week ending PRL 106, 167801 (2011) PHYSICAL REVIEW LETTERS 22 APRIL 2011

Ion Specificity and the Theory of Stability of Colloidal Suspensions

Alexandre P. dos Santos and Yan Levin Instituto de Fı´sica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil (Received 23 December 2010; published 20 April 2011) A theory is presented which allows us to accurately calculate the critical coagulation concentration of hydrophobic colloidal suspensions. For positively charged particles, the critical coagulation concentra- tions follow the Hofmeister (lyotropic) series. For negatively charged particles, the series is reversed. We find that strongly polarizable chaotropic anions are driven towards the colloidal surface by electrostatic and hydrophobic forces. Within approximately one ionic radius from the surface, the chaotropic anions lose part of their hydration sheath and become strongly adsorbed. The kosmotropic anions, on the other hand, are repelled from the hydrophobic surface. The theory is quantitatively accurate without any adjustable parameters. We speculate that the same mechanism is responsible for the Hofmeister series that governs stability of protein solutions.

DOI: 10.1103/PhysRevLett.106.167801 PACS numbers: 61.20.Qg, 64.70.pv, 64.75.Xc, 82.45.Gj

All of biology is specific. The ion channels which con- concentration of sodium thiocyanide is an order of magni- trol the electrolyte concentration inside living cells are tude lower than the CCC of sodium fluoride. Even more specific to the ions which they allow to pass. The tertiary dramatic is the variation of the CCC with the sign of structure of proteins is sensitive to both the pH and to colloidal charge. The CCCs predicted by the DLVO theory specific ions inside the solution. It has been known for over are completely invariant under the colloidal charge rever- a hundred years that while some ions stabilize protein sal. Therefore, changing the charge from, say, þ0:04 to solutions, often denaturing them in the process, others 0:04 C=m2 results in exactly the same CCC. This is lead to protein precipitation. In fields as diverse as bio- completely contradicted by the experiments, which find physics, biochemistry, electrochemistry, and colloidal sci- that, under the same charge reversal, the CCCs can change ence, ionic specificity has been known—and puzzled by as much as 2 orders of magnitude for exactly the same over—for a very long time. It has become known as the electrolyte. Similar dramatic effects are observed for pro- ‘‘Hofmeister effect’’ or the ‘‘lyotropic’’ series of electro- tein solutions for which some electrolytes are found to lytes, depending on the field of science. The traditional ‘‘salt-in’’ while others ‘‘salt-out’’ the same protein. In physical theories of electrolytes completely fail to account view of the fundamental importance of ion specificity for the ion specificity. The Debye-Hu¨ckel theory of electro- across so many different disciplines, there has been a great lytes and the Onsager-Samaras theory of surface tensions effort to understand its physical mechanisms [6–14]. A treat ions as hard spheres with a point charge located at the successful theory should be able to quantitative predict center [1]. The cornerstone of colloidal science, the the CCCs of colloidal suspensions and shed new light on Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of the specific ion effects so fundamental to modern biology. stability of lyophobic colloidal suspensions, is based on Construction of such a theory is the subject of the present an even simpler picture of hard spherelike colloidal parti- Letter. cles interacting with the pointlike ions through the Over the past 20 years, there has been a growing real- Coulomb potential. The DLVO theory showed that the ization that ionic polarizability—the effective rigidity of primary minimum of colloid-colloid interaction the electronic charge distribution—plays an important role potential—arising from the mutual van der Waals (disper- in coding the ionic specificity [15–18]. The work on sur- sion) attraction—is not accessible at low electrolyte con- face tension of electrolyte solutions showed that near an centrations because of a large energy barrier. When the air-water interface ions can be divided into two classes: electrolyte concentration is raised above the critical coagu- kosmotropes and chaotropes [19,20]. The kosmotropes lation concentration (CCC), the barrier height drops down remain strongly hydrated and are repelled from the inter- to zero, leading to colloidal flocculation and precipitation. face. On the other hand, chaotropes lose their hydration The DLVO theory, however, predicts that the CCC should sheath and redistribute their electronic charge so that it be the same for all monovalent electrolytes, which is remains mostly hydrated. This way, chaotropes gain clearly not the case [2–5]. In fact, it has been known for hydrophobic free energy at a small price in electrostatic a long time that the effectiveness of the electrolyte at self-energy. For hard nonpolarizable ions of the Debye- precipitating hydrophobic colloids follows the lyotropic Hu¨ckel-Onsager-Samaras theory, the hydrophobic forces series. For positively charged particles, the CCC are just too weak to overcome the electrostatic self-energy

