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Detailed Structural and Mechanical Response of Wet Foam to the Settling Particle Zefeng Jing,* Shuzhong Wang,* and Zhiguo Wang Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shanxi, China S Supporting Information *

ABSTRACT: Liquid foam, as a complex fluid, provides an observable prototype for studying a discrete fluid system. In this work, a numerical study on the settling behavior of a round particle in wet polydisperse foam has been conducted on the bubble scale. The local and nonuniform distribution of bubble pressure, as well as the localized plastic events, is presented. It shows a foam region of higher pressure in front of the settling particle due to the extrusion deformation of the bubbles applied by the particle. Additionally, the forces exerted on the particle by the disordered wet foam are measured during the sedimentation. It exhibits in particular a power-law dependence of the drag force caused by the bubble as a function of the foam quality. Moreover, sedimentation experiments are demonstrated to verify this power-law relation. The evolution of the components of drag force is demonstrated when a plastic event occurs in front of the settling particle. The result shows that both the contributions of the pulling force of foam films and the bubble pressure to the drag force decrease in that case. Likewise, the variation of both these contributions to the drag force is illustrated as well when a bubble in the wake detaches from the particle. These results assist in understanding the mesoscopic response of wet foam to a settling particle.

1. INTRODUCTION

Liquid foam, unlike colloids or polymers, provides a practical geometry as a model for studying complex fluids since its constituents, foam films, are experimentally easy to observe.18 Thus, some detailed studies concerning the particle moving in a foam, such as local plasticity,19,20 local elasticity,21 and the drag force components,21−24 have been conducted. The particles, much smaller than the bubble size, moving within the Plateau borders of foam have been mostly investigated in the froth flotation.25 Nevertheless, larger particles are able to squeeze or stretch the foam films as moving through the foam.24 Cantat and Pitois20 performed the quasi-static steady flow of a monodisperse dry foam around a fixed spherical bead whose size was a few times larger than the typical bubble size. Through image analysis, it was found that these plastic events26 occurred mainly within the first bubble layers around the bead. Le Goff et al.27 studied the motion of a sphere after its fast impact on liquid foam and found that foam’s elasticity started to have an effect on the sedimentation as the velocity of the sphere was below a threshold. They further interpreted the settling character using a visco-elasto-plastic model for foam, and yet there was no deeper research on the scale of foam film in their report. In fact, the elastoplasticity of foam originates from the mesoscopic film, which produces drag force to influence the sedimentation of particle.

Liquid foam, like an emulsion or colloid, is a multiphase system with the discrete phase dispersed in the continuous liquid phase.1−3 It can display simultaneously nonlinear viscous, elastic, and plastic behavior.4−6 This complex behavior is widely used in many industrial applications, such as enhanced oil recovery,7,8 ore flotation,9,10 and soft filtering materials.11 Particle settling in this fluid, similar to the classical Stokes experiment,12 is widespread in these applications. The motion of particle in these complex fluids has been extensively studied.13−17 Tabuteau et al.14,16 measured the drag force exerted on a sphere as it moved steadily through yieldstress Carbopol gels fluid. They found that the flow regimes observed for the falling sphere were analogous to those observed in creep tests for different applied stress levels. Gumulya et al.15 discussed the settling behavior of particles in shear−thinning thixotropic fluids by the numerical scheme. Their results showed that the flow field surrounding the settling sphere was highly localized, with distinct regions of disturbed/ undisturbed fluids. They also proposed that the extension of these regions depended on some rheological parameters of the fluid. Nevertheless, the more detailed studies, such as the link between the more detailed components of the settling drag force and the subtle change of the fluid in the localized region, are needed as the particle settles through the complex fluids, especially for the discrete fluid system. © XXXX American Chemical Society

Received: January 25, 2016

A

DOI: 10.1021/acs.langmuir.6b00281 Langmuir XXXX, XXX, XXX−XXX

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local distribution of pressure of bubbles around the settling particle. The evolution of drag force, as well as its detailed components, exerted on the particle by the disordered foam is also analyzed on account of the structural changes in the local bubbles surrounding the particle. In addition, the foam quality is extended to the lower level and the effect of foam quality on the detailed components of drag force is presented. At last, the T1 plastic events for the wet foam of different foam qualities are recorded during the sedimentation and its distribution characteristics are displayed as well.

