March, 1959
REVERSING PULSE TECHNIQUE IN ELECTRIC BIREFRINGENCE
423
THE REVERSING PULSE TECHNIQUE IN ELECTRIC BIREFRINGENCE BY IGNACIO TINOCO, JR.,AND KIWAMU YAMAOKA Department of Chemistry, University of California, Berkeley, California Received July 18, 1968
Equations for the birefringence of a macromolecular solution in a rapidly reversed electric field are derived. The equations are plotted for various values of the electrical parameters of the molecule and the advantage of the reversing pulse method for determining these parameters is discussed. The effect of a time dependent polarizability and of polydispersity is considered. A more quantitative interpretation of O’Konski, et al.’s experimental data is made.
Introduction The reversing pulse technique in electric birefringence, first introduced by O’Konski,l is an improved method for obtaining information about the electrical properties of macromolecules. An electric field is applied to a solution and the birefringence is measured in the usual manner12then the electric field is rapidly reversed in sign. Qualitatively we can say that if there is no change in birefringence on field reversal, then the orienting torque on the molecules must be entirely due to a fast induced dipole moment. A change in birefringence, however, indicates either some permanent dipole or slow induced dipole contribution to the orienting torque. More quantitative conclusions cannot be made until equations relating molecular parameters to the observed birefringence are available. These equations are derived in the present paper. They are presented graphically for various values of the molecular parameters. Direct comparison with experiment will thus give the desired quantitative description of the permanent and induced dipole moments of the molecules. These electrical properties in principle can be determined from the curves for the rise of the birefringen~e.~,~ However, as will be seen from the figures the reversing pulse method is more accurate and convenient.
Theory The derivation of the equations will only be sketched here as the formalism has been described before in a paper4 which we shall refer to as I. The model used will be discussed in more detail, polydispersity will be considered, and some errors in I will be corrected. A more logical notation will be used. A solution of non-interacting rigid macroniolecules in an electric field will be considered. The non-interaction implies that the solutions are so dilute that the macroscopic properties of the solution, such as refractive index, dielectric constant, etc., are essentially those of the solvent. The magnitude of the field, E , will be such that the potential energy of a molecule in the field will be less than kT. This means that only terms up to E 2 need to be considered in the angular distribution function of the molecule. The molecule is assumed to have roughly cylindrical symmetry (two rotary diffusion coefficients must be equal), but it need not be specified further. (1) C. T. O’Koneki and A. J. Haltner, J . Am. Chem. Soc., 79, 5634 (1957). (2) C. T. O’Konski and B. H. Zimm, Science, 111, 113 (1950). (3) H. Benoit, Ann. Phus., 6, 561 (1951). (4) I. Tinoco, Jr., J . Am. Chem. Soc., 17, 4486 (1955).
This does not necessarily imply an ellipsoid of revolution. A rod, a flat disk, a string of beads, a cylinder with square cross section, etc., are included. This assumption is less stringent than that used in flow birefringence theories. The orienting torque in flow birefringence is hydrodynamic and only a few simple shapes have been treated theoretically. I n the present experiment the orienting tcrque is electrical and is not dependent explicitly on the shape of the molecule. To relate the rotary diffusion coefficients to molecular dimensions, a specific model must be chosen, however. Consideration of the completely general, arbitrary shape with three unequal rotary diffusion coefficients would be extremely difficult. This is the unsymmetric top problem; it does not have a solution in closed form. The cylindrically symmetrical shape of the molecule imposes its symmetry on the polarizability. Both the electronic polarizability and the polarizability due to mobile protons on the surface of the protein6 or to the ion atmosphere1a2 should have the same symmetry as the molecule. We, therefore, characterize a macromolecule by a rotary diffusion tensor with principal values ell = 9 2 2 and (333.The rotary diffusion coefficient for rotation about the symmetry axis is 933. The polarizability tensor is specified by all = a22 and a33along these same principal axes. The permanent dipole vector of the molecule has a completely arbitrary direction; i t is specified by its components ( p l , p2, p 3 ) along the previously chosen axes. All three above parameters (e, a,p ) will depend on the solvent. The rotary diffusion coefficient 8 is inversely proportional to the viscosity of the solvent.6 The polarizability a is actually the polarizability increment of the solute, the difference in polarizability between solute and solvent. It is convenient to discuss a in two parts: CXE(electrical polarizability) and a. (optical polarizability). The value of a. simply depends on the shape of the molecule and the differences in refractive index of solvent (n) and those of the solute (nl = n2, n3). This relation for ellipsoids of revolution may be found in I. The value of QE may have various contributions. The electronic and atomic polarizability may be characterized by dielectric constants (el = e2, e3) and analogous equations for used. However, for a mobile ion contribution to the polarizability, general equations relating number and mobility of icns, and particle shape to (5) J. G. Kirkwood and J. B. Shumaker, Proc. Natl. Acad. Sei., 88 855 (1952). (6) This is not neawsarily true for mixed solvents.
