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Transient electric light scattering by large strongly elongated disperse particles
Transient Electric Light Scattering by Large Strongly Elongated Disperse Particles S. SOKEROV A~D M. STOIMENOVA , Institute of Physical Chemistry, Bulgarian Academy of Sciences, S O F I A 13, Bulgaria Received February 16, 1973; accepted May 17, 1973 The theory of transient electric light scattering is extended to larger particles, fulfilling the restrictions of the Rayleigh-Debye-Gans approximation. The equations derived are valid for rigid, strongly elongated, monodisperse particles, orientated under the simultaneous action of both their permanent and induced dipole moments. , The equation for the rise of the effect gives the possibility the particle electric moments to be determined. The equation describing the decay of electric light scattering differs from the one given by Wippler. It offers a method for obtaining the particle rotational diffusion coefficient. For small particles the equations derived coincide with the formulas of Nishinari-Yoshioka, Benoit, and Tinoco, describing transient electric birefringence. The dependence of the coefficient on p~rticle size in the formulas presented shows the limits of application of the small particle theory. INTRODUCTION
with a permanent dipole moment along the symmetry axis as a model for the particle Tinoco (2) extended Benoit's theory to include the effect of a transverse component of the permanent dipole moment. Benoit's equations for the rise process hold only for low fields. Making use of Schwarz's formulas (3) describing the variation of conductivity in an electric field, O'Konski el el. (4) obtained equations for high degrees of orientation. Further, Nishinari and Yoshioka (5) proposed formulas for arbitrary degrees of particle orientation. For the phenomenon of light scattering in an electric field, the transient process of disorientation was treated by Wippler (6) in the Rayleigh-Debye-Gans approximation and he gave an equation analogous to the one for electric birefringence. For electric light scattering by small anisodiametric particles (small compared to the wavelength of incident light) Stoylov (7) showed that all the equations describing the transient processes are completely analogous to those for electric birefringence.
On application of a rectangular electric pulse to a solution of anisodiametric particles the latter orientate themselves on account of the interaction of their electric moments with the applied field. After switching off the field Brownian motion causes disorientation of the particles. The processes of orientation and disorientation are connected with movement of the particles in a viscous medium. It is assumed that the times for rise and decay of the square electric pulse are much smaller than the characteristic time of particle orientation connected with particle dimensions and form. By a transient process of orientation we mean the change of any effect due to particle orientation (electric birefringence, electric light scattering, etc.) with the time until a steady-state regime is established. The theory of the transient electric birefringence upon application of a rectangular pulse was given by Benoit (1), who solved the diffusion equation for axially symmetric particles. He considered an ellipsoid of revolution 94
Journa~ of Colloid and Interface Science, Vol. 46, No. 1, January 1974
Copyright ~ 1974 by Academic Press. Inc. All rights of reproduction in any form reserved.
95
ELECTRIC LIGHT SCATTERING
Of interest is a theoretical treatment of the transient electric light scattering by anisodiametric particles, without restrictions on particle size, within the Rayleigh-Debye-Gans approximation. This would permit geometric and electric parameters of larger particles to be determined by the method of electric light scattering. THEORY
The solution is assumed to be monodisperse and sufficiently dilute so that interaction effects between the particles can be neglected. Further, it is assumed that the particles are rigid, optically isotropic, strongly elongated cylinders with a common axis of symmetry for their electric, optical and hydrodynamic properties. Such a model facilitates the calculations and describes comparatively well the optical behavior of the elongated particles. The approximation of Rayleigh-Debye-Gans (RDG) is utilized (8), its restrictive conditions being:
(4zr/X)llm- 1~i<<1;
Ira-
11 <<1,
where x is the wavelength of incident light in the solution, l is the length of the particle, and m is the relative refractive index (m = nv/ns; np is the refractive index of the particle and n~ that of the solvent). This approximation allows calculations to be carried out for comparatively large particles (their dimensions comparable with the wavelength of light and even greater) if their refractive index does not differ much from that of the medium. The effect of a relative change in the intensity of scattered light on application of an electric field to a solution of anisodiametric particles is given by IE
-
where IE and Io are the intensities of scattered light with and without a field, respectively, i(O, 9) is the intensity of light scattered by one particle, f(O, 9, t) is the orientation distribution function of the particles, 0 is the angle between the particle symmetry axis and the field direction, 9 is a spherical coordinate,f co • .. do~ shows integration over a sphere of radius unit, and t is the time. In the present model i(O, 9) and f(O, 9, t) are given as follows : The intensity of the light scattered by one particle (when the incident light is plane polarized) in the RDG approximation is given by (8) i(0,
2,.-7\ - ~ - ~ - /
sin R(0, @ =
[-kolsin (0'/2) sin 0 cos 9~ ko! sin (0'/2) sin 0 cos 9
,
E3]
R I S E PROCESS
The orientation distribution function f(O, 9,t) in the electric field is determined by the diffusion equation (I) :
i
Joy
V2f -t- - - div [-f grad
kT
Io
r21
IR(O', ¢)12/0,
where O' is the angle of observation (incident, scattered light); the other indexes are as explained above.
