Thermophoresis of a Spherical Particle: Reassessment, Clarification, and New Analysis

Figures & data

Table 1 Theoretical expressions for the functions Φ and Ψ. The current best estimates of the thermal creep, velocity slip, and temperature jump coefficients are K_tc = 1.10, C_m =1.13(2−α _m )/α _m , and C_e =2.17(2−α _e )/α _e

Authors and range of validity	Φ (Kn, Λ)	Ψ (Kn)
Kn → 0 (Epstein (1929) and Stokes (1851))		6π
(Brock (1962) and Bassett (1888))
(Waldmann (1959) and Epstein (1924))

Figure 1 Curves of −Φ/(2π) plotted against log₁₀(Kn) for various Λ using the interpolation formula of Talbot et al. (1980) with α_m = α_e = 1. All curves asymptote to Epstein's continuum limit as Kn→0.

Figure 2 The spherical polar (r, θ, φ) coordinate system used for the G13 analysis. There is axial symmetry around the z-axis (∂/∂φ=0).

Figure 3 Curves of −Φ/(2π) plotted against log₁₀(Kn) for various Λ using Equation (31a) (Brock's theory plus the temperature-driven part of the thermal stress). Reversed thermophoresis at high Λ is evident but the behavior at high Kn is nonphysical.

Figure 4 Curves of −Φ/(2π) plotted against log₁₀(Kn) for various Λ using Equation (32a) (Dwyer's theory with the pressure-driven heat flux included correctly). For Kn < 0.2, the curves are almost identical to those of Figure 3 but the behavior at high Kn is nonphysical.

Figure 5 Diagram to illustrate the calculation of the free-molecule distribution function f_FM .

Figure 6 Curves of −Φ/(2π) plotted against log₁₀(Kn) for a perfect thermally conducting sphere (Λ→∞). The calculations are from Sone and Aoki (1981, 1983) and Yamamoto and Ishihara (1988). Also shown is Equation (43) from Ivchenko and Yalamov, the first-order correction in

to Waldmann's free-molecule expression.

Table 2 Published experimental data for the thermophoretic force

Authors and source	Particle and gas	Λ	Kn
Saxton and Ranz (1952), Figure 3	Castor oil in air	9.5	0.05–0.12
Schmitt (1959), Figure 6	Silicone oil in air	7.5	0.11–2.00
Schadt and Cadle (1961), Figure 1b	TCP in air	10	0.14–0.76
Schadt and Cadle (1961), Figure 2b	NaCl in air	322	0.20–1.49
Schadt and Cadle (1961), Figure 3	Hg in air	581	0.45–3.60
Jacobsen and Brock (1965), Figures 4 and 5	NaCl in Ar	300	0.06–0.67
Tong (1975), Figure 6	Al in He	1519	0.01–0.32
Davis and Adair (1975), Figure 5	Cork in Ar	2.5	0.05–1.80
Li and Davis (1995a), Figure 11 (0905b)	PSL in air	5.4	0.06–5.00
Li and Davis (1995a), Figure 8 (0501a)	DOP in air	6	0.06–1.55
Li and Davis (1995a), Figure 10 (0510d)	DOP in air	6	0.05–5.00
Li and Davis (1995a), Figure 12 (0927b)	Glass in air	40	0.10–5.00
Li and Davis (1995a), Figure 12 (0726a)	Ni in air	4700	0.09–5.00
Li and Davis (1995b), Figure 6	PSL in CO2	10	0.10–3.50
Li and Davis (1995b), Figure 6	DOP in CO2	12	0.05–4.20
Li and Davis (1995b), Figure 6	Glass in CO2	75	0.12–4.00
Li and Davis (1995b), Figure 10	DOP in He	0.8	0.10–1.90

Figure 7 Curves of −Φ/(2π) plotted against log₁₀(Kn) for various Λ from Yamamoto and Ishihara (1988). The black circles are calculated from Equation (44) and Tables 1 and 2 of the original article. Also shown are the Epstein continuum limits.

