Transport of Nanoscale Latex Spheres in a Temperature Gradient

The transport of mass induced by a temperature gradient is commonly referred to as thermodiffusion or the Soret effect. Typically, mass transport in a...
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Langmuir 2005, 21, 5317-5323

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Transport of Nanoscale Latex Spheres in a Temperature Gradient Shawn A. Putnam* and David G. Cahill Department of Materials Science & Engineering, University of Illinois, Urbana, Illinois 61801 Received November 30, 2004. In Final Form: April 19, 2005 We use a micrometer-scale optical beam deflection technique to measure the thermodiffusion coefficient DT at room temperature (≈24 °C) of dilute aqueous suspensions of charged polystyrene spheres with different surface functionalities. In solutions with large concentrations of monovalent salts, J100 mM, the thermodiffusion coefficients for 26 nm spheres with carboxyl functionality can be varied within the range -0.9 × 10-7 cm2 s-1 K-1 < DT < 1.5 × 10-7 cm2 s-1 K-1 by changing the ionic species in solution; in this case, DT is the product of the electrophoretic mobility µE and the Seebeck coefficient of the electrolyte, Se ) (Q/C - Q/A)/2eT, DT ) -Se µE, where Q/C and Q/A are the single ion heats of transport of the cationic and anionic species, respectively. In low ionic strength solutions of LiCl, j5 mM, and particle concentrations j2 wt %, DT is negative, independent of particle concentration and independent of the Debye length; DT ) -0.73 ( 0.05 × 10-7 cm2 s-1 K-1.

Introduction The transport of mass induced by a temperature gradient is commonly referred to as thermodiffusion or the Soret effect. Typically, mass transport in a temperature gradient is characterized by either the Soret coefficient (ST) or the thermodiffusion coefficient (DT). The Soret coefficient is the ratio of DT to the diffusion coefficient Dc, ST ) DT/Dc, and can be defined within the framework of irreversible thermodynamics with the use of Onsager coefficients;1 e.g., at low particle concentration (c), the particle flux of a colloidal suspension subjected to a temperature gradient ∇T is J ) -c DT ∇T - Dc ∇c. For DT > 0, thermodiffusion drives particle motion from the hot to the cold region. Theoretical understanding of the thermodiffusion of particles is well-developed only in some limited cases. For instance, the theory is simple and complete for aerosol particles that are small in comparison to the mean free path of the gas; aerosol particles move toward regions of lower temperature due to the larger momentum transferred to the particles by gas molecules incident from the higher temperature region.2 The thermodiffusion of ions in aqueous electrolytes is also well-studied; the Soret coefficients of electrolytes have been documented for over 100 years. The theory is particularly well-developed for monovalent electrolytes3 and has been supported experimentally in several studies.4-7 For example, DT for a 1:1 salt is

[ (

)]

Dc ∂ ln γ( DT ) Q * 1+ 2 ∂ ln m 2kBT

-1

T

(1)

* To whom correspondence should be addressed. E-mail: [email protected]. (1) Onsager, L. Phys. Rev. 1930, 37, 405. (2) Waldmann, L.; Schmitt, K. H. Thermophoresis and diffusiophoresis of aerosols. In Aerosol Science; Davies, C. N., Ed.; Academic Press: New York, 1966. (3) Helfand, E.; Kirkwook, J. B. J. Chem. Phys. 1960, 32, 857. (4) Tanner, C. C. Trans. Faraday Soc. 1927, 23, 75. (5) Agar, J. N.; Turner, J. C. R. Proc. R. Soc. A 1960, 255, 307. (6) Snowdon, P. N.; Turner, J. C. R. Trans. Faraday Soc. 1960, 56, 1409. (7) Snowdon, P. N.; Turner, J. C. R. Trans. Faraday Soc. 1960, 56, 1812.

