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The interaction between charged aggregates in electrolyte solution. A Monte Carlo simulation study
\‘olurnc
108. number
CHEMICAL
6
THE INTERACTION A MONTE
CARLO
Bo SVENSSON
BETWEEN SIMULATION
27 April
17 July
LETTERS
AGGREGATES
IN ELECTROLYTE
1984
SOLUiION.
STUDY
and Bo JGNSSON
Yh_vsical Chetttisrr_r 2, Chetttical Rcceivcd
CHARGED
PHYSICS
Cettler.
198-l: in finnl form
P-0. Box 740. S-220
07 Lund. Sweden
26 hfnp 1984
T11c .\lonre Carlo simulalion ~ecbniquc 113sbeen used to calculate rhe electrostaric force acting upon a charged aggregate ~)utkk 3 similsrly charged wall. Contrary to intuition and e.xisIing electrostatic theories, the force is found to be attractive
for WIX rralktic values of the parameters dclerminin_e the system. High surface charge density, low temperature, low relaIIW permitlivity 2nd polyvslent neutralizing counterions are all factors that favour a net strraction between the wall and rl~c xxegate. In some cases the rcsuIIin_e electrostatic attractive force is found :o be an order of magnitude ordinary van dcr \V;lals artractiun applied in ibe DLVO theory of colloidal stability. Tbeartracrive interaction bcin:: due IU rorrcl;ltions between the counterions in the electric double layers.
larger
than
the
is inrerpreled
as
1. Introduction
wall and a spherical aggregate with charge density. The wall-aggregate
Intcracrions between charged aggregates play an nnpor~3n1 role in a variety ol’chemical systems. For tt_xainplc. the eleclrostatic interaction is a main com-
been calculated directly in the simulations and found to contain an attractive component, which is the dominating one in the “strong coupling regime”, that is with high surface charge density, multivalent countcrions, low E,Tvalues and at short distances_
poncnt in the stability ofcolloids [I,?] as well as of diNeI WI phases formed by amphiphilic molecules I_:]_ .L\~~slytical theories dealing with elecrrostatic inIcr2srions in solution usually focus on the ionic interxrions and assume a con0nuum model for the solWIII. In rile l’oisa~~l--Boltzn~ann (PB) equation. for example. 211 additional mean field approximation is
mtroduccd. sipccies
where the correlation between the ionic is neglccred (41. This latter approsimarion can
be tested via Xlonle Carlo (MC) simulations, which provide the exact answer within the given continuum model. Recent MC simulations have shown serious
dsficiencirs in the I’B approximation in systems with divalcnr ions [S.O]. In fact, the net electrostatic force brr\vecn rwo charged surfaces, with an intervening clcctrulytc solution, is sometimes found to be attractive [ 71. This is contrary IO the prediction of the DLVO theory, where the net electrostatic force is always repulsive
[ I,?].
In riie present
simulationsof 5so
work we report the results from MC rhe interaction betweena charged planar
the same surface mean force has
2. Model and methods The model
system
consisted
of two infinite
planar
walls, one of which had a surface charge density u, separated by a distance Z,,. The intervening diclcctric continuum contained the mobile co- and countcrions and spherical aggregates with the same surface charge density as the charged wall (see fig. 1). In the presence of both co- and counter-ions, it is necessary to assign a radius to the ions, while in a salt-free sysrem the counterions can be represented by point charges. The surface charge density on the wall was treated both as uniformly smeared out (model I) and as distinct charges(modc1 II), while the aggregate charge density was always uniformly smeared out. A uniform relative permittivity er was assumed, that is dielectric saturation 0 009-2614/84/S (North-Hallxnd
and image charge 03.00 Phv
effects
were neglected_
OElsevicr
Science
Pllhlichino
l%ririnn~
Publishers
B.V.
27 July 1984
CHEhlIICAL PHYSICS LETTERS
Volume 108, number 6
dard hard-sphere 0
oj
0
0
8 Q 0
Q
i !
Ra
Q---
0
0
i
!
