Elektrotehniški vestnih 77(2-3): 121-130, 2010 Electrotechnical Review, Ljubljana, Slovenija
Metal surface in contact with electrolyte solution -influence of spatial variation of dielectric constant
Klemen Bohinc1,2 *, Ekaterina Gongadze3 *Veronika Kralj-Iglič4, Ursula van Rienen3, Aleš Iglic1
1 Laboratory of Biophysics, Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia 2Faculty of Health Studies, University of Ljubljana, Poljanska 26a, 1000 Ljubljana, Slovenia
3 Institute of General Electrical Engineering, University of Rostock, Justus-von-Liebig Weg 2, 18059 Rostock, Germany
4Laboratory of Clinical Biophysics, Faculty of Medicine, University of Ljubljana, Lipičeva 2, 1000 Ljubljana, Slovenia
* equally share the first authorship ^ E-mail: ekaterina.gongadze@uni-rostock.de
Abstract. The interaction between a charged metal surface and an electrolyte solution causes the formation of an electrical double layer, which has been a subject of an extensive study for more than a century. The present paper provides a statistical mechanical description of orientational ordering of water molecules and excluded volume effect near charged metal surface. The results of statistical mechanical model are then included in generalized phenomenological Stern model by taking into account the spatial variation of the dielectric constant near the charged surface and the finite size of counterions.
Key words: orientational ordering of water, generalized Stern model, excluded volume effect, dielectric constant
Naelektrena kovinska površina v stiku z elektrolitsko raztopino -vpliv krajevne odvisnoti dielektrične konstante
Povzetek. Električna dvojna plast nastane kot posledica interakcije med naelektreno kovinsko površino in elektrolitsko raztopino. Pričujoči članek podaja statistično mehanski opis vpliva orientacije dipolov vodnih molekulin končnih velikosti molekul na električno dvojno plast. Napovedi statistično-mehanskega modela so v nadaljevanju na fenomenološki način vključene v posplošen Sternov model električne dvojne plasti z upoštevanjem prostorske odvisnosti dielektrične konstatne v bližini naelektrene kovinske površine.
Ključne besede: orientacijsko urejanje vodnih dipolov, Sternov model, končni volumni molekul, dielekrična konstanta
1 Introduction
The contact between a negatively charged metal surface and electrolyte solution results into rearrangement of the ion distribution and water reorientation near the metal surface and the formation of the so-called electrical double layer (EDL). Most of the models describing this phe-
nomenon assume that the dielectric constant in the whole system is constant. But actually close to the charged surface the water dipoles cannot move as freely as away from it and show a distinct preferential orientation in direction of the normal to the charged surface [1, 2, 3]. Also, due to accumulation of counterions near the charged metal surface [4] the water molecules are partially depleted from this region. All these result in spatial variation of dielectric constant near the charged surface [1, 33, 2]. Therefore, here we present a simple model of EDL, which takes into account the spatial variation of the dielectric constant. The orientational ordering of water molecules and the excluded volume effect near the charged metal surface are described within a statistical mechanical approach. The results of the statistical mechanical model are then included in a generalized Stern model via space dependency of the dielectric constant near the charged surface and the distance of closest approach for counterions. A possible application of the predicted results is also described.
Received 22 February 2010
Accepted 12 March 2010
2 Theory
2.1 Statistical mechanical description of excluded volume effect and orientation of water molecules near a charged metal surface in contact with an electrolyte solution
We consider a charged metal surface in contact with a solution of ions and the Langevin dipoles of a finite size. The metal surface is charged with surface charge density aef f. The lattice with an adjustable lattice site is introduced in order to describe the system of the Langevin water dipoles and salt ions. All lattice sites are occupied by ions or dipoles. For the sake of simplicity we assume that the volume of each ion is equal to the volume of a water molecule. Free energy of system F, measured in units of thermal energy kT, can be written as [1]
F kT
8TtIb
V®) dV
J[n+( r)lrn+(r) J (n(r,u>)
I
n0 n( r, cj)
+ n_(r) In
n_(r)
n o
In v 7 y ) dV nod
dV (1)
+ +
+ A
where the first term corresponds to the electrostatic field energy. Here
n(r, cj) ) — n+(r) — n_(r) dV,
V(x) = e0</>(x)/kT ,
(2)
is the reduced potential, where eo is the elementary charge and <j){pc) electrostatic potential. The Bjerrum length is equal to Ib = e^/AireokT, where eo the permittivity of the free space. The second line accounts for mixing free energy contribution of the positive and negative salt ions, n+ and n_ are the number densities of positively and negatively charged ions, respectively, while no is the bulk number density of positively and negatively charged ions, where we assume <j)(x oo) = 0. The third line accounts for the orientational and translational entropy contribution of water (Langevin) dipoles to the free energy, where now is a bulk number density of dipoles. The dipole distribution function is given by
n(r,u) = nd(r)P(r,uj)
(3)
where nd(r) is the number density of water dipoles and P(r, cj) is probability that dipoles located at r are oriented for the angle cj with respect to the normal to the charged metal surface. At any position r we require the normalization condition	= 1 to be fulfilled,
where the averaging over all angles cj is defined as:
FM
h J F('•")
d a
(4)
The last line in Eq.(l) is the constraint due to finite size of particles, ns being the number density of lattice sites: ns = l/flg, where as is the width of the single lattice site. Averaging over all angles uj in Eq.(3) gives the number density of water dipoles nw (r)
('n(r,u= (nd(r)P(r,uj)^ =
= nd(r)(p(r,a;)) = nd(r) . (5)
The charges of counterions, coions and water dipoles contribute to the average microscopic volume charge density
q(t) = e0 (n+(r) - n_(r)) - V P ,
(6)
where the polarization is given by P = ^p n(r, uo)j, p is the dipole moment of water molecules.
The free energy F = F(n+, n_, n) fully specifies the system. In thermal equilibrium F adopts minimum with respect to the functions n+(r), n_(r) and n(r,cj). The results of the variational procedure are
n+(r) = n0e	,
n-(r) = no e n(r,cu) = n0we
i
-p-V^/eo+A
(7)
(8) (9)
Inserting Eqs.(7)-(9) into the constraint (see the last line ofEq.(l)):
= n+(r) + n_(r) +	,
we can calculate the parameter A:
u	eo
where we take into account:
,-p-W/eo^ _
~	4:7T	~
1
2tt / (i(coscj)e_polv^l COSUJ/eo -1 _ 47r
- 60 sinh^M,
(10)
(11)
(12)
Po|W|
eo
where po is the size of the water dipole moments. In the above derivation we assume the azimuthal symmetry, where uj is the polar angle between the dipole moment and the axis perpendicular to the charged metal surface. Inserting the Boltzmann distribution functions Eqs.(7)-(9) into Eq.(6), we get the following expression for the volume charge density in the electrolyte solution:
Q = — 2 eo no eA sinh ^ —
(13)
where:
pe
-p-W/e0\