Vol. 63
IGNACIO TINOCO, JR.,AND KIWAMU YAMAOKA
424
(YE are not yet available. Relations among the apparent dipole moment in solution, p', the ideal, vacuum dipole moment p, and the solvent properties are quite complex. We will ignore the problem by using the definition p E r = p'E
where E , is the generally unknown, effective orienting field on the molecule and E is the measured, external field. Our subsequent equations will be written in terms of p'. For simple, generally unrealistic models such as spheres' or ellips o i d ~with ~ point dipoles a t their centers, equations for p/p' are available. These models are particularly inappropriate for large molecules whose dipole moments are primarily due to charges distributed over the periphery of the molecule. For example, inside a protein molecule with a conducting coatings the electric field would approach zero (a perfect conductor in an electric field has an inner field identically equal to zero) and produce no orienting torque on a point dipole inside. Molecules with a net charge introduce no difficulty as it is well known that the torque is equal to that on a molecule with a neutralizing charge a t the center of drag.6,g We have not discussed any orienting mechanism involving the conductivity of the solvent. These effects are complicated and will be omitted. Their inclusion would not change the form of our equations. Equations.-The electric pulse and resulting birefringence are considered in three regions: I, rise; 11, reverse and 111, decay. The rise and decay curves were first derived by Benoita; his model was extended in I. To obtain the birefringence, An, one must find the general angular distribution functionf(0, x, t ) as defined in 1.l0 Then for each region the appropriate conditions must be applied. These are: I, f ~ ( = t ~0) = 0, E = E; 11,f11 ( ~ I I= 0) = fi(t1 + a),E 7 -E; 111, ~III(~III= 0) = any f , E = 0. Multiplying f by an optical factor which is the same for all regions, one obtains An. At equilibrium in a static electric field Ans &(Pa - PI 4-n) A = 2~cEa/l5np Ans = equhbrium birefringence in electric field c = concentration in g./ml. g = (a0,33 - aOlll)/v ao,aa a0 11 = anistropy of optical polarizability v = voiume of solute particle E = fit?ldstrength in e.8.u. volt/cm. n = refractive index of solution p = density of solute pa (p'a/kT)' PI = Wl)' (pf*)*1/2(kT)2 4 ( f f E.? ~.li)/kT p' = effective dipole moment k = Boltxmann's constant T = absolute temperatiire 3
-
(1)
'
+
The dynamic birefringence curves are much more useful when one normalizes them by dividing by An,. Therefore, they will be presented thus (7) L. Onsager, J . Am. C k m . Soc.. 58, 1486 (1936). (8) c. T. O'Konaki, J . Chem. Phya., 23, 1559 (1965). (9) K. J. Mysels, ibid., 21, 201 (1958). (10) The second plus in ep. 4 of I, which definesf, should be a minus.