Figure 8 Curves of −Φ/(2π) plotted against log₁₀(Kn) for various Λ from Beresnev and Chernyak (1995). The black circles are calculated from Equation (45) and Table 1 of Beresnev and Chernyak. Also shown are the Epstein continuum limits.

Figure 9 The thermophoretic force data of (a) Schmitt (1959); (b) Schadt and Cadle (1961); (c) Jacobsen and Brock (1965); (d), (e), and (f) Li and Davis (1995a, 1995b). Theoretical predictions for the corresponding values of Λ: Beresnev and Chernyak (solid lines), Yamamoto and Ishihara (dotted lines).

Table 3 Published experimental data for the thermophoretic velocity

Authors and source	Particle and gas	Λ	Kn
Rosenblatt and LaMer (1946), Table III	TCP in air	10	0.05–1.50
Schmitt (1959), Figure 5	Oil in Ar, N2, CO2, H2	various	0.16–2.65
Prodi et al. (1979), Table 3	NaCl in air	309	0.08–0.64
Toda et al. (1998), Table 1	PMMA in air	7.9	0.004–0.027
Toda et al. (1998), Table 1	SiO2 in air	74	0.004–0.14
Santachiara et al. (2002), Figure 2	Wax in air	4.6	0.10–0.61
Santachiara et al. (2002), Figure 5	PSL in air	5.4	0.10–0.33
Santachiara et al. (2002), Figure 4	NaCl in air	309	0.14–0.56
Santachiara et al. (2002), Figure 2	Ag in air	22530	0.22–0.71
Prodi et al. (2006), Figure 2	Paraffin, wax, NaCl in air	various	0.04–0.90
Prodi et al. (2007), Figures 2 and 3	Paraffin in He, N2, Ar, Xe	various	0.05–0.90
Sagot et al. (2009), Figure 11	Vegetable oil in air	10	0.02–3.0

Figure 10 The thermophoretic velocity data of (a) Schmitt (1959), (b) and (c) Santachiara et al. (2002), (d) Prodi et al. (2007), and (e) Sagot et al. (2009). Theoretical predictions for the corresponding values of Λ: Beresnev and Chernyak (solid lines), Yamamoto and Ishihara (dotted lines).

Figure 11 Curves of −Φ/(2π) plotted against log₁₀(Kn) for various Λ. The white circles are from Yamamoto and Ishihara (1988), the black circles are from Beresnev and Chernyak (1995), and the solid lines are from the interpolation formula Equation (47) with C_int = 0.5.

Figure 12 Curves of −Φ/(2π) plotted against log₁₀(Kn) for various Λ using the interpolation formula Equation (47) with C_int = 0.5. For Kn < 0.2, the curves are almost identical to those of Figures 3 and 4. For comparison, the Talbot et al. (1980) prescription is also plotted.

Epstein , P. S. 1929 . Zur Theorie des Radiometers . Z. Physik. , 54 : 537 – 563 .

 Google Scholar

Brock , J. R. 1962 . On the Theory of Thermal Forces Acting on Aerosol Particles . J. Colloid Sci. , 17 : 768 – 780 .

 Google Scholar

Bassett , A. B. 1888 . A Treatise on Hydrodynamics (Vol. 2). Originally published by Cambridge, Deightion, Bell, 1888; Reprinted by Dover 1961. p. 270

 Google Scholar

Waldmann , L. 1959 . Uber Die Kraft Eines Inhomogen Gases auf Kleine Suspendierte Kugeln . Z. Naturforsch. , A14 : 589 – 599 .

 Google Scholar

Epstein , P. S. 1924 . On the Resistance Experienced by Spheres in Their Motion through Gases . Phys. Rev. , 23 : 710 – 733 .

 Google Scholar

Talbot , L. , Cheng , R. K. , Schefer , R. W. and Willis , D. R. 1980 . Thermophoresis of Particles in a Heated Boundary Layer . J. Fluid Mech. , 80 : 737 – 758 .

 Web of Science ®Google Scholar

Yamamoto , K. and Ishihara , Y. 1988 . Thermophoresis of a Spherical Particle in a Rarefied Gas of a Transition Regime . Phys. Fluids , 31 : 3618 – 3624 .

