A. Three-layer model of an ion in electrolyte
solution*
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In Sec. II.B, we used a model of the ion which is not fully selfconsistent: on the one hand, the ion is considered a point charge
with respect to the ion-solvent interactions (i.e., the cavity radius
Rcav is set to zero), but on the other hand, the ion-ion interactions
are modeled with a hard-sphere potential. For this reason, if one
sets LD = ∞ in Eq. (39), instead of obtaining the Born energy of a
charge of finite size, Eq. (11), one obtains the energy of a point
charge, Eq. (10).
The self-consistent model will assume that the ion is placed
into a cavity (r < Rcav, subscript “cav”), inside of which  = 0 and
Q = 0; the respective Coulomb-Ampere law reads:
 02cav  ei δ(r ) , r < Rcav.
(S1)
Outside the cavity (r > Rcav), there is a medium of dielectric
permittivity and quadrupolarizability Q, with free charges of
density  = -/LD2. However, these free charges cannot approach
the ion to a distance smaller than R = R+ + R-, so that near the ion
there is an ion-free, “insulating” layer (subscript “ins”)
(S2)
 2ins  L2Q 4ins  0 , R > r > Rcav;
The diffuse electric layer is situated at r > R (subscript “dif”):
(S3)
 2dif  L2Q 4dif  dif / L2D , r > R.
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The general solution of Eqs. (S1)-(S3) (which fulfils the
condition dif → 0 at r → ∞) is:
cav  0  X / r ;
exp  r / LQ 
exp  r / LQ 
A1
 A
 A
;
r
r
r
exp(r / lQ )
exp(r / lD )
dif  BD
 BQ
.
(S4)
r
r
These equations generalize Eq. (37). There are 8 integration
constant to determine: 0, X, A0, A1, A+, A-, BD and BQ. The three
conditions at r = Rcav (the boundary cav|ins) are derived in Ref.
[48]. The first one is the condition for continuity of the potential:
(S5)
cav (Rcav )  ins (Rcav ) .
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ins  A0 
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The second one is the Gauss law Dins, r ( Rcav )  Dcav, r ( Rcav )   S ,
35
which involves the quadrupolar moment density and E through
Eq. (2). In explicit form and after some simplifications, it reads

 Q d 2 Eins, r 
(S6)
  0 Ecav, r r  R  0 ,
  Eins, r 

cav
3 dr 2 r  R

cav
where Er = -d/dr. The last boundary condition reads48:
Qins, rr ( Rcav )  Qcav, rr ( Rcav )  0 ,
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(S7)
or in explicit form,
E 
 dE
(S8)
 Q  ins, r  ins, r 
0.
d
r
r r  Rcav

