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Salt Fractionation Effect for Spherical Macroions M. V. Smalley,*,† W. Scha¨rtl,† and T. Hashimoto†,‡ Hashimoto Polymer Phasing Project, ERATO, JRDC, 15 Morimoto-cho, Shimogamo, Sakyo-ku, Kyoto 606, Japan, and Department of Polymer Chemistry, Graduate School of Engineering, Kyoto University, Kyoto 606, Japan Received May 23, 1995. In Final Form: January 18, 1996X Ordered structures and void formation in systems of highly charged spherical macroions are investigated using the Sogami potential. The crucial feature of the treatment is the salt fractionation effect; Sogami theory combined with the Dirichlet boundary condition (constant surface potential) yields a definite prediction for the distribution of simple electrolyte between the macroion-rich and macroion-poor regions. The treatment is restricted to the extreme case of an equilibrium between an ordered structure and voids. The voids are found to be stable in the range between aκ ) 0.816 and aκ ) 3.05, where a is the radius of the particles and κ is the inverse Debye screening length. In this range, added simple electrolyte fractionates between the macroionic crystals and the voids, accumulating strongly in the voids. Added electrolyte does not fractionate outside this range, leading to an extraordinary regime below aκ ) 0.816 and to a homogeneous liquid-like structure above aκ ) 3.05. Within the range, the hypothesis that the relevant interaction κ in the crystals is determined solely by the fractionated simple salt leads to a definite prediction for the variation in the lattice parameter with added salt. This prediction is in quantitative agreement with recent USAXS measurements and with a physically realistic value for the surface potential of the charged spheres. It is noted that the standard DLVO theory of colloid stability has nothing whatsoever to say about these interesting phenomena.
Introduction One of the most interesting features of the behavior of dilute dispersions of charged spherical macroions is the rich variety of phase behavior they exhibit. As is well-known, monodisperse polymer latex particles form an ordered structure in dilute solutions. When ionic impurities are removed from the solutions, iridescence develops under appropriate conditions, which is due to Bragg diffraction of visible light due to an ordered arrangement of the particles, as proved originally by the analysis of Luck et al.1 Hachisu et al.2 demonstrated unequivocally the formation of the structure using an ordinary microscope, and a study by Ise et al.3 revealed that the interparticle distance in the ordered structure (2Dexp), which was directly evaluated from micrographs without any theoretical assumptions, was approximately equal to the theoretical distance (2D0) calculated from the known particle concentration by assuming a uniform distribution of particles in the solution, when the particles were not highly charged (about 103 analytical charges per particle). More interestingly, 2Dexp was found to be smaller than 2D0 at low particle concentrations for particles of high charge number (about 105 analytical charges per particle). In the early 1980s, Daly and Hasting,4 finding that the observed lattice constants in a study of these systems were “typically smaller than those anticipated from the prepared sphere concentration”, ascribed the observation to evaporation of solvent from their samples, but the relation 2Dexp < 2D0 has now been well established (see * Corresponding author. Current address: Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, U.K. † Hashimoto Polymer Phasing Project. ‡ Kyoto University. X Abstract published in Advance ACS Abstracts, April 1, 1996. (1) Luck, W.; Klier, M.; Wesslau, H. Ber. Bunsen-Ges. Phys. Chem. 1963, 67, 75. (2) Kose, A.; Ozaki, M.; Takano, K.; Kobayashi, Y.; Hachisu, S. J. Colloid Interface Sci. 1973, 44, 330. (3) Ise, N.; Okubo, T.; Sugimura, M.; Ito, K.; Nolte, H. J. J. Chem. Phys. 1983, 78, 536. (4) Daly, J. G.; Hasting, R. J. Phys. Chem. 1981, 85, 294.
ref 5, for a recent review) for dispersions in sealed containers. Sogami and Ise6,7 were the first to notice the true significance of such observations, namely that it points to the existence of long-range Coulombic attraction between the charged particles. Such a long-range attraction gives a natural explanation for experimental data, showing that the ordered regions form part of a two-state structure, existing in dynamic equilibrium with disordered regions of lower particle density. This coexistence of localized ordered and disordered regions was clearly demonstrated by Ito et al.,8 who used an ultramicroscope to directly observe the two-state structure, strongly resembling the coexistence of a liquid and a solid, in a deionized suspension. Even more convincing evidence of the long-range attraction has been given by Tata et al.,9,10 who reported a liquid-gas-like equilibrium in polystyrene latex suspensions. Although there have been both calculations and experiments indicating the possibility of ordering formations in dense hard sphere systems, the liquid-gas-like equilibrium definitely cannot exist without an attractive component in the pair potential.11 All of the observations in refs 2-5 and 8-10 show that the inside of a highly charged polyball system is highly inhomogeneous with respect to the particle distribution. The extreme example of this inhomogeneity has been found in recent experiments, that of void formation.5,12 Ito et al.12 studied the time evolution of highly purified polymer latex dispersions with a confocal scanning laser microscope. In such dispersions, which were initially homogeneous, the voids grew with time when the dispersions were kept standing. The fact that the voids, with diameters on the order of 50 µm, formed more quickly in (5) Dosho, S.; Ise, N.; Ito, K.; Iwai, S.; Kitano, H.; Matsuoka, H.; Nakamura, H.; Okumura, H.; Ono, T.; Sogami, I. S.; Ueno, Y.; Yoshida, H.; Yoshiyama, T. Langmuir 1993, 9, 394. (6) Sogami, I.; Ise, N. J. Chem. Phys. 1984, 81, 6320. (7) Sogami, I. Phys. Lett. A 1983, 96, 199. (8) Ito, K.; Nakamura, H.; Yoshida, H.; Ise, N. J. Am. Chem. Soc. 1988, 110, 6955. (9) Tata, B. V. R.; Rajalakshmi, M.; Arora, A. K. Phys. Rev. Lett. 1992, 69, 3778. (10) Tata, B. V. R.; Arora, A. K.; Valsakumar, M. C. Phys. Rev. E 1993, 47, 3404. (11) Kamenetzky, E. A.; Magliocco, L. G.; Panzer, H. P. Science 1994, 263, 207. (12) Ito, K.; Yoshida, H.; Ise, N. Science 1994, 263, 66.
