2182
n4. MATSUMOTO, H. WATANABE,
For systems undergoing structural relaxation, the single relaxation time formulation employed here will not be adequate and a more involved expression for the
AND
K.
YOSHIOKA
frequency dependence of the kinematic viscosity should be used in the calculations. This is a straightforward calculation which we do not discuss here.
The Transient Electric Birefringence of Rigid Macromolecules in Solution under the Action of a Rectangular Pulse and a Reversing Pulse
by Mitsuhiro Matsumoto, Hiroshi Watanabe, and Koshiro Yoshioka Department of Chemistry, College of General Education, Universityof Tokyo, Meguroku, Tolcyo, Japan (Received December 8, 1969)
The diffusion equation describing the rotational diffusion of axially symmetric rigid macromolecules in solution placed in an electric field is solved by the use of the perturbation method, and equations for the rise of the birefringence under the action of a rectangular electrical pulse and for the birefringence in a rapidly reversed electric field are derived from the angular distribution function thus obtained. Calculations are performed up to the fourth-order perturbation in the general case and up to the sixth-order perturbation in the case of pure permanent dipole orientation. The extent of applicability of the Benoit theory for the rise birefringence and the Tinoco-Yamaoka theory for the reverse birefringence is discussed. Further, the effect of polydispersity is considered. It is shown that the rise, reverse, and decay curves are related by a simple equation even for a polydisperse system.
Introduction When a rectangular electrical pulse is applied to a macromolecular solution, the birefringence produced changes with time; it rises, reaches a steady state, if the pulse duration is sufficiently long, and then decays.‘,* This pulse technique has been successfully used for investigating the electrical and hydrodynamical properties of rigid macromolecules in solution. Benoit‘ has developed a theory for the rise and decay of the electric birefringence. He considered an ellipsoid of revolution with a permanent dipole moment along the symmetry axis as a model for the macromolecule and solved the diffusion equation which describes the rotational diffusion of rigid macromolecules in an electric field by expanding the angular distribution function in a series of Legendre functions. He obtained an expression for the rise of the birefringence which is valid at low fields and showed that the mechanism of electrical orientation can be elucidated from the rise curve. Tinoco4 has extended Benoit’s theory to include the effect of a transverse component of the permanent dipole moment. He also considered the contribution of the fluctuating dipole moment due to proton migration to the rise of the birefringence. O’Eco11ski, Yoshioka, and Orttung5 have obtained equations for the rise of the birefringence a t infinitely high field strength in the cases of pure permanent dipole orientaThe Journal of Physical Chemiatry
tion and pure induced dipole orientation, following the treatment of Schwarz,6who pointed out that the effect of rotational diffusion can be neglected at infinitely high field strength. n’ishinari and Yoshioka’ have proposed a theory for the rise of the birefringence which holds for arbitrary field strength in the initial stage. They showed that it is possible to determine the permanent dipole moment and the anisotropy of electrical polarizability separately from measurements of the rise of the birefringence a t high fields. O’Konski and Haltner*have introduced the reversingpulse technique in birefringence measurements; a square pulse is applied to a macromolecular solution, and, after the steady-state birefringence is achieved, the field is rapidly reversed in sign. This technique is very (1) H . Benoit, Ann. Phys., 6, 561 (1951). (2) C. T . O’Konski and B. H. Zimm, Science, 111, 113 (1950). (3) For a review, see K. Yoshioka and H. Watanabe in “Physical Principles and Techniques of Protein Chemistry, Part A,” S. J . Leach, Ed., Academic Press, Inc., New York, N. Y . , 1969, p 335. (4) I . Tinoco, Jr., J . Amer. Chem. SOC.,77, 4486 (1955). (5) C. T . O’Konski, K. Yoshioka, and W. H. Orttung, J . Phys. Chem., 63, 1558 (1959). (6) G. Schwarz, 2. Phys., 145, 563 (1956). (7) K . Nishinari and K. Yoshioka, Kolloid-Z. Z . Polym., 235, 1189 (1969). (8) C. T . O’Konski and A. J . Haltner, J. Amer. Chem. SOC.,79, 5634 (1957).
TRANSIENT ELECTRIC BIREFRINGENCE OF RIGIDMACROMOLECULES useful for investigating the mechanisms of electrical orientation. Tinoco and Yamaokag have derived equations for the birefringence under the action of a reversing pulse and plotted them for various values of the electrical parameters of the macromolecule. Their equations are valid at low fields. In this paper we derive equations for the rise and reverse birefringences which hold even for higher fields. The angular distribution function of rigid macromolecules in an electric field is obtained by solving the diffusion equation by the use of the perturbation method. Calculations are performed up to the fourth-order perturbation in the general case and up to the sixth-order perturbation in the case of pure permanent dipole orientation. The results are also presented in graphs. The extent of applicability of the Benoit theory for the rise of the birefringence and the Tinoco-Yamaoka theory for the reverse birefringence is discussed. Further, the effect of polydispersity on the rise and reverse birefringence is considered, and a simple equation relating the rise, reverse, and decay curves which hold even for a polydisperse system is obtained.
