Order parameters

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An order parameter is some observable physical quantity that is able to distinguish between two distinct phases. The choice of order parameter is not necessarily unique.

Solid-liquid transition[edit]

Possible choices:

  • Fourier transform of the density
  • Shear modulus

Isotropic-nematic transition[edit]

The uniaxial order parameter is zero for an isotropic fluid and one for a perfectly aligned system. First one calculates a director vector [1]

Q_{\alpha \beta}
= \frac{1}{N}
\sum_{j=1}^{N} \left( \frac{3}{2}
\hat e_{j \alpha} \hat e_{j \beta}
-\frac{1}{2} \delta_{\alpha\beta}\right),~~~~~\alpha, \beta = x, y, z,

where Q is a second rank tensor, \hat e_{j} is a unit vector along the molecular long axis, and \delta_{\alpha\beta} is the Kronecker delta. Diagonalisation of this tensor gives three eigenvalues \lambda_+, \lambda_0 and \lambda_-, and n is the eigenvector associated with the largest eigenvalue (\lambda_+). From this director vector the nematic order parameter is calculated from [2]

S_2 =\frac{d \langle \cos^2 \theta \rangle -1}{d-1}

where d is the dimensionality of the system.

i.e. in three dimensions [3]

S_2 = \lambda _{+}= \langle P_2( n \cdot e)\rangle = \langle P_2(\cos\theta )\rangle =\langle \frac{3}{2} \cos^{2} \theta - \frac{1}{2} \rangle

where S_2 is known as the uniaxial order parameter. Here P_2 is the second order Legendre polynomial, \theta is the angle between a molecular axes and the director n, and the angle brackets indicate an ensemble average.

Tetrahedral order parameter[edit]

[4]

See also[edit]

References[edit]

Related reading