Langmuir 1999, 15, 4981-4988
4981
Effects of Salt Addition on the Ion Distribution Enclosed in a Cylinder and a Sphere Heng-Kwong Tsao Department of Chemical Engineering, National Central University, Chung-li, Taiwan 320, Republic of China Received October 27, 1998. In Final Form: April 15, 1999 The electric potential and ion distributions within a cylinder and a sphere at various salt concentration have been obtained analytically by solving the Poisson-Boltzmann equation in terms of series solutions. At a fixed ratio of co-ion to counterion concentrations at the center of the aqueous core (R < 1) or a given amount of added salt, the existence of the upper limit of the counterion concentration at the center has been observed regardless of the surface charge density. It is a consequence of counterion condensation on the charged surface. The upper limit depends on the value of R and increases rapidly when R f 1. At high surface charge density, the counterions are strongly attracted to the surrounding of the surface. A boundary layer with the thickness inversely proportional to the surface charge density is formed. Both the effects of curvature and salt addition hence become unimportant, and the surface potential is primarily determined by the surface charge density.
Surfactant-stabilized water in oil microemulsions or swollen reverse micelles have attracted much attention in recent years. They find applications in diverse fields, such as material sciences and biotechnology.1-3 These microdroplets can be used as microreactors for manufacturing nanosized particles, including inorganic materials4,5 such as semiconductor and organic particles like latexes.6 Moreover, enzyme-catalyzed synthesis reactions can proceed in the organic solvents by solubilization of enzymes in reversed micelles.7,8 The electrostatic contribution associated with the electric double layer in the aqueous core plays a significant role in the understanding of the thermodynamics and the structure of water in oil (W/O) microemulsions,9-13 aside from the bending free energy associated with the hydrophobic tails of the ionic surfactants. In ionic surfactant systems, the adjustment of the concentration of the added salt can result in phase transition and therefore is often used as an experimental control variables.1 The ionic strength in the aqueous core also affects the electrostatic interactions and possibly modifies the equilibrium size of the droplet and its ability to accommodate the hosted particles.11,13,14 Furthermore, the transport properties of W/O microemulsions, such as the electric conductivity, may be explained by the charge fluctuation model and are
also substantially influenced by the ion distribution in the aqueous phase.15-19 In this paper, it is our goal to study the effects of salt addition on the electric field and ion distributions enclosed in the water droplets. The ion distribution can be described by the sophisticated statistical mechanical theories, such as PY and HNC, and molecular simulations even at concentrated ionic solutions. However, the Poisson-Boltzmann equation (PB), despite its drawback (such as finite ion size effects), is perhaps the simplest model in depicting the diffuse electric double layer. The results are often accurate at low ion concentrations.20-22 In electrolyte solutions, the diffuse layers near a flat plate, outside a cylinder, and around a spherical particle have been extensively explored.22 Lately, a great deal of attention has been focused on the ion distribution inside a cylinder and a sphere as a result of intensive studies on the systems of reverse micelles, microemulsions, and vesicles.12,23-26 Despite the electric potential inside a cylinder and a sphere can be solved for the PB equation under the Debye-Hu¨ckel assumption,12 it is believed that no close solutions exist for high potential and spherical symmetry. For thin double layers κR . 1, asymptotic analytical solutions of the electric potential for cylindrical and spherical geometries have been obtained with addition of electrolytes.24,25 Here κ and R are respectively the inverse Debye length and the radius of the particle. Expanding the electric potential in
(1) Hunter, R. J. Foundations of Colloid Science II, Oxford: New York, 1989; Chapter 17. (2) Pileni, M. P. J. Phys. Chem. 1993, 97, 6961. (3) Shield, J. W.; Ferguson, H. D.; Bommarius, A. S.; Hatton, T. A. Ind. Eng. Chem. Fundam. 1986, 25, 603. (4) Pitit, C.; Pileni, M. P. Langmuir 1989, 92, 2282. (5) Stathatos, E.; Lianos, P.; Del Monte, F.; Levy, D.; Tsiourvas, D. Langmuir 1997, 13, 4295. (6) Hammouda, A.; Gulik, Th.; Pileni, M.P. Langmuir 1995, 11, 3656. (7) Goklen, K. E.; Hatton, T. A. Biotechnol. Prog. 1985, 1, 69. (8) Pitre, F.; Regnaut, C.; Pileni, M. P. Langmuir 1993, 9, 2855. (9) Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1983, 87, 2996. (10) Overbeek, J. Th., G.; Verhoeckx, G. J.; Bruyn, P. L. De; Lekkerkerker, H. N. W. J. Colloid Interface Sci. 1987, 119, 422. (11) Rahaman, R. S.; Hatton, T. A. J. Phys. Chem. 1991, 95, 1799. (12) Winterhalter, M; Helfrich, W. J. Phys. Chem. 1988, 92, 6865. (13) Bratko, D.; Luzar, A.; Chen, S. H. J. Chem. Phys. 1988, 89, 545. (14) Woll, J. M.; Hatton, T. A.; Yarmush, M. L. Biotechnol. Prog. 1989, 5, 57.
