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The role of electronic surface states in the charging of colloidal particles in electrolyte
The Role of Electronic Surface States in the Charging of Colloidal Particles in Electrolyte E. R. S M I T H I AND S. G. DAVISON Quantum Theory Group, Applied Mathematics Department, University of Waterloo, Waterloo,Ontario, Canada, N2L 3G1 Received January 17, 1977; accepted April 6, 1977 We discuss the role that electronic surface states may play in the establishment of a charged layer at the surface of a crystalline particle immersed in electrolyte. We propose that the surface charge is determined self-consistentlyby the change in the energy of the eIectronicsurface state caused by the electricfieldof the doublelayer set up by the surface charge. We examinea simplifiedmodel of particle charging, which gives surface potentials of roughly the correct magnitude A great deal of theoretical work has been done on the interaction of colloidal particles immersed in electrolyte (1, 2). The interaction usually consists of competing attractive dispersion forces and repulsive shielded electrostatic forces. In some systems, the repulsive forces have some other source, such as steric hindrance effects. In this letter, we discuss the origin of the charged surface layer on colloidal particles, which causes the damped electrostatic repulsive force. We describe this force via the electrostatic potential ~, which varies as = e;oe-~
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at distance x from an isolated surface in the linearized Gouy-Chapman theory of a charged surface immersed in electrolyte (1), where ~ is the inverse Debye length of the electrolyte and q~0 is the surface potential. The mechanism for the charging of a colloidal particle surface varies from system to system. For example, in many systems of biological interest, ionizable groups are fixed (3) or adsorbed (4) onto the surface, and their degree of ionization determined self-consist1Permanent address: Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia.
ently by their interaction with the electric field of the double layer, which the surface charge sets up. In some crystalline particles, the charge is thought to arise from specific adsorption of ions (for example, an AgI sol, which becomes positively or negatively charged according to which of the Ag+ or I - ions are in greater concentration in the electrolyte), and no charge results, if the necessary specific ions are not present in the electrolyte (5). Other crystalline particles will charge in pure water (e.g., quartz (5)) or in a nonspecific ionic solution. In these crystalline examples, the particle charge is considered to be an inherent property of the particle, which arises from individual charges located on cracks and impurities in the crystal structure. Regardless of whether a crystal has a planar or heterogeneous surface, and whether or not the surface has impurities adsorbed on it, the distribution of charge on its surface is, in general, determined by the electronic structure of the crystal. The surface of a crystal has localized electronic states associated with it (6), and we propose that it is the occupancy of these surface states which determines the surface charge. For the example of AgI quoted above, the species of ion adsorbed determines
471 Copyright (~ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 62, No. 3, December 1977 ISSN 0021-9797
472
SMITH AND
the nature of the electronic surface states, and thus the surface charge. Without specific ion adsorption, the surface states are apparently inappropriate for the purpose of establishing a surface charge. On the other hand, because some crystalline particles do charge in the absence of any observed specific ion adsorption, the surface states of pure crystals must, in other circumstances, be capable of establishing a charged surface layer. As electrons tunnel into or out of the electronic surface states, a surface charge results, and an electric double layer is established. The electric field set up by the double layer changes the energy of the surface states (7). With the potential of Eq. ~1~, with q~0 = 100 mV, and with K-1 = 10 A, the surface electric field is l0 s V/m. Such a field is known to affect the energies of the surface states (7). Indeed, a field of 109 V/m is the order of magnitude necessary to produce field emission (8). We suggest that the charging of the surface proceeds by electron tunneling, until the resulting electric field makes tunneling into or out of the surface states equally probable, thereby establishing the surface potential and charge self-consistently. Whether the surface bears adsorbed material or not, it is the electronic surface states which are crucial in determining the surface charge. The theory of surface states is then a useful tool with which to describe the surface charging process in colloidal particles. To make the idea discussed a little more concrete, we propose a simple model for the way in which a clean semiconducting crystal surface may become charged when it is immersed in electrolyte. We consider a system in which at least part of the band of surface states of the uncharged crystal is of higher energy than electrons in the bulk electrolyte (i.e., electrolyte at large distance from the surface). Electrons in the surface states tunnel out of the crystal, leaving the surface positively charged. The electric field set up by this surface charge, and its resultant double layer, changes the energy of the surface states until
DAVISON
a stage is reached where the top of the band of surface states has an energy equal to that of electrons in the bulk electrolyte. Electrons will then no longer tunnel out of the surface, and its surface charge and potential will be determined. The states at the top of the band of surface states will have no dispersion parallel to the surface. Thus, to evaluate the surface potential, we need only consider a one-dimensional model. In this model, the surface potential takes a value which makes the surface state energy equal to that of free electronic charges in the bulk electrolyte. We consider the colloidal particle as a semiconductor and represent it by the KronigPenney model potential (9). The electronic wavefunctions satisfy Schr6dinger's equation
(-h2/2m)~"(x) + V(x)~(x) = E~(x).
