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Temperature Dependence of the Soret Motion in Colloids :: Alois Wurger CPMOH, Universit e Bordeaux 1 and CNRS, 351 cours de la Lib eration, 33405 Talence, France Received January 16, 2009. Revised Manuscript Received February 16, 2009 We discuss thermally driven transport in both organic solvents and electrolyte solutions and, in particular, the variation of the thermophoretic mobility DT with temperature. We find that the T-dependence in organic solvents arises from the viscosity η(T), in qualitative agreement with data on polystyrene in ethylbenzene. A more subtle effect occurs for charged colloids in an electrolyte, where the strong correlation of DT with the thermal expansivity β of water is traced back to the thermoelectric effect. Our results provide an explanation for experiments on various colloidal suspensions and, in particular, for the observed change of sign of DT as a function of temperature.
I. Introduction When applying a thermal gradient to a complex fluid, one observes a flow of its constituents and, after reaching the steady state, a nonuniform distribution. This Soret motion, or thermophoresis, occurs in both ionic and uncharged systems; electric and dispersion forces result in transport velocities of comparable magnitude. The Soret effect has been shown to provide an efficient separation technique for macromolecules in organic solvents1 and has been used as a molecular trap for DNA in a microchannel with ambient flow.2a On a macroscopic scale, thermal forces may reshape 100 μm bicontinuous lyotropic crystals.3 In a colloidal suspension, a thermal gradient affects the solutesolvent interactions and drives the solute at a velocity u ¼ -DT rT
ð1Þ
In the stationary state of a closed system, the resulting particle flow is counterbalanced by the diffusion current with the Einstein coefficient D;4 most experimental techniques determine the density gradient in terms of the Soret coefficient ST = DT/D, whereas single-particle tracking gives the thermophoretic mobility DT.5 The thermophoretic mobility of charged colloids is to a large extent determined by the properties of the electrolyte solution. For negatively charged polystyrene beads, a change of sign was observed as a function of salt content and acidity.6 The underlying thermoelectric effect is strongest in acids and bases such as HCl and NaOH solutions but is reduced when adding salt. In the usual framework of Stokes hydrodynamics,7-12 the thermoelectric effect provides a quantitative explanation for the observed dependencies on pH and salinity.6,13-15 In dense suspensions, the coefficient DT is reduced by particle-particle interactions.16 (1) Giddings, J. C. Science 1993, 260, 1456. (2) (a) Duhr, S.; Braun, D. Phys. Rev. Lett. 2006, 97, 038103. (b) Duhr, S.; Braun, D. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 19678. (3) Pieranski, P. Liquid Crystals, preprint, 2009. (4) de Groot, S.; Mazur, P. Non-equlibrium Thermodynamics; North Holland Publishing: Amsterdam, 1962. (5) Piazza, R.; Parola, A. J. Phys.: Condens. Matter 2008, 15, 153102. (6) Putnam, S. A.; Cahill, D. G. Langmuir 2005, 21, 5317–5323. (7) Ruckenstein, E. J. Colloid Interface Sci. 1981, 83, 77–81. (8) Morozov, K. I. J. Exp. Theor. Phys. 1999, 88, 944–946. (9) Parola, A.; Piazza, R. Eur. Phys. J. E 2004, 15, 255–263. :: (10) Wurger, A. Phys. Rev. Lett. 2007, 98, 138301. :: (11) Fayolle, S.; Bickel, T.; Wurger, A. Phys. Rev. E 2008, 77, 041404. (12) Rasuli, S. N.; Golestanian, R. Phys. Rev. Lett. 2008, 101, 108301. :: (13) Wurger, A. Phys. Rev. Lett. 2008, 101, 108302. :: (14) Morthomas, J.; Wurger, A. Eur. Phys. J. 2008, 27, 425. :: (15) Morthomas, J.; Wurger, A. J. Phys.: Condens. Matter 2009, 21, 035103. (16) Piazza, R.; Guarino, A. Phys. Rev. Lett. 2002, 88, 2083021.
