Smart actuation of liquid crystal elastomer elements: cross-link density-controlled response

The microstructure of an elastomeric material consists of an amorphous arrangement of entangled chains joined at discrete points to form a network; the mechanics of this class of materials can be conveniently modelled by adopting a statistical-based approach describing the spatial chain arrangement [36–40].

Because of the disordered arrangement of the chains, the physical state of the network is well described by the entropic energy rather than by the standard deformation energy, the latter being typical of solids with an ordered microstructure such as crystalline ones. According to the freely-jointed chain model, a polymer chain is usually assumed to be made of $N$ rigid (Khun's) segments (monomers) of equal length $b$, arranged in the 3D space according to the so-called random-walk theory [39]. The maximum disorder degree of the network usually takes place in the stress-free state of the material; upon stretching, the chains are forced to elongate in the load direction and, because of the increasing order taking place during the deformation, the entropic energy decreases. The above-mentioned assumption referred to the conformation of a single chain, entails a limit for its maximum extension because of the rigidity of the constituting monomer segments; as a consequence, the maximum length of a stretched chain cannot overcome its contour length $Nb$, and the corresponding maximum stretch becomes equal to ${\lambda _{{\text{max}}}} = \sqrt N$. It has also been observed that the so-called end-to-end vector ${\boldsymbol{{r}}}$, identifying the relative position of the two chain's extremities, represents a suitable parameter identifying the chain's physical state [39].

By harnessing this concept, the statistical description of the chain arrangement provides a suitable tool for describing the physical state of the polymer. Let us now introduce the function ${\rho _0}\left( {\boldsymbol{{r}}} \right)$ describing the chain end-to-end vector distribution in the stress-free state of the material:

$$\begin{align}{\rho _0}\left( {\boldsymbol{{r}}} \right) & = {c_a}\,{\varphi _0}\left( {\left| {\boldsymbol{{r}}} \right|} \right),\nonumber \\ {\text{being}}\;{\varphi _0}\left( {\left| {\boldsymbol{{r}}} \right|} \right) & = {\left( {\frac{3}{{2\pi N{b^2}}}} \right)^{\frac{3}{2}}}{\text{exp}}\left( { - \frac{{3{{\left| {\boldsymbol{{r}}} \right|}^2}}}{{2N{b^2}}}} \right)\end{align} \tag{ 1 }$$

where ${c_a}$ represents the chain density (number of cross-linked chains per unit volume) while ${\varphi _0}$ is the normalized distribution function, usually assumed to be the 3D Gaussian characterized by mean value $\left| {\boldsymbol{{r}}} \right| = 0$ and standard deviation $b\sqrt {N/3} { }$. According to the above definition, the integration of the function ${\rho _0}\left( r \right)$ over the chain space (spanning all the possible chain lengths and orientations) must provide the number of chains per unit volume, ${c_a} = \left\langle {{\rho _0}} \right\rangle = \mathop \smallint \limits_{{\Omega }}^{} {\rho _0}{ }d{{\Omega }} = \mathop \smallint \limits_0^{2\pi } \mathop \smallint \limits_0^\pi \left( {\mathop \smallint \limits_0^{Nb} {\rho _0}\left( {\boldsymbol{{r}}} \right){ }{{\left| {\boldsymbol{{r}}} \right|}^2}dr{ }} \right)\sin \theta d\theta d\omega$, where the integral takes into account only the 'mechanically active' chains, i.e. those fully connected to the network that are able to contribute to the load carrying mechanism induced by an applied deformation. It is worth mentioning that the chain concentration ${c_a}$ is strictly related to the shear modulus $\mu$ of the material through the well-known expression: $\mu = {c_a}{ }{k_B}{ }T$, where ${k_B}$ and $T$ are the Boltzmann constant and the absolute temperature, respectively [39].

A simplifying assumption, often made in rubber elasticity, is the so-called affine deformation hypothesis: at the microscopic scale, the polymer chains deform as the material does at the continuum level. Hereafter, we are going to adopt such as assumption in the development of our theory. As a consequence, the energy density of the polymer is provided by the following integral, $\Psi = \mathop \smallint \limits_{{\Omega }} \rho \left( {{\boldsymbol{{r}}},t} \right){ }\psi \left( {\left| {\boldsymbol{{r}}} \right|} \right){ }d{{\Omega }}$, where the term $\psi \left( {\left| {\boldsymbol{{r}}} \right|} \right)$ represents the deformation energy per single chain, usually expressed as a function of the end-to-end vector distance $\left| {\boldsymbol{{r}}} \right|$; according to the Gaussian statistics, the deformation energy per single chain is expressed as $\psi \left( {\left| {\boldsymbol{{r}}} \right|} \right) = \frac{{3{k_B}T}}{{2N{b^2}}}{\left| {\boldsymbol{{r}}} \right|^2}$, while, according to the more realistic Langevin assumption, the chain energy is given by $\psi \left( {\left| {\boldsymbol{{r}}} \right|} \right) = N{k_B}T\left( {\beta {\mathcal L^{ - 1}}\left( \beta \right) + \ln \frac{{{\mathcal L^{ - 1}}\left( \beta \right)}}{{\sinh \left[ {{\mathcal L^{ - 1}}\left( \beta \right)} \right]}}} \right)$, being ${\mathcal L^{ - 1}}\left( \beta \right)$ the inverse of the Langevin function (with $\beta = \left| {\boldsymbol{{r}}} \right|/Nb$) which provides unlimited values when the chain length tends to its contour length, i.e. $\psi \left( {\left| {\boldsymbol{{r}}} \right|} \right) \to \infty$ when $\left| {\boldsymbol{{r}}} \right| \to {r_{{\text{max}}}} \cong Nb$) [36].