OL--
U] -
D Ot
or
4~ [ i(o, 9)/(o, 9, 8d~ ! J
ko4v2(m - 1~ 2 =
where k0 is the propagation constant, v is the volume of the particle, r is the distance from the center of the particle to the observer, R(O', 9') is a factor depending on particle geometry and orientation, and i0 is the incident light. In our case (thin rods in an electric field,, perpendicular to the plane of observation 1) the orientation factor can be written as (8) :
Io
-
9)
Pf(., 0
¢o
- l,
f i(O, 9)dco
D]
1 of(~, t)
D
Ol
1 The restriction of the electric field perpendicular to the plane of observation is not important. I t is not difficult to generalize the results for an arbitrary angle.
Journal of Colloid and Interface Science, Vol. 46, No. 1, January 1974
96
SOKEROV AND STOIMENOVA
where u -= cos O, t is the time, D is the rotational diffusion constant for rotation about the transverse axis of the particle, U - - - - u E cos 0 - - (6, cos 2 0 + & sin 2 0)E2/2 is the particle potential energy in the applied field, E is the field strength, u is the apparent permanent dipole moment in solution (when directed along particle s y m m e t r y axis), (~1 and & are the excess electric polarizabilities along the s y m m e t r y and transverse axes, respectively, k is the Boltzmann constant, T is the absolute temperature, and the operator/~ is defined by 02
P =
(1 -
process after time t, K = (2~-/X)l sin (0'/2) is a parameter proportional to the particle length, To = Si(2K)/K -- (sin K / K ) ~ is proportional to the scattered light intensity without field (8), and Si(x) = f o x (sin y/y)dy. The effect is expanded in the power series of both particle size and time. Substituting the corresponding power of the operator fi, carrying out the integration, and expressing the infinite series by tabulated functions we obtain the coefficients preceding the powers of time. Here is the series up to the second power of time :
0
u ~)
-
2u--
Ou2
~Ot
o~lr - -
To
0 -(1
- u=)(~ + 2vu) - -
+--
Si(x) .4 (x) = - -
t3 = tIE/k T, (h
We make use of the solution suggested by Nishinary and Yoshioka (5), which is an expansion of the distribution function in the power series of time:
f(u, l) = exp (Dlfi)f(u, 0), E4-] where f(u, 0) is the distribution function at t = 0. Since the distribution function must be normalized and the distribution is uniform in the absence of a field, we have f(u, O) = ~rc. Substituting the equations for i E21, and [33, and f [4-], in the general formula ~1~ for the effect a, and performing part of the calculation, we obtain 1 ~t r ~
--
~, (--1)"+'(2K)2"-2(2n -- 3)!!
X (2n-
~=o m ! 1)!! =
~
1)(2n
--
3)..-3.1;
Xa
8 x~
X2 '
3 sin x 8 xa 45 sin x
4
4
x ~"
F
4
M
M
Thus from the initial slope of the rise curve a versus t we can determine the anisotropy of electric polarizability, if D and K are known: (~1-
as) -
Cq
ET]
DE2 A(2K) \ dt /t~o"
Further the initial slope of the a/t versus 1 curve gives the value of the permanent dipole m o m e n t (along the s y m m e t r y axis): To]'
IS]
1
(2n --
4
45 cos x
kT[
(1 - - u ~ ) " - ' P " d u ,
3 sin x
X2
8 x
(2n) !(2~ - 2)!!
Draw f_~ E --
[6~
+
5 &(x) 5 cos x B(~) = - - +
~.
TO .=1
on
"r-lD2l 2+ . . . , 7"0 l
cos x 4- - -
X
6~)E~/kT.
-
T0-
where
where
=
(132 _ 3V)
+L
B(2K)
Ou
+2J-flu + "/(3u 2 -- 1)1,
~,
[A(2K)
A(2K)
Ou
(2n)!!
