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Multilayer Charged Structures in Nonpolar Dielectric Liquids
Journal of Colloid and Interface Science 230, 306–311 (2000) doi:10.1006/jcis.2000.6932, available online at http://www.idealibrary.com on
Multilayer Charged Structures in Nonpolar Dielectric Liquids V. A. Polyansky1 and I. L. Pankratieva Institute of Mechanics, Moscow State Lomonosov University, 1 Michurinskij pr., 119899 Moscow, Russia Received January 5, 2000; accepted Apirl 24, 2000
Theory of the multilayer charged structures adjacent to an electrode surface in nonpolar dielectric liquids with low conductivity under the action of an electric field is developed. Structures of this kind have been revealed by the probe measurements of the field strength in the vicinity of the flat electrode in hydrocarbon liquids. °C 2000 Academic Press Key Words: low-conducting nonpolar liquid; electric field; ion injection by interface; space electric charge.
1. INTRODUCTION
The conductivity of nonpolar technical liquid dielectrics is caused mainly by dissociation of a small impurity of electrolyte species and by the electrochemical injection of ions from the electrode. The progress in the study of the mechanism of interaction between the liquids of this kind and the field applied is stipulated in many aspects by solving the problem on how an electrization (a space charge generation) occurs in the medium. It is established theoretically that the electrization can occur in a strongly nonuniform field due to the field influence upon the dissociation rate of molecules (1–3). Experimental data show that an important role in the electrization is also played by the superficial electrochemical processes, which are accompanied by the injection of unipolar charge into the medium. The unusual layered structure of the near-electrode charged area has been registered by the probe measurements of the field strength in the vicinity of a flat electrode (4, 5). In this structure, the layer adjacent to the electrode has the same sign of the space charge as the electrode. Then, the layer with the opposite charge is situated and at last a quasi-neutral area exists. Under the field action on such a structure the pressure is redistributed in the liquid because the oppositely charged layer is forced to the electrode and the one with the same charge is repelled from it. As a result, there is a situation typical for the appearance of the Rayleigh–Taylor instability in the near-electrode area. The theoretical description of the near-electrode layers with alternate signs of the space charge has been undertaken in (5, 6). For this purpose in Ref. (5) an assumption was made that the mobility of the charge carriers changes in a special way with 1 To whom correspondence should be addressed. E-mail: pol@inmech. msu.su.
0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
moving off from the electrode. It was explained by the gradual transformation of the injected molecular ion into a cluster of the large size consisting of the ion and neutral molecules of the liquid. In Ref. (6) the bipolar layer was calculated in the framework of an electrohydrodynamical model of a medium with two type of ions without any additional assumptions of a nonphysical character. The processes of diffusion, drift of the ions in the electric field, and surface and bulk electrochemical reactions were taken into account, but only a qualitative agreement was obtained with experimental data. In particular, the characteristic thickness of the layers found theoretically falls far short of the experimental one. The whole structure with alternating signs of charge was located within the boundaries of the diffusion layer, the thickness of which in liquids is comparable with the probe diameter used for measurements. That distinction within several orders is connected with the use of a too simple model of the medium in (6). In Ref. (7) a more sophisticated model of a medium with three types of ions was offered for the modeling of the interaction of a dielectric liquid with a field. On the basis of this model, a quasi-stationary bipolar structure was constructed in the nearelectrode area with characteristic size close to the experimental one. However, further research has shown that the layer charged opposite to the electrode gradually dissipates under the action of the diffusion and the field migration of the ions in the time interval about several of the charge relaxation periods in the medium. In this paper we examine the structure of the near-electrode region in the framework of a refined model of the medium as compared with (7). 2. DESCRIPTION OF THE MODEL OF THE MEDIUM
We consider a multicomponent medium that consists of a carrying dielectric liquid and a small admixture of the electrolyte species always present in the technical liquids under investigation. While the species are dissolving, the weakly bonded ionic pairs are formed in the liquid. These ionic pairs dissociate and produce free ions, which provide a residual conductivity of the liquid. In the theoretical model of medium (7) all ions are combined into two kinds of ions: intrinsic positive (below they are marked by the subscript m = 1) and negative ions (m = 2). The
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307
third kind of ions (m = 3) included in the model is formed on the electrode as a result of the surface electrochemical ionization of neutral molecules of the liquid mixture considered. In particular, it may be the process of dissociation of the ionic pairs adsorbed on the liquid–metal interface. The problem of calculation of the motion of the medium components under the action of the field is solved in the onedimensional setting. The liquid is placed into a cell with two electrodes located at a distance L. We consider two configurations of the electrodes. The first one contains the plane electrodes with characteristic sizes that are much more then L. The second one consists of two concentric spheres. The radius of the inner sphere is much less than L. Choosing the coordinate axis x directed normal to the electrode surface, we write the equations of electrohydrodynamics (7) for the ion concentrations and the field in the form
We use the boundary conditions obtained from the balance of the fluxes of charged particles near the interface in the hypothesis that the surface electrochemical processes occur. For definiteness we assume that the ions of all kinds recombine on the electrode surfaces, but the surface ionization occurs only at one of the electrodes, for instance, at the anode. Under the action of repulsive forces the positive ions are rejected into the liquid. Obviously, the intensity of the flux of injected ions depends on the field strength that is generated near the electrode surface due to an external source and to a volume charge of the interelectrode gap. We assume a linear relation between the injected ion flux and the field strength at the electrode surface. With regard to aforementioned assumptions the boundary conditions on the electrode surfaces can be presented in the form
1 d p (x n 1 U1 ) = w − αn 1 n 2 , x p dx 1 d p (x n 2 U2 ) = w − αn 1 n 2 − βn 2 n 3 , x p dx 1 d p (x n 3 U3 ) = −βn 2 n 3 , x p dx dn m dF n m Um = −Dm − sign(em )bm n m , dx dx X X (em n m Um ), q = (em n m ), Js =
n m Um = K m n m , m = 1–3, x = x0 + L ,
m
n m Um = −K m n m , m = 1, 2, x = x0 , [3]
n 3 U3 = A3 E + B3 − K 3 n 3 , x = x0 , A3 = const, B3 = const, K m = const, m = 1–3, F(x0 ) = Fw = const > 0, F(x0 + L) = 0. [1]
[4]
Here, x0 is the anode coordinate and K m , A3 , and B3 are parameters of the superficial electrochemical processes. All constants present in the equations and the boundary conditions are supposed to be specified.
