Langmuir 1995,11, 1925-1933
1925
Approximate Expressions for the Surface Potentials of Charged Vesicles P. K. Yuet and D. Blankschtein* Department of Chemical Engineering and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received October 20, 1994. In Final Form: March 3, 1 9 9 P In the development of a theoretical descriptionof the formation and stability of spontaneous cationicanionic vesicles, one requires a rapid, yet reasonably accurate, computational method to calculate the electrostaticfree energy,gel,, of a charged vesicle. In turn, the evaluation ofgel, via the charging process requiresknowledge of the surface potentialsof the charged vesicle. With this in mind, we derive approximate expressions for the outer and inner surface potentials of a charged vesicle using the nonlinear PoissonBoltzmann (PB) equation. The derivation is carried out in two stages. The first stage is based on a generalization of the dressed-ionic micelle theory to the case of charged vesicles. Specifically, the PB equationis utilized to estimate the potential gradients at the outer and inner surfaces of the vesicle, which are then substituted in the two boundary conditions that describe the variation of the electric field across the boundaries. Combined with an expression relating the inner surface potential to the center-point potential,a set of three algebraic equationsis obtained. This set of equationscan then be solved numerically to calculatethe two surfacepotentials of the vesicle. In the second stage, by expanding around the surface potentials which correspond to a vesicle having an electrically neutral interior, the two surface potentials are expressed approximately in terms of the surface charge densities and other known vesicular characteristics such as the size of the vesicle. The resulting surface potentials can then be estimated directly and analyticallywithout resorting to any numerical procedure. In general, the surface potentials obtained by using the equations derived in the two stages are found to agree well with those obtained by a direct numerical integration of the PB equation. The approximate expressions for the vesicle surface potentials derived in this paper eliminate the need for a direct numerical integration of the PB equation, thus providing a much more efficient computational route. This, in turn, greatly facilitates the evaluation of the electrostatic free energy of a charged vesicle via the charging process.
I. Introduction Vesicles are widely used as model cells and are potentially useful as drug carriers.1*2 The formation of vesicles usually requires the input of energy, for example, in the form of sonication. However, vesicles can form spontaneously with some double-tailed surfactant^^-^ and in certain aqueous mixtures of cationic and anionic ~urfactants.~ To, ~understand the formation and stability of these spontaneously-forming vesicles, it is important to develop a theoretical description of spontaneous vesiculation, including the evaluation of the electrostatic free energy of the charged vesicles. A unilamellar vesicle is a spherical colloidal particle composed of an aqueous compartment enclosed by a bilayer, which is, in turn, surrounded by an aqueous environment(see Figure 1).The vesicular bilayer consists of an outer and an inner leaflet of amphiphilic molecules, with their hydrophilic moieties (hereafter referred to as “heads”) in the aqueous environment and their hydrophobic moieties (hereafter referred to as “tails”)aggregating to form the hydrophobic region of the bilayer. In the
* To whom all correspondence should be addressed. @
Abstract published in Advance ACS Abstructs, June 1, 1995.
(1) Lasic, D. D. Liposomes: From Physics to Applications; Elsevier: Amsterdam, 1993. (2) Gregoriadis, G., Ed. Liposomes as Drug Curriers;John Wiley & Sons: Chichester, 1988. (3) Ninham, B. W.; Evans, D. F.; Wei, G. J. J.Phys. Chem. 1983,87, 5020.
(4)Hashimoto, S.; Thomas, J. K; Evans, D. F.; Mukhejee, S.; Ninham, B. W. J. Colloid Interface Sci. 1983,95, 594. ( 5 ) Talmon, Y.;Evans, D. F.; Ninham, B. W. Science 1983,221. (6)Brady, J. E.; Evans, D. F.; Kachar, B.; Ninham, B. W. J. Am. Chem.SOC.1984,106,4279. (7) Kaler, E. W.; Murthy, A. K; Rodriguez, B. E.; Zasadzinski, J. A. N. Science 1989,245, 1371. (8) Kaler, E. W.; Herrington, K L.; Murthy, A. K; Zasadzinski, J. A. N. J. Phys. Chem. 1992,96,6698.
0743-7463/95/2411-1925$09.00/0
Region 3
(Outer Aqueous)
+
?+ i+
/Ri
I
Figure 1. Schematic diagram of a positively-charged vesicle showing the inner and outer aqueous regions, separatedby the hydrophobicregion composed of the surfactanttails. Thevesicle is assumed to be spherical, and the charges are assumed to be smeared on the surfaces at Ri and& the inner and outer radii, respectively. modeling of cationic-anionic vesicles, the free energy of vesiculationgneeds to be minimized with respect to many configurational variables, including the distribution of molecules between the outer and inner leaflets of the bilayer as well as the bilayer composition (that is, the mole fraction of one of the components in each leaflet). Consequently, the minimization procedure will invariably sample a large region in configuration space, which implies (9) The free energy ofvesiculation is the free-energychange associated with the formation of a vesiclefrom itsconstituent surfactant monomers in an aqueous environment.