where Q* is the molar heat of transport of the electrolyte, Dc is the isothermal diffusion coefficient, and γ( is the mean ionic activity coefficient on a molarity scale m.5-7 For monovalent electrolytes, the heat of transport of the electrolyte is Q* ) Q/C + Q/A, where Q/C and Q/A are the heats of transport for cationic and anionic species, respectively.3,5-7 The heat of transport associated with each ion is divided into two contributions:3 (i) a part that depends on ion-solvent interactions and (ii) a part that scales with the square root of concentration, c1/2, due to ion-ion interactions. Experimental research on the thermodiffusion of particles in liquids has been relatively limited.8-14 One reason for the relative lack of data is the long time-scales associated with the thermodiffusion of particle suspensions. For example, Jeon et al.9 measured thermodiffusion coefficients within the range 0.2 × 10-7 cm2 s-1 K-1 < DT < 0.8 × 10-7 cm2 s-1 K-1 for aqueous suspensions of both polystyrene (PS) and silica particles with diameters ranging between ≈100 and 300 nm. A typical electrophoretic mobility for such particles is µE ≈ 5 × 10-4 cm2 s-1 V-1. In contrast to electrophoresis experiments, where only ≈20 s is required for charged particles to drift 1 mm in an applied electric field of ≈10 V cm-1, in thermodiffusion experiments when DT ≈ 0.4 × 10-7 cm2 s-1 K-1, ≈70 h is required for the same particles to drift 1 mm in a temperature gradient of ≈10 K cm-1. Within the past 10 years, however, several groups have reported studies of nanoparticle systems. (We also note recent speculations on the role played by thermodiffusion of biological macromolecules in the origins of life.15) Blums et al.10 showed that hydrocarbon-based ferrocolloids have positive thermodiffusion coefficients, and Spill et al.12 (8) McNab, G. S.; Meisen, A. J. Colloid Interface Sci. 1973, 44, 339346. (9) Jeon, S. J.; Schimpf, M. E.; Nyborg, A. Anal. Chem. 1997, 69, 3442-3450. (10) Blums, E.; Odenbach, S.; Mezulis, A.; Maiorov, M. Phys. Fluids 1998, 10, 2155. (11) Schaertl, W.; Roos, C. Phys. Rev. E 1999, 60, 2020-2028. (12) Spill, R.; Ko¨hler, W.; Lindenblatt, G.; Schaertl, W. Phys. Rev. E 2000, 62, 8361-8368. (13) Lenglet, J.; Bourdon, A.; Bacri, J. C.; Demouchy, G. Phys. Rev. E 2002, 65, 031408. (14) Wiegand, S. J. Phys.: Condens. Matter 2004, 16, R357-R379. (15) Braun, D.; Libchaber, A. Phys. Rev. Lett. 2002, 89, 188103.

10.1021/la047056h CCC: $30.25 © 2005 American Chemical Society Published on Web 05/13/2005

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demonstrated that DT is a positive quantity independent of particle size for gold-doped microgels of different sizes and cross-link ratios. One of the most complete studies of the thermodiffusion of particles in a liquid is that of sodium dodecyl sulfate micelles in aqueous NaCl solutions,16 where DT in the dilute regime scaled as the square of the Debye length and DT was highly dependent on surfactant concentration at ionic strengths j200 mM. Several theoretical approaches have been proposed as explanations for thermally driven particle flows in liquids,17-23 but we are unaware of any theory that has been thoroughly tested by experiment. A purely electrostatic contribution to thermodiffusion in the limit of low surface potentials was proposed by Ruckenstien.17 Andreev18 has described thermodiffusion in terms of hydrodynamic fluctuations, where DT is predicted to scale as the square root of the particle radius due to the interaction of the particle with the thermal acoustic fluctuations in the liquid.18 Many of the theoretical descriptions are extremely involved; for example, Morozov22 has presented two elaborate theories for both ionic colloids and colloids stabilized with surfactant molecules, where analytic solutions only exist in the limit of particles that are large relative to the Debye screening length. Dhont24,25 has discussed how DT consists of two additive contributions, one due to specific particle-solvent interactions and a second due to particle-particle interactions. We give special attention to the description originally formulated by Derjaguin26-28 and then later reviewed by Anderson,19 where DT is proportional to the changes in the enthalpy of the fluid near the solid-liquid interface. We describe this theory in more detail below. Experimental Details Materials and Preparation of Aqueous Particle Suspensions. Aqueous particle suspensions of charge-stabilized PS spheres with different surface functionalities and particle diameters were purchased from Interfacial Dynamics Corporation (IDC).29 Characterization by the manufacturer consisted of transmission electron microscopy and conductometric titration.30 All spheres were negatively charged with either carboxyl, sulfate, carboxyl-sulfate, aldehyde-sulfate, sulfate-bromo, or chloromethyl functional groups. The term carboxyl-sulfate, for instance, is used to describe a particle surface that has both carboxyl and sulfate functional groups. The majority of our DT experiments used spheres with carboxyl functionality. Table 1 lists the IDC reported particle diameters, PS concentrations (cp), and titratable surface charge densities (σ) of the as-received carboxyl spheres. The PS concentrations of ≈4 g/100 mL were verified by index of refraction measurements with an (16) Piazza, R.; Guarino, A. Phys. Rev. Lett. 2002, 88, 208302. (17) Ruckenstein, E. J. Colloid Interface Sci. 1981, 83, 77-81. (18) Andreev, A. F. Sov. Phys. JETP 1988, 67, 117-120. (19) Anderson, J. L. Annu. Rev. Fluid Mech. 1989, 21, 61-99. (20) Giddings, J. C.; Shinudu, P. M.; Semenov, S. N. J. Colloid Interface Sci. 1995, 176, 454-458. (21) Morozov, K. I. J. Magn. Magn. Mater. 1999, 201, 248. (22) Morozov, K. I. In On the Theory of the Soret Effect in Colloids; Ko¨hler, W., Wiegand, S., Eds.; Springer-Verlag: Heidelberg, 2002; Vol. 584, Chapter 3, pp 38-60. (23) Bringuier, E.; Bourdon, A. Phys. Rev. E 2003, 67, 011404. (24) Dhont, J. K. G. J. Chem. Phys. 2003, 120, 1632. (25) Dhont, J. K. G. J. Chem. Phys. 2003, 120, 1642. (26) Derjaguin, B. V.; Sidorenkov, G. P. Dokl. Akad. Nauk SSSR 1941, 32, 622. (27) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface forces. In Surface Forces; Kitchener, J. A., Ed.; Consultants Bureau: New York, 1987; Chapter 11, pp 390-409. (28) Churaev, N. V.; Deryagin, B. V.; Zolotarev, P. P. Dokl. Akad. Nauk SSSR 1968, 183, 1139. (29) Interfacial Dynamics Corporation, Portland, OR; www.idclatex.com. (30) Glasstone, S. An Introduction to Electrochemistry; D. Van Nostrand Company, Inc.: New York, 1942; Chapter 2, pp 71-77.

