J. Phys. Chem. 1985,89, 5755-5758
5755
because t: reflects static properties while Ri reflects dynamic
the relation between R and Ri* is
+
1/R = x~/RI* X , / R ~ *
(25)
In the Ornstein-Zernike approximation, F, can be written as47948
F = (F/Ro)2
(26)
where Ro is proportional to the fourth moment of the direct correlation function for the concentration fluctuation and is almost independent of t e m p e r a t ~ r e . Combining ~~ eq 22,23, and 26, we obtain
L/L= ~F~/~R/R~
(27)
In the case of R = Ro, eq 27 becomes
L / L o = F1I2
(28)
D=
(29)
and hence ,l&’/2
In this case, L / L , cannot give much more information than F does. In Figure 8, the R values obtained from eq 23 are compared with the R , values obtained from eq 21 and 26. The figure demonstrates that Ro is nearly equal to R in the concentration range of 0.25 C w 2 C 0.6 at 23 OC and 0.15 C w2 C 0.6 at 42 O C . Therefore, we cannot obtain dynamic information from L / L , (and hence DIJ)at these concentrations. The same does not hold true of other solutions where R is not equal to Re Moreover, the above discussion is based on several assumptions described before. It does not seem easy, however, to obtain much information from L/Lo (or D,,) than F (or GI,) does at this stage. Finally, the “apparent” correlation length 6 obtained from eq 21 is plotted in Figure 9. It can be seen from the figure that 5 takes a maximum which is much larger than the molecular radii even at the temperatures far from the critical consolution point. These results are not inconsistent with the results in Figure 8 (47) The quantity F can be rewritten as F = limG S(q) where S(q) = p J j A x ( O ) A x ( r ) ) exp(iq.r) d’r/(x,x,) ( p is the num r density). For pure fluids, structure factor So(q) is defined so that So(0)becomes equal to 1 for an ideal gas. Then, we obtain So(0)= ( [ / R o ) for 2 critical fluids (see ref 49). For binary critical solutions, the same equation can be obtained if So(0)is replaced by S(0) (= F) because S(0) = 1 for an ideal solution. (48) Equation 26 is almost identical with eq 20 and 21 in ref 20 under the conditions that R = Roand the solution is dilute (F and Robecome equal to I and R in ref 20, respectively). (49) For example, Stanley, H. E. ‘Introduction to Phase Transition and Critical Phenomena”; Oxford University Press: London, 197 1.
Conclusion We have obtained the joint diffusion coefficients D,Jfor aqueous solutions of BE using data on QELS, PGNMR, and RBS. The absolute values of DtJ were found to be much larger than those for the reference system. We have also obtained the ratio L/Lo (= D 1 2 / D ~ 2 L/Lo ). also takes a maximum and increases with temperature. These results indicate that molecules of the same species have a tendency to diffuse together. The hydrodynamic radii calculated from the self-diffusion coefficients and viscosities are, however, smaller than each molecular radius. This suggests that there exist few long-lived clusters whose component molecules can diffuse together. The relation between L and F was also discussed. It was shown that L / L , is determined by only F i n a certain concentration range. This indicates that we cannot obtain as much information from L / L , (and hence from 4)than from F i n this concentration range. In general, it is difficult to analyze L/Lo or D, without any theoretical basis because these quantities are affected by the concentration fluctuations even for solutions where nonideality is not as strong as that of the present system. At the present stage, however, more data on D,, G,,and F a r e necessary for a quantitative discussion of this problem. Acknowledgment. I am grateful to Mr. Isao Kagawa and Mr. Koji Miura of Japan Electron Optics Laboratory Service for their help in the modification of the N M R spectrometer for this study, and Professor Hisashi Uedaira of Hokkaido University for his helpful advice on the PGNMR measurements. I also thank Professor Hiroyasu Nomura of Nagoya University and Professor Teizo Kitagawa and Professor Yasuo Udagawa of Institute for Molecular Science for giving me the oppourtinity to perform this work. The present PGNMR measurements were carried out at the Instrument Center of Institute for Molecular Science. I am indebted to all the members of the Instrument Center for permission to modify the N M R spectrometer. Registry No. BE, 11 1-76-2 (50) The 5 values calculated from eq 21 may be different from the correlation lengths usually defined as
5 = ( 1 / 6 ) 1 r 2 d r ) d 3 r / 1 e ( r ) d3r
d r ) = (Ax(O)Ax(r))
for solutions far from the critical point. It seems interesting, however, to study the concentration and temperature dependences of [. Recently, such a study has been reported for aqueous solutions of tert-butyl alcohol. See Euliss, G. M.; Sorensen, C. M. J. Chem. Phys. 1984, 80, 4767.