(15) Bratko, D.; Woodward, C. E.; Luzar, A. J Chem. Phys. 1991, 95, 5318. (16) Halle, B.; Bjo¨rling M. J. Chem. Phys. 1995, 103, 165. (17) Eicke, H.-F.; Borkovec, M.; Das-Gupta, B. J. Phys. Chem. 1989, 93, 314. (18) Hall, D. G. J. Phys. Chem. 1990, 94, 429. (19) Kallay, N.; Chittofrati, A. J. Phys. Chem. 1990, 94, 4755. (20) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976; Chapter 15. (21) Israelachvili, J. Intermolecular & Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1992; Chapter 12. (22) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersion; Cambridge University Press: Cambridge, 1989; Chapter 4. (23) van Aken, G. A.; Lekkerkerker, H. N. W.; Overbeek, J. Th. G.; de Bruyn, P. L. J. Phys. Chem. 1990, 94, 8468. (24) Lekkerkerker, H. N. W Physica A 1989, 159, 319. (25) Mitchell, D. J.; Ninham, B. W. Langmuir 1989, 5, 1121. (26) Daicis, J.; Fogden, A.; Carlsson, I.; Wennerstro¨m, H.; Jonsso¨n, B. Phys. Rev. E 1996, 54, 3984.
1. Introduction
10.1021/la981516n CCC: $18.00 © 1999 American Chemical Society Published on Web 06/09/1999
4982 Langmuir, Vol. 15, No. 15, 1999
terms of (κR)-1 up to (κR)-2, the surface potentials24 are found to be accurate within 2% as κR g 2. For W/O microemulsions and swollen reverse micelles without salt addition, there are still a great number of counterions, about O(100-1000), dissociated from the ionic surfactant shell for a droplet of radius about 6-60 nm. For counterions only, the electric potential for a cylinder has been solved exactly.27,28 In terms of a series solutions, the analytical expression for a sphere is also obtained. The accuracy of the solution can be systematically improved by taking into account higher order terms. An inherent feature in the Poisson-Boltzmann model is the occurrence of counterion condensation,29 which has also been found within an enclosed region for dimensions one to three.27 In addition, the existence of the upper limit of counterion concentration at the center of the aqueous core has also been observed. The latter is believed to be the consequence of the former. For a colloidal particle in an electrolyte solution, the concentrations of co-ions and counterions far from the particle are expected to be equal due to the condition of electroneutrality. However, unless an excess amount of inorganic salt is added, the concentrations of counterions and co-ions are expected to be different significantly from each other even at the center of the droplet. For instance, when salt is added, there are about 25 Na+(Cl-) ions for a 0.01 M NaCl solution in a 10 nm droplet. The ratio of co-ion to counterion concentration at the center of a droplet is about 0.8. How the distribution of ion concentration varies with increasing addition of salt is one of the major issues in this paper. In this study, the electrostatic field enclosed in a charged surface is investigated for various amount of salt addition based on the PB equation. Three systems are considered, including one, two, and three dimensions. The electrolyte solution between two planar plates can be considered as an enclosed system in one dimension instead of a simple model for two-particle interactions.21,30 The results are useful in studying lamellar liquid crystals formed by ionic amphiphiles31 and forces in soap films and clays. The electrolyte solutions within an infinitely long cylinder and a sphere can be regarded as two- and three-dimensional problems, respectively. The analytical solution of electric potential is presented in terms of the series solution. The ion distributions are hence obtained. The results depend on the properties at the center of the aqueous core, that is, the dimensionless counterion concentration (κ-L)2 and the ratio of co-ion to counterion concentrations R. The two quantities are related to the dimensionless surface charge density Λ and the amount of added salt (κsL)2. At high surface charge density, the asymptotic expression of the surface potential is obtained. The results are compared with the numerical solutions. The effects of added salt on the counterion condensation at the surface and the upper limit of counterion concentration at the center are also discussed. 2. Analytic Solution An aqueous core enclosed by an ionic surfactant shell is shown in Figure 1. The water pool contains both counterions coming off the inner surfaces and the added zs - zs, electrolytes. We consider three cases, including (27) Tsao, H.-K. J. Phys. Chem. B 1988, 102, 10243. (28) Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1987, 91, 338. (29) Ramanathan, G. V. J. Chem. Phys. 1988, 88, 2887. (30) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces, and Membrane; Addison-Wesley: Reading, MA, 1994; Chapter 5. (31) Jo¨nsson, B.; Wennerstro¨m, H. Chem. Scr. 1980, 15, 40.