At a generic time $t$, the configuration of the network is fully known through the corresponding distribution function $\rho \left( {{\boldsymbol{{r}}},t} \right)$ which, starting from the initial chain configuration quantified by the distribution function ${\rho _0}\left( {\boldsymbol{{r}}} \right)$, evolves because of the applied mechanical deformation and/or in a LCE, due to the self deformation induced by the change of the nematic order [32, 41].

Figure 1 sketches how the distribution function $\rho \left.\left( {\boldsymbol{{r}}} \right)\ \right($or equivalently its dimensionless counterpart $\varphi \left.\left( {\boldsymbol{{r}}} \right) \right)$ changes because of the change of the mesogens preferential orientation induced by cooling down or heating up the material.

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The current elastic energy density of a deformed network, evaluated with respect to its initial (stress-free) state, is expressed as:

$$\begin{align} {{\Delta }}\Psi \left( t \right) & = \Psi \left( t \right) - {\Psi _0} + A \nonumber \\ & = {c_a}\left\langle {\left[ {\varphi \left( {{\boldsymbol{{r}}},t} \right) - {\varphi _0}\left( {\boldsymbol{{r}}} \right)} \right]\psi } \right\rangle + p\left[ {\det \left( {\boldsymbol{{F}}} \right) - 1} \right] \end{align} \tag{ 2 }$$

where ${ }{\boldsymbol{{F}}} = \partial {\boldsymbol{{x}}}/\partial {\boldsymbol{{X}}} = \boldsymbol 1 + \partial {\boldsymbol{{u}}}/\partial {\boldsymbol{{X}}}$ is the deformation gradient tensor, $\boldsymbol 1$ is the second order identity tensor, and ${\boldsymbol{{u}}}$ is the displacement field, while $p$ is the hydrostatic pressure playing the role of a multiplier constraint required to fulfill the incompressibility condition (hereafter always assumed) expressed by $J = \det \left( {\boldsymbol{{F}}} \right) = 1$.

By introducing the distribution tensor ${\boldsymbol{{\mu }}}$, quantifying the number of chains with a given end-to-end distance and orientation in the 3D chain space [42], the previous energy density can be rewritten as follows:

$$\begin{equation}{{\Delta }}\Psi \left( t \right) = \frac{{3{c_a}{k_B}T}}{{2N{b^2}}}{\text{tr}}\left[ {{\boldsymbol{{\mu }}}\left( t \right) - {{\boldsymbol{{\mu }}}_0}} \right] + p\left[ {\det \left( {{\boldsymbol{{F}}}\left( t \right)} \right) - 1} \right]\end{equation} \tag{ 3 }$$

where ${{\boldsymbol{{\mu }}}_0} = \left\langle {{\varphi _0}\left( {{\boldsymbol{{r}}},t = 0} \right){\boldsymbol{{r}}} \otimes {\boldsymbol{{r}}}} \right\rangle$ is the distribution tensor in the initial stress-free state, while ${\boldsymbol{{\mu }}} = \left\langle {\varphi \left( {{\boldsymbol{{r}}},t} \right){ }{\boldsymbol{{r}}} \otimes {\boldsymbol{{r}}}} \right\rangle$ refers to the current deformed configuration.

In a standard polymer whose chain network has not any preferential direction, the chain orientations are distributed isotropically in the stress-free state and, by adopting the Gaussian statistics for the chain's energy, the distribution tensor assumes the simple expression ${{\boldsymbol{{\mu }}}_0} = \frac{{N{b^2}}}{3}\boldsymbol1$.