Static Electric Polarization of Simple Nonrigid Molecules Akio Morita* and Hiroshi Watanabe Department of Chemistry, College of Arts and Sciences, University of Tokyo, Komaba, Meguro- ku, Tokyo 153, Japan (Received: May 6, 1985; In Final Form: August 15, 1985)
The kinetic energy term has been taken into account for the calculation of electric polarization for simple nonrigid molecules. It is shown that even when the center of mass of a molecule with two weightless rigid rods connecting a central particle with two end particles is fixed on the origin of the internal coordinate, the deviation from the behavior of the rotational motion in the case where the central particle is fixed on the origin of the inernal coordinate is not significant except for a very restricted case. Various new theoretical results on the electric polarization are obtained.
Introduction Treatmepts of a rigid molecule in connection with dielectric investigated from various and elwtrmptial phenomena have aspects.’-2 However, it is surprising to realize that treatments (1) A. Morita, J . Chem. Phys., 76, 3198 (1982).
of even a simple molecule with an internal freedom require some new careful considerations arising from the kinetic energy term in the Maxwell-Boltzmann distribution f ~ n c t i o n . Although ~ this (2) H. Watanabe and A. Morita, qdu. Chem. Phys., 56, 255 (1984). (3) H. Watanabe and A. Morita, J. Phys. Chem., 89, 1787 (1985).
0022-3654/85/2089-5755$01.50/00 1985 American Chemical Society
5756 The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 Z'
m
Morita and Watanabe particles are connected by rods. For model B, neither the up and down nor the cooperative motion exists, because we assume that the two dumbbells are not in direct mechanical contact. Model B is also suitable for investigating the dipole-dipole coupling with two separate dipoles. We assume that a permanent dipole moment lies along each rod for both models A and B. We mention hinged polymers and apdihaloalkane as examples of models A and B, respectively.
Theoretical Formalism By using the generalized coordinates q l , q2, ..., qn for the description of the orientation of a molecule, an assembly of molecules, or groups of atoms, we calculate the ensemble average of a physical ...,q,), ( A ) , by the expression quantity A(ql,q2,
J J ...l AF exp(-V/kBT)
I Figure 1. The internal coordinate system for the three-particle model. Z'
(A)=
dq, dq2...dq,
1
l . . . l F eXp(-V/kBT) dq, dq ,... dq,
(1)
where V(ql,q2,...,q,,) is the potential, kB is the Boltzmann constant, T is the absolute temperature, and F(ql,q2,...,q,) is the factor resulting from the kinetic energy K(pl ,p2,...,pfl,q,q2,...,qn) where pi ( i = 1, 2, ..., n ) is the conjugate momentum for q1
,
F =C l f
...Sexp(-K/kBT) dp, dp2...dp,
(2)
with a constant C . We shall show how F may be calculated conveniently. To this end, by introducing an element vI of a vector 1) which is given by the following linear transformation 171
= cb,,(qldI24LJq,
(3)
I= I
Figure 2. The internal coordinate system for the two-dumbbell model. The distance between the centers of mass of two dumbbells is represented by I , and the origin of the coordinate is just halfway between the centers of mass of the dumbbells.