m
e1 > 0, e2 < 0, e3 > 0, µ ¶ 1 d dF p dF x = −q, E = − . εε0 p x dx dx dx Here, p is the index of symmetry: p = 0 and 2 correspond to the plane and spherical symmetry; n m , Um , Dm , bm , and em are the concentration, the velocity, the coefficient of diffusion, the mobility, and the charge of ions of the mth kind, respectively; F is the electric potential; E is the field strength; q is the space charge density; εε0 is the absolute dielectric permittivity of the medium; Js is the density of the total current. It is assumed that the injected ions are positive. On the right-hand side of the first three equations in [1], there are source terms that describe the bulk dissociation of the ionic pairs with rate w, recombination between the intrinsic ions, and recombination between the injected and intrinsic ions with recombination coefficients α and β, respectively. In the general case, the dissociation rate w depends on the concentrations of neutral molecules, on the temperature, and on the strength of the electric field. Here, we consider only the dependence on the electric field that can be written in the form (8) ¡ ¤ ¢ ±£ [2] w = w0 exp 2γ |E|1/2 , γ = e3/2 (εε0 )1/2 kT . Here, e is the elementary charge, k is the Boltzmann constant, T is the temperature, and w0 is the dissociation rate when no field is applied.
3. METHOD OF NUMERICAL SOLUTION OF THE PROBLEM
The finite-difference scheme with nonuniform mesh points along the x-coordinate is used in calculations. A nonlinear system of finite-difference equations is solved by an iterative method. All equations are integrated simultaneously. The equations in [1] and the boundary conditions [3, 4] are written in the dimensionless form. In the results below the characteristic parameters, which are marked by subscript zero, correspond to the dielectric liquid with the residual conductivity σ0 = 10−11 S/m and the ion diffusion coefficient D0 = 10−9 m2 /s. The numerical values of the remaining characteristic parameters can be calculated with the help of the following relations: the ion mobility b0 = eD0 /kT, T = 300 K; the equilibrium concentration of intrinsic ions n 0 = σ0 /(2eb0 ); the coefficients of recombination α0 = β0 = 2eb0 /εε0 ; the dissociation rate w0 = α0 n 20 . The x coordinate is scaled with the distance between the electrodes L. 4. NUMERICAL STUDY OF THE BIPOLAR STRUCTURES IN THE CASE γ = 0
We consider the case when the dependence of the dissociation rate on the electric field is not taken into account. The solution of the problem shows that a narrow range of parameters of the model is available in which the bipolar nearelectrode structure
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registered in experiments exists. Outside of this range the typical structures in the near-electrode area have two varieties. For weak injection the Debye layer opposite in sign adjoins directly to the electrode surface. The presence of this layer results in the increase of the field strength on the surface. For strong injection the whole of the near-electrode layer has the same sign of the charge as the electrode and the field at the electrode is less than the applied one. In the intermediate case the generation of the bipolar structure is due to the combination of several factors: the intensity of the injection, the migration of the injected ions in the field, and their recombination with the intrinsic ions. We examine the influence of these factors on the structure of the layer. In calculations we take the following values of the parameters: b1 = b2 = b0 , w = w0 , and α = α0 . The mobility of the injected ions b3 , the coefficient of recombination β, and the applied voltage are varied. In Fig. 1 the structure of the near-anode layer and its dependence on the mobility b3 are shown. The anode is placed at x ∗ = x/L = 0. We assume the coefficient of recombination β = 0.3α0 , the applied voltage Fw = 2000 kT/e, and the distance between the electrodes L = 0.005 m. One can see that for low values of b3 the positively charged layer is rather narrow (curve 1). The injected ions have enough time to recombine within a short distance from the injecting electrode due to the slow migration under the field action. With increasing b3 , the residence time of the injected ions inside the Debye layer of the negative intrinsic ions decreases. As a result, the recombination rate diminishes and the area of the positive space charge increases (curves 2–4). For large values of b3 /b0 (≥0.5) the re-
FIG. 1. The distribution of the space charge density q ∗ = q/q0 in the near-anode domain for different values of the injected ion mobility b3 /b0 and for Fw = 2000 kT /e, β = 0.3α0 . (1) b3 /b0 = 0.01; (2) b3 /b0 = 0.05; (3) b3 /b0 = 0.07; (4) b3 /b0 = 0.1; (5) b3 /b0 = 0.5. The line segment 6 shows the experimental value of the field maximum disposition (4, 5).
