Thermophoresis of a Micro-Particle in Gaseous Media with Effect of Thermal Stress Slip

Abstract

The present article pertains to a theoretical analysis for thermophoresis of a micro-particle in rarefied gas environment. Different from the conventional analyses, thermal stress effect in velocity slip boundary condition is taken into account. This interfacial thermal characteristic is always ignored in previous theoretical studies. The present theory with consideration of the thermal stress slip provides an appropriate modification and thus agrees well with the measured data of thermophoretic mobility of various particles. Order-of-magnitude analysis also qualitatively demonstrates enhancement of the thermal stress slip with reducing particle size or increasing characteristic temperature gradient. Finally, a parametric analysis shows that the thermal stress slip effect becomes more pronounced at conditions of high particle thermal conductivity and high Knudsen number as well.

1. INTRODUCTION

It has been well known that a micro-particle suspended in a gaseous medium with an existing temperature gradient experiences a thermophoretic force, which drives the particle in motion. This phenomenon is called thermophoresis. Extensive review and historical development of thermophoresis can be found in literature, e.g., Zheng (2002) and Piazza and Parola (2008). Thermophoresis is of considerable importance in a variety of practical applications, such as air cleaning, aerosol sampling, and particle contaminant control in the semiconductor industry (Chein and Liao 2005). It has been extensively investigated over the last few decades.

The physical interpretation of thermophoresis mechanism is based on the gas kinetic theory. The particle in a gaseous medium of non-uniform temperature field has hot and cold sides. The gas molecules in the region near the hotter side collide with the particle at higher momentum than the molecules in the cold region. It thus leads to a resultant force and in turn migration of the particle in the direction opposite to the temperature gradient (Senchenko and Keh 2007). The interaction between the gas molecules and the particle depends on a parameter termed Knudsen number (Kn), which is defined as the ratio of mean free path l of the gas to a characteristics length scale, e.g., particle diameter in dealing with phoretic motion. A number of theoretical studies on particle thermophoresis have been performed in slip-flow regime with Kn up to O(10⁻¹), for spherical particles (Epstein 1929; Brock 1962; Talbot et al. 1980; Keh and Chen 2003; Lu and Lee 2003; Keh and Chang 2006) and non-spherical particles (Williams 1986; Keh and Tu 2001; Keh and Ou 2004). In the previous theoretical studies of thermophoresis, the boundary condition employed in velocity and temperature solutions is simple slip condition.

In an earlier work of Sone (1972), it was demonstrated that a tangential velocity slip now named thermal stress slip can be stemmed from the variation in the normal gradient (∂T/∂n) of the gas temperature in the tangential direction along the gas-solid interface, i.e., ∂(∂T/∂n)/∂s, (n and s stand for normal and tangential directions, respectively). Recently, this thermal stress effect was also employed to develop a generalized second-order slip boundary condition (Lockerby et al. 2004) as a modification of the very earlier version of velocity slip condition proposed by Maxwell (1879). The slip condition with thermal stress has been employed in solutions of natural convection of rarefied gas between two non-coaxial cylinders (Aoki et al. 1998; Lockerby et al. 2004), theoretical analysis of particle photophoresis (Soong et al. 2009), etc. But it has not ever been considered in previous studies of particle thermophoresis.

For thermophoretic motion of spherical particle, with variation of the normal (radial) temperature gradient along the curved particle surface, the thermal stress slip effect should be taken into account, especially in the case of small particle. The present work is to revisit the problem of spherical particle thermophoresis with considering the thermal slip effect in gaseous slip-flows. Besides, the thermophoretic mobility with the modified slip-velocity boundary conditions is to be theoretically evaluated.

2. THEORETICAL ANALYSIS

Figure 1 shows a schematic diagram of an isotropic homogenous particle suspended in a gas medium of a prescribed temperature gradient ∇T ₀ with T ₀ denoting temperature field. A spherical coordinate system O(r, θ, ϕ) is fixed on the center of the particle. Since the thermophoretic motion of the particle is slow, the relative velocity of the gas around the particle and the forced convection is negligible, the energy transfer between the gas and particle is assumed to be dominated by heat conduction. The gas temperature distribution T_g and the temperature field inside the particle T_p are governed by the Laplace equation,

FIG. 1 Physical model of a micro-spherical particle thermophoresis in a gaseous medium.

