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Electrostatic energies in an ionic micellar solution in the mean spherical approximation
ELSEVIER
Physica A 231 (1996) 277-287
Electrostatic energies in an ionic micellar solution in the mean spherical approximation Y.C. Liu a'*, S.H. C h e n b aDepartment of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA bDepartment of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Abstract We present a method for calculating electrostatic energies of an ionic micellar solution under the conditions of quasi-spherical micelles and point-like counterions. It is based on a multicomponent mean spherical approximation (MMSA) theory with non-additive diameters. The partial structure factors can be evaluated analytically according to a method due to Khan and Ronis, and used to calculate the electrostatic energies. As an example, a micellar solution formed by Cesium Dodecyl Sulfate in water has been studied with small angle neutron scattering. The micellar size and effective charge are extracted from analysis of small angle neutron scattering data, and used to evaluate the internal energy and Gibbs free energy of the micellar solution.
1. Introduction Thermodynamics properties in an ionic micellar solution are usually formulated by partitioning the Gibbs free energy (or chemical potential) into hydrophobic, surface and electrostatic parts [1]. GTota 1 = G 1 q-- G 2 -t- G 3 ,
(1)
where G1 is the hydrophobic free energy, G2 the surface free energy, and Ga the electrostatic free energy. The formation and growth of micelle are due to a delicate balance a m o n g the three energies. The hydrophobic energy G1 describes the free energy advantage of segregation of the hydrophobic tails, and is best treated semi-empirically. A chemical potential difference of an alkane in hydrocarbon solvent and in water, obtained from accurate solubility measurements, varies linearly with he, the number of CH2 groups, by Ref. [2], #HC -- #aq = -- 10.2 -- 3.70 n¢ (KJ/mol). * Corresponding author. 0378-4371/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 5 ) 0 0 4 6 4 - 5
(2)
278
Y.C Liu, S.H. Chen / Physica A 231 (1996) 277-287
The hydrophobic interaction plays an important role in the formation of micelle and its equilibrium with monomeric surfactant molecules. The study of critical micelle concentration (c.m.c.) establishes a linear relation between nc and ln(cmc). The hydrophobic interaction is essentially determined by the chemistry of hydrocarbon tail of the surfactant molecule in aqueous solutions. The surface free energy G2 is mainly associated with the solvation energy of incorporation of the solvent molecules at the hydrophobic-hydrophilic interface, and with the steric interactions between the polar head groups. Dominated by the geometry and packing of the head groups, the surface energy is most responsible for the shape of the micelle. A study of the one-, two- and three-dimensional growth of micelle, or correspondingly the cylindrical, lamellar, and spherical shapes, has been discussed by Tanford in great details [2]. The third part is the electrostatic energy G3, which has been addressed the earliest and yet is the least understood. The electrostatic interaction is the sum of Coulombic attractions and repulsions among all the ionic species in a micellar solution. The role of counterions must not be ignored, in contrast with the hydrophobic and surface free energies. In a classic model an electrical double layer is formed by the ionic head groups and their counterions, which are partly bound to the hydrophilic surface of micelle and partly constitute a diffusive ion cloud surrounding the micelle. Isralachvilli used a simplified capacitor model [4] to calculate the electrostatic energy involving two adjustable parameters, namely, the surface charge density and separation distance between the plates. This treatment totally neglected the influence of the presence of other capacitors in the vicinity. Other similar treatments using the Debye-Huckel approximation were largely limited by the validity of the linearization of the Poisson-Boltzmann equation. In order to give an accurate expression of G3, one must know the intermicellar interaction as well as the micelle-counterion and counterion-counterion interactions. The major problem in a quantitative description lies in the difficulties in analytical and numerical evaluations of the interactions and correlations for all ionic species. The investigation of micelle-micelle interaction has yielded a formulation of onecomponent potential in a Yukawa form. The intermicellar potential has been successfully utilized in extraction of micellar structure factors and pair correlation functions from the intensity profiles in small angle neutron and X-ray scattering (SANS and SAXS) experiments. The counterion condensation and distribution can be calculated numerically by solving the Poisson-Boltzmann equation, or analytically by the multicomponent mean spherical approximation (MMSA) theory [3]. Khan and Ronis calculated excess reference free energy per unit volume by integration of thermodynamic gradients with respect to the total density Pv and a charging parameter #. The authors constructed contour maps of excess free energies of micelle suspensions and located the minimum energy points for the corresponding effective charges and non-additive diameters of the micelle. However, their approach failed to predict the correct effective charge and to estimate the volume fraction for micellar solutions.
Y.C Liu, S.H. Chen / Physica A 231 (1996)277-287
279
More recently, Senatore calculated the thermodynamics properties and the liquid-liquid transition in polyelectrolyte solutions with large size and charge asymmetry [5] based on the Khan-Ronis theory. In his work, comparison was made for the structure factors obtained from rescaled MSA and RY-HNC types of calculations. Thermodynamic properties were evaluated through the compressibility equation of state and the energy route from MSA. Senatore showed that energy obtained from MSA agrees reasonably well with HNC (hyper-netted chain) calculation. However, this work remains a theoretical exercise. The parameters used in the theory are not specified in terms of the physical parameters in a practical micellar system. Thus, a link between the theoretical framework and a micellar solution is not available. As shown in our previous publication 1-6] the MMSA theory can be effectively used to calculate the partial structure factors and to analyze SANS and SAXS intensity distribution functions. Under the conditions allowing rescaling the micelle diameter and the penetrable sphere effect, an analytical formulation has been given explicitly. In this paper, we give a full expression of electrostatic energy for spherical micelles based on the MMSA theory. We also demonstrate a method to apply the structural parameters extracted from experiments for evaluation of the internal energy and Gibbs free energy of a micellar solution.