0 1995 American Chemical Society
Yuet and Blankschtein
1926 Langmuir, Vol. 11, No. 6, 1995 that the electrostatic free-energy contribution to vesiculation needs to be evaluated repeatedly for many different combinations of the configurational variables. For this minimization procedure to be computationally efficient, therefore, it is essential to develop a fast and reasonably accurate method for calculating the electrostatic free energy of the charged vesicle. Several theoretical approaches may be utilized to estimate the electrostatic free energy of a charged vesicle. For example, the simple “capacitor”model has been used in the treatment of vesicles composed of zwitterionic surfactants.10-12 Other more rigorous approaches, such as those involving the calculation of the internal energy and entropy associated with demixing the ions in the aqueous environment,13J4can also be applied to charged vesicles. However, this requires knowing the distribution of ions as well as the potential profile around the vesicle. A more direct approach involves calculating the electrostatic free energy in terms of the reversible work required to charge the outer and inner surfaces of the vesicle s i m u l t a n e o ~ s l y . ~Mathematically, ~-~~ this charging process can be described as followsls
(1) where Gelecis the electrostatic free energy per vesicle,
d (.f,is the final outer (inner) surface charge density, ly,
is the electrostatic potential a t the outer (inner) surface, A, (Ai) is the outer (inner) surface area, and ilis the charging parameter, which is the fraction of the final charge density on the surface at any charging stage. Note that is the instantaneous outer (inner) surface charge density. For a thorough discussion of the physics behind this charging process, the reader is referred to ref 18. In the vesicular case, infinitesimal charges are moved from infinity to the outer and inner surfaces until the two surfaces reach their final charge densities. After each charging step, the system is allowed to equilibrate; that is, the ions in the outer and inner aqueous regions are allowed to attain their equilibrium distributions before another infinitesimal charge is brought onto the surfaces. In order to perform the integration in eq 1,one needs to calculate the surface potentials (qoand vi) a t any instantaneous surface charge densities characterized by 2. In principle, the surface potentials may be calculated by solving the Poisson-Boltzmann (PB) equation (see section 11). Unfortunately, an analytical solution of the PB equation in spherical geometry is not yet available, and therefore, an often tedious numerical integration procedure is r e q ~ i r e d . ~Consequently, ~,~~ it is quite prohibitive, from a computational standpoint, to utilize the PB equation in the aforementioned minimization procedure. In this respect, several approximate analytical solutions of the PB equation have been developed for a (vi)
Ad (24)
(10) Israelachvili, J . N.; Mitchell, D. J.; Ninham, B. W. J . Chem. Soc., Faraday Trans. 2 1976,72, 1525.
(11)Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. Biochim. Biophys. Acta 1977,470, 185. (12)Carnie, S.;Israelachvili, J. N.; Pailthorpe,B. A.Biochim.Biophys. Acta 1979,554, 340. (13)Gunnarsson, G.; Jonsson, B.; Wennerstrom, H. J . Phys. Chem.
1980,84,3114. (14)Marcus, R. A. J . Chem. Phys. 1956,23,1057. (15)Evans, D.F.; Mitchell, D. J.; Ninham, B. W. J . Phys. Chem. 1984,88,6344. (16)Bockris, J . O’M.; Reddy, A. K. N. Moclern Electrochemistry; Plenum: New York, 1970;Vol. 1. (17)Stigter, D.J . Colloid Interface Sci. 1974,47,473. (18)Verwey, E. J. W.; Overbeek, J . Th. G. Theory ofthe Stability of Lyophobic Colloids; Elsevier: New York, 1948. (19)Feitosa, E.; Neto, A. A,; Chaimovich, H. Langmuir 1993,9,702. Vanderkooi, G. J . Colloid Interface Sci. 1977,61,455. (20)Mille, M.;
single charged sphere in an electrolyte s o l ~ t i o n . ~In~ - ~ ~ particular, Evans, Mitchell, and Ninham (EMN) derived15,25~26 an analytical expression for the electrostatic free energy of ionic micelles in their development of the dressed-ionic micelle theory. Mitchell and Ninham later extended27this formulation to charged vesicles, where they assumed that the interior of the vesicle is electrically neutral and that the electrostatic potential at the center of the vesicle is zero. Although these assumptions simplify the mathematical complexities, they can be restrictive under conditions of low ionic strength andor small vesicle radius where the potential in the interior of the vesicle may not decay to zero at the center. In addition, the assumption of an electrically neutral interior implies that the electrostatic potential does not vary across the hydrophobic region, which is valid only when the vesicle has similar outer and inner surface charge densities. In this paper, we derive approximate relations between the surface potentials and the surface charge densities in order to calculate the electrostatic free energy of a charged vesicle. However, we do not make the assumptions of zero center-point potential and electroneutrality in the interior of the vesicle. Consequently, a solution strategy different from that of EMN is required, since the outer and inner surface potentials are coupled through the potential profile in the hydrophobicregion. The derivation of the approximate relations is presented in two stages. First, in section 11, we derive a set of three approximate algebraic equations describing the relations between the surface potentials, the center-point potential, and the surface charge densities, based on a generalization of the approach of EMN. The two boundary conditions at the outer and inner surfaces of the vesicle serve as the backbone of this derivation. This set of equations can then be solved numerically, and the resulting surface potentials can be used in eq 1to evaluate the electrostatic free energy of the charged vesicle. In the second stage, we introduce, in section 111,other approximations in order to obtain analytical expressions for the surface potentials. Using these analytical expressions, the surface potentials can be calculated directly without any numerical procedure, and therefore, Gelecin eq 1can be calculated much more efficiently. Accordingly,the derivation in the second stage represents an additional improvement on the efficiency of utilizing eq 1,as compared to simply utilizing the approach of EMN. A detailed derivation of the analytical expressions for the vesicle surface potentials is presented in the Appendix. The results of the calculations using the formulations derived in sections I1 and I11 are presented in section W , along with a comparison to the results obtained by a direct numerical integration of the PB equation. Finally, concluding remarks are presented in section V.