0031-9007=11=106(16)=167801(4) 167801-1 Ó 2011 American Physical Society week ending PRL 106, 167801 (2011) PHYSICAL REVIEW LETTERS 22 APRIL 2011 penalty of exposing the ionic charge to the low dielectric medium (water) is w. The Bjerrum length is defined as 2 air environment, forcing these ions to always stay in the B ¼ q =w, where ¼ 1=kBT, and is 7.2 A˚ , in water at bulk, contrary to simulations and experiments. room temperature. The system—two plates with an elec- It is natural to ask if the same mechanism is also respon- trolyte in between—is in contact with a salt reservoir at sible for the ionic specificity of the CCCs and for the concentration S. The number of cations and anions stability of protein solutions. The similarity between the N per unit area in between the surfaces, as well as their air-water interface and a hydrophobe-water interface, how- spatial distribution, will be calculated by minimizing the ever, is not so straightforward [21]. The simulations find grand-potential function. Note that we do not require the that, contrary to the air-water interface, the mean water charge neutrality inside the system—the external electro- density is actually larger near a hydrophobic surface than lyte will screen the excess charge. Of course, if the system in the bulk [22]. On the other hand, the interfacial water is is in vacuum, the charge neutrality will be enforced by the ‘‘softer’’ than bulk water, so that the hydration layer is very formalism developed below. compressible [22]. In particular, the simulations find that The grand potential per unit area is the energy cost of placing an ideal cavity of radius a near a Z L=2 þðxÞ hydrophobic surface is half that of having it in the bulk ðLÞ¼ dxþðxÞ ln 1 L=2 S [22]. For small cavities, the bulk energy cost scales with 3 Z L=2 the volume [21,23]asUbulk ¼ a , where ¼ 0:3kBT ðxÞ þ dxðxÞ ln 1 [24]. Moving an ion to the surface, therefore, gains L=2 S 3 Ucav ¼ a of hydrophobic energy. For kosmotropic Z L=2 Z L=2 2 w 2 ions, this energy is too small to compensate the loss þ dxEðxÞ þ dxþðxÞUþðxÞ 8 L=2 L=2 of the hydration sheath and for exposing part of the ionic Z L=2 2 charge to the low dielectric environment of the surface. þ dxðxÞUðxÞþ For highly polarizable chaotropic ions, the cavitational L=2 2ww energy gain compensates the electrostatic energy cost, þ2SL; (1) since these ions can shift most of their electronic charge to still remain hydrated. Furthermore, if there are no water where x is the distance measured from midplane, L is the molecules between an ion and a surface, dispersion separation between the plates, ðxÞ are the ionic concen- interactions—resulting from quantum electromagnetic trations, ¼ qN qNþ 2 is the deviation from the field fluctuations—come into play. Dispersion forces are charge neutrality inside the system, and UðxÞ are the strongly attenuated by water. For example, the Hamaker interaction potentials between the ions and the surfaces. constant for polystyrene in water is 7 times lower than it is The first two terms of Eq. (1) are the entropic free energy, in vacuum [25]. The London-Lifshitz theory predicts that the third term is the electrostatic energy inside the system, the dispersion interaction between an ion and a surface the sixth term is the electrostatic penalty for violating the decays with the third power of separation and is propor- charge neutrality, and the last term accounts for the work tional to the ionic polarizability. Although there is no that must be done against the pressure of reservoir. The exponential screening of dispersion force by the electro- electrostatic penalty (sixth term) is calculated by using the lyte, in practice, it is very short ranged and is relevant only Debye-Hu¨ckel equation (linearized Poisson-Boltzmann for chaotropic ions in almost a physical contact with a equation) and accounts for the screening of the excess hydrophobic surface; see Fig. 1. charge by the external reservoir. The inversepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (screening) ¼ 8 We shall work in the grand-canonical ensemble. The Debye length of the reservoir is w B S. The N ðxÞ colloidal particles will be treated as planar surfaces of equilibrium values of and the density profiles surface charge density with the electrolyte in between. are determined by numerically minimizing the grand- To account for the curvature we will use the Derjaguin potential function. ByR using the Gauss law, the electric 4q x 0 0 0 field is EðxÞ¼ ½þðx Þðx Þdx . Minimizing approximation [25]. The static dielectric constant of the w 0 the grand-potential function for a fixed N, we find L N ð a xÞeqðxÞUðxÞ ðxÞ¼ 2 ; RðL=2Þa 0 0 (2) 0 qðx ÞUðx Þ 2 0 dx e R x 0 0 where the electrostatic potential is ðxÞ¼ 0 dx Eðx Þ, is the Heaviside step function, a are the hydrated or unhydrated (bare) ionic radius, depending on whether the ion is a kosmotrope or a chaotrope. Substituting the ionic FIG. 1 (color online). Illustrative representation: While the densities into Gauss’ law yields an integral equation for the chaotropic ions lose part of their hydration sheath near a hydro- electric field which is then solved numerically. We proceed phobic surface and become adsorbed, the kosmotropic ions are as follows: For a trial number of cations and anions, N, repelled from the surface by the hydration layer. we calculate the electric field and the ionic density profile. 167801-2 week ending PRL 106, 167801 (2011) PHYSICAL REVIEW LETTERS 22 APRIL 2011