Contributions to the settling drag force exerted on the obstacle by the foam were decomposed into an elastic one and a fluid one by Dollet et al.21,28 Furthermore, the elastic contribution has two components: an elastic one caused by the network of foam films and a pressure one due to the bubble in contact with the obstacle.21 Nevertheless, these studies related to the drag force are generally limited to the foam of very high foam quality, and more detailed variation of these components of drag force is deficient when the films around the particle undergo structural change, such as the elastic-plastic deformation. Although Raufaste et al.29 involved the effect of foam quality, the value of foam quality was higher than 0.985 since lower values would lead to numerical problems in their simulation. It is worth mentioning that the forces of the settling circular discs applied by dry two-dimensional (2D), monodisperse foam were presented by Davies and Cox.23 They reported single as well as two circular discs settling through the two-dimensional dry foam. Subsequently, Davies et al.30 and Davies and Cox31 also developed quasi-static sedimentation of sphere in the three-dimensional (3D) dry foam. With the help of X-ray tomography, they demonstrated the validation of both the 2D and 3D simulations. In their simulations, the foam films that met the circular disc were always perpendicular to the edge of this disc. However, for the wet situation (see Figure 1), the

2. NUMERICAL METHOD In this study, the size of settling particle is assumed to be approximate to average bubble size. Therefore, the particle settling through the wet foam will not be considered to be a classical Stokes experiment owing to the discrete nature of wet foam. Davies and Cox23 proposed that the forces exerted on a settling particle in a foam sample should consist of the particle’s weight, the pulling force of films, the resultant pressure force from the contiguous bubbles, the viscous force and the frictional force. The frictional force, including the viscous component, caused by the liquid phase is generally proportional to the settling velocity of the particle when the motion of particle in a fluid is slow. According to Newton’s second law, the equilibrium equation of these forces can be given by d2S (⃗ t ) dS (⃗ t ) = Fp⃗ + Fn⃗ − G⃗ − λ 2 (1) dt dt where S(⃗ t) is the position of the settling particle at time t, G⃗ is the gravity of the particle, and λ is the friction coefficient. The total pulling force exerted on the particle by foam films in contact with the particle is given by23,32 m

m

Fn⃗ = γ ∑ (cos θi , sin θi) filmi

(2)

where m is the total number of films that meet the particle, γ is the line tension of each film i, and θi is the angle between the direction of the pulling force and the positive direction of the x axis. The total bubble pressure exerted on the particle can be written as23,32

Figure 1. Forces exerted on the settling particle. Each force is resolved into the x and y directions. The direction of the pulling force Fn⃗ is tangential to the bubble film in contact with the particle. The direction of pressure force Fp⃗ exerted on the particle is perpendicular to the particle’s edge.

n

Fp⃗ =

∑ bubblej

pj l j(cos θj , sin θj)

(3)

where n denotes the total number of bubbles that touch the particle, pj is the relative pressure of the bubble j, lj is the contact length between the particle and the bubble j, and θj is the angle between the direction of the pressure and the positive direction of the x axis. As we know, the drainage due to the gravity has an influence on the distribution of liquid in the foam. It can cause that the foam quality from the top to the bottom of the foam column is changing33 and thus the effect of given foam quality on the sedimentation will not be precise if the drainage is considered. Moreover, the drainage for a stable foam is generally inconspicuous in a certain amount of time. Therefore, based on the aim of this study, the gravity of the continuous liquid phase can be neglected to eliminate its effect on the foam drainage. On the other hand, although there is differential pressure between these bubbles as well as the liquid phase and the gas phase, the pressure of the continuous liquid phase is consistent. Otherwise, the liquid in different regions will adjust its own distribution to reach the pressure

liquid films move around the settling particle. There should be a different contact angle between the films and the particle’s edges. Ireland and Jameson24 explored the effect of this contact angle on the average settling drag force and found that the drag force increased with this contact angle. In addition, the foam structure is maintained by both the surface tension of foam films and the capillary pressure generated by the geometry of foam.23 The pressure distribution in this complex wet foam system not only is nonuniform but also changes with the elastic-plastic deformation of the films. Therefore, it is expected that the settling particle has an influence on the bubble pressure in the wet foam. In the present work, the aim is to further understand the mesoscopic response of wet foam to the settling particle in a deeper level. By using a numerical method, we focus on the sedimentation of a single particle in the wet foam and show B


