Diffusion Equation Study of Rotational Brownian Motion

20 Pages·1977·1.146 MB·English

by  McConnell, James

Diffusion Equation Study of Rotational Brownian Motion
Author(s): James McConnell
Source: Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical
Sciences, Vol. 77 (1977), pp. 13-30
Published by: Royal Irish Academy
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[ 13 3
2.
DIFFUSION EQUATION STUDY OF ROTATIONAL BROWNIAN MOTION
By JAMEMSC CONNELLM, .R.I.A.
Dublin Institute forA dvanced Studies*
[Received, 25 MAY 1976. Read, 8 NOVEMBER1 976. Published, 31 MARCH 1977.]
ABSTRACT
Sack's discussion of the rotationalB rownian motion of a sphericalp olar molecule
based on a diffusion equation is simplified. The solution of the equation for a
periodic external electric field is expressed in terms of elements of the reciprocal of a
matrix. If the reciprocal is expanded as a series of powers of a dimensionless para
meter, the results for complex polarisability agree with those of long calculations
of Ford, Lewis, McConnell and Scaife, and theym ay be extendedw ithout difficulty
to higher orders of approximation.
TABLEO FC ONTENTS
Page
1. INTRODUCTION ... ... ... 13
2. THE FOKKER-PLANCK EQUATION FOR THE ROTATING SPHERE ... ... ... ... 14
3. SOLUTIONS OF THE DIFFUSION EQUATION ... ... ... ... ... 18
4. RELAXATION EFFECTS IN DIELECTRICS ... ... ... ... ... 23
1. Introduction
A lengthy study of inertial effects in dielectric relaxation processes was made by
Sack in two papers (1957a, 1957b). In paper I he took the disk as a model of a
dipolar molecule and in II he took as models both the rotating needle and the rotating
sphere. His investigations were based on a diffusion equation, which is a Liouville
equation supplemented by Kramers terms.
Since Sack's calculations for the needle and the sphere, though condensed, are
both long and complicated, it has seemed worthwhile to seek alternative treatments
of the diffusion equation. This has been done for the needle by Ford (1975), who
employed expansions in associated Laguerre polynomials. We have found that
similar expansions produce no simplification for the spherical model. However, it
is possible to modify Sack's method so as to shorten the calculations considerably.
This is done in the present paper.
In the next section we give an alternative derivation of Sack's generalised Liouville
equation and elaborate some results from II. We then express the solution of the
equation for a periodic external electric field as a series, in which the coefficients are
*The subvention granted by the Dublin Institute for Advanced Studies towards the cost of
publication of papers by members of its staff is gratefully acknowledged by theR oyal IrishA cademy.
PROC. RI.A., VOL. 77, SECT. A t2I
14 Proceedings of the Royal Irish Academy
elements of the reciprocal of a matrix. The reciprocal is expanded as a power series
in a dimensionless parameter introduced in earlier investigations. In the final section
thism atrix method is applied to relaxation effects in dielectrics and, with substantially
less labour, previous results for complex polarisability are extended.
2. The Fokker-Planck equation for the rotating sphere
Sack obtained his LiouviMe equation by adapting to three-dimensional rotational
motion an equation of Kramers (1940, eq. (9)) for one-dimensional linear Brownian
motion. Since Kramers' derivation of his equation is open to objection, we shall
derive the diffusion equation independently putting down Langevin equations and
following standard procedures to go to the Fokker-Planck equation. To facilitate
comparison with Sack II we retain his notation expressing the components of angular
velocity of the rotating sphere (v9, v., vu) in terms of the Euler angles 0, q, X by
v0=OXv,, =4 sin 0, v5=-X+# cos 0. (1)
We take moving axes, numbered 1, 2, 3 in Fig. 1, for the rotating sphere with a dipole
of constant moment p along axis 2. The sphere, whose moment of inertia is I, is
subject to a frictional drag Ifi times the angular velocity, to a random driving couple
IdW
dt
FiG. I
MCCONNELL-Diffusion equation study of rotational Brownian motion 15
in no preferential direction and to an electric field of intensity F in the direction
0=0. W(t) is aW iener process. The components of angular velocity of the axes 1, 2,
3 are respectively (0, $ cos 0, $ sin 0) and the equations of motion are
I0 + I#k sin 0 -LF sin 0-I#d + I dtdWt 1
d dIV2
I- dd it +0 cos 0)=(-Ixx+@ cos 0)+I dt
d *si
I d( sinfO)-Icq= -If4sin0+I dW3
We express these in terms of vo, v,, v. by (1) as
sin
dvo -v.(v,-vv, cot 0) +fiv+ yE 0] dt+dW1
dv, = [v0(v7- v# cot 0) - fv4 dt + dW3 (2)
dvz= -flv dt + dW2.