Putnam and Cahill Table 1. Properties of the Suspensions of As-Received Carboxyl Spheres at ≈4 wt %a diameters (nm)

pH

g (µS cm-1)

σ (µC cm-2)

µE (10-4 cm2 s-1 V-1)

26 ( 6 34 ( 8 67 ( 9 90 ( 11 92 ( 15 130 ( 10

6.7 6.5 6.5 7.4 6.4 6.7

223 197 100 337 150 174

-0.5 -0.4 -0.6 -7.2 -0.1 -6.8

-4.5 ( 0.9 -5.1 ( 0.5 -5.2 ( 0.2 -5.3 ( 0.4 -4.9 ( 0.5 -6.2 ( 0.2

a The particle size distributions and titratable surface charge densities σ are values reported by IDC. The pH, ionic conductivities g, and electrophoretic mobilities µE are our measurements, where the µE measurements are for spheres diluted with 30 mM NaCl to a particle concentration of j0.3 wt %.

Abbe refractometer in conjunction with the predictions of effective medium theory31 (assuming n ) 1.591 for PS at 590 nm and 25 °C). Also provided in Table 1 are our pH, ionic conductivity (g), and electrophoretic mobility (µE) measurements of the as-received suspensions. We assume that the counterion in the as-received suspensions of carboxyl spheres is Na because the manufacturer dialyzed the suspensions in a ≈0.1 mM sodium hydroxide (NaOH) solution (≈9 pH) to ensure pH > 6.32 Particle suspensions were prepared by diluting as-received spheres (or spheres cleaned via dialysis) with either deionized (DI) water, monovalent salt solutions, or premixed buffering solutions. The ionic strengths of the prepared particle suspensions are based upon the pH and ionic conductivities. Reagent-grade monovalent salts and analytical-grade buffer chemicals were used. The buffers were made by (i) diluting the buffering chemicals to a desired concentration, (ii) adding salt if required, (iii) titrating with a conjugate base to the apparent pKa of the buffer, and then (iv) diluting with water to the final solution volume. For example, a ≈3 mM 3-(cyclohexylamino)-1-propanesulfonic acid (CAPS)-NaOH buffer was made by first titrating a ≈95 mL 6 mM CAPS-water solution with 0.1 M NaOH to the apparent pKa ≈ 10.5 and then diluting with water to 100 mL. Dialysis of Particle Suspensions. The as-received 26 nm carboxyl spheres were dialyzed with nominal 3500 molecular weight cutoff cellulose dialysis tubing to verify and replace the presumed Na counterion with Li and tetraethylammonium (TEA). The finial dialysis baths consisted of either ≈0.1 mM NaOH, LiOH, or TEA-OH. The dialysis procedure outlined here is similar to that employed by IDC.32 First, the tubes were cleaned with a ten-cycle process of heating in water to a boil and then rinsing with DI water, where the water was replaced after each heating cycle. Following the heat treatment of the dialysis tubes, the as-received spheres were dialyzed twice in baths of ≈0.1 mM NaOH for ≈15 h each. The Na counterion was replaced with Li by first dialyzing the spheres twice in ≈150 mM LiCl baths maintained at a pH of 9 (≈0.1 mM LiOH) for 15 h each. The spheres were then dialyzed again in two more baths of ≈0.1 mM LiOH for 15 h each. A similar dialysis scheme was used to replace the Na counterion with TEA, where the pH was adjusted with TEA-OH and the first two baths contained ≈60 mM of TEA-Cl. Measurement Technique and Raw Data. To measure the thermodiffusion coefficient of particle suspensions, we used a micrometer-scale apparatus based on the deflection of the path of a laser beam by a gradient in the index of refraction.33 Temperature gradients were produced by alternately heating a pair of Au thin-film lines fabricated by photolithography on a glass substrate. The width of the lines was 2b ) 7 µm, and the separation was 2a ) 25 µm. At low frequencies, f j Dc/(πa2) Hz, the periodic temperature gradients induced concentration gradients in the sample cell by thermally driven transport of the PS particles; that is, 1/f was the time required for PS particles to diffuse half the distance between the metal heaters. Prior to each experiment, the sample cell was dismantled and cleaned by (31) Landauer, R. Electrical conductivity in inhomogeneous media. In Electrical Transport and Optical Properties of Inhomogeneous Materials; Garland, J., Tanner, D., Eds.; American Institute of Physics: New York, 1978; Vol. 40. (32) IDC, May 2004, private communication. (33) Putnam, S. A.; Cahill, D. G. Rev. Sci. Instrum. 2004, 75, 2368.

