( DR+ os) (1.2)

MOLECULAR PHYSICS, 1987, VOL. 61, No. 4, 963-980

Nuclear spin relaxation induced by lateral diffusion on a fixed or freely rotating spheroidal surface

by BERTIL HALLE

Physical Chemistry 1, University of Lund, Chemical Centre, P.O.Box 124, S-22100 Lund, Sweden

(Received 20 January 1987; accepted 13 February 1987)

A theoretical method is presented for calculating the spectral density functions, which determine the N.M.R. relaxation behaviour, associated with lateral diffusion on the surface of a fixed or freely tumbling prolate or oblate spheroidal aggregate or macromolecule. Analytical results are given for the limits of fast or slow surface diffusion, while an efficient finite-difference algorithm is developed for the general case. The orientational order parameters that determine the N.M.R. line splittings from anisotropic systems composed of fixed spheroids are also obtained.

1. Introduction

Nuclear spin relaxation is a powerful experimental tool in colloid chemistry, capable of providing detailed structural and dynamic information about surfactant aggregates in isotropic micellar solutions and micro-emulsions as well as in lyotropic liquid crystalline phases [1, 2]. Whether one observes the resonance of nuclei belonging to the hydrocarbon chains or to the associated counterions or water molecules, the molecular motions inducing spin relaxation can usually be divided into fast local motions and slower global motions [3-8]. The global motions modu- late the residual spin-lattice coupling (usually a magnetic dipole-dipole or electric quadrupole interaction), which remains after partial orientational averaging by the local motions.

In many cases it is sufficient to consider only two kinds of global motion, viz. aggregate reorientation and surface diffusion (along the aggregate interface). If the aggregates are spherical and if the two kinds of motion are independent, then the spectral density associated with the global motions has the simple Lorentzian form

J(~o) = ~~ 1 + ((OZe) 2'

with the correlation time, zc, given by

(1.1)

( os) 1 _ 6 DR+ (1.2) Tr J "

Here D R is the aggregate rotational diffusion coefficient, Ds is the surface diffusion coefficient of the spin-bearing molecule and a is the aggregate radius.

A spin relaxation experiment performed at a single magnetic field strength provides information about the spectral density function at a few discrete frequencies only. Such limited experimental data usually cannot be used to discriminate among

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964 B. Halle

more elaborate (and realistic) microscopic models of structure and dynamics. However, by performing relaxation measurements at several fixed magnetic fields [9, 10] or by utilizing field-cycling techniques [11], it is possible to deduce the detailed shape of the spectral density function in the frequency range of interest. With this experimental possibility comes the need to calculate theoretical spectral densities for more detailed molecular models, incorporating, for instance, the effect of non- spherical aggregate shape.

Surfactant micelles are known to adopt non-spherical shapes as their size increases [123. Under certain conditions one obtains very long rod-like aggregates, whereas for relatively small departures from sphericity the aggregate shape is often modelled as an ellipsoid of revolution, i.e. a prolate or oblate spheroid [13-143. Spheroidal aggregates may also exist in microemulsions [153 and in cubic [163 and nematic [173 lyotropic liquid crystals. To derive quantitative information about aggregate shape and lateral mobility from spin relaxation experiments on such systems, one requires a theoretical method for calculating the spectral density function associated with lateral diffusion on a fixed (nematic phase) or freely tumbling (micellar solution) spheroidal surface. It is the aim of the present work to provide such a method.

The outline of the paper is as follows. In w 2 we formulate the physical model and derive a differential equation, the solution of which yields the spectral density for surface diffusion on a freely tumbling spheroid (isotropic systems). The limits of fast and slow surface diffusion are then considered in w 3. The modifications of the development in w 2 required for treating surface diffusion on a spheroid with fixed orientation (anisotropic systems) are described in w A finite-difference algorithm for solving the resulting differential equations is described in w 5 and is then used in w to obtain numerical results. Finally, in w the conclusions of this work are summarized.

2. Isotropic systems

The spin relaxation behaviour in an isotropic system is governed by the spectral density function [18]

fo J(og) = dz cos (a~z) G(z), (2.1)

with the reduced time autocorrelation function

G(z) = ( U}*(O)U~6(z))/( I Uo L 12) �9 (2.2)

U~ is the zeroth spherical component, expressed in a lab-fixed frame (L), of the second-rank irreducible spin-lattice coupling tensor, for example an intramolecular magnetic dipole-dipole or electric quadrupole interaction. Since we are concerned here with the effect of the global motions, the quantity Uo L should be interpreted as that part of the spin-lattice coupling that remains after partial orientational averaging by the fast local motions. (The local motions give rise to an additional, independent contribution to the spectral density.)

