Nematic Liquid Crystals with Tensorial Order Parameters

Abstract

This chapter goes beyond the ELP theory of LCs by modeling the microstructure of the liquid by a number of rank-i tensors $(i=1, \ldots ,n)$ (generally just one) with vanishing trace. These tensors are called alignment tensors or order parameters. When formed as exterior products of the director vector and weighted with a scalar and restricted to just one rank-2 tensor, the resulting mathematical model describes uniaxial LCs. The simplest extensions of the ELP model are theories, for which the number of independent constitutive variables are complemented by a constant or variable order parameter S and its gradient $\mathrm {grad}\,S$ paired with an evolution equation for it. We provide a review of the recent literature. Two different approaches to deduce LC models exist; they may be coined the balance equations models, outlined already in Chap. 25 for the ELP model, and the variational Lagrange potential models, which, following an idea by Lord Rayleigh (Strutt, Proc Lond Math Soc 4:357–368, 1873, [50]), are extended by a dissipation potential. The two different approaches may lead to distinct anisotropic fluid descriptions. Moreover, it is not automatically guaranteed in either description that the balance law of angular momentum is identically satisfied. The answers to these questions cover an important part of the mathematical efforts in both model classes. Significant conceptual difficulties in the two distinct theoretical concepts are the postulations of explicit forms of the elastic energy W and dissipation function R. Depending upon, how W and R are parametrized, different particular models emerge. Conditions are formulated especially for uniaxial models, which guarantee that the two model classes reduce to exactly corresponding mathematical models.

Appendices

Appendix 26.A   Lagrange Equations

26.1.1 26.A.1 Constraints of Coordinates

Consider a mechanical system (e.g., of mass points ${\mathcal P}_{i}$ $i=1,2, \ldots ,n$) whose motion can be determined by prescribing their coordinates $\{{\varvec{x}}_{1}, {\varvec{x}}_{2}, \ldots , {\varvec{x}}_{n} \}$ see Fig.  26.2 . The degree of freedom of a mechanical system is the number of independent coordinates needed to describe the motion of the system. Constraints or constraint conditions are equations between coordinates, which restrain the motion of the system. Rigid point systems with constants of the form

$$\begin{aligned} \forall i, j \quad {\varvec{x}}_{i}-{\varvec{x}}_{j} = {\varvec{r}}_{ij}= \mathrm {const.} \Longleftrightarrow {\varvec{x}}_{i}-{\varvec{x}}_{j}- {\varvec{r}}_{ij} = 0, \quad \text{ where } |{\varvec{r}}_{ij}|=\text{ const. } \end{aligned}$$

express rigid body motions. They have six degrees of freedom, three for translation and three for rotation. In analytical dynamics, one differentiates between two kinds of constraint conditions.

(i) Constraint conditions, which are expressible as equations between coordinates, are called holonomic.¹³

(ii) If constraints are not expressible in holonomic form, they are called anholonomic or skleronomic . If such a constraint condition is not explicitly expressible as a function also of time it is called skleronomic, else rheonomic .

Fig. 26.2

System of n mass points in ${\mathbb R}^{3}$ with positions ${\varvec{x}}_{i}$ $i=1, 2, \ldots ,n$ and distances $r_{ij}$

26.1.2 26.A.2 Generalized Coordinates

When holonomic constraints exist between the coordinates $\{{\varvec{x}}_{1}, {\varvec{x}}_{2}, \ldots , {\varvec{x}}_{n}\}$, these constraint conditions will reduce the degree of freedom. If these $\varLambda$ equations are independent of one another, the degree of freedom f will be $f = 3n - \varLambda$. In this case, the mechanical system can be described by the so-called generalized coordinates

$$\begin{aligned} q_{1}, q_{2}, \ldots , q_{f}. \end{aligned}$$

(26.192)