term has been calculated a long time ago by Gibbs4 in his classical work of statistical mechanics, and recently by other^,^ it seems that the concept has not been necessarily conveyed to studies in dielectric, electrooptical, and other phenomena in physical chemistry. This led to a difficulty that the probability of finding a molecule with a certain conformation depends upon the choice of the coordinate system3 Therefore, to avoid this logical and physical contradiction, we find it necessary to re-formulate the problem. The main aim of the present paper is to apply the general formulation with a convenient method to calculate the kinetic energy for a nonrigid molecule to the calculation of electric polarization in order to see how the existing results may be improved with the kinetic energy term. We have obtained the electric polarization for two simple models: (A) a three-particle system where two particles with equal masses attach to the central particle with two weightless rigid rods (see Figure 1) and (B) two equal dumbbells, each consisting of two unequal masses connected with a weightless rigid rod, with the center of mass for each dumbbell fixed in an internal coordinate system (see Figure 2). For model A, the translational motion is separated from the internal motion by fixing the origin on the center of mass, which results in an up or down motion as the angle of the two rods increases or decreases in order to keep the center of mass on the origin. In addition, there exists a cooperative motion due to the fact that motion of one of the particles necessarily causes that of other, because three
where q, represents the time derivative of qJ(t),we write n f l
=
'/zc~'IJ(q,,q~,"',qfl)17117, 1=I
,=I
We mention the relation between angular velocities and Eulerian angles (0, 4, $) as w, = 6 cos $ + 4 sin 8 sin $ w)
= -6 sin $ w,
=
+ 6 sin 0 cos J.
(4) (ox,wy, w,)
(5)
6 cos 8 + $,
It is obvious in eq 4 that a , = aJl,and it follows readily from eq 3 and 4 that
which allows us to write eq 6 by a matrix equation p = vAB
(7)
where matrices p, A, and B have elements p i , ul,, and b,, respectively. In carrying out the integral in eq 2, we must know the Jacobian J due to the transformation in eq 7 from the p space into the 7 space, which can be seen to be
Therefore, it follows that in view of
(2TkBT/IAI)'" (8) (4) J. W. Gibbs, "Elementary Principles in Statistical Mechanics", reprinted by Ox Bow Press, Wocdbridge, CT, 1981. (5) H. A. Kramers, J . Chem. Phys., 14, 415 (1946); N. GO and H. A. Scheraga, J . Chem. Phys., 51, 4751 (1969); M. Fixman, Proc. Natl. Acad. Sci. U.S.A.,71, 3050 (1974); M. Gottlieb and R. B. Bird, J . Chem. Phys., 65,2467 (1976). Especially for heuristic use, see 0. Hassager, J . Chem. Phys., 60, 2111 (1974); E. Helfand, J . Chem. Phys., 71, 5000 (1979). (6) S. Mizushima, Y . Morino, and K. Higasi, Sci. Pap. Inst. Phys. Chem. Res. (Jpn.), 25, 159 (1934).
one finds F = C ( ~ T ~ ~ T ) " / ' ~ A ( ~ / ~ ~ B I (9) In previous s t u d i e ~the , ~ expression for F has been calculated by regarding B as the identity matrix, resulting in the rather complicated and lengthy expression for A. However, it can be seen that if we find the transformation in eq 3, the calculation of F
Polarization of Simple Nonrigid Molecules
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5757
can be performed by obtaining IAI and IBI separately.
Simple Examples A . Three-ParticleSystem. In order to make the above general results more familiar and transparent, we first treat the simple example of a three-particle system as given in Figure 1 where m, M , and m are masses for particles 1 , 2 , and 3 , respectively. As shown in Appendix A, the kinetic energy arising from the internal motion is given by m12 K. = [M(cos2 a)w,Z + ( M + 2m sinZa)wY2+ In‘ M + 2m ( M + 2m)(sin2 a)wZ2 ( M + 2m cos2 a)iu2]( I O )
+
where V, is the potential due to the intramolecular interaction and = p E / k B T . The electric polarization P arising from the rotational motion of a molecule under the influence of the field is
where j = A or B. It follows by skipping the detailed derivations that for V, = 0
If we regard p , = ax,p 2 = wv,p 3 = uz,p4 = iu, q1 = 8, q2 = cp, q3 = $, and q4 = a , we find that
IBJ = C, sin 8
( 1 1)
IAI = Cz(a2- cos2 2a) cos2 a sin2 a
(12)
L2(f) - 1
+
where a = 1 ( M / m ) and C, and C2 are constants. Equation 11 originates from the motion of the internal coordinate with respect to the laboratory coordinate while eq 12 represents the effect due to the intramolecular motion on the internal coordinate system; the first term (a2- cos 2a) is due to the up and down motion to set the center of mass on the origin, and the second term (cos2 a sin2 a) is due to the rotational motion around the y’axis m , since a m it follows on the x’z’plane. In the limit of M that
-
-
IAl = C3 cosz a sin2 a
(13)
which is nothing but the result where M is at the origin at all times, and this case has been worked out previ~usly.~ It is interesting to note that if two rods are allowed to move, we find the expression identical with that in eq 13 as far as the part depending on a is concerned. B. Two-Dumbbell System without the Mechanical Contact between Them. In contrast to model A where two rods connect particles 1 and 3 mechanically with the central particle 2 so that the translational motion of one rod will necessarily affect the rotational motion of the other, we consider here a two-dumbbell system without such a mechanical contact shown in Figure 2. It is assumed that the center of mass of each dumbbell is fixed on the z’axis of the internal coordinate system with distance f L / 2 from the origin. Even though K looks complicated (see Appendix B), we find after working out lengthy calculations of the 6 X 6 determinant for IAl that
[AI = C4 sin2 O1 sin2 8,
(14)
where C4is a constant. It should be noted that IBJin this case is the same as that in eq 1 1 .