2. Electrostatic energy of two-component Ionic system
We focus on a two-component system composed of spherical macroions of charge ZM, and their monovalent counterions, while the latter are taken as point-like particles. In the absence of the salt particles, all the structure factors and the correlation functions of an ionic solution at a given temperature are determined explicitly by three parameters: the diameter cM, the effective charge ZM and the number density PM of the macroions. All other important parameters, such as the volume fraction of the macroions r/, Debye screening length, and the number density of the counterions Pc, can be deduced from the three parameters by internal relationships and the charge neutralization condition. Evaluation of electrostatic energy starts with summing the internal energies of all species in the solution. The total internal energy U and total Gibbs free energy Ga can be evaluated once the correlation functions are calculated with MMSA theory. The total configurational internal energy per unit volume of ionic solution can be written in terms of the pair correlation function of the particles and their corresponding pair-wise potentials [7]. U = 2 i,j PiPj
f
droij(r)uij(r),
(3)
where uq is the interaction potential between the ith and jth particle at the distance r. Under the conditions of mean spherical approximation, the direct correlation function
280
E C Liu, S.H. Chen / Physica A 231 (1996) 277 287
outside diameter a u is related to the interaction potential, gij(r) = 0
for rij < a i j ,
Cij(r ) = --
fluu(r ) for
rij > o'ij ,
(4)
where fl = 1~ks T and kB is the Boltzmann constant, and T the absolute temperature. So a dimensionless internal energy normalized by thermal energy can be written as
- flU = ~ ~. PiP, drgij(r)Cu(r),
, i,j
(5)
f
where gij(r)= hq(r)+ 1 is pair correlation function. The above equation can be decomposed into two parts. -- f l U =
~ PiP,
drCij(r) +
PiP,
drhij(r)Cij(r) .
(6)
Define h[j(Q) = ~ h u ( Q ) and cSj(O) = x~ipjci,(Q), where hij(Q) and cu(Q) are the Fourier transforms of hu(r ) and Cu(r ). The first part of the internal energy is
I = ~ .~. PiP, drCu(r) = ~ .~. ~ c T , ( Q t,J
¢
=
0).
(7)
/,J
For a two-component system containing macroions and counterions, I can be evaluated as 1
s
I = ~ [pMCMM(0) -4- 2 ~ c h c ( 0 )
+ pcc~¢(0)].
(8)
The leading terms of c~ diverge at Q = 0 in the way of 1/Q 2. The origin of the singularity lies in the divergent Coulomb potentials between the charged particles as their distances approach zero. However, it can be shown that lim
cs
PM MM(Q) -- lim
pocSo(Q)
--
1.
(9)
Thus, the divergent potentials are canceled out. The remaining term is I = ~
pM(1 - l/S(0)) + - f p M p o F M R 3 o
.
(10)
The second part of the internal energy can be evaluated by the direct correlation functions as I I = l ~2 i.j P'PJ f drhu(r)Cu(r) = ( 1~) 3 1 ~ 2i,jodQh~J(Q) ( cT,(Q).
(11)
Y.C. Liu, S.H. Chen/ Physica A 231 (1996) 277-287
281
For the two-component system, it can be evaluated by a three-dimensional integration of the macroion-macroion, macroion-counterion and counterion-counterion correlation functions. oo
II = ~
Go
Q MM(Q)MM(Q) Q + 2 0
Q Mc(Q) Mc(Q)dQ 0
oo
+
Q
co(Q) ¢¢(Q)dQ ,
(12)
0
where the direct correlation functions can be evaluated analytically following the equations given in the Appendix. Summing the two contributions, the total internal energy density is given by o0
-
flU = ~1 pM(1 -- 1/S(O)) +
~ pMpoFMR~¢+ ~ 21 I f
Q2h~M(Q)C~M(Q)dQ 0
ct3
oo
0
0
Given the internal energy, it is relatively easy to calculate other thermodynamics functions. Hoye and Stell [8] studied the expressions of the thermodynamics relations of MSA for simple fluids and then generalized their results to mixtures. The excess Gibbs free energy, or the excess chemical potential can be evaluated once the excess reference free energy is obtained from the internal energy. The total Gibbs free energy density of the system can be evaluated as - fiG = 1_2 x~PiCTi(O) - flU 2i,j
(14)
Considering the two parts described above in the total internal energy, it is simply - fig = 21 + II. The average Gibbs free energy per macroion particle GM can be evaluated as the following:
-/~GM - 24nr/MM -0
+
Q hcc(Q)ccc(Q)dQ
0
+
1--~-~
3
o The internal energy and Gibbs free energy in terms of kBT can be averaged by the total number of ionic species instead of the number of macroions.