In order to identify the contributions to the spectral density from different motional degrees of freedom, we introduce two further coordinate systems (see figure 1): the rotational diffusion tensor principal frame (R), which is fixed in the


Spin relaxation by surface diffusion

ZR

965

Z0

YR

(a)

ZR Zo

YR

(b) Figure 1. The prolate (a) and oblate (b) spheroids, showing labelling of semi-axes and orien-

tation of coordinate frames.

aggregate with the z R axis defined by the cylindrical symmetry axis of the aggregate, and the director frame (D), with the ZD axis (the local director) perpendicular to the aggregate surface and passing through the nucleus in question. (The local director may equally well be taken to point inward; the symmetry of the system ensures that G(z) is invariant to this transformation.) Assuming that the distribution function for


966 B. Halle

the orientation with respect to the aggregate surface of the molecule-fixed spin- lattice coupling principal frame possesses at least threefold rotational symmetry with respect to the local director, we may write

2 uL(~') E 2 2 D = DO",[DLR(Z)]D" ,o[~RD(r)]Uo, (2.3)

" ,= - 2

where the D2mq(D) are elements of the second-rank Wigner rotation matrix [19] and where flLR and flRD are the Euler angles specifying the instantaneous relative orientation of the indicated frames. Uo D is the residual spin-lattice coupling, partially averaged by the local motions.

By means of the transformation (2.3), we have factorized the time-dependence of uL(z) into two sets of fluctuating stochastic variables: DLR("C), which is modulated by aggregate rotation, and ~')RD('C), which is modulated by surface diffusion. Assuming that these two kinds of motion are statistically independent, we obtain from equations (2.2) and (2.3)

2 2

o(~) 5 Z Y~ 2, 2 = (Dom[f f lLR(O)]Doq[DLR(Z)])LR m = - 2 q = - 2

2 , 2 x (D",ornrD(0)]Dqo [flRO(t)])rD, (2.4)

where use has been made of the orthogonality of the rotation matrix [19]. The reorientational motion of the spheroidal aggregate is modelled as a free (no

external torque) rotational diffusion of a symmetric top. For this model [20]

2 * 2 ( Do",[~')LR(O)]Doq[QLR('C)])L R = t~",q 1 exp ( - t / t " , ) , (2.5)

with the rotational correlation times

z", = [(6 - mZ)DT + m2DL]- 1, (2.6)

where D E and D r are the longitudinal (rotation around the symmetry axis) and transverse (rotation of the symmetry axis) rotational diffusion coefficients of the spheroidal aggregate.

Substitution of equation (2.5) into equation (2.4) yields

2

" , = - - 2

where

By symmetry

exp (--Z/t",){(/52*[~RD(0)]/~20[~RD(Z)])RO +1 2 z (D,,o(nRD))RD I }, (2.7)

/5~0(~O ) = D~0(ff/RD ) 2 - - (Dm0(~)RD))RD. (2.8)

(D2mo(~')RD))RD = 6 m o ( P 2 ( x ) ) , (2.9)

where P2(x) is the second-rank Legendre polynomial with argument x = cos 0RD = ~R " 20- Hence

2 G(z) p exp (-Z/Zo) + ~ A2* A2 = (D",o[~RD(0)]D",o[f~RD(Z)])RO exp (-- z/z,,), (2.10)

" ,= - 2

with p -= (P2(x) ) 2.


Spin relaxation by surface diffusion 967

The surface diffusion correlation functions in equation (2.10), which clearly vanish as z ~ 0% may be written

^2* ^2 (Dm0[nRD(0)]D.,0[nRD(Z)])R D = dnOD f(noD) D,,,0(nRD)'2, 0

f 0 ^2 x dI'IRD f(I)RD, Z I nRn) D=o(I-IRD), (2.11)

where a zero index signifies the initial value of the stochastic variable. Because of the cylindrical symmetry with respect to the zR axis, equation (2.11) reduces to

^2* ^2 ;1 (D,.0EI)RD(0)]D,.0EflRD(Z)])RD = dxo f(xo) d~o(Xo)

1

x d(a dx f (x , (~, z I Xo) d~o(X) exp ( - im(a), (2.12) 1

where the reduced rotation matrix elements d~o(X) are defined as in equation (2.8) and where q$ denotes the azimuthal displacement (around the ZR axis) during time z.

The joint propagator in equation (2.12) may be expanded in a Fourier series

f (x , ~), z I xo) = ~ .= _ ~ f.(x, z I xo) exp (inc~), (2.13)

with

f0 2~ f,(x, V]Xo) = dq~ cos (n(~)f(x, r z lxo). (2.14)

Combining equations (2.10), (2.12) and (2.13) and performing the integration over ~b, we find

2 G(z) = p exp ( - z /%) + ~ exp (--z/z,.)

m = - - 2

The lateral motion of the spin is modelled as a free (no force field) translational diffusion on the surface of the spheroidal aggregate. The joint propagator f (x , ~b,z I Xo) then satisfies the surface diffusion equation

O---z f (x , e~, z I Xo) = DsV2 f (x , c~, Z lXo), (2.16)

where D s is the surface diffusion coefficient of the spin-bearing species and where V 2 is the angular part of the laplacian.