11. Implicit Relations between Surface Potentials and Surface Charge Densities In what follows, we model the charged vesicle as composed of three regions, which are separated by two charged surfaces (see Figure 1). Regions 1and 3 are the aqueous domains containing water and ions, and Region 2 is the hydrophobic domain made up of the surfactant tails. It is assumed that the ions can freely cross, but not (21)Bentz, J. J. Colloid Interface Sci. 1982,90,164. (22)Martynov, G.A. Colloid J. (USSR)1976,38,995. (23)Ohshima, H.; Healy, T. W.; White, L. R. J . Colloid Interface Sci. 1982,90,1. (24)White, L. R.J . Chem. Soc., Faraday Trans. 2 1977,73,577. (25)Evans, D.F.;Ninham, B. W. J . Phys. Chem. 1983,87,5025. (26)Mitchell, D.J.;Ninham, B. W. J . Phys. Chem. 1983,87,2996. (27)Mitchell, D.J.; Ninham, B. W. Langmuir 1989,5, 1121.
Surface Potentials of Charged Vesicles
Langmuir, Vol. 11, No. 6, 1995 1927
accumulate in, the hydrophobic Assuming that both the surfactant and the added salt are symmetric electrolytes having the same valence, z , and that the system is spherically symmetric, the nonlinear PB equation for each of the three regions can be written as follows 1. region l ( 0 Ir
l.atx=O
-&=o dY1
IRi)
d2Y, dx2
dY1 = sinh(y,) + -2 -
-
2. region 2 (Ri
The boundary conditions for the set of differential equations (2)-(4) can be written as follows
r
x dx
2. a t x = K&
Y1 = Y o
(9)
Y1 = Y 2
(11)
Y2 = Y 3
(13)
=xi
IR,)
d2Y2 2 dY2 -+--=o
dx
x
&2
(3)
3. region 3 ( r =- R,) d2Y3 + 2dY3 = sinh(y3) &2 x dx
(4)
j = 1-3
(5)
where yj = e z v / k T ,
4. as x X=KJ
(6)
-
00
dY3 --0 Kw=/
8nnoe2z2 E$T
dx
(7)
In eqs 2-7, vj is the electrostatic potential in region j (1,2, or 31, yj is the reduced potential in region j , r is the radial coordinate, K~ is the inverse of the Debye screening length, cw = 4~tr7~cO is the permittivity of water, where qw is the dielectric constant ofwater, and eo is the permittivity in vacuum, K is the Boltzmann constant, e is the elementary charge, Tis the absolute temperature, and no is the average ion concentration, which is the sum of the concentrations of the surfactant monomers and the added salt present in regions 1and 3.29 The homogeneous nature of eq 3 is a consequence of the assumption that there is no ion accumulation in the hydrophobic region (region 2), which implies that the charge density in the hydrophobic region is zero. Note that in utilizing the PB equation, we are making the usual assumptions of smeared surface charges and point-sized ions.16 The smearing of charges may be viewed as resulting from the rapid motion of the molecules within each leaflet of the bilayer and, therefore, from a timeaveraging point of view, appears to be a reasonable description. The assumption of point-sized ions does not appear to be too restrictive, as compared to the case of micelles, since the radius of Q cationic-anionic vesicle is typically larger than 300 A,7 whereas the size of a counterion is of the order of 1-2 A. In addition, for simplicity, we also assume that the dielectric constants, both in water and in the hydrophobic region, are constant and neglect other effects such as dielectric saturation (the reader is referred to ref 30 for a detailed discussion of these effects). (28)Tenchov, B. G.; Koynova, R. D.; Raytchev, B. D. J. Colloid Interface Sei. 1984, 102, 337. (29) Note that the equilibrium surfactant monomer concentration is typically not known apriori. In order to obtain the surfactant monomer concentration, from which no can then be computed, one needs to calculate the overall free energy of the vesicular solution iteratively subject to the constraint of surfactant mass balance. (30) Bottcher, C. J. F. Theory of Electric Polarization; Elsevier: Amsterdam, 1973;Vol. I.
Y3
=0
(15)
where a,,and ai are the charge densities at the outer and inner surfaces, respectively, cj = 4nr7jcois the permittivity in region j (1, 2, or 3), where ~j is the dielectric constant in region j , and yo is the reduced center-point potential, that is, the reduced potential at the center of the vesicle. Although regions 1and 3 contain ionic solutions, and may therefore have lower dielectric constants than pure water,16 we assume, again for simplicity, that €1 and €3 are equal to cw. Equations 10and 12 describe the variation of the electric field across the inner and outer surfaces, respectively, according to Gauss’ law. Equations 11and 13 state the continuity of the electrostatic potentials at the inner and outer surfaces, respectively. Equation 8 is the requirement of spherical symmetry, while eq 14 ensures that the entire system, that is, the charged vesicle and the aqueous solution in regions 1and 3, is electrically neutral. As mentioned earlier, the potential profiles in the three regions can be obtained by numerically integrating eqs 2, 3, and 4, subject to the boundary conditions given in eq 8 through eq 15. This integration starts at the center of the vesicle, that is, at x = 0, and the solution then propagates across the three regions until it approaches infinity. However, in the calculation of the electrostatic free energy, Gelec,using the charging process, the only relevant quantities are the two potentials at the outer (v,) and inner (vi) surfaces, as shown in eq 1. In other words, in order to evaluate Gelec,one does not need to know how the potential varies with the radial distance. Accordingly, a numerical integration of the PB equation will require unnecessary efforts spent on calculating the entire spatial potential profile. Indeed, eqs 10 and 12, which relate the potential gradients at the outer and inner surfaces through the surface charge densities, may be used to obtain the two surface potentials directly. More specifically, the PB equations presented in eqs 2-4 may be used simply to express the surface potential gradients
Yuet and Blankschtein
1928 Langmuir, Vol. 11, No. 6, 1995 in terms of the surface potentials, which can then be calculated by solving eqs 10 and 12.
obtain the following two equations
Following the derivation in ref 15, we obtain an approximate expression for the potential gradient a t X, in region 3 from eq 4;that is,
-1dY3
dxx.
