Predicting the Failure of a Thin Liquid Film Loaded with Spherical

Article pubs.acs.org/Langmuir

Predicting the Failure of a Thin Liquid Film Loaded with Spherical Particles Gareth Morris,* S. J. Neethling, and Jan J. Cilliers Rio Tinto Centre for Advanced Minerals Recovery, Department of Earth Science and Engineering , Imperial College London, London SW7 2AZ, U.K. ABSTRACT: A model is presented for predicting the failure of a thin liquid film stabilized by attached inert particles. A statistical analysis of roughly 3500 Surface Evolver1 simulations was used to identify the relationship between the packing density of the particles on the film, their contact angle distribution, and the capillary pressure required to rupture the film. The model presented allows a fast and simple method of calculating the range of pressures a thin film in a three-phase froth will fail at based upon three variables: the film loading, mean particle contact angle, and the standard distribution of contact angles round the mean. The predicted range of failure pressures can be used in simulations of bulk froth properties where bubble coalescence is an important factor governing the froth properties.

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INTRODUCTION When small particles attach to a thin liquid film, they can either increase its lifetime or actively destroy it. As many industrial processes rely on foams and thin films, investigation into the effects of particles on the film’s properties is an important step in developing better, fundamental models of film failure and more complex bulk foam simulations.2−5 Froth flotation is one such example; it is a process used by the mining industry to concentrate low grade ore using bubbles to collect particles of valuable mineral (90°), above which film failure occurs immediately. Brito e Abreu et al.19 developed a method of deriving the contact angle of particles using ToFSIMS. It is based upon measuring the species present at the solid surface that contribute to the particle contact angle. Brito e Abreu and Skinner20 then used this approach to derive the distribution of contact angles of particles recovered from batch flotation experiments. They found that a Gaussian distribution could be applied to the contact angles of the particles in both the concentrate and the tails. However, it should be noted that their sample size was not large enough to fit an appropriately reliable statistical distribution. The interior structure of a flotation froth is poorly understood and difficult to measure; often the interior properties are inferred from the upper surface of the froth. Wang and Neethling21 showed that even monodisperse foams have a wide range of bubble film sizes on the top surface, meaning that this is not a good measure of froth property. However, the need to understand the interior structure of the froth is still a hugely important step in modeling its bulk properties. Neethling and Cilliers6 developed a fundamentally based model of flotation froths, in which the bubble size evolution is calculated using the capillary pressure in the system and a user defined Pcrit to describe coalescence but accurate prediction of Pcrit based on knowledge of the properties of the particles attached to the film is still difficult. It is the bubble coalescence that governs the bubble size distribution within a

Figure 1. 2D analytical layout for spherical particles in a film (L = liquid, V = vapor, S = solid). R is the radius of curvature of the liquid− vapor interface, θ is the contact angle between the (tangent) of the LV interface and the solid surface at the three-point contact, and rp is the radius of the particle.