Electric Polarization In order to see how the kinetic energy term affects ensemble averages, we shall investigate the electric polarization as an example to compare with the previous results. If an electric field E is applied along the z axis of the laboratory coordinate system (x,y,z) and each rod in models A and B carries a permanent dipole moment k , the potential Vexarising from the interaction of E and the total dipole moment M is given by Vex= -M.E = -2kE cos 8 cos a
+E
(for model A with a
(19)
P = L ( t ) (for model B)
(20)
-
where L ( z ) is the Langevin function and Z,(z) is the modified Bessel function. It is noted that, as a 03, P in eq 18 tends to that for model B which is the usual expression for the polarization of two independent rigid molecules, whereas we obtain a new expression for model A with the correction resulting from the dependence of the center of mass on a. For special case where
V, = Yk~T(P1’112)/k~ = YkBT
= ~ ~ B T ( c8o, scos 82
COS
2c~
+ sin O1 sin 82 cos 2cp)
‘[
+
P = - 1 - L ( y ) + - L2(y) - 1 3aiY( 3 Y 9 (for model A with
‘( ::;)
p = 3
E
-
+
where 3u2
U2
VI
(for model A)
J2Rd# JRi2da cos a sin a X
1
(23)
P = - ( 1 - L ( y ) ) (for model B) (24) 3 It is again evident that a m for model A gives the same result as that for model B. If the dipole-dipole interaction is taken fully into account for model B, by writing V, = -u,kBT(cos 8, cos 82 sin 81 sin 8 2 cos 2cp) L’lkBT COS 81 COS 82
= - - and L3kaT
02
=L3kBT
We define partition functions QA and QB for models A and B, respectively, by
s,’”4
(21b)
a >> 1) ( 2 2 )
(for model A with a = 1 )
1--
-@[(COS 8 , + cos 8,) cos 0 + (sin 8, + sin 8,) sin 8 sin $ X cos cp + (sin 8 , - sin 8,) sin 8 cos # sin cp] (for model B)
8 d8
( 21a)
in which p, and p 2 are dipoles along the two rods, it follows that for E >
where
R = u,(l
+ 3x2)’/’
Morita and Watanabe
5758 The Journal of Physical Chemistry, Vol, 89, No. 26, I985 1.0
i I
1
30
15
Y
5 Figure 3. Saturation curves of the electric polarization P in the case of V, = 0 for model A. From the top to the lower curves, a = 10.0, 1.3, and 1 .o.
Figure 4. Low-field limits of the electric polarization with intramolecular interaction. For the curves with y < 0, the upper and lower broken curves are obtained from eq 25 and 26, respectively. From the top to the lower solid curves, a = 1.0, 1.1, and 10.0.
