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Electrokinetic phenomena in a concentrated suspension of soft particles
Colloids and Surfaces A: Physicochemical and Engineering Aspects 195 (2001) 129– 134 www.elsevier.com/locate/colsurfa
Electrokinetic phenomena in a concentrated suspension of soft particles Hiroyuki Ohshima * Faculty of Pharmaceutical Sciences and Institute of Colloid and Interface Science, Science Uni6ersity of Tokyo, 12 Ichigaya Funagawara-machi, Shinjuku-ku, Tokyo 162 -0826, Japan
Abstract A theory of electrokinetics in concentrated suspensions of soft particles (i.e. polyelectrolyte-coated particles) is reviewed with particular emphasis on an Onsager relation between sedimentation potential and electrophoretic mobility. A more general Onsager relation is derived on the basis of the thermodynamics of irreversible processes, which is an extension to high potentials. Expressions for colloid vibration potential (CVP) and electrokinetic sonic amplitude (ESA) are also derived. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Soft particle; Electrokinetic phenomena; Concentrated suspension
1. Introduction Electrokinetic behaviors of soft particles (i.e. colloidal particles coated with a polyelectrolyte layer) immersed in an electrolyte solution are quite different from those of hard particles with no surface structures. The electrophoretic mobility of soft particles is shown to have a non-zero limiting value at very high electrolyte concentrations [1–6]. Recently, the author [7 – 9] has extended a theory of electrokinetics in a dilute suspension of soft particles [1 – 6] to cover the case of concentrated suspensions on the basis of Kuwabara’s cell model [10,11] and of the Navier – Stokes –Poisson – Boltzmann approach. In a previous paper [8], we have derived an Onsager * Tel.: +81-3-3260-4272, ext. 5060; fax: +81-3-3268-3045. E-mail address: [email protected] (H. Ohshima).
relation between sedimentation potential and electrophoretic mobility in concentrated suspensions of soft particles. In the present paper, we derive a more general Onsager relation on the basis of the thermodynamics of irreversible processes. By analogy between the sedimentation potential and colloid vibration potential (CVP), we derive expressions for CVP and electrokinetic sonic amplitude (ESA).
2. Fundamental equations Consider a concentrated suspension of spherical soft particles moving with a velocity U in a liquid containing a general electrolyte in an applied electric field E or a gravitational field g. We assume that the particle core of radius a is coated with an ion-penetrable layer of polyelectrolytes with a
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H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 129–134
130
thickness d. The polyelectrolyte-coated particle has thus an inner radius a and an outer radius b a+ d (Fig. 1). We employ a cell model [10,11] in which each particle is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius of c such that the particle/cell volume ratio in the unit cell is equal to the particle volume fraction throughout the entire dispersion, viz, =(b/c)3. The origin of the spherical polar coordinate system (r, q, ) is held fixed at the center of one particle and the polar axis (q = 0) is set parallel to E or g. Let the electrolyte be composed of M ionic mobile species of valence zi and drag coefficient ui (i =1, 2,…, be the concentration (number M), and let n i density) of the ith ionic species in the electroneutral solution. We also assume that fixed charges are distributed within the polyelectrolyte layer with a density of zfix. We adopt the model of Debye –Bueche [12,13] in which the polymer segments are regarded as resistance centers distributed in the polyelectrolyte layer, exerting frictional forces on the liquid flowing in the polyelectrolyte layer. To solve the electrophoresis, electrical conductivity, and sedimentation problems, we need the solutions to the following two problems: (1) steady motion of soft particles with velocity UE in an applied electric field E and (2) steady motion of soft particles with velocity USED in a gravitational field g. For spherical soft particles, UE and USED are, respectively, parallel to E and g. Problem (1) corresponds to the electrophoresis and conductivity problems, and problem (2) to the sedimentation problem. The fundamental electrokinetic equations can be expressed as L(Lh −u 2h) = G(r),