On the particle surface, r=R, the normal heat flux is continuous and a temperature jump is considered, while in the far-field, the gas temperature gradient approaches the prescribed free-stream temperature gradient ∇T ₀. Considering the axi-symmetry nature, the temperature fields T_g and T_p in rθ-plane are solved by invoking the following boundary conditions,

where the radiative heat transfer is negligible, c_tj is the temperature jump coefficient, and k_g and k_p are the thermal conductivities of the gas and particle, respectively.

The fluid temperature distribution in spherical coordinates satisfying the above boundary conditions is

The Stokes flow around the spherical particle with fluid viscosity μ, the velocity vector v, and pressure field p can be described by the incompressible creeping flow equation.

In solution of the gas velocity field, the following second-order slip boundary condition based on the Maxwell relation with Burnett stress flux and heat flux (Lockerby et al. 2004) is considered, viz.,

where v stands for velocity, T_r for a reference temperature, Pr is the Prandtl number, and γ is the ratio of specific heat. The coefficient c_m is defined as c_m ≡(2−σ)/σ with σ denoting the tangential momentum accommodation coefficient, c_ts the thermal stress slip coefficient, and c_tc the thermal creep coefficient. With the characteristic quantities of temperature difference denoted by ΔT_c , the characteristic length L_c =R (particle radius), the reference velocity V ₀, the approximations ∂² v_s /∂n ²∼V ₀/R ² and ∂² T_g /∂n∂s∼ΔT_c /R ², and Pr = 0.7 and γ=1.4 for air, the ratio of the representative term of second-order velocity derivative, ∂² v_s /∂n ², to the thermal stress slip can be estimated as

The thermophoretic motion of the particle is slow; the Reynolds number Re=ρV ₀ R/μ is thus very low. In a typical example with the parameters Kn ∼O(10⁻¹), Re ∼O(10⁻³), ΔT_c /T_r ∼O(10⁻¹), the above ratio has the order-of-magnitude of 10⁻⁴, which means that the terms involving second-order derivative of velocity are quite smaller than the thermal stress slip. Thus, in the present analysis we adopt slip boundary condition with the second-order terms neglected but only thermal stress slip retained. Using the present nomenclature and coordinates, the boundary conditions can be depicted as

Where is the bulk-gas absolute temperature at the location of the particle center in the absence of the particle or the mean gas temperature in the vicinity of the particle (Senchenko and Keh 2007), and V ₀ the free-stream velocity.

Using the governing equations and boundary conditions above, the force exerted on the particle by the ambient gas can be determined by

The first term on the right hand side is the Stokes drag force with rarefaction correction, and the second term is the thermophoretic force (F_th ). This formula with thermal stress slip effect included is different from that proposed previously by Brock (1962) or Talbot et al. (1980). The values of coefficients c_m , c_tc , and c_tj were developed with kinetic theory for complete energy and momentum accommodations of Maxwellian/hard-sphere molecules (Keh and Chang 2009) but the thermal creep coefficient has been modified to be 1.17 by experimental and theoretical predictions (Talbot et al. 1980). Then, the coefficients, c_m =1.14, c_tc =1.17, and c_tj =2.18, which were extensively used in the previous studies of particle thermophoresis (Talbot et al. 1980; Lu and Lee 2003; Keh and Ou 2004; Chein and Liao 2005; Davis 2006; Keh and Chang 2009) are adopted in the present analysis. The thermal stress slip coefficient c_ts =1 was determined on the basis of the linearized Maxwell-Burnett boundary condition (Lockerby et al. 2004). For comparison, c_ts = 0 is taken for the cases with thermal stress slip ignored. Here, the thermal stress coefficient c_ts (1 or 0) is regarded simply as a switching index for consideration of the thermal stress slip effect. Poddoskin et al. (2001) demonstrated that the slip coefficients for monoatomic and polyatomic gases differ only slightly from one another. The theoretical analysis of Latyshev et al. (2005) also showed that slip coefficients depend on Prandtl number (Pr) considerably. For the gases of Pr around 2/3 (= 0.667), the slip velocity and the rate of thermophoresis in polyatomic gases differ from those in monoatomic gases insignificantly. Therefore, we adopted the coefficients based on monatomic gases for air (Pr ≃ 0.7) and applied the commonly accepted values in the analysis on particle thermophoresis.

