Surface interactions in simple electrolytes

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Surface interactions in simple electrolytes

Roland Kjellander, Stjepan Marčelja

To cite this version:

Roland Kjellander, Stjepan Marčelja. Surface interactions in simple electrolytes. Journal de Physique,

1988, 49 (6), pp.1009-1015. �10.1051/jphys:019880049060100900�. �jpa-00210767�

Surface interactions in simple electrolytes

Roland Kjellander ^and Stjepan Mar010Delja

Dept. ^of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, ACT 2601, ^Australia

(Requ le 2 octobre 1987, accepté le 30 novembre 1987)

Résumé. - Nous présentons des résultats et des concepts

qui proviennent ^de l’application rigoureuse des méthodes d’équation intégrale ^{de la} physique ^des liquides

problème des fluides ioniques qui

sont inhomogènes

voisinage de surfaces chargées. Cet article rassemble nos résultats récents obtenus par

l’approximation ^HNC anisotrope pour les interactions ^entre surfaces dans les électrolytes

^{et divalents.}

Par comparaison

la théorie de champ moyen de Poisson-Boltzmann, les différences principales ^sont

contribution attractive

interactions entre surface due

corrélations ion-ion et

répulsion causée par les

durs des ions. L’attraction est particulièrement forte dans les électrolytes divalents, ^{où la} répulsion ^de

double couche est spectaculairement ^réduite

transformée

attraction. Nous indiquons quelques conséquences ^{de cette} contribution

pressions ainsi que ^son lien

les interactions de Van der Waals. La taille des ions intervient dans la pression ^surtout pour les faibles distances ^entre surfaces, ^et principalement

pour les ions de grands rayons

à forte densité de charge.

Abstract.

We present

results and concepts which have emerged ^{from the} rigorous application ^of

the integral equation methods of liquid ^state physics ^{to the} problem ^of inhomogeneous ionic fluids in the

vicinity ^of charged surfaces. The report summarizes

recent results for surface interactions in mono- and divalent electrolytes ^obtained using ^the anisotropic Hypemetted ^Chain approximation. Compared ^{to the}

Poisson-Boltzmann mean-field theory ^the major differences

attractive contribution to the surface interaction due to ion-ion correlations and

repulsion from the ionic cores. The attraction is particularly strong in divalent electrolytes, ^{where the} double-layer repulsion ^is dramatically ^weakened

turned into

attraction. Some consequences of this pressure contribution and its connection ^to Van der Waals interactions

are

pointed ^out. The size of the ions affects the pressure predominantly ^at short surface separations, ^{and then}

to

substantial extent for large ^{ion radii}

^at high ^surface charge.

Classification

Physics ^Abstracts

82.70

1. Introduction.

The introduction of integral equation methods and computer simulations into liquid ^state physics ^has greatly ^{advanced the} theoretical understanding ^of

bulk solutions of simple electrolytes. ^{In the last three}

years, those methods have been fully applied ^{to the}

related problem ^of electrostatic interactions between

charged ^{surfaces or} between colloidal particles ⁱⁿ

ionic solutions. The specific problems ^associated

with accurate treatment of simple inhomogeneous

Coulomb fluids in the vicinity ^of charged ^surfaces

have now been _overcome, and a limited number of results has been obtained within the primitive ^model

of electrolyte solutions. The calculations of surface interactions have mainly ^been performed ^{in Lund} (Sweden) by Monte Carlo (MC) simulations [1-3]

and in Canberra (Australia) ^{via the} anisotropic Hypernetted ^Chain (HNC) approximation [4-7].

The accurate theoretical results are of particular

interest in view of the experimental information on

the interaction between charged surfaces immersed in aqueous electrolyte ^solutions [8] brought by ^the

introduction of the molecular force measuring ap-

paratus in Canberra some ten years ago. ^In evaluat-

ing ^surface interactions, ^the present calculations represent ^{the first} major progress beyond ^{the Pois-}

son-Boltzmann (PB) equation, ^used by Gouy ^and by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049060100900

1010

Chapman ^{at the} beginning ^{of the} century. ^The excellent reviews by ^Carnie and Torrie [9] give ^a good background of various double layer ^theories.

