Eu ophys. Le .,55 (3), pp. 425–431 (2001)
EUROPHYSICS LETTERS 1 Augus 2001
Modeling he elec opho e ic deposi ion
o colloidal pa icles
A. T. P´
e ez1, D. Sa ille2and C. So ia1
1Depa amen o de Elec ´onica y Elec omagne ismo, Facul ad de F´ısica
A da. Reina Me cedes s/n 41012 Se illa, Spain
2Depa men o Chemical Enginee ing, P ince on Uni e si y - P ince on, NJ 08544, USA
( ecei ed 18 Decembe 2000; accep ed in final o m 25 May 2001)
PACS. 82.45.-h – Elec ochemis y and elec opho esis.
PACS. 82.70.Dd – Colloids.
PACS. 47.65.+a – Elec ohyd odynamics and magne ohyd odynamics.
Abs ac . – This le e p esen s he esul s o nume ical simula ions o he buildup o a laye o
colloidal pa icles on an elec ode. In a low- equency elec ic field, pa icles suspended in a low-
conduc i i y liquid mig a e o one elec ode and hen o he o he . Du ing each cycle, deposi s
a e o med and dissipa ed. The cu en - ol age cha ac e is ics o he p ocess eflec p ope ies
o he suspension and he deposi ed laye . Using a flux co ec ed anspo (FCT) algo i hm,
he anspo equa ion o he pa icle phase is sol ed simul aneously wi h equa ions desc ibing
he elec ic field. The model ep oduces he main ea u es o he cu en - ol age ela ion.
In oduc ion. – Pa icles o a ious so s o en acqui e cha ge when imme sed in low-
conduc ing liquids. Al hough he ze a po en ials o such pa icles a e usually smalle han
hose encoun e ed in aqueous sys ems, he use o judiciously chosen addi i es con ibu es o
he cha ge on he pa icles and helps s abilize he suspension. Mo eo e , some su ac an
molecules o m in e se micelles ha s abilize ions and his, in u n, inc eases he conduc i i y
o he liquid. This leads o cha ged pa icles wi h mode a ely hick double laye s.
In his wo k we s udy he elec opho e ic deposi ion o colloidal pa icles suspended in
low-conduc ing liquids. The p oblem is ele an o undamen al as well as applied easons.
F om he undamen al poin o iew, he mechanics o pa icle agg ega ion is ela ed o he
in e pa icle o ces [1]. F om he applied poin o iew, elec opho e ic deposi ion is o en
used in liquid xe og aphy o inc ease image quali y [2].
In [1] a echnique o measu ing he cohesion o a laye o colloidal pa icles is desc ibed.
Acco ding o he p ocedu e, applica ion o a ce ain ol age causes pa icle buildup on one
elec ode. When he pola i y is e e sed and ol age inc eased, he laye b eaks. The ol ages
a which he laye is o med and b oken a e ela ed o he consolida ion and ensile s eng hs
o he laye . Submi ing he suspension o a e y low- equency al e na ing elec ic field causes
a laye o agg ega ed pa icles o o m, al e na i ely, on each elec ode. A hys e esis loop is
ob ained when he elec ic cu en is plo ed as a unc ion o ol age. This loop is associa ed
wi h he mig a ion o pa icles. In his le e we p esen a simple model ha ep oduces he
main ea u es o he mechanisms esponsible o he loop.
Model. – Conside a suspension o pa icles be ween wo pa allel elec odes a dis ance L
apa . The olume ac ion o pa icles is deno ed by φ. The elec opho e ic mobili y o he
pa icles is K(φ), so ha he ela i e eloci y o he pa icles o he liquid is =K(φ)E,E he
elec ic field. The conduc i i y σ(φ) is a unc ion o he olume ac ion. The elec odes a e
infini y la ge and we place he z-axis pe pendicula o hem in such a way ha z=0andz=L
c
EDP Sciences
426 EUROPHYSICS LETTERS
define he elec ode planes. A ime-dependen ol age, V( ), is applied ac oss he elec odes.
