IPrrrm~l cf IMolt~crrlar li1pi1ls. !%i (1993) 18>197 Elswicr Scirncc Publishcm B.V, Amsterdam

Application Methods to

of Krylov-Bogoliubov-Mitropolsky Relaxation Processes

James McConnell

School of Theoretical Physics: Dublin Institute for Advanced Studies 10 Burlington Road, Dublin 4, Ireland

Abstract

Mathematical methods devised by Kryiov, BogcSubov and Mitropol- sky for the solution of nonlinear differential equations have been adapted to the study of dielectric and nuclear magnetic relaxation processes. ‘l’his has led to 4uable results which are easy to comprehend and to empIoy.

1. INTRUDtiCTION

During the past twenty years attention has been directed to the importance of the study of the stochastic rotation operator for the in- vestigation of relaxation processes associai,ed wiih Brownian motion. It was found that while many of the results of such investigations could also be obtained by employing a Fokker-Planck equation, the use of the rota- tion operator coupled with a Langevin-type equation could provide both a clearer picture of the physica process and the possibility of carrying results to a higher order of approximation.

A defect in results so obtained is that they are often expressed in the form of a series whcse convergence properties are not evident. Conse- quently series expressions for dielectric permittivity or for nuclear mag- netic relaxation times may have li_mited validity. The investigations of

01674322 /93/!%.00 0 1993 - E&v&r Science Publishers B-V. AU rights rrser~ed

Krylov and Bogoliubov (ref. 1) and of Bogoliubov and MitropolskY (ref. 2) did much to improve this situation. For investigations of stochas- tic processes the adaptation of the Krylov, Bogoliubov and MitropolskY (KBM) theory by Van Kampen (ref. 3) may also be helpful.

In the next section the method of a*Jeraging (ref. 2) will be intro- duced and presented in a form suited to the investigation of relaxation processes. In section 3 the stochastic rotation operator is introduced, in section 4 the averaging method will be applied to the rotational Brown- ian motion of a spherical molecule, and in section 5 dielectric relaxation will be introduced. An alternative and more satisfactory treatment of the kinematical problem will occupy section 6. Then section 7 will be concerned with molecules which have no special symmetry. The problem of nuclear magnetic relaxation will be discussed briefly in section 8-

Before descending to details of calculation it may be opportune to explain how what we propose to do is related to the investigations of KBM. These investigators were concerned lvith the task of expressing the solution of a problem in a form that would present a clear picture of the evolution with time of the process under consideration. This Point

will be iIlustrated later in an example of dielectric relaxation.

In the present study the ideas of KBM will be employed not only to the above mentioned purpose but aiso to an objective which is perhaps

more fundamenta1, namely? to ink, .;tigate Lvhether the KBM method can

he used to provide a solution of a problem involving random v%riables; to be more precise, the problem of dielectric relaxation in dilute liquid SO-

htions caused by the rotational Brownian motion of molecules xs-hich are subject to thermal motion in a heat bath constituted by the neigtibouring molecules.

This then is the first problem to be faced. If a solution is found, lve shall have to enquire whether the form of the solution can be i-Proved by introducing the KBM method for nonstochastic variables.

2. THE AVERAGGIG METHOD

In future applications chapter 6 of ref. 2 on the method of averag- ing will be relevant. However for a theoretical physicist an unpoblished

185

account by G.W. Ford of the averaging method is simp!er to follow and is easily adapted to the discussion of probIems in physics and chemistry-

Let US therefore consider the

dx(t) dt

general problem of solving the equation

= EO(t)x(t), (2.u where x(t) is a random variable and might be an operator, E is a small constant dimensionless parameter and O(t) is a stochastic operator, that is, an operator which involves random variables. However O(t) is inde- pendent of E. We shall present the soIution of (2.1) in a simplified form of that proposed bv Ford, Lewis and ?vIcCcnnell (ref, 4). The solution of (2-l) will be slowly varying and, if lye co:np:y W. x(t) with its nonstochastic mean value < z(t) >, we may esyrcss the ::c:l~ition as a power series in E:

x(t) =< z(t) > +&l)(t) < z(t) > +&(2)(t) < x(t) > +...., (2-2)

where FtL 1 (t), F(*) (t), .___ are stochastic operators. If we take mean values of both sides of (2.2), the first term on the right hand side w-ill cancel the single term on the left hand side and equating to zero the coefficient of P we deduce that

< F’“)(t) >= 0. (n = 1,2, . ..) (2-3)

The nonstochastic mean < x(t) > v.-ill satisfy some nonstochastic differ- ential equation

d < =c(t) > dt

= A-P)(t) < x(t) > + &(2)(t) < x(t) > t..., (2.4)

where each Q(“)(t) is nonstochastic, so that

< nyt> >= n(qt). (n = 1,2, .1.-) (2.5)

In applications to relaxation processes our problem is usually to find < z(t) >. This is true for dielectric relaxation (ref. 5) and for nuclear magnetic relaxation except when this is caused by spin-rotational inter- action (ref. 6).