L(Lh)=G(r), Li =
a B r Bb
b Br B c
d 2u h dy zi i − i dr e r dr
u= k/p
(6)
where p is the viscosity, k is the coefficient characterizing frictional forces exerted on the liquid flow by the polymer segments in the polyelectrolyte layer, e is the elementary electric charge, mr is the relative permittivity of the solution, mo is the permittivity of a vacuum, y=e (0)/kT is the scaled equilibrium potential outside the particle core, k is Boltzmann’s constant and T is the absolute temperature. In Eqs. (1)–(3), functions i (r) and h(r) are related to the liquid velocity u(r) and the deviation lvi (r) of the electrochemical potential of ith ions due to E or g as
u(r)= (ur, uq, u ) 2 1 d = − hF cos q, (rh)F sin q, 0 r r dr lvi (r)= − zi ei (r)F cos q
(7) (8)
where F denotes E= E for problem (1) and g= g for problem (2). We denotei and h for problems ( j ) ( j= 1, 2) by (i j ) and h ( j ), respectively. According to Kuwabara’s cell model [11], we assume that at the outer surface of the unit cell (r= c) the liquid velocity is parallel to the electrophoretic velocity UE of the particle for problem (1) or to the sedimentation velocity USED for problem (2),
(1) (2) (3)
with L
2 d 1 d 2 d2 2 d r = 2+ − dr r dr r 2 dr r 2 dr
G(r)= −
e dy M 2 % z i n i exp( − zi y)i pr dr i = 1
(4) (5) Fig. 1. Spherical soft particles in concentrated suspensions.
u ·nˆ r = c− =
!
−UE cos q [problem (1)] −USED cos q [problem (2)]
(9)
o =
where nˆ is the unit normal outward from the surface of the particle core, UE = UE and USED = USED . We also assume that at the outer surface of the unit cell (r =c) the gradient of the electric potential is parallel to the applied electric field (which is absent for problem (2)), viz,
+
9l ·nˆ r = c− =
!
−E cos q 0
[problem (1)] [problem (2)]
(10)
l(r) being the deviation of electric potential (r) due to E or g and that the vorticity µ= e× u is zero at r =c. 3. Electrophoretic mobility The electrophoretic mobility v of the particle is defined by UE =vE. We derive an approximate mobility expression for soft particles in concentrated suspensions for the case where ub1, sb1, ud =u(b − a)1 and sd = s(b − a)1
(11)
and the relaxation effect can be neglected, that is, Dukhin’s number is small. The condition (Eq. (11)), which corresponds to the condition of thin Debye length and shielding length [12], is satisfied for most practical cases. We treat the case where the electrolyte is symmetrical with a valence z and bulk concentration n . In this case we obtain [7]
m m /s +DON/u d zfix v= r o o m f , + 2 1/sm +1/u a p pu
(12)
n n
with
d 2 1 a f , = 1+ a 3 2 b =
DON =
! " n ! " n
H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 129–134
2 1 1+ 3 2(1 + d/a)3
kT zfix ln + ze 2zen
!
(1 − ) 1− (1 + d/a)3
"n
zfix 2zen
2zen 1− zfix
2
zfix 2zen
zfix 2zen
131
1/2
+1
2
1/2
+1
(15)
where c (a/c)3 is the volume fraction of the particle core, DON is the Donnan potential in the polyelectrolyte layer, o is the potential at the boundary between the polyelectrolyte layer and the surrounding solution, which we call the surface potential of the soft particle, and
n
zfix 2zen
sm = s 1+
2 1/4
(16)
is the Debye–Hu¨ ckel parameter of the polyelectrolyte layer. It follows from Eq. (12) that v=
mrmo o/sm + DON/u zfix + 2, 1/sm + 1/u p pu
v=
2mrmo(1−) o/sm + DON/u zfix + 2, 3p 1/sm + 1/u pu
(17)
d a
da (18)
Note that the dependence of the mobility disappears for da and Eq. (17) agrees with the mobility expression for the dilute case [1–5]. This corresponds to the Smoluchowski limit sa1 for concentrated suspensions of hard spheres of radius a, in which case the volume fraction dependence of v also disappears [11].