0
0 0
0
I i i !
za
where the wall-aggregate (I
= -
wa
w,e
47TEoE,
Z WW
lation where all particles, including the aggregate_ are free to move in between the charged and uncharged walls. The aggregate concentration profile is then easily obtained during the simulation and hence also the potential of mean force. However, it turned out to be quite time-consuming to calculate the concentration profile for the aggregate, when the surface charge density exceeded ~0.16 C mm2_ This is probably due to the slow “diffusion” of the aggregate through configuration space during the course of the simulation. To circumvent this problem, the aggregate was held fried at different wall-aggregate separations 2, and the average force acting upon it was calculated explicitly in the simulation. In order to facilitate the force evaluation, a repulsive r-l2 term was introduced into the ion-aggegate
interaction,
instead
of a stan-
ener,,g
is
(?a’ + Z~)‘l” + a
( ( Saln
arcsin
(a’+Z.;) 3 11’-
(
(a’+Z,‘)’
1 +;
)
(21 -1)
and similarly for the wall-ion interaction U,vi_ The ion-ion interaction followed the primitive model, 7 rii>Ri+Rj, Uij = qiqje-/4iifTgEr’ij, =CO7
The MC simulations were performed in the NVT ensemble using the standard algorithm of Metropolis et al. [S]. In order to make the simulations more tractable, the aggregate concentration was chosen so that the MC box contained only one aggregate_ The size of the IMC box was 2a X 2a X Z,\,,\,= 64 X 64 X 128 A3 and periodic boundary conditions were applied in the two directions parallel to the walls. The system was equilibrated during IO4 configurations/ particle and at least the same number was generated for the analysis_ The most straightforward way to calculate the average force acting on the aggregate is to perform a simu-
interaction
a4 _ Zz _ 2a’Zz - 22, I
Fig. 1. The model system. Ions and a charged spherical aggregate between an equally charged wall and an uncharged wall. The solvent is regarded as a continuum and the relative permittivity is everywhere constant.
cut-off_ The total electrostatic
for model I can then be written:
rij~Ri+R-
I-
(3)
Here qi and Ri are the ion valency and the ionic radius, respectively. The ion-aggregate interaction is a slight modification of eq. (3):
(4 where R, defines the azregate radius, which was 12.8 .A throughout the simulations. In model II iJ_, U,\+and the interaction between the wall charges themselves (U,,,,\,) all followed the primitive model, eq. (3). The average force acting upon the aggregate is then simply the derivative F, =
-au,,,iaz, = -au,_iaz, -
~iKJj,laZ~.
(5)
One can note here that the wall-agiregate force is temperature independent in model I, with the smeared out surface charge density (see eq. (2)). In the second model, the force between the aggregate and the surface charges will be a Boltzmann average and depend on the correlation between surface charges, mobile ions and aggregate_ The minimum image approximation was used and all interactions were truncated at the edge of the MC box. This approximation was tested through simulations on larger systems, but no significant changes were found upon enlarging the system by a factor of four. To check the effects of the boundaries, simulationswere also performed
without periodic boundary conditions. 581
Volume
108, number
CHEMICAL
6
Instead, the system was confined to the MC box with the same dimensions as previously_ All interactions within the paralielepiped were taken into account. The aggregate was placed on the symmetry axis and the walls of the parallelepiped were treated as impenetrable. The results obtained in this model were virtually the same as with periodic boundary conditions_
3. Results In a recent
showed infinite
blCl simulation
that the electrostatic planar double layers
Guldbrand
et al. [7]
interaction between two becomes attractive in the
limit ofhigh surface charge density and divalent counterions. An interesting question is whether this is a property of this specific system or if it is a general phenomenon appearing also in other systems of different geometry. Part of the answer is found in table 1 where the aggregate net force is shown as a function of the wall-aggregate distance. Sufficiently far from the charged wall, the electrostatic force is always repulsive irrespective of the surface charge density. At intermediate distances and low surface charge density \ve still find a net repulsive force, as expected from the DLVO theory. However, when the charge density
Table
1
3). xting on rtn rrz_Srcgntc in ner clectroststic fbrcc units of 1013 N/moI aggregnre. Standard deviations are estimrrtcd IO 0.1. 0.1 and 0.2 X lOI N/mu1 aggregate for 0.0625, The corresponding disper0.125 and 0.35 C mm2 respectively. sion force. with rr iizrmllker constant .-I = I 0m2’ N m is shown in tlte last column (eq. (6)). Divalent counterions, cr = 77.3 2nd T= 301 I;
The avcr:~gc
becomes ~0. I6 C m-3,
Surface 0.0635
80 50 32 2-1 30 18 16
0.1) I 0.0 1 0.06 0.15 0.36 1.1
a) Negative
charge
density
0.115
0.0 1 0.05 0.0 1 -0.11 0.13 0.6’ 2.5
(C mm2)
a change in the temperature, as is evident from an analysis of the dimensions (see also ref. [7]). Model I may be criticized for being unbalanced in that it allows for the correlation between the mobile counterions but considers the wall charges as completely uncorrelated, that is uniformly smeared out. However, explicit simulations with er = 77.3 and di-
valent counterions show that the replacement of the smeared out surface charge with distinct charges, nobile in two dimensions. does not alter the aggregate force significantly. It might well be that for a sufficiently charged system there will appear a difference
sign rneilns an artmctive
aggre<c.