2 1+Xo cosh(y3,J2)
ria)[
x-2sinh-
+ 1]
(16)
where y3,, denotes the reduced potential (see eq 5 ) at x = X,, that is, at the outer surface of the vesicle. Two approximations are involved in deriving eq 16, and the reader is referred to ref 26 for complete details. One of the approximations, namely, the replacement of the first derivative, dy$dx, in eq 4 with the result from planar geometry, leads to an inconsistency which has already been discussed by H a ~ t e r and , ~ ~its validity can only be judged aposteriori. Similar approximationscan be applied to eq 2. Specifically, incorporating eqs 8 and 9, the approximate equation for the potential gradient at x =Xi in region 1 can be written as
where
gb,) =
C COS^ y1 - cash yo)
(18)
and y1,i denotes the reduced potential at x = Xi, that is, at the inner surface of the vesicle. The difference between eqs 17 and 16 originates from the different boundary conditions used in their derivation. To further simplify eq 17 by carrying out the integration, the functiong(y1) in the integrand is replaced by 2(cosh y1- 1);that is, the integration is carried out by treating yo as zero (see eq 18). This approximation should be valid for high ionic strengths, since in this case yo is essentially zero due to strong screening. Applying this approximation, eq 17 becomes
b3,o
[
2sinh(?2")
4na.e~ e2 1
(iiij
- Y1,i) =
2 1+-
Xocosh(y3,J2) + 1
-1
A+----(r G~KJZT c1 xi2 = ~0sh(y,,~/2) - cosh(yd2) 3p0
p'i)
~inh~(y,,~/2)
]
(23)
Note that eq 22 can be easily reduced to EMN's expression for a micelle (eq 11in ref 26). This can be seen as follows. In a micelle, region 1is also hydrophobic, which implies that there is no ion accumulation, and, therefore, the potential gradient at the inner surface in region 1 (at x =Xi) is zero. In addition, there is no charge on the inner surface; indeed, the inner surface does not exist in a micelle, and the outer surface corresponds to the aqueous/ hydrocarbon core interface. Consequently, dylldx and ai can be set to zero in eq 10, which results in dyddx at x = Xi being zero. From eq 20, then, ~ 3becomes , ~ equal to y~,i, and eq 22 becomes identical to eq 11in ref 26. Since the center-pointpotential,yo, is not known apriori, we have three unknowns, yo, y~,i,and y 3 , O ) and two equations, eqs 22 and 23. Consequently, we need one more relation in order to calculate the potentials at the outer and inner surfaces. This additional relation can be obtained by considering only the inner aqueous region (region 1). This region may be viewed as a spherical aqueous cavity surrounded by a charged surface of radius Ri. Several expressions can be found in the literature which describe the potential profile within such a cavity.28p32-33 The relation given by Tenchov and co-workers,28 which is adopted in the present work because of its simplicity, expresses the inner surface potential, yI,i, in terms of the center-point potential, yo; that is, where
2 sinh
k)[X, 1-
2 co~h(y,,~/2) - cosh(yd2) sinh2(y1,i/2)
]
sinh(Xi)
Yl = Xi
(19) m
Equation 3, which is a homogeneous differential equation, can be solved exactly to give the potential gradients at the two surfaces. Specifically,
an=
22n(22n- 1) 24n(2n)!
(27)
22"(22" - 1)
"= 3(2n + 1)2(2n)! We now have expressions for the four derivatives at the outer and inner surfaces, given by eqs 16,19,20, and 21. These four derivatives are related by the two boundary conditions, eqs 10 and 12, through the outer and inner surface charge densities. Therefore, substituting these expressions for the four derivatives in eqs 10 and 12, we (31)Hayter, J. B.Langmuir 1992,8,2873.
Equation 24 is accurate [or a surface charge density of up to about 2.4 x CIA2 (1.5 elnm2).28Note that in a cationic-anionic vesicle, because of the mixing of positive and negative charges, the surfacs charge density is C/A2 (0.1 e/nm2).34 typically less than 0.16 x (32) Curry, J.E.;Feller, S.E.; McQuamie, D. A. J. Colloid Interface Sei. 1991,143, 527. (33)Lampert, M. A.;Martinelli, R. U.Chem. Phys. 1984,88,399. (34) Chiruvolu, S.Microstructure ofvesicle-FormingSurfactant and Lipid Solutions. Ph.D. Thesis, University of California, Santa Barbara, 1994.