FIG. 2. The distribution of the dimensionless field strength E ∗ = E/(103 kT/eL) in the near-anode domain for different values of the injected ion mobility b3 /b0 . The values of the parameters are the same as those in Fig. 1.
gion with the negative charge disappears because the injected ions have no time to recombine inside the Debye layer. The near-electrode area becomes unipolar (curve 5). In Fig. 2 the distribution of the dimensionless field strength E ∗ = E/(103 kT/eL) is shown. The values of the parameters are the same as those in Fig. 1. One can see that the characteristic feature of the bipolar structures is the nonmonotonic alternation of the field in the near-electrode area with the maximum at the changing point of the charge sign (curves 1–4). This point leaves the anode with an increase of the mobility b3 . Note that the presence of the field maximum allows us to find out experimentally the near-electrode bipolar structures (4, 5). In Figs. 1 and 2 (and in Figs. 3 and 4) the line segment 6 shows the disposition of the field maximum registered in the experiment (4, 5). One can see that good agreement with the experimental data is given by curve 2 for which b3 /b0 = 0.05 and β = 0.3α0 . Figures 3 and 4 show the influence of the variation of the quantity β on the structure of the bipolar layer for b3 /b0 = 0.1 and Fw = 2000 kT/e. The distribution of the dimensionless space charge density q ∗ = q/q0 varies with decreasing β in the same manner as that of when b3 increases. For low values of β the layer becomes unipolar (Fig. 3, curves 1 and 2) and the distribution of the field strength becomes monotone (Fig. 4, curve 1). In Fig. 4 the values β/α0 in the curves are the same as those in Fig. 3. Comparing the results in Figures 3 and 4 with the experimental data (4, 5) (the line segment 6), we find that an agreement in the disposition of the points of the field maxima is given by the curve 4 where β/α0 = 0.5 and b3 /b0 = 0.1. In Refs. (4 and 5) it is noted that there is some quantitative difference between the field distributions in the vicinity of the
CHARGED STRUCTURES IN NONPOLAR DIELECTRIC LIQUIDS
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FIG. 3. The distribution of the space charge density q ∗ = q/q0 in the near-anode domain for different values of the kinetic parameter β/α0 and for Fw = 2000 kT/e and b3 /b0 = 0.1. (1) β/α0 = 0.01; (2) β/α0 = 0.1; (3) β/α0 = 0.3; (4) β/α0 = 0.5; (5) β/α0 = 1.0. The line segment 6 shows the experimental value of the field maximum disposition (4, 5).
FIG. 5. The distributions of the dimensionless ion concentrations n ∗m = n m /n 0 (curves: (1) n 1 /n 0 ; (2) n 2 /n 0 ; (3) n 3 /n 0 ) and the density of the space charge q ∗ = q/q0 (curve 4) along the interelectrode gap for Fw = 104 kT/e, b3 /b0 = 0.035 and β/α0 = 0.3. The coordinates of the anode and cathode are respectively x ∗ = 0 and x ∗ = 1.
anode and the cathode. In the framework of the model used such a difference can be explained by different kinetic and transport properties of the ions injected from the anode and the cathode. We can obtain the nonsymmetrical near-electrode distributions of the volume charge and the field if we take the quantities b3
and β in an asymmetric way. Obviously, the negative ions must be injected from the cathode. The increase of the voltage applied causes the extension of the bipolar layer and the rise of the field maximum. In Fig. 5 the distributions of the dimensionless ion concentrations n ∗m = n m /n 0 and the density of the space charge q ∗ = q/q0 are shown everywhere over the interelectrode gap for Fw = 104 kT/e, b3 /b0 = 0.035, and β/α0 = 0.3. The anode and the cathode are placed respectively at x ∗ = 0 and x ∗ = 1. With an increase of the applied field, the quasi-neutral area reduces and the thickness of the charged layers increases. The distribution of the dimensionless field strength E ∗ = E/(104 kT/eL) is depicted in Fig. 6. The field profile has two extrema. The change of the field in the vicinity of the maximum is caused by the near-anode bipolar structure. 5. BIPOLAR LAYERS IN A NONUNIFORM EXTERNAL FIELD
FIG. 4. The distribution of the dimensionless field strength E ∗ = E/(103 kT/eL) in the near-anode domain for different values of β/α0 . The values of the parameters are the same as those in Fig. 3.