Tsao
Figure 1. Co-ions and counterions in the aqueous core enclosed in an ionic surfactant shell. The system size is 2L.
two similarly charged planar surfaces, a hollow cylinder, and a spherical cavity. The electric potential φ* in the solution is described by the Poisson-Boltzmann equation. Since only difference in electric potential are physically meaningful, one can set φ(r)0) ) 0. As a result
1 d rm dr
( ) rm
dφ
)-
dr
n0c zce r0
(
exp -
∑
kT
0 ns,i zs,ie
i)(
)
zceφ*
r 0
-
(
exp -
)
zs,ieφ* kT
(1)
where n0c is the number density of counterions at r ) 0, which are of valency zc and dissociated from the surfaces. 0 ns,( are the respective number densities of the ionic species associated with the added salt at r ) 0. Note that 0 0 ns,+ is not equal to ns,. r is the dielectric constant of the solution and 0 the permittivity of a vacuum. k is the Boltzmann constant and T the absolute temperature. e stands for the fundamental charge, 1.6 × 10-19 C. Note that m ) 0, 1, and 2 represent rectangular, cylindrical, and spherical coordinates, respectively. Let 2L represents the characteristic size of the system, that is, the separation distance between two surfaces, the diameter of the cylinder or sphere. Without the loss of generality, we assume -zs,+ ) zs,- ) zc. Define
x ) r/L, and φ* ) zceφ/kT
(2)
The PB equation becomes
1 d m dφ x ) -(κ-L)2e-φ + (κ+L)2eφ dx xm dx
(
)
The Debye-Hu¨ckel parameter is defined as
(3)
Effects of Salt Addition on Ion Distribution
(κ(L)2 )
Langmuir, Vol. 15, No. 15, 1999 4983 ∞
n0( z2c e2 2 L r0kT
(4)
|
dφ
0 where the co-ion concentration n0+ ) ns,+ and counterion 0 0 0 concentration n- ) nc + ns,- at x ) 0. Equation 3 is a nonlinear ordinary differential equation. Fortunately, it can be solved analytically by the method of Frobenius. Assume that
φ(x) ) ln h(x)2
dx
)Λ)2
[
2x h
-
2
where R ) n0+/n0- ) (κ+L)2/(κ-L)2. R ) 0 corresponds to a system with counterions only. By definition, one has
h(x)0) ) 1
(7)
In addition, due to the symmetric condition at the center
|
|
dh dφ )0) dx x)0 dx x)0
(8)
Equation 6 can be solved by expressing the function h(x) in terms of an infinite series ∞
h(x) )
∑ a2k
and the condition of electroneutrality
nsLm+1 ) (m + 1)n0+
∫0L exp(-zs,+eφ*/kT)rm dr
∑ anxn
(9)
n)0
According the conditions 7 and 8, one has
a0 ) 1
(14)
or
4
(6)
(13)
k)0
(5)
+ (κ L) x(1 - Rh ) ) 0 ( ) ] + 2mh dh dx
dh dh dx dx2
2
∞
x)1
Note that 0 < h(x) e 1. Substituting eq 5 into 3 gives 2
∑ (2k)a2k k)1
∞
∫01 ( ∑ a2kx2k)2 xm dx
(κsL)2 ) (m + 1)(κ-L)2R
(15)
k)0
Here Λ ) σzceL/r0kT < 0 and (κsL)2 ) nsz2s e2L2/r0kT. Once one knows the amount of added salt ns and surface charge density on the shell σ, the ion concentration n0( can be determined from eqs 13 and 15. 3. Upper Limit of Counterion Concentration at the Center Though it would be relatively easy to tune the (κsL)2 at the condition of constant Λ in experiments, the systems of constant R but different values of (κ-L)2, in principle, can be obtained by changing both Λ and (κsL)2 appropriately. One would observe that (κ-L)2 approaches a limiting value, (κ-L) 2c , even when Λ goes to infinity. This result indicates that the system exhibits an upper limit of counterion concentration at the center of the cylinder or sphere for a fixed value of R t n0+/n0-. This situation corresponds to that the electric potential at the surface approaches -∞ when the surface charge density approaches infinity. That is ∞
and
h(x)1) ) 0 ) (1 - R) a1 ) 0
We insert eq 9 into eq 6 and collect the coefficients of the same power of x. Since the various powers of x in eq 9 are linearly independent, the equality can be established if and only if the coefficient of each power of x is zero.
