Effect of the Hydrophobic Force Strength on ParticleBubble Collision Kinetics: A DEM Approach Ya Gao1, Geoffrey M. Evans1, Erica J. Wanless2 and Roberto Moreno-Atanasio1 1 School of Engineering 2 School of Environmental and Life Sciences The University of Newcastle, Callaghan, NSW 2308, Australia Email:
[email protected] Abstract— The capture of solid particles by air bubbles is a complex process influenced by hydrodynamic and surface forces. Computer simulation provides an alternative way to experimental approaches to give an insight into the phenomenon of particle capture. In this paper, the threedimensional Discrete Element Method (DEM) has been applied to simulate the interactions between fine particles and a central bubble in a quiescent liquid. A system, consisting of 200 monodisperse silica particles and a stationary air bubble, was considered. Drag, buoyancy, hydrodynamic resistance, hydrophobic and gravitational forces have been included in the simulations. The attractive hydrophobic forces between the particles and the bubble was estimated through a single exponential decay law which depends on two parameters, K and λ. K is related to the maximum strength of the hydrophobic force, while λ is the length that indicates how fast the hydrophobic force decays with particle-bubble distance. The results have shown that a decrease in the ratio of maximum hydrophobic force to particle weight from 5.2×106 to 5.2 produced a decrease in the capture efficiency by 42.6% for λ=1 μm and by 12.5% for λ=1 nm. Keywords- Discrete hydrophobic force
Element
Method;
flotation;
I. INTRODUCTION Froth flotation is an important industrial process extensively used in the separation of mineral particles as well as in the treatment of wastewater. Flotation is generally described as a sequence of three sub-processes, namely, collision, attachment and detachment [1-5]. The improvement of the performance of the flotation process greatly depends on providing adequate understanding of the various sub-processes and detailed knowledge of the interaction forces between particles and bubbles. Therefore, study of interactions between solid particles and air bubbles in aqueous solutions has been the focus of an increasing amount of research in understanding froth flotation. However, due to the complexity of the flotation phenomenon, the principles governing the bubble-particle
interactions are not well understood despite many decades of research. The interacting forces between hydrophobic colloidal particles have been well explained by the classical DLVO theory [6, 7]. However, when studying the interaction between mineral particles and air bubbles the situation gets much more complicated due to the presence of non-DLVO surface forces, such as steric forces, hydration and hydrophobic forces [8, 9]. Undoubtedly, the hydrophobic force is considered as the most significant non-DLVO surface force that determines the capture of a particle by a rising bubble during flotation [10]. Hydrophobic interactions, on a molecular level, describe the relationships between water and hydrophobes which are nonpolar molecules that do not interact favorably with water molecules. This force is thought to appear due to the coalescence between nonpolar molecules distributed on approaching surfaces, because, in this way, the contact with water molecules can be reduced to lower the free energy [11]. The hydrophobic force has generally been found to be an attractive force that can increase with the macroscopic hydrophobicity of surfaces, and to be far stronger than the van der Waals attraction between surfaces [12]. However, the origin of the hydrophobic attraction is complicated and is still controversial at present. Israelachvili and Pashley (1982) [13] demonstrated that the interaction between hydrophobic solid surfaces was about ten times stronger than the maximum possible van der Waals force. After the pioneering work by these authors, a large amount of studies have focused on the determination of the strength of the hydrophobic force by using experimental techniques, such as surface forces apparatus (SFA) and atomic force microscope (AFM). Table 1 presents a brief review of recent progress in understanding the hydrophobic force in aqueous media. Despite the large efforts to determine the strength of the hydrophobic interaction, there is still no generally accepted mathematical expression of it. This is due to a number of factors such as: •
Most of the reported experimental data are within the range of hundreds of nanometers between the interacting surfaces. This is due to the working range
TABLE I.
BRIEF REVIEW OF HYDROPHOBIC FORCE
Author(s)
Hydrophobic Force, Fh
Technique
Notes and Symbol Definitions
Israelachvili and Pashley (1982) [13]
Single exponential decay law:
Surface Forces Apparatus (SFA)
Hydrophobic surfaces: hexadecyltrimethylammonium bromide monolayer-coated mica surfaces. Long-range force can act up to 10 nm. R is the radius of the curved surfaces, K is a constant, H is the separation distance and λ is the decay length (≈1 nm).