For thermophoretic motion at steady state, the resultant force acting on the particle is zero and the particle thermophoretic velocity, V_th , is of opposite direction of the fluid velocity V ₀ for transformation of the reference frame, i.e., V_th =−V ₀, and one has

Consider the viscosity given by (Kennard 1938), where is the mean molecular speed, then μ=4Mk_g /15k_B with k_B denoting the Boltzmann's constant and M the gas molecular mass. We obtain the thermophoretic force and velocity equations in alternative forms,

3. RESULTS AND DISCUSSION

3.1. Analysis of Order-of-Magnitudes

To check relative importance of the thermal stress slip, an order-of-magnitude analysis is performed. In Equation (8a), the first two terms, r∂(v _θ/r)/∂r+(∂v_r /∂θ)/r, in the square bracket on right hand side (RHS) are concerned with momentum exchange, the third one in the square bracket is the thermal stress slip term, , and the last term on RHS is thermal creep term. The parameter ℜ₁ is defined as the ratio of the thermal stress term to the momentum exchange term, and ℜ₂ as the ratio of the thermal stress term to the thermal creep, viz.,

Taking R as the characteristic length, V ₀ the characteristic velocity, and ΔT_c ∼R∇T ₀ the characteristic temperature difference in gas, the order-of-magnitudes of the above two ratios can be derived as follows,

Equation (14a) reveals that the thermal stress slip is proportional to the thermal parameter or to the prescribed temperature gradient |∇T ₀| of the background fluid. Since the coefficients c_m , c_ts , and c_tc are all of O(1), Equation (14c) shows that the ratio ℜ₂ has the order of Kn and the property of inversely proportional to the particle radius R, which demonstrates the increasing significance of thermal stress slip with reduction in particle size.

3.2. Comparison with Previous Predictions and Experimental Data

In slip-flow regime, Borck (1962) and Talbot et al. (1980) developed a theory for particle thermophoresis, in which the thermophoretic force is

Equation (15) can be used for thermophoresis in transition regime before more accurate solutions of the Boltzmann equation is available (Zheng 2002). Talbot et al. (1980) concluded that this formula can be applied over the whole range of Knudsen number with acceptable deviations. The previous theoretical solutions of linearized Boltzmann equation and the Bhatnager-Gross-Krook (BGK) model (Sone and Aoki 1981; Yamamoto and Ishihara 1988; Loyalka 1992) all yielded a similar form of thermophoretic force F_th as the product of the parameter group and a function of Kn and thermal conductivity ratio. Thereby, with dimensionless form and Equation (15), Davis (2006) proposed a thermophoretic force expression. In terms of the present notions, it is written as

With consideration of thermal stress slip effect, we develop a modified expression of thermophoretic force, i.e.,

where k*≡k_p /k_g is the particle-to-gas thermal conductivity ratio indicating the relative importance of the particle thermal conductivity in the thermophoresis. The present expression, Equation (17), is different from the previous one, Equation (16), with the presence of a new term, −c_tsc_mk*Kn.

To demonstrate the validity of the present theory, we try to compare our results with the previous predictions and experimental data. However, for lack of previous experimental data on thermophoresis in slip flow regime, the present theoretical predictions are compared with measurements of thermophoretic force in the range of 0.06⩽Kn⩽1. Figures 2–4 show comparisons of our predictions of dimensionless thermophoretic force with previous theory and measurements for three different kinds of particles. For dioctyl phthalate (DOP) particles of thermal conductivity k_p = 0.165 W/mK in gaseous medium (air) of k_g = 0.0262 W/mK, the present predictions agree well with the experimental data of Li and Davis (1995) in the range of Kn < 0.5. The theory with consideration of thermal stress slip is better than Equation (16) proposed by Davis (2006), which has noticeable deviation from the measurements and the deviations are more pronounced at high Kn. As Knudsen number approaches to the continuum limit, the theories with and without thermal stress slip coincide the measurements due to vanishing thermal stress slip effect. In Figure 3, for polystyrene latex (PSL) with thermal conductivity k_p = 0.195 W/mK in air, it can be observed that our predictions at Kn < 0.6 are closer to the experimental data (Li and Davis 1995; Santachiara et al. 2002) than the theoretical results of Davis (2006) without thermal stress slip. In Figures 2 and 3, data from Li and Davis (1995) are of a little abrupt change with increase in Kn and become closer to Davis's predictions (2006). So far we do not have any speculation about this tendency of the measurements.