The bipolar structure of the near-electrode domain calculated above is due to the ion injection in the uniform external electric field. To examine the influence of the nonuniformity of the applied field on the distribution of the space charge, we consider a liquid placed between two concentric spheres. Let the radius R0 of the inner sphere be much less than the radius R1 of the external one. In this case the field distribution near the inner sphere is strongly nonuniform. First, we consider the case γ = 0. The distribution of the space charge q ∗ is shown in Fig. 7 for Fw = 3 × 105 kT/e (curve 1), where the abscissa represents
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FIG. 6. The distribution of the dimensionless field strength E ∗ = E/ (104 kT/eL) along the interelectrode gap for Fw = 104 kT/e, b3 /b0 = 0.035, and β/α0 = 0.3.
the dimensionless distance from the inner sphere-anode placed at x ∗ = 0. The cathode is located at x ∗ = 1. One can see that the space charge distribution looks like that in the case of plane symmetry for a strong enough field (Fig. 5, curve 4). Some difference is connected with the presence of a narrow negatively charged diffusion layer adjacent to the inner electrode, which is
FIG. 7. The distribution of the space charge density q ∗ = q/q0 along the interelectrode gap in the case of the spherical symmetry ( p = 2, R0 /R1 = 0.01) for Fw = 3 × 105 kT/e, β = 0.3α0 , b3 = 0.01b0 . (1) = 0; (2) = 0.00318. The coordinates of the anode and cathode are respectively x ∗ = 0 and x ∗ = 1.
FIG. 8. The distributions of the dimensionless ion concentrations n ∗m = n m /n 0 along the interelectrode gap in the case of spherical symmetry ( p = 2, R0 /R1 = 0.01) for Fw = 3 × 105 kT/e, β = 0.3α0 , b3 = 0.01b0 (curves (1), (2), and (3) correspond to = 0; curves (4), (5), and (6) correspond to = 0.00318). (1) n 1 /n 0 ; (2, 5) n 2 /n; (3, 6) 0.1n 3 /n 0 ; (4) 0.1n 1 /n 0 . The coordinates of the anode and cathode are respectively x ∗ = 0 and x ∗ = 1.
due to the high value of the applied field strength on the inner electrode surface. Besides, the quasi-neutral domain is shifted to the external electrode, where the applied field decays because of the geometrical factor. In contrast to the plane case, the field strength changes monotonically throughout the interelectrode space. The distributions of the ion concentrations n ∗m are shown in Fig. 8 for Fw = 3 × 105 kT/e (curves 1–3). The applied field at the inner electrode surface E w = 3.02 × 107 kT/eL. The concentrations of intrinsic ions are far from the equilibrium values over all of interelectrode space. These results show that the nonuniformity of the field applied does not change essentially the type of the space charge generation in the interelectrode gap when only two mechanisms of electrization are taken into account, namely, the charge separation under the electric field and the charge injection from the electrode. We examine now one more mechanism of the space charge generation caused by the influence of the electric field on the dissociation rate of molecules of the electrolyte species dissolved in the liquid (γ 6= 0 in [2]). This mechanism gives essential contribution to the electrization of the interelectrode space in a strongly nonuniform electric field. A space charge is accumulated in the domain of strong field nonuniformity in the nonstationary process of nonequilibrium dissociation when an external voltage is applied. The nonuniformity of the field yields the nonuniformity of the volume source of ions. As a result, a disbalance of fluxes of ions coming into a volume element and going from it arises. Hence, in the vicinity of the anode the volume element is charged positively. A steady state occurs when the accumulated
CHARGED STRUCTURES IN NONPOLAR DIELECTRIC LIQUIDS
space charge changes the field so that the ion fluxes disbalance disappears (1). The distribution of the space charge q ∗ is shown in Fig. 7 (curve 2) for Fw = 3 × 105 kT/e and γ = 0.00318. This value of γ corresponds to a liquid with ε = 2.2 and T = 300 K. One can see that the whole interelectrode space is charged positively. The maximum value of the space charge density q ∗ in the vicinity of the inner sphere is more than 2 times larger than that in the case γ = 0 when the space charge is due to the injection only (curve 1). In Fig. 8 curves 4–6 show the ion distributions for γ = 0.00318. The distribution of the intrinsic positive ions is changed essentially. The growth of the injected ion concentration (curve 6) is connected with the drop of the recombination rate caused by the decrease of the intrinsic negative ion concentration (curve 5). 6. CONCLUSION
The model of the low-conducting multicomponent liquid used in the paper allows us to describe the complex near-electrode structures with alternation of the charge sign. We found the range of the parameters in which such structures exist and we obtained good agreement with experimental data (4, 5). Three mecha-
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nisms of space charge generation in a liquid were examined, namely, the charge separation in a field, the ion injection by the interface, and the nonequilibrium dissociation in a nonuniform field. ACKNOWLEDGMENT The work was supported by the Russian Foundation for Basic Research, Project 98-01-00109.
REFERENCES 1. Apfelbaum, M. S., and Polyansky, V. A., Magn. Hydrodynam. 1, 71 (1982) (in Russian). 2. Pankratieva, I. L., and Polyansky, V. A., J. Appl. Math. Mech. 49, 589 (1985). 3. Pontiga, E., and Castellanos, A., in “Proceeding of 12th International Conference on Conduction and Breakdown in Dielectric Liquids, Rome,” p. 122. 1996. 4. Stishkov, Yu. K., and Rychkov, Yu. M., Colloidal J. 41, 1204 (1978). 5. Rychkov, Yu. M., Izvestiya, A. N. BSSR, Ser. Phys.-Energ. Sci. 3, 104 (1985). [In Russian] 6. Vartanyan, A. A., Gogosov, V. V., Polyansky, V. A., and Shaposhnikova, G. A., J. Electrostat. 23, 431 (1989). 7. Pankratieva, I. L., and Polyansky, V. A., J. Appl. Mech. Techn. Phys. 36, 36 (1995). 8. Frenkel, Ya. N., J. Exp. Theor. Phys. 8, 1292 (1938) [in Russian].
