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Thermocapillary Coagulations of a Fluid Sphere and a Gas Bubble Shih H. Chen* Department of Chemical Engineering, Hwa Hsia College of Technology and Commerce, Chung-ho City, Taipei County 235, Taiwan Received September 23, 2002. In Final Form: March 10, 2003 This paper presents an analytical study of the thermocapillary motion of an adiabatic gas bubble and a liquid drop with constant temperature by using the method of reflections. The particles are allowed to differ in radius, and the droplet viscosity is arbitrary, which can be limited to a solid sphere. The Peclet and Reynolds numbers are assumed small, so that the temperature and flow fields are governed, respectively, by the Laplace and Stokes equations. The method of reflections is based on an analysis of the thermal and hydrodynamic disturbances produced by an insulated gas bubble and by a single drop with constant temperature, placed in an arbitrarily Laplacian temperature field. The results for two-sphere interactions are correct to O(r12-7), where r12 is the distance between the particle centers. The thermocapillary interactions between a gas bubble and a thermally uniform droplet are discussed in the situations of the prescribed temperature gradient parallel and/or normal to the particles’ center line. In addition, the axisymmetric particle interactions in a medium with uniform applied temperature are formulated. The effect of interactions on thermocapillary migrations of an insulated gas bubble and a thermally uniform droplet is generally stronger than that on the motions influenced by the interaction of two gas bubbles or droplets. Meanwhile, the particle behaviors are quite different from those we know.
Introduction When a liquid drop or a gas bubble is submerged in an immiscible liquid phase with a nonuniform temperature distribution, the fluid particle migrates from the colder region to the hotter region of the surroundings. This temperature-induced interfacial tension gradient driving the particle motion is known as thermocapillary migration, and it has been the subject of considerable investigations for many years, especially during the past two decades. Since this mechanism captures fluid particles on the hot surfaces, thermocapillary migration is useful in technical applications, such as processing of materials. Assuming small Peclet and Reynolds numbers, Young et al.1 pioneered the formulation for the migrating velocity of a spherical droplet of radius a in a fluid medium of viscosity η and thermal conductivity k. The droplet velocity U(0) is related to a uniform prescribed temperature gradient ∇T∞ by
U(0) )
a ∂γ 2 ∇T∞ (2 + ki/k)(2 + 3ηi/η) η ∂T
(
)
(1)
In eq 1, ηi and ki are the respective viscosity and thermal conductivity of the droplet; and ∂γ/∂T is the gradient of interfacial tension γ with the local temperature T. All physical properties are assumed to be constant except the interfacial tension, which is assumed to vary linearly with temperature on the scale of particle size. In practical situations of thermocapillary migration, fluid droplets are not isolated and move in the presence of neighboring particles2-15 and/or boundaries,16-24 and * E-mail:
[email protected]. (1) Young, N. O.; Goldstein, J. S.; Brock, M. J. J. Fluid Mech. 1959, 6, 350. (2) Meyyappan, M.; Wilcox, W. R.; Subramanian, R. S. J. Colloid Interface Sci. 1983, 94, 243. (3) Meyyappan, M.; Subramanian, R. S. J. Colloid Interface Sci. 1984, 97, 291. (4) Anderson, J. L. Int. J. Multiphase Flow 1985, 11, 813. (5) Feuillebois, F. J. Colloid Interface Sci. 1989, 131, 267.
this causes the particle motion to deviate significantly from that predicted by eq 1. Several important conclusions of these investigations were addressed. In the situation of interparticle interactions, (a) the particle interactions’ effects on thermocapillary migration are generally weaker than those on motion due to gravity; (b) the interaction effects will influence the migrating velocity of each droplet to be larger than that when it is isolated if the droplets are moving along their center line (however, the effects will decrease the droplets’ velocities when they translate normal to their center line); (c) the migration of each bubble in a collection of identical bubbles, which can be arbitrarily oriented, is unaffected by the presence of the others; (d) the influence of interactions on droplets is far greater on the smaller droplets than on the larger ones. On the other hand, there are some distinctive results in the situations of the boundary effects on the thermocapillary migration: (a) the effect of a solid plane is to reduce the thermocapillary migration velocity of a droplet moving normal (6) Keh, H. J.; Chen, S. H. Int. J. Multiphase Flow 1990, 16, 515. (7) Zhang, X.; Davis, R. H. J. Colloid Interface Sci. 1992, 152, 548. (8) Keh, H. J.; Chen, L. S. J. Colloid Interface Sci. 1992, 151, 1. (9) Satrape, J. V. Phys. Fluid 1992, A4, 1883. (10) Keh, H. J.; Chen, L. S. Chem. Eng. Sci. 1993, 48, 3565. (11) Wei, H.; Subramanian, R. S. Phys. Fluids 1993, A5, 1583. (12) Wang, Y.; Mauri, R.; Acrivos, A. J. Fluid Mech. 1994, 261, 47. (13) Golovin, A. A. Int. J. Multiphase Flow 1995, 21, 715. (14) Wei, H.; Subramanian, R. S. J. Colloid Interface Sci. 1995, 172, 395. (15) Leshansky, A. M.; Golovin, A. A.; Nir, A. Phys. Fluids 1997, 9, 2818. (16) Meyyappan, M.; Wilcox, W. R.; Subramanian, R. S. J. Colloid Interface Sci. 1981, 83, 199. (17) Meyyappan, M.; Subramanian, R. S. J. Colloid Interface Sci. 1987, 115, 206. (18) Chen, S. H.; Keh, H. J. J. Colloid Interface Sci. 1990, 137, 550. (19) Barton, K. D.; Subramanian, R. S. J. Colloid Interface Sci. 1990, 137, 170. (20) Ascoli, E. P.; Leal, L. G. J. Colloid Interface Sci. 1990, 138, 220. (21) Loewenberg, M.; Davis, R. H. J. Colloid Interface Sci. 1993, 160, 265. (22) Chen, S. H. Langmuir 1999, 15, 2674. (23) Chen, S. H. Langmuir 1999, 15, 8618. (24) Chen, S. H. J. Colloid Interface Sci. 2000, 230, 157.