Y.C. Liu, S.H. Chen / Physica A 231 (1996) 277-287
282
3. Electrostatic energy of ionic micellar solution
The MMSA theory provides a convenient way of calculating the free energies in an ionic system. The energy calculation works extremely well at large macroion--counterion charge asymmetry where numerical methods such as HNC often break down. At a very low charge ratio where the concentration of macroions is close to that of counterions, the accuracy of the calculation is limited by the validity of the Yukawa form of intermacroion potential. The partial structure factors and direct correlation functions can be evaluated analytically. The average internal energy and Gibbs free energy per particle (including macroions and counterions) is calculated as functions of temperature, and the diameter, charge and volume fraction of the macroion. As an example, the average internal energy and Gibbs free energy per particle are shown in Fig. 1 for an ionic system with macroion volume fraction of 10%, macroion diameter 50/~ at room temperature 25 °C. Fig. 2 shows a series of Gibbs free energies per particle at different macroion volume fractions, while the macroion diameter is fixed at 50 ~ and the temperature at 25 °C. It is clear that higher charge asymmetry corresponds to more negative Gibbs free energy. Similarly, a series of Gibbs free energies per particle at different macroion
0
-2
-4
-6 (-9 - 8 -10
-12 -14
-16
-18
t
I
60
t
do
Z Fig. 1. Average internal energy U (dashed line) and Gibbs free energy G (solid line) per particle in the unit of kBT as a function of macroion effective charge Z for an ionic system with macroion volume fraction of 10%, macroion diameter 50 A at room temperature 25 °C. Average energy is defined as total energy divided by total n u m b e r of particles, including macroions and counterions. Circles and crosses are values of internal energies and chemical potentials by Senatore using MMSA.
Y.C. Liu, S.H. Chen / Physica A 231 (1996) 277-287 0 .. -2
i ~~~.
J
i
i
i
i
283
i
""""...
68
~
(5 -10
" "- ~ ~ ~
-12
~ x
~
''"'".
~-~
~~~ ~
-14
~\~
~
~
-..
-16
"
-18 -20
J 20
30
' 40
' 50
6'0
70
' 80
9LO
100
Z Fig. 2. Gibbs free energy per particle G at various macroion volume fractions, while macroion diameter is fixed at 50 ~, and temperature at 25 °C. Solid line corresponds to macroion volume fraction of 10%. Dotted, dash-dotted and dashed lines correspond to macroion volume fractions of 1%, 5 % and 15%, respectively.
diameters at a fixed volume fraction of 10% and temperature of 25 °C is illustrated in Fig. 3. One can see that macroion of larger size has less negative Gibbs free energy. This indicates that a two-component system with larger macroions requires more counterions to stabilize and the stabilization energy can be evaluated quantitatively. The following scheme is suggested to obtain the accurate free energy of a real micellar system. The accuracy of energy evaluation is highly sensitive to the accuracy of micelle-micelle interaction potential. Although it is reasonable to histogram the free energy map and locate the configuration corresponding to the lowest energy, such a theoretical approach is unlikely to describe the real system. The failure of the theoretical approach is not because the assumptions and approximations made in the theory are poor. In fact, the problem is largely due to the complexity of the micellar system. The equilibrium configuration is at the lowest total free energy point as discussed in the introduction, rather than the most negative point of electrostatic energy alone. The most accurate energy calculation should be carried out based on a successful extraction of intermicellar structure factor from experiments. The accuracy of the theory sets in from the formulation of interparticle interactions and therefore determines the final accuracy of the energy evaluation. The internal energy and Gibbs free energy terms are formulated rigorously based on the mean spherical approximation through energy route. At conditions when MSA cannot hold, the energy formulation also breaks down. The assumption of point-like counterions simplifies the MMSA theory and allows analytical solutions of the direct
284
Y.C. Liu, S.H. Chen / Physica A 231 (1996) 277-287 0
:~-L ...............
i
i
1
i
i
-5
-10
(.9 -15
-20
-25 I
20
' 30
' 40
5'o
' 60 Z
7'0
8~0
90
100
Fig. 3. Gibbs free energy per particle G at various macroion diameters, while macroion volume fraction is fixed at 10%, and temperature at 25 °C. Solid line represents macroion diameter of 50 ~,. Dash-dotted, dashed and dotted lines correspond to macroion diameters of 30, 70, and 100 Pt, respectively.
correlation functions and partial structure factors. The analytical formulation has great advantages over the cumbersome numerical methods since it is suitable for fitting experimental data. On the other hand, due to the finite size of the small particles, the point-particle assumption may also introduce some inaccuracy especially in the counterion-containing partial structure factors at small Q range. The approach to calculating electrostatic free energy of micellar system is different from the approach to surface electrostatic energy based on a Poisson-Boltzmann equation [9, 10] because of the different reference zero-energy states. In the ionic liquid theory, the zero energy correspond to infinite separation of ions. However, in the Poisson-Boltzmann integration methods, the reference zero-energy state corresponds to a zero charge state equation. Finally, although a two-component system is discussed here, it should be pointed out that in a multi-component system the contributions of the partial structure factors and energy terms due to salt ions are entirely recoverable based on a similar scheme.
4. Electrostatic energy for cesium dodecyl sulfate miceilar solution We show the method of calculating the electrostatic internal energy and Gibbs free energy for a two-component system with macroions and their counterions, using the MMSA theory. As an example, an ionic micellar system is studied and used to demonstrate the methodology for evaluating the electrostatic free energy.
Y.C. Liu, S.H. Chen / Physica A 231 (1996) 277-287
285
2sI u 1.5
g
0.5
t
O0
0.05 '
0i1
0.15 '
o
0.2 '
0.25
0.3
[~/A]
Fig. 4. Small angle neutron scattering data of 3% cesium dodecyl sulfate micellar solution at 40 °C. Solid line represents fitted curve. Data analysis shows that micelle has an average diameter of 52 A and charge of 25e.