4. Average electric current and potential We define the macroscopic average electric field E and current i in the suspension of Np soft particles in an electrolyte solution of volume V as E= −
1− 1 − c
3
zfix kT ln + ze 2zen
2
i=
1 V
1 V
&
&
9(r) dV = −
V
1 V
&
9l(r) dV
V
(19) i(r) dV
(20)
V
(13)
where i is the current density at position r. The electrical conductivity K* of the suspension (for problem (1)) is defined by
(14)
i= K*E
1/2
+1
(21)
H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 129–134
132
The electric field E is different from the applied field E and these two fields are related to each other by continuity of electric current, viz, i =K*E = K E
(22)
where
we find from Eq. (25) that the sedimentation potential ESED is given by M
(2) % z 2i n i C i /ui
ESED = E i = 0 =
4yNp i = 1 M V % z 2i n i /ui i=1
M
K = % z 2i e 2n i /ui
(23)
i=1
is the conductivity of the electrolyte solution (in the absence of the particles). If we further put Ci = −
g
c 2 di −i r 3 dr
(24)
(2) 3 M z 2i e 2n i Ci g = 3 % b K i=1 ui
which, for low potentials, reduces to [8] ESED = −
r=c
where we have denoted Ci for problems ( j ) ( j= 1, 2) by C (i j ), then i may be written as [14]
(28)
(1− c) (cDzc + sDzs) vg (1+ c/2) K
(29)
Here s is the volume fraction of the polyelectrolyte segments within the suspension, Dzc =
M
Á  (1) % z 2i n i C i /ui à à 4yN i = 1 p ÃE − Eà [problem (1)] K N Á V à à % z 2i n à i /ui Ä Å i = 1 i = Í M Á  (2) à % z 2i n i C i /ui Ã Ã Ä K ÃE − 4yNp i = 1 gà [problem (2)] M V 2 à à % z n /u i i i Ä Å i=1 5. Electrical conductivity For problem (1), combination of Eqs. (22) and (25) yields M
Á % z 2n C (1)/u  K* à 3c i = 1 i i i i à =Ã1+ 3 à N K a 2 à % z i n i /ui Ã Ä Å i=1
−1
(26)
We find that for low potentials, the leading term of Eq. (26) is given by [9] K* 1−c = K 1+ c/2
(27)
which is the Maxwell’s relation [15] with respect to the volume fraction c of the particle core.
6. Sedimentation potential For problem (2), by setting i equal to zero,
(25)
zc − zo and Dzs = zs − zo, where zc, zs and zo are, respectively, the mass densities of the particle core, the polyelectrolyte segments and the electrolyte solution. We show below that the Onsager relation (Eq. (29)) is the special case of the more general Onsager relation between sedimentation potential and electrophoretic mobility derived on the basis of the thermodynamics of irreversible processes [16–19]. According to the thermodynamics of irreversible processes, we may generally write i= L11E+ L12g j= L21E+ L22g
(30)
for the average electric current density i and the average mass flux j, where Lij are constants. It follows from the Onsager relation that L12 = L21. Further we may write (see Eq. 26 of Ref. [16]) j= U(cDzc + sDzs)
(31)
H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 129–134
From Eqs. (30) and (31), we immediately obtain (see Eq. 27 of Ref. [16])
U E
=
g=0
i (cDzc +sDzs)g
(32)
E = 0
On the left-hand side of Eq. (32), the particle velocity U is related to the electrophoretic mobility v as U =vE. On the right-hand side of Eq. (32), the current density i is the sedimentation current iSED, which is given by Eq. (25). Eq. (32) thus yields ESED = −
(cDzc +sDzs) E vg K E
(33)
Since the two fields E and E are related to each other by Eq. (22), Eq. (33) can further be written as ESED = −
(cDzc +sDzs) K* vg K K
(34)