The conditions at r = R (the boundary ins|dif) are the same as
those used in Sec. III.A:
ins (R)  dif ( R) ;
dins / dr |r  R  ddif / dr |r  R ;
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d 2ins / dr 2 |r  R  d 2dif / dr 2 |r  R ;
d3ins / dr 3 |r  R  d3dif / dr 3 |r  R .
(S9)
The final 8th equation is the electroneutrality condition (38).
The solution of the linear system of equations (S5),(S6),(S8),
(S9) and (38) for the integration constants in Eq. (S4) is trivial,
but the result for 0 and the energy uel = ei0/2 is too complicated.
Therefore, we will only discuss several important features of the
solution (S4). The first one is that in the limit of Rcav → 0, it
simplifies to Eq. (37), which justifies the non-classical boundary
condition for finite potential, used in Sec. II (see also the
discussion in Ref. [48]). In the limit of infinitely dilute solution
(LD → 0), the result for uel following from Eq. (S4) simplifies to
the Born energy (11) (while the simplified Eq. (37) yields (10)).
At least for small ions, the final result stemming from Eq. (S4)
differs insignificantly from its simplified variant (39) obtained
under the assumption that Rcav = 0. With larger ions such as SO42, the deviations are more significant. This is demonstrated in Fig.
5, where the three layer model is compared to the simplified
equation (41). The simplified model (the sum of Eq. (41) and
(46)) always predicts slightly lower energy. In view of the
complicated form of the exact expression 0 and the fact that we
neglect various other effects which yield corrections of similar
order of magnitude as the difference between the three-layer
model and the simplified model in Sec. II.B, we believe that this
complicated correction is meaningless when theory is compared
to experimental data.
Let us finally comment on the interesting fact that the same
classical expression (36) will be obtained for the Debye-Hückel
energy from the three-layer model (Eqs. (S1)-(S3) with LQ = 0) or
the simplified model defined with Eqs. (34) and (5) with LQ = 0
(although the Born energies corresponding to the two models are
different). This means that Born energy (i.e., the ion-solvent
interaction energy) and Debye-Hückel (energy of the interaction
ion-ionic atmosphere) are not coupled in non-quadrupolarizable
solvents and there one can analyze both interaction energies
separately. In the case of non-zero quadrupolarizability of the
medium, this is not true anymore. Therefore, the diffuse layer and
the Born potential distribution must be rigorously considered
together. This coupling is rather weak, which allows us to use the
approximate equation (48) where the electrostatic contribution u0,i
+ uel,i of the two interaction energies is assumed additive.
Nevertheless, the fact that ion-solvent and ion-ion interactions are
coupled (or in other words, the structure of the ionic atmosphere
is interrelated with the structure of the dipole cloud around the
ion) is yet another fundamental property of our general
quadrupolar Coulomb-Ampere law (4), which has no analogue
within the classical Poisson-Boltzmann equation.
*A Referee of Ref. [48] contributed to a certain extent to the
considered model.
95
B. Maple 17 code for the calculation of the
integration constants in Eq. (37)
65
(paste the following code in the Maple worksheet and execute it
with the “!!!” icon)
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> restart;
> #solutions for the potentials at r<R
(phi1) and at r>R (phi2) - Eq (36):