Multilayer Charged Structures in Nonpolar Dielectric Liquids V. A. Polyansky1 and I. L. Pankratieva Institute of Mechanics, Moscow State Lomonosov University, 1 Michurinskij pr., 119899 Moscow, Russia Received January 5, 2000; accepted Apirl 24, 2000
Theory of the multilayer charged structures adjacent to an electrode surface in nonpolar dielectric liquids with low conductivity under the action of an electric field is developed. Structures of this kind have been revealed by the probe measurements of the field strength in the vicinity of the flat electrode in hydrocarbon liquids. °C 2000 Academic Press Key Words: low-conducting nonpolar liquid; electric field; ion injection by interface; space electric charge.
1. INTRODUCTION
The conductivity of nonpolar technical liquid dielectrics is caused mainly by dissociation of a small impurity of electrolyte species and by the electrochemical injection of ions from the electrode. The progress in the study of the mechanism of interaction between the liquids of this kind and the field applied is stipulated in many aspects by solving the problem on how an electrization (a space charge generation) occurs in the medium. It is established theoretically that the electrization can occur in a strongly nonuniform field due to the field influence upon the dissociation rate of molecules (1–3). Experimental data show that an important role in the electrization is also played by the superficial electrochemical processes, which are accompanied by the injection of unipolar charge into the medium. The unusual layered structure of the near-electrode charged area has been registered by the probe measurements of the field strength in the vicinity of a flat electrode (4, 5). In this structure, the layer adjacent to the electrode has the same sign of the space charge as the electrode. Then, the layer with the opposite charge is situated and at last a quasi-neutral area exists. Under the field action on such a structure the pressure is redistributed in the liquid because the oppositely charged layer is forced to the electrode and the one with the same charge is repelled from it. As a result, there is a situation typical for the appearance of the Rayleigh–Taylor instability in the near-electrode area. The theoretical description of the near-electrode layers with alternate signs of the space charge has been undertaken in (5, 6). For this purpose in Ref. (5) an assumption was made that the mobility of the charge carriers changes in a special way with 1 To whom correspondence should be addressed. E-mail: pol@inmech. msu.su.
0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
moving off from the electrode. It was explained by the gradual transformation of the injected molecular ion into a cluster of the large size consisting of the ion and neutral molecules of the liquid. In Ref. (6) the bipolar layer was calculated in the framework of an electrohydrodynamical model of a medium with two type of ions without any additional assumptions of a nonphysical character. The processes of diffusion, drift of the ions in the electric field, and surface and bulk electrochemical reactions were taken into account, but only a qualitative agreement was obtained with experimental data. In particular, the characteristic thickness of the layers found theoretically falls far short of the experimental one. The whole structure with alternating signs of charge was located within the boundaries of the diffusion layer, the thickness of which in liquids is comparable with the probe diameter used for measurements. That distinction within several orders is connected with the use of a too simple model of the medium in (6). In Ref. (7) a more sophisticated model of a medium with three types of ions was offered for the modeling of the interaction of a dielectric liquid with a field. On the basis of this model, a quasi-stationary bipolar structure was constructed in the nearelectrode area with characteristic size close to the experimental one. However, further research has shown that the layer charged opposite to the electrode gradually dissipates under the action of the diffusion and the field migration of the ions in the time interval about several of the charge relaxation periods in the medium. In this paper we examine the structure of the near-electrode region in the framework of a refined model of the medium as compared with (7). 2. DESCRIPTION OF THE MODEL OF THE MEDIUM
We consider a multicomponent medium that consists of a carrying dielectric liquid and a small admixture of the electrolyte species always present in the technical liquids under investigation. While the species are dissolving, the weakly bonded ionic pairs are formed in the liquid. These ionic pairs dissociate and produce free ions, which provide a residual conductivity of the liquid. In the theoretical model of medium (7) all ions are combined into two kinds of ions: intrinsic positive (below they are marked by the subscript m = 1) and negative ions (m = 2). The
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CHARGED STRUCTURES IN NONPOLAR DIELECTRIC LIQUIDS
307
third kind of ions (m = 3) included in the model is formed on the electrode as a result of the surface electrochemical ionization of neutral molecules of the liquid mixture considered. In particular, it may be the process of dissociation of the ionic pairs adsorbed on the liquid–metal interface. The problem of calculation of the motion of the medium components under the action of the field is solved in the onedimensional setting. The liquid is placed into a cell with two electrodes located at a distance L. We consider two configurations of the electrodes. The first one contains the plane electrodes with characteristic sizes that are much more then L. The second one consists of two concentric spheres. The radius of the inner sphere is much less than L. Choosing the coordinate axis x directed normal to the electrode surface, we write the equations of electrohydrodynamics (7) for the ion concentrations and the field in the form
We use the boundary conditions obtained from the balance of the fluxes of charged particles near the interface in the hypothesis that the surface electrochemical processes occur. For definiteness we assume that the ions of all kinds recombine on the electrode surfaces, but the surface ionization occurs only at one of the electrodes, for instance, at the anode. Under the action of repulsive forces the positive ions are rejected into the liquid. Obviously, the intensity of the flux of injected ions depends on the field strength that is generated near the electrode surface due to an external source and to a volume charge of the interelectrode gap. We assume a linear relation between the injected ion flux and the field strength at the electrode surface. With regard to aforementioned assumptions the boundary conditions on the electrode surfaces can be presented in the form
1 d p (x n 1 U1 ) = w − αn 1 n 2 , x p dx 1 d p (x n 2 U2 ) = w − αn 1 n 2 − βn 2 n 3 , x p dx 1 d p (x n 3 U3 ) = −βn 2 n 3 , x p dx dn m dF n m Um = −Dm − sign(em )bm n m , dx dx X X (em n m Um ), q = (em n m ), Js =
n m Um = K m n m , m = 1–3, x = x0 + L ,
m
n m Um = −K m n m , m = 1, 2, x = x0 , [3]
n 3 U3 = A3 E + B3 − K 3 n 3 , x = x0 , A3 = const, B3 = const, K m = const, m = 1–3, F(x0 ) = Fw = const > 0, F(x0 + L) = 0. [1]
[4]
Here, x0 is the anode coordinate and K m , A3 , and B3 are parameters of the superficial electrochemical processes. All constants present in the equations and the boundary conditions are supposed to be specified.