furce
-0.0017 -0.00-18 -0.0 I8 -0.059 -0.16 -0.34 -1.01 bct\veen
wall and
.
FE3
1
-10
0.03 0.06 -0.06 -0.64 -0.88 0.37 4.6
force
The attractive component of the force dominates at intermediate distances, while the total electrostatic force again becomes strongly repulsive at very short distances. These simulations were performed using model I with divalent point charges as counterions but without salt. In the case of monovalent counterions the force is everywhere repulsive for this set of parameters. However, as indicated earlier, the origin of the attractive force component is the correlation between the mobile ions. In the case of perfect correlation, that is a lattice, it is easily seen that the force will be attractive. Thus, one way to make the force more attractive should be to increase the electre static interaction by, for example, reducing the dielectric permittivity or the temperature. Fig. 2 shows indeed that the force becomes attractive also in the presence of monovalent counterions when the dielectric permittivity is less than ~40. The same is true for
- a(ldisp/az,
0.35
the net electrostatic
attractive.
becomes
0
Z,(X)
27 July 1984
PHYSICS LETTERS
1
___________i-___________~_____________~___________.
A
-I\
*
x
x
i 20
40
1
1
60
1
1
80
1
I
er
Fig. 7. The aggregate net force in units of I 013 N/moI aggregate as a function of the relative permittivity, 2, = 20 A,(r) monovalent and (X) divalent counterions. The standard deviation increases in both cxes from 0.7 to 1.2 X lOI3 N/moI ns cr is decreased from 77.3 to 10.
CHEhlICAL
Volume iOS, number 6
PHYSICS LETTERS
depend among other things on the polarisabihty of the media and are independent of the electrostatic interactions_ The dispersion attraction between two bodies immersed in a second medium can be estimated using an effective Hamaker constant (-4) [9] _ The interaction between a sphere of radius R, and an infinite planar and thick wall a distance 2, apart is
.
0
~__________________________________________________.
_I I
x
U,,
P
27 July 1984
Cs&cl)
Pig. 3_ The aggegate net force in units of 1 013 N/mol zgrcgate at different sait concentrations, cr = 50, Rj = 2 A, (A) monovalent ions and (X) divalent ions. The standard deviation is in both cases 0.3 X 1Cll3 N/mol.
between the two models. However, with such a low relative permittivity as Er = 10, no significant difference was found. So far we have only considered salt-free solutions. According to the simple DLVO theory we should expect a screening of the repulsive interaction when salt is added to the system. However, since the attractive force component, which we have shown to be important in the strong coupling regime, is neglected in the DLVO theory, we have no reason to expect the same salt dependence. Indeed in simulations with monovalent counterions and an added 1: 1 salt, the aggregate net force remains unaffected by the salt (fig. 3). With divalent counterions and a 2 : 2 salt there seems to be a weak initial reduction of the aggregate force. The concentration profiles of co- and counter-ions show that the negative and positive charges of the added salt are rather evenly distributed and mainly found between the aggregate and the neutral wall. Hence, the net charge distribution is only slightly affected by the salt. On adding a 2 : 2 electrolyte to a system with monovalent counterions, we would expect an interesting salt dependence, with an initially repulsive interaction, which by addition of divalent ions turns over and becomes attractive. In the DLVO theory of colloid stability, the electrostatic force is described with a mean field theory, for example via the non-linear Poisson-Boltzmann equation, and it is always repulsive_ The attractive interaction, which ultimately causes the colloid to coagulate is assumed to be due to dispersion forces. These forces
= -&A[&
+ ?a/(1
c~= R.J(Za - Ra) _
+ 101) -
1
ln(1 + XX)] , (6aj W)
Table 1 shows how the force - aUdti/aZa varies with the distance Z,. With high surface charge density it is smaller in magnitude than the electrostatic force at all intermediate distances_ At sufficiently short distances the continuum approximation, underlying both the electrostatic and the dispersion term, breaks down. Thus, the present results rais.2 some questions about the validity of the general picture of the interplay between attractive and repulsive forces and their origin in highly charged systems_
4. Conclusions Monte Carlo simulationshave shown that the interaction of two electrical double layers in general has an attractive component, which is non-existent in the usual mean field theory. The attractive component is dominating in the “strong coupling limit”, that is with high surface charge density, low dielectric permittivity, low temperature and/or with polyvalent counterions. The phenomenon is a general one, and can be thought of as a classical analogue of the wellknown London forces between rare-gas atoms. Unambiguous experimental evidence of this attractive force is rare, but some recent NMR studies of lamellar liquid crystals formed by ionic amphiphiles (with divalent counterions) [IO] can be interpreted in the light of the present simulations_ There also exist quite a number of experimental results both in colloid and biological chemistry [ 1 l] which do not fit the DLVO theory. Usually they are interpreted in terms of “specific interactions”, but might be explained by the present results in a purely electrostatic continuum model.
