Analysis of the Raman Frequency as an Order Parameter Close to the Melting Point in Biphenyl

Introduction

Aromatic compounds are important for industrial applications in polymer chemistry, nanotechnology, molecular biology and molecular energetic and in the condensed phase their phase transition equilibria becomes of fundamental importance [1, 2]. Biphenyl C6H5C6H5 is a crystalline no polar aromatic molecular compound in the series of para-polyphenyl crystals with each molecule consisting of two phenyl groups C6H5 connected by a single C-C bond in Para position [3]. It exhibits at low temperature two structural phase transitions from the high temperature, phase I to two incommensurate phases (phase II and III), which are the second order at TI=40K and the first order at TII=17K, respectively [4]. Transitions between those phases and the dynamic properties have been studied using various techniques such as absorption and fluorescence spectroscopy, Raman, neutron, NMR and EPR as also pointed out previously [3, 4, 5, 6, 7, 8, 9, 11, 12, 13]. Studies on the physical and chemical properties of biphenyl have also been reported in the literature. Some of those studies are the following: surface melting in biphenyl, chemical shift in the solid state of biphenyls, calorimetric study on two biphenyl liquid crystals, phase equilibria and heat capacity of diphenyl ether + biphenyl, electro-optic and dynamic studies of biphenyl benzoate ferroelectric liquid crystals and thermal conductivity epoxies containing substituted biphenyl mesogenic [14, 15, 16, 17, 18, 19]. At ambient pressures and temperatures between 40K and the melting point of biphenyl (~343K), biphenyl crystals in the high-temperature commensurate phase (CI) have a monoclinic structure with space group P21/a (𝐶2ℎ

5 ) with two molecules in the unit cell [3]. Biphenyl as one of the molecular solids melts from the solid state at Tm=343K. Its melting has been detected experimentally by obtaining the Raman spectra of low frequency modes at various temperatures. Differently from the other lattice modes, one low frequency mode at 70cm-1 at 104K decreases to 52cm-1 at the melting point (343K), which is unusual as observed experimentally [20].

𝟐𝑻𝒎 In this study, we analyze the temperature dependence of the Raman frequency shifts for the lattice modes of A and B, which increase as the temperature decreases as expected for the molecular solids. We also analyze the Raman frequency shifts of the 70cm-1 mode. By calculating the temperature dependence of the order parameter S from the mean field theory, we correlate between the Raman frequency shifts and the order parameter S for biphenyl below the melting temperature Tm.

Calculations and Results

We calculated the order parameter S as a function of temperature (Tm =343K) for biphenyl according to the mean field theory.

𝑻 , T<<Tm (1a) 𝑺= [𝟑 𝟏−

𝑺= 𝟎, T>Tm (1c) For this calculation of S, we used Equation (1a). We analyzed the observed frequency shifts of the Raman modes of A and B at various temperatures (below Tm). Since the order parameter S varies between 0 (T>Tm) and 1, the frequency was normalized as 𝜐/𝜐0 where 𝜐0 represents the maximum value of the Raman frequency. The order parameter was then fitted to the 𝜐/𝜐0 according to 𝝊 𝝊𝟎= 𝒂𝟎+ 𝒂𝟏𝑺+ 𝒂𝟐𝑺𝟐+ 𝒂𝟑𝑺𝟑 (2) Where a0, a1, a2 and a3 are constants. This fitting procedure was employed for the three lattice modes of A, B and 70cm-1 with the coefficients a0, a1, a2 and a3 as given in Table 1. Figures 1-3 give our calculated Raman frequencies of the lattice modes of A, B and 70cm-1, respectively close to the melting temperature (Tm=343K) in biphenyl. The observed Raman frequencies of those modes are also given in these figures [20].

Click to enlarge

Click to enlarge

In this study, we analyzed the experimental data for the Raman frequencies of the modes of A, B and 70cm-

1(at 104K) [20], as plotted in Figures (1-3), respectively at various temperatures below the melting point (Tm=343K) in biphenyl. By calculating the order parameter S as a function of temperature from the mean field theory (Equation 1), which was fitted to the observed frequency shifts 𝜐/𝜐0 (Equation 2), the Raman frequencies of the modes A, B and 70cm-1 were obtained as observed experimentally. The Raman frequency of those modes decrease with increasing temperature toward the melting temperature (Tm=343K). This is a drastic decrease for the modes A and B (Figures 1 and 2) as the melting point is approached, whereas decrease in the Raman frequencies for the 70cm-1 mode (Figure 3) is somehow different from those of the modes A and B. This critical behaviour of the 70 cm-1 lattice mode is unusual as observed experimentally, as also pointed out previously. For the modes A and B, a near linear decrease at low temperatures is followed by an increasing rapid decrease as the melting point is approached as stated above and the plots (Figures 1 and 2) Show that the frequency is going rapidly down, in a non linear way, near the melting point it reaches a very low value [20].

This is more apparent for the mode A (Figure 1) as compared to mode B (Figure 2). The fact that the frequency of a particular mode (mode A) of intermolecular vibration approaches zero, leads to the limit of stability of a crystalline phase in the pre-melting process [21]. Since the variation of the frequencies of phonon modes is directly related to the binding forces in a molecular crystal, the occurrence of certain phase changes should be accompanied by a decrease in frequency or decrease and vanishing of some force constants, as suggested by Cochran and Rosenstock, as also pointed out previously. In the case of biphenyl the occurrence of the phase change (melting) should be accompanied by a rapid decrease in the Raman frequencies of the mode A (Figure 1). Mode B can also be considered in the mechanism of the phase change toward the melting point (Figure 2). However, decrease in the Raman frequencies of the 70cm-1mode is continuous (not a rapid decrease) and it reaches about 52cm-1 at the melting temperature (Tm=343K) from the 70cm-1 (at 104K), as stated above. This indicates that the 70cm-1 mode is not associated with the phase change in the pre-melting process, which can be regarded as an internal mode as compared to the external modes (A and B) due to the torsion of the phenyl-phenyl bond [20, 22, 23].

Conclusions

Raman frequencies of the lattice modes of A, B and 70cm-1 in biphenyl near the melting point, were calculated by relating to the order parameter as predicted from the mean field theory. The observed frequencies of those Raman modes were fitted to the temperature dependence of the order parameter near the melting temperature (Tm=343K) in this molecular crystal. The mode frequencies decrease as the melting temperature is approached, which is more pronounced (rapid decrease) for the modes A and B compared to the 70cm-1 mode, as observed experimentally. This indicates that the phase change in the pre-melting process is associated with a rapid decrease of the frequencies of those Raman modes (A and B), in particular, mode A near the melting point in biphenyl.