Ion size effect on colloidal forces within the primitive model
The effect of ion size on the mean force between a pair of isolated charged particles in an electrolyte solution is investigated using Monte Carlo simulations within the framework of the primitive model where both colloidal particles and small ions are represented by charged hard spheres and the...
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irk-123456789-1196012017-06-08T03:06:09Z Ion size effect on colloidal forces within the primitive model Ravindran, S. Wu, J. The effect of ion size on the mean force between a pair of isolated charged particles in an electrolyte solution is investigated using Monte Carlo simulations within the framework of the primitive model where both colloidal particles and small ions are represented by charged hard spheres and the solvent is treated as a dielectric continuum. It is found that the short-ranged attraction between like-charged macroions diminishes as the diameter of the intermediating divalent counterions and coions increases and the maximum attractive force is approximately a linear function of the counterion diameter. This size effect contradicts the prediction of the Asakura-Oosawa theory suggesting that an increase in the excluded volume of small ions would lead to a stronger depletion between colloidal particles. Interestingly, the simulation results indicate that both the hard-sphere collision and the electrostatic contributions to the mean force are insensitive to the size disparity of colloidal particles with the same average diameter. Вплив розмірів іонів на потенціал середньої сили між парою ізольованих заряджених частинок досліджується за допомогою Монте Карло симуляцій в рамках примітивної моделі, в якій як колоїдні частинки, так і малі іони є представлені зарядженими твердими сферами, а розчинник трактується як неперервне діелектричне середовище. Показано, що короткодіюче притягання між однаково зарядженими макроіонами зменшується, якщо діаметр двовалентних контраіонів та коіонів збільшується, а максимальна сила притягання є приблизно лінійною функцією діаметру контраіонів. Такий вплив розмірів протирічить передбаченням теорії Асакури-Оосави, які вказують на те, що збільшення виключеного об’єму малих іонів веде до сильнішого притягання між колоїдними частинками. Відмічено, що комп’ютерні результати показують, що як твердосферні зіткнення, так і вклади електростатичних взаємодій до потенціалу середньої сили є нечутливими до відмінностей у розмірах колоїдних частинок, якщо середній розмір є однаковим. 2005 Article Ion size effect on colloidal forces within the primitive model / S. Ravindran, J. Wu // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 377–388. — Бібліогр.: 54 назв. — англ. 1607-324X PACS: 61.20.Ja, 82.70.Dd DOI:10.5488/CMP.8.2.377 http://dspace.nbuv.gov.ua/handle/123456789/119601 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The effect of ion size on the mean force between a pair of isolated charged
particles in an electrolyte solution is investigated using Monte Carlo simulations
within the framework of the primitive model where both colloidal
particles and small ions are represented by charged hard spheres and the
solvent is treated as a dielectric continuum. It is found that the short-ranged
attraction between like-charged macroions diminishes as the diameter of
the intermediating divalent counterions and coions increases and the maximum
attractive force is approximately a linear function of the counterion diameter.
This size effect contradicts the prediction of the Asakura-Oosawa
theory suggesting that an increase in the excluded volume of small ions
would lead to a stronger depletion between colloidal particles. Interestingly,
the simulation results indicate that both the hard-sphere collision and
the electrostatic contributions to the mean force are insensitive to the size
disparity of colloidal particles with the same average diameter. |
| format |
Article |
| author |
Ravindran, S. Wu, J. |
| spellingShingle |
Ravindran, S. Wu, J. Ion size effect on colloidal forces within the primitive model Condensed Matter Physics |
| author_facet |
Ravindran, S. Wu, J. |
| author_sort |
Ravindran, S. |
| title |
Ion size effect on colloidal forces within the primitive model |
| title_short |
Ion size effect on colloidal forces within the primitive model |
| title_full |
Ion size effect on colloidal forces within the primitive model |
| title_fullStr |
Ion size effect on colloidal forces within the primitive model |
| title_full_unstemmed |
Ion size effect on colloidal forces within the primitive model |
| title_sort |
ion size effect on colloidal forces within the primitive model |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119601 |
| citation_txt |
Ion size effect on colloidal forces within the primitive model / S. Ravindran, J. Wu // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 377–388. — Бібліогр.: 54 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT ravindrans ionsizeeffectoncolloidalforceswithintheprimitivemodel AT wuj ionsizeeffectoncolloidalforceswithintheprimitivemodel |
| first_indexed |
2025-07-08T16:14:53Z |
| last_indexed |
2025-07-08T16:14:53Z |
| _version_ |
1837096024089296896 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 2(42), pp. 377–388
Ion size effect on colloidal forces within
the primitive model
S.Ravindran, J.Wu∗
Department of Chemical and Environmental Engineering
University of California, Riverside, CA 92521, USA
Received October 20, 2004
The effect of ion size on the mean force between a pair of isolated charged
particles in an electrolyte solution is investigated using Monte Carlo sim-
ulations within the framework of the primitive model where both colloidal
particles and small ions are represented by charged hard spheres and the
solvent is treated as a dielectric continuum. It is found that the short-ranged
attraction between like-charged macroions diminishes as the diameter of
the intermediating divalent counterions and coions increases and the max-
imum attractive force is approximately a linear function of the counterion di-
ameter. This size effect contradicts the prediction of the Asakura-Oosawa
theory suggesting that an increase in the excluded volume of small ions
would lead to a stronger depletion between colloidal particles. Interesting-
ly, the simulation results indicate that both the hard-sphere collision and
the electrostatic contributions to the mean force are insensitive to the size
disparity of colloidal particles with the same average diameter.