583
Volume 108. number 6
CHEMICAL PHYSICS LETTERS
References U.V. Dcrjayin and L. Landau. Acta Phys. Chim. URSS 14 (1941) 633. E.J.W. Vcrwey and J.Th.G. Overbeck, Theory of lhe stability of tyophobic cotloids (Elsevier, Amsterdam. 1948). L. Guldbrclnd. U. Jbnsson and H. WcnnerstrBm, J. Cotloid Intcrfw Sci. 83 (1982) 532. H.JBnswrl. t I. Wcnncrstrbm and 1%.ltatlc. J. Phys. Chem. 84 (1980) 2179. tt. Wcnncrs~riim. 13.Jiinsson and I’. Linsc. J. Chcm. Phys. 76 (I 982) 4665.
584
27 July 1984
161 C.hl. Torrie and J.P. Valteau. J. Phys. Chcm. 86 (1982) 3251. I71 L. Guldbrand, B. Jcnsson, 11.WcnncrstrBm and P. Linsc, J. Chcm. Phys., to be published. 181 N. hlctropolis, A.W. Roscnbluth, M.N. Roscnbturh. A.H. Tcttcr and E. Tcllcr, J. Chem. Phys. 21 (1953) 1087. 191 H.C. Hamukcr. Physica 4 (1937) 1057. I to1 A. Khan. K. Pontctl, G. Lindblom und B. Lindman. J. Pbys. Chcrn. 86 (1982) 4266. 1111 C. Hui Bon Hou, E. Bfgard. P. Beaudry. P. Maorel, M. Grunberg-hfansgo nnd 1’. Douzou. Biochcmiswy 19 (1980) 3080.
108. number
CHEMICAL
6
THE INTERACTION A MONTE
CARLO
Bo SVENSSON
BETWEEN SIMULATION
27 April
17 July
LETTERS
AGGREGATES
IN ELECTROLYTE
1984
SOLUiION.
STUDY
and Bo JGNSSON
Yh_vsical Chetttisrr_r 2, Chetttical Rcceivcd
CHARGED
PHYSICS
Cettler.
198-l: in finnl form
P-0. Box 740. S-220
07 Lund. Sweden
26 hfnp 1984
T11c .\lonre Carlo simulalion ~ecbniquc 113sbeen used to calculate rhe electrostaric force acting upon a charged aggregate ~)utkk 3 similsrly charged wall. Contrary to intuition and e.xisIing electrostatic theories, the force is found to be attractive
for WIX rralktic values of the parameters dclerminin_e the system. High surface charge density, low temperature, low relaIIW permitlivity 2nd polyvslent neutralizing counterions are all factors that favour a net strraction between the wall and rl~c xxegate. In some cases the rcsuIIin_e electrostatic attractive force is found :o be an order of magnitude ordinary van dcr \V;lals artractiun applied in ibe DLVO theory of colloidal stability. Tbeartracrive interaction bcin:: due IU rorrcl;ltions between the counterions in the electric double layers.
