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Transcript
3.3 Electrokinetic phenomena 3.3.1 Introduction In 1808, the Russian chemist Ferdinand Fiodorovich Reuss, a colloid scientist, investigated the behaviour of wet clay. He observed that the application of a potential difference not only caused a flow of electric current, but also a remarkable movement of water towards the negative pole. The transport of a liquid through a porous medium soaked with the liquid itself, with a potential difference applied to the boundaries was subsequently called electroosmosis. In general, this term indicates the movement of a liquid, with respect to a stationary surface, that takes place inside porous media or within capillaries as an effect of an applied electrical field. The pressure necessary to counterbalance the osmotic flux is referred to as electroosmotic. In a second series of experiments, Reuss dipped two tubes filled with water in a layer of wet clay and then introduced two electrodes in the tubes. After having applied a constant potential difference (and therefore an electrical field), he observed a transport of clay particles towards the positive pole in addition to electroosmosis. With this experiment, transport phenomena of systems dispersed in a fluid triggered by an electrical field were shown for the first time. The phenomenon, called electrophoresis, would later Fig. 1. A, schematic illustration of an experiment in which a streaming potential is generated as an effect of the flux of a liquid through a porous sect; B, schematic illustration of Dorn’s experiment demonstrating the existence of a sedimentation potential. A B I VOLUME V / INSTRUMENTS see wide application in the selective separation of different components present in a colloidal dispersion, as well as be studied in all its different aspects. Based on the consideration that a flux of water is usually produced by a hydrostatic head, Reuss also performed an ingenious experiment, ‘opposite’ of the first of the two previously mentioned experiments. As illustrated in Fig. 1 A, he measured the electrical potential difference displayed at the boundary of a porous bed through which a fluid was flowing. In this way, he discovered that a flux of water though a porous membrane or a capillary generated a potential difference called the streaming potential. A fourth phenomenon, the opposite of electrophoresis, was later discovered by Friedrich Ernst Dorn. If quartz particles are permitted to fall in water, as shown in Fig. 1 B, it is observed that an electrical potential difference called sedimentation potential is located between two electrodes at different heights. The phenomena previously described are collectively called electrokinetic phenomena, classified according to Table 1. A peculiar aspect of this phenomena is represented by the fact that they emphasize coupled phenomena, where a certain effect can be caused not only by the force directly I 197 SURFACES AND DISPERSE SYSTEMS Table 1. Classification of electrokinetic phenomena Electrical forces Mechanical forces solid at rest solid in motion solid at rest solid in motion electroosmosis electrophoresis streaming potential sedimentation potential applied to it (for instance, the current generated by an electrical field), but also by forces associated with different effects (for instance, an electric current caused by a pressure difference which typically generates a fluid flow). An important confirmation of this aspect emerged 20 years after Reuss experiments when another group of phenomena was discovered presenting the same type of coupling: thermoelectric phenomena. In particular, Thomas Johann Seebek observed that by heating the ends of a bimetallic couple at different temperatures, an electrical potential difference was generated, whereas Jean-Charles-Athanase Peltier noticed that, inversely, a current transport through the couple caused a heat transfer from one junction to the other. This group of phenomena found a descriptive frame within the context of thermodynamics of irreversible processes. In order to illustrate this aspect, we will refer to the motion of a fluid (for instance, water) through a porous sect or a membrane, expressing the fluxes of electrical charges and of water molecules through the current intensity I and the water volumetric flux JV . Their coupling can be described by the following relationships: [1] I = L11∆ϕ + L12 ∆P JV = L21∆ϕ + L22 ∆P where Df indicates the electrical potential difference and DP the hydrostatic potential difference, whereas parameters L11, L22, L12, L21 are the phenomenological parameters. The Onsager reciprocity relationship is applied to the mixed terms Lij describing coupling by assuming L12⫽L21. From these expressions, it is possible to notice that even if no potential difference is applied (i.e. if Df⫽0), then simply the presence of a pressure difference can produce an electric current. On the other hand, if no pressure difference is applied, the presence of an electromotive force can still