We present a continuum theory of electrolytes composed of a waterlike solvent and univalent ions. First, we start with a density functional F for the coarse-grained solvent, cation, and anion densities, including the Debye–Hückel free energy, the Coulombic interaction, and the direct interactions among these three components. These densities fluctuate obeying the distribution exp(F/kBT). Eliminating the solvent density deviation in F, we obtain the effective non-Coulombic interactions among the ions, which consist of the direct ones and the solvent-mediated ones. We then derive general expressions for the ion correlation, the apparent partial volume, and the activity and osmotic coefficients up to linear order in the average salt density ns. Second, we perform numerical analysis using the Mansoori–Carnahan–Starling–Leland model [J. Chem. Phys. 54, 1523 (1971)] for three-component hardspheres. The effective interactions sensitively depend on the cation and anion sizes due to competition between the steric and hydration effects, which are repulsive between small–large ion pairs and attractive between symmetric pairs. These agree with previous experiments and Collins’ rule [Biophys. J. 72, 65 (1997)]. We also give simple approximate expressions for the ionic interaction coefficients valid for any ion sizes.

1.
R. A.
Robinson
and
R. H.
Stokes
,
Electrolyte Solutions
, 2nd ed. (
Dover
,
Mineola, NY
,
2002
).
2.
C. H.
Hamann
,
A.
Hamnett
, and
W.
Vielstich
,
Electrochemistry
(
Wiley-VCH
,
2007
).
3.
P.
Debye
and
E.
Hückel
,
Phys. Z.
24
,
185
(
1923
).
4.
D.
McQuarrie
,
Statistical Mechanics
(
Harper and Row
,
New York
,
1976
), Chap. 15.
5.
W.
Kunz
,
P.
Lo Nostro
, and
B. W.
Ninham
,
Curr. Opin. Colloid Interface Sci.
9
,
1
(
2004
).
6.
W.
Kunz
,
Curr. Opin. Colloid Interface Sci.
15
,
34
(
2010
).
7.
P. L.
Nostro
and
B. W.
Ninham
,
Chem. Rev.
112
,
2286
(
2012
).
8.
W.
Kunz
,
J.
Henle
, and
B. W.
Ninham
,
Curr. Opin. Colloid Interface Sci.
9
,
19
(
2004
); English translation of Franz Hofmeister’s historical papers.
9.
E.
Hückel
,
Phys. Z.
26
,
93
(
1925
).
11.
F. J.
Millero
,
Chem. Rev.
71
,
147
(
1971
).
12.
13.
G. N.
Lewis
and
M.
Randall
,
J. Am. Chem. Soc.
43
,
233
(
1921
).
14.
J. N.
Brönsted
,
J. Am. Chem. Soc.
44
,
938
(
1922
).
15.
E. A.
Guggenheim
,
Philos. Mag.
19
,
588
(
1935
).
16.
E. A.
Guggenheim
and
J. C.
Turgeon
,
Trans. Faraday Soc.
51
,
747
(
1955
).
17.
18.
K. S.
Pitzer
,
J. Phys. Chem.
77
,
268
(
1973
).
19.
B. A.
Pailthorpe
,
D. J.
Mitchell
, and
B. W.
Ninham
,
J. Chem. Soc., Faraday Trans. 2
80
,
115
(
1984
).
20.
O.
Redlich
,
J. Phys. Chem.
44
,
619
(
1940
).
21.
O.
Redlich
and
D. M.
Meyer
,