a2 ) -(κ-L)2(1 - R) a4 ) -(κ-L)4(1 - R)
1 4(1 + m)
(-1 + m) + (5 + 3m)R 32(1 + m)2 (3 + m)
(10)
(11)
a6 ) -(κ-L)6(1 - a) ×
(12) All the odd terms are zero; R2k+l ) 0 because they are proportional to a1. The values of a2k from a2 to a12 for m ) 0, 1, and 2 are listed in Table 1. Note that when R ) 0, the results reduce to the case of counterions only.27 The quantities (κ-L)2 and R (or n0() are related to the mean concentration of the added salt ns and surface charge density σ by the boundary condition at the surface, i.e., σ/r0 ) (dφ*/dr)r)a, or
(16)
Here the expression b2k(R;m) ) R2k/(l - R)(κ-L)2k has been used. The right hand side of eq 16 is a polynomial of (κ-L)2k c and can be solved to obtain the upper limit. Obviously, (κ-L)2k c varies with R. The limit of R ) 0 has already been discussed by the author in ref 27. The existence of the upper limit of the counterion concentration is related to the phenomenon of the counterion condensation on the charged surface. The above statements can be demonstrated by considering the case of two parallel plates, which can give an analytical expression of (κ-L)2c in terms of R. The exact solution of electric potential for m ) 0 has been expressed as the Jacobian elliptic functions in ref 31
(1 + 3m2 - 4m) + (-46 - 32m - 2m2)R + (68m + 61 + 15m2) R2 384(1 + m)3 (3 + m) (5 + m)
b2k(R;m)(κ-L) 2k ∑ c k)0
φ(x) ) 2 ln
[x
cos φ
1 - R sin2 φ
]
(17)
where φ is related to x through
κ-L
x2
x)
∫0φ
dθ
x1 - R sin2 θ
(18)
(κ-L) is related to the surface charge density Λ by the boundary condition
4984 Langmuir, Vol. 15, No. 15, 1999
Tsao
Table 1. The Coefficients a2k of Series Solutions for the Electric Potential within Two Parallel Plates, a Cylinder, and a Sphere (a) Parallel Plates (m ) 0)
b2k(R; m ) 0)
a2k (κ-L)2k(1 - R) a2
-
2
(κ-L) (1 - R) a4
1 4
1 - 5R 96
4
(κ-L) (1 - R) a6
-
(κ-L)6(1 - R) a8
1 - 46R + 61R2 5760
1 - 411R + 1731R2 - 1385R3 645120
8
(κ-L) (1 - R) a10
-
10
(κ-L) (1 - R) a12
1 - 3692R + 41838R2 - 88412R3 + 50521R4 116121600
12
1 - 33217R + 1003498R2 - 4349762R3 + 6081221R4 - 2702765R5 30656102400
a2k
b2k(R; m ) 1)
(κ-L) (1 - R)
(κ-L)2k(1 - R) a2
-
(κ-L)2(1 - R) a4
1 8
-
R 64
4
(κ-L) (1 - R) a6
(b) Cylinder (m ) 1)