Transport of Nanoscale Latex Spheres

Figure 1. Comparison between experimental data (symbols) and theoretical calculations (lines) for the deflection of the laser beam as a function of heater frequency for a 50-50 wt % mixture of dodecane and THN. Filled symbols are the in-phase (real) part of the beam deflection, and open symbols are the out-ofphase (imaginary) part. The temperature oscillations are ∆Tosc ≈ 0.3 K. first rinsing with toluene and DI water and then sonicating in heated baths of acetone and ethyl alcohol. After it was cleaned, the sample cell was rinsed with DI water and then dried under nitrogen. The sample cell was mounted on a two axis tilt stage and was heated from above. The thin-film lines were heated with high-frequency square-wave currents, f ) 6.1 kHz, to suppress electrokinetic effects. Heating from above ensures that density gradients induced by thermal expansion alone were always stabilizing; however, for DT < 0, particles will migrate to the top of the cell (hot regions) and create the possibility of convection driven by the Soret effect (the nominal density of PS at 20 °C is FPS ≈ 1.055 g cm-3). When we used large temperature excursions for the heaters, ∆Tosc > 0.3 K, we sometimes observed the effects of convection in our measurements at extremely low modulation frequencies, f j 10 mHz, through strong deviations of the measured beam deflection from the frequency dependence expected for diffusion and thermodiffusion. For ∆Tosc j 0.3 K, however, we did not observe significant discrepancies between experiment and our analytical model and concluded that convection does not play an important role in our experiments as long as the temperature excursion is not too large. Our apparatus and analysis methods have been validated previously in a published study of molecular PS dissolved in toluene,33,34 which yielded diffusion coefficients (Dc) and thermodiffusion coefficients (DT) within 10% of those published by Zhang et al.34 Here, we present another test using a benchmark solution of a 50-50 wt % mixture of dodecane (C12H26) and 1,2,3,4tetrahydronaphthalene (THN). Figure 1 is a comparison between the measured data for this mixture and the analytical model for the beam deflection. In this analysis, a four-step procedure was used to fit the analytical solution to the experimental data; i.e., the thermodiffusion coefficient (DT), diffusion coefficient (Dc), thermooptic coefficient (dn/dT), and thermal conductivity of the mixture (Λ) were taken as unknowns. An accurate determination of Λ and dn/dT is easily accomplished because the amplitude of the temperature oscillation is known. The magnitude of the maximum of the out-of-phase data at fth ) 150 Hz is directly correlated with dn/dT, and the location of the maximum is directly correlated with the thermal diffusivity Dth ) Λ/Cp because fth ≈ Dth/(π a2). Dc is then obtained by aligning the imaginary part of the low-frequency response; the frequency corresponding to the maximum in the imaginary part of the low-frequency response is fc ≈ Dc/(π a2) ≈ 1 Hz. Finally, DT is determined by adjusting the magnitude of the low-frequency response; DT is directly correlated with both the magnitude of the low-frequency maximum of the out-of-phase data at fc ≈ 1 Hz and the magnitude of the in-phase data at fc j 10 mHz. An accurate determination of DT is possible because dn/dc is determined independently via measurements with an Abbe refractometer; c dn/dc ) 0.061 at (34) Zhang, K. J.; Briggs, M. E.; Gammon, R. W.; Sengers, J. V.; Douglas, J. F. J. Chem. Phys. 1999, 111, 2270-2282.

