#### Abstract

Effective density of vibrational states in H-bonded liquids was measured by Raman scattering method. Actuality of a low-frequency part of the spectrum of the intermediate (fracton) region, which obeys a power law, indicates the correct application of the percolation model. The dependence of exponent on binary solutions concentration has been studied. Existence of correlation of the fractal structure parameter and dynamic viscosity has been noted.

#### 1. Introduction

In order to explain peculiarities of the vibrational spectra of amorphous materials (aerogels, polymers, and glass), in some cases it is more effective to use fractal geometry ideas [1], which reflect an existence of large-scale self-similarity of structural inhomogeneities. Therefore, introducing “fraction” concept, which describes intermediate vibrational excitation—between acoustic and optical phonons, we have succeeded in interpretation of low-frequency spectrum portion (10–100 cm^{−1}) as the distribution of density of vibrational states in the space with fractional dimension [2–4].

Noting [5, 6] that some molecular liquids from the viewpoint of the microstructure have common peculiarities with the amorphous medium, for example, in the water and some aqueous or nonaqueous solutions, due to the presence of hydrogen bonds between the molecules, is forming supramolecular spatial structure type of pseudopolymeric lattice with significant irregularities at microscopic scales regions—to several hundred nanometers. The existence of evidence of large-scale self-similarity within not less than two to three orders of magnitude, as well as the formal similarity of the low-frequency vibrational spectra of amorphous solids and liquids with hydrogen bonds [7], gives reason to consider the latter as a medium with fractal structure.

This paper, in terms of fractal geometry, describes the features of low-frequency Raman scattering of liquids with hydrogen bonds. A type of corresponding density of vibrational states functions has been defined. Factors, which modify liquid scale self-similarity structure, have been investigated.

#### 2. Inelastic Light Scattering by Fractal Media

Percolation theory gives understanding of geometrical and topological peculiarities of unordered media (first of all liquids whose molecules are able to form hydrogen bonds) as well as understanding of cause of the large-scale structure of the self-similarity of inhomogeneities.

Its application to the structure of water [8] contributed, in particular, an explanation of its thermodynamic characteristics in the bulk phase. Propagation of the percolation model for other liquids is possible with a certain amount of hydrogen bonds based on the certain molecule, and in case of a specific molecular geometry [9].

Let us discuss some consequences of this theory by example of water. In ordered phase, as crystal ice is, all potential hydrogen bonds are active and participate in formation of regular crystalline lattice. In liquid phase, some bonds are broken and pseudopolymeric lattice has defect structure. Its defectiveness degree is defined by number of active bonds, which depends on temperature. At some temperature value, (approximately 50°C for water [10]) lattice is fractionized; that is, single frame does not exist anymore. Critical number of bonds corresponds to so-called percolation boundary. When temperature is lower, liquid structure can be represented as an infinite percolation cluster; in cavities of such cluster, there are smaller clusters, small groups of connected molecules and separate molecules. The total number of active bonds defines clusters size distribution. The average size of the cavities determines the size of the structural inhomogeneities; it is called correlation length .

Cluster mass, when it is bigger than correlation length , is related to its size by ratio: where —Euclidean space dimensionality. For smaller dimensions , the ratio of the mass-size affects the geometry of the openwork and is typical for fractals: where corresponding dimensionality is called fractal one and such dimensionality can be fractional. Therefore, when disordered media structure dimensions are commensurate with correlation length and smaller, the structure is described in fractal geometry terms.

The intensity of the Raman scattering is observed in this case for the Stokes components of the spectrum can be written in form proposed by Shuker and Gammon [11]: where is coupling function of light with vibrational modes, is density of vibrational states, and —Bose-Einstein factor.

In accordance with Debye approximation, quantity of vibrational states in range is proportional to volume of reciprocal space corresponding area . Expansion of this term on space with arbitrary dimensionality (including fractional dimensionality) results in expression:

Inhomogeneity of the medium cause’s irregular (diffuse) nature of the propagation of vibrational excitations in it is described by the dispersion relation: where parameter is the index of anomalous diffusion. Substitution (5) into (4) allows obtaining a general form of the functions of density of vibrational states for the vibrational excitations in a fractal medium: where —spectral fractal dimensionality. Corresponding vibrational excitations are called fractal dimensionality. Fractal dimensionality represents medium fractal properties in relation to elastic wave propagation.

