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The dielectric constant in the equations of electrokinetics
190
LETTERS TO THE EDITOR :
: REFERENCES
1. C~ARL~,S,G. E., AND IVIAS0N,S. G., "The Coalescence of Liquid Drops with Flat Liquid/Liquid Interfaces," J . Colloid Sci. 15, 236 (1960). 2. NIST,S~N, L. E., unpublishe d. derivation. 3. ELTON, G. A. H., AND PICKNETT, R. G., Proceedings of the 2nd international Congress of Surface Activity, Vol. 1, p. 287. Butterworths, London, 1957. 4. NIELS]~N, L. E., WALL, R., AND ADAMS, G., J. Colloid Sci. 13, 441 (1958). 5. DEaJAGUIN, B. V., ANn KUSS~KOV,M., Acta Physicochim. U.R.S.S. 10, 25 (1939).
Plastics Division, Monsanto Chemical Company' Springfield, Massachusetts Received January 13, 1961
D . C . CHAPPELEAR
THE DIELECTRIC CONSTANT IN THE EQUATIONS OF ELECTROKINETICS The continued use of the dielectric constant in the equations of electrokinetics indicates that currently accepted concepts relating to electrical units have not been introduced into this field. For example, Smoluchowski's equation for the electrophoretic mobility, U (cm. ~ statvolt -1 sec.-l), of a particle moving with velocity, v (cm. sec.-~), under a potential gradient, X (statvolt cm.-~), is usually stated in the form: U-
v _ Di" X 4~ry '
where ~" is the electrokinetic potential (statvolt), ~ is the viscosity of the medium (poise = gram cm. -1 sec.-1), and D is the dielectric constant. It is clear from the units stated that the factor D/4~r must have units gram cm. sec. -2 statvolt -2 or (e.s.u. of charge) ~ gram -1 cm. -3 sec.2. Guggenheim (1) has suggested that these units be assigned to the factor 4~- since he defines D as a dimensionless ratio. Wood (2) has criticised this proposal and has assigned the necessary units to D. The dielectric constant can, however, be defined only as a dimensionless comparison ratio. The earlier solution to this difficulty regarded the ratio D/4~r as dimensionless, implying that the e.s.u, of charge had dimensions M 1/2 L 3/2 T -1. Several workers (1, 3) have argued strongly against this artificial reduction in the number of dimensions for electrical quantities from four to three. The introduction of the concept of the "permittivity of free space" enables a satisfactory four-dimensional system to be developed. The fundamental equation for the force, f (dynes = gram cm. sec.-2), between two charges ql, q2 (statcoulomb) separated a distance 4 (cm.) in a
LETTERS TO THE EDITOR
191
medium of permittivity K is: f _ ql q2 K d~"
The dimensions of K are thus identical with those of the factor D/4~r. The statcoulomb or e.s.u, of charge is defined b y assigning the value K = K0 = 1 statcoulomb ~ gram -1 cm. -3 sec. 2 to the permittivity of free space. The dielectric constant D is defined as K / K o (or some equivalent comparison ratio) and in the three-dimensional system, since K and K0 are arbi' trarily considered dimensionless, K and D become identical. To preserve the four dimensions it is necessary only to reintroduce K, with appropriate dimensions, in place of D. This same problem was discussed briefly by Grahame (4) in connection with the theory of electrocapillarity, and a similar solution was offered. The pertinence of Grahame's discussion to the theory of electrokinetics, however, does not seem to have been generally recognized. Guggenheim (1) has pointed out that only in those cases where K / K o is constant should it be called the dielectric constant. I t has been suggested (5) that this m a y not be the case in colloidal systems, and this constitutes another argument in favor of the use of K. A permittivity which varies with field strength does not present the contradiction in terms of the system currently in use. A further improvement may be made by introducing the M.K.S. system of units in accordance with the recommendations of the International Union of Pure and Applied Chemistry (1958) (6). In the "rationalized" (7) version of this system Smoluchowski's equation becomes U = K ~ / v , where U is in meter 2 volt -1 sec. -1, ~ is in volts, and ~ is in kg. meter -~ sec. -1. Here K is the absolute permittivity = DKo where K0 = 107/4~rc2 coulomb s kg. -~ meter -3 sec. 2 (or farad meter -~) ; c = velocity of electromagnetic waves i n vacuo -- 3 X l0 s meter sec. -~ The customary practice of recording ~ in volts rather than statvolts would facilitate the introduction of the M.K.S. system with its greater consistency. I wish to thank Professor A. E. Alexander for his guidance and encouragement in this work. I~EFERENCES ]. GUGGENHEIM,E. A., Trans. Faraday Soc. 36(1), 139 (1940). 2. WOOD, L. A., J. Am. Chem. Soc. 68, 432 (1946). 3. JAUNCE¥, G. E. M., AND LANGSDORF, A. S., "M.K.S. Units and Dimensions," p. 20. Macmillan, New York, 1940. 4. GRAHAME,I). C., Chem. Revs. 41, 441 (1947). 5. ROBINSON, L. B., J. Chem. Phys. 14, 721 (1946). BOLT, G. H., J. Colloid Sci. 10, 206 (1955).
