Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity
The potential of mean force (PMF) between like-charged colloidal particles immersed in aqueous electrolyte solution is studied using the integral equation theory. Solvent molecules are modeled as neutral hard spheres, and ions and colloidal particles are taken to be charged hard spheres. The Coulo...
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Інститут фізики конденсованих систем НАН України
2007
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| Цитувати: | Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity / R. Akiyama, N. Fujino, K. Kaneda, M. Kinoshita // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 587-596. — Бібліогр.: 31 назв. — англ. |
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irk-123456789-1189442017-06-02T03:03:24Z Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity Akiyama, R. Fujino, N. Kaneda, K. Kinoshita, M. The potential of mean force (PMF) between like-charged colloidal particles immersed in aqueous electrolyte solution is studied using the integral equation theory. Solvent molecules are modeled as neutral hard spheres, and ions and colloidal particles are taken to be charged hard spheres. The Coulomb potentials for ion-ion, ioncolloidal particle, and colloidal particle-colloidal particle pairs are divided by the dielectric constant of water. This simple model is employed to account for the effects of solvent granularity neglected in the so-called primitive model. The van der Waals attraction between colloidal particles, which is an essential constituent of conventional DLVO theory, is omitted in the present model. Nevertheless, when the electrolyte concentration is sufficiently high, attractive regions appear in the PMF. In particular, the interaction at small separations is significantly attractive and the contact of colloidal particles is stabilized. This interesting behavior arises from the effects of the translational motion of solvent molecules. Потенцiал середньої сили мiж одноiменно зарядженими колоїдними частинками, помiщеними у водний розчин електролiту, вивчається методами теорiї iнтегральних рiвнянь. Молекули розчинника моделюються твердими сферами, а iони та колоїднi частинки розглядаються як зарядженi твердi сфери. Кулонiвськi взаємодiї мiж iонами та колоїдними частинками включають дiелектричну сталу води. Ця проста модель використана для врахування ефектiв дискретностi розчинника, якими нехтують в примiтивнiй моделi. Ван дер Ваальсовим притяганням, яке є суттєвою рисою теорiї ДЛВО, в данiй моделi нехтуються. Тим не менше, притягальнi областi виникають в потенцiалi середньої сили при достатньо високих iонних концентрацiях. Зокрема, взаємодiя на малих вiдстанях є достатньо притягальною, щоб стабiлiзувати колоїдну систему. Ця поведiнка виникає за рахунок ефектiв трансляцiйного руху молекул розчинника. 2007 Article Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity / R. Akiyama, N. Fujino, K. Kaneda, M. Kinoshita // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 587-596. — Бібліогр.: 31 назв. — англ. 1607-324X PACS: 82.35.Rs, 82.35.Pq, 82.70.Dd, 83.80.Hj DOI:10.5488/CMP.10.4.587 http://dspace.nbuv.gov.ua/handle/123456789/118944 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
The potential of mean force (PMF) between like-charged colloidal particles immersed in aqueous electrolyte
solution is studied using the integral equation theory. Solvent molecules are modeled as neutral hard spheres,
and ions and colloidal particles are taken to be charged hard spheres. The Coulomb potentials for ion-ion, ioncolloidal
particle, and colloidal particle-colloidal particle pairs are divided by the dielectric constant of water.