It is worth mentioning that the distribution tensor can be thought as an extension of the so-called structure's tensor [43]. By harnessing the expression of the power density of the material:

$$\begin{align} {{\Delta }}\dot \Psi \left( t \right) & = {c_a} \langle \dot \varphi \left( {{\boldsymbol{{r}}},t} \right)\psi \rangle = \nonumber \\ & = \frac{{3{c_a}{k_B}T}}{{N{b^2}}}\left[ {{\boldsymbol{{\mu }}}\left( t \right) - {{\boldsymbol{{\mu }}}_0}} \right]:{\boldsymbol{{L}}}\left( t \right) + p\left( t \right){\text{ tr }}{\boldsymbol{{L}}}\left( t \right) = \nonumber \\ & = {\boldsymbol{{\sigma }}}\left( t \right)\,:{\boldsymbol{{D}}}\left( t \right) .\end{align} \tag{ 4 }$$

the Cauchy stress tensor ${\boldsymbol{{\sigma }}}\left( t \right)$ at the time $t$ can be easily identified to be expressed as [42]:

$$\begin{equation}{\boldsymbol{{\sigma }}}\left( t \right) = {J^{ - 1}}{\boldsymbol{{P}}}\left( t \right){{\boldsymbol{{F}}}^T}\left( t \right) = \frac{{3{c_a}{k_B}T}}{{N{b^2}}}\left[ {{\boldsymbol{{\mu }}}\left( t \right) - {{\boldsymbol{{\mu }}}_0}} \right] + { }p\left( t \right){\boldsymbol{{1}}}.\end{equation} \tag{ 5 }$$

In equation (4), ${\boldsymbol{{D}}}$ is the symmetric part of the velocity gradient tensor ${\boldsymbol{{L}}}$, namely ${\boldsymbol{{D}}} = {\text{sym }}{\boldsymbol{{L}}}$=$\left( {{\boldsymbol{{L}}} + {{\boldsymbol{{L}}}^T}} \right)/2$, being ${\boldsymbol{{L}}} = {\boldsymbol{{\dot F}}}{{\boldsymbol{{F}}}^{ - 1}}$, while the symbol: indicates the double contraction operator. In the case both damage (i.e. chain loss) and self-healing (i.e. chain gain) mechanisms do not take place, ${c_a}$ remains constant all along the whole mechanical deformation process. The evolution in time of the distribution function rate $\dot \varphi$, required to evaluate (4), can be determined from the chain density conservation relationship $\frac{D}{{Dt}}\mathop \smallint \limits_V \varphi \left( {{\boldsymbol{{r}}},t} \right){ }dV = 0$, stating that the time derivative of the chain concentration ${c_a}$ is zero (the number of chains in the volume $V$ is constant in time), i.e. ${\dot c_a} = {c_a}{ }\langle \dot \varphi \rangle = 0{ } \to \partial \varphi \left( {{\boldsymbol{{r}}},t} \right)/\partial t = - \left( {\nabla \varphi \otimes {\boldsymbol{{r}}}{ } + { }\varphi { }1} \right):{\boldsymbol{{L}}}$ [42].

Various models, based on the description of molecular force interaction, aimed at describing the behavior of liquid crystals have been proposed in the literature [41, 44].

The universally-recognized physics-based theory of Warner, Terentjev et al has had the merit to extend the classical rubber elasticity to nematic elastomers by accounting for the molecular anisotropy induced by the coupling of the mesogens reorientation to the elastomer network [31, 33, 34]. The physical state of a nematic elastomer is usually described by the director field, whose change, occurring even if the elasticity of the network partially hinders the mesogen reorientation, triggers the appearance of a spontaneous deformation.

Hereafter, by harnessing the statistical description of the network arrangement through the use of the above-mentioned distribution tensor, the mechanics of LCEs is developed based on the classical molecular rubber elasticity [31–34].

Differently from classical polymers having a random isotropic distribution of the chains orientation in the stress-free state (in such a case the distribution tensor ${{\boldsymbol{{\mu }}}_0}$ is characterized by a spherical symmetry and has three identical eigenvalues, ${\mu _{01}} = {\mu _{02}} = {\mu _{03}}$), polymers polymerized in the nematic state have the network's chains preferentially aligned with the nematic order of the mesogens. This implies that the distribution function ${\varphi _0}$ is non-isotropic, and a general expression for ${\varphi _0}$, to be used in place of equation (1), is the following [32]:

$$\begin{equation}{\varphi _0}\left( {\boldsymbol{{r}}} \right) = {\left( {\frac{3}{{2\pi Nb}}} \right)^{\frac{3}{2}}}{\left( {\frac{1}{{\det {\boldsymbol {\ell} _0}}}} \right)^{\frac{1}{2}}}\,{\text{exp}}\left[ { - \frac{{3\,{\boldsymbol{{r}}} \cdot \boldsymbol {\ell} _0^{ - 1}{\boldsymbol{{r}}}}}{{2Nb}}} \right]\end{equation} \tag{ 6 }$$

being ${\boldsymbol \ell _0} = {\ell _{0 \bot }}{ }\boldsymbol 1 + \left( {{\ell _{0\parallel }} - {\ell _{0 \bot }}} \right){ }{\boldsymbol{{n}}} \otimes {\boldsymbol{{n}}}$ the step-length tensor of the chain distribution; it quantifies the anisotropy of the polymer chains at the time of cross-linking and is expressed through the effective initial step lengths ${\ell _{0 \bot }}$ and ${\ell _{0\parallel }}$, measured parallel and perpendicular to the director ${\boldsymbol{{n}}}$, respectively (figure 2(a)). In the nematic state it happens to be ${\ell _{0\parallel }} \gg {\ell _{0 \bot }}$, while in a perfectly isotropic state the step length tensor reduces to ${\boldsymbol {\ell} _0} = b{\mathbf{1}}$ and the averaged square of the end-to-end distance becomes $\left\langle {r_i^2} \right\rangle = \frac{{N{b^2}}}{3},{ }i = 1,2,3$ [33, 39].