Once these are obtained, we substitute them back into the parameters. To further test the accuracy of the theory, we free energy functional (1). The trial values of N are varied have calculated the CCC for negatively charged colloidal until the minimum of the grand potential is found. In particles with ¼0:061 C=m2 in a suspension with practice, we couple the integral equation solver to a mini- NaCl [3]. In this case the experimental value of the CCC mization subroutine. was found to be 140 mM, while we obtain 134.2 mM. The van der Waals interaction between two surfaces is For chaotropic anions the situation is significantly more w 2 w given by [25] HðLÞ¼App=12L , where App is the complicated. As mentioned earlier, these ions can shed Hamaker constant for polystyrene in water [25]. To ac- their hydration sheath and adsorb to the colloidal surface, count for the finite radius of colloidal particles, we use the gaining hydrophobic free energy. In addition, diminished Derjaguin approximation [25]. The total interaction poten- hydration leads to strong dispersion interaction between tial between theR two latex spheres of radius Rc is then latex and a chaotropic anion. The London-Lifshitz theory 1 UtotðLÞ¼Rc L dl½ðlÞð1Þ þ HðlÞ. predicts that the ion-particle dispersion potential has the U ðxÞB½ 1 þ 1 B We begin by studying the colloidal suspensions with form dis ðL=2þxÞ3 ðL=2xÞ3 , where is the con- sodium salts of kosmotropic anions: IO3 , F , BrO3 , and stant related to ionic excess polarizability and the ioniza- Cl . For these salts both cations and anions are strongly tion potential, as well as the dielectric properties of hydrated and are repelled from the colloidal surface. Since polystyrene and water. Unfortunately, these quantities are the static dielectric constant of latex is less than that of not known. However, we can get a reasonable approxima- water, in addition to the hard-core repulsion at contact, the tion for B by using a simplified Hamaker theory [25]. ions also feel a repulsive charge-image interaction. To Within this approximation we find B ¼ 2Aeff=9, where explicitly calculate this potential one needs to resum an Aeff is the effective Hamaker constant for a metal- infinite number of images. This can be done by using the polystyrene interaction and is the ionic polarizability theory developed in Ref. [26], which yields the ion-image [18]. In vacuum, using a standardpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi relation for Hamaker interaction potential WimðxÞ. v constants, we obtain A ¼ AppAmm, where Amm and App For kosmotropic ions the latex-ion interaction potential eff are the constants for metal-metal and polystyrene- has a particularly simple form—hard-core repulsion at one v polystyrene interaction, respectively, resulting in Aeff ¼ hydrated ionic radius from each surface plus the ion-image 17:776 1020 J interaction. The hydrated radii were taken from .p Onffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the other hand, if the ion is fully Aw ¼ Aw Aw Aw Aw Nightingale [27] and are the same as used in our previous hydrated, eff pp mm, where mm and pp are the work on surface tensions [19,20]. The interaction potential Hamaker constants for metal-metal and polystyrene- Aw ¼ between the two colloidal particles, Utot, can now be polystyrene interaction in water [25], yielding eff 20 calculated; see Fig. 2. The potential has a primary mini- 6:245 10 J. A chaotropic ion near a latex surface, mum at L ¼ 0, followed by an energy barrier. The CCC is however, is only partially hydrated, so that the effective Av defined as the concentration S for which the barrier goes Hamaker constant should be intermediate between eff w down to zero. In Table I, we present the calculated values and Aeff, which we take to be the arithmetic average w v of the CCC for positively charged colloidal suspensions Aeff ¼ðAeff þ AeffÞ=2. and compare the theoretical results with the experimental As the ion moves away from the surface, it creates an measurements. A very good agreement between the additional cavity from which water molecules are ex- theory and experiments is found without any adjustable cluded. This once again carries a hydrophobic free energy cost. Since the volume of this new cavity is small, we can,