This simple model is employed to account for the effects of solvent granularity neglected in the so-called
primitive model. The van der Waals attraction between colloidal particles, which is an essential constituent of
conventional DLVO theory, is omitted in the present model. Nevertheless, when the electrolyte concentration
is sufficiently high, attractive regions appear in the PMF. In particular, the interaction at small separations is
significantly attractive and the contact of colloidal particles is stabilized. This interesting behavior arises from
the effects of the translational motion of solvent molecules. |
| format |
Article |
| author |
Akiyama, R. Fujino, N. Kaneda, K. Kinoshita, M. |
| spellingShingle |
Akiyama, R. Fujino, N. Kaneda, K. Kinoshita, M. Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity Condensed Matter Physics |
| author_facet |
Akiyama, R. Fujino, N. Kaneda, K. Kinoshita, M. |
| author_sort |
Akiyama, R. |
| title |
Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity |
| title_short |
Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity |
| title_full |
Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity |
| title_fullStr |
Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity |
| title_full_unstemmed |
Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity |
| title_sort |
interaction between like-charged colloidal particles in aqueous electrolyte solution: attractive component arising from solvent granularity |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118944 |
| citation_txt |
Interaction between like-charged colloidal particles in aqueous electrolyte solution: Attractive component arising from solvent granularity / R. Akiyama, N. Fujino, K. Kaneda, M. Kinoshita // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 587-596. — Бібліогр.: 31 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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2025-07-08T14:56:54Z |
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2025-07-08T14:56:54Z |
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1837091118216380416 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 4(52), pp. 587–596
Interaction between like-charged colloidal particles in
aqueous electrolyte solution: Attractive component
arising from solvent granularity
R.Akiyama1,2∗, N.Fujino1, K.Kaneda1, M.Kinoshita3
1 Department of Chemistry, Graduate School of Science, Kyushu University, Ropponmatsu, Chuo-ku,
Fukuoka, 810–8560, Japan
2 Institute for Molecular Science, Myodaiji, Okazaki, 444–8585, Japan
3 Institute of Advanced Energy, Kyoto University, Uji, Kyoto, 611–0011, Japan
Received August 8, 2007, in final form October 10, 2007
The potential of mean force (PMF) between like-charged colloidal particles immersed in aqueous electrolyte
solution is studied using the integral equation theory. Solvent molecules are modeled as neutral hard spheres,
and ions and colloidal particles are taken to be charged hard spheres. The Coulomb potentials for ion-ion, ion-
colloidal particle, and colloidal particle-colloidal particle pairs are divided by the dielectric constant of water.
This simple model is employed to account for the effects of solvent granularity neglected in the so-called
primitive model. The van der Waals attraction between colloidal particles, which is an essential constituent of
conventional DLVO theory, is omitted in the present model. Nevertheless, when the electrolyte concentration
is sufficiently high, attractive regions appear in the PMF. In particular, the interaction at small separations is
significantly attractive and the contact of colloidal particles is stabilized. This interesting behavior arises from
the effects of the translational motion of solvent molecules.
Key words: colloidal particle, electrolyte solution, potential of mean force, integral equation theory, DLVO
theory, solvent granularity
PACS: 82.35.Rs, 82.35.Pq, 82.70.Dd, 83.80.Hj
1. Introduction
The interaction between like-charged colloidal particles immersed in electrolyte solution is usu-
ally discussed using the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory [1]. In this theory the
interaction comprises the attractive van der Waals component WvdW and the repulsive electric
double layer component Wedl. WvdW(r) is strong, where r is the distance between centers of col-
loidal particles, and it is almost independent of the ionic strength of the electrolyte solution. On
the other hand, Wedl(r), which decreases almost exponentially with r, is strongly dependent on
the ionic strength. As the electrolyte concentration increases, the electrostatic repulsion between
colloidal particles is increasingly screened and the free-energy barrier for their cohesion becomes
lower. In solution containing electrolytes (e.g., NaCl) at sufficiently high concentration (∼ 0.1 M),
aggregation of colloidal particles takes place.
In the DLVO theory, there is no attractive part except the term WvdW(r), and the interaction
between like-charged colloidal particles is purely repulsive if we neglect WvdW(r). Here, we are
concerned with the case where WvdW(r) is absent in the model calculations. Kinoshita et al. ana-
lyzed the interaction between like-charged colloidal particles using the angle-dependent reference
hypernetted-chain (RHNC) integral equation theory combined with a molecular model for water
[2]. Though the term WvdW(r) is not considered, attractive regions appear in some of the inter-
actions calculated. This result suggests that an attractive component arises from a physical factor
other than WvdW(r). One of the essential differences between the DLVO theory and the analysis
made by Kinoshita et al. is that the solvent granularity is taken into account in the latter.
∗Corresponding author. E-mail: rakiyama@chem.rc.kyushu-u.ac.jp
c© R.Akiyama, N.Fujino, K.Kaneda, M.Kinoshita 587
R.Akiyama, et al.
On the other hand, large particles immersed in small particles attract each other. Suppose that
the number density of the small particles is much higher than that of the large particles, and the
small and large particles are hard spheres, in other words, there are no soft interactions (e.g., van
der Waals and electrostatic interactions) between the particles. In such a system, all the allowed
system configurations have the same energy and the system behavior is of purely entropic origin.
