Unexpectedly Large Decay Lengths of Double-Layer Forces in Solutions of Symmetric, Multivalent Electrolytes

Double layer forces acting between micron-sized silica particles are measured with the atomic force microscope (AFM) in solutions of symmetric, multivalent electrolytes. In particular, the following 2:2 electrolytes, CuSO4, MgSO4, and the 3:3 electrolyte LaFe(CN)6 were investigated. For the multivalent electrolytes, the measured decay lengths are substantially larger than the ones expected on the basis of simple Debye-Hückel (DH) theory. These deviations can be explained quantitatively by the formation of neutral ion pairs. The measured surface charge density decreases in magnitude with increasing valence. Both effects are caused by ion-ion correlations, which are not included in the classical DH theory. However, this theory remains applicable provided one considers the formation of ion pairs in solution and an effective surface charge density. This effective charge is substantially smaller in magnitude than the one of the bare surface. This reduction results from adsorption of counter-ions, which becomes stronger with increasing valence. These observations reveal that DH theory is applicable even in the presence of [...] SMITH, Alexander, et al. Unexpectedly Large Decay Lengths of Double-Layer Forces in Solutions of Symmetric, Multivalent Electrolytes. Journal of Physical Chemistry. B, Condensed Matter, Materials, Surfaces, Interfaces and Biophysical, 2019, vol. 123, no.


Introduction
Derjaguin, Landau, Verwey, and Overbeek (DLVO) stressed the importance of forces acting between solid surfaces across aqueous solutions already long time ago. [1][2][3] However, reliable measurements of such forces had to await the development of the surfaces forces apparatus (SFA). 4 More recently, the tools for such direct force measurement could be substantially expanded with total internal reflection microscopy (TIRM) 5,6 , optical tweezers combined with video microscopy 7,8 , and the colloidal probe technique based on the atomic force microscope (AFM). [9][10][11][12][13] Early measurements between mica surfaces have already demonstrated that at lower salt levels such forces are normally dominated by double layer interactions, which lead to a characteristic exponential decay at larger separation distances. 4,[14][15][16] Thereby, the range of this interaction decreases with increasing electrolyte concentration. This behavior can be rationalized with Debye-Hückel (DH) theory, which represents a linearized version of the Poisson-Boltzmann (PB) theory. In agreement with experiment, these theories predict that these forces decay exponentially at larger distances. 3 The corresponding decay length turns out to be identical to the Debye length, which indeed decreases with increasing electrolyte concentration.
However, recent theoretical work has cast doubt on the validity of the classical DH and PB theories, especially in systems containing multivalent ions. [17][18][19][20] These theories are based on a mean-field approximation, thereby neglecting correlations between the ions. Such ion-ion correlations have been claimed to be especially important near interfaces, but also in the bulk in the vicinity of highly charged ions. For example, such correlations may manifest itself by formation of fluid-like or crystalline ionic phases near interfaces. 19,21 In spite of these reservations, numerous experiments have confirmed that double layer forces agree with DH or PB theories even in the presence of multivalent ions. 6,11,15,16,[22][23][24][25] Moreover, the measured decay lengths match the expected Debye lengths too. In these systems, however, effects of ion-ion correlations should be important and mean-field DH or PB should not be necessarily applicable. This apparent contradiction calls for an explanation. This article sheds new light on this question through force measurements in symmetric, multivalent electrolytes.