m
e1 > 0, e2 < 0, e3 > 0, µ ¶ 1 d dF p dF x = −q, E = − . εε0 p x dx dx dx Here, p is the index of symmetry: p = 0 and 2 correspond to the plane and spherical symmetry; n m , Um , Dm , bm , and em are the concentration, the velocity, the coefficient of diffusion, the mobility, and the charge of ions of the mth kind, respectively; F is the electric potential; E is the field strength; q is the space charge density; εε0 is the absolute dielectric permittivity of the medium; Js is the density of the total current. It is assumed that the injected ions are positive. On the right-hand side of the first three equations in [1], there are source terms that describe the bulk dissociation of the ionic pairs with rate w, recombination between the intrinsic ions, and recombination between the injected and intrinsic ions with recombination coefficients α and β, respectively. In the general case, the dissociation rate w depends on the concentrations of neutral molecules, on the temperature, and on the strength of the electric field. Here, we consider only the dependence on the electric field that can be written in the form (8) ¡ ¤ ¢ ±£ [2] w = w0 exp 2γ |E|1/2 , γ = e3/2 (εε0 )1/2 kT . Here, e is the elementary charge, k is the Boltzmann constant, T is the temperature, and w0 is the dissociation rate when no field is applied.
3. METHOD OF NUMERICAL SOLUTION OF THE PROBLEM
The finite-difference scheme with nonuniform mesh points along the x-coordinate is used in calculations. A nonlinear system of finite-difference equations is solved by an iterative method. All equations are integrated simultaneously. The equations in [1] and the boundary conditions [3, 4] are written in the dimensionless form. In the results below the characteristic parameters, which are marked by subscript zero, correspond to the dielectric liquid with the residual conductivity σ0 = 10−11 S/m and the ion diffusion coefficient D0 = 10−9 m2 /s. The numerical values of the remaining characteristic parameters can be calculated with the help of the following relations: the ion mobility b0 = eD0 /kT, T = 300 K; the equilibrium concentration of intrinsic ions n 0 = σ0 /(2eb0 ); the coefficients of recombination α0 = β0 = 2eb0 /εε0 ; the dissociation rate w0 = α0 n 20 . The x coordinate is scaled with the distance between the electrodes L. 4. NUMERICAL STUDY OF THE BIPOLAR STRUCTURES IN THE CASE γ = 0
We consider the case when the dependence of the dissociation rate on the electric field is not taken into account. The solution of the problem shows that a narrow range of parameters of the model is available in which the bipolar nearelectrode structure
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registered in experiments exists. Outside of this range the typical structures in the near-electrode area have two varieties. For weak injection the Debye layer opposite in sign adjoins directly to the electrode surface. The presence of this layer results in the increase of the field strength on the surface. For strong injection the whole of the near-electrode layer has the same sign of the charge as the electrode and the field at the electrode is less than the applied one. In the intermediate case the generation of the bipolar structure is due to the combination of several factors: the intensity of the injection, the migration of the injected ions in the field, and their recombination with the intrinsic ions. We examine the influence of these factors on the structure of the layer. In calculations we take the following values of the parameters: b1 = b2 = b0 , w = w0 , and α = α0 . The mobility of the injected ions b3 , the coefficient of recombination β, and the applied voltage are varied. In Fig. 1 the structure of the near-anode layer and its dependence on the mobility b3 are shown. The anode is placed at x ∗ = x/L = 0. We assume the coefficient of recombination β = 0.3α0 , the applied voltage Fw = 2000 kT/e, and the distance between the electrodes L = 0.005 m. One can see that for low values of b3 the positively charged layer is rather narrow (curve 1). The injected ions have enough time to recombine within a short distance from the injecting electrode due to the slow migration under the field action. With increasing b3 , the residence time of the injected ions inside the Debye layer of the negative intrinsic ions decreases. As a result, the recombination rate diminishes and the area of the positive space charge increases (curves 2–4). For large values of b3 /b0 (≥0.5) the re-
FIG. 1. The distribution of the space charge density q ∗ = q/q0 in the near-anode domain for different values of the injected ion mobility b3 /b0 and for Fw = 2000 kT /e, β = 0.3α0 . (1) b3 /b0 = 0.01; (2) b3 /b0 = 0.05; (3) b3 /b0 = 0.07; (4) b3 /b0 = 0.1; (5) b3 /b0 = 0.5. The line segment 6 shows the experimental value of the field maximum disposition (4, 5).