To proceed further, we must specify whether the aggregate shape is prolate or oblate spheroidal. We denote the lengths of the symmetry semi-axis and the degen- erate semi-axis by a and b, respectively (see figure 1), and define the axial ratio, r ( 0 < r ~ < 1),as

rpr = b/a, (2.17 a)

rob = a/b. (2.17 b)


968 B. Halle

The angular spheroidal coordinate, t / ( - 1 ~ t / ~ 1), is related to x = cos 0RD through

xpr = rr/[1 - (1 - rS) r /2] - l / s , (2.18 a)

Xob = r/[r 2 + (1 -- r2)tls] -1/2 (2.18 b)

In the spherical limit (r = 1), equat ion (2.18) reduces to x = r/. The normalized uniform equilibrium distribution on the spheroidal surface is

fpr(r/) = [1 -- (1 -- r2)tlS]i/S/(r + A), (2.19 a)

fob(r/) = [ r 2 + (1 - - r2)r/211/2/(1 + B), (2.19b)

with

arccos r A = (1 -- rS) 1Is' (2.20 a)

B = 2(1 -- r2) 1/s In - - (1 r2)1/-----5_]" (2.20 b)

In the spherical limit, the distribution functions reduce to f(t/) - ~. Writ ing the surface diffusion equat ion (2.16) in prolate spheroidal coordinates,

multiplying by cos (mq~) and using the definition (2.14), we obtain

a 2 ~ ~ D--~s a--~ f"(q' z [r/o) = [1 -- (1 -- r2)r/s] - 1 ~qq[(1 -- r/s) ~ f,,(r/, z I r/o)]

m 2

r2(1 _ r/2) f.,(r/, z ] r/o). (2.21)

After separat ion of variables in equat ion (2.21), one obtains an ordinary differential equat ion in r/, the solution of which may be expressed as an eigenfunction expansion involving the angular prolate spheroidal wave functions, which, in turn, may be expanded in a basis of associated Legendre functions [21, 221. However, for arbi- t rary axial ratio, this approach leads to rather cumbersome expressions, the evalu- at ion of which requires extensive numerical computat ions. We have therefore adopted a different approach, wherein the problem is cast in a form suitable for solving by an efficient finite-difference algori thm [23, 24].

We thus introduce the quant i ty

Fro(q, r lqo) -f,.(r/, r[ r/o) exp (-r/r . , ) , (2.22)

and its Four ie r -Lap lace t ransform

ff,,(tl, o91 r/o) = dr exp (i09r)F,,,(r/, r I t/o). (2.23)

Together with equat ions (2.1) and (2.15), this leads to

p r o 2 f 1

J(09) - 1 + (09ro# + m = - 2 , I ' - I dtlo f (qo) d~o(Xo)

x f l , dr 1 d~o(X ) Re [ff,.(t/, 091 no)],

with x andf( t / ) given by equations (2.18) and (2.19), respectively.

(2.24)


Spin relaxation by surface diffusion

Combinat ion of equations (2.21) and (2.22) yields

E 1] ~--g F~(r/, ~lr /o) = ~ , . - FAr/ , z 1,7o),

where we have introduced the differential operator

Ds{ x 0 c3 m 2 } L f , ~ = ~ [1 - - (1-- rZ)r/2] - ~ ( 1 - - r / E ) ~ rE(1---~q E) .

969

(2.25)

(2.26)

Taking the Fourier-Laplace transform of equation (2.25) and using equation (2.23) and the initial condition F,,(~/, 0l r/o) = 6(r/-- r/o), we obtain

[ ' 1 -- - - + ico ff,,(r/, co [ r/o) = -- 6(r/-- r/o). (2.27) A(m Zm

We now define (for the prolate case)

f l -- r )r/o] dmo(Xo)fm(r/, col r/o), Qm( t l , co) = dr/oil (1 -- 2 2 1/2 n (2.28) 1

which, together with equations (2.19 a) and (2.24), yields

pz 0 2 f l J(co) - 1 + (coZo) 2 + (r + A) -1 ~', dr/d~o(X) Re [Q,,(~/, co)]. (2.29)

m=--2 , - 1

From equations (2.27) and (2.28) it follows that Qr,(r/, co) satisfies the ordinary differential equation

[ ' ] -- - - + ico Qm(r/, co) = - [ 1 - (1 - r2)r/2]'/2d~o(X). (2.30) "~m Tm

In w 5 we show how equation (2.30) may be solved by a finite-difference technique. In the oblate case, r + A in equation (2.29) is replaced by 1 + B, while equations

(2.26) and (2.30) should be replaced by

Ds { ~ "~gm ~ V I-F2 + (1 -- /'2)?] 2 ] - 1 ~ (1 - r/2) ~ -

and [ 1 ] - - - - "k ico Qm(r / , co) = - [ - r 2 q- ( 1 - - r2)q2ql/2d~o(X).