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the internal material than in material close to the glassdispersion interface showed that the void formation is not an artifact arising from the presence of the interface but a genuine bulk property of the dispersion. Understanding this structural inhomogeneity and the conditions which stabilize it is one of the major goals of the present paper. Model In addressing our goal, we will focus attention on the simple electrolyte distribution inside the dispersion and shall find that this is also highly inhomogeneous. We shall draw heavily on the Sogami-Ise interaction potential for spherical macroions6,7 and will solve the problem by making five strong assumptions about the system. (i) That the interaction between the macroions is governed by the Sogami potential, in particular, that the position of the minimum in the interaction potential, Rm, is given by6,7
Rm )
aκ coth aκ + 1 + [(aκ coth aκ + 1)(aκ coth aκ + 3)]1/2 κ (1)
where a is the radius of the particles and κ is the inverse Debye screening length. (ii) That the macroions maintain a constant surface potential, irrespective of the electrolyte concentration. In this case, the effective surface charge on the particles (not to be confused with the analytic surface charge, vide infra) is given by13
Z0 )
( ) ( ) Φs κ sinh 2πλB 2
(2)
where λB is the Bjerrum length, defined by
λB )
e2 4π0rkBT
(3)
with e the electronic charge, 0 the permittivity of free space, r the relative permittivity of the medium (dielectric constant), kB the Boltzmann constant, and T the temperature, and where Φs is a dimensionless surface potential, defined by
Φs )
eφs kBT
(4)
with φs the surface potential. (iii) That the two-state structure can be idealized as a perfect face centered cubic (fcc) lattice of macroions in equilibrium with pure macroion-free voids. In reality, there will, of course, be a few macroions lurking in the disordered regions, but we assume that their concentration is very small and so has a negligible effect on the equilibrium properties of the system. (iv) That the depth of the Sogami minimum is sufficient to localize the particles at the position
b ) Rm
(5)
where b is the nearest neighbor separation in the fcc lattice. (v) That the relevant interaction κ between the particles is determined solely by the background electrolyte in the macroionic phase; that is, the relevant κ to be inserted into eq 1 is given by
κ2m ) 0.107cm
(6)
where κm is the κ value in the macroionic crystal and cm the electrolyte concentration in the macroionic crystal, with κ expressed in units of inverse Angstroms and c expressed in moles/ liter. The aim of this paper is to investigate the consequences of these assumptions rather than to justify them. However, they seem to be physically realistic assumptions. Assumption i is a (13) Sogami, I. S.; Shinohara, T.; Smalley, M. V. Mol. Phys. 1992, 76, 1. (14) Derjaguin, B. V.; Landau, L. Acta Physiochem. 1941, 14, 633.
natural one given that, within the framework of mean field theory, Sogami et al.13 have derived an exact expression for the adiabatic potential of two charged plates immersed in an electrolyte solution with both Dirichlet and Neumann boundary conditions and have proved rigorously that the electric field energy induces a longrange weak attraction between the macroionic particles irrespective of the type of boundary conditions. The Sogami theory therefore has a rigorous basis in statistical mechanics, and the Sogami potential for spheres6,7 is the only serious candidate proposed to date for investigating phenomena such as void structures inside latex dispersions. In particular, the time evolution of the void structures from initially homogeneous dispersions12 points quite definitely to the existence of effective long-range attraction between the charged spheres. Interparticle interactions in charged colloids have traditionally been treated in terms of the purely repulsive DLVO (Derjaguin-LandauVerwey-Overbeek) potential,14,15 but if the pair interaction between the spheres were purely repulsive, there would be no possible mechanism for them to congregate into regions of higher density than the average. The Ostwald ripening mechanism observed in the growth of the high-density clusters5 also clearly indicates the existence of the attraction and validates the use of the Sogami potential. Assumption ii is a surprising one for some colloid scientists, especially those who favor the Nernst equation in describing the variation of surface potential with electrolyte concentration.16 There is, however, neither an experimental nor a theoretical basis for the use of the Nernst equation in colloid science.17 In the only experimental system, the n-butylammonium vermiculite system, for which the surface potential18 and salt fractionation effect19 have been independently determined, it has been proved experimentally that Φs is constant with respect to c. Furthermore, it has been proved theoretically20 that the Sogami theory with a constant surface potential gives excellent agreement with the observed salt fractionation effect in the n-butylammonium vermiculite system. In latex dispersions, Φs has not been determined experimentally as a function of c, so we make the natural choice Φs ) a constant with respect to c, on the basis of previous work.17-20 Assumption iii idealizes a liquid-gas-like equilibrium as a solid-gas-like equilibrium. As the high-density clusters and the voids have hugely different particle densities, this is not a severe approximation. After all, the density of an fcc lattice is not so different from the density of a liquid (on the order of a few percent). This approximation enables us to calculate the volume occupied by the high-density clusters and has the additional advantage that, in conjunction with assumption iv, it enables us to compare the interparticle distance in the clusters with those observed in recent USAXS (ultra-small-angle X-ray scattering) experiments on an fcc phase,21 as discussed in detail in a later section. Assumption iv is also a straightforward one. If the depth of the Sogami minimum did not greatly exceed the thermal energy, there would be no driving force for the formation of highdensity clusters in the first place. The spheres will therefore have a strong tendency to sit near the minimum in their pair interaction curve, and although many body effects will tend to contract the ordered regions,7 this effect will also be on the order of a few percent. We note that assumption iii will tend to overestimate the density of the dense regions (an fcc crystal is more dense than a liquid) and that assumption iv will tend to underestimate it (in reality, the fcc lattice will be contracted relative to the minimum position in the effective pair potential), so the errors in these two simple approximations will tend to cancel each other out in estimating the density of the dense phase. Assumption v contains two parts. First, we neglect the contribution of the counterions to the κ value in the dense regions. (15) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (16) Ettelaie, R. Langmuir 1993, 9, 1888. (17) Smalley, M. V. Langmuir 1995, 11, 1813. (18) Crawford, R. J.; Smalley, M. V.; Thomas, R. K. Adv. Colloid Interface Sci. 1991, 34, 537. (19) Williams, G. D.; Moody, K. R.; Smalley, M. V.; King, S. M. Clays Clay Miner. 1994, 42, 614. (20) Smalley, M. V. Langmuir 1994, 10, 2884. (21) Matsuoka, H.; Harada, T.; Yamaoka, H. Langmuir 1994, 10, 4423.
