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Fig. 2.19 Scheme of an all-solid lithium battery with PEO based electrolyte
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the dry state) discrete hydrophilic regions, separated by narrow hydrophobic channels. The hydrophilic regions are formed by a specific aggregation of the ionogenic side chains on the perfluorinated backbone, whereas the channels are composed of hydrophobic fluorocarbon chains (see the polymer formula above). The individual macromolecular chains are thus interconnected by the embedded clusters into a more complicated supramolecular network (Fig. 2.20). This, moreover, explains why Nafion and related materials are practically insoluble in water, in spite of the absence of regular interchain bonds as in classical cross-linked ion-exchange polymers (e.g. polystyrene sulphonate). By swelling with aqueous electrolyte, cations (and, to lesser extent, also anions) penetrate together with water into the hydrophilic regions and form spherical electrolyte clusters with micellar morphology. The inner surface of clusters and channels is composed of a double layer of the immobilized SO^ groups and the equivalent number of counterions, M + . Anions in the interior of the clusters are shielded from the SO^ groups by hydrated cations and water molecules. On the other hand, anions are thus
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Fig. 2.20 The Gierke model of a cluster network in Nafion. Dimensions are expressed in nm. The shaded area is the double layer region, containing the immobilized SO3 groups with corresponding number of counterions M + . Anions are expelled from this region electrostatically
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134 'trapped' inside the clusters owing to the high electrostatic barrier in the hydrophobic channel where the shielding effect is suppressed. This makes it clear why the channels are preferentially conductive for cations, i.e. the polymer membrane (despite the presence of free anions) is not anionconducting. Diffusion of cations in a Nafion membrane can formally be treated as in other polymers swollen with an electrolyte solution (Eq. (2.6.21). Particularly illustrative here is the percolation theory, since the conductive sites can easily be identified with the electrolyte clusters, dispersed in the nonconductive environment of hydrophobic fluorocarbon chains (cf. Eq. (2.6.20)). The experimental diffusion coefficients of cations in a Nafion membrane are typically 2-4 orders of magnitude lower than in aqueous solution.
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References Conway, B. E., Proton solvation and proton transfer in solution, MAE, 3, 43 (1964). Dekker, A. J., Solid State Physics, Prentice-Hall, Englewood Cliffs, 1957. Eisenberg, A. E., and H. L. Yeager (Eds), Perfluorinated Ionomer Membranes, ACS Symposium Series 180, ACS, Washington, 1982. Glasstone, S., H. Eyring, and K. J. Laidler, Theory of Rate Processes, McGrawHill, New York, 1942. Hertz, H. G., Self-diffusion in solids, Ber. Bunsenges., 75, 183 (1971). Hladik, J. (Ed.), Physics of Electrolytes, Academic Press, New York, a multivolume series, published since 1972. Huggins, R. A., Ionically conducting solid state membranes, AE, 10, 323 (1977). Inman, D., and D. C. Lovering (Eds), Ionic Liquids, Plenum Press, New York, 1981. Kittel, C , see page 104. Laity, R. W., Electrochemistry of fused salts, /. Chem. Educ, 39, 67 (1962).