In other words, these coordinates determine the values of $\{{\varvec{x}}_{1}, \ldots ,{\varvec{x}}_{n}\}$ uniquely as functions of time as follows:

$$\begin{aligned}&{\varvec{x}}_{1} = {\varvec{x}}_{1}\left( q_{1}, q_{2}, \ldots ,q_{f}, t\right) , \nonumber \\&{\varvec{x}}_{2} = {\varvec{x}}_{2}\left( q_{1}, q_{2}, \ldots ,q_{f}, t\right) , \nonumber \\&\vdots \quad \vdots \quad \vdots \nonumber \\&{\varvec{x}}_{n} = {\varvec{x}}_{n}\left( q_{1}, q_{2}, \ldots ,q_{f}, t\right) . \end{aligned}$$

(26.193)

These are 3n transformation equations between the dependent coordinates $\{x_{i}, y_{i}, z_{i}\}$ $i=1, 2, \ldots , n$ and the independent generalized coordinates $\{q_{1}, \ldots ,q_{f}\}$. Their total time derivatives $\{\dot{q}_{1}, \ldots , \dot{q}_{f}\}$ are called generalized velocities.

26.1.3 26.A.3 d’Alembert’s Principle, Principle of Virtual Work

A virtual displacement of $\{{\varvec{x}}_{i}$ $i=1, \ldots ,n\}$ of a mechanical system is a set $\{\delta {\varvec{x}}_{i}$ $i=1, \ldots ,n\}$ of instantaneous infinitesimal changes of the positions $\{{\varvec{x}}_{i}$ $i=1, \ldots ,n\}$ which are consistent with the existing forces and constraints. Here, the qualification “instantaneous” wants to emphasize that the displacement is performed, while the time is held fixed; this displacement is called “consistent with the applied forces and with the constraints,” because these displacements are kinematically force-freely admissible. Often, one also speaks of virtual velocities. This then leads to the Principle of Virtual Power . To derive it, let us start with the momentum equation in the form

$$\begin{aligned} {\varvec{F}}_{i} = \dot{\varvec{p}}_{i}, \quad (i=1, \ldots , n), \end{aligned}$$

(26.194)

in which ${\varvec{F}}_{i}$ are the external and internal forces and $p_{i}$ is the momentum corresponding to ${\varvec{F}}_{i}$. Multiplying both sides of (26.194) scalarly with $\delta {\varvec{x}}_{i}$ and summation over all indices $1, \ldots ,n$ yields

$$\begin{aligned} \sum _{i=1}^{n}\left( {\varvec{F}}_{i} - \dot{\varvec{p}}_{i}\right) \cdot \delta {\varvec{x}}_{i} = 0. \end{aligned}$$

(26.195)

In general, ${\varvec{F}}_{i}$ is the sum of the applied force ${\varvec{F}}_{i}^{(a)}$ and the constraint force ${\varvec{F}}_{i}^{(c)}$

$$\begin{aligned} {\varvec{F}}_{i} = {\varvec{F}}_{i}^{(a)} + {\varvec{F}}_{i}^{(c)} . \end{aligned}$$

(26.196)

The decisive additional assumption for the elimination of the constraint forces from the problem is the postulate that the virtual work of the constraint forces is null,

$$\begin{aligned} \sum _{i=1}^{n} {\varvec{F}}_{i}^{(c)} \delta {\varvec{x}}_{i} = 0 . \end{aligned}$$

(26.197)

This equation is the expression of the Principle of Virtual Work , or when expressed in virtual velocities the Principle of Virtual Power .

Combining (26.195) with (26.196), (26.197) yields

$$\begin{aligned} \sum _{i=1}^{n}\left( {\varvec{F}}_{i}^{(a)}-\dot{\varvec{p}}_{i}\right) \cdot \delta {\varvec{x}}_{i} = 0. \end{aligned}$$

(26.198)

This is known as d’Alembert’s Principle .