roughly proportional to the cosine of the particle contact angle. In both cases it was assumed that particles were in a single layer in the film, and in this case, the maximum sustainable contact angle is 90°. Above this and the particles actively break the film by drawing the opposite LV interface together, destroying it and reducing Pcrit to zero. γ ΔP = (1) R It was also shown through the analytical models of Denkov et al.8 and Ali et al.9 that when particles are evenly spaced they are quasi-stable but that if they are slightly perturbed the forces acting on them become unbalanced, and they will clump together on the film. This opens up large areas free from particle stabilization and reduces the film stability. In 2D the complex distortion of the LV interface is ignored, and the gaps between the particles cannot be accurately resolved. For these reasons it is necessary to expand the analysis into three dimensions (3D). In the 2D case particle agglomeration is represented as a simple drawing together of particles. However, in 3D this process is much more complicated as particle shape and size as well as interparticle friction can cause the pack to jam before it can reach a close-packed ordered state. This means the clumps are not the neat close-packed pattern that one would expect from the idealized case but are often quite loose aggregates with pockets of empty film formed when particles lock up in complex patterns. Kam and Rossen13 used an analytical approach to investigate the phenomena of an armored bubble in 2D. They found that the particles coating the bubble can help prevent its dissolution into the surrounding liquid by locking together and stabilizing the LV interface at zero or negative capillary pressures. Abkarian et al.11 built on the methods of Lauga and Brenner,12 who used Surface Evolver to investigate the effect of capillary forces and gas dissolution on the packing arrangements of particles attached to a small bubble in liquid in 3D. All found that as the volume of gas shrinks and the particles are drawn together the bubble takes on a polyhedral shape which represents a minimum-energy configuration. In their models they represented the particles as immiscible liquid droplets with a surface tension much higher that the LV interface, ensuring that a spherical shape was adopted without the need to define the particles shape or positions. The method presented here replaces the particle surface with an energy integrand representing the particle surface. While this requires a more complex initial model the simulation itself runs much more smoothly and quickly once it has been generated. In addition, flotation films, while starting small (90°) contact angles also increases. At the largest sample size (40 particles) the number of particles with a contact angle above 90° was only a problem for systems where θmid + 0.5θvar ≥ 90°, and the effects of this are covered in the Results and Discussion section. Several combinations of AFP, θmid, and θvar were chosen for simulations detailed in Table 1. A series of models for each combination in a particular row were simulated. For example, for a contact angle range of 60°, four sets of midpoint contact angles were used (30°, 40°, 50°, and 60°), and for each of these combinations of θvar and θmid, five packing densities were used, resulting in 20 sets of initial criteria (from the bottom row of Table 1). Each one of these initial criteria was then used to generate 20 models with a random contact angle distribution and particle positioning, resulting in 400 unique models for the bottom row of Table 1. All but one of the models used the same periodic cell edge length (L*) of 20; this was populated by 5, 10, 20, or 40 particles to give an AFP of 76.86, 36.86, 16.86, or 6.86, respectively. The final AFP of 3.25 was achieved by placing 40 particles in a periodic cell of edge length 16. The convergence of a Surface Evolver model for a given capillary pressure (P*) was established by monitoring the change in film shape. The use of the convex type of constraint

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RESULTS AND DISCUSSION Comparison with previous model results15 showed that the energy constraint method used in this article consistently predicts a lower Pcrit (Figure 3). This was due to the smoother and more reliable meshing used in the energy constraint method. By removing the solid surface of the particle, the mobility of the TPC was greatly increased and also less affected by the mesh resolution of the particles. This means that it was possible to reach lower energies (and produce thinner films) for a given P* than the models that included the particle surface. It is important to note that in the results presented here the trends are the same; only the constants have different values. When a comparison with a simple one particle model was conducted, the same film shape at the same pressure was found for the particle surface model and the energy constraint model. However, this required significant bespoke tailoring of the evolution procedure to the model. This was intractable for 998

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Figure 3. Pcrit as a function of particle packing density (AFP) for contact angles of 15° (left) and 75° (right) and θvar = 0°. The Surface Evolver models with all surfaces modeled tend toward higher predicted values of P*crit.

Figure 4. Left: critical capillary pressure (P*crit) as a function of inverse film area per particle (1/A*FP); the three cases shown represent a midpoint contact angle of 40° with three distributions θvar = 20°, 40°, and 60°. Right: P*crit as a function of cos θmid for four different AFP*. Error bars represent the 95% confidence interval.

Figure 5. Critical capillary pressure (P*crit) as a function of film area per particle (AFP*) for three different contact angle ranges: 20°, 40°, and 60°. Two cases are shown: the left is for a midpoint contact angle (θmid) of 40° and the right for one of 60°. 999

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Figure 6. (left) Averaged simulated critical capillary pressure as a function of predicted P*crit. Error bars represent the 95% confidence interval. (right) The individual simulated values of P*crit plotted against the predicted values for P*crit, as simulated in Surface Evolver. In both cases K = 0.775, A = 0.791, B = 0.113, and C = 0.812.

