Elektrotehniški vestnik 84(5): 221-234, 2017 Original Scientific Paper The internal and external dipole moment of a water molecule and orientational ordering of water dipoles in an electric double layer Mitja Drab1, Ekaterina Gongadze2, Luka Mesarec2, Samo Kralj3'4, Veronika Kralj-Iglic1, Aleš Iglic2 1 Laboratory of Clinical Biophysics, Faculty of Health Sciences, University of Ljubljana, Zaloška cesta 9, 1000 Ljubljana, Slovenia. 2 Laboratory of Biophysics, Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia. 3 Condensed Matter Physics Department, Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia. 4 Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, 2000 Maribor, Slovenia. t E-mail: ales.iglic@fe.uni-lj.si Abstract. The modelling of the electric double layer (EDL) in lattice statistics requires an understanding of the orientational ordering of water molecules in an ion solution and their contribution to the total free energy. Since water is a polar molecule with its own dipole moment, its total dipole moment arises from the internal and external contributions, the latter modified by the cavity field of neighbouring molecules. In the present paper, we first model a single water molecule as a rigid sphere with a point-like dipole at its center inserted into a continuum with a different dielectric permittivity. The internal and external dipole moment are derived in spherical coordinates. We also derive the energy of such a sphere in an electric field, bearing in mind the modified Langevin-Poisson-Boltzmann's (MLPB) equation used to predict the space-dependent relative permittivity in EDL for the case of point-like molecules. Keywords: electric double layer, water molecule, dielectric properties, cavity field, orientational ordering of water dipoles Notranji in zunanji električni dipolni moment molekule vode in orientacijsko urejanje vodnih molekul v električni dvojni plasti Proučevanje električnega dvojnega sloja v modelu mrežne statistične mehanike zahteva razumevanje orientacij vodnih molekul v ionski raztopini in njihov energijski prispevek k prosti energiji. Voda sestoji iž polarnih molekul, celotni električni dipolni moment molekule pa je vsota notranjih in zunanjih prispevkov, kjer so zadnji posledica polarizacije okoliških molekul. V pricujocem clanku najprej obravnavamo molekulo vode kot kroglo s tockastim elektricnim dipolom v njenem središcu, vstavljeno v kontinuum z razlicno dielektricno konstanto. Notranji in zunanji elektricni dipolni moment izpeljemo v sfericnih koordinatah. Prav tako izpeljemo energijo vodne molekule v zunanjem elektricnem polju. Na koncu izpeljemo tudi modificirani Langevin-Poisson-Boltzmannov model (MLPB) elektricne dvojne plasti za tockaste molekule. 1 Introduction At the interface between a charged surface and neighbouring electrolyte, the electric double layer (EDL) Received 3 August 2017 Accepted 15 November 2017 plays a crucial role (1-11). The properties of EDL may be strongly dependent on the spatial and orientational ordering of water molecules in this region (12-14). It has been shown that close to the charged surface, orientational ordering and depletion of water molecules may result in a strong decrease in the local permittivity of the electrolyte solution (5; 7; 8; 13; 15; 16). Considering the orientational ordering of water and finite size of molecules, Outhwaite and collaborators developed a modified Poisson-Boltzmann's (PB) theory of EDL composed of a mixture of hard spheres with pointlike dipoles and finite-sized ions (15; 19). Later, Szalai et al. (20) published a mean spherical approximation-based theory (21) that can reproduce simulation results for the electric field dependence of the dielectric permittivity of a dipolar fluid in a saturation regime. In addition, the problem was recently considered also within a lattice statistics model (13; 14). The dipole moment of an isolated water molecule is around 1.85 D (Debye is 3.336 10-30 Cm). In a solution, on the other hand, the dipole moment of a single water molecule differs from that of an isolated one since each water molecule is also polarized by the electric field of the neighbouring water molecules, creating an effective value of the dipole moment around 2.4 D - 2.6 D (23; 24). The effect of a polarising environment can be reproduced in the most simple way by introduction of the cavity field (25-27). By considering the cavity field at the centre of the water molecule (27; 28) and also the electronic polarizabil-ity of a water molecule (16; 27; 28), taken into account by portraying a single water molecule as a dielectric sphere with a point-like rigid dipole at its center, the magnitude of the external water dipole moment around 3 D was used within the mean-field lattice statistics approach (14; 16; 27). Neglecting the cavity field and electronic polarizability contribution within a similar lattice statistics, the theoretical models lead to much higher predicted values of the water external dipole moments (around 5 D) (17; 18). In the present paper we examine the derivation of the effective permittivity of a sphere placed in a homogenous electric field and inspect its spatial dependence in ionic solutions. The latter is achieved by dual means: by examination of a relative permittivity of water dipoles and by variational minimization of the Helmholtz's free energy of a pointlike ionic solution. 