Dynamical Landau–de Gennes theory for electrically-responsive liquid crystal networks

Liquid crystal networks combine the orientational order of liquid crystals with the elastic properties of polymer networks, leading to a vast application potential in the field of responsive coatings, e.g., for haptic feedback, self-cleaning surfaces, and static and dynamic pattern formation. Recent experimental work has further paved the way toward such applications by realizing the fast and reversible surface modulation of a liquid crystal network coating upon in-plane actuation with an AC electric field [Liu, Tito, and Broer, Nat. Commun. 8, 1526 (2017)]. Here, we construct a Landau-type theory for electrically-responsive liquid crystal networks and perform molecular dynamics simulations to explain the findings of these experiments and inform on rational design strategies. Qualitatively, the theory agrees with our simulations and reproduces the salient experimental features. We also provide a set of testable predictions: the aspect ratio of the nematogens, their initial orientational order when cross-linked into the polymer network, and the cross-linking fraction of the network all increase the plasticization time required for the film to macroscopically deform. We demonstrate that the dynamic response to oscillating electric fields is characterized by two resonances, which can likewise be influenced by varying these parameters, providing an experimental handle to fine-tune device design.

Figure 1

Simulation snapshots visualizing the problem geometry. Cross-linked mesogens are indicated in orange (gray), dipolar mesogens in green (light gray), and the main-chain polymer in blue (dark gray). The colored dots drawn on the beads indicate binding sites, which are used as adhesion points during in situ polymerization. The mesogens are prepared with the director field along the $y$ axis and reorient upon actuation with an electric field along the $x$ direction, increasing the LCN volume.

Figure 2

Schematic representation of the model mesogens. Cross-linked mesogens are indicated with orange (dark gray) ellipsoids and dipolar mesogens are indicated with green (light gray) ellipsoids. The scalar nematic order parameters of the cross-linked mesogens, $S_1$, and the director-field-aligned dipolar mesogens, $S_2$, are measured along the $y$ axis (dashed arrows), along which both mesogen species are prepared. The scalar nematic order parameter of the electric-field-aligned dipolar mesogens, $S_3$, is measured along the $x$ axis (dashed arrow), along which the electric field is applied. The population fraction $f^2$ denotes the fraction of dipolar mesogens that aligns with the electric field.

Figure 3

Equations of state for the scaled theory (a), (c) and MD simulations (b), (d) as a function of electric field strength. Panels (a) and (b) show the orientational order parameter for the cross-linked (dashed lines) and dipolar mesogens (solid lines), measured along the electric field axis. Panels (c) and (d) show a measure for the concomitantly generated free volume. Theory: Black curves denote stable solutions and green (gray) curves indicate stable solutions with saturated population of electric-field-aligned dipolar mesogens $f^2=1$. We denote the scaled scalar nematic order parameter, measured along the electric field axis, $s$, the scaled free volume order parameter $\tilde{\eta }$ and the scaled electric field strength $\sqrt{h}\propto \left|\underline{E}\right|$. The explicit scalings for these, as well as for our model parameters, are given in the main text at the end of Sec. 3. The values we use for our scaled model parameters are $\tilde{\kappa }=1.0$ for the elastic coupling of the cross-linked mesogens to polymer network, $\tilde{\lambda }=0.4$ for the excluded-volume-like coupling constant, $\zeta =0.03$ for the mesogen-volume coupling constant, $\widetilde{B_1}=5.0$ for the bulk-modulus-like free-energy penalty for the population fraction $f^2$ and $s_0=3.0$ for the initial degree of orientational order, crosslinked into the network. Simulations: the percentage volume increase is measured relative to the undeformed configuration prior to the application of the electric field. We work in terms of fundamental Lennard-Jones units, at fixed temperature $T=0.35$ and fixed pressure $p=1.0$, with time steps $dt=0.001$ and total simulation time $t_{\text{sim}}=300.0$; see the end of Sec. 3 for further details. Different colors denote different random network topologies and lines are guides to the eye.