Surface Potentials of Charged Vesicles
4,d,
For given values of no, Ri, and Ro,eqs 22,23, and 24 can be solved simultaneously to find yo, yl,i, and y3,o. Although a numerical procedure is still required because of the implicit nature of these equations, it is much less computationally intensive than that involved in the direct numerical integrations of the PB equations in the three regions, particularly when one of the boundaries is at infinity (see eqs 14 and 15). The three approximate algebraic equations, (22-24), thus represent the initial improvement i n the efficiency of using the PB equation for the calculation of the electrostatic free energy of a charged vesicle. It is important to emphasize at this point that it ~ , is the accuracy of the surface potentials, y1,i and ~ 3 , and not of the center-pointpotential, yo, that we are interested in. As shown in eq 1, only the surface potentials are involved in the calculation of the electrostatic free energy using the charging process, with the center-point potential never playing an explicit role. Consequently, in determining the validity of the approximate solutions with respect to the electrostatic free energy, the accuracy of the surface potentials, and not of the center-point potential, should be of primary importance. 111. Approximate Analytical Expressions for the Surface Potentials The computational efficiency of utilizing the PB equation may be further improved if one can obtain analytical expressions of y1,i and ~ 3 so, that ~ they can be evaluated directly from other known quantities such as the surface charge densities and the outer and inner radii of the vesicle. This section briefly discusses the derivation of such approximate expressions. A more detailed derivation can be found in the Appendix. Consider eqs 22 and 23. The hyperbolic functions appearing in these two equations may be linearized around the surface potentials which correspond to a vesicle having an electrically neutral interior and zero center-point potential. This reference configuration is chosen mainly for convenience. Indeed, as shown in the Appendix, such a vesicle will have the two surface potentials and the center-point potential completely decoupled, such that the two surface potentials can then be evaluated separately. By choosing this reference configuration, we are assuming that the surface potentials of a charged vesicle do not deviate much from those of the reference vesicle. Equations 22 and 23 thus become
and
respectively, where so,si,y , and the coefficientsdl, B1,A3, and B3 are given in the Appendix (see eqs A3, A4, A5, A13, A14, A10, and A l l , respectively). The outer surface potential, ~ 3 , can ~ , be expressed in terms of the inner surface potential, yl,i, using eq 29. The center-point potential, yo, can also be expressed in terms of y1,i by inverting eq 24. Substituting the resulting expressions
Langmuir, Vol. 11, No. 6, 1995 1929 Table 1. Comparison between the Approximate Solutions and the Numerical Integration of the PB Equations for no = 0.1 M (&Ri = 22.9)O
c/&
c/&
I I11
2.4
2.4 0.3 0.07 0.01 2.4 0.3 0.07 0.01 2.4 0.3 0.07 0.01
0 0 0 0 0 0 0 0 0 0 0 0
0.3
0.01
0 0 0 0 0 0 0
0 0 0 0 0
wo, mV
Vi, mV I1 I11
VC, m v 10214, 1021gf,-
I 132.5 132.3 132.3 132.3 39.8 39.2 39.1 39.0 2.6 1.7 1.5 1.4
132.7 132.6 132.6 132.6 39.9 39.3 39.1 39.1 2.5 1.7 1.5 1.4
132.4 132.3 132.3 132.3 39.8 39.2 39.0 39.0 2.6 1.7 1.5 1.4
I
I1
I11
131.4 37.4 10.0 2.0 131.3 37.0 9.5 1.5 131.3 36.8 9.3 1.3
131.1 37.3 10.0 2.0 131.1 36.9 9.5 1.5 131.0 36.8 9.3 1.3
131.4 37.4 10.0 2.0 131.3 37.0 9.5 1.5 131.3 36.8 9.3 1.3
a I: Numerical solution of eqs 22,23,and 24. 11: Solution of eqs 32 and 33. 111: Numerical integration of the PB equations (we =
k Tydez ).
in eq 30 and rearranging the terms, we obtain a polynomial in y1 = yl,i/Yl. Specifically,
P3'9i6
- 2P314- (B,FY~'X~ -
(AIF
1 ~ ,-2
+ BIE)XiYljjl - AIEXi = 0 (31)
where Yl is given in eq 25 and the coefficients PS,E, and F are given in the Appendix (see eqs A20,A22, and A23, respectively). Equation 31 may be solved numerically to obtain 51, or equivalently, y1,i. However, for small 91, we can neglect the fourth- and sixth-order terms in eq 31, and $1 can then be expressed approximately as follows
y1 M - (A1F + B$)Y1 f 2(B1FY,2 - 1/Xi) 1 (A@'+B$)Yi
-I[
1-
2 (BlFY12- 1/Xi)
+
'
(BIFY? - l/XJ
}"'
(32)
The selection of the or - sign in eq 32 is discussed in the Appendix followingeq A25. Oncey1 is known, ~ 3can, be obtained from eq 29; namely,
Since now y1,i and ~ 3 can , be ~ evaluated analytically for given outer and inner vesicle radii and surface charge densities, the integration in eq 1can be performed much more rapidly. In other words, at each charging stage, the instantaneous surface potentials can now be calculated analytically using eqs 32 and 33, rather than by solving the three algebraic equations, eqs 22,23, and 24 numerically. Equations 32 and 33 should therefore provide a much faster route, as compared to the numerical solution of eqs 22,23, and 24, in the calculation of the electrostatic free energy of a charged vesicle. However, as detailed in the Appendix, in going from the implicit relations described by eqs 22,23, and 24 to the analytical expressions given by eqs 32 and 33, we introduce more approximations, which, as shown in the next section, may cause a loss in accuracy in some cases.