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As pointed out by Schmitz,22 this is the only procedure consistent with using a pair potential that is independent of the macroion concentration. Second, we consider only the electrolyte trapped between the macroions in the dense regions. This is consistent with the idea that a pair potential describes a local interaction. An equivalent set of assumptions to i-v was introduced in ref 20 in a treatment of the properties of highly charged plate macroions. Our aim here is to investigate the consequences of these assumptions in the case of the structural inhomogeneity in a system of highly charged spherical macroions. Our first task is to determine how cm depends on ca, the experimentally controlled added salt concentration. A further simplification is that we consider first the case of infinite sol dilution, when the macroionic crystal is immersed in an infinite reservoir of electrolyte solution. In this case, expulsion of salt from the macroionic region has a negligible effect on the salt concentration, cex, in the fluid external to the region bounded by the macroions.
Calculation The position of the Sogami minimum is connected to the average electrolyte concentration in the macroionic crystal because it defines the average volume occupied per macroion. The inhomogeneous distribution of ions is described as the expulsion of a certain amount of ions of the same sign as the colloid, and the result is encapsulated in a quantity g that expresses the ratio of the co-ion deficit to the total double-layer charge.20,23 For a negatively charged sphere,
g)
Nde Z0
(7)
where Nde is the deficit of negative ions per sphere and Z0 is the total surface charge per unit area. As Z0 is known as a function of Φs (see eq 2) and g can be calculated in terms of the surface potential (see below), Nde can likewise be calculated and eq 1 then used to obtain the average number density of the deficit, nde. Since the number density of the deficit of negative ions in the macroionic crystal is defined by
nde ) nex - nm
(8)
where nex is the number density of simple ions in the voids (the subscript referring to the fluid which is external to the region bounded by the macroionic spheres) and nm is the average number density of the co-ions (negative ions) in the fluid bounded by the macroions, the calculation yields nm in terms of the experimentally measurable and controllable quantity nex. This in turn gives the ratio of these two quantities, the salt fractionation factor s
s)
nm nex
(9)
Note that this definition of s, which turns out to be more convenient for treating the spherical macroion problem, is the inverse of that used in refs 20 and 23 in the treatment of the plate macroion case. As for the plate macroion case, the large (negative) surface charge on the particles causes a deficit of co-ions in the region bounded by the macroions. Negative ions (e.g. chloride ions) are expelled, and the principle of electrical neutrality requires that an equal number of positive ions (e.g. sodium ions) are expelled too: salt is fractionated from the region bounded by the macroions. Unlike the plate macroion case, it is not so straightforward to solve the electrical integral necessary to obtain g. However, as shown in the Appendix, to a sufficient level (22) Schmitz, K. S. Macroions in Solution and Colloidal Suspension; VCH: New York, 1993. (23) Smalley, M. V.; Scha¨rtl, W.; Hashimoto, T. Langmuir, in press.
of approximation, the quantity g is given by the same formula in the spherical macroion case as it is in the plate macroion case, namely23
g)
(1 - e-Φs/2) Φs 2 sinh 2
( )
(10)
Likewise, we assume that the relationship between the surface charge and the surface potential (eq 2), which is exact in the plate macroion case, also holds in the spherical macroion case. Combining eqs 2, 7, and 10, the total number of ions expelled per unit area of surface of the particles is then given by
Nde ) gZ0 )
( )
κ (1 - exp(-Φs/2)) 4πλB
(11)
Since the potential is a decaying function of r in half of the unit cell, the number density of the deficit is given by
nde )
gZ 1 V 2 p
(12)
where Z is the absolute number of surface charges and Vp is the volume of the primitive unit cell. The total charge number Z is related to Z0 by
Z ) 4πa2Z0
(13)
and the assumption of the fcc structure gives us
Vp ) 0.707b3
(14)
Combining eqs 11-14 gives the following expression for the number density of the deficit.
