THEKERRCONSTANT OF RIGIDCONDUCTING DIPOLAR MACROMOLECULES
3243
Theory of the Kerr Constant of Rigid Conducting Dipolar Macromoleculesla by Chester T. O’Konskilb Department of Chemistry, University of California, Berkeley, California 94WO
and Sonja Krause Chemistry Department, Rensselaer Polytechnic Institute, Troy, New York 18181 (Received February 3,1989)
Using the Maxwell-Wagner boundary conditions, equations are derived for the electric field orientation effect and the Kerr constant of a dilute suspension of ellipsoids of arbitrary and anisotropic volume conductivities, dielectric constants, carrying a dipole. Equations developed earlier can be used to calculate, from counterion mobilities and concentration distributions, the parameters entering this theory. The treatment is applicable to large macromolecules or colloidal particles in aqueous solution, with ion atmosphere relaxation times short compared to the period of rotational diffusion, and may be used to calculate the anisotropy of conductivity, optical rotation, and dichroism in electrically oriented systems. Previous data for the specific Kerr constant of Tobacco Mosaic Virus (TMV) are employed to quantitatively test the theory, with the conclusion that the agreement is satisfactory. It is predicted that an insulating needlelike particle, ferroelectric along its long axis, will orient perpendicular to the electric field in a conducting solution.
Introduction Kerr, in his original studies of the electrooptic effect,Z reported that particles in liquid suspension were attracted into an electric field and sometimes formed a chain oriented along the field. Experiments followed on both electric dichroism and birefringence of various crystalline materials in suspension.at4 Simultaneous application of an electric and a magnetic field produced large nonadditive effects, which were considered6 due to permanent dipoles, following Langevin’se theory. Microscopic observations of the action of electric fields on colloidal particles have been Rodlike V206particles in an aqueous sol oriented with long axes perpendicular to the applied electric field, and an intermittent electric field produced changes in light scattering which depended upon the relative directions of applied field and the light beam. Birefringence of Vz06 sols also was observed.9 I n all these studies, it was supposed that the orientation effect was closely connected with hydrodynamic effects accompanying electrophoresis. The observations were qualitative and no mathematical theory was applied. Quantitative work on electric birefringence in colloids began with studies1° of electric dichroism, electric birefringence, and optical dispersion of these effects in VZOS and Au sols. Positive birefringence was found in Vz05 sols and negative birefringence in Au sols. Later, a wide variety of suspended particles was studied, l1 and the orientation effects were discussed in terms of dielectric anisotropy, but the basis for a negative electric birefringence in some systems and a positive electric birefringence in others was not understood. Subsequently electric birefringence of a large variety of
metallic and nonmetallic particles in suspension was studiedlZand an unusual birefringence buildup behavior, in which the birefringence reversed sign, was observed; again, no clear conclusions regarding orienting mechanisms emerged. Metallic sols generally gave negative birefringence, whereas in nonmetallic sols, both signs of birefringence frequently occurred. Electric birefringence studies in suspensions of clay minerals led to the conclusion that the orientation of the particles was primarily an electrical rather than a hydrodynamic phenomenon.18 An intensive study14 was made of (1) (a) This is paper VI of the series “Electric Properties of Macromolecules.” Presented in part a t the New York meeting of the American Chemical Society, Sept 11-16, 1960. Abstracts of Papers, 138th Meeting, No. 26,Division of Colloid Chemistry, p 10-1. This is an extension of part of the thesis submitted by S. Krause in partial fulfillment of the requirements for the Ph.D. in Chemistry, Sept 1957. (b) Inquiries preferably should be addressed to C. T.0. (2) J. Kerr, Phil. Mag., 50, 337 (1875). (3) (a) G.Meslin, C. R . Acad. Sci., 136, 888, 930 (1903); 148, 1179 (1909); J . Phys., 7, 856 (1908); (b) J. Chaudier, C . R . Acad. Sci,, 137, 248 (1903); 142, 201 (1906); 149, 202 (1909). (4) A. Cotton, H. Mouton, and P. Drapier, J . Chim. Phys., 10, 693 (1912); C . R . Acad. Sci., 157, 1063, 1519 (1913). (5) F. Pockels, Radium (Paris), 10, 152 (1913). (6) P. Langevin, ibid., 7, 249 (1910); C. R . Acad. Sci., 151, 475 (1910). (7) H. Siedentopf, 2. Wiss. Mikrosk. M., 29, 1 (1912). (8) H.R. Kruyt, Kolloid-Z., 19, 161 (1916). (9) H.Freundlich, 2.Elektrochem, 22, 27 (1916). (10) C. Bergholm and Y . Bjornstahl, Phys. Z.,21, 137 (1920). (11) S. Prooopiu, C . R . Acad. Sci., 172, 1172 (1921); Ann. Phys. (Paris), 1, 213 (1924). (12) Y.Bjornstahl, Phil. Mae., 2, 701 (1926). (13) C. E. Marshall, Trans. Faraday SOC., 26, 173 (1930). (14) J. Errera, J. Th. G. Overbeek, and H. Sack, J . Chim. Phys., 32, 681 (1935).