2 The energy of an internal POINT-LIKE DIPOLE IN A LOCAL FIELD Using the Laplace's equation, we can find the electric field configuration when a dielectric sphere is placed in a homogenous electric field. Such a uniform field can be attained by a parallel-plate capacitor if the linear dimensions of the plates are large enough with respect to the separation of the plates. Even for large separations between the plates, the field in the inner parts and away from the edges of the plates will be homogenous to a high degree of accuracy. If the size of the plates is increased to infinity and at the same time the distance between them is also increased to infinity for a constant surface charge density on the plates, a uniform electric field is created in the entire space. In this space we place a dielectric sphere. It is clear that as a result of polarization, the field strength near the sphere will change while it will remain unchanged at infinity. Let us determine the electric field strength in the entire space including the region inside the dielectric sphere. We assume that a sphere of radius R consists of dielectric with permittivity ew, while the surrounding space is filled with a dielectric with permittivity er (Fig. 1). In this part, we assume that er is constant through the space, since we are dealing with a single water molecule. Later on, when we will be considering the EDL, we will see that the permittivity has a spatial dependency. We assume that homogenous field E is directed along the x-axis. Figure 1. Model of the water molecule consists of a sphere with permittivity ew = n2 and permanent internal point-like dipole p at the center. For a homogenous dielectric with permittivity er, the Poisson equation states that V2^ = — P £Q£r (1) where ^ is the electric potential and p the charge density. Eq. (1) assumes a constant value of permimttivity er. In spherical coordinates, the Poisson equation can be written as follows: 1 d ,d 1 d --r2—- + r2 dr V dr J r2 sin (8) d8 1 + - sin (8) + (2) 32$ (8) da2 P £Q£r where r is the radial distance, a is the azimuthal angle and 0 is the zenith angle. There are no free charges considered in this chapter (p = 0), and Bfi/Ba = 0 on account of the axial symmetry. Hence the problem is reduced to the solution of the Laplace's equation 1 d ,d 1 d --r2 —- + r2 dr V dr j r2 sin (8) d8 sin (8) |8) =0. (3) The following conditions must be obeyed: 1) potential ^ is continuous and finite everywhere; 2) the normal components of electric displacement field vector D are continuous on the surface of the sphere; 3) the tangential components of vector E = —V^ are continuous on the surface of the sphere. The quantities corresponding to the inner region of the sphere are denoted by subscript 1, while those corresponding to the space outside the sphere are denoted by subscript 2. The general solution of Eq. (3) is given by = Air cos (8) + A2r-2 cos (8), = — Er cos (8) + B2r-2 cos (8), (4) (5) where Ai, A2 and B2 are constants, and E is the absolute value of the strength of the uniform electric field at infinity. Since and satisfy Eq. (3), they represent the correct electric potential if they satisfy all the conditions r r sin of the problem. Potential corresponds to the inner region of the sphere, while f 2 corresponds to the space outside it. It can be seen from Eq. (4) that ^ œ as r ^ 0. To avoid the singularity, we set A2 =0. The continuity equation for f at the boundary has the form AiRcos (0) = -ERcos (0) + B2R-2 cos (0), (6) which simplifies to Ai B2R — E. (7) The tangential component of electric field vector E on the surface of the sphere is Et Ee = - 1 df r~36 (8) r=R Condition E^ = E2e is satisfied if Eq. (6) is valid, i.e. Ai and B2 are connected through the relation in Eq. (7). The normal components of the electric field vector are given by Ein = - ^ |r=R = -Ai COs(0), (9) E2n = - ^ |r= R = E COS (0) + 2B R-3 COS (0). (10) dr Here, the electric field components are equal to Ein = Eir and E2n = E2r. We change the subscript from n to r to denote the usage of spherical coordinates. Furthermore, it follows from condition ewEir = er E2r that Ai = (E + 2B2R-^ . (11) The solutions of the system of Eqs. (7) and (11) are Ai Bo = + 2sy -E, (12) d3 R3 E. £w + 2e The potentials inside and outside the sphere are then (13) fi 3er 1 Er cos(0), (14) Er cos (0). (15) + 2e R3 r3 £w 2er The field inside the sphere is constant and parallel to the x-axis: Ec = - dfi dx 3er d(r cos(0)) ew + 2e, optical