xt\-ZZj,
h 0 + D(% C 7,, 0
= (2n)(2n -- 2)--.4.2. where ~t r is the value of the effect for the rise Journal of Colloid and Interface Science, V o l . 46, N o . 1, J a n u a r y 1 9 7 4
A'(2K) \ dt/,~oJ
"
[8]
ELECTRIC LIGHT SCATTERING If the diffusion equation is solved in the general case, when the direction of the permanent dipole moment is arbitrary, the equation [-6] is transformed into
97
or ( 1 - u 2) 02f(u, t) Ou2
Of(u, t) 2u-Ou 1 Of(u, t)
A (2K) at ~"= - %
DOt
TDll
/A (2K) [/312
/DI+D2\
B(2K) 2}
+ - - T
D1212+ "'',
[9-]
T0
f(u, 0
where 31 = O~rE)/kT, 52 = (g2E)/kT, St1 and ~ are the components of the permanent dipole moment along the symmetry and transverse axes, and D1 and D2 are the rotational diffusion constants for rotation about the transverse and symmetry axes, respectively. In this case formula [-9-] can only give an equation connecting the two components of the permanent dipole. In order to obtain its value and direction, we need a second equation for v~ and ~2 [we can obtain such a one, for example, for the steady-state process (9)]. DECAY PROCESS After the electric field is switched off the angular distribution function satisfies the simple diffusion equation (1): 1 Of V~f . . . . D Ot
at a =
-
with the initial condition that at t = 0 the function f(u, 0) is to be determined by the steady-state process. We utilize for f(u, t) the solution suggested by Benoit (1). It is an expansion of f(u, t) in a series of Legendre polynomials :
= k C~ exp [ - - n ( n + 1)Dt]Pn(u),
[10]
n=l
where U = COS 0,
c~ = ~-(2~ + 1) /(~, o) =
f
f(u, O)Po(u)du,
exp (-- U / k r ) o exp (-- U / k T ) d u
and P,~(u) is a Legendre polynomial. When the particles are fully orientated, their potential energy is given by U = --t*E --1(61E~). Substituting the above equation [10] for f(O, ¢, t) and [2].for i(O, ¢) in formula [1-] and integrating, we obtain the following expression for the effect ~ after the electric field is switched off (and if the disorientation begins from a state of full particle orientation) :
1 ~ ,~-1(--1)-+~+12(2K)2,, 2(n -- 1)!(2m -- 1)!!(4m + 1) e×p [--2m(2m + 1)Dt] E E To n=2 m~l (2n) !(2n -- 1)(n - - m - 1)!m!(2n + 1 ) ( 2 n - 1)... (2n + 2 m - 1)
[11]
-
The first several terms of the expansion in a power series of particle length are: 1 [ K: a'd = "ToL 9 e--6Dt
2K4( 10 e--2ODt 225 7 e - S ° t - - - 7 2K8
/20
- -
1--
127575 \11
e--6Dt ~
11
162 e-6Dr
-- --
4 e-2opt
143
_ _ e - - 2 O D t -~- _ _ e - - 4 2 D t
+
--
11
33
7 e-42Dr -- --
\ e-72Dr)
7
3 q -" "
.
[12-]
143
J o u r n a l o f Colloid and Interface Science,
Vol. 46, No. 1, January 1974
98
SOKEROV AND STOIMENOVA
Expansion of a? t in a power series of time exponential yields: L(2K) ect d - -
- -
M(2K) e - - 6 D t ~ - -"
N(2K)
To
To
[13]
e- e D t + . . . ,
- e--2ODt ~- - - -
%
where 5 S
)+
+3
81 ( S i ( x ) c o s x
+ 325/Si(x)
cos x
5sinx
70cosx
3
3
21 sin x 5
xa
70sinx
.4 + 3 294 cos x 5
x4
;)
9
'
1071sin x l0
x5 2772 sin x
2772cosx
2561)
---~
5
x~
5
x7
25
From Eq. [11] one can obtain the expansion of a t d in the power series of time as follows:
1[
2K2 ( 8) ( 2 0 8 at a = - - 1 -- 7-'0-Dt + 2 K z Jr_ _ [~4 DZt2 __ 4K 2 _}._ _ _ _ To 3 5 45
The initial slope of the decay curve a d versus t (when K is known) allows one to determine the rotational diffusion constant for rotation about the particle transverse axis: D=
3T0 [d(--a*d) 1 2K 2L
7;
-It~o"
Another method for determining the rotational diffusion coefficient is by measuring the effect of disorientation for time t = 1 / 6 D [-the terms of the series after exp (--7/3) can be neglected] : L(2K) o~td - -
To Xexp(-1)
1-{- L ( 2 K ) exP
--
q- -N(2K) exp (-- 6) q- -.- 1 . [15-] rX2K)
1{4 @
8)
21
K 6 Data
]
+ .-..