phi1:= A0+Ap*(exp(+r*sqrt(1/lQ^2+1/lD^2))1)/r-Am*(exp(-r*sqrt(1/lQ^2+1/lD^2))-1)/r;
phi2:= BD*exp(-r/lD)/r+BQ*exp(-r/lQ)/r;
#r-components of the field E:
Er1:= -diff(phi1,r): Er2:= -diff(phi2,r):
#charge density
rho:=-epsilon*phi2/LD^2;
#self potential
phi0:= limit(phi1,r=0);
> #relation between lD, lQ, LD and LQ - Eq
(19):
LD:= sqrt(lD^2+lQ^2);
LQ:= sqrt(1/(lD^2+lQ^2))*lQ*lD;
> #verification that phi1 and phi2 are
solutions of the generalized Coulomb-Ampere
law - Eq (5):
de1:= 1/r^2*diff(r^2*Er1,r) - LQ^2/r^2*diff(
r^2*diff(1/r^2*diff(r^2*Er1,r),r) ,r):
simplify(%);
de2:= 1/r^2*diff(r^2*Er2,r) - LQ^2/r^2*diff(
r^2*diff(1/r^2*diff(r^2*Er2,r),r) ,r)1/epsilon*rho:
simplify(%);
> #boundary conditions - 4 equations for
continuity and the condition for
electroneutrality:
bc1:= subs(r=R, phi1-phi2=0):
bc2:= subs(r=R, diff(phi1-phi2,r)=0):
bc3:= subs(r=R, diff(phi1-phi2,r,r)=0):
bc4:= subs(r=R, diff(phi1-phi2,r,r,r)=0):
eln:= ei+int(rho*4*Pi*r^2,r=R..infinity)=0
assuming R>0,lD>0,lQ>0:
> #solution for A0, Ap, Am, BQ, BD:
sol:=
simplify(solve({bc1,bc2,bc3,bc4,eln},{A0,
Ap, Am, BQ, BD})) assuming
lD>0,lQ>0,lD>lQ,R>0:
assign(sol);
> #activity coefficient
lngamma:= ei/2/T*simplify(phi0ei/4/Pi/epsilon/LQ) assuming
lD>0,lQ>0,lD>lQ,R>0:
#the result simplifies to Eq (38)
> #limit for diluted solutions
LQ:='LQ': LD:='LD':
lD:= LQ/sqrt(1/2-1/2*sqrt(1-4*LQ^2/LD^2));
lQ:= LQ/sqrt(1/2+1/2*sqrt(1-4*LQ^2/LD^2));
#in first approximation, it yields the
limiting Debye-Hueckel:
convert(series(lngamma,LD=infinity,3),polyno
m):
lngammaDH:= simplify(%) assuming LQ>0;
> #extended Debte-Huckel model as the limit
at LQ = 0
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lngammaDHe:=
simplify(limit(lngamma,LQ=0,right)) assuming
LD>0,R>0;
> #parameter values
with(ScientificConstants):
T:= GetValue(Constant('k'))*(273.15+25):
e:= GetValue(Constant('e')); ei:=e;
epsilon:=
78.303*GetValue(Constant('epsilon[0]'));
LD:= sqrt(epsilon*T/2/e^2/c);
c:= GetValue(Constant('N[A]'))*1e3*cM;
LQ:=2.1e-10;
R:=2.35e-10;
> #plot of the potential - Fig. 3
plot([
[1e10*r,1e3*evalf(subs(cM=0.01,phi2)),r=2.35
e-10..2e9],[1e10*r,1e3*evalf(subs(cM=0.01,phi1)),r=0
..2.35e-10],
[1e10*r,1e3*evalf(subs(cM=1,phi2)),r=2.35e10..2e9],[1e10*r,1e3*evalf(subs(cM=1,phi1)),r=0..2
.35e-10],
[2.35,y,y=0..48]
],r=0..20,phi,color=[
ColorTools:-Color([0., 0.65,
0.85]),ColorTools:-Color([0.9, 0.1, 0.1]),
ColorTools:-Color([0., 0.65,
0.85]),ColorTools:-Color([0.9, 0.1, 0.1]),
black],thickness=[2,1,2,1,1],linestyle=[soli
d$4,dot],axesfont=[Times,Roman],tickmarks=[4
,5]);
C. Summary of the initial parameters of the
electrolytes used for the calculations
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For the convenience of the reader, a summary of the
parameters involved in the calculations in Sec. III.B is provided
here.
We used two different regression formulas for the
experimental activity coefficients presented in Fig. 4-Fig. 6. For
all aqueous electrolytes, the following equation is used15,24:
3
Z Z A k C
n
(S10)
lg       D is m    n  k isCm  ;
1  B k isCm
n 1
 osm  1 
A Cm
1  B Cm
  0Cm  1Cm exp(1Cm )  kCm2 ; (S11)
here A = 1.294 (kg/mol)1/2; B = 3.2 (kg/mol)1/2; 0 = 0.1058
kg/mol; 1 = 0.5578 kg/mol; 1 = 2 kg/mol; k = 0.083476
(kg/mol)2. The activity coefficient was calculated accordingly
through Gibbs-Duhem relation.
Table S I. Values of the parameters of the regression formula (S10) for
the activity coefficient of the aqueous electrolytes, used in Fig. 4 and Fig.
5.
salt
1
B
2
3
i
Cmax
Ref.
(kg/mol)1/2 kg/mol (kg/mol)2 (kg/mol)3 mol/kg
ii
NaF
1.280
-0.0180
1
[24]
KCl
1.295