g-p-W/eo^
f pe~P'w/e° dQ
fe-P-vv/eodn
Polfi f du
u e
-p0\V^\u/e0
f due~P°|V*|«/e0
-1
W
■poj^(coth(po|W|/eo)
eo
Po|W|
)
(14)
In the last step of the above derivation we assume the az-imuthal symmetry and defined u = cos uo. Using Eq.(12) it follows from Eq.(14):

The function T(u) is defined as:
x „/ x sinhiz T{u) = C(u)-
(15)
(16)
where C(u) = (coth(u) — l/u) is the Langevin function. Function C(po | V\P|/eo) describes the average magnitude of dipole moments at given r. Inserting the volume charge density (13) into the Poisson equation =
—AttIbq/cq we get [1]:
A^ = 871-/5 n0 n.
sinh^
H '
- AitIb n0w ns — V-eo
W ^(p0|W|/e0)
L|W|
H
where function H, related to the finite particle size, is given by
H = 2n0 cosh*+^sinh^|W|
Po|W|
eo
(18)
The differential equation (17) has two boundary conditions. The first boundary condition is obtained by integrating the differential equation (17) [1]:
W| 5 = -47dB
a W
/1 7
- 47TIb ns now —
eo
(19)
W ^(p0|W|/e0)
|W|
U
where ^^ is the vector normal to the surface S. The condition of electro-neutrality of the whole system was taken into account. The second boundary condition is V\F|oo = 0.
In the case of one large charged metal surface being in contact with the electrolyte solution, the differential equation (Eq.(17)) reduces to [1]:
VP" = AttIb ns (2 no
sinh^ H
— no w ' , eo ax
with boundary conditions
V\x = 0) = 4 J—
eo
po d rT(po\*'\/eo)
ns n0w Po
H
°eff ~
Hvom/eo)
(20)
U
x=0.
and
00) = 0 .
(21)
(22)
The polarization (where x-axis is perpendicular to the charged surface) is proportional to the electric field strength
P = eo(eeff - 1)E ,
(23)
where ee// is the dielectric constant. From Eq. (23) the dielectric constant can be calculated
6eff
1 +
e0E
(24)
On the other hand, the polarization can be defined also via the expression for the volume density of dipole moments P = ^p nin which we insert Eqs. (11) and (15) to get:
(17)
P = -po n0w ns
W
T