[2]
Inside the crystal (x ~ 0), the potential is a linear array of delta functions :
E33 The solution of [-2], with E3~ substituted for V(x), which decays exponentially for x large and negative, may be written as
$~(x) = Ao{sinp[x + (n + 1)a] e-*sinp(x + na)}e -~* [4~ -
-
on the interval --(n + 1)a < x < -- ha. The parameters p and ~bare given by (9) cosh ~b = cos pa -J- (mX/p]~2) sin pa, p~ = 2mE/h 2. [SJ In the electrolyte (the region x >_ 0), we model the potential V(x) using Eq. [-1J, i.e.,
x being the coordinate normal to the surface, - q the electronic charge, q0 the electrostatic surface potential, and V0 the potential energy in the electrolyte bulk away from the surface at x --- 0. Equation [6-] in [2"] gives
(-hV2m)~"(x)
Journal of Colloid and Interface Science, VoL 62, No. 3, December 1977
SURFACE STATES IN CHARGING COLLOIDAL PARTICLES
473
potential upon the ionic concentration is a defect of the model, since such a variation of ~2 = 2 m ( E Vo)/h 2, r 2 = 2mq¢o/h 2, ~0 Mth ionic concentration is not normally = ~e -'~'/2, [8-] observed. However, since clean semiconductor and Eq. [77 becomes crystals do not bear surface charges when they are in vacuum, the result for zero ion concen(Z/4)~b"(~) + (Z/4()~'(() tration is reassuring. There are two obvious features of our model which m a y have lead to which has solutions this defect. The first is the one-dimensional nature of the model; the actual band structure ~ ( x ) = AJ2i~/,(2(/K) + BJ_2i~/~(2~/K). E10~ of the surface states may be expected to change We note that the order of these Bessel func- the nature of the results somewhat. The other tions is imaginary for E > V0 and real for is the use of a very simplified continuum picE < V0. For the case of interest to us, when ture for the electrolyte. The whole success of E = V0, the order is zero. Noting that ~b(~) the model of the semiconductor depends on must be bounded as x becomes large, we find using a potential model which has strong variations on an interatomic scale. Thus, it is ~b(~) = CJo(2~/K). [117 probably inadequate to ignore all the atomic The solution of Schr6dinger's equation and its scale potential variation in the electrolyte. derivative must be continuous at x -- 0 (the However, it is difficult to suggest an alternasurface of the crystal) where ~ = r. Together tive, though recent work on interfaces of soluwith the condition E = Vo, these conditions tions with crystals may be a useful approach (11, 12). The first step in a program to improve give an equation for the electrostatic potential this model is to obtain a better understanding at the crystal surface, viz., of the surface states in the one-dimensional 2p(cos pa -- e-C) 2rJl(2r/~) system, and this work is at present under way F12-] (13). Ic sin .Oa KJo(21"/K) In spite of the above qualifications, we Following Modinos (10), we choose the believe that our