This article aims to develop a model of P*crit based on the assumption that AFP, θmid, and θvar are known and then return a range of values for the P*crit. However, if as in the case for θvar = 60° and θmid = 60° there is the likelihood of the film containing a particle with θ > 90°, a proportion of the films will fail immediately. This issue will be dealt with later in this article. Building on eq 8 and adding in the θvar variable, the value of P*crit is predicted using eq 9, where K, A, B, and C are constants fitted to the data by minimizing the sum of the square of the differences between P*crit as predicted by the models and P*crit as predicted by eq 9.

larger scale models containing more particles as well as computationally more expensive. Comparing the averaged values of P*crit for a given set of conditions showed that the same trends as those previously reported, with P*crit inversely proportional to the AFP and proportional to cos(θmid) (Figure 4). However, it can also be seen in Figure 4 that there was only a small variation in P*crit with the distribution of contact angles over a given θmid. The values plotted have 95% confidence intervals included based on 20 simulations of each combination of variables. The P*crit was only noticeably lower in Figure 4 when θvar was 60°, implying that a fairly narrow range of contact angles (θvar < 40°) has little effect on the average P*crit of a loaded film. In two dimensions (2D) the capillary pressure required to rupture a film is a simple geometric consideration9 shown in eq 7. This was expanded to eq 8 in 3D15 for particles with a single contact angle, where K is a fitted constant of 2.31. P*crit =

P*crit =

2 cos θ *2 − 1 SPP

(7)

K cos θ AFP*

(8)

* = Pcrit

K cos θmid A cos θvar B * AFP

C

(9)

The predicted P*crit values plotted against averaged values of the simulated P*crit are shown in Figure 6 (left) (K = 0.775, A = 0.791, B = 0.113, and C = 0.812). It is clear that there is a linear trend between the two. If the averaged values were replaced with the actual values, as in Figure 6 (right), it can be seen that as the average P*crit increases the distribution of values for simulated P*crit widens. There were some values out of sequence and close to zero for the simulated P*crit that can be seen in Figure 6 (right). They represent the models where one of the particles has been attributed a θ > 90° which broke the film immediately. Models were more likely to have a particle with θ > 90° when they have a high θmid and θvar; however, due to the high average θ of the other particles, they also tended to have lower P*crit. Models with lower θmid and θvar also tended to have higher P*crit which is why the zero value P*crit were only present for model sets with low average P*crit. Plotting the predicted values of P*crit alongside the averaged values again showed good correlation between them (Figure 7). If the standard deviation of P*crit is plotted as a function of the average pressure for each set of variables, two trends arise as

If we compare the individual values of P*crit for each model as a function of AFP* over a range of θvar as in Figure 5, it can be seen that the spread of the values returned by the numerical simulations increases as the A FP* decreases (and P*crit increases). It was also noticeable that for a θmid of 60° and θvar of 60° there were a number of models that had a P*crit of, or close to, zero. These were the models that had one or more particles with contact angle of, or approaching, θ = 90°. In this case the particles in question act as foam breakers and draw the opposite sides of the film together, destroying it almost immediately. 1000

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Figure 7. A comparison of predicted and simulated values of P*crit over a range of variables using the same fitted values as in Figure 6. P*crit is shown as a function of AFP*.

gradient set. The larger range of simulated P*crit arising in the steeper case are due to the few models containing particles with θ > 90°. These individual cases have a P*crit of zero and skew the standard deviation. They do, however, show the necessity to take into account active film destruction by “foam breaking” particles when modeling a froth that has a number of particles with high contact angles. Both cases show that as P*crit increases so too does the range of values it is spread over, for a given case. This distribution of P*crit about the values predicted using eq 9 will be discussed next. The distribution of the simulated P*crit about the predicted value was analyzed in two sets: one in which the upper bound of the contact angle range (θmax) was 90° and one for all other values. This reflects the different trends in standard deviation seen in Figure 8. Figure 9 shows the normalized distribution of the percentage difference between the simulated P*crit and predicted one using eq 9. A Gaussian distribution has been fitted to both sets of data with A = 0.122, B = 0.069, and C = 0.161 for case A and A = 0.09, B = 0.158, and C = 0.203 for case B. It can be seen in Figure 9B that just under 4% of the values have a difference of 1 from the predicted value of P*crit. These represent the cases where the model failed at P* = 0, due to particles with contact angle above 90° being present in the film.