The presence of a large particle generates a volume from which the centers of the small particles
are excluded. When two large particles contact each other, the excluded volumes overlap, and the
total volume available to the translational motion of the small particles increases by this amount.
This increase leads to a gain in the free energy of the small particles and an attractive interaction
is thus induced between the large particles. In the original discussion by Asakura and Oosawa, the
large particles are colloidal particles and the small particles are macromolecules [3,4]. Other studies
[5–15] have considered solute molecules as the large particles immersed in small particles forming
the solvent. In these studies the crucial importance of the solvent granularity and the translational
motion of solvent molecules is emphasized.
In the present article, we analyze the interaction between like-charged colloidal particles in
electrolyte solution using the integral equation theory. Solvent molecules are modeled as neutral
hard spheres, and ions and colloidal particles are taken to be charged hard spheres. The Coulomb
potentials for ion-ion, ion-colloidal particle, and colloidal particle-colloidal particle pairs are divided
by the dielectric constant of water. This simple model is best suited to the exclusive investigation
of the effects due to the solvent granularity, as opposed to the detailed but rather complicated
model employed by Kinoshita et al.[2].
2. Model and theory
We consider a pair of colloidal particles immersed in a simple model of electrolyte solution
which comprises neutral hard spheres and charged hard spheres corresponding to solvent molecules
and 1-1 ions, respectively. The colloidal particles are modeled as like-charged hard spheres or
uncharged (neutral) hard spheres. The Coulomb potentials between ion-ion, ion-colloidal particle,
and colloidal particle-colloidal particle pairs are divided by 78.5, i.e., the dielectric constant of
water. The potential parameters for ions are those developed by Pettitt and Rossky [16] and
later modified by Kinoshita and Harano [17]. The solvent molecules and the colloidal particles are
represented by “V” and “M” respectively. Major parameters for the pair potentials are summarized
in table 1. The diameter of the solvent spheres is taken to be the effective diameter of water
Table 1. Partial charges and diameters of ions, solvent molecule[17], and colloidal particles. Qj
denotes the charge of particle j. σjj denotes the diameter of the hard-sphere interaction between
particles of j. “V” and “M” denote the solvent molecule and the colloidal particles. QM is −5, 0,
or +5.
Site Qj(e) σjj(Å)
Na+ 1 2.26
K+ 1 3.14
Cs+ 1 3.72
Cl− −1 3.92
Br− −1 4.19
I− −1 4.54
V 0 2.80
M QM 16.8
molecules σVV = 2.8 Å. The diameter of colloidal particles is set at σMM = 6σVV = 16.8 Å. The
pair potential is expressed as
uij(r) = ∞ (r 6 σij), (1)
=
QiQj
78.5r
(r > σij), (2)
588
Interaction between charged colloidal particles
where the usual mixing rule, σij = (σii + σjj)/2 (i, j = A, C, V, M), is employed and “A” and
“C” denote anions and cations, respectively. Most of the calculations are carried out for φt = 0.383
where φt = Σl=A,C,V φl, φl = (πρlσ
3
ll)/6, and ρl is the number density of l (φV = 0.383 corresponds
to ρV = 0.033333 Å−3 that is the number density of pure water at 1 atm and 298K).
Computer simulations are not practically applicable to systematic calculations when the size
asymmetry of the particles is high. Moreover, they become much more difficult for a system in-
cluding ions at finite concentration due to the problems arising from establishing estimations of the
bulk equilibrium concentration of ions in each simulation. The integral equation theory we employ,
however, is free from these considerable difficulties.
The ion-ion, solvent-ion, and solvent-solvent correlation functions for the bulk electrolyte solu-
tion are first calculated using the Ornstein-Zernike (OZ) equation [18]
hij(r) = cij(r) + cil(r) ∗ ρlhlj(r), (3)
coupled with the hypernetted-chain (HNC) closure[18]
hij(r) = exp [−uij(r)/kBT + hij(r) − cij(r)] − 1, (4)
where hij(r) (i, j, l = A, C, V) is the total correlation function, cij(r) is the direct correlation
functions, ρi is the number density, kB is the Boltzmann constant, uij(r) is the pair potential, and
* denotes the convolution product.