Vibrational excitations with wavelengths that are longer than a certain correlation length are not sensitive to medium fractality and, therefore, they have linear dispersion: . Critical wavelength corresponds to the so-called crossover frequency. Vibrational excitations with lower frequency are acoustic phonons, with higher frequency—acoustic fractons. Crossover frequency is revealed as a peculiarity in vibration spectra. In particular, it is proved [12] that, for water at room temperature, cm^{−1}, which corresponds to nm with dispersion law for phonons taken into consideration.

According to [11], the function is defined as the average of the dielectric response of the modulation of the vibrational modes with frequencies in the vicinity of . However, , where is correlation length and is effective dimensionality of space, which generally complies with fractal dimensionality.

Since the power-law scaling is correct both for wavelength and correlation length, with dispersion relation (5), taken into account for , we have power-law dependence on the frequency, which allows us to write a function of the effective density of vibrational states as the following:

The exponent of , which is equal to the slope of the linear portion of the spectrum fracton in double logarithmic scale, is sensitive to the relative molecules orientation and defines thermal vibrational excitations distribution in disordered medium. In general, the index is determined by the fractal dimension of the instantaneous network of hydrogen bonds and can therefore be defined as an integral parameter of the supramolecular structure of the medium.

#### 3. Experiment

We investigated low-frequency (10–100) cm^{−1} Raman spectra portions for a range of liquids and some binary aqueous solutions. All samples complied with chemically pure substances; dilution of volume concentration error did not exceed 1%. Investigations have been carried out by means of DFS-24 spectrometer (Fastie-Ebert double diffraction monochromator with a resolution of 5 cm^{−1}). Argon laser emission with 514 nm wavelength and 100 mW output power was used as an excitation source.

#### 4. Results and Discussion

The main problem, which appears when vibrational density of states of low-frequency spectra is changed, is a correct extraction of structureless wanted signal from intensive area of Rayleigh scattering. Normally, spectrum transformation, based on expression (3), is used for this purpose. Finally, the following intensity distribution is obtained: where is intensity distribution in initial Raman spectrum and is Bose-Einstein factor.

Thus, the reduced intensity is equal to the effective density of vibrational states:

Figure 1(a) shows the initial Raman spectra of liquids with hydrogen bonds, in Figure 1(b), are reduced spectra shown in double logarithmic scale. Analysis of the figures shows that, in all cases, the portion (10–50) cm^{−1} on a logarithmic scale is linear; hence the reduced dependence of the intensity (and the effective density of vibrational states) describes the frequency of an exponential function. The exponent which is equal to the slope of the straight portion is equally introduced above structural parameter .

**(a)**

**(b)**

The value of structural parameter for water () coincides with the measurement results [12] and for values ****close to the amorphous structure of tetrahedral environments bonds in particular aerogels [3].

The absolute value of the parameter is a visual characteristic of the structure of the hydrogen bonds. Because of the lack of a universal algorithm for calculating the communication function , the problem of finding the numerical correspondence between the structural parameter and the fractal dimension of a disordered medium in general seems to be very complicated.

The authors cited above [12] were able to establish such a connection to structure of the liquid water. For this purpose, the outcomes of neutron scattering experiment and computer simulation were additionally used. We can expect that similar bond will be defined for other liquids provided that corresponding data are available.

At the same time the structural parameter can be quite effective and intuitive and provide the new information in case of comparison of low-frequency vibrational spectra for liquids with different molecular organization, but with the same type of connection. With that end, in view, concentration dependencies of structure parameter in binary aqueous solutions have been investigated.

