1. Introduction
The supercritical CO
2 is widely used in many technologies such as film deposition under high pressure, synthesis of nanoparticles by laser ablation, laser-induced etching of hard materials, impregnation of substances into solid substrates, particle coating, foaming of plastics, etc. The processes involving supercritical fluids (SCF) such as CO
2 or H
2O are sustainable, environmentally friendly, cost efficient and offer the possibility of obtaining new products. This broad technological development has stimulated relevant fundamental research of the most efficient conditions for industrial applications. Significant progress has been achieved in understanding the physical properties of supercritical fluids [
1,
2], largely due to the improvement of experimental techniques [
3,
4]. However, there are several questions regarding the physical micro- and macroscopic behavior of supercritical fluids.
In the vicinity of the critical point, at temperatures from 300 to 350 K and pressures in the range of 65–75 bars, the physical properties of CO
2 drastically change. In particular, the density increases from the values specific for dense gas (about 120 kg·m
−3) to the values typical for compressible liquid (about 750 kg·m
−3) [
5]. The density variations in the vicinity of the critical point correlate with other macroscopic properties (isobaric heat capacity, molecular diffusivity, viscosity, nonlinear refractive index, etc.) that reflect changes at the molecular level, such as cluster formation [
6]. For example, supercritical fluids in the critical point vicinity feature gas-like viscosities. The kinematic viscosity shows a minimum in the vicinity of the critical point [
7,
8]. Thereby, in the practical aspect carrying out processes in SCF may result in energy savings and efficient separation [
8].
Previously, it was believed that the supercritical state is homogeneous. Recent research has made it possible to identify states of matter with different liquid or gaseous properties even under supercritical conditions, namely liquid-like (LL) and gas like (GL) SCF [
9]. A change in the structure of the SCF from liquid-like to gas-like occurs when the representing point in the p-T diagram intersects the coexistence line of these medium states. This so-called Widom line was first identified experimentally in [
9], where the authors demonstrated the inadequacy of the classical representation of the supercritical fluid as a homogeneous medium. (this area was called the “ridge” or “extension curve”). Afterwards this region was renamed to the Widom delta [
10]. In the p-T diagram below the critical point, the vapor–liquid equilibrium line separates the two phases (liquid and gas) [
10]. Above the critical point the vapor–liquid equilibrium line broadens into a two-dimensional region, named the “Widom delta”. In the Widom delta, the SCF can be represented as a mixture of coexisting LL and GL (gas-like) molecules forming intertwined domains [
10]. In the (p, T)-diagram, the Widom delta can be represented as a ridge with a top (Widom line). The Widom line is determined by the maximum density fluctuations [
2,
11,
12] or as a set of points where extremum of some thermodynamic parameters appear [
13,
14]. In the Widom delta, several physical parameters (e.g., adiabatic compressibility, heat capacities, thermal expansion coefficient, speed of sound, thermal conductivity) demonstrate violation of the monotonic dependence on density. The strong adjustability (small change in pressure or temperature leads to the large change of physical parameters) and anomalous behavior of physical parameters in this region make it an object of interest for technological applications. For example, the maximal molecular mobility of CO
2 is achieved in the Widom delta, which is essential when using scCO
2 as a solvent [
15]. Both the anomalous behavior of the parameters and phase transitions are a manifestation of van der Waals interactions of atoms and molecules. They are manifested, pronounced at low temperatures and high pressures. Wherein depending on the temperature and pressure, the formation of clusters, condensation and crystallization is possible. In this regard, one of the possible applications is the use of SCF as a working fluid in supersonic jets. This is a powerful method of modern nanochemistry and nanophysics, that allows efficient adiabatic cooling of the jet substances. The cooling depth and pressure at any point of the jet are controlled by the temperature and pressure in the nozzle chamber (stagnation parameters). This allows, by adjusting the stagnation conditions, to obtain jets containing large aggregates of molecules (clusters, microdroplets and microcrystals). The corresponding jet expansion model (as, for example, [
13]) will allow calculating the distribution of clusters, their sizes and jet parameters in the flow field. Nowadays, active research with cluster jets obtained during the supersonic expansion of a supercritical fluid as laser targets for the generation of neutrons and proton beams are carried out [
14,
15].