El4]
DISCUSSION The equations obtained for transient electric light scattering by elongated particles within the RDG approximation extend considerably the application of the method to a wider range of particle size, and allow the precision in determining the particle electric and geometric parameters to be increased. The determination of particle electric parameters by utilizing the steady-state effect at low degrees of orientation yields an error due to the dispersion of the results at low fields. The above formulas for the rise process are obtained without restrictions on field strength and are valid for arbitrary degrees of orientation. Thus one is able to determine the value of electric polarizability at various orienting fields and to study the probable existing saturation of that value. The formula for the decay process is valid in the case when particle disorientation begins
J o u r n a l o f C o l l o i d a n d I n t e r f a c e Sc i e nc e , Vol. 46, No. 1, J a n u a r y 1974
ELECTRIC
LIGHT
from a state of full orientation along the field direction. It differs from the equation given by Wippler (6), and even in the case of a monodisperse solution the effect is described by a sum of time exponentials. Thus, using only one curve of the effect of full orientation and disorientation, one can determine the anisotropy of electric polarizability by the formula: (~l
-
&) -
2kT
K2
3E~ A ( 2 ~ )
/(% ]
×
An analogous equation can be written for the permanent dipole moment, if the latter is directed along the particle symmetry axis. The value of K and hence l can be obtained by the steady-state effect, which in the case of full particle orientation is given by a~ = (1 - %)/To (where % is a function only of K) (7). The combination of l, obtained by a~, with D, obtained by the decay process, allows one to determine the particle axial ratio p, and diameter b with the help of Broersma's formula (10). Of particular interest is the case when the particle permanent dipole moment has an arbitrary direction. If #1 and g2 are the components of the permanent dipole along the particle symmetry and transverse axes respectively, we can write formula [-8~ for this case as follows : / D 1 Jr- D2 \
)-
K~T 2
99
SCATTERING
for ellipsoids of rotation. Using Perrin's formulas for the case of cylinders one obtains an error not higher than 15% (10). According to Perrin the quantity (Di q- D2) 2D1
(2 In 2p
-- 2)
is a function only of particle axial ratio. For p varying from 5 to 20 the quantity (D1 Jr- D2)/2D1 changes in the interval 8.7 to 73.5. If the quantity ~ 2 _ #22(D~ Jr_ D2)/2D1 has a positive value, we can consider the permanent dipole as being directed nearly along the particle symmetry axis, since its component along this axis is at least [(D~ q-D~)/2DI-1 *~ times longer than the one along the transverse axis. In the case of a negative value of the above quantity, as we mentioned, a second equation is necessary for determining the permanent dipole moment. Using ~z~, we can introduce (4) the quantity "degree of orientation"
= [TO/(1 -
-T0)]~,.
The formulas for q~t are obtained by replacing the coefficients F ( K ) / T o in the equations for at by F ( K ) / ( 1 %). Expand~ing the coefficients F ( K ) / ( 1 -- To) in the power series of K and neglecting the terms higher than K 2 we obtain the following values of these coefficients for small particles : A (2K)/(1 -- %) = 2/5, B (2K)/(1 -- ¢0) = -- 1/35, L(2K)/(1 -
To
@2 _ in 2p)
TO) = 1,
M(2K)/(1 - To) = 0, N ( 2 K ) / ( 1 -- %) = O.
A2(2K) dt /e+0/ The quantity (Dt + D2)/2D1 can be estimated utilizing the formulas of Perrin (11, 12)
Substituting the above values in formulas E6~, E9], and 1-13], we obtain respectively the equations of Nishinari-Yoshioka (5), Tinoko (2) at t--+ ~o, and Benoit (1) for the electric transient birefringence. The latter formula, [133, in this case coincides with Wippler's equation (6) for the decay of electric light scattering. These
Journal of Colloid and Interface Science, Vol. 46, No. 1, J a n u a r y 1974
100
SOKEROV AND STOIMENOVA
results once more confirm that when particles are small enough (compared to the wavelength of incident light), the "degree of orientation" for the different electooptic phenomena is described b y the same equations (7). The correlation of the coe~cients in the formulas for small and for large particles allows one to estimate the error which is made when particle parameters are determined utilizing the theory for small particles. The estimation shows that the values obtained for D will always be greater than the real ones, while those for 7 and 1 will be smaller. The equations valid for small particles yield an error of about 30% for D and -y at K = 2, and for l at K = 4. For particles of higher K the equations presented in this paper should be used. Otherwise the values of the experi-
mentally determined parameters an order from the real ones.
can differ
REFERENCES 1. BENOIT,H, Ann. Phys. 6, 651 (1951). 2. TINOCO,I., Jf~., ar. Amer. Chem. Soc. 77, 4486 (1955). 3. SC~IWAI~Z,G., Z. Physik 145, 563 (1956). 4. O'KoNsK1, C. T., YOSHIOKA,K., AND O~TUNG, W. H., J. Phys. Chem. 63, 1558 (1959). 5. Nlsmi,VA~i, K., AND YOSItlOKA, K., Kolliod-Z. Z. Polym. 235, 1189 (1969). 6. WIPPL~,X,C., J. Chem. Phys. 53, 321 (1956). 7. STOYLOV, S. P., Advan. Colloid Interface Sci. 3, 45 (1971). 8. VAStDE HULST, H. C., "Light Scattering by Small Particles," p. 104. Inostrannaja Literatura, Moscow, 1961. 9. SOKEROV, S., A?CDSTOIMXNOVA,M., unpublished results. 10. B~OERS~A,S., Y. Chem. Phys. 33, 1626 (1960). 11. PER~IN, F., J. Phys. Radium 5, 497 (1934). 12. NEURA:C~,H., AN9 BAILEY,K., "The Proteins," p. 291. Moscow, 1956.
