LOW-FREQUENCY DIELECTRIC DISPERSION IN
COLLOIDAL SUSPENSIONS OF UNCHARGED
INSULATING PARTICLES
C. Grosse1, J.J. López-García2, J. Horno2.
1
Departamento de Física, Universidad Nacional de Tucumán and
Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina.
2
Departamento de Física, Universidad de Jaén, Jaén, Spain.
1
INTRODUCTION
Theory of the dielectric response of suspensions of uncharged spherical
particles:
J.C. Maxwell
static conductivity
["Electricity and Magnetism", Clarendon Press, Oxford, Vol. 1, 1892]
K.W. Wagner
complex permittivity
[Arch. Elektrotech. 2, 371 (1914)]
C.T. O´Konski
surface conductivity
[J. Phys. Chem. 64, 605 (1960)]
A. García, C. Grosse, P. Brito
diffusion effects
[Journal of Physics D, 18, 739 (1985)]
2
All these treatments coincide in that a suspension of uncharged
suspended particles undergoes
just one high-frequency dispersion
usually in the 1 MHz to 100 MHz range.
On the contrary, for charged suspended particles, experimental,
theoretical, and numerical studies developed in the last 40 years show that
the suspension undergoes
an additional dispersion at low frequencies
usually in the 100 Hz to 100 kHz range.
3
In this work we present rigorous analytical results that show:
the LFDD is also to be expected in suspensions of uncharged particles
when the diffusion coefficients of the two types of ions are different.
While this general case is usually considered in studies dealing with charged
particles, it was apparently omitted when the particles are uncharged.
For charged particles, the effect of different diffusion coefficients is
obscured by the presence of the usual LFDD.
For uncharged particles, it leads to the appearance of a LFDD in a frequency
range where no dispersion process is classically expected.
4
THEORY
We consider a spherical uncharged insulating particle represented by:
R
radius of the particle
εi
absolute permittivity
The surrounding electrolyte solution is characterized by:
εe
absolute permittivity
z±
ion valences
D±
ion diffusion coefficients
C0±
ion equilibrium number concentrations
5
When an electric field E ( t ) = Eeiω t is applied to the system, the ion
concentrations and the electric potential are determined by:
Equations for the ion flows:
D±
∇Φ
j = − D ∇C ∓ C z e
kT
Continuity equations:
±
∂
C
∇⋅ j± = −
∂t
Poisson equation:
±
±
(
∇ Φ =−
2
±
± ±
)
z + C + − z −C − e
εe
To first order in the applied field, there is no field-induced liquid flow
because there is no equilibrium volume charge surrounding the particle.
Therefore, neither the fluid velocity term in the first equation nor the Navier
Stokes equation need to be considered.
6
Referring the ion concentrations to their equilibrium values:
C ± = C0± + δC ±
and keeping only linear terms in E :
+ + + 2
+ + − 2


C
z
z
e
C
z z e
i
ω
2
+
+
0
0
∇ δC = 
+ + δ C −
δ C−
ε e kT
D 
 ε e kT
− − + 2
− − − 2

C
z
z
e
C
z z e
iω  −
2
−
+
0
0
∇ δC = −
+ − δ C
δC +
ε e kT
D 
 ε e kT
(
∇ Φ =−
2
)
z +δ C + − z −δ C − e
εe
This system can be analytically solved in terms of two Helmholtz terms:
7
δ C±
ρ ( R −r ) 1 + ρ r
σ ( R −r ) 1 + σ r
±
±
∓
∓
= Nz 1 ∓ z β Ku e
+ Nz α ∓ z K v e
2
2
E cosθ
ρ r
σ 2r 2
(
)
(
)
 K d R3

ρ ( R −r ) 1 + ρ r
σ ( R −r ) 1 + σ r
=  2 − r  + ( A + β ) Ku e
+ ( B + 1) K v e
2
2
2 2
E cosθ  r
r
r
ρ
σ

Φ
ρ 2 =  a + d − (d − a )2 + 4bc  2
σ 2 =  a + d + (d − a) 2 + 4bc  2
α =  d − a − (d − a ) 2 + 4bc 
β =  a − d + (d − a ) 2 + 4bc 
( 2b )
c = ∆a
d = Qa + κ 2