"~m Tm (2.32)

3. Limits of slow and fast surface diffusion If surface diffusion is much slower than aggregate rotation, then the Euler angles

[~RD(Z) in equation (2.10) do not change significantly on the time-scale on which the exponentials exp (-Z/Zm) decay. We can then set ~RD(0) = ~RD(Z) in equation (2.10) to obtain, with equation (2.1),

2 CmZm (3.1)

J(co) = ~ 1 + ((.OTto) TM m=O


970 B. H a l l e

The coefficients , c , , , wh ich s u m u p to uni ty , a re g iven by

cm =- (2 - 6, .o)([d2o(X)32> = (2 - 5,,o) I t dqf(r l )[d2o(X)] z. (3.2) j - 1

U s i n g e q u a t i o n s (2.18)-(2.20) a n d e v a l u a t i n g the in tegra l , we f ind for the p r o l a t e case

(1 - - 8 r 2 - - 20r4)A + (1 + 22r 2 + 4r4)r c~ = 4(1 - - r 2 ) Z ( r -k- A) ' (3.3 a)

3r2[(1 + 2r2)A - 3r] c l - ( 1 - - r2)2(r + A) ' (3.3b)

3[(1 - 4r2)A + (1 + 2rZ)r] (3.3 c) c2 = 4(1 - rZ)2(r + A)

F o r the o b l a t e case, we f ind s imi l a r l y

(4 + 22r 2 + r 4 ) - (20 + 8r 2 - r4)B Co = 4(1 - r2)2(1 + B) ' (3.4 a)

3[(2 + rZ)B - 3r 2] c , = ( 1 - - r2)2(l + B) ' (3.4b)

3r2[(2 + r 2) - (4 - r2)B] c2 = 4(1 -- r2)2(1 + B) ' (3.4c)

wi th A a n d B def ined by e q u a t i o n (2.20). These coeff ic ients a re s h o w n in f igure 2 as func t ions of the ax ia l ra t io , r, of p r o l a t e a n d o b l a t e sphe ro ids . In the sphe r i ca l l imi t (r = 1), c o = 1/5 a n d c~ = c 2 = 2/5. F o r p r o l a t e s in the l imi t r = 0, c o = 1/4, c l = 0 a n d c2 = 3/4, as e x p e c t e d for an inf in i te ly l ong cy l inder . F o r ob l a t e s in the l imi t r = 0, Co = 1 a n d c~ = c2 = 0, as e x p e c t e d for a p l a n e of inf in i te extent .

T h e l im i t i ng fo rm given b y e q u a t i o n s (3.1)-(3.4) is a c t u a l l y n o t va l id in the c o m p l e t e ab sence of surface d i f fus ion (D s = 0), for we have t ac i t ly a s s u m e d t h a t

Figure 2.

.6

Cm .4

.2

i l i AT IE

.2 .4 .6 .8

I I I I

PROLATE

.8 .6 .4 .2 0'

The coefficients c m in the spectral density for slow surface diffusion versus the axial ratio, r, of the spheroid. The value of m is given by each curve.



surface diffusion is fast on the time-scale of spin relaxation. If this were not the case, the spins would not have time to sample all positions on the surface and the observed spin relaxation behaviour would be a superposition of that for nuclei with different local director orientations x = cos 0ao. For each orientation the spectral density is given by (3.1) with

c,, = (2 - - 6mo)[d20(x)] 2. (3.3)

This is a well-known result, first derived by Woessner [25]. If, on the other hand, surface diffusion is much faster than aggregate rotation,

then the Euler angles ~RD(Z) in equation (2.10) will have become completely randomized by the time the exponentials exp (-r/%) have decayed significantly. Consequently, we may set 1/% = 0 in the sum over m in equation (2.10) to obtain, with equation (2.1),

J((o) = JLR(<9) + J,~D(09), (3.5)

where

and

JLR(CO) -- pro 1 + (09%) 2, (3.6)

2(0 JRD(Og) = ~ dr cos (C0r)(/52~[flRD(0)]/)20[ftRD(Z)])RD . (3.7) m = - 2 ,

JsD(Og) is the contribution to the spectral density from surface diffusion and is also given by the sum over m in equation (2.29), provided that one sets 1/% = 0 in equation (2.30). This contribution vanishes in the limit of infinitely fast surface diffusion.

The lorentzian contribution J L s ( ( O ) to the spectral density represents the effect of modulation of the axially symmetric residual spin-lattice coupling Uo R = (P2(x) )U D by isotropic rotation with rotational diffusion coefficient DT. This contribution is independent of D s, the effect of surface diffusion being only to further reduce the effective spin-lattice coupling.

The coefficient p in equation (3.6) is given by

with

p - (PE(X)) 2, (3.8)

F1 (P2(x)) = | dqf(tl)P2(x ). (3.9)

d - 1

Using equations (2.18)-(2.20) and evaluating the integral, we find

(4r 2 - 1)A - (1 + 2r2)r (P2(x))pr - 2(1 -- r2)(r + A) ' (3.10a)

(2 + r 2 ) - ( 4 - re)B <PE(X))~ = 2(1 -- rE)(1 + B) ' (3.10b)

with A and B defined by equation (2.20). The dependence of the coefficient p on the axial ratio r, as given by equations (3.8) and (3.10), is shown in figure 3 for prolates


972 B. Halle

Figure 3.