Chem. Rev.
64
,
221
(
1964
).
22.
B. E.
Conway
,
R. E.
Verrall
, and
J. E.
Desnoyers
,
Trans. Faraday Soc.
62
,
2738
(
1966
).
23.
J. E.
Desnoyers
,
M.
Arel
,
G.
Perron
, and
C.
Jolicoeur
,
J. Phys. Chem.
73
,
3346
(
1969
).
24.
J. C.
Rasaiah
and
H. L.
Friedman
,
J. Chem. Phys.
48
,
2742
(
1968
).
25.
E.
Waisman
and
J. L.
Lebowitz
,
J. Chem. Phys.
52
,
4307
(
1970
).
27.
J.-P.
Simonin
,
L.
Blum
, and
P.
Turq
,
J. Phys. Chem.
100
,
7704
(
1996
).
28.
W.
Ebeling
and
M.
Grigo
,
J. Solution Chem.
11
,
151
(
1982
).
29.
Y.
Levin
and
M. E.
Fisher
,
Physica A
225
,
164
(
1996
).
30.
M. E.
Fisher
,
J. Stat. Phys.
75
,
1
(
1994
);
G.
Stell
,
J. Stat. Phys.
78
,
197
(
1995
).
31.
D. N.
Card
and
J. P.
Valleau
,
J. Chem. Phys.
52
,
6232
(
1970
).
32.
J. M.
Romero-Enrique
,
G.
Orkoulas
,
A. Z.
Panagiotopoulos
, and
M. E.
Fisher
,
Phys. Rev. Lett.
85
,
4558
(
2000
).
33.
P. S.
Ramanathan
and
H. L.
Friedman
,
J. Chem. Phys.
54
,
1086
(
1971
).
34.
L.
Blum
,
J. Chem. Phys.
61
,
2129
(
1974
).
35.
J.
Perkyns
and
B. M.
Pettitt
,
J. Chem. Phys.
97
,
7656
(
1992
).
36.
Y. V.
Kalyuzhnyi
,
V.
Vlachy
, and
K. A.
Dill
,
Phys. Chem. Chem. Phys.
12
,
6260
(
2010
).
37.
I. S.
Joung
,
T.
Luchko
, and
D. A.
Case
,
J. Chem. Phys.
138
,
044103
(
2013
).
38.
S.
Weerasinghe
and
P. E.
Smith
,
J. Chem. Phys.
119
,
11342
(
2003
).
39.
B.
Hess
,
C.
Holm
, and
N.
van der Vegt
,
J. Chem. Phys.
124
,
164509
(
2006
).
40.
B.
Hess
,
C.
Holm
, and
N.
van der Vegt
,
Phys. Rev. Lett.
96
,
147801
(
2006
).
41.
I.
Kalcher
and
J.
Dzubiella
,
J. Chem. Phys.
130
,
134507
(
2009
).
42.
L.
Vrbka
,
M.
Lund
,
I.
Kalcher
,
J.
Dzubiella
,
R. R.
Netz
, and
W.
Kunz
,
J. Chem. Phys.
131
,
154109
(
2009
).
43.
B.
Klasczyk
and
V.
Knecht
,
J. Chem. Phys.
132
,
024109
(
2010
).
44.
M.
Fyta
and
R. R.
Netz
,
J. Chem. Phys.
136
,
124103
(
2012
).
45.
M.
Kohns
,
M.
Schappals
,
M.
Horsch
, and
H.
Hasse
,
J. Chem. Eng. Data
61
,
4068
(
2016
).
46.
N.
Naleem
,
N.
Bentenitis
, and
P. E.
Smith
,
J. Chem. Phys.
148
,
222828
(
2018
).
47.
J. G.
Kirkwood
and
F. P.
Buff
,
J. Chem. Phys.
19
,
774
(
1951
).
48.
B.
Widom
and
R. C.
Underwood
,
J. Phys. Chem. B
116
,
9492
(
2012
).
49.
K.
Koga
,
V.
Holten
, and
B.
Widom
,
J. Phys. Chem. B
119
,
13391
(
2015
).
50.
C. A.
Cerdeiriña
and
B.
Widom
,
J. Phys. Chem. B
120
,
13144
(
2016
).
51.
W. G.
McMillan
and
J. E.
Mayer
,
J. Chem. Phys.
13
,
276
(
1945
).
52.
R.
Evans
and
T. J.
Sluckin
,
Mol. Phys.
40
,
413
(
1980
).
53.
54.
M. Z.
Bazant
,
M. S.
Kilic
,
B. D.
Storey
, and
A.
Ajdari
,
Adv. Colloid Interface Sci.
152
,
48
(
2009
).
55.
D.
Ben-Yaakov
,
D.
Andelman
,
R.
Podgornik
, and
D.
Harries
,
Curr. Opin. Colloid Interface Sci.
16
,
542
(
2011