I. Rise 5811 f % A n / h n e = 1 - [Rp3/2(p3-- pl 4-q)]e-Zent [6011/(5011- 033)1 IpI/(p3 - pl ~?le-(~ll+ea3)~ f [P3/2 171 - Q - 6811~1/(58n - 0m)][l/(?h PI q)]e-fiellt (21 (b) 5% = 0 3 3 A = 1 \3~s/2(pa - pi g)],-'e111 [ ( p 3 / 2 pi - Q Gplellt)l(pa - PI q)le-Gelit (3) 11. Reverse (a) 5e11 # 8 3 8 A = 1 [3p3/(p3 - pl -I- g)]e-2ellt Ii2el1/~5ol1 - e3d][pi/(p3 p1 q)le-(ell+e=)t [3p3 i ~ ~ ~ ~ p ~833)][1/(pa / ( 5 e ~ ~PI p)e-6ellt (a) A
+ +
-
++
-
Ib)
+
+
+ +
+
+
-
-
-
+
+- +
(4)
5011 = 1 3 ~ ~
A = 1
-
+- +
[.3pa/(p3 p l g)]e-2ellt [ ( 3 p 3 12plel1t)/(p3- pl
111. Decay
+ q)le-6e111
A = e-6eiit
(5)
(6)
The previous equations are for the simplest case: a monodisperse system with fast induced dipole effects. If there are slow induced dipole effects, the equations must be modified. . A polarizability due to mobile ions which redistribute themselves over the surface of the molecule in a time of the same order of ma,gnitude as l/e would cause these effects. The previous equations can be modified by adding the following terms: IA and IIA, rise and reverse (add to eq. 2,3,4, or 5 ) (a)
601175
# 1; 601171 # 1
- [6011~3~i.s8/(6F)1i~a - l)(pr - pi -k q)le-t/r~
~6011~1qi,ii/(6011~1 - l)(pt - pi f q ) l e - t / T 1 +-6011~1qi,11/(601171 [6011~3qi.a3/(6011~a - 1) - 1)lIl/pa - PI + ~le-~e111
(b) 6911~8= 1; 601171 = 1 [6011t(gi.sa q i ~ l ) / b 3- p1
-
-
qi,jj aE,jj
=
=
+ q)le-Gelrt + (1 + -
aiE,jj/kT
aeE,jj
Q = qs
(7)
(8)
j = 1 or 3
e-t/Ti)
~r'E.jj ~i,33
Pisll
aE,s3; OLE,^^ = ionic component,sof the electrical polarizability along the symmetry and transverse axes, respectively 73, 71 = relaxation times for these components q., a e --~ electronic or atomic cont,ributions.
If there is an ionic polarizability with a relaxation time short compared to the time of rotation of the molecule, then eq. 2-5 are correct as they stand. The influence of polydispersity will be discussed in the succeeding section. Results The general equations in the preceding section are more useful if one considers special cases. If either eg3= ell or e33 >>el1,the equations are simplified considerably. The previous equations can be written as 1B. Rise ea3 >>el1or 8 3 3 = ell A = 1 - (3/2)p/(p l)e-Zeiit (l/2)(p - 2)Ap l)e-6elll IIB. Reverse Baa > >0 1 1 or 0 3 8 = 0 1 1 A = 1 [3p/(p l)][e-6'3118 e-2eiitJ If eas= 0 1 1 , then 13 = 60 = (PS pd/q If 0 a s >>%I, then P = Bp = p3/(q pi)
++
+
+
+
-
-
(9) (10)
-
To use with eq. 9 and 10, eq. 1 can be written in the form
REVERSING PULSE TECHNIQUE IN ELECTRIC BIREFRINGENCE
March, 1959 A% = &(Bo An, = AdB,
+ 1)q
425
(11)
4- l)(q - PI) t 12) Eq. 9 and 10 are plotted in Figs. 1,2 fcr various values of p. The rise curves have been given before by Benoita; however, the advantage of the reversing pulse method is immediately apparent from the figures. The reversing pulse birefringence is much more sensitive to p than the rise, particularly for positive values of 0. For example, a P of 0.1 would cause an easily apparent 10% dip in the birefringence on reversing the field, but would give 81, t . a rise curve practically indistinguishable from P = 0. Fig. 1.-The rise and reverse birefringence plotted us. As seen from the figures the position and magnitude of the various minima and maxima give a 811t for various positive values of p. If the rotary diffusion equal, = ell,then 0 = PO = (pa - pl)/o. quick measure of the molecular parameters. The coefficients are thenfl = Pp = p d ( q - p , ) . appropriate equations, obtained from the deriva- I f €)aa>>eI1, tives of eq. 9 and 10 for the time t, a t the extremum and the value of A(A,) at this time are 20 or
IC. Rise 0 3 8 >>eIlor Bs3=
ell 15
10
A. 05
0
Eq. 13 shows that for the rise curve t, depends on both p and 8, and that there are no maxima or minima for positive values of 0. For the reverse curve, however, t, is independent of p. Therefore, : both ell (eq. 16) and p (eq. 18) can be obtained conveniently and rapidly. More accurate values can, of course, be obtained from a lengthy complete analysis of the reverse and decay curves. Values of t, and A, cannot be obtained explicitly from the general equations with an arbitrary value of ea3.These values mould depend on both rotary diffusion coefficients and all electrical parameters. However, special cases are again of interest. If p l = 0, thenPo = pp = p 3 / q , and independent of the ratio 033/el1eq. 9 and 10 can be used instead of eq. 2-5. If q = 0, then the birefringence will depend on the ratios e33/e11 and 3/pl. Figure 3 shows the results for one value o p3/p1 and five values of ea3/el1. We notice that in general we have to know the ratio of the rotary diffusion coefficients before we can interpret the rise and reverse curves. If there are slow induced dipole terms, many new combinations of effects become possible. We shall , consider the case where a slow polarizability along the symmetry axis is the only important orienting factor. Figure 4 gives the results for four values of 6Ol1r3. Equations 2, 4, 7 and 8 were used with pt = p3 = qe = qi,ll = 0. It is apparent tha,t a slow induced dipole can give results qualitatively like a permanent dipole. The shapes of the birefringence curves, however, are different. For the reverse curve t, and Am are given by
P
IID. Reverse (only (a) 6 e i i ~ sf 1
important)
-1 0
:
5
0
-
E
Fig. 2.-The
Reversed 8111.
