J . Phys. Chem. 1985,89, 997-1001
997
Dynamical Behavior of Solutes In Smectics Giorgio Moro and Pier Luigi Nordio* Institute of Physical Chemistry, The University, Padua, Italy (Received: October 19, 1984)
The full diffusion equation is solved for solutes in smectic phases, subjected to McMillan-type potentials. We then discuss the effects of the roto-translational coupling induced by the potential on the diffusion parameters measured by field gradient NMR and on the spectral density functions inferred from magnetic resonance relaxation experiments.
I. Introduction When compared with the large number of papers published on the dynamical behavior of molecules in nematic liquid crystals, the amount of work devoted to smectic phases appears surprisingly little. Among the most detailed and informative of them, some concern neutron scattering experiment^,'-^ some other the relaxation behavior of spin probes in ESRS and NMR! respectively, and a few more the diffusion of small solutes parallel and perpendicular to the smectic planes.'J Apart from the ascertainment that the translations across the planes are hindered and characterized by high activation energies, all other motional features are considered to be essentially the same as in nematics. A little thought shows, however, that this simple picture cannot be completely satisfactory. The smectic layers are almost invariably composed of alternating regions of rigid cores and relatively flexible chains. Therefore, the phase is characterized not only by density waves but also by a nonuniform orientational order, this being expected to be higher near the mesogen cores than in the chain region. Therefore, the motion of particles traveling normally to the layers will have the effect of modulating the intensity of the orientational pseudopotential acting on them and modifying continuously the characteristics of the rotational motions, which are very sensitive to the presence of external torques. In conclusion, a new mechanism of roto-translational coupling must be operative in the smectic phases. In fact, this is included in the common form of McMillan potential^,^-'^ which are not factorizable into the angular and spatial variables. In a previous paper," we derived an exact expression for the diffusion parameters characterizing the translational motions of spherical solutes in uniaxial phases. In that treatment, the coupling with the rotations was avoided because of the symmetry of the problem. An account of the generalization to molecules of arbitrary shape was also given,12again directing the attention only to translational diffusion. W e shall consider here the general problem of translational and rotational diffusion of a probe molecule in spatially nonuniform smectic phases, with particular emphasis on the effects of such motions on physical properties which are commonly inferred from magnetic resonance experiments. One of these is certainly the diffusion coefficient, determined by field gradient NMR. Other relevant quantities are the rotational correlation times, responsible for the spin-relaxation behavior. We shall refer here to 2H or I3CN M R relaxation or ESR line shapes of typical paramagnetic probes, which are ex(1) Dianoux, A. J.; Heidemann, A.; Volino, F.; Hervet, N. Mol. Phys. 1976, 32, 1521. (2) Volino, F.; Dianoux, A. J. Mol. Phys. 1978, 36, 389. (3) Richardson. R. M.: Leadbetter. A. J.: Bonsor. D. H.: KrUaer. - G. J. Md.'Phys. 1980, 40, 741. (4) Richardson, R. M.; Leadbetter, A. J.; Frost, J. C. Mol. Phys. 1982, 45, 1163. (5) Lin, W. L.; Freed, J. H. J . Phys. Chem. 1979, 83, 379. (6) Selwyn, L. S.;Vold, R. L.; Vold, R. R. J. Chem. Phys. 1984,80, 5418. (7) Kriiger, G. J. Phys. Rep. 1980, 82, 230. (8) Mweley, M. E.; Loewenstein, A. Mol. Cryst. Liq. Cryst. 1982, 90,117; 1983, 95, 5 1. (9) McMillan, W. L. Phys. Rev.A 1971, 4, 1238; 1972, 6 , 936. (10) Humphries, R. L.; Luckhurst, G. R. Mol. Phys. 1978, 35, 1201. (1 1) Moro, G.;Nordio, P. L.; Segre, U. Chem. Phys. Lett. 1984,105,440. (12) Moro, G.; Nordio, P. L.; Segre, U. Mol. Cryst. Liq. Cryst., in press.