Experimental
Materials. Silica particles with diameter 5.2 μm (Bangs Laboratories Inc., USA) were attached to tipless cantilevers (MicroMash) using epoxy glue. Similar particles were also sprinkled over quartz substrates and together with cantilevers then heated to 1150 °C for 3 hours to firmly attach the particles. As previously reported, this sintering process also shrinks the particles to 4.4 μm, and significantly reduces their surface roughness. 26 Immediately prior to experiment, substrates and cantilevers were rinsed thoroughly with ethanol and ultrapure water. After drying in a stream of nitrogen, they were placed in air plasma for at least 15 minutes. Electrolyte solutions were prepared from NaCl, CsCl, CuSO 4 ·5H 2 O (Sigma-Aldrich, >99.9%) and MgSO 4 (Acros Organics, 97% anhydrous) with ultrapure Milli-Q water (Millipore).
LaFe(CN) 6 was prepared by precipitation reaction as described by Bhat et al. 27 In brief, concentrated solutions of LaCl 3 and K 3 Fe(CN) 6 (Sigma-Aldrich � 99%) were mixed together in equimolar amounts.
The LaCl 3 solution was added dropwise with constant stirring at 25 °C. The mixture was then left in the dark overnight, and the resulting precipitate was washed with ultrapure water over a Buchner funnel to remove any traces of KCl. The red-brown crystals were stored in a desiccator over silica gel for several weeks before use. X-ray powder diffraction confirmed that the salt was LaFe(CN) 6 ·4 H 2 O.
The amount of crystal water was also confirmed with thermal gravimetric analysis (TGA). The concentrations of all solutions used were verified by atomic absorption spectroscopy (AAS) or complexometric titrations. The measured concentrations agreed with the nominal ones within <2%.
The solution pH in the experiments was 5.8±0.3.

Direct Force Measurements.
Forces between silica particles were measured using a closed-loop AFM (MFP-3D, Asylum Research) mounted on an inverted optical microscope (Olympus IX 73). The particle on the cantilever was aligned above a particle deposited on the substrate to within 50 nm using the horizontal scanner of the AFM. After centering a pair of particles, at least 200 approach-retraction cycles were measured at velocity 300 nm/s. For each pair the diffuse layer potential was verified to lie within about 10% of the average of the sample. This method assures that the particle pair is close to symmetric.
The deflection signal was converted to force-separation profiles by subtracting the baseline in the absence of any forces, and by fitting the constant compliance region where the particles are in intimate contact at high forces. The spring constant of the cantilever was measured using the method described by Sader et al. 28 , which relies on the lateral dimensions of the cantilever and its frequency response. The forces obtained from repeated approach curves between a particular pair of particles were averaged, typically leading to a force resolution of about 2 pN and down to distances of about 0.2 nm.