The interaction between colloidal particles is discussed in terms of the potential of the mean
force (PMF) which is calculated from the correlation functions for the bulk electrolyte solution in
the following manner. A colloidal particle is immersed in the solution, and the colloidal particle-
solvent and the colloidal particle-ion correlation functions, hMj and cMj , are calculated using the
OZ-HNC theory. The correlation function for colloidal particles TMM(k) = HMM(k) − CMM(k) is
then calculated via
TMM(k) =
∑
l=A,C,andV
ρlHMl(k)ClM(k), (5)
where TMM(k),HMl(k), and ClM(k) are the Fourier Transforms of tMM(r)(= hMM(r) − cMM(r)),
hMl(r), and clM(r), respectively, and k is the wave number. The PMF Φ(r) is obtained as
Φ(r)
kBT
=
uMM (r)
kBT
− tMM(r), (6)
where uMM(r) is the pair potential between colloidal particles. Equation (6) comes from the HNC
approximation. Due to the presence of Coulomb potentials, care must be taken in the numerical
treatment within the Fourier transformation. More details are described in earlier publications
[19–24].
3. Results and discussion
3.1. Effects of electrolyte concentration on potential of mean force
Figure 1 shows the effects of electrolyte (NaCl) concentration on the PMF between like-charged
colloidal particles (σMM = 6σVV, QM = 5e). The PMFs at smaller separations and at larger
separations are plotted in figure 1(a) and in figure 1(b), respectively. The concentration tested is
in the range from 1 · 10−5 M to 1 M and φt is fixed at 0.383. Since the concentration of colloidal
particles is zero due to the infinite dilution limit, the concentration of counter ions, which neutralize
the charges of colloidal particles, is also zero. The concentration of counter charged ions, which
screen the charges of colloidal particles, is equal to the electrolyte concentration. The curve for
1 · 10−4 M is almost indistinguishable from that for 1 · 10−5 M. This is because the electrolyte
concentration is extremely low and the screening of the electrostatic repulsion is rather weak in
these cases. However, adding more electrolytes to the solution leads to an increasingly stronger
589
R.Akiyama, et al.
-5
0
5
10
15
16 17 18 19 20 21 22 23 24
Φ
(r
)
/k
B
T
r/10-10m
(a)
0.01
0.1
1
10
100
20 30 40 50 60 70 80 90 100
rΦ
(r
)
/k
B
T
/
10
-1
0 m
r/10-10m
(b)
Figure 1. Potential of mean force between like-charged colloidal particles (σMM = 6σVV, QM =
+5e) immersed in our model electrolyte solution. The electrolyte is NaCl. Solid line: 1 M, Long-
dash line: 1 · 10−1 M, Short-dash line: 1 · 10−2 M, Dotted line: 1 · 10−3 M, Long dash-dot line:
1 · 10−4 M, and Short dash-dot line: 1 · 10−5 M. (a) Linear plot for smaller separations. (b)
Logarithmic plot for larger separations. The solid line is omitted in (b).
screening. As the electrolyte concentration becomes higher, the PMF curve exhibits a downward
shift (i.e., shifts in more attractive direction).
As shown in figure 1(a), the PMF curves exhibit an oscillation with a period corresponding to
the diameter of the solvent molecules. When the electrolyte concentration is less than 1·10−2 M, the
PMFs are always repulsive and the colloidal particles should be dispersed. When the concentration
is 1·10−1 M, on the other hand, an attractive region appears in the immediate vicinity of the contact
position and the contact value of the PMF is about −1.18kBT (−0.70 kcal/mol). In the case of
the 1 M solution, the contact value reaches −2.85kBT (−1.69 kcal/mol) which is much larger than
kBT . Therefore, the potential of mean force calculated for the present model qualitatively resembles
that for the DLVO theory, with regard to the dependence on the electrolyte concentration, and
a dimer of like-charged colloidal particles can be stabilized in spite of the absence of WvdW, the
attractive component in the DLVO theory. As for the PMFs at sufficiently large separations, the
Poisson-Boltzmann picture is acceptable, because the curves plotted in figure 1(b) asymptotically
approach straight lines [25]. Thus, the behavior of the PMF is similar to that predicted by the
DLVO theory at a sufficiently large separation.