Claesson et al. (1986) [17]
Fh / R = K exp( − H / λ )
SFA
Hydrophobic surfaces: dioctadecyldimethlammonium bromide monolayer-coated mica surfaces. Measurable range: about 30 nm.
Claesson and Christenson (1988) [18]
Double exponential decay law:
SFA
Hydrophobic surfaces: uncharged hydrocarbon and fluorocarbon monolayer-coated mica surfaces. Very long-range force has measurable range around 80 nm. λ=2-3 nm; λ*=13 nm (hydrocarbon) and16 nm (fluorocarbon).
Atomic Force Microscope (AFM)
Hydrophobic interaction between dissimilar surfaces: octadecyltrichlorosilane (OTS) coated glass sphere and silica plate surfaces. λ =2-32 nm. θ1, θ2 are contact angles of two surfaces. a=-7.0, b=-18.0
Ducker et al.(1994) [15]
AFM
Hydrophobic surfaces: air bubble surface and OTS monolayer-coated silica particle surface. Long-range attractive force exists between hydrophobic particle and bubble.
Ishida and Higashitani (2006) [16]
AFM
Hydrophobic surfaces: silica particle surface coated with OTS. A strong long-range attraction between hydrophobic surfaces with nanobubbles (not referred as “hydrophobic attraction”). A short-range attraction force between hydrophobic surfaces without nanobubbles (genuine hydrophobic attraction).
Fh / R = K exp( − H / λ )
Fh / R = K exp( − H / λ ) + K * exp( − H / λ* )
Yoon et al.(1997) [14]
Power law:
Fh / R = − K / H 2 cos θ1 + cos θ 2 K = − exp( a + b) 2
of AFM measurement technique. Existence of the hydrophobic force beyond these distances has not been explored yet [14-16]. •
Particle manipulation prior to experiments (hydrophobizing and drying) may introduce nanobubbles on to the particle surfaces, which may affect the validity of the analysis [19].
•
Bubble deformation due to the approaching particle before the formation of the three-phase contact (TPC) line also poses a problem that can lead to incorrect results [4].
Although many researchers [18, 20-22] have made significant contributions to address these problems, there are still many aspects of the hydrophobic interaction that are not well understood. These aspects include its origin, range and strength description.
The use of computational methods in the analysis of particle-bubble systems has received increasing attention in recent years. Discrete Element Method (DEM), as one of numerical methods in particle technology, serves as a powerful tool for investigating the detailed phenomena in the particlebubble interaction as well as describing the interaction forces of different kinds. In such a way, the DEM model allows us to probe the internal state of the system and understand the fundamental particle-bubble interactions underlying the complex, macro-scale response. In 2012, Maxwell et al. [23] developed a DEM code written in Fortran 77 for studying the particle-bubble interactions in monodisperse and polydisperse systems together with the particle sliding time over the bubble surface. In their work, the hydrophobic interaction was assumed to follow a relationship with the inverse of the particle-bubble surface distance. Afterwards, Moreno-Atanasio [24] analysed the effects of three different models for
hydrophobic force reported in the literature on the particle capture efficiency. The work presented here builds upon the previous findings by Maxwell et al. (2012) and Moreno-Atanasio (2013). The objective of the present paper is to investigate the influence of the strength of the hydrophobic force on the kinetics of capture of particles by the bubble. The hydrophobic attraction is described by a single exponential law which takes particle hydrophobicity and decay length into account. Additionally, the buoyancy force is introduced in the simulation which was excluded in the studies mentioned above. Van der Waals, electrical double-layer and hydrophobic forces between particles have not been incorporated into the DEM simulations. This is due to the fact that these forces will interfere with the elucidation of the influence of the particlebubble hydrophobic force on the capture efficiency. However, the role of the particle-particle interactions on particle capture by a central bubble will be addressed in future. II.
