Approach to Estimation of Hamaker Constant as Taking Hofmeister

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Approach to Estimation of Hamaker Constant as Taking Hofmeister Effects into Account Yaxue Luo, Xiaodan Gao, Rui Tian, and Hang Li J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b12830 • Publication Date (Web): 10 Apr 2018 Downloaded from http://pubs.acs.org on April 10, 2018

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The Journal of Physical Chemistry

Approach to Estimation of Hamaker Constant as Taking Hofmeister Effects into Account Yaxue Luo, Xiaodan Gao, Rui Tian, Hang Li* Yaxue Luo and Xiaodan Gao comcontributed equally to the work College of Resources and Environment & Chongqing Key Laboratory of Soil Multi-scale Interfacial Process, Southwest University, Chongqing 400715, China * To whom correspondence should be addressed: E-mails: [email protected] Phone: 086-023-68250674; Fax: 086-023-68250444.

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Abstract: The Hamaker constant is an important material parameter for both the description of particle interaction and the prediction of colloidal stability. Existing methods based on the classic DLVO theory ignore the Hofmeister effects and would derive flawed results. In this paper, approaches to the estimation of Hamaker constant with and without consideration of the Hofmeister effects were respectively suggested, where the Hamaker constant was obtained based on the mathematic relationship between electrostatic repulsive and van der Waals attractive interaction at critical coagulation concentration (CCC) by using dynamic light scattering technique. It indicated that the montmorillonite particles aggregation kinetics in the presence of Li+, K+ and Cs+ exhibited remarkable Hofmeister effects and the CCC values show: Li+ (277.2 mM) > K+ (80.3 mM) > Cs+ (27.2 mM). Without consideration of Hofmeister effects, completely distinct Hamaker constants for the same material were obtained from aggregation kinetics of the three cations (6.70, 17.4 and 35.7 × 10–20 J for Li+, K+ and Cs+ system, respectively). Obviously it is unacceptable because the obtained Hamaker constant values should be the same for the same montmorillonite. In contrast, by taking the Hofmeister effects into account, consistent Hamaker constants (6.20, 6.09 and 6.75 × 10–20 J for Li+, K+ and Cs+ system, respectively) could be derived and reached in good agreements with results reported in literatures. Our study proved that Hofmeister effects deeply affect the solid/liqud interface process of nano-/micro-sized particles, and only by taking the Hofmeister effects into account could we get the reliable Hamaker constant and correctly describe particle interaction.


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1. Introduction The Hamaker constant is an important material parameter for the description of particle interaction and it is widely applied in many scientific and engineering fields, especially in the environmental science and technology. Particle interaction of colloids is a significant process in environment, significantly affecting the fate and transport of nanoparticles along with ions and contaminants substance.1 The colloidal stability is controlled by the balance of the repulsive and attractive interaction forces, and thereby deep understanding of internal forces between particles at the nanoscale is essential for the precise control of the transport, sequestration, and mitigation of environmental pollutant. The van der Waals attractive force is one of the major interaction forces ubiquitous between nanoparticles in solutions. Within the equations that describe the van der Waals force, the accurate calculation of the van der Waals force between particles is dependent on the value of Hamaker constant, A, a parameter that is material dependent.2 Hence, an accurate determination of Hamaker constant is needed for the purpose of precisely describing the interfacial phenomena and predicting the aggregation or dispersion behavior of colloids. A number of approaches and techniques have been put forward to measure the Hamaker constant. The theoretical value of Hamaker constant could be calculated using the Lifshitz theory as the dielectric or spectral properties of the given material in a wide frequency spectrum known in advance.3,4 However, difficulties for both the obtainment of full-spectrum data for most materials and the accurate determination of related