R(5 - 9R) 4608
(κ-L)6(1 - R) a8
R(1 - 9R + 9R2) 36864 R2(104 - 325R + 225R2) 7372800 R2(234 - 2038R + 3825R2 - 2025R3) 530841600
-
8
(κ-L) (1 - R) a10 (κ-L)10(1 - R) a12 (κ-L)12(1 - R)
b2k(R; m ) 2)
a2k (κ-L)2k(1 - R) a2
(c) Sphere (m ) 2)
-
(κ-L) (1 - R) a4
1 12
-
(κ-L)4(1 - R) a6
1 + 11R 1440
-
5 - 118R + 257R2 362880
-
107 + 727R - 15167R2 + 17213R3 261273600
-
2671 + 7180R + 341778R2 - 1375300R3 + 1058231R4 172440576000
-
258803 + 59677R - 10159266R2 + 165445306R3 - 370063297R4 + 215426457R5 376610217984000
2
6
(κ-L) (1 - R) a8 8
(κ-L) (1 - R) a10 10
(κ-L) (1 - R) a12 12
(κ-L) (1 - R)
(1 - R)
κ-L
sin φ
x2 cos φx1 - R sin2 φ
|
x)1
)-
Λ (19) 2
Equation 17 shows that the potential φ becomes singular when φ f π/2. This limit is corresponding to (κ-L)2c with x ) 1. Equation 18 now becomes a complete elliptic integral of the first kind and one can obtain32 (32) Abramowitz, M., Stegun, I. A., Eds. Handbook of Mathematical Functions; Dover: New York, 1970.
(κ-L) 2c ) 2
[
∫0π/2
dθ
x1 - R sin
2
π2 2
]
2
θ
)
[∑[ ∞
k)0
] ]
(2k - 1)!! (2k)!!
2
2
Rk
(20)
where (2k)!! ) 2‚4‚6‚...‚2k and (2k - 1)!! ) 1‚3‚5‚...‚(2k 1). Note that (κ-L)2c f ∞ when R f 1. For a cylinder and a sphere, the upper limits of counterion concentrations also exist and can be calculated
Effects of Salt Addition on Ion Distribution
Langmuir, Vol. 15, No. 15, 1999 4985
per unit area are accumulated. The foregoing statements can be justified later by using eq 23. Multiplying both sides of eq 3 by dφ/dx gives
d dφ 2 m dφ 2 d +2 ) 2(κ-L)2 (e-φ + Reφ) (22) dx dx x dx dx
( )
( )
Carrying out the integration and using eq 13 yields
1 2 Λ ) (κ-L)2 (e-φs + Reφs - 1 - R) 2
∫01 mx (dφ dx )
2
dx (23)
Note that eq 23 can also be obtained from the force balance including both osmotic pressure and electric stress22 or the contact value theorem generalized to curved geometries.33 The second term on the right hand side of eq 23 is O(-Λ) and can be neglected in comparison with other O(Λ2) terms. The surface potential can then be obtained
Figure 2. Upper limit of counterion concentration at the center (κ-L)2c is a function of the ratio of co-ion to counterion concentration at the center R for various geometry.