10.1021/la020805f CCC: $25.00 © 2003 American Chemical Society Published on Web 04/26/2003
Thermocapillary Coagulations
to the plane; however, this solid wall can be an enhancement factor on the motion of a droplet migrating parallel to the wall; (b) the droplet velocity of thermocapillary migration nearby a free surface can be either greater or smaller than that when the surface is absent, depending on the relative conductivity and the relative distance from the surface; (c) the boundary characteristics of the plane strongly influence the droplet motion as the plane migrates parallel to the wall; (d) in general, the boundary effects on thermocapillary migration are quite complicated and relatively weak in comparison with that on sedimentation. The above summaries of particle interactions and boundary effects are addressed by the neglect of convective effects and particle deformations. Recently, the thermocapillary migration of droplets considered the effects of convective transport,25,26 and deformations of the interface27,28 were studied. Investigations of thermocapillary migration are usually focusing on situations of a prescribed constant temperature gradient. When a fluid sphere is immersed in a thermally uniform phase, the particle will not move, since there is no thermocapillary effect occurring at the droplet-fluid interface, whether the particle temperature is initially higher or lower than that of the surrounding fluid. However, the fluid particle can be driven by thermocapillarity if there is another particle with a higher or lower temperature submerged nearby the previous one, since the spherically symmetric temperature distribution is destroyed by each particle and this generates a thermal gradient around both particles. This thermocapillary motion is quite different from those driven by an applied temperature gradient, and it usually occurs in nature and during manufacturing processes. Golovin13 investigated the thermocapillary interactions between a gas bubble and solid particle with constant temperature by using bipolar coordinates. The numerical results showed that the influence of particle interactions on the particle motion is unlike those driven by a prescribed temperature gradient. The objective of the present study is to discuss the thermocapillary interactions between a gas bubble and a fluid droplet, for which the radii may be different. We extend the work done by Golovin13 to more general situations, and a different mathematical analysis is applied. The gas bubble is thermally insulated, while the viscosity of the thermally uniform droplet is arbitrary, which can be limited to the situation of a solid particle or a gas bubble. The quasi-steady energy and momentum equations applicable to the system, no matter whether there exists a prescribed temperature gradient or not, are solved by using the method of reflections. The results for two-sphere interactions are analytically formulated and are correct to O(r12-7), where r12 is the distance between the particle centers. Thermocapillary Migration of a Single Sphere When one applies the method of reflections to obtain the interactions between two spherical particles, it is essential to realize the thermal and hydrodynamic behaviors of a single particle in an arbitrary temperature field TA(x). We consider that a spherical particle of radius a exists in a surrounding fluid. The particle could be a solid sphere and/or a liquid droplet with constant tem(25) Lavrenteva, O. M.; Leshansky, A. M.; Nir, A. Phys. Fluids 1999, 11, 1768. (26) Lavrenteva, O. M.; Nir, A. Phys. Fluids 2001, 13, 368. (27) Haj-Hariri, H.; Shi, Q. Phys. Fluids 1997, 9, 845. (28) Rother, M. A.; Davis, R. H. J. Colloid Interface Sci. 1999, 214, 297.
Langmuir, Vol. 19, No. 11, 2003 4583
perature Ti. This is reasonable if the suspended particles have a high thermal conductivity or their particle sizes are small. The particle could also be a gas bubble with zero heat flux at the interface, that is, a thermally insulated bubble. All the physical properties of the particle and the surrounding fluid are taken to be constant except the interfacial tension of the bubble, which is assumed to be linear with temperature. The instantaneous center of the particle is positioned at x0, and the relative position vector is defined as r ) x - x0. Although x0 changes with time, the problem can be dealt with as a quasi-steadystate process if both Peclet and Reynolds numbers are small. It is assumed that a|∇TA|/Ti , 1. Since the boundary conditions of the fluid velocity field are coupled with the temperature gradient at the particle surface, it is necessary to determine the temperature distribution first. Temperature Field. The energy equation governing the temperature distribution, T(x), for the external fluid satisfies the Laplace equation
∇2T ) 0
(2)
∇2
It is obvious that TA ) 0. The boundary conditions at the particle surface (r ) a) require that the temperature continuity for liquid drops (or solid particles) and the zero heat flux for gas bubbles must occur. In addition, the fluid temperature has to approach the prescribed field far away from the particle. Thus, one has
r)a
rf∞
T ) T i,
for solid sphere or liquid drop (3a)
er‚∇T ) 0, for gas bubble
(3b)
T f TA
(3c)
where er is the radial unit vector in the spherical coordinates. The boundary condition (eq 3a) is reasonable for a drop with a thermal conductivity that is large compared with that of the surrounding fluid; the droplet would likely maintain a uniform temperature while heat is being transferred in or out. A general solution to eq 2 that satisfies eq 3c is modified by Anderson4 ∞
T ) TA +
() a
∑ m)1 r
m+1
Sm[‚]λm
(4)
Here, λm is a polyadic constant, the symbol [‚] represents m scalar products applying the inner nesting convection, and the mth-order polyadic Sm is a surface harmonic defined as
Sm ) rm+1(∇...∇)m(r-1)
(5)
Substitution of eq 4 into eq 3a gives
λ0 ) [Ti - (TA)0], λ1 ) a(∇TA)0, 1 λ2 ) - a2(∇∇TA)0, ... (6) 6 where the subscript 0 to variables inside parentheses denotes evaluation at x ) x0. By Substituting eq 6 into eq 4 together with ∇2TA ) 0, we get
T ) TA +
(ar)[T - (T ) ] - (ar) r‚(∇T ) 1a rr:(∇∇T ) + O(a r ∇∇∇T ) 2( r ) 3
i
A 0
A 0
5
7 -4
A 0
A 0
(7)
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Equation 7 describes the temperature profile surrounding a liquid drop or a solid particle in which the internal temperature distribution is kept uniform. If the particle is a gas bubble, the boundary condition in eq 3a should be replaced by eq 3b, so that
T ) TA +
1a3 1a5 r‚(∇TA)0 + rr:(∇∇TA)0 + 2r 3r O(a7r-4∇∇∇TA)0 (8)
()
()
