1.1 Theory of rotational diffusion
1.1.6 The Debye model for rotational diffusion of free particles
In this chapter we consider non-interacting spherical Brownian particles with a time-dependent orientation specified by the unit vector u(t) with spherical polar coordi-b
nates Ω = (θ, ϕ), the end of which remain on a sphere of unit radius. The vector is embedded rigidly in the particle and rotates with it. The reorientation of the particle is then equivalent to a two-dimentional random walk of the end-point of u(t)b u(0) on the surface of the unit sphere.b
Debye (1929) developed a model for reorientation processes based on the
assump-Figure 1.2: u(0)b and u(t)b are unit vectors representing the orientation angles of the symmetry axis of a cylindrically symmetric molecule at times 0 and t, respectively.
The locus of all the possible vectors u(t)b is the surface of a sphere of unit radius (a unit sphere). The reorientation of the molecule can be regarded as a trajectory on the surface of the unit sphere. A random walk trajectory gives rise to rotational diffusion.
tion that collisions are so frequent in a liquid that Brownian particle can only rotate through a very small angle before suffering a reorientation collision. Any assembly of Brownian particles initially oriented along some direction uc0 behaves such that each molecule follows a different trajectory on the surface of a unit sphere. In the beginning this assembly is represented by a cloud of points which is very intense in the direction uc0, but with time particles reorient and the cloud spreads out, finally covering the sphere uniformly. The basic assumption of the Debye theory is that
1.1. THEORY OF ROTATIONAL DIFFUSION. 29
the cloud diffuses on the surface of the unit radius. The equation that governs this motion is the diffusion equation.
∂P(r, t)
∂t =D∇2P(r, t) (1.52)
where P(r, t) is the concentration of particles at the point r = u on the surface of the unit sphere at the time t.
Because of the spherical symmetry - the points diffuse on the surface of a sphere, it is most convenient to solve equation (1.1) in spherical polar coordinates (r, θ, ϕ), where r= 1. The Laplacian∇2r in spherical polar coordinates is:
∇2r = For fixed r= 1 all derivatives with respect to r vanish and we have:
∇2r = 1 If P(u, t)d2u is the fraction of particles with orientation u in the solid angle d2u= sinθdθdφ at time t and substituting (Eq. 1.54) into (Eq. 1.52) we get rotational diffusion equation (Debye equation), which is a special type of the Smoluchowski equation: where Θ is the rotational diffusion coefficient.
The differential operator in Eq.(1.55)is the angular momentum operator of quantum mechanics. So, the rotational diffusion equation can be written as
∂P(u, t)
∂t =−ΘIb2P(u, t) (1.56)
Since spherical harmonics Ylm(θ, φ) ≡Ylm(u) are eigenfunctions of Ib2 and Ibz corre-sponding to the eigenvalues l(l+ 1) and ml respectively we have
Ib2Ylm(u) =l(l+ 1)Ylm(u) l = 0,1,2, ...∞ (1.57) IbzYlm(u) = mlYlm(u) ml =−l, ..,0, ...+l
These functions form a complete orthogonal set spanning the space of functions of where Ib2 is an operator acting only on u. The particular solution of Eq.(1.56) subject to the initial condition Equation (1.62) can be written as
P(u,0) =X
lm
exp(−l(l+ 1)Θt)Ylm(u0)Ylm∗ (u) (1.63) This particular solution of the diffusion equation can be interpreted as the transition probability for a particle to have orientationuat timet given that it had orientation u0 initially. Let us take
Ks(u, t|u0,0) =X
lm
Ylm(u0)Ylm∗ (u)exp(−l(l+ 1)Θt) (1.64) where Ks is a transition probability
Ks(u, t|u0,0) =Y00∗(u)Y00(u0) = 1
4π (1.65)
This means that the particles eventually become uniformly distributed on the sur-face of the unit sphere.
The correlation functions required in light scattering are of the formDYl∗′m′(u(0)Ylm(u(t))E.
1.1. THEORY OF ROTATIONAL DIFFUSION. 31
They can be written as
DYl∗′m′(u(0)Ylm(u(t))E=
Z
d2u0
Z
d2uYlm(u)Gs(u, t;u0,0)Yl∗′m′(u0) (1.66) where Gs(u, t;u0,0)d2u0d2u is the joint probability of finding a particle with orien-tation u0 ind2u0 initially and u in d2u at time t. Gs can be expressed in terms of Ks and the probability distribution function p(u0) of the initial orientation as
Gs(u, t;u0,0) = Ks(u, t|u0,0)p(u0) (1.67) In an equilibrium ensemble of particles is in uniform distribution and p(u0) = 1/4π.
Combining this with Eq.(1.70), Eq.(1.49) and Eq.(1.50) we obtain:
DYl∗′m′(u(0)Ylm(u(t))E=Fl(t)δl,l′δm,m′ (1.68) with
Fl(t) = 1
4πexp(−l(l+ 1)Θt) (1.69)
The required correlation functions are
Fmm(2)′ =F2(t)δm,m′ = 1
4πexp(−6Θtδm,m′) (1.70) In previous chapter we foundFs(q, t) = exp−q2Dtfor translational diffusion. Com-bining this equation with Eq.(1.70), Eq.(1.49) and Eq.(1.50) we obtain combined rotational and translational diffusion
IV Vα (q, t) =hNi
½
α2+ 4
45β2(exp−6Θt)
¾
exp(−q2Dt) (1.71) IV Hα (q, t) = 1
15hNiβ2exp(−6Θt)exp(−q2Dt) (1.72)
Chapter 2
Magnetic and optical properties of liquid crystalline colloidal particles
2.1 Liquid crystals
Figure 2.1: Schematic illustration of the solid, liquid crystal, and the liquid phases.
The elliptical shapes represent molecules.
Liquid crystal is a special phase of some organic substances, whose mechanical properties and the symmetry properties are intermediate between those of a liquid and those of a crystal. The classification of mesophases is essentially based on their symmetry. There are four major classes: nematics, smectics, cholesterics and colum-nar phases. Nematic liquid crystals consist of rod-like molecules which tend to align with their axis in one direction. The main properties of nematics are as follows.
1. The centers of gravity of the molecules have no long-range order. The correlations in the positions between the centers of gravity of neighbouring molecules are similar to those existing in a conventional liquid. In fact, nematics do flow like liquids.
33
2. There is some order in the direction of the molecules; they tend to be parallel to some common axis, labelled by a unit vector (”director”) n. This is reflected in all macroscopic tensor properties: for instance, optically a nematic is a uniaxial medium with the optical axis along n. The difference between refractive indices measured with polarization parallel or normal to n is large: 0.2.
3.The direction of n is arbitrary in space. In practice it is imposed by the minor forces (such as the guiding effect of the walls of the container). This is a situation of the broken rotational symmetry.
4. The states of director n and -n are indistinguishable.
5. Nematic phases occur only with materials that do not distinguish between right and left.