Figure 4

Time traces of the generated free volume for the scaled theory (a) and MD simulations (b). The dashed line (a) indicates a short-time approximation obtained by linearizing the theory. Theory: we denote the scaled free volume order parameter $\tilde{\eta }$, the scaled time $\tilde{\tau }=\tau /\left[{\xi }^2/{\mathrm{\Gamma }}_{\eta }{B}_0^{2}C\right]$ and we apply a scaled electric field strength $\sqrt{h}=18$. See Appendix pp1 for the full scaling procedure for these, as well as for our model parameters. Parameter values used: $\tilde{\kappa }=1.0, \tilde{\lambda }=0.4, \zeta =0.03, \widetilde{B_1}=5.0, s_0=3.0, \delta f{\left(0\right)}^2={10}^{-16}$ for the initial fraction of field-aligned dipolar mesogens, $\tilde{\gamma }=0.001$ for the free volume relaxation parameter, and ${\tilde{\mathrm{\Gamma }}}_{s_1}=0.05, {\tilde{\mathrm{\Gamma }}}_{s_2}={\tilde{\mathrm{\Gamma }}}_{s_3}=0.1, {\tilde{\mathrm{\Gamma }}}_f=0.01$ and ${\tilde{\mathrm{\Gamma }}}_{\tilde{\eta }}=1.0$ for the scaled kinetic coefficients. Simulations: we work in terms of fundamental Lennard-Jones units, at fixed temperature $T=0.35$ and fixed pressure $p=1.0$, with time steps $dt=0.001$ and electric field strength $E=30.0$; see the end of Sec. 3 for further details.

Figure 5

Resonance frequencies for steady-state free volume gain as a function of electric field strength (a), (b) and the corresponding microscopic mechanisms (c), (d). (a) Steady-state gain in the scaled free volume order parameter $\tilde{\eta }$ as a function of the scaled driving frequency $\tilde{\omega }=\omega \left[{\xi }^2/{\mathrm{\Gamma }}_{\eta }{B}_0^{2}C\right]$, for a scaled electric field strength $\sqrt{h}=7$; the colored symbols denote resonance frequencies. (b) Scaled resonance frequencies $\tilde{\omega }$ as a function of the scaled electric field $\sqrt{h}\propto \left|\underline{E}\right|$ in the regimes of low electric field strength (blue circles) and high electric field strength (red squares). The filled symbols correspond to the field strength used in panel (a). The black curve denotes the numerically evaluated plasticization time at the same field strength (divided by 4 to correct for oscillation of the AC electric field), with the corresponding estimate Eq. (15) indicated with the dashed line. See Appendix pp1 for the full scaling procedure and see Fig. 4 for the used parameter values. (c, d) Schematic representation of the reorientation of the dipolar mesogens during a full oscillation of the electric field in the low-field regime (c) and the high-field regime (d). The dipolar mesogens are indicated by filled ellipsoids, with the dashed outlines denoting their previous orientation. The corresponding response of the free volume order parameter, which is also an important aspect of the resonances, is not shown. See the main text for an explanation.

Figure 6

Steady-state free volume gain as a function of electric field strength and driving frequency for theory (a), (c), MD simulations (b), and experiments (d). Theory: we denote the scaled free volume order parameter $\tilde{\eta }$, the scaled driving frequency $\tilde{\omega }=\omega \left[{\xi }^2/{\mathrm{\Gamma }}_{\eta }{B}_0^{2}C\right]$ and the scaled electric field strength $\sqrt{h}\propto \left|\underline{E}\right|$. See Appendix pp1 for the full scaling procedure and see Fig. 4 for the used parameter values. Simulations are in terms of fundamental Lennard-Jones units, at fixed temperature $T=0.35$ and fixed pressure $p=1.0$, with time steps $dt=0.001$, total simulation time $t_{\text{sim}}=10.0$ and the driving frequency $f$ in terms of ${\left(100\times dt\right)}^{-1}$; see the end of Sec. 3 and Ref. [35] for further details. Panels (b) and (d) are reprinted from Ref. [35] under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/.

Figure 7

Equations of state for the scaled “minimal” theory (a), (c) and MD simulations (b, d) as a function of electric field strength. (a) and (b) show the orientational order parameter for the cross-linked (dashed lines) and dipolar mesogens (solid lines), measured along the electric field axis. Panels (c) and (d) show a measure for the concomitantly generated free volume. Theory: Black curves denote stable solutions and green curves indicate stable solutions with saturated population of electric-field-aligned dipolar mesogens $f^2=1$. We denote the scaled scalar nematic order parameter, measured along the electric field axis, $s$, the scaled free volume order parameter $\tilde{\eta }$ and the scaled electric field strength $\sqrt{h}\propto \left|\underline{E}\right|$. The explicit scalings for these, as well as for our model parameters, are given in the main text at the end of Sec. 3. The values we use for our scaled model parameters are $\tilde{\lambda }=0.4$ for the excluded-volume-like coupling constant, $\zeta =0.03$ for the mesogen-volume coupling constant, $\widetilde{B_1}=5.0$ for the bulk-modulus-like free-energy penalty for the population fraction $f^2$ and $s_0=3.0$ for the initial degree of orientational order, crosslinked into the network. Simulations: the percentage volume increase is measured relative to the undeformed configuration prior to the application of the electric field. We work in terms of fundamental Lennard-Jones units, at fixed temperature $T=0.35$ and fixed pressure $p=1.0$, with time steps $dt=0.001$ and total simulation time $t_{\text{sim}}=300.0$; see the end of Sec. 3 in the main text for further details. Different colors denote different random network topologies and lines are guides to the eye. (To be compared with Fig. 3 in the main text; the range plotted here for the scaled free volume order parameter $\tilde{\eta }$ is larger than in the main text.)

