2000
J . Phys. Chem. 1988, 92, 2000-2007
are relatively small compared with the ideal reference terms, so these theories cannot be accurate very near the critical point of water. In applying these theories to solutes that have strong attractive forces for the solvent, the problem becomes worse because the ideal term has the opposite sign to the correction term and both go to infinity at the critical point of the solvent. In spite of these clear problems with these theories, they do give large maxima and minima in the correct places, so they may be useful in fitting experimental data, provided that the range of data fit does not come too close to the critical point of the solvent. In a preliminary attempt to fit the present data to what is essentially the scaled particle theory appr~ximation,'~ we found that the maxima and minima in heat capacity and volumes were not predicted at exactly the right temperatures, so we could not get good fits to the data. The predicted temperatures of the maxima were about 2 K too high for Cp,#and 5 K too low for V$. The magnitude of the maxima and minima could be fit qualitatively. At this point we do not know whether the failure to get the right temperatures for the maxima is due to the fact that our results are for finite concentrations and the theory is for infinite dilution, or whether there are errors in the theory or even temperature errors in the measurements. As expected, the theory could give quite reasonable fits if the data were limited to results below the maximum and only a reasonable number of parameters were necessary to give these fits. It seems probable that the theory would also be reasonably good at predicting thermodynamic properties above the maxima and minima. It has been surprising that scaled particle theory and perturbation theory work so well for water where many anomalous effects occur. There has been some controversy over the reason for this success. Fernandez-Prini et al.36conclude that "The dissolution of small nonpolar solutes in water does not involve significant reorientation of the neighboring water molecules. If this were the case the hard-sphere perturbation theory would not be capable of describing the thermodynamics of the process over a wide temperature range." Pierotti30 points out that the use of the (36) Fernandez-Prini, R.; Crovetto, R.; Japas, M. L.; Laria, D. Arc. Chem. Res. 1985, 18, 207.
experimental value of the thermal expansion coefficient introduces implicit information about the liquid structure of the solvent into the theory. The ability of these theories to predict the correct qualitative behavior near the critical point is another example of the implicit introduction of information about the liquid structure. Just as the use of experimental densities at high temperatures builds in some of the properties of a reference fluid with a critical point, the use of experimental densities at low temperatures builds in some of the anomalous properties of water at low temperatures. In particular it builds in a temperature of maximum density ( a = 0 at 4 "C) which is due to the open three-dimensional structure of water with its tendency to coordinate tetrahedrally. Eley3' showed that the anomalous a of water allows prediction of the high heat capacities of nonpolar solutes in water. This does not mean that this process is not accompanied by significant reorientation of the water molecules. We note that simulations using model potentials for the solute and solvent show significant reorientations about isolated nonpolar solutes and pairs of nonpolar sol~tes.~*-~~
Acknowledgment. This research was supported by the National Science Foundation under Grants CHE8009672 and CHE8412592. We also thank Dorothy E. White for helpful discussions about experimental technique, Patricia S. Bunville for preparing the figures, and William E. Davis for help with some of the calculations. Registry No. Ar, 7440-37-1;Xe, 7440-63-3;ethylene, 74-85-1. (37) Eley, D. D. Trans. Faraday SOC.1938, 35, 1281. (38) Geiger, A.; Rahman, A.; Stillinger, F. H. J . Chem. Phys. 1979, 70,
263.
(39) Pangoli, C.; Rao, M.; Berne, B. J. J . Chem. Phys. 1929, 71, 2975, 2982. (40) Rapaport, D. C.; Scheraga, H. A. J . Phys. Chem. 1982, 86, 873. (41) Zichi, D. A.; Rossky, P. J. J . Chem. Phys. 1985, 83, 797. (42) Awicki, J. C.; Scheraga, H. A. J . Am. Chem. SOC.1977, 99, 7413. (43) Swaminatham, S.; Harrison, S . W.; Beveridge, D. L. J . Am. Chem. SOC.1978, 100, 3255. (44) Battino, R. Solubility Data Series; Clever, H. L., Ed.; Paragon Press: New York, 1980; Val. 4, p 2. (45) Bradbury, E. J.; McNulty, R.; Savage, R. L.; McSweeney, E. E. Ind. Lng. Chem. 1952, 4 4 , 2 11.