For low potentials, by substituting Eq. (27) into Eq. (34), we obtain Eq. (29). We thus see that the general Onsager relation (Eq. (34)) reproduces Eq. (29) at low potentials. There is an alternative definition of electrophoretic mobility for concentrated suspensions [18]: UE =v*E, where E is the average electric field in the suspension. It follows from the continuity condition (Eq. (22)) of electric current, i.e. K E= K*E that v*/v = K*/K (in the dilute case, v*=v). In terms of v*, Eq. (34) is simply written as ESED = −
(cDzc +sDzs) v*g K
(35)
133
ing formula for the CVP in a concentrated suspension of spherical soft particles immediately results: (1− c) (cDzc + sDzs) vD( , )9p (1+ c/2) zoK (36) Here vD( , ) is the dynamic electrophoretic mobility of spherical soft particles (which depends on the frequency and on the particle volume fraction ) and 9p is the pressure gradient due to the sound wave. In the dilute case of spherical soft particles, the -dependence of the dynamic mobility of a soft particle is negligible when k b1 (where k =(i+ 1)( zo/2p)1/2) [20]. Also for typical spherical soft particles with d«a, the static mobility of soft particles is given by Eq. (17), which does not depend on the particle volume fraction . In such cases, vD( , ) may be replaced by the static mobility for the dilute case. Eq. (36) thus reduces to CVP =
CVP = ×
(1− c) (cDzc + sDzs) (1+ c/2) zoK
!
"
n
mrmo o/sm + DON/u z + fix2 9p p 1/sm + 1/u pu
(37)
The above result can also be applied to other electroacoustic phenomena. When an oscillating electric field E exp(− i t) is applied to a suspension of spherical soft particles (where is the angular frequency of the applied field and t is time), a macroscopic sound wave is generated in the suspension. The amplitude of this sound wave, which is called electrokinetic sonic amplitude (ESA), can be expressed as (1−c) (cDzc + sDzs)vD( ,c,s)E (1+ c/2) (38)
7. Colloid vibration potential (CVP) and electrokinetic sonic amplitude (ESA)
ESA = B
When a suspension of charged colloidal particles is irradiated with a sound wave, a macroscopic electric field is generated in the suspension. This field is called the colloid vibration potential (CVP). The mechanism of the generation of CVP is essentially the same as that for the sedimentation potential in a suspension of charged colloidal particles in a gravitational field. Thus, by analogy between the sedimentation potential and CVP [19], the follow-
where B is the instrument factor and E= E . For typical soft particles with da when k b1, Eq. (38) reduces to ESA = B ×
(1− c) ( Dz + sDzs) (1+ c/2) c c
!
"
n
mrmo o/sm + DON/u z + fix2 E p 1/sm + 1/u pu
(39)
134
H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 129–134
Note that for typical soft particles, the strong dependence of CVP and ESA on the volume fraction c of the particle core arises mostly from the conductivity ratio (1−c)/(1 +c/2) (or K*/K ) rather than the volume fraction dependence of the dynamic mobility itself.
References [1] [2] [3] [4] [5]
H. Ohshima, J. Colloid Interface Sci. 163 (1994) 474. H. Ohshima, Adv. Colloid Interface Sci. 62 (1995) 189. H. Ohshima, Colloids Surf. A 103 (1995) 249. H. Ohshima, Colloid Polym. Sci. 275 (1997) 480. H. Ohshima, in: H. Ohshima, K. Furusawa (Eds.), Electrical Phenomena at Interfaces, 2nd Edition, Marcel Dekker, New York, 1998 Ch. 2. [6] H. Ohshima, J. Colloid Interface Sci. 228 (2000) 190.