()
nde ) 0.398κ(1 - exp(-Φs/2))
a2 b3
(15)
We can now use the Sogami result for the position of the minimum between spherical macroions (see eq 1) to express b3 via eq 5. In order to avoid cumbersome expressions, we write this as
b)
f(aκ) κ
(16)
where the numerator in eq 1, which is a function of the product aκ only, has been labeled f(aκ). Combining eqs 15 and 16 yields
z2 nde ) 0.398κ2(1 - exp(-Φs/2)) 3 f(z)
(17)
where we have used the further abbreviation z ) aκ. The relationship between κ and λB for a simple uni-univalent electrolyte solution (the only case treated here) is
( )
κ2 )n 8πλB
(18)
where n is the number density of the simple ions, which is equal to nex in the case of infinite sol dilution. Using eq 18 to express κ2 transforms eq 17 into
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z2 nde ) 70.7nex(1 - exp(-Φs/2)) 3 f(z)
Langmuir, Vol. 12, No. 10, 1996 2343
(19)
where the value for λB appropriate for water at 25 °C has been used. Using the notations
γ ) 70.7(1 - exp(-Φs/2))
(20)
and
t(z) )
z2 f(z)3
(21)
enables us to write eq 19 in the compact form
nde ) γt(z)nex
(22)
As in the case for the plate macroion problem,20,23 the number density of the co-ion deficit is proportional to the overall co-ion number density. Substitution of this result into eq 8 and elimination of nde gives nm, so eq 9 is solved by
s(z) )
nm ) 1 - γt(z) nex
(23)
where the salt fractionation factor for spherical macroions has been relabeled s(z) to emphasize that it is a zdependent quantity. Before proceeding to a discussion of the function t(z), we note the following about the γ-factor. First, of course, it tends to zero as Φs tends to zero. There is no salt fractionation for zero surface potential (or charge). Second, for small surface potentials, the number density of the defect is linearly proportional to the surface potential. Third, in the limit of massive surface potentials, γ takes on the limiting value of 70.7. This may seem an odd looking number, but we shall show below that the maximum value of the function t(z) is 0.0126, so it implies that the maximum value of nde in this model is equal to 0.891nex. In other words, the number density of the deficit is guaranteed to be less than the number density of salt molecules. This is a satisfactory result. It is not the perfect Donnan equilibrium obtained for the plate macroion case,20,23 but in view of the brutality used in adapting eq 11 directly from the plate macroion case, it is sufficiently close to give us confidence in the method. Note that in this limit, the number density of co-ions in the fcc crystal is 0.109 times its value in the supernatant fluid, corresponding to a ratio nex/nm of 9.17. For a very highly charged system at maximum salt fractionation, there can be nearly ten times as much salt in the voids as there is in between the macroions. Since Φs must anyway be large in order to localize the macroions in a cubic structure, we should expect a strong salt fractionation effect in any system exhibiting iridescence. We concentrate now on the properties of the function t(z) ) z2/f(z)3, which depends only on the dimensionless variable z ) aκ. Some illustrative values of the function t(z) are given in Table 1, and t(z) is plotted against z in Figure 1. Since s(z), the z-dependent salt fractionation factor, is a linear function of t(z), the following discussion applies to both functions. The most remarkable feature of t(z) is that it has a turning point as a function of z, at z = 3. That this must be so is readily revealed by explicit differentiation (see below) but is most simply revealed by using the high z expansion
coth z = 1
(24)
Figure 1. Function t(z) plotted against the dimensionless variable z ) aκ. Table 1. Some Illustrative Values of the Functions f(z), t(z), and d(z) in Terms of the Dimensionless Variable z ) aK z
f(z)
t(z)
d(z)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 5.0 6.0 7.0 8.0
4.828 4.835 4.856 4.890 4.937 4.997 5.070 5.154 5.250 5.356 5.471 5.729 6.020 6.336 6.672 7.025 7.956 8.929 9.920 10.92 12.93 14.94 16.94 18.95
0 8.85 × 10-5 3.49 × 10-4 7.70 × 10-4 1.33 × 10-3 2.00 × 10-3 2.76 × 10-3 3.58 × 10-3 4.42 × 10-3 5.28 × 10-3 6.11 × 10-3 7.66 × 10-3 8.98 × 10-3 0.0101 0.0109 0.0155 0.0124 0.0126 0.0125 0.0123 0.0116 0.0108 0.0101 9.40 × 10-3
0 1.76 × 10-3 3.43 × 10-3 4.94 × 10-3 6.21 × 10-3 7.22 × 10-3 7.93 × 10-3 8.35 × 10-3 8.51 × 10-3 8.44 × 10-3 8.17 × 10-3 7.24 × 10-3 6.04 × 10-3 4.80 × 10-3 3.66 × 10-3 2.68 × 10-3 9.70 × 10-4 5.99 × 10-5 -3.98 × 10-4 -6.22 × 10-4 -7.65 × 10-4 -7.55 × 10-4 -7.00 × 10-4 -6.34 × 10-4
Within this approximation, which is a good one for z = 3 (for z ) 3, coth z ) 1.005), we have
f(z) ) z + 1 + x(z + 1)(z + 3)
(25)
and the square root term can be approximated by z + 2, giving us the very simple approximation
f(z) ) 2z + 3
(26)
The function
t(z) )
z2 (2z + 3)3
(27)
is trivially easy to differentiate, yielding
d(z) )
dt(z) 2z(3 - z) ) dz (2z + 3)4
(28)
which has a turning point at z ) 3, as stated. Note that we have labeled the derivative of t(z) with respect to z as a new function d(z), since it will turn out that this function has an important physical significance in its own right. It is also instructive to examine the low z limit of t(z). In this case we can expand the exponentials to give
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coth z )
ez + e-z 1 + z + 1 - z 1 ) ) ez - e-z 1 + z - 1 + z z
Smalley et al.