We use he ollowing scales: L o leng h, K0V0/L o eloci y, V0, he maximum applied
ol age, o elec ic po en ial, σ0V0/L o cu en densi y and V0/L2 o cha ge densi y. He e
is he dielec ic cons an and he e e ence alues o mobili y and conduc i i y co espond
o he limi s a infini e dilu ion: K0=K(0) and σ0=σ(0). The equa ion ha de e mines
he olume ac ion φ(z, ) o pa icles is, in non-dimensional o m [3]:
∂φ
∂ +∂
∂z(φ(1 −φ)K(φ)E)= 1
Pe
∂
∂z Kh(φ)∂(φZ(φ))
∂z ; (1)
he ac o (1 −φ) akes in o accoun he back flow o he fluid necessa y o compensa e
o he ad ance o pa icles owa ds he elec ode. The Pecle numbe Pe is K0V0/D0.The
pa icle diffusion coefficien depends on he olume ac ion, and can be ela ed o he osmo ic
p essu e as D(φ)=D0Kh(φ)d(φZ(φ))
dφwi h Z(φ) he non-dimensional osmo ic p essu e Z(φ)=
Π/(nkBT)(nis he numbe densi y o pa icles, kBBol zmann’s cons an and T he absolu e
empe a u e). Fo e y dilu e sys ems Z→1 and he diffusion coefficien is cons an , D=D0.
Bu as he olume ac ion ends o a ce ain maximum packing ac ion, he osmo ic p essu e
di e ges, eflec ing he ac ha he pa icles canno be u he comp essed.
The coefficien Kh(φ) is he hyd odynamic mobili y and, al hough i s ela ion o he elec ic
mobili y is s aigh o wa d in he case o ions, his is no he case o colloidal pa icles (1).
The elec ic field equa ions a e
dE
dz=ρ, τ
τ
∂ρ
∂ +d(σE)
dz=0,V( )=1
0
Edz, (2)
whe e we assume ha quan i ies a y only in zand E(z) s ands o he zcomponen o he
elec ic field. All he quan i ies appea ing in hese equa ions a e dimensionless. τ and τ
deno e, espec i ely, he elec ic elaxa ion ime o he liquid (τ =/σ0) and he ime o
fligh o pa icles be ween he elec odes in a e y dilu e suspension (τ =L2/K0V0).
We ocus on he p ocess o laye o ma ion, so he p oblem is one-dimensional and he
fields depend only on z. Acco dingly, he liquid mo ion is educed o he back flow −φ and
he equa ion o mo ion o he liquid need no be sol ed.
Fo a model suspension o TiO2pa icles in mine al oil, ypical alues o he quan i-
ies in ol ed a e [1]: =1.6×10−11 F/m, σ0=10
−8S/m, K0=3×10−10 m2/Vs and
D0=8×10−14 m2/s (no e ha K0/D0e/kBT). In he expe imen s ypical alues o V0
and La e 100 V and 1 mm, espec i ely. These gi e Pe = 400,000 and τ /τ =5×10−4.
The e o e, he diffusion e m in eq. (1) and he ime-depending e m in he cha ge conse a ion
eq. (2) a e negligible.
Wi h hese app oxima ions he equa ions become
∂φ
∂ +∂
∂z(φ(1 −φ)K(φ)E) = 0 (3)
and
j( )=σ(z)E(z),V( )=1
0
Edz, (4)
whe e j( ) defines he elec ic cu en densi y.
Be o e going u he , wo impo an poin s dese e a en ion. The fi s one conce ns he
diffusion e m in eq. (1). Fo la ge Pecle numbe s his e m is negligible, excep when he
osmo ic p essu e g adien becomes o o de Pe. This should be he case in he bounda y
(1)I we we e o s udy he sedimen a ion p oblem unde g a i y o ce, we would ha e =Kh(φ)U0wi h U0
he S okes eloci y.
A. T. P´
e ez e al.:Modeling he elec opho e ic deposi ion e c. 427
be ween he suspension and he agg ega ed laye . The e, he osmo ic p essu e inc eases o
p e en concen a ions beyond he maximum packing. Howe e , he ex en o he egion whe e
diffusion mus be aken in o accoun is o o de 1/Pe and we can sa ely di ide he domain
in o wo egions: he concen a ed laye wi h φ=φm, whe e φmis he maximum olume
ac ion, and he suspension wi h φ<φ
m. This can be easily accommoda ed by making
K(φ)→0 o φ→φm. Al hough he e is no physical eason o imposing his condi ion on
K, i assu es ha he ne flux o pa icles owa ds he concen a e is ze o and i p oduces he
co ec eloci y o he discon inui y. Also he de ails o he laye o ma ion a e insensi i e o
he pa icula o m o K(φ).