A Poisson-Boltzmann Approximation for Strongky Interacting Macroionic Solutions Clifford E. Woodward* and Bo Jonsson Department of Physical Chemistry 2, Chemical Centre, LTH, S-221 00 Lund, Sweden (Received: July 7 , 1987; In Final Form: October 14, 1987)
Disregarding many (23)-body interactions, the effective pair potential between macroions is derived as the difference in free energy between adding two macroions at fixed separation and at infinite separation to the dispersion, assuming other macroions do not respond to the added particles. The free energy difference is obtained by using the overlap approximation, within the framework of a free energy functional, obtained by simplifying a full Poisson-Boltzmann treatment of the dispersion. The resulting pair potential has a screened Coulombic form, but with an apparently renormalized macroion charge and Debye screening length. The model thus gives the appearance of ion binding, but no binding is assumed. The theory gives excellent agreement with experimental structure factors for dilute dispersions of micelles and colloids, where the usual DLVO potential fails, provided the surface charge density and volume fraction are not too large.
1. Introduction Solutions containing highly charged aggregates are frequently encountered in chemistry. Micellar solutions, colloidal dispersions, and biomolecules are a few examples of such systems, which in many cases are of great technical importance. Theoretical work in this area has been stimulated by recent progress in liquid-state physics and, nowadays, it is usual for a report on structural data determined by, for example, light or neutron scattering, to include a theoretical analysis of the structure factors based on some
integral equation approximation.'s2 In these analyses it is usual that a McMillan-Mayer3 approach is adopted, where the macroions only are treated as a fluid interacting via an effective potential, which is just the free energy of the mobile species in (1) Medina-Noyola, M.; McQuarrie, D. A. J . Chem. Phys. 1980,73, 6279. Senatore, G.; Blum, L. J . Phys. Chem. 1985, 89, 2676. Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 109. (2) Hansen, J. P.; Hayter, J. B. Mol. Phys. 1982, 46, 651. (3) McMillan, W. G.; Mayer, J. E. J . Chem. Phys. 1945, 13, 276.
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Poisson-Boltzmann Approximation for Macroionic Solutions the system. These species, whose positions (and orientations) are averaged, should rigorously include all the small ions as well as the solvent. When the latter merely screens electrostatic interactions with a dielectric constant we have the primitive model.4 The resulting effective potential between the macroions is, in principle, a function of all macroion positions, which is usually approximated as a sum over one- and two-body interactions. The inclusion of higher body interactions would rapidly make this approach impractical. Alternatively, a multicomponent mixture approach may be taken, where all ionic species are treated equally, within the primitive model. Here, of course, the integral equation theories for mixtures find immediate application, and the hypernetted chain (HNC),j and HNC/Percus-Yevick6 approximations, as well as the mean spherical approximation (MSA)’ and rescaled-MSA (RMSA),*s9have been put forward as candidates for the determination of accurate structural information. However, these calculations remain time consuming and their reliability is not beyond doubt. For example, in the case of the H N C approximation, accurate predictions in 1: 1 and 1:2 electrolytesI0 cannot guarantee similar success for highly asymmetric macroion solutions. Though, recent simulations” support the quantitative accuracy of the H N C approximation for interparticle structures in the case of small low-charged micelles with monovalent counterions, extensive calculations by BelloniI2 have shown that the H N C becomes unstable for more strongly coupled systems, Le., when macroion-ounterion correlations become stronger. The instability seems to be due to the approach of the spinodal region in an apparent gas-liquid-like phase transition. This is surprising in the context of standard DLV013 theory, which always predicts an effective repulsion between macroions at constant charge and in the absence of van der Waals forces. The source of the necessary attraction is probably the soluation effect described by Patey14 in his application of the H N C approximation to the case of two macroions in salt. Recent simulations of a macroion interacting with a charged wall, in the presence of salt,15do show the existence of an attractive electrostatic force due to counterion~uctuations.’6 The existence of this attractive force between two planar electric double layers has also been supported by simulations” and by calculations using a nonuniform version of the H N C appro~imation.l~~’~ It appears, however, to be of a qualitatively different nature to the attractive force predicted by Patey. Furthermore, work by Teubner20 indicates that the onset of the attraction observed in the H N C theory is an artifact of the approximation. Modifications to the H N C closure, using the Percus-Yevick approximation for macroion-counterion correlations,6 have ex-