Primitive model electrolytes consist of charged hard-sphere ions immersed in a continuum dielectric medium. The use of the primitive model for electric double layer systems ^{is a} slight generalization ^{of the}

usual modelling ^{of such} systems ⁱⁿ physical ^{or colloid} chemistry : point charge ^{ions in a} dielectric con-

tinuum. In the conventional ^treatment

the DLVO

theory [10, 11]

surface interactions are calculated

on the basis of the PB equation ^{with the addition of}

the Van der Waals force. The DLVO theory ^{is still}

in widespread practical use, but since it is based on

the mean-field PB treatment of the ion interactions

-

neglecting ion-ion correlations

in some circum- stances it gives ^erroneous predictions. ^{The aniso-} tropic HNC and the simulation studies give ^accurate

results within the primitive ^model. However, ^{in a} description ^{of the} inhomogeneous systems ^{on this} level it is still not feasible to go beyond ^the simple

Coulomb and spherically symmetrical ^{core interac-}

tions and explicitly include the structure of aqueous solvent. As a result, the interaction of surfaces in aqueous solutions of electrolytes ^{at small} separations

remains poorly understood.

In this paper ^we shall present ^{some of our recent} results for the interaction between charged plates

immersed in electrolyte ^{solutions as} calculated using

the anisotropic HNC method. For simplicity, ^we

shall restrict ourselves to systems ^{where the dielectric}

constant of the plates ^is equal ^to that of the solvent,

i.e. we neglect image charge ^effects. However, ^the anisotropic HNC method is capable ^of handling ^the image charge interactions without any further

approximations [4, 5], ^{and this} topic ^{will be} briefly

discussed in the concluding section in conjuction

with Van der Waals forces.

The HNC closure was selected because of its known accuracy in .calculations for particles ^interact- ing with Coulomb potentials. ^The accuracy of the calculated density profiles and pressures has been checked by comparison ^{with the MC} simulations [3, 4, 7]. ^While relatively ^{few suitable} MC results are

available, ^{in all cases we have found} good agreement between the HNC and the simulation results.

2. Theory and method.

The HNC approximation ^{for the} pair distribution function 9 ij (r, r’ ) between ionic species ⁱ and j ^{is to}

set

where /3 = (kB T)- 1, kB ^is Boltzmann’s constant, ^T is the temperature, Uij ^{is the} pair potential, hij = gij - 1 ^and Cij ^is the direct correlation function. The

functions hij ^and cij ^are (by definition) ^related by ^the

Ornstein-Zernike equation

where nk (r ) is the number density ^of species ^k.

These equations ^are used in the standard HNC

theory ^for homogeneous systems, ^{in what case n is} independent ^{of r and all} pair ^{functions are} isotropic,

i.e. only depend on r - r’ . ^For inhomogeneous

systems, like the diffuse double layer, ^the density

varies from point ^to point, ^{and the} pair ^{functions are} anisotropic. ^{The set of} equations (1), (2) ^{is no} longer complete ^{and has to be} supplemented by ^an equation

for the ion density distribution. In the anisotropic

HNC approximation, ^as adopted by Kjellander ^and Mar6elja [4, 5], equations (1) ^and (2) ^{are used as} they stand, ^{and the} density distribution is determined

by

where j

and where’ = exp (,B tk i )/A 3 ^{is the} activity ^of species i, JL the chemical potential, Ai the thermal

wavelength, t/J av (r) the average electrostatic poten- tial at r, t/J 00 ^the potential ^at infinity ^{and ues the}

electrostatic pair potential between the ions. The function ffINc(r) gives the contributions from the

pair correlations of the ions. Incidentally, ^{we remark}

that without f HNC ^we would obtain the density

distribution of the PB approximation (in ^{which case} 4’a, ^{should be evaluated in the same} approximation).

It is noted that the explicit expression ^for frNc(r)

given ^above originates [5] ^{from a} general formula for the chemical potential ^{evaluated in the HNC} ap-

proximation. ^{Therefore its} validity ^is restricted to that case.

