Thermal-Diffusive Behavior of a Dilute Solution of Charged Colloids

The Soret coefficient of the charged colloids is measured as a function of the ... In the stationary state, the colloidal mass fluxes induced by the t...
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Langmuir 2008, 24, 2426-2432

Thermal-Diffusive Behavior of a Dilute Solution of Charged Colloids Hui Ning,† Jan K. G. Dhont,‡ and Simone Wiegand* Forschungszentrum Ju¨lich GmbH, IFF-Weiche Materie, D-52428 Ju¨lich, Germany ReceiVed NoVember 12, 2007. In Final Form: December 7, 2007 Thermal diffusion of a dilute solution of charged silica colloidal particles (Ludox) is studied by a holographic grating technique. The Soret coefficient of the charged colloids is measured as a function of the Debye screening length and the surface charge density of the colloids. The latter is varied by means of variation of the pH. The experimental Soret coefficients are compared with several theoretical predictions. The surface charge density is independently obtained from electrophoresis measurements, the size of the colloidal particles is obtained from electron microscopy, and the Debye length is calculated from ion concentrations. The only adjustable parameter in the comparison with theory is therefore the intercept at zero Debye length, which measures the contribution to the Soret coefficient of the solvation layer and possibly the colloid core material.

I. Introduction Thermal diffusion, or thermophoresis, of colloids in a liquid medium describes the movement of colloids due to the inhomogeneity of the temperature distribution. In the stationary state, the colloidal mass fluxes induced by the temperature gradient and the concentration gradient cancel. The ratio of the concentration gradient ∇c and temperature gradient ∇T that comply with such a stationary state is characterized by the Soret coefficient ST, which is defined as ST ) DT/D, where DT is the thermal diffusion coefficient and D is the mass diffusion coefficient. Here, the thermal diffusion coefficient is defined such that the contribution of the thermal gradient to the equation of motion for the colloid concentration c is taken equal to cDT∇2T. The sign of Soret coefficient determines the temperature gradient induced migration direction of the colloidal particles. For positive values of the Soret coefficient, colloids move to the cold side, while for negative Soret coefficients colloids will enrich at the warm side. Thermal diffusion for macromolecules is of interest because of its potential to be applied for polymer characterization,1 and the investigation of this effect might contribute to the understanding of some fundamental aspects related to life science.2,3 The first observation of this effect4 dates back one and a half century. Up to this date, however, there exists no satisfactory theory that explains the thermal-diffusive behavior of simple liquids, while for colloids a few theories have been proposed recently. The development of new experimental techniques during the past decade made it possible to measure the Soret coefficients of macromolecular systems. In particular, experimental data became available for charged colloids,5-8 and a few theories9-11 for the contribution of the electrical double * To whom correspondence should be addressed. E-mail: s.wiegand@ fz-juelich.de. Website: http://www.fz-juelich.de/iff/personen/S.Wiegand/. † Website: http://www.fz-juelich.de/iff/personen/H.Ning/. ‡ Website: http://www.fz-juelich.de/iff/personen/Dhont_J_K_G/. (1) Giddings, J. C.; Caldwell, K. D.; Myers, M. N. Macromolecules 1976, 9, 106. (2) Braun, D.; Libchaber, A. Phys. ReV. Lett. 2002, 89, 188103. (3) Piazza, R.; Iacopini, S.; Triulzi, B. Phys. Chem. Chem. Phys. 2004, 6, 1616. (4) Ludwig, C. Sitzungsber. Akad. Wiss. Wien, Math.-Naturwwiss. Kl. 1856, 20, 539. (5) Duhr, S.; Braun, D. Phys. ReV. Lett. 2006, 96, 168301. (6) Piazza, R.; Guarino, A. Phys. ReV. Lett. 2002, 88, 208302. (7) Putnam, S. A.; Cahill, D. G. Langmuir 2005, 21, 5317. (8) Ning, H.; Buitenhuis, J.; Dhont, J. K. G.; Wiegand, S. J. Chem. Phys. 2006, 125, 204911. (9) Ruckenstein, E. J. Colloid Interface Sci. 1981, 83, 77.

