Electrostatic Double Layer Force: Part III

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 Electrostatic Double Layer Force: Part III Dr. Pallab Ghosh Associate Professor Department of Chemical Engineering IIT Guwahati, Guwahati 781039 India Joint Initiative of IITs and IISc Funded by MHRD 1/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 Table of Contents Section/Subsection Page No. 3.4.1 Stern layer 3 3.4.2 Electrostatic double layer around spherical particles 6 3.4.3 Electrostatic double layer repulsion between two surfaces 7 11 3.4.3.1 Derjaguin approximation 11 3.4.4 Zeta potential 12 3.4.5 Limitations of the Poisson Boltzmann equation 15 Exercise 17 Suggested reading 18 Joint Initiative of IITs and IISc Funded by MHRD 2/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 3.4.1 Stern layer This part of the electrostatic double layer is concerned with the layer of adsorbed ions on the surface. It is also known as the compact part of the double layer. The GouyChapman theory predicts that the ions can approach the surface charge without limit. Let us consider an aqueous solution of NaCl of 1 mol/m 3 concentration. The surface potential is 300 mv and the temperature is 298 K. The concentration of counterions near the surface, assuming the absence of the Stern layer, would be, 19 e 0 1.60210 0.3 5 cc exp 1 exp 1.18 10 kt mol/m 3 23 1.38110 298 Since the ions have finite dimensions, the concentration at the surface found by the above calculation is too high! Such a high value was obtained because of the assumption that the ions have point charge and they can approach the surface without any limit. Furthermore, the value of surface potential, 0, can be several hundreds of millivolts but the value of d is much lower than this value as a consequence of the Stern layer. The result of the above calculation shows that the GouyChapman theory becomes inadequate when a considerable part of the space charge should be present very close (e.g., 0.5 nm) to the interface. Stern did not distinguish between the inner and outer Helmholtz planes, but he mentioned the possibility that such a distinction might be necessary. Let be the number of adsorbed ions per unit area of the surface, 0 be the number of ions per unit volume of solution, be the maximum number ions that can be on unit area of the surface and 0 be the maximum number of ions for which there is space in the unit volume of solution. If the course of a single ion is followed for some time, it will be found for a fraction of time,, at the surface and for a fraction of time, 0, in the solution. If no work were required to move an ion from the interior of the solution to a place Joint Initiative of IITs and IISc Funded by MHRD 3/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 at the inner Helmholtz plane, 0 would be the ratio of the number of free places on the surface to those in the solution, i.e., 0 0 0 If we consider not one ion but all the ions, it is evident that, (3.4.1) (3.4.2) 0 0 If the amount of work needed to move an ion from the interior of the solution to a free place at the surface is not zero but an amount equal to then the right side of Eq. (3.4.2) needs to be multiplied by the Boltzmann factor. Therefore, we have, exp (3.4.3) 0 0 0 kt Except in very concentrated solutions, 0 0. With this simplification, Eq. (3.4.3) becomes, exp 0 0 kt Equation (3.4.4) can be solved for to obtain, 1 0 0exp kt Experimental evidence suggests that the term 0 0 exp kt 1 1947). Therefore, Eq. (3.4.5) simplifies to, 0 exp 2rion0exp 0 kt kt where r ion is the radius of the non-solvated ion. (3.4.4) (3.4.5) (Grahame, (3.4.6) The work,, is composed of the electrical potential at the inner Helmholtz plane and the adsorption potential of the ions. Stern considered the possibility of simultaneous adsorption of both cations and anions. However, there is hardly any Joint Initiative of IITs and IISc Funded by MHRD 4/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 general experimentally-unambiguous procedure for dividing up in this manner [see Hunter (2005)]. The potentials at the inner and outer Helmholtz planes can be written in terms of the charge density. The total thickness of the compact layer is, d d1 d2. The potential drop in the IHP is, 0d 1 0 1 (3.4.7) 1 0 and the potential drop in the OHP is, d d 2 1 d (3.4.8) 2 0 where 1 and 2 are the average values of the dielectric constant in the respective regions. Therefore, the potential changes linearly with distance. The charge balance requires that, 0 1 d 0 (3.4.9) A complete solution of potential distribution in the double layer can give the values of 0, 1, d, 0, 1 and d. In a simpler situation where 1 0, we have, d d 0 d (3.4.10) c 0 where c is the average dielectric constant of the