Journal of Colloid and Interface Science, Vol. 46, No. 1, January 1974
with a permanent dipole moment along the symmetry axis as a model for the particle Tinoco (2) extended Benoit's theory to include the effect of a transverse component of the permanent dipole moment. Benoit's equations for the rise process hold only for low fields. Making use of Schwarz's formulas (3) describing the variation of conductivity in an electric field, O'Konski el el. (4) obtained equations for high degrees of orientation. Further, Nishinari and Yoshioka (5) proposed formulas for arbitrary degrees of particle orientation. For the phenomenon of light scattering in an electric field, the transient process of disorientation was treated by Wippler (6) in the Rayleigh-Debye-Gans approximation and he gave an equation analogous to the one for electric birefringence. For electric light scattering by small anisodiametric particles (small compared to the wavelength of incident light) Stoylov (7) showed that all the equations describing the transient processes are completely analogous to those for electric birefringence.
On application of a rectangular electric pulse to a solution of anisodiametric particles the latter orientate themselves on account of the interaction of their electric moments with the applied field. After switching off the field Brownian motion causes disorientation of the particles. The processes of orientation and disorientation are connected with movement of the particles in a viscous medium. It is assumed that the times for rise and decay of the square electric pulse are much smaller than the characteristic time of particle orientation connected with particle dimensions and form. By a transient process of orientation we mean the change of any effect due to particle orientation (electric birefringence, electric light scattering, etc.) with the time until a steady-state regime is established. The theory of the transient electric birefringence upon application of a rectangular pulse was given by Benoit (1), who solved the diffusion equation for axially symmetric particles. He considered an ellipsoid of revolution 94
Journa~ of Colloid and Interface Science, Vol. 46, No. 1, January 1974
Copyright ~ 1974 by Academic Press. Inc. All rights of reproduction in any form reserved.
95
ELECTRIC LIGHT SCATTERING
Of interest is a theoretical treatment of the transient electric light scattering by anisodiametric particles, without restrictions on particle size, within the Rayleigh-Debye-Gans approximation. This would permit geometric and electric parameters of larger particles to be determined by the method of electric light scattering. THEORY
The solution is assumed to be monodisperse and sufficiently dilute so that interaction effects between the particles can be neglected. Further, it is assumed that the particles are rigid, optically isotropic, strongly elongated cylinders with a common axis of symmetry for their electric, optical and hydrodynamic properties. Such a model facilitates the calculations and describes comparatively well the optical behavior of the elongated particles. The approximation of Rayleigh-Debye-Gans (RDG) is utilized (8), its restrictive conditions being:
(4zr/X)llm- 1~i<<1;
Ira-
11 <<1,
where x is the wavelength of incident light in the solution, l is the length of the particle, and m is the relative refractive index (m = nv/ns; np is the refractive index of the particle and n~ that of the solvent). This approximation allows calculations to be carried out for comparatively large particles (their dimensions comparable with the wavelength of light and even greater) if their refractive index does not differ much from that of the medium. The effect of a relative change in the intensity of scattered light on application of an electric field to a solution of anisodiametric particles is given by IE
-
where IE and Io are the intensities of scattered light with and without a field, respectively, i(O, 9) is the intensity of light scattered by one particle, f(O, 9, t) is the orientation distribution function of the particles, 0 is the angle between the particle symmetry axis and the field direction, 9 is a spherical coordinate,f co • .. do~ shows integration over a sphere of radius unit, and t is the time. In the present model i(O, 9) and f(O, 9, t) are given as follows : The intensity of the light scattered by one particle (when the incident light is plane polarized) in the RDG approximation is given by (8) i(0,
2,.-7\ - ~ - ~ - /
sin R(0, @ =
[-kolsin (0'/2) sin 0 cos 9~ ko! sin (0'/2) sin 0 cos 9
,
E3]
R I S E PROCESS
The orientation distribution function f(O, 9,t) in the electric field is determined by the diffusion equation (I) :
i
Joy
V2f -t- - - div [-f grad
kT
Io
r21
IR(O', ¢)12/0,
where O' is the angle of observation (incident, scattered light); the other indexes are as explained above.