In addition we have from (1)
dO= v6ldt, do sin 0 dt, dX = (v1- v# cot 0) dt. (3)
We employ (2) and (3) to establish the Fokker-Planck equation for the probability
density w specified in configuration-angular velocity space by the coordinates 0, v0,
v4,,v . and the time t, the probability density being clearly independento f the angles q
and X. We write
Y =0,Y2k, Y3 X,Y 4=V69 )Y5= V+, Y6 =vz (4)
and express (2) and (3) as
dy1 =y4dt, dy2 = ;no dt, dY3=(-cot 0 Y5+Y6) dt
F
i
dY4 =-[v(v2-v.* cot 6) + flv + dt + dWi (5)
dy5 = [v0(v- - Vu cot 0)- fivo] dt + dW 3
dy6= -vv dt + dW2.
In general the equations
m
dXi(t)=Afi(, X) dt+ E G(t, X) dW,Q) (i= 1, 2, .... d)
1=1
lead to the Fokker-Planck equation (cf.A rnold 1974, pp. 156 - 159)
aw X 8 d d 02
Ft + Yi (JXt,y ) w)-Ic2 y.oy.((GG')iiw)O, (6)
PROC. R.I.A.5 VOL. 7 7, SECT. A [2AJt
16 Proceedings of the Royal Irish Academy
where G' is the transpose of the matrix G and c2 is determined by
<W i(t) Wj(t')> = i c min (t, t'), (7)
the brackets on the left hand side denoting ensemble average. However, in establish
ing (6) it is assumed that the volume element in the y-coordinates is dy1 dy2.. .d Yd,
whereas for (4) it is
sin y1 dy1 dy2 dy3 dy4 dy5 dy6.
Then on integrating by parts, with g a smootb bounded function with continuous
partial derivatives and vanishing outside a bounded domain,
{dtj' dO sin 0 4 wvo= -J'dt vofg ,- (w sin 0) dO
=-J'dtvo gJsinO (T+ w cot 0) dO,
whereas
Jdt Jd 4, wv =-{dt v, ygo do.
Hence in writing down (6) with fi, GCi given by (5) we replace
8w 8w
by + cot 0.
The Fokker-Planck equation is therefore
8w 8w / 8w 8w N F 8w
i+?vo yg- v p- --sin. 0 -3,Bw
gp0+(vt,PcotO-vz)Kv.
I
(8)
/ 8w Z(w w 8ww 2 82w v 2w 82w\ 8V0
_ft oy ++ y+ vy t-Ic 2 + 7+ 8V2}?
To obtain the equation in the form given by Sack we use the relation (Lewis,
McConnell and Scaife 1976, eq. (12))
c2 kT
2/- I'
where k is the Boltzmann constant and T the absolute temperature. This relation
could be derived independently by taking for w in (8) the steady state solution given
below in (12). On substituting for I c2 equation (8) becomes
8w aw / w 8w N ,uF 8w
- -
at+ vo1+ (v, cot 8vvp, jz) vo )- sin
8( kT8aw 8 kT8w 8a kTaw\ (9)
+
whc g =efel -Kvowth+ 7r}e v. I cmief liw w$Wthw+ +-y I 3vzg } Tfl Vz w+-j-~ I )v
which agrees with II eq. (3.8) combined vvith 1 (2.3).
MOCONNELL-Diffusion equation study of rotational Brownian motion 17
For future reference we repeat more explicitly Sack's transformation of equation
(9). Writing
O-V01, V=-V2, Vz=V3, U=-U1, UO=U2, Uz=U3
3 3
V1I2 =v2, Z u12=U2
I-1
~1=1
we make the Fourier transformation
r+ 00+ or+ Go3
P(O, uo, ut,, uZ,0 =J jJw (O vo, v4t, v", t) exp (-i Eu, V) dvl dv2 dV3, (10)
- 00 - o00
which transforms (9) to
04D @2FD a a a ii
7 +? im j-i coto T -u.y- -D - sin uo ?