parameters, e.g., dielectric permittivity, polarizabilities, make the estimation of Hamaker constant hard to achieve. The surface force apparatus (SFA)5 and atomic force microscope (AFM)6–8 are powerful instruments introduced to estimate the Hamaker constant, whereas the results are sensitive to the presence of impurities9 and require complex sample preparations. Quite a few methods based on the classic Derjaguin-Landau-Verwey-Overbeek (DLVO) theory,10 such as the widely applied attachment efficiency approach11–13 and the so-called yield stress-zeta potential technique9 have been presented recently, in which the Hamaker constant is obtained by the mathematic relationship between the electrical double layer repulsive interaction and van der Waals attractive interaction at critical coagulation concentration (CCC) or isoelectric point (IEP). However, those approaches for Hamaker constant estimation include the following flaws: 1) In the aggregation attachment efficiency approach, the initial increase in the hydrodynamic radius of particles was assumed to be linear with respect to time. Thus the experiment data of aggregate growth were treated to fit a straight line in the initial stage of aggregation with slope of the straight line used to calculate the aggregation rate constant. 14–17 However, numerous previous dynamic light scattering (DLS) results have indicated that the aggregates growth was actually not linear with respect to time but followed a power law.12, 14–18 2) The surface potential is a crucial parameter in relation to assessing the repulsive electrostatic interaction, whereas zeta potential (ζ) is always used as a surrogate of surface potential in the application of the approach,12, 13, 19, 20 despite the fact that there


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are obvious differences between the two. The shear plane is demonstrated to be far away from surface but close to the Gouy plane with zeta potential found to be up to 3 to 6 times lower than surface potential.21, 22 Low23 has determined the swelling pressure of montmorillonite in aqueous solution and the theoretical values predicated by zeta potential completely deviated from the experimental data (Figure 1),23 while the swelling pressure estimated by surface potential exactly match the experimental data (Figure 1).21 Consequently, taking zeta potential as the surrogate of surface potential would result in the repulsive electrostatic interaction and obtained Hamaker constant underestimated (Aestimated < Areal). 3) More importantly, current approach based on the classic DLVO theory neglects the influence of Hofmeister effects on ion-surface interaction and particle interaction. Hofmeister effects, observed in protein precipitation experiments 120 years ago, significantly affect a series of interface interactions, and a Hofmeister series of Cs+ > Rb+ > K+ > Na+ > Li+ for monovalent cations has been reported in colloidal particles aggregation according to their decreasing coagulation ability.24,25 Correct estimation of Hamaker constant of materials based on the classic DLVO theory requires that there is only classic Coulomb interaction energy in ion-surface interaction to yield the correct electrostatic repulsive interaction. Whereas ions would bear additional non-Coulomb interaction in the presence of Hofmeister effects, and thus results based on the classic DLVO theory couldn’t reflect the real Coulomb interaction and the obtained repulsive electrostatic interaction and Hamaker constant would be overestimated (Aestimated > Areal). The above discussions show that, at present, a more reliable and widely applicable



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approach is needed for Hamaker constant estimation. In this paper, a new approach for determination of the Hamaker constant through the modified DLVO theory as taking Hofmeister effects into account using DLS measurement was developed and validity of the new approach was verified. 2. Theory 2.1. Approach to Hamaker constant estimation based on the classic DLVO theory without consideration of Hofmeister effects The classic DLVO theory which treats particle interaction in terms of a balance of van der Waals attraction, εvdW, and electrical double layer repulsion, εEDL, could be expressed as,

∆E ( f i ) = ε EDL + ε vdW

(1)

where ∆E(fi) is activation energy of particle aggregation; fi (mol L–1) is the concentration of ith ion species in bulk solution; The electrostatic repulsion between colloidal particles depends on the potential at the overlapping position of two EDLs for the adjacent two particles, φ(λ/2), and it can be calculated by eq 2, 26

π

2

1 1 +   e 2   2 

2 Z i Fϕ ( λ / 2 ) RT

2

 3 +  e 8

4 Z i Fϕ (λ / 2 ) RT

Z i Fϕ 0 − Z i Fϕ ( λ / 2 ) − Z i Fϕ ( λ / 2 )  1 2 RT = λκe 2 RT (2)  − arcsin e 4 

where Zi is the ion valence and F (C mol–1) is Faraday constant; λ (m) is the distance between two adjacent particle surfaces; R (J mol–1 K–1) and T (K) are the gas constant and absolute temperature, respectively. κ (m–1) is the Debye-Hückel parameter, and