As was noted above, the most interesting properties of the SCF (as adjustability, high molecular mobility and strongly nonlinear behavior) are achieved in the Widom delta that is caused by the cluster formation that drastically distinguishes SCF from liquids or gases. The cluster formation can be significantly increased in a mixture of SCFs [
16]. The mixtures find different applications in technology. First of all, the combination of scCO
2 and ethanol can be used as a solvent, moreover in most applications scCO
2 serves as a solvent [
17]. It was shown in [
16] that the critical point of a binary mixture can be significantly shifted from the critical point of pure medium. As we show in [to be published in Russian Journal of Physical Chemistry B] in the binary mixture of CO
2 and ethanol, a new Widom delta (corresponding to the mixture critical point) is formed. Thereby, to identify the current state of matter, it is especially important to have a robust and simple diagnostic tool. The clustering of molecules that occurs in supercritical fluid leads to the modification in the molecular refraction, an increase of the nonlinear refractive index [
18] and the offset of the rotational–vibrational transitions [
19]. Specific modifications in the SCF microstructure can be detected by optical techniques such as Raman spectroscopy. Moreover, the diagnostics can be applied for non-stationary processes such as laser ablation as a real-time feedback.
One of the most commonly used diagnostic tools is a vibration spectroscopy. IR and Raman spectroscopy are also widely used to investigate different samples from inorganic materials to complex organic structures [
20]. Previously, Raman and coherent anti-stokes Raman scattering (CARS) spectroscopy were applied for scCO
2 investigations [
19,
21,
22]. However, the anomalous properties of the Raman spectra in the Widom line vicinity were not highlighted. These methods can be used for both qualitative identification and complex quantitative analysis. The applications of the vibrational spectroscopy include, but are not limited to, the characterization of chemical bonding structure of a molecule, orientation, crystallinity [
20]. The vibration spectroscopy is a complementary technique required to completely measure the vibrational modes of a molecule. Some vibrational modes may be active in both IR and Raman, or only in one spectroscopy [
20]. One of the possible approaches to increase the sensitivity of the spectroscopy is to use nonlinear techniques such as stimulated Raman scattering or CARS [
23].
The optical diagnostics is not limited by spectroscopic methods [
24]. We recently demonstrated that by determining the linear and nonlinear refractive indices, the relative concentration of clusters and the average number of molecules in a cluster can be estimated [
18]. Moreover, the nonlinear refractive index measurements are one of the most sensitive approaches due to nonlinear (for instance, n
2 has a quadratic dependence on the cluster concentration) processes [
25].
The main aim of the work was to demonstrate the influence of CO
2 fluid microstructure (from weakly interacting molecules in the gas phase to clusters in SCF) on the optical properties of the medium. The present article is organized as follows. In
Section 2, we will briefly present the main methods applied in the work. In the
Section 3, it will be followed by the obtained results and their discussion. Finally, the summary and conclusions will be presented in
Section 4.
3. Methods
In the manuscript, we use optical methods (nonlinear refractive index and molecular refractive index measurements, Raman spectroscopy) to obtain the dependence of macroscopic physical properties (nonlinear refractive index, molecular refraction, vibration spectrum) on pressure and temperature. Then, we compare these dependencies with retrieved from the molecular dynamic simulation. After verification of the applied numerical model, we use MD as a tool for visualization of scCO2 fluid microstructure and the clustering characterization. The influence of supercritical state in the cell on the formation of supersonic cluster jets is studied using the Mie scattering technique.
3.1. Raman Spectroscopy
Raman spectroscopy is a well-proven technique that does not require expensive equipment and complex facilities, such as a femtosecond laser; thereby, it is widely used in different fields of science and technology [
30]. We used Raman spectroscopy to retrieve the dependence of Raman-active vibration modes spectrum on pressure and temperature. Raman spectroscopy was performed using the classical scheme of Raman spectroscopy with passing through the SCF cell. The required pressure was set using a high-pressure generator connected to the SCF cell through a needle control valve that was used to adjust the pressure.
The SCF cell was heated using a PID thermoregulator (TPM-210, Owen, Moscow, Russia), with connected band heater and a K-type thermocouple, that regulates the temperature in the cell with an accuracy of 0.1 K. The measurement of the pressure and temperature inside the SCF sell was carried out using a thermocouple inserted inside the cell in a stainless steel capillary with a diameter of 0.5 mm and a digital pressure sensor (P-30, Wika, Frankfurt, Germany). The thermocouple and pressure sensor were connected to a digital meter (TPM-200, Owen, Moscow, Russia), which continuously recorded parameters. The accuracy measurement for temperature was 0.1 K and for pressure 0.1 bar.
