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Foamability of Liquid Particle Suspensions: A Modeling Study† Krishna Vijayaraghavan,‡ Alex Nikolov,‡ Darsh Wasan,*,‡ and Douglas Henderson§ Department of Chemical and Biological Engineering, Illinois Institute of Technology, Chicago, Illinois 60616, and Department of Chemistry and Biochemistry, Brigham Young UniVersity, ProVo, Utah 75202
Recent experiments on the foamability of liquid particle suspensions (with a particle size varying from nanometers to micrometers in diameter and in the absence of any surfactants) have shown that the foamability correlates well with the degree of particle coverage at the air-liquid surface. Furthermore, these experiments have also revealed foam inhibition due to the aggregation or clustering of particles in the bulk at high particle concentrations. The present study is aimed at theoretically modeling the observed phenomena. Theoretical analyses based on Monte Carlo simulations and the integral equations of statistical mechanics are used to model the particle aggregation phenomena in the bulk. The radial distribution functions, which show the particle packing density, are analyzed to obtain the effective energy of interaction between the particles. The maximum in foaminess due to the adsorption of particles on the air-liquid surface and clustering of particles in the bulk is explained. The modeling results are in good agreement with the experimental observations on foaminess with small particles. Introduction Foam is a dispersion of gas bubbles in a liquid system stabilized by surfactants but is usually unstable due to its large surface area. Bubbles in foams are separated from each other by a liquid film or lamella-forming foam cell (Figure 1). The lamella thickness is on the order of a few nanometers to a hundred micrometers.1-5 Foams are classified according to the shape of the bubble: wet foams have spherical bubbles and dry foams have polyhedral bubbles. Figure 1 shows typical wet and dry foams. Wet foams have a low gas volume fraction (roughly around 20-30%) while dry foams have a large gas volume fraction (>80%).5,6 Foam formation is accompanied by two other processes, foam drainage and foam lamellae rupture. A surfactant, particle, or protein provides the necessary barrier against bubble coalescence. The stability of the foam is governed by number of factors, such as the viscosity of the bulk solution, concentration of surfactant/proteins or particles, and a number of external forces.3-13 When two bubbles approach each other, the coalescence of the bubbles depends on the stability of the lamella separating the two bubbles. If the bulk solution is very viscous or if the thick film contains a large amount of particles, the bubbles are very “kinetically” stable and often do not coalesce. This can be compared to a very viscous honey, where a small number of bubbles are very stable and remain in the same position for a long period of time. Surface-active agents (surfactants) or polymers are often used to stabilize foams. Surfactants or polymers can be replaced by particles with modified wettability characteristics,4,7,10-23 since such particles are easy to replicate or reproduce, relatively nontoxic, and cheap. Particle-stabilized foams are used in many industrial applications, such as when minerals are separated from ores, food, and agricultural products, electrophoresis coating, † This paper is in honor of Professor J. B. Joshi, with whom Prof. Wasan shares a friendship lasting more than three decades. One of the authors, Krishna Vijayaraghavan, is also a student from the same university for which Professor Joshi is the director. * To whom all correspondence should be addressed: E-mail:
[email protected]. Tel.: 312-567-3001. Fax: 312-567-3003. ‡ Illinois Institute of Technology. § Brigham Young University.