Key words: colloids, electrostatic interactions, Monte Carlo
PACS: 61.20.Ja, 82.70.Dd
1. Introduction
Talking about how a pair of particles interact with each other in a solvent, prob-
ably no other person knows it better than Doug Henderson, a contemporary legend
in the field of liquid-state theory. A thorough review of his contributions to this area
is not attempted here because that would easily take several volumes! Thanks to his
mounds of classical publications, [1–23] we now have much improved understanding
of colloidal forces, in particular, regarding the effects of solvation, electrostatics, ex-
cluded volume, and interactions between colloidal particles in a polymerizing solvent.
Theoretical description of the electrostatic interactions between a pair of charged
particles in an electrolyte solution often starts with the Poisson-Boltzmann equation,
a mean-field theory that neglects the size effect and the correlation of small ion distri-
∗To whom all correspondence should be addressed. E-mail: jwu@engr.ucr.edu
c© S.Ravindran, J.Wu 377
S.Ravindran, J.Wu
butions [7]. The Poisson-Boltzmann equation, as well as its approximated analytical
solution given by the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory (1940s),
provides a semi-quantitative description of the solvent-mediated potential between
charged colloidal particles. However, as early as 1930s, the mean-field description
was questioned by prominent researchers including Langmuir and Kirkwood [24].
Langmuir argued that because of the intervention of small ions of opposite charge
or counterions, there could exist an electrostatic attraction between like-charged par-
ticles at certain solution conditions. The electrostatic attraction between like charges
directly contradicts the prediction of the Poisson-Boltzmann equation, which states
that the average solvent-mediated electrostatic potential between two macroions of
similar charge is always repulsive. Kirkwood and Schumaker developed an analytical
expression for the attractive force and demonstrated that the fluctuations in the net
charges of macroions result in an attractive potential. In the late 1960s, Oosawa
demonstrated that the correlated fluctuations of the small ion distributions around
two macroions could lead to an electrostatic attraction [25]. Moreover, Sogami and
Ise proposed a variation of the DLVO theory based on the Gibbs free energy. As
far as it predicts a ubiquitous long-range attraction between like charges [26], the
Sogami-Ise theory has ensured lots of controversies ever since it was published.
Molecular simulations were applied to the study of interaction between infinitely-
large, parallel charged plates first by Guldbrand et al. [27], and a few years later by
Valleau et al. [28]. Both groups indicated that in contrast to the prediction of the
DLVO theory, the electrostatic force between like-charged plates turns out to be al-
most everywhere attractive in the presence of multivalent counterions. Electrostatic
attraction between like-charged macroions of other geometries such as cylinders and
spheres has also been demonstrated in recent years using either Monte Carlo or
molecular dynamic simulations [29–39]. These simulation results have settled down
substantial controversies in the colloid literature on attraction between like charges
and more importantly, provide benchmark data for the development of an improved
theory for colloidal interactions.
Amid numerous theoretical efforts toward an improved theory for colloidal forces,
the integral equation approaches pioneered by Henderson and coworkers might be
most promising [1,5,6,9,11,14,16,17,20–22,40–44]. Some other methods have also
been proposed in the literature. For instance, both anisotropic hypernetted-chain
theory [45] and density functional theory [46,47] have been applied to the calculati-
on of the interaction force between charged plates. After much computational effort,
both theories were found in good agreement with simulation results. However, due to
the numerical difficulties, extension of similar approaches to macroions of other ge-
ometry has not been reported. For interaction between cylindrical colloidal particles,
Ha and Liu proposed a field-theory approach taking into account the electrostatic at-
traction. These authors indicated that, as elucidated by earlier investigations [25,27],
the attraction between like charges resembles the van der Waals attractions between
spherical molecules of nonpolar species such as argon [48]. However, the field-theory
calculation was followed by criticisms of Levin and others [49,50]. For interaction
between spherical macroions, Netz and Orland proposed a correction term to the
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Ion size effect on colloidal forces within the primitive model
DVLO theory based on a third-order cumulant expansion [51]. This additional term
introduces attractive contributions but vanishes for interaction between two isolated
colloidal particles in an electrolyte solution.
