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A Simple Statistical Mechanical Approach to the free Energy of the Electric Double Layer Including the Excluded Volume Effect Veronika Kralj-Iglič, Aleš Iglič To cite this version: Veronika Kralj-Iglič, Aleš Iglič. A Simple Statistical Mechanical Approach to the free Energy of the Electric Double Layer Including the Excluded Volume Effect. Journal de Physique II, EDP Sciences, 1996, 6 (4), pp.477-491. <10.1051/jp2:1996193>. <jpa-00248310> HAL Id: jpa-00248310 https://hal.archives-ouvertes.fr/jpa-00248310 Submitted on 1 Jan 1996 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Phys. J. II France (1996) 6 477-491 Mechanical Approach to the Layer Including the Excluded Statistical A Simple Double the Electric Kralj-Iglit (~) Veronika APRIL and 1996, PAGE 477 Free Energy of Elsect Volume Iglit (~,*) Ale§ (~) Institute of Biophysics, Medical Faculty, Lipiteva 2, 61000 Ljubljana, Slovenia Slovenia Engineering, Trialka 25, 61000 Ljubljana, (~) Faculty of Electrical (Received 29 May 1995, A free the simple analytic energy of a single of solutions of surfaces 1995, accepted and interfaces January1996) 9 theory ensemble Classical Abstract. for November 14 Thermodynamics Thermodynamics PACS.82.60.Lf PACS.82.65.Dp PACS.05.20.Gg sion revised statistical electric approach mechanical double layer. is applied Electrostatic to interactions derive are an expres- considered field while the finite size of particles constituting the elecby including the excluded effect. calculated free volume The univalent electrolyte is compared to the corresponding free energy obtained by the energy of an Poisson-Boltzmann theory. It is shown that the excluded considerably volume effect increases the energy- of the field while the corresponding electrostatic decrease of the entropic contribution contribution pronounced. As the entropic the of the electrostatic is less is larger than energy field and as the effects both partially cancel, the free energy contributions obtained by the on Poisson-Boltzmann approximation regarding to the excluded volume effect. theory is an excellent by of the means trolyte solution electrostatic mean considered is Introduction 1. electrolyte solution is in with the charged plane, accumulated counterions contact are plane and coions are depleted from this region, thereby creating a diffuse electric double layer. The electric double layer is used to describe electrostatics of liquid crystals, the cellular membranes, phospholipid bilayers and metals in contact with the electrolyte solution. The simplest description of the electric double layer is obtained by the Poisson-Boltzmann theory [1,2j and its extensions such as the Stern-Gouy-Chapman model [3j which introduces the of closest approach of the ions to the charged plane. distance these theories Within ions considered be dimensionless surface considered uniformly point charges, charge is be to to are smeared described with the plane, the electrolyte solution uniform is continuum over as a If the the near dielectric originating neglected. the system. [4,5], and quantity © Author phase Les free for (ditions of ions which free The electrostatic (*) charges from A potential of the constant, energy transitions energy of correspondence de Physique mean force is the charged and is also of is used of the the interest to lipid electric describe to surface the scope interaction [6]. Within layer has (e-mail: [email protected]) 1996 be the while and is the molecules double taken been electrostatic mean direct of this ion-ion work is the potential interactions free energy are of layers the Poisson-Boltzmann theory from derived thermodynamic of the two electric double JOURNAL 478 PHYSIQUE DE applying either charging process or the or by summing energy of the expressions being equivalent [5j. grounds, three order In to charged plane the ions, between were approach mech~nical was electric field integration of the energy of the entropic contribution, the and all theoq-, finite size of ions, image effects, direct interactions, and discreteness of charge forming included in the description of the electric double