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The variation of DT with temperature, however, is poorly understood. Aqueous solutions of charged macromolecules, micelles, and nanoparticles show a strikingly similar behavior as a function of temperature: An inverse Soret effect (DT < 0) occurs below about 10 C; upon increasing the temperature, the coefficient DT changes sign and seems to saturate well above 50 C.2b,17-21 As noticed by several authors, the mobility DT is strongly correlated with the thermal expansivity β of water;18,22 yet, so far there is no satisfactory explanation for this observation. Regarding high polymers in organic solvents, Brochard and de Gennes pointed out that hydrodynamic interactions are of little significance, and as a consequence, DT is independent of the molecular weight.23 Despite the experimental data,24-32 the underlying van der Waals forces between solute and solvent have been given little attention, and so far, there is no consistent theoretical description for thermal transport driven by dispersion forces. In a previous article, we briefly discussesd thermal diffusion of short polymers, and in particular the role of the molecular weight.33 In the present work, we address the temperature dependence of the Soret effect of dilute suspensions. In section II, we derive the thermophoretic mobility arising from dispersion forces, and recall in section III the electric double layer contribution. Section IV gives a detailed comparison with data for polystryene in ethylbenzene and charged nanoparticles in an electrolyte solution. In section V, we discuss various aspects of the thermophoretic mobility, and summarize our results in section VI. (17) Iacopini, S.; Piazza, R. Europhys. Lett. 2003, 63, 247–253. (18) Iacopini, S.; Rusconi, R.; Piazza, R. Eur. Phys. J. E 2006, 19, 59–67. (19) Putnam, S. A.; Cahill, D. G.; Wong, G. C. L. Langmuir 2007, 23, 9221– 9228. (20) Braibanti, M.; Vigolo, D.; Piazza, R. Phys. Rev. Lett. 2008, 100, 108303. (21) Ning, H.; Dhont, J. K. G.; Wiegand, S. Langmuir 2008, 24, 2426–2432. (22) Brenner, H. Phys. Rev. E 2006, 74, 036306. (23) Brochard, F.; de Gennes, P.-G. C. R. Acad. Sci. Paris Serie II 1981, 293, 1025. (24) Giddings, J.; Caldwell, K.; Myers, M. N. Macromolecules 1976, 9, 106. (25) Schimpf, M.; Giddings, J. Macromolecules 1987, 20, 1561. :: (26) Rossmanith, P.; Kohler, W. Macromolecules 1996, 29, 3203. (27) Rauch, J.; W., K. Phys. Rev. Lett. 2002, 88, 185901. (28) Chan, J.; Popov, J.; Kolisnek-Kehl, S.; Leaist, D. J. Solution Chem. 2003, 32, 197. (29) Wiegand, S. J. Phys.: Condens. Matter 2004, 16, 357. (30) Kita, R.; Wiegand, S.; Luettmer-Strathmann, J. J. Chem. Phys. 2004, 121, 3874. :: (31) Hartung, M.; Rauch, J.; Kohler, W. J. Chem. Phys. 2006, 125, 214904. :: (32) Stadelmaier, D.; Kohler, W. Macromolecules 2008, 41, 6205. :: (33) Wurger, A. Phys. Rev. Lett. 2009, 102, 078302
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II. Dispersion Forces If there has been significant progress in the comprehension of charged colloids, little is known about the Soret motion driven by dispersion forces. Here, we evaluate the thermophoretic mobility arising from the van der Waals interaction of a large particle with the solvent. Following ref 34, we use, as an essential simplification, the boundary approximation, which is valid if the curvature radius of the surface is significantly larger than the interaction length. Moreover, we suppose fluctuations to be negligible35 and the solute particles to be significantly larger than the solvent molecules.33 The fluid velocity field v close to a solid surface is determined by the solution of the stationary Stokes equation ηr2 v ¼ rP - f
ð2Þ
with the hydrostatic pressure P and the force density f exerted by the solute particle on the surrounding fluid. The latter is given by f ¼ -crj
ð3Þ
where j is the interaction potential of the surface with a solvent molecule, and cðrÞ ¼
X
Æδðr - ri Þæ
ð4Þ
Figure 1. Schematic view of the spherical particle in a thermal gradient (a) and the boundary layer approximation (b). The fluid velocity relative to the particle surface increases and attains the value vB and distances beyond the interaction range.