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Figure 2. Comparison between experimental data (symbols) and theoretical calculations (lines) for the beam deflection as a function of heater frequency for 26 nm carboxyl spheres in aqueous buffer solutions (≈2 wt %). The circle symbols correspond to spheres dispersed in a ≈1.5 mM CAPS-NaOH buffer, and the triangle symbols correspond to spheres dispersed in a ≈50 mM CAPS-NaCl buffer. The amplitude of the temperature oscillations is ∆Tosc ≈ 0.3 K. c ) 50 wt % (C8H12). The procedure yields thermodiffusion, diffusion, and thermooptic coefficients within 10% of those published by Wittko et al.;35 i.e., DT ) 0.61 ( 0.02 × 10-7 cm2 s-1 K-1, Dc ) 6.5 ( 0.2 × 10-7 cm2 s-1, and dn/dT ) -4.6 ( 0.2 × 10-4 K-1. The measured beam deflection data (symbols) for 26 nm carboxyl latex spheres dispersed in ≈1.5 mM CAPS-NaOH buffers (pH ) 10.5 ( 0.4) are shown in Figure 2 with comparisons to the calculated value of ∆θ (lines). The calculated values of ∆θ are one parameter fits of the thermodiffusion coefficient DT; that is, the diffusion coefficient is calculated from the Stokes-Einstein relation, Dc ) kBT/6π η RH, and all other model parameters are taken from the literature or are measured independently.33 The data shown in Figure 2 are for two different experiments; the circles are data for the 26 nm spheres in a ≈1.5 mM CAPSNaOH buffer (DT ) -0.89 ( 0.04 × 10-7 cm2 s-1 K-1), and the triangles are data for the 26 nm spheres in a ≈50 mM CAPSNaCl buffer (DT ) 0.26 ( 0.03 × 10-7 cm2 s-1 K-1). The concentration of NaCl in the 50 mM CAPS-NaCl buffer is ≈48.5 mM. Characterization by Light Scattering and Electrophoresis. A commercial Malvern 3000HS Zetasizer was used for dynamic light scattering (DLS) and electrophoresis measurements. We used DLS to verify that stable particle suspensions were used in our DT experiments. All light scattering measurements were conducted at particle concentrations j0.02 wt %. Electrophoresis experiments were preformed with modulated electric fields of ≈24 V/cm at 2 kHz. The electrophoretic mobilities listed in Table 1 are for as-received suspensions diluted with 30 mM NaCl to a particle concentration of j0.3 wt %. Experiments were carried out on 26, 34, 90, and 92 nm carboxyl spheres as a function of pH and ionic strength, where the pH was measured before and after each µE experiment. Surprisingly, the electrophoretic mobility of the 26 nm carboxyl spheres was nearly constant (µE ) -4.5 ( 0.9 × 10-4 cm2 s-1 V-1 in all experiments with a pH ranging from 1.7 to 11.0 and NaCl concentrations e 100 mM). In addition, µE for the 26 nm spheres was independent of electrolyte species (NaCl, TEA-Cl, and LiCl) and the composition of the buffer (3 mMsCAPS and citric acid). The ionic strengths tested for the 26 nm spheres were ≈3, 10, and 50 mM and the 3 mM buffers ranged in pH from 3.1 to 10.5. The electrophoretic mobilities of 34, 90, and 92 nm carboxyl spheres were measured as a function of pH in ≈30 mM NaCl solutions. The 34 nm spheres showed almost no dependence on pH; i.e., µE ) -5.1 ( 0.5 × 10-4 cm2 s-1 V-1, and a slight dependence on pH was observed with the 90 and 92 nm spheres. For example, µE for the 90 and 92 nm spheres was nearly constant (-5.3 ( 0.4 × 10-4 and -4.9 ( 0.5 × 10-4 cm2 s-1 V-1) at a pH > 6 and monotonically decreased to -2.5 ( 0.5 × 10-4 and -3.5 ( 0.6 × 10-4 cm2 s-1 V-1 when the pH was reduced to ≈3. At a pH j 3, the 90 and 92 nm spheres started to flocculate. (35) Wittko, G.; Ko¨hler, W. Philos. Mag. 2003, 83, 1973.

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Figure 3. Thermodiffusion coefficient DT of the negatively charged PS spheres with different surface functionalities in water (≈2 wt %). The labels for each the data pointse.g., carboxyl (26 nm)sdescribe the surface functionality and particle diameter.