26.1.4 26.A.4 Derivation of the Lagrange Equations

Consider now virtual displacements of the generalized coordinates $\delta {\varvec{q}}_{i}$ $(i=1, \ldots ,f)$. If the functions (26.193) are differentiable, which we will assume, we may write

$$\begin{aligned} \delta {\varvec{x}}_{i}=\sum _{i=1}^{f}\frac{\partial {\varvec{x}}_{i}}{\partial q_{j}} \delta q_{j}. \end{aligned}$$

(26.199)

A differentiation with respect to time is missing in this expression because the virtual displacements are instantaneously performed. Substitution of (26.199) into d’Alembert’s Principle yields with $\dot{{\varvec{p}}}_{i}= m_{i} \ddot{{\varvec{x}}}_{i}$

$$\begin{aligned} \sum _{i,j} {\varvec{F}}_{i} \cdot \frac{\partial {\varvec{x}}_{i}}{\partial q_{j}} \delta q_{j} - \sum _{i,j}m_{i} \ddot{{\varvec{x}}}_{i} \cdot \frac{\partial {\varvec{x}}_{i}}{\partial q_{j}} \delta q_{j} = 0. \end{aligned}$$

(26.200)

Remarks:

• The quantity

  $$\begin{aligned} Q_{j} : = \sum _{i=1}^{n} {\varvec{F}}_{i} \cdot \frac{\partial {\varvec{x}}_{i}}{\partial q_{j}} \quad (j=1, \ldots ,f) \end{aligned}$$

  (26.201)

  is known as jth generalized force .

• The reader can easily verify the following formulae:

  • (1)
    $$\begin{aligned} {\varvec{v}}_{i} = \dot{\varvec{x}}_{i} = \sum _{j=1}^{f}\frac{\partial {\varvec{x}}_{i}}{\partial q_{j}} \dot{q}_{j} + \frac{\partial {\varvec{x}}_{i}}{\partial t}\quad \Longrightarrow \quad \frac{\partial {\varvec{v}}_{i}}{\partial \dot{q}_{j} }= \frac{\partial {\varvec{x}}_{i}}{\partial q_{j}}, \end{aligned}$$

    (26.202)
  • (2)
    $$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d} t}\left( \frac{\partial {\varvec{x}}_{i}}{\partial q_{j}}\right)= & {} \sum _{k}\frac{\partial ^{2} {\varvec{x}}_{i}}{\partial q_{j} \partial q_{k}} \dot{q}_{k} + \frac{\partial ^{2}{\varvec{x}}_{i}}{\partial q_{j} \partial t} \nonumber \\= & {} \frac{\partial }{\partial q_{j}}\left( \sum _{k}\frac{\partial {\varvec{x}}_{i}}{\partial q_{k}} \dot{q}_{k} + \frac{\partial {\varvec{x}}_{i}}{\partial t}\right) =\frac{\partial {\varvec{v}}_{i}}{\partial q_{j}} , \end{aligned}$$

    (26.203)
  • (3)
    $$\begin{aligned}&\sum _{ij}m\ddot{{\varvec{x}}}_{i}\cdot \frac{\partial {\varvec{x}}_{i}}{\partial q_{j}} \delta q_{j} \nonumber \\= & {} \sum _{i,j}\left\{ \frac{\mathrm {d}}{\mathrm {d}t}\left( \frac{\partial }{\partial \dot{q}_{j}} \frac{m_{i} {\varvec{v}}_{i}\cdot {\varvec{v}}_{i}}{2} \right) - \frac{\partial }{\partial q_{j}} \left( \frac{m_{i} {\varvec{v}}_{i}\cdot {\varvec{v}}_{i}}{2}\right) \right\} \delta {q}_{j} \nonumber \\= & {} \sum _{j}\left\{ \frac{\mathrm {d}}{\mathrm {d}t}\left( \frac{\partial }{\partial \dot{q}_{j}} \frac{\sum _{i} m_{i}{\varvec{v}}_{i}\cdot {\varvec{v}}_{i}}{2}\right) - \frac{\partial }{\partial q_{j}} \left( \frac{\sum _{i}m_{i} {\varvec{v}}_{i} \cdot {\varvec{v}}_{i}}{2}\right) \right\} \delta {q}_{j} \nonumber \\= & {} \sum _{j}\left\{ \frac{\mathrm {d}}{\mathrm {d}t}\left( \frac{\partial T}{\partial \dot{q}_{j}}\right) - \frac{\partial T}{\partial q_{j}}\right\} \delta q_{j} , \end{aligned}$$

    (26.204)

    where T is the kinetic energy.