Note that the leading order in T decays as r-3 for the gas bubble situation, while that decays as r-1 for the liquid droplet (or solid particle) situation. This is due to the fact that the internal temperature distribution of the droplet is assumed constant, and this results in droplet acting as a heat source (or sink) for the drop temperature higher (or lower) than surrounding’s. In real situations, the particle temperature is time-dependent; however, a particle with high heat capacity in a quasi-steady situation could be viewed as a heat reservoir in a short time period. The tangential component of the temperature gradient at the particle surface, ∇ sT ()(I - erer)‚(∇T)r)a, where I is unit dyadic), is demanded in a later derivation. After differentiating eq 7 at r ) a together with the Taylor expansion of ∇TA about x ) x0 and eliminating the normal component, one obtains
∇sT ) 0 + O(∇∇∇TA)0
(9)
for the particle with constant temperature, and
flow inside the droplet; γ is the local interfacial tension for the droplet; and U is the instantaneous migrating velocity of the droplet to be determined. The surface temperature gradient, ∇sT, can be obtained from eqs 9 and 10. Note that ∂γ/∂T is assumed as a constant on the scale of particle radius. A solution for the velocity field can be constructed from Lamb’s general solution as outlined by Brenner.29 The fluid velocity is completely specified when the polyadic coefficients and βm in the following formulas are calculated using the value of the velocity field on the particle surface (vs): ∞
er‚vs )
2
ηi∇ vi - ∇pi ) 0
(11a)
∇‚vi ) 0
(11b)
η∇2v - ∇p ) 0
(11c)
∇‚v ) 0
(11d)
where vi and v are the fluid velocities for internal and external flows of the droplet, respectively; pi and p are the corresponding dynamic pressures. Due to the continuity of the fluid velocities and the thermocapillarity along the droplet-fluid interface as well as the surrounding fluid at rest far from the sphere, the boundary conditions for the flow fields are
r)a
rf∞
v ) vi
(12a)
er‚(v - U) ) 0
(12b)
∂γ (I - erer)er:(τ - τi) ) - ∇sT ∂T
(12c)
vf0
(12d)
Here, τ ()η[∇v + (∇v)T]) and τi ()ηi[∇vi + (∇vi)T]) are the viscous stress tensors for the external flow and for the
(13a)
∞
∑ βm[‚]Sm
-a∇‚vs )
(13b)
m)1
Here the boundary condition in eq 12a is automatically satisfied. The force F exerted by the fluid on the surface r ) a is given by
F ) 2πηa(3R1 + β1)
(14)
Since the temperature field produces no bulk forces in the fluids, the migrating velocity U is obtained by setting the above expressions for F as zero. The translational motion is described by the coefficients Rm and βm. It can be obtained from eqs 12b and 13a as
3 5 ∇sT ) (I - erer)‚(∇TA)0 + a(I - erer)er:(∇∇TA)0 + 2 3 O(∇∇∇TA)0 (10) for the gas bubble situation. Velocity Field. On the basis of the low Reynolds numbers encountered in thermocapillary migration, the fluid flows inside and outside the droplet satisfy the Stokes equations
∑ Rm[‚]Sm m)1
R1 ) -U
(15a)
Rm ) 0 for m > 1
(15b)
The coefficient βm must be determined by the boundary condition in eq 12c. Applying the expressions for τ and τi in terms of Rm, βm, and ∇sT obtained from eqs 9 and 10, one can get
β1 )
1 U 1 + η*
β2 ) 0
(16a) (16b)
for the liquid drop (or solid particle) at constant temperature, and
β1 ) U + β2 ) -
a ∂γ (∇TA)0 η ∂T
(
)
1 a2 ∂γ (∇∇TA)0 3 η ∂T
(
)
(17a) (17b)
for the insulated gas bubble. Here, η* ()ηi/η) denotes the internal-to-external viscosity ratio of the droplet. The forcefree characteristic of this problem is applied with eqs 15a, 16a, and 17a (β1 ) -3R1 ) 3U); thus, the thermocapillary migrating velocity is found to be
U ) 0 (for liquid drop) U)
∂γ a (∇TA)0 (for gas bubble) 2η ∂T
(
)
(18a) (18b)
as the general result for an imposed temperature field TA (29) Brenner, H. Chem. Eng. Sci. 1964, 19, 519.
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Langmuir, Vol. 19, No. 11, 2003 4585
as long as ∇2TA ) 0. Obviously, the migrating velocity of a liquid droplet (or solid particle) is zero. This is due to the assumption of uniformity of the internal temperature, which results in a zero induced surface tension gradient to drive the particle motion. On the other hand, the translating velocity of a gas bubble is proportional to the particle size and the prescribed temperature gradient evaluated at the bubble center. Equation 18b can also be simplified from the article by Anderson.4 The corresponding flow fields inside and outside the particle can be obtained by expressions given by Happel and Brenner.30 The volume-averaged internal velocity, vi, which equals U, is not needed in the following derivation. While the external flow field is
v ) 0 (for liquid droplet)
( )( ) ( )[ ( )
∂γ 1a 4 η ∂T 1a ∂γ 3 6 η ∂T
v)
(19a)
a 3 (3erer - I)‚(∇TA)0 r a 3rrr a5 rrr + 2Ir - 5 2 :(∇∇TA)0 + r r2 r r O(a7r-5∇∇∇TA)0 (for gas bubble) (19b)
( )(
)]
the perturbed flow field produced by the motion of a thermally uniform droplet (or solid particle) is zero, since the thermocapillarity driving the particle motion at the particle surface is zero. On the other hand, the leading order in v for the thermocapillary migration of a thermally insulated gas bubble decays as r-3 rather than a Stokeslet (force source) or Stresslet (force dipole). This is a characteristic of the “phoretic motion”, which is the movement of particles caused by driving forces interacting with the surface of each particle. In application of thermocapillary migration, a prescribed flow field usually exists. Due to the linearity in thermocapillary migration, the motion of a freely suspended droplet under an arbitrarily applied temperature gradient ∇TA and flow field vA in an unbounded fluid can be obtained by combining eqs 18 and the modified Faxen’s laws for a liquid drop30,31
U ) Ut + Uh ) Ut + (vA)0 +
η* a2(∇2vA)0 2(2 + 3η*)
(20)
where Ut denotes the thermocapillary migration velocity predicted by eq 18; Uh means the hydrodynamic velocity evaluated by Faxen’s law. In the limitation of η* ) 0 and η* f ∞, eq 20 predicts the general behavior of the gas bubble and solid particle, respectively. Interactions in a Bubble-Droplet Pair We now consider the quasi-steady low-Peclet-number interactions of two spherical particles of radii a1 and a2 in thermocapillary migration. Particle 1 is specified as a gas bubble (which is the main character in the particle -interaction), and particle 2 is a liquid droplet (which can be limited to a solid sphere as η* f ∞) with constant internal temperature Ti2 and viscosity ηi2. They are oriented in an arbitrary direction to the prescribed temperature gradient E∞ ()∇T∞, where T∞(x) is the (30) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Nijhoff: Dordrecht, The Netherlands, 1983. (31) Batchelor, G. K. An Introduction to Fluid Dynamics; Cambridge University Press: New York, 1967.