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Figure 2. (a) A settling round particle in a two-dimensional wet foam of foam quality Φg = 0.92 and area disorder36 μ2(A) = 0.0432. The periodic boundary condition is maintained in the y direction. The vertices of films that join either wall are fixed to implement a no-slip boundary condition. (b) The trajectory of the center coordinate (x0, y0) of the round particle as it settles through the foam channel. The initial center coordinate of the particle whose area is 0.0265 is at the coordinate (1.172, 3.889). The contact angle between the film and the particle’s edge is set to 10.284°.

configuration of a wet foam with required liquid fraction is built first and some more details have been described by Jing et al.26 One bubble is then selected as the settling particle. In the simulation, the particle’s edges are constrained to keep a circle of constant size and thus the corresponding constraint can be written as

equilibrium. Accordingly, the pressure of the liquid phase, without regard to the relative pressure pj of the bubble, distributes evenly around the particle (i.e., its contribution to Fp is 0). In the actual computation, the pressure pj is the relative pressure of the bubble j to the continuous liquid phase. Consequently, as described in eq 3, the total contribution of pressure exerted on the particle is all due to this relative pressure pj of the bubble. In this case, the forces exerted on the particle in the x and y directions by the foam films and bubble pressure can be referred to as the lift force Fx and drag force Fy,34 respectively: Fx = Fnx + Fpx

(4)

Fy = Fny + Fpy

(5)

(x − x0)2 + (y − y0 )2 = r0 2

where (x0, y0) is the center coordinate of the circle and r0 is the radius. Subsequently, the structure is relaxed toward a quasiequilibrium configuration by minimizing the free energy26 of the foam. The two main methods of energy minimization in the surface evolver are the gradient descent method and the conjugate gradient method. In the simulation, the initial quasiequilibrium configuration of the settling model is obtained by combining these two methods, and then the initial forces exerted on the particle are measured. The center coordinate (x0, y0) of the particle is moved according to eqs 8 and 9 in the x and y directions of the resultant force applied by both the wet foam and its own weight, respectively.

The motion of the particle in the wet foam is assumed to be slow and steady. As a result, the acceleration term can be negligible23 and then eq 1 is simplified to the following equation 1 dS (⃗ t ) = Fp⃗ + Fn⃗ − G⃗ ξ dt

(7)

(6)

where ξ = 1/λ denotes the effective time scale of the motion. The quasi-static “surface evolver” method35 is adopted in our simulation. The main assumption of the quasi-static model is that the time scale at which the foam returns to quasiequilibrium is shorter than all other time scales.23,26 Likewise, the default system of units in the surface evolver is adopted and thus the default unit of each parameter can be omitted to make the simulation more convenient. The tension of air−liquid−air interfaces γ1 is set to 0.990 and the tension of air−liquid interfaces γ2 = 0.497, which are consistent with the parameters in another paper of ours.26 To ensure that the motion is not brought to a halt by the foam, the gravity of the particle is kept at G = 9.0. To generate the settling model, the equilibrium

Δx = ξ(Fnx + Fpx)

(8)

Δy = ξ(Fny + Fpy − G)

(9)

In the simulation, to avoid the numerical problems, ξ is set to a small value of 2 × 10−4. The process of energy minimization is repeated by the conjugate gradient method until the energy of the system has converged to at least six significant figures after the decimal point, and then the forces exerted on the settling particle are measured. Similarly, the motion of the particle is implemented again, and then the quasi-equilibrium configuration is again obtained by the conjugate gradient method after the particle is moved to a new position. These processes are continued until the particle settles to the bottom of the foam channel. Therefore, these simulations consist of a C


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Figure 3. Bubbles are colored depending on their relative pressure. The pressure increases with the order blue, light blue, cyan, green, light green, light red, and red. Each figure represents the part of pressure field of the foam of Φg = 0.92 under different iteration numbers. The black circle denotes the settling particle.

