Collective thermoelectrophoresis of charged colloids

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by  Arghya Majee

Collective thermo-electrophoresis of charged colloids
Arghya Majee and Alois Wu¨rger
Laboratoire Ondes et Mati`ere d’Aquitaine, Universit´e Bordeaux 1 & CNRS,
351 cours de la Lib´eration, 33405 Talence, France
Thermally driven colloidal transport is to a large extent due to the thermoelectric or Seebeck
effect of the charged solution. We show that, contrary to the generally adopted single-particle
picture,thetransportcoefficientdependsonthecolloidalconcentration. Forsolutionsthataredilute
in the hydrodynamic sense, collective effects may significantly affect the thermophoretic mobility.
Our results provide an explanation for recent experimental observations on polyelectrolytes and
charged particles, and suggest that for charged colloids collective behavior is the rule rather than
the exception.
4
PACSnumbers: 66.10.C,82.70.-y,47.57.J-
1
0
2
I. INTRODUCTION macroions, collective effects may occur at low dilution,

n thatis, forparticledispersionwithnegligiblepairpoten-
a Transport in macromolecular or colloidal dispersions tial and polymer solutions where neighbor chains do not
J
is mainly driven by interface forces [1–4]. Because of the overlap.
9
2 rathershort-rangedflowpatterninducedinthesurround- Thermally driven motion of charged colloids is very
ing fluid, these forces do not result in hydrodynamic in- sensitivetothesolventcomposition. Frompreviouswork

] teractions,incontrasttodiffusionandsedimentation. As it emerges that two rather different mechanisms con-
t
f a consequence, nearby beads hardly see each other, and tribute to the velocity [15, 16],
o
their transport velocity is independent of concentration
s
. [5]; for the same reason, free-solution electrophoresis of
t
a polyelectrolytesdoesnotdependonthemolecularweight
m [6,7].Similarresultshavebeenobtainedforthermaldif-
u=−µ ∇T +µE. (2)
fusion of high polymers [8–11]. T
-
d Recent experiments on thermophoresis in charged col-
n loids, however, dress a rather different picture and in-
Thefirsttermarisesfromthelocalparticle-solventinter-
o
dicate that the single-particle description fails in several
c actions in a non-uniform temperature. As first pointed
[ instances: Contrary to expectation, the transport veloc- out by Ruckenstein [17], the temperature gradient de-
ity due to a temperature gradient,
formstheelectricdoublelayerandinducesapressuregra-
1
v dient opposite to ∇T. The resulting thermoosmotic sur-
u=−D ∇T, (1)
7 T face flow toward higher T drives the particle to the cold
5 side;theoverallpictureissimilartoelectroosmoticeffects
was found to depend on the volume fraction of particle
5 in an electric field [18]. The coefficient µ ∝εζ2/ηT de-
dispersions and on the chain length N of macromolecu- T
7 pends on the ζ-potential, and the solvent permittivity ε
. lar solutions: (i) Data on sodium polystyrene sulfonate
1 andviscosityη; differentprefactorsoccurinthelimitsof
(NaPSS) [12] and single-stranded DNA [13] at constant
0 smallandlargeparticles[17–25]. Thisformagreesrather
polymer content but variable N, reveal that the mobil-
4 well with the observed salinity dependence [26], yet fails
1 ity DT becomes smaller for larger molecules; e.g., in the in view of the strong variation with T reported for vari-
: rangefrom50to48000basepairs,thatofDNAdecreases
v ous systems [12, 13], thus suggesting the existence of an
by a factor of 5. These findings are obtained at low con-
i additional, so far poorly understood contribution to µ .