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
H. Ohshima, J. Colloid Interface Sci. 225 (2000) 233. H. Ohshima, J. Colloid Interface Sci. 229 (2000) 140. H. Ohshima, J. Colloid Interface Sci. 229 (2000) 307. S. Kuwabara, J. Phys. Soc. Japan 14 (1959) 527. S. Levine, G.H. Neale, J. Colloid Interface Sci. 47 (1974) 520. P. Debye, A. Bueche, J. Chem. Phys. 16 (1948) 573. J.J. Hermans, H. Fujita, Koninkl. Ned. Akad. Wetenschap. Proc. Ser. B 58 (1955) 182. H. Ohshima, J. Colloid Interface Sci. 208 (1998) 295. J.C. Maxwell, A Treatise on Electricity and Magnetism, vol. 1, Dover, New York, 1954 Art. 313. S.R. de Groot, P. Mazur, J.Th.G. Overbeek, J. Chem. Phys. 20 (1952) 1825. J.Th.G. Overbeek, J. Colloid Sci. 8 (1953) 420. A.S. Dukhin, V.N. Shilov, Yu. B. Borkovskaya, Langmuir 15 (1999) 3452. H. Ohshima, A.S. Dukhin, J. Colloid Interface Sci. 212 (1999) 449. H. Ohshima, J. Colloid Interface Sci. 233 (2001) 142.
Electrokinetic phenomena in a concentrated suspension of soft particles Hiroyuki Ohshima * Faculty of Pharmaceutical Sciences and Institute of Colloid and Interface Science, Science Uni6ersity of Tokyo, 12 Ichigaya Funagawara-machi, Shinjuku-ku, Tokyo 162 -0826, Japan
Abstract A theory of electrokinetics in concentrated suspensions of soft particles (i.e. polyelectrolyte-coated particles) is reviewed with particular emphasis on an Onsager relation between sedimentation potential and electrophoretic mobility. A more general Onsager relation is derived on the basis of the thermodynamics of irreversible processes, which is an extension to high potentials. Expressions for colloid vibration potential (CVP) and electrokinetic sonic amplitude (ESA) are also derived. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Soft particle; Electrokinetic phenomena; Concentrated suspension
1. Introduction Electrokinetic behaviors of soft particles (i.e. colloidal particles coated with a polyelectrolyte layer) immersed in an electrolyte solution are quite different from those of hard particles with no surface structures. The electrophoretic mobility of soft particles is shown to have a non-zero limiting value at very high electrolyte concentrations [1–6]. Recently, the author [7 – 9] has extended a theory of electrokinetics in a dilute suspension of soft particles [1 – 6] to cover the case of concentrated suspensions on the basis of Kuwabara’s cell model [10,11] and of the Navier – Stokes –Poisson – Boltzmann approach. In a previous paper [8], we have derived an Onsager * Tel.: +81-3-3260-4272, ext. 5060; fax: +81-3-3268-3045. E-mail address: [email protected] (H. Ohshima).
relation between sedimentation potential and electrophoretic mobility in concentrated suspensions of soft particles. In the present paper, we derive a more general Onsager relation on the basis of the thermodynamics of irreversible processes. By analogy between the sedimentation potential and colloid vibration potential (CVP), we derive expressions for CVP and electrokinetic sonic amplitude (ESA).