FIG. 2 Comparison of the present theory with measured thermophoretic velocity of DOP (dioctyl phthalate) particle in air with k*=6.298.

FIG. 3 Comparison of the present theory with measured thermophoretic velocity of PSL (polystyrene latex) particle in air with k*= 7.443.

FIG. 4 Comparison of the present theory with measured thermophoretic velocity of NaCl particle in air with k*= 244.3.

For a NaCl particle of thermal conductivity k_p = 6.4 W/mK in air, Figure 4 presents the predictions of dimensionless thermophoretic force and compared with the previous experimental data (Schadt and Cadle 1961; Jacobsen and Brock 1965) and theoretical predictions (Davis 2006). The present predictions with thermal stress slip effect are more comparable with experimental data and the qualitative features about the thermal stress slip presented in Figures 2 and 3 are similar. The deviation between the theories with (present) and without (Davis 2006) thermal stress slip effect is enlarged as Kn increases or, equivalently, the particle size reduces.

3.3. Parametric Analysis of Thermophoretic Mobility

With V_th normalized as , it can be expressed in a dimensionless form, viz.

To further understand the thermal stress slip effect on thermophoretic mobility, we perform a detailed parametric analysis. In Equation (18), the dimensionless thermophoretic velocity V* _th is a function of two major parameters. The Knudsen number, Kn≡l/R, characterizes the rarefaction of the gas and, by its definition, it is also an indication for the influence of the particle size. The other is the dimensionless thermal conductivity of the particle, k*≡k_p /k_g . Figure 5 presents the influences of the thermal conductivity ratio k* on the thermophoretic velocity V* _th at various conditions of Kn with and without considering thermal stress slip. It can be observed that V* _th monotonically decreases with k* for all cases. It agrees with the conclusions of previous works (Ohwada and Sone 1992; Keh and Tu 2001). Physically, a particle of high thermal conductivity is able to spread heat over the particle and reduce the local temperature gradient along the particle surface. The thermal creeping flow and thus the thermophoretic velocity are suppressed but, relatively, the thermal stress flow becomes dominant.

FIG. 5 Variation of dimensionless thermophoretic velocity with relative thermal conductivity of the particle at various conditions of Knudsen number.

Figure 5 demonstrates that the influence of thermal stress slip is diminished as k* reduces but becomes more remarkable at higher values of k*. Since the thermal source in the field is the temperature gradient of the gas medium, the particle has the surface temperature gradient, (∂T_p /∂θ) _w , in the same direction as that in gas field. As k* increases, the non-uniformity of the temperature distribution along the particle surface is smeared out and, through interactions at the particle-gas interface with temperature jump condition, the gas side temperature near the interface has the same tendency, i.e., reduction in (∂T_g /∂θ) _w . This will raise the temperature on the cold side of the particle and increase the local value of the normal temperature gradient (∂T_g /∂r) _w on the cold side. Therefore, it leads to an increase in the tangential variation of the gradient (∂T_g /∂r) _w or an enhancement of the thermal stress slip ∂(∂T_g /∂r) _w /∂θ. As to the thermal stress slip effect at various conditions of Kn, both Figures 5 and 6 demonstrate that the effect on V* _th is more pronounced at high Kn or small radius of the particle just as that disclosed by the order-of-magnitude analysis.

FIG. 6 Variation of dimensionless thermophoretic velocity with Knudsen number at various conditions of relative thermal conductivity of the particle.

4. CONCLUDING REMARKS

In the present study, we have modified the conventional thermophoresis theory with consideration of the thermal stress slip and developed novel analytical expressions for predictions of thermophoretic mobility. Through comparison with the measured data for various particles, the present theory has shown its capability in appropriate prediction of the thermophoretic mobility. Order-of-magnitude analysis as well as the theoretical expressions developed in the work both demonstrate enhancement of the thermal stress slip with reducing particle size. The present theory reveals that the thermal stress slip effect becomes more pronounced at conditions of high particle thermal conductivity and high Knudsen number as well. The theoretical expressions and the physical insights obtained in this work are useful in further understanding of the physical mechanism of the thermophoresis with strong interfacial thermal interactions.

This study was supported by National Science Council of the Republic of China (Taiwan) through the Grant NSC-98-2221-E-035-068-MY3 (2009-2012).