Figure 1. Geometric sketch for thermocapillary interactions of two spheres.
undisturbed temperature field of the surrounding fluid with viscosity η). The particles are assumed to be sufficiently close to interact thermally and hydrodynamically with each other, but they are far from the boundary walls and, thus, the surrounding fluid can be regarded as unbounded. Let e be the unit vector pointing from the center of particle 1 to the center of particle 2 and r12 denote the center-to-center distance between the particles, as illustrated in Figure 1. E∞ is assumed to be constant over distances comparable to r12, and the fluid at infinity is at rest. The effects of Brownian motion are ignored. For (a1 + a2)/r12 , 1, a method of reflections30 is used to solve the two-particle problem. Due to the linear characteristic governing eqs 2 and 10 as well as boundary conditions 3 and 11, the solutions of the temperature and velocity fields for eqs 2 and 10 can be decomposed into a sum of fields, which depend on increasing powers of r12-1:
T ) T1(1) + T2(2) + T1(3) + T2(4) + ...
(21a)
v ) v1(1) + v2(2) + v1(3) + v2(4) + ...
(21b)
where the subscripts 1 and 2 represent the reflections from particle 1 and particle 2, respectively, and the superscript (i) denotes the ith reflection from either of the particle surfaces. Hence, the translational velocities of the particles can also be expressed in terms of a series:
U1 ) U1(0) + U1(2) + U1(4) + U1(6) + ...
(22a)
U2 ) U2(1) + U2(3) + U2(5) + U2(7) + ...
(22b)
where U1(i) is related to T2(i) and v2(i) by eq 24 for i ) 2, 4, 6, ..., while U2(i) is related to T1(i) and v1(i) for i ) 1, 3, 5, .... Obviously, the unperturbed linear temperature field gives
U1(0) ) A1E∞
(23a)
where
A1 )
( )
a1 ∂γ1 2η ∂T
(23b)
The initial temperature gradient ∇T1(1) and velocity field v1(1), which correspond to the thermocapillary migration of particle 1 isolated in an unbounded fluid under the prescribed field ∇T∞, are easily obtained from eqs 8
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and 19b for r1 > a1 as
() ()
∇T1(1) ) E∞ + v1(1) ) -
1 a1 3 (I - 3er1er1)‚E∞ 2 r1
1 a1 (I - 3er1er1)‚U1(0) 2 r1
(24a)
3
(24b)
where er1 ) r1/r1 (r1 is the position vector relative to the center of particle 1, and r1 ) |r1|). Note that v1(1) is irrotational and satisfies the Laplace equation. The contributions of ∇T1(1) and v1(1) to the velocity of particle 2 are determined from eq 20. It is found that
( ) ( )
Substituting ∇T2(2) and v2(2) ()v2t(2) + v2h(2)) into eq 24, we obtain the contributions to the velocity of particle 1 due to the reflected fields from particle 2:
(∇T2(2))r2)-r12e )
( )
1 a1 3 ]r1)r12e ) (I - 3ee)‚U1(0) 2 r12
[v1
(v2(2))r2)-r12e ) -
U2
( )
1 a1 3 )(I - 3ee)‚U1(0) 2 r12
∇T2
() () ( )(
() ( )
)
r2r2r2 2Ir2 - 5 :(∇∇T1(1))r1)r12e + 2 r2 O(a27r2-5∇∇∇T1(1)) (27)
On the other hand, the solution for v2h
() ( )
() ()
2 + 5η* a2 e e e :(∇v1(1))r1)r12e + 2(1 + η*) r22 r2 r2 r2
5
-3
(1)
O(a2 r2 ∇∇v1 ) (28b) where (eri,eθi,eφi) are the unit vectors in the spherical coordinates, the origin of which is positioned at the center of particle i; v1(1) is obtained by eq 24b.