We have studied a micellar solution made of 3% cesium dodecyl sulfate surfactant in deuterated water at 40 °C using SANS and SAXS. Careful analyses of SANS and SAXS data show that the micelle is composed of about 100 cerium dodecyl sulfate molecules with a dense hydrocarbon core and a hydrated outer-layer composed of about 10 solvent molecules per sulfate head group. The micelle has an average diameter of about 52 ~ and an average charge of - 25e. The volume fraction of the micelles in the solution is about 3%. SANS data and fitted curve of the micellar solution are shown in Fig. 4. The intermicellar structure factor SMM(Q)was extracted from the SANS spectrum. Micelle-counterion and counterion-counterion structure factors SMc(Q)and See(Q)were further calculated using the formulae in the Appendix, and proved to be reasonably accurate by additional SAXS experiment on the same system. The partial structure factors obtained are illustrated in Fig. 5. Once the effective charge, diameter and volume fraction of the micelles are known, electrostatic internal energy and Gibbs free energy of the entire micellar system can be calculated easily. Especially, the Gibbs free energy (or chemical potential) per ionic species turns out to be - 1.52kBT. The total electrostatic Gibbs free energy of the 3% cesium dodecyl micellar solution is - 0.0676 KJ/L. Acknowledgements
This research is supported by a grant from the Materials Science Division of USDOE.
Y.C Liu, S.H. Chen I Physica A 231 (1996) 277-287
286 1.8
i
1.6l'.... 1.4 ". 1.2
o..p./,,
0.I/ ',, 0.2ri
~
0
0.05
0.1
0.15
O
0.2
ILIA]
0.25
0.3
0.35
0.4
Fig. 5. Partial structure factors of cesium dodecyl sulfate micellar solution. Solid line, dashed line and dotted line represent micelle-micelle structure factor SMM(Q),micelle-counterion structure factor SM0(Q) and counterion-counterion structure factor See(Q),respectively.
Appendix. Partial structural factors for multi-component system An ionic system can be reduced into a three-component system composed of macroions (M), counterions (c) and neutral ions (n). Define Sij(Q) = 6ij + hTi(Q) with hTj(Q) = x ~ p j h u ( Q ) and c~j(Q) = x/~picij(Q). The partial structure factors are evaluated as: (Note that S(Q) = SMM(Q)), SM,(Q) = Chn(Q)S(Q) ,
(A.1)
Snn(Q) = 1 + [c~,(Q)] 2 S(Q),
(A.2)
Q2 SM~(Q) - k2 + Q2 C ~ ( Q ) S ( Q ) ,
(A.3)
Q2 S~,(Q) - k2 + Q2 c~(Q)c~n(Q)S(Q) ,
(A.4)
soo(Q) = k~ + Q------~+ k~ ¥ - &
(g.5)
[C~,~(Q)]~s(o,
c~c(Q) = 4rcx//~Mpc [FuR~d(QRMo) - LBZMZ~ c o s ( g R u j O 2 ) ] ,
(A.6)
Y.C. Liu, S.H. Chen / Physica A 231 (1996) 277-287
clan(Q) = 4~
(A.7)
p~Mp~F.R~j(QRM~),
C~M(Q) = 1 -- 1 / S ( Q ) -
(
287
)
[c~c(Q)] 2 + [c~.(Q)] 2 ,
(A.8)
where sin(x) - x cos(x) y(x) =
x~
(A.9) ,
F. = - 1/(1 - t/M.),
(A.IO)
kLBZMZc[2y exp(kRM¢)Cosh(kRM¢) - 1] F u - 27exp(kRMo)[sinh(kRMJ -- kRM~cOsh(kRMc)] + 1 + kRM~'
(A.I1)
with r/Mn = 4~pMR~o/3 and ~ = v e x p ( - k)/k, where k = X/4rtLBp~ZEaMM is an inverse Debye screening length, trMM is macroion-macroion diameter, and RM~ is macroion-counterion radius. LB = e2/ekB T is the Bjerrum length. Once S(Q) is known from analysis of scattering data, all the other partial structure factors and the corresponding partial correlation functions can be calculated. The macroion-macroion diameter trMM is extended by the rescaling, and the macroion-counterion diameter equals to the radius of the micelle. The counterions can come to the actual micellar surface and the non-additivity condition allows RMc ~< ½. This reflects the so-called penetrable sphere effect [6].
References [1] H. Wennerstorm and B. Lindman, Physics Reports, 52 (1979) 1. [2] C. Tanford, Hydrophobic Effects: Formation of micelles and Biological Membranes (Wiley, New York, 1973). [3] S. Khan, T.L. Morton and D. Ronis, Phys. Rev. A 35 (1987) 4295; S. Khan and D. Ronis, Mol. Phys. 60 (1987) 637. [4] J. IsralachviUi, Intermolecular and Surface Forces (Academic Press, New York, 1992). [5] G. Senatore, in: Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution, eds., S.H. Chen, J.S. Huang and P. Tartaglia, NATO ASI Series C369 (Kluwer Academic Publishers, Dordrecht, 1992) pp. 175. [61 Y.C. Liu, C.Y. Ku, P. NoLostro and S.H. Chen, Phys. Rev. E 51 (1995) 4598. [7] J.P. Hansen and I.R. McDonold, Theory of Simple Liquids (Academic Press, New York, 1988). [8] J.S. Hoye and G. Stell, J. Chem. Phys. 67 (1977) 439; L. Blum and J.S. Hoye, J. Phys. Chem. 81 (1977) 1311. [9] D.F. Evans and B.W. Ninham, J. Phys. Chem. 87 (1983) 2996. [10] Y.S. Chao, E.Y. Sheu and S.H. Chert, J. Phys. Chem. 89 (1985) 4862.