larger
than
the
is inrerpreled
as
1. Introduction
wall and a spherical aggregate with charge density. The wall-aggregate
Intcracrions between charged aggregates play an nnpor~3n1 role in a variety ol’chemical systems. For tt_xainplc. the eleclrostatic interaction is a main com-
been calculated directly in the simulations and found to contain an attractive component, which is the dominating one in the “strong coupling regime”, that is with high surface charge density, multivalent countcrions, low E,Tvalues and at short distances_
poncnt in the stability ofcolloids [I,?] as well as of diNeI WI phases formed by amphiphilic molecules I_:]_ .L\~~slytical theories dealing with elecrrostatic inIcr2srions in solution usually focus on the ionic interxrions and assume a con0nuum model for the solWIII. In rile l’oisa~~l--Boltzn~ann (PB) equation. for example. 211 additional mean field approximation is
mtroduccd. sipccies
where the correlation between the ionic is neglccred (41. This latter approsimarion can
be tested via Xlonle Carlo (MC) simulations, which provide the exact answer within the given continuum model. Recent MC simulations have shown serious
dsficiencirs in the I’B approximation in systems with divalcnr ions [S.O]. In fact, the net electrostatic force brr\vecn rwo charged surfaces, with an intervening clcctrulytc solution, is sometimes found to be attractive [ 71. This is contrary IO the prediction of the DLVO theory, where the net electrostatic force is always repulsive
[ I,?].
In riie present
simulationsof 5so
work we report the results from MC rhe interaction betweena charged planar
the same surface mean force has
2. Model and methods The model
system
consisted
of two infinite
planar
walls, one of which had a surface charge density u, separated by a distance Z,,. The intervening diclcctric continuum contained the mobile co- and countcrions and spherical aggregates with the same surface charge density as the charged wall (see fig. 1). In the presence of both co- and counter-ions, it is necessary to assign a radius to the ions, while in a salt-free sysrem the counterions can be represented by point charges. The surface charge density on the wall was treated both as uniformly smeared out (model I) and as distinct charges(modc1 II), while the aggregate charge density was always uniformly smeared out. A uniform relative permittivity er was assumed, that is dielectric saturation 0 009-2614/84/S (North-Hallxnd
and image charge 03.00 Phv
effects
were neglected_
OElsevicr
Science
Pllhlichino
l%ririnn~
Publishers
B.V.
27 July 1984
CHEhlIICAL PHYSICS LETTERS
Volume 108, number 6
dard hard-sphere 0
oj
0
0
8 Q 0
Q
i !
Ra
Q---
0
0
i
!
0
0 0
0
I i i !
za
where the wall-aggregate (I
= -
wa
w,e
47TEoE,
Z WW
lation where all particles, including the aggregate_ are free to move in between the charged and uncharged walls. The aggregate concentration profile is then easily obtained during the simulation and hence also the potential of mean force. However, it turned out to be quite time-consuming to calculate the concentration profile for the aggregate, when the surface charge density exceeded ~0.16 C mm2_ This is probably due to the slow “diffusion” of the aggregate through configuration space during the course of the simulation. To circumvent this problem, the aggregate was held fried at different wall-aggregate separations 2, and the average force acting upon it was calculated explicitly in the simulation. In order to facilitate the force evaluation, a repulsive r-l2 term was introduced into the ion-aggegate
interaction,
instead
of a stan-
ener,,g
is
(?a’ + Z~)‘l” + a
( ( Saln
arcsin
(a’+Z.;) 3 11’-
(
(a’+Z,‘)’
1 +;
)
(21 -1)
and similarly for the wall-ion interaction U,vi_ The ion-ion interaction followed the primitive model, 7 rii>Ri+Rj, Uij = qiqje-/4iifTgEr’ij, =CO7
The MC simulations were performed in the NVT ensemble using the standard algorithm of Metropolis et al. [S]. In order to make the simulations more tractable, the aggregate concentration was chosen so that the MC box contained only one aggregate_ The size of the IMC box was 2a X 2a X Z,\,,\,= 64 X 64 X 128 A3 and periodic boundary conditions were applied in the two directions parallel to the walls. The system was equilibrated during IO4 configurations/ particle and at least the same number was generated for the analysis_ The most straightforward way to calculate the average force acting on the aggregate is to perform a simu-
interaction
a4 _ Zz _ 2a’Zz - 22, I
Fig. 1. The model system. Ions and a charged spherical aggregate between an equally charged wall and an uncharged wall. The solvent is regarded as a continuum and the relative permittivity is everywhere constant.
cut-off_ The total electrostatic
for model I can then be written:
rij~Ri+R-
I-
(3)
Here qi and Ri are the ion valency and the ionic radius, respectively. The ion-aggregate interaction is a slight modification of eq. (3):
(4 where R, defines the azregate radius, which was 12.8 .A throughout the simulations. In model II iJ_, U,\+and the interaction between the wall charges themselves (U,,,,\,) all followed the primitive model, eq. (3). The average force acting upon the aggregate is then simply the derivative F, =
-au,,,iaz, = -au,_iaz, -
~iKJj,laZ~.