frequencies of the external field can therefore be written as: (17) — n . Furthermore, when the medium surrounding the cavity has a dielectric constant much larger than the square of the refractive index, er >> n2, an approximation may be used (16): Ec 3 E 2 Ec 3 E. 2 (18) It has to be noted that so far we have neglected the reaction field, i.e. the field due to the point-like dipole in the center of the cavity. This reaction field is directly proportional to the strength of dipole Ereaci . Similarly, the potential must behave like that of an ideal dipole in the limit r ^ 0. These conditions must be obeyed solving the Laplace equation Eq. (3). The general form of the solution is / n \ fM)=£ Ai r; + -¿J P; (cos(0)). (21) ;=o ^ r ' Here, P; are the Legandre's polynomials of order l. Inside the sphere, we have E. (16) We call field Ec the local cavity field, while the outside field is labeled by E. The local field is the sum of the external field and the field created by the bound charges appearing on the surface of the sphere. If the dielectric constant of sphere ew is in the range of optical frequencies, then it is equal to the square of the refractive index of the cavity (n2 ). The permittivity of the electronic induced moment of the water molecule at ¿i(r, 0) = + E Air1 Pi (cos (0)), (22) 4newr2 z—' w ;=0 and outside, it follows ^M) = E G^l) P'(cos (0)). (23) i=0 ^ ' Applying continuity condition Eq. (19) and equating coefficients of P;, we get p B +AiR = r1 , (1=:) (24) A, R1 B; (l — 1). (25) w w We can therefore express B; in terms of A;: P , A Tj3 Bi B; 4n£w = B2;+1A1 + AiR3, (l =1) (26) (l =1). (27) Using the condition defined by Eq. (20) and equating the coefficients as before, we get P 2nR3 + £w Ai ■;-i 2B -£r 2BT, (l =1) (28) , (l = 1).(29) ew A; 1R2;+1 = —er (l + 1)A;R2;+1. (30) The only way this can be satisfied is if A; =0 for l = 1, so we get A; = B; =0 for l = 1. For the l = 1 case, Eq. (26) and Eq. (28) yield A1 = B1 = P 2nR3e. 1 f — 1 f + 2 3p 4ner \£f +2 (31) (32) Here, e / = e w/er marks the ratio between the permittivities. The potentials are ¿iM) = &M) = pcos (0) 4newr2 cos (0) 1 + 2 ^ f — 1 R3 V ef + 2 4nerr2 \ e f +2 3p ,(33) (34) The denominator of Eq. (34) includes product e ref = e w = n2, so based on Eq. (34), we can define external dipole moment pe in the vector form as (16) Pe 3 n2 +2 P. (35) The energy of an internal point-like dipole in a local field is We = —p • Ec. (36) Substituting from Eq. (18), we can express the dipole energy as (16) We 3 2 + n2 We = YPoE cos(w), poE cos (0), (37) (38) where p0 is the magnitude of vector pe and angle w is supplementary to 0, as shown in Fig. 2. Constant y equals (16) Y = 3 (2+n21 . (39) 2 3 Ec e\m V0C Substituting for B; (Eq. (27)) into the right-hand side of Eq. (29) gives Figure 2. Relation between angles d and u. The water internal dipole moment is marked by p, the local cavity field is given by Ec and points in the opposite direction of the potential gradient by V0c. 3 Space-dependent dielectric PERMITTIVITY IN THE ELECTRIC DOUBLE LAYER In general, the relative permittivity is given as P e r = 1 + e oEc (40) where P is the polarization of water dipoles. The total polarization is given by: P Peleci + |Porient|, (41) where Pe;eci is the electronic polarization and Porjeni is the orientational polarization. Considering equations Eq. (40) and Eq. (41), the permittivity can be written as __Peleci |Porient| er — 1 + + e oEc e oEc (42) Electronic polarization is connected to the refractive index as n2 = 1 + Pe;eci/e0Ec, which yields 2 , |Porient| e r = n2 +-- e oEc (43) In this formulation, the electronic polarization contribution is described by refractive index n. In our case of a negatively charged planar surface (a < 0), the projection of polarization vector Porieni points in the direction opposite to the direction of the x-axis, so the Porieni is considered to have magnitude which points in a negative direction. So far we have assumed that er is a constant quantity, but this is not the case with the electric double layer, where the ordering of water molecules directly affects the spatial dependence of the dielectric permittivity. Orientational polarization Porieni is now (29; 31): /2 + n2N Porient(x) = n^(x)po ( -3- ) (œs(w))w, (44) where in this paper we assume nw (x) = nw is the constant (bulk) number density of water in the solution E 2 3 and po is the magnitude of the vector defined in Eq. (35): Po = |Pe|. (45) Factor (cos is the ensemble average of the orientation of water molecules over the solid angle, defined as (cos(w))w = f cos (w) exp (—^ypqE cos (w)) d^ f exp (—^ypoEcos (w)) d^ (46) where w is the angle between the dipoles and potential gradient, as shown in Figure 2. Boltzmann's factor fi equals fi = 1/kT, where kT is the thermal energy. The element of solid angle d^ is d^ = 2n sin (w)dw. (47) Eq. (47) assumes rotational symmetry with respect to the azimuthal axis of