.8

.6

.4

.2

I I I I

.2 .4 ,6 .8

I I I I

PROLATE

.8 .6 .4 ,2 0

The coefficient p in the spectral density for fast surface diffusion versus the axial ratio, r, of the spheroid.

and oblates. In the spherical limit p = 0, since then surface diffusion averages the residual spin-lattice coupling to zero, so that aggregate rotation has no effect. In the cylindrical limit (prolate with r = 0) p = 1/4, corresponding to the well-known fact that surface diffusion around the cylinder axis reduces the magnitude of the spin- lattice coupling by a factor of two. In the planar limit (oblate with r = 0) surface diffusion has, of course, no averaging effect, and p = 1.

The range of validity of the limiting formulae given in this section depends on the relative magnitude of the rotational correlation times T~ and the characteristic diffusion times a2/Ds and b2/Ds . In some cases, however, the criteria of validity obtained by comparing these quantities are unnecessarily strict. As an example, consider the uniaxial growth of a prolate spheroid, for example a surfactant micelle with increasing aggregation number. If the slow surface diffusion limit is valid for the spherical aggregate, i.e. if Zo "~ b2/Ds, then the limiting formulae (3.1)-(3.3) for the zero-frequency spectral density J(0) will actually remain accurate for all aggregate sizes with the same minimum dimension b. This can be understood by noting that, for large aggregates (a >> b), only the m = 0 term in equation (3.1) contributes significantly to J(0) (since D T ,~ D L and, hence, Zo >> zx, "~2)" Although surface diffusion is no longer slow compared to transverse rotation, it modulates only a very small fraction of the residual spin-lattice coupling (since, for long prolates, (P2(x)) is close to its infinite-cylinder limit of - 1 /2 ) and therefore contributes negligibly to J(O).

4. Anisotropic systems

The N.M.R. spectrum from an anisotropic system is split into two or more peaks, the frequency separation of which depends on the anisotropic orientational distribution of the spin-bearing molecules. In the high-field approximation, the split- ting is proportional to the motionally averaged static spin-lattice coupling and, hence, to (P2(x)) (cf. equations (2.3) and (2.9)). For the case of a spheroidal aggre-


Spin relaxat ion by surface diffusion 973

gate (or a collection of similarly oriented spheroids), this quantity is given by equation (3.10).

The spin relaxation behaviour in an anisotropic system is governed by the spectral density functions

f; Jk(f.O) = dz COS ((.OZ)Gk(Z), k = 0, 1, 2, (4.1)

with the ' reduced' time autocorrelation functions

Gk(Z) = (0L*(0)0L(z))/( [ U D 12). (4.2)

0L(z), which is that part of uL(z) which fluctuates at a rate higher than the static spin-lattice coupling frequency and thus is effective in inducing spin relaxation, may be written

2

t~L(z) (-- 1) k Z 2 ~2 o = D _ km(~LR)Dmo [flRD(Z)] Uo, (4.3) m = - 2

with/)~O(~RD) defined by equation (2.8). Assuming that the orientation ~'~LR of the spheroid remains stationary on the

time-scale defined by the static spin-lattice coupling, we obtain from equations (4.2) and (4.3)

2 2

= D_km(~-~LR)D_kq(~'~LR)(Drno[~'~RD(O)']Dqo[~"~RD('C)]~R D . ( 4 . 4 ) m = - 2 q = - 2

According to a symmetry theorem derived by Wennerstr6m [7], the cylindrical symmetry around the zR axis implies that the correlation functions in (4.4) vanish unless m = q. Hence

2

Gk(Z; 0LR) = ~ [d~m(COS 0LR)]2Gm(z), (4.5) m = - - 2

with

am(T ) ~ 2 , "2 = (Dr,,O[f~RD(O)]DmO[~RD(Z)])R D . (4.6)

A comparison with equation (2.10) shows that Jm(e~) = ~ dz cos (mz)Gin(z) is given by the mth term in the sum of equation (2.29), provided that we set 1/z~ = 0 in equations (2.30) and (2.32).

On the basis of equation (4.5), it is possible to determine separately the three different spectral densities Jo(~), Jl(~o) and J2(t.o) by studying the 0LR dependence of the relaxation behaviour of a macroscopically oriented sample. In w 6 we present numerical results for Jm(og), which is equal to Jk(r 0LR = 0) and, hence, corresponds to a sample orientation where the spheroid symmetry axis is parallel to the external magnetic field.

5. Finite-difference algorithm In this section we describe a numerical algorithm, based on the finite-difference

technique as previously employed in similar problems [23, 24], for solving the ordinary differential equations (2.30) and (2.32), and thus, by way of equation (2.29), for computing the spectral density function for surface diffusion on a fixed or freely tumbling spheroid.