OL--
U] -
D Ot
or
4~ [ i(o, 9)/(o, 9, 8d~ ! J
ko4v2(m - 1~ 2 =
where k0 is the propagation constant, v is the volume of the particle, r is the distance from the center of the particle to the observer, R(O', 9') is a factor depending on particle geometry and orientation, and i0 is the incident light. In our case (thin rods in an electric field,, perpendicular to the plane of observation 1) the orientation factor can be written as (8) :
Io
-
9)
Pf(., 0
¢o
- l,
f i(O, 9)dco
D]
1 of(~, t)
D
Ol
1 The restriction of the electric field perpendicular to the plane of observation is not important. I t is not difficult to generalize the results for an arbitrary angle.
Journal of Colloid and Interface Science, Vol. 46, No. 1, January 1974
96
SOKEROV AND STOIMENOVA
where u -= cos O, t is the time, D is the rotational diffusion constant for rotation about the transverse axis of the particle, U - - - - u E cos 0 - - (6, cos 2 0 + & sin 2 0)E2/2 is the particle potential energy in the applied field, E is the field strength, u is the apparent permanent dipole moment in solution (when directed along particle s y m m e t r y axis), (~1 and & are the excess electric polarizabilities along the s y m m e t r y and transverse axes, respectively, k is the Boltzmann constant, T is the absolute temperature, and the operator/~ is defined by 02
P =
(1 -
process after time t, K = (2~-/X)l sin (0'/2) is a parameter proportional to the particle length, To = Si(2K)/K -- (sin K / K ) ~ is proportional to the scattered light intensity without field (8), and Si(x) = f o x (sin y/y)dy. The effect is expanded in the power series of both particle size and time. Substituting the corresponding power of the operator fi, carrying out the integration, and expressing the infinite series by tabulated functions we obtain the coefficients preceding the powers of time. Here is the series up to the second power of time :
0
u ~)
-
2u--
Ou2
~Ot
o~lr - -
To
0 -(1
- u=)(~ + 2vu) - -
+--
Si(x) .4 (x) = - -
t3 = tIE/k T, (h
We make use of the solution suggested by Nishinary and Yoshioka (5), which is an expansion of the distribution function in the power series of time:
f(u, l) = exp (Dlfi)f(u, 0), E4-] where f(u, 0) is the distribution function at t = 0. Since the distribution function must be normalized and the distribution is uniform in the absence of a field, we have f(u, O) = ~rc. Substituting the equations for i E21, and [33, and f [4-], in the general formula ~1~ for the effect a, and performing part of the calculation, we obtain 1 ~t r ~
--
~, (--1)"+'(2K)2"-2(2n -- 3)!!
X (2n-
~=o m ! 1)!! =
~
1)(2n
--
3)..-3.1;
Xa
8 x~
X2 '
3 sin x 8 xa 45 sin x
4
4
x ~"
F
4
M
M
Thus from the initial slope of the rise curve a versus t we can determine the anisotropy of electric polarizability, if D and K are known: (~1-
as) -
Cq
ET]
DE2 A(2K) \ dt /t~o"
Further the initial slope of the a/t versus 1 curve gives the value of the permanent dipole m o m e n t (along the s y m m e t r y axis): To]'
IS]
1
(2n --
4
45 cos x
kT[
(1 - - u ~ ) " - ' P " d u ,
3 sin x
X2
8 x
(2n) !(2~ - 2)!!
Draw f_~ E --
[6~
+
5 &(x) 5 cos x B(~) = - - +
~.
TO .=1
on
"r-lD2l 2+ . . . , 7"0 l
cos x 4- - -
X
6~)E~/kT.
-
T0-
where
where
=
(132 _ 3V)
+L
B(2K)
Ou
+2J-flu + "/(3u 2 -- 1)1,
~,
[A(2K)
A(2K)
Ou
(2n)!!