IV. Results and Discussions The surface potentials obtained by solving eqs 22, 23, and 24 (solution I) and by using eqs 32 and 33 (solution 11) are shown in Tables 1-3 for three different ionic strengths. The results of a direct numerical integration
~
Yuet and Blankschtein
1930 Langmuir, Vol. 11, No. 6, 1995
Table 2. Comparison between the Approximate Solutions and the Numerical Integration of the PB Equations for no = 0.01 M (K,Ri = 7.2)O
2.4
0.3
0.01
a
2.4 0.3 0.07 0.01 2.4 0.3 0.07 0.01 2.4 0.3 0.07 0.01
1.3 1.3 1.3 1.3 0.8 0.8 0.8 0.8 0.1 0.08 0.06 0.05
0.95 0.95 0.95 0.95 0.7 0.7 0.7 0.7 0.1 0.08 0.06 0.05
191.4 191.3 191.2 191.2 90.1 89.2 88.7 88.5 10.4 7.3 5.7 5.0
192.1 191.9 191.9 191.8 91.0 90.0 89.5 89.3 10.4 7.3 5.7 5.0
191.4 191.3 191.2 191.2 90.1 89.1 88.6 88.4 10.4 7.3 5.7 5.0
190.2 83.5 28.8 6.9 190.1 82.8 27.4 5.3 190.1 82.3 26.2 3.9
189.7 82.9 28.8 6.9 189.6 82.2 27.3 5.3 189.5 81.7 26.1 3.9
189.6 83.5 28.8 6.9 190.1 82.8 27.4 5.3 190 82.2 26.2 3.9
I: Numerical solution of eqs 22,23, and 24. 11: Solution ofeqs 32 and 33. 111: Numerical integration ofthe PB equations (vc= kTydez).
Table 3. Comparison between the Approximate Solutions and the Numerical Integration of the PB Equations for no 0.001 M (&Ri = 2.3)"
vc, m v 10214, CIA2 2.4
0.3
0.01
Vi,
mV
vo,
=
mv
10216,, CIA2
I
I11
I
I1
I11
I
I1
I11
2.4 0.3 0.07 0.01 2.4 0.3 0.07 0.01 2.4 0.3 0.07 0.01
58.5 58.5 58.5 58.5 44.2 44.1 43.9 43.9 17.5 13.9 11 8.8
36.6 36.6 36.6 36.6 34.5 34.5 34.4 34.4 17.9 14.6 11.7 9.6
250.6 250.5 250.4 250.3 148.6 147.6 146.9 146.5 40.5 31.4 24.2 19.3
251.6 251.4 251.4 251.3 151.2 150.2 149.4 149 41.6 31.3 24.3 19.5
250.6 250.5 250.4 250.3 148.6 147.6 146.9 146.5 42.4 33.4 26.1 21
249.4 139.9 66.2 19.4 249.3 139.1 63.6 15.4 249.2 138.2 60.3 10.5
248.5 137.6 65.1 19.6 248.4 136.9 62.5 15.6 248.3 136 59.4 10.5
249.3 139.7 66 19.4 249.3 139 63.4 15.4 249.2 138.1 60.2 10.5
*
a I: Numerical solution of eas 22, 23, and 24.11: Solution of eqs 32 and 33, and 111: Numerical integration of the PB equations (qc= kTydez).
of eqs 2, 3, and 4 (solution 111) are also shown in the corresponding tables. As emphasized earlier, we are interested mainly in the outer and inner surface potentials, since they are directly involved in the calculation of the electrostatic free energy (see eq 1).Accordingly, attention should be focused on the accuracy of vo= kTy3,dez and vi = kTy1,iIez. The center-point potential, vc = kTydez, is also shown for completeness. Note that in using the analytical expressions, eqs 32 and 33, one need not evaluate the center-point potential, and therefore, vc corresponding to solution I1 is not shown in the tables. The following typical parameter values were used in all the calculations: T = 25 "C, R, = 265 A, Ri = 220 A, qw = 78.54, and qz = 2.5.19 As shown in Tables 1and 2,at the higher ionic strengths (no = 0.1 and 0.01 M), the agreement between the approximate solutions (I and 11) and the results of the numerical integration (111)are excellent. The largest error is less than 1%. This is to be expected because, as explained in the Appendix, most of the approximations made in the present formulation are based on large Xi = K ~ values, R ~ that is, large vesicle radii or high ionic strengths (see eq 7). In particular, at the higher ionic strengths, the charges on the inner surface are screened so strongly that the center-point potential,yo, is essentially zero for a range of charge densities that covers 2 orders of magnitude. Consequently, the approximations used to effect eq 19, namely,yo = 0 and the linearization involved in the derivation of the analytical expressions, introduce negligible errors in this case, and therefore, both formulations provide excellent agreement. The effect of the approximations becomes more noticeable as the ionic strength decreases. With the ion concentration, no, equal to 0.001 M (see Table 3), one begins to observe small
discrepanciesin the inner surface potentials at low surface charge densities. The largest error in vi is about 8%,which occurs at af, and .f equal to 0.01 x CIAz. The effect of the ionic strength depends very much on the inner surface charge density. Ifthe inner surface charge density is high, the importance of yo diminishes because in this case y1,i is large compared to yo, and the dependence of yo in eq 23 becomes negligible. Therefore, even when there is significant discrepancy in yo, one still obtains very good agreement in the two surface potentials. As the inner surface charge density becomes very low (for example, CIAz), the magnitude of the center-point 0.01 x potential becomes comparable to that of the inner surface potential, y1,i. The approximation involved in eq 19 may no longer be valid, and one sees a deviation in the inner surface potential from the results of a direct numerical integration. Note that, as shown in eq 33, the accuracy of vi also determines that of vo. As can be seen in Table 3, the values of v0,corresponding to solutions I and 11, are in very good agreement with those corresponding to solution 111, even at low ionic strengths. The electrostatic free energy per molecule, gelec,calculated as gelec= GeleJNs,where N , is the total number of amphiphilic molecules in the vesicle, is shown in Table 4 for two ionic strengths (0.01 and 0.001 M). Note that N , is calculated as N , = (A, Ai)/al,where a1 is the average y e a per molecule, and is assigned a typical value of 67 A2,35and Gelecis calculated by carrying out the integration in eq 1 numerically. As before, the agreement between the approximate formulations (I and 11) and the direct integration of the PB equation (111)is very good, with the
+
(35)Israelachvili, J. N. ZntermoZecuZar and Surface Forces, 2nd ed.; Academic: London, 1991.