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At the time of cross-linking we assume the state of the polymer is stress-free; it can be easily shown that, in this case, the distribution tensor is related to the step-length tensor by the following simple relation: ${{\boldsymbol{{\mu }}}_0} = \frac{{Nb}}{3}{\boldsymbol {\ell} _0}$ [45].

In its principal directions' frame of reference the distribution tensor can be expressed as ${{\boldsymbol{{\mu }}}_{0p}} = {{\boldsymbol{{R}}}^T}{{\boldsymbol{{\mu }}}_0}{ }{\boldsymbol{{R}}} = \mathop \sum \limits_{i = 1}^3 {\mu _{0i}}{{\boldsymbol{{m}}}_i} \otimes {{\boldsymbol{{m}}}_i}$, being ${\boldsymbol{{R}}}$ a second order rotation tensor and ${{\boldsymbol{{m}}}_i}$ the versor of the $i-{\text{th}}$ principal direction, while (${\mu _{01}},{\mu _{02}},{\mu _{03}}$) are the three eigenvalues of ${{\boldsymbol{{\mu }}}_0}$. When the principal directions are superposed to the frame of reference's axes, ${{\boldsymbol{{\mu }}}_{0p}}$ assumes the simple diagonal form ${{\boldsymbol{{\mu }}}_{0p}} = {\mu _{0i}}{ }{\delta _{ij}},{\text{ }}i,j = 1,2,3$.

It is worth mentioning that, when the principal directions of the distribution tensor are aligned with the coordinate axes, the eigenvalues (${\mu _{01}},{\mu _{02}},{\mu _{03}}$) are related to the standard deviation of the end-to-end distance components measured along the axes, namely ${\mu _{0i}} = \left\langle {r_{0i}^2} \right\rangle ,{\text{ }}i = 1,2,3$, i.e. when the mesogen units are preferentially aligned along the 3rd Cartesian axis: ${\mu _{01}} = \left\langle {r_{01}^2\left( t \right)} \right\rangle = \frac{{N{b^2}}}{3}\left[ {1 - Q\left( t \right)} \right],{\text{ }}{\mu _{02}} = \left\langle {r_{02}^2\left( t \right)} \right\rangle = \frac{{N{b^2}}}{3}$ $\left[ {1 - Q\left( t \right)} \right]$ and ${\mu _{03}} = \langle r_{03}^2\left( t \right)\rangle = \frac{{N{b^2}}}{3}\left[ {1 + 2Q\left( t \right)} \right].$

In the frame of reference's principal directions, the distribution tensor can be graphically represented by an ellipsoid whose axes are the principal directions. A mechanical deformation, as well as the order imposed by the nematic mesogens, have the effect of inducing the elongation of such an ellipsoid in a preferential direction (i.e. that of the stretch or that of the nematic director) and, because of the material incompressibility constraint, a contraction in the perpendicular directions arises.

The so-called order parameter at the time $t$, $Q\left( t \right)$—expressing the dispersion of the angle $\theta \left( t \right)$ formed by the mesogen units with respect the their average direction identified by the director ${\boldsymbol{{n}}}$ (figure 2(a))—is defined as $Q\left( t \right) = \left\langle {\frac{3}{2}{{\cos }^2}\theta \left( t \right) - \frac{1}{2}} \right\rangle$, and the corresponding distribution tensor becomes ${\boldsymbol{{\mu }}}\left( t \right) = \frac{{Nb}}{3}{\boldsymbol{{\ell }}} \left( t \right) = \frac{{Nb}}{3}b\left[ {\left( {1 - Q\left( t \right)} \right){\mathbf{1}} + 3Q\left( t \right){ }{\boldsymbol{{n}}} \otimes {\boldsymbol{{n}}}} \right]$. In particular, when the order parameter (whose possible values range in the interval $- 1/2\, \le Q \le 1$) is equal to $Q = 1$, the nematic order of the mesogens is perfect, and they are ideally aligned along a well defined direction, while when $Q = 0$ they are randomly oriented such as in a standard isotropic arrangement of polymeric chains. When $0 < Q < 1$, the mesogens are characterized by an intermediate degree of alignment, where the degree of the orientation dispersion increases as $Q \to 0$. Finally, the value $Q = - 1/2$ of the nematic order parameter indicates that all the rod molecules lay on the $x,y$ plane.