simplified model shows the values a = 4.0 A, X = 6.0 eV A, and V0 = 0.75 importance of the role that electronic surface eV for our semiconductor parameters, and states can play in the charging of colloidal take K = 0.1 A -~, which corresponds to 0.1 M particles, and suggest that further work on the electrolyte, then we obtain q'0~---80 mV, problem is well worthwhile. which is an entirely reasonable value for the ACKNOWLEDGMENTS surface potential of such a colloidal particle. If we call z~,j the j t h zero of J~(z), then the The work reported here was partially supported by right-hand side of [12"] ranges from 0 to + oo the Australian Research Grants Committee and the on (0, zo.1) and from --oo to 0 on (Zo,l, z1.1). National Research Council of Canada. The value of the left-hand side of [-12-] is REFERENCES usually large (it is --5.12 for the case con1. VERWEY, E. J. W., AND OVERBEEK, J. TH. G., sidered), which leads to the approximate solu"Theory of the Stability of Lyophobic Colloids." tion 2r/K = zo.1, and an electrostatic surface Elsevier, Amsterdam, 1948. potential 2. HAYDON, D. A., in "Recent Progress in Surface ®o ~-- (zo,~h~)2/Smq. E133 Science" (J. F. Danielli, K. G. A. Pankhurst, and We now make the substitutions
-
We note that as ~ ~ 0 (i.e., the charge concentration in the electrolyte goes to zero) q~0--~ 0 as (z0,~K)2. This dependence of surface
A. C. Riddiford, Eds.). Academic Press, New York, 1964. 3. NINtIAM, B. W., AND PARSEGIAN,V. A., J. Theor. Biol. 31, 405 (1971).
Journal of Colloid and Interface Science, VoL 62, No. 3, December 1977
474
SMITH AND DAVISON
4. BAI~OUCH,t~., P.~RRAM, S- V~T.,AND SMITH, E. R., -Proc. Roy. Soc., Set. A 334, 49, 59 (1973). 5. See Ref. (1) for a discussion. 6. DAVlSON,S. G., ANDLEVlNE, 7. D., in "Solid State Physics" (H. Ehrenreich, F. Seitz, and D. Turnbull, Eds.), Vol. 25. Academic Press, New York, 1970. 7. DAVISON, S. G., AND TAN, K. P., Surface Sci. 27, 297 (1971).
8. SHEPHERD,W. ]3., AND PERIA, W. T., Surface Sci. 38, 461 (1973). 9. SXESI~ICKA,M., Progr. Surface Sci. 5, 157 (1974). 10. MOmNOS, A., Surface Sci. 41, 425 (1974). 11. PERRA~, J. W., AND SMITH, E. 1~., to appear. 12. BLUM,L., J. Phys. Chem., to appear. 13. DAVlSON,S. G., SMITH, E. R., AND PARENT, L.G., to appear.
Journal of Colloid and Interface Science. Vol. 62. No. 3, December 1977
Eli
at distance x from an isolated surface in the linearized Gouy-Chapman theory of a charged surface immersed in electrolyte (1), where ~ is the inverse Debye length of the electrolyte and q~0 is the surface potential. The mechanism for the charging of a colloidal particle surface varies from system to system. For example, in many systems of biological interest, ionizable groups are fixed (3) or adsorbed (4) onto the surface, and their degree of ionization determined self-consist1Permanent address: Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia.