METHODOLOGY
A. General Model Description The most important feature of discrete element models (DEMs) is that all particles are explicitly considered as individual bodies in their interactions and movements. The soft sphere model [25], which allows minor deformations between contacting bodies, is adopted in the simulations presented here. This model was first proposed by Cundall and Strack (1979) for quasi-static deformation of a particle bed [26]. The contact forces are simply computed using linear elastic spring model in the normal direction. Elastic properties were also assigned to the bubble as described in the work by Attard and Miklavicic, or Goldman [27, 28]. For the DEM simulations, computer code in C language was developed. A system made of 200 monodisperse silica particles and an air bubble was employed for the numerical computation. The initial state of the system is shown in Fig. 1. The bubble with immobile surface was fixed to be stationary at the center of the working space throughout the simulation
process. The primary particles were randomly generated around the central bubble within a maximum distance from the bubble surface equal to 50μm. These particles were constrained to not overlap with each other or with the bubble at the beginning of the simulations. All particles and the bubble were assumed to be spherical in shape. The particle velocities were initially set to zero. The properties of water were considered to describe the fluid phase. The computation was performed on the basis of quiescent flow of the liquid (water). So the analysis of the influence of the relative strength of the hydrophobic interactions can be directly studied. B. Equation of Particle Motion and Numerical Integration Scheme The motion of the particles is described by the equation of Newton’s second law. For the ith particle, the motion equation can be written as
mi a i = Fi .
(1)
where ai is the acceleration of the center of the particle i; mi, its mass; Fi, the resultant force. The resultant force, in the present work, consists of the long-range forces (gravitational, hydrophobic force), hydrodynamic forces (drag, buoyancy, resistant force), and the elastic contact force. In the discrete element method, the particle and particlebubble interactions are calculated cyclically and the evolution of the system is determined using a time dependent explicit solution method. The half-step leapfrog Verlet (LFV) integration scheme is used in the simulation due to its high accuracy (mathematically), stability and computational efficiency [29]. In this scheme, the velocity is calculated at each half time step, which is an important modification based on the Verlet algorithm. The translational motion at the nth time step is given by:
a ni =
Fin , mi
(2)
v ni +1 / 2 = v ni −1 / 2 + a ni ∆t , .
(3)
x ni +1 = x ni + v ni +1 / 2 ∆t , .
(4)
1 v in +1 = v ni +1 / 2 + a ni ∆t. . 2
(5)
where v is the particle velocity and x the displacement. The solution proceeds in small time steps Δt during which the acceleration of the particles is assumed to be constant. Every time step is started with computing the particle accelerations from Eq. (2), followed by performing the numerical integration of accelerations over the time increment to obtain the particle velocities at next half time step (Eq. (3)). Figure 1. Visualisation of the initial state of the system. (Particles are coloured in yellow and the bubble in blue.)
Further, the displacements are calculated using Eq. (4) after which the velocity at the next full time step is obtained by Eq. (5). Having determined new positions and velocities for all the particles, the program repeats the cycle of updating resultant forces and particle accelerations. C. Forces between Particle and Liquid Phase The particles used in the systems are small (Stokesian) particles. In this case, the forces related to the motion of a Stokesian particle falling under gravity through a quiescent liquid can be modeled by applying the forces involving: •
Buoyancy forces, Fbuoy
Fbuoy = − m f g .
(6)
Where mf is the mass of liquid displaced by the particle, g is the acceleration due to gravity. •
Drag force, Fdrag
Fdrag = −6πµR p v .
(7)
where μ is the liquid’s viscosity, Rp is the particle radius, v describes the relative velocity of the particle with respect to the liquid. •
Hydrodynamic resistance force, Fresis
6πµR p2 r r v⋅ . H r r
(8)
Where r is the vector that joins the centre of the bubble with the centre of the particle, H is the distance between particle and bubble surfaces. The value of hydrodynamic resistance force tends to reach infinity as surface distance approaches zero. In order to avoid this numerical problem, a cut-off distance (Hcut-off) 0.5 nm has been established. This number is somehow arbitrary but takes into account the limit between classical and quantum levels. Thus, this force was evaluated at zero for any distances smaller than Hcut-off. D. Forces between Particle and Bubble •
Fh = KR p exp ( − H / λ )
Hydrophobic force, Fh There is a large variety of models (see Table 1.) in the literature available for hydrophobic forces between bubbles and particles. However, there is no universally accepted expression. In the present research, the exponential form was applied as it depends on two
r r
(9)
where K is related to the maximum strength of the hydrophobic force, while λ is the length that indicates how fast the hydrophobic force decays with particlebubble distance. The cut-off value of Hcut-off is also used here. The hydrophobic force approaches its maximum value when surface distance is less than 0.5 nm. A particular focus in the present work is placed on the effects of changing the values of K and λ on the capture efficiency. E. Normal Contact Force The elastic force Fne is modeled using Hooke’s law which is proportional to the normal contact stiffness kn and to the overlap of the two spherical surfaces δrn , i.e.
Fne = k n δrn .
(10)
The overlap is calculated as
δrn = d − R1 − R2 .