X centration where the molecular mean distance is much T
r larger than the gyration radius. (ii) Regarding particle The present work deals with the second term in (2),
a suspensions, experiments on 70-nanometer silica beads which accounts for the Seebeck effect of the charged so-
[14] and 26-nm latex spheres [15] in a weak electrolyte, lution or, in other words, for electrophoresis in the ther-
showthatatavolumefractionof2%, D issignificantly moelectric field E with the mobility µ = εζ/η. Due to
T
reduced with respect to the zero-dilution value. their temperature dependent solvation forces, ions mi-
In the present work we show that these experimental gratealongoroppositetothethermalgradient.Asacon-
findings arise from an interaction mechanism that has sequence, surface charges develop at the cold and warm
been overlooked so far, i.e., the collective thermoelectric boundaries of the vessel and give rise to a macroscopic
response of the composite system. By treating the salt electric field E = −ψ∇T/T; see Fig. 1. The thermopo-
ions and the dispersed colloid on an equal footing, we tential parameter ψ is related to the Seebeck coefficient
find that both the thermoelectric field and the mobil- S = −ψ/T; for electrolytes S attains values of several
ity D vary with the colloidal concentration. Depend- 100µV/K, which is by one to two orders of magnitude
T
ing on the electrolyte strength and the valency of the larger than in common metals [27].
2
II. THERMOPHORETIC MOBILITY A. The steady state
Eqs. (3) and (4) provide the currents as functions of
We consider a dispersion of negatively charged parti-
thegeneralizedthermodynamicforces,thatis,ofthecon-
cles or macromolecules of valency −Z and concentration
centration and temperature gradients [28]. We are inter-
n, in a monovalent electrolyte solution of ionic strength
ested in the steady state characterized by
n withaconstanttemperaturegradient∇T. According
0
to the general formulation of non-linear thermodynam-
J =0=J. (5)
ics the currents of colloid and salt ions are linear func- ±
tionsofgeneralizedforces[28];thelattercanbeexpressed For later use we give of a resulting relation for the elec-
through thermal and concentration gradients. The cur- tric field. Inserting the drift velocity (2) and superpos-
rent of colloidal macroions is given by ing the three equations (5) such that the concentration
gradients result in the gradient of the charge density,
∇ρ = e∇(n −n −Zn), and collecting terms propor-
+ −
tional to E and ∇T, one has
J =−D∇n+nu, (3)
2n α −2n α −ZnTµ /D∇T ∇ρ
E =e + + − − T + , (6)
where the first term on the right-hand side accounts for εκ2 T εκ2
normal diffusion and the second one for transport with with the shorthand notation κ2 = e2(n + n +
+ −
the drift velocity (2). ZnTµ /D)/εk T.
T B
Thedensitiesofsmallionsaccountforthecounterions Inordertodeterminethefourunknowns∇n ,∇n,E,
±
releasedbythecolloidalparticlesandtheaddedsalt. the thethreeequations(5)needtobecompletedbyafourth
salinity. The mobile ion currents condition; it is provided by Gauss law
divE =ρ/ε (7)
(cid:18) ∇T eE (cid:19) which relates E and the charge density ρ=e(n+−n−−
J =−D ∇n +2n α ∓n (4)
± ± ± ± ± T ±k T Zn), and thus closes the above set of equations.
B
comprise normal diffusion with coefficients D , thermal
± B. Small-gradient approximation
diffusionwiththereducedSoretparametersα ,andelec-
±
trophoresiswiththeHu¨ckelmobilityformonovalentions.
The above Eqs. (5) and (7) are non-linear in the con-
In (3) and (4) we have added an electric field term; it is
centrationsandthuscannotbesolvedastheystand. The
important to note that E is not an external field but
arises from the kinetics of the mobile charges and is pro-
portional to the applied temperature gradient. A similar
phenomenon occurs in a non-uniform electrolyte, where
the electric field is proportional to the salinity gradient
andtothedifferenceoftheionicdiffusioncoefficientsD
±
[2, 29].
Thenumbersα describethedriftofpositiveandneg-
±
ative salt ions in a temperature gradient. The values for
the most common ions have been determined by Agar
from thermopotential measurements of electrolyte solu-
tions [31]; our notation and Agar’s “heat of transport”
Q∗ are related through α =Q∗/2k T. Typical values
± ± ± B
range from α≈0 for Li+ to α≈3 for OH−; those of the
FIG. 1: Thermoelectric effect in a colloidal suspension of
most common ions are given in Table I. charged particles in salt solution. In the example presented,
theSoretparametersaresuchthatnegativeandpositiveions
accumulate at the cold and warm boundaries, respectively.