Noting that vx is independent of x, the parallel component of Stokes’ equation becomes η
d2 vx dP dc ¼ ¼ - j dx dx dz2
ð5Þ
This relation is integrated with Stokes boundary conditions, that is, with zero velocity at the solid surface and zero shear stress beyond the boundary layer, vx|z=0 = 0 = ∂zvx|z=¥. Thus, one finds for the boundary velocity vB ¼
1 dc η dx
Z
¥
dzzj
ð6Þ
0
i
the number density, with the statistical average Æ...æ. Since dispersion forces are of short range, we use the boundary layer approximation, which consists of separating the length scales given by the particle size R and the interaction range a0, :: which is on the order of a few angstroms. Then, the van der Waals interaction of a solvent molecule with the flat surface reads jðzÞ ¼ -
Vs H 6πz3
where z is the distance from the surface, Vs the solvent molecular volume, and H an effective Hamaker constant.36 In boundary layer approximation, the normal and parallel components of Stokes’ equation decouple from each other, in terms of local coordinates x and z parallel and perpendicular to the particle surface. Close to the surface, the fluid velocity has a tangential component vx only, whereas the normal velocity is zero, vz = 0; with 32vz = 0 the normal componenent of Stokes’ equation reads dP dj ¼ fz ¼ -c dz dz Noting that c does not vary as a function of z, this is readily integrated P ¼ P0 -cj with the constant hydrostatic pressure P0 of the bulk solvent. Now, we consider the tangential component of Stokes’ equation. Since the corresponding component of the van der Waals force is zero, ∂xj = 0, the force density vanishes, fx = 0.
Its overall form is similar to the expression obtained in eq 9 of ref 34 for a large particle interacting with a dilute molecular solute; note that the latter depends exponentially on the interaction potential j, whereas eq 6 is linear. In the present case, the bulk solvent density c is not influenced by the particle but depends only on temperature according to cðrÞ ¼ cð1 -βr3rTÞ Here, c = Vs-1 is a reference value and β ¼ -
the thermal expansivity. Inserting the atomic distance a0 as the lower bound of the integral and projecting the temperature gradient parallel to the surface, Tx = ex 3 3T, we find vB ¼
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βH Tx 6πa0 η
ð7Þ
This boundary velocity occurs, e.g., in a micropore with a temperature gradient parallel to the pore axis. For a spherical particle, the parallel component varies along the surface with the polar angle according to Tx = Tx sin θ. The resulting particle velocity vB = vB sin θ has the opposite sign and is given by the orientational average over the surface34 u ¼ -ÆvB ex æ With Æ(rT 3 ex)exæ = (2/3)3T, one has ðdfÞ
DT (34) Anderson, J. L. Annu. Rev. Fluid Mech. 1989, 21, 61–99. (35) Astumian, R. D. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 3. (36) Russel, W.; Saville, D.; Schowalter, W. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989.
