Conductivity at the Interface between Two Dielectrics Due to Image Forces
Korobeynikov S. M.1, Melekhov A. V.2, Soloveitcik Yu.G.1, Royak M.E.1, Charalambakos V.P.3
and Agoris D.P.3
1
Novosibirsk State Technical University,
20, K. Marx av., Novosibirsk, 630092, Russia
e-mail: kor_ser_mir@ngs.ru
2
Institute of Laser Physics SB RAS,
13, Institutskaya str., Novosibirsk, 630090, Russia
3
University of Patras, Patras-Rio, Gr 26500, Greece
PACS 77.30; 73.25; 47.65
Abstract.
In the present paper, the processes of recombination and dissociation of ion pairs in liquid
dielectrics near the boundary of the other dielectrics (solid, gas, or liquid) are considered.
Different ratio of the dielectric permittivities is proposed. It has been shown that if the
permittivity of the liquid is less than the permittivity of the other dielectrics dissociation
increases and recombination decreases. In the case of inverse ratio the recombination increases
and the dissociation decreases. The experimental results of the electrical conductivity
measurements in films, foams, and also the charge of colloidal particles, surfaces and bubbles are
discussed.
1. INTRODUCTION
Study of the charge carriers generation, movement and disappearance in two phase
medium is important for electrical insulation [1], colloidal and electrical chemistry [2],
electrohydrodymamics (EHD) [3] and other related fields. In the field of electrical insulation
there is a problem of contact of two dielectrics where one of them is liquid. While devices like
transformers and partition insulator are working and the oil is moving along the paper the oil
electrization takes place. The stability of different colloidal systems like suspensions, foams,
films is closely connected with the creation and behavior of the charges at the interfaces. EHD
device converses the electrical energy into kinetic or hydrostatic energy of liquid. Usually EHD
devices (pumps, heat exchanger, etc) have at least three systems: the first one produces space
charge in dielectric liquids, the second one moves the liquid due to electrical force, and the third
one removes space charges. The functioning of the second device is not a problem because it is a
special HV electrode system but other ones concern to the problem of charge generation and
recombination.
The aim of the present paper is the study of charge creation due to image forces and
surface conductivity at the interface between the liquid dielectrics and the dielectrics with
another permittivity.
2. RECOMBINATION AND DISSOCIATION NEAR DIELECTRICS WITH
DIFFERENT PERMITTIVITIES
As it was shown earlier [4] the energy of ion pair dissociation W in the layer close to the
interface of the dielectric with higher permittivity could be several times less than the energy of
dissociation in the bulk of liquid. In the case of arbitrary ratio of permittivities of coexisting
phases the obtained expression is following:

( e' / e)  ( R1  R2 ) 
Ws  Wb 1 

(( R1  R2 ) 2  4 R1 R2 ) 

(1)
Here index b is related to the bulk dissociation, index s – to the surface one, R1, R2 – radii
of ions of the ion pair (fig.1), e – ion charge, e' – image ion charge
e  (1   2 )
e 
(1   2 )
(2)
1 – dielectric permittivity of liquid, 2 - dielectric permittivity of another coexistent dielectrics.
It is clear that in the case of lower permittivity of the second dielectrics in comparison with the
permittivity of the first one, the energy of the ion pair dissociation is more than the energy of the
ion pair dissociation in the bulk of liquid.
The ratio of the liquid dielectric permittivity to the permittivity of the solid dielectrics is very
important for the dissociation-recombination equilibrium processes.
The formulas, obtained in [4] for 2/1>>1, can be modified to find equilibrium constants in the
close to boundary region for arbitrary ratio 2/1.
k RS 

R12
k DS 
4 ( D1  D2 )
e
2 1/ 2
-( LB / r  ( e ' / e ) LB /( r 2  R12
) )
(3a)
/r 2 dr )
( D1  D2 )
RB
R12
e
2 1/ 2
LB / r  ( e ' / e ) LB /( r 2  R12
)
2
r dr  (
RB
r
e
2 1/ 2
-( LB / r  ( e ' / e ) LB /( r 2  R12
) )
2
/r dr )
(3b)
D1, D2 – diffusion coefficients of ions, R12- sum of ion radii, kDS – constant of dissociation, kRS –
constant of recombination, LB is doubled so called Bjerrum radius
RB 
e2
801  kT
The symbols used correspond to the previous paper [4]. Fig.2 and 3 demonstrate calculated
o
o
dependences of kRS/kR and kDS/kD on the ratio 2/1 for transformer oil (R1=1 A , R2=5 A ,
o
o
o
o
Lb=250 A , 1= 2.3, D1  D2  1010 m2 s 1 ) and for ester (R1=1 A , R2=5 A , Lb=140 A , 1=4). One
can see that the recombination constant doesn’t change very much and the change is similar for
the both liquids. The dissociation constant changes more radical, formal treatment could lead to
the difference between cases 2/1>>1 and 2/1<<1 more the 21 orders of magnitudes. Here one
should note that dielectric permittivity can’t be less than 1, therefore constant values at 2/1<
1/2.3 for transformer oil and 2/1< ¼ for ester are meaningless and they should not be
considered. If this circumstance is taken into account the constant change at transition from 2=1
to 2=  is to be less. So the change of the dissociation constant is to be 13 orders of magnitude
approximately, five orders in the case of 2<1, the other 8 orders - at 2>1 .