ently by their interaction with the electric field of the double layer, which the surface charge sets up. In some crystalline particles, the charge is thought to arise from specific adsorption of ions (for example, an AgI sol, which becomes positively or negatively charged according to which of the Ag+ or I - ions are in greater concentration in the electrolyte), and no charge results, if the necessary specific ions are not present in the electrolyte (5). Other crystalline particles will charge in pure water (e.g., quartz (5)) or in a nonspecific ionic solution. In these crystalline examples, the particle charge is considered to be an inherent property of the particle, which arises from individual charges located on cracks and impurities in the crystal structure. Regardless of whether a crystal has a planar or heterogeneous surface, and whether or not the surface has impurities adsorbed on it, the distribution of charge on its surface is, in general, determined by the electronic structure of the crystal. The surface of a crystal has localized electronic states associated with it (6), and we propose that it is the occupancy of these surface states which determines the surface charge. For the example of AgI quoted above, the species of ion adsorbed determines
471 Copyright (~ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 62, No. 3, December 1977 ISSN 0021-9797
472
SMITH AND
the nature of the electronic surface states, and thus the surface charge. Without specific ion adsorption, the surface states are apparently inappropriate for the purpose of establishing a surface charge. On the other hand, because some crystalline particles do charge in the absence of any observed specific ion adsorption, the surface states of pure crystals must, in other circumstances, be capable of establishing a charged surface layer. As electrons tunnel into or out of the electronic surface states, a surface charge results, and an electric double layer is established. The electric field set up by the double layer changes the energy of the surface states (7). With the potential of Eq. ~1~, with q~0 = 100 mV, and with K-1 = 10 A, the surface electric field is l0 s V/m. Such a field is known to affect the energies of the surface states (7). Indeed, a field of 109 V/m is the order of magnitude necessary to produce field emission (8). We suggest that the charging of the surface proceeds by electron tunneling, until the resulting electric field makes tunneling into or out of the surface states equally probable, thereby establishing the surface potential and charge self-consistently. Whether the surface bears adsorbed material or not, it is the electronic surface states which are crucial in determining the surface charge. The theory of surface states is then a useful tool with which to describe the surface charging process in colloidal particles. To make the idea discussed a little more concrete, we propose a simple model for the way in which a clean semiconducting crystal surface may become charged when it is immersed in electrolyte. We consider a system in which at least part of the band of surface states of the uncharged crystal is of higher energy than electrons in the bulk electrolyte (i.e., electrolyte at large distance from the surface). Electrons in the surface states tunnel out of the crystal, leaving the surface positively charged. The electric field set up by this surface charge, and its resultant double layer, changes the energy of the surface states until
DAVISON
a stage is reached where the top of the band of surface states has an energy equal to that of electrons in the bulk electrolyte. Electrons will then no longer tunnel out of the surface, and its surface charge and potential will be determined. The states at the top of the band of surface states will have no dispersion parallel to the surface. Thus, to evaluate the surface potential, we need only consider a one-dimensional model. In this model, the surface potential takes a value which makes the surface state energy equal to that of free electronic charges in the bulk electrolyte. We consider the colloidal particle as a semiconductor and represent it by the KronigPenney model potential (9). The electronic wavefunctions satisfy Schr6dinger's equation
(-h2/2m)~"(x) + V(x)~(x) = E~(x).
[2]
Inside the crystal (x ~ 0), the potential is a linear array of delta functions :
E33 The solution of [-2], with E3~ substituted for V(x), which decays exponentially for x large and negative, may be written as
$~(x) = Ao{sinp[x + (n + 1)a] e-*sinp(x + na)}e -~* [4~ -
-
on the interval --(n + 1)a < x < -- ha. The parameters p and ~bare given by (9) cosh ~b = cos pa -J- (mX/p]~2) sin pa, p~ = 2mE/h 2. [SJ In the electrolyte (the region x >_ 0), we model the potential V(x) using Eq. [-1J, i.e.,
