Interparticle force between like-charged colloidal systems: a numerical study

Colloids and Surfaces A: Physicochemical and Engineering Aspects 182 (2001) 299– 304 www.elsevier.nl/locate/colsurfa

Interparticle force between like-charged colloidal systems: a numerical study Takamichi Terao *, Tsuneyoshi Nakayama Department of Applied Physics, Hokkaido Uni6ersity, Sapporo 060 -8628, Japan Received 7 August 2000; accepted 2 November 2000

Abstract We investigate numerically the interparticle force between like-charged colloidal systems confined by a cylinder. The interactions between colloidal particles are repulsive with smaller electrostatic coupling. On the contrary, the attractive interaction dominates with larger electrostatic coupling. These results clarify that the fluctuation effect beyond the Poisson–Boltzmann equation plays an important role, and the strength of the electrostatic coupling in an aqueous solution is an essential parameter to determine the effective potential between like-charged colloidal particles. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Interparticle forces; Colloidal suspensions; Primitive model; Computer simulations

1. Introduction It is of fundamental interest to understand the effective interaction between like-charged colloidal particles in an aqueous solution, because a vast number of industrial and natural processes depend on controlling the interaction between colloidal particles [1 – 10]. In the framework of continuum theory, effective potential between charged colloidal particles have been described by the Derjaguin – Landau – Verwey – Overbeek (DLVO) theory, which predicts the screened Coulomb repulsion between charged colloidal particles in an aqueous solution [1]. In the DLVO * Corresponding author. Tel.: +81-11-7166175. E-mail address: [email protected] (T. Terao).

theory, the effective interaction between particles UDLVO(r) is given by UDLVO(r)=



Z 2e 2 e sa 4pm 1+ sa



2 − sr

e

r

(1)

where Z, e, s, a, m and r denote the surface charge of colloidal particles (macroions), the elementary charge of an electron, the inverse of Debye – Hu¨ckel screening length, a radius of colloidal particles, the dielectric coefficient of the medium, and the center-to-center distance between two colloidal particles, respectively. In general, the interaction between colloidal particles is of importance to determine the physical properties of various colloidal systems [11 –13]. Recently, a lot of works are devoted to clarify the counterion condensation and the attractive

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T. Terao, T. Nakayama / Colloids and Surfaces A: Physicochem. Eng. Aspects 182 (2001) 299–304

interaction between charged colloidal particles, which are inconsistent with the traditional DLVO theory [14 –26]. Neu [27] and Sader and Chan [28] have proved analytically that the nonlinear Poisson –Boltzmann equation can only yield repulsion and the numerical result by Bowen and Sharif [29] is not correct. These results have suggested that the attractive interaction between like-charged colloids in an aqueous solution is an essentially fluctuation-based phenomenon, where the effect of fluctuations is neglected in the theoretical treatment based on Poisson – Boltzmann equation [1,27,28]. In this paper, the highly charged colloidal system under geometrical confinement and its added-salt effect are numerically clarified, where a lot of recent works have been devoted to soft matter systems under confined geometries [30]. We perform Monte Carlo simulation of charged colloidal particles confined by a cylindrical pore, and clarify the effective force F(r) between a pair of colloidal particles in an aqueous solution.

2. The model We adopt the primitive model of strongly asymmetric electrolytes to describe the colloidal suspensions [6,7]. The solvent enters into this model by its dielectric constant m, which reduces the Coulomb interaction. We consider two spherical macroions with the surface charge − Ze (Z\ 0), where e is the elementary charge of an electron. In

addition, Nc counterions are dispersed carrying an opposite charge qe (q \0). The number of counterions Nc is determined by the condition of global charge neutrality such as − 2Z+ Ncq= 0.