generate a water flux by electroosmosis. Moreover the following relationships are valid: [2] J I = L12 = V = L21 ∆P ∆ϕ = 0 ∆ϕ ∆P = 0 Actually, when other information is unavailable, the previous equations are also unable to provide the amount of electric or volumetric flux, as thermodynamics alone is not sufficient to calculate the values of the Lij coefficients. In order to deal with this problem, it is necessary to extensively investigate the influence that electric charges on the surfaces have on the behaviour of fluids in contact with them, as illustrated below. solid and a liquid phase. As such, they apply to systems where the ratio is high between the interphase area and the volume, such as capillaries and porous materials soaked in liquids or dispersions of solid particles in a liquid. In fact, electrokinetic processes are due to the opposite charges on the solid particles and in the liquid. On the solid, the charge is generated by the presence of ions on the surface due to either their selective adsorption from the solution or the ionization of molecules present on the surface itself. These phenomena do not arise in liquids characterized by small values of the dielectric constant, such as chloroform, diethylether and carbon disulphide. On the other hand, these phenomena are observed in polar liquids such as acetone, alcohols and especially water. At the boundary, there is a segregation of positive or negative electric charges perpendicular to the surface itself. For example, on a silica surface in contact with an aqueous solution, there are some hydroxylic groups, derived from SiO2 hydration, that forms silic acid. This compound causes the following ionic dissociation: + 2− H 2SiO3 ⫺ ⫺SiO3 + 2H 䉴 䉳 producing negative charges on the surface that exert an attractive action on hydrogen ions in the solution, forming an electric double layer (Fig. 2). Another example is silver H⫹ H⫹ SiO32⫺ 198 SiO32⫺ SiO32⫺ A H⫹ H⫹ H⫹ H⫹ SiO32⫺ SiO32⫺ H⫹ H⫹ H⫹ H⫹ SiO32⫺ SiO32⫺ H⫹ Electrokinetic phenomena arise from the polarization process that takes place at the contact surface between a H⫹ H⫹ SiO2 [3] 3.3.2 Formation and structure of the electrical double layer H⫹ H⫹ H⫹ SiO32⫺ SiO2 H⫹ SiO32⫺ SiO32⫺ SiO32⫺ H⫹ H⫹ H⫹ H⫹ H⫹ B Fig. 2. Formation of an electric double layer on a silica surface: A, plane surface; B, spherical surface. ENCYCLOPAEDIA OF HYDROCARBONS ELECTROKINETIC PHENOMENA iodide particles suspended in a solution of potassium iodide, the molecules of which are adsorbed on the surface. The study of the characteristics of electrical double layers was conducted by various authors who investigated this problem at different levels. The first and most significant studies, credited to G. Gouy and D.L. Chapman, date back to the beginning of the Twentieth century. These two authors described the surface as an infinite surface on which a continuous electric charge is distributed in contact with a solution containing point-like ions having opposite charges. At an infinite distance from the surface, the electrical potential identifies with that of the solution, whereas when close to the surface, the potential gradually varies until it assumes the values corresponding to the surface itself. In this zone, two regions can be identified. One region includes the ions adsorbed on the surface and the other, called the diffuse region, encompasses the ions present in the solution, whose distribution is determined by the conflict between electrostatic interactions to which they are subjected as well as random thermal movements. In general, ion adsorption produces an electrostatic energy barrier that hinders particle coagulation with the formation of a precipitated phase which is more stable from a thermodynamic point of view. In conclusion, at every interphase there is an electrical double layer present, originating from the asymmetry of the force field involved. In order to describe the characteristics of the double layer in quantitative terms, it is appropriate to refer to a plane surface by simulating the layer of adsorbed ions with a continuous charge distribution. This charge interacts with ions of the opposite charge in the solution. This attraction brings about their accumulation near the surface as opposed to thermal agitation which promotes an even distribution in the solution. Therefore, the distribution of i ions can be expressed by the Boltzmann law: Z y( z ) e Ci ( z ) = Ci0 exp − i k BT where Ci (z) indicates their concentration at distance z from the surface and Ci0 is the concentration in the bulk of the solution; Zi represents the charge of ion i, y(z) its potential energy, kB is the Boltzmann constant, T the temperature and e the electric charge. By combining the previous equation with the Poisson equation, one obtains: infinite distance from the surface, the solution itself must be electrically neutral, there is [8] ∑C Z 0 i i =0 i and it is possible to derive [9] y( z ) = y 0 e − χ z where y0 is the value of the potential at the surface. Parameter c is expressed