## 2 Electro-Optic Properties of Polymers
### Transient Properties  
At the end of the poling procedure, when temperature has been lowered to the ambient one and the poling field has been turned off, the orientational distribution \(f(\Omega)\) is no longer at thermodynamic equilibrium and will relax via an orientational diffusion process, slowly decreasing the magnitude of the induced electro-optic properties. The distribution function is then time-dependent \(f(\Omega,\,t)=f(\theta,\,t)\). The same happens if we consider the inverse problem, when an isotropic material is subject, since a certain time, to a poling field. In order to describe these time-dependent phenomena, we introduce the rotational diffusion equation (Smoluchowski-Einstein) for a molecular dipole [33]:  
\[\frac{1}{D}\frac{\partial f\big{(}\theta,t\big{)}}{\partial t}=\frac{1}{\sin \!\left(\theta\right)}\frac{\partial}{\partial\theta}\Bigg{[}\sin\!\left( \theta\right)\!\!\left[\frac{\partial}{\partial\theta}\,f\big{(}\theta,t \big{)}+\frac{1}{kT}\frac{\partial U}{\partial\theta}\,f\big{(}\theta,t\big{)} \right]\!\Bigg{]} \tag{18}\]  
where \(D\) is a diffusion constant, related to the orientational mobility of the nonlinear dipoles in the polymer matrix. However the behavior expected from the model showed above is not always in strong agreement with experimental data, because in the model, it is assumed that the material homogeneously acts on the mobility of the chromophores via a determined diffusion constant \(D\). In reality, the polymer matrix is an amorphous inhomogeneous medium and each chromophore feels a different environment with a own diffusion constant \(D_{i}\). The values of the diffusion constants are statistically distributed with a mean value and variance. For example in the guest-host system the chromophores manly feel two time constants [36], the faster one due to the rotation of the single chromophores around their original position, while the slowest one is due to the relaxation of the polymer host matrix. In side-chain polymers, usually we do not observe a single or double time constant exponential relaxation, but a stretched exponential with a homogeneous distribution of time constants with a given spread. In this case, in order to describe the time dependence of the electro-optic coefficient for an inhomogeneous matrix, the following expression is often used:  
\[r_{zz}\big{(}t\big{)}=r_{zz}\big{(}\infty\big{)}+\Delta r_{zz}\big{(}E_{p} \big{)}\cdot\Phi\big{(}t\big{)} \tag{19}\]  
where \(r_{zz}(\infty)\) is the steady-state electro-optic response, \(\Delta r_{zz}(E_{p})\) is the induced electro-optic coefficient for a poling field \(E_{p}\) and \(\Phi(t)\) is the Kohlrausch-Williams-Watts (KWW) stretched exponential function given by [37, 38, 39]:  
\[\Phi\big{(}t\big{)}=\exp\!\left[-\left(\frac{t}{\tau}\right)^{\beta}\right] \tag{20}\]Where \(\beta\) is a stretching parameter describing the deviation from the simple exponential law. The value of \(\beta\) is always between 0 and 1, where 0 represents a wide distribution of time constants and 1 represents a Dirac delta distribution leading to a single exponential relaxation, representative of the [PERSON] model. For \(\beta\
eq 1\), the \(\tau\) time constant alone is not meaningful and its average must be taken into account:  
\[\left\langle\tau\right\rangle=\int\limits_{0}^{\infty}\exp[-\left(\frac{t}{ \tau}\right)^{\beta}]dt=\frac{\tau\,\Gamma(1/\,\beta)}{\beta} \tag{21}\]  
where \(\Gamma(x)\) is the Euler gamma-function \(\Gamma(x)=\int\limits_{0}^{\infty}t^{x-1}e^{-t}dt\). The average time constant \(<\)\(\varepsilon>\) depends on temperature with different laws:  
For temperature above or equal to \(T_{0}\), a characteristic temperature of the material that is close to, but lower than, the glass transition, the matrix is in a quasi-fluid state and the chromophores can move easily.