(29)
such that f(z) ) 4.83, t(z) ) z2/113, and d(z) ) z/56.3. This shows that d(z) f 0 as z f 0, which is the second stationary value of the function t(z). Between the two stationary values of t(z) at z ) 0 and z = 3, there is a point of inflection at z = 0.8. In order to identify this point more closely, we differentiate the full form of t(z) with respect to z, giving d(z) )
[
{
z 2 - z coth z + 3z2 csch2 z + f(z)4 2
]}
6 - z coth2 z + 2z coth z + 3z3 coth z csch2 z + 6z2 csch2 z
x(z coth z + 1)(z coth z + 3)
(30)
Some representative values of the function d(z), which has a turning point at z ) 0.816 and which becomes negative at z ) 3.05, are given in Table 1, and the function is plotted in Figure 2. We now use the functions t(z) and d(z) to attack the physics of the problem. Three Regions of Spherical Macroion Colloid Stability The interpretation of the physics contained in the function t(z) and its derivative d(z) is not entirely straightforward. Superficially, it would seem that the salt fractionation factor s(z) should vary continuously with added salt, and the consequences of such an interpretation are currently under investigation. However, let us consider what happens as the system develops from its initial state, which is homogeneous with respect to the particle distribution. As the dense regions form, due to the interparticle attraction, salt is fractionated into the voids and, in keeping with the adiabatic approximation that underlies the Sogami potential,13 local equilibrium with respect to the new salt distribution is rapidly established. The κ value in the macroionic region becomes smaller than the average global value and that in the voids becomes larger. The original variable z ) aκ, while still well-defined as a global average over the inhomogeneous distribution, is no longer well-defined locally. The macroion interaction has now become mediated by zm ) aκm, but zm has not yet attained its equilibrium value. As a local process, salt fractionation will develop as a function of zm rather than of z and will continue until dt/dzm ) 0, when there is no further driving force for redistribution of the simple ions. This in turn suggests that the system can only attain stability at dt/dz ) 0 or d(z) ) 0 when z is varied. This interpretation of the physics, which we now pursue, simplifies the problem considerably because there are only two possibilities, namely the points z ) 0 (no salt fractionation) and z ) 3.05 (maximum possible salt fractionation at the particular value of the surface potential). Which of these two points is found by the system at variable z (salt can be added to the system in a continuous fashion) would depend on which side of the point of inflection at z ) 0.816 the global salt concentration lies. This divides the behavior of the spherical macroion system as a function of electrolyte concentration into three regions. (i) The range 0 < aκ < 0.816. In this range, the particles will tend to congregate in the Sogami minimum and form ordered structures, but any ordered regions would be unable to fractionate salt into the disordered regions. As a result of this, adding salt might weaken the interaction between the macroions in the ordered regions and lead to an increase in the lattice spacing 2Dexp as salt is added to the system, until the critical value of aκ ) 0.816 is reached. (ii) The range 0.816 < aκ < 3.05. In this range, the particles will tend to congregate in the Sogami minimum
Figure 2. Function d(z) plotted against the dimensionless variable z ) aκ.
and form ordered structures, and the maximum possible salt fractionation effect would occur. Therefore salt which is added to the system will fractionate into the disordered regions, increasing the osmotic pressure on the ordered regions. As a result of this, adding salt will strengthen the interaction between the macroions in the ordered region and the lattice spacing 2Dexp will decrease as salt is added to the system, until the critical value of aκ ) 3.05 is reached. (iii) The range aκ > 3.05. Beyond this point d(z) becomes negative: the system can no longer maintain the salt fractionation effect. As a result, salt molecules rush into the ordered structure, melting it. According to this model, aκ ) 3.05 corresponds to the maximum possible salt concentration for maintainence of the two-state structure. At higher salt concentrations, the particles become homogeneously distributed throughout the bulk of the dispersion volume. Note that this liquid-like state is still a “secondary minimum” state. Of course, at higher salt concentrations still, coagulation will occur, namely the particles will collapse into a primary minimum state. This transition is beyond the scope of the present calculations. The most interesting region from the point of view of its peculiar inhomogeneity with respect to both the particle distribution and the salt distribution is the intermediate region (ii). This will be the region of the void formation reported by Yoshida et al.5,12,24 According to the model presented here the voids will be salt bubbles. This is a perfectly logical conclusion and one which is in accord with the Donnan equilibrium discussed previously.23 We note that Mirnik25 has concluded that the separation of a system into a colloidal crystal part and a part with disordered particles is a consequence of electrolyte being expelled from the ordered part. However, Mirnik25 described the interparticle interactions as purely repulsive, and in these circumstances the interparticle distance should always be determined by the inverse cube root of the volume fraction, irrespective of salt concentration, in contradiction to the experimental results. Such an approach cannot yield a definite prediction for the range of stability of the voids. Delville26 has also suggested that the local concentration of small ions may be higher in the voids than in the surrounding regions, without making any definite predictions. We now discuss our prediction. First note that it will be a pretty crude prediction. Because we have used linearized theory, we get little information on the behavior of the system as a function of the surface charge on the particles. Our main prediction is thus in terms of the particle radius, a, and the inverse Debye screening length, κ. Since κ2 is proportional to the (24) Yoshida, H.; Ise, N.; Hashimoto, T. Langmuir, in press. (25) Mirnik, M. Croat. Chem. Acta 1993, 66, 509. (26) Delville, A. Langmuir 1994, 10, 395.