974 B. Halle

The q-space is discretized by introducing N equidistant grid points

q k = - - I +(k--�89 k = 1 , . . . , U , (5.1)

each point being centred on an interval of width

A = 2/U. (5.2)

The differential equations (2.30) and (2.32) may now be expressed in matrix notation a s

Here U is the N x N unit matrix, while Q~ and P," are column vectors with elements Q,,(qk, 0)) and

(P,")k = -- [1 -- (I -- rE)q2]'/2dZo(xk), (5.4 a)

(P,,)k = -- [ rz + (1 -- r2)q 2] X/Zd~o(xk), (5.4 b)

for prolates and oblates, respectively. In equation (5.4), x k is obtained from equation (2.18), with q replaced by qk.

L., in equation (5.3) is the tridiagonal N • N matrix representation of the operator s given by equations (2.26) or (2.31). This matrix may be diagonalized by a similarity transformation

D m ---- T,, L m T , ~ 1. ( 5 . 5 )

On combining equations (5.3) and (5.5), we get

Q , , , = T ~ I [ D , , , + ( i 0 ) - I ~ u ] - I T m P m , (5.6, :,~/ A

so that equation (2.29) may be written

J(0)) - PZo 1 + (0)z0) 2

2 N N N + (r + A)-1A ~ ~ ~ ~, d~o(Xi) (Tm 1)ij[(Om)JJ - - 1/'~m'](Tm)jk

m = - 2 i = 1 j = l k = l [(n,)j j - - 1/'Cm] 2 + 0) 2 (Pm)k" (5.7)

The numerical problem thus amounts to finding the eigenvalues and the eigen- vectors of the matrix L,". This has to be done only once to obtain the entire power spectrum J(0)). The desired numerical accuracy is achieved by increasing the number, N, of grid points.

We now construct the matrix representation Lm of the differential operator 0,%~ on the grid (5.1). The angular derivatives at the kth grid point of a function A(q) are approximated by central differences as follows:

d A(q) k 1 d-~ = ~-~ [A(k + 1) -- a(k - 1)], (5.8 a)

d2 A(q) k 1 dtl2 = - ~ [A(k + 1) -- 2A(k) + A(k - 1)3. (5.8 b)



Combina t i on of equat ions (2.26) and (5.8) yields for the ' i n t e r n a l ' grid points (k = 2, . . . , N - - I )

&a m A(r/) k Os(1 -- r/~ + At/k) A(k -- 1) = ( a A ~ l 7 - -0 - r2)~l~]

D s [ 2(1 -- r/2) m2 1 a 2 LA2[1 7 ( ~ __--;2)/12] + r2(l _ r/k2 )~A(k)

Os(1 -- r/k z -- Ark) + (aA---~m--(]---- r--~-~] A(k + 1). (5.9)

Equa t ion (2.31) leads to a similar expression for the oblate case. The matr ix representa t ion of the opera to r Lf,, can now be writ ten

Lm = W + B,,, (5.10)

where B m is a diagonal N x N matr ix with elements

m2Ds (B,,)u = -- 5k' b2(-~ - - }2) ' (5.11)

which holds true also for oblate spheroids. The N x N matr ix W is t r idiagonal with the following non-zero elements:

Wk, k-1 = Os(1 -- r/2 + Arlk)/d, (5.12a)

Wkk = -- 2Os(1 -- ~l~)/d, (5.12 b)

Wk, k+, = Os(1 -- qk z -- arlk)/d, (5.12 c)

where

d = (aA)2[1 - (1 - r2)~/2], (5.13 a)

d = (bA)2[r 2 + (1 - r2)1/2], (5.13 b)

for prolates and oblates, respectively. Equa t ion (5.12) is valid only for the internal grid points k = 2 . . . . . N - 1. The

remaining four elements are obta ined f rom the probabi l i ty conservat ion require- ment

N

Wk t = 0 , l = 1 . . . . . N. (5.14) k = l

Hence

~ 1 ~ - - ~ 1 ~ ~ 2 = - ~ 2 - ~ 2 ,

~ , N - 1 = - - ~ - I , N - 1 - - ~ - 2 , N - 1 ,

~ N = - ~ - I , N .

(5.15 a)

(5.15b)

(5.15c)

(5.15d)

6. Calculations

In this section, we present numerical results for the spectral density functions associated with surface diffusion on a fixed or freely tumbl ing spheroidal aggregate.


976 B. Halle

In all calculations, the spheroids are derived from a sphere of radius 2nm by deformation at constant volume. The surface diffusion coefficient is Ds = 10-11_10 9 m 2 S-1, the lower value being representative of lateral motion of typical surfactants [26] and the higher value of bulk diffusion of water and simple electro- lytes.