A=
D− − D+
+ +
Def =
− −
D z +D z
a ( ∆ − Qβ )
ρ
2
( 2c )

b = ∆z + z − a
a = iω Def
∆=

B=
a ( ∆α − Q )
σ
2

D+ D− ( z + + z − )
+ +
− −
D z +D z
κ=
8

z + z − ( z + + z − )e 2 N
ε e kT
Q=
D+ z − + D− z +
D+ z + + D− z −
N=
C0+
z
−
=
C0−
z+
The unknown coefficients K d , Ku , and K v are determined using the
usual boundary conditions:
(1)
εi
Φ ( R)
R
dΦ
= εe
dr
r=R
obtained from continuity of the electric potential:
Φi ( R ) = Φ ( R )
and of the radial component of the displacement:
dΦi
εi
dr
(2),(3)
dδ C ±
−
dr
r=R
C0± z ± e dΦ
∓
kT dr
r=R
dΦ
= εe
dr
=0
r=R
ions from the electrolyte solution cannot penetrate inside the particle.
9
r=R
The results so obtained are:
−3ε iα RGσ
Ku =
Den
Kv =
3ε i RGρ
Den
ε e RGρ Gσ (αβ − 1)

ε i ( B − Aα ) RGρ Gσ − 

+ε i ( B + 1) Hσ Gρ − α ( A + β ) H ρ Gσ  

Kd =
Den
ε e RGρ Gσ (αβ − 1)

Den = ε i ( B − Aα ) RGρ Gσ + 2 

+ε i ( B + 1) Hσ Gρ − α ( A + β ) H ρ Gσ  
1 + xR
Hx = 2 2
x R
2 + 2 xR + x 2 R 2
Gx = −
x 2 R3
10
For D + = D − = D , these expressions simplify to:
δ C±
σ ( R −r ) 1 + σ r
∓ ±
= ∓ Nz z K v e
E cosθ
σ 2r 2
 K d R3
 κ2
σ ( R −r ) 1 + σ r
=  2 − r  + 2 Kve
2 2
E cosθ  r
r
σ
σ

Φ
3ε i R
ε i BRGσ + 2 [ −ε e RGσ + ε i ( B + 1) Hσ ]
σ2 =κ2 +
ε i BRGσ − [ −ε e RGσ + ε i ( B + 1) Hσ ]
Kd =
ε i BRGσ + 2 [ −ε e RGσ + ε i ( B + 1) Hσ ]
iω
B=−
Dσ 2
Kv =
iω
D
which reduce to those previously presented in the simplest case: z + = z − = 1.
11
DISCUSSION
For D + = D − field-induced ion concentration profiles extend to a
distance of the order of 1 κ from the surface of the particle.
For D + ≠ D − they extend much further away at low frequencies, leading to
volume charge densities at distances of the order of the
radius of the particle.
These far-reaching charge densities are responsible for the appearance of an
additional low-frequency dispersion.
12
Analytical results calculated for:
R = 10−7 m
εi = 2 ε0
ε e = 80 ε 0
D + = 2 × 10−8 m2/s
D − = 2 × 10−10 m2/s
z+ = 1
z− = 2
C0+ = 2 × 1022 m-3
N = 1022 m-3
ke =
(
Nz + z − e 2 z + D + + z − D −
kT
C0− = 1022 m-3
) = 2.52 ×10−3 S/m
13
C = 1.66 mM
κ R = 2.29
(
D = z + D+ + z − D−
Field-induced charge density
)(
z+ + z−
)
D+= D-= 6.8 10-9m2/s, z+=1, z-=2, ee=80 e0, ei=2 e0, ke=2.52 10-3 S/m, R=10-7m, kR=2.29
2
modulus of charge density / E (C/m V)
1.E+14
1.E+13
1.E+12
1.E+11
1.E+10
1.E+09
1.E+08
100 MHz
1.E+07
1.E-07
2.E-07
10 MHz
3.E-07
1 MHz
4.E-07
1 Hz - 100 kHz
5.E-07
6.E-07
distance to center (m)
14
7.E-07
8.E-07
9.E-07
1.E-06
D+ ≠ D−
Field-induced charge density
1.E+13
2
modulus of charge density / E (C/m V)
1.E+14
D+= 2 10-8, D-=2 10-10m2/s, z+=1, z-=2, ee=80 e0, ei=2 e0, ke=2.52 10-3S/m, R=10-7m, kR=2.29
100 kHz
1.E+12
10 kHz
1.E+11
1 kHz
1.E+10
100 Hz
1.E+09
1.E+08
100 MHz
10 MHz
1 MHz
10 Hz
1 Hz
1.E+07
1.E-07
2.E-07
3.E-07
4.E-07
5.E-07
6.E-07
distance to center (m)
15
7.E-07
8.E-07
9.E-07
1.E-06
For D + ≠ D − and low frequencies, the field-induced charge density extends
to far greater distances than for D + = D − :
order of R rather than 1 κ
This leads to the appearance of an additional dispersion.
These distances are so large that:
build up of the charge densities requires a long time so that the
dispersion phenomenon appears at low frequencies.
significant contribution to the dipolar coefficient, despite the small
values of the charge densities involved.
16
We now show the permittivity and conductivity increment spectra
∆ε =
ε − εe
p
 ´
ke " 
= 3ε e  K d +
Kd 
ωε e