pesticide manufacture, cosmetics, and pharmaceuticals.9-23 The future promises to favor the use of particles in stabilizing foams and bubbles. Metallic foams, obtained by using particles instead of conventional surfactants, are replacing polyurethane foams.24-26 Lightweight metal foams provide additional rigidity to automobiles. Ceramic industries use particles to form sintered foams, enabling the manufacture of the macroporous ceramics used in catalysis, tissue engineering, and insulating materials. Foaming in the presence of particles occurs by two mechanisms: by the layering of the particles in the lamella or the narrow region between the two bubbles, and by the adsorption of particles at the air-water interface which provides a steric type of stabilization and prevents the coalescence of bubbles.7,10,11,18-22 Different types of particles contribute to foaming, namely hydrophilic, partially wetted (hydrophilic or hydrophobic) or biphilic. Figure 2 shows the different types of particles and their role in foam stability. Many research groups have studied the role of particle wettability or biphilicity on lamella or foam stability and numerous papers have been written on this subject.22,26-28 Many researchers have shown that hydrophilic particles inside the lamella tend to be multilayered (Figure 2) and the lamella thins in a stepwise manner; there exists an optimum size of the film below which the particles show no tendency to leave the film.2,4,14,29 Bindal et al.4 found that the number of stepwise film thickness transitions increased with the particle concentration and decreased with an increase in the particle size. In other words, the foam stability increased with the increasing concentration but decreased with the particle size. Although one can correlate the foam stability with the concentration of particles, the literature demonstrates discrepancies with respect to the particle size.30-33 The foam stability in the presence of partially wet particles (i.e., biphilic) is due to adhering of the particles at the air-water interface. Figure 2 shows the orientation of solid particles at the air-water interface. Hydrophobic particles have a contact angle of more than 90° with respect to the water phase while hydrophilic particles have a contact angle less than 90°. However, when the particles are too hydrophobic (contact angle greater than 160°) or too hydrophilic (contact angle less than 20°), the foams are not stable and the bubbles coalesce.5,7,12
10.1021/ie801741q CCC: $40.75 2009 American Chemical Society Published on Web 04/07/2009
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Figure 1. Types of foam.
Figure 2. (a) Orientation of particles at the air-water interface; (b) stability of air bubbles in the presence of biphilic solid particles; and (c) stability of film by layering of hydrophilic particles.
Partially wet (biphilic) or surface-modified and amphiphilic particles have also been identified in the figure and been found to dramatically influence foam formation and stabilization.10,11,18-21,34-37 Binks et al.21 found that Janus particles, having one part hydrophilic and the other part hydrophobic, also stabilized foams. The particles prevent the coalescence of air bubbles due to the steric stability, so the foams are stable. Recently, Hunter et al.37 discussed the uses of particles in stabilizing foams and emulsions. They highlighted the similarities between foams and emulsions, and the methods in which
the particles act as stabilizing agents. Horozov38 also reviewed recent developments in the field of particle-stabilized aqueous foams and foam films. Experimental Details Recently, we presented experimental data on foam formation in the presence of crystalline-type particles.19 In these experiments, a three-phase foam was generated by the evaporation of a saturated NaCl solution in which the initial concentration of
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Figure 3. Foaminess during evaporation of saturated NaCl solution and the surface concentration as a function of bulk concentration of particles.5
Figure 4. Schematic representation of foam formation enhanced by the amphiphilic particle adhered at the air-water surface vs particle concentration: (a) surface coverage of about 20 vol %, (b) surface coverage of about 36 vol %, (c) micrograph depicting air bubble laden with particles, (d) schematic of clustered particles in the bulk and a photograph showing the flocculated particles on the walls of the fleaker.