layer. A statistical [7-9j. initiated basing on the expression for a molecular Hamiltonian the surlrount interactions N°4 temperature field electric II Poisson-Boltzmann solvent and structure and of the-b,alk electrolytes were double extended to the development of the modified Poisson-Boltzmann theory [10-12j, cluster expansion theory of the double layer [13j and integral equation theories [14-17j. predictions with computer simulations Comparison of theoretical [18-29] indicates that the Poisson-Boltzmann interface theory satisfactorily describes the ion density and monotonous electrolyte over a wide range of surface charge and electrolyte potential pmfiles of univalent j12;16,18,19,22-24j while oscillatory behavior of the ion profile concentration concentration and charge which inversion in highly charged surfaces, highly charged electrolyte may occur [20-22,24j were explained by applying the theories based and high electrolyte concentrations of theories The nonuniform layer problem resulting on the fluids in the [11,12,16,22,23j. Hamiltonian molecular layers pr~dicted by The of the interaction electric two double theory agrees well with the results of the Monte for dilute electrolyte, large separations and low surface charge [27j, Carlo simulations univalent while oscillatory effects and between like-charged surfaces which may in hi)hly attraction occur surfaces, highly charged electrolyte and high electrolyte [27,29,30] charged concentrations were obtained by integral equation theories [29,32j. based on the molecular Hamiltonian often give» a numerical, However, the theories are little form. transparent simplicity providing theory can and Poisson-Boltzmann the Therefore still transparency remarkably good a describe fit features the the Poisson-Boltzmann widely used [33-37j. Also, of the and interpretation for it was that are experimental that determined layer providing double formalisms related of the to often Poisson-Boltzmann the the due data, fitted parameters are physical parameters [38j. not as If the surfa~~ charge density is high, the the and the accumulate counterions surface near Poisson-Boltzmann theory may there the overestimate counterion concentration to an extent for that it becomes unphysical [37j. In this work the corresponding accounted corrections were field approach. electrostatic A simple by including an excluded volume effect within the mean free statistical mechanical procedure is proposed in which the expression for the electrostatic the and consistently related electostatic potential and solvent distribution ion to to energy, contributing terms is transparent as the analytical functions, is derived. The origin ofindividual form of the free energy is retained along the derivation. The obtained free energy is compared with the free energy of the Poisson-Boltzmann theory and the conditions for the validity of the considered latter 2. studied. are Theory MINIMIZATION 2.I. wITii over real the THE are~L an density a species of extends OF CHARGED A is in ions. in the and THE bearing positive ENERGY A at its system surface OF is THE described uniformly ELECTROLYTE in which distributed a SOLUTION plane charge at x with IN CONTACT 0 extending surface charge = the molecules electrolyte solution consisting of solvent and M bounded by the area A in the x occupies a volume 0 plane and direction. Boundary effects are neglected. It is taken that the charged with contact The FREE PLANE. solution x electrostatic plane and the charges of ions in the solution create a mean occupied by the solution, while on the other side of the charged plane ix = field < 0) in the- the volume electrostatic N°4 ENERGY FREE field is OF ELECTRIC THE LAYER DOUBLE 479 zero. The excluded effect is volume included description by requiring the in that the volume of the system is a sum of the volumes of all the constituting particles. Thereby in calculation of introduced. All sites in this lattice is entropy a lattice with an adjustable lattice constant whole the occupied. deriving the expression for the free energy of the system F subject to the local thermodyequilibrium (Appendix A) the methods of statistical