Surface Potentials of Charged Vesicles
Langmuir, Vol. 11, No. 6, 1995 1931
Table 4. Electrostatic Vesicular Free Energies per Molecule, gelao = GeledNs,Calculated using Equation '1 no = 0.01 M no = 0.001 M 1021af, CIA2 2.4
0.3
0.01
102'4, CIA2
I
I1
I11
I
I1
I11
2.4 0.3 0.07 0.01 2.4 0.3 0.07 0.01 2.4 0.3 0.07 0.01
5.534 2.422 2.29 2.28 3.353 0.246 0.116 0.107 3.241 0.137 0.0092 0.00035
5.532 2.434 2.302 2.292 3.341 0.246 0.117 0.108 3.227 0.136 0.0092 0.00036
5.533 2.422 2.29 2.279 3.353 0.246 0.116 0.107 3.241 0.137 0.0092 0.00036
7.767 3.467 3.222 3.197 4.769 0.473 0.231 0.207 4.553 0.261 0.0225 0.00115
7.761 3.488 3.247 3.222 4.742 0.473 0.235 0.212 4.521 0.256 0.0223 0.00117
7.766 3.467 3.222 3.197 4.769 0.473 0.232 0.208 4.552 0.261 0.0226 0.00122
The surface potentials in eq 1are calculated using I (numerical solution of eqs 22, 23, and 241, I1 (solution of eqs 32 and 331, and I11 (numerical integration of the PB equations).
largest error being about 4%. Based on these results and other calculations, the two approximate formulations should be applicable fof Xi L 2.3 (corresponding to no x 0.001 M for Ri = 220 A), for a range of suorfacecharge densities from 0.01 x to 2.4 x C/A2. The limit for Xi also depends on the inner radius, Ri, of the vesicle. WithXi 2 2.3, the error in the surface potentials is within 10% for a value of Ri down to about 40 A (no x 0.03 M). In general, the surface potentials and the electrostatic free energies obtained with the analytical expressions (eqs 32 and 33) are slightly less accurate than those obtained from the numerical solution of eqs 22,23, and 24, mainly because of the additional linearization involved. However, for most cationic-anionic vesicular systems, where the ion concentration is typically of the order of 0.01 M, the errors in the electrostatic free energies calculated using the analytical expressions are within I% of those obtained by a direct numerical integration of the PB equation (see Table 4), which is quite acceptable considering the improved efficiency associated with using the analytical expressions.
described here can also be applied to mixed micelles. Work along these lines is also underway.
Acknowledgment. This research was supported in part by the National Science Foundation (NSF) Presidential Young Investigator (PYI)Award to Daniel Blankschtein, the National Institute of Health (NIH) Biotechnology Training Grant to Pak Yuet, and NSF Grant No. DMR-84-18778administered by the Center for Materials Science and Engineering at MIT. Daniel Blankschtein is grateful to Kodak and Unilever for providing PYI matching funds. Appendix. Derivation of Analytical Expressions for the Surface Potentials In this Appendix, we derive approximate analytical expressions for the outer and inner surface potentials of a charged vesicle. Consider eqs 22 and 23 presented in section 11. Specifically, so -
V. Concluding Remarks
r(, - yl) = 2 sinh - 1+ 2 x,2 ("2") [ Xocosh(y3/2) + 1
We have derived a set of algebraic equations which relates the outer and inner surface potentials of a charged vesicle to the surface charge densities. By solving this set of equations, one can obtain the surface potentials at any surface charge densities, which can then be used to cosh(y3/2) - cosh(yd2) evaluate the electrostatic free energy of a charged vesicle (A2) via the charging process. This set of implicit equations sinh2(yl/2) can be reduced to two analytical expressions for the surface potentials by linearizing around the surface potentials where which correspond to a vesicle configuration characterized by an electrically neutral interior. The accuracy of these approximate equations is quite high, provided that K ~ R ~ is larger than 2.3. The accuracy of the analytical expressions is generally lower, although they still provide good agreement in the evaluation of the electrostatic free (A4) energy for typical cationic-anionic vesicular systems. The main advantage of these approximate equations, which is also the central point of this paper, is that they provide a faster and yet reasonably accurate method to calculate (A51 the surface potentials of a charged vesicle for given surface charge densities, as compared t o the direct numerical integration of the Poisson-Boltzmann equation. This, in Note that the surface potentials ~ 3and , y1,i ~ are abbreviated turn, greatly facilitates the evaluation of the electrostatic asy3 and y1, respectively, for clarity. Equation A1 can be free energy of a charged vesicle. rewritten as We are currently utilizing the results of this paper in a molecular-modeling study of spontaneously-forming (36) EquationA1 is an approximate expression obtained by linearizing the square-root term. cationic-anionic vesicles. Note that the methodology
]
~~
Yuet and Blankschtein
1932 Langmuir, Vol. 11, No. 6, 1995 so - +(y,
- y,) = 2 sinh
B , = cash($)
