3.3.1 Introduction see wide application in the selective separation of different components present in a colloidal dispersion, as well as be In 1808, the Russian chemist Ferdinand Fiodorovich Reuss, studied in all its different aspects. a colloid scientist, investigated the behaviour of wet clay. He Based on the consideration that a flux of water is usually observed that the application of a potential difference not produced by a hydrostatic head, Reuss also performed an only caused a flow of electric current, but also a remarkable ingenious experiment, ‘opposite’ of the first of the two movement of water towards the negative pole. The transport previously mentioned experiments. As illustrated in Fig. 1 A, of a liquid through a porous medium soaked with the liquid he measured the electrical potential difference displayed at itself, with a potential difference applied to the boundaries the boundary of a porous bed through which a fluid was was subsequently called electroosmosis. In general, this term flowing. In this way, he discovered that a flux of water indicates the movement of a liquid, with respect to a though a porous membrane or a capillary generated a stationary surface, that takes place inside porous media or potential difference called the streaming potential. within capillaries as an effect of an applied electrical field. A fourth phenomenon, the opposite of electrophoresis, The pressure necessary to counterbalance the osmotic flux is was later discovered by Friedrich Ernst Dorn. If quartz referred to as electroosmotic. particles are permitted to fall in water, as shown in Fig. 1 B, it In a second series of experiments, Reuss dipped two is observed that an electrical potential difference called tubes filled with water in a layer of wet clay and then sedimentation potential is located between two electrodes at introduced two electrodes in the tubes. After having applied different heights. a constant potential difference (and therefore an electrical The phenomena previously described are collectively field), he observed a transport of clay particles towards the called electrokinetic phenomena, classified according to positive pole in addition to electroosmosis. With this Table 1. experiment, transport phenomena of systems dispersed in a A peculiar aspect of this phenomena is represented by fluid triggered by an electrical field were shown for the first the fact that they emphasize coupled phenomena, where a time. The phenomenon, called electrophoresis, would later certain effect can be caused not only by the force directly
Fig. 1. A, schematic illustration of an experiment AB 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.
II
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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 solid and a liquid phase. As such, they apply to systems electrical field), but also by forces associated with different where the ratio is high between the interphase area and the effects (for instance, an electric current caused by a pressure volume, such as capillaries and porous materials soaked in difference which typically generates a fluid flow). An liquids or dispersions of solid particles in a liquid. In fact, important confirmation of this aspect emerged 20 years after electrokinetic processes are due to the opposite charges on Reuss experiments when another group of phenomena was the solid particles and in the liquid. On the solid, the charge discovered presenting the same type of coupling: is generated by the presence of ions on the surface due to thermoelectric phenomena. In particular, Thomas Johann either their selective adsorption from the solution or the Seebek observed that by heating the ends of a bimetallic ionization of molecules present on the surface itself. These couple at different temperatures, an electrical potential phenomena do not arise in liquids characterized by small difference was generated, whereas Jean-Charles-Athanase values of the dielectric constant, such as chloroform, Peltier noticed that, inversely, a current transport through the diethylether and carbon disulphide. On the other hand, these couple caused a heat transfer from one junction to the other. phenomena are observed in polar liquids such as acetone, This group of phenomena found a descriptive frame alcohols and especially water. within the context of thermodynamics of irreversible At the boundary, there is a segregation of positive or processes. In order to illustrate this aspect, we will refer to negative electric charges perpendicular to the surface itself. the motion of a fluid (for instance, water) through a porous For example, on a silica surface in contact with an aqueous sect or a membrane, expressing the fluxes of electrical solution, there are some hydroxylic groups, derived from