2. Fundamental equations Consider a concentrated suspension of spherical soft particles moving with a velocity U in a liquid containing a general electrolyte in an applied electric field E or a gravitational field g. We assume that the particle core of radius a is coated with an ion-penetrable layer of polyelectrolytes with a
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H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 129–134
130
thickness d. The polyelectrolyte-coated particle has thus an inner radius a and an outer radius b a+ d (Fig. 1). We employ a cell model [10,11] in which each particle is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius of c such that the particle/cell volume ratio in the unit cell is equal to the particle volume fraction throughout the entire dispersion, viz, =(b/c)3. The origin of the spherical polar coordinate system (r, q, ) is held fixed at the center of one particle and the polar axis (q = 0) is set parallel to E or g. Let the electrolyte be composed of M ionic mobile species of valence zi and drag coefficient ui (i =1, 2,…, be the concentration (number M), and let n i density) of the ith ionic species in the electroneutral solution. We also assume that fixed charges are distributed within the polyelectrolyte layer with a density of zfix. We adopt the model of Debye –Bueche [12,13] in which the polymer segments are regarded as resistance centers distributed in the polyelectrolyte layer, exerting frictional forces on the liquid flowing in the polyelectrolyte layer. To solve the electrophoresis, electrical conductivity, and sedimentation problems, we need the solutions to the following two problems: (1) steady motion of soft particles with velocity UE in an applied electric field E and (2) steady motion of soft particles with velocity USED in a gravitational field g. For spherical soft particles, UE and USED are, respectively, parallel to E and g. Problem (1) corresponds to the electrophoresis and conductivity problems, and problem (2) to the sedimentation problem. The fundamental electrokinetic equations can be expressed as L(Lh −u 2h) = G(r),
L(Lh)=G(r), Li =
a B r Bb
b Br B c
d 2u h dy zi i − i dr e r dr
u= k/p
(6)
where p is the viscosity, k is the coefficient characterizing frictional forces exerted on the liquid flow by the polymer segments in the polyelectrolyte layer, e is the elementary electric charge, mr is the relative permittivity of the solution, mo is the permittivity of a vacuum, y=e (0)/kT is the scaled equilibrium potential outside the particle core, k is Boltzmann’s constant and T is the absolute temperature. In Eqs. (1)–(3), functions i (r) and h(r) are related to the liquid velocity u(r) and the deviation lvi (r) of the electrochemical potential of ith ions due to E or g as
u(r)= (ur, uq, u ) 2 1 d = − hF cos q, (rh)F sin q, 0 r r dr lvi (r)= − zi ei (r)F cos q
(7) (8)
where F denotes E= E for problem (1) and g= g for problem (2). We denotei and h for problems ( j ) ( j= 1, 2) by (i j ) and h ( j ), respectively. According to Kuwabara’s cell model [11], we assume that at the outer surface of the unit cell (r= c) the liquid velocity is parallel to the electrophoretic velocity UE of the particle for problem (1) or to the sedimentation velocity USED for problem (2),
(1) (2) (3)
with L
2 d 1 d 2 d2 2 d r = 2+ − dr r dr r 2 dr r 2 dr
G(r)= −
e dy M 2 % z i n i exp( − zi y)i pr dr i = 1
(4) (5) Fig. 1. Spherical soft particles in concentrated suspensions.
u ·nˆ r = c− =
!
−UE cos q [problem (1)] −USED cos q [problem (2)]
(9)
o =
where nˆ is the unit normal outward from the surface of the particle core, UE = UE and USED = USED . We also assume that at the outer surface of the unit cell (r =c) the gradient of the electric potential is parallel to the applied electric field (which is absent for problem (2)), viz,
+
9l ·nˆ r = c− =
!
−E cos q 0
[problem (1)] [problem (2)]
(10)
l(r) being the deviation of electric potential (r) due to E or g and that the vorticity µ= e× u is zero at r =c. 3. Electrophoretic mobility The electrophoretic mobility v of the particle is defined by UE =vE. We derive an approximate mobility expression for soft particles in concentrated suspensions for the case where ub1, sb1, ud =u(b − a)1 and sd = s(b − a)1
(11)
and the relaxation effect can be neglected, that is, Dukhin’s number is small. The condition (Eq. (11)), which corresponds to the condition of thin Debye length and shielding length [12], is satisfied for most practical cases. We treat the case where the electrolyte is symmetrical with a valence z and bulk concentration n . In this case we obtain [7]
m m /s +DON/u d zfix v= r o o m f , + 2 1/sm +1/u a p pu
(12)
n n
with
d 2 1 a f , = 1+ a 3 2 b =
DON =
! " n ! " n
H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 129–134
2 1 1+ 3 2(1 + d/a)3
kT zfix ln + ze 2zen
!