(2)

The interactions between particles are given by Á Vmm(r)=Í 2 2 ÄZ e /4pmr

for r0 2rm for r\ 2rm,

Á Vmc(r)=Í 2 Ä− Zqe /4pmr

Vcc(r)=

!

q 2e 2/4ymr

for r0rm + rc for r\ rm + rc,

for r0 2rc for r\2rc,

(5)

where Vmm(r), Vmc(r) and Vcc(r) represent the pair potential on macroions (m) and counterions (c), and rm and rc are the radii of colloidal particles and counterions, respectively. In the following, we consider two negatively charged colloidal particles confined by a cylindrical pore (Fig. 1). In this case, the linear system size in z-direction is taken to be L, and the radius of cylindrical pore is defined as Ld (− L/20z 0 L/2, x 2 + y 2 0 Ld). In this case, the two macroions are placed on the positions R1 = (0,0,− r/2) and R2 = (0,0,r/2), where r(L) denotes the center-to-center interparticle distance.

Fig. 1. Geometry of colloidal particles in an aqueous solution confined by a cylinder.

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3. Numerical results We perform Monte Carlo simulation to study the effective force between like-charged colloidal particles confined by a cylinder. All simulations are performed at constant temperature and volume. We start with an arbitrary counterion configuration which does not penetrate into the two colloidal particles. The positions of two macroions are fixed and not moved. It takes 2× 104 Monte Carlo steps (MCS) to get the system into equilibrium, and 1× 105 MCS to take the canonical average after the equilibrium. We also take sample average over 60 samples for each run. In the following, the temperature T and the relative dielectric constant of water mr are taken to be T=300 K and mr =78, respectively. These are realistic values for actual colloidal particles which are experimentally studied. The radii of macroions rm and microion rc are rm =10 nm and rc = 2.8 A, , respectively. We impose the hardwall condition in the x-, y-directions and the periodic-boundary condition in the z-direction. Comparing the results calculated by minimum-image convention with those by Lekner summation technique [31] including periodic image charges, we have checked that boundary effects are not relevant in the following calculations. The Lekner summation technique has been originally developed for the periodic three-dimensional systems, but later it is also applied to quasi-two-dimensional as well as quasi-one-dimensional systems [31 –34]. For the detail of Lekner summation technique, see Ref. [31]. In the following calculations, the image charge effect is neglected for simplicity. The effective force F1(r) on the first colloidal particle can be obtained by F(r)= F dir(r) +F ind(r).

(6)

In Eq. (6), F dir(r) is the direct Coulomb repulsion such as F dir(r) = −9





Z 2e 2 1 , 4ym r

(7)

and F ind(r) is the indirect part induced by counterions written by

Fig. 2. (a) Effective force F(r)/F0 between two colloidal particles versus the interparticle distance r. The surface charge Z is taken to be Z =450 and (b) Effective force F(r)/F0 between two colloidal particles versus the interparticle distance r. The surface charge Z is taken to be Z=600.





Zqe 2 1 F ind(r)= 9 % , j 4ym rj

(8)

where rj is the distance between the first macroion and the jth microion. In this paper, we neglect the influence of the force caused by the depletion effect. This is because the main interest of this study is to explain the long-range attractive nature, but the range of the depletion force becomes very short-ranged on highly charged colloidal particles (Z/q1), which is proportional to q/Z [16]. Fig. 2 shows the dimensionless effective force F(r)/F0 between a pair of colloidal particles confined by a cylinder, where F0 is defined to be