Results Data for nanoscale PS spheres with similar diameters (d ≈ 24 ( 5 nm) but widely varying surface chemistries are shown in Figure 3. All data are for as-received suspensions diluted with DI water to a particle concentration of ≈2 wt %. Particle surface chemistry seemingly plays a key role in the thermodiffusion of nanoscale latex spheres. Because the carboxyl spheres show the largest value of |DT|, the remainder of our study focused on the changes in DT for the carboxyl spheres produced by different electrolytes, ionic strengths, and pH. Figure 4a shows our measurements of DT as a function of ionic strength for the as-received 26 nm carboxyl spheres dispersed in the CAPS-NaCl and citric acid-NaCl buffers. At low ionic strength, j5 mM, the composition of the buffer has a strong influence on DT. On the other hand, at high ionic strengths, ≈100 mM, the buffer has no significant influence on DT; i.e., DT is constant and independent of pH when the NaCl concentration is ≈100 mM, DT ) 0.36 ( 0.03 × 10-7 cm2 s-1 K-1. Figure 4b,c shows how DT is affected by changing the salt to LiCl or to TEA-Cl. Again, DT is independent of the buffer composition (pH) when the salt concentration is J100 mM. At low ionic strength, ≈2 mM, the compostion of the electrolyte has a strong influence on DT; for example, DT changes from DT ) -1.01 ( 0.05 × 10-7 cm2 s-1 K-1 in the CAPS-LiOH buffer to DT ) -0.53 ( 0.03 × 10-7 cm2 s-1 K-1 in the CAPS-TEA-OH buffer. Data acquired at low ionic strengths in Figure 4, j5 mM, are complicated because a variety of ions are present in the solution at comparable concentrations. For this reason, we replaced the Na counterions in the as-received suspensions with Li or TEA by dialysis and then studied the thermodiffusion coefficient of the spheres in unbuffered electrolytes; see Figure 5. At high ionic strengths, data for DT show the same strong dependence on the composition of the electrolyte. At low ionic strength, DT converges to a common value in NaCl and LiCl solutions; in TEACl solutions, DT is more positive. At low ionic strength, changing the particle concentration cp has a significant effect only for suspensions that contain TEA ions, and changing the ionic strength does not change DT significantly for LiCl concentrations j5 mM (DT ) -0.73 ( 0.05 × 10-7 cm2 s-1 K-1). Figure 6 shows our measurements of DT for the asreceived carboxyl spheres diluted with DI water for various diameters. Clearly, the large negative value of DT for the

Figure 4. Thermodiffusion coefficient DT as a function of ionic strength (bottom axis) and Debye length κ-1 (top axis) for asreceived 26 nm carboxyl spheres diluted with citric acid (pH ≈ 3.3) and CAPS (pH ≈ 10.5) buffers. All data are for ≈2 wt % suspensions stabilized with buffer chemical concentrations of ≈1.5 mM, where NaOH, LiOH, and TEA-OH are the conjugate bases used in the preparation of each respective buffer. The figure labels (a) NaCl, (b) LiCl, and (c) TEA-Cl list the monovalent species added to the ≈1.5 mM buffer in each case.

Figure 5. Thermodiffusion coefficient DT as a function of ionic strength (bottom axis) and Debye length κ-1 (top axis) for dialyzed 26 nm carboxyl spheres diluted with monovalent salt solutions. The open symbols are for dialyzed suspensions at particle concentrations of cp ≈ 2 wt %, and the filled symbols are for cp ≈ 0.6 wt %.

26 nm carboxyl spheres does not persist in the other specimens. For the data in Figure 6, the ratios of the sphere radii to the Debye screening length κRH for the 26, 34, 67,

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E)

Figure 6. Thermodiffusion coefficient DT as a function of the particle diameter of as-received carboxyl spheres diluted with DI water.

90, 92, and 130 nm spheres are ≈1.6, 1.7, 2.7, 5.0, 3.2, and 4.3, respectively. The titratable surface charge densities of the spheres σ are provided in Table 1: Note that σ for the 92 nm spheres is ≈70 times less than that of the 90 and 130 nm spheres. Discussion Particle Transport Driven by Thermally Generated Electric Fields. As shown in Figures 4 and 5, the thermodiffusion of charged latex spheres is strongly influenced by the electrolyte concentration and composition. Moreover, the data in Figures 4 and 5 at high ionic strengths, J100 mM, show that DT is independent of salt concentration but still depends on the composition of the salt. These observations suggest that DT of the PS spheres is controlled by the thermodiffusion of the salt ions when the concentration of the ions is large. We therefore consider the electric fields that are generated in an electrolyte subjected to a temperature gradient.3,5-7 This effect has been discussed by Derjaguin3,26 for thermoosmosis of an electrolyte in a porous medium, and we derive the result here for the simpler case of a bulk electrolyte. At steady state, the single ion particle fluxes for a salt such as NaCl in a temperature gradient are Na Na JNa ) JNa E + JT + JD ) 0

(2)

Cl Cl JCl ) JCl E + JT + JD ) 0

(3)

and

Na Na where in eq 2, JNa E , JT , and JD are the particle fluxes of the Na cations driven by electric fields,

JNa E )

eDNa Na c E kBT

(4)

temperature gradients,

JNa T

)-

Q/a DNa kBT

2

cNa ∇T

(5)

and concentration gradients, Na ∇cNa JNa D ) -D

(6)

By subtracting the single ion particle flux of the Na cations (JNa) from the single ion particle flux of Cl anions (JCl), we find that the electric field generated by the thermodiffusion of the electrolyte is