3.2. Forces Exerted on the Particle. The forces exerted on the settling particle by the foam versus the iteration number are illustrated in Figure 4. It can be obviously seen that the drag

sequence of quasi-equilibrium configurations. Because of the set area constraint of each bubble in the foam sample, both the area of each bubble and the liquid fraction of the whole foam will remain unchanged when the quasi-equilibrium configuration of the foam is obtained at each iteration. On the basis of the aim of this study, it should be noted that both Ostwald’s ripening and bubble coalescence that lead to the change of bubble size are not involved in the simulation since evolution of these phenomena for relatively stable foam is slow and can be negligible in practice.27 The computational time of the simulation for each wet foam sample is more than 1 month. Figure 2 shows the settling model and the corresponding trajectory of the particle descending under its own weight in the wet foam channel. As shown in Figure 2, this trajectory has slight lateral fluctuation from the vertical direction due to the asymmetric structure of the wet foam. This structure leads to uneven distribution of the forces exerted on the settling particle. The nonuniform force will be presented in the next section.

Figure 4. Evolution of the drag and lift forces exerted on the particle by the foam of Φg = 0.92 versus the iteration number. These dashed lines denote two average values of the drag and lift forces. To eliminate any artificial error in the initial structure, the average value is calculated from the 101th iteration.

3. RESULTS AND DISCUSSION 3.1. Bubble Pressure around the Particle. Particle squeezes or stretches the films when settling through the foam, which gives rise to local change in the characteristic parameters of a foam sample. The relative pressure of a bubble to the continuous liquid phase can be given by the surface evolver method as the Lagrange multiplier of the area constraint on each bubble.26 Figure 3 demonstrates the local and nonuniform distribution of bubble pressure in the wet foam. As shown in Figure 3, in the wake of the settling particle, the localized liquid fraction increases, while in front of the particle, the pressure of the localized bubbles is also increased and the region of foam with higher pressure extends to several bubbles. These results agree with the case of dry foam in the experiment of Davies et al.30 As the particle settles in the wet foam, the bubbles in front of the particle are squeezed, which causes the increase in the elastic deformations of these bubbles as well as the intermittent T1 topological events. These elastic deformations make the mean curvature K of these bubbles increase. Consequently, according to the Laplace−Young law Δp = 2γK, the relative pressure level of these bubbles increases. At the back of the particle, because of increasing localized liquid fraction, the bubbles become more rounded, and accordingly, the elastic deformation between these bubbles decreases. Therefore, compared to the bubbles in front of the particle, there is a region of lower pressure in the bubbles at the back of the particle.

and lift forces fluctuate around their corresponding average values, which is similar to the trace of force applied by the foam in the experimental result of Cantat and Pitois20 as well as Ireland and Jameson.24 These fluctuations are a result of the gradual evolution of the foam films, including the movement of the films surrounding the particle (elastic loading) and the T1 topological event (plastic rearrangement). The average value of lift force approximates to zero, which indicates that the horizontal displacement is small as the particle settles through the wet foam. To show the bubbles’ response in more detail, the contributions of foam films as well as bubble pressure to the drag and lift forces are illustrated in Figure 5. As shown in Figure 5a, the average value of Fpy is higher than the one of Fny, which indicates that the contribution of bubble pressure to the drag force is greater. For the lift force, these average values of its components are smaller and their deviations from the zero point are a result of the asymmetric foam structure. The sedimentation will cause the displacements of bubbles and the changes in the thickness of liquid lamellae separating the bubbles in the front and wake of the particle (see Supporting Information), which influences the forces acting upon the particle. In general, it is more interesting to take into consideration the response of the forces to the topological change (T1 event) occurring near the particle in detail. Figure D


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Figure 6. Bubbles surrounding the settling particle at different iteration numbers: (a) the 1185th iteration and (b) the 1190th iteration. These two pictures show the parts of the foam of Φg = 0.92 as the particle settles in the foam channel. The red arrow denotes the direction of gravity. (c) The corresponding contributions of the pulling force of films and the bubble pressure to the drag force Fy during 1180−1190 iterations. The separation between bubbles a and b occurs at the 1186th iteration and the T1 event between bubbles c and d appears at the 1187th iteration.