When the case of a temperature-driven order change is concerned, i.e. $\theta \left( t \right) = \theta \left( {T\left( t \right)} \right)$, the distribution tensor describing the chain distribution in space becomes itself a function of the temperature. The above-defined distribution tensor can be related to the so-called nematic order tensor ${\boldsymbol{{Q}}}\left( t \right)$ (also called the de Gennes order tensor [32, 46]), through the relation:

$$\begin{align} {\boldsymbol{{Q}}}\left( t \right) & = \frac{{Q\left( t \right)}}{2}\left( {3{\boldsymbol{{n}}} \otimes {\boldsymbol{{n}}} - {\mathbf{1}}} \right) = \frac{1}{2}\left( {\frac{{\boldsymbol {\ell} \left( t \right)}}{b} - {\mathbf{1}}} \right) = \nonumber \\ & = \frac{1}{2}\left( {3\frac{{{\boldsymbol{{\mu }}}\left( t \right)}}{{N{b^2}}} - {\mathbf{1}}} \right) .\end{align} \tag{ 7 }$$

As mentioned in the introduction, the nematic order of LCEs is strictly related to the temperature of the material: for $T < {T_{NI}}$ the order parameter is $Q = {Q_0}$, being ${Q_0}$ the value of the nematic order at the time of cross-linking assumed to occur in the nematic state, while if $T > {T_{NI}}$ then $Q \to 0$, i.e. the chain configuration tends to become isotropic.

The nematic order-temperature dependence is influenced by the physical-chemical properties of the mesogens and by the cross-link density, and can be determined from experimental tests [47]. Typically, LCEs are responsive to a temperature change because the initial order of the mesogen units is progressively destroied when the material is heated above the so-called nematic-isotropic transition temperature ${T_{NI}}$. This process is fully reversible and provides a noticeable actuation force [32]. In the present study, we assume the nematic order parameter to be related to the temperature change by the relationship: $Q\left( T \right) = {Q_0}{ }g\left( T \right)$, with $g\left( T \right) = {\left[ {1 + \exp \frac{{T - {T_{NI}}}}{c}} \right]^{ - 1}}$, being $c$ a material-dependent parameter. The function $g\left( T \right)$ quantifies the way the transition of the order parameter, from ${Q_0}$ (nematic state) to the isotropic one $Q\rightarrow 0$, occurs as the temperature increases beyond the transition temperature ${T_{NI}}$ (figure 2(b)).

The precise tuning of such a transition function is important if the nematic-isotropic transient response has to be determined, while it is not so relevant if only the final self-deformation arising because of the temperature change is sought.

Dealing with LCE materials, in general it must be considered that the distribution tensor ${\boldsymbol{{\mu }}}\left( t \right)$ evolves and changes in time because of: (a) the applied mechanical deformation, and (b) the spontaneous deformation induced by the nematic-isotropic transition triggered by an external stimulus. The evolution of the distribution tensor can thus be expressed through its time rate ${\boldsymbol{{\dot \mu }}}\left( t \right)$ splitted into the following contributions:

$$\begin{equation}{\boldsymbol{{\dot \mu }}}\left( t \right) = {{\boldsymbol{{\dot \mu }}}_{\boldsymbol{{F}}}}\left( t \right) + {{\boldsymbol{{\dot \mu }}}_n}\left( t \right)\end{equation} \tag{ 8 }$$

where the term ${{\boldsymbol{{\dot \mu }}}_{\boldsymbol{{F}}}}$ indicates the stress distribution tensor rate induced by the mechanical deformation, while ${ }{{\boldsymbol{{\dot \mu }}}_n}$ is the rate associated with the change of the nematic order. Explicitly, the above rates can be evaluated as follows [7, 45]:

$$\begin{align}{{\boldsymbol{{\dot \mu }}}_{\boldsymbol{{F}}}}\left( t \right) = {\left. {\frac{{\partial {\boldsymbol{{\mu }}}\left( t \right)\,}}{{\partial t}}} \right|_n} & = \left\langle {\varphi \left( t \right)\,{\boldsymbol{{r}}} \otimes {\boldsymbol{{r\,}}}} \right\rangle {\boldsymbol{{L}}}\left( t \right) = {\boldsymbol{{L}}}\left( t \right)\,{\boldsymbol{{\mu }}}\left( t \right) \nonumber \\ & \quad + {\left[ {{\boldsymbol{{L}}}\left( t \right)\,{\boldsymbol{{\mu }}}\left( t \right)} \right]^T}\end{align} \tag{ 9a }$$