In the left panel, vertical dashed lines indicate the thick-
TABLEI:ReducedSoretcoefficientα ofseveralsaltionsat ness of the surface layers of about one Debye length λ. This
±
room temperature. The values of the heat of transport Q∗ schematicviewexaggeratesthesurfacelayers,whicharemuch
±
are taken from Ref. [31]. The parameters α are calculated thinner in real systems. The right panel shows the spatial
±
from α =Q∗/2k T. variation of the net charge density ρ, the thermoelectric field
± ± B
E,andthethermopotentialU;dashedlinesindicatethezero
Ion H+ Li+ K+ Na+ OH− Cl−
oftheordinate. Notethenon-zerosurfacechargesatthecold
Q∗i (kJ/Mol) 13.3 0.53 2.59 3.46 17.2 0.53 and hot boundaries. The present paper discusses the bulk
α 2.7 0.1 0.5 0.7 3.4 0.1 behavior only, where ρ=0 and where E is constant.
i
3
salt and colloid concentrations vary very little through D. Zero-dilution limit
the sample; the relative changes δn/n and δn /n be-
± ±
tween the hot and cold boundaries are proportional to We briefly discuss the case of a very dilute suspension
the reduced temperature difference δT/T. Since in ex- where the colloidal charges are negligible for the electro-
periment, the ratio δT/T is much smaller than unity, we static properties. Putting n → 0 in the electric field (8)
may safely replace the concentrations n and n± in the we have ψ0 =−(α+−α−)kBT/e and
coefficients of (6) with constants n¯ and n¯ ; the latter
±
k ∇T
are defined as the colloidal and salt concentrations at E =(α −α ) B .
∇T =0. 0 + − e
Formally, this small-gradient approximations corre- This expression has been used previously in [15, 16, 30,
spondstoneglectingtermsthatarequadraticinthesmall 32]. Note that the parameter κ−1 reduces to the usual
quantities ∇n±, ∇n, E, and ∇T. This approximation exprerssion of the Debye screening length.
has been used, more or less explicitly, in previous works InsertingthethermoelectricfieldE inthedriftvelocity
onthethermoelectriceffect[30,31]andinrecentapplica- (2) and comparing with (2) defines the thermophoretic
tions in colloidal thermophoresis [15, 16, 32]. Moreover, mobility
various works on the osmotic flow driven by externally
εζψ
imposed gradients of charged solutes resorts to the same D0 =µ + 0. (9)
approximation, albeit with the salinity change ∇n in- T T ηT
0
stead of the temperature gradient [2, 29, 33]. Not surprisingly it is independent of the colloidal con-
centration. Theparameterψ andthemacroscopicther-
0
mopotential U =ψ δT/T between the hot and cold ves-
0
C. Bulk thermoelectric field sel boundaries, are given by the steady state of the elec-
trolyte solution. With the numbers of Table I, one finds
the values ψ = −15 mV and +70 mV for NaCl and
The above relations (5)-(7) describe both the bulk 0
NaOH solutions, respectively. Thus one expects D0 to
properties of a macroscopic sample and boundary effects T
change its sign upon replacing one salt by the other [16].
such as the surface charges that develop at the hot and
Thisisconfirmedbyaveryrecentstudyonsodiumdode-
cold boundaries; see Fig. 1. The thickness of the surface
cylsulfate(SDS)micelles, wherethe electrolytecomposi-
layer is given by the Debye length and thus in the range
tion NaCl OH was varied at constant ionic strength
between one and hundred nanometers. This is much 1−x x
[32]; increasing the relative hydroxide content x from 0
smaller than the sample size. Thus we discard surface
to 1 resulted in a linear variation of the Soret coefficient
effects and discuss the bulk behavior only; a full evalua-
S and a change of sign at x≈ 1 [32].