charges and of water molecules through the current intensity SiO2 hydration, that forms silic acid. This compound causes I and the water volumetric flux JV . Their coupling can be the following ionic dissociation:
described by the following relationships: ᭤ 2−+ Ϫ᭣ + HSiO23 Ϫ SiO 3 2H [1] IL=+∆∆ϕ LP 11 12 producing negative charges on the surface that exert an [2] =+∆∆ϕ JLV 21 LP 22 attractive action on hydrogen ions in the solution, forming an where Df indicates the electrical potential difference and electric double layer (Fig. 2). Another example is silver DP the hydrostatic potential difference, whereas parameters L , L , L , L are the phenomenological parameters. The ϩ 11 22 12 21 H ϩ ϩ Onsager reciprocity relationship is applied to the mixed H H Hϩ ϩ terms L describing coupling by assuming L ϭL . From H ϩ ij 12 21 2Ϫ 2Ϫ H 2Ϫ these expressions, it is possible to notice that even if no SiO3 SiO3 SiO3 potential difference is applied (i.e. if Dfϭ0), then simply the presence of a pressure difference can produce an electric SiO2 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 A following relationships are valid:
I J ϩ [3] ==L V = L H ϩ ∆∆P 12 ϕ 21 H ∆ϕ =0 ∆P=0 ϩ ϩ ϩ H H 2Ϫ H ϩ Actually, when other information is unavailable, the SiO3 H Ϫ Ϫ previous equations are also unable to provide the amount of SiO2 SiO2 3 3 ϩ electric or volumetric flux, as thermodynamics alone is not Hϩ H sufficient to calculate the values of the L coefficients. In Ϫ ij 2Ϫ SiO2 order to deal with this problem, it is necessary to extensively SiO3 SiO2 3 investigate the influence that electric charges on the surfaces ϩ ϩ H H have on the behaviour of fluids in contact with them, as 2Ϫ 2Ϫ SiO3 SiO3 illustrated below. 2Ϫ ϩ Hϩ Hϩ SiO3 H Hϩ ϩ Hϩ H 3.3.2 Formation and structure of the electrical double layer B
Electrokinetic phenomena arise from the polarization Fig. 2. Formation of an electric double layer on a silica surface: process that takes place at the contact surface between a A, plane surface; B, spherical surface.
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iodide particles suspended in a solution of potassium iodide, infinite distance from the surface, the solution itself must be the molecules of which are adsorbed on the surface. electrically neutral, there is The study of the characteristics of electrical double 0 = [8] ∑CZii 0 layers was conducted by various authors who investigated i this problem at different levels. The first and most and it is possible to derive − χ significant studies, credited to G. Gouy and D.L. Chapman, [9] yy()z = 0e z date back to the beginning of the Twentieth century. These two authors described the surface as an infinite surface on where y0 is the value of the potential at the surface. which a continuous electric charge is distributed in contact Parameter c is expressed by with a solution containing point-like ions having opposite 8πe2 charges. At an infinite distance from the surface, the [10] χ 2 = ∑ CZ02 εkT ii electrical potential identifies with that of the solution, B i whereas when close to the surface, the potential gradually Therefore, the potential decreases exponentially; the 1/c varies until it assumes the values corresponding to the term has the dimensions of a length and represents the width surface itself. In this zone, two regions can be identified. where the surface double layer is basically located. One region includes the ions adsorbed on the surface and the By applying [5] and using the Debye-Hückel other, called the diffuse region, encompasses the ions present approximation, the following expression for the charge in the solution, whose distribution is determined by the density as a function of the coordinate z is obtained: conflict between electrostatic interactions to which they are 2 ε d y ε − χ subjected as well as random thermal movements. In general, [11] r =− =− χ 20y e z π 2 π ion adsorption produces an electrostatic energy barrier that 44dz hinders particle coagulation with the formation of a An important parameter in this analysis is the surface
precipitated phase which is more stable from a electric charge density, referring to the unit area s0. Using thermodynamic point of view. In conclusion, at every the previous equation, s0 is expressed by: interphase there is an electrical double layer present, ∞ ∞ ε d 2y originating from the asymmetry of the force field involved. [12] σ =−r()zdz =dz = 0 ∫ ∫ π 2 In order to describe the characteristics of the double 0 0 4 dz layer in quantitative terms, it is appropriate to refer to a ε dy εχy0 =− = plane surface by simulating the layer of adsorbed ions with a 4π dz 4π continuous charge distribution. This charge interacts with 0 ions of the opposite charge in the solution. This attraction It is possible to observe that the surface potential y0 is brings about their accumulation near the surface as opposed related to both the surface charge density and the ionic to thermal agitation which promotes an even distribution in composition of the medium. For example, if c increases, the