Substituting (26.201)–(26.204) into (26.200) d’Alembert’s Principle leads to

$$\begin{aligned} \sum _{j}\left\{ \frac{\mathrm {d}}{\mathrm {d}t}\left( \frac{\partial T}{\partial \dot{q}_{j}}\right) - \frac{\partial T}{\partial q_{j}} - Q_{j}\right\} \delta q_{j} = 0, \quad \forall \delta q_{j}. \end{aligned}$$

(26.205)

Consequently,

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\left( \frac{\partial T}{\partial \dot{q}_{j}}\right) - \frac{\partial T}{\partial q_{j}} = Q_{j} \quad (j = 1, \ldots , f). \end{aligned}$$

(26.206)

These equations are sometimes called the Lagrange equations. Regularly, this denotation is, however, only used, if the forces ${\varvec{F}}_{i}$ are derivable from a potential $V(q_{1}, \ldots , q_{f},t) = V({\varvec{x}}_{1}, {\varvec{x}}_{2}, \ldots , {\varvec{x}}_{n})$ according to

$$\begin{aligned} {\varvec{F}}_{i} = -\mathrm {grad}\,_{i}(V). \end{aligned}$$

(26.207)

The generalized forces can then be written as

$$\begin{aligned} Q_{j} = - \sum _{i}\left( \mathrm {grad}\,_{i} V \right) \cdot \frac{\partial {\varvec{x}}_{i}}{\partial q_{j}} = - \frac{\partial V}{\partial {\varvec{x}}_{i}}\cdot \frac{\partial {\varvec{x}}_{i}}{\partial q_{j}} = -\frac{\partial V}{\partial q_{j}}. \end{aligned}$$

(26.208)

Introducing the Lagrange function

$$\begin{aligned} L = T - V \end{aligned}$$

(26.209)

and using (26.208) and the fact that V does not depend upon $\dot{q}_{i}$ we deduce from (26.205)

$$\begin{aligned} \left( \frac{\mathrm {d}}{\mathrm {d} t}\left( \frac{\partial L}{\partial \dot{q}_{j}}\right) - \frac{\partial L}{\partial q_{j}}\right) \delta q_j = 0, \quad \forall \delta q_j, \end{aligned}$$

or

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d} t}\left( \frac{\partial L}{\partial \dot{q}_{j}}\right) - \frac{\partial L}{\partial q_{j}} = 0, \quad (j=1, \ldots ,f), \end{aligned}$$

(26.210)

which is equivalent to (26.206).

Example: Consider a pearl with mass m tied to a vertical circular wire, which rotates with constant angular velocity $\omega$ around a vertical axis, see Fig.  26.3 . The pearl can freely move along the wire. Let the radius of the circle of the wire be R and let the gravity vector in the vertical plane be ${\varvec{g}}$ positive downward. The motion of the pearl can be described by the angle $\varphi$. So, this angle serves as the only generalized coordinate. For the pearl, treated as a mass point, we have

$$\begin{aligned} {\varvec{v}}= & {} R\dot{\varphi }{\varvec{e}}_{\varphi } + r\omega {\varvec{e}}_{\perp } = R\dot{\varphi }{\varvec{e}}_{\varphi }+R\omega \cos _{\varphi }{\varvec{e}}_{\perp },\\ T= & {} \frac{m}{2}R^{2}\dot{\varphi }^{2}+\frac{m}{2}R^{2}\omega ^{2}\cos ^{2}\varphi ,\\ V= & {} mgz = mgR\sin {\varphi },\\ {\mathcal L}= & {} T - V = \frac{m}{2}R^{2} \left( \dot{\varphi }^{2}+\omega ^{2}\cos ^{2}\varphi \right) - mgR\sin \varphi . \end{aligned}$$