layer to the thermal diffusion coefficient of charged colloids have been proposed. Colloids can be charge stabilized in polar solvents. The thermophoresis of charged colloids was extensively investigated for several systems, such as silica particles, DNA, proteins, ionic surfactants, and carboxyl modified polystyrene spheres, with different experimental methods.5-7 It was found that the thermaldiffusive behavior of charged colloids depends on several parameters such as the salinity,7,12,13 temperature,14 mutant variants of the protein that is studied,15 particular type of electrolyte that is used,7 surface charge density,13 colloidal concentration,12 and size of the colloids.5 In some cases, modification of those parameters changes the behavior from thermophobic to thermophilic. A distinction should be made between thermophoresis in very dilute systems and that in concentrated systems. In highly diluted dispersions, the interactions between the colloid particles and surrounding solvent, and also the formation of the double layer, dominate the physical properties of the colloids, while in concentrated dispersions the behavior is complicated by colloid-colloid interparticle electrostatic repulsion.16-18 The Soret coefficient at high dilution, where colloid-colloid interactions are negligible, is referred to as the single-particle Soret coefficient. Recently, different theoretical approaches for the single-particle Soret coefficient of charged colloids have been developed. Ruckenstein9 suggests that the origin of the thermophoretic motion of a colloid is caused by a “microscopic Marangoni effect”. Bringuier and Bourdon10 proposed an expression for ST in terms of the derivative of total internal energy with respect to the temperature. Wu¨rger et al.19 worked out a theory that starts from the Navier-Stokes equation with additional forces that arise from the presence of a temperature gradient. In the theories by Dhont11 and Braun,13 the Soret coefficient is derived by (10) Bringuier, E.; Bourdon, A. Phys. ReV. E 2003, 67, 011404. (11) Dhont, J. K. G.; Wiegand, S.; Duhr, S.; Braun, D. Langmuir 2007, 23, 1674. (12) Piazza, R. Philos. Mag. 2003, 83, 2067. (13) Duhr, S.; Braun, D. P. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 19678. (14) Iacopini, S.; Rusconi, R.; Piazza, R. Eur. Phys. J. E 2006, 19, 59. (15) Putnam, S. A.; Cahill, D. G.; Wong, C. L. Langmuir 2007, 23, 9221. (16) Dhont, J. K. G. J. Chem. Phys. 2004, 120, 1642. (17) Wiegand, S. J. Phys.: Condens. Matter 2004, 16, R357. (18) Ning, H.; Kita, R.; Kriegs, H.; Luettmer-Strathmann, J.; Wiegand, S. J. Phys. Chem. B 2006, 110, 10746. (19) Fayolle, S.; Bickel, T.; Le Boiteux, S.; Wu¨rger, A. Phys. ReV. Lett. 2005, 95, 208301.

10.1021/la703517u CCC: $40.75 © 2008 American Chemical Society Published on Web 02/07/2008

Thermal-DiffusiVe BehaVior of Ludox Particles

considering the electrostatic energy that is necessary to build up the double layer. These theoretical approaches lead to explicit expressions for the Soret coefficient, which makes a comparison between theory and experiment feasible. To test the validity of the available theories, we measured the thermal-diffusive behavior of charged silica colloids (Ludox TMA) at high dilutions using a holographic grating technique. At high salt content, the charged colloids as well as the salt contribute to the holographic grating signal, which indicates a thermophoretic motion of both salt ions and colloids. The separation in time scales for salt diffusion and colloid diffusion, however, allows us to measure the pure colloid Soret coefficient. Furthermore, the Debye length, the surface charge density, and the radius of the colloids are measured independently. This allows for an unambiguous test of the available theories for the doublelayer contribution to the single-particle Soret coefficient. This paper is organized as follows. In section II, we describe the experimental details. Section III is a summary of the expressions for the Soret coefficient as derived by Ruckenstein, Wu¨rger, Dhont, and Braun. We will present the characterization and our thermal diffusion forced Rayleigh scattering (TDFRS) results on Ludox colloidal particles in section IV and discuss the comparison with theoretical expressions.

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Figure 1. Dependence of the refractive index of the NaCl/water solution on the weight fraction of NaCl (a) and refractive index of Ludox particle dispersions as a function of the weight fraction of Ludox particles (b) at 25 °C. The solid lines are linear fits to the data.