Considering the dynamic aspects of the thinning of the intervening liquid film between an air bubble and a particle surface, the short-range hydrodynamic resistance force is present in the system. The expression of this force is shown as Eq. (8), known as the Taylor equation, which gives a good agreement with the experimental data within the range of one bubble radius [30].
Fresis = −
parameters and therefore it provides a large versatility in the study of the attraction between hydrophobic bodies [13, 17].
(11)
where d is the distance between two spheres (particle-particle or particle-bubble) centers, and R1 ,R2 are their respective radii. Eq. (10) is only used in the case of overlap less than zero, otherwise the elastic contact force is equal to zero. F. Computer Simulation Parameters Computer simulations were performed utilizing the properties of the materials and fluid shown in Table 2. The parameters of these materials were defined to be as close as possible to the physical experiments performed [31]. Four different values of decay length, λ, between 1 μm and 1 nm were selected, since this range covers any possible values reported in the literature [4], and therefore the effect of the hydrophobic force strength on the kinetics of capture of particles by the bubble could be better understood.
TABLE II.
COMPUTER SIMULATION PARAMETERS
Particles Number of particles
200
Particle density
2.6 × 10-3 kg/m3
Particle radius
3.3 × 10-5 m
Particle stiffness
100 N/m
Bubble Number of bubbles
1
Bubble radius
1.0 × 10-3 m
Bubble stiffness
5.0 × 104 N/m
Liquid Water density
1.0 × 103 kg/m3
Water viscosity
1.0 × 10-3 Pa•s
Hydrophobic Force Constant K
6.1 × 102, 6.1, 6.1 × 10-2, 6. 1 × 10-4 N/m
Decay length, λ
10-6, 10-7, 10-8,10-9 m
Figure 2. The nomalised number of particle-bubble contacts as a function of time for three individual systems consist of 200, 300 and 650 particles. The decay length λ = 1 μm, constant K = 6.1 × 102 N/m.
Others 5 mm × 5 mm × 5 mm
Space dimention Time step
1 ns
Run time
1s
III.
RESULTS AND DISCUSSIONS
A. Influence of Particle Concentrations and Initial Distributions on Particle-Bubble Contacts In this section we present the investigations considering the effect of the particle concentrations and their random distributions around the central bubble on the number of collisions against the bubble as a function of time. The results of three different simulation runs corresponding to 200, 300 and 650 particles in the systems are shown in Fig. 2. Each simulation was carried out under the same initial conditions and using the same values of the decay length and constant K in Eq. (9) (10-6 m and 6.1 × 102 N/m respectively). It is observed from Fig. 2 that the simulations for the three different particle concentrations produce almost identical results. This suggests that the 200 particle simulation is an appropriate approximation particularly for 650 particle system in which the particle concentration is very close to experimental system. It is worth noting that due to the limitation of the size of DEM systems the larger number of particles involved, the longer the computational time required to complete a specific given value of real time. Therefore, a system made of 200 particles was found to be suitable for the task of estimating the kinetics of collision of particles against the bubble. Understanding of the influence of the initial random distribution of particle positions on the collision results is desirable. Fig. 3 shows the comparison of the effects of three different initial particle positions on the number of particlebubble contacts. Although the randomly generated particle positions before the simulation are different, the variations
Figure 3. The nomalised number of particle-bubble contacts as a function of time for three individual systems of different initial particle positions. The decay length λ=1 μm, K=6.1 × 102 N/m.
between the configurations are only noticeable at the beginning of the simulations. However, the final number of particlebubble contacts is insensitive to the initial condition. B. Effect of the Hydrophobic Force The influence of the hydrophobic force strength (using Eq. (9)) on the attachment of particles was investigated by changing the values of constant K and decay length λ. Four different values of K, which determines the maximum strength of the hydrophobic force, ranging between 6.1 × 102 and 6.1 × 10-4 N/m, were used in the simulations. These values correspond to the maximum value of the hydrophobic forces equal to 2.0 × 10-2 and 2.0 × 10-8 N respectively. Four different values of the decay length λ equal to 1.0 × 10-6 m, 1.0 × 10-7 m, 1.0 × 10-8 m and 1.0 × 10-9 m were considered.
a
b
c
d
Figure 4. Effect of hydrophobic force strength on the normalised number of particle-bubble contacts as a function of time.
Figure 5. Effect of values of K on the normalised number of particle-bubble contacts as a function of decay length λ.