1 dc c dT
¼
βH 9πa0 η
ð8Þ
The boundary layer approximation ceases to be valid for very small solutes. In this case, one has to use the atom-atom van der DOI: 10.1021/la9001913
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Waals potential j(r) = -C/r6 and to calculate the fluid flow in :: the Huckel limit of electrophoresis;37 then, the transport velocity ) is given by -vB, and the resulting coefficient D(df T = βH/(6πa0η) 33 is by a factor of 3/2 larger than eq 8. Since the distance parameters a0 cannot be calculated precisely but are an effective quantitity that accounts for the effect of the molecular structure in the continuous potential j, we discard corrections to the boundary layer approximation and use eq 8 throughout this paper. For the sake of simplicity, the thermal conductivities of solute ξP and solvent ξS are supposed to be the same. Indeed, for insulating materials and polymers, the well-known factor 3ξS/ (2ξS + ξP) differs little from unity. As a noteworthy exception, the large heat conductivity of metal particles significantly reduces their transport velocity.38
III. Electric Double-Layer Forces The electric double layer is treated in Poisson-Boltzmann mean field theory, relying on boundary layer hydrodynamics and small Peclet number of mobile ions.34 Proceeding in analogy to the motion driven by an electric field, one finds the thermophoretic mobility13 ðelÞ
DT
¼
kB ^ Cv 12πηl B
ð9Þ
where the dimensionless quantity 2 ζ^ C^ ¼ ^ζ þ 8ðR þ τ -3Þlncosh -3^ζδR 4
ð10Þ
is a function of the reduced zeta potential ζˆ = eζ/kBT. The surface electrostatic potential37 ζ ¼
2kB T arsinhð2πσl B λÞ e
depends on the number of elementary charges per unit surface σ, the Debye length λ = (8πn0 l B)-1/2 with the salinity n0, and the Bjerrum length l B = e2/4πεkBT. For weakly charged systems (2πσl Bλ , 1), one has ζ = eσλ/ε, whereas in the opposite limit, the surface potential ζ ∼ (4kBT/e)ln(2πσl Bλ) varies only logarithmically with the system parameters. The above result is generally valid, within the framework of continuous fluid mechanics. At very high salinity and surface charges, hydration and specific ion effects become important; for Debye lengths well below 1 nm, macroscopic hydrodynamics breaks down in the thin boundary layer. However, most colloidal systems are in the range 2πσl Bλ ∼ 1, where eq 10 is valid. The coefficients R and δR describe the electrolyte Soret coefficient Sel = R/T and thermoelectric field E¥ = δRkBTx/e. For a NaCl solution, one has R ¼
Q/Na
Q/Cl
þ 2kB T
, δR ¼
Q/Na -
Q/Cl
2kB T
ð11Þ
where the heat of transport Qi* expresses the tendency of a given ion to diffuse to the cold (Qi* > 0) or to the warm (Qi* < 0). Absolute values of the ionic coefficients are difficult to measure; literature values for NaCl at 25 C scatter significantly, the heats (37) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry; Dekker: New York, 1997. (38) Giddings, J. C.; Shinudu, P. M.; Semenov, S. N. J. Colloid Interface Sci. 1995, 176, 454.
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Figure 2. Comparison of the three contributions to the dimensionless factor C^ as given in eq 10. If the parameters τ, R, and δR are on the order of unity, the middle term 8 ln cosh(ζˆ /4) is of little significance.
of transfer reported in ref 39, QNa* = 3.5 kJ/M and QCl* = 0.5 kJ/M, result in R = 0.8 and δR = 0.6, whereas ref 40 gives a larger value for QCl*. The term proportional to τ = -(T/ε) dε/dT accounts for the variation of the dielectric constant ε with temperature; at 25 C, it takes the value τ = 1.4 and increases slightly with T. The remaining terms are due to the explicit temperature dependence of the electrostatic potential and charge density. An inverse Soret effect (DT < 0) occurs if the product δRζ takes a sufficiently positive value. In physical terms, R < 0 means a higher salinity in warmer regions of the solution; a negative value of δR results in an excess positive charge at higher T, which in turn attracts negatively charged colloidal particles (DT < 0) and repels those with positive valency (DT > 0). Figure 2 shows that, for τ and R on the order of unity, the second term in eq 10 is of little significance; it will be discarded in the following. Inserting the surface potential ζ and defining the thermoelectric potential ψ¥ ¼ -δR
kB T e
ð12Þ
the remainder reads ðelÞ
DT ¼
εζ ζ þ ψ¥ ηT 3
ð13Þ
This expression shows how the surface and thermoelectric potentials ζ and ψ¥ determine the sign of DT; for a negatively charged particle ζ < 0, an inverse Soret effect occurs if ψ¥ + (1/3)ζ > 0, that is for sufficiently negative δR. Note the prefactor is given by the Helmholtz-Smoluchowski electrophoretic mobility εζ/η. The two terms in eq 13 arise from different physical mechanisms. The first one, proportional to ζ2, results from the flow of the electric energy density εE2 in a temperature gradient, whereas the second one is driven by the thermoelectric field. (39) Agar, J. N.; Mou, C. Y.; Lin, J. J. Phys. Chem. 1989, 93, 2079–2082. (40) Sokolov, V. N.; Safonova, L. P.; Pribochenko, A. A. J. Solution Chem. 2006, 35, 1621.