κ = 8πF 2 f i /εRT ; ε is the dielectric constant (8.9 × 10–9 C2 J–1 m–1 for water); φ0 is surface potential and can be calculated by solving the classic Poisson-Boltzmann


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equation for 1:1-type electrolytes.27

ϕ 0 (classic ) = −

κσ fi

= 1+

2 RT  1 − u  In  Zi F 1+ u 

4 4 − 1 + u 1 + e −1u

(3)

(4)

where σ (C m–2) is the surface charge density; u is a temporary parameter defined as u = tanh(ZiFφ0/RT). Then, the electrostatic repulsive pressure PEDL (λ) (Pa) between two adjacent particles was calculated as,28

  ZFϕ (λ/ 2 )   PEDL (λ ) = 2 RTf i cosh   - 1  RT  

(5)

Correspondingly, the van der Waals attractive pressure Pvdw(λ) (Pa) was estimated as,29 PvdW (λ ) = -

A −3 λ 6π

(6)

where A (J) is the effective Hamaker constant. Thus the energy barrier ∆W (kT) for particle aggregation can be expressed by eq 7,

∆W =

S kT

∫

b

a

PDLVO (λ )dλ =

10 −9 (b − a ) b ∫a [PEDL (λ ) + PvdW (λ )]dλ kT

(7)

where a (m) and b (m) are the lower and upper limits when P(λ) = 0 of the p(λ) ~ λ curves, respectively, and just in the domain of a < λ < b, a potential barrier is presented; S (m2) represents the average area to go through a repulsive space from λ = a to λ = b, in which 10–9 m is the approximate thickness of hydrated montmorillonite plate. However, a particle of Brownian motion would have an average instantaneous kinetic energy in suspension, and Einstein equation gave a description of the velocity of



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the particle,

v=

[∆x(t )] 2 /t =

(8)

2D / t

[∆x(t)] 2 is the mean square displacement in one

where v is the velocity of the particle;

dimension of a free Brownian particle during time t; D is the diffusion constant and can be calculated using D = kT/η, in which k is Boltzmann constant and η is the Stokes friction coefficient. Li et al.30 have experimentally verified that the real velocity of a particle with mass m in suspension is expressed as v2 = kT/m, and thereby the average kinetic energy of the Brownian particle is 0.5 kT. However, 0.5 kT is a maximum value of the kinetic energy for the Brownian particle, which could’t be reached in most cases. Li et al.31 has found that the real kinetic energy of the Brownian particle at aggregation is merely 0.2 kT. Therefore, at fi = CCC, the real activation energy should be expressed as, ∆E (CCC ) = ∆ W (CCC ) − 0.2 kT = 0

∆W (CCC ) =

S kT

∫

b

a

PDLVO (λ )dλ = 0.2 kT

(9) (10)

Combining eqs 2–6 and 10, the electrostatic repulsive and van der Waals attractive interaction between particles at CCC in various electrolyte solutions could be accurately calculated by introducing different assumed Hamaker constants, A. The value of A which produced an energy barrier of 0.2 kT for particles aggregation is the Hamaker constant of the particle.

2.2. Estimation of Hamaker constant as taking Hofmeister effects into account However, the classic DLVO theory which underpins colloid and interface science for hundreds of years is shown to be flawed because it is unable to explain the ion


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specificity in colloidal particles interaction.32 As Hofmeister effects are considered, an additional energy must be taken into account in the ion-surface interaction.