II. Experimental Section A. Sample Preparation. The Ludox TMA dispersion was donated by Grace Davison. The concentration of the original stock dispersion was 33.8% by weight, and the pH was around 7.0 at 25 °C. A small amount of Na2SO4 (0.036%) was present in the stock solution. The solutions were obtained by diluting the stock dispersion with deionized water (Milli-Q). Sodium chloride (Merck KGaA, >99.5%) was added into the dispersion to adjust the salinity. To adjust the pH of the dispersion, and thereby the surface charge density of the Ludox, we added a certain amount of NaOH (Merck KGaA, >99%). The actual pH values were measured with an ORION pH meter. The samples for the TDFRS measurements were prepared by filtering the solution into the sample cell using a Nylon filter (Carl Roth GmbH) with a mesh size of 0.45 µm. B. Sample Characterization. 1. Thermal Diffusion Forced Rayleigh Scattering (TDFRS). The experimental setup of TDFRS has been described in detail elsewhere.17,18 In brief, an interference grating is written by a solid-state laser operating at a wavelength of λ ) 488 nm. The grating is read by the diffraction efficiency of a He-Ne laser operating at λ ) 632.8 nm. The TDFRS measurements are carried out at a temperature of T ) 25.0 °C. The temperature of the sample cell is thermostatically controlled by a circulating water bath with an uncertainty of 0.01 °C. It should be noted that a small amount (roughly 10-6 by weight) of inert dye, basantol yellow, is added into the solution to convert the optical grating into a temperature grating. The absorption band of the dye shows no shift with the addition of the Ludox particles, which indicates that the dye does not adsorb at the surface of the colloid.20 The contribution from the dye to the ionic strength is less than 1% for the samples with the lowest salt concentration. For the ternary solutions under consideration, the normalized heterodyne TDFRS signal ζhet can be analyzed by a doubleexponential expression ζhet(t) ) 1 + -1 ∂n ∂n ∂T c1,c2 ∂c1

() ( ) () ( ) ∂n ∂T

-1

c1,c2

c1(1 - c1) STf (1 - exp(-q2Df t)) +

c2,T

∂n ∂c2

c2 (1 - c2) STs (1 - exp(-q2Ds t)) (1) c1,T

(20) de Gans, B. J.; Kita, R.; Mu¨ller, B.; Wiegand, S. J. Chem. Phys. 2003, 118, 8073.

Figure 2. (∂n/∂T)c1,c1 as a function of NaCl content for different Ludox concentrations: (4) NaCl water solutions with φ ) 0%, (9) Ludox dispersions with φ ) 0.1%, and (O) Ludox dispersions with φ ) 0.5%. The solid line is a linear fit to all data. where the subscripts f and s stand for fast and slow mode, respectively. The decay times and the amplitudes of the two modes depend on two diffusion coefficients, Df and Ds, and two amplitude coefficients, STf and STs, respectively. Furthermore c1 and c2 are the concentrations of NaCl and Ludox particles, respectively. To calculate the Soret coefficient, the refractive index increments (∂n/∂ci)T (i ) 1,2) and (∂n/∂T)c1,c2 of the colloidal dispersions and NaCl solutions are required. These increments are measured separately by using an Abbe refractometer and a Michelson interferometer at a wavelength of 632.8 nm.21 As can be seen from Figure 1, the refractive index of NaCl aqueous solutions and Ludox dispersions linearly increases with concentration. The refractive index increment (∂n/∂ci)T was obtained from the slope as 0.176 ( 0.001 for the NaCl/water solution and 0.072 ( 0.002 for the Ludox/water solution. The values for (∂n/∂T)c1,c2 of the Ludox particles in NaCl solution are plotted in Figure 2 as a function of the NaCl concentration. As can be seen, (∂n/∂T)c1,c2 is independent of the Ludox concentration and decreases with increasing NaCl content as (∂n/∂T)c1,c2 ) (-1.044.03cNaCl) × 10-4 K-1, where cNaCl is the weight fraction of NaCl. At high salt content, the (∂n/∂T) values of Ludox dispersions show pronounced deviation from the fitted line. The error bar in Figure (21) Ko¨hler, W.; Scha¨fer, R. In New DeVelopments in Polymer Analytics II; Schmidt, M., Ed.; Advances in Polymer Science Series 151; Springer: Berlin, 2000; pp 1-59.

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2 for the (∂n/∂T)c1,c2 data is the standard deviation as obtained from at least three measurements. 2. Electrophoresis. Electrophoresis measurements were performed using a commercial Malvern 2000 Zetasizer. The electrical field was 29 V/cm, and the temperature is 25 °C. The samples for electrophoresis measurements were prepared in the same way as the TDFRS samples. The electrophoretic mobility µE of the charged particles was obtained from the frequency of the oscillatory contribution to the scattered intensity autocorrelation function. For very low salinity, according to Hu¨ckel’s formula, the colloid surface charge Q can be calculated from V Q µE ) ) E 6πRη

(2)

where V is the velocity of the particle, E is the electrical field strength, R is the radius of the colloidal particles, and η is the shear viscosity.22 For moderate salinity, the electrophoretic mobility can be expressed in terms of the zeta potential ζ as µE )

ζ f (κR) η

(3)

with  being the dielectric constant. The function f (κR) is in good approximation equal to23 f (κR) )

(

)

2 1 1+ 3 2(1 + {2.5/κR[1 + 2 exp(-κR)]})3

(4)

where κ is the reciprocal Debye length. The zeta potential ζ can be expressed as a function of the surface charge as ζ)

Q 1 4πR 1 + κR

(5)

for moderate surface potentials, where the Debye-Hu¨ckel approximation is valid.