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Figure 3. Thermal diffusion coefficient DT of polystryrene in ethylbenzene, as a function of temperature. The data are taken from ref 24 for molecular weights 19 800 (o), 51 000 (0), 97 200 (4), 160 000 (3). The curve is given by eq 8 with a temperature dependent viscosity as discussed in the main text.
the slope of η-1 is close to that of the experimental data. At higher temperature up to 75 C, the measured DT hardly changes and thus does not follow the increase of the inverse viscosity. This discrepancy at higher T is probably related to the properties of the solute-solvent interface, which in eq 8 are subsumed in the minimum distance a0. The temperature dependence of DT is given by that of the inverse viscosity η-1 ∼ e-B/T. In a physical picture, this factor accounts for thermally activated diffusive jumps of adjacent molecules, which in turn determine the shear viscosity of the bulk solvent. The short-ranged thermal force in eq 5, however, is equlibrated by friction of the first solvent layer around the solute molecule; it is by no means clear that the frictional force between solute and solvent follows the same Arrhenius law as η-1. This illustrates the limits of the macroscopic hydrodynamics approach. B. Charged Colloids. Charged colloids in aqueous solutions are subject to both electric and dispersion forces. We start by comparing absolute values of the corresponding contributions (df ) to the thermophoretic mobility, D(el) T and DT . With the para-3 meters η = 10 Pa s, lB = 0.7 nm, and the dimensionless prefactor C^ ∼ 4, the electric part takes the value ðelÞ
IV. Comparison with Experiment Here, we compare eqs 8 and 13 with experimental findings on polystyrene in ethylbenzene and charged nanoparticles in water. For both systems, we discuss the theoretical expressions in view of the measured data and, in particular, their temperature dependence. A. Polymers in Organic Solvents. We start by comparing the thermophoretic mobility DT for various polymer/solvent systems.24-27 The mobility measured in different solvents covers the range DT = 3-16 μm2/Ks for polystyrene (PS); 13-16 μm2/ Ks for poly(methyl methacrylate) (PMMA); 4-7 μm2/Ks for polyisoprene (PI).29 The analysis of these data in ref 31 shows that the dependence on the solvent arises mainly from the viscosity; indeed, ηDT is constant for a given polymer. On the other hand, one notes that polymers of large and heavy building blocks (such as PMMA) have a larger DT than those made of small repeat units (like PI); so far, there is no satisfactory explanation for this observation. Here, we discuss the temperature dependence of the coefficient DT and compare eq 8 with data on polystyrene in ethylbenzene.24 Figure 3 shows measured values between 0 and 75 C at different molecular weights. The solid curve is given by eq 8 with the parameters H/a0 = 2 10-10 J/m and β = 10-3 K-1. The Hamaker constant H and the thermal expansivity β depend little on temperature. By contrast, the viscosity changes significantly with temperature; we use the phenomenological law η ¼ AT eB=T
ð14Þ
where T is the absolute temperature; the parameters A = 1.9 10-8 Pa 3 s/K and B = 1400 K are determined from the values reported for η in 10-3 Pa 3 s at different temperatures: 0.78 (10 C); 0.68 (20 C); 0.43 (60 C). With parameters H ∼ 6 10-20 J and a0 ∼ 3 10-10 m that are consistent with known properties of polystyrene solutions, the theoretical expression 8 reproduces the order of magnitude of measured values DT ∼ 10 μm2/Ks. More importantly, the temperature dependence of the viscosity acocunts, at least qualitatively, for the increase with T. Indeed, between 5 and 40 C, Langmuir 2009, 25(12), 6696–6701
DT ∼ 2 10 -12 m2 =Ks
ð15Þ
The term due to dispersion forces depends on the polystyrenewater Hamaker constant H ∼ 1.4 10-20 J, the thermal expansivity of water β = 0.2 10-3 K-1, and the characteristic distance a0 ∼ 0.5 nm ðdfÞ
DT ∼ 0:2 10 -12 m2 =Ks
ð16Þ