The setup used a semiconductor-pumped, Q-switched, pulsed solid-state laser, TEM00, M
2 < 1.2, the pulse energy instability with respect to the root-mean-square deviation was <2%. Laser radiation (527 nm, 1 kHz, 250 mW) was focused into a supercritical cell (length 10.5 cm) by a lens with a focal length of 50 mm. After passing through the cell, the transmitted radiation was collimated and passed through a set of filters that cut off the radiation at the fundamental wavelength. The Raman signal was recorded using a fiber QE-PRO spectrometer (Ocean Insight, Rochester, USA) spectral resolution about 0.8 cm
−1. The experimental setup is presented in
Figure 7.
3.2. Nonlinear Refractive Index and Molecular Refractive Index Measurements
To reveal the behavior of the nonlinear properties of CO
2 in the Widom delta, we measured the nonlinear refractive index for different pressures and temperatures. For measurement of the nonlinear refractive index (n
2), we used a method presented in [
18]. In the framework of the method, the spectral broadening of the laser pulses transmitted through a medium with Kerr nonlinearity is registered. After passing through a nonlinear medium with a known Kerr response, the spectrum of laser pulses broadens due to the self-phase modulation.
where Δ
ωout and Δ
ω are the root-mean-square spectral width of the output and input impulses;
ω0 is the central frequency of the laser pulse,
I is the intensity,
ϕmax is the phase shift of the pulse after passing through the nonlinear medium:
The ratio of the transmitted spectrum bandwidth to the initial spectrum gives a phase shift that is recalculated to the n
2. In these experiments, the radiation of a femtosecond Cr:Forsterite laser system (140 fs, 10 Hz, energy up to 400 µJ) was used. The obtained values of the n
2 were normalized on the density of the fluid. To obtain the linear refractive index (n) of the SCF, we measure the third-order autocorrelation function of the femtosecond pulse passed through supercritical cell. The main laser pulse on the fundamental frequency that passed through the SCF was mixed with its second harmonic generated in the KDP (Potassium Dihydrogen Phosphate) crystal, that gave the autocorrelation function. By varying the time delay, the change in optical path was determined with a change in pressure. After normalizing the obtained data on the value of the refractive index at 1 bar we retrieve the change in the refractive index with pressure. The molecular refraction can be represented as:
where M is the molecular weight and
ρ is the density.
3.3. Mie Scattering
The Mie scattering technique [
31,
32]] is used to study the influence of supercritical conditions on the size and density of molecular clusters in molecular jets in the supersonic jets produced for various applications. The pressure and temperature in the supercritical cell were limited (85 bar and 100 °C). The expansion of the supercritical fluid from the supercritical cell is performed through a conical supersonic nozzle (opening angle 10 deg, length 25 mm, critical section 500 μm and outlet diameter 4.7 mm). The nozzle connects a pulsed gas valve with a vacuum chamber (pressure about 35 mTorr). The scattering signals ratio from the ensemble of particles at the angles
SF(
θ) and
SB(180 −
θ) is related to the average radius of particles in the jet a and the distribution function of the number of monomers in the cluster
f(
nc) as:
We used the software package base on the source code [
32], where size distribution is taken into account to calculate
SF/SB ratio for various average cluster radius
R (6). By comparing the measured
SF/SB ratio with the ratio calculated from the program code, the distribution-averaged radius
R was determined. Symmetric angles were chosen in a manner that the size of the recorded scattering area remains unchanged. The function
f(
n) has the form of a log-normal distribution, the width of the distribution corresponds to the mean radius [
31]. The cluster radius and the number of particles in the cluster are related by the ratio:
R = rN
1/3, where
r is the monomer radius. The verification of the program was carried out on the literature data [
5,
33,
34]. A detailed description of the applied method can be found in [
34].