In order to deduce the value of < x(t) obtain expressions for s?(‘)(t), Q(*)(t) etc. (2.2) into (2.1) and equai;e the coefficients

> from (2.4) . To do this of E 1 E2 1 e3 ‘) -*-

we must first we substitute on both sides

thus getting a sequence of equations

W)(t) + W(t) = 0(t)

W)(t) + W(t ) = owe - W’(t)SW(t)

. . . . . . . . . . . . . . . . . . . . . .

(2.6)

From (2.3), (2.5) axi (2.6) we deduce that

R(‘)(t) =< O(i) >

and from (2.5j, (2.5) and (2.7) that

SP)(t) =< o(t)F+-l)(t) > I

(2.8)

(2.9>

From (2.6) and i2.8) it follows that

t F(l)(t) = J (O(b)- < O(h) >)a 0

and then, from (2.9),

S2t2)(t) = s

t[< o(t)o(t,) > - < 0(t) >< O(k) >I&. 0

The values of SIt3)(t) and S2t4)( t) have been calculated and suits are to be seen in section 5.4 of ref. .5. The expression for --

the re- W)(t)

involves a triple integral and the integrand contains 26 terms. To avoid further complication we shall suppose that, unless explicitly stated to the contrary, Ott) is a centred Gaussian stochastic operator and that

< o(t,)o(~,>....o(t,] > vanishes for odd values of r.

3. THE STOCHASTIC ROTATION OYERATOR

We retclrn to the problem posed at the end of section 1 concerning dielectric relaxation resulting from rotational Brownian motion of liquids. Fixing our attention on one solute rigid molecule, which for the moment we suppose to have no particular symmetry, we take a molecular coordi- nate frame with origin at the centre of mass and axes in the directions of the principal axes of inertia. We call this frame the inertial system. Omitting considerations of translational motion we denote by R(t) the rotation operator which specifies the orientation of the molecule at time t relative to its orientation at time zero. On account of the thermal motion R(t) is stochastic but R(o) is clearly the identity operator ~9. It is easy to show (rei. 5, eq. (iJ.8)) that

where Ji , Jz, .I” are the infinitesimal generators of rotation and w(t) is the

Wt) dt

= -i(J -w(t))R(t>, (3-l)

angular velocity of the molecule. The values of the compcnezts ~1, ~2, ~3

of w(l) may be obtained by solving the Euler-Langevin equations !:zf- 5, sections 7.1 and II-I). Then making the substitutions

z(t) - R(fj T d(f) - -i(J - w(t))

we can employ results of the previous section.

4. THE SPI-IERICAL MOLECULXR MODEL

For a simple application of these ideas we take a spherical molecule whose moment of inertia about a diameter is I. lsre assume that the angular velocity w(t) satisfies the Langevin equation

r Wt) dwt) ’ dt

= -IBw(tj +I dt ,

where fB is the coefficient of rotational friction, Iv is the driving couple and W(t) is a Wiener process. The components w1 (t), ~r;g (t), ~3 (t)

are centred Gaussian random \rariaMes and consequently the two prop- erties of Q(t) postulated at the end of the previous section are satisfied. Hence (ref. 5, p. 91) all Qczn- ‘l(t) vanish, and so (2.4) combined with (3.2) yields

where the calculation of the ~*~S’2(~~)(t) is elementary but usually lengthy.

On integrating (4.1) one finds for the mean value of the rotation operator (ref. 5, eq. (9, 2.28))

< R(t) >= csp {

- J2[r KT 10) (t)

kT 4

+(,I v-4 “‘[L) t S::“(t) + [-4J2 + 161@)(t)

+[-10J2 + 381]1A”(t))] + -..}, (4.2)

where the values of the I(“‘)(l)‘s are given in Appendis D of ref. 5_

That the result (4.2) is correct has been conlCirmed by calculations based on a Fokker-Planck equation that were made by P.S. Hubbard (ref. 7)- However his approximations lvere not carried out quite as far as those in eq. (4.2).