Figure 5. Variation of the contributions of the pulling force of films and the bubble pressure to (a) the drag force Fy and (b) the lift force Fx versus the iteration number. These dashed lines denote the average values of these corresponding forces.

6a,b demonstrates the foam structure around the settling particle before and after a T1 event between bubbles; a and b appears in front of the settling particle, respectively. As shown in Figure 6c, both these contributions of the pulling force of films and the bubble pressure to the drag force decrease as this T1 event appears. On the one hand, the T1 event changes the local structure of foam. By comparison of the foam structure near the particle in Figure 6b,a, it can be obtained that two bubbles (i.e., the bubbles a and b in Figure 6) are squeezed to both sides of the particle and the thickness of liquid lamellae separating these two bubbles increases. The direction of pulling force of the films in contact with the particle changes, which makes the downward pulling force of these films increase. The negative contribution of the pulling force to the drag force increases and thus the Fny decreases. On the other hand, the interaction between these two bubbles disappears as the T1 event occurs. The bubbles become more rounded and the mean curvature K of these bubbles decreases accordingly. The pressure of these two bubbles decreases, according to Laplace− Young law. Therefore, as displayed in Figure 6c, the contribution of the bubble pressure to the drag force decreases as well. Additionally, the mean curvature of bubble a decreases further, owing to the separation between bubbles c and d at the 1187th iteration, which will also cause further reduction in the pressure of bubble a. Thus, as illustrated in Figure 6c, the pressure component Fpy decreases further. Parts (a) and (b) of Figure 7 display the foam structure around the particle before and after the bubble in the wake

detaching from the particle, respectively. From Figure 7c, as the detachment is triggered, the contribution of the pulling force of foam films to the drag force decreases, while there is no obvious change in the contribution of bubble pressure. In Figure 7a,b the pulling force exerted on the particle by the bubble e disappears owing to the detachment. As a result, the contribution of the pulling force decreases. In addition, according to eq 3, the contact length between the particle and the bubble e is so small that the contribution of bubble pressure changes little when this detachment happens. 3.3. Effect of Foam Quality on the Drag Force. In general, we are more concerned with the drag force compared to the smaller lift force. The foam quality is one of the important parameters that influence the foam’s rheology,36 and it should have an effect on the settling particle accordingly. To show the effect of foam quality on these forces, the evolution of average values of drag and lift forces is displayed in Figure 8. From Figure 8a, it is shown that the drag force increases gradually with the increase of foam quality. For the foam with lower foam quality, the particle has bigger space to pass these bubbles. Accordingly, the interaction between the particle and these bubbles weakens. The relationship between the drag force and foam quality, according to Figure 8a, is in accordance with the power-law model. The model can be given by E


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Figure 7. Bubbles around the settling particle at different iteration numbers: (a) the 1645th iteration and (b) the 1650th iteration. (c) The corresponding contributions of the pulling force of films and the bubble pressure to the drag force Fy during 1645−1655 iterations. The bubble e in the wake of detaching from the particle at the 1650th iteration.

Fy = 0.1441(1 − Φg )−0.5615

Figure 8. (a) Evolution of the average values of drag and lift forces with respect to the foam quality. (b) The contributions of pulling force of films and bubble pressure to the corresponding drag force Fy for different foam qualities. The values of Fny predicted by eq 15 are also plotted and are generally higher than the ones provided by these wet foams.

(10)

To further demonstrate experimentally the relationship between the drag force and foam quality, several experiments were conducted. On the basis of our laboratory apparatus7 (foam rheological test system and static suspended sand experiment system), we used “3 wt % surfactant (quaternary ammonium surfactant) + 0.8 wt % NaSal + water” as the base fluid of foam and CO2 as the gas phase to form stable foam. The ceramsite, which is generally used as the fracturing proppant for an oilfield, was adopted as the rounded particle. Its density is close to 2000 kg/m3 and the equivalent diameter is about 0.7 mm. All tests were conducted at the temperature 40 °C and pressure 6.5 MPa to make the bubble closer in size to the particle. First, to measure frictional force caused by the liquid phase, the sedimentation of particle in the base fluid (Φg = 0) was tested. Accordingly, it can be obtained that ⎛ dS (⃗ t ) ⎞ ⎟ − G⃗ = 0 − λ⎜ ⎝ dt ⎠BF

where

dS (⃗ t ) dt

( )