The Journal of Physical Chemistry, Vol. 74, No. 17, 1070
CHESTER T. O’KONSKIAND SONJAKRAUSE
3244
electric birefringence in VzOssols in sine wave fields as a function of frequency, and it was recognized that the electric field would anisotropically deform the diffuse double layer around a charged colloidal particle, with the largest polarization occurring along the long axis of the particle. A rise in birefringence with frequency, followed by a decay toward zero, were observed. The Kerr effect was divided into a positive component, presumably arising from the double-layer polarization, and undergoing a dispersion around lo7cps, and a negative part which was explained in terms of a permanent moment perpendicular to the long axis, and having an orientational dispersion. Electric birefringence in solutions of tobacco mosaic virus (TMV) was first interpreted as arising from a permanent dipole moment.16 Electric birefringence in bentonite sols was studied16-18with the conclusion that the complex behavior of the birefringence could not be readily explained, and complications arising in interpreting orientation effects were stressed.’9 The nature of the orienting forces on polyions was examined in a series of quantitative studies of aqueous solutions of TN!V.zO-zz Electrolyte concentration effects showed that ion atmosphere and solvent conductivity were important.20 From comparisons of birefringence transients, dispersion, and magnitude with theory, it was concluded that in low-frequency electric fields the TRiIV orients predominantly because of a very large induced polarization of the mobile counterions, and this was substantiated by the electric birefringence saturation studies.22 Permanent dipole effects may be significant even in the presence of large ionic polarizabilities and evidence for a permanent moment contribution to orientation of polyethylene sulfonate ions was found.z2 It was also suggestedlZ3from studies of the anisotropy of electric conductance, that an ionic polarization effect was important in the orientation of polyphosphate ions by electric fields. Quantitative studies of the Kerr constant of solutions of bovine serum albumin revealed effects of pH and of ionic ~ t r e n g t h . ~ ~ ~ ~ ~ ~ ” I n this contribution we derive equations for the orientation effect of an electric field on a rigid conducting ellipsoid carrying a permanent dipole moment in a conducting medium. Such a model can be used to represent the electric polarization of a rigid polyelectrolyte, as discussed earlier in relation to dielectric and conductivity properties. 2 4 The present derivation is made for Maxwell-Wagner boundary conditions corresponding to steady-state conduction, which occurs at low frequencies. Expressions for the electric energy as a function of orientation and for the electric birefringence are obtained. From these equations, other orientation dependent properties can readily be evaluated for the same model. The Journal of Physical Chemistry, Vol. 74, No. 17, 1970
Theory In previous t h e o r i e -27 ~ ~ of~ ~electric ~ ~ birefringence of macromolecular solutions, the equations for the electric free energy of the macromolecule as a function of its orientation in an applied electric field were based upon earlier theory developed by application of Laplace’s equation to electrically insulating systems. The local (“internal”) field acting upon the macromolecule was expressed in terms of the dielectric constant of the medium and the static electric properties of the macromolecule. It is clear that this procedure is not valid for conducting media containing conducting particles. In theories of the dielectric and the conductivity prope r t i e ~ , ~a ~consideration ,~* of boundary conditions shows that the electric field distribution will depend upon the solvent and macromolecule conductivities, or, in general, on the charge transport properties of the system. In addition, it was shown experimentallyz1that changes of the conductivity of the solvent medium indeed affect the extent of orientation of polyelectrolyte macromolecules, and that the electric field orientation effect undergoes a