wi ( total) = Z i Fϕ 0 + wi (add) = γi Z i Fϕ 0

(11)

where wi(total) is the total ion-surface interaction energy of the ith ion species; ZiFφ0 is the classic Columb interaction energy; wi(add) represents the additional interaction energy of Hofmeister effects; γi is effective ionic charge coefficient of the ith ion species. The origin of the Hofmeister effects and wi(add) is still under debate so far, however. Current explanations of Hofmeister effects include the ionic sizes, ionic hydration, dispersion force or ionic non-classical polarization etc.33, 34 Note that, we do not need to figure out the origin of Hofmeister effects here since no matter which aforementioned effects produce the additional interaction energy of cations, its contribution to the total ion-surface interaction energy, wi(total), could be reflected in modification of the charge number. As a result, the cationic apparent charge changes from Zi into γiZi in the presence of Hofmeister effects as eq 11 indicated. Additionally, the differences in wi(add) of cations will certainly lead to the different adsorption ability of cations with charged surfaces, which could be quantitatively characterized by the effective ionic charge coefficient, γi, obtained from exchange equilibrium. Thus, γi can be taken as a parameter used to modify the classic Coulomb interaction energy due to the additional interaction and to quantitatively reflect the strength of Hofmeister effects of the ith ion species. On the other hand, the different adsorption affinity of cations with surface, arising



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from the different additional interaction energy, would make the surface potential of charged particles changed in the presence of Hofmeister effects. As taking into account the different additional interaction energy of equivalent cations, we replaced Zi by γiZi in eqs 3 and 4, the modified surface potential that considering the Hofmeister effects could be rewritten as, ϕ0 m = −

2 RT  1 − u  ln   γi Z i F  1 + u 

(12)

Then, solving eq 12 in combination with eqs 2, 5, 6 and 10 with the effective charge coefficient of cation, γi, known in advance, the modified particle interaction energies and the Hamaker constant as taking the Hofmeister effects into account in different cations systems could be obtained. Here, we will show how the γi derived from exchange adsorption experiments permit valid Hamaker constants to be calculated in the presence of Hofmeister effects. In aggregation process of a given charged particle, there is ∆E(fi) = 0 at CCC, and thus eq 1 indicates that reliable estimation of the van der Waals interaction energy, εvdW, depends on the correct electrostatic repulsive interaction energy, εEDL, in cases of different equivalent cations. For equivalent cations with identical classic Coulomb interactions, cation with large wi(add) and large γi means a high adsorption affinity with charged surface to more effectively screen surface charge. As a result, the electric fields around particles and electrostatic repulsive interaction energy barrier, εEDL, for particle aggregation would remarkably decrease. As electrostatic repulsive interaction in cases of different equivalent cations could be modified by γi, correct van der Waals interaction could be obtained to permit valid Hamaker constants estimation.


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3. Results and discussion 3.1. Estimation of Hamaker constant without consideration of Hofmeister effects The aggregation kinetics of montmorillonite particles, with a surface charge density of 0.1586 C m–2, in LiNO3, KNO3 and CsNO3 solutions given by Tian et al.35 were presented in Figure 2. It shows that marked differences could be observed in the growth curves of the effective hydrodynamic diameters of aggregates in the presence of the three monovalent cations, clearly demonstrating the presence of Hofmeister effects. It can be seen that Cs+ exhibited the fastest aggregation rates and followed by K+ and Li+. For example, at concentration of 30 mmol L–1, the effective hydrodynamic diameters of montmorillonite aggregates at 60 min were 332.9, 2110 and 2819 nm in LiNO3, KNO3 and CsNO3 solutions, respectively. Based on the study of Jia et al.36, the CCC value of colloidal particles aggregation could be obtained by the defined total average aggregation rate (TAA rate) by DLS measurements. The TAA rate is derived from the following equations.

1 v~T ( f i ) = t0

∫

t0

0

1 v~(t , f i )dt = t0

∫

t0

0

1 t   t ∫0 v(t , f i )dt  dt

(13)

~

where vT(fi) (nm min–1) is the TAA rate from t = 0 to a given time t = t0, and the upper limit of t0 can be the ending time of the aggregation process; v(t, fi) (nm min–1) is the real-time aggregation rate defined as the growth rate of the effective hydrodynamic ~

diameter of aggregates measured by DLS; v(t, fi) (nm min–1) is the average growth rate of aggregates from t = 0 to an arbitrary time t. Since v(t , f i ) =

dDh (t ) , the average growth rate at a given fi from t = 0 to t = t was dt

expressed as,



D (t ) − Dh (0) 1 t dDh (t ) 1 D (t ) v~ (t , f i ) = ∫ dt = ∫ dDh (t ) = h D 0 ( 0 ) t dt t h t