0.08 NaF 0.06 TABLE I. Comparison between the calculated CCCs and the NaCl experimental values. The colloidal surface charge density is 0.04 NaBr 2 c þ0:04 C=m NaI from Ref. [5]. Judging from the spread of experi- /R 0.02 10 mM

tot mental data, the experimental error is approximately .

U 0 Ions Theory (mM) Experiments (mM) -0.02 IO 85 [5] -0.04 3 79.5 F 78 80 [5] 0204060 80 100 BrO * L (Å) 3 71.7 Cl 70 70 [5], 90 [3] NO FIG. 2 (color online). Particle-particle interaction potentials in 3 31 36 [5], 35 [3] units of kBT=A, for suspensions with sodium-halide salts. The ClO3 25.08 * colloidal surface charge is 0:04 C=m2. The electrolyte concen- Br 24.3 45 [5] ¼ 20 mM tration is S . While suspensions containing NaF, ClO4 20.9 * NaCl, and NaBr at this concentration are stable—there is an I 14.8 18 [5] energy barrier for the primary minimum—suspension with NaI SCN 8.625 12 [5], 27 [3] is unstable. 167801-3 week ending PRL 106, 167801 (2011) PHYSICAL REVIEW LETTERS 22 APRIL 2011 once again, use the volumetric scaling of the cavitational [1] Y. Levin, Rep. Prog. Phys. 65, 1577 (2002). free energy to estimate its cost. This solvophobic free [2] M. Bostro¨m, D. R. M. Williams, and B. W. Ninham, Phys. 3 87 energy will grow linearly as UsolðxÞ¼a=2 þ Rev. Lett. , 168103 (2001). 2 [3] T. Lopez-Leon, A. B. Jodar-Reyes, D. Bastos-Gonzalez, ð3=4ÞaðL x aÞ up to the maximum separation and J. L. Ortega-Vinuesa, J. Phys. Chem. B 107, 5696 5a=3 from the surface, after which distance it will be (2003). zero—the cavitational energy will take its bulk value. [4] T. Lopez-Leon, M. J. Santander-Ortega, J. L. Ortega- Thus for chaotropic ions, in addition to the ion-image Vinuesa, and D. Bastos-Gonzalez, J. Phys. Chem. C interaction, the solvophobic and the dispersion energies 112, 16 060 (2008). must be taken into account: UðxÞ¼WimðxÞþUsolðxÞþ [5] J. M. Peula-Garcia, J. L. Ortega-Vinuesa, and D. Bastos- UdisðxÞ. Finally, unlike the kosmotropes—which are lim- Gonzalez, J. Phys. Chem. C 114, 11 133 (2010). ited by their hydration—the chaotropes can come to the [6] K. D. Collins and M. W. Washabaugh, Q. Rev. Biophys. colloidal surface up to one bare ionic radius. The ionic 18, 323 (1985). 303 radii and polarizabilities for I , NO3 , Br , ClO3 , and [7] B. C. Garrett, Science , 1146 (2004). [8] W. Kunz, P. Lo Nostro, and B. W. Ninham, Curr. Opin. ClO4 are the same as used in our previous work on surface Colloid Interface Sci. 9, 1 (2004). tensions [19,20]. The CCCs are once again obtained by 34 requiring zero energy barrier for the primary minimum. [9] K. D. Collins, Methods , 300 (2004). [10] Y. Zhang and P.S. Cremer, Curr. Opin. Chem. Biol. 10, Table I compares the theoretical predictions with the ex- 658 (2006). periment. The good agreement in this case, however, might [11] L. M. Pegram and M. T. Record, J. Phys. Chem. B 111, be fortuitous considering the crudeness of our treatment of 5411 (2007). the dispersion interaction. In order to test that this is not the [12] R. Zangi, M. Hagen, and B. J. Berne, J. Am. Chem. Soc. case, we have calculated the CCC for a sodium-nitrate salt 129, 4678 (2007). in suspension of negatively charged colloidal particles of [13] M. Lund, L. Vrbka, and P. Jungwirth, J. Am. Chem. Soc. ¼0:061 C=m2, the CCC of which is known experi- 130, 11 582 (2008). mentally to be 170 mM [3]. The present theory predicts [14] N. Schwierz, D. Horinek, and R. R. Netz, Langmuir 26, 163 mM, suggesting that the good agreement between the 7370 (2010). 