medium in the compact region. Therefore, we have four unknowns, viz. 0, d, 0 and d, and three equations [Eqs. (3.3.44), (3.4.9) and (3.4.10)]. Another equation can be derived from the imposed external e.m.f. [see Hunter (2005)], i.e., where 0 dipole (3.4.11) is the Galvani potential difference between the surface and the interior of the solution, and the polar molecules at the surface. dipole is the potential difference due to the orientation of Joint Initiative of IITs and IISc Funded by MHRD 5/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 The Stern theory may be considered as an improvement over the GouyChapman theory. It distinguishes between the total double layer potential 0 and the potential at the OHP d. The experimental data show that 0 can be of the order of several hundreds of millivolts where the GouyChapman theory will be inapplicable. However, the value of d rarely exceeds 100150 mv, which has been verified experimentally (Verwey and Overbeek, 1948). It has been suggested that the adsorbed ions in the Stern layer can impart repulsion between two approaching surfaces. This provides stability to the colloids at high concentration of salt. As the thickness of the Stern layer increases (by increasing the size of the adsorbed ions on the interface), two approaching surfaces repel each other at a rather large separation. The separation between the two approaching surfaces can be as large as 2 nm (Claesson et al., 1984). The stability of some colloidal dispersions at high salt concentrations, swelling and repeptization can be explained by the Stern layer stabilization (Frens and Overbeek, 1972). 3.4.2 Electrostatic double layer around spherical particles The electrostatic double layer around spherical particles has a great importance in the stability of colloidal particles. Let us derive the expression of potential distribution with the DebyeHückel approximation (i.e., low potential). The differential equation governing the variation of with r is given by, 1 d 2 d 2 r 2 r dr dr The solution to this differential equation is, exp (3.4.12) I1exp r r I2 r r (3.4.13) where I 1 and I 2 are the constants of integration. Putting the boundary condition: 0 as r, we obtain I 2 0. Therefore, Eq. (3.4.13) becomes, Joint Initiative of IITs and IISc Funded by MHRD 6/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 If I 1 exp r r (3.4.14) d at r Rs d (where R s is the radius of the sphere and d is the thickness of the Stern layer) we have, I1 Rs d d exp Rs d (3.4.15) Putting the value of I 1 in Eq. (3.4.14) we obtain, Rs d d exp rrs d r (3.4.16) When the potential is not low, numerical solution of the differential equation is necessary. 3.4.3 Electrostatic double layer repulsion between two surfaces The repulsive force between two surfaces begins to develop when they approach each other so closely that the double layers on their surfaces overlap. This repulsion opposes the approach of the surfaces. Let us consider two infinitely large planar charged surfaces separated by a distance with the solution of electrolyte between them as shown in Fig. 3.4.1. Fig. 3.4.1 The overlap of diffuse double layers when two charged flat surfaces approach each other. Joint Initiative of IITs and IISc Funded by MHRD 7/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 The concentration of the electrolyte in the bulk of the solution is n. There is a reservoir of infinitely large amount of solution outside the plates, which balances the change in volume when the distance between the plates is altered. The reservoir is beyond the field of the double layers where the potential is zero and the hydrostatic pressure is P. Let us assume that the Stern layers are absent and only the diffuse parts of the double layers interact. Let us further assume that the surface potential is 0, as shown in Fig. 3.4.1. The potential distributions in the individual double layers (in absence of the other) are shown by the dotted curves. The full curve represents the potential distribution when both the double layers overlap. The profile indicates that the potential reaches its minimum value at x 2. At this point, d dx 0. At equilibrium, the hydrostatic pressure gradient and the force on the space charge balance each other at every point of the solution phase (Verwey and Overbeek, 1948), dp d 0 (3.4.17) For a symmetric electrolyte of valence z, we have, ze ze zen exp exp kt kt From Eqs. (3.4.17) and (3.4.18) we obtain, (3.4.18) ze ze ze dp zen exp exp d 2zen sinh d kt kt (3.4.19) kt Integrating we get, P m m ze dp 2zen sinh d (3.4.20) P 0 kt At x 2, the hydrostatic pressure is P m and the potential is m. Inside the reservoir, the hydrostatic pressure is P and the potential is zero. Therefore, we obtain, Joint Initiative of IITs and IISc