Results of these investigations reveal that there are at least two parameter behavior scenarios, which depend on concentration. Glycerol-, ethanol-, and acetone-water solutions display -like dependence somehow or other (Figure 2(a)). At the same time similar water-ethylene glycol solution dependence has no peculiarities (Figure 2(b)).

**(a)**

**(b)**

It should be noted that, in the work [12], the concentration dependence of was not investigated, ****a conclusion which only was made about the effect of ethanol on the position of the crossover frequency, but not on the value of structural parameter. Thus, we observed manifestation of the structural peculiarities of the binary solutions in the low-frequency Raman spectra for the first time.

In our opinion, overall picture of the formation of the intermediate (fracton) region of the Raman spectrum of binary solutions can be as the following. Medium polarization fluctuations are defined by contributions from each molecule. Molecular polarizability, which is defined by molecular symmetry, is added with polarizability of its intermolecular bond (hydrogen bonds in our case), which, in turn, depends on symmetry of these bonds.

In cases when symmetry of molecular bonds does not comply with symmetry of compact package, in essence, hydrogen bonds must have breaks even when thermal agitations are absent. If bond net is formed of molecules of different types, combined package will depend not only on symmetry of molecules, but also on their proportion.

In general, hydrogen bonds fractal dimension can be expressed by means of binary solution concentration as a function of specific number of hydrogen bonds and package type: where and are number of hydrogen bonds in the first and second class molecules, respectively and is package parameter for specified volume concentration of the first and the second class molecules, which represent a result of molecular mix with different symmetry.

In order to be able to analyze available concentration dependencies, let us assume that function for all liquids is linearly dependent on concentration [13]. This assumption allows a qualitative assessment of the structural peculiarities of the solutions.

Potentially glycerol molecular can form six hydrogen bonds, but its relatively complicated structure impedes this; finally the average number of bonds in one molecule does not exceed two. Ethanol has the same number of bonds. Water molecule can form four hydrogen bonds.

Type of dependence of structure parameter on concentration for glycerol, ethanol, and acetone in some degree (Figure 2(a)) testifies existence of different structure phases in solutions: depleted, enriched, and intermediate, which is typical for micellar systems.

In the first and in the second cases, solution structure is realized as a coexistence of matrix-dissolvent and other liquids nanodimentional associates. For glycerol-water solutions, such partitioning is confirmed by high-frequency electrometry. Vibration excitations, which are propagated in dissolvent percolation, dominate at power spectrum formation as compared with excitations, localized in associates. In intermediate phase, some kind of solution components structure competition takes place: fractal dimension of hydrogen bonds general net becomes quite sensitive to concentration.

It is worth to mention that concentration, which corresponds to the most radical restructuring of water-glycerol solution hydrogen bonds net (in concentration dependence such radical restructuring corresponds to point of inflexion), coincides with anomaly density of this solution. When volume glycerol concentration is approximately equal to 40%, the solution density deviates from a sum of partial densities of its components as much as possible. Therefore, a reason for anomaly density decrease is restructuring of hydrogen bonds net, caused by a change of molecules package conditions.

Similar considerations can be applied for ethanol and acetone aqueous solutions where anomaly densities can be expected as well (possibly, less obvious).

Ethylene glycol molecule, as well as water molecule, is able to create four bonds. When solution concentration is changed, molecules of other class are gradually incorporated into percolation cluster, which is built of molecules of the same class. Cluster fractal dimension is steadily changed in accordance with symmetry of both molecular—constituent and configuration of hydrogen bonds. Therefore structure competition is absent, which is obvious from structure parameter linear concentration dependence.

Acetone has no hydrogen bonds that are able to create percolation cluster. At least, sufficient nonlinearity of spectrum low-frequency portion (Figure 1(b)) denotes inapplicability of percolation model to pure acetone. Add-on of small water volume fraction (5%) into acetone facilitates hydrogen bonds net formation, which creates a structure with fractal properties. In the low-frequency Raman spectrum appears a straight “fractons” portion.

For comparison purpose, spectra of some other liquids without hydrogen bonds have been measured and analyzed. As it can be seen from Figure 3, benzene and carbon tetrachloride, as well as pure acetone, have no linear section in low-frequency area of effective density of vibrational states spectrum.