192
LETTERS T O T H E
EDITOR
A., A_NDHAYDON, D. A., Report of discussion on "Electrical Double Layer in Colloid Science." Nature 183, 78 (1959). 6. Commission on Symbols and Terminology, Section of Physical Chemistry, I.U.P.A.C., Copenhagen, 1958. 7. ~V[A.RRIOTT,~-I., AND CULLEN, A. L., P r o c . I n s t . E l e c . Engrs. (London) 97(1), 245 (1950). KITCH]~NER,J.
:
i
Division of Soils, . . . . C.S.I.R.O., Canberra, Australia Received November 15, 1960
R.J.
HUNTER
STREAMING POTENTIAL AND TURBULENCE
We have measured E/P and Ely relationships for two Pyrex glass capillaries and found definite indications of a change in slope when the Reynolds number is approximately 2000. These results contrast with those of Bocquet, Sliepcevich, and Bohr (1) and also with those of Rutgers, de Smet, and de Moyer (2) using aqueous solutions, although the latter authors detected a strong turbulence effect when using a benzene solution of Zndi-i-propyl-salicylate. Streaming potential measurements were carried out on two Pyrex capillaries, capillary I having 1 = 33.3 cm., r = 0.057 cm., and capillary I I having 1 = 24.5 cm., r = 0.104 cm.; with these dimensions the factor b y which the fluid velocity must be multiplied to give Reynolds number is 12.5 for I and 23.3 for II. The capillaries were cleaned with hot chromic acid, washed with conductivity water, and dried for 5 hours at 200°C. before measurements. The streaming measurements were carried out using an experimental setup similar to that used by Bocquet et al. with compressed nitrogen as the driving force and a Keithley Model 600 electrometer for the potential measurements. The electrodes consisted of 1/~ inch internal diameter silver tubes appropriately treated internally to form Ag-AgC1 electrodes; these tubes were 3/~ inch long and of the same external diameter as the Pyrex tubes to which they were butted and affixed with an external seal of Tygon tubing. The fluid flow rates were of course actually measured and converted to linear velocities from the capillary dimensions. The results shown in Figs. 1 and 2 are for twice distilled water of conductivity 2.8 × 10-6 ohm -1 cm. -I in the case of capillary I (second distillation from alkaline permanganate), and once distilled water of conductivity 8.6 X 10-~ for capillary II. Calculation of the zeta potential b y the Helmholtz-Smoluchowski equation from the slopes of each portion of the E/P curve of capillary I gives
LETTERS TO THE EDITOR :
: REFERENCES
1. C~ARL~,S,G. E., AND IVIAS0N,S. G., "The Coalescence of Liquid Drops with Flat Liquid/Liquid Interfaces," J . Colloid Sci. 15, 236 (1960). 2. NIST,S~N, L. E., unpublishe d. derivation. 3. ELTON, G. A. H., AND PICKNETT, R. G., Proceedings of the 2nd international Congress of Surface Activity, Vol. 1, p. 287. Butterworths, London, 1957. 4. NIELS]~N, L. E., WALL, R., AND ADAMS, G., J. Colloid Sci. 13, 441 (1958). 5. DEaJAGUIN, B. V., ANn KUSS~KOV,M., Acta Physicochim. U.R.S.S. 10, 25 (1939).
Plastics Division, Monsanto Chemical Company' Springfield, Massachusetts Received January 13, 1961
D . C . CHAPPELEAR
THE DIELECTRIC CONSTANT IN THE EQUATIONS OF ELECTROKINETICS The continued use of the dielectric constant in the equations of electrokinetics indicates that currently accepted concepts relating to electrical units have not been introduced into this field. For example, Smoluchowski's equation for the electrophoretic mobility, U (cm. ~ statvolt -1 sec.-l), of a particle moving with velocity, v (cm. sec.-~), under a potential gradient, X (statvolt cm.-~), is usually stated in the form: U-
v _ Di" X 4~ry '
where ~" is the electrokinetic potential (statvolt), ~ is the viscosity of the medium (poise = gram cm. -1 sec.-1), and D is the dielectric constant. It is clear from the units stated that the factor D/4~r must have units gram cm. sec. -2 statvolt -2 or (e.s.u. of charge) ~ gram -1 cm. -3 sec.2. Guggenheim (1) has suggested that these units be assigned to the factor 4~- since he defines D as a dimensionless ratio. Wood (2) has criticised this proposal and has assigned the necessary units to D. The dielectric constant can, however, be defined only as a dimensionless comparison ratio. The earlier solution to this difficulty regarded the ratio D/4~r as dimensionless, implying that the e.s.u, of charge had dimensions M 1/2 L 3/2 T -1. Several workers (1, 3) have argued strongly against this artificial reduction in the number of dimensions for electrical quantities from four to three. The introduction of the concept of the "permittivity of free space" enables a satisfactory four-dimensional system to be developed. The fundamental equation for the force, f (dynes = gram cm. sec.-2), between two charges ql, q2 (statcoulomb) separated a distance 4 (cm.) in a
LETTERS TO THE EDITOR
191
medium of permittivity K is: f _ ql q2 K d~"