$$\begin{equation}{{\boldsymbol{{\dot \mu }}}_n}\left( t \right) = {\left. {\frac{{\partial {\boldsymbol{{\mu }}}\left( t \right)\,}}{{\partial t}}} \right|_{\boldsymbol{{F}}}}\! = 2\frac{{N{b^2}}}{3}\left[ {{\boldsymbol{{\dot Q}}}\left( t \right)\! - {\boldsymbol{{W}}}\left( t \right){\boldsymbol{{Q}}}\left( t \right) + \!{\boldsymbol{{Q}}}\left( t \right)\,{\boldsymbol{{W}}}\left( t \right)} \right] \\[6pt] \end{equation} \tag{ 9b }$$

where the notations ${\left. \blacksquare \right|_{\boldsymbol{{n}}}}$, ${\left. \blacksquare \right|_{\boldsymbol{{F}}}}$ indicate that the quantity $\blacksquare$ is evaluated at constant nematic order and at constant deformation, respectively, while ${\boldsymbol{{W}}} = {{{\scriptstyle 1}/{\scriptstyle 2} }}\left( {\nabla {\boldsymbol{{\dot u}}} - \nabla {{{\boldsymbol{{\dot u}}}}^T}} \right)$ is the spin tensor [27]. The distribution tensor at the current time instant $t$ can be evaluated by integrating the above-defined rates over the time interval $\left( {0,\,t} \right)$ as ${\boldsymbol{{\mu }}}\left( t \right) = {\boldsymbol{{\mu }}}\left( 0 \right) + \mathop \smallint \limits_0^t \left[ {{{{\boldsymbol{{\dot \mu }}}}_{\boldsymbol{{F}}}}\left( \tau \right) + {{{\boldsymbol{{\dot \mu }}}}_n}\left( \tau \right)} \right]d\tau$.

By assuming that the order parameter is related to the environmental temperature while the principal directions of ${\boldsymbol{{Q}}}$ do not change, the rate of the nematic order tensor ${\boldsymbol{{\dot Q}}}\left( t \right){ }$ is expressed by

$$\begin{align}{\boldsymbol{{\dot Q}}}\left( t \right) & = \frac{1}{2}\frac{{\dot {\boldsymbol{{\ell }}} \left( t \right)}}{b} \nonumber \\ {\text{being}}\;\dot {\boldsymbol{{\ell }}} \left( t \right) & = b\dot Q\left( t \right)\,\left[ {3\,{\boldsymbol{{n}}} \otimes {\boldsymbol{{n}}} - \boldsymbol{1}} \right] \nonumber \\ {\text{with}}\;\dot Q\left( t \right) & = \frac{{\partial Q\left( T \right)}}{{\partial T}}\dot T. \end{align} \tag{ 10 }$$

By considering a generic deformation, quantified by the deformation gradient ${\boldsymbol{{F}}}$ taking place while the material undergoes a nematic order change due to a temperature variation, the energy density increment (evaluated with respect to the initial stress-free state for which ${\Psi _0} = 3\mu /2$), is [45]:

$$\begin{align} {{\Delta }}\Psi \left( t \right) & = \frac{{3\mu }}{{2N{b^2}}}{\text{tr}}\left( {{\boldsymbol{{\mu }}}\left( t \right) - {{\boldsymbol{{\mu }}}_0}} \right) \nonumber \\ & = \frac{\mu }{2}\left[ {{\text{tr}}\left( {{\boldsymbol{{F\,}}}{\boldsymbol{\ell} _0}{{\boldsymbol{{F}}}^T}{{\boldsymbol{{\ell }}} ^{ - 1}}} \right) - 3} \right] \end{align} \tag{ 11 }$$

where the affine deformation hypothesis, mathematically expressed by ${\boldsymbol{{r}}} = {\boldsymbol{{F}}}{{\boldsymbol{{r}}}_0}$, has been adopted; equation (11) provides an extension of the deformation energy of the classical rubber elasticity theory [39]. Since the order tensor in the spatial configuration is related to that in the reference domain through [32] ${\boldsymbol{{Q}}} = {J^{ - 1}}{\boldsymbol{{F}}}{{\boldsymbol{{Q}}}_0}{{\boldsymbol{{F}}}^T}$, the distribution tensor in the current (deformed) configuration is related to the initial step length tensor as follows: ${\boldsymbol{{\mu }}} = \frac{{Nb}}{3}{\boldsymbol{{F}}}{{\boldsymbol{{\ell }}} _0}{{\boldsymbol{{F}}}^T}$, being ${{\boldsymbol{{\mu }}}_0} = \frac{{Nb}}{3}{{\boldsymbol{{\ell }}} _0}$.

In this section, we compare the results provided by the present micromechanical model with an experimental case reported in [52]. The uniaxial contraction of an element made of a single LCE layer synthesized by using different cross-linker contents, observed during a thermoelastic experiment by heating the material above the ${T_{NI}}$, is studied. We analyze three different LCE strips whose matrix has been prepared by combining nematogen units (C411U8) with a cross-linker (TAC-4) at three different percentages, leading to different values of the material's Young's modulus; according to [52], it has been assumed to be ${E^{\left( A \right)}} = 3\,{\text{MPa}}$, ${E^{\left( B \right)}} = 8.5\,{\text{MPa}}$ and ${E^{\left( C \right)}} = 14.5\,{\text{MPa}}$ corresponding to $5;10;15\,\% \,{\text{mol}}$ content of T-4 cross-linker, respectively.