FIG. 2. The distribution of the dimensionless field strength E ∗ = E/(103 kT/eL) in the near-anode domain for different values of the injected ion mobility b3 /b0 . The values of the parameters are the same as those in Fig. 1.
gion with the negative charge disappears because the injected ions have no time to recombine inside the Debye layer. The near-electrode area becomes unipolar (curve 5). In Fig. 2 the distribution of the dimensionless field strength E ∗ = E/(103 kT/eL) is shown. The values of the parameters are the same as those in Fig. 1. One can see that the characteristic feature of the bipolar structures is the nonmonotonic alternation of the field in the near-electrode area with the maximum at the changing point of the charge sign (curves 1–4). This point leaves the anode with an increase of the mobility b3 . Note that the presence of the field maximum allows us to find out experimentally the near-electrode bipolar structures (4, 5). In Figs. 1 and 2 (and in Figs. 3 and 4) the line segment 6 shows the disposition of the field maximum registered in the experiment (4, 5). One can see that good agreement with the experimental data is given by curve 2 for which b3 /b0 = 0.05 and β = 0.3α0 . Figures 3 and 4 show the influence of the variation of the quantity β on the structure of the bipolar layer for b3 /b0 = 0.1 and Fw = 2000 kT/e. The distribution of the dimensionless space charge density q ∗ = q/q0 varies with decreasing β in the same manner as that of when b3 increases. For low values of β the layer becomes unipolar (Fig. 3, curves 1 and 2) and the distribution of the field strength becomes monotone (Fig. 4, curve 1). In Fig. 4 the values β/α0 in the curves are the same as those in Fig. 3. Comparing the results in Figures 3 and 4 with the experimental data (4, 5) (the line segment 6), we find that an agreement in the disposition of the points of the field maxima is given by the curve 4 where β/α0 = 0.5 and b3 /b0 = 0.1. In Refs. (4 and 5) it is noted that there is some quantitative difference between the field distributions in the vicinity of the
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FIG. 3. The distribution of the space charge density q ∗ = q/q0 in the near-anode domain for different values of the kinetic parameter β/α0 and for Fw = 2000 kT/e and b3 /b0 = 0.1. (1) β/α0 = 0.01; (2) β/α0 = 0.1; (3) β/α0 = 0.3; (4) β/α0 = 0.5; (5) β/α0 = 1.0. The line segment 6 shows the experimental value of the field maximum disposition (4, 5).
FIG. 5. The distributions of the dimensionless ion concentrations n ∗m = n m /n 0 (curves: (1) n 1 /n 0 ; (2) n 2 /n 0 ; (3) n 3 /n 0 ) and the density of the space charge q ∗ = q/q0 (curve 4) along the interelectrode gap for Fw = 104 kT/e, b3 /b0 = 0.035 and β/α0 = 0.3. The coordinates of the anode and cathode are respectively x ∗ = 0 and x ∗ = 1.
anode and the cathode. In the framework of the model used such a difference can be explained by different kinetic and transport properties of the ions injected from the anode and the cathode. We can obtain the nonsymmetrical near-electrode distributions of the volume charge and the field if we take the quantities b3
and β in an asymmetric way. Obviously, the negative ions must be injected from the cathode. The increase of the voltage applied causes the extension of the bipolar layer and the rise of the field maximum. In Fig. 5 the distributions of the dimensionless ion concentrations n ∗m = n m /n 0 and the density of the space charge q ∗ = q/q0 are shown everywhere over the interelectrode gap for Fw = 104 kT/e, b3 /b0 = 0.035, and β/α0 = 0.3. The anode and the cathode are placed respectively at x ∗ = 0 and x ∗ = 1. With an increase of the applied field, the quasi-neutral area reduces and the thickness of the charged layers increases. The distribution of the dimensionless field strength E ∗ = E/(104 kT/eL) is depicted in Fig. 6. The field profile has two extrema. The change of the field in the vicinity of the maximum is caused by the near-anode bipolar structure. 5. BIPOLAR LAYERS IN A NONUNIFORM EXTERNAL FIELD
FIG. 4. The distribution of the dimensionless field strength E ∗ = E/(103 kT/eL) in the near-anode domain for different values of β/α0 . The values of the parameters are the same as those in Fig. 3.