by [10] χ2 = 8π e2 ε k BT ∑ Ci0 Z i2 i Therefore, the potential decreases exponentially; the 1/c term has the dimensions of a length and represents the width where the surface double layer is basically located. By applying [5] and using the Debye-Hückel approximation, the following expression for the charge density as a function of the coordinate z is obtained: [11] r=− ε d 2y ε 2 0 − χz =− χ ye 4π dz 2 4π An important parameter in this analysis is the surface electric charge density, referring to the unit area s0. Using the previous equation, s0 is expressed by: [12] ∞ ∞ 0 0 σ 0 = − ∫ r( z ) dz = ∫ ε d 2y dz = 4π dz 2 =− ε dy εχ y0 = 4π dz 0 4π It is possible to observe that the surface potential y0 is related to both the surface charge density and the ionic composition of the medium. For example, if c increases, the double layer is compressed, and therefore either s0 increases or y0 decreases. [4] d 2y 4π r =− dz 2 ε which relates electrical potential to the volumetric charge density r Z y( z ) e r=e Ci0 Z i exp − i [6] k BT i and the following non-linear differential equation can be derived: Z y( z ) e d 2y 4π Z i eCi0 exp − i =− [7] 2 k BT ε dz i where e is the dielectric constant of the liquid. An approximated solution of [7], that attributes the dependency of the potential on the z coordinate, can be easily obtained by considering that Ziey(z)ⲐkBT⬍⬍1 (the Debye-Hückel approximation), and using exp(⫺x)⬇1⫺x for x⬍⬍1. Moreover, considering that in the bulk of the solution, i.e. at [5] ∑ ∑ VOLUME V / INSTRUMENTS 3.3.3 The Stern layer. Electrokinetic potential The approach described up to here, where ions are depicted as charged points resulting in a homogeneous distribution of surface charge, was improved by the theory proposed by Otto Stern, who assigned a defined volume to ions, so that the distance of their centres from the surface cannot be less than the radius, as illustrated in Fig. 3. Furthermore, this theory accounts for the fact that at short distances from the surface, there can be chemical interactions between the ions and the atoms of the surface itself, associated with the adsorption phenomena, which occur when the ions reach a distance from the surface comparable to the bond lengths. Essentially, the double layer is divided in two parts separated by a plane (the Stern surface) located at a distance approximately equal to the radius of the hydrated ions. The potential ranges from the value y0 at the surface to the value yd at the Stern surface, and then decays to zero in the diffused layer, in agreement with [9]. If the Stern layer is described as a condenser of width d, the surface charge density can be expressed by the following relationship: [13] ε σ 0 = (y 0 − yδ ) δ 199 SURFACES AND DISPERSE SYSTEMS performing measurements of an electrokinetic nature that involve the relative motion of the solid surface with respect to the liquid. Taking a plane surface encountered by a current of fluid in laminar motion, it is possible to define an ideal plane parallel to it where the shear stress is located and where a rapid viscosity variation occurs. Actually, the real position of this plane is unknown, since it is necessary to add solvent molecules to surface ions. However, it is reasonable to assume that this plane is located very close to the Stern plane and for this reason, the z potential is only marginally smaller than yd (see again Fig. 3). It is often assumed that the values of z and yd are the same, a small error that can, however, become significant for high ionic concentrations. surface Stern plane surface of shear 3.3.4 Theory of electrokinetic phenomena The theory of electrokinetic phenomena involves both the description of the electrical double layer and the fluid dynamics description of motion, and it is therefore relatively complicated. In order to formulate a model exemplifying the situation, it is necessary to refer to a liquid layer of length l containing electrolytes in contact with a plane surface under the influence of an electrical field parallel to the surface itself. Single ions will tend to move dragging the solvent under the influence of an electrical force X, defined as X⫽Df/l. This force is balanced by the friction force present in the liquid. Therefore, in case of a stationary laminar motion, each liquid layer of thickness dz (Fig. 4) moves parallel to the surface with a velocity u that depends on z but remains constant over time. An expression showing the balance between the electrical force acting on the volume and the friction force due to viscous forces is as follows: diffuse layer Stern layer potential y0 yd z 0 d distance z Fig. 3. Schematic representation of the formation of an electric double layer according to Stern with the corresponding graph of the electrostatic potential. [14] where m is the viscosity. Introducing the Poisson equation [5] in this expression, it is possible to derive: [15] Electrokinetic phenomena originate from the fact that a liquid moving tangentially to a surface does not drag along the