(25)
Inserting Eq.(25) into Eq.(24) and taking into account the definition E = —V0, we can calculate the dielectric constant:
jrfMyy)
-1 .	A 1 PO	\ eo J
eeff = l + nowns^lB-- |w| ^
(26)
In the case of charged planar metal surface Eq.(26) reads [1]:
eeff(x) = 1 + now ns 47tIb —
eo
T

(27)
In the approximation of small electrostatic energy and small energy of dipoles in the electric field compared to thermal energy, i.e. small and smallpo\^'\/eo, Eq.(20) can be expanded in the Taylor series up to the third order to get [1]:
vl/7 =
+ + è)^3 +
f/2

(28)
where

___ + ___ (29)
AttIbtiq 3no eo 3ns eo
The corresponding boundary condition Eq.(21) expanded up to the third order is:
Fig. 1 shows the spatial variation of dielectric constant, calculated according to Eq.(32). The dipole moment of a single water molecule was chosen to be 5 Debyes (D) in order to reach the dielectric constant of pure water 78.5 far away from the charged metal surface. The bulk water concentration (tiow/Na) was chosen 55 mol/1, where Na is Avogadro number.
®'(0) =
eo

, (30)
where

(31)
while the dielectric constant can be expressed as [1]:
2 \
teff(x) where
' 47tIb 1 H--now
B1
n5 V 6ns 10 ) \eo
(32)
(33)
Hereafter Eqs.(28)-(32) were used to calculate the spatial profile of the dielectric constant.
Figure 1. Dielectric constant close to the charged metal surface. Dipole moment of water po = 5D, bulk concentration of water tiqw/Na = 55 mol/1, surface charge density cre// = 0.05 As/m2. The width of a single lattice site a s = 0.318 nm. Bulk concentration of salt is tio/Na = 0.1 mol/1.
Slika 1. Krajevna odvisnost dielektrične konstante v bližini naelektrene kovinske plošče. Modelini parametri so: dipolni moment vodnih molekul po = 5D, koncetracija vodnih molekul daleč stran od plošče tiqw/Na = 55 mol/1, površinska gostota naboja na kovinski plošči cre// = 0.05 As/m2, širina mrežnega mesta a = 0.318 nm. Koncetracija soli daleč stran od plošče no/Na = 0.1 mol/1 (polna črta). Širina mrežnega mesta as = 0.318 nm.
Figure 2. Number densities of counterions (n+) and water molecules (rid) as a function of the distance from the charged metal surface. Bulk concentration of salt: full line no /Na = 0.1 mol/1 and dashed line uq/Na = 0.2 mol/1. Model parameters: dipole moment of water po = 5D, bulk concentration of water tiqw/Na = 55 mol/1, metal surface charge density cre// = 0.05 As/m2. The width of a single lattice site as = 0.318 nm.
Slika 2. Številska gostota protionov (n+) in vodnih molekul (rid) kot funkcija razdalje od naelektrene kovinske plošče. Koncetracija soli daleč stran od plošče : no/Na = 0.1 mol/1 (polna črta) in no/Na = 0.2 mol/1 (prekinjena črta). Modelni parametri so: električni dipolni moment vodnih molekul po = 5D, koncentracija vodnih molekul daleč stran od plošče now/Na = 55 mol/1, površinska gostota naboja aeff = 0.4 As/m2. Širina mrežnega mesta as = 0.318 nm.
Fig. 2 shows the number densities of counterions and water molecules as a function of the distance from the charged metal surface. The results are given for three different bulk concentrations of salt. The number density of counterions decreases with increasing distance from the charged metal surface. The number density of water molecules increases with the increasing distance from the charged metal surface and reaches a plateau value far away from the charged surface. Near the charged surface, the number density of coions is negligible compared to the number density of counterions. The thickness of EDL increases with the decreasing bulk concentration of salt.
The average cosine of the angle co between the dipole vector of the Langevin dipoles and the axis perpendicular
to the metal surface is given by equation:
cosüj
cos üü e~Po IIcos ^
e—po\^'\ cosw/eo ^
(34)
where
means the averaging over all angles uo
weighted by the Boltzmann factor. The average cosine
^cos (jO^j , as a function of the distance from the charged
surface for different surface charge densities and bulk counterion number densities, is shown in Fig. 3. Fig. 3 shows that the dipole moment vectors at the charged metal surface are predominantly oriented towards the surface. Far away from the charged metal surface all orientations of dipoles are equally probable, therefore < coscj >b = 0 (see Fig. 3). The absolute value of < cos lj >b increases with increasing aeff corresponding to stronger orientation of water dipoles. Due to the stronger screening, the absolute value of < cos lj > b is decreasing with increasing n0.
Figure 3. Average cosine of angle of the Langevin dipoles (A,B) and polarization (C,D) as s function of the distance from the charged metal surface. Figures A and C: no /Na = 0.1mol/l, surface charge densities from bottom to top are cre// = 0.05 As/m2 and cre// = 0.01 As/m2. Figures B and D: cre// = 0.05 As/m2, bulk salt concentrations from bottom to top follows as no/Na = 0.1mol/l, no/Na = 0.2 mol/1. The width of a single lattice site as = 0.318 nm (from [1]).
Slika 3. Krajevna odvisnost povprečnega cosinusa nagnjenosti dipolov vodnih molekul in polarizacije v bližini naelek-trene kovinske plošče. Sliki A in C: no/Na = 0.1 mol/l, površinska gostota naboja na kovinski plošči od spodaj navzgor: aeff = 0.05 As/m2 in 0.01 As/m2. Sliki B in D: aeff = 0.05 As/m2, Koncetracija soli daleč stran od plošče od spodaj navzgor no/Na = 0.1 mol/l in 0.2 mol/l. Širina mrežnega mesta as = 0.318 nm (iz [1]).