x being the coordinate normal to the surface, - q the electronic charge, q0 the electrostatic surface potential, and V0 the potential energy in the electrolyte bulk away from the surface at x --- 0. Equation [6-] in [2"] gives
(-hV2m)~"(x)
Journal of Colloid and Interface Science, VoL 62, No. 3, December 1977
SURFACE STATES IN CHARGING COLLOIDAL PARTICLES
473
potential upon the ionic concentration is a defect of the model, since such a variation of ~2 = 2 m ( E Vo)/h 2, r 2 = 2mq¢o/h 2, ~0 Mth ionic concentration is not normally = ~e -'~'/2, [8-] observed. However, since clean semiconductor and Eq. [77 becomes crystals do not bear surface charges when they are in vacuum, the result for zero ion concen(Z/4)~b"(~) + (Z/4()~'(() tration is reassuring. There are two obvious features of our model which m a y have lead to which has solutions this defect. The first is the one-dimensional nature of the model; the actual band structure ~ ( x ) = AJ2i~/,(2(/K) + BJ_2i~/~(2~/K). E10~ of the surface states may be expected to change We note that the order of these Bessel func- the nature of the results somewhat. The other tions is imaginary for E > V0 and real for is the use of a very simplified continuum picE < V0. For the case of interest to us, when ture for the electrolyte. The whole success of E = V0, the order is zero. Noting that ~b(~) the model of the semiconductor depends on must be bounded as x becomes large, we find using a potential model which has strong variations on an interatomic scale. Thus, it is ~b(~) = CJo(2~/K). [117 probably inadequate to ignore all the atomic The solution of Schr6dinger's equation and its scale potential variation in the electrolyte. derivative must be continuous at x -- 0 (the However, it is difficult to suggest an alternasurface of the crystal) where ~ = r. Together tive, though recent work on interfaces of soluwith the condition E = Vo, these conditions tions with crystals may be a useful approach (11, 12). The first step in a program to improve give an equation for the electrostatic potential this model is to obtain a better understanding at the crystal surface, viz., of the surface states in the one-dimensional 2p(cos pa -- e-C) 2rJl(2r/~) system, and this work is at present under way F12-] (13). Ic sin .Oa KJo(21"/K) In spite of the above qualifications, we Following Modinos (10), we choose the believe that our simplified model shows the values a = 4.0 A, X = 6.0 eV A, and V0 = 0.75 importance of the role that electronic surface eV for our semiconductor parameters, and states can play in the charging of colloidal take K = 0.1 A -~, which corresponds to 0.1 M particles, and suggest that further work on the electrolyte, then we obtain q'0~---80 mV, problem is well worthwhile. which is an entirely reasonable value for the ACKNOWLEDGMENTS surface potential of such a colloidal particle. If we call z~,j the j t h zero of J~(z), then the The work reported here was partially supported by right-hand side of [12"] ranges from 0 to + oo the Australian Research Grants Committee and the on (0, zo.1) and from --oo to 0 on (Zo,l, z1.1). National Research Council of Canada. The value of the left-hand side of [-12-] is REFERENCES usually large (it is --5.12 for the case con1. VERWEY, E. J. W., AND OVERBEEK, J. TH. G., sidered), which leads to the approximate solu"Theory of the Stability of Lyophobic Colloids." tion 2r/K = zo.1, and an electrostatic surface Elsevier, Amsterdam, 1948. potential 2. HAYDON, D. A., in "Recent Progress in Surface ®o ~-- (zo,~h~)2/Smq. E133 Science" (J. F. Danielli, K. G. A. Pankhurst, and We now make the substitutions
-
We note that as ~ ~ 0 (i.e., the charge concentration in the electrolyte goes to zero) q~0--~ 0 as (z0,~K)2. This dependence of surface
A. C. Riddiford, Eds.). Academic Press, New York, 1964. 3. NINtIAM, B. W., AND PARSEGIAN,V. A., J. Theor. Biol. 31, 405 (1971).
Journal of Colloid and Interface Science, VoL 62, No. 3, December 1977
474
SMITH AND DAVISON
4. BAI~OUCH,t~., P.~RRAM, S- V~T.,AND SMITH, E. R., -Proc. Roy. Soc., Set. A 334, 49, 59 (1973). 5. See Ref. (1) for a discussion. 6. DAVlSON,S. G., ANDLEVlNE, 7. D., in "Solid State Physics" (H. Ehrenreich, F. Seitz, and D. Turnbull, Eds.), Vol. 25. Academic Press, New York, 1970. 7. DAVISON, S. G., AND TAN, K. P., Surface Sci. 27, 297 (1971).
8. SHEPHERD,W. ]3., AND PERIA, W. T., Surface Sci. 38, 461 (1973). 9. SXESI~ICKA,M., Progr. Surface Sci. 5, 157 (1974). 10. MOmNOS, A., Surface Sci. 41, 425 (1974). 11. PERRA~, J. W., AND SMITH, E. 1~., to appear. 12. BLUM,L., J. Phys. Chem., to appear. 13. DAVlSON,S. G., SMITH, E. R., AND PARENT, L.G., to appear.
Journal of Colloid and Interface Science. Vol. 62. No. 3, December 1977