Langmuir 2005, 21, 5317-5323
5317
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.; Köhler, 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.
5318
Langmuir, Vol. 21, No. 12, 2005
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;
Köhler, 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.
Langmuir, Vol. 21, No. 12, 2005 5319
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.; Köhler, W. Philos. Mag. 2003, 83, 1973.
5320
Langmuir, Vol. 21, No. 12, 2005
Putnam and Cahill
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,
Transport of Nanoscale Latex Spheres
Langmuir, Vol. 21, No. 12, 2005 5321
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
JNa
∇cNa
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 ĥ(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 ĥ(y), ĥ(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.
5322
Langmuir, Vol. 21, No. 12, 2005
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 ĥ(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 Ĥ ) ∫y ĥ(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 ĥ(y). Many mechanisms could,
in principle, contribute to ĥ(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 ĥ(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,
ĥE(y) )
1
T ∂ 2
E (y)
1+
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., Ĥ ) -0.05ζ2. Ĥ for the 26 nm carboxyl
spheres at ≈2 mM is then Ĥ ≈ -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 <50 nm; see
Figure 6. To our knowledge, all theoretical models, with
exception to description by Andreev,18 predict that DT
should be independent of particle radius. (In contrast with
the behavior of solid particles, the thermodiffusion coefficient of a liquid droplet suspended in a second fluid is
expected to increase linearly with particle radius and
linearly with the temperature dependence of the free
energy of the interface; see the introduction to ref 19. The
conclusion of Anderson’s article also contains a lengthy
discussion of why a simply thermodynamic analysis of
the free energy of the interface is not always valid for
phoretic transport of a solid particle.) Unfortunately, for
the data shown in Figures 3 and 6, we cannot clearly
distinguish between the effects of particle size from the
effects of particle surface structure and chemistry.
Conclusions
With large concentrations of monovalent salts, >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
Langmuir, Vol. 21, No. 12, 2005 5323
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