From the above mentioned Young's modulus values, the corresponding chain concentrations $c_a^{\left( A \right)}$, $c_a^{\left( B \right)}$ and $c_a^{\left( C \right)}$ have been evaluated through the standard rubber elasticity relationship (see section 2.1) $c_a^{\,\,\left( i \right)} = {\mu ^{\left( i \right)}}{k_B}T$ with $i = A,B,C$, figure 5. In the numerical model, we simulate a LCE monodomain strip, with initial order parameter ${Q_0} = 0.5$, having the mesogen units aligned with the element's longitudinal axis in order to provide a contraction of the sample when heated above the transition temperature ${T_{NI}} = 65\,^\circ {\text{C}}$; the parameter governing the phase transition has been assumed to be $c = 8$. Since we consider the maximum actuation achievable after the nematic-isotropic transition has occurred (i.e. at ${T_b} \gg {T_{NI}}$), the exact ${T_{NI}}$ value and its dependence on the cross-linker content, as emphasized in [52], is neglected.

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We assume the material, whose optimal cross-link density ${\bar c_a}$ corresponding to the optimal shear modulus ${\bar \mu _a} = 1\,{\text{MPa}}$ is assumed, to be in region II (see figure 3), while the parameter ${\alpha _{II}}$ in equation (13b) is assumed to be $0.65$. Since the value of the maximum cross-link density ${c_t}$ hindering the self-deformation is not known, for each of the three investigated materials we consider five different values of such a cross-link density, namely $c_t^{\,\left( 1 \right)}$, $c_t^{\,\left( 2 \right)}$, $c_t^{\,\left( 3 \right)}$, $c_t^{\,\left( 4 \right)}$ and $c_t^{\,\left( 5 \right)}$ (see figure 5); the above listed quantities have been determined through the above-mentioned cross-link density-shear modulus relationship, by assuming the following values of the shear modulus $\mu _t^{\,\left( 1 \right)} = 3\,{\text{MPa}}$, $\mu _t^{\,\left( 2 \right)} = 5\,{\text{MPa}}$, $\mu _t^{\,\left( 3 \right)} = 6\,{\text{MPa}}$, $\mu _t^{\,\left( 4 \right)} = 10\,{\text{MPa}}$ and $\mu _t^{\,\left( 5 \right)} = 20\,{\text{MPa}}$. The results of the numerical simulations are reported in figure 5 where the experimental data are also shown; it appears that, if the value of ${c_t}$ decreases, the contraction deformation reduces in the considered range of cross-link densities, $c_a^{\left( A \right)} - c_a^{\left( C \right)}$. It can be observed that the maximum cross-link density $c_t^{\,\left( 3 \right)}$ is the one that best fits the experimental results.

It is important to emphasize that when ${c_a}$ is greater than ${c_t}$ the mesogen effectiveness disappears and no deformation takes place in the material; this is the case of the material characterized by the smallest ${c_t}$, namely $c_t^{\,\left( 1 \right)}$, whose response is represented by empty triangles in figure 5: the bottom right portion of the curve goes to zero because the cross-link density in such a region overcomes the maximum cross-link density $c_t^{\,\left( 1 \right)}$ of the material.

In this section we simulate numerically the self-deformation shown by a bi-layer plate made of two superposed LC layers with identical thickness, each one with its own cross-link concentration, $c_a^{\left( 1 \right)}$ in the lower layer and $c_a^{\left( 2 \right)}$ in the upper layer.

By applying a temperature rate $\dot T = 2.083{ }^\circ {\text{C }}{{\text{s}}^{ - 1}}$, the temperature of the bottom surface of the lower layer is made to increased from the room temperature ${T_b} = 25{ }^\circ {\text{C}}$ up to ${T_b} = 150{ }^\circ {\text{C}}$.

The temperature evolution within the element is determined by solving the heat conduction problem by adopting the thermal conductivity and the specific heat to be $\kappa =$0.8 W mK⁻¹ and $C =$1050 J Kg⁻¹ K⁻¹, respectively, while the sizes of the bi-layer cantilever beam, restrained along the edge marked with ① in figure 6, are $L = 8{\text{ mm}}$, $h = t = 0.5{\text{ mm}}$ (figure 6).

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In order to exclude further deformations not related to the phase change of the LCE, the thermal expansion of the material is assumed to be negligible in the temperature range considered.