( ) ( )
3 2 1 a1 a2 A1∆Ti2 1 a1 a2 (0) ee‚U e1 2 r 4 2 r 5 r12 12 12 3 3(2 + 5η*) a1 a2 ee‚U1(0) 2(1 + η*) r126
3
( )
3 3 1 a1 a2 (I + 3ee)‚U1(0) + O(r12-7) (31a) 2 r 6 12
()
( ) [ ( ) ( )] ( ) ( )
3 a13a2 A1∆Ti2 1 a1 (0) (I - 3ee)‚U1 e+ U2 ) 2 r12 r12 r124 3 3 a15a2 A1∆Ti2 4η* a1 a2 + 2 e2 + 3η* r 6 r12 r126 12
a25 3η*(2 + 5η*) (3er2er2er2 - eθ2eθ2er2 10(1 + η*)(2 + 3η*) r24 eφ2eφ2er2 - eθ2er2eθ2 - eφ2er2eφ2):(∇v1(1))r1)r12e +
()
a2 A1∆Ti2 a2 3 e(I - 3ee)‚U1(0) r12 r12 r12
3
is
3
v2h(2) ) -
( )
3 3 1 a1 a2 (I + 3ee)‚U1(0) + (r12-8) (30) 2 r 6 12
U1 ) U1(0) +
(28a) (2)
( )
3 3 3 3(2 + 5η*) a1 a2 1 a1 a2 (0) ee‚U ee‚U1(0) 1 6 2 r 4 2(1 + η*) r 12 12
Owing to the fact that the governing equation (eq 10) is linear, the first reflected velocity field from particle 2, v2(2), can be decomposed into two parts v2t(2) and v2h(2), which are the reflected velocities to the incident temperature field T1(1) and to the velocity field v1(1), respectively. The solution for v2t(2) satisfying the Stokes equations is given by
v2t(2) ) 0
()
where ∆Ti2 ) (Ti2 - T∞). Following the derivations of eqs 26-29, one can obtain U2(3), U1(4), U2(5), U1(6), U2(7), .... For conciseness, detailed derivations of the higher reflected temperature and velocity fields and their corresponding particle velocities are omitted. The translational velocity of thermocapillary migration of bubble 1 and the induced moving velocity of droplet 2 can be expressed as
a2 3 (I - 3er2er2)‚(∇T1(1))r1)r12e r2 5
(29c)
a2 A1∆Ti2 a2 3 e(I - 3ee)‚U1(0) r12 r12 r12
U1(2) )
(26)
(1) a2 Ti2 - (T1 )r1)r12e )er2 r2 r2
1 a2 2 r2
O(r12-8) (29b)
which, when combined, give
The first reflected temperature gradient field from particle 2, ∇T2(2), can be derived from using eqs 8 and 25a as (2)
( )
3 3 3(2 + 5η*) a1 a2 ee‚U1(0) + 2(1 + η*) r126
(∇2v2(2))r2)-r12e ) 0 + O(r12-8)
(25b)
where r1 ) r12e represents the position of the center of particle 2. Thus (1)
() ( )
3 3 3 1 a1 a2 1 a1 a2 ee‚E (I + 3ee)‚E∞ + O(r12-8) ∞ 2 r 4 2 r 6 12 12 (29a)
1 a1 3 (I - 3ee)‚E∞ (25a) [∇T1(1)]r1)r12e ) E∞ + 2 r12 (1)
()
a2 ∆Ti2 a2 3 e(I - 3ee)‚E∞ r12 r12 r12
6
a13a23 r126
ee‚U1(0) +
3 3 1 a1 a2 (I + 3ee)‚U1(0) + O(r12-7) 2 r 6 12 (31b)
As expected, both the particles will move with the velocity that would exist in the absence of the other for any arbitrary orientation of the particles as r12 f ∞ (U1 )
Thermocapillary Coagulations
Langmuir, Vol. 19, No. 11, 2003 4587
U1(0); U2 ) 0). In eq 31, the direction of thermocapillary migration of particle 1 (gas bubble) or of particle 2 (liquid droplet) is notedly deflected by the other, unless the temperature gradient prescribed is either parallel or perpendicular to the line of particle centers. The result of eq 31 can be expressed in terms of dimensionless mobility tensors Mij and defined as follows: 2
Ui )
Mij‚Uj(0) ∑ j)1
(i ) 1 or 2)
(32)
where Uj(0) is determined by eq 23. Comparing eqs 31 and 32, the mobility tensors can be further expressed as
Mij ) Mij(p)ee + Mij(n)(I - ee)
(33)
U1
Here, the mobility coefficients Mij(p) and Mij(n) are
() ( ) ( ) ( ) () ( ) () ( ) ( ) ( ) ( ) () ( )
M11(p) ) 1 +
( ) a1a2 r122
∆Ti2* + 2
a2 r12
A1(∆Ti2/a1)e
3
3
-
1 a1 a2 2 r 4 12
U2
4 2 3 3 1 a1 a2 10 + 19η* a1 a2 ∆T * + O(r12-8) (34a) i2 6 2 r 6 2(1 + η*) r12 12
M11(n)
M21(p)
a1 ) r12
a2 )1r12 3
-
a14a2 r125
3
3 3 1 a1 a2 + O(r12-8) (34b) 2 r 6 12
∆Ti2* - 4
a13a23 r126
+
4 3 a16a2 4η* a1 a2 ∆T * + 2 ∆Ti2* + O(r12-8) i2 7 2 + 3η* r 7 r 12 12 (34c)
M21(n)
1 a1 )2 r12
3
3 3 1 a1 a2 + + O(r12-8) (34d) 2 r 6 12
M12(p) ) M12(n) ) M22(p) ) M22(n) ) 0
tion surrounding the gas bubble is uniform (thus, the interfacial gradient is zero). However, if another particle (particle 2), whose temperature is different from that of the surroundings (either higher or lower), is placed near particle 1, the uniform or symmetric temperature distribution around each particle will be disturbed, and then particle 1 will migrate according to the thermocapillary effect, which induces the flow field to drive the motion of particle 2. This phenomenon frequently exists in the manufacturing processes or in the natural environment. In this situation, the temperature distribution of the external fluid is axisymmetric and leads the two interacting particles to move nonrotationally along their center line. Here, it is convenient to denote the particle velocity in its dimensionless form
(34e)
A1(∆Ti2/a1)e
)
( ) ( ) ( )[ ( ) ( )] a1a2
-
r122
a14a2
)-
r125
4 2 1 a1 a2 + O(r12-8) 2 r 6 12
+
(36a)
4 3 4η* a1 a2 + 2 + 3η* r 7 12
2
a16a2
+ O(r12-8) (36b)
r127
The particular situation of the interaction between a solid particle and a gas bubble, which has been discussed by Golovin13 using the method of bipolar coordinates, can be limited by setting η* f ∞ of particle 2. On the other hand, the dimensionless migrating velocity can be further reduced as
U1
( ) ( ) ( ) ( )
A1(∆Ti2/a)e U2 A1(∆Ti2/a)e
)-
)
a r12
a r12
5
+
2
-
1 a 2 r12
6
+ O(r12-8) (37a)
2(2 + 5η*) a 2 + 3η* r12
7
+ O(r12-8) (37b)
for the specified case of two identical spheres (a1 ) a2 ) a).