Figure 6. Effect of buoyancy force on the normalised number of particle-bubble contacts as a function of time.
Fig. 4(a-d) presents the normalised number of particlebubble contacts as a function of time. Each curve was obtained from separated simulation carried out under the same physical conditions. Three different regimes can be observed for all cases plotted in Fig. 4(a-d). This is in agreement with the results reported by Moreno-Atanasio [24] as expected. The number of particle-bubble contacts during the first regime increases quite slowly with the time. This is a consequence of the initial stationary state of the particles and the gravitational force being a dominant force in this stage. Therefore, the particles require a certain critical time to reach the bubble surface. The second regime is characterized by a rapid increase in the number of particle-bubble collisions. The number of contacts then stays constant, which represents the third stage of the particle-bubble capture evolution. Due to the insufficient hydrophobic force to overcome the gravitational force some particles do not collide against the bubble, as highlighted by Moreno-Atanasio (2013). When comparing the four plots in Fig. 4, the effect of the magnitude of the constant K and decay length λ on the capture of particles by the bubble was found to be dramatic, especially for the case which λ is equal to 1.0 × 10-6 m. The number of contacts decreases with decreasing the decay length λ and parameter K. Moreover, the higher the value of the parameter λ or K, the faster the transition into the second and final regimes. It can be concluded from the simulations that the strength of the hydrophobic force is a significant factor influence the particle capture efficiency. Fig. 5 exhibits the normalized number of particle-bubble contacts as a function of decay length. The data represent the ultimate values of the number of contacts for the cases of K equal to 6.1×102, 6.1, 6.1×10-2 and 6.1×10-4 N/m.. As shown in Fig. 5, although decreasing λ produces a decrease in the number of particle-bubble contacts for each case, it exerts a much smaller influence on the capture efficiency. For each decrease in strength of K (or the ratio of maximum value of hydrophobic force to the gravity) by two order of magnitude, the capture efficiency decreased by 42.6% for λ=1×10-6, 15.6% for λ=1×10-7, 8.5% for λ=1×10-8 and 12.5% for λ=1×10-9. Interestingly, each line seems to show an exponential relationship between capture efficiency and decay length λ. However, this relationship needs to be further investigated.
more preponderant role on the capture of particles than the buoyancy force. However, the buoyancy force resulted in a clear delay in the evolution of the number of contacts. This is due to the fact that the buoyancy force which acts on the same line with gravity but in the opposite direction may slow down the particle moving downwards to collide against the bubble. This phenomenon obviously affects more significantly on those particles which are located around the upper hemisphere of the bubble. When the particles slide around the bubble to the bottom hemisphere, most of them will fall due to gravity independently of any buoyancy force. CONCLUSIONS Three-dimensional DEM computer simulations of particlebubble interactions in a quiescent medium have been reported. Any conclusions drawn from the simulation data have been restricted to a monodisperse particle system. The qualitative behaviour of the 200 particle system observed here has been shown to be qualitatively correct for higher particle concentrations which are close to the cases reported in the literature. The hydrophobic force between the particle and the bubble was estimated through a single exponential decay law which depends on a constant K related to the strength of the hydrophobic force and a decay length λ. It has been shown that increasing the value of K or decay length λ can produce higher particle capture efficiency. Despite this, it is observed that the evolution of the number of particle-bubble contacts slows down by introducing the buoyancy force of particles, and gets faster by increasing the magnitude of the hydrophobic force parameters (K and λ). The results presented in this paper constitutes the first numerical analysis of the influence of the strength of the hydrophobic interaction using the description of the single exponential law on the collision of particles against bubble, and it is expected to be helpful for the future determination of generally valid expression of hydrophobic force. REFERENCES [1]
C. Effect of the Buoyancy Force The final step in our study was to investigate the role of the buoyancy force on the capture of particles. A comparison between cases of the presence and absence of buoyancy force for two different values of the strength of the hydrophobic interactions is presented in Fig. 6. The results show that the trend of the curves does not significantly change and the number of contacts reaches almost the same value in both scenarios of λ = 10-6 m, K = 6.1 × 102 N/m and λ = 10-6 m, K = 6.1 N/m. This is due to the fact that the magnitude of the buoyancy force is very small (the value of the buoyancy force normalized to the particle weight is equal to 0.38). In contrast, the ratios of maximum hydrophobic forces to the particle weight in these two cases are 5.2 × 106 and 5.2 × 104. Therefore, it is expected that the hydrophobic attraction would have a
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