Fig. 26.3

Pearl moving on a permanently rotating circular wire

Because the system has only one degree of freedom, we have only a single Lagrange equation:

$$\begin{aligned} \frac{\partial {\mathcal L}}{\partial \dot{\varphi }}= & {} mR^{2}\dot{\varphi }, \quad \frac{\mathrm {d}}{\mathrm {d}t}\left( \frac{\partial {\mathcal L}}{\partial \dot{\varphi }}\right) = mR^{2} \ddot{\varphi },\\ \frac{\partial {\mathcal L}}{\partial \varphi }= & {} - mR^{2}\omega ^{2} \sin \varphi \cos \varphi - mgR \cos \varphi \\= & {} - \frac{m}{2}R^{2}\omega ^{2} \sin (2\varphi ) - mgR\cos \varphi . \end{aligned}$$

Substituting these expressions into the Lagrange Equation (26.210) yields

$$\begin{aligned}&mR^{2}\ddot{\varphi }+\frac{m}{2}R^{2}\omega ^{2} \sin (2\varphi )+mgR\cos \varphi = 0,\\ \Longrightarrow&\ddot{\varphi }+\frac{\omega ^{2}}{2}\sin (2\varphi ) + \frac{g}{R} \cos \varphi = 0,\\ \Longrightarrow&\underbrace{ \ddot{\varphi }\dot{\varphi }}_{(\dot{\varphi }^{2}/2)^{\displaystyle \cdot }} + \frac{\omega ^{2}}{2}\underbrace{ \sin (2\varphi )\dot{\varphi } }_{-{\textstyle \frac{1}{2}}[\cos (2\varphi )]^{\displaystyle \cdot }} + \frac{g}{R}\underbrace{\cos (\varphi ) \dot{\varphi }}_{(\sin \varphi )^{\displaystyle \cdot }} = 0 \\ \Longrightarrow&\left\{ \frac{\dot{\varphi }^{2}}{2} - \frac{\omega ^{2}}{4}\cos (2\varphi ) + \frac{g}{R} \sin \varphi \right\} ^{\displaystyle \cdot } = 0, \\&\frac{\dot{\varphi }^{2}}{2} - \frac{\omega ^{2}}{4} \cos (2\varphi ) + \frac{g}{R} \sin \phi = \mathrm {const.} = c, \\ \Longrightarrow&t - t_{0} = \int _{\varphi _{0}}^{\varphi } \frac{\mathrm {d}\bar{\varphi }}{\left[ c+\textstyle {\frac{\omega ^{2}}{2}}\cos (2\bar{\varphi })-\textstyle {\frac{g}{R}} \sin {\bar{\varphi }}\right] ^{1/2}}, \end{aligned}$$

which is the solution of the equation of motion.

Appendix 26.B   Implications of the Frame Indifference Requirement of the Free Energy as a Function of Tensorial Order Parameters

Let W be the free energy. In the main text, it is assumed to be a function of the director ${\varvec{n}}$ and its gradient, $\mathrm {grad}\,{\varvec{n}}$. Satisfaction of the frame indifference requirement has been expressed as the statement (26.45) or (26.46). If W depends on a set of rank-i tensors $(i=2, \ldots ,n)$ its frame indifference is expressed as Eq. (26.44). Here, we begin with a W-function that depends only on a rank-2 tensor and its gradient: $W = W(\pmb {\mathbb O}, \mathrm {grad}\,\pmb {\mathbb O})$. Invariance of W under Euclidian transformations (rigid body motions) then implies

$$\begin{aligned} DW = W\left( {\mathbb O}_{i^{*}j^{*}}, {\mathbb O}_{i^{*}j^{*}, k^{*}}\right) - W\left( {\mathbb O}_{ij}, {\mathbb O}_{ij,k}\right) {\mathop {=}\limits ^{!}} 0, \end{aligned}$$

(26.211)

in which

$$\begin{aligned} {\varvec{x}}^{*}={\varvec{R}}{\varvec{x}} + {\varvec{b}}^{*}, \quad x^{*}_{k^{*}} = R_{k^{*} k}x_{k} + b^{*}_{k^{*}}, \quad x_{k} = (x^{*}_{k^{*}} - b^{*}_{k^{*}})R_{k^{*}k}.\qquad \end{aligned}$$

(26.212)

