Calculation of the Streaming Potential Near a Rotating Disk with Rotational Elliptic Coordinates

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1 Portugaliae Electrochimica Acta 6/4 (8) PORTUGALIAE ELECTROCHIMICA ACTA Calculation of the Streaming Potential Near a Rotating Dis with Rotational Elliptic Coordinates F.S. Lameiras * and E.H.M. Nunes Centro de Desenvolvimento da Tecnologia Nuclear, Comissão Nacional de Energia Nuclear, Av. Antônio Carlos, 6.67, Belo Horionte, Minas Gerais, Brail Received 3 rd January 8; accepted th March 8 Abstract The calculation with rotational elliptic coordinates of the streaming potential in the vicinity of a dis-shaped sample rotating in an electrolytic solution is presented. The measurement of this streaming potential is used to determine the eta potential of planar surfaces. Rotational elliptic coordinates are favored in relation to integral transform methods because only simple mathematical methods are employed to explain the theory of this technique. Keywords: eta potential, planar surfaces, rotating dis, streaming potential, rotational elliptic coordinates. Introduction A new method and a new apparatus to measure the eta potential of a planar surface, ζ, based on the measurement of the electric potential in the vicinity of a rotating dis were described in three previous papers [,,3]. A sample in the shape of a dis is fixed to a rotating cylindrical support immersed in a fluid with electric charges. The rotation of the dis generates a radial movement of electric charges in the diffuse layer adjacent to the sample s face, resulting in an electric potential distributed in the space near the sample. This electric potential can be measured and is proportional to the eta potential. The advantage of this new method is the experimental simplicity, which allows the construction of a reliable equipment. In the first two papers [,] the authors tried to employ elliptic rotational coordinates to develop the theory of the method, but they found some disagreements between theory and practice. In the third paper [3] they corrected * Corresponding author. address: fernando.lameiras@pq.cnpq.br

2 F.S. Lameiras and E.H.M. Nunes / Portugaliae Electrochimica Acta 6 (8) the theoretical flaws by using the Hanel transform according to Nanis and Kesselman [4]. However, this solution is complex and difficult to explain to most users of this new technique. The authors derived the Legendre s equation by applying the rotational elliptic coordinates to the Laplace s equation, but in the corrected application the Chebyshev s equation is derived. This paper presents the solution of the Chebyshev s equation to this problem, which is derived with simple mathematical methods and agrees with the solution of the Hanel transform. Theory Fig. shows the situation where the face of a dis is immersed in a fluid where there are negative and positive electric charges. Let the face of the dis be made of a ind of material whose surface has affinity for negative electric charges. A layer of negative electric charges adheres to this surface. After that, a layer of positive charges is attracted to this layer of negative charges. However, because of thermodynamic fluctuations, this layer presents disorganiation. Another more disorganied layer with negative charges approaches this surface. Successive layers with progressive disorganiation will follow, and after a distance d, the disorganiation between negative and positive charges will be the same as observed in the fluid. ample rotating dis disco rotativo ρ(r,) r fluid velocity ϕ Figure. Scheme of the assembly used in the measurement of the eta potential using a rotating dis. d ϕ = 37

3 F.S. Lameiras and E.H.M. Nunes / Portugaliae Electrochimica Acta 6 (8) The separation between positive and negative charges near the face of the dis generates electric dipoles, whose electric field can be observed. If the dis is rotated, according to the principles of the viscous fluids dynamics, the fluid layer in contact with the dis will rotate at the same velocity as the dis. The layer near the dis will rotate at a lower velocity. The difference between the velocities of these two layers is proportional to the viscosity of the fluid. The velocity of the successive layers decreases in the same way, as shown in Fig.. The rotating movement of the dis generates an outward radial movement of the fluid towards the edge of the dis. The velocity v r of this movement is given by [5,6,7] v =γ r r () /.53Ω 3/ = ν, where Ω is the angular velocity of the dis, and ν is the γ inematic viscosity of the fluid. r and are cylindrical coordinates. This radial movement generates a displacement of electric charges in the vicinity of the dis because, as shown in Fig., there are more free positive electric charges than free negative electric charges. This generates a radial electric current. The charges move outward to the edge of the dis, where they leave this layer and flow bac through the liquid toward the dis. Let ρ(r,) be the electric charge density in the fluid. This density will be different from ero only in the vicinity of the dis. Therefore, ρ(r,) will rapidly approach ero when increases. The radial electric current through a wall of fluid of radius r and height is According to Gauss s law, ir = ρ(r, )vr π r () ρ(r, ) = ε ϕ, where ϕ is the electric potential and ε is the electric permittivity of the fluid. ϕ = ( r) r ( r ϕ r) + ϕ is the Laplacian of the electric potential in cylindrical coordinates with cylindrical symmetry ( ϕ θ = ). ϕ r has a small variation for r < R (R is the radius of the dis). Thus, i r ϕ ε γ r π r (3) The radial electric current can be obtained through the integration of (3), i ϕ ϕ = π r εγ = π r εγ [ ϕ] r d (4) This radial current is confined to a layer near the face of the dis. The physical quantity 37

