Role of collective interactions in self-assembly of charged particles at liquid interfaces.
This paper is dedicated to Jacob Masliyah, whose outstanding contributions to understanding the role of electrokinetic phenomena in bitumen extraction from Canadian oil sands have been recognized with numerous professional awards and honours. Our research interests have overlapped in the area of interfacial and colloidal phenomena in chemical engineering, and one of the authors of the present paper (Wasan) greatly values Jacob Masliyah's friendship.
Nanometre- to micrometre-sized charged particles at liquid interfaces self assemble to form various interfacial colloidal domains varying from liquid-like to crystalline ordering (Binks and Horozov, 2006; Bergstrom, 2006). Two-dimensional (2-D) self-assembly of colloidal particles at interfaces is a simple and robust method to fabricate patterned materials with desirable structural, optical, electrical, and magnetic properties, such as bio-sensors, catalysts, photonic devices, and memory chip displays (Velikov and Velev, 2006).
Finely divided solid particles can assist in emulsion formation and/or improve its stability. In fact, in a series of papers (Menon and Wasan, 1986a, b; Menon et al., 1988, 1989), we reported the technological factors affecting the stability of solid-stabilized emulsions. We also developed a new method for measuring the film tension between coalescing water drops and a particle-covered planar oil/water interface; we evaluated the effect of the interfacial solid concentration for hydrophobic solids, such as particles filtered from shale oil. Such charged colloidal particles tend to form 2-D periodic structures at the liquid-liquid interface, and the inter-particle interaction energy (potential of mean force) exhibited multiple maxima and minima. The origin of the oscillatory decay of collective energy curve is the particle excluded volume effect (i.e., the particles' structure in the confines of the liquid-liquid surfaces resulting in both the attractive depletion and structural energy barrier). It was suggested that the repulsive electrostatic interactions between charged colloids are the basis for the formation of such 2-D periodic structures (Menon et al., 1988).
Theoretically, Kalia and Vashishta (1981) and Zangi and Rice (1998) studied the melting transition of a colloidal monolayer using dipole-dipole interactions and a Marcus-Rice potential, respectively, with molecular dynamics (MD) simulations. Similarly, Terao and Nakayama (1999) studied the crystallization of 2-D colloidal systems by means of Monte Carlo simulations. Sun and Stirner (2001) used molecular dynamics to study the compression of 2-D polystyrene particles at an oil-water interface through monopole-monopole Coulombic interactions. Moncho-Jorda et al. (2002) used molecular dynamics to study the coagulation of polystyrene latex particles dispersed at the air-salt solution interface interacting via a potential model of both dipole-dipole and monopole-monopole Coulombic interactions.
During the last ten years, there have been numerous studies on the formation and stability of foams and emulsions containing finely divided solids (Bindal et al., 2001; Binks and Lumsdon, 2001; Binks, 2001; Dai et al., 2005; Danov et al., 2004, Danov and Kralchevsky, 2006; Dickinson, 2006; Du et al., 2003; Grozenbach et al., 2006; Levine et al., 1989a, b; Tambe and Sharma, 1993; Tarimala and Dai, 2004; Vijayaraghavan et al., 2006). Professor Masilyah has made significant contributions to this area (Lopetinsky et al., 2006; Yan and Masliyah, 1993, 1994; Yan et al., 2001).
We have had a long-standing interest in the formation of two-dimensional self-structuring phenomenon of particles in thin liquid films confined between two drops or foam bubbles which are relevant to the formation and stability of particle-stabilized foam and emulsion systems (Nikolov and Wasan, 1992; Bindal et al., 2001; Sethumadhavan et al., 2004; Vijayaraghavan et al., 2006). These studies have revealed the formation of particle layers between the film surfaces and the in-layer 2-D hexagonal or cubic structure (i.e., the colloid crystal-like structure) when the film thickness is of the order of one or two particle layers (Chu et al., 1994). This ordered in-layer structure formation provides the structural energy barrier needed for stabilizing foams and emulsions.