).
56.
F.
Fogolari
,
A.
Brigo
, and
H.
Molinari
,
J. Mol. Recognit.
15
,
377
(
2002
).
57.
J. J.
Bikerman
,
Philos. Mag.
33
,
384
(
1942
).
58.
I.
Borukhov
,
D.
Andelman
, and
H.
Orland
,
Phys. Rev. Lett.
79
,
435
(
1997
).
59.
V.
Kralj-Iglič
and
A.Iglič
,
J. Phys. II
6
,
477
(
1996
).
60.
A.
Onuki
,
Phase Transition Dynamics
(
Cambridge
,
2002
); In this book, discussions are given on the fluctuation variances in Sec. 1.3 and on the steric interaction in polymer solutions in Sec. 3.5.
61.
G. A.
Mansoori
,
N. F.
Carnahan
,
K. E.
Starling
, and
T. W.
Leland
,
J. Chem. Phys.
54
,
1523
(
1971
).
62.
P. M.
Biesheuvel
and
M.
van Soestbergen
,
J. Colloid Interface Sci.
316
,
490
(
2007
).
63.
P.
Zhang
,
N. M.
Alsaifi
,
J.
Wu
, and
Z.-G.
Wang
,
Macromolecules
49
,
9720
(
2016
).
64.
N. F.
Carnahan
and
K. E.
Starling
,
J. Chem. Phys.
51
,
635
(
1969
).
65.
R.
Okamoto
and
A.
Onuki
,
Eur. Phys. J. E
38
,
72
(
2015
).
66.
R.
Okamoto
and
A.
Onuki
,
J. Phys.: Condens. Matter
28
,
244012
(
2016
).
67.
R.
Okamoto
and
A.
Onuki
,
J. Chem. Phys.
149
,
014501
(
2018
).
68.
W.
Kunz
,
K.
Holmberg
, and
T.
Zemb
,
Curr. Opin. Colloid Interface Sci.
22
,
99
(
2016
).
70.
K. D.
Collins
,
G. W.
Neilson
, and
J. E.
Enderby
,
Biophys. Chem.
128
,
95
(
2007
).
71.
K. D.
Collins
,
Q. Rev. Biophys.
52
,
e11
(
2019
).
72.
R.
Schurhammer
and
G.
Wipff
,
J. Phys. Chem. A
104
,
11159
(
2000
).
73.
T. M.
Herrington
and
C. M.
Taylor
,
J. Chem. Soc., Faraday Trans. 1
78
,
3409
(
1982
).
74.
F. J.
Millero
,
J. Chem. Eng. Data
15
,
562
(
1970
).
75.
K.
Sadakane
,
A.
Onuki
,
K.
Nishida
,
S.
Koizumi
, and
H.
Seto
,
Phys. Rev. Lett.
103
,
167803
(
2009
).
76.
A.
Onuki
,
S.
Yabunaka
,
T.
Araki
, and
R.
Okamoto
,
Curr. Opin. Colloid Interface Sci.
22
,
59
(
2016
).
77.
S.
Yabunaka
and
A.
Onuki
,
Phys. Rev. Lett.
119
,
118001
(
2017
).
78.
N.
Tasios
,
S.
Samin
,
R.
van Roij
, and
M.
Dijkstra
,
Phys. Rev. Lett.
119
,
218001
(
2017
).
79.
N.
Bjerrum
and
K.
Dan
,
Vidensk. Selsk. Mat.-Fys. Medd.
7
,
1
(
1926
).
80.
D. E.
Smith
and
L. X.
Dang
,
J. Chem. Phys.
100
,
3757
(
1994
).
81.
L.
Degrève
and
F. L. B.
da Silva
,
J. Chem. Phys.
110
,
3070
(
1999
).
82.
Y.
Marcus
and
G.
Hefter
,
Chem. Rev.
106
,
4585
(
2006
).
83.
S. A.
Hassan
,
J. Phys. Chem. B
112
,
10573
(
2008
).
84.
C. J.
Fennell
,
A.
Bizjak
,
V.
Vlachy
, and
K. A.
Dill
,
J. Phys. Chem. B
113
,
6782
(
2009
).
85.
J.
Zwanikken
and
R.
van Roij
,
J. Phys.: Condens. Matter
21
,
424102
(
2009
).
86.
N. F. A.
van der Vegt
,
K.
Haldrup
,
S.
Roke
,
J.
Zheng
,
M.
Lund
, and
H. J.
Bakker
,
Chem. Rev.
116
,
7626
(
2016
).
87.
R. M.
Adar
,
T.
Markovich
, and
D.
Andelman
,
J. Chem. Phys.
146
,
194904
(
2017
).
88.