xt\-ZZj,
h 0 + D(% C 7,, 0
= (2n)(2n -- 2)--.4.2. where ~t r is the value of the effect for the rise Journal of Colloid and Interface Science, V o l . 46, N o . 1, J a n u a r y 1 9 7 4
A'(2K) \ dt/,~oJ
"
[8]
ELECTRIC LIGHT SCATTERING If the diffusion equation is solved in the general case, when the direction of the permanent dipole moment is arbitrary, the equation [-6] is transformed into
97
or ( 1 - u 2) 02f(u, t) Ou2
Of(u, t) 2u-Ou 1 Of(u, t)
A (2K) at ~"= - %
DOt
TDll
/A (2K) [/312
/DI+D2\
B(2K) 2}
+ - - T
D1212+ "'',
[9-]
T0
f(u, 0
where 31 = O~rE)/kT, 52 = (g2E)/kT, St1 and ~ are the components of the permanent dipole moment along the symmetry and transverse axes, and D1 and D2 are the rotational diffusion constants for rotation about the transverse and symmetry axes, respectively. In this case formula [-9-] can only give an equation connecting the two components of the permanent dipole. In order to obtain its value and direction, we need a second equation for v~ and ~2 [we can obtain such a one, for example, for the steady-state process (9)]. DECAY PROCESS After the electric field is switched off the angular distribution function satisfies the simple diffusion equation (1): 1 Of V~f . . . . D Ot
at a =
-
with the initial condition that at t = 0 the function f(u, 0) is to be determined by the steady-state process. We utilize for f(u, t) the solution suggested by Benoit (1). It is an expansion of f(u, t) in a series of Legendre polynomials :
= k C~ exp [ - - n ( n + 1)Dt]Pn(u),
[10]
n=l
where U = COS 0,
c~ = ~-(2~ + 1) /(~, o) =
f
f(u, O)Po(u)du,
exp (-- U / k r ) o exp (-- U / k T ) d u
and P,~(u) is a Legendre polynomial. When the particles are fully orientated, their potential energy is given by U = --t*E --1(61E~). Substituting the above equation [10] for f(O, ¢, t) and [2].for i(O, ¢) in formula [1-] and integrating, we obtain the following expression for the effect ~ after the electric field is switched off (and if the disorientation begins from a state of full particle orientation) :
1 ~ ,~-1(--1)-+~+12(2K)2,, 2(n -- 1)!(2m -- 1)!!(4m + 1) e×p [--2m(2m + 1)Dt] E E To n=2 m~l (2n) !(2n -- 1)(n - - m - 1)!m!(2n + 1 ) ( 2 n - 1)... (2n + 2 m - 1)
[11]
-
The first several terms of the expansion in a power series of particle length are: 1 [ K: a'd = "ToL 9 e--6Dt
2K4( 10 e--2ODt 225 7 e - S ° t - - - 7 2K8
/20
- -
1--
127575 \11
e--6Dt ~
11
162 e-6Dr
-- --
4 e-2opt
143
_ _ e - - 2 O D t -~- _ _ e - - 4 2 D t
+
--
11
33
7 e-42Dr -- --
\ e-72Dr)
7
3 q -" "
.
[12-]
143
J o u r n a l o f Colloid and Interface Science,
Vol. 46, No. 1, January 1974
98
SOKEROV AND STOIMENOVA
Expansion of a? t in a power series of time exponential yields: L(2K) ect d - -
- -
M(2K) e - - 6 D t ~ - -"
N(2K)
To
To
[13]
e- e D t + . . . ,
- e--2ODt ~- - - -
%
where 5 S
)+
+3
81 ( S i ( x ) c o s x
+ 325/Si(x)
cos x
5sinx
70cosx
3
3
21 sin x 5
xa
70sinx
.4 + 3 294 cos x 5
x4
;)
9
'
1071sin x l0
x5 2772 sin x
2772cosx
2561)
---~
5
x~
5
x7
25
From Eq. [11] one can obtain the expansion of a t d in the power series of time as follows:
1[
2K2 ( 8) ( 2 0 8 at a = - - 1 -- 7-'0-Dt + 2 K z Jr_ _ [~4 DZt2 __ 4K 2 _}._ _ _ _ To 3 5 45
The initial slope of the decay curve a d versus t (when K is known) allows one to determine the rotational diffusion constant for rotation about the particle transverse axis: D=
3T0 [d(--a*d) 1 2K 2L
7;
-It~o"
Another method for determining the rotational diffusion coefficient is by measuring the effect of disorientation for time t = 1 / 6 D [-the terms of the series after exp (--7/3) can be neglected] : L(2K) o~td - -
To Xexp(-1)
1-{- L ( 2 K ) exP
--
q- -N(2K) exp (-- 6) q- -.- 1 . [15-] rX2K)
1{4 @
8)
21
K 6 Data
]
+ .-..