used [39j. Starting namic mechanics are from energies of individual particles and treating the particles as independent and indistinguishable the free energy of the system F is obtained (A.16), are In W~' F where W~~ )eeo = = il E~(x)A + /~ l~j~ kT dx is the ix) nj In A ii) dx, nsq of the energy ~~ ~j~ field electrostatic IA. ii), k is the Boltz- nj is the density of the number of particles of the j-th species (A.13), where j solvent 0 denotes molecules and j ,M ions of the j-th 1,2,. species, ns is the density of the number of lattice sites (A.14), q) is the partition function of the particle of the j-th species subject to no electrostatic variable (A.6-A.7), e is the permitT is the constant, mann temperature, = tivity of strength. solution, Integrations = the permittivity of the free space and E is the electrostatic field performed over the extension of the system in the x direction (d). At the distance d from the charged plane the effects of the charged plane can longer be no perceived by virtue of the screening of its electrostatic field by the counterions. However, the explicit expressions for the functions nj(x), j JI and E(x) are 0,1,2,.. known. To derive explicit expressions for these functions, we use the condition for the free not minimum thermodynamic equilibrium of the whole systein, so that in the energy to be at its the eo is are = , bF taking . into the is conditions the that while constant following the account the total (2) 0, = conditions: number particles of each of the species in the whole system performed, variation is ~ /nj x)A dx = Aj, J = j3j M, o,1, 2,. , . the validity of the Gauss law at any x, ~ ~~° " e0 ~ Vjnj(X), (~) J"1 where . the eo is the elementary charge and vj valence of the ion of the j-th species, condition, that all of the lattice sites are occupied (A.2). Using the density of the of particles and lattice sites (A.13-A.14), it follows that for any x number M ~s ~llj(X). " j=0 (5) JOURNAL 480 problem isoparametric Above reduced is / ~ F + ~j the to functional a ljAj PHYSIQUE DE II ordinary N°4 problem by constructing variational ~ L(n(x), E(x))A dx, = (6) o ~=o ~~~~~ ~~ LjnjX), EjX) lj, j = M, 0,1, 2,. j660E~fi) " of from variables the ~~ l~~ ~~ ~~~~ ~ )~)~ ~~~ nM). Eliminating two Lagrange multipliers and n (no ni, (4) and (5) the variation is performed by solving a system the are = ., of the ~~ + ~~ constraints equations Euler ) j 0, = (8) M, 2, 3,. = llJ , ~ ~/~Ej lx ~~' ~~~ 0~ At the d the distance charge average density irolume well as the as electrostatic mean field are zero Ed ~~ = = dx (10) 0, ~ and the potential 4ld is constant. chose We 4ld Taking into gives after account some the above the calculation no(x) = ill) 0. condition the particle distribution solution of the equations of system ~~ = (8) and (9) functions, (12) ~ , ~(n~d/nod) exp(-v~eo4l(x) /kT) 1 + z=1 i~j ix) ~~(~~~~°~~ = 1 + ~~~~ "~~°~~~~~~~~ j = M, 1, 2,. , (13) , ~j(n~d/nod) exp(-v~eo4l(x) /kT) z=1 and the differential equation for 4l(x) M -eon~ ~2@ ~jvj(njd/nod) exp(-eovj4l(x)/kT) ~~~ ~~~~ $ M eeo 1 + ~j(njd/nod) exp(-eovj4l(x)/kT) J"~ ' FREE N°4 njd is the of the where OF density of number density the ENERGY particles of of molecules solvent far DOUBLE ELECTRIC THE from the j-th species the charged plane LAYER at x 481 According d. = (5) to is M n0d ~njd. ns " (~5) j=1 order In distance x, M and 4l on the the explicit dependencies of nj, j 0,1, 2,. equation (14) subject to the conditions (10) and ill is solved. The charged boundary condition at x by an additional 0, is taken into account obtain to = ., differential the plane at x 0 = = ~~ ~~~~ ~o ~ EQUILIBRIUM 2.2. order In should to be defined. reference there It chosen were seems system are hitherto appropriate electrostatic diminished be in (which field to to which extent an = 0, 1, 2,. free Helmholtz the formalism the energy of each species as the reference electric solution, field in the electrostatic an If the coions. Therefore reservoir. of the of the excess the in reference counterions over the the even system coions) thermal reservoir in system and the are that the temperature of the temperature, so system well. in