III. Theoretical Descriptions In this section, we will very briefly discuss the theoretical models of Ruckenstein, Dhont, Wu¨rger, and Braun and present the corresponding explicit expressions for the contribution of the electrical double layer to the single-particle Soret coefficient. A. Ruckenstein’s Model.9 By considering the electrostatic contribution to the interfacial tension in a temperature gradient, Ruckenstein proposed the following expression for the singleparticle Soret coefficient of a charged colloid with a thin double layer:

{ ( ) } 2

ST )

2

DT 1 3π 4πlB σ R ) 1+ + A(T) D T 4 e κ2lB3

(6)

where σ ) Q/4πR2 is the surface charge density, lB ) βe2/4π is the Bjerrum length (0.71 nm for water at room temperature), with  being the dielectric constant of water at ambient temperature, and A(T) is the additive contribution from the solvation layer and the core material of the colloid. B. Dhont’s Model.11 The theory starts from the idea of force balance on a charged colloidal particle, considering the temperature dependence of the reversible work Wdl necessary to build up the double layer. The thermal diffusion coefficient DT of a single charged colloid is found to be equal to (22) Henry, D. C. Proc. R. Soc. London, Ser. A 1931, 133, 106. (23) Ohshima, H. J. Colloid Interface Sci. 1994, 168, 269.

1 ∂Wdl DT ) D 0 kBT ∂T

(7)

where D0 is the Einstein mass-diffusion coefficient. The reversible work to form an electrical double layer, within the DebyeHu¨ckel approximation, is equal to

Wdl )

QΦs 2

(8)

where, as before, Q is the bare charge on the colloidal surface and Φs is the surface potential. For moderate surface potentials, Φs is to a good approximation equal to the zeta potential ζ in eq 5. By substituting eqs 8 and 5 into eq 7, the Soret coefficient of a single charged colloid is found to be given by

{ ( )

2 2 DT 1 1 4πlB σ 1 κR4 × ST ) ) 1+ D T 4 e (1 + κR)2 lB3

dln  2 1 + )] + A(T) (9) [1 - dln T( κR }

where dln /dln T ) -1.34 for water at room temperature. This expression is valid for an arbitrary thickness of the double layer. The contribution stemming from the temperature dependence of the surface charge to the Soret coefficient, which is given in the original work,11 is not considered here. Note the different dependence of the Soret coefficient on the radius of the particle and the Debye length in eqs 9 and 6. C. Wu1rger’s and Braun’s Model. Wu¨rger calculated the Soret coefficient on the basis of earlier work of van Kampen24 in terms of the internal energy of the double layer19 and derived essentially the same result using a hydrodynamic model.25 Later, Braun13 calculated the Soret coefficient for a charged colloid from the analogy of the colloid’s surface and its double layer with an electric capacitor. Both theories are limited to thin double layers. These two different approaches lead to the same expression for the single-particle Soret coefficient,

{ ( )

]} + A(T)

2 2 DT 1 1 4πlB σ R2 dln  ) 1+ 1ST ) 3 D T 4 e dln T κlB

[

(10)

This result agrees with eq 9 in the case of thin double layers where κR . 1.

IV. Results and Discussion In this section, we shall shortly describe the phase behavior of our Ludox suspensions and the characterization of the Ludox particles and discuss the electrophoresis experiments. The experimental thermal-diffusion data are presented and compared with the above-discussed theories. In addition, we also compare these theories with two data sets for other charged colloids taken from refs 13 (polystyrene (PS) spheres) and 6 (sodium dodecyl sulfate (SDS) micelles). A. Sample Characterization. 1. Stability. Ludox particles, which contain mainly SiO2, are dispersed in water. The surface charge density of the Ludox particles is almost independent of the colloidal concentration, and the dispersions can be kept stable in a plastic container for at least several months. By weighing the evaporated amount of water from a given volume of the (24) van Kampen, N. G. Stochastic Processes in Physics and Chemistry; Elsevier: New York, 1981. (25) Fayolle, S.; Bickel, T.; Wu¨rger, A. 2007, arXiv:0709.0384v1 [condmat.soft]. arXiv.org e-Print archive. http://arxiv.org/abs/0709.0384v1.