Theory The time dependence of the birefringence for a dilute solution of rigid macromolecules under the action of a rectangular pulse and a reversing pulse will be considered. We assume that the macromolecule has a common axis of symmetry for its electrical, optical, and hydrodynamic properties and the permanent dipole moment is along this axis. When an electric field is applied to the solution, a torque is exerted on the macromolecule and the angular distribution function of orientation changes with time. The angular distribution function f (e, t ) , which depends on time t and the angle 6 between the symmetry axis of the macromolecule and the field direction, should satisfy the diffusion equation
metry and transverse axes, respectively. the notations
M = -p’E sin 0 - ( a l - a2)E2sin 0 cos 6
(3)
where E is the field strength, p’ is the apparent permanent dipole moment in solution, and a1 and a2 are the excess electrical polarizabilities along the sym-
(4) (54
and 7 = [(ai - az)/2kT]E2 =
cE’
(5b)
we obtain from eq 2 and 3
d((1 bU
- u2) bf - (1 - u2) x ( P + 27u)f} =
e
(6)
To solve this partial differential equation we attempt to separate variables by the substitution
(7)
f(u, t) = g(t)h(u)
Introducing this into eq 6 and dividing by g(t)h(u), we obtain dh du
h du
(P
+ 27u)hJ
=
1 1 dg
e-&
(8)
The right side of eq 8 is a function of t alone and the left side is a function of u alone. Hence the value of the quantity to which each side is equal must be a constant, which we call - A . Equation 8 can then be written as two differential equations, namely
(9) and dh
(1 - u2) x
The equation for g(t) can be integrated a t once to give g(t) = (constant) exp( -xet)
(11)
whereas the equation for h(u) cannot be analytically integrated in general. However, eq 10 is transformed into the self-adjoint form by multiplying it by exp( -Pu - yu2), namely d -{(I du
I n the present model the torque is given by
Introducing
u = cos e P = (p’/kT)E = bE
du
where 8 is the rotational diffusion constant for rotation about the transverse axis, M is the torque, { is the rotational frictional coefficient, and the symbol V 2 is the Laplacian operator. On introducing Einstein’s relation 8 = k T / { , where k is the Boltzmann constant and T is the absolute temperature, eq 1 becomes
2183
-
u*) exp(--pu
(67u2
-
+ 2Pu - 27) exp(-Pu X exp(-Pu
- y u 2 ) h+ - yu2)h = 0
(12)
(9) 1. Tinoco, Jr., and K. Yamaoka, J. Phys. Chem., 63, 423 (1959).
Volume 74, Number 10 M a y 14, 1970
2184
M. MATSUMOTO, H. WATANABE, AND K. YOSHIOKA
This equation is a kind of the Liouville equation with two singular points at u = 1 and u = -1. Denoting the solution by h, for X = X, and by h, for X = ,A, we multiply the equation for h, by h, and the equation for h, by h, and subtract the one from the other. Integrating the resulting equation from -1 to 1, we obtain the reht'ion 1
- h,)J
(A,
-1
- y u 2 ) du
h,h, exp(-0u
=
0 (13)
+ a,)b, + n(n + l)c, 0 + + 6,}P, +
(bu
--{
d (1 du
,)%}
(CZ(U4
+ ru2)/2}
y,(u) = M u ) exp{ - ( P U
%)+ [buE + {:( + 3e)u2 - (:
n(n
(14)
into eq 10 and referring to eq 6a and 5b, we find that y, must satisfy the equation
- u2)
n1
(16)
du
For the solution of this equation to be finite in the range -1 5 u 5 1, X, = n(n 1) with n = 0, 1, 2, , . . . The function yn corresponding to X, is the Legendre function of degree n, P,(u). We can obtain an approximate solution of eq 15 by the use of the perturbation method. Let us expand the eigenvalue A, and the eigenfunction yn in terms of E
+
+ 1) + a,E + &E2 + y,E3 + 6,E4 + .... Pn(u) + an(u)E + bn(u)E2+
A, = n(n
=
C A h , exp(-AX,W n=O
+ dn(u)E4+ .-..