Thus, the contribution driven by dispersion forces is by 1 order of magnitude smaller than the electrostatic one. The following discussion of experiments on charged particles thus neglects the former, and is based on eq 9 only. In Figure 4, we compare the temperature dependence of the electrostatic contribution in eq 9 with experimental data for charged polystyrene beads in 4 mM/L NaCl solution.18 The measured thermophoretic mobility shows an almost linear variation with temperature and becomes negative below 5 C. The theoretical lines are calculated from eq 13 with constant values for the number density of elementary charges at the particle surface, σ = -0.05 nm-2, the Debye length λ = 5 nm, and the Bjerrum length lB = 0.7 nm. As shown in Figure 2, the contributions to DT involving τ and R are small in the experimentally relevant range and thus have been neglected. The results of Figure 4 rely essentially on the assumption of a linear temperature dependence of the thermoelectric coefficient δRðTÞ ¼ 0:8 þ
0:025 ðT -298 KÞ K
ð17Þ
This form and the resulting relation between the coefficients DT and δR are supported by various measurements of the ionic Soret coefficients in different electrolytes, as discussed in the following. The Soret effect of binary electrolytes shows intricate dependencies on ion size, salinity, and temperature,41,42 that arise from a superposition of electrostatic interactions, thermal (41) Caldwell, D. R. J. Phys. Chem. 1973, 77, 2004–2008. (42) Gaeta, F. S.; Perna, G.; Scala, G.; Bellucci, F. J. Phys. Chem. 1982, 86, 2967–2974.
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1.3 10-3 Pa 3 s at 10 C to 0.65 10-3 Pa 3 s at 40 C;43 this reduction by a factor of 2 corresponds to a logarithmic derivative d ln η/dT ∼ -0.02 K-1. The straight line in Figure 4 is calculated with constant η = η0, where η0 is the value at 30 C. As an illustration, we plot the second curve with the temperature-dependent viscosity η(T).43 The differences between the two curves are of little significance, since the viscosity provides an overall factor to DT that does not affect the relative magnitude of its contributions. Finally, we note that, without the thermoelectric contribution, DT is strictly positive.7-12 Thus, Figure 4 gives strong evidence that the temperature dependence of DT observed for various colloids2,17-21 arises from the thermoelectric effect of the electrolyte. The mobility DT is zero at the temperature where the expression in parentheses in eq 13 vanishes. Previous work suggested that the change of sign could arise from a negative contribution due to dispersion forces;21 this possibility is ruled out by the sign and the numerical value of eq 16. Figure 4. Temperature dependence of the thermophoretic mobility of polystryene beads. The full symbols are data for 30 nm polystyrene beads, measured by Iacopini et al.18 The curves represents eq 9 with the Debye length λ = 5 nm and the number density of elementary surface charges σ = -0.05 nm-2; the latter value is slighlty larger than -0.04 nm-2 given in ref 18. The temperature dependence of the straight line arises from the thermoelectric coefficient δR; the second curve accounts moreover for the variation of the viscosity, as explained in the main text.
expansion, and hydration effects. Measurements on 0.5 M/L NaCl solution give a constant slope dR ¼ 0:03 K-1 ð0:5 M=L NaClÞ dT between 0 and 25 C, and smaller values at higher temperatures.41 NaCl and KCl solutions show an irregular feature at about 100 mM/L at 30 C, the position of which depends slightly on temperature.41,42 Outside this domain, however, available experiments indicate a monotonic increase of R with temperature. As an indication of the experimental uncertainty, we compare measured values for HCl, R = 2.8,39 and R = 3.40 With the ionic heats of transport of ref 39, QNa* = 3.5 kJ/mol and QCl* = 0.5 kJ/mol at room temperature, one finds R = 0.8 and δR = 0.6 for NaCl solution, which is close to δR = 0.8 used in eq 17. The assumption on the temperature derivative of δR can be restated in terms of the ionic transport quantities introduced in eq 11: The slopes of R(T) and δR(T) are similar, if QNa* and QCl* show a similar temperature dependence; the latter condition seems to be plausible since the Qi* are to a large extent determined by the solvation energies of positive and negative ions in water. Summarizing the discussion of eq 17, we note that the constant δR = 0.8 at room temperature is close to the measured value;39 the derivative dδR/dT = 0.025/K similar to that measured for the Soret coefficient, dR/dT = 0.03/K, is supported by their relation to the heat of transport. The thermoelectric coefficient of eq 17 leads to a linear variation of DT with temperature and provides a good fit to the experimental data. The change of sign occurs where the two terms in eq 13 cancel each other, (1/3)ζ + ψ¥. Note that the linear law for R(T) holds only approximately, especially at higher T, and a similar behavior is expected for δR(T); thus, it is not surprising that the measured DT of various systems increases only weakly above 30 C.18 Because of its significant variation with T, the viscosity of water η(t) needs to be discussed in some detail. It decreases from 6700 DOI: 10.1021/la9001913