Figure 8

Time traces of the generated free volume for the scaled “minimal” theory (a) and MD simulations (b). The dashed line (a) indicates a short-time approximation obtained by linearizing the theory. Theory: we denote the scaled free volume order parameter $\tilde{\eta }$, the scaled time $\tilde{\tau }=\tau /\left[{\xi }^2/{\mathrm{\Gamma }}_{\eta }{B}_0^{2}C\right]$ and we apply a scaled electric field strength $\sqrt{h}=18$. See Appendix pp1 for the full scaling procedure for these, as well as for our model parameters. Parameter values used: $\tilde{\lambda }=0.4, \zeta =0.03, \widetilde{B_1}=5.0, s_0=3.0, \delta f{\left(0\right)}^2={10}^{-16}$ for the initial fraction of field-aligned dipolar mesogens, $\tilde{\gamma }=0.001$ for the free volume relaxation parameter, ${\tilde{\mathrm{\Gamma }}}_f=0.01$ and ${\tilde{\mathrm{\Gamma }}}_{\tilde{\eta }}=1.0$ for the scaled kinetic coefficients. Simulations: we work in terms of fundamental Lennard-Jones units, at fixed temperature $T=0.35$ and fixed pressure $p=1.0$, with time steps $dt=0.001$ and electric field strength $E=30.0$; see the end of Sec. 3 in the main text for further details. (To be compared with Fig. 4 in the main text; the range plotted here for the scaled free volume order parameter $\tilde{\eta }$ is larger than in the main text.)

Figure 9

Resonance frequencies for steady-state free volume gain as a function of electric field strength (a), (b) and the corresponding microscopic mechanisms (c), (d). (a) Steady-state gain in the scaled free volume order parameter $\tilde{\eta }$ as a function of the scaled driving frequency $\tilde{\omega }=\omega \left[{\xi }^2/{\mathrm{\Gamma }}_{\eta }{B}_0^{2}C\right]$, for a scaled electric field strength $\sqrt{h}=7$; the colored symbols denote resonance frequencies. (b) Scaled resonance frequencies $\tilde{\omega }$ as a function of the scaled electric field $\sqrt{h}\propto \left|\underline{E}\right|$ in the regimes of low electric field strength (blue circles) and high electric field strength (red squares). The filled symbols correspond to the field strength used in panel (a). The black curve denotes the numerically evaluated plasticization time at the same field strength (divided by 4 to correct for oscillation of the AC electric field), with the corresponding estimate Eq. (15) (in the main text) indicated with the dashed line. See Appendix pp1 for the full scaling procedure and see Fig. 7 for the used parameter values. (c, d) Schematic representation of the reorientation of the dipolar mesogens during a full oscillation of the electric field in the low-field regime (c) and the high-field regime (d). The dipolar mesogens are indicated by filled ellipsoids, with the dashed outlines denoting their previous orientation. The corresponding response of the free volume order parameter, which is also an important aspect of the resonances, is not shown. See the main text for an explanation. (To be compared with Fig. 5 in the main text; the range plotted here for the scaled free volume order parameter $\tilde{\eta }$ is unchanged from the main text.)

Figure 10

Steady-state free volume gain as a function of electric field strength and driving frequency for “minimal” theory (a), (c), MD simulations (b) and experiments (d). Theory: we denote the scaled free volume order parameter $\tilde{\eta }$, the scaled driving frequency $\tilde{\omega }=\omega \left[{\xi }^2/{\mathrm{\Gamma }}_{\eta }{B}_0^{2}C\right]$ and the scaled electric field strength $\sqrt{h}\propto \left|\underline{E}\right|$. See Appendix pp1 for the full scaling procedure and see Fig. 8 for the used parameter values. Simulations are in terms of fundamental Lennard-Jones units, at fixed temperature $T=0.35$ and fixed pressure $p=1.0$, with time steps $dt=0.001$, total simulation time $t_{\text{sim}}=10.0$ and the driving frequency $f$ in terms of ${\left(100\times dt\right)}^{-1}$; see the end of Sec. 3 in the main text and Ref. [35] for further details. Panels (b) and (d) reprinted from Ref. [35] under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/. (To be compared with Fig. 6 in the main text; the range plotted here for the scaled free volume order parameter $\tilde{\eta }$ is larger than in the main text.)