A{ (1 - u2) p.,) + (bu + a,)P, du n(n
+ a , ) ~ ,+ n(n + l)b,
The Journal of Physical Chemistry
c A,h, m
f(u, 0) =
n=O
=
m
C n=O
A,Y, exp((pu
(18)
+ yu2)/2} = 4,1
(22)
The solution of eq 10 for X = 0 gives h,, and it is found that
f(u,a ) = Aoho
exp(0u
=
2a
= 0 (19a)
0 (19b)
SI,
+ yu2)
+ y u 2 )du
(24)
exp(@u
For a reversing pulse, the angular distribution function is obtained by replacing E by - E (P by -0) in the preceding treatment and applying the initial condition n=O
=
(21)
where A,'s are constants. For the rise process, An's are determined so as to satisfy the initial condition
+ + l)a,
+ ru2)/2) exp(--)r,W
(17)
du
(bu
Any, exp((flu
From eq 22 and the orthogonality of yn we obtain
Introducing eq 17 and 18 in eq 15 and equating the terms of E , E2, E3, and E4 to zero, we obtain a set of differential equations
du
= n=O
du
+1
m
f(u,t ) m
c,(u)E3
2n
I n this work we retained all terms up to E4. It is to be noted that a, = y n = 0. Thus, the general solution of eq 6 is given by
In the limit as E +. 0, eq 15 reduces to
=
(19d)
0
yn2du =
J-1
+ c)]E2 +
+ l)d,
These equations are solved by expanding a,(u), b,(u), c,(u), and dn(u)in terms of Legendre functions. The coefficients of expansion are subject to the condition
-1
yn(u)
= 0
242)
+
+
du
(19c)
du
as long as h, and h, are finite a t u = 1 and u = - 1. Equation 13 indicates that h, exp{ -(flu yu2)/2] and h, exp{ -(Pu y u 2 ) / 2 } are mutually orthogonal and that A, and X, are the eigenvalues. Making the substitution
-{(l d du
=
2rJ
exp(0u -1
+ y u 2 ) du
TRANSIENT ELECTRIC BIREFRINGENCE OF RIGIDMACROMOLECULES
It is found that
+ yu2) exp(-Pu + w2) du
exp(-Pu
f(u,a ) = Aoho =
1
2rJ-1
2185
Xi' = 3P2
+ ~3 (
(26)
XZ'
=
3P2
+
2 6 4 71f127) ~
1 + +4P4 + 9PZY)
+ 6p27)
2 175
Birefringence. The electric birefringence is given
X3' = -(llP4
by lo
(32b) (32c)
(324
At limiting low fields, the equation for the rise of the birefringence reduces to where c is the concentration in grams per cubic centimeter, p is the density of the solute, n is the refractive index of the solution, and gl - g2 is the optical anisotropy factor. Substituting eq 21 into eq 27 and expanding the exponential functions up to the terms of E4, we can perform the integration. The final result for the rise of the normalized birefringence, A ( t ) , is =1
1 - xxo exp(-Xlet) +
XZ
X3
- exp( -X38t) xo exp( -Xx,et)- xo
+ p2/5 - 47/5 = 6 + p2/7 - 47/7 = 12 + 2P2/15 - 87/15 = 2
Xp A3
X o = p2
2 + 27 - 21 -(p4
3 x1 = 32-p2 - 350 -(9P4
1 2
Xp = -(Pz
- 47)
+ ,(1a4
(29b)
(29c)
- 2pz7 - 2 7 9 (30a)
- p27)
+3P22 r exp( -2et) + 3P2 exp(-68t) P 2 + 27
~
P2
(28)
(294
(33)
(34)
These equations coincide with those of Benoit' and Tinoco and Y a m a ~ k a . In ~ these cases the normalized birefringence vs. 8t curves are solely determined by
r 5p2/27
=
b2/2c =
pf2/(a1
- az)lcT
(35)
independent of the field strength. I n the special case of pure permanent dipole orientation we performed the calculation of the birefringence up to the terms in E6 (sixth-order perturbation). For the rise of the birefringence, we have A(t) = 1 -
'($2
XO 2
49
-27 4 350'
(30b)
- l l p 2~ 87') ( 3 0 ~ )
65 -p2 1
- ----84)et} 1 - -(-p4 1
7
The normalized birefringence in a rapidly reversed electric field is given' by
xz ' Xa' x-,o exp(--X~@t)- XO' exp( - M t )
+ 2 r ) exp( -602)
and the equation for the birefringence in a rapidly reversed field reduces to A(t) = 1 -
where An(t) is the birefringence a t time t, An( a ) is the birefringence for 1 4 a , &e., the steady-state birefringence, and XI, Xp, ha, X o , X1, Xz, and X3, are functions of P and y A1
P 2 - 47 2(P2
1 Xo 175
4116
--P) 19 157,500
+
+2 +
exp{
- (12
10
37,913 -I- 427,314,888
$2
19 222,750 (31)
where X,I XZ, and X 3 are the same as expressed by eq 29, and
(2o -I- 77'' where
(10) A. Peterlin and
H.A. Stuart, 2. Phys.,
112, 129 (1939).