V. Discussion A. Size Dependence. Contrary to the Einstein coefficient D = kBT/6πηR that varies with the inverse radius R, the thermophoretic mobility DT is independent of the particle size. From the theory of electrophoresis,37 there are two approximation schemes for calculating DT. The boundary layer approximation, which corresponds to the Helmholtz-Smoluchowski mobility, applies to particles that are larger than the Debye :: length λ;7,34 the Huckel limit applies to the opposite case of small solutes. When expressing the thermophoretic mobility in terms of the surface potential of charged particles, one finds that both schemes differ merely by factors on the order of unity.12,14 The above statement holds true for thermophoresis in nonpolar systems. Because of the power law behavior of the van der Waals interaction, eq 8 is determined by the immediate vicinity of the solute; in other words, the effective interaction range of dispersion forces is comparable to the size of the solvent molecules. Indeed, the parameter a0 appearing in the denominator of eq 8 describes an effective molecular spacing, which is required quite generally when integrating a van der Waals potential in a continuous phase.36 This length a0 may depend on temperature and molecular details; yet, it is not related to the size of the solute particle. Moreover, general considerations on the hydrodynamic flow show that, for high polymers, DT is independent of the molecular weight.23 ) B. Surface Roughness. The expression for D(df T depends on the cutoff parameter a0 that describes the minimum distance of the molecular units of the solvent and solute particles. Such parameters are necessary quite generally when neglecting the atomic structure and reducing the atom-atom potential C/r6 to an interaction potential of homogeneous phases.36 In a static problem, a0 is the mimimum atomic distance and takes values of :: a few angstroms. For shear flow, however, a0 is the distance of the effective zero-velocity plane. The charges of colloidal particles are, in many cases, carried by molecular end groups such as sulfate or carboxyl, or ionic surfactants grafted at the surface. A schematic view is given in Figure 5, which provides a simple picture for the fluid flow along the rough surface; the plane of the boundary condition vx = 0 is shifted from the particle surface into the fluid by the typical size of the molecular groups attached at the surface. This effect could be described by a negative slip length.15 (43) Lide, D. R. CRC Handbook of Chemistry and Physics; CRC Press:Boca Raton, 2005.
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of polymer thermophoresis is determined by the solvent viscosity. By contrast, thermal expansion of water varies strongly and even changes sign at 4 C; yet, because of the small absolute value of β, it contributes little to colloidal transport. As to (ii) and charged colloids, the present work suggests a much more subtle effect, based on the thermoelectric response δR(T) of the electrolyte solution, that leads to the surprising temperature dependence of D(el) T observed for proteins, micelles, DNA, and nanoparticles.18 Caldwell pointed out that the Soret coefficient of a 0.5 M/L NaCl solution is proportional to the expansivity of water; Figure 5 of ref 41 shows an almost perfect linear relation between R and β, with the slope
Figure 5. Schematic view of the average velocity profile close to a surface colloidal particle. The solvent between the grafted molecules does not flow; thus, there is an effective minimum distance a0 for van der Waals forces, which is significantly larger than the usual value of a few angstroms.