tion including surface effects is given in the Appendix. T 2
In a macroscopic sample the net charge density van-
ishes because of the huge electrostatic energy. With
E. Collective effects on the electric field E
ρ =0,
bulk Now we derive the main result of this paper, that is,
the dependence of E and D on the colloidal concentra-
Gauss’ law imposes (7) imposes a constant electric field; T
tion and, in the case of polyelectrolytes, on its molecular
its explicit expression is readily obtained from (6)
weight. Astwoimportantparameterswedefinetheratio
of the colloidal charge density and the salinity,
∇T
E =−ψ , (8)
T
Zn¯
φ= , (10)
with the shorthand notation for the coefficient of ∇T/T n
0
and the ratio of colloidal electrophoretic mobility µ and
2n¯ α −2n¯ α −Zn¯Tµ /D
ψ =−e + + − − T diffusion coefficient D,
εκ¯2
k T |µ|
and κ¯2 =e2(n¯ +n¯ +Zn¯Tµ /D)/εk T. Note that we ξ = B . (11)
+ − T B e D
haveusedthesmall-gradientapproximationandreplaced
In the following we assume a negative surface poten-
the colloidal and ion concentrations with their mean val-
tial. For typical colloidal suspensions, the charge ratio
ues.
is smaller than unity, φ ∼ 0.1, whereas the parameter ξ
Although it is not always mentioned explicitly, the
may exceed 102.
argument of zero bulk charge density has been used
Rewriting the coefficient ψ in (8) in terms of the di-
in previous works on the Seebeck effect of electrolytes
mensionless quantities φ and ξ, we have
[15, 16, 30–32] and, more generally, for colloidal trans-
port in non-equilibrium situations involving thermal or 2(1+φ)α −2α −φTµ /Dk T
ψ =− + − T B . (12)
chemical gradients [2, 29, 33]. 2+φ+φξ e
4
Eq. (12) shows how the thermoelectric field arises from III. COMPARISON WITH EXPERIMENT
the competition of the Soret motion of the mobile ions
and the colloidal solute. In the low-dilution limit φ→0 We discuss Eq. (13) in view of recent experiments on
the first term in the numerator reduces to (α+ − α−) colloidal suspensions. At relevant values of the charge
which corresponds to the response of the electrolyte so- ratio (φ∼0.1) the numerator hardly differs from that of
lution discussed in previous work [15, 16, 32]. the dilute case. Thus in the following we focus on the
The φ-dependent term in the numerator becomes rel- reduction of D due to the denominator.
T
evant where φ∼D/Tµ and, in particular, may change
T
the sign of ψ and thus of the thermoelectric field. With
typicalvaluesTµT ∼10−9 m/s2 onehasD/TµT =10−3 A. Polyelectrolytes
for micron-size particles (and polyelectrolytes of a gy-
ration radius.of 1 µm), and D/Tµ = 10−1 for 10-
T We start with experimental findings on polyelec-
nanometer beads. This means that, at typical colloidal
trolytes at constant volume fraction but variable molec-
densities, the thermoelectric field is essentially deter-
ular weight. In their study of 2 g/l of NaPSS in a 100
minedbythemacroions. Thedenominatorin(12)results
mM/l NaCl solution, Iacopini et al. found a significant
inanoveralldecreasewhenaugmentingthecolloidalcon-
variation with the chain length [12]: Fig. 2a shows the
centration. data measured at 30◦ C for molecules of 74, 160, and
360repeatunits,withanoveralldecreaseofthemobility
by 40 percent. The same factor has been found in the
F. Collective effects on the mobility D
T temperature range from 15 to 35◦ C.