Discussion Since it is difficult to work out the electric polarization in an arbitrary field strength analytically if intramolecular interaction is included even for the present simple models, we have treated two extreme cases: (a) high fields with 6 = 0 and (b) low fields with V, given in eq 21. For model A in both cases a and b, analytical expressions for P in the special values of a = and 1 are obtained. The electric polarization P for an arbitrary value of a should lie between those at a = and 1. The saturation curve in case a is shown in Figure 3 from which it is seen that P deviates only slightly from the usual Langevin function in the intermediate field strength around 2 < ,$ < 15. The presence of the (a2- cos2 ~ L Y ) I / ~term in the partition function QA influences slightly the saturation curve in the very restricted range of 0 < M / m < 0.5, and its effect is to shift the saturation to higher fields. The lighter the central particle is, the more difficult the saturation becomes. Equation 18 directly indicates that, for small and large (, no contribution from the second term on the right due to the motion of the mass center within the internal coordinate is expected. Plots of 3P/,$for the case of ,$ 1 the mass center correction hardly contributes both to the saturation curve and to the polarization in low fields. Two broken curves are obtained for comparison from the well-known result of Mizushima, Morino, and Higasi
Appendix A: Calculation of the Kinetic Energy due to the Internal Motion for Model A On representing the bond length between particles 1 and 2 by I, we easily find that x i ’ = -I sin 01 x2’= 0 x3:= I sin oi Y,’ = 0 ”>’ = 0 Y3, = 0 (A-1 1
Q)
~3
z,’ = I , cos CY
z 2 ’= - I ,
3
1-
. .+
Pi = ?(’
0
x f;
we find Kin,in eq 10. This approach is simpler than the previous methods.
Appendix B Calculation of the Kinetic Energy Due to the Internal Motion for Model B Again on representing the length of a dumbbell by I‘ and the masses for particles 1, 2, 3, and 4 by m’, M‘, m‘, and M’, respectively, we find that x,’ = -1,’ sin 8 , cos cp x,/ = 12’sin 8 , cos cp y,’ = - l l r sin 8 , sin cp y2’ = I,/ sin 8 , sin cp L z,’ = 4 - I,’ cos 8 , z; = - + I,/ cos 8, 2
y3’ = I,’ sin O2 sin cp
xq/ = I2‘sin O2 cos cp y4’= -12‘ sin 8, sin
(a
L = _ _L - I,’ cos 8 2 zq/= - - + 12‘ cos 82 2 where 1,‘ = M’l‘/(M‘+ m’) and 1; = m’l’/(M‘+ m’). By using a similar procedure for obtaining Kin,in Appendix A, we find after lengthy calculations that K. = -1- m’Mrlr2j ( w z + ) 2 sin2 8, + (0, - + ) 2 sin2 8’ In‘ 2 m‘ M 2(w, + +) sin O1 cos O1 (w, cos (a + wy sin cp) 2(w, - +) sin 82 cos 8’ (w, cos cp - w y sin cp) + 6,’ + 62’ + w;[cos2 8 , + cos2 6, + (sin2 8, + sin2 8,) sin2 cp] + wyZ[cos2 8 , + cos2 8’ + (sin2 8 , + sin2 8,) cos2 cp] + 2(8, + 6,)w, cos cp - 2(6, - B2)w, sin cp - 2w,wy sin cp x cos cp (sin2 8 , - sin2 0 2 ) )+ y4(m’+ M?L2(w,2 + wyZ) zji
and from eq 25 with the potential due to the dipoledipole coupling for model B. We have shown that eq 26 is not a proper expression for P.3 The difference between P from eq 26 and that for model A in y for the same value of 3 P / l is significant around y = 4. It should be noted that P for model A with a = m, corresponding to the case where the central particle is fixed on the origin of the internal coordinate, becomes identical with that for model B. In this limit of model A, rotation of two rods around the central particle does not cause the cooperative motion in view of the fact that the central particle cannot move during rotation. This rotation in fact is identical with that for two separate dumbbells, each of whose centers of mass is fixed in space or on the internal coordinate, and its analytical treatments are easier than model A with an arbitrary a.
= 1 , cos 01
(‘4-2) substituting eq 5 and A-1 in eq A-2, and calculating ii,which is put into
x3’ = -1,’ sin 82 cos cp
(26)
z3
where 1, = M I / ( M + 2m) and l2 = 2ml/(M + 2m). By using the relation between vectors 7, in the laboratory coordinate and ?/ in the internal coordinate
2
!( -)
cos oi
+
+
