil Foundation, Arc0 Oil and Gas Co., and Shell Research and Development Corp. is also gratefully acknowledged.
Induced Circular Dichroism in Nonlntercalative DNA-Drug Complexes. Sector Rules for Structural Applications Mikael Kubista,* Bjorn Akerman, and Bengt Norden* Department of Physical Chemistry, Chalmers University of Technology, S-412 96 Goteborg, Sweden (Received: November 5, 1987)
An algorithm based on an idealized coupled-oscillator model for calculating the induced circular dichroism (CD) of an achiral chromophore externally bound to a chiral macromolecule has been developed and applied to provide CD sector rules for nonintercalativeDNA complexes. For example, a drug molecule bound in either of the grooves of DNA is predicted to exhibit positive induced CD for electric dipole allowed transitions (in the near-UV or visible region) whose transition moments are pointing along the groove and negative CD for transitions with moments perpendicular to the helix axis. A fair agreement is noticed with experimental CD data of nonintercalators such as netropsin, hoechst 33258, 4’,6-diamidino-2-phenylindole, and two triphenylmethane dyes.
Introduction Optical activity, generally measured as circular dichroism (CD), induced in achiral chromophores upon interaction with a chiral substrate has been studied for a variety of systems and has for a long time been used as a tool for revealing binding between drugs and biopolymers. However, despite its great relevance, the problem of relating induced CD to binding geometry has not yet been given any general solution in either theoretical or empirical terms. One reason is the complicated electric and magnetic nature of optical activity which requires molecular geometries and wave functions to be known or represented with high order of accuracy. Thus relatively small symmetry-breaking perturbations could give considerable contributions to optical activity. The C D induced in an achiral chromophore upon interaction with a chiral substrate is usually ascribed to one (or several) mechanism(s), each responding to a more or less idealized situation: the chromophore molecule may be perturbed to adopt a chiral (equilibrium) conformation or it may (with or without retained achiral geometry) gain optical activity through electric and/or magnetic interact i o n ~ . ’ - ~Using perturbation theory the predicted C D may be classified as arising from p k c o ~ p l i n g , p~ .m~ c o ~ p l i n g ,or ~ . the ~ one-electron mechanism.2-6 All these mechanisms are included in cakulations that have been made lately on the CD of DNA (1) Tinoco, I, Jr. Adu. Chem. Phys. 1962, 4 , 113. (2) Schellman, J. A. Acc. Chem. Res. 1968, I , 144. (3) Charney, E. The Molecular Basis of Optical Activity; Wiley: New York, 1979. (4) Mason, S. F. Molecular Optical Activity and the Chiral Discriminations;Cambridge Univ. Press: Cambridge, U.K., 1982. (5) Kirkwocd, J. G . J . Chem. Phys. 1937, 5, 479. (6) Condon, E. U.; Altar, W.; Eyring, H. J. Chem. Phys. 1937, 5 , 753.
0022-3654/88/2092-2352$01.50/0
and other biopolymer^.'^^ However, a limiting factor has been lack of detailed knowledge about the electronic states of the interacting moieties, such as directions and magnitudes of electric and magnetic transition moments. Whereas the electric moments may be readily determined by using polarized-light spectroscopy on oriented molecules,9-1z the magnetic dipole transitions are difficult to obtain experimentally and one is referred to quantum mechanical calculations which are generally hard to correlate in this respect with experiment. For this reason the pop coupling, Le., interactions between electric dipole allowed transitions in different, achiral chromophores, chirally disposed relative to each other, is the only mechanism that has found considerable structural applications in biopolymers. Still, the knowledge about electric transitions too is far from complete, and mostly limited to near-UV transition^.^^,^^ The most important case of coupling is the “exciton CD” of interacting, identical chromophores (Le., degenerate transition^).^ Exciton C D was early observed for dye molecules upon association to helical biopolymers (e.g., acridine orange on poly(g1utamic acid)),14evidencing interactions between adjacently bound dye molecules with chiral orientation relative to each other.15 Exciton CD, including vibronic exciton CD,I6 (7) Rizzo, E.; Schellman, J. A. Biopolymers 1984, 23, 435. (8) Woody, R. W. In The Peptides; Academic: Orlando, FL, 1985; Vol. 7, Chapter 2. (9) NordBh B. Appl. Spectrosc. Rev. 1978, 14, 157. (10) Matsuoka, Y.; Norden, B. J. Phys. Chem. 1982, 86, 1378. (1 1) Schellman, J. A,; Jensen, H. P. Chem. Reu. 1987, 87, 1359. (1 2) Michl, J.; Thulstrup, E. W. In Specrroscopy with Polarized Light; VCH Verlagsgesellshaft: Weinheim, West Germany, 1986. (13) Novros, J. S.;Clark, L. B. J . Phys. Chem. 1986, 90, 5666. (14) Stryer, L.; Blaut, E. R. J . .4m. Chem. SOC.1961, 83, 1411.
0 1988 American Chemical Society
