1558
CHESTER T. O’KONSKI, KOSHIRO YOSHIOKA AND WILLIAM H. ORTTUNG
Vol. 63
ELECTRIC PROPERTIES OF MACROMOLECULES. IV. DETERMINATION OF ELECTRIC AND OPTICAL PARAMETERS FROM SATURATION OF ELECTRIC BIREFRINGENCE I N SOLUTIONS BY CHESTERT. O’KONSKI,KOSHIRO YOSHIOKA AND WILLIAM H. ORTTUNG’ Department of Chemistry, University of California, Berkeley
11.
Received March 8, 1969
The theory of electric birefringence in solutions of rigid axially symmetric dipolar molecules is extended to strong fields. The saturation behavior is computed for various ratios of permanent to induced moment contributions to the birefringence. By fitting the experimental birefringence saturation results to a theoretical curve, the electrical parameters and the optical anisotropy factor of the molecule can be separately determined. The birefringence buildup in strong fields and the decay are treated. Experimental saturation data are presented for a rigid polyelectrolyte (tobacco mosaic virus), a rigid polypeptide (poly-y-benzyl-L- lutamate) and a flexible polyelectrolyte (sodium polyethylene sulfonate). The first exhibits saturation characteristic o$ a large electrical polarizability, and the second and the third most closely fit permanent dipole moment saturation. Molecular constants were obtained, and are in satisfactory agreement with values obtained by other methods, where data are available.
Introduction The recent development of the transient electric birefringence method has facilitated the characterization of rigid macromolecules through measurements of their rotational diffusion constant,2-6 and through determination of their electrical properties in those cases where their optical properties could be determined from other measurement^.'^^ In principle, it is possible to subject a dilute solution of macromolecules to an electric field of sufficient intensity to obtain complete orientation. Under these conditions, measurement of the birefringence is a direct measure of the optical anisotropy of the molecules with respect to the axis of orientation. Then, from measurements of the Kerr constant obtained in the ordinary way, one may compute the pertinent eiectrical anisotropy factor, which is of interest in the elucidation of the macromolecular structure. I n the actual case, complete orientation of the macromolecules cannot be achieved because of the disorienting effect of Brownian motion, and it becomes necessary to extrapolate the measurements by some suitable method. As will be shown, the nature of the birefringence saturation curve depends upon the electric polarization mechanism which is responsible for the orientation. Therefore, it is necessary to calculate saturation functions for appropriate models so that experimental results may be interpreted. In this article it will be shown that the theory of the saturation of electric birefringence can be put into a convenient form for interpretation of saturation data on axially symmetric macromolecules. The early theoretical papers on electric birefringence considered mainly the effect of small applied fields.s Game considered arbitrarily large (1) Bell Telephone Laboratories Fellow, 1957-1958. (2) H. Benoit, Ann. phys., 6 , 561 (1951); J. chim. phys., 49, 617 (1952). (3) (a) C. T. O’IConski and B. H. Zimm, Science, ill, 113 (1950). (4) C. T. O’Konski and A. J. Haltner, J . Am. Chsm. Sor., 78,3604 (1956). (5) I. Tinoco, ibid., 77,3476 (1955). ( 6 ) C. T. O’Konski and A. J. Haltner, ibid., 79, 5634 (1957). (7) I. Tinoco, ibid.. 79, 4336 (1957). (8) (a) P. Langevin, Le Radium, 7 , 249 (1910); Compt. rend., 161, 475 (1910); (b) M. Born, Ann. Phueik, 66, 177 (1918). (9) R. Gans, ibid., 6 4 , 481 (1921).
