C ONCLUSIONES

A subsequent model used to describe the anisotropic surface interaction between a solute molecule and a nematic host is the surface tensor approach, introduced by Ferrarini et al.²⁹⁷ In this method, the molecular surface is used as an input variable, as in Equation (5.31), but rather than considering a one-dimensional variable describing the molecular shape, such as the circumference, it considers the molecular surface as a whole, from which a tensor defining the orientations of surface elements may be unambiguously constructed irrespective of molecular symmetry or complexity. The potential energy, U, of a given structure with a defined surface, S, at an orientation defined by Euler angles β and γ within the laboratory frame, is defined by Equation (5.32), where ψ is the angle between the normal to any point on the molecular surface and the host director, ε is a parameter representing the orienting strength of the host medium, k_B is the Boltzmann constant and T is the temperature of the system. Equation (5.32) was originally derived from the standard form of surface anchoring free energy of macroscopic surfaces.²⁹⁸

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B

, cos d

U P S

k T

  



If the molecular surface is considered to comprise a set of small discrete planar surface elements, Equation (5.32) may be considered in terms of each of these surface elements tending to align parallel to the director.²⁹⁹ The variation in energy of a single vector normal to a surface element with the angle made with the director according to Equation (5.32) is shown in Figure 5.6. If a set of normal vectors to all of the surface elements is considered, then the minimum energy orientation corresponds to that which minimises the projection of these vectors along the director.

Figure 5.6 Variation in energy, U/kBT, for a vector normal to a surface element (shown above) with angle, ψ, against the director, n.

For a rod-like solute with D∞h symmetry, the energy is at a minimum when the molecular axis, in this case the symmetry axis, is parallel to the director because at this orientation the majority of the solute surface faces perpendicular to the director, resulting in P₂(cos ψ) = −0.5 for these perpendicular vectors.

For a molecular surface composed of discrete surface elements, s, with normal unit vectors, s, as described above, a surface tensor, t, may be constructed, as defined by Equation (5.33), where sx, sy, and sz are the Cartesian components of the unit vectors (i.e. the direction cosines) associated with each surface element in the molecular frame.³⁰⁰

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Equation (5.33) may be expressed in a more practical form as a summation of vectors associated with a finite number of discrete surface elements of area, a_s, given by

Diagonalising t results in three eigenvectors corresponding to three perpendicular molecular axes that define the orientations of maximum surface vectors, minimum surface vectors and an orthogonal axis.³⁰¹ The eigenvalues associated with these vectors give the projection of the surface area along these axes. For example, in the case of anthracene, the structure of which is shown in Figure 5.7, the van der Waals surface may be triangulated, and a normal vector assigned to each triangular surface element, shown in Figure 5.7.

Figure 5.7 The surface of anthracene optimised at the B3LYP/6-31g(d) level (left), the triangulated surface with randomly coloured normal unit vectors shown for each surface element (centre), and the cuboid (rotated slightly out of the plane of the page for clarity) with face areas proportional to the eigenvectors of t (right).

Continuing with the example of anthracene, the resulting diagonalised tensor of eigenvalues, obtained from the tensor, t, from Equation (5.34) in the Cartesian frame

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shown in Figure 5.7 is given by Equation (5.35). The trace of the tensor, Tr(t) = −192.3 Ų, corresponds to the negative of the total surface area, S, of the molecule.

52.0 0.0 0.0

The eigenvector associated with the minimum eigenvalue, the z axis in the case above, represents the molecular axis with the maximum perpendicular surface vectors, corresponding to the molecular alignment axis. The eigenvector associated with the maximum eigenvalue, the y axis in the case above, corresponds to the molecular axis with the minimum perpendicular vectors, and the eigenvector associated with the intermediate eigenvalue is orthogonal to the other two axes, therefore the x axis in the case above. This representation of a molecular surface is intuitive, and allows a molecule of any shape or symmetry to be considered as a cuboid with surface areas corresponding to the inverse eigenvalues of t, also shown for the example of anthracene in Figure 5.7. The alignment of the solute may therefore be treated with the simplifications associated with this symmetry, introduced in Sections 5.1.1 and 5.1.2.

The mathematical treatment of Equation (5.32) expressed in Cartesian form is expressed as Equation (5.36),^{299, 301} where the values of T^2,n are the components of the spherical surface tensor, T, that can be derived from the Cartesian surface tensor, t (see below), incorporating the integration of the molecular surface described in the molecular frame, and D²_0,2(β,γ) describes the molecular orientation in the lab frame. The derivation of Equation (5.36) from Equation (5.32) is shown in Appendix A3.3.

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The Cartesian form of the surface tensor, T, may be obtained from t according to Equation (5.37),³⁰² where 1 is the identity matrix. The derivation of Equation (5.37) is shown in Appendix A3.4.

The interpretation of T is less intuitive than that of t, but the components of T are still related to the shape of the molecule: a large positive value of Tii indicates a large surface area perpendicular to axis i, such that the molecule has a tendency to align along the axis i, whereas a large negative value corresponds to a small surface area perpendicular to axis i. For the surface of a rod-like molecule, two values will be negative and one positive, however for a “disk-like” molecule, two values will be positive and one negative. Continuing the example of anthracene above, the Cartesian form of the surface tensor, T, obtained using Equation (5.37), is given by Equation (5.38).

14.8 0.0 0.0 from t in Equation (5.37), is traceless and diagonal, the spherical components, T^0,0, T^1,m and T^2,±1, are equal to 0, and the other components may be simplified from the expressions for the general case, given in Appendix A3.5, to give Equations (5.39) and (5.40).

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In the example of anthracene above, the spherical components of T are T^2,0 = 38.7 Ų, and T^2,2 = T^2,−2 = 30.6 Ų.

Applying Euler’s formula, Equation (5.41) simplifies to yield Equation (5.42).

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The orientational distribution function, f(β,γ), may be obtained from the orientational potential energy, U(β,γ), given by Equation (5.42) using a standard Boltzmann distribution to give Equation (5.43).³⁰³ The components of the Saupe ordering matrix may then be determined from this distribution, as described in section 5.1.1.

To conclude the example of anthracene used in this section, the orientational distribution function calculated by using the surface tensor method and Equation (5.43), with an orienting strength of ε = 0.04 Å⁻² shown to be typical for nematic liquid crystals,³⁰¹ is shown in Figure 5.8. For clarity, plots of f(β) for angles of γ = 0˚ and γ = 90˚ are also shown in Figure 5.8. Using Equations (5.19), (5.15) and (5.16), the order parameters of Szz = 0.267 and Sxx – Syy = 0.329 were obtained from the calculated distribution function, and are comparable to those determined from experiment.³⁰⁴

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