the solution. Therefore, the distribution of i ions can be double layer is compressed, and therefore either s0 increases expressed by the Boltzmann law: or y0 decreases. Zzey() [4] Cz()=− C0 exp i ii kT B 3.3.3 The Stern layer. where Ci(z) indicates their concentration at distance z from Electrokinetic potential 0 the surface and Ci is the concentration in the bulk of the solution; Zi represents the charge of ion i, y(z) its potential The approach described up to here, where ions are depicted energy, kB is the Boltzmann constant, T the temperature and as charged points resulting in a homogeneous distribution of e the electric charge. By combining the previous equation surface charge, was improved by the theory proposed by with the Poisson equation, one obtains: Otto Stern, who assigned a defined volume to ions, so that the distance of their centres from the surface cannot be less d 2yr4π [5] =− than the radius, as illustrated in Fig. 3. Furthermore, this 2 ε dz theory accounts for the fact that at short distances from the which relates electrical potential to the volumetric charge surface, there can be chemical interactions between the ions density r and the atoms of the surface itself, associated with the adsorption phenomena, which occur when the ions reach a 0 Zzeiy() [6] r =−eCZ∑ exp distance from the surface comparable to the bond lengths. ii kT i B Essentially, the double layer is divided in two parts separated and the following non-linear differential equation can be by a plane (the Stern surface) located at a distance derived: approximately equal to the radius of the hydrated ions. The 2 potential ranges from the value y0 at the surface to the value d y 4π 0 Zzey() [7] =−∑ ZeC exp − i y at the Stern surface, and then decays to zero in the 2 ε i i kT d dz i B diffused layer, in agreement with [9]. where e is the dielectric constant of the liquid. An If the Stern layer is described as a condenser of width d, approximated solution of [7], that attributes the dependency the surface charge density can be expressed by the following of the potential on the z coordinate, can be easily obtained by relationship: considering that Z ey(z)րk TϽϽ1 (the Debye-Hückel i B ε approximation), and using exp(Ϫx)Ϸ1Ϫx for xϽϽ1. 0 [13] σ =−()yyδ Moreover, considering that in the bulk of the solution, i.e. at 0 δ
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surface performing measurements of an electrokinetic nature that Stern plane involve the relative motion of the solid surface with respect surface of shear 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.
3.3.4 Theory of electrokinetic phenomena
The theory of electrokinetic phenomena involves both the diffuse layer description of the electrical double layer and the fluid Stern layer dynamics description of motion, and it is therefore relatively complicated. In order to formulate a model exemplifying the y0 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 XϭDf/l. This force is balanced by the friction force present y in the liquid. Therefore, in case of a stationary laminar
potential d z 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: 0 du du du2 d z Xdzr = µµµ − = dz distance [14] 2 dz zdz+ dz z dz Fig. 3. Schematic representation of the formation where m is the viscosity. Introducing the Poisson equation [5] of an electric double layer according to Stern in this expression, it is possible to derive: with the corresponding graph of the electrostatic potential. Xdε 2y du2 [15] −=µ 4π dz2 dz2 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 y0 generating a potential difference. If, conversely, an electrical u(z) field is applied, the positive or negative charges generated in the double layer tend to migrate toward the electrodes with z 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 y(z) static equilibrium condition. Unfortunately it cannot be directly verified by experiments. In fact, there are no d dz z methods capable of measuring yd as they would require placing an electrode on the plane which passes through the Fig. 4. Graph showing electrostatic potential centre of the first layer of adsorbed atoms. It is possible, and fluid velocity for a liquid in contact with a plane surface however, to determine another quantity, close to yd, called and in tangential motion with respect to it, as a function the electrokinetic potential z or simply the zeta potential, by of the coordinate z perpendicular to the wall.