In the present paper, we examine the role of long-range collective interactions between similarly charged particles in a multi-particle system to form 2-D structures at liquid-gas and liquid-liquid interfaces. We carry out Monte Carlo simulations using long-range repulsive electrostatic interactions including both the screened Coulombic and dipole-dipole contributions.
MODELLING AND SIMULATIONS
Modelling Inter-Particle Interactions at Interfaces Pieranski (1980) was the first to conduct experiments on particle structuring at the water/air interface, and found that nanometer-sized (245 nm) spheres formed ordered structures. He suggested that the dipole-dipole interactions were responsible for the formation of 2-D ordered structures. Later, Aveyard et al. (2002) measured the long-range repulsive force between two charged micrometre-sized polystyrene particles at an oil/water interface with the laser tweezer method. They showed that the long-range repulsive force is due to the dipole-dipole interactions induced by the presence of a very small residual electric charge at the particle/oil interface. There are also a number of published studies on long-range attractive interactions between like-charged micrometre-sized interfacial particles; the authors have argued that the capillary force or electrically induced capillary interactions are responsible for the particle ordering phenomenon at the liquid interfaces (Kralchevsky and Denkov, 2001; Earnshaw, 1986; Fernandez-Toledano et al., 2006; Foret and Wurger, 2004; Martinez-Lopez et al., 2000; Megens and Aizenberg, 2003; Nikolaides et al., 2002; Nikolaides, 2001; Stamou et al., 2000; Tata and Ise, 1998).
Stillinger (1961), in studying the discrete adsorbed charge effect, found an integral expression for the electrostatic interaction between point charges at an electrolyte interface by solving the linearized Poisson-Boltzmann equation. The expression for the interaction between two particles at an interface between a non-polar fluid (e.g. air, oil) and polar liquid (e.g. water) is
[u.sub.P-B](r) = [(Ze).sup.2]/[epsilon]r y([kappa]r) (1)
[epsilon] = [[epsilon].sub.w] + [[epsilon].sub.n]/2, (1a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1b)
where Ze is the charge on each interfacial particle, r is the distance between them, [[kappa].sup.-1] is the Debye screening length, and the effective dielectric constant, [epsilon], is the average of that of the polar liquid, [[epsilon].sub.w] and non-polar fluid, [[epsilon].sub.n]. The screening correction term, y([kappa]r), is the integration of the first kind of Bessel function [J.sub.0] (x).
Hurd (1985) analyzed Stillinger's (1961) interfacial particle pair potential and extracted two asymptotic forms. He showed that the electrostatic interaction between charged interfacial colloidal particles is dominated by the screened-Coulomb contribution, which decays exponentially at small separations and by an algebraic dipole-dipole interaction at large separations:
u ~ [u.sub.dipole] + [u.sub.coulomb]
= 2[(Ze/[epsilon][kappa]).sup.2] [r.sup.-3] + 2[(Ze).sup.2]/[epsilon]r [[epsilon].sup.2]/([[epsilon].sup.2] - 1) exp(- [kappa]r) (2)
Hurd's results established Pieranski's suggestions (Pieranski, 1980) concerning dipole-dipole interactions at large separations between interfacial colloidal particles on a firm theoretical basis, and clarified the effect of the electrostatic interactions. In our Monte Carlo simulations, we use Hurd's pair interactions to explore the effect of multi-particle collective interactions on the particle structure formation at liquid interfaces. The role of the capillary force will not be considered in our analysis because the meniscus curvature induced by the gravitational force around the nanometre-sized particles is negligible.