Table 1. Settling Velocity of the Ceramsite in the Foam of Different Foam Qualities foam quality −3

settling velocity / × 10 cm/s

0

0.82

0.85

0.88

0.92

0.95

2.11

0.91

0.89

0.84

0.78

0.70

wet foam is slow and steady, the acceleration term can be negligible and thus the equilibrium equation is dS (⃗ t ) 0 = Fp⃗ + Fn⃗ − G⃗ − λ dt

(12)

Therefore, the drag force due to the bubble can be written as Fy = Fny + Fpy = G − λ

(11)

dS (⃗ t ) dt

(13)

According to eq 13, we can obtain the value of drag force which can be expressed as the following equation:

denotes the settling velocity in the base fluid

BF

and its value is shown in Table 1. Then we measured the settling velocities of the ceramsite, the values of which are also presented in Table 1, in the foam of different foam qualities. Since the motion of the particle in the

Fy = 0.9715λ(1 − Φg )−0.1246

(14)

This is also a power-law dependence of the drag force due to the bubble on the foam quality, which is consistent with the F


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Figure 9. Position of each T1 topological event as the particle settles through the foam channel during (a) 100−500 iterations, (b) 600−1000 iterations, and (c) 1100−1500 iterations for the foam of Φg = 0.92. The plus signs denote the positions of T1s and the circles denote the location that the particles move in these iteration intervals.

simulation results of the two-dimensional case. And yet, the parameters of this power-law relationship are dependent on characteristic parameters of a foam sample as well as the settling particle. In addition, as shown in Figure 8a, there is no obvious relationship between the lift force and foam quality. Its values fluctuate around zero for different foam qualities. This is attributed mostly to the disordered structure of the foam. Raufaste et al.29 proposed that the contribution of the pulling force of foam films to the drag force exerted on a round particle could be expressed by Fny =

γd0 0.516 0.25 Ab (1 − Φg )

This is also a power-law relationship, which is in agreement with eq 15. Nevertheless, the power exponent is different and may be dependent on several other parameters, such as the disorder level36 of a foam sample and the contact angle37 between these foam films. The evolution of the contribution of bubble pressure to the drag force with foam quality is presented in Figure 8b as well. It shows that the contribution of pressure also increases with the foam quality. The foam will become more compact as the foam quality is increased. From Figure 3, it is obtained that there is a region of higher pressure in front of the settling particle. The extrusion deformation of the bubbles applied by the particle is more severe due to the more compact foam structure. Accordingly, the relative pressure of the bubbles in front of the particle is higher for the foam of higher foam quality. The higher bubble pressure will provide more contribution to the drag force. 3.4. Positions of T1 Events. The local bubbles will move when the particle settles through the foam channel. This forms a fluidized region of the foam. It is within this region that the particle applies the greater force on these bubbles and, as a consequence, it is also the region where the deformation rate of the foam films is at its maximum. Accordingly, the elastic deformation and plastic rearrangement of these bubbles will occur frequently. As shown in Figure 3, there is a major change in the bubbles behind and in front of the settling particle. The topological rearrangement, also called T1 event, is fundamental to the flow of foam. The plasticity of foam originates from this rearrangement. It is expected that more T1 events are triggered in this fluidized region. Sun and Hutzler19 used a hybrid latticegas model to investigate the plasticity of the foam as a small rigid disk was pushed through the foam in a 2D channel. They found that T1 events mainly occurred within five bubble layers surrounding the disk and the average number of plastic events per bubble was a function of liquid fraction. In our simulation, the T1 event can be traced by recording the coordinate where two bubbles separate when the T1 event takes place. Figure 9 shows the positions of T1 events as the particle settles through

(15)