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Boundary conditions reflect the values that the potential along its z axis. In this situation, the Poisson equation [5] and the velocity of the fluid must have in the bulk of the fluid and the force balance [14] should be modified at distance d from the surface which limits the zone in which accordingly. If r indicates the radial coordinate the liquid is stationary. Therefore, they can be written as: perpendicular to the capillary axis with its origin on it, y ϭ0 u ϭu du/dz ϭ0 for zϪ᭤ϱ considering the cylindrical symmetry of the system, the e two equations take this form: y ϭz u ϭ0 for z ϭd 14Ѩ Ѩyr π where u is the velocity in the bulk of the fluid. [21] r =− e Ѩ Ѩ ε Integrating [15] is relatively easy and, accounting for rr r boundary conditions, it permits the following relationships: µ Ѩ Ѩu −∇P = r z − X εζX [22] z r [16] u = rrѨ Ѩr e 4πµ Generally speaking, in [22], which corresponds to [14], a 4πµu ζ = e pressure gradient along the cylinder axis was also [17] ε X introduced. This approach allows for a unified vision for the ϭ The liquid flow is given by the product ueA JV, where A complementary phenomena: electroosmosis and an electric is the area of the layer. Keeping in mind that the field current generated by a liquid flow. intensity X is given by the ratio between the potential Previous equations should satisfy the above mentioned ϭ Ϫ ϭ difference and the length l of the medium where this boundary conditions for r r0 d, whereas for r 0, i.e. on Ѩ րѨ ϭ difference is applied, and using the Ohm relationship to the cylinder axis, uz and y are both finite, with ( y r)0 0. express the current intensity where the electric conductivity Although the integration is relatively complex, by applying k of the medium is introduced, then: the Debye-Hückel equation, it can be performed analytically, ∆ϕ permitting the following expression for the current intensity [18] IA= k and liquid flow rate, respectively: l 2 I σ IRIR( ) ( ) and equation [17] becomes: [23] s =+−X k δ 1 0020 + πr 2 µ IR2 ( ) 4πµkJ 0 1 0 [19] ζ = V εI εζ IR( ) + 20∇P This relationship, known as the Helmholtz- 4πµ ( ) IR10 Smoluchowski equation, gives a linear relationship between the liquid flow rate and the zeta potential and, J Xεζ IR( ) r2 [24] V = 20−∇0 P together with [16], plays an important role in the study of πr2 48πµIR( ) π electrokinetic phenomena. Since it does not contain the 0 10
characteristic geometric parameters of the system under I0, I1 and I2 are the modified Bessel functions of the first ϭ investigation, this expression offers a way to derive the type of the zeroeth, first and second order, R0 r0 c, whereas value of the zeta potential directly from the measured sd is the surface charge calculated through equation [12], 0 values of JV and I. Its validity was confirmed by using yd rather than y . experimental results, showing that the current intensity is The previous equations offer a general solution to the ,proportional to the volumetric flow. problem, and specifically refer to electrophoresis if ٌPϭ0 Actually, Ohm’s law is not strictly valid for the systems and to a streaming potential if Xϭ0. 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. 3.3.5 Electroosmosis Accounting for both effects and in the case of a pipe, it is
possible to define an effective conductivity keff through this As seen above, electroosmosis appears as a flux of a liquid relationship containing electrolytes when a potential difference Df is applied to it. Ignoring the surface curvature, fluid motion =+b [20] kkkeff s can be described by equation [16], and the liquid flow can be A expressed as: where k is the conductivity in the bulk of the fluid and ks A∆ϕεζ JuA== the conductivity on the surface, A the pipe area and b its [25] Veo πµ perimeter. Based on this last relationship, previous 4 l
equations should be appropriately modified, even if for where ueo is now defined as electroosmotic velocity. This capillaries with a relatively large radius, the correction equation can be obtained with [24] if ٌPϭ0 and if a high needed due to the effect of surface conductivity is value is assigned to the duct radius so that the radius ր basically negligible. I2(R0) I1(R0) tends to unity. Expression [25] is also applied The approach just described refers to a simplified for the calculation of fluxes through porous materials. In the