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Figure 3. (a) TEM image of the Ludox particles and (b) the corresponding size distribution of colloidal particles.

stock dispersion, the density of Ludox particles was found to be equal to F ) 2.19 g/mL (assuming that the volumes of water and Ludox particles are additive). With additional salt, the repulsive force between the charged particles decreases due to more effective screening of the surface charges, so that generally the stability of the dispersion is reduced. Salt-induced gelation of the Ludox dispersion occurs only for samples with a volume fraction larger than φ ) 0.5% at a salt content larger than cNaCl ) 0.0117 wt. Dispersions with a volume fraction of φ ) 0.1% or less are stable up to cNaCl ) 0.0234 wt. All of the measurements were conducted in the isotropic stable phase, far away from the gelation concentration. 2. DLS and TEM. The size of the Ludox particles was measured by transmission electron microscopy (TEM) and dynamic light scattering (DLS). We used the same TEM and DLS instruments as described in our previous work.8 Figure 3 shows a TEM picture of the Ludox particles (a) and their size distribution (b). The histogram was obtained by counting more than 400 particles. The distribution results in a number-average radius of 〈RTEM〉 ) 12.3 ( 2.4 nm. We also performed DLS measurements at a volume fraction of 0.1% without additional salt. The hydrodynamic radius Rh as calculated by the Stokes-Einstein relation is 9.5 ( 0.8 nm, which is smaller than the 〈RTEM〉 value. For a polydisperse dispersion, Rh should be larger than 〈RTEM〉 by a factor of (1 + PD.I)5, where PD.I is the polydispersity index.26 A possible reason for this discrepancy is that, without additional salt, even at such low volume fraction of 0.1%, we still cannot completely avoid the repulsive interparticle interactions, which results in a larger diffusion coefficient.27 To suppress the electrostatic interactions between particles, we added a small amount of NaCl into the dispersion. The DLS measurement on the Ludox dispersions with the same concentration (φ ) 0.1%) and additional NaCl (cNaCl ) 0.0117 wt, which leads to λDH ∼ 1 nm) results in Rh ) 15.6 nm with PD.I ) 0.03. Therefore, the number-averaged radius determined from DLS is 13.4 nm, which agrees within the error bars with the TEM result. B. Electrophoresis. The surface charge of the Ludox particles was characterized by electrophoresis. Figure 4a shows the electrophoretic mobility µE of Ludox particles without additional salt at low concentrations. In the dilute regime, the electrophoretic mobility is independent of the colloidal composition. The average mobility, -3.4 × 10-4 cm2 s-1 V-1, is obtained from eq 2, which corresponds to a charge of 7.014 × 10-18 C/particle. The negative sign of the mobility means that the colloids carry a negative surface charge. The errors indicated in the figure correspond to 1 standard deviation determined from at least three measurements. The large error bars occurring at very low concentrations are due (26) Thomas, J. C. J. Colloid Interface Sci. 1987, 117, 187. (27) Bowen, W. R.; Mongruel, A. Colloids Surf., A 1998, 138, 161.

Figure 4. Electrophoretic mobility, µE, of Ludox particles as a function of (a) volume fraction, φ, and (b) Debye length, λDH. The dashed lines are guides to the eye.

to the weak scattering intensity. The electrophoretic mobility was also measured as a function of salinity. In Figure 4b, the electrophoretic mobility is shown as a function of the Debye length λDH. The electrophoretic mobility increases with increasing Debye length. The surface charge on the silica particles is due to a partial dissociation of surface hydroxide groups.28 The electrophoretic mobility of Ludox dispersions with a volume fraction φ ) 0.5% of Ludox was measured at various pH values. µE is plotted in Figure 5a as a function of pH. As can be seen, the amplitude of the mobility |µE| decreases with increasing pH. This behavior of the mobility can be understood from eqs 3-5. These equations show that the mobility is determined by the surface charge Q as well as the Debye length λDH. By adding sodium hydroxide to the solution to increase the pH, two counterbalancing effects occur. With increasing pH, the surface charge increases due to the stronger dissociation of the hydroxide groups on the surfaces of the silica particles, which leads to an increase of the mobility. At the same time, the salinity increases, leading to a decrease of the Debye length which in turn leads to a decrease of the mobility. In the case where the increase of the surface charge dominates, we observe that |µE| increases, while in the case where the decrease of the Debye length dominates, |µE| decreases. The Debye lengths were calculated by taking into account the concentrations of Na+ and OH- in the dispersion, which are listed in Table 1. The values of f(κR) and Q are calculated from eqs 4 and 3, respectively. In Figure 5b, the surface charge density σ is plotted as a function of pH. One can observe that σ increases (28) Iler, R. K. The Chemistry of Silica: Solubility, Polymeriztion, Colloid and Surface Properties, and Biochemistry; Wiley: New York, 1979.