 Web of Science ®Google Scholar

Saxton , R. L. and Ranz , W. E. 1952 . Thermal Force on an Aerosol Particle in a Temperature Gradient . J. Appl. Phy. , 23 : 917 – 923 .

 Web of Science ®Google Scholar

Schmitt , K. H. 1959 . Untersuchungen an Schwebstoffteilchen im Temperaturfeld . Z. Naturforsch. , 14a : 870 – 881 .

 Google Scholar

Schadt , C. F. and Cadle , R. D. 1961 . Thermal Forces on Aerosol Particles . J. Phys. Chem. , 65 : 1689 – 1694 .

 Web of Science ®Google Scholar

Jacobsen , S. and Brock , J. R. 1965 . The Thermal Force on Spherical Sodium Chloride Aerosols . J. Colloid Sci. , 20 : 544 – 554 .

 Google Scholar

Tong , N. T. 1975 . Experiments on Photophoresis and Thermophoresis . J. Colloid Interface Sci. , 51 : 143 – 151 .

 Web of Science ®Google Scholar

Davis , L. A. and Adair , T. W. 1975 . Thermal Force on a Sphere . J. Chem. Phys. , 62 : 2278 – 2285 .

 Web of Science ®Google Scholar

Li , W. and Davis , E. J. 1995a . Measurement of the Thermophoretic Force by Electrodynamic Levitation: Microspheres in Air . J. Aerosol Sci. , 26 : 1063 – 1083 .

 Web of Science ®Google Scholar

Li , W. and Davis , E. J. 1995b . The Effects of Gas and Particle Properties on Thermophoresis . J. Aerosol Sci. , 26 : 1085 – 1099 .

 Web of Science ®Google Scholar

Beresnev , S. and Chernyak , V. 1995 . Thermophoresis of a Spherical Particle in a Rarefied Gas: Numerical Analysis Based on the Model Kinetic Equations . Phys. Fluids , 7 : 1743 – 1756 .

 Web of Science ®Google Scholar

Rosenblatt , P. and LaMer , V. K. 1946 . Motion of a Particle in a Temperature Gradient: Thermal Repulsion as a Radiometer Phenomenon . Phys. Rev. , 70 : 385 – 395 .

 Web of Science ®Google Scholar

Prodi , F. , Santachiara , G. and Prodi , V. 1979 . Measurements of Thermophoretic Velocities of Aerosol Particles in the Transition Region . J. Aerosol Sci. , 10 : 421 – 425 .

 Web of Science ®Google Scholar

Toda , A. , Ohnishi , H. , Dobashi , R. and Hirano , T. 1998 . Experimental Study on the Relation between Thermophoresis and Size of Aerosol Particle . Int. J. Heat Mass Transfer , 41 : 2710 – 2713 .

 Web of Science ®Google Scholar

Santachiara , G. , Prodi , F. and Cornetti , C. 2002 . Experimental Measurements on Thermophoresis in the Transition Region . J. Aerosol Sci. , 33 : 769 – 780 .

 Web of Science ®Google Scholar

Prodi , F. , Santachiara , G. , Travaini , S. , Vedernikov , A. , Dubois , F. , Minetti , C. and Legros , J. C. 2006 . Measurements of Phoretic Velocities of Aerosol Particles in Microgravity Conditions . Atmos. Res. , 82 : 183 – 189 .

 Web of Science ®Google Scholar

Prodi , F. , Santachiara , G. , Di Matteo , L. , Vedernikov , A. , Beresnev , S. A. and Chernyak , V. G. 2007 . Measurements of Thermophoretic Velocities of Aerosol Particles in Microgravity Conditions in Different Carrier Gases . J. Aerosol Sci. , 38 : 645 – 655 .

 Web of Science ®Google Scholar

Sagot , B. , Antonini , G. and Buron , F. 2009 . Annular Flow Configuration with High Deposition Efficiency for the Experimental Determination of Thermophoretic Diffusion Coefficients . J. Aerosol Sci. , 40 : 1030 – 1049 .

 Web of Science ®Google Scholar