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(14)

where Dh(0) (nm) and Dh(t) (nm) are the effective hydrodynamic diameter of the primary particles (t = 0) and the aggregates at time t, respectively. Dh(0) and Dh(t) can be directly determined by the DLS measurements. Based on the experimental data shown in Figure 2 and eq 14, the average growth rate of aggregates at arbitrary time of the aggregation process could be obtained and the ~

derived fitted equation of v(t, fi) ~ fi was used to calculate TAA rate of particles within various electrolyte solutions by eq 13. Then Figure 2 can be transformed into the TAA rates curves displayed in Figure 3. The electrolyte concentration at the intersection point of the two distinct stages in the ~

curves of v T(fi) vs. fi would be the CCC value36 and thereby the CCC values of montmorillonite particles aggregation in the presence of Li+, K+ and Cs+ were equal to 277.2, 80.3 and 27.2 mmol L–1, respectively. Once the CCC values were obtained, the sum of DLVO forces pressure at fi = CCC could be calculated by taking the Hamaker constant, A, as the single unknown parameter with combination of eqs 2–6. Then relationships between the net DLVO force pressure, PDLVO(λ), and λ at CCC, calculated by different assumed Hamaker constants in Li+, K+ and Cs+ systems were plotted in Figure 4. It clearly shows that: 1) with increasing Hamaker constant, the net repulsive pressures between particles sharply decreased; 2) for a given assumed Hamaker constant (e.g. A = 9 × 10–20 J), the PDLVO(λ) between particles in cases of the three different monovalent ions were quite different and decreased in the order as Cs+ > K+ > Li+. Based on the above results of the net


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DLVO forces pressure, we could deduce that obvious differences would be observed in the derived Hamaker constants in Li+, K+ and Cs+ systems with the Hamaker constants showing Cs+ > K+ > Li+. Then values of the areas of closed curves PDLVO(λ) vs. λ shown in Figure 4 were utilized to estimate the energy barrie, ∆W, of montmorillonite particle aggregation in LiNO3, KNO3 and CsNO3 solutions via eq 7. The results of the obtained energy barrier at different assumed Hamaker constants were presented in Figure 5. It shows that the energy barrier of 0.2 kT were respectively derived using Hamaker constants of 6.70 × 10–20, 17.4 × 10–20 and 35.7 × 10–20 J for montmorillonite particles aggregation in cases of Li+, K+ and Cs+ systems, which indeed followed the order predicted above. Evidently, completely distinct Hamaker constants for the same montmorillonite material were derived from the aggregation kinetics of three monovalent ions as Hofmeister effects weren't taken into account. It was out of line with the facts. Moreover, the pronounced discrepancy in the determined Hamaker constants clearly implied that we couldn’t get a reliable Hamaker constant without consideration of the Hofmeister effects.

3.2. Estimation of Hamaker constant as taking Hofmeister effects into account. In the presence of Hofmeister effects, the reliable estimation of Hamaker constant of materials depends on the correct obtainment of electrostatic repulsive interaction in cases of various cations. Clearly, within the equations that describe the modified electrostatic repulsive interaction (eqs 12, 2 and 5), γi is the crucial parameter that determines the strength of additional interaction energy of the ith ion species, and further determines the strength of electric field around particles and energy barrier for