Monte Carlo Simulations
The Metropolis algorithm (Metropolis et al., 1953) is used to obtain the equilibrium in a canonical ensemble (constant NVT ensemble) (Hansen and McDonald, 1997). The use of the periodic boundary conditions eliminated the boundary effects. The basic simulation cell length, l, was calculated according to [l.sup.2] = (n[pi][d.sup.2])/(4A) to get the required particle area fraction A, where d is the diameter of the 2-D Monte Carlo particles and is the same as the diameter of the particles used in Pieranski's (1980) or Nikolaides' (2001) experiment. The particle number was fixed at n = 484, as commonly used in a regular Monte Carlo simulation (Sethumadhavan et al., 2004). According to the analysis of the 2-D size-dependent properties proposed by Zollweg and Chester (1992), the difference in the pressure for 2-D systems with n = 256 and n = 16384 is smaller than 0.1%. By performing simulations for the same set of parameters with n = 250, 432, and 1024, Tata and Ise (1998) also found that the results are the same within statistical error for n [greater than or equal to] 432. The system was allowed to evolve with Monte Carlo trajectories. Approximately 30 000 sweeps were initially discarded (one sweep is an attempted move per particle). The radial distribution function g(r) was monitored to ascertain if equilibrium had been reached. Our averaging was done over 30 000 sweeps. The distance interval of [DELTA]r = 0.01d is used in the g(r) calculations as in the standard method (Rahman, 1964).
RESULTS AND DISCUSSION
Particle Self-Assembly at an Air-Water Interface
We first compare Hurd's asymptotic form (Equation (2)) with Stillinger's complete solution (Equation (1)) for repulsive electrostatic interactions between the air-water interfacially constrained latex particles in Pieranski's experiments (Figure 1). Pieranski studied the 2-D interfacial particle system: polystyrene spheres (d = 0.254 [micro]m) were trapped at an air-water interface due to the surface energy (Figure 1a). A drop of the latex suspension was introduced to a cylindrical test tube with a microscopic glass slide at the bottom (Figure 1b). He observed a monolayer of interfacial particles with 2-D crystalline and 2-D liquid structures at the area fractions of A = 0.04 and 0.02, respectively, as seen in the microscopic photographs (Figure 1c). Here, the area fraction is defined as A = (n[pi][d.sup.2])/(4S) for n particles of diameter d in the system of area S. For the purpose of calculation, the integral in Equation (1b) was transformed into a series by splitting up the range of integration between the zeros of [J.sub.0] (x) and performing each integral by Simpson's rule. The resulting alternating-sign was terminated after 20 terms, and the remainder was estimated by Euler's transform using the next 10 terms.
[FIGURE 1 OMITTED]
The pair potential of Stillinger's integrated form and Hurd's asymptotic expression, along with its dipole and Coulomb contribution, are plotted in Figure 2. Hurd's asymptotic potential (dipole + Coulomb) is in good agreement with Stillinger's complete integral potential. The contribution of the Coulomb repulsion is larger than that of the dipole repulsion when the radial distance is smaller than 20d; the potential energy is dominated by the screened Coulomb repulsion at the interparticle distance of 6d for the liquid-like interfacial colloid structure observed by Pieranski for the area fraction of A = 0.02 (Figure 1c).
Figures 3 and 4 show the results of the Monte Carlo simulations for the particle configurations; the corresponding radial distribution functions [g(r)] for the air-water interfacial particles at the area fractions of A = 0.02 and A = 0.04, respectively. Due to the poor quality of the images in Pieranski's paper, we could not make a direct comparison of the computed radial distribution functions with the experimental measurements. However, snapshots of the particle configuration obtained from the Monte Carlo simulations appear qualitatively similar to those observed by Pieranski for disordered (A = 0.02) and ordered (A = 0.04) structures, respectively.
The potential of the mean force is calculated from the radial distribution function [g(r)] via the Boltzmann relation,
w(r)/kT = -ln[g(r)] (3)
where k it the Boltzmann constant and T is the temperature.
Figure 5 compares the potential of the mean force for the two area fractions. The energy barrier difference between a 2-D disordered structure (at A = 0.02) and a crystal-like structure (at A = 0.04) is 2.4 kT. That is, for the charged nanoparticles used by Pieranski, the structure changes from liquid-like to a solid at a low energy difference and at a very low area fraction.