Unlike the DLVO theory, most modern approaches for representing the colloidal
forces explicitly take into account both the size and valence of small ions. Conse-
quently, the overall potential between colloidal particles includes at least two parts,
one due to the particle size and the other due to the electrostatic interactions.
Whereas most previous investigations have been focused on the electrostatic effects,
much less attention is given to the role of ion diameter on the overall force between
colloidal particles and the interplay between the excluded volume and electrostatic
interactions of intermediating small ions. For an electrostatically neutral system, a
fair understanding of the excluded-volume effect is provided by the phenomenologi-
cal theory of Asakura and Oosawa (AO) based on a simple geometry argument [52].
The AO theory predicts that the excluded-volume effect introduces an effective at-
traction between colloidal particles within the range of the size of the intermediating
species. For colloidal particles surrounded by neutral species, the prediction of the
AO theory is in semi-quantitative agreement with more recent molecular simulati-
ons [53]. While the entropically driven excluded effect is well understood at least
for uncharged systems, it is much less apparent if the intermediating species are
also charged or coupled with other force fields. In principle, both integral equation
theories and density functional theories are capable of treating the excluded-volume
effects and electrostatics separately. Regrettably, most previous investigations as-
sumed that the diameter of small ions is fixed (round 4.25 Å), appropriately equal
to the diameter of a solvated ion in water. Quite recently, the ion size effect on the
diffuse potential and ion distributions of planar electric double layers was investi-
gated using Monte Carlo simulation and the HNC/MSA integral equation theory
[54]. A major conclusion from this work is that for realistic hydrated ions of 2:1
electrolytes, the decrease of the diffuse potential with an increasing surface charge
density as predicted by the integration theories was not observed in Monte Carlo
simulations. A systematic investigation on the effect of ion size on colloidal forces
was not explored.
In this work, we report the results from Monte Carlo simulations examining
the effect of the excluded volume of intermediating small ions on colloidal forces
within the primitive model. Special attention is given to the interplay between the
electrostatic and excluded-volume contributions to the overall forces.
2. Simulation techniques
We consider the mean force between a pair of uniformly charged colloidal par-
ticles dispersed in a symmetric electrolyte solution containing divalent counterions
and coions. In the primitive model, the colloidal particles as well as the counteri-
ons and coions are represented by hard spheres of different diameters and valences,
and the solvent is represented by a dielectric continuum. The NVT-ensemble Monte
Carlo simulation is used to calculate the collision and electrostatic forces between
379
S.Ravindran, J.Wu
the isolated colloidal particles intermediated by small ions [33]. Specifically, for each
separation of colloidal particles, NVT simulation is applied to a cubic simulati-
on box containing 80 counterions, 60 coions, and two identical negatively charged
macroions. The macroions are placed along the box diagonal in order to minimize
the boundary effect. The three-dimensional periodic boundary conditions are appli-
ed to all simulations and the long-ranged Coulomb interactions are calculated using
the Ewald sum method. Due to the charge of macroions, there is a small difference
in the concentrations of counterions and coions in order to keep the condition of
electrostatic neutrality.
Table 1. The diameters of small ions (σs) and macroions (σ1 and σ2) and the
valence of macroions for the various runs performed to calculate the force between
two macroions with charge fixed at the center.
Runs σ1( Å ) σ2( Å ) Z1 Z2 σs ( Å )
A 20 20 –20 –20 2
B 20 20 –20 –20 4
C 20 20 –20 –20 5
D 20 20 –20 –20 6
E 18 22 –18 –22 5
F 20 20 –20 –20 5
G 16 24 –16 –24 5
Throughout this work, the concentration of electrolyte is fixed at C = 0.06 M
and the permittivity “ε” of the solvent and solutes corresponds to that of water
at ambient conditions. The box length is fixed at 118.4 Å, approximately 20 times
of the Debye length of small ions (≈ 6.2 Å). As a result, the correlation between
macroions in image simulation boxes is negligible, i.e., the system mimics the in-
teraction between colloidal particles at an infinite dilution. Table 1 summarizes the
simulation parameters for the different conditions investigated in this work. Further
details on the implementation of the Monte Carlo simulation are provided in our
previous publications [30,33–35,37].