electric fields for pure induced moment and pure permanent moment orientation. He obtained a birefringence saturation function only for pure permanent dipole orientation. Peterlin and Stuart‘” developed a theory of birefringence for insulating suspensions and obtained a birefringence saturation equation for induced moment orientation. In this research, birefringence saturation in the case of mixed orienting mechanisms is treated and compared with the pure permanent moment and the pure induced moment orientation behavior. Tables and graphs of saturation functions and methods of analyzing experimental data are presented. Approximate equations are developed for the birefringence transient resulting from the application of a saturating square voltage pulse. Saturation measurements have been made in this Laboratory on poly-ybenzyl-L-glutamate” (PBLG), tobacco mosaic virus (TMV) and sodium polyethylene sulfonate12 (NaPES). The first of these displayed a saturation behavior in agreement with permanent dipole moment orientation. The dipole moment and the optical anisotropy factor of the molecule have been evaluated and compared with results from other types of measurements. For TMV, the birefringence saturation confirms an induced polarization mechanism and the optical and electrical properties were determined. With respect to saturation, it was found that PES ions behave as if they have a permanent dipole moment. A tentative explanation is offered for this. Very few experimental results on electric birefringence saturation could be found in the literature. Some relatively weak saturation effects previously observed3 are discussed. Theory The electric anisotropy of a molecule in a solution causes the interaction energy of the molecule with the applied field to depend upon the angle between the molecular axis and the applied field; hence, the angular distribution of molecular directions becomes non-random in the presence of the field. In the (10) A. Peterlin and H. A. Stuart, Z . Physdb, 113, 129 (1939): also “Hand- und Jahrbuch der ohemisohen Physik,” Bd. 8, Abt. l B , Leipzig, 1943. (11) C. T. O’Konski and W. H. Orttung, J . A m . Chem. SOC.,in press. (12) K. Yoshioks and C. T. O‘Konski, J . Polymer Sci., unpublished.
r
ELECTRIC PROPERTIES OF MACROMOLECULES
Oct., 1959
following calculation rigid molecules are considered which have an axis of symmetry for their electric, optical and hydrodynamic properties. The assumption of a hydrodynamic axis of symmetry is used only in the calculation of transient effects. This assumption is less restrictive than the assumption of a specific model, such as a cylinder or an ellipsoid of revolution. Further, it is assumed that the solution is so dilute that interaction effects can be neglected. The fundamental equation for the birefringence of a dilute solution of axially symmetric particles has been given by Peterlin and StuartlO
Field.-For the steady state, the distribution function is given by
(3)
where we have introduced the notation u = COS e ,B = pEBl/kT y =
3 COS^^ 2
-
1
X 2~ sin 0 dB (1)
where
The integral in the denominator of eq. 3 may be evaluated as
$':
e8u+ru2
= volume fraction of particles An = n11 = n1, the birefringence
du = e-B2/4r
n
gl
-
f(e) e
2 referring to the symmetry and transverse axes, respectively = angular orientation distribution function = angle between particle symmetry axis and the axis of anisotropy, in this case, the field direction
To compute the orientation distribution function, an expression for the energy of interaction of the particle with the external electric field E is required. For an axially symmetric particle with dipole moment p along the symmetry axis in the solvent, the dipole interaction energy may be written
u1=
- p ~ COS ~ l
e
(3a)
e(d/ru+P/2&)2
+8/2d/y
E(t) =
E
For ellipsoids of revolution in insulating media, B1 has been evaluated13 BI = 1/11
+ - ea)l1/4~eo] (€1
where L1 is an elliptic integral, and €1 and EO are dielectric constants of particle and solvent, respectively. If the solution and ellipsoid are conducting, the internal field function is the same a t low frequencies except that the specific conductivities, ~1 and KO, replace €1 and The induced dipole interaction energy may be written 1 U2 = - 9 (011 - az)E* COS' e (2b) where, for insulating media,'O the coefficients a1 and a2are the excess polarizabilities of the particle in the solvent. For conducting media the coefficients involve the conductivities as well as dielectric properties; they have been derived for steadystate conduction.14 The total interaction energy is
u
=
u1 + Ua
(2c)
Steady-state Birefringence in a Saturating
(13) Lord Rayleigh, Phil. Mag., [ 5 ] 44, 28 (1897). (14) C. T. O'Konski and 9. Krause, THW.TOURNAT., in preparation.
du ( 6 )
(7)
where tl = P I 2 4 7 - d? and f2 = P I 2 4 7 respectively. Using the definition (9)
+ 47, (9)
eX'dX
the integral may be written e8u+rua
du =
( t -~ ~) ( t ~ )(io) i
( i / f i ) e - ~ 2 / 4 [ ~~
= an internal field function = Elht/El = E cos e = component of external field along the
symmetry axis Elink= resulting internal field a t the dipole, with field and dipole assumed to be colinear
A.