There is no derived previous in solution over n~. obtain we the The reservoir; nj,r~~ solution in Applying the sections. (1), field electrostatic mean uniformly number density distributed electrolyte of the counterions such M, occupying the sites with the ., by using of negligible. is the described is system particles with a large reservoir contact system to be in dimensions much larger than phase whose are Both, the electric double layer and the reference consequence describing the electric double layer surroundings having a constant and of the reservoir is kept constant as and all reservoir species of particles are j the of the d. In with over a amount bulk a with contact is the same reference distributed. corresponding to double layer electric of the the of particles still be consider to contain LAYER. DOUBLE layer double solution. amount would ELECTRIC THE electric to of excess same there the over an particles considered OF of the energy homogeneously therefore for all species of extension is the contain to layer described particles were ENERGY system is required the if these is The FREE free distributed double the electrostatic homogeneously electric double layer system the ELECTROSTATIC the species each In obtain free energy the contact = reservoir expression of the the njd, for reservoir ~ F~~~ = ~j kT v,e~ where The of the V~~~ is the system and supersystem electrostatic field volume of the the reservoir F~P consists and the distribution electrostatic the functions whole (12) together of the J"0 referred are contribution contribution of the J = energy of the should be taken and the as a supersystem solution under [39j. the Free influence energy of the (18) F + F~~~ free (13) to of the reservoir, system and nsq reservoir. F~P While deriving equilibrium of Ii 7) ~~§ dV, njd In system, into equilibrium requirement of thermodynamic by inserting the equilibrium electrostatic potential (obtained by the account JOURNAL 482 solving the PHYSIQUE DE II N°4 equation (14)) into the expression iii. differential After rearranging some we obtain M W~' F~P / kT + = I + ~j ~~~ M d ~ ° 4l(x) /kT njvj eo nod ~_~ + n~ In ~ ~"~ ~j l + ~~~ A dx exp(-vjeo4l(~) /kT) nod ~_~ M kT~jNjT In + ~~§, where taken it is supersystem (19) ~~QJ j=0 into that account the numbers of each of the species of the particles M the in constant, are NjT /~ nj(x) " / A dx + njddv, j 0,1, 2,. = (20) M. , v~m The free electrostatic between the free with the the electrostatic energy the electric supersystem the free is same defined of the energy species NjT> j that 4l 0, of field is layer F~' double and F~P particles of the negligibly small so number same of energy of the supersystem M, 0,1, 2,. = difference the as reference but which in , = jlel j~~j j/sp j/sp ref' the Since reservoir density of the taken is particles of redistribution the particles of number of the electrostatic field causing very large, the presence solution in the original system only negligibly changes the be to in the in reservoir njd> j ,M, 0,I,2,. = that so follows it (19) from M l~$ ~ kT = NjT nsq~ J= ~ £$~ By inserting of the solution vjnj(x) from the influence under (4) it F~~ where W~~ is the fi4l(x) eeo dx order to calculate 4l ix F~' Electrostatic free potential depend the particles in the solution free energy (23) + ~j entropic contribution ~~~ ~_ ~ kTn~ In dependence of the be electrostatic energy electrostatic (24) A dx. M F~' the should the 1l0d + ~j J=1 charged plane that is and F~~t is the field 1 + In field W~~ + F~~t = electrostatic ~2q~ d = electrostatic ~M of the energy F~~~ (19)-(22) from follows of the (22) ~~§ In known. well as on n~. This as the Note the is ~ ~ exp(-vjeo4l(x) /kT) ~°~ potential by solving the electrostatic obtained particle density of that ~ the distribution the number contribution f/ distance (12) which kTn~ In can from the equation (14). differential functions of sites the on and (13) and be occupied (I + £$1 ~) 0 the by Adx FREE N°4 originates reflects entropic electrostatic 2.3. VERY field, the DILUTE If it is assumed for x It should solution and that choice the of the The thermic WITH very PLANE. CHARGED THE everywhere