Equations (1)-(3) together ^with the condition of

electroneutrality ^{form a} complete ^{set of} equations

that can be solved numerically. ^{For an} open system in equilibrium ^{with a bulk salt solution, the mean}

chemical potential JL:t from the bulk is inserted in

equation (3) (for ^{details cf. Ref.} [7]). ^{We note that}

in planar geometry, for instance for a liquid ^between

two walls, ^the density only depends ^{on the coordi-}

nate perpendicular ^{to the walls,} n (r) = n (z) (pro-

vided the ion-wall potential ^{is also a function of}

only). ^The pair ^functions depend ^{on three} indepen-

dent coordinates, e.g. the z-coordinate of the ^two

particles ^{and their} separation. ^{If the two surfaces are} equal, ^{which we shall assume,} the function n (z) ^is symmetrical around the midplane.

We solve [5] ^the problem ⁱⁿ practice by dividing

the space between the surfaces _{into many} layers, ^and evaluating ^{the structure of the} equivalent physical

system : ^a multi-species two-dimensional fluid,

where each species corresponds ^{to one} layer ^{of the}

actual physical system. Typical calculations may involve some hundred layers, ^with denser subdivi- sions closer to the surfaces. In the limit _{of many,} very thin layers, ^one obtains solutions for the

inhomogeneous ^{fluid with the same} accuracy ^as that

normally ^{achieved in the} corresponding ^homo-

geneous fluid calculations. The method has been described in our earlier publications [5, 7]. It suffices to say here that equations (1)-(3) ^{are solved} by ^a

double iterative procedure. ^{Given a} starting profile,

for example ^{from the PB} theory, equations (1), (2)

are solved by iteration for the pair correlation functions. These correlation functions are inserted in

equation (3) ^{and a new} profile is determined itera-

tively. ^{The new} profile is then inserted in equa- tions (1), (2) and the whole procedure ^is repeated

until full self-consistency ^{is attained. The} long-range

tails and the discontinuities caused by the Coulomb and the hard core potentials (in both real and Fourier space) ^are subtracted off and treated analyti- cally.

Once the density profile ^{and the} pair distribution functions have been calculated, various ther-

modynamical quantities ^{such as} the pressure between the walls and the free _energy, as well as the local and the average electrostatic potentials ^are easily ^deter-

mined. The internal pressure due ^to the ions is given by

where ni (0) is the concentration at the midplane

between the surfaces, Pei ^{is the} electrostatic force per unit ^{area across} the midplane ^{due to} the ion-ion interactions (this ^{is non-zero} since each ion on one

side affect the ion distribution on the other side) ^and Pcorc is the pressure component ^{due to core-core}

contacts across the midplane. Explicit formulae for these components ^{can be found} in reference [7].

The net pressure between the surfaces equals P net = Pint - Pbulk, ^where P bulk ^{is the} pressure of ^the bulk electrolyte with which the system ^{is in} equilib-

rium. The bulk pressure is analogously composed ^of

an ideal contribution kB T £ ^{npulk and an} electrostatic

i

and a core contribution, which should all be calcu- lated within the same approximation ^as the internal pressure. We accordingly ^have

Note that in the PB approximation, ^which neglects pair correlations, ^{the last two brackets in} equation (5) ^are identically ^{zero, and} only ^the first, ideal contribution remains.

3. Density profiles.

The ion density profiles ^show large deviations from the mean-field results only for rather high ^surface charge densities. Near the surfaces, ^the density profile ^{can have a} secondary peak indicating layering

due to the crowding ^{of the} counterions. This

phenomenon ^was first found in simulation studies for a single ^surface [12], ^{and has since been con-}

firmed by ^various integral equation techniques [7, 13]. ^The anisotropic ^HNC calculations for two surfaces [7], ^{show that the} secondary peak ^increases

in size at short surface separations.