where Results and Discussion
∆Ti2 ∆Ti2* ) a1e‚E∞
(35)
It should be noted that the particle-interaction coefficients Mij(p) and Mij(n) are independent of the interfacial gradient. The thermocapillary behavior of the two interacting particles is dominated by the gas bubble, which can be easily figured out from eq 34e. The internal temperature of particle 2 is assumed to be uniform, and this makes the induced interfacial gradient of particle 2 zero, which drives the thermocapillary effect of the particle. The hydrodynamic effect is the only mechanism that induces the motion of particle 2, which is quite different from the thermocapillary effect driving the motion of particle 1 (influenced thermally and hydrodynamically). Particle 2 can be limited as a solid particle by setting η* f ∞, though the results of eqs 31 and 34 are derived on the basis of the couple interactions of a gas bubble and a liquid drop. A gas bubble (particle 1) with a thermally insulated property suspended in an immiscible liquid phase, where the prescribed temperature distribution is uniform (without temperature gradient, E∞ ) 0), does not undergo thermocapillary motion, since the temperature distribu-
There are two reasons driving the thermocapillary interactions between two particles in a temperature gradient field. First, each particle disturbs the local temperature field experienced by the other. Second, the movement of each particle drags the surrounding fluid, which causes convection and moves the other. The leading term of the translational velocity of a gas bubble is O(r12-1) due to the temperature disturbance influenced by the other particle (solid sphere or liquid drop with constant temperature) decaying with O(r12-1), which implies that particle 2 acts as a heat source (or sink). Thus, the influence of particle interactions on the thermally insulated gas bubble undergoing thermocapillary migration is much stronger than that caused by the two-gas-bubble thermocapillary interactions2-4,6 in an applied thermal gradient, since the leading term of the latter is O(r12-3). On the other hand, the leading terms of the solid sphere (or liquid drop) with constant temperature decay with O(r12-3). This is because there is no thermocapillary effect at a solidfluid interface or a surface with constant temperature, and the particle motion is driven by the hydrodynamic disturbance invoked by the gas bubble in thermocapillary migration. In a special situation of the interactions
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Figure 3. Dimensionless velocity of a gas bubble versus log η* with a2/a1 as a parameter for λ ) 0.8 (∇T∞ ) constant, ∆Ti2 ) 0, and particles moving along the line of their centers).
Figure 2. Dimensionless velocities of (a) a gas bubble and (b) a solid particle versus separation parameter λ with a2/a1 as a parameter (∇T∞ ) constant and ∆Ti2 ) 0). Solid lines and dashed lines, respectively, represent the direction of ∇T∞ parallel and perpendicular to the line of particle centers.
between an insulated gas bubble and a thermally uniform solid particle submerged in a medium of constant temperature, our analytical asymptotic results agree with the predictions evaluated by Golovin13 using bipolar coordinates when the pair is not too close ((a1 + a2)/r12 e 0.8). The dimensionless translational velocities for various cases of two thermocapillary interacting particles are displayed in Figures 2-5. It can be seen that the effect of particle interactions on the dimensionless velocity is increased with the increase in separation parameter λ ()(a1 + a2)/r12). The larger particle is obviously less influenced by the interaction effect than the smaller one. Figure 2 illustrates the migrating velocities of particle 1 (gas bubble) and particle 2 (solid sphere), which are normalized by the velocity of a single gas bubble in
thermocapillary motion, at a constant prescribed temperature gradient parallel and/or perpendicular to the particles’ center line. The temperature difference between the solid sphere and the surrounding fluid is considered to be zero (∆Ti2 ) 0); thus, the temperature perturbation caused by particle 2 will be zero simultaneously. The migrating velocity of the gas bubble is generally increased by the influence of the particle interactions as λ increases when the applied temperature gradient is parallel to the particles’ center line, and its velocity is higher than when it is isolated. However, this tendency will be quite different if the bubble radius is larger than that of the solid sphere. The gas bubble moves slightly faster than when it is isolated, as λ increases at the beginning and then the bubble velocity decreases, and this decreasing effect could make the bubble velocity smaller than its isolated velocity as the two particles get closer. The velocity of the solid sphere is similar to that of the gas bubble, as illustrated in Figure 2b. Both the interacting particles move in the same direction along the prescribed temperature gradient in axisymmetric thermocapillary migration. The particle velocities of thermocapillary migration for the case of the applied temperature gradient normal to the particles’ center line are also displayed in Figure 2 by dashed lines. It shows that the influence of particle interactions in asymmetric thermocapillary motion decreases the bubble velocity as the gap between the pairs becomes small. This trend is obviously significant, as the gas bubble is smaller than the solid sphere. The disturbed flow field reflected from the gas bubble markedly influences the velocity of the solid sphere, and the two interacting particles move in opposite directions. The hydrodynamic feature of the phoretic motion (such as thermocapillary motion, electrophoresis, and thermophoresis) can explain this phenomenon. The gas bubble pushes the external fluid in front of the bubble, while the fluid in the rear of the bubble moves into the region vacated by the bubble. However, the flow patterns of the external fluid around the equator of the bubble move opposite to the direction of particle movement because of the physiochemical properties at the interface, and this
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Figure 5. Dimensionless velocity of a gas bubble versus log η* with a2/a1 as a parameter for λ ) 0.8 (∇T∞ ) 0 and ∆Ti2 ) constant * 0).
Figure 4. Dimensionless velocities of (a) a gas bubble and (b) solid particles versus separation parameter λ with a2/a1 as a parameter (∇T∞ ) 0 and ∆Ti2 ) constant * 0).