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F0 kBT/uB (uB e 2/4pmkBT is the Bjerrum length). In the following, F(r) is defined to be F(r) F1(r)(R1 −R2)/ R1 −R2 , where R1 and R2 are the positional vectors of the first and the second colloidal particles, respectively. This means that a positive value of F(r) implies repulsion, and a negative value indicates attraction. On the charge of counterions q, we consider three different cases such as monovalent ions (q= 1) and multivalent ones (q = 2, 3) to clarify the effect of the strength on electrostatic couplings. Fig. 2(a) and (b) are the calculated results with the surface charge Z=450 and 600, respectively. The values of surface charge (Z =450, 600) are much larger than those in most of previous studies [14]. Solid squares, solid circles and solid triangles denote the results with q =1, 2, and 3, respectively. In these calculations, the system size L and the radius of the cylinder Ld are set to be L = 150rm and Ld = 2.5rm, respectively. In Fig. 2(a) and (b), the effective forces between colloidal particles are repulsive with smaller electrostatic coupling. With larger Z and q, on the contrary, a pair of colloidal particles shows an attraction. We observe attractive interaction between highly charged colloidal particles for strong electrostatic couplings. To observe the relationship between the strength of electrostatic couplings and the screening effect, we show snapshots of two colloidal particles (Z= 600) and counterions in an aqueous solution in Fig. 3. The profiles of electric doublelayers with q = 1, 2 and 3 are displayed in Fig. 3(a –c), respectively. There is a difference between monovalent case and multivalent ones: In the case of repulsive interaction (Fig. 3(a)), colloidal particles are surrounded by diffusing counterions. In the case of attractive interaction (Fig. 3(b) and (c)), counterions are tightly bound to macroion surfaces and form quasi-two-dimensional counterion layers. We also calculate the effective interaction between charged colloidal particles in high-salt content, as well as in low-salt content. Fig. 4 shows the results of the interparticle force F(r) for colloidal particles (Z = 600) confined by a cylinder, with additional salt ions. Solid squares and open squares denote the results with the additional salt densities zs =0.0 and 10 − 4 M, respectively. Fig. 4(a) and (b) display the results

with q=1 and 2, respectively. In Fig. 4(a), the effective force F(r) becomes small for zs = 10 − 4 M, since the screening effect due to microions becomes dominant and the screening length is small. In Fig. 4(b), we can see that the attraction vanishes in high-salt content. Our results reflects the microscopic origin of attractive force between negatively charged colloidal particles. We can explain the attraction between like-charged colloidal particles based solely on Coulomb interaction with non-linear screening effect. The continuum mean-field approximation [27 –29] is not appropriate to de-

Fig. 3. Snapshot of colloidal particles and surrounding counterions: (a) q = 1, (b) q =2 and (c) q = 3.

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smaller electrostatic couplings. On the contrary, the attractive interaction dominates with larger electrostatic couplings. It has been demonstrated that the effective interaction between highly charged colloidal particles in a low-salt content obeys non-DLVO behaviors. It has been also shown that such attractive interaction diminishes in high-salt content. We have clarified that fluctuation effects beyond the Poisson –Boltzmann theory play an important role, and the strength of the electrostatic coupling is an essential parameter to determine the effective potential between like-charged colloidal particles in an aqueous solution. Rouzina and Bloomfield have theoretically analyzed the attraction between like-charged planer surfaces at short separation [19]. Linse and Lobaskin have applied their theory into a spherical geometry, and proposed a criterion that the attraction can happen when the coupling parameter Y obeys Y:2 or larger [14], where Y is defined to be Y

Fig. 4. (a) The effective force F(r) with monovalent counterions (q = 1) and (b) The effective force F(r) with divalent counterions (q =2).

scribe this problem, but the primitive-model approach including the spatial and temporal density fluctuation of counterions is necessary, which is a characteristic feature of fluctuationinduced attractions [35].

  Z 4pq

1/2

q 2uB . rm + rc

(9)

Our results have confirmed the above prediction with highly charged systems (Z= 600) in a confined geometry. In Refs. [14,19], it is suggested that the attraction operates on a length scale compared to the distance between neighboring counterions at the charged surfaces. However, in our simulation results with large value of Z, the attraction observed is much long-ranged. This result suggests that the range of attraction between like-charged colloids is not fully understood yet, and this point is a future problem.

Acknowledgements 4. Conclusion In conclusion, we have performed Monte Carlo simulation to study the effective interaction between like-charged colloidal particles confined by a cylinder. The effective forces between colloidal particles are repulsive with

This work was supported in part by a Grantin-Aid from the Japan Ministry of Education, Science, and Culture for Scientific Research. The authors thank the Supercomputer Center, Institute of Solid State Physics, University of Tokyo for the use of the facilities.

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