The second ema k conce ns he ime-dependen elec ical phenomena. Neglec ing he
fi s e m in he cha ge conse a ion equa ion is accu a e only i he empo al a ia ion o he
applied field is o o de o , o slowe han, he d i o pa icles be ween bo h elec odes. Fo
example, i he ol age is V( ) = sin(2π /T) he app oxima ion is alid only o T≥1. I we
we e in e es ed in as e ansien s, he empo al de i a i e mus be aken in o accoun .
Quali a i e beha io . – Conside eq. (3) wi h E= 1 and he ollowing ini ial and bound-
a y condi ions: φ(z=1, )=φm,φ(z, =0)=φ0i 0 <z<1. The ini ial condi ion
ep esen s a homogeneous dis ibu ion o pa icles h ough he whole gap.
The solu ion o eq. (3) is ob ained by he me hod o cha ac e is ics. Hence, φis cons an
along he lines dz
d =d (φ)
dφ,(5)
whe e he flux unc ion is (φ)=φ(1 −φ)K(φ). I is mo e con enien o e o mula e he
bounda y and ini ial condi ions p oblem as an ini ial alue p oblem [4]: φ(z, =0)=φ0i
0<z<1, φ(z, =0)=φmi 1 <z, φ(z, =0)=0i z<0.
Figu e 1 shows he cha ac e is ic lines. Since he olume ac ion becomes mul i- alua ed
a egions whe e he cha ac e is ics in e sec , shock-like discon inui ies de elop. These dis-
con inui ies mo e wi h eloci y gi en by Us=( (φ+)− (φ−))/(φ+−φ−), whe e he signs
±deno e he sides o he discon inui y. One discon inui y de elops a he bounda y be ween
he clea liquid below, and he suspension abo e. I mo es wi h eloci y U1= (φ0)/φ0=
(1 −φ0)K(φ0). The o he one is he bounda y be ween he laye o agg ega ed pa icles and
he suspension. This eloci y is nega i e, and i s absolu e alue is U2= (φ0)/(φm−φ0)=
(1 −φ0)φ0K(φ0)/(φm−φ0), whe e i has been used ha (φm)=0.
When bo h discon inui ies mee , a new (s a iona y) discon inui y o ms be ween he ag-
g ega e and he clea liquid. The final hickness o he agg ega e, δ =φ0/φm, is eached a
= c=1/(U1+U2). In he case K(φ)=1−φ/φmi is c=1/(1 −φ0). No e ha o <
c
he e a e h ee egions o cons an olume ac ion: he laye wi h φ=φm, he suspension
wi h φ=φ0and he liquid wi h φ= 0. Figu e 2 is a schema ic iew o hese egions.
Clea ly he ime o o ma ion o he laye is ela ed o he pa icle mobili y. Measu ing
he elec ic cu en du ing he ansien , i is possible, expe imen ally, o de e mine his ime
and he elec opho e ic mobili y. Va ying he ini ial olume ac ion e eals i s dependence
on φ. We ecall ha he s anda d echniques o measu ing he elec opho e ic mobili y a e
usually alid o dilu e suspensions. The measu emen o he elec ic cu en du ing he laye
o ma ion o diffe en olume ac ions p o ides an al e na i e echnique.
As a second p oblem conside he case when a laye has al eady been o med be ween z=0
and z=δ0and a cons an ol age is applied. In his case he ini ial condi ions a e: φ(z, =
0) = φmi 0 <z<δ
0,φ(z, =0)=0i δ0<z<1, φ(z, > 1)=φmi 1 <z,φ(z, =0)=0
i z<0, whe e 1=(1−δ0)/ (0) is he ime a which he fi s cha ac e is ic emana ing
om z=δ0 eaches he uppe elec ode. A his momen he laye s a s o build up a his
elec ode. Figu e 3 shows a schema ic iew o he cha ac e is ics. F om he poin z=δ0,
428 EUROPHYSICS LETTERS
z
z=1
= c
Fig. 1
V
liquid
suspension
agg ega e
Fig. 2
z
z=1
z=
δ
0
=
2
=
1
=
3
Fig. 3
Fig. 1 – Cha ac e is ic cu es and associa ed discon inui ies o he o ma ion o a laye om a
homogeneous suspension.