Funded by MHRD 8/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 ze P 2 cosh m m P ktn 1 EDL kt (3.4.21) Pm P is the excess pressure at the midpoint of the separation between the surfaces. This is, therefore, the repulsive force per unit area that opposes the approach of the two surfaces. Now, we need to determine the unknown potential, m. If the surfaces are quite far apart, the overlap between the double layers is moderate. In that case we can use the following approximation known as linear superposition approximation (Gregory, 1975), m 1 2 2 x 2 (3.4.22) From Eq. (3.3.39) with d 0 and d 0 (since the Stern layer is assumed to be absent) we have, 4kT ze tanh 0 exp 2 2 x ze 4kT (3.4.23) Therefore, from Eqs. (3.4.22) and (3.4.23) we obtain, 8kT ze tanh 0 m exp 2 ze 4kT (3.4.24) As a consequence of the assumption that the interfaces are far apart, the potential, ze m, is low. Let us expand the term, cosh m kt, in Eq. (3.4.21) in Maclaurin series as follows. 2 4 ze 1 1 cosh m ze 1 m zem kt 2! kt 4! kt (3.4.25) Since m is small, we can neglect the third and higher terms of the series. This gives, 2 ze 1 cosh m ze 1 m kt 2! kt Therefore, from Eq. (3.4.21) we get, (3.4.26) Joint Initiative of IITs and IISc Funded by MHRD 9/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 2 ze m EDL ktn kt (3.4.27) Substituting the value of m from Eq. (3.4.24) in Eq. (3.4.27) we get, 2 ze 0 EDL 64kTn tanh exp 4kT (3.4.28) Equation (3.4.28) can be used for calculating the repulsive double layer pressure (also known as positive disjoining pressure due to electrostatic double layer) between two flat surfaces. It is evident that the double layer repulsion depends on electrolyte concentration. If binding of counterions takes place, the surface potential, 0, is reduced significantly. This would considerably reduce the repulsion. Example 3.4.1: Calculate the variation of disjoining pressure, EDL, with the distance between two planar surfaces in 10 mol/m 3 aqueous NaCl solutions for surface potentials of 50 mv and 75 mv at 298 K. Calculate the profiles between 2 nm and 10 nm separations. Explain your results. Solution: From Eq. (3.4.28) we have, 2 e 0 EDL 64RTc tanh exp 4kT Putting R 8.314 J mol 1 K 1, T 298 K, c 10 mol/m 3, 23 k 1.381 10 J/K in the above equation, we get, 4 2 EDL 158.56510 tanh 9.732 0 exp, 19 e 1.60210 C and 1 3 nm In this equation, 0 is in V, is in nm and EDL is in Pa. The variation of EDL with at 0 50 mv and 0 75 mv for these two concentrations are shown in Fig. 3.4.2. It can be observed from this figure that the repulsive pressure is low when the surface potential is low. Joint Initiative of IITs and IISc Funded by MHRD 10/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 Fig. 3.4.2 Variation of EDL with the separation between the surfaces. 3.4.3.1 Derjaguin approximation The Derjaguin approximation can be applied to calculate the electrostatic double layer repulsion force between bodies having curved surfaces using the interaction energy for the planar double layers. Let us determine the interaction energy from Eq. (3.4.28) by integrating its right side with respect to. This gives, 1 2 ze 0 ktn pp EDL 64 tanh exp 4kT Therefore the repulsive force between two spheres of radius ss pp 1 2 ze 0 FEDL Rs EDL 64 RskTn tanh exp 4kT The energy of interaction between two spheres is given by, ss 2 2 ze 0 EDL 64 RkTn s tanh exp 4kT R s is given by, (3.4.29) (3.4.30) (3.4.31) Note that the Derjaguin approximation is applicable when the radius of the sphere 1 R s is much larger than the thickness of the double layer. Joint Initiative of IITs and IISc Funded by MHRD 11/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 3.4.4 Zeta potential The zeta potential is the potential at the surface of shear, which lies close to the outer Helmholtz plane. Due to the presence of Stern layer, the value of - potential is less than the surface potential, 0, but close to the potential at the OHP (i.e., d ). The value of -potential has considerable importance in interfacial engineering because it provides an estimate of 0. Often it is the only available measure for characterizing the double layer properties. -potential is frequently used as a measure of the stability of a colloidal system. If the -potential is low (say 5 mv) the colloids coagulate rapidly. On the other hand, if the -potential is high (say 50 mv) the colloids have good stability. Let us derive the relationship between -potential and electrophoretic mobility for a dilute solution in which is small. For a dilute solution, the potential around a charged particle can be expressed by the equation for potential of an isolated ion whose charge is Q. This is given by, Q (3.4.32) 40r where r is the distance from the charge, Q. Let us now return to the