(5)
One can note here that the wall-agiregate force is temperature independent in model I, with the smeared out surface charge density (see eq. (2)). In the second model, the force between the aggregate and the surface charges will be a Boltzmann average and depend on the correlation between surface charges, mobile ions and aggregate_ The minimum image approximation was used and all interactions were truncated at the edge of the MC box. This approximation was tested through simulations on larger systems, but no significant changes were found upon enlarging the system by a factor of four. To check the effects of the boundaries, simulationswere also performed
without periodic boundary conditions. 581
Volume
108, number
CHEMICAL
6
Instead, the system was confined to the MC box with the same dimensions as previously_ All interactions within the paralielepiped were taken into account. The aggregate was placed on the symmetry axis and the walls of the parallelepiped were treated as impenetrable. The results obtained in this model were virtually the same as with periodic boundary conditions_
3. Results In a recent
showed infinite
blCl simulation
that the electrostatic planar double layers
Guldbrand
et al. [7]
interaction between two becomes attractive in the
limit ofhigh surface charge density and divalent counterions. An interesting question is whether this is a property of this specific system or if it is a general phenomenon appearing also in other systems of different geometry. Part of the answer is found in table 1 where the aggregate net force is shown as a function of the wall-aggregate distance. Sufficiently far from the charged wall, the electrostatic force is always repulsive irrespective of the surface charge density. At intermediate distances and low surface charge density \ve still find a net repulsive force, as expected from the DLVO theory. However, when the charge density
Table
1
3). xting on rtn rrz_Srcgntc in ner clectroststic fbrcc units of 1013 N/moI aggregnre. Standard deviations are estimrrtcd IO 0.1. 0.1 and 0.2 X lOI N/mu1 aggregate for 0.0625, The corresponding disper0.125 and 0.35 C mm2 respectively. sion force. with rr iizrmllker constant .-I = I 0m2’ N m is shown in tlte last column (eq. (6)). Divalent counterions, cr = 77.3 2nd T= 301 I;
The avcr:~gc
becomes ~0. I6 C m-3,
Surface 0.0635
80 50 32 2-1 30 18 16
0.1) I 0.0 1 0.06 0.15 0.36 1.1
a) Negative
charge
density
0.115
0.0 1 0.05 0.0 1 -0.11 0.13 0.6’ 2.5
(C mm2)
a change in the temperature, as is evident from an analysis of the dimensions (see also ref. [7]). Model I may be criticized for being unbalanced in that it allows for the correlation between the mobile counterions but considers the wall charges as completely uncorrelated, that is uniformly smeared out. However, explicit simulations with er = 77.3 and di-
valent counterions show that the replacement of the smeared out surface charge with distinct charges, nobile in two dimensions. does not alter the aggregate force significantly. It might well be that for a sufficiently charged system there will appear a difference
sign rneilns an artmctive
aggre<c.
furce
-0.0017 -0.00-18 -0.0 I8 -0.059 -0.16 -0.34 -1.01 bct\veen
wall and
.
FE3
1
-10
0.03 0.06 -0.06 -0.64 -0.88 0.37 4.6
force
The attractive component of the force dominates at intermediate distances, while the total electrostatic force again becomes strongly repulsive at very short distances. These simulations were performed using model I with divalent point charges as counterions but without salt. In the case of monovalent counterions the force is everywhere repulsive for this set of parameters. However, as indicated earlier, the origin of the attractive force component is the correlation between the mobile ions. In the case of perfect correlation, that is a lattice, it is easily seen that the force will be attractive. Thus, one way to make the force more attractive should be to increase the electre static interaction by, for example, reducing the dielectric permittivity or the temperature. Fig. 2 shows indeed that the force becomes attractive also in the presence of monovalent counterions when the dielectric permittivity is less than ~40. The same is true for
- a(ldisp/az,
0.35
the net electrostatic
attractive.
becomes
0
Z,(X)
27 July 1984
PHYSICS LETTERS
1
___________i-___________~_____________~___________.
A
-I\
*
x
x
i 20
40
1
1
60
1
1
80
1
I
er
Fig. 7. The aggregate net force in units of I 013 N/moI aggregate as a function of the relative permittivity, 2, = 20 A,(r) monovalent and (X) divalent counterions. The standard deviation increases in both cxes from 0.7 to 1.2 X lOI3 N/moI ns cr is decreased from 77.3 to 10.