the insoluble solids (NaCl particles) was 2.3 wt %.19 The hydrophilic NaCl crystals were partially hydrophobized using modifier cetyl trimethyl ammonium bromide (CTAB). At low volume percentages (e.g., 5-35 vol %) NaCl crystals (the particles) are cubic and well dispersed in the aqueous phase with particle sizes ranging from 20- 40 µm. At the beginning of boiling, the particle concentration is low and fewer particles are expected to attach to the bubbles, so there is less foaminess (Figure 3). With time, more particles form and the foaminess sharply increases. At a particle volume fraction of about 40 vol %, the foaminess has a well-defined maximum (200 vol %), after which even the particle concentration in the bulk continues to increase and the foaminess sharply decreases. The foaminess is due to particles attached to the air bubble surface (see the inset photos in Figure 3). To monitor the particle attachment to the bubble surface, the following experiment was conducted. The boiling was stopped, the foam collapsed, and most of the particles attached to the foam lamellae spread on the air/
water surface. We estimated the particles attached (surface coverage) to the air bubbles at various particle concentrations in the bulk at this point. The surface coverage is defined as the ratio of the area occupied by the particles to the total area. The data for the particle surface (air/water) coverage versus the particle bulk concentration are plotted in Figure 3. The curve has a maximum at 40 vol% particle bulk concentration, the same concentration at which there is maximum foaminess. The fact that the two curves plotted in Figure 3 have similar trends led us to conclude that what governs foaminess is the state of particle flocculation in the bulk. As the particle concentration in the bulk increased during boiling, we observed that the particles tend to flocculate and form larger clusters. The flocculation is more visible at higher particle concentrations (e.g., 40 vol % and higher). The amphiphilic NaCl crystals (particles) flocculate and form clusters with sizes from 100-300 µm (Figure 3). Photographs shown in Figure 3 depict the flocculation of amphiphilic NaCl particles
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in the bulk and the attachment of amphiphilic NaCl particles to the air bubbles. These observations reveal that the clustered particles are too heavy to attach to the bubbles generated during boiling. The photographs show that at high particle concentrations in the bulk, only a few particles attach to the air bubbles. We now present a simplified theoretical analysis using the Ornstein-Zernike integral equation (Percus-Yevick approximation) of statistical mechanics and Monte Carlo simulations to model the observed particle aggregation phenomenon in the bulk. We based our analysis on hard-sphere particles with the aim of rationalizing the experimental findings. Theoretical Analysis. We model the phenomenon of foaming in the presence of fine, partially hydrophobized particles, which depends on two processes: the attachment of particles to the air bubble surface and the flocculation (i.e., aggregation) of particles in the bulk. A schematic of the phenomenon is depicted in Figure 4. Aggregation in the Bulk. The phenomenon of decreasing foaminess due to the aggregation or clustering of particles in the bulk has been observed by several investigators.10-12,17-19 For example, Dickinson et al.12 reported that once particles aggregated in the bulk, the foaminess decreased due to the nonavailability of the particles in preventing the coalescence of bubbles. The degree of foaminess depends on the balance between the particle surface coverage at the air/water surface and the extent of the particle aggregation in the bulk. A net maximum in foaminess is expected as a result of the two opposing phenomena.17-19 In the following discussion, we rationalize the experimental observations on the particle aggregation phenomenon in the bulk by solving the Percus-Yevick equation that is based on the Ornstein-Zernike (O-Z) integral equation of statistical mechanics and also with Monte Carlo simulations. By following the work of Smith, Henderson et al.39 we obtained the radial distribution function for monodispersed hardsphere particles in the bulk by solving the O-Z equation N
hij(r12) ) cij(r12) +
∑ ∫ F h (r k ik
13)ckj(r32)
dr3
(1)
k)1
where h(r) is the total correlation function that accounts for the direct and indirect particle-particle interactions in the multiparticle system and is related to the radial distribution function, g(r), by the equation h(r) ) g(r) - 1
(2)
c(r) is the direct correlation between the two particles, N is the number of different types of particles in the system, and rk is the density of the particle type k. Different approximations have been suggested to solve the O-Z relation and one among them is the Percus-Yevick relation, [[h(r) + 1][1 - exp(U(r)/kT)] ) C(r)]
(3)
where U(r) is the hard sphere interparticle interaction energy, C(r) is the direct correlation function between the particles, k is the Boltzmann’s constant, and T is the temperature. The effective energy of interaction between the particles or the potential of the mean force, W(r), is calculated from the radial distribution function using the equation, W(r) ) -kT ln g(r)
(4)
Figure 5 shows the radial distribution function for several volume fractions of particles in the bulk. The first maximum
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Figure 5. Radial distribution function g(r) for various volume fractions in the bulk calculated from the O-Z integral equation.
Figure 6. Particle interaction energy W(r)/kT calculated from the PY g(r) for various particle volume fractions.