Unless differently stated, for all the LCE parts we assume the nematic-isotropic transition temperature to occur at ${T_b}/{T_{NI}} = 0.5;0.75;2$, while the parameters describing the transition from the initial nematic (the mesogens are assumed to be initially aligned with the $X$direction) to the final isotropic state are assumed to be ${Q_0} = 0.3$ and $c = 20$ (see the expression for the function $g\left( T \right)$, section 2.2). The chain concentrations of the double network, expressed through the corresponding shear moduli ${\bar \mu _a},{\mu _t}$, are as follows: ${\bar \mu _a} = {\bar c_a}\,{k_B}\,T = 0.4\,{\text{MPa}}$, ${\mu _t} = {c_t}\,{k_B}\,T = 6\,{\text{MPa}}$ while the exponent in equation (13a ) has been adopted to be ${\alpha _{II}} = 2$.

In figure 6, the deformed pattern of the middle line of the bi-layer element is shown for two dimensionless temperatures of the bottom surface, namely ${T_b}/{T_{NI}} = 0.75$ (figure 6(a)) and ${T_b}/{T_{NI}} = 2.0$ (figure 6(b)) by varying the relative cross-link density, $c_a^{\left( 1 \right)}/c_a^{\left( 2 \right)}$, of the composing layers. In this figure, as well as in the following ones, geometrical quantities are plotted by using the dimensionless coordinates $x/L$ and $y/h$, where $x$ and $y$ represent the current coordinates evaluated in the deformed configurationt (see the adopted frame of reference in figure 6(b)). The greater the difference of the cross-link density of the two layers, the larger the bending actuation shown by the element; when $c_a^{\left( 1 \right)}/c_a^{\left( 2 \right)} = 1$ the element shows a simple contraction without any bending actuation. In figure 7, the deformed shapes of the LCE element and the corresponding values of the relative order parameter field, $Q\left( T \right)/{Q_0}$, for $c_a^{\left( 1 \right)}{ }/c_a^{\left( 2 \right)} = 1.04$ (left column) and $c_a^{\left( 1 \right)}{ }/c_a^{\left( 2 \right)} = 2.50$ (right column) are displayed.

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In this section, we consider the actuation shown by a bi-layer LCE element (the same boundary conditions assumed in the previous example are considered) whose constituent layers have different thicknesses and different chain concentrations.

The bottom to top cross-link density ratio is assumed to be equal to $c_a^{\left( 1 \right)}/c_a^{\left( 2 \right)} = 1.25$ and $2.5$, while the layer's relative thickness ratio is assumed to be equal to ${h^{\left( 1 \right)}}/h = 0.5;\,0.8;\,0.9$ (figure 8). It can be appreciated that the bending actuation increases when the layers' thickness tends to be identical, while a large thickness ratio is not effective in producing actuation.

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In the limit case of a very large value of the ${h^{\left( 1 \right)}}/{h^{\left( 2 \right)}}$ ratio, the deformed shape tends to that of a single layer, as previously shown in figure 6 (simple contraction).

As was observed in the previous parametric example, due to the small size of the element, for a fixed relative thickness ratio a higher chain concentration ratio $c_a^{\left( 1 \right)}/c_a^{\left( 2 \right)}$ enables a larger bending actuation (see figures 8(a) and (b)).

Finally, in figure 9 the deformed shapes of the element for two thickness ratios are displayed; the colour quantifies the relative order parameter $Q\left( T \right)/{Q_0}\,$corresponding to the dimensionless temperature ${T_b}/{T_{NI}}\,$assigned at the bottom of the elemen. It is worth noticing that, due to the small size of the element, the temperature field is almost constant within the domain so the order parameter is uniform throughout the beam.

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The actuation of a cantilever beam having a graded distribution of the cross-link density, is considered in this section. The main parameters involved in the simulations are the same used in the previous examples; however, the beam is now assumed to have a fixed cross-link density at its top edge, ${c_a}^{\left( T \right)} = {\bar c_a}$, while the cross-link density varies linearly through the thickness of the element ($Y$-direction), up to the maximum value reached at the bottom edge where it assumes the value ${c_a}^{\left( B \right)} = \alpha \,{\bar c_a}$ ($2.5 \le \alpha \le 10$), being ${\bar c_a}$ the cross-link density corresponding to the shear modulus $\bar \mu = 0.4{\text{ MPa}}$. The cross-link density gradient parameter is thus defined as $\kappa = \left( {\alpha - 1} \right)/h$. The mesogen units are assumed to be initially aligned with the $X$-direction, while the initial order parameter has been assumed to be equal to ${Q_0} = 0.3$ (figure 10).

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Greater chain density gradients produce more pronounced self-deformation actuation, with an increase of the actuation amplitude up to four times by amplifying the cross-link density gradient parameter from $\kappa = \left( {\alpha - 1} \right)/h = 1.5/h$ up to $\kappa = 9/h$.

It can be appreciated that creating a graded cross-link density distribution across the thickness—nowadays easily obtainable by modern 3D printing technologies—can be used to encode the element's morphing response into the material's microstructure.