dispersion which can be explained in terms of a time dependent ion atmosphere polarization. In a theoretical study of the electric free energy of liquid droplets and their deformation by electric fields,z9 it was shown that the free energy depends strongly on conductivities, as well as on dielectric constants, in electrically conducting systems. Rigid polyelectrolytes may be represented as rigid conducting ellipsoids, for which closed solutions exist for Maxwells’ equations. At frequencies which are low compared with ionic relaxation frequencies, the condition of steady-state conduction may be assumed. In a recent extension of the theory of electric birefringence in saturating fields, an internal field function B1 was defined, and it was pointed out that the equations should explicitly include the
(15) M. A. Lauffer, J . Phys. Chem., 42, 935 (1938). (16) H. Mueller and B . W. Sakmann, Phys. Rev., 56, 615 (1939). (17) H . Mueller, ibid., 55, 508, 792 (1939). (18) F. J. Norton, ibid., 55, 668 (1939). (19) W. Heller, Rev. Mod. Phys., 14, 390 (1942). (20) C. T. O’Konski and B. H . Zimm, Science, 111, 113 (1950). (21) C. T. O’Konski and A . J. Haltner, J . Amer. Chem. Soc., 79, 5634 (1957). (22) C. T. O’Konski, K. Yoshioka, and W. H. Orttung, J . Phys. Chem., 63, 1558 (1959). (23). (a) M . Eigen and G. Schwarz, 2. Phys. Chem. (Frankfurt a m Mazn), 9, 318 (1956); (b) J . CoZZoid Sci., 12, 181 (1957); (0) G. Schwarz, 2. Phys., 145, 563 (1956); (d) 9. Krause and C. T. O’Konski, J . Amer. Chem. SOC.,81, 5082 (1959) ; (e) C. I. Riddiford and B. R . Jennings, ibid., 88, 4359 (1966). (24) C. T. O’Konski, J . Phys. Chem., 64, 605 (1960). (25) A. Peterlin and H. A. Stuart, 2.Phys., 112, 129 (1939). (26) H. Benoit, Ann. Phys. (Paris), 6, 561 (1951). (27) I. Tinoco, J. Amer. Chem. SOC.,77, 4486 (1955). (28) H. Fricke, J. Phys. Chem., 57, 934 (1953). (29) C. T . O’Konski and F. E. Harris, ibid., 61, 1172 (1957).
THEKEERCONSTANT OF RIGIDCONDUCTING DIPOLARMACROMOLECULES conductivity. 22 The range of applicability of the present theory will be discussed below. We consider the following model; a rigid ellipsoid with axes a, b, and c, with anisotropic dielectric constants e,, e b , and e E , and anisotropic conductivities K,, K b , and K,, along the respective axes. Let the ellipsoid be immersed in a solvent of low-frequency dielectric constant B and conductivity K , in an external homogeneous electric field of intensity E. The first step is to compute the components of the internal electric field intensity, E,, Eb, and E,. Introducing the coordinate system of Morse and Feshbach, with Eulerian angles 8, +, and 4, we can find the external field components in the directions of the ellipsoidal axes.3o
E,
=
E cos e
(14
E, = -E sin e cos
E,
=
+
Ob)
E sin e sin $
(14
For each external field component, the internal field intensity is determined by the condition for steadystate current flow, namely, V - J = 0 where J is the current density given by J = KE. Boundary conditions for the external field components, designated by the subscript 1, are El, -.L E,, EII,+ EbJand El, + E , at infinity. It is assumed that the usual boundary conditions of Maxwell-Wagner polarization a t low frequencies apply, namely, J l and El1 are continuous a t the interface. The electric field problem is mathematically identical with the magnetic field problem for ellipsoids treated by RiIaxwell. Applying the known solution31 of the magnetic field problem and substituting K,, K b , K,, K , and J for the permeabilities and the magnetic induction, respectively, the magnitudes of the internal field components are obtained
E, Eb
=
E,
=
B,E
COS
e
(24
-BbE Sin 6 COS 9
(2b)
B,E sin e sin rL
(2c)
=
where B, ( j = a, b, or c) is the internal field function. I n the present case, B,, Bb, and B, are computed for steady-state conduction and are functions of the conductivities only, uiz., Ej(int)/El(ext)
Bj
=
[l
+
(K,/K
- l)Aj]-l
(3a)
A , is the depolarization factor depending only upon
where
R,
= [(s
+ a2)(s+ b2)(s + c2)]’/’
(3~) The A , can be determined in closed form only for ellipsoids of revolution and must be evaluated by numerical methods in the general case. For ellipsoids of revolution, where b = c