Physica A 231 (1996) 277-287
Electrostatic energies in an ionic micellar solution in the mean spherical approximation Y.C. Liu a'*, S.H. C h e n b aDepartment of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA bDepartment of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Abstract We present a method for calculating electrostatic energies of an ionic micellar solution under the conditions of quasi-spherical micelles and point-like counterions. It is based on a multicomponent mean spherical approximation (MMSA) theory with non-additive diameters. The partial structure factors can be evaluated analytically according to a method due to Khan and Ronis, and used to calculate the electrostatic energies. As an example, a micellar solution formed by Cesium Dodecyl Sulfate in water has been studied with small angle neutron scattering. The micellar size and effective charge are extracted from analysis of small angle neutron scattering data, and used to evaluate the internal energy and Gibbs free energy of the micellar solution.
1. Introduction Thermodynamics properties in an ionic micellar solution are usually formulated by partitioning the Gibbs free energy (or chemical potential) into hydrophobic, surface and electrostatic parts [1]. GTota 1 = G 1 q-- G 2 -t- G 3 ,
(1)
where G1 is the hydrophobic free energy, G2 the surface free energy, and Ga the electrostatic free energy. The formation and growth of micelle are due to a delicate balance a m o n g the three energies. The hydrophobic energy G1 describes the free energy advantage of segregation of the hydrophobic tails, and is best treated semi-empirically. A chemical potential difference of an alkane in hydrocarbon solvent and in water, obtained from accurate solubility measurements, varies linearly with he, the number of CH2 groups, by Ref. [2], #HC -- #aq = -- 10.2 -- 3.70 n¢ (KJ/mol). * Corresponding author. 0378-4371/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 5 ) 0 0 4 6 4 - 5
(2)
278
Y.C Liu, S.H. Chen / Physica A 231 (1996) 277-287
The hydrophobic interaction plays an important role in the formation of micelle and its equilibrium with monomeric surfactant molecules. The study of critical micelle concentration (c.m.c.) establishes a linear relation between nc and ln(cmc). The hydrophobic interaction is essentially determined by the chemistry of hydrocarbon tail of the surfactant molecule in aqueous solutions. The surface free energy G2 is mainly associated with the solvation energy of incorporation of the solvent molecules at the hydrophobic-hydrophilic interface, and with the steric interactions between the polar head groups. Dominated by the geometry and packing of the head groups, the surface energy is most responsible for the shape of the micelle. A study of the one-, two- and three-dimensional growth of micelle, or correspondingly the cylindrical, lamellar, and spherical shapes, has been discussed by Tanford in great details [2]. The third part is the electrostatic energy G3, which has been addressed the earliest and yet is the least understood. The electrostatic interaction is the sum of Coulombic attractions and repulsions among all the ionic species in a micellar solution. The role of counterions must not be ignored, in contrast with the hydrophobic and surface free energies. In a classic model an electrical double layer is formed by the ionic head groups and their counterions, which are partly bound to the hydrophilic surface of micelle and partly constitute a diffusive ion cloud surrounding the micelle. Isralachvilli used a simplified capacitor model [4] to calculate the electrostatic energy involving two adjustable parameters, namely, the surface charge density and separation distance between the plates. This treatment totally neglected the influence of the presence of other capacitors in the vicinity. Other similar treatments using the Debye-Huckel approximation were largely limited by the validity of the linearization of the Poisson-Boltzmann equation. In order to give an accurate expression of G3, one must know the intermicellar interaction as well as the micelle-counterion and counterion-counterion interactions. The major problem in a quantitative description lies in the difficulties in analytical and numerical evaluations of the interactions and correlations for all ionic species. The investigation of micelle-micelle interaction has yielded a formulation of onecomponent potential in a Yukawa form. The intermicellar potential has been successfully utilized in extraction of micellar structure factors and pair correlation functions from the intensity profiles in small angle neutron and X-ray scattering (SANS and SAXS) experiments. The counterion condensation and distribution can be calculated numerically by solving the Poisson-Boltzmann equation, or analytically by the multicomponent mean spherical approximation (MMSA) theory [3]. Khan and Ronis calculated excess reference free energy per unit volume by integration of thermodynamic gradients with respect to the total density Pv and a charging parameter #. The authors constructed contour maps of excess free energies of micelle suspensions and located the minimum energy points for the corresponding effective charges and non-additive diameters of the micelle. However, their approach failed to predict the correct effective charge and to estimate the volume fraction for micellar solutions.
Y.C Liu, S.H. Chen / Physica A 231 (1996)277-287
279
More recently, Senatore calculated the thermodynamics properties and the liquid-liquid transition in polyelectrolyte solutions with large size and charge asymmetry [5] based on the Khan-Ronis theory. In his work, comparison was made for the structure factors obtained from rescaled MSA and RY-HNC types of calculations. Thermodynamic properties were evaluated through the compressibility equation of state and the energy route from MSA. Senatore showed that energy obtained from MSA agrees reasonably well with HNC (hyper-netted chain) calculation. However, this work remains a theoretical exercise. The parameters used in the theory are not specified in terms of the physical parameters in a practical micellar system. Thus, a link between the theoretical framework and a micellar solution is not available. As shown in our previous publication 1-6] the MMSA theory can be effectively used to calculate the partial structure factors and to analyze SANS and SAXS intensity distribution functions. Under the conditions allowing rescaling the micelle diameter and the penetrable sphere effect, an analytical formulation has been given explicitly. In this paper, we give a full expression of electrostatic energy for spherical micelles based on the MMSA theory. We also demonstrate a method to apply the structural parameters extracted from experiments for evaluation of the internal energy and Gibbs free energy of a micellar solution.