Particle Self-Assembly at the Oil-Water Interface
We next calculate the radial distribution function and the potential of the mean force using Monte Carlo simulations to quantify the effect of the collective particle interactions on the self-assembly of interfacial colloids at the oil-water interface. In our analysis we consider the data obtained by Nikolaides et al. (2002) and Nikolaides (2001), who observed the fluorescent-labelled PMMA beads (diameter = 1.5 [micro]m) suspended at the decalin-water interface. A water drop of PMMA suspension was injected into a microscopic chamber. The particles self-assembled on the surface of the water droplet and were arranged in an ordered structure (called "loose hexagonal packing" by the authors) at the area fraction of A = 0.054 (which is the only particle area fraction presented by the authors besides the special seven-particle system). The inter-particle distance identified from the first peak position in g(r) in Nikolaides et al.'s paper is 5.7 [micro]m. This average inter-particle distance of 5.7 (3.8d) corresponds to a 2-D particle area fraction of A = 0.054.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
Hurd's repulsive potential (with a particle charge of Z = 2 x [10.sup.4]e and a Debye screening length of [[kappa].sup.-1] = 0.44 [micro]m) was used in the Monte Carlo simulations at A = 0.054. This value was estimated using the electrophoresis data reported by Perez and Lemaire (2004) and Scholtmeijer (2005), taking into consideration the relative particle size and the effective media dielectric constant used in the Nikolaides et al. experiments (Nikolaides, 2001; Nikolaides et al., 2002). The Debye length was calculated as [[kappa].sup.-1] = 0.44 [micro]m, based on the electro-neutrality of the system (assuming the counter ions for the charged particles are only [H.sup.+] or [OH.sup.-] and no salts are inside the suspension).
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
The resultant snapshots of the particle configuration and the corresponding radial distribution function are shown in Figure 6. There is good agreement between the data of Nikolaides et al. and the Monte Carlo simulations. In both cases, the oscillatory decay curves of g(r) have the same characteristic first peak amplitude of 3.3 and peak position of 5.7 [micro]m. The particles have a liquid-like structure characterized by the oscillatory decay profile of g(r) curve in both the experimental observations and simulations.
The particle snapshots taken during the Monte Carlo simulations are very similar to the snapshots of the PMMA latex particles observed experimentally.
It should be noted that particles with valences above 10 000 may not be entirely described by the Debye-Huckel approximation. Close to the particle surface, the counter ions form a strongly charged diffuse layer (Gouy Chapman layer) that influences the electrostatic properties. Moreover, there is a recent claim that a strong attraction does occur for particles with high valences at a curved interface (Wurger, 2006).
Effect of Particle Charge
We did a parametric study to see the effect of the particle charge on the particle structure and particle interactions. The ionic strength of the water phase is a dependent variable of the particle charge based on the electro-neutrality, as described previously. The calculated Hurd potential curves with three different charges, Z = 2 x [10.sup.4] e ([[kappa].sup.-1] = 0.28 [micro]m), Z = 5 x [10.sup.4] e ([[kappa].sup.-1] = 0.28 [micro]m) and Z = 2 x [10.sup.5] e ([[kappa].sup.-1] = 0.14 [micro]m), are shown in Figure 7. The higher particle charge creates a stronger repulsive particle potential.
We conducted Monte Carlo simulations at the 2-D particle area fraction A = 0.054 of the Nikolaides et al. experiment, using the Hurd repulsive potentials with the above charges. Figure 8a shows the radial distribution function with the radial distance scaled by the particle effective diameter, the geometric diameter plus two times the Debye screen length (d + 2 [[kappa].sup.-1]). The scaled inter-particle distance (the first peak position) decreases with the decreasing particle charge because of the increasing Debye screen length and effective particle diameter. The g(r) shows an oscillatory decay curve at Z = 2 x [10.sup.4] e. However, a bump occurs in the second peak of g(r) at Z = 5 x [10.sup.4] (shown by the arrow in Figure 8a), which grows and splits at Z = 2 x [10.sup.5]. The split of the second peak of g(r) into two bumps with the increasing particle charge is characteristic of the 2-D hexagonal packing structural formation. It indicates that the particles have a 2-D structural transition from liquid-like to crystal-like when the particle charge increases from Z = 2 x [10.sup.4] to Z = 2 x [10.sup.5]. A similar ordered structure formation was observed at the air-water interface upon increasing the charge from Z = 1000e ([[kappa].sup.-1] = 0.2 [micro]m) to Z = 5000e ([[kappa].sup.-1] = 0.7 [micro]m).