influences the solid particle to move in reverse. This phenomenon is different from the general concept for the interactions of thermocapillary migration between the two fluid spheres, which has been well discussed.2-11 Figure 3 illustrates how the variation of the viscosity of particle 2 can influence the motion of the neighboring gas bubble (particle 1) if particle 2 is considered as a liquid droplet with constant internal temperature in axisymmetric thermocapillary migration for the case of λ ) 0.8 and ∆Ti2 ) 0. The increase of the droplet viscosity (particle 2) slightly decreases the migrating velocity of the gas bubble (particle 1); that is, the viscosity of particle 2 is an insignificant physical property to affect the motion of the gas bubble when the two particles are interacting along their center line. The droplet velocity is not influenced by the variation of its viscosity in this specified situation (λ ) 0.8 and ∆Ti2 ) 0), which can be examined from eq 31b or eq 34c (since M21(p) and M22(p) are not functions of η*); the droplet velocities normalized by the
isolated velocity of particle 1 are 0.1402, 0.0476, and 0.0074 with a2/a1 being 0.5, 1, and 2, respectively. For the asymmetric thermocapillary interactions of a gas bubble and a droplet that migrate owing to the prescribed temperature gradient perpendicular to the particles’ center line as λ ) 0.8 and ∆Ti2 ) 0, the particle velocities are not a function of the internal-to-external viscosity ratio. The variation of η* has no influence on the particle velocities. The dimensionless velocities of the gas bubble are 0.9796, 0.9340, and 0.8469 for a2/a1 ) 0.5, 1, and 2, respectively, while those of the droplet are -0.0744, -0.0300, and -0.0080. When a fluid particle is submerged in an immiscible phase with uniform prescribed temperature, the particle would not move whether its temperature is higher or lower than that of the surrounding phase. There is no existing thermocapillary effect at the particle surface in this situation. However, the spherically symmetric temperature distribution in the single-particle system will be disturbed by introducing a neighboring particle, and the previous fluid particle would then move due to thermocapillarity, since the thermal uniformity at the particle surface is destroyed. The particle interactions affect the temperature field, which makes the thermocapillary motion of the fluid sphere (though there is no prescribed temperature gradient). In addition, the movement of the fluid particle influences the hydrodynamic behavior of the neighboring one. Figures 4 and 5 describe the particle motions in this situation (∇T∞ ) 0 and ∆Ti2 ) constant * 0). Here, the translational velocities of the two particles are normalized by a constant velocity A1∆Ti2e/a1. Again, the interacting effects remarkably influence the motions of the pair when the gap between the particle surfaces gets small, as illustrated in Figure 4. However, this interacting effect on the dimensionless velocity of the gas bubble is not a monotonic function of a2/a1 at a specified value of λ. It is quite different from the general knowledge of thermocapillary motion resulting from the literature,2-15 which describes the effects of pair interaction on the particle motions influencing the smaller particle more significantly than the larger one. On the contrary, the influence of particle interactions on the motion of the solid
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Figure 6. Plots of the moving direction for the thermocapillary interaction of two particles as (∇T∞ ) 0 and a1 < a2).
sphere is gradually increased as a2/a1 decreases. This is because the motion of the gas bubble is affected by the coupling influence of the thermal and hydrodynamic effects, but the motion of the solid sphere is driven by the hydrodynamic effect only. In this particular case of interactions between a constant-temperature solid particle and an insulated gas bubble in an initially thermal uniform medium, our results agree with the results of Golovin,13 which have been formulated by the method of bipolar coordinates. The bubble velocity is not a function of the internalto-external viscosity ratio of the thermally uniform neighboring droplet when there is a nonprescribed temperature gradient and ∆Ti2 is constant, as shown in eq 36a. Figure 5 plots the dimensionless velocity of particle 2 (a liquid droplet with constant temperature) depending on the variation of its internal-to-external viscosity ratio (η*) as λ ) 0.8, ∇T∞ ) 0, and ∆Ti2 ) constant * 0. It can be seen that the influence of an increase of the particle viscosity is going to slow the translational speed of the droplet in all the considered situations, and the corresponding velocities of the neighboring gas bubble in thermocapillary migration are 0.1393, 0.1580, and 0.1415 for a2/a1 ) 0.5, 1, and 2, respectively. This phenomenon is quite different from the previous situation of axisymmetric thermocapillary interaction for ∇T∞ * 0 and ∆Ti2 ) 0. The phenomenon of decreasing surface tension with an increase of temperature, ∂γ/∂T < 0, satisfies most of the liquid-gas or liquid-liquid interfaces, though there are some alcohols with high molecular weight whose surface tensions increase with an increase of temperature in certain temperature ranges. Figure 6 qualitatively denotes the direction of motion under the interaction of a gas bubble and a solid sphere in the case of a1 < a2, ∇T∞ ) 0, and ∂γ/∂T < 0. Certainly, the illustration is not exact quantitatively; it is a figure only to display the direction of motion. The motion of the gas bubble is always in the same direction, which only depends on the positive or negative value of ∆Ti2, whatever the gap between the
Chen
particles is. However, the motion of the solid particle is much more complicated. When Ti2 < T∞, the solid particle moves in the opposite direction of the gas bubble migration when the gap between the interacting couple is large; the two particles repel each other at small separation parameter λ. The solid sphere will translate following the gas bubble when the surface-to-surface distance is small. The above characteristic implies that a particular separating distance between the pair exists to make the solid sphere motionless. On the contrary, the two particles attract each other for the case of Ti2 > T∞ when λ is small; then, the movement of the solid particle becomes slow and it stops at a particular distance; finally, the gas bubble pushes the solid particle to move in the same direction along their center line as λ approaches 1. Similar to the situation of Ti2 < T∞, there also exists a specific separation distance to hold the solid particle stationary under the pair interaction in the situation of Ti2 > T∞. One might be interested in what the specific distance will be as particle 2 reaches its maximum speed under the particle interactions of thermocapillary migration in a medium with zero applied thermal gradient. The following description is based on the situation of ∆Ti2 > 0, and the corresponding case for ∆Ti2 < 0 is the same in magnitude of the velocity but opposite in direction of translation. The solid sphere (or liquid drop) is influenced by the interacting effect to move toward the gas bubble, which also migrates toward the solid particle, at the beginning. The velocity of particle 2 increases as the gap between the pair gets small, and the magnitude of the dimensionless velocity (|a1U2/A1∆Ti2e|) reaches the maximum when the separation parameter satisfies