-5
0
5
10
15
16 17 18 19 20 21 22 23 24
Φ
(r
)
/k
B
T
r/10-10m
Figure 2. Line A (Long-dash): Potential of mean force between uncharged colloidal particles
(σMM = 6σVV, QM = 0) immersed in our model electrolyte solution (NaCl, 1 · 10−1 M, φt =
0.383). Line B (Short-dash): Potential of mean force between like-charged colloidal particles
(σMM = 6σVV, QM = +5e) immersed in our model electrolyte solution (NaCl, 1 · 10−1 M,
φt = 0.383). Line C (Solid): Potential of mean force between like-charged colloidal particles
(σMM = 6σVV, QM = +5e) immersed in pure solvent(φt = 0.383).
590
Interaction between charged colloidal particles
The PMF between uncharged colloidal particles is also calculated and shown in figure 2 as line
A which is characterized by a large, negative contact value (−4.63kBT ). This attractive behavior
originates from the translational motion of solvent molecules. Line C in figure 2 illustrates the PMF
between like-charged colloidal particles in pure solvent where there is no screening of the electro-
static repulsion by ions. Line C is similar to the long dash-dot and short dash-dot lines in figure 1.
As the electrolyte concentration becomes higher, the PMF approaches line A with the oscillatory
behavior qualitatively unchanged. The contact values of the PMFs for 1·10−1 M (line B in figure 2),
1 M, and line A are, respectively, −1.18kBT , −2.85kBT , and −4.63kBT . It seems that the screening
is not complete near the colloidal particle even when the concentration of electrolyte is 1.0 M.
A charge neutrality curve q(r) for the charged colloidal particle is calculated to represent the
screening in terms of the pair correlation function between the colloidal particle and the ion. The
definition of q(r) is
q(r) = QM + 4π
∫ r
0
∑
i=A,C
QiρigMi(r
′)r′2dr′, (7)
where r is the distance from the center of the colloidal particle, Qi is the charge of i (i is the
anion Cl− and the cation Na+), ρi is the number density of i, and gMi(r) is the radial distribution
function between the colloidal particle and i. The radial distribution function gMi(r) is obtained
using the OZ-HNC theory. The function q(r) is shown in figure 3. It is clearly dependent on the
electrolyte concentration, and the colloidal particle is increasingly covered by counter charged ions
as the electrolyte concentration becomes higher.
0
1
2
3
4
5
6
7
0 10 20 30 40 50
q(
r)
/e
r/10-10m
Figure 3. Charge neutrality curve q(r) for the charged colloidal particle (σMM = 6σVV, QM =
+5e) immersed in our model electrolyte solution. The electrolyte is NaCl. Solid line: 1 M, Long-
dash line: 1 · 10−1 M, Short-dash line: 1 · 10−2 M, Dotted line: 1 · 10−3 M, Long dash-dot line:
1 · 10−4 M, and Short dash-dot line: 1 · 10−5 M.
This function shows the screening behavior for a colloidal particle caused by the electrolyte.
For example, q(r1) = 0 means that the charge of the colloidal particle is completely screened at r1.
The plot for 1.0 M shows that the colloidal particle charge is almost completely screened by ions
at r = 18.0 ·10−10 m, and the distance is slightly larger than σMM. However, the colloidal particles
are not completely screened when they contact each other in figure 1(a) and the attraction arising
from the solvent molecules and ions is reduced.
Here, the plot for 1.0 M (the solid line in figure 3) has an interesting peak of about q(r) = 6.2/e
at around 10.0 · 10−10 m, although there is no peak in other plots because of Coulomb repulsion
between the cations (Na+) and the positively charged colloidal particles. This peak means that the
small cations (Na+) contact the colloidal particle, even though the colloidal charge is also positive.
This indicates that the anions (Cl−) are positioned with the cations on the surface of the colloidal
particle in the case of the high electrolyte concentration. Therefore, the cations are found in the
first solvation shell of the colloidal particles. This phenomenon will be discussed in the next paper.
Based on the results described above, we draw the following conclusions. The behavior of the
PMF between like-charged colloidal particles can be understood by the combination of the screened
591
R.Akiyama, et al.
repulsive electrostatic interaction and an attractive component originating from the translational
motion of solvent molecules. When the concentration is less than 1 · 10−2 M, the ion distribution
roughly obeys the Poisson-Boltzmann picture and the Debye length is much larger than the size
of solvent molecules. In this situation, two like-charged colloidal particles repulsively interact with
each other through the electric double layer. On the other hand, when the concentration is larger
than 1 · 10−2 M, the Debye length is comparable or smaller than the size of solvent molecules. In
this situation, the attractive component arising from the solvent granularity comes into play as the
plot for 1 ·10−1 M in figure 1(a). The addition of more electrolytes to the solution leads to stronger
suppression of the electrostatic repulsion, making the attractive component more explicit.
