КОЛЛОИДНЫМ ЖУРНАЛ, 2007, том 69, № 1, с. 25-32
МАТЕРИАЛЫ XIII МЕЖДУНАРОДНОЙ КОНФЕРЕНЦИИ "ПОВЕРХНОСТНЫЕ СИЛЫ"
УДК 541.18
FUNDAMENTAL ASPECTS OF ELECTROSTATIC INTERACTIONS AND CHARGE RENORMALIZATION IN ELECTROLYTE SYSTEMS
© 2007 r. Roland Kjellander
Dept. of Chemistry, Göteborg University, SE-412 96 Göteborg, Sweden e-mail: rkj@chem.gu.se Поступила в редакцию 15.09.2006 г.
The consistent inclusion of ion-ion correlations and molecular solvent effects in electrolyte theory can be expressed in a physical formalism where the particles acquire a renormalized charge density and where they interact electrostatically via a generalized screened Coulomb potential. The latter usually decays for large distances r like a Yukawa function exp(-Kr)/r, where 1/к is the decay length (normally different from the Debye length), but for smaller r the screened Coulomb potential is a more complicated function. The resulting electrostatic theory, "Yukawa electrostatics", differ in many important aspects from ordinary (unscreened) Coulomb electrostatics. In the present paper we give illustrations and explanations of some important differences between Coulomb and Yukawa electrostatics. The effective "Yukawa charge'" of a particle differs from the ordinary Coulombic charge. Furthermore, contributions from multipoles of all orders contribute in general to the leading asymptotic term in the potential for large r, which decays like exp(-Kr)/r. Thus, the electrostatic potential from, for example, an electroneutral molecule with an internal charge distribution has generally the same range as the potential from an ion. Some implications of these facts are pointed out. The presentation is based on exact statistical mechanical analysis where all particles are treated on the same fundamental level, but the main focus lies on physical consequences and interpretations of the theory.
I. INTRODUCTION
The screening of electrostatic interactions in electrolyte systems has been an important topic ever since the appearance of the Gouy-Chapman [1, 2] theory for electric double layers and the Debye-Huckel theory [3] for electrolyte solutions in the early 20th century. Both are based on the Poisson-Boltzmann (PB) approximation that ever since has had many uses in the treatment of electrolyte systems, in particular in colloid and surface science and related fields. The DLVO theory [4, 5] of surface forces is a prominent example. The PB approximation is a mean field theory that is based on the primitive model of electrolyte solutions where the solvent is treated as a dielectric continuum. This approximation still maintains an important role in applications, despite that more sophisticated approaches of the treatment of electrostatic screening have appeared during its life time. Kirkwood's contributions starting in the 1930-s were among the earliest. The development of integral equation theories starting around 1960 was an important step. Density functional methods should also be mentioned here. Some of these approximate theories can be solved analytically, while some must be solved numerically. Due to the great increase in efficiency of computers, the focus has shifted from more or less analytical approaches to purely numerical ones. In particular, simulation techniques have made a huge impact on our understanding of electrolyte systems in the primitive model and, to a lesser extent, for systems with molecular solvent. It has become increasingly more ap-
parent during the last decades that effects of many-body ionic correlations and discrete solvent must be understood in more detail. Both of these effects are entirely neglected in the PB approximation.
Approximate analytical theories are useful, partly since they provide formulas that can be analysed mathematically and thereby give insight into the behaviour of physical quantities when, for example, the state of the system is changed. The disadvantage is that these formulas are approximate and therefore not reliable in general. Simulations give accurate and reliable results (if correctly performed), but since they only provide numbers one usually has to do a series of simulations to obtain the changes in physical quantities and thereby a qualitative picture of the behaviour of the system. Thus, analytical approximations are physically transparent but unreliable, while simulations are reliable but not so transparent.
There remains a need to understand basic aspects of the statistical mechanics of electrostatic interactions and screening by some method that, in the best of worlds, should be both fully reliable and transparent. It is, however, probably futile to hope for a treatment of the exact statistical mechanics for fluids, that satisfies this. Still one can obtain exact results in certain limits, like low densities, high temperatures or large distances. Such results are very useful since they provide firm handles on which the understanding of many important aspects of the system can be secured.