Fig. 2 – Schema ic ep esen a ion o he o ma ion o a laye unde an applied ol age.
Fig. 3 – Cha ac e is ic cu es and associa ed discon inui ies o he ans e o a laye om one
elec ode o he o he .
= 0 emana es a an o lines ha fill he gap be ween he cha ac e is ics, wi h slopes (0)
and (φm). This s uc u e is analogous o a a e ac ion wa e in gases. Time 2=δ0/ (φm)
co esponds o he a i al o he cha ac e is ic emana ing om z=δ0a he lowe elec ode.
F om he e a shock sepa a ing clea fluid om suspension de elops. On he o he hand, a shock
sepa a ing he laye o agg ega e om he suspension de elops om z=1a = 1.Now,
bo h discon inui ies a e cu ed, as a consequence o he con inuous a ia ion o he olume
ac ion. A a ce ain ime 3, o which he analy ical exp ession is a he cumbe some, bo h
discon inui ies me ge in he final one ha sepa a es he clea liquid om he agg ega e.
Mobili y and conduc i i y o concen a ed suspensions. – The elec opho e ic mobili y
o pa icles has been a subjec o esea ch o many yea s and exp essions o he mobili y
o sphe ical pa icles wi h double laye s o a bi a y hickness in e y dilu e suspensions a e
a ailable. Howe e , he e a e no simple closed o m exp essions o concen a ed suspensions.
Al hough many pape s deal wi h he elec opho e ic mobili y o concen a ed suspensions,
he e is no gene al ag eemen on he ela ion be ween mobili y and olume ac ion. I ap-
pea s ha hyd odynamic and elec ical in e ac ions balance each o he o he case o e y hin
double laye s [5,6]. In ha case he elec opho e ic mobili y will be insensi i e o pa icle con-
cen a ion up o mode a e alues o he olume ac ion. Fo human e y h ocy es, Zukoski and
Sa ille [7] ound ha his is he case up o olume ac ions close o 1 o de o mable pa icles.
The si ua ion is simila o he conduc i i y whe e, s ic ly speaking, he Maxwell exp es-
sion applies only o dilu e suspensions o uncha ged pa icles. The effec i e conduc i i y o
dilu e suspensions o cha ged pa icles has been add essed by Sa ille [8]. In his wo k he
effec o he coun e ions and non-specific adso p ion is aken in o accoun . The exp ession o
he conduc i i y is σ=σ0(1+αφ), whe e he coefficien α anges be ween 1 and 60 depending
on he ze a po en ial and he double-laye hickness. A ough es ima e o he conduc i i y o
he suspension is
σ=σ0+nczceKc+npzpeK0,(6)
whe e ncis he a e age numbe densi y o coun e ions, zc he alence o he coun e ions, Kc
hei mobili y, np he densi y o pa icles, zp he numbe o elemen a y cha ges pe pa icle
and K0, as abo e, he elec opho e ic mobili y o he pa icles a infini e dilu ion. Fo a
suspension o TiO2pa icles in mine al oil [1] σ0=10
−8S/m, Kc=5×10−10 m2/Vs,
K0=3×10−10 m2/Vs, np=3×1018φm−3,zc=1andzp= 30. In oducing nc=(zp/zc)np
A. T. P´
e ez e al.:Modeling he elec opho e ic deposi ion e c. 429
012345
z
0
1
0.1
0.2
0.4
0.4
0.3
0.3
0.2
0.1
0.0
0.0
0.0
0.4
0.1
0.1
Fig. 4 – Cu es o cons an olume ac ion o he ans e o a laye om one elec ode o he o he .
Fo 1 < <2 a egion wi h a con inuous a ia ion o φis obse able.
we ob ain he es ima e σ=σ0(1+1.2φ). We ha e o no e ha in his case we a e dealing wi h
a non-aqueous liquid and he alues o ze a po en ial ob ained in such sys ems a e smalle
han in aqueous sys ems.