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
The potential of the mean force calculated using the Boltzmann relation is shown in Figure 8b. The energy barriers for 2-D particle structural transitions with charges of Z = 2 x [10.sup.4] and Z = 5 x [10.sup.4] are 2.5 kT and 4.5 kT, respectively. Therefore, the energy for particle crystallization (i.e., the energy barrier difference between a 2-D liquid-like and a crystal-like structure) is about 2.0 kT. It is similar to the value of 2.4 kT, at which latex particles crystallize at the air-water interface.
We have presented results of simplified analyses using Monte Carlo simulations to examine the role of inter-particle collective interactions responsible for inducing 2-D self-assembly of colloids at interfaces. We used Hurd's asymptotic pair potential, which combines the screened Coulombic and dipole-dipole interactions. It is noteworthy that the results of our Monte Carlo simulations qualitatively predict the particle configurations observed by both Pieranski (1980) (for air-water surfaces) and Nikolaides et al. (2002) (for oil-water interfaces). The calculated radial distribution functions reveal that the 2-D structure of charged particles at liquid interfaces changes from liquid-like to crystal-like at a low area function (from A = 0.02 to 0.04 in Pieranski's experiments for the air-water surfaces).
The particle charge greatly affects the particle ordering at liquid interfaces. Upon increasing the particle charge from Z = 1000e to 5000e, the interfacial colloid structure at the air-water surface changes from disordered to highly ordered (i.e., crystal-like) for a low area function of A = 0.02 (in Pieranski's experiments). These calculations are not shown here. A similar phenomenon was observed at an oil-water interface when the particle charge was varied 2 x [10.sup.4] to 5 x [10.sup.4] (see Figure 8). The energy (i.e., potential of the mean force) difference between the 2-D liquid-like and crystal-like structures is only 2.0 kT for the oil-water interface, which is similar to the value of 2.4 kT for particles to crystallize at the air-water interface. It is also similar to the value of 1.8 kT reported for latex particles to form crystalline ordered structures in the confined boundaries of a wedge film (Wasan and Nikolov, 2003).
From a practical point of view, the interfacial particle crystallization energy can be used to quantify the particle interactions which render the energy barrier against droplet or bubble coalescence in an emulsion or a foam system containing particles. By changing the particle charge, we can affect the collective interactions and the 2-D structural transitions and, thereby, control the emulsion and foam stability.
This work was supported in part by the National Science Foundation under grant CTS-0553738.
NOMENCLATURE 2-D two-dimensional A particle area fraction d diameter of particles g(r) radial distribution function [J.sub.0] (x) first kind of Bessel function k Boltzmann constant l cell length n number of particles T temperature u potential of interaction [u.sub.dipole] dipole-dipole interaction [u.sub.coulomb] Coulomb interaction w(r) potential of mean force y distance coordinate Ze particle charge Greek Symbols [epsilon] dielectric constant [[epsilon].sub.w] dielectric electric constant for water [[epsilon].sub.n] dielectric electric constant for non-polar fluid [[kappa].sup.-1] Debye screening length
Manuscript received February 28, 2007; revised manuscript received May 25, 2007; accepted for publication May 25, 2007.
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Stanley Wu, Alex Nikolov and Darsh Wasan *
Illinois Institute of Technology, 10 West 33rd Street, Chicago, IL, U.S.A. 60616
* Author to whom correspondence may be addressed. E-mail address: firstname.lastname@example.org
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|Author:||Wu, Stanley; Nikolov, Alex; Wasan, Darsh|
|Publication:||Canadian Journal of Chemical Engineering|
|Date:||Oct 1, 2007|
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