Volume 74, Number 10 May 14, 1970
2186
M. MATSUMOTO, H. WATANABE, AND K. YOSHIOKA
For the birefringence in a rapidly reversed field, we have
The birefringence in a rapidly reversed field for a polydisperse system is given by
78 269 + 7 p 4 4--P6) X xo ( 17s 8750 exp{ - ( 2 + 1 - &~4)et} + +( 3P2 +
3Cc,p12 exp( -2eit)
A(t) = 1 - 73p2
4 7-64
+
659
- (6
exp{
+1 7
~
1
2 4 x ~ 4 )
=
-
+ 2yJ
&Pr2
+
i
x instead of eq 34. From eq 42 and 40a we obtain A R ( ~ )= 1 - 3A~’(t/3)
+ 3h~’(1)
(43)
In the case of pure permanent dipole orientation, eq 41 and 43 reduce to
10
(
37,913 o4)et> exp{ - 2o + ZP2 + 427,314,888
(37)
3 1 AB(^) = 1 - -A~(t/3) -k AD(^) 2
(44)
A R ( ~ )= 1 - 3A~(t/3)-k 3 A ~ ( t )
(45)
and
where
XO’ =
1 1 ++ -0 6 14 504 p4
p2
Polydispersity. For a polydisperse system, the rise or buildup of the birefringence a t limiting low fields is given by
respectively. On the other hand, in the case of pure induced dipole orientation, we have AB(^) = 1 - AD(^)
and
(47) Thus, in these special cases, the rise and reverse curves can be obtained from the decay curve even for a polydisperse system, as long as the interactions between macromolecules are negligible. I n general, AB(^), A R ( ~ ) ,and AD(^) are related by a simple equation. Namely, from eq 39, 41, and 43, we obtain AB(t) =
i
i
instead of eq 33, if it is assumed that all components have the same density and the same optical anisotropy factor. This assumption will hold for helical polypeptides, for instance. On the other hand, the decay of the birefringence is given by
+
ccl(P12 2ri) e x p ( - - 6 ~ 4 i
AD(o
CC&2
=
i
= AD'(^>
+ 27,)
+ AD'(^)
(39)
where A=’(t) and AD'(^) are defined by ~ c & exp(-60&) 2 AD'(^) = ’ CCd(Pt2 27,) i
+
(404
Cct(27,)ex~(--6ett) i
AD’(t) =
CC,(&* + 2y,) i
(40b)
Substituting eq 40a and 40b into eq 38, we obtain 3
AB(^) = 1 - - A~’(t/3) 2 The Journal of Physical Chemistry
+ -21 AD'(^) -
AD'(^)
(41)
(46)
2Ag(t)
1
- A R ( ~ )= 1 - 2&3(t)
(48)
Results and Discussion Equation 28 for the rise of the normalized birefringence is plotted for various values of P2 and y in Figures 1 and 2. The broken curves in these figures indicate the normalized rise curves at limiting low fields (Benoit’s equation, eq 33). When the ratio r ( = p 2 / 2 y ) is positive and small (e.g., r = 0.75), a negative deviation from Benoit’s equation is noticed at higher fields. On the other hand, when r is positive and large (e.g., r = 2.5), a positive deviation is noticed. When r is negative, the deviation from Benoit’s equation is more pronounced than in the case of r 2 0, as shown in Figure 2. In order to express the deviation quantitatively, - + ,et , = 0.5 and 1 are given the values of ~ ( t ) / [ ~ ( t ) ] ~ for in Table I. Equation 31 for the normalized birefringence in a rapidly reversed field is plotted for various values of p2 and y in Figures 3 and 4. The broken curves in these figures indicate the normalized reverse curves a t
2187
TRANSIENT ELECTRIC BIREFRINGENCE OF RIGIDMACROMOLECULES
1
1
0 . 8 , b'=0.533( r50.75 ) n
h
4-
c)
Y
Y
Q
Q
0 Y' 0
I
I
I
I
\
0
1
1
1
1
1
2
1
2
et
et Figure 3. Normalized reverse birefringence plotted us. et for various values of 02 and y( 2 0).
Figure 1. Normalized rise birefringence plotted us. Bt for various values of p2 and y ( 2 0).
LTinoco-Yamaoka( r = - 2 # 5 )
'I
W
Figure 4. Normalized reverse birefringence plotted us. et for various values of 02 and y(