(4) Friedman, H. L. J . Chem. Phys. 1960, 32, 1 1 34. (5) Rasaiah, J. C. J . Chem. Phys. 1972,56, 3071. Ng, K. C. J . Chem. Phys. 1974,61,2680. Larsen, B. J . Chem. Phys. 1978,68,4511. Elkoubi, D.; Turq, P.; Hansen, J. P. Chem. Phys. Lett. 1977, 52, 493. Rogers, F. J. J . Chem. Phys. 1980,73,6272. Beresford-Smith, B.; Chan, D. Y. C. Chem. Phys. Lett. 1977, 52, 493. Belloni, L. Chem. Phys. 1985, 99, 43. (6) Bratko, D.; Friedman, H. L.; Zhong, En Ci. J . Chem. Phys. 1986,85, 377. (7) Blum, L. Mol. Phys. 1975, 30, 1529. Hiroike, K. Mol. Phys. 1977, 33, 1195. (8) Belloni, L. J . Chem. Phys. 1986, 85, 519. (9) Sheu, E. Y.; Wu, C. F.; Chen, S. H.; Blum, L. Phys. Reu. A 1985,32, 3807. (10) Rasaiah, J. C.; Friedman, H. L. J . Chem. Phys. 1968, 48, 2742. Friedman, H . L.; Ramanathan, P. S. J . Phys. Chem. 1970, 74, 3756. Ramanathan, P. S.;Friedman, H. L. J . Chem. Phys. 1971, 54, 1086. (11) Linse, P.; Jonsson, B. J. Chem. Phys. 1983, 78, 3167. (12) Belloni, L. Chem. Phys. 1985, 99, 43. (13) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Sfability of Lyophobic Colloids; Elsevier: Amsterdam, 1945. (14) Patey, G. N . J . Chem. Phys. 1980, 72, 5763. (15) Svensson, B.; Jonsson, B. Chem. Phys. Lett. 1984, 108, 580. (16) Oosawa, F. Biopolymers 1968, 6, 1633. (17) Guldbrand, L.; Jonsson, B.; Wennerstrom, H.; Linse, P. J . Chem. Phys. 1983, 78, 3167. (18) Kjellander, R.; Marcelja, S. J . Chem. Phys. 1985, 82, 2122. (19) Kjellander, R.; Marcelja, S. Chem. Phys. Lett. 1986, 127, 402. (20) Teubner, M. J . Chem. Phys. 1981, 75, 1907.
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 2001 tended the region of applicability, if only in an ad hoc manner. No such problems appear to arise in the case of the rescaled MSA.2,9 However, though they can give significantly different predictions, there is no compelling reason to choose one of these approaches over the other. In any case, concerning practical applications, the multicomponent models cannot provide rapid calculations, with the exception of the MSA, which is largely unreliable, and a disputed version of the RMSA.8*9For this reason the one-component models remain the simpler and preferred approach for the analysis of experiment. The one-component approach originated with the development ~ is firmly based on the of the DLVO pair p ~ t e n t i a l ’which Poisson-Boltzmann approximation. This potential has been “improved” by various workers2’ and, more recently, has been rederived by Beresford-Smith et and Bellonis from integral equation theory. The simplest form for this potential is the screened Coulomb potential Veff(r) = (XZ,)2/c(1
-- a ;
r 5 a
+ UK/2)
exp[-K(r - u)]/r;
r
> (1)
where Z , is the macroion charge and K is the inverse Debye screening length, which is determined by both counterions and added salt. Simulations have shown that this potential can predict the macroion structure remarkably for weakly coupled systems. The potential is usually derived from a linear response theory or as an asymptotic result and its popular and persistent use is no doubt linked to the fact that an analytic solution to the MSA can be found for the resulting one-component The parameter X has been given an almost bewildering number of expressions depending upon boundary conditions or the approximations used in the derivation of eq 1.8,22,25 When used to fit experimental data, this parameter is usually included in an effectiue charge XZ., If this is less than the expected stoichiometric charge, which is usually the case for highly charged macroions, the reduction is often ascribed to counterion binding.28 Recently, Beresford-Smith et a1.22proposed a nonlinear theory for strongly interacting macroionic systems. They showed that purely electrostatic effects can cause an apparent renormalization of the true mcroion charge in eq 1. This fact is also implicit in the asymptotic results of Oshima et a1.26 The Debye length, however, was to be determined by using the full stoichiometric counterion (and salt where present) density. This is apparently at odds with recent experiment^^^^^^ where both the macroion charge and the Debye length were varied consistently in order to fit structural data under conditions of low added electrolyte. From these experiments one could then argue that, since counterions are removed from the screening process, and one does not apparently find this from electrostatic arguments, a binding mechanism must be at work. Though binding obviously occurs in many systems, there are others where the concept is counter to chemical intuition. Furthermore, the extent of binding (due to some chemical mechanism) can only be properly estimated once the purely physical effects are accounted for. In this paper we elucidate (nonchemical) mechanisms that can give rise to apparent renormalisation of both the macroion charge and the Debye length. We develop a free energy functional theory, using the PoissonBoltzmann approximation. In principle we could utilize more accurate approximations incorporating, for example, excluded volume and ion correlation effect^,^^,^^ but we felt the Poisson(21) Levine, S.; Jones, J. E. Kolloid-Z. 1968, 230, 306. Bell, G.M.; Levine, S.; McCartney, L. N. J . Colloid Interface Sci. 1970, 33, 335. (22) Beresford-Smith, B.; Chan, D. Y. C.; Mitchell, D. J . J . Colloid Interface Sci. 1985, 105, 216. (23) Svenssun, B.; Jonsson, B. Mol. Phys. 1983, 50, 489. (24) Waisman, E. Mol. Phys. 1973, 25, 45. (25) Sogami, I. Phys. Left.A 1983, 96, 199. Sogami, I.; Ise, N. J . Chem. Phys. 1984, 81, 6320. (26) Oshima, H.; Healy, T. W.; White, L. R. J . Colloid Interface Sci. 1982, 90, 17. (27) Chao, Y . S.;Sheu, E.Y . ;Chen, S. H. J . Phys. Chem. 1985,89,4862. (28) Benedouch, D.; Chen, S. H.; Koehler, W. C. J . Phys. Chem. 1983, 87, 2621. (29) Nordholm, S. Aust. J . Chem. 1984, 37, 1 .