rise and reverse birefringence plotted rtq The ratio e88/011equals
eI1tfor varioua negative values of a. lor
m.
A,
(b)
60117~
=
1 - (6e1173)V(1-6e~~~a) (20)
1
tm = l/6e11 Am = (1 l/e)
(21) (22)
-
Figure 4 and eq. 19-22 will be useful if, for a monodisperse system with 833>>011 or Bar = 011,ell obtained from the minimum in the reverse curve does not agree with ell obtained from the decay curve. A non-zero value of 7 3 thus indicated, its magnitude could be estimated by direct comparison]withFig. 4. Polydispersity.-All the previous equations have referred to monodisperse systems. Unfortunately these systems are very hard to find in macromolecular chemistry. For a polydisperse system the analysis of birefringence curves becomes diffiand cult. We will consider the case where e33>>e11 the polydispersity is mainly in the length of similar molecules. Good examples are such systems as synthetic polypeptides or some TMV samples. For these molecules all pi = p and gi = g. We will not include slow induced dipole effects. The equations now become Ane =
(23)
Ane,! i
= Agci
(P3
- PI
P)i
(24)
IGNACIO TINOCO, JR.,AND KIWAMU YAMAOKA
426
VOl. 63
Im= (In 3)/4
(28)
The value of is a complicated average of the 8 i involving the electrical properties of the molecules. If t, is substituted into eq. 27 we obtain the value A at the extremum. 3 A m r l -
cipai~i i
C ci(p3 i
+ a)i
Bi = (3)-0i/2 11 - (3)-0i/]
25v
..
cipS,i
.
Fig. 3.-The rise and reverse birefringence plotted vs. O1,t for p = 0, p s / p 1 = I/*, and various valiies of @&I.
@,It.
Fig. 4.-The rise and reverse birefringence plotted vs. Bl1t for a time dependent polarizability along the symmetry axis of the molecule and no other orienting factors. The value of 7 ) characterizes the relaxation time of the polarizability.
Rise, reverse, or decay An =
(30)
Calculation shows that when 0.5 >(p3 and p l ) see. and 7 3 S 5 X Using the present equations we can make the following statements. Assuming that a 2% change
(11) The subscript 11 haa been dropped from the equationa ia this aeation for simplicity.
(12) C. T. O'Konski and A. J. Haltner, J . Am. Chem. Soc., 18, 3604 (lQ66).
From eq. 23 and 24 it follows that the equilibrium birefringence for a polydisperse system gives a weight average value of the electrical parameters.
wi(p3 =
- ~1
i
+ g)i
wi
(26)
i
wi = weight of component i
Analysis of the reverse curve can be made by an approximate method. We can write the polydisperse reverse equation asll IIE. Reverse ~ ~ p ~ , ~ [ e -6 0c-Zeit] it
3 A = - - - - - - An
- 1 +
i
c i ( ~ 3-
AlZe,i
i
i
PI
+ g)i
(27)
-
. I
(
March, 1959
GASCHROMATOGRAPHY WITH HYDROGEN AND DEUTERIUM
in birefringence on field reversal would be noticeable, eq. 17 gives -0.02