${\varvec{R}}$ is an orthogonal transformation [${\varvec{R}}{\varvec{R}}^{T}={\varvec{I}}$]. Cartesian tensor notation is used with indices $i, j, k, \ldots$ and $i^{*}, j^{*}, k^{*}, \ldots$ in the original and in the rotated coordinates, respectively. ${\varvec{x}}^{*}$ is the position of the point ${\varvec{x}}$ measured in the rotated coordinates given by ${\varvec{R}}$ and translation by ${\varvec{b}}^{*}$. Below we shall restrict attention to infinitesimal rotations,

$$\begin{aligned} R_{i^{*}m}= \delta _{i^{*}m}+\varOmega _{i^{*}m}, \quad \Vert \varOmega _{i^{*}m}\varOmega _{j^{*}m}\Vert \ll 1, \end{aligned}$$

(26.213)

for which only linear terms in ${\varvec{\varOmega }}$ are accounted for. It is then easily shown that

$$\begin{aligned} \delta _{i^{*}j^{*}}\equiv & {} R_{i^{*}m} R_{j^{*}m} = (\delta _{i^{*}m}+\varOmega _{i^{*}m})(\delta _{j^{*}m} +\varOmega _{j^{*}m}) \end{aligned}$$

(26.214)

$$\begin{aligned}\approx & {} \delta _{i^{*} j^{*}} + \varOmega _{i^{*} j^{*}} + \varOmega _{j^{*} i^{*}}\nonumber \\ \Longrightarrow&\varOmega _{i^{*}j^{*}} = - \varOmega _{j^{*}i^{*}}. \end{aligned}$$

(26.215)

Our next step is evaluation of $\pmb {\mathbb O}$ and $\mathrm {grad}\,\pmb {\mathbb O}$ in the rotated coordinates:

• $$\begin{aligned} {\mathbb O}_{i^{*}j^{*}}= & {} R_{i^{*}m} R_{j^{*}m} {\mathbb O}_{mn} = (\delta _{i^{*}m}+\varOmega _{i^{*}m})(\delta _{j^{*}m}+\varOmega _{j^{*}m}) {\mathbb O}_{mn}\nonumber \\= & {} \left( \delta _{i^{*}m}\delta _{j^{*}m}+\delta _{i^{*}m}\varOmega _{j^{*}n}+ \delta _{j^{*}n}\varOmega _{i^{*}m}\right) {\mathbb O}_{mn} \nonumber \\= & {} {\mathbb O}_{i^{*}j^{*}}+{\mathbb O}_{i^{*}n}\varOmega _{j^{*}n} +{\mathbb O}_{m j^{*}}\varOmega _{i^{*}m}, \end{aligned}$$

  (26.216)
• $$\begin{aligned} {\mathbb O}_{i^{*}j^{*}, k^{*}}= & {} R_{i^{*}m} R_{j^{*}n} {\mathbb O}_{mn,k} \frac{\partial x_{k}}{\partial x^{*}_{k^{*}}}{\mathop {=}\limits ^{({6.B2})}} R_{i^{*}m} R_{j^{*}n}R_{k^{*}k}{\mathbb O}_{mn,k} \nonumber \\= & {} \left( \delta _{i^{*}m}\delta _{j^{*}m} + \delta _{i^{*}m} \varOmega _{j^{*}n} + \delta _{j^{*}n}\varOmega _{i^{*}m}\right) \left( \delta _{k^{*} k} + \varOmega _{k^{*}k}\right) {\mathbb O}_{mn, k} \nonumber \\= & {} \left( \delta _{i^{*}m}\delta _{j^{*}n}\delta _{k^{*}k} + \delta _{i^{*} m} \delta _{k^{*} k} \varOmega _{j^{*}n}\right. \nonumber \\&\left. + \delta _{j^{*} n} \delta _{k^{*} k} \varOmega _{i^{*}m}\varOmega _{i^{*}m} + \delta _{i^{*}m}\delta _{j^{*}n}\varOmega _{k^{*} k}\right) {\mathbb O}_{mn, k} \nonumber \\= & {} {\mathbb O}_{i^{*}j^{*},k^{*}} + {\mathbb O}_{i^{*}n,k^{*}}\varOmega _{j^{*}n} + {\mathbb O}_{m j^{*},k^{*}}\varOmega _{i^{*}m} \nonumber \\&+ {\mathbb O}_{i^{*} j^{*},k^{*}}\varOmega _{k^{*}k} .\qquad \end{aligned}$$