In calculating the average colloidal forces for each separation of colloidal particles,
we assume that the system attains equilibrium after approximately 5 ·105 moves per
particle and up to a million moves per particle are used for sampling the hard-
sphere collision and electrostatic forces. All calculations were performed using IBM
SP RS/6000 computers from the National Energy Research Scientific Computing
Center (NERSC).
3. Results and discussion
Figure 1a presents the overall (F ), hard-sphere (Fhs), and electrostatic (F
cc
) mean
forces between a pair of macroions of valance ZM = −20 and diameter σ = 20 Å
in an aqueous solution of 2:2 electrolyte solution. Here lB stands for the Bjerrum
380
Ion size effect on colloidal forces within the primitive model
(a) (b)
Figure 1. (a) The total force (◦) as well as the hard-sphere contribution (�)
and the electrostatic contribution (M) for the interaction between two identical
macroions in a 2:2 electrolyte solution (Run C). (b) The relative internal energy
as a function of the center-to-center separation between macroions. All lines are
for the guidance of the eye.
length, defined as the average separation between two unit charges where the elec-
trostatic potential is equal to the thermal energy kT. For an aqueous solution at
ambient condition, lB = 7.8 Å. The divalent counterions and coions have the same
diameter of 5 Å. As observed in our previous simulations, [30,33–35,37] the overall
force between like-charged macroions is repulsive near the contact, turns into at-
traction when the surface-to-surface distance is of the diameter of the counterions,
and monotonically decays at larger separations. The maximum attraction occurs
approximately at a surface-to-surface separation that is sufficient to accommodate
a monolayer of counterions.
The appearance of the short-ranged attraction is due to the correlated fluctu-
ations of small ion distributions that are ignored completely in the DLVO theory.
While the origin of this attraction is now well understood, at least qualitatively,
and the force profile or potential of mean force can be semi-quantitatively captured
using the aforementioned integral equation theories or density functional theories,
quantitative descriptions of the hard-sphere collision and electrostatic mean forces
for spherical macroions in an electrolyte solution remains theoretically challenging
even within the primitive model. Different from the predictions of the AO theory, the
hard-sphere collision is purely repulsive at essentially all ranges of inter-particle sep-
arations. For interactions between colloidal particles surrounded by neutral species,
the intermediating species distribute almost uniformly around the colloidal particles
and an entropic depletion attraction is induced due to the overlap of the depletion
layers of the individual particles. For macroions dispersed in an electrolyte soluti-
on, however, the counterions as well as coions are crowded in the space between
the macroions, leading to an osmotic repulsion. Intuitively, the short-ranged hard-
sphere repulsive force can be understood by the fact that a higher concentration of
small ions in the space between macroions causes more collisions of small ions, and
thereby driving the macroions apart.
381
S.Ravindran, J.Wu
The electrostatic contribution of the mean force is also repulsive near the contact
and turns into attraction at small separations of macroions. The strong repulsion
near the contact of macroions is mainly due to the bare charge effect and the short-
ranged attraction is due to the fluctuation of charge distributions. The effect of
electrostatic energy on the mean forces is most clearly seen in the plot of relative in-
ternal energy as a function of distance, ∆E(r), as depicted in figure 1b. Here ∆E(r)
is defined as the internal energy of the system when the two macroions are kept
at a center-to-center distance r minus that of the system when the macroions are
infinitely apart. Apparently, only the electrostatic interactions directly contribute
to ∆E(r). Interestingly, ∆E(r) is always negative at all separation of macroions,
indicating that small separation of macroions is energetically favorable but entrop-
ically unfavorable. It other words, the repulsive potential between similar charged
macroions is mainly due to the entropy penalty for the localization of small ions.
(a) (b)
(c) (d)
Figure 2. The total (a), hard-sphere (b), and electrostatic (c) force profiles be-
tween two identical macroions in a 2:2 electrolyte solution with microions of
different sizes. (Runs A-D) (d) The correlation between the maximum attractive
(or minimum) force between macroions and the diameter of surrounding small
ions.
A comparison of the force and energy profiles indicates that there is a direct cor-
relation between the ranges of attractive and repulsive mean electrostatic forces and
the slope of ∆E(r). It suggests that the mean-electrostatic force between macroions
is mainly determined by the energy effect. The attraction between like charges is
382
Ion size effect on colloidal forces within the primitive model
most probable when the macroions are dispersed in electrolyte solutions containing
multi-valent counterions. For instance, at ambient conditions of aqueous solutions,
the mean force between like charged particles is always repulsive if the counterions
are monovalent and the strength of attraction increases with the counterion valence.