dilute particles. of motion CONTACT IN is 483 electrostatic energy of the energy. field in the system while of the electrostatic influence of of particles under the distribution the SOLUTION electrolyte the LAYER emphasized be free the effect volume ELECTROLYTE that for accounts excluded DOUBLE ELECTRIC THE supersystem. F~~t contribution the any reference OF important for the expression for the strength and the distribution is state field W~' the the from reference ENERGY the in that I-e- system, M ~jnj(x) < no(x), (25) j=i the distribution ion functions taking denominator and distribution functions nj(x) the account njd = from equation (13) by neglecting the approximation nod Gt n~, to yield the obtained are into exp(-vjeo4l(x)/kT), j 1, 2,. = sum in the Boltzmann (26) M. , Considering above the (25), equation (14) condition equation ~ ~j ~ ~ (/~ " "jnjd exPl transforms -vj eo Poisson-Boltzmann the into 41x)/kT) 127) o The in free electrostatic energy F~' is obtained by considering In(I approximation the + x) Gt x expressions (23, 24), the F~' = W~~ + /~ oco ~ ~ j kT£(nj 4l(x) ix) (28) A dx. njd J" The f/ £$ term calculate F~' the kTnjd ~ A d~ in dependence (28) originates of the from the potential electrostatic reference on the supersystem. In from the distance order to charged plane 4l(x) should be known. This is obtained by solving the Poisson-Boltzmann equation (27). The expression (28) is equivalent to the expressions proposed by Marcus [5]. (23, 24), particle In electrostatic free distribution expressions for the contrast to energy functions (12) and (13) and differential electrostatic equation for the potential (14), the corresponding Poisson-Boltzmann expressions (26-28) do not depend on the density of the number of sites n~. This that in the Poisson-Boltzmann theory the particles in the solution means of particles. considered dimensionless there is no limit of the and concentration are as upper It should be kept in mind that this is strictly true only within the assumption (25) that the of ions is very If the surface charge density small everywhere in the system. concentration high, attracted of is the the charged plane and therefore counterions in vicinity (a( many are (25) condition plane is very CASE 2.4. for UNIVALENT OF illustration. coions. of the solvent Far be there may violated even if the of concentration ions far charged the from low. from number of molecules the are charged plane the solution coions is ELECTROLYTE. there In the are denoted equal and by nod solvent A of system molecules, density of the denoted by are univalent univalent number nd while of electrolyte counterions counterions the density of considered is and and the univalent the density number of JOURNAL 484 DE PHYSIQUE N°4 II 8o 70 < 3 60 no 40 ' 30 20 10 net o 0.0 0.1 X [arm (full lines) and the correFig. I. Density profile of counterions and of solvent molecules net no Poisson-Boltzmann sponding density profile calculated using the theory (broken line). The counterion density of number of available lattice sites n~ (calibrated to the of pure water) is marked concentration (dotted line). The values of model parameters are (a( oA As/m~, T 310 K,e 78.5, MH~O = kg/kmol, 18 pH~o results The compared are trolyte where = = " kg/m~ 1000 # results the to of the theory for univalent elecyields analytical expressions Poisson-Boltzmann simplicity that d cc. ion density profiles [4j as it is taken for the sake of - This free potential and well as for the electrostatic profiles The density of the and of the solvent molecules presented counterions in [6]. are energy calculated by using Figure I (full lines). The corresponding density profile of the counterions theory is also shown (broken line). Close to the charged plane there is a the Poisson-Boltzmann considerable excluded volume effect on the density profile of the and on the solvent counterions The of the comparable to the of molecules. concentration counterions is there concentration for the the electrostatic molecules solvent that so the concentration of the solvent molecules deviates significantly charged plane. By using the Poisson-Boltzmann theory the