3. NUMERICAL SIMULATION
In the previous section the consideration from the physical point of view was carried out, it
was assumed that the forces acting on charges are considered to depend on the distance between
the charges. The eccentricity of the forces was negligible. Numerical simulation permits to take
into account the forces between ions and images of other ions more accurately.
Let's construct the numerical simulating procedure for the two charges dissociation process
when there is a plane separating two half-spaces with different dielectric permittivities: (of the
upper half-space - 1=2.3, of the lower half-space 2 =  ). Let's consider, that the first charge
o
o
has radius 1A , the second - 5 A . When the first charge position is fixed, the second charge
center distribution function f = f (r , z ) can be found from the equation of continuity of
r
(
current density i = - D1 + D2
r
grad f + LB f F :
)(
)
r
div grad f + LB f F = 0 ,
(
r
)
(4)
r
where F = F (r , z ) - total force operating on the second charge, r , z - cylindrical coordinates
of the second charge center when the first charge is located on R axis. The equation (4) is valid
in area W with the following boundary: G1 - geometrical locus of the points describing the
location of the second charge center when the second charge contacts the first charge, G2 geometrical locus of the points describing the location of the second charge center when the
second charge touches two half-spaces with different dielectric permittivity separating plane (i.e.
o
G2 - part of plane z = 5 A ), G3 - axis of symmetry r = 0 , G4 - remote boundary
o
o
r = 1000 A , G5 - remote boundary z = 1000 A . Let's consider that on the boundaries G1 ,
G2 and G3 the condition of the normal currents non-flowing is
(
)
in = - (D1 + D2 ) ¶ f ¶ n + LB f Fn = 0 ,
(5)
and on the remote boundaries G4 and G5 the condition
f = 0
(6)
is . Furthermore, in one of boundary G1 points (with maximum r , in our case it is the point of
G1 and G2 intersection) let’s set a value
f = 1.
(7)
Let's solve a boundary problem for the current balance equation (4) using finite element
method (FEM). To derive equivalent variational formulation let’s multiply the first and the
second members of the equation (4) on a trial function   r, z  and integrate results over the
calculated area W. Applying the Green formula (integration by parts) and taking into
consideration the boundary conditions (6)-(7) we’ll take:
r
grad f ×grad Y + LB f F ×grad Y d W= 0
ò(
W
)
(8)
To build finite-element approximation let’s decompose W on the triangular finite elements
and also let’s define the piecewise linear finite basic functions Y i on them. Let’s represent the
function f as a linear combination of these basic functions f =
å
q j Y j . By substituting a
trial function Y in turn for all basic functions Y i , a set of the finite-element equations will be
obtained:
å
j
r
q j ò grad Y j ×grad Yi + LB Y j F ×grad Yi d W= 0
(
)
(9)
W
Thus, we have got a set of the linear algebraic equations (SLAE)
å
aijq j = bi ,
(10)
j
where
matrix
cells
are
defined
ò(
aij =
or
from
the
ratio
r
grad Y j ×grad Yi + LB Y j F ×grad Yi d W, or from the boundary conditions
)
W
(6,7), and the right-part member components bi all are zero, except the one component
corresponding to node, where condition (7) is . To solve this SLAE LU-factorization method
taking into account the special profile SLAE matrix data storage format was used.
The dissociation problem was solved using mesh containing 130 015 nodes and 257 842
triangles. In addition, the mesh was essentially condensed near the boundaries G1 and G2 . The
numerical solution accuracy was checked by some mesh subdivisions. Fig.4. displays the
allocation of function f (r , z ) near the boundary G1 . The axes are graduated in Angstrom unit.
To calculate the dissociation constant, the function f (r , z ) obtained as a solution of the
boundary problem was scaled with constant g :
f g (r , z ) = f (r , z ) g ,
(11)
where
g=
ò f (r , z )d W
(12)
WB
and WB - intersection of the calculated area W with a sphere of radius R B
òf
g
(i.e.