2. Electrostatic energy of two-component Ionic system
We focus on a two-component system composed of spherical macroions of charge ZM, and their monovalent counterions, while the latter are taken as point-like particles. In the absence of the salt particles, all the structure factors and the correlation functions of an ionic solution at a given temperature are determined explicitly by three parameters: the diameter cM, the effective charge ZM and the number density PM of the macroions. All other important parameters, such as the volume fraction of the macroions r/, Debye screening length, and the number density of the counterions Pc, can be deduced from the three parameters by internal relationships and the charge neutralization condition. Evaluation of electrostatic energy starts with summing the internal energies of all species in the solution. The total internal energy U and total Gibbs free energy Ga can be evaluated once the correlation functions are calculated with MMSA theory. The total configurational internal energy per unit volume of ionic solution can be written in terms of the pair correlation function of the particles and their corresponding pair-wise potentials [7]. U = 2 i,j PiPj
f
droij(r)uij(r),
(3)
where uq is the interaction potential between the ith and jth particle at the distance r. Under the conditions of mean spherical approximation, the direct correlation function
280
E C Liu, S.H. Chen / Physica A 231 (1996) 277 287
outside diameter a u is related to the interaction potential, gij(r) = 0
for rij < a i j ,
Cij(r ) = --
fluu(r ) for
rij > o'ij ,
(4)
where fl = 1~ks T and kB is the Boltzmann constant, and T the absolute temperature. So a dimensionless internal energy normalized by thermal energy can be written as
- flU = ~ ~. PiP, drgij(r)Cu(r),
, i,j
(5)
f
where gij(r)= hq(r)+ 1 is pair correlation function. The above equation can be decomposed into two parts. -- f l U =
~ PiP,
drCij(r) +
PiP,
drhij(r)Cij(r) .
(6)
Define h[j(Q) = ~ h u ( Q ) and cSj(O) = x~ipjci,(Q), where hij(Q) and cu(Q) are the Fourier transforms of hu(r ) and Cu(r ). The first part of the internal energy is
I = ~ .~. PiP, drCu(r) = ~ .~. ~ c T , ( Q t,J
¢
=
0).
(7)
/,J
For a two-component system containing macroions and counterions, I can be evaluated as 1
s
I = ~ [pMCMM(0) -4- 2 ~ c h c ( 0 )
+ pcc~¢(0)].
(8)
The leading terms of c~ diverge at Q = 0 in the way of 1/Q 2. The origin of the singularity lies in the divergent Coulomb potentials between the charged particles as their distances approach zero. However, it can be shown that lim
cs
PM MM(Q) -- lim
pocSo(Q)
--
1.
(9)
Thus, the divergent potentials are canceled out. The remaining term is I = ~
pM(1 - l/S(0)) + - f p M p o F M R 3 o
.
(10)
The second part of the internal energy can be evaluated by the direct correlation functions as I I = l ~2 i.j P'PJ f drhu(r)Cu(r) = ( 1~) 3 1 ~ 2i,jodQh~J(Q) ( cT,(Q).
(11)
Y.C. Liu, S.H. Chen/ Physica A 231 (1996) 277-287
281
For the two-component system, it can be evaluated by a three-dimensional integration of the macroion-macroion, macroion-counterion and counterion-counterion correlation functions. oo
II = ~
Go
Q MM(Q)MM(Q) Q + 2 0
Q Mc(Q) Mc(Q)dQ 0
oo
+
Q
co(Q) ¢¢(Q)dQ ,
(12)
0
where the direct correlation functions can be evaluated analytically following the equations given in the Appendix. Summing the two contributions, the total internal energy density is given by o0
-
flU = ~1 pM(1 -- 1/S(O)) +
~ pMpoFMR~¢+ ~ 21 I f
Q2h~M(Q)C~M(Q)dQ 0
ct3
oo
0
0
Given the internal energy, it is relatively easy to calculate other thermodynamics functions. Hoye and Stell [8] studied the expressions of the thermodynamics relations of MSA for simple fluids and then generalized their results to mixtures. The excess Gibbs free energy, or the excess chemical potential can be evaluated once the excess reference free energy is obtained from the internal energy. The total Gibbs free energy density of the system can be evaluated as - fiG = 1_2 x~PiCTi(O) - flU 2i,j
(14)
Considering the two parts described above in the total internal energy, it is simply - fig = 21 + II. The average Gibbs free energy per macroion particle GM can be evaluated as the following:
-/~GM - 24nr/MM -0
+
Q hcc(Q)ccc(Q)dQ
0
+
1--~-~
3
o The internal energy and Gibbs free energy in terms of kBT can be averaged by the total number of ionic species instead of the number of macroions.