countewhere the excluded rion concentration is higher than the corresponding volume concentration effect is considered. that in the vicinity of the charged plane the It can be noted counterion calculated by using the Poisson-Boltzmann theory exceeds the of concentration concentration lattice sites thereby proving to be unphysical. The corresponding coion density profile available the values which are due to strong repulsion by the charged plane at least two orders attains of magnitude smaller in the region considered that the of the contribution coions to the so of the ion and solvent density profiles as effect is negligible. The deviation excluded volume field from the theory can therefore be attributed well as of the electrostatic Poisson-Boltzmann mainly to steric effect of counterions and solvent molecules in a small region in the vicinity of the charged plane. from its It can value be far effective the in seen plane strongly screen thickness number of from the Figure its of the counterions I that the electrostatic which counterions field. The screening layer xi/2, electric double net (calculated relative which to its accumulated are can be is the value described distance far from near the charged by introducing the density of the charged plane nd) where the ENERGY FREE N°4 drops to half of its one value at x OF act calculated ~~~ ~~~ while (0) is density according to the ~ of the the nd of number incti°j # counterions potential is ~~~~ ~ at x to 2n~~n~~~~~~l~~o4l/kT) ~~ by solving calculated is (Eq. (30)) is free energy is /kT)) ~ ~~' ~~~~ equation differential the ~~~~)~~~~~~~~~)~~~)~) ~~~~ ' (15) nod Equation (31) electrostatic The 0 = eeonod according 129j Rdj/2> l + ~ where 485 expressions (23, 24), ~ ~~° ~~ ~~~~ electrostatic LAYER 0, = nctiXi/2) where DOUBLE ELECTRIC THE # n~ (32) 2nd. numerically using the Itunge Kutta method, while the integration of of the performed numerically by using the Simpson method. The extension determined by the condition that its twofold increase system in the ~ direction d had been The density of the did not the value of F~~ beyond the prescribed number of increase error. of pure determined by the pwNA /MH~O, where pw is the sites n~ was concentration water n~ molecules and NA is the Avogadro number. density of water, MH~O is molar mass of water Within the Poisson-Boltzmann theory the effective thickness of the electric double layer can analytically obtained be F~~ solved = ~~~~ ~ ~ where I /~z is the (~/(1 + (~/(1 + exp(-eo4l(0)/kT)) /2 exp(-eo4l(0) /kT)) /2 + exp(eo4l(0)/2kT)) 1)(1 1)(1 + exp(eo4l(0) /2kT) ~' Debye length ~~~~ ~ 4l(0) ~ ~~ ~~~ = In I + ((a( /c)~ + (a( /c) (35) , eo c = 8kTeeoNAnd (36) thickness of the electric double layer ~i/2 in dependence on the absolute value of (full line). density of shown The corresponding the plane Figure 0 is in 2 (a( area x Poisson-Boltzmann theory is also shown dependence of ~i/2 on (a(, calculated by using the (broken line). Both higher charge for small the plane bearing decreasing since (a( curves are effective. ofcounterions and the screening is therefore in its vicinity larger number attracts more excluded volume effect is taken the Within the presented theory where the into account, Poisson-Boltzmann Howthickness of the diffuse layer is increased with respect to the case. excluded volume imposes an upper limit on the density of the number of counterions, ever, the especially significant near the charged plane where accumulated which becomes counterions are (Fig. 1). This imposes a limit also on the screening of the electrostatic field. As more couninfluence of the the charged plane with further terions attracted increase of (a(, the are near excluded effect minimal volume with increasing (a(, so that ~i/2> after reaching its increases model, there value, begins to increase with increasing (a( (Fig. 2). In the Poisson-Boltzmann The the effective charge = JOURNAL 486 PHYSIQUE DE 00 0.5 The 2. effective thickness of the electric 10 is = upper no layer x1j2 double charge area density of the x 0 plane (a( for effect (full line) and the corresponding dependence (broken line). The values of other parameters the