(1 − ) 1− (1 + d/a)3
"n
zfix 2zen
2zen 1− zfix
2
zfix 2zen
zfix 2zen
131
1/2
+1
2
1/2
+1
(15)
where c (a/c)3 is the volume fraction of the particle core, DON is the Donnan potential in the polyelectrolyte layer, o is the potential at the boundary between the polyelectrolyte layer and the surrounding solution, which we call the surface potential of the soft particle, and
n
zfix 2zen
sm = s 1+
2 1/4
(16)
is the Debye–Hu¨ ckel parameter of the polyelectrolyte layer. It follows from Eq. (12) that v=
mrmo o/sm + DON/u zfix + 2, 1/sm + 1/u p pu
v=
2mrmo(1−) o/sm + DON/u zfix + 2, 3p 1/sm + 1/u pu
(17)
d a
da (18)
Note that the dependence of the mobility disappears for da and Eq. (17) agrees with the mobility expression for the dilute case [1–5]. This corresponds to the Smoluchowski limit sa1 for concentrated suspensions of hard spheres of radius a, in which case the volume fraction dependence of v also disappears [11].
4. Average electric current and potential We define the macroscopic average electric field E and current i in the suspension of Np soft particles in an electrolyte solution of volume V as E= −
1− 1 − c
3
zfix kT ln + ze 2zen
2
i=
1 V
1 V
&
&
9(r) dV = −
V
1 V
&
9l(r) dV
V
(19) i(r) dV
(20)
V
(13)
where i is the current density at position r. The electrical conductivity K* of the suspension (for problem (1)) is defined by
(14)
i= K*E
1/2
+1
(21)
H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 129–134
132
The electric field E is different from the applied field E and these two fields are related to each other by continuity of electric current, viz, i =K*E = K E
(22)
where
we find from Eq. (25) that the sedimentation potential ESED is given by M
(2) % z 2i n i C i /ui
ESED = E i = 0 =
4yNp i = 1 M V % z 2i n i /ui i=1
M
K = % z 2i e 2n i /ui
(23)
i=1
is the conductivity of the electrolyte solution (in the absence of the particles). If we further put Ci = −
g
c 2 di −i r 3 dr
(24)
(2) 3 M z 2i e 2n i Ci g = 3 % b K i=1 ui
which, for low potentials, reduces to [8] ESED = −
r=c
where we have denoted Ci for problems ( j ) ( j= 1, 2) by C (i j ), then i may be written as [14]
(28)
(1− c) (cDzc + sDzs) vg (1+ c/2) K
(29)
Here s is the volume fraction of the polyelectrolyte segments within the suspension, Dzc =
M
Á  (1) % z 2i n i C i /ui à à 4yN i = 1 p ÃE − Eà [problem (1)] K N Á V à à % z 2i n à i /ui Ä Å i = 1 i = Í M Á  (2) à % z 2i n i C i /ui Ã Ã Ä K ÃE − 4yNp i = 1 gà [problem (2)] M V 2 à à % z n /u i i i Ä Å i=1 5. Electrical conductivity For problem (1), combination of Eqs. (22) and (25) yields M
Á % z 2n C (1)/u  K* à 3c i = 1 i i i i à =Ã1+ 3 à N K a 2 à % z i n i /ui Ã Ä Å i=1
−1
(26)
We find that for low potentials, the leading term of Eq. (26) is given by [9] K* 1−c = K 1+ c/2
(27)
which is the Maxwell’s relation [15] with respect to the volume fraction c of the particle core.