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Table 2. Data of Matsuoka et al.21 ca (M)
a
Rmin (Å)
Rmin/ab
aκm
0 2 × 10-6 7 × 10-6 10-5 2 × 10-5 4 × 10-5 7 × 10-5 10-4
4750 4990 5100 5100 4860 4530 4140 melting
5.03 5.28 5.40 5.40 5.14 4.79 4.38
1.11 1.05 1.02 1.02 1.08 1.19 1.36
c
a
Observed experimental results. b Observed experimental results in terms of the particle radius. c aκ value obtained by fitting the observed interparticle separation to the Sogami minimum (see eq 1).
electrolyte concentration, it is immediately apparent that a z-range of 3.05/0.816 ) 3.74 corresponds to a c-range of approximately 14. In other words, the voids will be stable over a little over one decade of added electrolyte concentration. Which decade this is depends on the particle radius. Since the voids have so far only been observed by real space microscopy, the experiments have only been conducted for fairly large particles. The smallest particles studied to date24 have a ) 0.35 µm, for which the κ-range for void stability should be 2.3 × 10-4 to 8.7 × 10-4 Å-1, corresponding to a c-range between 5 × 10-7 and 7 × 10-6 M. Unfortunately, these values are too small for reliable in situ measurement of c to be possible, and it is highly unlikely that even with hyperefficient ion exchange resins it would be possible to achieve a background electrolyte concentration less than 5 × 10-7 M. It will therefore not be possible to reach a definite judgment until void formation is observed with smaller particles. Of more interest are the recent ultra-small-angle X-ray scattering studies of Matsuoka et al.21 on polystyrene sulfonate latex sphere dispersions, discussed below. Data of Matsuoka et al.21 Matsuoka et al.21 studied spherical macroions of particle radius a ) 945 Å, so the anticipated range for observation of the salt fractionation effect is 8.6 × 10-4 to 3.2 × 10-3 Å-1 in κ, corresponding to 7 × 10-6 to 10-4 M in ca, a range which is just experimentally controllable. In a USAXS study, the salt fractionation does not show up in void formation but rather in its effect on the ordered regions giving rise to diffraction. The essential experimental facts were as follows. (1) The particle radius, a, was 945 Å. (2) The volume fraction of the particles in the condensed matter system, φ, was 0.038; i.e., the particles occupied 3.8% of the volume. (3) An fcc structure was observed at all salt concentrations. (4) The interparticle spacing, 2Dexp, given as a function of ca, was as given in Table 2. It is immediately apparent from Table 2 that there is indeed a changeover in the behavior of 2Dexp vs ca around 7 × 10-6 M and that the ordered structure does indeed disappear at 10-4 M, so we seem to have some qualitative insight into the properties of the system. Before considering these fascinating results in more detail, we note that the preceding calculations have been carried out in the limit of low particle concentrations, such that the excluded salt has a negligible effect on the added salt concentration. We shall therefore have to modify our calculations slightly to allow for a feedback effect of the excluded salt on the final salt concentration in the voids. This is easily done using the principle of conservation of salt molecules, as shown in the following. In order to proceed quantitatively, we first consider facts 1-3. For a space-filling fcc structure (i.e., if the ordered structure filled the whole dispersion homogeneously), Matsuoka et al.21 noted that the nearest neighbor interparticle spacing would be 5100 Å, as observed around ca ) 10-5 M. They denoted this result by 2D0 ) 5100 Å. We first repeat this calculation.
The usual unit cell for the fcc structure is taken to be a cube with lattice parameter d ) x2b, where b is the nearest neighbor separation. For b ) 5100 Å, d ) 7200 Å and the volume of the nonprimitive cubic unit cell is d3 ) 3.75 × 1011 Å3. This cell contains four atoms, so the volume of the primitive unit cell, Vp, is equal to 9.38 × 1010 Å3. The volume, Vs, of a sphere of radius a ) 945 Å is 3.53 × 109 Å3, so
Vs ) 0.038 Vp
(31)
which corresponds to the bulk volume fraction, as required. Since the lattice parameter is known to three significant figures, the volume occupied by the macroionic region is likewise known to this accuracy. The total number of moles of salt added to the system, Nt, is a known quantity and must be equal to the sum of the number of moles in the macroionic region and the macroion-free region:
Nt ) cm(Vm - 0.038) + cexVex
(32)
where cm and cex represent the salt concentrations in the macroionic and void regions, respectively, and Vm and Vex represent their respective volumes. It is convenient to write the salt concentrations as molarities and to express the volumes of the two phases in liters, considering the total volume of the dispersion to be 1 L. We now determine the salt fractionation factor, s, by fitting the observed interparticle spacing, b, to the minimum position of the Sogami potential, Rm, using the postulate that the relevant interaction κ between the particles is determined solely by the background electrolyte in the macroionic phase.20 We choose the point in the middle of the concentration range studied, at an added salt concentration of 4 × 10-5 M, to be the fitting point. In this case (see Table 2) the approximation Rm ) b implies aκm ) 1.19, so κm ) 1.26 × 10-3 Å-1, with κm the κ value in the macroionic crystal. This determines cm via eq 6 as 1.48 × 10-5 M. Now cex is the only unknown in eq 32 expressing the conservation of the number of salt molecules, since Vm ) 0.700 1 and Vf ) 0.300 1 have been determined from the d-spacing in the macroionic region and Nt is the known quantity of added salt. Note that Nt ) 4 × 10-5 × 0.962 ) 3.85 × 10-5, because of the finite volume occupied by the macroions. Equation 32 then gives cex ) 9.49 × 10-5 M, and we have solved for the salt fractionation factor, s ) cm/cex ) 0.156. There is 6.4 times as much salt in the voids as in the region occupied by the macroionic crystal. We have now solved for the unknown γ in eq 23. Since by hypothesis, the system is sitting at the minimum of t(z) defined by z ) 3.05, for which t(z) ) 0.0126, we have γ ) 67. Substitution of this value into eq 20 gives Φs/2 ) 2.95 or φs/2 ) 74 mV, corresponding to a surface potential of 150 mV to 2 significant figures. This value accords well with the value of the analytic surface charge, Za ) 3.2 µC/cm2, given by Matsuoka et al.21 We can see this by substituting Φs/2 ) 2.95 and κm ) 1.26 × 10-3 Å-1 (the value of κ in the macroionic region at the fitting point: see Table 2) into eq 2, which gives Z0 ) 2.68 × 10-4 Å-1 ) 0.429 µC/cm2. The fit therefore corresponds to
Z0 ) 0.13 Za
(33)
This value has been reported in the literature as the one with which the Sogami potential fits the data on thermal compression of colloidal crystals,27 and is in agreement with the number of effective charges on latex (27) Ise, N.; Smalley, M. V. Phys. Rev. B 1994, 50, 16722.