Figure 2a depicts the mean forces between macroions for 4 different diameters
of the intermediating counterions and coions while all other solution conditions re-
main the same. As the diameter of small ions increases, the mean force becomes
more repulsive near the contact and the shallow attractive well shifts outward and
eventually diminishes. Due to the strong coupling of the electrostatic and excluded-
volume interactions, the size effect is entirely different from the prediction of the
AO theory. Intuitively, one expects that the stronger repulsion at short separations
should be attributed to the higher probability of collision as the size of small ions
grows. However, a close inspection of figures 2b and 2c for the collision and elec-
trostatic contributions of the mean forces indicates that the size mainly affects the
range rather than the strength of the repulsion. The electrostatic forces exhibit a be-
havior similar to that for the overall forces. Most surprisingly, it appears that there
is a linear correlation between the maximum attraction and the size of small ions,
as shown in figure 2d. We suspect that the linear relation is due to the relatively
small change of the ion diameter investigated in this work.
Figure 3. The force profiles between macroions of different diameters in a 2:2
electrolyte solution.(Runs E-G) Here σ stands for the average diameter.
Intuitively, one may expect that at similar solution conditions, the average force
would be sensitive to the size and charge density of interacting macroions. To test
383
S.Ravindran, J.Wu
this speculation, we calculate the mean forces between three pairs of macroions of
different sizes but the same average diameter (Runs E-G). Figure 3 presents the
force profiles as well as the excluded volume and electrostatic contributions. Here
the diameter of divalent small ions is fixed at 5 Å. Apparently, the overall force
profile as well as the individual excluded volume and electrostatic contributions are
essentially independent of the disparity in macroion size and charge as long as the
average values are fixed. Even though the behavior is consistent with the prediction
of the DLVO theory, this observation is somewhat counterintuitive because even at
the same average diameter and charge the total surface area of macroions varies with
the disparity of the macroion pairs, leading to the change in both the probability of
collision and the charge density at the macroion surface. Therefore, we suspect that
the results might be quite different when the macroions are highly asymmetric in
both valence and size.
4. Conclusions
Monte Carlo simulations are applied to the investigation of the effect of ion size
on the mean force between a pair of macroions in various electrolyte solutions. We
find that both the electrostatic and hard-sphere collisions are sensitive to the ion
size. In divalent electrolyte solutions, the short-ranged attraction between macroions
diminishes as the diameter of intermediating small ions increases. This size effect
is totally different from the prediction of the AO theory, which is applicable only
to interactions between colloidal particles surrounded by neutral species. Within a
small variation of the size of intermediating small ions, the maximum attractive force
between like charges linearly correlates with the ion diameter. This suggests that
the negligence of the ion size will enhance the attraction due to the fluctuation of
small ion distributions. The simulation results also suggest that at similar solution
conditions, the mean force between like-charged macroions appears relatively insen-
sitive to the size and charge disparity of macroions as long as the average values are
fixed.
Acknowledgement
The financial support from the University of California Energy Institute and
generous allocation of computing time from the National Energy Research Scientific
Computing Center (NERSC) are gratefully acknowledged.
384
Ion size effect on colloidal forces within the primitive model
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S.Ravindran, J.Wu
Вплив іонних розмірів на взаємодії між колоїдами в
рамках примітивної моделі
С.Равідран, Дж.Ву
Університет Каліфорнії, Ріверсайд, США
Отримано 20 жовтня 2004 р.
Вплив розмірів іонів на потенціал середньої сили між парою
ізольованих заряджених частинок досліджується за допомогою
Монте Карло симуляцій в рамках примітивної моделі, в якій як
колоїдні частинки, так і малі іони є представлені зарядженими
твердими сферами, а розчинник трактується як неперервне діелек-
тричне середовище. Показано, що короткодіюче притягання між
однаково зарядженими макроіонами зменшується, якщо діаметр
двовалентних контраіонів та коіонів збільшується, а максимальна
сила притягання є приблизно лінійною функцією діаметру кон-
траіонів. Такий вплив розмірів протирічить передбаченням теорії
Асакури-Оосави, які вказують на те, що збільшення виключеного
об’єму малих іонів веде до сильнішого притягання між колоїдними
частинками. Відмічено, що комп’ютерні результати показують, що як
твердосферні зіткнення, так і вклади електростатичних взаємодій
до потенціалу середньої сили є нечутливими до відмінностей у
розмірах колоїдних частинок, якщо середній розмір є однаковим.
Ключові слова: колоїди, електростатичні взаємодії, Монте Карло
PACS: 61.20.Ja, 82.70.Dd
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