El4]
DISCUSSION The equations obtained for transient electric light scattering by elongated particles within the RDG approximation extend considerably the application of the method to a wider range of particle size, and allow the precision in determining the particle electric and geometric parameters to be increased. The determination of particle electric parameters by utilizing the steady-state effect at low degrees of orientation yields an error due to the dispersion of the results at low fields. The above formulas for the rise process are obtained without restrictions on field strength and are valid for arbitrary degrees of orientation. Thus one is able to determine the value of electric polarizability at various orienting fields and to study the probable existing saturation of that value. The formula for the decay process is valid in the case when particle disorientation begins
J o u r n a l o f C o l l o i d a n d I n t e r f a c e Sc i e nc e , Vol. 46, No. 1, J a n u a r y 1974
ELECTRIC
LIGHT
from a state of full orientation along the field direction. It differs from the equation given by Wippler (6), and even in the case of a monodisperse solution the effect is described by a sum of time exponentials. Thus, using only one curve of the effect of full orientation and disorientation, one can determine the anisotropy of electric polarizability by the formula: (~l
-
&) -
2kT
K2
3E~ A ( 2 ~ )
/(% ]
×
An analogous equation can be written for the permanent dipole moment, if the latter is directed along the particle symmetry axis. The value of K and hence l can be obtained by the steady-state effect, which in the case of full particle orientation is given by a~ = (1 - %)/To (where % is a function only of K) (7). The combination of l, obtained by a~, with D, obtained by the decay process, allows one to determine the particle axial ratio p, and diameter b with the help of Broersma's formula (10). Of particular interest is the case when the particle permanent dipole moment has an arbitrary direction. If #1 and g2 are the components of the permanent dipole along the particle symmetry and transverse axes respectively, we can write formula [-8~ for this case as follows : / D 1 Jr- D2 \
)-
K~T 2
99
SCATTERING
for ellipsoids of rotation. Using Perrin's formulas for the case of cylinders one obtains an error not higher than 15% (10). According to Perrin the quantity (Di q- D2) 2D1
(2 In 2p
-- 2)
is a function only of particle axial ratio. For p varying from 5 to 20 the quantity (D1 Jr- D2)/2D1 changes in the interval 8.7 to 73.5. If the quantity ~ 2 _ #22(D~ Jr_ D2)/2D1 has a positive value, we can consider the permanent dipole as being directed nearly along the particle symmetry axis, since its component along this axis is at least [(D~ q-D~)/2DI-1 *~ times longer than the one along the transverse axis. In the case of a negative value of the above quantity, as we mentioned, a second equation is necessary for determining the permanent dipole moment. Using ~z~, we can introduce (4) the quantity "degree of orientation"
= [TO/(1 -
-T0)]~,.
The formulas for q~t are obtained by replacing the coefficients F ( K ) / T o in the equations for at by F ( K ) / ( 1 %). Expand~ing the coefficients F ( K ) / ( 1 -- To) in the power series of K and neglecting the terms higher than K 2 we obtain the following values of these coefficients for small particles : A (2K)/(1 -- %) = 2/5, B (2K)/(1 -- ¢0) = -- 1/35, L(2K)/(1 -
To
@2 _ in 2p)
TO) = 1,
M(2K)/(1 - To) = 0, N ( 2 K ) / ( 1 -- %) = O.
A2(2K) dt /e+0/ The quantity (Dt + D2)/2D1 can be estimated utilizing the formulas of Perrin (11, 12)
Substituting the above values in formulas E6~, E9], and 1-13], we obtain respectively the equations of Nishinari-Yoshioka (5), Tinoko (2) at t--+ ~o, and Benoit (1) for the electric transient birefringence. The latter formula, [133, in this case coincides with Wippler's equation (6) for the decay of electric light scattering. These
Journal of Colloid and Interface Science, Vol. 46, No. 1, J a n u a r y 1974
100
SOKEROV AND STOIMENOVA
results once more confirm that when particles are small enough (compared to the wavelength of incident light), the "degree of orientation" for the different electooptic phenomena is described b y the same equations (7). The correlation of the coe~cients in the formulas for small and for large particles allows one to estimate the error which is made when particle parameters are determined utilizing the theory for small particles. The estimation shows that the values obtained for D will always be greater than the real ones, while those for 7 and 1 will be smaller. The equations valid for small particles yield an error of about 30% for D and -y at K = 2, and for l at K = 4. For particles of higher K the equations presented in this paper should be used. Otherwise the values of the experi-
mentally determined parameters an order from the real ones.
can differ
REFERENCES 1. BENOIT,H, Ann. Phys. 6, 651 (1951). 2. TINOCO,I., Jf~., ar. Amer. Chem. Soc. 77, 4486 (1955). 3. SC~IWAI~Z,G., Z. Physik 145, 563 (1956). 4. O'KoNsK1, C. T., YOSHIOKA,K., AND O~TUNG, W. H., J. Phys. Chem. 63, 1558 (1959). 5. Nlsmi,VA~i, K., AND YOSItlOKA, K., Kolliod-Z. Z. Polym. 235, 1189 (1969). 6. WIPPL~,X,C., J. Chem. Phys. 53, 321 (1956). 7. STOYLOV, S. P., Advan. Colloid Interface Sci. 3, 45 (1971). 8. VAStDE HULST, H. C., "Light Scattering by Small Particles," p. 104. Inostrannaja Literatura, Moscow, 1961. 9. SOKEROV, S., A?CDSTOIMXNOVA,M., unpublished results. 10. B~OERS~A,S., Y. Chem. Phys. 33, 1626 (1960). 11. PER~IN, F., J. Phys. Radium 5, 497 (1934). 12. NEURA:C~,H., AN9 BAILEY,K., "The Proteins," p. 291. Moscow, 1956.
Journal of Colloid and Interface Science, Vol. 46, No. 1, January 1974