At low to medium surface charge densities, ^the

deviations of the profiles ^{from the} PB results are

larger ^{for coions} than for counterions on a relative

scale, ^{but on an absolute scale} the differences are not large. Very good theoretical profiles ^{have been}

obtained earlier with a simpler approximation : ^the

modified PB equation [14, 15]. Comparison ^{of those}

results with extensive MC simulations is available in references [12] ^and [15, 16]. While those earlier workers had the technical framework to examine surface interactions, ^this apparently ^{has not been} attempted.

In contrast to density profiles, the surface-surface interactions are very sensitive ^to the deviations of the ionic fluid structure from the predictions ^{of the}

mean-field theory. Whilst the overall ionic density

may be very close ^to the mean-field value, ^the correlation between ions will significantly change ^the

force balances in the system, which leads to a strong attractive contribution to the surface-surface interac- tion. In cases where the bulk reservoir solution contains dissolved salt, ^this sensitivity is further enhanced by ^the fact that the net pressure ^is the difference between the internal pressure (between

the surfaces) and the external pressure (in ^the bulk).

4. Surface interactions in monovalent electrolytes.

We begin ^the examination of surface interactions in the presence of monovalent ions by considering ^one component electrolytes : ^solutions containing only

counterions. The system ^{without coions is numeri-} cally faster and simpler ^to evaluate. The results are

also of relevance at low salt concentrations when the surface separation is short. The coions are then

largely excluded from the space between the ^sur- faces. Under such conditions, the internal pressure between the surfaces in the presence of salt is hardly distinguishable ^from the pressure in ^cases where counterions are the only ^ion species ^{in the} system.

The interaction between the surfaces is affected by

1012

two major contributions which are not present ^{in the} PB theory : ^the attractive effect of electrostatic correlations and the repulsive pressure of the hard

cores. Each of these terms can dominate the other,

and the pressure ^can be either larger ^or smaller than the corresponding ^{PB value} (Fig. 1). ^{The hard cores}

are only important ^at small surface separations ^and

Fig. 1. - The pressure

function of separation ^be-

tween two charged ^surfaces interacting

^electro- lyte phase

calculated in the PB and anisotropic ^HNC

theories. The pressure P/RT, where R is the gas ^constant and T the temperature, is measured in mol dm-3 [M]. ^The

surfaces have

uniform surface charge density ^of

elementary charge per 0.6 nm2. The electrolyte phase

consists of monovalent counterions with radii 2.3 A. The dielectric constant is 78.358 and the temperature ^{298.15 K} (this ^also applies ^to all other cases below unless explicitly

stated otherwise). Throughout this paper the surface

separation is defined

the distance between the points ^of

closest approach of the ion centers to the two surfaces.

Fig. ^2.

Interaction between the

surfaces

in

figure 1 for various counterion radii. The pressures

calculated within the anisotropic ^HNC theory ^and

shown relative to the PB values by plotting ^{the ratio}

between the HNC and the PB results.

relatively high ^surface charges. ^{This is} illustrated in

figure 2, ^{where at short} separations the increase in

repulsion compared ^{to the PB} theory ^is clearly

attributable to the effect of ion radius.

Next, ^{we treat} _systems with added salt, ^{i,e. in} equilibrium ^{with a bulk} electrolyte ^solution (reser- voir). ^From figures 3 and 4 it can be seen that

Fig. 3. - Interaction (P/RT[M ] ) ^{between two} plates

with uniform surface charge density ^of

elementary charge per 0.85 nM2 in the anisotropic HNC and the PB theories. The plates

immersed in

1.0 M 1:1 electro-

lyte solution which consists of ions with the radius of 2.125 A. The temperature is 298.0 K and the dielectric constant 78.5.

Fig. ^4.