particle aggregation. Li et al.31 has estimated the absolute effective charge coefficient of Na+ (γNa = 1.18) based on the dynamic light scattering measurements of activation energies for montmorillonite particle aggregation. Then by treating Na+ as a reference cation, the γi values of Li+, K+ and Cs+ could be obtained from cation exchange equilibrium experiments. Liu et al.37 has obtained the relative effective charge coefficients of Li+ and K+ from Li+/Na+ and K+/Na+ exchange equilibrium as taking Na+ as a reference, and γLi/Na and γK/Na were equal to 0.901 and 1.646, respectively. Thus we have γLi = γNa × γLi/Na = 1.18 × 0.901 = 1.063; γK = γNa × γK/Na = 1.18 × 1.646 = 1.942. In addition, a value of γCs/K = 1.475 was estimated by Li et al.38 from K+/Cs+ exchange equilibrium, and thus we could get γCs = γK × γCs/K = 1.942 × 1.475 = 2.864. Clearly, the remarkable discrepancy in the absolute effective charge coefficient of Li+, K+ and Cs+ indicated that the additional interaction energy of the three monovalent cations are quite different. Ion species with a larger γi value are ought to bear stronger additional interaction energy to more effectively compress the diffuse double layer and screen surface charges around particles. As a result, it can reduced the energy barrier of aggregation and induced fast aggregation starting at quite lower concentrations than ion species with a smaller one, and thus for γCs > γK > γLi, there should be CCC(Cs) < CCC(K) < CCC(Li). It can be seen that, the obtained γi rationally explain the Hofmeister series experimentally observed in the CCC values and colloidal stability in cases of the three monovalent cations. With γLi, γK and γCs obtained, the modified net DLVO forces pressure, PDLVO(λ), in Li+, K+ and Cs+ systems as Hofmeister effects presented were quantitatively calculated


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with combination of eqs 2, 12, 5 and 6, shown in Figure 6. By integrating the closed curves of PDLVO(λ) vs. λ in Figure 6, the modified energy barrier, ∆W, at different assumed Hamaker constants in the presence of Hofmeister effects were estimated via eq 7 (Figure 7). Comparing Figure 5 with Figure 7, we can see that, obvious decreases in the obtained ∆W were detected after the contributions of the electrostatic repulsive interaction were modified by γi. For example, at a Hamaker constant of 5 × 10–20 J in the presence of Li+ system, ∆W was slightly reduced from 0.586 kT to 0.427 kT as Hofmeister effects are considered. Unlike Li+, sharp declines in the energy barriers for aggregation were observed in the presence of Cs+, e.g. at a Hamaker constant of 9 × 10– 20

J, ∆W was reduced from 6.83 kT to 0.0778 kT as taking Hofmeister effects into

account. It’s clear that the derived electrostatic repulsive interaction and energy barrier for particle aggregation without consideration of Hofmeister effects were overestimated and a calibration of electrostatic repulsive interaction and energy barrier of particle aggregation was required to yield a reliable Hamaker constant in the presence of Hofmeister effects. Figure 7 clearly shows that the energy barrier of 0.2 kT were respectively derived using Hamaker constant of 6.20 × 10–20, 6.09 × 10–20 and 6.75 × 10–20 J for montmorillonite particles aggregation in Li+, K+ and Cs+ systems as Hofmeister effects were considered. It can be seen that by using the known effective charge coefficient, γi, to modify the classic Coulomb interaction energy of the three alkali metal ions, the obtained Hamaker constants from the ion-specific aggregation kinetics in cases of Li+, K+ and Cs+ were satisfyingly consistent. In addition, the obtained values of Hamaker



constants of montmorillonite derived from the new approach all fell within the range of results (2.25-6.659 × 10–20 J) previously reported in literatures determined by different methods,39–41 which further certified the reliability of Hamaker constants of montmorillonite particles estimated by the new approach. The above discussions strongly indicated that the Hofmeister effects are ubiquitous and profoundly influence nanoparticles aggregation behavior, and only by taking the Hofmeister effects into account could we get the reliable Hamaker constant. However, despite that classic DLVO theory fails for biology and colloid systems as the Hofmeister effects presented, the classic DLVO theory is conditionally correct and could be appropriately applied as the additional interaction are weak enough to be reasonably ignored. Among all the ions with non-valence electron, Li+ is the smallest with an ionic radius of 0.9 Å,42 and its static polarization of merely 0.028 Å3 is the weakest in inorganic cations and is much lower than that of K+ (0.795 Å3) and Cs+ (2.354 Å3).43 Additionally, the estimated absolute effective charge coefficient of Li+, γLi, is merely 1.063 ≈ 1, and thereby it is rationally believed that Li+ only bears the classic Coulomb interaction in ion-surface interaction. As a result, the classic DLVO theory could be approximatively applied to Li+ system. On the other hand, the derived Hamaker constants calculated from Li+ system with and without consideration of Hofmeister effects are very close to each other (6.20 × 10–20 and 6.70 × 10–20 J). Thus, our study suggested that Li+ could be employed for further application of the approach for Hamaker constant estimation.