6.1. Isotropic systems The rotational diffusion coefficients were calculated for a rigid spheroid

immersed in an incompressible medium of viscosity t /with a perfect stick boundary condition. The result for prolates is [27, 28]

D L = K(1 -- r2S)/(1 - r2), (6.1 a)

D T = Kr2[(2 -- r2)S -- 1]/(1 - r '), (6.1 b)

with

kBT K - 4~/V" (6.2)

V is the volume of the spheroid and

S = _ ( l _ r 2 ) _ l / 2 1 n [ ! +(1 r-- r2)1/2]. (6.3)

For oblates, r should be replaced by 1/r in equation (6.1), while equation (6.3) is replaced by

S - r(1 - r2) - 1/2 arccos r. (6.4)

In all calculations, we have taken T = 298 K and q -- 8.9 x l0 -4 kg (ms)-1. Figure 4 shows the spectral density function for freely tumbling spheroids of

different axial ratios (or eccentricities) in the limit of slow surface diffusion, as given by equations (3.1)-(3.4). With increasing eccentricity the spheroid tumbles more slowly; consequently J(0) increases and the dispersion is shifted to lower frequencies. The non-lorentzian shape of the dispersion is most clearly seen for prelates, for which D E and D T differ more than for isoeccentric oblates. The sharp reduction of D T with increasing eccentricity accounts for the lower dispersion frequency for prelates. However, because of the small coefficient (see figure 2) weighting Zo = (6DT)- 1, prelates have smaller J(0) than isoeccentric oblates.

Figure 5 shows the effect of surface diffusion for freely tumbling spheroids of axial ratio r = 0.5. For a sphere, J(co) vanishes in the limit of infinitely fast surface diffusion, whereas, for infinite eccentricity (r = 0), J(~o) is identical in the limits of slow and infinitely fast surface diffusion (see figures 2 and 3). For finite eccentricities, surface diffusion reduces the value of J(0). Oblates with r = 0-5 have slightly larger J(0) than isoeccentric prelates, since surface diffusion can average out more of the residual spin-lattice coupling in the latter case. For larger eccentricities, however, prelates have larger J(0) on account of their slower tumbling. The shape of the dispersion is strictly lorentzian in the limit of infinitely fast surface diffusion and is very nearly lorentzian in the limit of slow surface diffusion (for an oblate with r = 0.5, the three rotational correlation times z,, differ by merely 15 per cent). For finite D s values the dispersion is more extended. This is seen here most clearly for



? o

- )

PROLATE

2 .1

.5

6 7 8

Log ( ~ / r a d s ~)

Figure 4.

8

v

3

2

I .5

I

0 I J

6 7 8 9

I.og (cO/tad s -1)

Spectral density function for freely tumbling spheroids in the limit of slow surface diffusion�9 The axial ratio is given by each curve.

D s = 10 -9 m 2 s-1 , where surface diffusion gives rise to an extended dispersion well above the (nearly lorentzian) ro ta t ional dispersion a round ~ = 10 8 rad s - 1.

6.2. Anisotropic systems

We now consider the spectral density functions J,,(~o) = Jk(09; 0LR = 0), given by the real pa r t of the Four i e r -Lap lace t ransform of Gin(z) in equat ion (4.6). These functions determine the spin re laxat ion induced by lateral diffusion over the surface of an or ientat ional ly fixed spheroid. In the spherical limit (r = 1), where the system is, in fact, isotropic a l though there is no aggregate rotat ion, J,,(co) is independent of


978 B. Halle

. 8

.6

if)

s ~ " . 4

v

.2

10 -11 PROLATE

1

lO -9

7 8 9

log ( (O/rad s -1)

Figure 5.

.8 ~ OBLATE 10 -11

.6

Z E, lO

.4 -)

.2 10 -9

0 I I 7 8 9

[0g (~/rad s -1)

Spectral density function for freely tumbling spheroids of axial ratio r = 0'5. The value of the surface diffusion coefficient, Ds/m 2 s - 1, is given by each curve.

m and equal to (1/5) rcl[1 + ((OZe)2"], with rr = a2/(6Ds). In the cylindrical limit (prolate with r = 0), only the diffusion a round the z R axis contributes and Jm(o~) = 6,,2(318)~/[1 + (o9z~)2], with zr = b2/(4Ds). In the planar limit (oblate with r = 0), surface diffusion does not modula te the residual spin-lattice coupling and Jm(09) = O.

Figure 6 shows the spectral density functions for spheroids of axial ratio r = 0.5 and with a surface diffusion coefficient D s = 1 x 10 - lo m 2 s - 1 The zero-frequency spectral densities, J,,(0), are inversely propor t iona l to D s. The m = 0 contribution, Jo(o~), is not affected by azimuthal diffusion (around the z R axis); it reflects only the modula t ion of the angle 0RD. On a prolate surface, 0RD is randomized more slowly than q~, whereas the reverse is true for an oblate (see figure 1). A m o n g the three different J~(og) functions, Jo(e~) therefore disperses at lowest (highest) frequency for prolates (oblates).



1.5

1.0

==

.5

1.5

I 0 2 PROLATE

7 8

tog (~/rad s -I)

Z I (D

Figure 6. oids of axial ratio r = 0.5 and with surface diffusion coefficient D s The value of rn is given by each curve.

1 OBLATE

0

1.0

2

.5

0 1 l ,

7 8 g

log (c~/rad s -I)

Spectral density functions, Jm(og) = Jk(co; 0LR = 0), for orientationally fixed spher- = I x 1 0 - 1 ~ -1 .