shown in Figure 8. The standard deviations that fall on the steeper gradient correspond to sets that have an upper limit of 90° (θmid + (θvar/2) = 90°), all other cases fall on the lower

Figure 8. Showing the standard deviation of P*crit values as a function of average P*crit for that range. 1001

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Figure 9. Gaussian distribution of the percentage difference between simulated P*crit and the predicted values for (A) all values with a θmax < 90° and (B) all values where θmax = 90°.

and organized structures would begin to appear which may affect the predicted P*crit in ways this model cannot predict. This model encompasses a wide range of AFP* but those found in a flotation froth would be expected to be close to the minimum values of AFP* presented. At larger values of AFP* users should be aware that particle aggregation would occur and alter the values of P*crit from those predicted by the model.

The normal distribution of contact angles means that it was possible for any model to contain a particle with a θmax of 90°. However, this was strongly dependent upon the size of the set or the number of particles in the film. As the largest number of particles in a film was 40, the inclusion of a particle with a contact angle above 90° only occurred for models with a θmax approaching 90°. This resulted in ∼4% of the models with a P*crit = 0 due to particle bridging failure. If these models were increased in size to the many thousands of particles found on a flotation film, the normal distribution would mean that the presence of at least one particle with θ > 90° is inevitable. Even if the models with a P*crit of 0 were discounted, the standard deviation from the predicted P*crit for models with a θmax of 90° was still around 3 times larger than those for other combinations of variables. Simulations of bulk froth properties involving particles with contact angles approaching 90° should therefore be treated as a special case. Initially, the wider normal distribution of P*crit about the predicted value should be used to predict coalescence. In addition, there will also be a portion of films that need to be treated as actively destroyed at P* = 0. The actual percentage of films that fail at P* = 0 will be proportional to the percentage of particles in the distribution that have a contact angle above 90°. An in-depth analysis of the effect of a small number of high contact angle particle acting as film breakers within a froth is beyond the scope of this article; however, a simple approach would be to treat the proportion of films that break at P* = 0 within the froth as equal to the proportion of particles that have a θ > 90°. However, flotation froths rarely contain particles with such high contact angles so this is unlikely to be a problem in their simulation. Returning to the more regular cases of θmax < 90°, the Gaussian distribution fitted to the predicted P*crit shown in Figure 9 can be used in conjunction with eq 9 to generate a range of values for film coalescence within simulations of flowing froth if θmid, θvar, and AFP are known. Finally, it should be noted that a much greater range of particle packing densities have been modeled and used in this analysis than would appear in a flotation froth. The higher values of AFP* were included to show the fit of the model over a greater range. The failure pressure of films with smaller values of AFP* can be generated but the user should be aware that as one decreases AFP* much further beyond 2, very closely packed

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CONCLUSIONS A model for predicting the failure of a thin liquid film coated in spherical particles has been presented. Both the mean critical capillary pressure and distribution of critical capillary pressures about the mean can now be calculated based upon knowledge of the particle packing density on the film, the average contact angle of the particles, and the standard deviation of θ from the mean. This assumes a Gaussian distribution of contact angles in the film and also returns a Gaussian distribution of critical capillary pressures. The discrete dynamics between large numbers of particles within a thin film in the lead up to coalescence is a complex topic that remains the focus of much investigation. However, this article provides a statistical method of predicting the failure pressure of a particle laden film, allowing the individual particle effects on the film to be discounted. The statistical model can therefore be used in the bulk simulations of three-phase froths and foams, providing greater accuracy in assumptions on bubble coalescence while removing the need to discretely simulate the particle−film interaction, which is at present computationally intractable for a froth of any meaningful size.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (G.M.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The author thanks Rio Tinto for funding this research which was carried out at the Rio Tinto Centre for Advance Minerals Recovery at Imperial College London. 1002

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