In the original paper,3a2 and a3 can be different, while they have been equated in most subsequent papers.

89.
D. G.
Archer
and
P.
Wang
,
J. Phys. Chem. Ref. Data
19
,
371
(
1990
).
90.
D. P.
Fernández
,
A. R. H.
Goodwin
,
E. W.
Lemmon
,
J. M. H. L.
Sengers
, and
R. C.
Williams
,
J. Phys. Chem. Ref. Data
26
,
1125
(
1997
).
91.
L. G.
Hepler
,
J. Phys. Chem.
61
,
1426
(
1957
).
92.
P.
Mukerjee
,
J. Phys. Chem.
65
,
740
(
1960
).
93.
J.
Padova
,
J. Chem. Phys.
39
,
1552
(
1963
).
94.
V.
Mazzini
and
V. S. J.
Craig
,
Chem. Sci.
8
,
7052
(
2017
).
95.
J. P.
O’Connell
and
A. E.
DeGance
,
J. Solution Chem.
4
,
763
(
1975
).
96.
97.
R. L.
de Carvalho
and
R.
Evans
,
Mol. Phys.
83
,
619
(
1994
).
98.
J. P.
Hansen
and
I. R.
McDonald
,
Theory of Simple Liquids
(
Academic
,
1986
).
100.
M.
Lund
,
B.
Jagoda-Cwiklik
,
C. E.
Woodward
,
R.
Vácha
, and
P.
Jungwirth
,
J. Phys. Chem. Lett.
1
,
300
(
2010
).
101.
T. T.
Duignan
,
D. F.
Parsons
, and
B. W.
Ninham
,
Phys. Chem. Chem. Phys.
16
,
22014
(
2014
).
102.
A. M.
Smith
,
A. A.
Lee
, and
S.
Perkin
,
J. Phys. Chem. Lett.
7
,
2157
(
2016
).
103.
R. M.
Adar
,
S. A.
Safran
,
H.
Diamant
, and
D.
Andelman
,
Phys. Rev. E
100
,
042615
(
2019
).
104.
S. W.
Coles
,
C.
Park
,
R.
Nikam
,
M.
Kansdusč
,
J.
Dzubiella
, and
B.
Rotenberg
,
J. Phys. Chem. B
124
,
1778
(
2020
).
105.
P. G.
Kusalik
and
G. N.
Patey
,
J. Chem. Phys.
86
,
5110
(
1987
).
106.
K. E.
Newman
,
J. Chem. Soc., Faraday Trans. 1
85
,
485
(
1989
).
107.
W.
Olivares
and
D. A.
Mcquarrie
,
J. Phys. Chem.
66
,
1508
(
1962
).
108.
V.
McGahay
and
M.
Tomozawa
,
J. Non-Cryst. Solids
109
,
27
(
1989
).
109.
F. J.
Millero
,
G. K.
Ward
,
F. K.
Lepple
, and
E. V.
Hoff
,
J. Phys. Chem.
78
,
1636
(
1974
).
110.
Y.
Luo
and
B.
Roux
,
J. Phys. Chem. Lett.
1
,
183
(
2010
).
111.
W. J.
Hamer
and
Y. C.
Wu
,
J. Phys. Chem. Ref. Data
1
,
1047
(
1972
).
112.
R. D.
Shannon
,
Acta Crystallogr., Sect. A
32
,
751
(
1976
).
113.
C. B.
Stubblefield
and
R. O.
Bach
,
J. Chem. Eng. Data
17
,
491
(
1972
).
114.
H. G.
Glasbrenner
and
H.
Weingärtner
,
J. Phys. Chem.
93
,
3378
(
1989
).
115.
I. S.
Joung
and
T. E.
Cheatham
 III
,
J. Phys. Chem. B
113
,
13279
(
2009
).
116.
J. L.
Aragones
,
E.
Sanz
, and
C.
Vega
,
J. Chem. Phys.
136
,
244508
(
2012
).
117.
T.
Yagasaki
,
M.
Matsumoto
, and
H.
Tanaka
,
J. Chem. Theory Comput.
16
,
2460
(
2020
).
118.
R.
Okamoto
and
A.
Onuki
,
Phys. Rev. E
82
,
051501
(
2010
).
119.
A.
Onuki
and
R.
Okamoto
,
Curr. Opin. Colloid Interface Sci.
16
,
525
(
2011
).
120.
P.
Drude
and
W.
Nernst
,
Z. Phys. Chem.
15
,
79
(
1894
).
121.
L. D.
Landau
and
E. M.
Lifshitz
,
Electrodynamics of Continuous Media
(
Pergamon
,
1984
).
122.

The polarization energy of a water molecule around an ion is μ0|E(r)| ∼ 3kBT/r2 (r in Å) outside the hydration shell in ambient water, where μ0 = 2.3 D. The polarization saturates for r3 Å.

123.
Y. Z.
Wei
,
P.
Chiang
, and
S.
Sridhar
,
J. Chem. Phys.
96
,
4569
(
1992
).
124.
R.
Buchner
,
G. T.
Hefter
, and
P. M.
May
,
J. Phys. Chem. A
103
,
1
(
1999
).
125.
A.
Levy
,
D.
Andelman
, and
H.
Orland
,
J. Chem. Phys.
139
,
164909
(
2013
).
126.
J.
Vincze
,
M.
Valiskó
, and
D.
Boda
,
J. Chem. Phys.
133
,
154507
(
2010
).
127.
V.
Talanquer
,
C.
Cunningham
, and
D. W.
Oxtoby
,
J. Chem. Phys.
114
,
6759
(
2001
).
128.
N. F.
Carnahan
and
K. E.
Starling
,
AlChE J
18
,
1184
(
1972
).
129.
R. A.
Fine
and
F. J.
Millero
,
J. Chem. Phys.
59
,
5529
(
1973
).
130.

In the original paper,61 another quantity y2 also appears. In Eq. (E1), it is removed from the relation y2 = 1 − y1y3.

131.
Y.
Marcus
and
G.
Hefter
,
J. Sol. Chem.
28
,
575
(
1999
).
You do not currently have access to this content.