by numerical or truncated series solutions. The variation of the value of (κ-L)2c with R for different geometry is shown in Figure 2. The upper limits are in the order of 2 2 2 > (κ-L)c,m)1 > (κ-L)c,m)1 for the same R. Since (κ-L)c,m)2 the volume-to-surface ratio is in the order (L)m)0 > (L/ 2)m)1 > (L/3)m)2, the sphere has the highest concentration of counterions at the center due to its lowest available volume for a given size L. 4. Asymptotic Analysis Two asymptotic limits will be discussed in this section: high surface charge density -Λ . 1 and low charge density -Λ j O(1). For the former, we derive the analytical expressions of the potential and the ion concentrations at the surfaces, i.e., φs and ns(, respectively. The surface potential is useful in evaluating the electrostatic free energy. Under the Debye-Hu¨ckel approximation, the latter case can be analyzed by solving the linearized PB equation. 4.1. (-Λ). 1: Counterion Condensation. First, let us show that at high charge density the effect of curvature on the electric potential distribution near the surface is small. Expanding the potential in terms of the Taylor series about x ) 1 gives
1 φ(y) ) φs - Λy - [mΛ + (κ-L)2(e-φs - Reφs)]y2 + 2 d3φ y3 (21) O 3 dx x)1
[( ) ]
Here y ) 1 - x and eqs 3 and 13 have been used. When -mΛ , (κ-L)2e-φs, the correction associated with curvature (the term involving m) can be neglected. In addition, to ensure that the O(y2) term is small compared to the O(y) term, one must have
y,
1 ,1 -Λ
Consequently, (-Λ)-1L can be regarded as the thickness of the boundary layer where an O(σ) amount of counterions
φs = -ln Λ2 + ln 2(κ-L)2 + O
(mΛ)
(24)
For -Λ . 1, the leading behavior of the cylinder and sphere is just like that of two planar plates. The next order term is a function of both Λ and R and also varies with m due to the curvature effects. This term denotes the contribution associated with counterions at the center. When Λ is large enough, (κ-L)2 approaches (κ-L)2c , which depends on the geometry (m) and R only. In other words, when Λ2 . (κ-L)2, the effects of curvature and salt become unimportant and the surface potential is mainly determined by the surface charge density. According to eq 24, the counterion and co-ion concentrations at the surface (ns- and ns+) can then be estimated as
ns- ) n0-e-φs =
1 σ2 m 1+O 2 r0kT Λ
[
( )]
(25)
and
ns+ ) Rn0-eφs =
r0kT
(κ-L)4
(zseL)
Λ2
2R 2
(26)
This result shows that in the limit -Λ . 1, the leading behavior of the concentration of counterions at the surface is always proportional to the square of the charge density regardless of the amount of salt added and geometry. On the other hand, the concentration of co-ions at the surface depends seriously on R in addition to the surface charge density. 4.2. (-Λ) j O(1): Debye-Hu 1 ckel Approximation. When φ j O(1), an approximate solution of the PB equation can be obtained under the Debye-Hu¨ckel approximation. The exponential term in the PB equation 3 can then be linearized to give
1-R 1 d m dφ x ) K2 φ m dx dx 1 +R x
(
)
[
]
where K t (κ-L)(1 + R)1/2. By defining φ ) 0 at x ) 0 and subjecting to the boundary conditions 8 and 15, the solutions are given as follow: For m ) 0, (33) Wennerstro¨m, H.; Jo¨nsson, B.; Linse, P. J. Chem. Phys. 1982, 76, 4665.
4986 Langmuir, Vol. 15, No. 15, 1999
φ(x) )
Tsao
1-R [1 - cosh Kx] 1+R
where (κ-L) and R are related to Λ and (κsL) by
1-R K sinh K ) -Λ 1+R and
2R sinh K 2 - (1 - R) 1+R K
[
(κsL)2 ) (κ-L)2
]
For m ) 1,
1-R [1 - I0(Kx)] 1+R
φ(x) ) with
1-R KI (K) ) -Λ 1+R 1 and
[
Figure 3. Surface potential φs for a sphere is a function of the surface charge density -Λ for R ) 0.1, 0.5, 0.95. The comparison between the results of the series and numerical solutions is made.