95 theory and experiment is not a coincidence. [15] L. Perera and M. L. Berkowitz, J. Chem. Phys. , 1954 The case of thiocyanide SCN is somewhat more com- (1991). [16] L. X. Dang and D. E. Smith, J. Chem. Phys. 99, 6950 plicated since this ion can not be modeled as a sphere but is ˚ ˚ (1993). rather a cylinder of radius 1.42 A and length 4.77 A [28]. [17] S. J. Stuart and B. J. Berne, J. Phys. Chem. A 103, 10 300 3 The transverse polarizability of SCN is taken to be 3:0 A (1999). [29]. It is energetically favorable for this ion to be adsorbed [18] Y. Levin, Phys. Rev. Lett. 102, 147803 flat onto colloidal surface. The cavitational energy can be (2009). calculated similarly as above. With these parameters we [19] Y. Levin, A. P. dos Santos, and A. Diehl, Phys. Rev. Lett. find that SCN is very strongly adsorbed at the colloidal 103, 257802 (2009). surface; see Table I. Finally, we note that, for negatively [20] A. P. dos Santos, A. Diehl, and Y. Levin, Langmuir 26, charged hydrophobic surfaces, the CCCs follow the re- 10 778 (2010). 437 versed Hofmeister series. For example, for a suspension [21] D. Chandler, Nature (London) , 640 (2005). ¼0:04 C=m2 [22] R. Godawat, S. Jamadagni, and S. Garde, Proc. Natl. of particles with the CCCs range from 106 IO SCN Acad. Sci. U.S.A. , 15 119 (2009). 70 mM for 3 to 239 mM for . Unfortunately, no [23] K. Lum, D. Chandler, and J. D. Weeks, J. Phys. Chem. B experimental data are available in this case. 103, 4570 (1999). We have presented a theory which allows us to quantita- [24] S. Rajamani, T. M. Truskett, and S. Garde, Proc. Natl. tively predict critical coagulation concentrations. For posi- Acad. Sci. U.S.A. 102, 9475 (2005). tively charged particles, the CCCs follow the Hofmeister [25] W. B. Russel, D. A. Saville, and W. R. Schowalter, series. The only significant deviation is Br. If the experi- Colloidal Dispersions (Cambridge University Press, mental data are correct, this would suggest that the values of Cambridge, England, 1989). polarizability for halogen ions quoted in the literature are [26] Y. Levin and J. E. Flores-Mena, Europhys. Lett. 56, 187 overestimated. This seems to be consistent with the recent (2001). 63 conclusions of ab initio simulations [30]. For negatively [27] E. R. Nightingale, Jr., J. Phys. Chem. , 1381 charged particles, the Hofmeister series is reversed. In view (1959). [28] Y. Iwadate, K. Kawamura, K. Igarashi, and J. Mochinaga, of the success of the present theory, it is now reasonable to J. Phys. Chem. 86, 5205 (1982). hope that a fully quantitative understanding of the [29] P. B. Petersen, R. J. Saykally, M. Mucha, and P. Jungwirth, Hofmeister effect for protein solutions might be in sight. J. Phys. Chem. B 109, 10 915 (2005). This work was partially supported by the CNPq, INCT- [30] B. A. Bauer, T. R. Lucas, A. Krishtal, C. Van FCx, and by the U.S. AFOSR under Grant No. FA9550-09- Alsenoy, and S. Patel, J. Phys. Chem. A 114, 8984 1-0283. (2010).

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