R,
=
3245 (s
+ b2)(s+ u ~ ) ” ~
= A,. If the ellipsoids of revolution also have and K b = K ~ then , BB = B,. Values of A , and A b for ellipsoids of revolution with p = 1/200 to 200 have been tabulated. 32 Dipole Interaction Energy. We consider a dipole moment, p, oriented a t any angle with respect to the axes of the ellipsoid with components pa, EL,b, and p C along the respective axes. The orientation with respect to an external coordinate system is again specified by the Eulerian angles 0, +, and 4. The dipole interaction energy term is a generalization of that given earlier.27
and Ab eb
= e,
U1
=
-vfE
=
-paEo
- pi$& - /AGE,
(4)
except that E,, Eb,and E , are given by eq 2a, 2b, and 2c where a corresponding function of the dielectric constants, B , ( E ) ,replaces B,(K), defined by eq 3. An apparent “dipole moment,” which we call p‘, was defined, with components paf = Ba(e)pa, pbf = &,(e) p b , for ellipsoids of revolution. Since B,(B) and & ( E ) are internal field functions, this constituted a way of modifying an earlier treatmenLZ6 I n both cases, the internal field was not explicitly calculated. Here we explicitly include the internal field function in the equations, reserving the term dipole moment and the symbol EL, for the actual permanent electric moment. It should be noticed that p = (pa2 pc2)”*isthe permanent dipole moment of the solvated macromolecule in the particular solvent system. It is the resultant of the internal charge distributions arising from local polarization effects in solvent and macromolecule and the rearrangement of charges within the ion atmosphere by the dipole field. If the macromolecular unit under consideration carries a net charge, the dipole moment must be defined with respect to the hydrodynamic center of the system, for it is not invariant with respect to choice of origin, and the orientation occurs about this center. Polarization Energy. The energy associated with the induced polarization of the ellipsoid and the surrounding medium can be found by evaluating the integral
+ +
Uz
(‘/B?T)JD.E dv
(5)
over all space. The external medium must be included because the external field is modified by insertion of the ellipsoid. To find how energy changes with orientation in a constant external field, one can compute the energy change as a function of orientation on immersing the ellipsoid into the medium. If the homogeneous field E (30) P. M . Morse and H. Feshbach, “Methods of Theoretical Physics,” Vol. 1, LMcGraw-Hill, New York, N. Y.,1953, p 28. (31) J. C. Maxwell, “A Treatise on Electricity and Magnetism,” Vol. 1, 3rd ed, Clarendon Press, Oxford, 1892,p 66. (32) Reference 24, Table I, p 612.
The Journal of Physical Chemistry, Vol. 74, N o . 17, 1970
CHESTERT. O’KONSKIAND SONJAKRAUSE
3246
U
is maintained in the medium a t large distances from the ellipsoid, insertion causes the energy change
U2 =
‘S
-
8T
+ ecEc2- €E2)dvz +
+
(eaEa2
2- (El2 - E2) dv1
8T
(6)
where the first integral is taken over the volume of the ellipsoid and the second over the surrounding medium. E is the magnitude of the field in the absence of the ellipsoid, E1 is the magnitude of the field outside the ellipsoid with the ellipsoid in place, and E,, E,, and E, are defined above. For the present calculation we assume a dilute system of ellipsoids. For the conduction problem, the first integral can be evaluated easily on substitution of the eq 2 and 3. The second integral would be difficult to evaluate directly, but an indirect method2$ can be used. Since this integral depends only upon the external field distribution, its value may be found from the solution for the corresponding electrostatic problem in which the fields are the same. I n that case the total energy U’ may be obtained as an integral over the volume in the interior of the ellipsoid only.** For an anisotropic dielectric ellipsoid of zero conductivity within an insulating medium, the energy is
U‘
=
+
vE2(( e - ea)B,(e) cos2 e (E
+
- e,)B,(c) sin26’ sin2 # ] / S T
(7) where v is the volume of the ellipsoid. Employing this with the result for the first integral of eq 6, we find for the electrostatic case (e