]
I1(K) 2R 1 - (1 - R) (κsL) ) (κ-L) 1+R K 2
2
where In(x) is the modified Bessel functions of the first kind of order n. For m ) 2, the electric potential is
φ(x) )
1-R sinh Kx 11+R Kx
[
]
with
1 - R K cosh K - sinh K ) -Λ 1+R K and
(κsL)2 ) (κ-L)2
[
]
K cosh K - sinh K 3R 2 - (1 - R) 1+R 3 K3
In general,
(κsL)2 )
R [2(κ-L)2 + (m + 1)Λ] 1+R
Obviously, (κ-L)2c does not exist in this limit. 5. Discussion and Conclusions In this work, the effects of salt addition on the ion distribution within an enclosed surface are studied. Three cases have been considered, including two charged plates for one-dimension, a cylinder for two-dimensions, and a sphere for three-dimensions. Applying the method of Frobenius, the nonlinear Poisson-Boltzmann equation is solved in terms of series solutions. The accuracy can be improved systematically by keeping higher order terms. The distribution of electric potential and the concentration profile of ions are obtained and depend on four important, dimensionless parameters in the system: (κ-L)2, R, Λ, and (κsL)2. Once two of them are specified, the rest are determined. Usually, Λ and (κsL)2 can be tuned experimentally by adding cosurfactant and salt. Nevertheless, it is convenient to adjust (κ-L)2 and R in doing mathematical calculations. A typical water droplet of radius about 10 nm in W/O microemulsions contains O(1500) ionic surfactants and
an approximately equal amount of counterions in the aqueous core. If the surface possesses a high charge density, such as σ ) 0.2 C/m2 at 25 °C as a result of complete dissociation of counterions, then -Λ ≈ 112. A larger droplet may have higher values of -Λ because it is proportional to L. However, if the degree of dissociation of the counterions from the surfactant shell is about 20%, then -Λ ≈ 22. Addition of nonionic surfactant or cosurfactant may reduce the surface charge density furthermore. Consequently, an analytical expression of the electric potential, which is accurate from low to high charge densities, is useful in studying the microemulsion systems. The series solutions of the electric potential gives exact results for x , 1, and the error increases as x f 1. Thus, the accuracy of them can be illustrated by the surface potential φs. Figure 3 illustrates the variation of φs with the surface charge density Λ at different values of R for a sphere. When R j 0.9, a truncated series solution, which keeps terms in eq 9 for h(x) up to O(x12), agrees quite well with the numerical solution. However, to obtain accurate results for -Λ . 1 as R f 1, higher order terms, such as R14, R16, etc., must be evaluated. Though the DebyeHu¨ckel approximation is expected to be valid as |φs| , 1, Figure 4 shows that it gives a quite satisfactory result for φs j -2, which corresponds to Λ ) -20 for R ) 0.9999. The asymptotic expression 24 for φs is in excellent agreement with the numerical result for -Λ . 1 even when R f 1. This result indicates that when (-Λ) . 1, the curvature effect becomes unimportant and the surface potential is mainly determined by the surface charge density. The effects of salt addition on the surface potential are also revealed in Figures 3 and 4. For a given Λ, the surface potential (-φs) decreases with increasing R. This result indicates that the addition of salt lowers the electrostatic interactions due to the screening effects. By comparing the differences among different values of R in Figure 3, one can note that the effect of salt addition is not significant when R is not so close to unity. The variation of the counterion concentration with the position within a sphere of R ) 0.7 for various surface charge density is shown in Figure 5. (κ-L)2x and (κ+L)2x
Effects of Salt Addition on Ion Distribution
Figure 4. Variation of the surface potential φs with the surface charge density -Λ for different approaches, including the numerical result, asymptotic expression, and Debye-Hu¨ckel theory.
Figure 5. Variation of the dimensionless concentration of counterions (κ-L)2 with the dimensionless position x within a sphere at R ) 0.7 for various surface charge density.
represent respectively the dimensionless concentrations of counterions n-(x) and co-ions n+(x) scaled by r0kT/ (zceL)2. As -Λ j O(1), the concentration profile is quite uniform and the Debye-Hu¨ckel approach is adequate. At a specified value of R, the counterion concentration increases as the charge density increases by adding an appropriate amount of inorganic electrolytes. It is worth mentioning that in order to maintain R ) 0.7 with increasing (-Λ), one has to put in salt incrementally when (-Λ) < 8.35. However, when (-Λ) > 8.35, R is kept constant for increasing (-Λ) by slowly decreasing amount of added salt. As -Λ J O(100), the concentration near the charged surface increases several order of magnitude. At high surface charge density, as discussed in section 4.1, most counterions are attracted to a boundary layer near the surface with the thickness ∼(-Λ)-1 to balance the
Langmuir, Vol. 15, No. 15, 1999 4987
Figure 6. Dimensionless concentration profiles of co-ions and counterions within a sphere at Λ ) -100 for R ) 0.50 and 0.99.