$Tll
Making the substitution x = dYu one obtains
where BI El
(4) (5)
- az)E2/2kT
(01
C,
= index of refraction of the soh. gz = optical anisotropy factor,IO the subscripts 1 and
1,559
A table of E(t) was given by Terrill and Sweeny.'& Introducing eq. 3 into eq. 1, one has where
m , r )=
3 2
u2ePu+ru2
$-+11
e8u+w2
du
du
- -I
(12)
Making the substitution expressed by eq. 7, we can evaluate the integral in the numerator. Thus
$
X e X 2 dX =
- exa 1 2
From eq. 12, 10 and 13 one obtains (15) H. M. Terrill and L. Sweeny, J . Franklin Inat., 237, 496 (1944); ass, 220 (1944).
CHESTERT. O'KONSKI, KOSHIRO YOSHIOKA AND WILLIAM H. ORTTUNG
1560 1.o
c =
0.8
6 @/E rBi/kT ? / E 2 (ai - an)/2kT
Vol. 63 (19b) (19c)
The specific Kerr constant, as defined by Peterlin and Stuart,lO and employed in this Laboratory,e is
'F: 0.6 6.
g 0.4 0.2
Introducing eq. 19a and 15 0
4
8
12 16 20 24 28 32 36 Pa 2r. 27 for Fig. 1.-Degree of orientation, y), us. p = 0, p2 = y, p2 = 2y, fJ2 = 4y, pa = 97, b2 = 167, and y = 0.
+
+
@(e,
(b) Pure Permanent Moment Orientation. p >> r.-Then W6,O) = Q = 1 - 3 ( ~ 0 t h O- l/P)/B (21) The limiting form for low fields may be obtained from eq. 17. For limiting high fields
1.2
*
+
+ i.0
,A
q 3 = 1 - 3/a 3 / P 2 (22) (c) Pure Induced Moment Orientation. y >> @.-Then
e4
+ 0.8
a*
m
5 0.6 +
6 0.4
G
4 0.2 0
4
8
12 P2
16
20 24
28 32 36
+ 2fr . 27) us. pa + 27.
Qiy
Fig. 2.-15@(p, y)/(B2 @(Ply)=
This function may be computed from a table of e-%'(t) given by Miller and Gordon.lB Equation 17 gives the form for small values of y. For large values of y (y > 10)
3 -x 4Y
== 1
- 3/(2r - 1)
(24)
Equation 24 was obtained from the work of Schwarz17 who has investigated the anisotropy of conductance of solutions produced by electric fields. The limiting form of eq. 14 for high fields is W , r ) = 1 - 3/03 2Y) (25) Values of 9~and 15 as functions of p are given in Table I, and those of @r and 159.,/2y as functions of d y are given in Table IT. 0 ( & ~ ) 2y) are plotted against P2 and 15Q(P,7)/(P2 27 (which is equal to (b2 2c)E2) in Fig. 1 and Fig. 2, respectively, for some special cases. 150 (p,?)/(p2 27) is auseful function because of the relationship to the experimental quantity
+
P2/2y
-
l]
- 1/2
(14)
+(P,y) is a useful function to consider because as E +- 00, CP + 1. If the steady-state birefringence at infinite field strength is denoted by An,, then
1 An,
2a
7 = 2 (m
-
s2)
+
+
+
+
Using An., eq. 11 can be put in the form An/An, = *(Ply)
(16)
+(Ply) is an orientation factor, which may be called
the degree of orientation. Three special cases of eq. 14 are of interest. (a) Weak Fields.-D,r