3.2. Effects of ionic sizes on dimerization free energy of colloidal particles
It is known through experiments that the addition of electrolytes often causes a conformational
transition of biopolymers. For a protein with many positively charged groups on the side chains,
this addition induces a transition from an unfolded (fairly extended) conformation to a molten-
globule-like conformation [26–28]. The ability to induce this transition is strongly dependent on
anion species and follows, for monovalent anions, the order: I− > Br− > Cl− which is the reverse
Hofmeister series. That is, for example, I− is capable of inducing the transition at a lower concentra-
tion than Br−. In the molten-globule-like conformation, the positively charged groups are close to
one another, in marked contrast with the case of the unfolded conformation where they are fairly
separated. Another interesting example is found in the DNA structural change which is known
as the right-to-left (B-Z) transition of d[(G-C)] polymers and oligomers, induced by increasing
the electrolyte concentration [29,30]. This structural transition, however, is strongly dependent on
cation species and follows, for monovalent cations, the order: Na+ > K+ > Rb+ > Cs+ which is the
Hofmeister series. A major geometric difference between B-DNA and Z-DNA is that the negatively
charged phosphates are much closer together in Z-DNA. In the salting-out phenomenon observed
for neutral proteins, the ability to decrease the solubility follows the orders Cl− > Br− > I− and
Na+ > K+ > Cs+.
Interactions between like-charged groups in biopolymers play an essential role in the confor-
mational transition described above. In cases where the electrolyte concentration is very low, the
biopolymers take the conformations in which the charged groups are well separated due to elec-
trostatic repulsion. When the concentration becomes sufficiently high, the electrostatic repulsion
is strongly screened, allowing the charged groups to come closer together. The salting-out phe-
nomenon for neutral proteins, however, should be related to interactions between their uncharged
groups: As these interactions become more attractive, protein aggregation should be promoted
which leads to a decrease in protein solubility. For these reasons, the analysis of the effects of
ionic species on the interactions between charged and uncharged colloidal particles gives physical
insights into the behavior of biopolymers.
In the following treatment, we consider two sets of electrolytes: (i) NaCl, KCl, and CsCl; and (ii)
NaCl, NaBr, and NaI to examine the effects of cation species and anion species, respectively, on the
dimerization free energy Φmin/kBT , which is defined as the contact value of the PMF between two
colloidal particles. The electrolyte concentration is chosen to be 1 · 10−1 M. We consider colloidal
particles with charges of +5e and −5e and those without charges.
First, we fix φt at 0.383. The relation between Φmin/kBT and the cation species and that
between Φmin/kBT and the anion species are illustrated in figure 4(a) and (b), respectively. It is
observed that Φmin/kBT for colloidal particles with negative charges is considerably dependent
on the cation species while no significant dependence on the ionic species is clear in the other
cases. In particular, the addition of Na+ leads to a much larger, negative value of Φmin than
K+ and Cs+. On the other hand, Φmin/kBT for positively charged colloidal particles is relatively
independent of the anion species. This result can be understood in the following way: Firstly,
small counter-charged ions can get closer to the colloidal particles than larger ones, leading to
a larger stabilization by electrostatic attractive interactions. As a result, more counter charged
ions bind to the colloidal particles and colloidal charges are screened more strongly. Secondly, the
local density of counter charged ions in contact with colloidal particles increases as the size of the
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Interaction between charged colloidal particles
counter charged ions becomes smaller, leading to a stronger attractive interaction between colloidal
particles, arising from the enhanced effects due to the translational motion of counter charged ions.
However, these physical factors are not so important once the cation size becomes sufficiently large,
and the results for K+ and Cs+ are not very different. Here we emphasize that when the colloidal
charges are negative, the PMFs in cases of sufficiently high electrolyte solution are considerably
more attractive than those shown in figure 1(a).