The effective spacing hardly modifies the above treatment of the electric double layer. The charges are situated on the molecular end groups and thus close to the plane vx = 0. Moroever, for a weak or moderate electrolyte solution, the Debye length is significantly larger than a nanometer; thus, the detail of the surface roughness and charge distribution do not affect the properties of the electric double layer. C. Electric vs Dispersion Forces. The thermophoretic mobility of uncharged systems in organic solvents is on the order of DT ∼ 10-11 m2/Ks and thus exceeds typical values for charged colloids in aqueous solutions; see, e.g., Figures 3 and 4. On the other hand, according to eqs 15 and 16, the term driven by dispersion forces is rather insignificant for charged systems and, in particular, much weaker than in organic solvents. Accordingly, the numercial value given in eq 16 for charged colloids is 1 or 2 orders of magnitude smaller than those reported for polymers in organic solvents. Several effects contribute to this large factor. First, the thermal expansivity β of water is about five times smaller than that of organic solvents such as toluene. Second, the Hamaker constant of polystyrene in water, H ∼ 1.4 J/M, is several times smaller than that in toluene. Third, as discussed above, the surface groups grafted on polystyrene particles result in an effective parameter a0, which may significantly exceed the actual molecular spacing. D. Role of Solvent Thermal Expansion. As noted by several authors,18,22 there is a close correlation of the temperature dependence of DT of charged colloids and the thermal expansion coefficient β of water. The striking similarity of the data on various charged colloidal suspensions has been discussed in ref 18 and related to a solvent mass flow; ref 22 pointed out that DT is correlated with the product of solvent diffusion and expansion coefficients. The present work derives the relation between the thermophoretic mobility and the thermal expansivity by treating the solute-solvent interactions in the framework of low Reynolds number hydrodynamics. We find that DT depends in two different ways on the thermal expansivity, (i) directly through the contribution 8 accounting for dispersion forces, and (ii) in a hidden manner through the thermal response of the electrolyte, that is, the Soret and thermoelectric coefficients R and δR. Regarding (i), eq 8 establishes, for uncharged systems, a linear relation between DT and β. Since the expansion coefficient of organic solvents varies little with T, the temperature dependence
Langmuir 2009, 25(12), 6696–6701
dR ¼ 9:1 ð0:5 M=L NaClÞ dβ With the assumption that δR and R show a similar temperature dependence, one finds that the thermoelectric coefficient δR is a linear function of the expansivity; thus, eq 17 is a direct consequence of the nonuniform density variation of water. Moreover, the large factor dR/dβ enhances the electric contribution by about 1 order of magnitude, which explains why dispersion forces are rather irrelevant in aqueous solution. Finally, we note that in deriving eq 7 we have neglected the thermal expansion of the solute surface. This is well-justified for solid particles, with their expansivity being by several orders of magnitude smaller than that of liquids. This is not the case for the bicontinuous lyotropic crystals studied in ref 3, where surfactant bilayers play the role of the solid. Since both the density and the orientational order of such bilayers depend strongly on temperature, they could significantly affect the thermokinetic phenomena in complex systems.
VI. Summary and Conclusion We briefly summarize the main results of this work. (i) Equation 8 confirms that thermophoresis driven by dispersion forces does not depend explicitly on the solute volume, but varies linearly with the solvent thermal expansivity. (ii) The fit of Figure 3 with data on polystyrene in ethylbenzene24 provides a qualitative agreeemnt with the theoretical expression. It provides strong evidence that the temperature dependence of DT arises from the viscosity η(T ); deviations at higher T indicate the limits of the macroscopic hydrodynamics approach. (iii) Regarding charged colloids in an electrolyte solution, the temperature dependence of DT is determined by the thermoelectric response of the electrolyte. The correlation with the thermal expansivity of water β is traced back to the fact that the ionic Soret effect in salt solutions is proportional to β. With plausible values for the thermoelectric coefficient δR, eq 13 provides a good fit to data on charged polystyrene beads in 4 mM/L NaCl solution,18 as shown in Figure 4. The variation of the viscosity η(T) turns out to be of minor relevence. (iv) A detailed analysis reveals that dispersion forces lead to a large thermophoretic mobility in organic solvents. By contrast, they are rather insignificant in aqueous solution, mainly because of the smaller thermal expansivity and Hamaker constant; as a consequence, electric double layer forces prevail. Acknowledgment. I thank Yves Leroyer for helpful discus:: sions and Dominik Stadelmaier and Werner Kohler for valuable exchange on the thermal properties of ethylbenzene.
DOI: 10.1021/la9001913
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