whole double layer. Only a portion is free to move with it while another part remains anchored to the solid. In this way, a charge separation parallel to the interface is created generating a potential difference. If, conversely, an electrical field is applied, the positive or negative charges generated in the double layer tend to migrate toward the electrodes with the opposite sign. If the solid is at rest, a movement of the liquid phase takes place, as occurs in electroosmosis. If the solid, however, is composed of a dispersion of particles, these tend to move as in electrophoresis. The double layer theory described up to here and widely used to interpret surface phenomena, refers in any case to a static equilibrium condition. Unfortunately it cannot be directly verified by experiments. In fact, there are no methods capable of measuring yd as they would require placing an electrode on the plane which passes through the centre of the first layer of adsorbed atoms. It is possible, however, to determine another quantity, close to yd, called the electrokinetic potential z or simply the zeta potential, by 200 d 2u du du − µ = µ 2 dz X rdz = µ dz z dz z + dz dz − X ε d 2y d 2u =µ 2 2 4π dz dz y0 u(z) z y(z) d dz z Fig. 4. Graph showing electrostatic potential and fluid velocity for a liquid in contact with a plane surface and in tangential motion with respect to it, as a function of the coordinate z perpendicular to the wall. ENCYCLOPAEDIA OF HYDROCARBONS ELECTROKINETIC PHENOMENA Boundary conditions reflect the values that the potential and the velocity of the fluid must have in the bulk of the fluid at distance d from the surface which limits the zone in which the liquid is stationary. Therefore, they can be written as: y ⫽0 u ⫽ue y ⫽z u ⫽0 du/dz ⫽0 for z⫺⬁ 䉴 for z ⫽d where ue is the velocity in the bulk of the fluid. Integrating [15] is relatively easy and, accounting for boundary conditions, it permits the following relationships: ε Xζ [16] ue = 4πµ 4πµue [17] ζ = εX The liquid flow is given by the product ueA⫽JV, where A is the area of the layer. Keeping in mind that the field intensity X is given by the ratio between the potential difference and the length l of the medium where this difference is applied, and using the Ohm relationship to express the current intensity where the electric conductivity k of the medium is introduced, then: ∆ϕ l and equation [17] becomes: [18] [19] I = kA ζ= 4πµkJV εI This relationship, known as the HelmholtzSmoluchowski equation, gives a linear relationship between the liquid flow rate and the zeta potential and, together with [16], plays an important role in the study of electrokinetic phenomena. Since it does not contain the characteristic geometric parameters of the system under investigation, this expression offers a way to derive the value of the zeta potential directly from the measured values of JV and I. Its validity was confirmed by experimental results, showing that the current intensity is proportional to the volumetric flow. Actually, Ohm’s law is not strictly valid for the systems under investigation since the larger concentration of ions in the diffused part of the double layer results in higher electric conductivity as compared to the bulk of the fluid. Accounting for both effects and in the case of a pipe, it is possible to define an effective conductivity keff through this relationship [20] keff = k + b k A s where k is the conductivity in the bulk of the fluid and ks the conductivity on the surface, A the pipe area and b its perimeter. Based on this last relationship, previous equations should be appropriately modified, even if for capillaries with a relatively large radius, the correction needed due to the effect of surface conductivity is basically negligible. The approach just described refers to a simplified geometrical situation where no effect is associated with the surface curvature. In order to examine this aspect more in depth, it is convenient to refer to a capillary with a radius equal to r0 and a length equal to l, containing a liquid to which a potential difference Df is applied, acting VOLUME V / INSTRUMENTS along its z axis. In this situation, the Poisson equation [5] and the force balance [14] should be modified accordingly. If r indicates the radial coordinate perpendicular to the capillary axis with its origin on it, considering the cylindrical symmetry of the system, the two equations take this form: [21] 1 ⭸ ⭸y 4π r r =− r ⭸r ⭸r ε [22] −∇Pz = µ ⭸ ⭸u z r − Xr r ⭸r ⭸r Generally speaking, in [22], which corresponds to [14], a pressure gradient along the cylinder axis was also introduced. This approach allows for a unified vision for the complementary phenomena: electroosmosis and an electric current generated by a liquid flow. Previous equations should satisfy the above mentioned boundary conditions for r⫽r0⫺d, whereas for r⫽0, i.e. on the cylinder axis, uz and y are both finite, with (⭸yⲐ⭸r)0⫽0. Although the integration is relatively complex, by applying the Debye-Hückel equation, it can be performed analytically, permitting the following expression for the current intensity and liquid flow rate, respectively: [23] [24] 2 σ δ k X = + µ π r02 Is ( ) ( ) ( ) I 0 R0 I 2 R0 + 1 − I12 R0 εζ I 2 R0 + ∇P 4πµ I1 R0 ( ) ( ) JV X εζ I 2 ( R0 ) r02 − ∇P = 2 π r0 4πµ I1 ( R0 ) 8π I0, I1 and I2 are the modified Bessel functions of the first type of the zeroeth, first and second order, R0⫽r0 c, whereas sd is the surface charge calculated through equation [12], using yd rather than y0. The previous equations offer a general solution to the problem, and specifically refer to electrophoresis if ⵜP⫽0, and to a streaming potential if X⫽0. 