(r , z )d W= 1 )
WB
Let’s consider the function
LB
f R (r ) =
e
r
- 1
LB
e R12 - 1
1
× R,
g
RB
g R = 4p ×ò
R 12
LB
e
r
- 1
LB
×r 2dr » 7.2 ´ 10- 29 m 3 , (13)
e R12 - 1
that defines a probability density of the second charge distribution to the first charge in
homogeneous space with a relative permittivity e . Fig.5 shows the graphs of behavior of the
function f
g
function f
R
along the boundaries G2 (curve 3) and G3 (curve 2) in comparison with the
(r )
g
(curve 1). Calculated using function f (r , z ) the dissociation constant
kDS =
ò i dS
n
» 2.4 ´ 105 s - 1 ,
(14)
Sg
where S g - surface of the cylinder, that completely contains the boundary G1 , differs from the
corresponding dissociation constant kD » 3.8 ´ 10
- 6
s - 1 for homogeneous space almost 11
orders of magnitude.
4. MEASUREMENTS OF SURFACE CONDUCTIVITY
DC measurements of the surface conductivity were carried out. Ceramics based on
BaTiO3 and CaTiO3 having 2=40, >1013 Ohmcm according to the producer’s data were used.
Rectangular-shaped samples were of 2.5 x 1 cm2 x 0.7mm. The surface was polished.
Transformer oil with 1=2.3, =1012 Ohmcm was used as a liquid dielectrics. Measurements of
the surface conductivity were performed by means of the direct current voltage power supply
B5-50 with the voltage from 1 V till 300 V and nanoamperemetre EMG-1352. The samples with
the attached coaxial cables were screened by the copper box. The background noise of the
measuring device was several picoamperes.
The resistance of the liquid was measured in a special microcell with the rectangular
electrodes 2 x 1 cm made of stainless steel. Measurements of the surface conductivity were
carried out in the ad lib cell made of two polish surfaces of ceramics with two plate-shaped foil
metal electrodes between them (fig.6). Tightly pressed each other the plates had the clearance
which size determined by the thickness of the foil that the electrodes were made of.
Bulk resistance Rv and surface resistance Rs were determined from the ratios:
d
l 
d
Rs   s
2l
Rv   v
(15)
where v is the specific space resistance of the liquid, s is the specific surface resistance,  - foil
electrode thickness, d - distance between electrodes, l - their length. The bulk ceramic resistance
that is two orders more than the oil’s one was neglected. The ratio of the bulk resistance to the
surface one depends on the electrode thickness. Changing  the part responsible for the surface
conductivity could be separated from the whole resistance that is parallel connection of the bulk
resistance and the surface one.
Testing of the method was realized by the measuring of the standard transformer oil using
electrodes of 12 m, 60 m and 200 m thickness. The width of every electrode was 1 mm, the
length was 2.5 cm, the inter electrode gap -1mm, the material- stainless steel, gap voltage 100 V.
The testing results are shown in the fig.7. The surface resistivity could be found from the graph
having been directed the electrode thickness to zero. It can be realized that the resistance value
is 2.11012 Ohm. Recounting obtained results for the case of the surface resistivity one can obtain
1014 Ohm. To estimate the surface conductivity from the side of ceramic sample and its possible
contribution to the conductivity of the whole system the following measurements were carried
out. All above-cited measurements were made in the air. Obtained surface conductivity was
about 1015 Ohm.
As far as the solid body conductivity and the surface conductivity from the side of the
solid body contribution to the whole conductivity according to given experiment is only a
methodological error their combined contribution was estimated and this one was unimportant.
Our efforts to input organic compounds (C4H9)3NC2H5Br, (C4H9)4NCl, ((C2H5)2N)4PBr,
(C6H5)3P C2H5Br were unsuccessful. Their solubility was so negligible that there were not any
visible changes or changes of conductivity. The surface conductivity was also unvarying. This
fact could be an additional confirmation of the measured value s  1014 Ohm.
A new attempt to measure the surface conductivity in the more polarized conductive
liquid isoamyl alcohol C5H11OH was made. Its dielectric permittivity is 16, the specific
resistance is 106 Ohmcm. The surface conductivity measured with the method described above
is s  4108 Ohm.