pressible in terms of single-particle purely rotational correlation functions. Even so, the roto-translational coupling due to the nonuniform spatial structure of the smectic phase gives rise to particular effects absent in the normal nematic phases, except perhaps in those which are undergoing nematicsmectic transitions. 11. Theory As already pointed out in the precedent paper," the translational diffusion parameter D(u) which is actually measured along the arbitrary laboratory direction u, from pulsed field gradient N M R spin-echo decays, is defined in terms of the particle di~placement'~ Ar(t) = r ( t ) - r(0) as
D(u) = lim( lu.Ar(t)12)/ 2 t
(1)
I--
This definition pertains to the N M R experiments due to their long time scales. In fact, during a millisecond pulse a small probe molecule with a typical diffusion constant of lod cm2 s-l is allowed to travel a distance equal perhaps to 1000 molecular lengths; therefore, the N M R experiments probe motions across many molecular layers, which imply the passage over potential barriers. The rotational behavior of the molecules is fully described in terms of correlation functions, on the basis of the complete set spanning the Euler angle space, of the Wigner functions Dpq(pP(")
n2
(cufly):
qqw= ( q , ( t ) * q p , ( o ) ) = Jdr JdQ 13L$,~*(n) exp(-kt)SD$Q)
P(r,Q)
(2)
In this expression, 6f& denotes the deviationJrom the thermal average (Dpq).The time evolution operator R is chosen as the roto-translational diffusion ~ p e r a t o r ' ~for * ' ~particles subjected to the anisotropic pseudopotential
V ( r , Q ) / k T = -In P(r$)
(3)
P(r,n) being the equilibrium distribution function satisfying the equation AP(r,n) = o
(4)
In general, P(r,Q) will be not factorizable into the spatial and angular variables, and a McMillan form9,10will be considered appropriate for the anisotropic potential. The diffusion operator is written as15 A = A R -+ A T = M . D R * P M P 1- V*DT(Q).PVP'
(5)
where P is a shorthand notation for P(r,Q),M are generators of infinitesimal rotations about the molecular axes, and V is the gradient operator referred to a laboratory frame having the z axis coincident with the mesophase director n; DR and DT are the rotational and translational diffusion tensors, which in isotropic fluids could be inferred from the molecular geometry by hydro(13) Douglas, D. C.; McCall, D. W. J . Phys. Chem. 1958, 62, 1102. (14) Hwang, L. P.;Freed, J. H. J . Chem. Phys. 1975, 63, 118. (15) Moro, G.; Segre, U.; Nordio, P. L. In 'NMR of Liquid Crystals"; Emsley, J. W., Ed.; Reidel Publishing Co.: Dordrecht, Holland, 1985.
0022-3654/85/2089-0997$01.50/0 0 1985 American Chemical Society
998
The Journal of Physical Chemistry, Vol. 89, No. 6, 1985
Mor0 and Nordio
dynamic argument^.'^,^' The orientation dependence of DTwhen expressed in a laboratory frame is evident. We observe that a term arising from hydrodynamic rototranslational (RT) coupling should also be included, but it is a simple matter to show1*that its effect is expected to be negligible in comparison with the R T coupling induced by a nonfactorizable distribution function P(r,Q). The statistical average given in eq 1 for particles undergoing rotational and translational Brownian motions can also be conveniently expressed in terms of the Laplace transform F,(k,s) of the correlation function F,(k,t): D(U) = l i m ( ~ ~ / 2 ) { - ( a / a k ) ~ Fk,s)J ,( r-0
(6)
F,(k,t) = l d r S d Q exp(-ikr.u) exp(-At) exp(ikr.u)P(r,Q) (7)
In the following, we shall consider the case of a uniaxial smectic A phase. The anisotropic potential is independent of the spatial ( X J ) variables, and after some algebra one obtains
where &k) =
RR + RTZ+ kZDT,,(Q) - ik (9)
with
a
ATZ - -DTz,(Q)P az
a
az
In eq 9 and 10, the components of DTin the laboratory frame appear according to following definitions DTzu=
X
DTzjuj
(1 l a )
e
2
.4
.6
.8
r/d
I
Figure 1. Orientational order parameter as function of z / d . The parameters A , B, and C of set I in Table I have been used.
matrices of dimension of the order of 100 are sufficient. The spectral densities Jiq(w), Fourier Laplace transforms of rotational correlation functions $&f) given by eq 2, are calculated from the continued fraction generated by the Lanczos a1g0rithm.l~ In a similar way is calculated the translational diffusion parameter D(u) since the second term in eq 12 can be written as the zerofrequency spectral density associated with the function DT,, a/az (VIk7‘). All the calculations were done on a PDP 11/24 minicomputer. An approximate but very efficient procedure consists of projecting out the “fast” rotational motions and then solving the translational problem. This approximation is expected to be valid for small solute probes, i.e. of dimension do