2.2 Generalized Stern model. Influence of spatial variation of dielectric constant
The Stern Model [5] was the first attempt to incorporate steric effects by combining the Helmholtz [8] and Gouy-Chapman [6, 7] model. Helmholtz treated the double layer mathematically as a simple capacitor, based on a physical model in which a layer of ions with a single layer of solvent around each ion is adsorbed at the surface. Gouy [6] and Chapman [7] considered the thermal motion of ions and pictured a diffuse double layer composed of ions of opposite charge (counterions) attracted to the surface and ions of the same charge (coions) repelled by it embedded in a dielectric continuum described by the Poisson-Boltzmann (PB) differential equation [9, 10, 4, 11] . In its simple version the Stern Model [5] consists of the inner Helmholtz plane (IHP), where the coions are bound near the surface due to specific adsorption, and the so-called outer Helmholtz plane (OHP) of hydrated counterions at the distance of the closest approach (b), and a diffuse double layer.
In our generalized Stern model the electrolyte solution consists of water molecules, monovalent cations and anions (Fig.4). As mentioned above that the dielectric constant profile close to the charged surface (Fig.l) is mainly determined by two opposing mechanisms: the depletion of water dipoles at the charged surface due to accumulated counterions (Fig.2) and the decrease in orientational ordering of the water dipoles as a function of the increasing distance from the charged membrane surface (Fig.3).
Also water molecules in the electrolyte solution can better organize their hydrogen bonding network without ions, therefore it is favourable that ions which disrupt the hydrogen bonded water network are moved from the bulk towards the charged membrane surface [2]. In accordance with the predictions given in Fig.2 in our generalized Stern model, the dielectric constant of the solution is approximately described by step function (Fig.5). In this way, the orientational ordering of water molecules near the charged metal surface (Figs.3 and 4 ) is taken into account phenomenologically.
In our generalized Stern model, the hard core interactions between the cations (counterions) and the negatively charged metal surface with effective surface charge density aef f is taken into account by means of the distance of the closest approach b < a (see Fig.4). In Fig.4 a is defined as the region of strong water orientation, where the dielectric constant substantially differs from the bulk value. The parameter b defines the distance of the closest approach for counterions.
Similarly as in the Stern model [5], the charge density in the different layers can be written as (see Eq.(6))
q(x) =	náx)
(35)
water molecules
specifically bound negatively charged ions
O G© ©O©©
©^-^	nncitr
o© °
negatively charged hydrated ions
positively charged hydrated ions
region of preferentially oriented water molecules
The ions are assumed to be distributed according to the Boltzmann distribution:
riiix) — rii{oo) • exp(—vi eo (t>{x) /kT) . (37)
According to the results given in Fig. 2 the dielectric constant of the solution is approximately described by the step function (see Fig.5):
s(x) =
x < a x > a
(38)
0 a
x[m]
Figure 5. Model of the dielectric constant in electrolyte solution with respect to the distance from the charged surface. The value of ei = 78.5 corresponds to the bulk value, while €2, which is in the range of 10 - 60, and a are the model parameters of the generalized Stern model (see Fig.4).
Slika 5. Shematični prikaz krajevne odvisnosti dielektrične konstante elektrolitske raztopine v bližini naelektrene kovinske plošče. Vrednost ei = 78.5 ustreza vrednosti daleč stran od naelektrene kovinske plošče, 62 v območju blizu naelektrene kovinske plošče je parameter posplošenega Sternobega modela (glejte sliko 4).
where for monovalent ions, the valence v i is
v+ = 1 , v — — — 1 .	(36)
By inserting Eqs.(35)-(38) into the Poisson equation, we obtain the Poisson-Botzmann (PB) differential equations (see also [9]) corresponding to the three different regions:
d20 dx2
^ • exp(e0<Kx)/kT) , 0 <x<b
• sinh(eo0(x)/kT) < x < a (39) ^f ' sinh(eo0(x)/^T) , a < x < 00
Figure 4. Schematic figure of the generalized Stern model of electric double layer near the negatively charged metal surface, where b is the distance of the closest approach for the counteri-ons and a is the region of strong water orientation. The effective surface charge density cre// accounts also for specifically bound (adsorbed) negatively charged ions (coions). Slika 4. Shematični prikaz pospošenega Sternovega modela električne dvojne plasti v bližini naelektrene kovinske plošče, kjer je b razdalja priblžanja centrov kationov, a pa označuje področje močne ureditve vodnih dipolov. Efektivna površinska gostota naboja na kovinski plošči cre// zajema tudi prispevek specifično vezanih koionov.
čeffA
The boundary condition at x = 0 is consistent with the condition of electroneutrality of the whole system:
d<j) dx