(Q/Na - Q/Cl) ∇T 2eT

(7)

where we have assumed that (∇c/c)Na ≈ (∇c/c)Cl. For the lack of a better term, we refer to this prefactor (Q/C Q/A)/2eT as the Seebeck coefficient of the electrolyte (Se), by analogy with the electronic Seebeck coefficient of solid state physics.36 Figure 7 is a plot of the thermodiffusion coefficient DT of as-received carboxyl 26 nm spheres as a function of Se of the electrolyte. The single ion heats of transport are taken from literature values6 and are listed in Table 2. The data points are for LiOH, LiF, LiCl, NaF, NaCl, NaBr, and TEA-Cl electrolytes at concentrations J80 mM. The Seebeck coefficient Se couples DT to µE remarkably well; see Figure 7. We stress that this prediction for charged latex particles in concentrated electrolytes, DT ) -Se µE, does not imply DT ) 0 at low ionic strength. Many mechanisms may contribute to DT. By writing DT ) -Se µE, we are only predicting the component of DT induced by the differences in the thermodiffusion coefficients of the cations and anions in the electrolyte. Interparticle Interactions. Dhont24,25 has recently emphasized the importance of short-range particleparticle interactions on the thermodiffusion coefficient. Because our system may include long-range electrostatic interactions between particles, our experiments cannot directly address the predictions of Dhont’s theory. Nevertheless, we have tested the dependence of DT on the particle concentration cp; see Figure 8. The changes in DT for latex spheres in LiCl solutions of constant ionic strength are small and comparable to the uncertainties of the measurement; therefore, we conclude that the particle concentration does not play an important role in determining the thermodiffusion coefficient of these PS spheres when cp j 2 wt %. Intrinsic Thermodiffusion Coefficient of the Carboxyl Spheres. In low ionic strength LiCl electrolytes, j5 mM, where the thermally generated electric fields are particularly small, the thermodiffusion coefficient is independent of both ionic strength and particle concentration, DT ) -0.73 ( 0.05 × 10-7 cm2 s-1 K-1; see Figures 5 and 8. In addition, at low ionic strength, j2 mM, DT is not effected by Na or Li counterions; see Figure 5. We therefore refer to this value of DT for as-received spheres in water (or spheres in low ionic strength solutions of LiCl) as the intrinsic thermodiffusion coefficient of those spheres. To understand the origin of this intrinsic thermodiffusion coefficient, we consider the theory for thermophoretic transport originally proposed by Derjaguin26 and later reviewed by Anderson.19 In this theory, when the particle radius is large as compared to the thickness of the interfacial layer, the thermodiffusion coefficient of the particle is predicted to be

DT ) -

[

2Λl 2 η T 2Λl + Λp

]∫

∞

0

y hˆ (y) dy

(8)

where η is the viscosity of the fluid, Λp is the thermal conductivity of the particle, Λl is the thermal conductivity of the liquid, the integral is the 1st moment of the local specific enthalpy increment hˆ (y), hˆ (y) ) h(y) - h∞, and h(y) is the enthalpy density at a distance y from the particle surface.19 (36) Ashcroft, N. W.; Mermin, N. D. Solid State Physics; Saunders College Publishing: New York, 1976.

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Putnam and Cahill

Figure 7. Thermodiffusion coefficient DT of the as-received 26 nm carboxyl spheres as a function of the Seebeck coefficient of the electrolyte Se ) (Q/C - Q/A)/2eT, where Q/C and Q/A are the heats of transport for cationic and anionic species, respectively. The dashed line represents the independently measured electrophoretic mobility of the as-received spheres multiplied by Se; i.e., DT ) - Se µE, where the electric field generated by the thermodiffusion of the electrolyte is E ) Se∇T and µE ) -4.5 × 10-4 cm2 s-1 V-1.

Figure 8. DT as a function of particle concentration cp for the LiCl-dialyzed 26 nm carboxyl spheres at constant ionic strength (≈2.0 ( 0.3 mM). Table 2. Single Ion Heats of Transport at 10 mM and 25.3 °C, Taking Q/Cl ) 0 meV6 cation

Q/C (meV)

anion

Q/A (meV)