7.0×10-5 3.60×10-3 -1.95×10-4
0
0
5
[24]
Na2SO4
1.280
-0.06081 5.17×10-3
-1.3×10-4
4
[15]
MgSO4
1.022
-2.78×10-4 1.74×10-3
0
3
[15]
(i) Eq. (S10) is valid up to Cm = Cmax. (ii) All solutions are aqueous.
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The parameters of the empirical (Cel) dependence (43) used
for the curve-fitting of the experimental data for the static
dielectric permittivity of salt solutions are summarized in Table S
II.
Table S II. Parameters of Eq. (43) for the dependence of the dielectric
permittivity of electrolyte solutions on the molarity of the solution.
salt
1
2
3
4
6
(l/mol)1/2 l/mol (l/mol)3/2 (l/mol)2 (l/mol)3
ii
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i
Cmax
Ref.
mol/l
NaF
0
-5.84
0
-4.3
0
0.9
[20]
KCl
0
-12.4
0
0.95
0
4
[20]
Na2SO4
4.4
0
-28.1
12.5
0
1.6
[58]
MgSO4
14.7
0
0
-23.4
7.6
1.1
[20]
NaBr in
CH3OH
0
-47
0
119
-144
0.5
[20]
(i) Eq. (42) is valid up to Cel = Cmax. (ii) All solutions with the exception
of NaBr in CH3OH are aqueous.
The radii Rcav and R, as well as the partial molar volumes of
the electrolytes used for the calculations in Sec. III.B are given in
Table S III. Note that the dependence of the mass density of the
Table S III. Cavity radii Rcav+ and Rcav- of the cation and the anion,
minimal distances R of approach between two counterions and partial
molar volumes Vel of the electrolytes at infinite dilution. All data are
calculated by using the crystallographic radii and the single ion partial
molar volumes selected by Marcus62.
salt
where AD is the Debye-Hückel coefficient in water (-0.5108
kg1/2/mol1/2 at 25˚), and the empirical parameters B and n are
summarized in Table S I. For NaBr in methanol, we used Pitzer’s
regression formula for the osmotic coefficient data from Ref.
[20]:
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methanol solution on Cel was neglected, due to the absence of
data. For the relation between molality and molar concentration
in methanol, we used Cm = Celm, where m was assumed equal to
the mass density of pure methanol, 791.8 kg/m3.
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i
i
Rcav+ =
Rcav- =
gV1/3(R+ + Lcav+) gV1/3(R- + Lcav-)
R = R + + R-
Vel
Å
Å
Å
l/mol
NaF
1.90
1.48
2.35
2.4
KCl
2.30
2.02
3.19
26.8
Na2SO4
1.90
2.57
3.32
11.6
MgSO4
1.56
2.57
3.02
-7.2
NaBr in
CH3OH
ii
ii
2.98
-
1.90
2.19
(i) Lcav+ = 0.69Å, Lcav- = 0 Å, gV = 1.39, cf. Ref. [48]. (ii) Cf. also
supplementary material D.
D. Size parameters of the ion cavity in methanol
solutions – comparison with experimental
solvation energies
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In this section, we will use the data collected by Kelly et al. 63 to
test our formula for the cavity radii, Eq. (44), and to defend our
choice of the values of the parameters Lcav+, Lcav- and gV. The data
of Kelly et al. 63 is insufficient to repeat the detailed optimization
procedures used in Ref. [48] (data for only six simple ions in
methanol is available). It nevertheless is enough to test our
assumptions.
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FIG. S 1. Comparison between the experimental solvation energies of
various ions in methanol assembled by Kelly et al.63 and the theoretical
equation (S12). We used the comparison to determine the solvation
energy of H+ in methanol as the single fitting parameter.
We use the data63 for the conventional solvation energies 0+
of the anions and 0+ of the cations. Our model (11) of the Born
energy predicts for their dependence on the crystallographic radii
Ri the formulae:

1  3LQ / Rcav 
ei2  1 1
Δ0+  Δ0,H+ 
 
;
2
2
8πRcav    0  1  3LQ / Rcav   3LQ / Rcav  
Δ0   Δ0,H 
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(S12)
where 0,H+ is the solvation energy of the conventional ion, H+.
We will determine the value of 0,H+ from the data, by
minimizing the dispersion
 Δ2 (Δ0,H ) 
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 Δ
i
0 i ,th
(Δ0,H ; Ri )  Δ0i ,exp ( Ri ) 
2
; (S13)
N f
here 0i,th is the calculated solvation energy from Eqs. (S12) for
an ion of crystallographic radius Ri (using Lcav+ = 0.69 Å, Lcav- = 0
and gV = 1.39 according to our assumption that the dielectric
lengths and the packing factor of water and methanol are similar);
0i,exp is the respective experimental value from Ref. [63]; N is
the number of data points (N = 6) and f is the number of free
parameters (f = 1). Note that Eq. (S13) does not involve a term
for the standard states in gas and in methanol solution 48, due to
the choice of Kelly et al.63 for the standard state in the gas phase.
The value of LQ in Eq. (S12) is unimportant for the final result
since the 1/ term in Eqs. (S12) is negligible in comparison with

30

1  3LQ / Rcav 
ei2  1 1
 
 ,
2
8πRcav    0  1  3LQ / Rcav   3L2Q / Rcav
 
1/0 (as it is with water48). Nevertheless, we used the value LQ =
1.1 Å obtained in Sec. III. B for methanol.
The optimization of  yield for the best value of the
solvation energy of H+ in methanol the value 0,H+ = -1052±12
kJ/mol. The value is by 4±12 kJ/mol higher than the hydration
energy of H+, -1056 kJ/mol62, which is another manifestation of
the fact that the first solvation shells of simple ions in water and
in methanol are similar. Kelly et al.63 also obtained almost equal
solvation energies of H+ in methanol and in water (0,H+ is by
12±10 kJ/mol higher in methanol; note that they preferred another
value for the hydration energy of H+). From the TATB
approximation, for the same difference the value 9 kJ/mol was
obtained64. The comparison between Eq. (S12) and the
experimental data is shown in FIG. S 1. The good agreement
confirms the validity of the assumed values of Lcav+, Lcav- and gV.
We can conclude that our model for the cavity size, Eq. (44),
provides not only a good first approximation for the hydration
energies but also allows for a fast determination of the
conventional energies of accuracy comparable to the existing
methods.
E. List of symbols
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Ccr
critical concentration for potential oscilations, Ccr =
kBT/2kise2LQ2
Cel electrolyte concentration (in units [m–3] or [M])
Ci ith ion concentration, Ci = iCel
Cm electrolyte molality [mol/kg]
D electric displacement field
E electric field intensity
e elementary charge
ei ith ion charge, ei = Zie
kis ratio between ionic strength and Cel, kis = (+Z+2+-Z-2)/2
kB Boltzmann constant
Lcav dielectric length, Lcav+ = 0.69 Å and Lcav- = 0
LD Debye length, LD2 = kBT/2kise2Cel
lD first characteristic length of the quadrupolar Debye-Ampere
law, Eqs. (18)-(19)
lIm imaginary part of lD at LD < 2LQ, Eqs. (26)
LQ quadrupolar length, LQ2 = Q/3
lQ
second characteristic length of the quadrupolar DebyeAmpere law, Eqs. (18)-(19)
lRe real part of lD at LD < 2LQ, Eqs. (26)
Mel electrolyte molecular mass
Ms molar mass of the solvent
P macroscopic polarization vector, P = QE
Q macroscopic quadrupolarization tensor, Q=Q(E - U∙E/3)
R distance of closest approach between counterions, R = R+ + RRi crystallographic or Pauling radius of the bare cation/anion
Rcav,i cavity radius of the ith ion, Rcav,i = gV1/3(Ri + Lcav)
r spherical radial coordinate
T temperature
u0 self-energy (Born energy) of the ion
u0∞ Born energy u0 at infinite dilution
uel full electric energy of the ion, u0 + kBTlnDH
Vel partial molar volume of the electrolyte
xi molar fraction of the ith ion
Zi dimensionless charge of the ion
P macroscopic polarizability of a medium, P =  – 0
Q macroscopic quadrupolarizability of a medium
± bulk mean activity coefficient, (++-)ln±=+ln++-lnBorn self-salting-out (Born energy or ion-medium interaction)
contribution to the activity coefficient; kBTlnBorn = u0 – u0∞
DH
Debye-Hückel (ion-ion interaction) contribution to the
activity coefficient
i activity coefficient of the ith ion
x contribution to the activity coefficient originating from the
choice of standard concentration, Eq. (51)
 Dirac delta-function
 absolute dielectric permittivity of the electrolyte solution
0 absolute dielectric permittivity of vacuum
∞ absolute dielectric permittivity of pure solvent
i stoichiometric number of the ith ion
 free charge density of the electrolyte solution
55
m mass density of the solution
m0 mass density of pure water
 electrostatic potential
0 electrostatic potential at r = 0