eff

(40)
The validity of the Gauss's law at x = b and x = a, respectively, is fulfilled by the following equations:
dx
62 dx
b-
dx
b+
61 dx
(41)
(42)
a+
Due to the screening effect of the negatively charged metal surface caused by the accumulated cations, we assume that far away from the charged metal surface the strength of electric field Ex tends to zero:
dx
= 0
(43)
Equations (39) are rewritten in dimensionless form:
K • exp(^) , 0 < £ < £ 2K • sinh(^) , £ < £ < 1 (44) 2L • sinh(^) , 1 < £ < 00
d^2
where the reduced potential = eo^(x)/kT (Eq.(2)) and the reduced length:
a
(45)
and the constants are defined as:
K =
2 2 e^npa
e2 e0kT
L =
2 2 egnpflr
eieo kT
(46)
Respectively, the boundary conditions for the dimension-less case are

ueffae o e2eokT
df
C2

ei
<M dÇ
i+

= 0

= J 4Ksiiih(y)d9
dU
These transformations lead to
where
' +	0 < C < £
y/D + 4K cosh^, I < £ < 1 -V8Lsmh(f) , 1 < Î < oo

D = C-2Kexp ( vi/ ( —
Now we proceed with the solution of Eq. (55) considered separately in each of the three intervals. In the interval 0 < £ < ^ we can get an analytical solution of Eq.(55) by rearranging it as:

dm
(47)
(48)
(49)
(50)
^C + 2K exp (\F) For the sake of simplicity, let u to be equal to
u = + and we obtain
Kexp(^)
du =
\/C + 2K exp
dm
and
u2 - C = 2Kexp (m).
(58)
(59)
(60)
(61)
Integrating Eq. (58) gives:
In addition to Eqs.(47)-(50) we consider also continuity of the electric potential at x=b and x=a. Hereafter, we take a closer look at the derivation of the solutions of Eq.(44). Equation (44) is multiplied at both sides by 2^
/
J u
u2 du
u2 — C
(62)
(51)
by taking into account the continuity of the electric potential at x=b and x=a, and integrated to get respectively: / d (?) = / 0 - ^ < ^ (52)
u(0)
i Vc+u(o) y/c-u r n
VČ VČ-u(o) ' Vč+u ° ^ u
(arctan	arctan ) C < 0
In order to obtain we can transform Eq. (62) as:
=	C>0 (63)
- < £ < 1 (53) a
exp And for C < 0
= (arctan And then u is equal to:
u	u(0) \
—== — arctan —■==
VW\ VW\)
(64)
Jd(jjP) = /4Lsinh($)d$ !<£<oo (54)
J C>0
M={ V^Cftanfal + arctan-jg) C<0
(55)
where
P =
(VČ + m(0)) - (\/Č - u(0)) exp (ygg) (VČ-W(0))exp(VČi)
(56)
(57)
=
Now we refer to Eq. (59) and rewrite Eq. (65) as: VCQ C> 0
(65)
(66)
(67)
V/C + 2Kexp(1') =
VW\ tan(çl + g2) C < 0
where
Q = (v/Č + 7)~(v1q-7)exp(v/Če) ^ (6g)
MČi-7)expMČ[0
7= v/C + 2^exp(i'(0)) ,
q'2 = arctan ■
7
Finally, from Eq. (68) we receive ^ as:
[é(Q2-i)] c> o
^ (tan2 (ql + q2) - 1)
(69)
(70)

C < 0
(71)
^r = yjD + 4/v cosh 4'
(72)
is solved numerically.
In the interval 1 < £ < oo, we rearrange the corresponding equation from Eq.(55) as:
di =
Integrating Eq.(73)

VŠL sinh ^
(73)
f«— J1	V^-Ac
gives the following solution:


^(i) sinh ^
(74)