  (26.217)

With the normalization $W({\mathbb O}_{ij}=0$ ${\mathbb O}_{ij,k}=0)=0$ and employing first-order Taylor series expansion one may write

$$\begin{aligned} DW\approx & {} \Bigg \{\frac{\partial W}{\partial {\mathbb O}_{i^{*}j^{*}}} \left( {\mathbb O}_{i^{*}j^{*}}+{\mathbb O}_{i^{*}n} \varOmega _{j^{*}n} + {\mathbb O}_{m j^{*}} \varOmega _{i^{*} m} - {\mathbb O}_{i^{*} j^{*}} \right) \nonumber \\&+ \frac{\partial W}{\partial {\mathbb O}_{i^{*} j^{*}, k^{*}}} \left( {\mathbb O}_{i^{*}j^{*}, k^{*}}+{\mathbb O}_{i^{*}n,k^{*}} \varOmega _{j^{*} n} \right. \nonumber \\&\left. + {\mathbb O}_{m j^{*}, k^{*}} \varOmega _{i^{*} m} + {\mathbb O}_{i^{*} j^{*}, k} \varOmega _{k^{*} k} - {\mathbb O}_{i^{*} j^{*}, k^{*}}\right) \Bigg \} \nonumber \\&= \frac{\partial W}{\partial {\mathbb O}_{i^{*} j^{*}}} \left( {\mathbb O}_{i^{*}n} \varOmega _{j^{*} n}+ {\mathbb O}_{m j^{*}}\varOmega _{i^{*} m}\right) \nonumber \\&+ \frac{\partial W}{\partial {\mathbb O}_{i^{*} j^{*}, k^{*}}}\left( {\mathbb O}_{i^{*} n, k^{*}}\varOmega _{j^{*} n} + {\mathbb O}_{m j^{*}, k^{*}}\varOmega _{i^{*} m} + {\mathbb O}_{i^{*} j^{*}, k}\varOmega _{k^{*} k} \right) \nonumber \\&{\mathop {\equiv }\limits ^{!}} 0 . \end{aligned}$$

(26.218)

This expression is linear in the skew-symmetric rank-2 tensor ${\varvec{\varOmega }}$ and can be written in the following new form by reshuffling indices.¹⁴ Such a reshuffling yields

$$\begin{aligned} DW= & {} \Bigg \{ \frac{\partial W}{\partial {\mathbb O}_{i^{*} k{*}}} {\mathbb O}_{i^{*} k}+ \frac{\partial W}{\partial {\mathbb O}_{k^{*} i{*}}} {\mathbb O}_{k i^{*}} + \frac{\partial W}{\partial {\mathbb O}_{i^{*} k^{*}, \ell ^{*}}} {\mathbb O}_{i^{*} k, \ell ^{*}} \nonumber \\&+ \frac{\partial W}{\partial {\mathbb O}_{k^{*} i^{*}, \ell ^{*}}}{\mathbb O}_{i^{*} k, \ell ^{*}} + \frac{\partial W}{\partial {{\mathbb O}_{i^{*} j^{*}, k}}} {\mathbb O}_{i^{*} j^{*},k^{*}}\Bigg \} \varOmega _{k^{*} k} \equiv 0 . \end{aligned}$$

(26.219)