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(b)
Figure 4. Dimerization free energy between colloidal particles immersed in our model electrolyte
solution. The electrolyte concentration is 0.1 M. The total packing fraction of the solution φt is
fixed at 0.383. Solid: QM = 0, Dash: QM = +5e, Dots:QM = −5e. (a) The dependence on cation
species (the electrolytes tested are NaCl, KCl, and CsCl). (b) The dependence on anion species
(the electrolytes tested are NaCl, NaBr, and NaI).
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0.4030.3930.3830.3730.363
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(b)
Figure 5. Dimerization free energy between colloidal particles with respect to φt. The electrolyte
concentration is 0.1 M. Solid: QM = 0, Dash: QM = +5e, Dots:QM = −5e. (a) NaCl. (b) KCl.
Next, φt is varied as the major parameter. The relations between Φmin/kBT and φt observed for
NaCl and KCl are illustrated in figure 5(a) and (b), respectively. In all cases Φmin/kBT takes a larger
negative value as φt becomes higher. This behavior is expected because the attractive interaction
induced by the translational motion of solvent molecules and ions is roughly proportional to the
number density of the solution. It is known for real aqueous electrolyte solutions that φt becomes
higher with added electrolytes due to electric striction (i.e., φt is much higher than 0.383 for any
of NaCl, KCl, CsCl, NaBr, and NaI solutions) [31]. Further, as the ionic size becomes smaller, the
electric striction becomes stronger and φt becomes larger. If this effect is taken into consideration,
Φmin/kBT in figure 4(a) and (b) shifts to more negative values furthermore, this shift being larger
for smaller ions.
From the above discussions, we can draw the following conclusions. The contact of uncharged
colloidal particles is further stabilized as the electrolyte concentration becomes higher, and the
ability of electrolytes to stabilize the contact follows the Hofmeister series for both cations and
593
R.Akiyama, et al.
anions. The ability of electrolytes to screen the electrostatic repulsion between negatively charged
colloidal particles is strongly dependent on cation species and also follows the Hofmeister series.
These are consistent with the experimental observations known for the conformational transition
of biopolymers with many negatively charged groups and the salting-out phenomenon for neutral
proteins. As for the contact of positively charged colloidal particles, however, our result indicates
that it is further stabilized as the anion size decreases, which conflicts with the experimental
data supporting the inverse Hofmeister series. This contradiction stems from the simplification of
the solvent model. In fact, the inverse Hofmeister series was well reproduced by Kinoshita and
Harano [17] who employed a more detailed model for water, SPC/E, in which H and O sites carry
partial charges. In this model the hydration of ions can be described. Moreover, the strength of the
hydration becomes stronger as the ionic size decreases. In the previous paragraph, we mentioned
that more counter charged ions tend to bind to the charged colloidal particles as the size of the
counter charged ions become smaller (factor 1). However, for the counter charged ions to reach
the binding, they should be dehydrated. This dehydration penalty becomes larger as the ionic size
decreases (factor 2). The screening ability of counter charged ions is determined by the interplay of
the two opposing factors. Kinoshita and Harano showed that factor 1 dominates for cations whereas
factor 2 is more important for anions. Our simple model in the present study is not capable of
elucidating factor 2.
4. Concluding remarks
This article deals with the interaction (PMF) between like-charged colloidal particles immersed
in a simple model of aqueous electrolyte solution using the OZ-HNC theory. The solvent molecules
are modeled as neutral hard spheres, and colloidal particles, anions, and cations are taken to be
charged hard spheres. Uncharged (neutral) colloidal particles are also considered. The Coulomb
potentials between ion-ion, ion-colloidal particle, and colloidal particle-colloidal particle pairs are
divided by the dielectric constant of water. This simple model is employed to exclusively address the
effects due to solvent granularity (i.e., the translational motion of solvent molecules) and serves as a
first-step improvement of the so-called primitive model. The effects of electrolyte concentration and
1-1 electrolyte species (NaCl, KCl, CsCl, NaBr, and NaI) on the interaction between uncharged,
positively charged, and negatively charged colloidal particles are analyzed and discussed in detail.
When the electrolyte concentration is sufficiently low, the conventional picture that like-charged
colloidal particles interact through the electric double layer is acceptable in the sense that the
interaction is always repulsive. When the electrolyte concentration is sufficiently high (e.g., higher
than 1 · 10−1 M), however, regions where the interaction is attractive appear. In particular, the
interaction near the contact can be strongly attractive with the stabilization of the dimer of colloidal
particles though the direct van der Waals attraction is not incorporated in the model. The attractive
interaction in our model is induced by the solvent granularity.
