Article pubs.acs.org/JPCB
On the Characterization of Inhomogeneity of the Density Distribution in Supercritical Fluids via Molecular Dynamics Simulation and Data Mining Analysis Abdenacer Idrissi,†,* Ivan Vyalov,§ Nikolaj Georgi,§ and Michael Kiselev‡ †
Université de Lille 1 Sciences et Technologies, LASIR UMR8516 59655, Villeneuve d′Ascq Cedex, France Institute of Solution Chemistry of the Russian Academy of Sciences, Ivanovo, Russia § Max-Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D 04103 Leipzig, Germany ‡
ABSTRACT: We combined molecular dynamics simulation and DBSCAN algorithm (Density Based Spatial Clustering of Application with Noise) in order to characterize the local density inhomogeneity distribution in supercritical fluids. The DBSCAN is an algorithm that is capable of finding arbitrarily shaped density domains, where domains are defined as dense regions separated by low-density regions. The inhomogeneity of density domain distributions of Ar system in sub- and supercritical conditions along the 50 bar isobar is associated with the occurrence of a maximum in the fluctuation of number of particles of the density domains. This maximum coincides with the temperature, Tα, at which the thermal expansion occurs. Furthermore, using Voronoi polyhedral analysis, we characterized the structure of the density domains. The results show that with increasing temperature below Tα, the increase of the inhomogeneity is mainly associated with the density fluctuation of the border particles of the density domains, while with increasing temperature above Tα, the decrease of the inhomogeneity is associated with the core particles. ne of the most important properties of supercritical fluids (SCF) is the local density inhomogeneity distribution. The inhomogeneity is described in various terms such as formation of clusters with various sizes and as fluctuation of the local density. From the microscopic point of view the formation of these inhomogeneities in the fluid is associated with the increase of the thermodynamic response functions such as the isobaric thermal expansion αP and the isothermal compressibility KT. As a consequence, the density of supercritical fluids can be varied continuously and markedly from gas-like to liquid-like values with a small change in pressure or/and temperature. This feature makes SCF an attractive alternative to liquid solvents for the development of new chemical processes. In several previously reported experimental and theoretical studies the importance of the local density inhomogeneity distribution in the determination of SCF properties has been clearly emphasized.1−12 Particularly, it was demonstrated that the rate of increase of the solubility is maximum at the temperature where large inhomogeneity density fluctuation occurs.13−17 Furthermore, the characterization of the local density inhomogeneity distribution is relevant to the interpretation of experimental studies (vibration spectroscopy, NMR, ...) aimed at understanding molecular interactions in supercritical fluids and which provide information on the immediate environment surrounding a probe molecule. For instance, the position and the width of a given vibration mode are sensitive to the change in the local structure. Apart from this, reactions are likely to be influenced by short-ranged rather than by long-ranged density fluctuation.
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We need then to characterize the local density and particularly the number of molecules contained within a specific distance from a probe molecule. In general, radial distribution functions, accessible by both Xray and neutron scattering measurements and molecular dynamics simulations, provide an important tool to study the local structure in liquid systems. However, since these functions are averaged over the entire system, they do not provide sufficient details on the nature of the local environment of the individual molecules in such systems. In other words, when analyzing the short-range structure by means of the radial distribution functions one has to face the problem that subtle changes of the local structure of certain, e.g., hydrogen bonding atoms are washed out when averaging over the entire statistical mechanical ensemble, and this could hinder the understanding of the structure of the local environment of the molecules.18−21 Attempts to describe theoretically the local density inhomogeneity distribution have been based on algorithms in which threshold values were introduced to decide whether a particle belongs to the same density domain or not. As a consequence several approaches were used. In the earliest one, the single linkage clustering algorithms were used in order to study the density inhomogeneity.3,22 There, the decision whether a given particle belongs to a particular spatial domain (with high or low Received: May 17, 2013 Revised: August 27, 2013 Published: September 9, 2013 12184
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and the spatial extent that they occupy, rt. These values may be estimated from the analysis of the first solvation shell around a probe atom, the spatial extent of which is defined by the position of the first minimum in the radial distribution function. Using the coordination number at this distance we then can estimate nt and rt that we need to use DBSCAN algorithm. However, when the temperature increases, the position of the first minimum becomes poorly defined (Figure 1a) and thus