The solid line represents collective effects arising from
Now we determine the steady-state thermophoretic
the denominator of Eq. (13). It has been calculated
mobility. Plugging the value of the electric field E given
with the double-layer term in the small-bead limit, as-
in (8) into the drift velocity (2) and comparing with (1),
suming the monomer to be small as compared to the
we get
Debye length (R<λ) [22–24],
D0
D = T . (13) dε ζ2
T 1+ φ ξ µ =− ,
2+φ T dT 3η
where D0 is defined by Eq. (9) albeit with a modified
T and with the Hu¨ckel-limit electrophoretic mobility µ =
parameter 2εζ/η. InsertingthediffusioncoefficientD =k T/6πηR
3 B
(1+φ)α −α k T andtheBjerrumlength(cid:96) =e2/4πεk T in(11),wehave
ψ =− + − B . (14) B B
0 1+φ/2 e
e|ζ| R
ξ = . (15)
The mobility and its dependence on the ratio φ consti-
k T (cid:96)
B B
tute the main result of this paper. According to (9), the
sign of D is determined by the competition of the bare The theoretical curve of Fig. 2a is calculated with the
T
mobility µ and the Seebeck term proportional to ζψ . parameters ζ = −27 mV, nN = 10 mM/l, and φ =
T 0
Sinceφ<1inmostcases,thenumeratorof(13)israther 0.1. Its variation arises only from the gyration radius
similar to the dilute case discussed above (9). R=(cid:96)N1−νNν; we have used the usual exponent ν = 3,
K 5
A much more striking variation arises from the de- the size of a monomer (cid:96) = 0.4 nm, and the number of
nominator of (13). For typical values of the charge ratio monomers per segment NK = 10. The dashed line in-
φ∼0.1, collective effects set in where 1φξ ∼1, in other dicates the mobility in the short-chain limit. The theo-
2
words where ξ is of the order of 20. For high polymers reticalexpression(13)providesagooddescriptionofthe
(N =103...106) and colloidal particles in the range from reduction of DT with increasing chain length.
ten nanometers to a micron, the parameter ξ takes val- Asasecondexample,DNAin1mM/lTrisbuffershows
ues between 10 and 103. This simple estimate suggests a similar behavior; its mobility decreases by a factor of
collective effects to occur in many systems. A detailed 5 over the range from N =50 to 48500 base pairs per
comparisonwithexperimentisgiveninthefollowingsec- molecule [13]. The overall DNA content was kept con-
tion. stant, Nn=50µM/l, with a charge ratio φ=0.05. Eqs.
Inthelimitofzerodilutionφ→0onereadilyrecovers (13)and(15)provideagoodfittothesedata,albeitwith
theexpression(9). Theoppositecaseofasaltfreesystem a somewhat too small exponent ν ≈ 0.4. In view of this
leads to discrepancy one should keep in mind the rather complex
electrostatic properties of polyelectrolytes.
D0
D = T , (φ→∞) ThereductionobservedforbothNaPSSandDNAcan-
T 1+ξ
notbeexplainedbyhydrodynamiceffects. Interchainin-
with ψ determined by the counterions only. In view teractions are of little significance because of the low di-
0
of the large values of ξ mentioned above, one expects a lution. Indeed, the effective volume fraction of the poly-
strong reduction of the mobility in the salt free case. mercoilshardlyattainsafewpercent, nR3 ∼10−2; thus
5
nearbychainsdonotoverlapandleaveboththeviscosity colloidal volume fraction; at 3% D is reduced to less
T
and the diffusion coefficient unchanged. Regarding hy- than half of the zero-dilution value. The negative sign of
drodynamic interactions of beads of the same molecule, the measured D indicates that the thermoelectric con-
T
it is known that they enhance the electrophoretic mobil- tributionεψ ζ/ηT to(9)overtakestheRuckensteinterm
0
ity in (2) and (9) with increasing chain length. Yet this [16]
effect occurs for short polyelectrolytes and saturates for
chainslongerthanthesizeofthescreeningcloud[35];for εζ2
µ = .
the examples studied here, it would enhance D in the T 3ηT
T
range N < 40. We conclude that hydrodynamic effects
The negative surface potential ζ implies that the ther-
mayruledoutasanexplanationforthereductionshown
mopotentialparameterofthesulpho-rhodaminesolution
inFig. 2a. Finallywediscusselectrostaticsingle-particle
is positive, ψ >0.
effects. The electrophoretic mobility in saltfree solution 0
The curves in Fig. 2b are calculated from (11) and
has been found to decrease slightly at higher concentra-
(13) with ψ =10 mV, which is comparable to common
tion, because of the increase of the overall ionic strength 0
saltsandweakerthanthevaluesofNaOHandtetraethy-
and the shorter screening length [36, 37]. In the present
lammonium [15, 32]. The rather small D suggests that
case, however, the weight fraction of the polyelectrolyte T
the particles are weakly charged; we use Z = 30 and
is constant, and so is the overall charge density. Thus
ζ = −10 mV. The dashed line gives the mobility D0 in
the electrostatic properties of the solution are the same T
the zero-dilution limit, whereas the solid lines are given
for different chain lengths.
by (13).