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Figure 3. Function s(z) plotted against the dimensionless variable z ) aκ, when the γ-factor has been adjusted to fit the data of Matsuoka et al.21 at an added electrolyte concentration of 4 × 10-5 M.
spheres with SO3H groups determined by transference measurements.28 Furthermore, if the ζ potential can be related to the surface potential, φs ) 150 mV is typical for highly charged polystyrene latex spheres in dilute electrolyte solutions.29 The fit therefore corresponds to a realistic surface potential. This enables us to plot s(z) as a function of z for this system, as shown in Figure 3. We can now calculate the d-spacing as a function of the added salt as a continuous function of ca, as there will always be 6.4 times as much salt in the void region. The fraction of the two-state structure occupied by the macroions is given by
Vm f(z)3 ) Vsf 51003κ3
(34)
where Vsf is the volume of the space-filling structure with Rm ) 5100 Å. The volume occupied by the macroion-free region, Vex, is given by Vex ) 1 - Vm and the salt concentration in the macroion-free region, cex, is given by cex ) 6.41cm, so the equation of conservation of salt, eq 32, becomes
Nt ) cm(Vm - 0.038) + 6.41cm(1 - Vm)
(35)
where Vm is given by eq 34. The most straightforward method for solving this equation for Vm is to solve first for κm, the Debye screening length in the macroionic phase, using κm2 ) 0.107cm. Trivial algebra yields
6.30κm3 - 0.103caκm - (4.03 × 10-11)f(aκm)3 ) 0 (36) where ca is expressed in moles/liter and κm is in inverse Angstroms. This is not a simple cubic function because it includes the cube of the slowly varying function f(aκm), but eq 36 is easy to solve numerically. The interparticle spacing in the macroionic region, Rm, is then given immediately by eq 1 for any value of ca. The full Sogami prediction for the d-spacings in range ii is shown in Figure 4. It gives an excellent reproduction of the observed data. As an example of the complete set of parameters specifying the solution for a given value of ca, let us consider the point ca ) 70 µM. In this case, κm ) 1.44 × 10-3 Å-1, Rm ) 4140 Å, Vm ) 0.535 L, cm ) 1.95 × 10-5 M, Vex ) 0.465 L, and cex ) 1.23 × 10-4 M. These parameters satisfy the conservation of salt equation, reproducing the input parameter ca ) 70 µm, and give an excellent reproduction of the observed interparticle spacing of 4140 Å. Note also (28) Ito, K.; Ise, N.; Okubo, T. J. Chem. Phys. 1985, 82, 5732. (29) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: New York, 1981.
Figure 4. Fit to the data of Matsuoka et al.21 in the range of electrolyte concentrations between 10-5 and 10-4 M. The squares are the experimental points, and the continuous curve is the Sogami prediction, fitted to the data at 4 × 10-5 M. The horizontal line is the prediction of DLVO theory.
that cex is equal to 1.76 times the added salt concentration, illustrating clearly that the exclusion of salt from the macroionic region has a strong feedback effect on the salt concentration in the voids. The model gives an excellent estimate of both the salt concentration at the melting transition and the salt concentration at the observed maximum in 2Dexp. It also gives an excellent reproduction of the observed behavior of 2Dexp vs ca in range ii. It does not, however, provide us with any insight into the behavior of 2Dexp vs ca in range i. The unexpected increase of the interparticle spacing with ca in this regime can be qualitatively explained by assuming that the macroions are unable to expel salt for aκ < 0.816, but the quantitative behavior remains a challenge to theory. Discussion Although the interpretation of the behavior of t(z) and d(z) in terms of two fixed stability points, that of no salt fractionation and that of maximum salt fractionation, provides some agreement with the data of Matsuoka et al.,21 this agreement should be treated with some caution. Another perfectly natural interpretation of the physics contained in s(z) is that salt fractionation develops continuously in accordance with eq 23. This latter interpretation is currently under investigation. Another reason for caution is that there is no direct proof for salt fractionation in spherical macroion systems. In the best characterized plate macroion system, the n-butylammonium vermiculite gels, the equilibrium is a static one and the macroion-free phase can be removed with a pipet, enabling its electrolyte concentration to be measured by standard analytical techniques.19 However, in the polystyrene latex sphere system, the equilibrium is a dynamic one, and although the voids are huge (50 × 50 × 150 µm3)12 and stable (for more than 60 min),24 they cannot be removed by pipet for standard analysis and direct in situ measurements would be very difficult. In this context, it is worth re-emphasizing that the systems which can be directly imaged with a microscope should also display void stability at extremely low electrolyte concentrations. The precise control of the concentrations of simple salt at such low levels is not in principle impossible, but neither is it at all easy. Shifting the electrolyte regime to more experimentally convenient values implies using particles which cannot be directly imaged. It is therefore hard to imagine how we could directly detect the salt fractionation effect in the system studied by Matsuoka et al.21 Other data on the salt dependence of the d-spacings in charged polyball systems is scarce, in spite of the great attention which various features of these systems have