are the of N~4 lAs/m2j Id Fig. II nd 0.I = calculated same dependence in the on absolute mol/I considering the excluded using the Poisson-Boltzmann Figure in as value volume theory I. density of the number of ions and there is a decrease monotonous of (a( For high values of (a( the use of Poisson-Boltzmann theory profile is not justified. Namely, the condition (25) is not concentration the vicinity of the charged plane. limit of the of xi /2 with the increase for calculating the ion fulfilled in Figure 2 that xi/2 using both models are very small meaning that the charged plane is almost insensitive details the charged plane. to the near (with or without electrostatically condense the the charged plane counterions near way finite constraint) does not affect the effective potential far from the surface. This volume difference in be also seen in Figure I where the density profiles calculated with counterion models rapidly decreases. It can behavior The the can both also be far from in seen the free energy F~' (a) and both contributions to F~' energy W~' F~~~ (b) in dependence on the of the electrostatic field and the entropic contribution absolute value of the charge area density of the x 0 plane (~( (full lines). The corresponding dependencies calculated by using the Poisson-Boltzmann theory are also shown (broken lines). All the electrostatic field in the system is stronger if increasing with increasing (~( as curves are volume (~( is higher. Since the screening of the electric field is less effective when the excluded Figure 3 the shows electrostatic = effect is taken electrostatic theory. Boltzmann When into solution rearrangement the F~~ to (which this is the near where the the effect sum latter on field the of concentration effect F~~~, so prevailing. that of W~~ and F~~~) the effect partly is further much on cancels into free itself is the much volume energy the ions of ions is in of the (Fig. 3a). Poisson- the Solution. further However, in the W~' is electric the Poisson- vicinity the in on the extended than higher effect of the energy solution. pronounced less excluded the and by using rearranging of counterions The solution calculated energy rearrangement extends charged plane the into reflects considered, the is electrostatic ions F~~~ contribution effect the of the charged plane, contrary volume as model, Boltzmann entropic The excluded the the further into account, the field protrudes field W~' is higher than the corresponding of therefore double layer ENERGY FREE N°4 OF THE b a ~ ~ ~ < ~ 0. o-o DOUBLE ELECTRIC , ~ LAYER 487 JOURNAL 488 PHYSIQUE DE II N°4 Acknowledgments authors The indebted are Appendix dr. Vlachy V. fruitful for discussion. A Energy Free prof. to Electrolyte ofthe Solution in Contact with Charged the equilibrium and taking into account expression for the free energy within the thermodynamic Assuming local particles in the solution the Plane energies of the individual field approximation mean derived. is The divided system is into cells of equal volume, V~ IA-I A A~ = direction. that in the ~ It is assumed, of the change macroscopic properties system over In the particular cell chosen, there and Nj ions of No solvent molecules are volume effect is included in the description so that j I, 2,..., M. The excluded of distributed Nj sites equal volume in the cell, all sites being occupied, are over A~ is where comparing the dimension to the of cell the distance A~ is small appreciably. j-th Species, which particles the = M ~j Nj (A.2) N]. = j=o accordance In (A.2) to assumed, is also The solution particle in partition qm,j the volume of the proportional cell is particles described the cell is using the function q of the mj-th particle that = volume the of the ~jexp (-e~,~~ /kT) m~ , methods of the of particles the j N]. It [39j. The is jA.3j M, o,1, 2,. = sites mix. mechanics statistical N~, of the number while j-th species 1, 2,. = the to conserved is , , ~ where k is the energy states Since the to Boltzmann and T is the constant mj-th particle em,j~. individualion is charged, there an of the given energy Em, Km, " jz jz (#(rm,j) + ej to other contributions the and potential