The cluster concentration
nc was estimated on the basis of Beer’s law according to the method presented in [
35], based on the approximation when all particles in the jet have an average radius
R:
The ratio I/I0 corresponds to the decrease of the probe laser beam intensity after passing through a scattering medium with the length h. The scattering efficiency C was calculated using the Mie program code for a single cluster with an averaged radius. A cw diode laser with a wavelength of 445 nm and a power of 100 mW was applied as a probe beam. The laser beam propagates at a distance of 3 mm from the exit of the nozzle. The probe laser radiation scattered by the cluster jet was captured using an optical fiber placed in a vacuum chamber. The fiber-optic line transfers the scattered light from the cluster jet inside the vacuum chamber to an external PMT. In the experiment, the scattering signal was recorded at angles of 30° and 150° to the laser beam. The laser radiation transmitted through the cluster jet was analyzed using a photodiode, which makes it possible to measure the energy expended in scattering.
3.4. Molecular Dynamics
In the framework of molecular dynamics (MD), the evolution of a system in time is obtained by integrating the equations of atoms’ motion. The movements of atoms are calculated based on the classical mechanics. The interatomic interaction forces are presented in the form of the classical potential forces. We used the LAMMPS software package (Sandia National Labs, New Mexico, USA,
https://lammps.sandia.gov) for MD simulation [
36]. The simulation was carried out for 10,000 atoms, the interatomic interaction potential is COMPASS [
37]. The potential is a “class II” ab initio-based force field, based on the consistent force field (CFF), that is successfully used for modeling both organic and inorganics molecules even at high (up to 200 MPa) pressures [
38]. The usage of the potential was verified by the modeling of such macroscopic parameters as density and Raman spectrum for pure CO
2 and ethanol. The coincidence with table data [
5] is better than 1% in all ranges of pressures and temperatures discussed in this article except the critical and sub-critical where the accuracy of the MD is about 3%. The modeling was carried out in several stages. In the first stage, the system was brought into thermodynamic equilibrium (for given p and T) using a combination of Langevin thermostat (fix langevin), Berendsen barostat (fix pressure/berendsen) and constant NVE (constant volume, number of particles and energy) integration to update position and velocity for every atom each timestep (0.1 fs) for 10 ns. For each timestep we calculated the density, volume, temperature and enthalpy of the system. Further, at a fixed volume and temperature (fix nvT), temperature, the velocity autocorrelation function was calculated (time step 0.001 fs, 2,000,000 steps). The Fourier transform (in a script written in Python) over the original autocorrelation function gives the rotational–vibrational spectrum of the media. The nonlinear refractive index was retrieved using the approach presented in [
25]. During simulation of the binary mixture the fraction of CO
2 molecules to ethanol molecules is 3:1. At the start of simulation, the components of the mixture were separated.
4. Conclusions
We applied optical-based methods for quasi-static characterization nano-and microstructure of scCO2 and binary scCO2-containing ethanol as an admixture. Maximal clustering of the supercritical fluid in the Widom delta leads to the anomalous (violation of the monotonic dependence on the density) behavior of the molecular refraction, nonlinear refractive index and vibrational–rotational CO2 spectrum. The scCO2 in the Widom delta is preferable for applications when the high-density fluctuations or molecular mobility can play the important role in such processes as laser ablation and nanoparticle generation. We showed that optical macroscopic parameters significantly depend on microscopic properties of the fluid that is also followed from the molecular dynamics simulation. Namely, there are extrema in the dependence of molar refraction and nonlinear refractive index on pressure in the Widom delta. Moreover, using Raman spectroscopy, we demonstrated the broadening of the normalized on the density Raman-active lines in the Widom delta. The Mie scattering demonstrates that in the Widom delta the clustering in the molecular jets manifest itself stronger. In combination, the proposed methods give opportunity to characterize the microscopic properties of the CO2 in the broad range of pressures and temperatures including supercritical state. The applications of the proposed approach are not limited by the pure CO2, but also can be applied for CO2-containig mixtures. It is shown that three species-specific deltas can be observed in the CO2-ethanol mixture, and the maximum value of the nonlinear refractive index could be reached in the species-specific delta.
Finally, we want to note that the other optical approaches are required for investigation of the non-stationary process’s dynamics initiated, for example, by a powerful ultrashort laser pulse in subcritical carbon dioxide or water. It is possible that the nonlinear optical techniques (such as femtosecond phase-modulation in the pump-probe regime, or picosecond CARS-technique) are necessary to retrieve the cluster “life time” during laser excitation. These techniques also provide the spatial and temporal resolution in the pump-probe experiments.
We believe that the proposed approach will be especially useful in modern supercritical technologies. We believe that optical methods will continue to play an important role in determining fundamental processes in complex supercritical liquid systems.