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Langmuir, Vol. 12, No. 10, 1996 2347
received during the past decade. The earliest attempt to make a systematic study appears to be that of Ise et al.,30 who remarked that “the interparticle distance in the ordered structure could be observed only in the limited range of simple salt concentrations”. We anticipate that this will be a general finding and that it is unlikely that future data will exceed the range or the accuracy of that discussed in the preceding section. It would, of course, be highly desirable to obtain similar data for spheres of different radii and surface charge densities. The main weakness of the calculations presented here is the naive assumption that we can consider the dispersions to be perfect fcc crystals in equilibrium with macroion-free voids, a ‘gas-solid’-type equilibrium which is unlikely to be realized in practice. There are other weak points too, such as in setting the interparticle separation exactly equal to the minimum position in the Sogami potential and in calculating the final interparticle separation in terms of a macroion phase κ, κm, determined solely by the fractionated added electrolyte, which assumes that neither the macroions nor the counterions contribute to the screening length. Although this approach is consistent with the use of a macroion-macroion potential independent of the macroion concentration,22 it is not necessarily correct. In view of these many failings, the quantitative agreement with the Matsuoka et al.21 data shown in Figure 4 may be fortuitous, but qualitatively we have definitely given some insight into the inhomogeneous nature of charged colloidal dispersions. The strength of our calculations is that they offer experimentalists some guidance where standard theory has nothing whatsoever to offer. It is conceivable that if the segregated distribution of the particles is taken a priori, the salt fractionation effect could stabilize the two-state structure due to a purely osmotic effect, even if the interparticle interaction were purely repulsive.31 However, a purely repulsive potential, such as the DLVO potential, offers no plausible mechanism by which the particles could phase separate from an initially homogeneous distribution, as observed experimentally,5,12,24 and the prediction of DLVO theory14,15 for the system studied by Matsuoka et al.21 is the horizontal straight line drawn at d ) 5100 Å in Figure 4. DLVO theory always predicts a homogeneous space-filling structure and has nothing to say about these interesting phenomena. The whole concept of salt fractionation, experimentally proved for clays,19,20 is quite alien to the standard theory of colloid stability and a natural consequence of Sogami theory.
by the presence of other macroions, then Φ is a function of r only:
(A2)
dV ) r2 sin ϑ dϑ dφ dr
(A3)
and using
transforms eq A1 into
∫∫sin ϑ dϑ dφ ∫ar (eΦ(r) - 1)r2 dr g) ∫∫sin ϑ dϑ dφ ∫ar (eΦ(r) - eΦ(r))r2 dr 0
0
(A4)
where a is the radius of the particle and r0 is the radius of an effective spherical primitive unit cell. The angular integrals in eq A4 cancel out and using
dr )
dr dΦ dΦ
(A5)
to change the integration variable transforms eq A4 into
g)
dr dr ∫Φ0 (eΦ(r) - 1)r(Φ)2 dΦ s
dr dr ∫Φ (eΦ(r) - e-Φ(r))r(Φ)2 dΦ 0
(A6)
s
where we have used Φ(a) ) Φs and Φ(r0) ) 0, the latter corresponding to the neglect of the midplane potential in the one-dimensional case.23 In general, the integrals in this expression cannot be solved analytically. However, the crude approximation
dr s dr × Φs dΦ ) f( ) ∫0Φ f(Φ) dΦ 2 dΦ Φ /2
|
Φ
s
(A7)
s
can be used provided that the integrand f(Φ) is a smoothly varying function.23 Applying this approximation to both the numerator and denominator in eq A6 gives
Acknowledgment. We would like to thank Professors Norio Ise, Ikuo Sogami, and Hideki Matsuoka and Dr. Hiroshi Yoshida for many stimulating discussions. In particular, we thank Prof. Sogami for the derivation presented in the Appendix, Prof. Matsuoka for providing us with a preprint of ref 21, and Dr. Yoshida for providing us with the latest results on void stability.
|
dr × Φs dΦ Φs/2 g) dr (eΦs/2 - e-Φs/2)r2 × Φs dΦ Φs/2 (1 - e-Φs/2)r2
|
(A8)
and the cancellation of terms in eq A8 yields
Appendix Quantity g in Three Dimensions. In three dimensions, the quantity g is given by the integral
∫∫∫(1 - e-Φ(r)) dV g) ∫∫∫(eΦ(r) - e-Φ(r)) dV
Φ(r) ) Φ(r)
(A1)
g)
(eΦs/2 - e-Φs/2)
(A9)
To the same level of approximation, the quantity g is the same for the three-dimensional case as it is for the onedimensional case.23 It can be rewritten as
where V is the volume of the primitive unit cell and Φ(r) is the dimensionless electric potential function. If we assume that the spherical symmetry of the simple ion cloud surrounding the macroion is not seriously disturbed (30) Ise, N.; Ito, K.; Okubo, T.; Dosho, S.; Sogami, I. J. Am. Chem. Soc. 1985, 107, 8074. (31) Delville, A. Langmuir, to be published.
(1 - e-Φs/2)
g)
LA950400D
(1 - e-Φs/2) Φs 2 sinh 2
( )
(A10)