In wha ollows we suppose ha bo h K(φ)andσ(φ) a e known. Lacking well-es ablished
exp essions, we ha e chosen simple, easonable o ms o hese unc ions. Since he conduc-
i i y has mo e influence on he cu en han he mobili y du ing pa icle sedimen a ion, we
paid mo e a en ion o he ole o he conduc i i y.
Nume ical solu ion. – We sol ed eq. (3) o he olume ac ion using a Flux Co ec ed
T anspo (FCT) algo i hm, which is widely used o sol e p oblems in fluid dynamics ha
in ol e shocks and discon inui ies. The e is a a ie y o FCT algo i hms and since ou p ob-
lem is a ansien , one-dimensional p oblem, we ha e used he one adap ed by Mo ow and
C am [9]. To simula e a condi ion o ze o flux a he bounda ies we ha e in oduced wo
alse nodes a each bounda y and imposed symme ical olume ac ions he e wi h espec
o z=0andz=1.
To compu e he elec ic field i is necessa y o e alua e he in eg al in eq. (4), aking in o
accoun he spa ial dependence o he conduc i i y induced by he a ia ion o he olume
ac ion. The in eg al is compu ed using a cubic spline in e pola ion o he alues o σ(φ(z)).
In all he compu a ions shown below, he dis ance be ween nodes is ∆z=0.01 and he ime
s ep used ∆ =0.001. The unc ions K(φ)andσ(φ) can be a ied as desi ed.
Deposi ion om homogeneous suspension. – To es o he o e all algo i hm, we sol ed
he p oblem o a homogeneous suspension subjec o a cons an ol age wi h K(φ)=1−φ/φm
and σ(φ) = 1. The maximum and ini ial alues o he olume ac ion a e φm=0.4and
φ0=0.1. The con ou plo o he olume ac ion as a unc ion o zand co esponds exac ly
o ha expec ed om fig. 1. The wo expec ed discon inui ies a e clea , one sepa a ing he
laye o he agg ega e om he uni o m suspension, he o he sepa a ing he suspension om
he clea liquid. Since c=1/(1 −φ0), we p edic a ime o 1.1 o he me ging o he wo
ini ial discon inui ies, in o al ag eemen wi h he nume ical solu ion. The final sedimen has
a hickness o 0.25, co esponding o φ0/φm.
T ansi o a laye be ween wo elec odes. – A second calcula ion simula es he ans e
o a laye o pa icles om one elec ode o he o he when a cons an ol age is applied.
Keeping he same mobili y dependence, we used σ=1+αφ0, wi h α= 5. The ini ial laye is
be ween z=0andz=0.25, wi h he maximum concen a ion φ=0.4. Figu e 4 is a con ou
plo o he olume ac ion in his case. As expec ed, wo cu ed discon inui ies me ge in o

430 EUROPHYSICS LETTERS
012345
j
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
Fig. 5
0 2 4 6 8 101214161820
z
0
1
0.0
0.1
0.0
0.0
0.0
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
0.4
0.0
0.0
0.4
0.4
0.1
0.1
0.1
0.3
Fig. 6
Fig. 5 – Elec ic cu en as a unc ion o ime o he ans e o he laye om one elec ode o
he o he .
Fig. 6 – Cu es o cons an olume ac ion du ing a cycle o a ime-dependen ol age V( )=
sin(2π /T).
he final s eady-s a e discon inui y, ha ep esen s he laye o pa icles o med on he o he
elec ode. A an o cons an olume ac ion cu es emana es om z=0.25 a = 0 acco ding
o he pic u e desc ibed abo e. This an o cu es esul s, o ∼1, in a con inuous a ia ion
o he olume ac ion be ween 0 and 0.4 h ough he gap. La e he laye begins o buil
on o he uppe elec ode and o =2.5 he final s eady-s a e is eached. Figu e 5 shows he
cu en densi y as a unc ion o ime. The cu en when he laye is o med is 1.2, acco ding
o eqs. (4). The maximum cu en eco ded is close o he alue 1.5, co esponding o a
homogeneous suspension. The coefficien αde e mines he con ibu ion o pa icles o he
cu en . The g ea e α, he highe he cu en peak. The elec ic field is highe whe e he
conduc i i y is lowe . When he laye is o med on o he uppe elec ode, he elec ic field
is cons an in he wo egions, he clea liquid egion and he laye , bu has diffe en alues
on each zone. The discon inui y o he elec ic field gi es place o a su ace cha ge densi y a
he in e ace. This su ace cha ge densi y will play a majo ole in he elec ohyd odynamic
s abili y o he ad ancing pa icle on .