5. DIELECTRIC RELASATIOX

%Ye shall now consider how the \Talue of < R(t) > calculated for molecules of any shape may be employed in the study of dielectric relax- ation of polar molecules in dilute solution in a nonpolar solvent. If n(t) is a unit vector in the direction of the dipole moment of a molecule at time t, the complex permittivity e(w) for orientational polarization is given by (ref. 5, p. 30)

E(W) - Ew =

l

- iw El - coo SW < (n(0) - n(t)) > emiwtdt!

0

(5.1)

189

where E s = e(o) and coo is the reIati_:e permittivity at frequencies SO high that orientational polarization is no longer effective. Equation (5.1) may also be expressed as

44 - cc+3 Oc, = 1 _ id J < R(t) >bo emi"'&, Es - E, 0

(5-2)

where < R(1) >bo denotes the oo - element of the matrix representative of < R(1) > with reference to the basis 17r, -1, Y>o,Yrr (ref. 5, section 12.5; ref. 6, Appendix B).

Returning to the case of the spherical molecule we deduce from (4.2) that

< R(t) >b, = exp {

-2[- kyIqt) + ( yp(t) + (y(p(t) + 413t))

+C-- k;)q(I:8’(t) -I- 81i8’(t) + 819t) + 181k8’(t))] + ___.}_(5_3)

If the exponential in (5.3) is expanded into a series, successive terms will have opposite signs and consequently it will be diffic_ult to discuss the

35) _ convergence properties of the series. Moreover the Iq terms become very lengthy and the process of integration is rather tedious- Thus in the final words of the Introduction “we shall ha\-e to enquire whether the

form of the solution can be improved by introducing the KB31 method for nonstochastic variables”.

6. A?PLLCATION OF THE KBM METHOD TO XONSTOCHASTIC VARIABLES

We have from (4.1) that

where we put

Q(t) = c2W)(t) + e4c2j4)(t) + &P)(t) i e8Q (8)(t) + “., (6-2)

190

and so

-$ < R(t) >= $-2(t) < R(t) >, R(o) = E (W

where E denotes the identity cperator. We write the solution of (6.3) as

< R(t) >= V(t)Lqt), (6-4)

where U(t) satisfies the equations

‘;;“’ = GU(t) , u(o) = E. (6.5)

In equations (6.4) and (6.5) G is independent of the time and V(L) pro- vides a small correction to the approximate value U(t) of < R(t) >.

Substituting (6.4) into (6-3) we see that V(t) satisfies

dV(t)

dt = R(t)V(t) - V(t)G. (6-6)

In order to find a solu’,ion of (6.6) we follow the example in (6.2) to

expand G and V(t) as follows:

G = e2Gt2) + E4G(4) + E6G(6) + ___ (6.7)

V(t) = E + E2v”‘(t) + Ew4)(t) + E6w)(t) + “., (6-W where VI*)(t), V(‘)(t), __._ must be bounded for all values of t and must vanish for t = 0. On substituting (6.7) and (6.8) into (6.6) and equating coefficients of E2, E4, e” etc. we obtain

dVt2)(t)

dt = 52[*)(t) - G(*)

dVt4) (t)

dt = Rf4)(t) - Gt4) -j- (Q(‘)(t) - G(‘)(t))V(*)(t)

(6-9)

(6.10)

dv(6)

dt = Q(@(t) - Gt6) + (st’4’(t) - G(4))V[2)(t)

+(il(*) (t) - G(2))Y(4) (t). (6.11)

191

Since 12’(*)(t) is bounded for large t and since V(')(o) vanishes, we deduce that

R(2)(Oo) = G2, -[+)ii) = J

t(sI(*)(tl) - R(2)(w))dtl. 0

Proceeding along step-by-step we calculate t'G[*), e4Gt4), ?Gg, e*Vc*)(t), E-lW)(t),A (63{t). The final result is (ref. 5, section 12.3)

< R(t) >= [E + #(l - c?- .rzt ) + y2{J$ - (Bt + l)e-Bt - +-=,1

where y is a dimensionless quantity defined by

kT Y =m- (6-U)

Equation (6.13) is vaIid for all times.