potential distribution in the double layer of a spherical particle discussed in Section 3.4.2. From Eqs. (3.4.14) and (3.4.32), under the limit of 0, we get, Q I1 (3.4.33) 40 1 Since the value of is large in dilute solutions, the surface of shear may be assumed to lie very close to the surface of the particle, because this difference is small in comparison with the thickness of the double layer. Therefore, the - potential is approximately equal to the surface potential (i.e., at is small, we obtain, r Rs ). Since Joint Initiative of IITs and IISc Funded by MHRD 12/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 Q (3.4.34) 40Rs The charged colloid particles move in an electric field towards the oppositely charged electrode like a single ion. The electrical force on a colloid particle is, FE QE, E electric field (3.4.35) If the particle moves with a low velocity, u, for which Stokes law is applicable, the viscous force on the particle is, FV 6 Rsu (3.4.36) where is the viscosity of the liquid. The steady state velocity is obtained when FE FV, i.e., QE 6 Rsu (3.4.37) Therefore, u E Q 6 R (3.4.38) s The term, ue, is termed electrophoretic mobility. Substituting Q from Eq. (3.4.34), we get, u E 20 (3.4.39) 3 This equation is known as Hückel equation, which is valid in dilute solutions (where is small). It is valid for spherical particles for which R s 0.1. In nonaqueous solutions, the ionic concentration is low and the Hückel equation is applicable. Therefore, we can measure the -potential from Eq. (3.4.39) by measuring the electrophoretic mobility, which is explained in the next example. Example 3.4.2: The measured electrophoretic mobility of a spherical particle is m 2 V 1 s 1 in water at 298 K. Calculate the -potential using the Hückel equation. 5 10 8 Solution: At 298 K, 3 1 10 Pa s. Substituting the values of and 0 in Eq. (3.4.39) we get, Joint Initiative of IITs and IISc Funded by MHRD 13/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 3 8 3110 510 12 278.58.85410 Therefore, the -potential is 108 mv. 0.108 V = 108 mv If the value of is large (i.e., small 1, thin double layer), the -potential can be calculated from Smoluchowski s equation, u 0 (3.4.40) E The details of the derivation of Smoluchowski s equation have been presented by 1 Hiemenz and Rajagopalan (1997). A small value of corresponds to a high electrolyte concentration. It has been found that Eq. (3.4.40) is valid when R s 100. Equations (3.4.39) and (3.4.40) are valid at the two extremes of the electrolyte concentration. In many interfacial applications, the -potential needs to be measured at intermediate concentrations. The size of the particles is another important parameter which decides the applicability of these two equations. Henry s equation or Ohshima s equation (see Lecture 4 of Module 1) can be used for the intermediate values of Rs. In aqueous solutions, the ph and salt concentration are the two most important parameters that affect the -potential. The -potential can change from positive to negative by the variation of ph. The ph at which the -potential becomes zero is termed isoelectric point. The -potential versus ph graph gives information about the stability of the colloidal system. The Debye length depends on salt concentration. At the higher salt concentrations, the Debye length decreases. The valence of the ions has a very important effect on the magnitude of the Debye length. A trivalent ion, such as Al +3, will compress the double layer to a greater extent than a monovalent ion such as Na +. If the counterions bind specifically on the surface groups, the - potential can be dramatically affected even at low concentrations. The Joint Initiative of IITs and IISc Funded by MHRD 14/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 information on the effect of concentration of a component of a colloidal formulation on the -potential can give valuable insight in developing a product which will have maximum stability. The influence of known contaminants on the -potential of a sample can be a powerful tool in formulating the product to resist coagulation. 