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Figure 5. pH dependence of the (a) electrophoretic mobility, µE, and (b) surface charge density, σ, of Ludox particles. The solid line in (b) is a linear fit to the data.

Figure 6. Normalized heterodyne signal, ζhet, of a Ludox dispersion with φ ) 0.1% at (O) low salinity (cNaCl ) 6 × 10-4 g/mL) and (9) high salinity (cNaCl ) 2.3 × 10-2 g/mL). Table 1. Parameters of the Ludox Dispersions as a Function of pH Value pH

[Na]+/ 10-4 M

[OH]-/ 10-6 M

λDH/ nm

κR

f(κR)

Q/ 10-18 C

7.15 7.74 7.95 8.36 8.77 9.25 10.01

0.52 1.1 2.5 5.1 10 25

0.54 8.9 2.3 5.9 18 100

59.4 41.8 26.9 18.9 13.3 8.4

0.20 0.29 0.45 0.63 0.89 1.42

0.67 0.67 0.67 0.68 0.69 0.70

7.01 ( 0.1 7.28 ( 0.3 7.54 ( 0.2 8.20 ( 0.4 9.38 ( 0.4 10.59 ( 0.3 11.82 ( 0.5

essentially linearly with increasing pH as σ ) (-29.4 + 7.03 × pH) × 103 e/µm2. C. Thermal-Diffusive Behavior. TDFRS measurements were performed at volume fractions of 0.1% and 0.5% at 25 °C. Figure 6 shows a typical normalized heterodyne TDFRS signal ζhet(t) as a function of time for low and high salinity. The rapid increase of ζhet(t) is due to the establishment of the temperature gradient, and the slower decay reflects the formation of a concentration gradient due to thermophoretic motion. In principle, our system is a ternary solution: Ludox particles dispersed in water with added NaCl. The contribution from the thermal diffusion of NaCl can be identified at high salinity as the fast mode of the concentration part of the TDFRS signal (signal with 9 in Figure 6). A separation of the NaCl contribution to the concentration part of the TDFRS signal is only significant at high salt contents

(cNaCl > 0.1 M), which corresponds to a small Debye length (κ-1 < 1 nm). For low salinity, the salt contribution to the TDRFS signal can be neglected. From the independently determined refractive index increments (∂n/∂c) and (∂n/∂T) in section II, we know that, for both NaCl and Ludox particles, a positive concentration plateau implies migration to the cold side, while a negative concentration plateau implies enrichment at the warm side. The TDFRS signal was fitted with eq 1, and the obtained Soret coefficients of the Ludox particles are plotted in Figure 7 a. The plot shows that, in the probed Debye length range, the Soret coefficients of the Ludox particles are negative, which corresponds to movement of the Ludox particles to the warm side. The errors correspond to one standard deviation obtained from at least three measurements. Negative Soret coefficients were also observed by Rusconi29 for more concentrated Ludox particle dispersions (φ ) 1.12 wt %). The data from Rusconi are included in the plot in Figure 7a. We found that, for small Debye lengths, the experimental results from the present TDFRS study are consistent with the results from Rusconi, who employed a thermal lens setup. At larger Debye lengths, the data from Rusconi (the star symbol in Figure 7a) deviate from our results. This is due to colloid-colloid interactions, which indeed become more pronounced for larger Debye lengths. The negative sign of ST is due to contributions from the hydration layer (and possibly the core of the colloidal material).13 From Figure 7a, it can be seen that the Soret coefficient increases with increasing Debye length and reaches a shallow maximum at λDH ≈ 5 nm. In the range λDH < 5 nm, the Soret coefficients for the Ludox particles are identical within the error bars for both concentrations φ ) 0.1% and 0.5%, which implies that we are indeed in the diluted regime where colloid-colloid interactions can be neglected. A clear concentration dependence has been observed in refs 6 and 29 at higher concentrations. An increase of ST with increasing Debye length was also observed for other charged systems, such as charged polystyrene spheres13 and also ionic surfactant micelles (SDS).6 These data are plotted in Figure 7b and c. The lines in Figure 7 correspond to the different theoretical models discussed in section III. In fitting the data for the Ludox particles, the only fitting parameter is the intercept A(T) at zero Debye length, which is due to the solvation layer and core material contributions to the Soret coefficient. For Ludox particles, in the Debye length range 1 nm < λDH < 5 nm, we find that eq 9 agrees with the experimental results for ST. The experimental data deviate systematically at smaller Debye lengths (λDH < 1 nm) and above λDH > 5 nm. The deviations at smaller Debye lengths (λDH e 1nm) are probably due to the high salinity, which leads to a violation of the Debye-Hu¨ckel approximation where the finite extent of ions is neglected. For large Debye lengths (λDH > 6 nm), the surface potential is high (eΦs/kBT > 0.9), so that linearization of the Poisson-Boltzmann equation is probably no longer allowed. Also, the prediction by Wu¨rger and the capacitor model of Braun in eq 10 (the dotted line) agrees reasonably well in the intermediate Debye length range. In Figure 7b and c, we can observe that the models of Dhont, Wu¨rger, and Braun are in good agreement for the PS and SDS systems as well. As mentioned before, Dhont’s result reduces to that of Wu¨rger and Braun for thin double layers. Apparently, the finite extent of the double layer does not have a pronounced effect on the numerical value of the Soret coefficient for the Ludox particles and SDS micelles. The theoretical prediction by Ruckenstein in eq 6 (the dash-dotted lines) deviates significantly from the experimental data in the entire λDH range. Note that in Ruckenstein’s prediction (29) Rusconi, R.; Isa, L.; Piazza, R. J. Opt. Soc. Am. B 2004, 21, 605.