In addition to the explicit concentration dependence
in terms of the parameter φ, one has to take into ac-
B. Colloidal particles
count that, even at moderate colloidal volume fraction,
the Einstein coefficient D is not constant. Indeed, co-
Now we discuss the concentration dependent mobility
operative diffusion of charged particles arises from the
D that has been reported for dispersions of solid parti-
T electrostatic pair potential Φ(r) and, to a lesser extent,
cles inweak electrolytes. Ghofraniha etal. studied silica
from hydrodynamic interactions [38]. To linear order in
particles (R = 35 nm) in a 30 µM/l solution of the neg-
the concentration, the virial expansion for the Einstein
atively charged dye sulpho-rhodamine B [14]. The data
coefficient reads as
shown in Fig. 2b reveal a significant decrease with the
D =D (1+2nB), (16)
0
with the parameter
1(cid:90) (cid:16) (cid:17)
B = dV 1−e−Φ/kBT .
2
Forhardspheresthevirialcoefficientisgivenbythepar-
ticle volume, B =4V with V = 4πR3. The electrostatic
3
pair potential results in an effective interaction volume
V = 4π(R+ χλ)3, where λ is the Debye length and χ a
3 2
numericalfactor[26,32,34];forsmallandhighlycharged
particles in a weak electrolyte, the repulsive forces may
enhancethevirialcoefficientbyoneortwoordersofmag-
nitude. On the other hand, hydrodynamic interactions
contribute a negative term B/V ∼ −6.5 and reduce the
Einstein coeffient accordingly [38]. Our discussion of the
data of Ref. [14] is restricted to volume fractions up to
3%; at higher concentration the measured D saturates
FIG. 2: Comparison with measured data. (a) Variation
andthelinearapproximationceasestobevalid. Inunits
of D with the chain length N of a polyelectrolyte at fixed
T of the particle volume V, the measured virial coefficient
volume fraction. The data on 2 g/l NaPSS in a 100 mM/l
NaCl solution at 30◦ C are taken from Iacopini et al. [12]. reads B/V =20 [14]; the best fit of the mobility data is
The solid line is calculated from Eq. (13) with the parame- obtainedwithB/V =14. Thisvalueismuchlargerthan
ters as given in the main text. The dependence on N arises that of hard spheres and thus indicates the importance
from the gyration radius R. (b) Volume fraction dependence of electrostatic repulsion. The concentration of mobile
of D of a dispersion of 70-nm silica beads in a solution of charge carriers n = 30µM/l leads to a screening length
T 0
30µM/l sulpho-rhodamine B. The data are from Ghofraniha of about 50 nm. With χ∼2 in the above expression for
etal. [14];thefitcurvesareobtainedfrom(13)and(16),with the effective volume, one finds a virial coefficient close
different values of the reduced virial coefficient B/V, where
to the measured value. As an illustration of the effect
V = 4πR3 is the particle volume.
3 of collective diffusion on DT, we plot Eq. (13) for these
6
three values: Though the variation of D with B is not R. Piazza, D. Braun, and N. Ghofraniha are gratefully
T
neglegible,itissignificantlyweakerthanthatofthether- acknowledged.
moelectric effect. V. APPENDIX
As a second experiment we mention data by Putnam
and Cahill on latex beads of radius R = 13 nm in an The thermoelectric field (8) has been derived by us-
electrolytesolutionof2mM/lionicstrength[15]; varying ing the charge neutrality of the bulk of a macroscopic
the volume fractions from 0.7 to 2.2 wt%, these authors sample. Here we give a derivation based on the steady
observedareductionofDT byabout10percent. Witha state, Gauss’ law, and the electrostatic boundary condi-
valency of Z ∼50 one finds that, at the highest particle tions. Resortingtothesmall-gradientapproximation,we
concentration n = 4µM/l, the charge ratio φ does not replace the coefficients in (6) by their mean values and
exceed 10 percent. thus have
Finally we address the concentration dependence ob-
served by Guarino and Piazza for the Soret coefficient ∇T ∇ρ
E =−ψ + .