surface charge. This boundary layer becomes thinner as the surface charge density increases. Nevertheless, the concentration profile is still nearly uniform in the proximity of the center. Its value (n0-) does not increase substantially with increasing -Λ. In fact, further increasing the surface charge density does not have any effect on n0 when (κ-L)2 reaches the upper limit (∼17). The existence of the upper limit of counterion concentration at the center of an enclosed space is the consequence of the phenomenon of counterion condensation at the surface.29 The effects of salt addition on the ion distributions within a sphere are depicted in Figure 6 for Λ ) -100. Both the counterion and co-ion concentrations for R ) 0.99 are higher than those for R ) 0.5. It indicates that more salt is added for the former case. In the neighborhood of the surfaces, the concentrations of counterions for both cases increase rapidly and converge as x f 1. This result is consistent with eq 25. The concentration of counterions at the surface depends on the square of the surface charge density only, which is the same for both cases. On the other hand, the concentrations of co-ions at the surface also depend on R. Equation 26 shows that the co-ion cocentration at the surface is proportional to R(κ-L)4, which can be evaluated from Figure 2 and approaches R(κ-L)4c at large (-Λ). As R increases, the co-ion concentration can increase significantly due to the screening effect. Intuitively, one would expect that the counterion concentration everywhere in the system grows with increasing surface charge density W/O salt addition because of the entropy of mixing with the solvent. Nonetheless, when the surface charge density is high enough, it has been proved that without salt addition, there exists the upper limit of counterion concentration at the center of two parallel plates, a cylinder, and a sphere.27 In this study, we found that if salt is added, the upper limits also exist for a specified R. The relation between (κcL)2 and (-Λ) for various R is demonstrated in Figure 7 for a sphere. Figure 7 shows clearly that the surface charge increases very quickly when (κ-L)2 approaches (κ-L)2c at a given R. For a known amount of salt addition, the system also exhibits the existence of the upper
4988 Langmuir, Vol. 15, No. 15, 1999
Figure 7. Variation of the surface charge density -Λ with the dimensionless concentration of counterions at the center of the system (κ-L)2 at various values of R for a sphere.
Figure 8. Variation of R with the surface charge density -Λ in a sphere for different amounts of added salt (κsL)2.
limit of counterion concentration regardless of the surface charge density. The upper limits increase with the amount of salt addition. Note that R varies with -Λ and approaches a limiting value as -Λ f ∞. This result indicates that for a given size L (e.g., the radius of a sphere), the concentration of counterion at the ceneter cannot exceed the value of (κ-L)2c despite of the surface charge density. When -Λ J O(100), the counterion concentration at the center varies only with L-2. For a fixed amount of salt addition, one would expect that R decreases with increasing -Λ since more counterions are released to the aqueous core. However, an interesting phenonmenon has been observed. As illustrated in Figure 8, there exists a minimum value of R at Λmin. When -Λ > (-Λ)min, R increases very slowly to
Tsao
Figure 9. Relations between the mean salt concentration (κsL)2 and R in a sphere for different surface charge densities.
a limiting value R∞. In other words, for a given amount of added salt, the system exhibits a limiting value of the counterion concentration at the center (κ-L)2c , which is corresponding to R∞, with increasing surface charge density (-Λ . 1). Note that one is unable to attain the condition of R ) 1 for z:z salt addition because of dissociation of counterions from the inner surfaces of the surfactant shell. However, when a large amount of salt, e.g., -Λ , (κsL)2, has been put in, R f 1. To keep a constant value of R, the amount of salt that has to be added grows with increasing surface charge density as (-Λ) e (-Λ)min. When one continues increasing -Λ, however, a slowly decreasing amount of salt is needed to maintain a constant R. Eventually, only a fixed amount of salt is required regardless of the surface charge density. This is a consequence of counterion condensation on the surface. If 0.01 M 1:1 salt is added in a 10 nm aqueous core, one obtains R ≈ 0.8 at a high enough surface charge density. The variation of the mean salt concentration with R for various surface charge densities is depicted in Figure 9. This representation may be useful in practice. Note that the values of R are in the order (R)Λ)-10 < (R)Λ)-30 < (R)Λ)-100 < (R)Λ)-2 for a known amount of salt addition. This result merely reflects the conclusion illustrated in Figures 8 and 9. It is worth noting that to apply our results to the microemulsion systems, especially high surface charge density, further refinement such as the presence of a Stern layer and incomplete dissociation from the ionic surfactant shell have to be accounted for. For a given size L, if the surface charge density σ continues increasing, the assumption of point ions in the PB equation will break down, particularly near the surface. To understand the degree of accuracy and limitations of the PB equation, the approaches which take into account the finite size of ions, such as molecular simulations, are necessary and currently studied. Acknowledgment. This research is supported by National Council of Science of Taiwan under Grant No. NSC 88-2214-E-008-010. LA981516N