^Some examples of the interaction (relative ^to

the PB pressure) ^{between two} uniformly charged plates ⁱⁿ

1:1 electrolyte solutions. The three systems ^{have the} following parameters (listed ^from top ^to bottom) : ^{0.1 M} electrolyte, ^0.6 nm2 per unit surface charge and ion radii 2.3 A ; ^0.5 M, ^0.714 nm2 and 2.3 A ; 1.0 M, ^0.85 nm2 and 2.125 A (same

ⁱⁿ Fig. 3) respectively. ^The symbols

at the end of the 0.5 M

signify points ^{with much}

lower relative accuracy than the full

The net

pressure for these large separations ^is

very small

difference between two large ^numbers.

beyond ^{the short} separation region where hard core

effects are important, ^the double-layer repulsion ^is

much smaller than that expected from the PB

theory. The effect is most dramatic at higher ^salt

concentrations. In figure ^{3 we have selected a surface} charge ^{of 0.188 C} m-2, ^{which for} separations ^above

0.85 nm corresponds ^{to the constant} potential example presented by Lozada-Cassou and Hender-

son [17]. The calculated interaction _{is very} different from that of reference [17], ^{which uses a} simpler theory, ^a singlet ^HNC theory, where the direct correlation function for the inhomogeneous ^electro- lyte ^{solution is} approximated by its bulk values.

For concentrations up ^to about 0.1 M, the interac- tion at large separations appears ^{as a} roughly exponential law, ^shifted by ^an approximately ^con-

stant factor with respect ^to the PB values. For the 0.1 M _case, figure ^4, this behaviour is evident at

separations larger than about 2 _nm, where the ratio between the HNC and the PB results is nearly

constant. At 0.5 M and 1 M electrolyte ^concen- trations, ^{the PB} predictions ^are hardly ^{ever valid as}

seen from the examples presented ⁱⁿ figure ^4.

5. Surface interactions in divalent electrolytes.

In case of divalent ions, ^our currently ^available

results are restricted to one-component electrolytes.

The reason is the increased difficulty ⁱⁿ achieving

sufficient numerical accuracy in calculations of the

rapidly varying anion-cation correlation functions.

While this does not greatly affect the calculation of ion density profiles, the interaction values are not

yet sufficiently reliable ; ^{but work on the} improve-

ments is currently in progress. Nevertheless, ^{we can}

say that in the ^case of 1 : 2 and 2 : 2 electrolytes ^we

still find the attractive double-layer interaction which is the most interesting feature of the results

presented below. It should be noted that excepting

very large ^surface separations, ^the one-component electrolyte ^{results are not} substantially changed

when millimolar amounts of salt are added.

In figure 5, ^we begin ^the examination of the interaction by gradually increasing the surface

charge density ^{from a} very low initial value. Com-

pared ^{to the PB} theory prediction ^the repulsion ^is substantially reduced, and the ion radius which

equals ^2.125 A has no significant ^{effect on the}

results. On further increasing the surface charge density (Fig. 6) ^the repulsive ^{force is} overcome, and at moderate surface separations ^{the net result is a} double-layer attraction. This behaviour of the double

layer interaction at increasing coupling strength ^has previously been established by ^MC simulations [1]

and anisotropic ^HNC calculations [4]. ^{In a} separate

publication [18], ^{we have} argued ^{that this attractive} double-layer interaction is responsible for the very limited swelling ^{of calcium} clays ^{in water.} They

Fig. ^5.

Interaction within the anisotropic ^HNC theory (relative ^{to the PB} pressure) ^between plates with various surface charges. ^The one-component electrolyte ^consists

of divalent counterions with radii 2.125 A. The values of the surface charge

(from ^the top ^{to the} bottom)

elementary charge per 100 nm2, ⁴⁰ nM2 , ²⁰ nm2, ¹⁰ nm2,

4 nM2 and 2.5 nM2 respectively.

Fig. ^{6. - Same}

ⁱⁿ figure 5, ^{but for}

elementary charge per 2.5 nm2 (same

^{the last}

^of Fig. 5), 2 nm2, ^{1.35 nm2 and 0.6 nm2. Note the} change ^{in the} separation ^scale.

remain stable at surface-surface separations ^between

the clay platelets ^of approximately ^{0.6 to 1.4 nm.}

Furthermore, the appearance of attractive double

layer ^{forces when} changing from monovalent to divalent counterions in some surfactant-water sys- tems can explain ^the phase behaviour of such systems [19].

But even the reduced double-layer repulsion ^found

at larger ^surface separations and lower surface

charges ^has important practical consequences. In

figure 7, ^{we have} presented the interaction at the

constant surface separation ^{of 25 nm as a} function of

surface charge. If the values of the interaction

1014

Fig. ^7.

