Distribution functions are vital in identifying the type of phase present in a
simulation. For liquid crystals there are several spatial and orientational distribution functions which when considered collectively may uniquely characterise a phase.
One of the most fundamental distribution functions is the pair distribution function g2(r/jry)- For isotropic homogeneous systems this function depends
only on the interparticle separation Yy =
r;-
- 17, and is referred to as the radialdistribution function, denoted g(r) [11]. A definition convenient for use in computer simulations is given by Allen and Tildesley [3];
where Vol is the volume of the system.
The radial distribution function represents the probability of finding a particle a given distance away from a specified particle with respect to that same probability in an ideal gas. Thus, in a hypothetical pure solid at absolute zero for example, a plot of g(r) would consist of a series of infinitely tall vertical lines (representing the 5 functions of [11.67]), corresponding to the location of the lattice sites, in all directions with respect to a given site. In reality, however these delta functions would be broadened due to thermal excitation, into sharply
peaked Gaussians about a mean corresponding to the lattice site location [33]. The peaks in a liquid are even more diffuse and overlap because of the continuous relatively large scale motion of the particles with respect to one another. However structure should still be present in g(r) at short range representing the shells of nearest neighbours surrounding each particle. At long range this order diminishes and the radial distribution function converges to the ideal gas value of 1. This will not of course occur for the solid since the periodicity of the lattice sites will manifest itself as a continuously oscillating g(r) even at long range.
The distribution function g(r) is related to experimental structure characterisation techniques such as X-ray diffraction and neutron scattering. [12]. Like the mean square particle displacement, the g(r) can yield valuable anisotropic information about a phase when resolved into components parallel and perpendicular to the instantaneous system director h(7):
£| (l)=
(ll Z 8(r W • “M - % W • “W)\;
giO i) =
^
2 3 ( z Z s (rM x “ M - rv W x “M ) \ • p.68]The components of g(r) are important in distinguishing between an orientationally ordered phase such as the nematic phase, and a higher additionally translationally ordered phase, for example a smectic-A phase is characterised by the existence a one dimensional density wave parallel to the director. The function g|(/|) is particularly sensitive to this density wave. Additionally any evidence of structure in gj_(/l) may indicate the onset of two-dimensional
translational order, such as in a columnar liquid crystal, or it may indicate that the
system has indeed formed a genuine crystal.
A quantitative measure of angular correlations is provided by the second rank orientational correlation coefficient G2(r) [33, 52];
where P2 is the second Legendre polynomial. G2(r) shows very short ranged order in the isotropic phase of a molecular fluid, quickly decaying to zero after a few molecular separations. In an orientationally ordered phase though, G2(r) decays to a limiting value equal to the square of the second rank orientational
The calculation of an order parameter is essential in quantitatively classifying and identifying phases and phase transitions in many materials [53]. The order parameter may take many different forms. For example the magnetisation in a ferromagnetic material or the electric polarisation in a Ferroelectric [53]. The lowest category of liquid crystal, in terms of degree of symmetry breaking from the isotropic phase is the uniaxial nematic. In the nematic phase molecules tend to align themselves with a preferred direction; the nematic director
n
and the phase has point symmetry group D^h. Thus in our simulations it is convenient to identify with a single molecule potential matrix property A, say, defined as [52];[11.69]
order parameter (P2)2 (see below) [33, 52].
where uz is a unit vector parallel to the molecular symmetry axis, i.e. u!? = (0,0,1), and thus;
Ab = 00 00 00
0 0 1
p .7 i]
In space-fixed coordinates the average components of A are given by;
M-it
1=1 a' p’ [11.72]where a, p and a 1, P' range over the space-fixed (non primed) and body fixed (primed) Cartesian indices x, y and z, and the i?. are the components of the rotation matrix that rotates Ab into the space-fixed frame. The right hand side of [K72] is non-zero only when a'= P' = z, thus [11.72] reduces to;
(4 p ) = K % ) = Q xp+fsaP. [n.73]
Equation [11.73] defines the components of the ordering matrix, the so called Q tensor [33], 8 is the Kronecker 8. The rotation matrix that diagonalises Q leads
to a symmetric and traceless tensor that defines the director frame. Q has three real eigenvalues denoted X+, XQ and X_. Normally the largest eigenvalue, X+ is taken to be the value of (P2). The corresponding eigenvector yields the direction of the system director
n
in the laboratory fixed frame. The remaining eigenvalues are small and of opposite sign and correspond to the degree ofbiaxiality present in the phase. This particular method only allows 1 > {P2)> 0,
because the largest positive value, X+ is always chosen as the eigenvalue to represent the magnitude of the order parameter. The order parameter (P2) can take values 0 > (i^ )> -0 -5 however, which correspond to the principle
symmetry axes of the particles tending to lie orthogonal to the system director on average. Thus some workers have chosen to call (P2) the eigenvalue most different from the other two. There are some problems associated with this method applied to disordered phases. When the eigenvalues are small and approximately equal, a consistent choice of the eigenvalue corresponding to (P2) is difficult to make [54]. However, with small values of (P2), indicating that a
system is not orientationally ordered, the director of the phase has no meaning, and it is therefore not appropriate to identify the corresponding eigenvalue.
In later simulations involving multisite HGBLR models we have found it useful
to monitor the orientational ordering of three mutually perpendicular axes fixed in the molecule. This facilitates the identification of a phase where the principle molecular symmetry axes are found to be lying in a plane orthogonal to the director thereby eliminating the problem of consistently choosing a positive eigenvalue in assigning the director. More details are to be found in chapter V.