However, the assumption of eq 15 is that the obstacle diameter, as well as the obstacle−wall distances, is larger than the bubble diameter, and the foam tends to dry (the limit Φg →1). To show the difference of eq 15 applied to the wet foam, the corresponding values, according to eq 15, are demonstrated in Figure 8b as well. It can be seen that the values predicted by eq 15 are generally higher than the ones presented in these investigative wet foams. On the one hand, this can be attributed to more compact films for the dry foam. The films in contact with the particle increase with the increase of foam quality. As a result, the pulling force of films will increase accordingly. On the other hand, in our simulation, the size of the particle is close to the average bubble size, which is also inconsistent with the assumption of eq 15. Smaller particle size results in fewer films in contact with the particle. Hence, the pulling force of films is lower in our research. In addition, as displayed in Figure 8b, since increasing foam quality is close to the limit Φg, the deviations between the values of Fny and the ones predicted by eq 15 become gradually smaller with the increase of foam quality. On the basis of these simulations, the relationship between the contribution of the pulling force of films to the corresponding drag force and the foam quality can be expressed byny Fny = 0.06715(1 − Φg )−0.4771

(16) G


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the T1 event appears in front of the settling particle. Likewise, as the bubble in the wake of the particle detaches from the particle, the contribution of the pulling force of the films to the drag force decreases as well. The effect of the foam quality on the settlement behavior in the wet foam has shown some regularities. The settling drag force of the particle and its corresponding components increase generally with the foam quality. In addition, the power-law relationship between the drag force due to the bubble and the foam quality is numerically and experimentally obtained, and it can also apply to the association between the contribution of pulling force of films to the drag force and the foam quality. At last, the T1 plastic events are recorded during the sedimentation. Most T1 events appear near the particle and extend to several bubble layers around this particle. For the foam of Φg ≤ 0.88, the T1 events extend to the sixth bubble layers and thus the fluidized region surrounding the particle is larger than the foam of Φg > 0.88. In regard to the settlement of multiparticles in wet foam, it can be expected that the particles interact with each other, and the response of the microscopic structure of wet foam should be significantly different. These will certainly be involved in the future.

the foam during different iteration intervals. It can be obviously seen that most T1 events appear around this settling particle. To explore the fluidized region of the foam caused by the settling particle, the distance of a T1 event from the boundary of the particle is introduced and can be given by T1d =

(xi − x0)2 + (yi − y0 )2 − r0

(17)

where (xi, yi) denotes the coordinate where the T1 event occurs. The distribution of the distance T1d for different foam qualities is displayed in Figure 10. Because of the disordered

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b00281. Displacement fields of bubbles during the sedimentation. Analysis of changes in the thickness of liquid lamellae separating the bubbles in the front and wake of the falling particle (PDF)

Figure 10. Distribution of the distance T1d between T1 topological events and the edge of the particle. The distance T1d is normalized by the average bubble diameter db.

and discrete structure of foam, there is no obvious relationship between the foam quality and the number of T1 events. Nevertheless, for the foam of Φg > 0.88, the most T1 events are triggered within the first five bubble layers around the particle, which is in accordance with the result of Sun and Hutzler,19 while the T1 events extend to the sixth bubble layers for the foam of Φg ≤ 0.88, which means that the fluidized region surrounding the particle is larger.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail (Z.J.): [email protected]. *E-mail (S.W.): [email protected]. Notes

The authors declare no competing financial interest.

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4. CONCLUSIONS The structural and mechanical response of wet foam to a settling particle has been investigated by the quasi-static method. The evolution of the more detailed components of drag force is presented due to structural changes of the films surrounding the particle, and lower foam quality is extended when compared to the available literature.23,29 These obtained results can reveal the physical phenomenon of a particle settling through the wet foam in the meso-level, suggesting some revelations to the study of the real physics. Significantly, it contributes to deeper understanding in the engineering field where the particle, such as proppant, descends through the foam fluid. As the particle settles through the wet foam, there is a region of lower pressure in the bubbles at the back of the particle compared to the bubbles in front of the particle. Moreover, a slight fluctuation of the settling trajectory in the horizontal direction is triggered due to the asymmetric structure of the wet foam. Meanwhile, this disordered structure of foam also gives rise to the irregular evolution of the forces exerted on the settling particle. Both the contributions of the pulling force of films and the bubble pressure to the drag force decrease when

ACKNOWLEDGMENTS We thank K. Brakke for assistance with the surface evolver method and S. J. Cox for the prompting. We are also grateful to C. C. Feng, X. R. Luo, and Xe. J. Zhang for fruitful discussions.

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REFERENCES

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