Mo ion unde a low- equency al e na ing ol age. – Finally, we conside he simula ion
o a comple e cycle o sinusoidal ol age. This case is o in e es because his a ia ion o he
V( )
-1 0 1
j ( )
-1
0
1
Fig. 7
V (kV)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
I(
µ
A)
-0.50
-0.25
0.00
0.25
0.50
Fig. 8
Fig. 7 – Elec ic cu en as a unc ion o ol age du ing a pe iod o a low- equency elec ic field.
Fig. 8 – Measu ed cu en s. applied ol age o a xe og aphic ink. The pe iod o he signal is 100 s,
i s ampli ude 250 V. (Adap ed om [1].)
A. T. P´
e ez e al.:Modeling he elec opho e ic deposi ion e c. 431
elec ic cu en was used expe imen ally o de e mine he mechanical s esses o he agg e-
ga es [1,2]. S a ing wi h a laye o maximum olume ac ion on one elec ode, we apply a
ol age V( ) = sin(2π /T). A e he ini ial ansien , he laye will be ans e ed back and
o h om one elec ode o he o he . Figu e 6 shows cons an olume ac ion lines o a
comple e cycle. In his plo we used K(φ)=(1−φ/φm), σ=1+10φ,φm=0.4, δ0=0.25
and T= 20. The successi e o ma ion o he laye a bo h elec odes is clea in he figu e.
The dis inc ea u e om hese simula ions is ha he cha ac e is ics s a a = 0 wi h a
ze o slope, co esponding o a ze o elec ic field. Figu e 7 is a plo o he cu en as a unc ion
o he applied ol age. The loop obse ed in his cu e has he same quali a i e beha io as
hose obse ed expe imen ally (see fig. 8). When he ol age is inc easing, he laye is being
buil and an addi ional con ibu ion o he cu en is no iceable. Once he laye is o med,
he cu en is linea ly dependen on he ol age. When he ol age becomes nega i e, he
pa icles begin o mig a e owa ds he o he elec ode, making an addi ional con ibu ion o
he nega i e cu en . This con ibu ion disappea s once he laye is assembled.
The ampli ude o he loops is ela ed o he coefficien αin he conduc i i y unc ion.
Fo α= 1 he loop is sca cely no ed, whe eas o α= 10 i is clea ly isible. Taking in o
accoun ha he co ec ion o he liquid conduc i i y is αφ and ha φis o o de 0.2, his is
equi alen o saying ha , o ha e a no iceable effec on he cu en , he con ibu ion o he
pa icles o he effec i e conduc i i y mus be g ea e han he liquid conduc i i y i sel . Fo
TiO2pa icles in mine al oil used in [1] he loop is small, in ag eemen wi h ou es ima e o
α∼1 o his suspension. On he o he hand, he xe og aphic ink used in he same wo k
p esen s he loop plo ed in fig. 8, indica ing a g ea e alue o α o his suspension.
Conclusion. – We p esen ed a simple model o he elec opho e ic deposi ion o colloidal
pa icles. The model elies on he knowledge o he mobili y K(φ), and conduc i i y σ(φ).
The model ep oduces he loops in he cu en - ol age cha ac e is ics ha a e ob ained expe -
imen ally when a suspension o colloidal pa icles is subjec ed o a low- equency AC ol age.
To ob ain he wide loops obse ed expe imen ally, we showed ha he con ibu ion o he
pa icles o he conduc i i y o he suspension has o be g ea e han he liquid conduc i i y
i sel . I is also shown ha K(φ) is ela ed o he ime o o ma ion o a laye o pa icles
om a homogeneous suspension; we sugges how simple deposi ion expe imen s may be used
o de e mine his ela ionship. To b ing heo y and expe imen in o close ag eemen will
equi e be e knowledge o how mobili y and conduc i i y depend on olume ac ion.
∗∗∗
This wo k was ca ied ou wi h financial suppo om Minis e io de Ciencia y Tecnolog´ıa,
BFM2000-1056.
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