3 O(Dy -O u2kT
--f EiZ ___- (D.
au,
The probability density wo for a steady state is given by the Boltzmann distribution:
w = const. exp{ - [IV2+ V(Q)]/kT}, (12)
where the potential energy V obviously depends only on the Euler angle 0. From (10)
the Fourier transform ( of wo is given by
(FO(0u,e , u#, uz, t)=const. exp (-V(0)/kT) exp (-kTu2/2I). (13)
We lighten the calculations by elimination of the second exponential putting
L'(0i,0 , it, u, t)= exp (kTu2/2I0)( 0, uou, , u,t). (14)
Then (11) leads to the differential equation for P
0T a2' i8 F ap _
Aj+i @ --j kTj +mFsin0Wj
- 3 (15
a a kT aT alp ' aT
+tau -au" I (cot it. aujuo 1J=-fl?ui (5
which is the diffusion equation with which we shall be chiefly concerned.
The equilibrium solution To of the last equation is seen from (13) and (14) to be
given by
0 = const. exp (- V(0)/kT).
The V(O) due to the external field F is - piF cos 0 and (I2) becomes
1 I 3/2 puF 1v2 ALFco0s
o 8=ni 2nrkT1'- F e 2T e ,E (16)
kT sinhu
where the constant multiplying factor has been chosen such that
r27 02X IC r+OD+ o+ oo
S0 X 0{ o 0s if fdo0 00 oco 00 f-cn0 0
18 Proceedings of the Royal Irish Academy
From (16)w e deduce that
pP cosO kTua
1 kT e 2i
pFe
kT sinh kT
and from (14) that
F -cos
1 pFe kT
P (0, u9, u,, uz) 2 -
kTsinh kT
If now we suppose that
kT
is sufficiently small that it is permissible to take the linear approximation
uFcos 0
+ kT
for the exponential, we write
co
879 Po(Q ue, uuz)=l + kT * (17)
3. Solutions of the diffusion equation
Sack's investigations of relaxation effects in dielectrics were based on two obser
vations:
a. the polarisation of the dielectric due to the external electric field is proportional
to
(2n (2 Jr it
lim dXl d4J sin Ocos OT(O, u6, uis, u,, t) dO,
uO0 o0o 1 0
b. the value of ' in the linear approximation for the equilibrium case is given by
(17).
Since Sack was immediately interested in TF, he studied the solution of (15) rather
than that of the Fokker-Planck equation (9). He confined his attention to the
linear approximation and so was guided by (17) in his search for a solution of (15).
We shall likewise confine our attention to the linear approximation. For the
solution of (15) we try
8n2' = + cos 0 XI (u2, t) + sin 0 uo Z2(u2, t) + (cos 0 UZ2+ sin 0 u_2u4)X 3(U2, t) (18)
where XI, X2z X3 are of order
F
kT
MCCONNELLD iffusion equation study of rotational Brownian motion 19
and termso f higher order in this dimensionless quantity are neglected. This equation
is equivalent to 11 (3. 14), if we put
01= xl(u 2, t), 02=ux2(U2, t), #3=u2x3(u, t)
and note the expressions given in 1I (2.22) and (3.16) for #. 02' 03 when F=Foe't
This is in fact the case with which we shall be almost entirely concerned. Sack
obtained 11 (3.14) by expanding T in u-space as a series of spherical harmonics.