(or peak) increases with the increasing concentration of particles. The second maximum is not clearly observed for the low volume fraction of 0.3. However, the second peak becomes pronounced with an increase in the volume fraction. The radial distribution function for various volume fractions of hard-sphere particles (3-D) in the bulk was also verified using Monte Carlo simulations. For the bulk, hard spheres were used to obtain an equilibrium in a canonical (NVT) ensemble. The length of the cell was varied as L3 ) nφd3/6φ, where d is the diameter of the particles, n is the number of particles in the cell, and φ is the volume fraction. The volume fraction is the ratio of the volume occupied by the spheres to the total volume (V) of the system and is given by the following expression: volume fraction ) φ )
Fπd3 6
(5)
where F ) N/V. The number of the particles in the cell for the Monte Carlo simulations was fixed at 1024 for regions other than the phase transition region, where it was fixed at 1500. The results obtained using both O-Z integral equation and the Monte Carlo methods were comparable. Figure 6 shows a plot of the effective energy of interaction between particles. The contributions from the repulsive energy and the attractive depletion are illustrated in the inset with arrows. The magnitude is the energy contribution from the secondary minima to the maxima for the repulsive energy and maxima to primary minima for the attraction energy. For a volume fraction of 0.45, the magnitude of the repulsive energy is about 0.75kT while the magnitude of the depletion attraction energy is 1.9kT. Figure 7 shows that the particle depletion attraction at a particle volume fraction higher than 0.45 is greater than 1.5kT, and one can expect particle aggregation to initiate at this point. In the actual system, due to the biphilic (amphiphilic) nature of the particles, the aggregation is expected to begin at a lower particle concentration.
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Acknowledgment This work was supported by the Office of Science, U.S. Department of Energy, grant no. DE-FG02-05ER64004. Note Added after ASAP Publication: The version of the paper that was published online April 7, 2009 had errors in the description and presentation of eq 5. The corrected version of this paper was reposted to the Web April 9, 2009. Literature Cited
Figure 7. Depletion attraction energy well vs particle bulk volume fraction. Insets are schematics of the system at various volume fractions as obtained using snapshots of our Monte Carlo simulations.
The insets in Figure 7 also show the schematic of the packed concentration calculated from the Monte Carlo simulations as mentioned above with an increasing volume fraction of particles in the bulk. Our experimental results indicating a decrease in foaminess after about 40 vol % are in good agreement with the simulations and theory. It should be noted that the theory is for a monodisperse, hard-sphere system, and that our experimental system contained particles polydisperse in size and irregular in shape. Also, foaming is a dynamic process whereas the theoretical results are based on an equilibrium analysis. Another major factor contributing to the decrease in the foaminess at higher particle concentrations is the increase in the viscosity in the bulk. Phan et al.40 experimentally determined the viscosities of hard-sphere suspensions and found that beyond a concentration of 46 vol%, the viscosity shows a sudden increase in the region of the liquid-solid phase transition. Beyond this value, the system becomes very viscous. As the system becomes more viscous, the incorporation of air becomes difficult. Dickinson et al.12 and Binks et al.21 also arrived at this conclusion. Conclusions The foaming power of small, solid, partially hydrophobized particles increases with an increase in the concentration in amphiphilic or partially wetted particles, and a maximum in foaminess is observed experimentally at around 40 vol % in the bulk. Theoretical analyses based on the integral equation of statistical mechanics using hard-sphere particles and the Monte Carlo simulations were used to explain the experimental observations. The foaminess is directly correlated to the coverage of the particles at the surface. The surface coverage increases as the number of particles in the bulk increases, but due to the clustering (aggregation) of particles in the bulk, the coverage eventually decreases. It has been found that beyond a particle concentration about 40 vol %, fewer particles attach to the bubble and approach the air-water interface owing to the clustering of amphiphilic particles in the bulk. This causes a reduction in the surface area covered by the particles and simultaneously decreases the resistance to the coalescence of bubbles, resulting in a decline in foaminess. Therefore, we are able to rationalize our experimental results using a simplified theoretical analysis. This model should be applicable to a number of other foaming systems containing finely divided particles.
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ReceiVed for reView November 14, 2008 ReVised manuscript receiVed February 25, 2009 Accepted February 26, 2009 IE801741Q