 ´ ωε e " 
k − ke
= 3ke  K d −
∆k =
Kd 
p
ke


where
p
ke =
volume fraction of particles
(
Nz + z − e 2 z + D + + z − D −
kT
)
conductivity of the electrolyte solution
17
Permittivity and conductivity increments
+
-
-3
-7
z =1, z =2, ee=80 e0, ei= 2 e0, ke=2.52 10 S/m, R=10 m, kR=2.29
-3.7824E-03
-115.49
-3.7826E-03
-115.50
relative permittivity increment
-3.7830E-03
-115.51
-3.7832E-03
-115.52
-3.7834E-03
-3.7836E-03
-115.53
-3.7838E-03
-115.54
-3.7840E-03
-3.7842E-03
-115.55
-3.7844E-03
-115.56
1.E+02
1.E+03
1.E+04
1.E+05
Frecuency (Hz)
18
1.E+06
1.E+07
-3.7846E-03
1.E+08
conductivity increment (S/m)
-3.7828E-03
Classical solution
Permittivity and conductivity increments
z =1, z =2, ee=80 e0, ei= 2 e0, ke=2.52 10 S/m, R=10 m, kR=2.29
+
-
-3
-7
-115.49
-3.7824E-03
-3.7826E-03
-115.50
relative permittivity increment
-3.7830E-03
-115.51
-3.7832E-03
-115.52
-3.7834E-03
D+= D-=6.8 10-9 m2/s
-115.53
-3.7836E-03
-3.7838E-03
-115.54
-3.7840E-03
-3.7842E-03
-115.55
-3.7844E-03
-115.56
1.E+02
1.E+03
1.E+04
1.E+05
Frecuency (Hz)
19
1.E+06
1.E+07
-3.7846E-03
1.E+08
conductivity increment (S/m)
-3.7828E-03
Classical solution
Permittivity and conductivity increments
+
-
-3
-7
z =1, z =2, ee=80 e0, ei= 2 e0, ke=2.52 10 S/m, R=10 m, kR=2.29
-3.7824E-03
-115.49
-3.7826E-03
-115.50
relative permittivity increment
-3.7830E-03
-115.51
-3.7832E-03
-115.52
-3.7834E-03
+
-
-9
2
D = D =6.8 10 m /s
-115.53
-3.7836E-03
-3.7838E-03
-115.54
+
-8
-
D =2 10 , D =2 10
-10
-3.7840E-03
2
m /s
-3.7842E-03
-115.55
-3.7844E-03
-115.56
1.E+02
1.E+03
1.E+04
1.E+05
Frecuency (Hz)
20
1.E+06
1.E+07
-3.7846E-03
1.E+08
conductivity increment (S/m)
-3.7828E-03
Classical solution
CONCLUSIONS
When the diffusion coefficient values differ, far-reaching field-induced
charge densities appear around uncharged suspended particles.
These charge densities modify the dipolar coefficient leading to a lowfrequency dielectric dispersion.
Despite the small amplitude of the whole dispersion for low permittivity
particles in aqueous electrolyte solutions, the relative change of the
dispersion can be large.
21
Potential divided by dipolar potential
(
D = z + D+ + z − D−
)(
z+ + z−
)
D+= D-= 6.8 10-9m2/s, z+=1, z-=2, ee= 80 e0, ei= 2 e0, ke=2.52 10-3S/m, R=10-7m, kR=2.29
1.004
1.002
2
1.000
10 MHz
3
|(F(r )+Er )/(K dR E /r )|
1 MHz
100 kHz
0.998
1 Hz - 10 kHz
0.996
0.994
0.992
0.990
1.E-07
2.E-07
3.E-07
4.E-07
5.E-07
6.E-07
distance to center (m)
22
7.E-07
8.E-07
9.E-07
1.E-06
D+ ≠ D−
Potential divided by dipolar potential
+
-8
-
D = 2 10 , D =2 10
-10
2
+
-
-3
-7
m /s, z =1, z =2, ee=80 e0 , ei=2 e0, ke=2.52 10 S/m, R=10 m, kR=2.29
1.004
1 MHz
1.002
2
|( F(r )+Er )/(K dR E /r )|
10 MHz
1.000
100 Hz
3
1 kHz
0.998
100 kHz
10 kHz
0.996
0.994
0.992
0.990
1.E-07
2.E-07
3.E-07
4.E-07
5.E-07
6.E-07
distance to center (m)
23
7.E-07
8.E-07
9.E-07
1.E-06