7. Conclusions

We have presented above a method for calculating the spectral density functions, and thus the spin relaxation rates, associated with surface diffusion on a fixed or freely tumbling spheroid. Analytical results were obtained for the limits of fast or slow surface diffusion as well as for the orientational order parameters that determine the N.M.R. line splittings from anisotropic systems composed of fixed spheroids. These results should be of value for interpreting N.M.R. data from nuclei belonging to laterally diffusing molecules or ions in micellar solutions, microemulsions and liquid crystals as well as in isotropic solutions and ordered phases of spheroidal macromolecules.


980 B. Halle

In previous work [29, 30] dealing with the effect on N.M.R. line splittings of orientational averaging by translational diffusion on the surface of finite non- spherical aggregates, only hemi-sphere capped cylinders and hemi-toroid rimmed lamellae have been considered. The spin relaxation behaviour has also been analysed for these aggregate shapes [31]. In that work, however, approximations were introduced in the dynamical description, the effect of which is difficult to assess. The present work provides exact results for a wider class of aggregate shapes.

I am grateful to Olle S6derman for inspiration and to the Swedish Natural Science Research Council for financial support. This work was not supported by any military agency.

References [1] LINDMAN, B., S()DERMAN, O., and WENNERSTROM, H., 1986, Surfactant Solutions, edited

by R. Zana (Marcel Dekker), p. 263. [2] EMSLEY, J.W. (editor), 1985, Nuclear Magnetic Resonance of Liquid Crystals (D. Reidel). [3] WENNERSTROM, H., LINDBLOM, G., and LINDMAN, B., 1974, Chem. scripta, 6, 97. [4] UKLEJA, P., PIRS, J., and DOANE, J. W, 1976, Phys. Rev. A, 14, 414. ['5] FREED, J. n., 1977, J. chem. Phys., 66, 4183. [6] WENNERSTROM, H., LINDMAN, B., SODERMAN, O., DRAKENBERG, T., and ROSENHOLM,

J. B., 1979, J. Am. chem. Soc., 101, 6860. [7] HALLE, B., and WENNERSTR6M, H., 1981, J. chem. Phys., 75, 1928. [8] HALLE, B., WENNERSTRt)M, n., and PICULELL, L., 1984, J. phys. Chem., 88, 2482. [9] SODERMAN, O., WALDERHAUG, n., HENRIKSSON, U., and STILBS, P., 1985, J. phys. Chem.,

89, 3693. [10] S()DERMAN, 0., HENRIKSSON, U., J. chem. Soc. Faraday Trans. I (in the press). [11] NOACK, F., 1986, Prog. N.M.R. Spectrosc., 18, 171. [12] WENNERSTR6M, H., and LINDMAN, B., 1979, Phys. Rep., 52, 1. [13] BIRDI, K. S., 1985, Pro9. colloid polym. Sci., 70, 23. [14] BERR, S. S., CAPONETTI, E., JOHNSON, J. S., JONES, R. R. M., and MAGID, L. J., 1986, d.

phys. Chem., 90, 5766. [15] NORTH, A. N., DORE, J. C., McDONALD, J. A., ROBINSON, B. H., HEENAN, R. K., and

HOWE, A. M., 1986, Colloids Surf, 19, 21. [16] FONTELL, K., FOX, K. K., and HANSSON, E., 1985, Molec. Crystals liq. Crystals, 1, 9. [17] FORREST, B. J., and REEVES, L. W., 1981, Chem. Rev., 81, 1. [18] HUBBARD, P. S., 1969, Phys. Rev., 180, 319. [19] BRINK, D. M., and SATCHLER, G. R., 1968, Angular Momentum, 2rid edition (Clarendon

Press). [20] BERNE, B. J., and PECORA, R., 1976, Dynamic Light Scatterino (Wiley). [21] NIVEN, C., 1880, Phil. Trans. R. Soc., 171, 117. [22] FLAMMER, C., 1957, Spheroidal Wave Functions (Stanford University Press). [23] HWANG, L.-P., and FREED, J. H., 1975, J. chem. Phys., 63, 4017. [24] HALLE, B., 1987, Molec. Phys., 60, 319. [25] WOESSNER, D. E., 1962, J. chem. Phys., 37, 647. [26] LINDBLOM, G., and WENNERSTROM, H., 1977, Biophys. Chem., 6, 167. [27] EDWARDES, D., 1892, Q. Jl pure appl. Math., 26,70. [28] PERRIN, F., 1934, d. Phys. Radium, Paris, 5, 497. [29] CHARVOLIN, J., and HENDRIKX~ Y., 1985, Nuclear Magnetic Resonance of Liquid Crystals

(D. Reidel), p. 449. [30] ERIKSSON, P.-O., LINDBLOM, G., and ARVIDSON, G., 1985, J. phys. Chem., 89, 1050. [31] ZUMER, S., and VILEAN, M., 1985, J. Phys., Paris, 46, 1763.


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