^{The HNC} and the PB values for the interaction

(P /RT[mM]) ^{between two} uniformly charged plates ^at

the constant separation ^{of 25}

function of the

per unit surface charge. ^The

component electrolyte

consists of divalent counterions with radii 2.125 A. Note,

for example, that the HNC pressure between ^two surfaces with

surface charge ^{of 12.6} nm2 _{per unit} charge ^{is the}

same as

the PB prediction for surfaces with 25.0 nm2 _per

unit charge.

measured in some experiments ^are lower than those

expected from the PB theory, ^one may be inclined to

incorrectly interpret the deviation as resulting ^{from a}

lower surface charge ^{due to ion} binding ^{to the}

surface (see ^the example given ^{in the} caption ^of Fig. 7). ^{As we have seen,} the decreased repulsion ^is

a consequence ^{of the} ion-ion correlations. Of course, this does ^not rule out ion adsorption ⁱⁿ particular ^{cases, but it means that the amount of}

such adsorption ^{will be} overestimated if determined

by ^a comparison ^{based on} the PB pressure.

6. Concluding ^remarks.

Accurate results for surface-surface interactions in

electrolyte ^{solutions have so far been obtained} using

the Monte Carlo simulation method and the aniso-

tropic ^HNC approximation described in this report.

In practice, ^{these numerical methods are} currently

limited to calculations of the pressure in systems where the surface separation ^{is not too} large. ^If

one adopts ^the Debye-Huckel approximation

Cij(rl’ ^{r2) = -} Uij(rl’ ^{r2)/kB T for the inho-}

mogeneous fluid between the surfaces, the asymp- totic forms of the surface-surface interaction at large separations may be calculated analytically [20].

While this work is still in progress, the results obtained so far are very encouraging ^{for the extreme}

surface separation regime which is inacessible to the numerical methods.

Within a wider point ^of view, the attractive correlation contribution to the surface-surface inter- action discussed in this work is a part ^{of the static}

(zero frequency) ^{term of} the Van der Waals force.

This fact becomes apparent ^{when one discusses} interaction in the presence of electrostatic images,

where consistent results are obtained only ^{when the}

zero frequency Lifshitz interaction between the dielectric media is added to the contribution calcu- lated from the electrolyte ^between the surfaces [20- 23]. The Lifshitz theory only ^considers the Van der Waals interaction due to correlations in polarization

fluctuations in the dielectrics. When ions are present,

one has to consider the additional electrostatic fluctuation forces due to correlations in charge

fluctuations (ion-ion correlations). Furthermore, ^the charge ^{and the} polarization fluctuations are interde-

pendent, ^{as described} by ^the image charge ^interac-

tions. To obtain fully consistent theoretical treat- ment, all of these interaction mechanisms have to be included on an equal footing. ^{In the} anisotropic

HNC theory ^{this can be done without} any other

approximations [5]. In the results [22-23], ^{the static}

contribution to the Van der Waals force between the dielectric media is screened by ^the electrolyte ^{as a}

consequence of the cancellation [20-22] ^{of the static}

Lifshitz term by ^{a term due to the} ion-image charge

interactions.

The need to extend Van der Waals interactions to include the response of the electrolyte ^{was well} appreciated ^some 15 years ago [24]. ^{We have now an}

accurate method to actually carry ^out that _pro- gramme for the primitive ^model electrolytes. ^What perhaps ^{was not obvious before the} present ^calcu- lations is that the effect of ion correlations can be

extremely dramatic, ^and qualitatively change ^the picture ^{based on} the classical mean-field theories.

Note added in proo f : ^The attractive double layer

interactions for divalent ions have recently ^been

confirmed experimentally by ^direct measurements of the force between mica surfaces in 0.15 M CaCl2

solution. Both the strength and the range of the attraction agree with HNC calculations for 1:2 elec-

trolytes. (R. ^M. Pashley, ^R. Kjellander, ^S. Marcelja

and J. P. Quirk, ⁱⁿ preparation).

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