Results and Discussion
We use the AFM to study forces between silica particles of about 5 µm in diameter. A scheme of the experiment is shown in Fig. 1a. In particular, we investigate 2:2 electrolytes, namely CuSO 4 and MgSO 4 , and the 3:3 electrolyte LaFe(CN) 6 at pH 5.6. While the exponential decay of the double layer force is observed in all these systems, the decay length is substantially larger than the one expected from the DH theory. This behavior can be explained quantitatively by the formation of ion-pairs. In fact, formation of ion pairs is a consequence of ion-ion correlations.
where h is the smallest surface separation, R the particle radius, 0 ε is the permittivity of vacuum, ε the dielectric constant, eff ψ is the effective surface potential, and eff κ is the effective inverse decay length. The latter parameter is normally assumed to be identical to the inverse Debye length D κ , which can be evaluated for a symmetric z:z electrolyte from the relation 3 whereby e is the elementary charge, tot c is the total molar concentration of the electrolyte, A N the Avogadro's number, and whereby k is the Boltzmann constant and T the absolute temperature. We use ε = 80 as appropriate for water at 25°C. We have fitted the profiles shown in where H is the Hamaker constant. Since a power law decays more slowly than an exponential, the van der Waals force dominates the force profile at larger distances. At very high salt concentrations, the repulsive double layer force disappears completely, and the force profile is entirely dominated by the attractive van der Waals force, see Fig. 3.
In order to analyze the experimental data quantitatively, we rely on classical DLVO theory. This theory surmises that the overall force F can be approximated by the sum of the contributions of the van der Waals and double layer forces, namely 3 Thereby, we express the van der Waals force with eq. (3), while the double layer force is approximated with the DH-like expression given in eq. (1). This description further relies on the Derjaguin approximation, but this approximation is expected to be highly accurate as the size of the colloidal particles is much larger than the range of the interactions. 3 The Hamaker constant is evaluated from the force profiles at high salt concentrations. As shown in The fitted effective inverse decay length eff κ is shown in Fig. 4a, where it is plotted relative to the inverse Debye length D κ given by eq. (2). This ratio is unity within experimental error for both 1:1 electrolytes investigated, namely NaCl and CsCl. For multivalent electrolytes, however, this ratio is substantially below unity, and gets as low as 0.7 for the 2:2 electrolytes, and even 0.3 for the 3:3 electrolyte. This ratio decreases with increasing salt concentration. Interestingly, analogous deviations were revealed by TIRM in 2:2 electrolytes, albeit only mM concentrations were investigated. 6 The authors of the latter study ascribed these deviations, probably incorrectly, to experimental noise. No deviations in the decay length were reported in earlier force measurements in 2:2 electrolytes 22 , but this conclusion was possibly justified given the lower force resolution in these experiments. In asymmetric 1:z and z:1 electrolytes, measured decay length agreed reasonably well with the expected ones. 6,11,15,16,[22][23][24][25] Deviations in the decay length were equally reported in highly concentrated electrolytes and ionic liquids containing monovalent ions. [31][32][33] In dilute electrolytes, the deviations seem to be most pronounced in multivalent, symmetric electrolytes studied here. In the present situation, we interpret these deviations as follows. Multivalent ions in symmetric electrolytes are known to form ions pairs, which form according to the chemical equilibrium 35  The results of these calculations are compared with the relative effective inverse decay length in Fig.   4a, while the corresponding ionization fractions are shown in Fig. 4b. Indeed, the simple chemical equilibrium given in eq. (5) accounts for the observed dependence very well. Thereby, the value of the association constant K has to be fitted to the experimental data. The respective values are given in Table 1. The two measured values for the 2:2 electrolytes are identical within experimental error. One also observes that the values extracted from the force measurements compare favorably with earlier studies, which were based on conductivity, dielectric spectroscopy, or osmotic measurements. [35][36][37][38][39][40][41][42] While effects of ion pairing were also discussed in 1:1 electrolytes 43 , we cannot detect any deviations of this kind through force measurements in the present system. Ion pairing was also invoked to explain deviations in the decay lengths highly concentrated monovalent electrolytes and ionic liquids, albeit solvent structuring could also be involved. [31][32][33] Formation of ions pairs is also less pronounced in asymmetric z:1 and 1:z electrolytes 36 , and for this reason, deviations from the expected Debye lengths were not reported in these systems, or they were minor. 6,11,15,16,[22][23][24][25]  Recently, several authors have suggested that ion-ion correlation effects may induce a variation of the effective inverse decay length eff κ with the concentration. 34,44,45 For a symmetric z:z electrolyte, Kjellander and Mitchell proposed that at sufficiently low concentrations the leading correction is 34 where D κ is the inverse Debye length given by eq.
. Similar corrections as given in eq. (8) were also proposed in the literature. 44,45 However, the latter approaches predict 0 a > , which is in obvious disagreement with experiment. Note that the chemical equilibrium treatment does not lead to the correction given in eq.