Several authors [5–15] have emphasized the crucial importance of attractive interactions driven
by the translational motion of solvent molecules, namely solvent granularity, in biological systems:
It plays a crucial role in protein folding, protein aggregation, and in a variety of self-assembling
processes. The only force which could predominate over the entropically driven interaction, is the
strong electrostatic force between highly charged groups in biomolecules. However, in these cases,
the electrolyte concentration is about 0.15 M, which is sufficiently high to screen the electrostatic
force. In other words, the electrolytes such as NaCl in biological systems are likely to play a major
role in the entropically induced interactions.
The stabilization of contact between uncharged colloidal particles is progressively enhanced
as more electrolytes are added to the solution, and this effect becomes stronger as the cation
or anion size decreases (result 1). The ability of electrolytes to screen the electrostatic repulsion
between negatively charged colloidal particles is strongly dependent on cation species and becomes
higher as the cation size decreases (result 2). The ability of electrolytes to screen the electrostatic
repulsion between positively charged colloidal particles becomes higher as the anion size decreases
while its dependence on anion species is much weaker (result 3). Only result 3 is inconsistent with
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Interaction between charged colloidal particles
the analysis made by Kinoshita and Harano [17] who employed a more detailed model for water,
and with the experimental observations known for conformational transition of biopolymers. More
counter charged ions tend to bind to the charged colloidal particles as the size of the counter
charged ions becomes smaller (factor 1). However, the counter charged ions should be dehydrated
to reach the binding. This dehydration penalty becomes larger as the ionic size decreases (factor
2). The ability of counter charged ions to enable stronger binding and screening of the colloidal
charges is determined by the interplay of the two opposing factors. Kinoshita and Harano showed
that factor 1 dominates for cations whereas factor 2 is more important for anions. Our simple
model in the present study is not capable of elucidating factor 2.
Acknowledgements
We would like to acknowledge thought-provoking discussions on solution chemistry with Prof.
Fumio Hirata. We thank Mr. Y. Karino for fruitful discussions and our local network adminis-
tration. This work was supported by grants from the Japanese Ministry of Education, Science,
Sports and Culture (No.18031028 and No. 15076203), by NAREGI Nanoscience Project, and by
the Sumitomo Foundation.
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Взаємодiя мiж одноiменно зарядженими колоїдними
частинками у водних розчинах електролiтiв: притягальна
компонента, що виникає за рахунок дискретностi
розчинника
Р.Акiяма1,2, Н.Фужiно1,К.Канеда1, М.Кiношiта3
1 Хiмiчний факультет, Кюшю унiверситет Фукуока, Японiя
2 Iнститут молекулярних наук, м. Оказакi, Японiя
3 Iнститут прикладної енергетики, унiверситет м. Кiото, Японiя
Отримано 8 серпня 2007 р., в остаточному виглядi – 10 жовтня 2007 р.
Потенцiал середньої сили мiж одноiменно зарядженими колоїдними частинками, помiщеними у во-
дний розчин електролiту, вивчається методами теорiї iнтегральних рiвнянь. Молекули розчинника
моделюються твердими сферами, а iони та колоїднi частинки розглядаються як зарядженi твердi
сфери. Кулонiвськi взаємодiї мiж iонами та колоїдними частинками включають дiелектричну сталу
води. Ця проста модель використана для врахування ефектiв дискретностi розчинника, якими не-
хтують в примiтивнiй моделi. Ван дер Ваальсовим притяганням, яке є суттєвою рисою теорiї ДЛВО,
в данiй моделi нехтуються. Тим не менше, притягальнi областi виникають в потенцiалi середньої
сили при достатньо високих iонних концентрацiях. Зокрема, взаємодiя на малих вiдстанях є доста-
тньо притягальною, щоб стабiлiзувати колоїдну систему. Ця поведiнка виникає за рахунок ефектiв
трансляцiйного руху молекул розчинника.
Ключовi слова: колоїдна частинка, розчин електролiту, потенцiал середньої сили, теорiя
iнтегральних рiвнянь, теорiя ДЛВО, дискретнiсть розчинника
PACS: 82.35.Rs, 82.35.Pq, 82.70.Dd, 83.80.Hj
596
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