S = D /D of SDS micelles [26]. Its decrease with the T ε κ¯2
T T
SDS content, is well described by collective diffusion ac-
FromGauss’slaw(7)onehas ∇ρ/ε=∇2E andthusob-
cording to (16). In a very recent measurement, Vigolo
tains a differential equations for the thermoelectric field
etal. varytheelectrolytecompositionNaCl OH and
1−x x
E with a constant inhomogeneity −(ψ/T)∇T,
thusthethermaldiffusionparameteroftheanionin(14),
α = (1−x)α +xα [32]. The observed linear de-
− Cl OH ∇2E ∇T
pendenceofS onxconfirmsthecrucialroleofthether-
T E− =−ψ .
mopotential. Unfortunately there are no mobility data κ¯2 T
formicelles;thusatpresentitisnotpossibletodetermine
The solution E = E +E consists of two contribu-
whether their D is subject to collective effects similar inh h
T tions. The inhomogeneous term E = −(ψ/T)∇T ac-
to those of polyelectrolytes and solid beads. inh
countsforthemacroscopicSeebeckeffect. Theremaining
one E is related to surface charges at the cold and hot
h
boundaries of the sample. The homogeneous equation
IV. SUMMARY AND CONCLUSION
∇2E =κ¯2E is solved by the exponential function,
h h
In summary, charged colloids in a non-uniform tem- E =A eκ¯z+A e−κ¯z,
h + −
perature show collective transport behavior mediated by
the Seebeck effect of both colloidal and salt ions. For where z is the coordinate in the direction of the temper-
large particles and macromolecules, cooperative effects ature gradient. Its range is −1L ≤ z ≤ 1L with the
2 2
set in at rather low concentration, where hydrodynamic sample size L.
interactions are absent and where the charge ratio φ is Theelectrostaticboundaryconditionsrequirethatthe
much smaller than unity. The criterion for the onset of electric field vanishes at z = ±1L. Putting E = 0 and
2
collective behavior, φξ ∼ 1 in (13), involves the ratio solving for the coefficients of E , one readily finds A =
h ±
of the electrophoretic mobility and the Einstein coeffi- −1E /cosh(κ¯L/2) and the thermoelectric field
2 inh
cient; by contrast, the criterion for cooperative diffusion,
(cid:18) (cid:19)
Bn∼1,dependsonthepairpotentialofthesoluteparti- ψ cosh(κ¯z)
E =− ∇T 1− .
cles. The discussed examples suggest that the collective T cosh(κ¯L/2)
thermoelectric effect is generic for colloids at ordinary
concentrations. Thisissuecouldberelevantformicroflu- BothE andthecorrespondingchargedensityρareillus-
idic applications of thermophoresis. trated in the right panel of Fig. 1. The field vanishes
Weconcludewitharemarkonthethermoelectricfield attheboundariesandreachesitsconstantbulkvalue(8)
giveninEq. (12). Bothitsmagnitudeanditssigncanbe within a few screening lengths κ¯−1. The parameter κ¯−1
tuned by chosing the appropriate electrolyte and adjust- takes values in the range between 1 and 100 nanometers
ingthechargeratio. Withathermalgradientoflessthan andthusismuchsmallerthanthesizeofsampleL. Even
oneKelvinpermicron, E mayattainvaluesof100V/m. in microfluidic devices, κ¯L is in general larger than 103.
Thusthethermoelectriceffectcouldbeusedforapplying In real systems a more complex picture may emerge
electric fields in microfluidic devices. Local laser heating from the surface roughness of the boundaries, the solute
would permit to realize almost any desired spatiotempo- size, and surface charges of other origin. Note that such
ral electric-field pattern. additional effects do not affect the bulk electric field (8)
Helpful and stimulating discussions with D.G. Cahill, and thus are irrelevant for the results of this paper.
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