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Linford, R. G., Electrochemical Science and Technology of Polymers, Vol. 1 & 2,
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Elsevier, London, 1987 and 1990. MacCallum, J. R., and C. A. Vincent (Eds), Polymer Electrolyte Reviews, Elsevier, London, 1987. Vashista, P., J. N. Mundy, and G. K. Shenoy (Eds), Fast Ion Transport in Solids, North-Holland, Amsterdam, 1979. Vincent, C. A., The motion of ions in solution under the influence of an electric current, /. Chem. Educ, 53, 490 (1976). 2.7 2.7.1 Transport in a Flowing Liquid Basic concepts
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Processes involving transport of a substance and charge in a streaming electrolyte are very important in electrochemistry, particularly in the study of the kinetics of electrode processes and in technology. If no concentration gradient is formed, transport is controlled by migration alone; convection
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Fig. 2.21 Prandtl and Nernst layer formation at the boundary between a solid plate and flowing liquid (c* = 0)
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has no effect. The opposite situation where concentration gradients are formed during transport has interesting characteristics; if transport through migration can be neglected (e.g. at a sufficiently high concentration of an indifferent electrolyte), transport is controlled by diffusion in the streaming liquid, i.e. convective diffusion is involved. To a major part this chapter will consider processes occurring in a steady state, i.e. where dcjdt = 0. It will be simultaneously assumed that the liquid is incompressible. A change in its velocity occurs close to the boundary with a solid phase or with another liquid. This is because the molecules of a liquid in direct contact with the solid phase have the same velocity as the solid phase. In other words, their velocity relative to the solid phase is zero. It is therefore convenient to use a reference frame that is fixed with respect to the solid phase. Viscosity forces act on the layers of the liquid that are further apart from the phase boundary, forming a velocity distribution in the liquid, depicted in Fig. 2.21 for liquid flowing along a solid plate. A tangent drawn from the origin to the curve of the dependence of the tangential velocity component, vx, on the distance y from the plate intercepts the straight line vx = Vo at a distance <50. The quantity 50 is termed the thickness of the hydrodynamic or Prandtl layer. It should be pointed out that <50 is a function of the distance from the leading edge of the plate, /. As / increases, the quantity <50 also increases; the liquid also moves in a direction perpendicular to the plate, away from the plate with velocity vy. If diffusion occurs towards the surface of the plate, the motion of the liquid at a distance from the surface inside the hydrodynamic layer is sufficient to maintain the original concentration in the bulk of the solution, c . A decrease of the concentration to a value c* at the phase boundary
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136 (in Fig. 2.21 c* = 0) occurs only in a thin layer within the Prandtl layer, with thickness 6 < d0. The concentration c* is attained in the immediate vicinity of the interface and is determined by its properties (e.g. for transport to an electrode, the electrode potential is decisive see Section 5.3). The quantity S is termed the thickness of the Nernst layer. For a steady-state concentration gradient in the immediate vicinity of the phase boundary we have
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The steady-state distribution of the concentration of the diffusing substance at the phase boundary is termed the Nernst layer.t For the material flux (cf. Eq. 2.5.18) we have Jy=^r (c - c*) = K(C - c*) (2.7.2)
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where D is the diffusion coefficient and K = D/6 is the mass transfer coefficient (this term is employed in chemical and electrochemical engineering). 2.7.2 The theory of convective diffusion The material flux during convective diffusion in an indifferent electrolyte (grad 0 0) can be described by a relationship obtained by combination of Eqs (2.3.18) and (2.3.23): J = Jdif + Jconv = -D grad c + cv (2.7.3) where Jdif is the diffusion and Jconv the convection material flux. Equation (2.2.10) for ip = c and the incompressibility condition divv = 0 (2.7.4) yield -^ = DV2c - v grad c or, for the steady state, DV2c - v grad c = 0 In Cartesian coordinates, this equation becomes (d2c d2c d2c\ ( Be dc y \dx dy dz I \ dx dy (2.7.6) 3c\ dz) (2.7.5)
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f Nernst studied the theory of heterogeneous reactions around the year 1900 and assumed that the flow of a liquid along a phase boundary with a solid phase involves formation of an immobile liquid layer at this surface. A steady-state concentration gradient is then formed within this layer as a result of diffusion in the limited space (see Eq. 2.5.18). In fact, the diffusion process has the character of convective diffusion everywhere as a result of the velocity distribution in the hydrodynamic layer.