(7).
Let us now discuss the measured effective potentials, which were converted to the more commonly used diffuse layer potential dl ψ . For a symmetric electrolyte, the respective relation reads 3 dl eff 4 tanh 4 z e z e kT Thus, by knowing the effective potential eff ψ one can obtain the diffuse layer potential dl ψ by inverting eq. (9). The diffuse layer potential reflects the electric potential at the origin of the diffuse layer. From force experiments in symmetric geometries, one can only extract the magnitude of the potential. The negative sign follows from the fact that silica is negatively charged through the dissociation of the surface silanol groups. This fact is also well established by independent methods, such as electrophoresis or streaming potential measurements. 46,47 The resulting diffuse layer potentials dl ψ versus the total electrolyte concentration are shown in Fig. 5.  Table 1.
One observes that the diffuse layer potential increases with increasing salt, which results from progressive screening. This trend can be quantified with the Grahame equation 3 where σ is the surface charge density compensating the diffuse layer charge. In this expression, one must use the concentration of the free ions, which can be calculated by means of the mass action law and the fitted association constants given in Table 1. The resulting charge densities are also given in this  25 Analogous trends of diffuse layer potentials with concentration and valence were also reported for positively and negatively charged latex particles in the presence of divalent, trivalent, and tetravalent counterions. 11,24 For ions of higher valence, the adsorption can be so important that a charge reversal is induced. 11,12,24 However, such a charge reversal is not observed here. Under present conditions, the bare charge density of the water-silica interface is about -30 mC/m 2 , as can be estimated by potentiometric titrations. 46,48,49 The magnitude of the latter value is substantially larger than the magnitude of the surface charge densities obtained from the present force measurements (Table 1). This difference originates from the charge compensation by the adsorbed and condensed counter-ions close to the interface.
In the light of the present results, let us now address the question why the DH or PB theories remain applicable to systems containing multivalent ions, in spite of the fact that such mean-field approach is expected to fail. [17][18][19] Levine and coworkers have already answered this question by analyzing their Monte Carlo simulation results. 20 They have investigated the concentration profiles around charged macroions in electrolyte solutions containing multivalent ions interacting through Coulombic and hard-sphere forces only. These results were compared with PB theory. It was found that the PB theory fails seriously, if the electrolyte is assumed to be fully dissociated and if the bare surface charge is used. However, PB theory was found to work well, provided two modifications were introduced.
First, the formation of ion pairs in solution was considered. Second, a renormalized surface charge density was introduced, which was chosen such that the large distance behavior of the concentration profiles was reproduced. The latter aspect was already pointed out earlier. 50 Since their model does not introduce any other interactions than electrostatic, ion pairing and charge renormalization are caused by ion-ion correlation effects.
Our approach to interpret our experimental data is not any different. By fitting an effective decay length, one considers the formation of ion pairs. The respective association constants are found to agree with independent reports. The fitted effective charge density is nothing but a renormalized charge density, which is chosen such that the long distance decay of the force profile is reproduced.
These charge densities vary with the type of ions, which clearly indicates their effective character.
Close to the interface, the ionic concentration is substantial, and ion-ion correlation effects will be important. , the precise choice of this cut-off is unimportant, provided the ion-pairs are sufficiently stable. Exactly the same relation as given in eq.
(11) can also be obtained from a statistical mechanical evaluation of the association constant in the classical approximation. Since ions are predominantly in direct contact, a situation that is referred to as contact ion pairs, the distance of closest approach min r can be estimated from the sum of the ionic radii that are known from X-ray and neutron diffraction. 52,53 The relevant ionic radii are summarized in Table 2. With these distances, one can use Bjerrum's theory to calculate the association constants, and which are given in Table 1. Indeed, this theory predicts the right trend of the measured constants with valence, albeit the calculated constants are about factor of 2 larger than the experimental ones.
This discrepancy could be due to the fact that not only contact ion pairs are present, but some of these pairs are separated by water molecules, leading to the formation of solvent-separated or solventshared ion pairs. 36 The distance between ions in these ion pairs is larger, which leads to a lower binding constant. Moreover, ions are not just charged hard-spheres, and the assumed Coulomb law will become inaccurate close to contact, especially since the dielectric continuum picture is no longer appropriate and other forces are equally present, particularly, dispersion and solvation interactions. In spite of its simplicity, the predictions of the Bjerrum theory are most reasonable, and they confirm that Coulombic forces and, thus, ion-ion correlations are indeed important in the formation of these pairs.

Conclusion
To conclude, we have presented results of direct force measurements between negatively charged micron-sized silica particles in solutions of symmetric, multivalent electrolytes, in particular, 2:2 and 3:3 electrolytes. We find that the measured force profiles can be quantitatively interpreted with DLVO theory, by taking van der Waals and double layer interactions into account. The latter can be accurately modeled with the DH theory, provided two effects are taken into account. First, the formation of ion pairs must be considered, as their presence increases the decay length of the electrostatic interactions substantially. Second, the diffuse layer potential of the interface must be properly adjusted, as this value is influenced by effects of ion adsorption. Both modifications are a consequence of ion-ion correlations, which become especially important in multivalent symmetric electrolytes.