1
V7^ "" V tanh 4
: 111
4~) ■
(75)
By transforming Eq.(75), we get the final result for in the form:

^ = 4 tanh 1(tanh
V 4
exp
(V^L(l-O)) • (76)
In the interval | < £ < 1, the corresponding equation fromEq. (55):
Figure 6. Electric potential 0 as a function of the distance from the charged surface (x) calculated using the statistical mechanical model (full line) and generalized the Stern model (circles) for (je// = —0.02 As/m2, the bulk salt concentration is no /Na = 0.1mol/l. Model parameters within the statistical mechanical description: width of a single lattice site as=0.318 nm, dipole moment of water p0 = 5D. In the generalized Stern model 62 = 30, ei = 78.5, b = 0.01 nm and o = 0.1 nm. Slika 6. Elektrostatski potencial (f) kot funkcija razdalje od naelektrene kovinske plošče (x) izračunan v okviru statistično mehanskega modela (polna črta) in posplošenega Sternovega modela (krogci) za vrednosti efektivne površinske gostote naboja na kovinski plošči cre// = —0.02 As/m2 (spodnja slika). Koncetracija soli daleč stran od kovinske plošče no/Na = 0.1mol/l. Modelni parametri statistično mehanskega modela: širina mrežnega mesta as= 0.318 nm, dipolni momemnt vodne molekule po = 5D, koncentracija vode daleč stran od kovinske plošče now/NA = 55 mol/l. Modelni parametri posplošenega Sternovega modela e2 = 30, d = 78.5, b = 0.01 nm in a = 0.1 nm.
0
-0.5
^ -1
-1.5
-2o
0.5 x [nm]
The space dependency of the dielectric constant near a charged metal surface is considered in both models, within the statistical mechanical approach and also within the generalized Stern model, where the space dependency of dielectric constant is approximated by a simple step function. The corresponding parameters 62 and a (see Fig.5) in the generalized Stern model are determined by
Figure 7. Electric potential 0 as a function of the distance from the charged surface (x) calculated using the generalized Stern model for cre// = — 0.4 As/m2 and three values of the dielectric constant e2 : 10, 20, 30 (see Fig.4). Model parameters are: bulk salt concentration is no/Na = 0.15mol/l, a = 78.5, b = 0.36 nm and a = 0.72 nm.
Slika 7. Elektrostatski potencial 4> kot funkcija razdalje od naelektrene kovinske plošče (x) izračunan v okviru posplošenega Sternovega modela za efektivno površinsko gostoto naboja na kovinski plošči cre// = —0.4 As/m2 in tri vrednosti dielektrične konstante e2 :10, 20, 30 (see Fig.4). Ostali modelni parametri so: koncetracija soli daleč stran od kovinske plošče no/Na = 0.15mol/l, ei = 78.5, b = 0.36 nm in a = 0.72 nm.
fitting the space dependency of electric potential calcu-
lated by using the statistical mechanical model (Fig.6).
The inference from Fig.6 is that both approaches are in a good agreement, when we choose a small value for the distance of closest approach b. This is justified by the fact that in the statistical mechanical approach the excluded volume effect is taken into account namely by allowing the centers of ions and water molecules to approach x=0 plane. Both models complement each other, since the GS model is not restricted to small values of the surface charge, making it a good supplement to the statistical mechanical approach (see Fig.7).
Fig.7 presents the distribution of the electric potential for values of the dielectric constant of 10, 20 and 30. It becomes clear that the absolute value of the electric potential decreases with the increase in the dielectric constant.
3	Conclusions and Discussion
The presented results might be important for the improvement of the biocompatibility of the implant surfaces. Namely, for the clinical success of an implant, a profound knowledge of the interaction between the biomaterial and the cells is needed [12]. The functional activity of cells in contact with the biomaterial is determined by the material characteristics of the surface as well as the surface topography [13]. As described in this work, the contact between negatively charged metal surface and electrolyte solution results into rearrangement of the ion distribution and water orientational ordering near the metal surface. Thus, the surface electric potential is modified, which may among others assist the protein adhesion and the proliferation of the osteoblasts. Most of the models describing this phenomenon assume that the dielectric constant in the whole system is constant.
Therefore, in this work the orientational ordering of water dipoles and the excluded volume were explicitly taken into account in the statistical mechanical model. It was shown that the dipole moment vectors of water molecules at the charged metal surface are predominantly oriented towards the negatively charged surface while all orientations of water dipoles far away from the charged metal surface are equally probable. Due to accumulation of counterions near the metal surface, we predicted that the dielectric constant is there significantly reduced.
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Klemen Bohinc received his B.Sc. and M.Sc. degrees in Physics from Department of Physics, University of Ljubljana. He received his PhD degree in 2001 from the Faculty of Electrical Engineering. He is an Assistant Professor at the Faculty of Health Sciences, University of Ljubljana. He is a member of the Laboratory of Biophysics, Faculty of Electrical Engineering. His area of interest includes electrostatics, statistical physics and biomechanics.
Ekaterina Gongadze received her B. Sc. degree from the Faculty of Industrial Engineering at the Technical University of Sofia in 2006 and her M. Sc. degree from the Faculty of Computer Science and Electrical Engineering of the University of Rostock in 2008. She is currently working as a Junior Researcher in the Research Training Group Welisa at the Faculty of Computer Science and Electrical Engineering at the University of Rostock. Her research interests are numerical the modelling and simulations of electrical double layer.
Veronika Kralj-Iglic received her Diploma and Ph.D. degrees in physics and M.Sc. degree in biophysics from the Department of Physics, University of Ljubljana. She is an associate professor of biophysics at the University of Ljubljana. Her research interests are in electrostatics and statistical physics of biological membranes, biomechanics of membranes. She is the Head of the Laboratory of Clinical Biophysics at the Faculty of Medicine, University of Ljubljana.
Ursula van Rienen received her Diploma in Mathematics from the Reinische-Friedrich-Wilhelms-University, Bonn, her Ph.D. degree in Mathematics and Scientific Computing from Darmstadt University of Technology (TUD) and her Habilitation in the fields Theoretical Electrical Engineering and Scientific Computing from the Faculty of Electrical Engineering and Information Technology at the TUD. Since 1997 she has been a Professor of Theoretical Electrical Engineering at the University of Rostock and one of the leaders of the Research Training Group Welisa (www.welisa.uni-rostock.de). Presently, she is the Vice-Rector for Research at the University of Rostock. Her research work is focused on computational electromagnetics with various applications ranging from biomedical engineering to accelerator physics.
Aleš Iglic received his Diploma and Ph.D. degrees in physics and M.Sc. degree in biophysics from the Department of Physics, and Ph.D. degree in electrical engineering from the Faculty of Electrical Engineering, all from the University of Ljubljana. Since 2007 he has been holding the position of a full professor at the Faculty of Electrical Engineering. He is also a visiting professor of the Research Training Group Welisa at the University of Rostock and the Head of Laboratory of Biophysics at the Faculty of Electrical Engineering. His research interests are in electrostatics and statistical physics of biological membranes.