5. DISCUSSION
Experimental data. Let’s estimate the electrical parameters of the surface layer by mean of the
measured values of the surface resistance. One takes the mobility of the charge carriers near the
interface equal to the bulk one ~10-4 cm2/(Vs). So for the case of ceramics in transformer oil one
could get the surface charge carriers density 109 1/cm2. Taking into consideration the fact that
the charge carriers are in the layer of 10 A depth the concentration has been approximately 10 16
1/cm3. For comparison: according to the measurements of the bulk conductivity the ionic
concentration was 1011 1/cm3. In the case of isoamyl alcohol one can also see the great difference
between ionic concentrations in the bulk of liquid and close to the ceramics surface.
Thus the theoretical model, computations and experimental results show that near the
boundary between two dielectrics special the layers which conductivity differs from the bulk
conductivity should be formed. There are some discrepancies in the rate of dissociation: the
numerical simulation leads to less change of near interface dissociation rates than the physical
consideration. Nevertheless both in the model and in computations the layer conductivity near
the interface and in the bulk of liquid is to be rather different.
What properties of these layers one should expect?
Boundary with dielectrics of higher permittivity. In these cases 2/1>>1, so as the
dissociation increases, the recombination decreases, image forces attract charges from the bulk
of liquid. The ion concentration could increase, in comparison with the bulk one, near the metal
surfaces (e.g. electrodes) and some dielectric samples with the higher permittivity. What kind of
the layer structure could be then? It could be the double electrical layers, when the sizes of ions
in dissolved dissociation pollution in the form of ion pair differ one another. In the case of equal
sizes of the ions the layer has both positive and negative ions. In the electrical insulation systems
the presence of dielectrics with the other permittivity could be dangerous. It leads to the
additional conductivity, the energy losses, electric field redistribution and therefore probability
of breakdown increasing in this region. Moreover, if in the contact of liquid dielectrics with the
solid one, the double layer is formed, then at the forced movement of liquid along the solid
boundary the charges are to be separated and the induced charge is to appear on some parts of
the system. This fact could explain some problems in streaming electrification of the oil in
transformers [1].
Boundary with dielectrics of lower permittivity. In these cases 2/1<<1, so dissociation
decreases, recombination increases, image forces repulse charges from the boundary to the bulk
of liquid. The ion concentration decreases in comparison with the bulk one, near some dielectric
samples with the lower permittivity (e.g. bubbles, foams, thin films etc). Experiments in the
conductivity measurements in the thin water films show that the thinner film is – the less
o
conductivity is [6]. This effect appears if the film is thinner than 400 A . It was explained by the
charge carriers mobility decreasing in the films. In our opinion it is doubtful supposition. Both in
[6] and in other papers, it was shown that the charge carriers mobility in the dilute part of the
double electric layer is the same as the bulk mobility. As the size of the dilute part (tens of
Angstrom) is less than the thickness of the film, it is obviously that this effect should not be
taken into consideration for explanation.
In our opinion one of the reasons of such kind of behavior could be electrolyte dissociation
decrease in the film due to the image forces. In accordance with (1), the image charge is equal
0.98 of the ion charge. Besides, in this case one should take into account the influence of two
bounds between the liquid and another dielectrics – the air. Moreover images of the images are
to form the infinite row of influenced charges here, which value weakly decreases with the
transition to the next charge in the row. The action of every charge could lead to the decrease of
the dissociation degree in the film. The less film thickness is – the less dissociation degree is.
This consequence has qualitative coincidence with the data of the work [6].
The charge of the surface. By the influence of image forces in some two-phase systems
one could explain the appearance and the sign of the interface charge. For this case one should
take in account the structure of the double electric layer. It is assumed that the surface doesn’t
content active groups; liquid doesn’t have surfactants and specific adsorbed ions. If the ions of
the ion pair have equal sizes the double layer could not be formed. Another situation is to happen
if the ions have different sizes. Then if 2/1>>1 the smaller ions are attracted to the surface,
while the bigger ions form the second plate of the double electric layer. So the surface will
acquire the charge of the smaller ion. In the case when 2/1<<1, ions repulse from the interface.
In the equilibrium the current that attributes with repulsion is to be compensated by the diffusion
current and the induced current, that appears due to the field of the formed double electric field.
It is obviously that the diffusion current in the case of smaller ions balances the repulsion current
at the less gradient, because of the less diffusion coefficient. Therefore the ion concentration
profiles could form higher concentration of the bigger ions near the surface than the
concentration of the smaller ions. It is clear that the surface acquires the charge which sign
corresponds to the sign of the bigger ion charge.
What kind of ions is to take part in these processes in the real two-phase system? It is well
known that independently of the components nature, in the absence of electrolytes, the
component with higher permittivity obtains positive charge in the two-phase system [6-9].