Thermal-DiffusiVe BehaVior of Ludox Particles

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Figure 7. (a) Soret coefficients of Ludox particles, (O) φ ) 0.1%, (9) φ ) 0.5%, and (f) φ ) 1.12%,29 as functions of the Debye length, λDH. (b) Soret coefficients of polystyrene spheres. Data are taken from ref 13. (c) Soret coefficients of SDS micelles. Data are taken from ref 6. All the measurements were performed at 298 K. Solid lines are fits of the data to eq 9, dotted lines are fits to eq 10, and dash-dotted lines are fits to eq 6. Table 2. Parameters of the Charged Colloids in Figure 7

Ludox (pH ) 7.0) polystyrene polystyrene polystyrene SDS micelles

R/nm

σ/104eµm-2

A(T) from fit of eq 9/K-1

12.3 100 250 550 2.7

2.4 0.45 0.45 0.45 28

-0.16 0.35 0.11 3.01 7.1 × 10-4

ST ∝ R/κ2, while the other theories predict that ST ∝ R2/κ (for thin double layers). Also, the dependence of the dielectric constant on temperature (signified by the terms ∝ dln /dln T) is absent in Ruckenstein’s expression. The intercept A(T) is around -0.16 K-1 for the Ludox particles (see Table 2). For the PS particles, the fitting parameter A(T) can be a function of the size of the particles, since A(T) is the contribution to the Soret coefficient stemming from the hydration layer13 and possibly the colloid core material. The colloid core material can contribute due to the response of the internal degrees of freedom of the core material to a change in temperature. Note that, except for Ruckenstein’s prediction, the theories predict that the thermophoretic mobility (which is proportional to DT) is not independent of the size of the colloidal particles, in contrast with the electrophoretic mobility. Electrophoresis is driven by electric forces that act on the surface charges of the colloidal particles. The main driving force for thermal diffusion (as far as the double layer is concerned) is due to the change of the internal energy of the double layer due to temperature changes. The origins of the driving forces for electrophoresis and thermodiffusion are thus of a different nature. Therefore, the colloid size dependence of the electrophoretic and thermophoretic mobilities shows a different behavior with respect to the size dependence of the colloid. For Ludox particles, variation of the pH leads to a change of the surface charge density. The thermal-diffusive behavior of the Ludox particle was measured at various pH values with a fixed Debye length equal to λDH ) 2.8 nm, where eΦs/kBT ) 0.5. This is in the range where the experiments agree with the theoretical predictions in eqs 9 and 10. Figure 8 shows that in a pH range between 7.0 and 9.5 both the Soret coefficient and thermal diffusion coefficient increase with increasing pH. This implies that the colloidal particles become more thermophobic with increasing surface charge density, as predicted by all the theories discussed above. In contrast, an experimental study on the thermophoresis of lysozyme with various zeta potentials did not show such a pronounced surface charge dependence.15 The