3.3.5 Electroosmosis As seen above, electroosmosis appears as a flux of a liquid containing electrolytes when a potential difference Df is applied to it. Ignoring the surface curvature, fluid motion can be described by equation [16], and the liquid flow can be expressed as: [25] JV = ueo A = A∆ϕεζ 4πµl where ueo is now defined as electroosmotic velocity. This equation can be obtained with [24] if ⵜP⫽0 and if a high value is assigned to the duct radius so that the radius I2(R0)ⲐI1(R0) tends to unity. Expression [25] is also applied for the calculation of fluxes through porous materials. In the case of a circular duct with a radius equal to r0, and accounting for the fact that there is a conductivity difference between the surface layer and the liquid bulk, taking into account equation [20], the [25] should be modified as follows: 201 SURFACES AND DISPERSE SYSTEMS [26] JV = Aεζ I 2k 4πµ k + s r particle moves at constant velocity. In developing this analysis, Erich Armand Hückel derived the following expression of mobility: 0 Electroosmosis can be adopted for the solution of practical problems, related in particular to the dehydration of porous materials. To this purpose, a humid mass is located between two electrodes connected to the opposite poles of an external source of electric current. The current causes a transfer of water towards the cathode where it is removed by pumping. In the meantime, the anodic mass is compressed. The electroosmotic phenomenon is ideal for reconditioning humid rooms, particularly in restoration operations. In practice, two electrodes, positive and negative, are installed in the walls and in the floor connected with an electric generator that delivers a continuous low tension electrical current. These applications, however, are rare since this is a slow and expensive process. Techniques involving the use of electroosmosis to remove water from oil were also studied. 3.3.6 Electrophoresis Particles representing a dispersed phase in a liquid whose charge is due to the formation of a double layer on the surface, show the behaviour described in Fig. 5 when subjected to the action of an electric field. Under the influence of this field, each particle moves together with the compact layer of ions present in the double layer, whereas the diffuse ionic atmosphere tends to move in the opposite direction. Choosing a system of coordinates fixed on the particle, and neglecting the geometrical factors related to its shape, the situation is identical in principle to that previously seen when describing electroosmosis and therefore, in first approximation, it is legitimate to apply equation [16]. This is confirmed by many experiences revealing the existence of a proportionality between the electrical force and particle velocity, which in this case is indicated by uep, called the electrophoretic velocity. Another parameter is also used called the electrophoretic mobility (v), equal to the particle velocity under the influence of a unit force: uep εζ = [27] v = X 4πµ An alternative way to tackle the problem consists in focusing attention on a spherical charged particle, which moves in a fluid as an effect of an electrical field. In stationary conditions, the force of electrical nature acting on the particle is balanced by viscous friction forces, which can be expressed by Stokes’ law, where the Fig. 5. Variation of ion distribution as an effect of the motion of a charged particle in a liquid. 202 [28] v= εζ 6πµ which, as seen, differs from [27] for a numerical factor equal to 2/3. The previous equation was derived by assuming that the electrical conductivity of the particle is equal to that of the medium and that its dimensions are small with respect to the width of the double layer. It is included in the description of electrophoretic phenomena occurring in non-aqueous solutions. In a more detailed analysis, D.C. Henry explicitly accounted for the geometrical shape of the particle in the configuration of the electric field that forms around it. In a quantitative approach, Henry suggested modifying equation [16] as follows: [29] uep = ε Xζ r f πµ δ where a complicated function ( f ) of the r/d ratio between the particle radius and the width of the diffuse layer appears. If a constant value equal to 0.25 is attributed to it, [29] identifies with [16]. Another situation, called the relaxation effect, is associated with