# Probing Globular Protein Self-Assembling Dynamics by Heterodyne Transient Grating Experiments
## 3 Results and Discussion  
In the HD-TG experiments, the pump fields interacted with the material, inducing into the sample a density grating; this is build up by both a weak absorption of the laser pulses and the electrostriction effects. The absorption was due to the presence in the sample of harmonic/combination vibrational bands having frequencies resonant with the laser wavelength, 1064 nm. The electrostriction was due to the migration of the induced dipole in the regions of the maximum of the electric field gradient. These mentioned effects produced two standing acoustic waves characterized by a phase difference of \(\pi/2\). Both the standing waves decayed in time with the exponential law, \(e^{-\Gamma_{A}t}\). The laser absorption phenomena induced in the sample a third interaction effect, the thermal grating. This is static grating that decays through the heat diffusion processes; represented by the exponential law, \(e^{-\Gamma_{H}t}\). The thermal diffusion time constant, \(\tau_{H}=\frac{1}{\Gamma_{H}}\), was typically much longer than the damping time of the acoustic oscillation, \(\tau_{A}=\frac{1}{\Gamma_{A}}\). The \(S^{\mathrm{HD}}\) can be found using the generalized hydrodynamic equations and can be expressed as follows [28; 30; 41]:  
\[S^{\mathrm{HD}}=A\;e^{-\Gamma_{A}t}cos(\omega_{A}t)+B\;e^{-\Gamma_{A}t}sin( \omega_{A}t)+C\;e^{-\Gamma_{H}t}.\]  
This expression has been utilized to perform the fitting of the measured HD-TG signals. The fitting parameters are: The acoustic angular frequency, \(\omega_{A}\), the acoustic damping rate, \(\Gamma_{A}\), the thermal decay rate, \(\Gamma_{H}\), and the corresponding amplitude constants, \(A\), \(B\) and \(C\). In this work, we focused our attention on the speed of sound, \(C_{s}=\omega_{A}/q\), and the acoustic damping rate, \(\Gamma_{A}\). In particular, we focused on the trend of \(C_{s}\) and \(\Gamma_{A}\) as a function of temperature. The relative uncertainties associated with these values were those related to the fitting procedure, which were \(\Delta\omega_{A}/\omega_{A}\leq 0.1\%\) and \(\Delta\Gamma_{A}/\Gamma_{A}\leq 1\%\), respectively. In the HD-TG measures performed in this work, the angle between the pump laser beams was \(10.4^{\circ}\) and this condition defined a wave vector of \(2.1\) um\({}^{-1}\), which corresponded to a sound wavelength of \(3\) um. Figure 1 shows the linear log plot of the HD-TG signal and the relative fit of the 120 mg/mL lysozyme solution at room temperature. Figure 2 shows the linear-log plot of the HD-TG decay signals and their relative fit of both lysozyme solutions and of the gel system, at room temperature and at 60 \({}^{\circ}\)C. Figure 2 shows that at room temperature (left panel of Figure 2). The damping rate of the acoustic oscillation was higher in the solution with the highest protein concentration (\(\Gamma_{A}=14.7\) MHz), and even higher in the gel system (\(\Gamma_{A}=20.0\) MHz). Such behavior was related to the increase of the sample viscosity, going from the diluted solution to the gel matrix, and to the amount of the relaxation channels that an acoustic wave comes across propagating in the sample. Furthermore, scattering processes could start to play a role during the propagation of ultrasonic waves in these complex materials. The increase of damping rates reflected the increase of the system complexity. With regard to the 120 mg/mL HEWL solution, the damping rate of the acoustic oscillation at 60 \({}^{\circ}\)C (right panel of Figure 2) was lower compared to that at 23 \({}^{\circ}\)C (left panel of Figure 2). In fact, at high temperature, a decrease of solution viscosity occurred, due to the increase of the thermal motions of the molecules. Concerning the concentrated solution and the gel matrix at 60 \({}^{\circ}\)C, their HD-TG decay signals were comparable.