I‘
= J(E12
- E’)dv =
VE~[COS’ 0 ((E sin2 e cos2 + { ( E sin2 e sin2 +{ ( 6
-
-
+ - e,~,z(e)/ej + + 11
E,)Ba(E)/e
-. eaBa’(e)/E}
Eb)Bb(e)/E
- ~JB~(E)/E
~~~~‘(e)/e)
+
+
- K)B,(K)/K} sin2 e cos2 I){( K , / K eb/e)B02(K)
+
(Kb
sin2 e sin2 +[( ( K J K
+
-
- K)Bb(K)/K} - ec/~)Bc2(~)
+
(Kc
(90
(gc
-
- K)Bc(K)/Kj1/87r
(9)
The total electric free energy change on inserting particle in the system is The Journal of Physical Chemistry, Vol. 74, No. 17, 1970
5t
-
- gb)(pa -
gC>(pb
-
pC
+
pb
&bc)
+ +
&ab)
+
- ga)(Pc- Pa + &dl= X,,C,nE2
(11)
where g , g j = optical anisotropy factor, defined by Peterlin and Stuart,34 C, = volume fraction of the ellipsoids, Pa = / , L , ~ B , ~ /PO~ ~=T p~b ,2 B b 2 / k 2 T 2 ,P, = pc2Bc2/k2T2, and Qtj
=
EV{(K~/K
- et/r)Bt2 + ( ~ -t K ) B ~ /K - ( K j - K)Bj/K]/47rkT
(12) and B,, B,, and B, are identical with B a ( ~Bb(K), ), and B,(K)above. Here K,, is the specific Kerr constant. The coefficients of the form ( K ~ / K - € ( / E ) indicate the magnitude and polarity of free charges which accumulate a t the interface, I n insulating systems, no free charges accumulate so these terms cannot occur. The equation is not applicable to insulating systems, but it is interesting to note that if the terms related to free charges are omitted, and if dielectric constants are substituted for conductivities in the remaining terms in brackets, one obtains, for ellipsoids of revolution where Bb
U2 = - e ~ E 2e([(K,/K ~ ~-~ E2, / E ) B . ~ ( K ) (K,
An = (2T/15n)C0E2[(gG
(8)
The value of the integral over the external volume for steady-state conduction is now obtained by substituting K,, K ~ ,K~~ and K for ea, ebl eo, and e in eq 8. Introducing this into eq 6 and evaluating the first integral by direct substitution, the polarization energy of the ellipsoid in the field becomes
(10)
u 2
with U1 and U2given by eq 4 and 9. The SteadpState Orientation and the K e n Constant. We may employ (4),(9), and (10) and the Boltzmann equation to calculate the steady-state orientation distribution function.22 From it, various properties of the electrically oriented system such as anisotropy of optical refraction, optical rotation, dielectric constant, and conductivity may be computed. We now evaluate the optical birefringence in weak electric fields (U > I / A a
-
Qab A E V / ~ T ~ T A ,
(14)
It follows that
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trolyte, which is only about l o + sec for pure water), we obtain
This applies when a >> b, K, > Kb, and predicts alignment of the particles along the field, as expected, when e, = E, > 1, the equation simplifies to
- gb)/30n2kTAa (15) in agreement with the earlier result,21 if there is no K,,
eV(ga
permanent dipole moment. The condition K , / K b >> l / A a means that very high ionic or internal conductivities are required for this limiting law to hold for highly elongated macromolecules. As observed earlier,21it predicts a Kerr constant four times too high for TMV. A further observation is that whenever K , / K >> 1 and K b / K >> 1, the internal field functions for long, thin macromolecules (a/b >> 1) are given approximately by Ba A
K/(AaKa
Bb
2K/(K,
+
K)
(164
+ 2K)
(16b)
This means that in general the dipole orientation terms P, in the electric orientation function (eq 12) are suppressed. This suppression effect will be less for the dipole moment along the long axis of a rod-like macromolecule than a perpendicular component, because A , becomes small for high axial ratio. It means also that all numerical values for the dipole moments of polyelectrolyte macromolecules must be reevaluated in the light of the present theory. Further, the theory of electric polarization of polyelectrolyte macromolecules must be extended to a similarly versatile model before low-frequency dielectric increments of polyelectroyte macromolecules may be reliably interpreted. With moderately conducting prolate ellipsoids of high axial ratio, the internal field functions simplify. Then when K , / K >> E , / € and K , / K >> E J E , it can be shown for the nondipolar cases that
This applies when a >> b, K, = Kb and K , / K