Li+ Na+ TEA+ H+

0.4 30 123 133

FClBrOH-

34 0 0.9 172

Anderson considered the case of a thermally insulating particle (Λp ) 0). Therefore, we have added the bracketed term in eq 8 to account for perturbations in the temperature field surrounding the particle due to differences between Λp and Λl.37 This correction does not change the prediction significantly for the case of PS in water: at ≈25 °C, ΛPS ) 1.6 × 10-3 W cm-1 K-1,38 Λwater ) 6.1 × 10-3 W cm-1 K-1,39 and the bracketed term is ≈0.88. Equation 8 suggests that DT is sensitive to the thermodynamics of solid-liquid interfaces. For example, if the particles are hydrophobic, meaning that the surrounding water has no chemical affinity for the particle surface, then hˆ (y) > 0 and thus DT < 0.19 At this qualitative level, DT for PS latex spheres corresponds well with the predictions of eq 8; that is, hydrophobic spheres in water have negative thermodiffusion coefficients (DT < 0). The intrinsic thermodiffusion coefficient for our 26 nm spheres corresponds to H ˆ ) ∫y hˆ (y) dy ≈ 6.1 meV nm-1. However, (37) Henry, D. C. Proc. R. Soc. A 1931, 133, 106. (38) Dashora, P.; Gupta, G. Polymer 1996, 37, 231. (39) Nikogosyan, D. N. Properties of Optical and Laser-Related Materials: A Handbook; John Wiley and Sons Ltd.: Chichester, 1997.

because DT for our PS spheres seemingly depends strongly on factors such as ionic strength, particle size, or even particle surface chemistry and structure, the hydrophobic argument alone is not sufficient; see Figures 3, 5, and 6. The difficulty in evaluating eq 8 quantitatively is to have an accurate model for hˆ (y). Many mechanisms could, in principle, contribute to hˆ (y); for example, polarization of water molecules by the electric fields of the double layer27,28,40 or chemical interactions with the surface might change the enthalpy significantly.28 The specific interactions of the ions and water molecules with the surface are difficult to evaluate reliably. We therefore in this publication examine only the electrostatic contributions to hˆ (y). The variations in the local enthalpy density due to the polarization of water molecules by the electric field of the double layer have been evaluated previously.28 In this case,

hˆ E(y) )

1 T ∂ 2 1+ E (y) 2  ∂T

(

)

(9)

where  ) r 0 is the dielectric constant of the liquid, E is the electric field, and [1 + (T/) (∂/∂T)] ≈ -0.40 for water at T ) 283-293 K.27,28 We approximate the electric field by E(y) ) κζexp(-κy) and extract an estimate for the ζ-potential ζ from our measurements of the electrophoretic mobility.41 This approach suggests that the electrostatic contribution to DT is determined by the ζ-potential of the suspension; i.e., H ˆ ) -0.05ζ2. H ˆ for the 26 nm carboxyl spheres at ≈2 mM is then H ˆ ≈ -1.4 meV nm-1, which corresponds to DT ≈ 0.17 × 10-7 cm2 s-1 K-1. Thus, the predicted thermodiffusion coefficient for the 26 nm spheres by this mechanism has the wrong sign and is ≈4 times less in magnitude than observed. Perhaps the greatest obstacle associated with achieving a theoretical description for the thermodiffusion of our latex spheres is the fact that DT appears to depend on particle size. DT decreases with increasing particle size for the carboxyl spheres with diameters 100 mM, the thermodiffusion coefficient (DT) of charged latex spheres in aqueous suspensions is controlled by the Seebeck coefficient of the electrolyte (Se) and can be accurately predicted if the electrophoretic mobility of the particle and the heats of transport of the anion and cation are known. In electrolytes of low ionic strength, however, thermodiffusion of latex spheres is a complex phenomenon: Particles with different surface chemistries and (40) Levine, S. Proc. Phys. Soc. A 1951, 44, 781. (41) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: New York, 2002.

Transport of Nanoscale Latex Spheres

diameters display a wide range of values for the thermodiffusion coefficient, -0.6 × 10-7 cm2 s-1 K-1< DT < 0. Electrolytes such as LiCl with small values of Se simplify the experimental behavior and enable controlled studies of the changes in DT with particle concentration and ionic strength, free from the effects of thermally generated electric fields in the electrolyte. We observe only small changes in DT with changes in particle concentrations in the range 0.6 < c < 2.0 wt %. Increasing the ionic strength quenches the thermodiffusion. The magnitude of thermodiffusion coefficient |DT| for 26 nm diameter latex spheres decreases by a factor of 2 at a LiCl ionic strength of I ) 20 mM. Unfortunately, we cannot provide even a qualitative explanation for the magnitude and sign of DT. The strong dependence of DT on the ionic strength of LiCl electrolytes argues against any explanation based solely on molecularscale interactions at the PS/water interface. Also, the negative value of DT does not appear to be consistent with

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a purely electrostatic mechanisms: This conclusion is based on approximate calculations of the changes in enthalpy of water molecules in the double layer and the well-established observation that individual ions generally have positive heats of transport. Thermodiffusion is nevertheless a sensitive tool for probing nanoparticle suspensions when compared to traditional methods of characterization by electrophoresis. For example, in the full spectrum of our experimental conditions, the electrophoretic mobility of the latex spheres varies by only (20% while the thermodiffusion coefficient spans positive and negative values over the entire range |DT| < 10-7 cm2 s-1 K-1. Acknowledgment. This work is based upon work supported by the STC Program of the National Science Foundation under Agreement No. CTS-0120978. LA047056H