The bipolar structure of the near-electrode domain calculated above is due to the ion injection in the uniform external electric field. To examine the influence of the nonuniformity of the applied field on the distribution of the space charge, we consider a liquid placed between two concentric spheres. Let the radius R0 of the inner sphere be much less than the radius R1 of the external one. In this case the field distribution near the inner sphere is strongly nonuniform. First, we consider the case γ = 0. The distribution of the space charge q ∗ is shown in Fig. 7 for Fw = 3 × 105 kT/e (curve 1), where the abscissa represents
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FIG. 6. The distribution of the dimensionless field strength E ∗ = E/ (104 kT/eL) along the interelectrode gap for Fw = 104 kT/e, b3 /b0 = 0.035, and β/α0 = 0.3.
the dimensionless distance from the inner sphere-anode placed at x ∗ = 0. The cathode is located at x ∗ = 1. One can see that the space charge distribution looks like that in the case of plane symmetry for a strong enough field (Fig. 5, curve 4). Some difference is connected with the presence of a narrow negatively charged diffusion layer adjacent to the inner electrode, which is
FIG. 7. The distribution of the space charge density q ∗ = q/q0 along the interelectrode gap in the case of the spherical symmetry ( p = 2, R0 /R1 = 0.01) for Fw = 3 × 105 kT/e, β = 0.3α0 , b3 = 0.01b0 . (1) = 0; (2) = 0.00318. The coordinates of the anode and cathode are respectively x ∗ = 0 and x ∗ = 1.
FIG. 8. The distributions of the dimensionless ion concentrations n ∗m = n m /n 0 along the interelectrode gap in the case of spherical symmetry ( p = 2, R0 /R1 = 0.01) for Fw = 3 × 105 kT/e, β = 0.3α0 , b3 = 0.01b0 (curves (1), (2), and (3) correspond to = 0; curves (4), (5), and (6) correspond to = 0.00318). (1) n 1 /n 0 ; (2, 5) n 2 /n; (3, 6) 0.1n 3 /n 0 ; (4) 0.1n 1 /n 0 . The coordinates of the anode and cathode are respectively x ∗ = 0 and x ∗ = 1.
due to the high value of the applied field strength on the inner electrode surface. Besides, the quasi-neutral domain is shifted to the external electrode, where the applied field decays because of the geometrical factor. In contrast to the plane case, the field strength changes monotonically throughout the interelectrode space. The distributions of the ion concentrations n ∗m are shown in Fig. 8 for Fw = 3 × 105 kT/e (curves 1–3). The applied field at the inner electrode surface E w = 3.02 × 107 kT/eL. The concentrations of intrinsic ions are far from the equilibrium values over all of interelectrode space. These results show that the nonuniformity of the field applied does not change essentially the type of the space charge generation in the interelectrode gap when only two mechanisms of electrization are taken into account, namely, the charge separation under the electric field and the charge injection from the electrode. We examine now one more mechanism of the space charge generation caused by the influence of the electric field on the dissociation rate of molecules of the electrolyte species dissolved in the liquid (γ 6= 0 in [2]). This mechanism gives essential contribution to the electrization of the interelectrode space in a strongly nonuniform electric field. A space charge is accumulated in the domain of strong field nonuniformity in the nonstationary process of nonequilibrium dissociation when an external voltage is applied. The nonuniformity of the field yields the nonuniformity of the volume source of ions. As a result, a disbalance of fluxes of ions coming into a volume element and going from it arises. Hence, in the vicinity of the anode the volume element is charged positively. A steady state occurs when the accumulated
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space charge changes the field so that the ion fluxes disbalance disappears (1). The distribution of the space charge q ∗ is shown in Fig. 7 (curve 2) for Fw = 3 × 105 kT/e and γ = 0.00318. This value of γ corresponds to a liquid with ε = 2.2 and T = 300 K. One can see that the whole interelectrode space is charged positively. The maximum value of the space charge density q ∗ in the vicinity of the inner sphere is more than 2 times larger than that in the case γ = 0 when the space charge is due to the injection only (curve 1). In Fig. 8 curves 4–6 show the ion distributions for γ = 0.00318. The distribution of the intrinsic positive ions is changed essentially. The growth of the injected ion concentration (curve 6) is connected with the drop of the recombination rate caused by the decrease of the intrinsic negative ion concentration (curve 5). 6. CONCLUSION
The model of the low-conducting multicomponent liquid used in the paper allows us to describe the complex near-electrode structures with alternation of the charge sign. We found the range of the parameters in which such structures exist and we obtained good agreement with experimental data (4, 5). Three mecha-
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nisms of space charge generation in a liquid were examined, namely, the charge separation in a field, the ion injection by the interface, and the nonequilibrium dissociation in a nonuniform field. ACKNOWLEDGMENT The work was supported by the Russian Foundation for Basic Research, Project 98-01-00109.
REFERENCES 1. Apfelbaum, M. S., and Polyansky, V. A., Magn. Hydrodynam. 1, 71 (1982) (in Russian). 2. Pankratieva, I. L., and Polyansky, V. A., J. Appl. Math. Mech. 49, 589 (1985). 3. Pontiga, E., and Castellanos, A., in “Proceeding of 12th International Conference on Conduction and Breakdown in Dielectric Liquids, Rome,” p. 122. 1996. 4. Stishkov, Yu. K., and Rychkov, Yu. M., Colloidal J. 41, 1204 (1978). 5. Rychkov, Yu. M., Izvestiya, A. N. BSSR, Ser. Phys.-Energ. Sci. 3, 104 (1985). [In Russian] 6. Vartanyan, A. A., Gogosov, V. V., Polyansky, V. A., and Shaposhnikova, G. A., J. Electrostat. 23, 431 (1989). 7. Pankratieva, I. L., and Polyansky, V. A., J. Appl. Mech. Techn. Phys. 36, 36 (1995). 8. Frenkel, Ya. N., J. Exp. Theor. Phys. 8, 1292 (1938) [in Russian].