CHEhlICAL
Volume iOS, number 6
PHYSICS LETTERS
depend among other things on the polarisabihty of the media and are independent of the electrostatic interactions_ The dispersion attraction between two bodies immersed in a second medium can be estimated using an effective Hamaker constant (-4) [9] _ The interaction between a sphere of radius R, and an infinite planar and thick wall a distance 2, apart is
.
0
~__________________________________________________.
_I I
x
U,,
P
27 July 1984
Cs&cl)
Pig. 3_ The aggegate net force in units of 1 013 N/mol zgrcgate at different sait concentrations, cr = 50, Rj = 2 A, (A) monovalent ions and (X) divalent ions. The standard deviation is in both cases 0.3 X 1Cll3 N/mol.
between the two models. However, with such a low relative permittivity as Er = 10, no significant difference was found. So far we have only considered salt-free solutions. According to the simple DLVO theory we should expect a screening of the repulsive interaction when salt is added to the system. However, since the attractive force component, which we have shown to be important in the strong coupling regime, is neglected in the DLVO theory, we have no reason to expect the same salt dependence. Indeed in simulations with monovalent counterions and an added 1: 1 salt, the aggregate net force remains unaffected by the salt (fig. 3). With divalent counterions and a 2 : 2 salt there seems to be a weak initial reduction of the aggregate force. The concentration profiles of co- and counter-ions show that the negative and positive charges of the added salt are rather evenly distributed and mainly found between the aggregate and the neutral wall. Hence, the net charge distribution is only slightly affected by the salt. On adding a 2 : 2 electrolyte to a system with monovalent counterions, we would expect an interesting salt dependence, with an initially repulsive interaction, which by addition of divalent ions turns over and becomes attractive. In the DLVO theory of colloid stability, the electrostatic force is described with a mean field theory, for example via the non-linear Poisson-Boltzmann equation, and it is always repulsive_ The attractive interaction, which ultimately causes the colloid to coagulate is assumed to be due to dispersion forces. These forces
= -&A[&
+ ?a/(1
c~= R.J(Za - Ra) _
+ 101) -
1
ln(1 + XX)] , (6aj W)
Table 1 shows how the force - aUdti/aZa varies with the distance Z,. With high surface charge density it is smaller in magnitude than the electrostatic force at all intermediate distances_ At sufficiently short distances the continuum approximation, underlying both the electrostatic and the dispersion term, breaks down. Thus, the present results rais.2 some questions about the validity of the general picture of the interplay between attractive and repulsive forces and their origin in highly charged systems_
4. Conclusions Monte Carlo simulationshave shown that the interaction of two electrical double layers in general has an attractive component, which is non-existent in the usual mean field theory. The attractive component is dominating in the “strong coupling limit”, that is with high surface charge density, low dielectric permittivity, low temperature and/or with polyvalent counterions. The phenomenon is a general one, and can be thought of as a classical analogue of the wellknown London forces between rare-gas atoms. Unambiguous experimental evidence of this attractive force is rare, but some recent NMR studies of lamellar liquid crystals formed by ionic amphiphiles (with divalent counterions) [IO] can be interpreted in the light of the present simulations_ There also exist quite a number of experimental results both in colloid and biological chemistry [ 1 l] which do not fit the DLVO theory. Usually they are interpreted in terms of “specific interactions”, but might be explained by the present results in a purely electrostatic continuum model.
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References U.V. Dcrjayin and L. Landau. Acta Phys. Chim. URSS 14 (1941) 633. E.J.W. Vcrwey and J.Th.G. Overbeck, Theory of lhe stability of tyophobic cotloids (Elsevier, Amsterdam. 1948). L. Guldbrclnd. U. Jbnsson and H. WcnnerstrBm, J. Cotloid Intcrfw Sci. 83 (1982) 532. H.JBnswrl. t I. Wcnncrstrbm and 1%.ltatlc. J. Phys. Chem. 84 (1980) 2179. tt. Wcnncrs~riim. 13.Jiinsson and I’. Linsc. J. Chcm. Phys. 76 (I 982) 4665.
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