Y.C. Liu, S.H. Chen / Physica A 231 (1996) 277-287
282
3. Electrostatic energy of ionic micellar solution
The MMSA theory provides a convenient way of calculating the free energies in an ionic system. The energy calculation works extremely well at large macroion--counterion charge asymmetry where numerical methods such as HNC often break down. At a very low charge ratio where the concentration of macroions is close to that of counterions, the accuracy of the calculation is limited by the validity of the Yukawa form of intermacroion potential. The partial structure factors and direct correlation functions can be evaluated analytically. The average internal energy and Gibbs free energy per particle (including macroions and counterions) is calculated as functions of temperature, and the diameter, charge and volume fraction of the macroion. As an example, the average internal energy and Gibbs free energy per particle are shown in Fig. 1 for an ionic system with macroion volume fraction of 10%, macroion diameter 50/~ at room temperature 25 °C. Fig. 2 shows a series of Gibbs free energies per particle at different macroion volume fractions, while the macroion diameter is fixed at 50 ~ and the temperature at 25 °C. It is clear that higher charge asymmetry corresponds to more negative Gibbs free energy. Similarly, a series of Gibbs free energies per particle at different macroion
0
-2
-4
-6 (-9 - 8 -10
-12 -14
-16
-18
t
I
60
t
do
Z Fig. 1. Average internal energy U (dashed line) and Gibbs free energy G (solid line) per particle in the unit of kBT as a function of macroion effective charge Z for an ionic system with macroion volume fraction of 10%, macroion diameter 50 A at room temperature 25 °C. Average energy is defined as total energy divided by total n u m b e r of particles, including macroions and counterions. Circles and crosses are values of internal energies and chemical potentials by Senatore using MMSA.
Y.C. Liu, S.H. Chen / Physica A 231 (1996) 277-287 0 .. -2
i ~~~.
J
i
i
i
i
283
i
""""...
68
~
(5 -10
" "- ~ ~ ~
-12
~ x
~
''"'".
~-~
~~~ ~
-14
~\~
~
~
-..
-16
"
-18 -20
J 20
30
' 40
' 50
6'0
70
' 80
9LO
100
Z Fig. 2. Gibbs free energy per particle G at various macroion volume fractions, while macroion diameter is fixed at 50 ~, and temperature at 25 °C. Solid line corresponds to macroion volume fraction of 10%. Dotted, dash-dotted and dashed lines correspond to macroion volume fractions of 1%, 5 % and 15%, respectively.
diameters at a fixed volume fraction of 10% and temperature of 25 °C is illustrated in Fig. 3. One can see that macroion of larger size has less negative Gibbs free energy. This indicates that a two-component system with larger macroions requires more counterions to stabilize and the stabilization energy can be evaluated quantitatively. The following scheme is suggested to obtain the accurate free energy of a real micellar system. The accuracy of energy evaluation is highly sensitive to the accuracy of micelle-micelle interaction potential. Although it is reasonable to histogram the free energy map and locate the configuration corresponding to the lowest energy, such a theoretical approach is unlikely to describe the real system. The failure of the theoretical approach is not because the assumptions and approximations made in the theory are poor. In fact, the problem is largely due to the complexity of the micellar system. The equilibrium configuration is at the lowest total free energy point as discussed in the introduction, rather than the most negative point of electrostatic energy alone. The most accurate energy calculation should be carried out based on a successful extraction of intermicellar structure factor from experiments. The accuracy of the theory sets in from the formulation of interparticle interactions and therefore determines the final accuracy of the energy evaluation. The internal energy and Gibbs free energy terms are formulated rigorously based on the mean spherical approximation through energy route. At conditions when MSA cannot hold, the energy formulation also breaks down. The assumption of point-like counterions simplifies the MMSA theory and allows analytical solutions of the direct
284
Y.C. Liu, S.H. Chen / Physica A 231 (1996) 277-287 0
:~-L ...............
i
i
1
i
i
-5
-10
(.9 -15
-20
-25 I
20
' 30
' 40
5'o
' 60 Z
7'0
8~0
90
100
Fig. 3. Gibbs free energy per particle G at various macroion diameters, while macroion volume fraction is fixed at 10%, and temperature at 25 °C. Solid line represents macroion diameter of 50 ~,. Dash-dotted, dashed and dotted lines correspond to macroion diameters of 30, 70, and 100 Pt, respectively.
correlation functions and partial structure factors. The analytical formulation has great advantages over the cumbersome numerical methods since it is suitable for fitting experimental data. On the other hand, due to the finite size of the small particles, the point-particle assumption may also introduce some inaccuracy especially in the counterion-containing partial structure factors at small Q range. The approach to calculating electrostatic free energy of micellar system is different from the approach to surface electrostatic energy based on a Poisson-Boltzmann equation [9, 10] because of the different reference zero-energy states. In the ionic liquid theory, the zero energy correspond to infinite separation of ions. However, in the Poisson-Boltzmann integration methods, the reference zero-energy state corresponds to a zero charge state equation. Finally, although a two-component system is discussed here, it should be pointed out that in a multi-component system the contributions of the partial structure factors and energy terms due to salt ions are entirely recoverable based on a similar scheme.
4. Electrostatic energy for cesium dodecyl sulfate miceilar solution We show the method of calculating the electrostatic internal energy and Gibbs free energy for a two-component system with macroions and their counterions, using the MMSA theory. As an example, an ionic micellar system is studied and used to demonstrate the methodology for evaluating the electrostatic free energy.
Y.C. Liu, S.H. Chen / Physica A 231 (1996) 277-287
285
2sI u 1.5
g
0.5
t
O0
0.05 '
0i1
0.15 '
o
0.2 '
0.25
0.3
[~/A]
Fig. 4. Small angle neutron scattering data of 3% cesium dodecyl sulfate micellar solution at 40 °C. Solid line represents fitted curve. Data analysis shows that micelle has an average diameter of 52 A and charge of 25e.