2002
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988
Boltmann functional was capable of capturing the physical aspects we wished to address in this work. As in the theory of Beresford-Smith et al., we use the overlap approximation3’to determine an effective macroion potential in the presence of an average background of other macroions and mobile ions. However, an important difference in our approach is that we include some measure of mobile ion-macroion correlations in the background, making use of the well-known cell model.32 The net effect is to increase the Debye length, K - ~ ,which taken together with a decrease in the effective surface charge, has the appearance of a binding mechanism. In section 2 of this paper we discuss our theoretical approach in more detail. We compare our predictions for the macroion structure with experiment in section 3, using simple integral equation approximations, and section 4 contains some concluding remarks.
2. Theory The usual method of obtaining the effective many-body potentials, appropriate for a charged macroion solution, in the presence of added salt, requires one to find the change in free energy upon successive additions of macroions to the salt solution.13 This procedure cannot be applied when the concentration of the counterions is of similar magnitude to the added salt, e.g., in ion-exchanged dispersions. Beresford-Smith et ai. (BS),, have shown that for these systems the dispersion, in essence, provides its own salt but the macroions do not take part in the screening. Their approach for small macroion charge, in the absence of added salt, involves Debye-Huckel-like approximations; the counterion distribution then responds linearly to the perturbing macroions. In this approximation, the free energy is a bilinear functional of the counterion distribution, and the total free energy (for a given configuration of macroions) can be written as a sum of one- and two-body effective potentials. The two-body potential is of screened Coulomb form, identical with eq 1, for point ions ( u = 0), with the bare charge, X = 1. For finite-sized macroions, approximations must be made leading to X values different from unity. A similar approach has been used by S ~ g a m iwho , ~ ~obtained the same two-body contribution to the Helmholtz free energy as BS. However, his one-body term was inadequate due to neglect of counterion-counterion correlations in the Poisson-Boltzmann approximation. Nevertheless, as both approaches predict the same pair potentials they also predict the same macroion structure. 2.1. The Effective Pair Potential. Consider a dispersion consisting of N monodispersed macroions (m) with charge 2, in some volume V, together with their counterions (c) and possibly other ionic species (labeled by a = 1, 2 , ...), with density and charge n, and Z,, respectively. Both N and V are assumed to be very large so that we are essentially at the thermodynamic limit. All ionic interactions are treated within the primitive model. For a given configuration of macroions, which we denote by { R ]= ( R l , R,, ..., R N ) ,A,,(RI,Rz,...,R N ) ,is the Helmholtz free energy of the remaining mobile species (counterions + salt). In the usual one-component fluid approach one assumes this to be a sum of one- and two-body potentials, i.e., N
A ~ ~ ~ ( R I , R ~ , .=. .Cvefr(l)(Ri) ,RN) + i= I
n: .v
X3C Verf(Ri,Rj) ilo5 K s-') of their micrometer-size droplets from an aerosol. Their thermal behavior on reheating was studied by differential scanning calorimetry and the heats of devitrification were determined. The NaC1-H20 glasses devitrify to cubic ice in three steps with one exotherm at about the same temperature as pure glassy water and the other two at higher and lower temperatures, respectively. Their peak areas depend in a systematic way on NaCl concentration. The ethylene glycol-H,O glasses devitrify in two steps to cubic ice with peak onset temperatures and areas again depending systematically on solute concentration. Stepwise devitrification is interpreted in terms of differencesin the mobilities of the water molecules near the glass-liquid transition. For the NaCI-H,O glasses, their origin is related to differences in the mobilities of the water molecules at higher temperatures. The thermal behavior on reheating a hyperquenched glass containing 50% ethylene glycol is compared with that of the slow-cooled glass: only the former devitrifies in two steps and its T, is shifted by about 2' to higher temperature.