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137 The assumption of a steady state corresponds to laminar flow of the liquid. This is fulfilled only under certain conditions (limited velocity of the liquid, smooth phase boundary between the flowing liquid and the other phase, etc.). Otherwise, turbulent flow occurs, where the local velocity depends on time, pulsation of the system, etc. Mathematically the turbulent flow problem is a very difficult task and it is often doubtful whether, in specific cases, it is possible to obtain any solution at all. Liquid flow originates from the effect of various forces. In forced convection, these are mechanical forces acting, e.g. on a piston forcing the liquid into a vessel, or a stirrer transferring the impulse to a liquid in a vessel, etc. In this manner, pressure gradients are formed in the solution, resulting in motion of the liquid. Natural {free) convection results when density changes are produced in the solution as a result of concentration changes produced by transport processes. It is the force of gravity that causes natural convection and produces hydrostatic pressure gradients in solution that are different from conditions in a liquid of constant density that is at rest. The relationship between the motion of a viscous liquid and the mentioned forces is given by the Navier-Stokes equation + (v grad)v = - - grad/ + vV2v + g ^ P Po^ (2.7.8) dt p p where p is the variable liquid density, p 0 is the density of the liquid at rest, v is its kinematic viscosity (v = r]/p), gis the acceleration of gravity and p is the difference between the total pressure and the hydrostatic pressure. The right-hand side of this equation describes the pressure gradient for forced convection, the effect of viscosity forces and the effect of gravitation during natural convection. For sufficiently intense forced convection, the effect of natural convection can be neglected. These conditions are assumed in the subsequent discussion. As it is assumed that the liquid is incompressible, the continuity equation (2.7.4) is valid. In forced convection, the velocity of the liquid must be characterized by a suitable characteristic value V{), e.g. the mean velocity of the liquid flow through a tube or the velocity of the edge of a disk rotating in the liquid, etc. For natural convection, this characteristic velocity can be set equal to zero. The dimension of the system in which liquid flow occurs has a certain characteristic value /, e.g. the length of a tube or the longitudinal dimension of the plate along which the liquid flows or the radius of a disk rotating in the liquid, etc. Solution of the differential equations (2.7.5), (2.7.7) and (2.7.8) should yield the value of the material flux at the phase boundary of the liquid with another phase, where the concentration equals c*. If a rigorous solution to this problem is at all possible, it consists of two parts: (a) Solution of the hydrodynamic part of the problem in order to obtain a
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relationship for the flow velocity as a function of the spatial coordinates (i.e. solution of Eq. (2.7.8) with suitable boundary conditions) and (b) Solution of Eq. (2.7.6) or Eq. (2.7.7) on the basis of a known relationship for v, or for vx, vy and vz This approach is possible only if density gradients are not formed in the solution as a result of transport processes (e.g. in dilute solutions). Otherwise, both differential equations must be solved simultaneously a very difficult task. Two examples will now be given of solution of the convective diffusion problem, transport to a rotating disk as a stationary case and transport to a growing sphere as a transient case. Finally, an engineering approach will be mentioned in which the solution is expressed as a function of dimensionless quantities characterizing the properties of the system. As a rotating disk is a very useful device for many types of electrochemical research, convective diffusion to a rotating disk, treated theoretically by V. G. Levich, will be used here as an example of this type of transport process. Consider a disk in the xz plane, rotating around the y axis with radial velocity co (see Fig. 2.22). If the radius of the disk is sufficiently larger
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Fig. 2.22 Distribution of streamlines of the liquid at the surface of a rotating disk. Above: a side view; below: a view from the direction of the disk axis. (According to V. G. Levich)
139 than the thickness of the hydrodynamic layer, then the change of flow at the edge of the disk can be neglected. Solution of the hydrodynamic part of the problem, carried out by von Karman and Cochrane, indicates that the thickness of the hydrodynamic layer is not a function of the distance from the centre of the disk y; its value in centimetres is given by the relationship
(2.7.9) The velocity of the liquid in the direction perpendicular to the surface of the disk in centimetres per second is described by the equation vy = -0.51co3/2y-y2y2 (2.7.10)
It will be assumed that the concentration gradient in the direction perpendicular to the surface of the disk is much greater than in the radial direction. Then Eq. (2.7.7) reduces to the form dc _ The boundary conditions are