assumed, It is to the states relative calculated chosen its through runs possible all ions energy are infinitely is constant. that the a ofits contribution mj " Nj, 1, 2,. j , energy " (A.4) M, 1, 2,. , electrostatic the to the distant We chose electrostatic reference potential apart that so o. 4l~~f field in the the system energy ej4l~~f. In the reference field in this vanishes electrostatic case = system does qm,j q$ = ~ exp influence not Therefore, by inserting (AA) into (A.3) electrostatic potential energy can therefore be Km,j~. of ions, the potential electrostatic j-th species, 4l(rm,j) the potential of the electrostatic potential energy of the ion situated at rm,j, and potential energy The electrostatic energy of a given state. Km,j~ ion is is 4~~ef) ion of the all I state, where ej is the charge of the field, ej (4l(rm,j) 4l~ef) the ofthe Index temperature. of the and written (-ej4l(rm,j)/kT) the summing before contributions over the all energy sum, (A.5) , ENERGY FREE N°4 OF DOUBLE ELECTRIC THE 489 LAYER where qS,j ~jexp (-Km,j~/kT) = m~ , 1, 2,. = j N~, 1, 2,. = jA.6j M. , , ~ function Partition of the qmoo £exp qs~o = molecule solvent is j-K~z~o~/kTj, = mo IA-?) No. 1, 2,. = , ~ The and heat solution in the number constant cell is chosen considered be to with system a particles No> Aii,N2> of all species of the NM> immersed is , bath, that so correlations its temperature described are T is through effect of the effect, the particles in the cell are considered chosen species are also qS,j e constant. Since it is assumed, that field and electrostatic mean explicitly considered to to be equal with respect qj, m~ N~, 1, 2,. = indistinguishable. Assuming that the system taking into account all possible nonequivalent canonical partition function of the cell is ' o,1, 2,. l~fly jnj ' a excluded the volume particles of The a I(m,j~, states jA.8j ii , thermodynamic particles in "local" the is in and p~ = energy , and Qcj~c j all to l~~ in particle-particle the independent. be volume constant which distributions equilibrium cell, the of the the fifC( ~J jNj ~ s gj ' J4 J~~ fl j~~ J"° J. where N = NM). Using (A.5)-(A.8), equation (A.9) (No> Ni>N2> can be written as , M Q~lvc, N, Tj exp = ~c, j-Awe'/kT) fljqjjN> IAioj S' J=° fl , ~ J"° N J. where ~j e14l(ri), AW~' = IA-it) i summing over Knowing solution in the the all ions (of canonical chosen cell all the species) partition in function cell. the Q~> we obtain the Helmholtz AF number of sites of the energy (A.12 kT In Q~. = Inserting (A.10) into (A.12), using IA-I) and (h.2), applying the Stirling introducing the density of the number of particles of the j-th species nj as of the free AF, approximation well as the and density n~, ' nj j = M, 2, 0. 1, = , (A.13) , fifC n~ IA.14) ~ = V~ , JOURNAL 490 PHYSIQUE DE II N°4 obtain we M Awe( ~ ~ ~ ~T ~ ~ In y~~ ~~ ~=o Equation (A.15) gives the expression for the chosen cell, where the densities of the number of .,M. j=0,1,2,. The contributions the over of all the cells of the extension which system d is W~' + kT free Helmholtz ions solution energy of the of position, I.e. nj functions are the constitute performed (h.~~) A A~. ~~~J system to obtain (xl In summed, are the free I.e. the in the energy the in i~j(~), = integration field mean approximation F = /~ ~( o While created in a calculating W~~, by all other ions medium that it is and consider we expression (A.11) ions To in the be can into e. where eo is free energy electrostatic particles in permittivity the of the field the to free all the charges ions at are regarding distributed to volume with the site of assumed the to infinite a given ion is be immersed self energy charge density pe(r) of so into /~ E~(~)A ~eeo 2 (A.17) d~, ~ and space expressed by (A.16) system due of the = potential The problem the transformed the well. as solution (h.16) A d~. nsqj that account avoid W~~ ~~~~ ~=o charged plane the permittivity with point charges, taken nj in the E is the includes and system electric interactions the field of entropy strength. ions of with mixing Thus the the mean of all the system. 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