Introduction Dilute aqueous solutions can now be vitrified completely by rapid cooling, or hyperquenching, of micrometer-size droplets, obtained from an aerosol, on a cryoplate. 1-3 The infrared spectra of vitrified dilute alkali metal nitrate and perchlorate solutions have been reported as a first appli~ation.~ In this work, the thermal behavior on rewarming hyperquenched aqueous solution glasses, investigated by differential scanning calorimetry (DSC), is reported. NaCl was chosen as one of the solutes because of its physiological importance, and the devitrification of the glassy water component to cubic ice was studied as a function of NaCl concentration. Ethylene glycol (EG) was chosen as the other solute because of its importance as a cryoprotectant. The EG-H20 system is of particular interest because the whole range from pure HzO to pure EG can be vitrified, and for concentrated solutions the effects of hyperquenching and slow cooling can be compared. On rewarming hyperquenched glasses, both types of solute show the same general behavior, namely the phase transition of the glassy water component to cubic ice occurring in three or two distinct steps, respectively. Their separated peak areas depend in a systematic way on solute concentration. This is discussed in terms of differences in the mobility of the water molecules in hyperquenched aqueous solution glasses near their glass liquid transition.
-
Experimental Section The vitrified samples were prepared as described in earlier reports. Briefly, aerosol droplets were transferred through a ( 1 ) Mayer, E. J . Appl. Phys. 1985, 58, 663. (2) Hallbrucker, A.; Mayer, E. J . Phys. Chem. 1987, 91, 503. (3) Mayer, E. J . Phys. Chem. 1986, 90,4455.
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small opening (200-pm diameter), with nitrogen as carrier gas, into a high-vacuum system, accelerated by supersonic flow, and deposited on a cryoplate at 77 K. The aerosol was led through a glass tube (40 cm long, 25-mm i.d.) to the orifice. Solutions with 5% and 15% (w/w) EG (Fluka puriss. p.a.) were nebulized with an ultrasonic nebulizer (Engstrom, Model NB 108; operation at 3 MHz gives droplets with 13-pm diameter) under conditions described recently in a DSC study of vitrified pure liquid water.2 The concentrated (50%) solution and pure EG could not be nebulized by ultrasonics. From these liquids an aerosol was made with a retouching air brush (Grafo One bar of working pressure gave a nitrogen flow rate of 1.3 L min-' and 0.6 mL m i d of liquid consumption. The latter method gives a much larger distribution of droplet sizes than ultrasonic nebulization. However, we seem to have obtained some fractionation in favor of the smaller droplets by spraying into the glass tube that guides the aerosol to the orifice, because hyperquenched samples of pure liquid water contained nearly the same amount of vitrified material as samples from aerosols made by ultrasonic nebulization. The dilute NaC1-containing solutions were likewise nebulized with the ultrasonic nebulizer and the concentrated solutions with Grafo I. The aerosols were deposited on an X-ray low-temperature sample holder for simultaneous determination of the percent crystalline ice in the quenched samples and of the diffraction patterns associated with the exotherms during reheating. Optimally quenched samples of vitrified pure liquid water, made as described in ref 2 and 3, contain consistently 5 & 1% crystalline, mainly cubic, ice. All hyperquenched samples with EG as solute, including even the 5% solution, were completely amorphous. Q4s5
(4) Bachmann, L.; Schmitt, W. W. Proc. Narl. Acad. Sci. U.S.A.1971,
68, 2149. (5) Plattner, H.; Bachmann, L. Int. Reo. Cytol. 1982, 7 9 , 237.
0 1988 American Chemical Society