Figure 8. Dependence of (a) the Soret coefficient, ST, and (b) the thermal diffusion coefficient, DT, of Ludox particles on pH. Solid lines in (a) are fits of the data to eq 9, dotted lines are fits to eq 10, and dash-dotted lines are fits to eq 6. The dashed line in (b) is a guide to the eye.

reason for this is still unclear. The curves in Figure 8a relate to the different theoretical models using the surface charge density that is measured independently by means of electrophoresis, as discussed in part B of section 4. All models discussed here show good agreement with the experimental results in the pH range from 7.0 to 9.5. The pH dependence of the Soret coefficient was also studied for charged polystyrene carboxyl spheres (≈2 wt %) in the pH range between 3.3 and 10.5, using a micrometer-scale beamdeflection device, in ref 7. For Debye lengths larger than 1 nm, the thermal diffusion coefficient of the PS spheres was found to become smaller with increasing pH at constant salinity, contrary to what we found for the Ludox particles. This is probably due to the fact that the surface charge density of the PS particles is independent of the pH, since the surface groups are essentially fully dissociated. For the Ludox particles, however, the weak acidic epoxy groups are only partly dissociated, which renders the surface charge density pH dependent. As mentioned before, a clear contribution from NaCl on the TDFRS signal is observed at high salinity (see Figure 6). Although in the past the thermal-diffusive behavior of charged colloids

2432 Langmuir, Vol. 24, No. 6, 2008

Ning et al.

V. Conclusion

Figure 9. Soret coefficient of NaCl (9) measured in NaCl/water solutions and (O) measured in NaCl/Ludox/water ternary solutions.

was also measured at high salinity (cNaCl up to 500 mM),12 this is the first time that the thermophoretic motion of NaCl could be extracted from the signal of the ternary mixtures consisting of charged colloids, water, and salt. The measured Soret coefficients of NaCl are shown in Figure 9. It can be seen that the Soret coefficient is positive, which means that NaCl migrates to the cold side, in accordance with ref 30. Furthermore, ST decreases with increasing salt content. The Soret coefficients of salt in binary NaCl/water solutions and ternary NaCl/Ludox/ water solutions are in quantitative agreement. In the past, Tanner measured the Soret effect of NaCl in water by using a beam deflection setup,30 but a quantitative comparison of data is not possible because his data were obtained at much higher temperatures. The electrostatic energy to separate one pair of Na+ and Clions over a distance equal to the Bjerrum length (0.71 nm for water) is by definition equal to 1 kBT. This shows that macroscopic charge separation of many ions over larger distances cannot occur. Therefore, there will be no electric field due to thermal diffusion of Na+ and Cl- that would act with a force on the colloids. (30) Tanner, C. Trans. Faraday Soc. 1927, 23, 75.

Single-particle thermal diffusion in a dilute dispersion of charged colloids (Ludox particles) was measured, as a function of both the Debye length and pH. The independence of the Soret coefficient of the concentration in the investigated concentration range shows that the measured Soret coefficients relate to singleparticle diffusion. Similar to the thermal-diffusive behavior of many other systems of charged colloids, a negative Soret coefficient for Ludox particles was found at 25 °C, which implies diffusion to warm regions. In addition to the contribution to the Soret coefficient stemming from the double layer, there are contributions from, for example, the hydration layer. The latter is, too, a good approximation, independent of the Debye length. In comparing theories concerned with the double-layer contribution to ST with experiments where ST is measured as a function the Debye length, the latter contributions give rise to a shift of the entire experimental curve. The negative value of Soret coefficients for Ludox particles is due to the large negative contribution from the hydration layer and possibly the colloid core material. We found a reasonable agreement between our experimental data, as well as with other existing data from the literature, and the theoretical predictions by Dhont, Wu¨rger, and Braun. The dependence of the Soret coefficient both on the Debye length and on the surface charge density is well described by these theories. Deviations are found for high salinity and extended double layers, where the Debye-Hu¨ckel approximations on which these theories are based are no longer valid. At high salinity, a slight decrease of the Soret coefficient with increasing Debye length is found. There are no theories available yet, which go beyond the Debye-Hu¨ckel approximation, that could explain this behavior. Acknowledgment. The authors would like to thank Gerhard Na¨gele, Christoph Go¨gelein and Johan Buitenhuis for fruitful discussions. The authors are grateful to Grace Davison for the donation of Ludox particles. This work was partially supported by the Deutsche Forschungsgemeinschaft grant Wi 1684. LA703517U