density domains) is made on defined criteria. The second approach consists of much more advanced grid-based algorithms, where the data space (coordinates) is first split into a set of cells and then such a partitioning is used to quantify the various density domains.23−25 All these previously cited algorithms share the drawback associated with the necessity to introduce an arbitrary threshold parameter to decide whether the particle belongs to a density domain or not. In the first group, such a parameter is the distance (orientation or energy) between a probe particle and the nearest one; in the second group there is also an empirically chosen threshold to define the grid.23,26−28 Another disadvantage arises from the fact that the threshold parameter defined for a given thermodynamic state will not be suitable under other conditions, which makes it impossible to study the inhomogeneity of density distribution dependence over a wide range of thermodynamic conditions. Thus, in order to quantify the inhomogeneity in the density distribution of particles, particularly when approaching the critical conditions, one needs to have objective criteria chosen in a unambiguous way to decide whether a particle belongs to a density domain or not. In this paper we offer a new method to analyze the inhomogeneity of density domain distributions of particles in sub- and supercritical conditions. This method combines molecular dynamics simulation and DBSCAN algorithm (Density Based Spatial Clustering of Application with Noise).29 The density-based clustering approach is a methodology that is capable of finding arbitrarily shaped clusters, where clusters are defined as dense regions separated by low-density regions. This algorithm relies on the definition of two parameters: the first one is the distance rt which defines the spatial extent around a probe particle and the second one is nt which defines the number of particles that have to occur within rt distance. The density domain is then a spatial region which contains a certain number of particles, nt, within a distance rt. This algorithm also allows us to distinguish the core particles of the density domain which are defined as a particles that have more than nt neighbors within rt distance. It can also distinguish the border particles that have less than nt neighbors within rt distance. These border particles should be located in a neighborhood of core particles. A noise particle is any particle that is neither a core particle nor a border particle.29 In order to test our approach, we carried out molecular dynamics simulations (MDS) at various thermodynamic points along the 50 bar isobar of an argon system. The Lennard-Jones parameters of the potential model are ε/kB = 116.79 K and σ = 3.3952 Å.30 The critical parameters of this model are Tc = 153 K, Pc = 5.21 MPa, and ρc = 13.32 mol·L−1, which are close to the corresponding experimental values 150.8 K, 4.87 MPa, and 13.30 mol·L−1, respectively. Simulations were performed with the DLPOLY package.31The equations of motion have been integrated using the leapfrog algorithm, employing an integration time step of 2.0 fs. A typical simulation run was composed of two stages: first, the system of 3375 Ar atoms was equilibrated during 20 ns at a given temperature and pressure; second, in the 20-ns-long production stage 40 000 sample configurations, separated from each other by 50 fs long trajectories, have been recorded to disk for further analyses. The DBSCAN algorithm was applied to the coordinates of Ar obtained from the MDS at each thermodynamic point. As explained previously, to successfully apply DBSCAN algorithm to our problem we need somehow to characterize the average density in terms of the minimum number of atoms, nt,
Figure 1. (a) Temperature dependence of the Ar−Ar radial distribution functions. For clarity, plots of the radial distribution functions are shifted upward from the initial temperature 90 K by 0.5. The dashed vertical line indicates the shift of the first maximum of the radial distribution function. The small arrow indicates the position of the first minimum, which becomes poorly defined as the temperature increases. (b) Temperature dependence of ΔrAr−Ar(n,T) as a function of ⟨rAr−Ar(n,T)⟩. ⟨rAr−Ar(n,T)⟩ is the average distance of the nth neighbor of a probe Ar atom at temperature T and ΔrAr‑Ar(n,T) is the corresponding fluctuation. The large gap in the values of ΔrAr−Ar(n,T) is associated with the occurrence of maximum of the thermal expansion response function. Legend as in (a).
the number of particles within this distance as well. In order to rationalize the choice of these parameters, we propose to use the radial nearest neighbor distribution approach, which allows us to determine the two input parameters in a rational way. In this approach,21,32−38 the neighbors of a central atom are sorted by distance into the first neighbors; second neighbors, etc. Separate radial distribution functions, pα−β(n,r) may be defined for each set of nearest neighbor atoms β (indicated by n), and at distance r from the central atom α. It is obvious that an average is done over all choices of central atom and that the corresponding radial distribution function (rdf) gα−β(r) is equal to ∑npα−β(n,r). We then calculate from these distributions the 12185
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average distance ⟨rα−β(n,T)⟩ between a reference atom α and an atom β belonging to the nth neighbor class pα−β(n,r), as well as the distance standard deviation Δrα−β(n,T) which characterizes the distance fluctuation between a reference atom α and an atom β. As pointed out in our previous works,18,38,39 the position of the maximum in the fluctuation ΔrAr−Ar(n,T) coincides with the position of the minimum of the radial distribution function. It is then suggested here to use the position of the maximum of the fluctuation and the number of particles at this position to define in a rational way the two parameters rt and nt, respectively. It should be mentioned that as the distance rt characterizes the spatial extent of the first coordination shell, the discovered density domains using the DBSCAN algorithms will incorporate the local density fluctuation within this distance, and the results obtained in this paper reflects the short-range structural changes surrounding a probe particle.1 The temperature dependence of ΔrAr−Ar(n,T) as a function of the average distance ⟨rAr−Ar(n,T)⟩ is given for each studied temperature in Figure 1b. We checked that the occurrence of the large gap in the values of ΔrAr−Ar(n,T) between 150 and 160 K is independent of the box size. We then carried out MDS for large box size containing 10 000 and 32 000 Ar atoms. The same behavior was observed for the fluctuation of the distance ΔrAr−Ar(n,T). Hereafter, the temperature at which this gap occurs is referenced as Tα. Furthermore, the existence of this gap was also found for CO2,38 and NH340 potentials near their respective critical temperature. This behavior is associated with the large increase of thermal expansion of the system near the critical temperature. After applying the DBSCAN algorithm on the coordinates of Ar particles given