# Reversal of Photoinduced Bending of Crystals Due to Internal Refraction of Light
###### Abstract
The corresponding contribution to \(\varepsilon\) is determined by the value and sign of the coefficient of thermal expansion (CTE), which is most often positive. This phenomenon was described in [27; 28], called there the photothermal effect. Since the CTE value usually does not exceed \(10^{-4}\) K\({}^{-1}\), corresponding strains caused by usual light sources turn out to be one or two orders of magnitude lower than the strains caused by photochemical transformations. For this reason, the photothermal bending is usually masked by the stronger effect from photoisomerization. Due to the fact that photothermal strains and their change with time are associated not with the degree of transformation, but with the resulting temperature profiles in the bulk of the crystal, the photothermal bend is clearly observed only on crystals whose thickness significantly exceeds the depth of light penetration. In this case, the weak contribution to the crystal bending from the transformation of a thin surface layer, the thickness of which slowly increases according to the transformation kinetics, can be surpassed by a stronger bending from the temperature gradient, which appears and develops much faster, according to the kinetics of heat transfer from the surface heated by the irradiation. In this case, the direction of bending itself can differ in sign if CTE and transformation strains have opposite signs. This was confirmed in [29], where rapid but weak reversal bending at angles less than \(1^{\circ}\) was observed on thick salicideneaniline crystals. In the case of thin crystals considered in this study, the heat releases more uniformly in the bulk of the crystal, which further reduces temperature gradients and the contribution of the photothermal effect, which, therefore, can be neglected here. For a more explicit formulation of the direction of crystal bending caused by photochemical reaction, one can consider an expression for the crystal bending rate \(\mathrm{d}R^{-1}/\mathrm{d}t\). For the cases of small transformation strain we can use the linear approximation of the interconnection between the strain and the transformation degree [24];  
\[\varepsilon(\mathbf{r})\ =\ \varepsilon_{0}C(\mathbf{r}), \tag{4}\]  
where \(\varepsilon_{0}\) is a linear coefficient of the transformation strain. Then the bending rate is defined by Equation (5);  
\[\frac{dR^{-1}}{dt}\ =\ -\frac{\varepsilon_{0}}{\int\limits_{S}}\,x\frac{dC( \mathbf{r})}{dt}ds. \tag{5}\]  
The transformation kinetics can be described as a first order reaction, according to Equation (6) [1; 24];  
\[\frac{dC(\mathbf{r})}{dt}\ =\ I_{0}\varphi\mu v\frac{I(\mathbf{r})}{I_{0}}(1-C( \mathbf{r})) \tag{6}\]  
\(I_{0}\) is the intensity of the light incident on the crystal in terms of photons flux (photon\({}^{*}\)cm\({}^{-2}\)s\({}^{-1}\)), \(I(\mathbf{r})/I_{0}\) is the bulk distribution of the relative intensity of light inside the crystal, \(\varphi\) is the quantum yield of the photochemical reaction, \(\mu\) is the absorption coefficient in the Bouguer-Beer-Lambert (BBL) law (cm\({}^{-1}\)), \(v=1/C_{0}\) the volume per one molecule in the crystal (cm\({}^{3}\)), \(C_{0}\) is molecule concentration (cm\({}^{-3}\)). Then the starting bending rate \(\Omega_{0}\) is defined by Equation (7);  
\[\Omega_{0}\ =\ \frac{dR^{-1}}{dt}\bigg{|}_{t\ =\ 0}\ =\ -\frac{\varepsilon_{0}I_{0} \varphi\mu v}{\int\limits_{S}}\,x\frac{I(\mathbf{r})}{I_{0}}ds, \tag{7}\]  
from which it follows that the direction of the bend, that is, the sign of \(\Omega_{0}\), is uniquely determined by the sign of the deformation coefficient \(\varepsilon_{0}\) and the sign of the relative intensity moment, the integral in Equation (7). The latter is normally considered negative as the absorption of light should lead to a negative intensity gradient \(dl/dx<0\).