Because the tensor $\varOmega _{k^{*} k}$ is skew-symmetric, the rank-2 tensor $F_{k^{*} k}$ in the curly bracket of this expression must be symmetric, or its skew-symmetric part must vanish. This can be expressed as $\varepsilon _{p k^{*} k}F_{k^{*} k}= 0$ or

$$\begin{aligned}&\varepsilon _{p k^{*} k} \Bigg \{ \Bigg (\frac{\partial W}{\partial O_{i^{*} k^{*}}}{\mathbb O}_{i^{*} k} + \frac{\partial W}{\partial {\mathbb O}_{k^{*} i^{*}}} {\mathbb O}_{k i^{*}} \nonumber \\&\qquad \quad + \frac{\partial W}{\partial {\mathbb O}_{i^{*} k^{*}, \ell ^{*}}} {\mathbb O}_{i^{*} k, \ell ^{*}} + \frac{\partial W}{\partial {\mathbb O}_{i^{*} j^{*}, k}}{\mathbb O}_{i^{*} k, k^{*}}\Bigg ) \nonumber \\&\qquad \quad + \frac{\partial W}{\partial {\mathbb O}_{i^{*} k, \ell ^{*}}} {\mathbb O}_{i^{*} k, \ell ^{*}}\Bigg \}\equiv 0 . \end{aligned}$$

(26.220)

Introducing the multi-indices I and $I_{k ^{j}}$ as defined in (26.14), this expression can alternatively be written as

$$\begin{aligned}&\varepsilon _{p k^{*} k}\Bigg \{\sum _{i^{*} = 1}^{2} \Bigg ( \frac{\partial W}{\partial {\mathbb O}_{{{I^{*}}_{i^{*}}}^{k^{*}}}} {\mathbb O}_{{{I^{*}}_{i^{*}}}^{k^{*}}} +\frac{\partial W}{\partial {\mathbb O}_{{{I^{*}}_{i^{*}}}^{k^{*}}, \ell ^{*}}} {\mathbb O}_{{{I^{*}}_{i^{*}}}^{k}, \ell ^{*}}\Bigg ) \nonumber \\&\qquad \quad + \frac{\partial W}{\partial {\mathbb O}_{I^{*}, \ell ^{*}}} {\mathbb O}_{I_{i^{*}}^{k}, \ell ^{*}}\Bigg \} \equiv 0 . \end{aligned}$$

(26.221)

Sonnet and Virga [44] must have performed the analogous computation for a free energy function W which is a function of a finite number of rank-i tensors $\pmb {\mathbb O}$ ($i = 1,2, \ldots ,n$). The frame indifference postulate is then Eq. (26.44). Invariance of W under an infinitesimal rigid body rotation is then expressible as a statement analogous to (26.219), explicitly as

$$\begin{aligned}&\varepsilon _{pk^{*}k}\Bigg \{\sum _{i^{*}=1}^{n}\Bigg (\frac{\partial W}{\partial {\mathbb O}_{{{I^{*}}_{i^{*}}}^{k^{*}}}} {\mathbb O}_{{{I^{*}}_{i^{*}}}^{k}} + \frac{\partial W}{\partial {\mathbb O}_{{{{I^{*}}_{i^{*}}}^{k^{*}}},\ell ^{*}}} {\mathbb O}_{{{I^{*}}_{i^{*}}}^{k},\ell ^{*}}\Bigg )\nonumber \\&\qquad \quad + \frac{\partial W}{\partial {\mathbb O}_{I^{*}, k^{*}}} {\mathbb O}_{I^{*}, k^{*}}\Bigg \}=0. \end{aligned}$$

(26.222)

With the above proof of the frame indifference as a function of the rank-2 order parameters $\pmb {\mathbb O}$, it is quite natural, how (26.221) can be proven, e.g., by the reader.

Appendix 26.C   Euclidian Invariance of ${\mathop {{\varvec{Q}}}\limits ^{\diamond }}$

We prove here the frame indifference of the co-deformational derivative of the rank-2 order parameter

(26.223)

where $\sigma$ is a scalar constitutive parameter or a constant. Let ${\varvec{R}}$ be an orthogonal second rank tensor, so that ${\varvec{R}}{\varvec{R}}^{T} = {\varvec{I}}$. Note, moreover, that ${\varvec{Q}}$ is a deviator by definition. Then

(26.224)

which demonstrates objectivity of the quantity $({\cdot })$ in (26.223). Next,

(26.225)

demonstrating Euclidian invariance of the last term on the right-hand side of (26.223). It follows that the co-deformational derivative of ${\varvec{Q}}$ is an objective rank-2 tensor.