geometrical situation where no effect is associated with case of a circular duct with a radius equal to r0, and the surface curvature. In order to examine this aspect more accounting for the fact that there is a conductivity difference in depth, it is convenient to refer to a capillary with a between the surface layer and the liquid bulk, taking into
radius equal to r0 and a length equal to l, containing a account equation [20], the [25] should be modified as liquid to which a potential difference Df is applied, acting follows:
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AIεζ particle moves at constant velocity. In developing this [26] J = V 2k analysis, Erich Armand Hückel derived the following 4πµ k + s expression of mobility: r 0 εζ v = Electroosmosis can be adopted for the solution of [28] πµ practical problems, related in particular to the dehydration 6 of porous materials. To this purpose, a humid mass is which, as seen, differs from [27] for a numerical factor equal located between two electrodes connected to the opposite to 2/3. The previous equation was derived by assuming that poles of an external source of electric current. The current the electrical conductivity of the particle is equal to that of causes a transfer of water towards the cathode where it is the medium and that its dimensions are small with respect to removed by pumping. In the meantime, the anodic mass is the width of the double layer. It is included in the description compressed. The electroosmotic phenomenon is ideal for of electrophoretic phenomena occurring in non-aqueous reconditioning humid rooms, particularly in restoration solutions. operations. In practice, two electrodes, positive and In a more detailed analysis, D.C. Henry explicitly negative, are installed in the walls and in the floor accounted for the geometrical shape of the particle in the connected with an electric generator that delivers a configuration of the electric field that forms around it. In a continuous low tension electrical current. These quantitative approach, Henry suggested modifying equation applications, however, are rare since this is a slow and [16] as follows: expensive process. Techniques involving the use of εζX r u = f electroosmosis to remove water from oil were also studied. [29] ep πµ δ where a complicated function ( f ) of the r/d ratio between 3.3.6 Electrophoresis the particle radius and the width of the diffuse layer appears. If a constant value equal to 0.25 is attributed to it, [29] Particles representing a dispersed phase in a liquid whose identifies with [16]. charge is due to the formation of a double layer on the Another situation, called the relaxation effect, is surface, show the behaviour described in Fig. 5 when associated with the disturbance of the symmetry of the subjected to the action of an electric field. Under the electrical field in the diffuse layer. The cause is due to influence of this field, each particle moves together with the polarization decreasing the effective value of the electric compact layer of ions present in the double layer, whereas force which, in turn, influences the parameters the diffuse ionic atmosphere tends to move in the opposite determined by the electric force. This results in a direction. Choosing a system of coordinates fixed on the retardation due to the opposite flux of counterions which particle, and neglecting the geometrical factors related to its causes additional friction. Actually, thanks to the shape, the situation is identical in principle to that previously electrical conductivity increase in the double layer and seen when describing electroosmosis and therefore, in first the diffusive processes in it, the system tends to recover approximation, it is legitimate to apply equation [16]. This is the symmetrical configuration, requiring a relaxation confirmed by many experiences revealing the existence of a time which depends on the electrokinetic potential, the proportionality between the electrical force and particle product of the electric conductivity multiplied by the ion