4. Conclusion


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Hofmeister effects have a significant impact on the solid/liqud interface process of particles in nanoscale. In this paper, we presented approaches to the estimation of the Hamaker constant with and without consideration of the Hofmeister effects, respectively. It indicated that without consideration of Hofmeister effects, completely distinct Hamaker constants for the same montmorillonite material were obtained from the ion-specific particles aggregation kinetics in the presence of Li+, K+ and Cs+, which disaccord with the facts. Whereas, as taking the Hofmeister effects into account, consistent Hamaker constants could be derived and good agreements with results reported in literatures were achieved. The results showed that only by taking the Hofmeister effects into account could we get a reliable Hamaker constant and make it possible for both the correct description of particle interaction and the prediction of colloidal stability.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 41530855 and Grant No. 41501241) and the Graduate Research Program of ChongQing (Grant No. CYS2015047).

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Figure captions Figure 1.

The relationship between the swelling pressure, P, and the interlayer

distance, λ, as determined experimentally (●experimental data from Low (1981)) and by double-layer theory using different values of potential in EDL (▬▬ theoretical curve with φ = φ0; ▬□▬ theoretical curve with φ = ζ).

Figure 2.

The variations in the effective hydrodynamic diameters of montmorillonite

particles with time in LiNO3, KNO3 and CsNO3 solutions.

Figure 3.

The variations in TAA rates of the montmorillonite particles with electrolyte

concentrations in LiNO3, KNO3 and CsNO3 solutions.

Figure 4.

Relationships between PDLVO(λ) and λ at CCC, where curves in different

colors represent the net DLVO forces pressure calculated by different assuming Hamaker constants with unit of 10–20 J.

Figure 5.

Values of the estimated energy barrier, ∆W, at different assuming Hamaker

constants without consideration of Hofmeister effects.

Figure 6.

Relationships between the modified PDLVO(λ) and λ at CCC, where curves in

different colors represent the calculated sum of DLVO forces pressure of different assuming Hamaker constants with unit of 10–20 J.

Figure 7.

Values of the estimated energy barrier ∆W of Li+, K+ and Cs+ systems at

different assuming Hamaker constants as taking Hofmeister effects into account.



Figure 1


Page 24 of 31

Hydrodynamic diameter (nm)

3500

Li

3000 10 30 50 70 100 150 200 300 500

2500 2000 1500

3500

+

1000

+

K

3000 10 20 30 40 50 70 100 150 200

2500 2000 1500 1000 500

500 0

20

40

Time (min)

0

60

3500


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Page 25 of 31

+

Cs

3000 3 5 7 10 15 20 30 40 50

2500 2000 1500 1000 500 0

20

40

60

Time (min)

Figure 2


20

40

Time (min)

60


80

Li

150

+

+

K

v~T ( f 0 ) = 0.113 f 0 + 122

120

−1

TAA rates (nm min )

−1

TAA rates (nm min )

v~T ( f 0 ) = 0.0229 f 0 + 71.0

60 v~T ( f 0 ) = 0.295 f 0 - 4.56

40

20

90 v~T ( f 0 ) = 1.66 f 0 - 8.48

60 30

277.2 0

100

200

300

80.3 400

500

0

−1

Electrolyte concentrations (mmol L ) 150 −1

TAA rates (nm min )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 31

Cs

40

v~T ( f 0 ) = 0.800 f 0 + 98.3

90 v~T ( f 0 ) = 4.92 f 0 - 13.5

60 30 27.2 0

10

20

30

40

120

160

200 −1

Electrolyte concentrations (mmol L )

+

120

80

50 −1

Electrolyte concentrations (mmol L )

Figure 3


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Figure 4



Figure 5


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Figure 6



Figure 7


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TOC graphic


Approach to Estimation of Hamaker Constant as Taking Hofmeister

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