6. Sedimentation potential For problem (2), by setting i equal to zero,
(25)
zc − zo and Dzs = zs − zo, where zc, zs and zo are, respectively, the mass densities of the particle core, the polyelectrolyte segments and the electrolyte solution. We show below that the Onsager relation (Eq. (29)) is the special case of the more general Onsager relation between sedimentation potential and electrophoretic mobility derived on the basis of the thermodynamics of irreversible processes [16–19]. According to the thermodynamics of irreversible processes, we may generally write i= L11E+ L12g j= L21E+ L22g
(30)
for the average electric current density i and the average mass flux j, where Lij are constants. It follows from the Onsager relation that L12 = L21. Further we may write (see Eq. 26 of Ref. [16]) j= U(cDzc + sDzs)
(31)
H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 129–134
From Eqs. (30) and (31), we immediately obtain (see Eq. 27 of Ref. [16])
U E
=
g=0
i (cDzc +sDzs)g
(32)
E = 0
On the left-hand side of Eq. (32), the particle velocity U is related to the electrophoretic mobility v as U =vE. On the right-hand side of Eq. (32), the current density i is the sedimentation current iSED, which is given by Eq. (25). Eq. (32) thus yields ESED = −
(cDzc +sDzs) E vg K E
(33)
Since the two fields E and E are related to each other by Eq. (22), Eq. (33) can further be written as ESED = −
(cDzc +sDzs) K* vg K K
(34)
For low potentials, by substituting Eq. (27) into Eq. (34), we obtain Eq. (29). We thus see that the general Onsager relation (Eq. (34)) reproduces Eq. (29) at low potentials. There is an alternative definition of electrophoretic mobility for concentrated suspensions [18]: UE =v*E, where E is the average electric field in the suspension. It follows from the continuity condition (Eq. (22)) of electric current, i.e. K E= K*E that v*/v = K*/K (in the dilute case, v*=v). In terms of v*, Eq. (34) is simply written as ESED = −
(cDzc +sDzs) v*g K
(35)
133
ing formula for the CVP in a concentrated suspension of spherical soft particles immediately results: (1− c) (cDzc + sDzs) vD( , )9p (1+ c/2) zoK (36) Here vD( , ) is the dynamic electrophoretic mobility of spherical soft particles (which depends on the frequency and on the particle volume fraction ) and 9p is the pressure gradient due to the sound wave. In the dilute case of spherical soft particles, the -dependence of the dynamic mobility of a soft particle is negligible when k b1 (where k =(i+ 1)( zo/2p)1/2) [20]. Also for typical spherical soft particles with d«a, the static mobility of soft particles is given by Eq. (17), which does not depend on the particle volume fraction . In such cases, vD( , ) may be replaced by the static mobility for the dilute case. Eq. (36) thus reduces to CVP =
CVP = ×
(1− c) (cDzc + sDzs) (1+ c/2) zoK
!
"
n
mrmo o/sm + DON/u z + fix2 9p p 1/sm + 1/u pu
(37)
The above result can also be applied to other electroacoustic phenomena. When an oscillating electric field E exp(− i t) is applied to a suspension of spherical soft particles (where is the angular frequency of the applied field and t is time), a macroscopic sound wave is generated in the suspension. The amplitude of this sound wave, which is called electrokinetic sonic amplitude (ESA), can be expressed as (1−c) (cDzc + sDzs)vD( ,c,s)E (1+ c/2) (38)
7. Colloid vibration potential (CVP) and electrokinetic sonic amplitude (ESA)
ESA = B
When a suspension of charged colloidal particles is irradiated with a sound wave, a macroscopic electric field is generated in the suspension. This field is called the colloid vibration potential (CVP). The mechanism of the generation of CVP is essentially the same as that for the sedimentation potential in a suspension of charged colloidal particles in a gravitational field. Thus, by analogy between the sedimentation potential and CVP [19], the follow-
where B is the instrument factor and E= E . For typical soft particles with da when k b1, Eq. (38) reduces to ESA = B ×
(1− c) ( Dz + sDzs) (1+ c/2) c c
!
"
n
mrmo o/sm + DON/u z + fix2 E p 1/sm + 1/u pu
(39)
134
H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 129–134
Note that for typical soft particles, the strong dependence of CVP and ESA on the volume fraction c of the particle core arises mostly from the conductivity ratio (1−c)/(1 +c/2) (or K*/K ) rather than the volume fraction dependence of the dynamic mobility itself.
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