velocity, which in this case is indicated by uep, called the dimensions, and the valence of the electrolytes present in electrophoretic velocity. Another parameter is also used the system. called the electrophoretic mobility (v), equal to the particle An important field of application of electrophoresis is velocity under the influence of a unit force: the separation of high molecular weight complex organic compounds in a solution. The difference between their uep εζ [27] v == migration velocities is due to their charges and dimensions. X 4πµ This approach proved very useful in biochemistry, An alternative way to tackle the problem consists in particularly for the classification of proteins in blood focusing attention on a spherical charged particle, which plasma. There are several different analytical chemistry moves in a fluid as an effect of an electrical field. In techniques that perform this operation. 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 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 Fig. 5. Variation of ion distribution as an effect of the motion capillary with a radius r0 and a length l, this current has the of a charged particle in a liquid. following form:
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r0 = π 3.3.8 Sedimentation potential [30] Iurrdrs 2 ∫ ()()r 0 During the deposition of solid particles in a fluid, the where r(r) usually expresses the charge density as a function presence of a potential difference can be measured as shown of the radial coordinate r, while u(r) represents the liquid in Fig. 1 B. In this case, pressure is replaced by gravity, and velocity as a function of r, which can be expressed by the the force acting on a spherical particle of radius r is Poiseuille equation, by integrating the Navier-Stokes therefore represented by the following relationship: equation for the stationary laminar motion in a circular duct: 4 mg=−π r3() d d g ∆P [37] 0 [31] ur()=− (rr22 ) 3 4µl 0 where d and d0 are the density of the solid and the fluid, By neglecting the effect of curvature, r(r) can be respectively; g is the acceleration of gravity; and m is the expressed by the equation [11]. In this case, the integration apparent mass of the particle. of [30] is relatively easy, resulting in the following By replacing the previous equation with [33] and in the expression: case of a swarm of particles with density n (number of εζ∆ particles per unit volume), the following expression is = PA [32] Is obtained for the electrical potential difference measured 4πµl between two electrodes located at the ends of a column of By expressing the current intensity using Ohm’s law, it is height l: possible to derive the following equation for the difference in rd3()− dgnεζ potential measured at the ends of the capillary: [38] ∆y = 0 l 3µk εζ∆P [33] ∆ϕ = Actually, the application of this relationship is not always πµ 4 k easy, as real systems are normally polydispersed, and
In stationary conditions, current Is must be balanced by a furthermore, particles are not spherical. An additional current I which returns the charge to the system through the complication is due to the difference of the velocities for the liquid bulk and its surface, shown by various particles. ∆ϕ Even though they have not found specific industrial [34] =+(ππ2 ) applications, sedimentation potentials are of remarkable I rrkk2 0 s l interest as they take parte in atmospheric discharges. ϩ ϭ Since Is I 0 in a stationary condition, equation [33] is more appropriately written as: εζ∆P Bibliography [35] ∆ϕ = 2k 4πµ k + s Fridrikhsberg D.A. (1986) A course in colloid chemistry, Moscow, r0 Mir. Finally, it is interesting to observe that by applying Newman J.S. (1973) Electrochemical systems, Englewood Cliffs (NJ), thermodynamics of irreversible processes and accounting for Prentice-Hall. the Onsager reciprocity condition, on the basis of equation Shaw D.J. (1970) Introduction to colloid chemistry and surface [3], it is possible to obtain chemistry, London, Butterworths. I ∆P Voyutsky S.S. (1978) Colloid chemistry, Moscow, Mir. [36] s = ∆ϕ JV Sergio Carrà ϭ Since JV Aue, if equation [16], obtained by applying the Dipartimento di Chimica, Materiali Helmholtz-Smoluchowski model, is attributed to ue, it is e Ingegneria Chimica ‘Giulio Natta’ easy to verify that the expression of Is identifies with [32], Politecnico di Milano thereby confirming the compatibility of the two approaches. Milano, Italy
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