The dimensions of K are thus identical with those of the factor D/4~r. The statcoulomb or e.s.u, of charge is defined b y assigning the value K = K0 = 1 statcoulomb ~ gram -1 cm. -3 sec. 2 to the permittivity of free space. The dielectric constant D is defined as K / K o (or some equivalent comparison ratio) and in the three-dimensional system, since K and K0 are arbi' trarily considered dimensionless, K and D become identical. To preserve the four dimensions it is necessary only to reintroduce K, with appropriate dimensions, in place of D. This same problem was discussed briefly by Grahame (4) in connection with the theory of electrocapillarity, and a similar solution was offered. The pertinence of Grahame's discussion to the theory of electrokinetics, however, does not seem to have been generally recognized. Guggenheim (1) has pointed out that only in those cases where K / K o is constant should it be called the dielectric constant. I t has been suggested (5) that this m a y not be the case in colloidal systems, and this constitutes another argument in favor of the use of K. A permittivity which varies with field strength does not present the contradiction in terms of the system currently in use. A further improvement may be made by introducing the M.K.S. system of units in accordance with the recommendations of the International Union of Pure and Applied Chemistry (1958) (6). In the "rationalized" (7) version of this system Smoluchowski's equation becomes U = K ~ / v , where U is in meter 2 volt -1 sec. -1, ~ is in volts, and ~ is in kg. meter -~ sec. -1. Here K is the absolute permittivity = DKo where K0 = 107/4~rc2 coulomb s kg. -~ meter -3 sec. 2 (or farad meter -~) ; c = velocity of electromagnetic waves i n vacuo -- 3 X l0 s meter sec. -~ The customary practice of recording ~ in volts rather than statvolts would facilitate the introduction of the M.K.S. system with its greater consistency. I wish to thank Professor A. E. Alexander for his guidance and encouragement in this work. I~EFERENCES ]. GUGGENHEIM,E. A., Trans. Faraday Soc. 36(1), 139 (1940). 2. WOOD, L. A., J. Am. Chem. Soc. 68, 432 (1946). 3. JAUNCE¥, G. E. M., AND LANGSDORF, A. S., "M.K.S. Units and Dimensions," p. 20. Macmillan, New York, 1940. 4. GRAHAME,I). C., Chem. Revs. 41, 441 (1947). 5. ROBINSON, L. B., J. Chem. Phys. 14, 721 (1946). BOLT, G. H., J. Colloid Sci. 10, 206 (1955).
192
LETTERS T O T H E
EDITOR
A., A_NDHAYDON, D. A., Report of discussion on "Electrical Double Layer in Colloid Science." Nature 183, 78 (1959). 6. Commission on Symbols and Terminology, Section of Physical Chemistry, I.U.P.A.C., Copenhagen, 1958. 7. ~V[A.RRIOTT,~-I., AND CULLEN, A. L., P r o c . I n s t . E l e c . Engrs. (London) 97(1), 245 (1950). KITCH]~NER,J.
:
i
Division of Soils, . . . . C.S.I.R.O., Canberra, Australia Received November 15, 1960
R.J.
HUNTER
STREAMING POTENTIAL AND TURBULENCE
We have measured E/P and Ely relationships for two Pyrex glass capillaries and found definite indications of a change in slope when the Reynolds number is approximately 2000. These results contrast with those of Bocquet, Sliepcevich, and Bohr (1) and also with those of Rutgers, de Smet, and de Moyer (2) using aqueous solutions, although the latter authors detected a strong turbulence effect when using a benzene solution of Zndi-i-propyl-salicylate. Streaming potential measurements were carried out on two Pyrex capillaries, capillary I having 1 = 33.3 cm., r = 0.057 cm., and capillary I I having 1 = 24.5 cm., r = 0.104 cm.; with these dimensions the factor b y which the fluid velocity must be multiplied to give Reynolds number is 12.5 for I and 23.3 for II. The capillaries were cleaned with hot chromic acid, washed with conductivity water, and dried for 5 hours at 200°C. before measurements. The streaming measurements were carried out using an experimental setup similar to that used by Bocquet et al. with compressed nitrogen as the driving force and a Keithley Model 600 electrometer for the potential measurements. The electrodes consisted of 1/~ inch internal diameter silver tubes appropriately treated internally to form Ag-AgC1 electrodes; these tubes were 3/~ inch long and of the same external diameter as the Pyrex tubes to which they were butted and affixed with an external seal of Tygon tubing. The fluid flow rates were of course actually measured and converted to linear velocities from the capillary dimensions. The results shown in Figs. 1 and 2 are for twice distilled water of conductivity 2.8 × 10-6 ohm -1 cm. -I in the case of capillary I (second distillation from alkaline permanganate), and once distilled water of conductivity 8.6 X 10-~ for capillary II. Calculation of the zeta potential b y the Helmholtz-Smoluchowski equation from the slopes of each portion of the E/P curve of capillary I gives