rotation. After integrating, Eq. (46) simplifies to (16): (cos(w))u = -£(fi7p oE), (48) where L is the Langevin's function defined as L(u) = coth(u) - 1. (49) u The orientational polarization thus becomes (16): f 2 + n2 \ Porient(x) = -nwpo ( —3— j L(fiYPoE). (50) Considering Eq. (43), the dielectric permittivity is now in general spatially dependent (16): £r (x) = n + 2 . nwpo f 2 + n2\ £(^7poE(x)) £o 3 Ec(x) (51) In the approximation of a small electrostatic energy and small energy of dipoles compared to the thermal energy, the relative permittivity may be expanded using the Taylor's series. For small x, the Langevin's function is well approximated by 3 u u3 L(u) «---. v 7 3 45 In the limit of vanishing the electric field strength (E ^ 0) and zero potential (^ ^ 0), the above equation gives the Onsager's expression for the bulk permittivity (16; 25; 29; 31): 2 /2 + n2 n2 + 3 nwpM 2eo . (53) Using Eq. (51) for space-dependent relative permittivity er(x), the Modified Langevin-Poisson-Boltzmann's (MLPB) equation (29; 31) used in calculating the ion concentrations in the EDL becomes d dx d4c £0£r (x)—— dx = 2eono sinh (eo^), (54) where eo is the elementary charge. The macroscopic (net) volume charge density of coions and counterions is written in the form: Pfree(x) = eon+(x)-eon_(x) = -2eono sinh (eo^cfi). (55) Here, n+(x) and n_(x) are the number densities of counterions and coions and no is the bulk number density of counterions and coions in the electrolyte solution. The boundary conditions for the MLPB equation are: ——(x = 0) = --— , dx £o£r (x = 0) ' and dx"(x = 0. (56) (57) The spatial dependence of permittivity er (x) is plotted in Fig. 3 for T = 298 K, nw = 55 mol/l and po = 3.1 Debye (29). 4 Minimization of the free energy The same result for the spatial dependence of solution permittivity er as described by Eq. (51) can be obtained also by using the minimization of the Helmholtz's free energy. Our model considers the electrolyte solution as consisting of a mixture of point-like ions in a water solution, where the water molecules are considered as the Langevin's dipoles. It is assumed that the ions and water molecules create an average electric field within electrolyte Ec(x). We can write the Helmholtz's free energy of the system F as n+ (x) F = 2 n2£o no n_(x) E2(x) dV + kT J (n+(x)ln — (n+ — n0)j dV + /^n_(x)ln — (n_ — n0)j dV + J(A+n+ + A_n_) dV+ + y nw (P(x,w)ln P(x,w))u dV+ (52) + J nwn(x) ((P(x,w))u — 1) dV . (58) The thermal energy is given by kT. The first term in the above equation is the energy from mean local electric field Ec. The second and third term account for the mixing entropy free energy contribution of pointlike counterions and coions. The fourth term gives the constraint of a constant number of particles in the system, where A+ and A_ are the global Lagrange's multipliers for counterions and coions. The fifth term is the free energy that corresponds to orientational entropy of the Langevin's dipoles, while the last term gives the local constraint for orientation of dipoles. P(x, w) is the probability that the Langevin's dipole located at x is oriented at angle w with respect to the normal to 2 £ r the charged surface (Fig. 2). The brackets denote the average: 1 1 r (F(x,w)), = yo F(x,w) da (59) Here, w is the angle between internal dipole moment vector p and n^ = V^/|V^| (see Fig. 2), while d^ marks the element of the solid angle as defined in Eq. (47). We perform a variation on Helmholtz free energy F Eq. (58), so that JF = 0 (see Appendix A). If we express the local field as local potential Ec(x) = —^C(x) and perform a variation on the electrostatic term, we obtain J I e°n' I A'2 2 C dV eon2 "2" 2^Ci(&) dV. (60) We can rearrange this term if we consider the rules of differenting a function product = - ^c ^c . The integral in Eq. (60) thus yields (61) J 4>'A4>'c) dV = -J &,<*(&') dV, (62) where the first term on the right-hand side equals 0 at infinity, since we pose that the electric potential there is constant and equal to 0. Taking into account the Poisson's law &'(*) = - P(x) eon2 (63) we can transform the variation by to the variation by average microscopic volume charge density p(x), which is the sum of the contributions of the local net ion charges and the dipole moments, represented by polarization Porjeni (16; 45). p(x) = eo(n+ - n_) - V • Porieni, (64) Performing the variation on this charge distribution gives Jp(x) = eo[Jn+(x) — Jn_(x)] — J (V • Porieni), (65) simplifying the electrostatic energy term to the integral / ^cJpdV = ^c(eo[Jn+—Jn_]—J (V • Porieni)) dV. (66) The variation of the mixing entropy of ions and their constraints of Eq. (58) gives j^x=kw dV inj ln + A+ ) + + v no + ' + in_ ( ln — + A_ - no - (67) For the subscript and superscript nomenclature, see Appendix A. Combining positive ion differentials ¿n+ from the electrostatic terms, mixing terms and costraints, we arrive at the variational integral iF+ . ei,mix,c dVin+(x) kT ( ln + A+ no + + ^ceo 0. (68) Since the integrand has to be zero (see Appendix A), we arrive at the expression for the distribution of positively charged ions n+(x). The derivation for negatively charged ions is practically identical. Both distributions are (69) n+(x) = no exp ^ - ßeo^c - A+j, n_(x) = no exp ^ßeo^c - . (70) Here, ß = 1/kT. The boundary conditions state that ^c(x ^ to) = 0 and n+ (x ^ to) = no, which renders A+ = A_ = 0. We may now turn our attention to the variation of the dipole orientation. Water molecules polarization Porjeni is given by Porieni (x) = n„ (P(x, w)),pn0, (71) where p = |p| is the internal point-like dipole magnitude, nw is the concentration of water molecules in the bulk of the solution and (P(x,w))w is defined by Eq. (59). As mentioned earlier (after Eq. (43)), Porieni points in the direction opposite to the direction of the x-axis for a < 0, hence Porieni(x) is considered negative. We may transform the polarization part of the integral in Eq. (66) so that — i ¿cj( V • Porieni) dV = / (V^c) (JPorieni) dV. (72) Considering Eq. (71), we may write JPorieni = (nBP JP(x, w))w . (73) The variation of the free energy of the Langevin's dipoles in Eq. (58) gives JF P = kTn dV (iP(x, w) lnP(x, w)+iP(x, w)), (74) while the variation of the local constraint for orientation of the Langevin's dipoles yields iFCP = kTnw dV in(x)(P(x, w)), + + n(x)(iP(x, w)), - in(x) (75) Combining Eqs. (72-75) and varying by P(x,w), we arrive at the equation (see Appendix A): jf p J dV [(nwiP(x,w)(V^c • p+ ln P(x, w) + n(x) + 1 v + ß J)o The variation requires the integrand to be zero, which gives us the Lagrange's equation for the dipole orientation. After substituting the gradient of the local potential field with the local electric field (considering |Ec| = |V^c|) and evaluating the dot product V^c • p = Ecp cos (w) (see Fig. 2), the equation reads Ecpcos(w) + fcT^ In P(x,w) + n(x) + l) = 0. (77) Solving for P(x,w) gives P(x,w) = A(x)exp( - ,0£cpcos(w)), (78) where A(x) = exp(-n(x) - 1). Substituting local electric field Ec by electric field E (Eq. (18)) and dipole moment p by pe (Eq. (35)) gives .. . , N ( „3E (2 + n2\ . P(x, w) = A(x) exp - ^—3— Jpo cos (w)J, (79) where p0 is the magnitude of pe. The final result is expressed using constant 7 defined in Eq. (39): P(x, w) = A(x) exp ( - cos (w)) . (80) We can now evaluate the average internal dipole moment by integrating by mean orientations, Porient (x) = nwp(cOS (w))w ' 2 + n2 -nw Po 3 L (£7Epo) (82) into Eq. (64), we get the expression for the average microscopic volume charge density in an electrolyte solution: p(x) = —2e0n0 sinh (,0eo^c(x))+ + (2 + n2 ' d _ )) (83) + nwPo I —3— I dX (L (PYEPo)). Inserting the above expression for average microscopic volume charge density p(x) into the Poisson's equation, A'U \ P(X) n2£o (84) we get ^C(x) = n2eo — nw po 2eono sinh (^eo^c(x)) — 2 + n2 d — dx (L (^YEPo)) (85) where ^C(x) is the second derivative of the electric potential ^c(x) with respect to x. As mentioned, this Langevin's PB differential equation is subject to two boundary conditions. The first boundary condition arises from the electro-neutrality of the system. If we integrate Eq. (85) once over the whole system, we get (see Appendix B), ^C(x = 0) = — 1 n2£o a + nwPo 2 + n2 (86) •£(^7poE|x=o) . Here we also consider the second boundary condition, which states that the electric potential far away from the charged surface is constant ^C(x ^ ro) = 0, rendering = 0. Based on Eq. (82) and Eq. (43), we can express the effective permittivity of the electrolyte solution (er) in contact with the planar charged membrane surface as \Porient \ n2 + p(cos (w))w = po (2+3n ) (cos (w))w /cT (2+3W^ P0 cos (w) exp (-^7^p0 cos (w)) d^ /; exp (-^7^p0 cos (w)) d^ = -P^L (^7Ep0). (81) Here, L represents the Langevin's function. In our derivation we assume an azimuthal symmetry. Inserting the Boltzmann's distribution functions of ions (Eqs. (69), (70)) and the expression for polarization (Eq. (81)), 2, po f 2 + n2^ L(£7Epo) n + nw— ' ' (87) eo V 3 J Ec Eq. (85) can be rewritten in a more general form as: ' 2 + n2' V- eon2V^c (r) + nwpo 3 V • (88) n0 L (^7Epo) where n^ = Eq. (88) that V- = 2eono sinh (^eo^c(r)), V^c/|V^c| = V^c/Ec. It follows from eo n2 + po f 2 + n2\ L C07Epo) \ eo 3 Ec = 2eono sinh (^eo^c(r)). Eq. (89) can finally be rewritten in the form V • [eoer(r)V^c(r)] = — Pfree(r), JV^c(r) (89) (90) where p/ree(r) is the macroscopic (net) volume charge density of coions and counterions, while er (r) is defined by Eq. (87). A more general version of the boudary condition (Eq. (86)) is V^c(r = rSUrf ) =--2— n2eo an^ + n^nwpo 2 + n2 r=surf) . (91) 1 3 r n w Rearranging, it follows that V^c(r = rSurf) 1 + nwpo f 2 + n V^c(r = rsurf) n2eo \ 3 C(hPoE (r)| r=surf) J n0. n2£o (92) Evaluating the second expression on the left-hand side of Eq. (92) gives 10 V^e(r = rsurf ) 1 V^e(r = rsurf) |V^e(r = rsurf )| V^e(r = rsurf ) 1 Ee(r = rsurf ) (93) Combining this simplification with Eq. (82), Eq. (92) becomes (n V^e (r = rsurf )£r (r = rsurf ) = - —. (94) £o Here we also take into account the expression for £r (Eq. (87)). We see that the term inside the square brackets on the left-hand side of Eq. (92) is precisely the definition of the relative permittivity on the surface of charged membrane £r (r = rsurf) (Eq. (87)), yielding the general result V^e(r = rsurf) = - Jn0 £o£r(r = rsurf) (95) 5 Discussion In the past, treatments of cavity and reaction fields and the structural correlations between water dipoles in the Onsager's (25), Kirkwood's (26) and Frohlich's (27) models were limited to cases of relatively small electric field strengths, far away from the saturation limit of polarization and orientational ordering of water molecules. High magnitudes of the electric field strength in a saturation regime were later taken into account in (16; 20; 28; 32) and other similar works. Besides the saturation in polarization/water dipole orientation at high magnitudes of the electric field strength (13; 16; 28; 32), the important thing to consider in the EDL studies is also the saturation in the counterion concentration near the charged surface due to the finite size of ions (steric interactions), first considered in the Wicke-Eigen's model (also called the Bikerman's model) and their modifications (3; 6; 33-38). Generalization of the Kirkwood-Onsager-Frohlich's theory with the intention of taking into account the saturation of polarization at high magnitudes of the electric field strength was first performed by Booth (28). The generalization of the Kirkwood-Onsager-Frohlich's theory in the saturation regime in the EDL theory (16; 29; 32) was recently discussed in (40). -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 1.5 x / nm 1.5 x / nm 2 3 x / nm 0.4 0.6 x / nm Figure 3. Average orientation ((cos ), electric field magnitude (E), potential magnitude (0) and relative permittivity (er) in the electric double layer as a function of the distance from a negatively charged surface x = 0 calculated within the MLPB model. The bulk concentrtion of counterions and coions no = 0.1 mol/l and a = -0.3 As/m2. 0 2 n 0 0 2 3 6 5 2 0 0 2 3 0 0 4 5 80 75 70 65 60 55 0 Recently Lopez-Garcia et al. (40) applied the 19th century Maxwell's mixture formula (39) describing the ions as dielectric spheres within an oversimplified macroscopic thermodynamic phenomenological model with many free model parameters to fit the experimentally estimated dependence of the differential capacitance on the electric potential drop. One among many totally unjustified and oversimplified assumptions in (40) is the assumption of a surface charge density-independent relative permittivity in the inner (Stern) layer which is clearly not the case. Due to orientational ordering of water dipoles, the relative permittivity of the Stern's layer depends on the electric field strength, i.e. on the surface charge density (a) of the electrode (41; 42). Fitting the model curves with many free model parameters to the experimental points (40) cannot prove that the inner (Stern) layer capacitance and permittivity are a-independent. The decrease in the relative permittivity close to the charged surface (electrode) is obviously partially the consequence of orientational ordering of water dipoles close to the saturation regime or in the saturation regime as shown theoretically in (10; 13; 15-17; 1921; 32; 36; 41), a fact totally neglected in (40). Simulations of EDL clearly showed that orientational ordering of water molecules in an electrolyte solution is increased in the direction towards the charged (electrode) surface, including the Stern's and diffuse layers (43), amounting to a decrease in the relative permittivity in this direction, which is in accordance with theoretical predictions (see (29) and references therein). Therefore, the phenomenological approach in (40), with the a-independent permittivity in the inner (Stern) layer and assumed decrease in the electrolyte solution permittivity in a diffuse layer due to the presence of the dielectric spherical ions with the assumed lower relative permittivity than that of the bulk water cannot contribute to a better understanding of the physics of the electric double layer, since the basic physical mechanism of the decrease in the relative permittivity, i.e. the orientational ordering of water molecules in Stern and diffuse layer, is totally neglected. Lopez-Garcia et al. (40) claimed that close to the electrode, almost all water molecules belong to water shells around the ions while the free water molecules are excluded. This is certainly not true, namely the results of simulations clearly predicted the increased water ordering in the direction towards the charged surface (including the region close to the charged surface) (43) even for high salt concentrations, in