The value of < R(t) >LO is obtained by t&ingj = I SO that J2 = 2E, and then

+Y3 ; - (B2tz + 8Bt + 15)e-*t + (@ + 5)e-2Bt _ ,,_,,,}

+---I exp[-2yB I + 2 IT+ 25

$2 + zY3 + --- >

t]. (6.15j

To calculate the complex relati\*c permittivity we write

%y(! + $y + 2.3

;Y2 + 3673 + ._.) = Gr (6.16)

W = Bw’.

Then (5.2): (6.15) and (6.16) yield

c(w) - E, = l-i13w’ 9

6:s - cot 1+“i(2-2e--EL)+y2(;j-Ge_8’-_2Bte-BL+~e-2B1)

5 -- ge

-3BL} + . ..] x exp[--B(Gr + iw’)t]dt. (6.17)

On performing the integrations one obtains from (6.i7)

E(W) - Eoo G - I 2 2

c, - EC0 = Gr + iw’ - zw Gr + iti’ - G1 + 1 + iw’ I

+ y2 [ 9

6 2

- - G1 : iw’ GI + 1 + iw’ (Gr + I + iw’)2 + 3- 2 1 Gr+2+i&

I 5 8 2

- - G1 + 1 + iw’ (G1 + 1 + id)2 (G1 + 1 + is’j3

5 2 5

+- 5 G1 + 2 + id +(G2 - 1 7 2 + is’)2 Gl+3+iwi

(6.18)

Thus the application of the KBLI method provides an expression for E(W) without excessive computation.

When t = 0 is substituted into (6.13): then clearly < R(t) > is equal to the identity operator. When the dimensionless quantity Bt tends to infinity, eq. (6.13) reduces to

Bym_ < R(t) >= - [E + rJ2 + yz($P + fJ3,

+y3( TJ2 + fJ4 + $P) + ._*I exp(-yBJ2t)- (6.19)

In the Debye theory (ref. 5, section 9.3)

< R(t) >= exp(-PyBt). (6.20)

On comparing (6.20) with (6.19j we see that the Debye limit is not just the same as the long time limit but that it requires the additional condition that y dcfincd b_v (6.14) satisfies y << 1. This condition is satisfied for our prcscnt problem of a molecule undergoing random rotational motion in a dilute nonpolar solvent (ref. 8)_

7. TIIE .-1SY5lIIETRIC 1IOLECULAR MODEL

The method employed for a spherical molecule may be estended to cover the cast of an asymmetric molecuIe (ref. 9). 1Vorking in the inertial system of coordinates we define e by

where Il,Iz, f3 are the principal moments of inert.ia and 11 B1,12 B2, Is& are the coxesponding coefficients of rotational friction. Each component of angular velocity ~~(1) is expressed as a power series:

Wi(t) = s!c)(t) -i- EXil)(t) + E2si2)(t) + e3ui3)(t) + ____

and &i(t) is a centred and, in general, ncn-Gaussian (3.1) now gives

variable. Equation

dRdl” = [OK + ELK + e3K0(t) + ___]R(t),

w-here K(*)(t) = -i(J * d”)(t))_

The calculation of < R(t) > is quite onerous but it follows along the lines of (6.2) - (6.13). It is concluded that

< R(t) >=

x esp

194

where ijk is a cyclic permutation of 123. In (7.1)

L)!‘) = kT s IiBi’

O!” z is a lengthy expression of iower order than 0: I),

It ~vould be very laborious to go to a higher order of approximation.

If we write

and if n1,n2,n3 are the direction cosines of the dipole ais referred to the inertial systein. the complex pcrmittivity E(W) is given by (ref. 5, p. 215)

c w --cm _

l I-CgO -

+ n;

L)l f L32flr; [ D~(Dl+D2CE1)+ruD~~) + D1(D,+D2fB2;+-D~)’

D1 +L’2+ISl iru a] +D2+B31+- I- ( 7.2)

The averaging method is also applicable to many nuclczr magnetic processes (ref. 6). Thus, for example, if the relaxation is caused by in-

tramolecular dipole-dipole interaction of two like spins I, the longitudinal relaxation time Tr and the transverse relasatican time T2 are given by (ref. 6, eq. (4.58), (4.59), (6.1))

1 - = I(1 + 1) Tl 1 fj(wo) ‘- Yj(2”,) )

1 - = I(1 t 1) 7‘2 1

2j(Oj -t +.Q) + $24 1

(8-l)

(8-2)

lY5

Ir these equations y’ is the gyromagnetic ratio of a nucleus which con-

tains a spin 1 and wo is the product of y’ and the intensity of a prescribed constant magnetic field in a fixed direction. In (8.3) T is the distance be- tween the interacting dipoles, and the polar 8’ and azimuthal angle $ specify the orientation of the dipole-dipole axis with reference to the in- ertial frame of one of the molecules co:lLaining a dipole. It is clear from (8.3) that all that is required to complete the calculation of T: and Tz is the value of < R(t) >. As we have seen, this may be done by employ- ing the averaging method. Moreover the same procedure will provide the values of Tr and Tz when the relaxation mechanism is quadrupole interaction (ref_ 10) or anisotropic chemical shift (ref- 1 I).