An exact statistical mechanical approach that goes some way towards the goal of physical transparency is Dressed Ion Theory (DIT) [6, 7] for electrolytes in the primitive model and its generalizations Dressed Molecule Theory (DMT) [8, 9] and Dressed Interaction Site Theory (DIST) [10, 11] for systems with molecular solvent and solutes. In DMT the molecules are rigid but can have any shape, size and internal charge distribution, while in DIST they are modeled as flexible molecules in the Reference Interaction Site Model (RISM). These approaches build on a line of research that was highly developed already in the 1970s and concerned the exact analysis of statistical mechanics of electrolyte solutions and of dielectric screening in polar liquids, see for example the review by Stell et al. [12]. For a review of DIT and its applications for bulk electrolytes and double layer systems, see ref. [13].
In the current paper we will see how DIT and DMT can be approached and thereby formulated in a way that is similar to the PB approximation. We will thereby initially use results from the PB approximation as aids for understanding the corresponding DIT and DMT results. The latter are exact, i.e. no approximations have been made in the statistical mechanical treatment of the system for given interaction potentials between the molecules. A substantial part of the paper is devoted to an exposition and explanation of the results of the DMT analysis of electrolytes. We will see how fairly simple arguments can be used to give a physical understanding of some mathematical results of DMT, in particular multipolar expansions of electrostatic interactions in electrolyte solutions with a molecular polar solvent [14]. Thereby, we will see how and why important differences appear for electrostatics in non-electrolytes on one hand and electrolytes on the other.
II. THE SCREENED COULOMB POTENTIAL
Let us denote the Coulomb potential from a point charge q as V[9](r), where r is the distance from the charge. In vacuum the potential is yM(r) = q/(4ne0r) = = q9coul(r), where £0 is the permittivity of vacuum and 9Coul(r) is the potential from a unit charge. In electrolyte theory it is common to adopt the primitive model where the solvent is modelled as a dielectric continuum. Then, in pure solvent the potential from a point charge q immersed in the solvent is ^(r) = q/(4rt£0£,.r) = = q9Coul(r)/er, where er is the relative permittivity (dielectric constant) of the solvent. For a system with a molecular solvent this is correct for large r only. We then have [15, 16]
V[ q]( r )
4ne0 £r r
q % oui ( r )
£r
(pure solvent), (1)
when r —» ra. This holds irrespectively of the magnitude of q. For small distances the r dependence is more complicated due to the molecular nature of the solvent. Before we treat systems with molecular solvent let us,
however, continue with primitive model electrolytes, where the ions are charged hard spheres and the solvent is a dielectric continuum.
The electrostatic potential yM(r) from a point charge q immersed in the electrolyte originates from the charge q itself and the ion atmosphere that is formed around it by attraction of counterions and repulsion of coions. The charge density of the atmosphere will be
denoted p^r) and constitutes a response of the electrolyte solution due to the field from q. The total charge due to the source charge q at the origin will be denoted
p[ 9]( r) and it equals the sum of p^r) and the point charge q itself. We then have from Coulomb's law
V[g](r12) = I dr3p[g](r13 №coul( 32 ), (2)
where ^Coul(r) = ^CoulW/Er
The ion atmosphere makes the electrostatic potential screened, and in the linearized Poisson-Boltzmann (LPB) approximation the potential is given by yM(r) = = qexp(-KDr)/(4n£0£rr) for all r values, where 1/kd is the Debye length. We can write this as
¥[q]( r ) = q^ Coul ( r )(LPB),
(3)
where ^Coul is the "unit" screened Coulomb potential of the PB theory defined as
^Coul( r ) =
4 rt£0£r r
( PB),
(4)
i.e. a Yukawa function. The Debye parameter kd is defined from kD = «qjV(£0£rkBT), where n¡ is the number density and q¡ is the charge of ionic species i, kB is Boltzmann's constant and T is the absolute temperature.
In the full (non-linear) PB approximation the potential V[9](r) decays like a Yukawa function for large r only, while for small r the distance dependence is quite different. In the tail region (for large distances) we have
¥[q]( r )
■q0^Coui(r) when r
(5)
where in general the coefficient q0 is not equal to q. This coefficient can be interpreted as the effective value of the charge and it depends non-linearly on the magnitude of q. In the limit q —- 0 the potential becomes small everywhere and linearization is adequate (linear response), i.e. the PB result turns into the LPB result (3) for all r. Accordingly, we have q0 —► q when q —► 0 and
lim
q ^ 0
^ = *Coul( r )
(6)
for all r values. Equation (6) can be used as a general definition
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