4 F.S. Lameiras and E.H.M. Nunes / Portugaliae Electrochimica Acta 6 (8) ζ ϕ (5) = [ ϕ] is defined as the eta potential. The current i R increases with the square of r until the maximum value i = πεr γζ is reached. R The radial current increment in a ring of the layer of fluid near the surface of the dis is ir = 4πεγζ r r (6) This current increment must be supplied by the fluid below the ring. Thus, the current density perpendicular to the surface of the dis, j, is j i r = = πr r εγ ζ (7) The negative sign taes to account that j is in the opposite direction of the coordinate axis. This equation shows that the current density returning to the layer of fluid near the face of the dis is uniform when r R. In a volume element sufficiently distant from the face of the dis, there is a charge balance. Thus, according to Gauss s law, ϕ =. This equation can be solved using the elliptical coordinates ξ and η, where r R[ ( + ξ )( η )] / = and = Rξη. Considering a solution such as ϕ ( ξ, η) = M ( ξ ) P( η), one obtains Chebyshev s differential equations (with real argument for η and imaginary argument for ξ). Solutions convergent for ξ and η are obtained considering M j ( ξ ) = j = a j ξ and P ( η) = j j = c j η, where j is an integer. The general solution is where ( η;b) = [ a M ( ξ;b) + a M ( ξ;b) ][ c P ( η;b) c P ( η;b) ] ϕ ξ; + (8) M M P j ξ [ ][ b ( j 4) ] L [ b ] ( j )! ( ξ;b) = b ( j ) j+ ξ [ ][ b ( j 3) ] L [ b ] ( j + ) ( ξ;b) = b ( j ) ( η;b) = ( j ) j η [ b] [( j 4) b] L [ b] ( j )!! (9a) (9b) (9c) 37

5 F.S. Lameiras and E.H.M. Nunes / Portugaliae Electrochimica Acta 6 (8) P ( η;b) = ( j ) j+ η [ b] [( j 3) b] L [ b] ( j + )! (9d) The first boundary condition is that the function ϕ(r;)=ϕ(ξ;η;b) must provide a current density perpendicular to the face of the dis equal to j = εγζ ϕ ϕ j = = = Rη ξ One obtains from (9) that ξ =., where is the electric conductivity of the fluid. ' ' ( ) ( ) [ ( ) ( )][ c P η;b + cp η;b ] a M ;b + a M ;b η ε γ R = () [ ]η This equation is true if c P ( η;b) + cp ( η;b) is constant. In order to fulfill this condition, c = and b =, because P ( η;) = η. Moreover, M ' ( ;) =, M ' ( ;) =, and M ( ξ;) = ξ. The solution is given by ( η;) = a c η M ( ξ;) ηξ ϕ ξ; () The second boundary condition is that the streaming potential is maximum at the center of the dis (r=;=) or (η=;ξ =) and is equal to, so that ϕ ( ξ; η;) = η[ M ( ξ;) ξ] () Discussion Equation () is true where ϕ =, that is, where there is electric charge balance in a volume of fluid. This situation exists for > d from the face of the dis (Fig. ). In the region between = and = d, ϕ, that is, there is a net electric charge. The radial electric current is confined to this region. d is related to the Debye s distance in an electrolyte [8]. In Fig., d is shown as amplified for didactical purposes, because d<<<r. The electric potential ϕ(ξ;;) can be measured by putting an electrode in (,r)=(rξ;) and another one in a position >> R, where ϕ(,r) =. Along the axis of the dis (η=), equation () gives that ( ξ;;) ϕ (3) = M ( ξ;) ξ The right hand side of (3) gives the same values for small ξ as the approximated function employed by Sides, 373

6 F.S. Lameiras and E.H.M. Nunes / Portugaliae Electrochimica Acta 6 (8) ( ξ;;) ϕ ξ + ξ (4) ϕ(ξ;;) is typically around. mv near the dis and falls as ξ growths. This streaming potential is not very high relative to the noise observed in practice. So, ξ should be as close as possible to ero in order to eep the streaming potential as high as possible to decrease the error of measurement. Another good procedure is to give an identifiable pattern to the rotation of the dis (e. g., a sinusoidal pattern) to separate the streaming potential from the noise. Conclusion This paper corrects the approaches previously published by using elliptic rotational coordinates to calculate the streaming potential near a rotating dis immersed in a fluid with electric charges. The rotation of a dis whose radius is R, immersed in a fluid where there are negative and positive electric charges, generates an electric potential difference between a position along the axis of the dis and another position distant from the dis given by (for R) ϕ R = M ( ξ;) R (5) where ϕ(/r) is the electric potential difference, ε is the electric permittivity of the fluid, is the electric conductivity of the fluid, ζ is the eta potential of the / surface of the dis, R is the radius of the dis,.53ω 3/ γ = ν, where Ω is the angular velocity of the dis and ν is the inematic viscosity of the fluid, and [ ( j ) ] ( j 4) [ ] ( j) M = L R! R j (6) ϕ, ν and can be measured, whereas Ω, R and can be controlled. In these conditions, it is possible to calculate the eta potential with equation (5) by measuring the electric potential in a position along the axis of the dis. Acnowledgments To CNPq (Processes n. 477/4- and 358/6-) and to CAPES. References. P.J. Sides and J.D. Hoggard, Langmuir (4) J.D. Hoggard, P.J. Sides and D.C. Prieve, Langmuir (5) P.J. Sides, J. Newman, J.D. Hoggard and D.C. Prieve, Langmuir (6)

7 F.S. Lameiras and E.H.M. Nunes / Portugaliae Electrochimica Acta 6 (8) L. Nanis and W. Kesselman, J. Electrochem. Soc. 8 (97) Th. Von Kármán, Z. Angew. Math. Mech. (9) W.G. Cochran, Proc. Cambridge Philos. Soc. (934) M.H. Rogers and G.N. Lance, J. Fluid Mech. 7 (96) R.J. Golston and P.H. Rutherford, Introduction to Plasma Physics, Institute of Physics Publishing, Philadelphia,

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