XO For small y3, one may approximate tanh(yd4) by the leading-order term, yd4. This approximation may seem rather restrictive at first glance, since the difference between tanh(y) andy reaches 30%aty = 1, corresponding to an outer surface potential of only about 100 mV. Since tanh(y) approaches a constant value of 1 as y increases, the error of this approximation is expected to scale as y a t large y. However, sinh(y) scales as the exponential of y at large y, which implies that sinh(yd2) will become the dominant term on the right-hand side of eq A6 as y3 becomes large. As a result, the accuracy in the approximation of tanh(yd4) will not be important. Indeed, using Xo= 1, the largest error in evaluating the righthand side of eq A6, due to this approximation of tanh(y3/4),is only about lo%, which occurs a t y3 x 5. The function sinh(y3/2) can also be linearized around the potential yi, which corresponds to a vesicle having an electrically neutral interior. The corresponding equation for this condition can be written as follows so = 2 sinh -
1
+ io cosh(yi/2) + 1
]‘I2
+
(Zs2
= [(l-
q + $IU2
(A81
(A161
Xi
Yo2 =
(A17) One can now invert eq 24 in section 11, which yields yo
=yl - Pd13
(A18)
91 = YlNl
(-419)
where
P3
- 1)1/2~
+ -2
The center-point potential, yo, can now be expressed in terms of y1 and y3 from eq 30, which yields
(A7)
where yi is the outer surface potential when the interior of the vesicle is electrically neutral. Note that this is equivalent to settingy3 equal to y1 in eq Al.36 The solution to eq A7 is given in ref 15. Specifically,
yi = 2 ln[z,
2,
=~
3
~
(-420)
1
with YI and Y3given in eqs 25 and 26, respectively. Note that eq A18 is valid for small 91, which should be a good approximation for largexi, since Y1increases exponentially withxi. Substituting eqs A18 and 33 in eqAl7, one obtains
G1-
= (E
+ FYal)Xi(Al+ B I Y a l )
(A21)
where
where
[ + i)” +
z3 = c o s h b ) = (1
-
-
XO
(A91
Using this linearization and the approximation for tanh(yd4)discussed above, one obtains eq 29 in section 111, where
2 F = - [1 Xi2 y + X o + B.&:
- (7
- Xi + BIX:)
1
(A23)
After some rearrangement, one obtains a polynomial in
91 as given in eq 31 in section 111.
A, = 2 s i n h g ) - y i cash(?-) Yi
(A10)
B, = c o s h p )
(All)
For eq A2,we first approximate the function cosh(yd2) by expanding around yo = 0. To retain the dependence on yo, we keep the second-order term (the first nonvanishing term that contains yo). Equation A2 can then be written as si
+ x(y - y,) = 2 s i n h g ) - 4 tanhc) + x: Xi Yo2 (A121 2Xi sinh(yJ2)
Applying approximations similar to those discussed above, one obtains eq 30 in section 111, where
A , = 2 s i n h g ) - yy c o s h g )
(A131
In order to obtain a simpler analytical solution, let us neglect the fourth- and sixth-order terms in eq 31 for the moment. This truncation should be valid for smally1,an approximation which was already used in the inversion of eq 24. It is noteworthy that by truncating the polynomial to only the second-order term, we are, indeed, applying the linearized Poisson-Boltzmann equation to the interior of the vesicle, which has the solution yo = J1. To make full use of Tenchov’s expression, which contains the first nonlinear correction to the linear solution, however, one needs to keep at least the fourth-order term in eq 3 1,and this will undoubtedly complicate any attempt to find an analytical expression. Nevertheless, the truncated polynomial can be written as
912+
+ BiJW’iYi (BlFY12Xi- 1)
-1
+
A,EXi (B1FYl2Xi- 1)
=O (A24)
The solution to eq A24 is given by eq 32 in section 111. The root ofthis quadratic equation is chosen to yield the correct limit as Xi and X, approach infinity. One can show that as Xi and X, approach infinity, eq 32 can be reduced to glY1 = y1,i = -EIF,which corresponds to the solution to
Surface Potentials of Charged Vesicles
Langmuir, Vol. 11, No. 6, 1995 1933
eqs 29 and 30 given in section 111. Specifically, consider eq 32 in section I11
y, x -
+ BJDY,
*
2(BlFYl2- llXi)
Note that the terms ylXo2and ylXp are inversely proportional to the bilayer thickness and are, therefore, finite. As X,, Xi m, one can express eqs 29 and 30 as
-
As Xi approaches infinity, the square-root term in eq A25 can be approximately written as
or
so-
- Y , , ~ )M
x,2
A, + y3,$,
(A321
From eqs A29 and A33, y3,o and y1,i can be expressed as
-
Substituting the square-root term in eq A25 by the limiting expression given in A27, with the limitxi one obtains 00,
- si - A l
Y1,i -
or Y1,i
-
E
(A291
Note that in going from the expression in A26 to that in A27, we have assumed that the numerator in A27 is positive, and in this case, we choose the positive root in eqA25 (the root with a positive sign in front ofthe squareroot term) to obtain eq A29. If, however, the numerator in A27 is negative, as may be the case when si is negative, then the negative root in eq A25 should be used in order to obtain eq A29, since in this case the term (B1E - AlF) in A27 is positive. To show that eq A29 is indeed the solution to eqs 29 and 30 given in section 111, we first rewrite E and F (see eqs A22 and A23) in the limit X,, Xi that is,
-
B,
+ ylX?
+ B , ylx? + ylXi2
3,0
(-435)
Substituting eq A34 in eq A35 and performing some rearrangement, one obtains
X?
00;
LA9408228
(y1xJii>2 yIX:+B,
]
Y , , ~= (si- A,)
+