agreement with our theoretical predictions (compare Fig.6 in (43) and Fig.4 in (29)). For example, for the surface charge density magnitude 0.16 As/m2 there is practically no difference in orientational ordering and space distribution of water dipoles close to the charged surface between water with and without NaCl (of concentration 500 mmol/l) (43). In general, for the magnitudes of the surface charge density up to around 0.3 As/m2, where the mean-field approach can still be justified, there is only a weak quantitative influence of salt on the profile of orientational ordering of water dipoles in Stern and diffuse layers, but no qualitative influence (43). Note that the multi-layering of water predicted in simulations (43) cannot be predicted within our mean-field approach (16; 32) as well also not in (40). In (40) it is also claimed that in model (32), the space occupied by the ions behaves just as a vacuum which is not true since the model assumes that the permittivity of hydrated ions is equal to n2 (considered also in (16; 32) and other works), where n is the refractive index of water (see also (27; 28)). This means that it assumes that the electronic polarizability of hydrated ions is equal to electronic polarizability of water (16; 32). This is obviously a very rough approximation, but certainly in (32), the space occupied by ions is not considered a vacuum. In (40), it is further argued that in (32) the magnitude of the external dipole moment is 3.1 D rather than 'generally accepted' 1.85 D (similarly as in Fig. 3) and that this is done in order to recover the water permittivity when the ionic concentrations vanish. Value 3.1 D is certainly not selected in order to recover the water permittivity when the ionic concentrations vanish, but rather in order to get the bulk permittivity around 78.5 for the vanishing electric field in the bulk solution which can be clearly understood from the expansion of the permittivity as a function of the magnitude of the electric field strength and electric potential (see for example the similar expansion of the permittivity given in (16)). The authors of (40) also overlook that in (32) and other similar works there is no referencing to the Onsager's model, using instead the generalized Kirkwood-Onsager-Frohlich's theory which can also be used in the saturation regime of the water dipole orientational ordering and polarization. The dipole moment of a water molecule in liquid water differs from that of an isolated water molecule because each molecule is further polarized by the electric field of its neighbours which can be taken into account by means of the cavity and reaction fields (13; 16; 27; 28) and correlations between water dipoles (27; 28; 44). The magnitude of the external dipole moment of an isolated water molecule is indeed around 1.85 D which is a smaller value than the dipole moment of a water molecule in clusters (p0 = 2.7 D), and the average dipole moment of a water molecule in the bulk solution (po = 2.4-2.6 D) (24). The value 3.1 D in (32) is considerably smaller than the corresponding value 4.79 D used in models (17) and (18), which both did not take into account the cavity field and the electronic polarizability of water molecules (27) as it was later Figure 4. Schematic figure of the electric double layer showing cations, anions and water molecules in contact with a negatively charged planar surface. Distance b denotes the distance of the closest approach (also called the outer Helmholtz's plane), approximately equal to the hydration radius of the counterions (cations in our case) (41). considered in (16; 29; 32). Including the structural correlations between water dipoles (direct interactions term) (28) would further decrease the magnitude of the external dipole moment to the value around 2.0 D as shown in Fig.8 in (14). However, the Booth's model does not consider the excluded volume effect and non-homogeneous distribution of water molecules as it is the case in the electric double layer close to the charged surface (16; 32). Therefore Booth's model, which considers the direct interactions and structural correlations between water molecules (28), can be applied in a simple way only in the Langevin Poisson-Boltzmann's model for the point-like molecules (13; 14; 29), but not also in (16; 32), where the finite size of molecules in an electrolyte solution is taken into account. In (40), the authors also argued that in (32) the same value of the magnitude of the external dipole moment of water molecules (i.e. 3.1 D) was used for all values of the electric field, including the strong electric field in the saturation regime. As it is explained above, a somewhat larger value of pa determined experimentally in water clusters is mainly the consequence of neglection of the direct interactions between water dipoles (28) which are however not crucially dependent on the electric field. The polarization in (32) depends on the electric field strength due to the electronic polarizability term (see the first term in the expression for the relative permittivity (Eq. 41) and due to the orientational ordering term (see the second term in Eq. 41). Note that in the case of considering the space-dependent number density of water molecules nw (x) in Eq. (44), the Eq. (53) for bulk permitivitty would tranform into (32) £r = n2 + 2 [2 + n2\ [ns — (a+ + a_)no]p0^ , (96) 3 ) 2ea where we took into account that in the bulk of the solution (32): ,(x ^ to) = ns — a+no — a_no. (97) Here, ns is the number density of lattice sites in the EDL model which takes into account the finite size of molecules and a+ and a_ are the numbers of lattice sites occupied by a single positive and a single negative hydrated ion, respectively (32). We assume that a single water molecule occupies just one lattice site, therefore ns/NA = 55 mol/l. Eq. (96) predicts the linear decrement of the relative permitivitty in bulk solution with salt concentration. Experiments show linear decrement from er « 78.5 at zero NaCl concentration to er « 58 at 2 mol/l NaCl concentration (46), corresponding to (a+ + a_) « 7.5 in Eq. (96). 6 Conclusion In the present paper we present a thorough derivation of the relation between the external and internal dipole moment of a water molecule. In the model, the water molecule is considered as a rigid sphere with internal point-like dipole p inserted into a homogenous electric field. We see that the external dipole moment depends on the local field arising from the contribution of the point-like dipole at the center of the sphere, since the net charge is accumulated at the interface of two dielectrics with differing permittivities, and the outside cavity field arising from the contributions of other water molecules surrounding the sphere. We calculate the energy of such a dielectric sphere (with a point-like dipole in the centre) in an electric field. Furthermore, considering orientational ordering of water molecules in the electric double layer, we derive the spatial dependency of the relative permittivity within the MLPB model. At small values of the electrostatic field, we see that such an approximation limits to the Onsager's expression for the bulk permittivity. As shown in (43), an addition of NaCl even at high concentrations (0.5 M) affects the spatial and orientational arrangement adopted by the water molecules only for very high magnitudes of surface charge density a. Lower magnitudes of the surface charge densities, which still allow for the mean-field approach, barely affect the ordering of the water layers, as already mentioned above. A stronger salt effect on the orientation angle of the water molecules is predicted for higher magnitudes of the surface charge density (43), however, the average orientation angle of n the water dipoles profile is not qualitatively changed after the addition of NaCl even at a concentration of 0.5 M for low enough surface charge densities (43). For example, for the surface charge density 0.16 As/m2, there is only a very small quantitative difference in the water average orientation profile between the two cases with and without NaCl (43). Even at the magnitude of the surface charge density 0.48 As/m2, the shape of the average orientation angle profile is not changed substantially after the addition of NaCl, but only decreased in values (43). Based on these results, we conclude that considering the orientational ordering of water dipoles in Stern and diffuse layers (10; 13; 1517; 19-21; 29; 32; 36; 41) is essential to realistically capture the basic physical properties of EDL and cannot be neglected as it is done in some oversimplified 19th century based macroscopic phenomenological models of EDL (40), where the statistical physics orientational ordering of water dipoles in EDL is not taken into account. Appendix Appendix A Helmholtz's free energy F, defined in Eq. (58), is defined as a sum of independent contributions, labelled as F = F+' ' + Fmix + Fmix + Fc' + FCP. (98) First term F, + -P accounts for the electrostatic contri- F ±,-,P Fel + F- + Fp . (99) Likewise holds for multiple subscripts. A compact expression including all three integrals for counterions would read F ± = F ± + F± el,mix,c el 1 mi + F+. (100) Eq. (98) can be rewritten SF = SFXmix,c + SFXl,mix,c + 5FPmix,c + SF? = (101) We see that the total variation becomes the sum of variations of four independent functions: the number density of counterions (Sn+(x), corresponding to SF+lmixc), the number density of coions (Snx(x), corresponding to SF- mix c), the probability of the Langevin's dipoles orientation (SP(x, w), corresponding to SFplmix c) and the local constraint of the Langevin's dipoles orientation function (Sn(x), corresponding to SFn). Bearing in mind the derivations from the body of the text (Eqs. (60-75)), Eq. (101) expands into S F = dVSn±(x) kT[ ln n±M+ X+) +