Photodynamics of stress in clamped nematic elastomers

We describe the complex time dependence of the buildup of force exerted by a clamped photoelastomer under illumination. Nonlinear (non-Beer) absorption leads to a bleaching wave of a significant cis isomer dye concentration deeply penetrating the solid with a highly characteristic dynamics. We fit our experimental response at one temperature to obtain material parameters. Force-time data can be matched at all other temperatures with no fitting required; our model provides a universal description of this unusual dynamics. The description is unambiguous since these are clamped systems where gross polymer motion is suppressed as a possible source of anomalous dynamics. Future experiments are suggested.

Figure 1

Schematic of a nematic elastomer with regular mesogenic molecules (white rods) and photoactive molecules (gray). In the dark all photoactive molecules are found in their linear trans state (left panel). Upon illumination some photoactive molecules undergo a transition to the bent cis state (right panel), which would lead to contraction if the elastomer were freely suspended. Such contraction is prevented by the clamps, resulting in a force $f$ on the clamps.

Figure 2

Light intensity versus the reduced depth $x/d_{\mathrm{t}}$ for various reduced times $t/\tau$. The dash-dotted lines represent the case with the presence of back photoconversion ($\alpha =36$, $\beta =1$, and $\eta =3$ in this example), while the solid lines correspond to the absence of back photoconversion ($\alpha =36$ and $\beta =0$); note that even a small value of $\beta$ (compared to $\alpha$) significantly affects the variation of $\mathcal{I}$ with depth. In the limit of large $t/\tau$ one achieves the stationary regime described by Eq. (15).

Figure 3

The cis number fraction $n_{\mathrm{c}}$ against the reduced depth $x/d_{\mathrm{t}}$ for various reduced times $t/\tau$. The dash-dotted lines represent the case $\alpha =36$, $\beta =1$, and $\eta =3$, while the solid lines represent the case $\alpha =36$ and $\beta =0$. In the large time limit one approaches the solution presented by the second equation of (14).

Figure 4

Approximate thickness $x_{\mathrm{c}}/d_{\mathrm{t}}$ of the cis layer against the reduced time $t/\tau$ for $\alpha =36$ and $\beta =0$ (solid line) and $\alpha =36$, $\beta =1$, and $\eta =3$ (dash-dotted line). Both curves reach their stationary values after a moderate time (measured in units of $\tau$), which coincide with corresponding values (thin solid lines) obtained from Eq. (22) as explained in the main text.

Figure 5

Normalized stress as a function of time. Theoretical predictions are represented by solid lines, while experimental data, taken from [8], are represented by different symbols corresponding to different temperatures. For better visibility the five curves corresponding to temperatures $T>45{}^{\circ }\mathrm{C}$ are shifted along the ${\overline{\sigma }}_{\mathrm{n}}$ axis by the values 0.05, 0.1, 0.15, 0.2, and 0.25, respectively. Only the $T_0=45{}^{\circ }\mathrm{C}$ curve is fitted, while all other curves are theoretical.