Separately, for the water-glycerol solution, structural parameter in the concentration range of 0–10% was measured. According to [14] in the water-alcohol solutions at low concentrations, there is an additional maximum of light scattering, nature of which is still not completely clear.

In our experiments, we observed considerable variation of value that was dependent on the concentration of glycerol in water (Figure 4). The concentration at which there was the standard deviation maximum is close to the values obtained in the work [14]. It is well known that thermodynamic equilibrium in the water-glycerol solution is achieved for a very long time compared with the characteristic molecular times and, depending on the concentration, can vary from hours to days [6, 14]. At the same time, dependence of the light scattering signal can have complex, sometimes, oscillatory behavior [6, 14].

We performed a set of measurements of the structural parameter for the same series of hermetically sealed samples of water-glycerol solutions at different concentrations during 8 days (Figure 5).

According to our data, the watery-glycerol solution at any time and at any concentration corresponds to the percolation model, since the relevant reduced spectra have a line form in the log-log scale. However, as can be seen from Figure 5, the structural parameter changes with time. For concentrations of 4, 6, and 8% these changes have an alternating pattern.

In the framework of the percolation model, the dynamics of equilibrium establishment in the solution can be explained by the restructuring of the percolation cluster associated with the involvement of additional hydrogen bonds from glycerol. It can be explained by a slow exchange of glycerol molecules between nanodimentional associates (micelles) and matrix dissolvent.

The dynamic nature of the supramolecular structure of liquids with hydrogen bonds causes the dependence of the effective spatial dimension of the percolation cluster on the lifetime of the individual hydrogen bonds. The same parameters (lifetime and average number of hydrogen bonds per one molecular) define liquid dynamic viscosity.

Calculations based on the experimental data of values ****of the structural parameter correlated with the lifetime of the hydrogen bonds and the dynamic viscosity (Table 1). The existence of unique correspondence between these parameters enables discussion of used approach as an experimental foundation for development of spectroscopic viscosity definition method for many types of liquids with hydrogen bonds.

Summing up, in our interpretation of the low-frequency Raman spectrum of the liquids and, in particular, the water, we based on the paper [12]. Fracton excitations concept has been successfully used to describe the low-frequency vibrational spectrum of amorphous media [2–4]. Nevertheless, for liquids, it is more common to use other interpretations, based on mode-coupling theory [15].

At the same time, our approach does not contradict with it. We present the two straight sections of the density function of vibrational states in the log-log scale which correspond to the dispersion branches of acoustic phonons and fractons. In the other interpretation, these low-frequency parts of the spectrum correspond to the low-frequency wings of two contours—the slow and fast relaxation [16].

#### 5. Conclusions

In this work by the method of Raman scattering effective density of vibrational states in liquids with hydrogen bonds is measured, as well as their binary solutions at room temperature.

The existence of a low-frequency part of the spectrum of the transition (fracton) region indicates the correct application of the percolation model and the concept of fractals to describe the structure of liquids. Angular coefficient for spectrum transfer section in a double logarithmic scale defines distribution of thermal vibrational excitations in nonuniform mediums and can be used as structure integral parameter.

The results of binary aqueous solutions investigations have revealed two scenarios of structure parameter behavior depending on concentration, which comply with two types of liquids. For glycerol and ethanol solutions (as well as for acetone in some degree) competition of hydrogen bonds nets structures. Concentration critical values conform (for glycerol) or predict (for ethanol and acetone) an existence of other physicochemical properties anomalies with corresponding concentration dependencies.

The long-time supramolecular structure forming for water-glycerol solution was investigated. The slow dynamics of equilibrium establishment can be explained by the restructuring of the percolation cluster associated with the involvement of additional hydrogen bonds from glycerol molecule.

The existence of unique correspondence between structural parameter, the average lifetime of hydrogen bonds, and dynamic viscosity can be used as an experimental foundation for development of spectroscopic viscosity definition method for many types of liquids with hydrogen bonds.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.