( ){ [
λmax ) 1 +
( ) ]}
a2 14 2η* a2 a1 5 2 + 3η* a1
2
-1/2
+1
(38)
for a general description. In the limitation of η* f ∞, eq 38 describes the situation of particle 2 being a solid sphere. Equation 38 is derived from eq 36b together with the relation dU2/dλ ) 0. After arriving at the most rapid velocity, the movement of particle 2 will then be slowing down and the two particles keep attracting each other under the thermocapillary effects. Then, particle 2 will stop moving toward the gas bubble at a certain separation distance (called λmin ), and it furthermore turns back to translate in the same direction of the gas bubble, which always migrates in the same direction but with the magnitude of the speed influenced by the interaction. Finally, particle 2 is heading in the opposite direction of its movement at the beginning, and particle 1 (gas bubble) follows it; both particles translate with the same velocity. The separation distance at which particle 2 turns back or, say, stops can be evaluated by eq 36b after setting U2 ) 0. That is
( )[
λmin ) 1 +
() ]
a2 4η* a2 a1 2 + 3η* a1
2
-1/2
+2
(39)
Note that λmin does not physically imply the minimum distance between the pair because the gas bubble is still approaching particle 2; it is just to denote the distance at which particle 2 begins to move in the opposite direction. The dimensionless distances of λmax and λmin do not always exist, as shown in Table 1; however, λmax can be obviously found if λmin exists. Table 1 lists the values of λmax and λmin in some specified situations. It shows that the “turn back” feature of the droplet (particle 2) can always be found when particle 2 is much smaller than the gas bubble
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Table 1. Values of Separation Distance (λ) for the Maximum and Minimum of |a1U2/A1∆Ti2e| as E∞ ) 0 and ∆Ti2 * 0 η*
a2/a1
λmax
λmin
η*
a2/a1
λmax
λmin
0.0
0.1 0.2 0.5 1.0 2.0 5.0 10.0 0.1 0.2 0.5 1.0 2.0 5.0 10.0 0.1 0.2 0.5 1.0 2.0 5.0 10.0 0.1 0.2 0.5 1.0 2.0 5.0 10.0
0.6574 0.7171 0.8964
0.7778 0.8485
10
0.7778 0.8484
100
0.6553 0.7083 0.8337 0.9376 0.9583 0.8794 0.8249 0.6552 0.7078 0.8303 0.9270 0.9385 0.8558 0.8018 0.6552 0.7078 0.8299 0.9258 0.9363 0.8531 0.7991
0.7754 0.8381 0.9864
0.6573 0.7170 0.8953
0.1 0.2 0.5 1.0 2.0 5.0 10.0 0.1 0.2 0.5 1.0 2.0 5.0 10.0 0.1 0.2 0.5 1.0 2.0 5.0 10.0
0.01
0.1
1
0.6571 0.7159 0.8868
0.7775 0.8471
0.6561 0.7115 0.8547
0.7763 0.8418
∞
0.9761 0.7753 0.8375 0.9824
between the particle surfaces is small; the bubble is heading in front when Ti2 < T∞, and quite the reverse is true when Ti2 > T∞. This phenomenon is remarkable if the particle size of the droplet (or solid particle) is much smaller than that of the gas bubble. Generally speaking, the particle interactions of a gas bubble and a constanttemperature droplet in thermocapillary migration are different from those of two insulated gas bubbles or droplets. Acknowledgment. This research was supported by the National Science Council of the Republic of China under Grant NSC90-2214-E-146-001. Nomenclature
0.9487 0.7752 0.8374 0.9820
0.9456
(particle 1). The values of λmax and λmin get smaller as η* increases; in other words, the “turn back” feature of particle 2 happens earlier if the viscosity of particle 2 increases. In general, λmax and λmin are not monotonic functions of a2/a1. Conclusions The analytical formulation of the thermocapillary interactions between a thermally insulated gas bubble and a thermally uniform droplet is studied in this article. The prescribed temperature distribution can be linear or uniform. A method of reflections, which is correct to O(r12-7), has been applied to obtain the temperature distributions and flow fields inside and outside the particles. The mobility functions (or dimensionless velocities) of both the gas bubble and the liquid drop (or solid particle) in the thermocapillary migration are examined. The thermocapillary behavior of the gas bubble is generally the same as the corresponding situation influenced by the interaction between two gas bubbles in a prescribed temperature gradient. However, the hydrodynamic behavior of the liquid drop (or the solid particle) with constant temperature is quite different from the results as expected, whether the prescribed temperature gradient is zero or not. When the parts of the interacting pair are far from each other and there is no applied temperature gradient, the two particles attract each other, as the temperature of the drop (Ti2) is higher than the surrounding temperature (T∞), while the couple repel each other as Ti2 < T∞. However, the pair migrates one after the other if the gap
a, a1, a2 ) particle radii (m) e ) unit vector pointing from the center of particle 1 to the center of particle 2 er, eθ, eφ ) unit vectors in the spherical coordinate system E∞ ) uniform applied temperature gradient (K m-1) F ) force exerted by the fluid on the particle (N) I ) unit dyadic k ) thermal conductivity of the external fluid (W m-1 K-1) ki ) thermal conductivity of the particle (W m-1 K-1) k* ) ki/k Mij ) mobility tensor defined in eq 33 Mij(n), Mij(p) ) dimensionless mobility functions defined in eq 34 p ) dynamic pressure of the external flow (N m-2) pi ) dynamic pressure of the flow inside the droplet (N m-2) r ) position vector pointed from the particle center (m) r ) |r| (m) r12 ) distance between the centers of particle 1 and 2 (m) Sm ) surface harmonic polyadic T ) temperature distribution of the external fluid (K) TA ) prescribed temperature distribution (K) Ti ) temperature distribution inside the particle (K) T∞ ) applied linear temperature field (K) U ) instantaneous velocity of particle (m s-1) U(0) ) velocity of a droplet in isolated thermocapillary migration (m s-1) v ) velocity distribution of the external fluid (m s-1) vi ) velocity distribution of the fluid inside the droplet (m s-1) vs ) surface velocity of the particle (m s-1) x ) position vector (m) x0 ) position vector of the particle center (m) Greek Letters Rm, βm ) polyadic coefficient (m s-1) γ ) interfacial tension (N m-1) ∆Ti2 ) Ti2 - T∞ (K) η ) viscosity of the external fluid (kg m-1 s-1) ηi ) viscosity of the fluid inside the droplet (kg m-1 s-1) η* ) ηi/η λ ) (a1 + a2)/r12 λm ) polyadic constant (K) τ ) viscous stress tensor of the external fluid (N m-2) τi ) viscous stress tensor of the fluid inside the droplet (N m-2) LA020805F