3.4.5 Limitations of the Poisson-Boltzmann equation The theory of electrostatic double layer is based on the PoissonBoltzmann theory. A large number of studies have shown that the main features of the charge distribution outside a highly-charged surface are described well by the PoissonBoltzmann equation. However, this theory has some limitations which are discussed below. Theories developed using the PoissonBoltzmann equation (e.g., the DLVO theory discussed in Lecture 5 of this module) sometimes deviate from experimental observations due to these limitations. o The PoissonBoltzmann equation proves to be insufficient near the interface due to the assumption that the ions are point charges and they can approach the interface without limit. However, the ions cannot move closer to an interface than a certain distance. The finite size of the ions was ignored in the PoissonBoltzmann theory. o When two charged surfaces approach each other, due to the ion correlation effect, the counterions concentrate towards the charged wall reducing the overlap of the double layers. The mobile counterions in the diffuse double layer form a highly polarizable layer at the surface. The fluctuations in the ion clouds of the two surfaces lead to an attractive force which is similar to the van der Waals force. It becomes significant for divalent counterions at high surface charges and at short separations. Therefore, deviations from PoissonBoltzmann equation are observed (Guldbrand et al., 1984). o The work required to bring an ion from infinity to a position where the potential is was assumed to be ze i. This ignores the effect of the ion on the Joint Initiative of IITs and IISc Funded by MHRD 15/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 rearrangement of all other ions and dipoles. Therefore, this work-term in the PoissonBoltzmann equation is approximate. o The discrete surface charges were assumed to be smeared out. Therefore, the attractive forces arising due to the discreteness of the charges were ignored. o A charge interacts with a surface because of the field reflected by the surface. Suppose that the charge is located at a distance from the surface. The reflected field is same as if there were an image charge at a distance 2 from the charge (see Fig. 3.4.3) (Onsager and Samaras, 1934). The contribution from image force was ignored. Fig. 3.4.3 The interaction of a charge with a surface due to the image effect. o The short-range solvation forces were not accounted in the PoissonBoltzmann theory. Joint Initiative of IITs and IISc Funded by MHRD 16/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 Exercise Exercise 3.4.1: The electrophoretic mobility of a 1 m diameter spherical particle is 3 10 8 m 2 V 1 s 1 in aqueous medium containing NaBr at 10 mol/m 3 concentration at 298 K. Calculate the -potential. Exercise 3.4.2: Calculate the electrostatic double layer repulsive force between two spheres of 1 m radius in aqueous NaCl solution (concentration = 10 mol/m 3 ) at 10 nm separation. The surface potential is 100 mv and the temperature is 298 K. Exercise 3.4.3: When the surface potential is low, show that the repulsive disjoining pressure between two planar surfaces is given by, 2 2 EDL 200 exp Exercise 3.4.4: Answer the following questions clearly. (a) Explain how you will calculate the potentials in the Stern layer. (b) Give two examples where the effect of Stern layer is important. (c) Write the equation for disjoining pressure between two flat plates due to electrostatic double layer. (d) Explain how you will calculate the double layer repulsion between two spheres if the double layer interaction energy between two flat surfaces is given. (e) Explain how the -potential can be calculated using Hückel and Smoluchowski equations. Under what conditions are these equations applicable? (f) What is electrophoresis? (g) What is the effect of ph on zeta potential? (h) Write three major limitations of the PoissonBoltzmann equation. Joint Initiative of IITs and IISc Funded by MHRD 17/18

NPTEL Chemical Engineering Interfacial Engineering Module 3: Lecture 4 Suggested reading Textbooks P. C. Hiemenz and R. Rajagopalan, Principles of Colloid and Surface Chemistry, Marcel Dekker, New York, 1997, Chapter 11. P. Ghosh, Colloid and Interface Science, PHI Learning, New Delhi, 2009, Chapter 5. R. J. Hunter, Foundations of Colloid Science, Oxford University Press, New York, 2005, Chapters 7 & 8. Reference books A. W. Adamson and A. P. Gast, Physical Chemistry of Surfaces, John Wiley, New York, 1997, Chapter 5. E. J. W. Verwey and J. Th. G. Overbeek, Theory of the Stability of Lyophobic Colloids, Dover, New York, 1999, Parts I & II. J. Lyklema, Fundamentals of Interface and Colloid Science, Vol. 2, Academic Press, London, 1991, Chapter 3. J. N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1997, Chapter 12. Journal articles D. C. Grahame, Chem. Rev., 41, 441 (1947). G. Frens and J. Th. G. Overbeek, J. Colloid Interface Sci., 38, 376 (1972). J. Gregory, J. Colloid Interface Sci., 51, 44 (1975). L. Guldbrand, B. Jönsson, H. Wennerström, and P. Linse, J. Chem. Phys., 80, 2221 (1984). L. Onsager and N. N. T. Samaras, J. Chem. Phys., 2, 528 (1934). O. Stern, Z. Elektrochemie und Angewandte Physikalische Chemie, 30, 508 (1924). P. Claesson, R. G. Horn, and R. M. Pashley, J. Colloid Interface Sci., 100, 250 (1984). Joint Initiative of IITs and IISc Funded by MHRD 18/18