by the MDS, the snapshots of the largest discovered density domains are given in Figure 2 at four temperatures 90, 150, 180, and 210 K. In each discovered density domain, Ar atoms are represented by the same color; the strong color defines the core particles, the light color defines the border particles, while the white color defines the noise particles. We calculated the probability distribution of the largest density domain (highest number of particles, Np forming the density domain). The average number of Ar atoms ⟨Np⟩ in the largest density domain and the fluctuation of the size, ΔNp, of the large density domain were calculated. The behavior of these two parameters as a function of temperature is plotted in Figure 3. This figure shows that at low temperature (high macroscopic density) most Ar atoms are in a large-sized density domains. One can notice that the mean number of Ar atoms ⟨Np⟩ drops dramatically near the critical temperature, and for subsequent temperatures, only small-sized density domains are observed. The largest fluctuation ΔNp in the size of the large density domain occurs near Tα. Furthermore, using the coordinates of Ar atoms belonging to the largest density domain, the DBSCAN algorithm defines the coordinates of border and core Ar atoms. In order to quantify the density of the core and the border Ar atoms belonging to the discovered largest density domain, we used the Voronoi ̈ polyhedral (VP)18,39 analysis by considering the coordinates of core and border particles, to calculate the number density distributions of the core and border of Ar atoms in the discovered density domains. Indeed, the Voronoi tessellation is constructed by associating a single Voronoi cell with each Ar atom that corresponds to the section of the simulation box which is closer to that Ar atom than any other. This makes it straightforward to compute the VP volume. The reciprocal volume of the VP, ρ =
Figure 2. Snapshot of the largest density domain obtained by applying the DBSCAN algorithm to the Ar atoms coordinates at T = 90, 150, 180, and 210 K. Each color is associated with a discovered density domains: the strong and the light color of the same domain define the core and border Ar atoms, respectively. The white color represents noise particles.
Figure 3. Temperature dependence of the average number, ⟨Np⟩ of Ar atoms in the largest cluster and the corresponding fluctuation ΔNp.
1/V, can serve as a measure for the local density around the central particle. We calculated the VP statistical distributions of the local density as well as the average value and the standard deviation of these distributions. The average value and the standard deviation of the local density are denoted here as ⟨ρ⟩ and Δρ, respectively. The Voro++ package41 was used for this purpose: one advantage of this algorithm is that it allows us to calculate the Voronoi cells of Ar atoms near the border of these large density domains with a high accuracy. The behavior of the average number density values ⟨ρC⟩ and ⟨ρB⟩ as well as the corresponding fluctuation ΔρC and ΔρB are given in Figure 4a,b, respectively. The first part shows that the rate of decrease of ⟨ρC⟩ and ⟨ρB⟩ is fast near Tα. Furthermore, the standard deviation, ΔρC and ΔρB, passes through a maximum near this temperature. The occurrence of this maximum is evidence that the density distribution of core and border Ar atoms in the 12186
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at supercritical conditions in order to characterize the local structure in these systems and to help to interpret our spectroscopic data on these systems.
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AUTHOR INFORMATION
Corresponding Author
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[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
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REFERENCES
The Institut du Développement et des Ressources en Informatique Scientifique (IDRIS), the Centre de Ressources Informatiques (CRI) de l′Université de Lille, and the Centre de Ressource Informatique de Haute−Normandie (CRIHAN) are thankfully acknowledged for the CPU time allocation. This project was supported by the Marie Curie program IRSES (International Research Staff Exchange Scheme, GAN°247500).
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Figure 4. (a) Average local density of the core and border Ar atoms as determined by Voronoi polyhedral analysis. (b) Fluctuation of the local density of the core and border Ar atoms as determined by Voronoi polyhedral analysis.
largest density domains becomes increasingly inhomogeneous upon approaching the Tα temperature, and it becomes less inhomogeneous when the temperature increases above Tα. These results suggest that the local density inhomogeneity is associated with the concomitant sharp decrease of the number of Ar atoms forming the large density domains and with the maximum of the fluctuation of this number. Furthermore, the analysis of local density of border and core Ar atoms of these large density domains shows that, when the temperature increases below Tα, the increase of density inhomogeneity is associated mainly with the increase of the density fluctuation of the Ar border atoms, while above Tα, the decrease of the inhomogenity is associated mainly with the decrease of the density fluctuation of the core Ar atoms. It is obvious that the behavior of a vibration mode, or the chemical shift of a probe molecule, is mainly determined by its nearest neighbor molecules and then its local structure. This information is of fundamental importance, as it allows us to determine the influence of the nearest neighbors on the behavior of a reference molecule, particularly the relaxation processes as obtained from vibration spectroscopy, that are mainly associated with the influence of the predominantly interacting neighboring molecules around a reference molecule. The main difficulty is to define a statistical function which characterizes the local structure. Our results point out that combining MD simulation with the DBSCAN algorithm provide detailed insight into changes of the local structure in Ar atomic system. Work is in progress to apply this approach to a molecular system such as carbon dioxide, ammonia, and water 12187
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