Where from could the ions in the absence of electrolytes appear? In our opinion the water
molecules play the part of the ion provider. After dissociation of the water molecules the ions H+
and OH- present in the liquid. Their sizes differ significantly, hydroxyl ion is two times bigger
than the proton one. It is doubtless that some two-phase systems contain water as contamination,
which removing requires special hard efforts. Therefore water molecules are in the liquid.
Consequently the proton is to lead to the positive charging of the particles with higher 2 while
the hydroxyl ion is to lead to the negative charging of species with lower 2.
Particularly, the bubbles are to have got negative charge. This fact was registered in
experiments in water [6,7], and freon [9]. The explanation, proposed in [7] is based on the
decreasing dielectric permittivity near the boundary with the air due to decrease of the liquid
density close to the surface layer. But the fact of density decrease is insignificant, and related
decrease of the permittivity is assumed insignificant. Nevertheless, the mechanism of the
negative surface charge creation due to decrease of the permittivity is ideologically similar to our
mechanism. In reality the presence of the coexistence phase with low 2 is equivalent to decrease
of permittivity from the position of decreasing of the charges interaction energy.
A series of experiments in the fields of dielectric composite materials, colloid chemistry,
boiling and cavitation show this effect. Experimental data [8] point that in composition
“ferroelectric ceramic dust – liquid”, the ceramics which permittivity is 10000, obtains the
positive charge, but the liquid dibutylphthalat which permittivity is 7, obtains the negative
charge. In another experiment [9] it is shown that the bubbles in boiling freon have the negative
charge. Special experiments on measurements of -potential of the bubble in water (1~80) show
that both the bubble and the surface of glass cell (2~5) acquire the negative charge.
6. CONCLUSION
As a result of consideration one could propose the new mechanism of the surface charge and the
surface conductivity creation due to the action of image forces near the interface between liquid
dielectrics and the dielectrics with another permittivity. One can suppose that it is possible to
control the dissociation-recombination equilibrium due to inserting another dielectrics with
different permittivity into liquid, e.g. non-soluble liquid, gas etc.
ACKNOWLEDGEMENTS
This work was supported by Russian Foundation for Basic Researches (grant N 01-0216932) and NATO Fellowship Program.
REFERENCES
[1] M. A. Brubaker, J. K. Nelson, “A Parametric Study of Streaming Electrification in a FullScale Core-Form Transformer Winding Using a Network-Based Model”. IEEE Trans. on Power
Delivery, Vol.15, pp. 1188-1200, 2000.
[2] I.A. Razilov, F. Gonzalez-caballero, A.V. Delgado, S.S. Dukhin, “The Influence of pH on
Low-temperature Dielectric Dispersion. The Participation of Admixture in the Polarization of the
Stern Layer”. Colloid Journal of the Russian Academy of Sciences, Vol. 58, pp. 222-229, 1996.
[3] J. Seyed-Yagoobi, J. Didion, J.M. Ochterbeck, J.Allen, “Thermal Control and Enhancement
of Heat Transport Capacity of Two-Phase Loops with Electrohydrodynamic Conduction
Pumping”, Fifth Microgravity Fluid Physics and Transport Phenomena Conference, NASA
Glenn Research Center, Cleveland, OH, CP-2000-210470, pp. 542-565, August 9, 2000.
[4] S. M. Korobeynikov, A. V. Melekhov, G. G. Furin, V. P. Charalambakos and D. P. Agoris
“Mechanism of surface charge creation due to image forces”. Journal of Phys.D.: Applied
Physics, Vol.35, p.1193-1196, 2002.
[5] E. Swayne, John Newman, C. Radke, “Surface Conductivity and Disjoining Pressure of
Common Black Films Stabilized with Sodium Dodecyl Sulfate”. Journal of Colloid and Interface
Science, Vol. 203, pp. 69-82, 1998.
[6] A. Graciaa, G. Morel, P. Saulner, et al, “The -Potential of Gas Bubbles”
Journal of
Colloid and Interface Science. Vol. 172, pp. 131- 136, 1995.
[7] R. Schechter, A. Graciaa, J. Lachaise, “The Electrical State of a Gas\Water Interface Source”.
Journal of Colloid and Interface Science, Vol. 204, pp398-399, 1998.
[8] E. M. Belokurov, V.M. Kopylov, S.M. Korobeynikov et al, “Study of dielectric Media with
Relatively High Permittivity”, Colloid Journal. V. 63, pp. 395-402, 2001.
[9] Asch V. “ELECTROKINETIC PHENOMENA IN BOILING "FREON-113”, Journal of
Applied Physics V.37, P. 2654-2656, 1966.
FIGURES
http://sermir.narod.ru/tryd/figures.htm