This is no longer true when the nuclear magnetic relaxation is due to spin-rotational interaction (ref. 12). In order to see whether the KBM method might be applied to spin-rotational relaxation we take a spherical molecule, fix our attention on one nucleus in the molecule and consider the interaction of the magnetic moment of this nucleus with the magnetic field produced by the rotation of the molecule. If I is the spin operator of the nucleus and tiJ the angular momentum of the molecule, the spin-

rotational Hamiltonian

FLG = hI-c-5, (8-4)

where C is a three-by-three rea1 tensor. We take an inertial system of coordinates with the third axis passing through the nucleus. To keep the problem as simple as possible we assume that the third axis is an n-fold axis of symmetry C, with n 1 3.

The longitudinal Tl and transverse T’- relaxation times are given by (ref. 6: eq. (lo-i), (lO.i9), (LO.Zl), (10.26))

where

a> = 2 Q&w) + c(-iw)]

(8.5)

(8-6)

c(s) =p $ 2 2 (-)mbnpbmu

w,v=l m,n=-1

4 n,--m (8.7)

q= i&) f (8.8)

The immediate problem is no\v the evaluation of < R(t)w,(t)~~,(o) >.

Since for the spherical molecuIar model u;(t) is a centrcd Gaussian ran- dom variable, we deduce from (2.2) with z(f) E R(t) that

< R(t)w,(t)L&(o) >

=< (E + E*Fyt) + Ev+*)(t) + . ..)LJ&)~LJ(O) >< R(t) > - (8.9)

The subsequent caIculations are very lengthy and we shall give the result only for the extreme narrowing approsimatian where j(~u.~) in (8.5) is replaced by j(o). The final resuIt is (ref. 12, eq. (3.40))

1 1 -=-= Tl 'r2

(8.10)

where I is the moment of inert.ia of the molecule about a diameter of the molecule, IB is the coeficicnt of rotational friction: y is defined in (6.14) and

cl1 = c33 : Cl = c,, = 4722.

The rotational diffusion approximation of (8.20) obtained by putting -I equal to zero is

1 1 21kT(C;f + 2CT)

r,=r,= 3h2B -

9. coscLc’sIox

The i .?portance of mathematical methods used by Krylov, Bogoli- ubov and 1Iitropolsky in their solutions of nonlinear differential equations

197

involving stochastic or nonstochastic dependent variables is illustrated by esamples related to rotational Brownian motion. Some implications for dielectric relasation and for nuclear magnetic relaxation are investigated.

REFERENCES I.

2.

3.

4.

5.

6.

7. 8. 9.

10. 11.

N. Krylov and N. Bogoliubov, Introduction to Non-Iinear Mechanics, Prixeton University Press, Princeton, New Jersey, 1947. N-N. Bogoliubov and Y.A. %IitropoIsky, Asymptotic Methods in the Theory of Xon-linear OscilIations, Gordon and Breach Inc., Xew E’ork, 1961. N-G. Van Kampen, Stochastic Differential Equations, Physics Reports 24 C, 171-228, 1976. G-U’. Ford, J.T. Lewis and J. McConneII, Proc. R. Ir. Xcad. 76-1, 117- 1.13, 1976. J- McConnell, RatatiDnal Brownian Motion and Dielectric Theory, Academic Press, London, 1980. J. McConnell, The Theory of Xuclcdr &Iagnetic Relaxation in Liquids, Cambridge University Press, Cambridge, 1987. P.S. Hubbard, P+_ Rev. A6, 2421-2433, 1972. K-F. Herzfeid, J. Am. Chem. Sot- 86, 3468, 1964. G-W. Ford, J-T. Lewis and J. McConnell, Phys. Rev. A 19, 907-919, 1979. J. McConnell, Physica 117-A, 251-264, 1983. J. McConnell, Physica 127A, 152-172, 1984.

12- J. McConnell, Physica I1 IA, 85-113, 1982.