the disturbance of the symmetry of the electrical field in the diffuse layer. The cause is due to polarization decreasing the effective value of the electric force which, in turn, influences the parameters determined by the electric force. This results in a retardation due to the opposite flux of counterions which causes additional friction. Actually, thanks to the electrical conductivity increase in the double layer and the diffusive processes in it, the system tends to recover the symmetrical configuration, requiring a relaxation time which depends on the electrokinetic potential, the product of the electric conductivity multiplied by the ion dimensions, and the valence of the electrolytes present in the system. An important field of application of electrophoresis is the separation of high molecular weight complex organic compounds in a solution. The difference between their migration velocities is due to their charges and dimensions. This approach proved very useful in biochemistry, particularly for the classification of proteins in blood plasma. There are several different analytical chemistry techniques that perform this operation. 3.3.7 Streaming potential When an electrolytic solution flows through a capillary due to pressure difference, the presence of an electrical potential difference is measured between two electrodes located at the ends of the duct. This streaming potential occurs when the fluid flow transports the ions present in the mobile part of the electric double layer that forms close to the capillary surface. The potential difference Df along the capillary axis therefore generates a streaming current Is. For a cylindrical capillary with a radius r0 and a length l, this current has the following form: ENCYCLOPAEDIA OF HYDROCARBONS ELECTROKINETIC PHENOMENA r0 [30] I s = 2π ∫ u(r ) r(r )dr 3.3.8 Sedimentation potential 0 where r(r) usually expresses the charge density as a function of the radial coordinate r, while u(r) represents the liquid velocity as a function of r, which can be expressed by the Poiseuille equation, by integrating the Navier-Stokes equation for the stationary laminar motion in a circular duct: [31] [37] ∆P 2 u (r ) = (r0 − r 2 ) 4 µl By neglecting the effect of curvature, r(r) can be expressed by the equation [11]. In this case, the integration of [30] is relatively easy, resulting in the following expression: εζ∆PA Is = [32] 4πµl By expressing the current intensity using Ohm’s law, it is possible to derive the following equation for the difference in potential measured at the ends of the capillary: [33] ∆ϕ = εζ∆P 4πµk In stationary conditions, current Is must be balanced by a current I which returns the charge to the system through the liquid bulk and its surface, shown by ∆ϕ π r 2 k + 2π r0 ks l Since Is⫹I⫽0 in a stationary condition, equation [33] is more appropriately written as: εζ∆P [35] ∆ϕ = 2k 4πµ k + s r [34] I= ) ( 0 Finally, it is interesting to observe that by applying thermodynamics of irreversible processes and accounting for the Onsager reciprocity condition, on the basis of equation [3], it is possible to obtain [36] I s ∆P = JV ∆ϕ Since JV⫽Aue, if equation [16], obtained by applying the Helmholtz-Smoluchowski model, is attributed to ue, it is easy to verify that the expression of Is identifies with [32], thereby confirming the compatibility of the two approaches. VOLUME V / INSTRUMENTS During the deposition of solid particles in a fluid, the presence of a potential difference can be measured as shown in Fig. 1 B. In this case, pressure is replaced by gravity, and the force acting on a spherical particle of radius r is therefore represented by the following relationship: mg = 4 3 π r (d − d0 )g 3 where d and d0 are the density of the solid and the fluid, respectively; g is the acceleration of gravity; and m is the apparent mass of the particle. By replacing the previous equation with [33] and in the case of a swarm of particles with density n (number of particles per unit volume), the following expression is obtained for the electrical potential difference measured between two electrodes located at the ends of a column of height l: [38] ∆y = r 3 ( d − d0 ) gnεζ l 3µk Actually, the application of this relationship is not always easy, as real systems are normally polydispersed, and furthermore, particles are not spherical. An additional complication is due to the difference of the velocities for the various particles. Even though they have not found specific industrial applications, sedimentation potentials are of remarkable interest as they take parte in atmospheric discharges. Bibliography Fridrikhsberg D.A. (1986) A course in colloid chemistry, Moscow, Mir. Newman J.S. (1973) Electrochemical systems, Englewood Cliffs (NJ), Prentice-Hall. Shaw D.J. (1970) Introduction to colloid chemistry and surface chemistry, London, Butterworths. Voyutsky S.S. (1978) Colloid chemistry, Moscow, Mir. Sergio Carrà Dipartimento di Chimica, Materiali e Ingegneria Chimica ‘Giulio Natta’ Politecnico di Milano Milano, Italy 203