We have studied a micellar solution made of 3% cesium dodecyl sulfate surfactant in deuterated water at 40 °C using SANS and SAXS. Careful analyses of SANS and SAXS data show that the micelle is composed of about 100 cerium dodecyl sulfate molecules with a dense hydrocarbon core and a hydrated outer-layer composed of about 10 solvent molecules per sulfate head group. The micelle has an average diameter of about 52 ~ and an average charge of - 25e. The volume fraction of the micelles in the solution is about 3%. SANS data and fitted curve of the micellar solution are shown in Fig. 4. The intermicellar structure factor SMM(Q)was extracted from the SANS spectrum. Micelle-counterion and counterion-counterion structure factors SMc(Q)and See(Q)were further calculated using the formulae in the Appendix, and proved to be reasonably accurate by additional SAXS experiment on the same system. The partial structure factors obtained are illustrated in Fig. 5. Once the effective charge, diameter and volume fraction of the micelles are known, electrostatic internal energy and Gibbs free energy of the entire micellar system can be calculated easily. Especially, the Gibbs free energy (or chemical potential) per ionic species turns out to be - 1.52kBT. The total electrostatic Gibbs free energy of the 3% cesium dodecyl micellar solution is - 0.0676 KJ/L. Acknowledgements
This research is supported by a grant from the Materials Science Division of USDOE.
Y.C Liu, S.H. Chen I Physica A 231 (1996) 277-287
286 1.8
i
1.6l'.... 1.4 ". 1.2
o..p./,,
0.I/ ',, 0.2ri
~
0
0.05
0.1
0.15
O
0.2
ILIA]
0.25
0.3
0.35
0.4
Fig. 5. Partial structure factors of cesium dodecyl sulfate micellar solution. Solid line, dashed line and dotted line represent micelle-micelle structure factor SMM(Q),micelle-counterion structure factor SM0(Q) and counterion-counterion structure factor See(Q),respectively.
Appendix. Partial structural factors for multi-component system An ionic system can be reduced into a three-component system composed of macroions (M), counterions (c) and neutral ions (n). Define Sij(Q) = 6ij + hTi(Q) with hTj(Q) = x ~ p j h u ( Q ) and c~j(Q) = x/~picij(Q). The partial structure factors are evaluated as: (Note that S(Q) = SMM(Q)), SM,(Q) = Chn(Q)S(Q) ,
(A.1)
Snn(Q) = 1 + [c~,(Q)] 2 S(Q),
(A.2)
Q2 SM~(Q) - k2 + Q2 C ~ ( Q ) S ( Q ) ,
(A.3)
Q2 S~,(Q) - k2 + Q2 c~(Q)c~n(Q)S(Q) ,
(A.4)
soo(Q) = k~ + Q------~+ k~ ¥ - &
(g.5)
[C~,~(Q)]~s(o,
c~c(Q) = 4rcx//~Mpc [FuR~d(QRMo) - LBZMZ~ c o s ( g R u j O 2 ) ] ,
(A.6)
Y.C. Liu, S.H. Chen / Physica A 231 (1996) 277-287
clan(Q) = 4~
(A.7)
p~Mp~F.R~j(QRM~),
C~M(Q) = 1 -- 1 / S ( Q ) -
(
287
)
[c~c(Q)] 2 + [c~.(Q)] 2 ,
(A.8)
where sin(x) - x cos(x) y(x) =
x~
(A.9) ,
F. = - 1/(1 - t/M.),
(A.IO)
kLBZMZc[2y exp(kRM¢)Cosh(kRM¢) - 1] F u - 27exp(kRMo)[sinh(kRMJ -- kRM~cOsh(kRMc)] + 1 + kRM~'
(A.I1)
with r/Mn = 4~pMR~o/3 and ~ = v e x p ( - k)/k, where k = X/4rtLBp~ZEaMM is an inverse Debye screening length, trMM is macroion-macroion diameter, and RM~ is macroion-counterion radius. LB = e2/ekB T is the Bjerrum length. Once S(Q) is known from analysis of scattering data, all the other partial structure factors and the corresponding partial correlation functions can be calculated. The macroion-macroion diameter trMM is extended by the rescaling, and the macroion-counterion diameter equals to the radius of the micelle. The counterions can come to the actual micellar surface and the non-additivity condition allows RMc ~< ½. This reflects the so-called penetrable sphere effect [6].
References [1] H. Wennerstorm and B. Lindman, Physics Reports, 52 (1979) 1. [2] C. Tanford, Hydrophobic Effects: Formation of micelles and Biological Membranes (Wiley, New York, 1973). [3] S. Khan, T.L. Morton and D. Ronis, Phys. Rev. A 35 (1987) 4295; S. Khan and D. Ronis, Mol. Phys. 60 (1987) 637. [4] J. IsralachviUi, Intermolecular and Surface Forces (Academic Press, New York, 1992). [5] G. Senatore, in: Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution, eds., S.H. Chen, J.S. Huang and P. Tartaglia, NATO ASI Series C369 (Kluwer Academic Publishers, Dordrecht, 1992) pp. 175. [61 Y.C. Liu, C.Y. Ku, P. NoLostro and S.H. Chen, Phys. Rev. E 51 (1995) 4598. [7] J.P. Hansen and I.R. McDonold, Theory of Simple Liquids (Academic Press, New York, 1988). [8] J.S. Hoye and G. Stell, J. Chem. Phys. 67 (1977) 439; L. Blum and J.S. Hoye, J. Phys. Chem. 81 (1977) 1311. [9] D.F. Evans and B.W. Ninham, J. Phys. Chem. 87 (1983) 2996. [10] Y.S. Chao, E.Y. Sheu and S.H. Chert, J. Phys. Chem. 89 (1985) 4862.