On substituting (18) into (15) it is found that 8X2 the left-hand side of (15) is equal
to
Cos i( +2iZ2+2iXa2 (u2 + U2)-j x2 (UO + U))
ikT
+sin 0 (a -2i I iX3)
OX2
+ (cos 0 uz2 +sin 0 uz Xu) -X3-2i sin 0 u zX2 2i Uz2 COS 0
.20ax2ik T. ikT 6 2 ikT OS0U2
+2 2iUu Cos 0 2+ sinO0uuo I2+ -cos Ou X2A I Co 054X2+'
-=CCOo s0o(Q 0 pt1++2 ix22i 2XU22 + 2i TUx22 -i ikcTuU 2 xX 2)
+ sin 0 (X2- 2i DX'ikT _ipF+F (19)
i T
+(Cos0 V2+sin0uzu X -+j2i, 2+
We introduce the dimensionless variable z by
kTu2
21 (20)
There will be no danger of confusing this z with the subscript in u. or v:. We then
deduce that (19) equals
cos 0 ( ,+2i X2 +2iz X2-2iz X2
at xa- +x >h(1
+ sin 0 + ) (21)
OX2ikTX,ikrTx ikTF
+(cos 0 U, 2 sn u";a3 OX _ikT DX2 ikT )
20 Proceedings of the Royal Irish Academy
Similarly for 8n2 the right-hand side of (15)
3
O OW O
-S2f2lZui-= -82fl - = -l16nr2flz
Now from (18)
g8nz2a LTOc=ozO so?k '+siJanrxO ,+ s~n~U~~ ~~o~TaOz(o 0e@ X0 22 O?Uaud?lI)+. (CO0S U 2+i 0 z X)3a
X3[2 cos 0 u z+sino (+ 8*
so that
- 16.n2z= Z -2/1 jcos Oz axj , u.9 sin z9 X+2 IX2
+(cos 0 u.2 +sin 0 u u4)( a++3)}+ * (22)
On equating the coefficients of cos 0, is0 sin 0, cos 0 U z2+sin 0 uz u,p in (21) and (22)
we obtain the equations determining Xi, X2, X3
x
0Xj+ 2i X2+ 2iz X2--2iz x2=-2fiz (23)
0Xat22_ 'kika@TXDt0+X 1 ikT i4sF 0X2
I xl-I -+ix =-2fl z-F-fX2 (24)
0X3 ikT 0X2 ikT 0X3
_Ik Oz?+ IkT X2 21 z0 z-2PX3 (25)
The solutions of these equations when substituted into (18) give the W(0, uo, u,A,u i2, t)
resulting in linear approximation from the field F. We now specialise to the case
where F= Fo e"'t and try the solutions
X1(U2,t )=eiwt > Ar zr,x 2(u2, t)=elw >1 Br Zr,
rO
r0o
oD (26)
X3(u2, t)=efitt E Cr zr.
r=O
Equation (23) yields
(icoQ+2r)A r+2i(r+1)Br-2iBr,_1=0. (27)
Equation (24) gives
ikT ikT
i#F0
ic Brzr- T E rAr zr + I1E Ar Z-y
+iZ C,r zr+2p z rB4+r PE BrZ =o.
MCCONNELL-Diffusion equation study of rotational Brownian motion 21
On account of the presence of
ipFo
it is necessary to distinguish between r=0 and r= 1, 2,.
ikT ikT ipFo
(iw+fl)Bo+ I Ao+iCo- I A1l - , (28)
ikT ikT
(iw+2flr+f) Br+ -I Ar+iC- y(r+ 1)A ,+1=O.(r 1I2*..) (29)
Lastly, from (25)
ikT ikT
(io 2flr + 2f0)C r + Br-- (r+1) Br+1 (30)
On eliminating Ar, Ar+ 1, Cr between (27), (29) and (30) we get
2 2r+2 2r+3
(iw+ 2flr +fl)(BcO +Byr+jTkLT io + 2r B -1 +\ic(w + 2fJr +3 i1 + 2)r + 2
(r + 1)(2r + 1)
iwic?+o 22fSlrB?+22flp Br+l =0-.(r=1,s2,.* )
When r= 0 and =l 0 we cannot eliminate Ar from (27). We shall therefore assume
that co# 0, which implies that our subsequent calculations do not apply to the case of
a constant field Fo. Then for r= 0 equation (31) will be true provided that we put
ipFo
I
on the right-hand side, in accordance with (28). On introducing the dimensionless
parameters cv'a nd y by
cc kT
cv= gp sv y=~ Ip(32)
(31) and its related equation for r= 0 may be expressed as
_2 / 2r+2 2r+3 0
(ico'F+ 2r +1 )BF+v[1+2 Br_ 1 + ico' + 2r + 2r Br
(33)
(r + 1) (2r+ 5) IiF0
iwl+2r+2 BBr +1 '1= rO
Let us put
La, = (ico'+ 2n + 1) bnl (34)
2 ( 2n+2 2n+3 \ (n+1)(2n+5)
Mn-ico+ 2n ico61 1+\t2n+ iw +2n+2 2 ico'+ 2n+2
so that (33) becomes
ipFo
(Ln I+ yMn 1) Bz = 1 ?