Structural Universalities in a Two-Dimensional Yukawa Fluid

The structural properties of a two-dimensional fluid in a wide range of the screening parameter κ are considered by example of a Debye–Hückel (Yukawa) system. The behavior of structural indicators appears universal and is independent of the screening parameter κ. This property makes it possible, in particular, to easily and noninvasively determine the key parameters of the interparticle interaction from the configuration of particles observed in experiments with complex (dusty) and colloidal plasmas.

The behavior of strongly correlated two-dimensional systems is one of the most important problems of condensed matter physics. Such two-dimensional and quasi-two-dimensional systems have been intensively studied beginning with pioneering theoretical works [1–6], for which a Nobel Prize in physics was awarded recently. In those studies, the main attention was paid to the type and properties of a two-dimensional “crystal–liquid” transition. The physics of this transition is much richer than the physics of melting of three-dimensional systems, in particular, because of the existence of an intermediate hexatic phase, which is characterized by the presence of a quasi-long range orientational order [7–14]. The properties of a two-dimensional liquid (particularly, far from the melting curve) are much less studied. In this work, we consider the structural properties of a two-dimensional liquid using a model two-dimensional system with the Yukawa (Debye–Hückel) repulsive potential as an example. The pair interaction between particles in the Yukawa system is described by the potential

$$U(r) = (Q{\text{/}}r)\exp ( - r{\text{/}}\lambda ),$$

(1)

where Q is the charge of a particle and λ is the screening length. It is known that the phase state of this system is described by two parameters: coupling parameter Γ = Q²exp(–κ)/(TD) and the screening parameter κ = D/λ, where D is the average distance between particles in the system and T is the temperature of particles. The Yukawa system is often used to describe experiments in a dusty (complex) plasma and in colloidal systems (see, e.g., [15, 16]). In this work, the main tool to obtain configurations of particles is the classical molecular dynamics method implemented for the canonical (NVT) ensemble with a Nosé–Hoover thermostat and periodic boundary conditions [17]. The typical number of microparticles in the system under study is N ~ 10⁵.

The two-dimensional Yukawa system at zero temperature and any screening parameters κ is a crystal with triangular (hexagonal lattice) and six nearest neighbors of each atom, which are located at the vertices of a regular hexagon. Defects in which the number of the nearest neighbors differs from six appear in this system at finite temperature. Dislocation pairs of five to seven defects (where one particle has five nearest neighbors and the other has seven nearest neighbors) and/or a dislocation quadruple consisting of four defects of different forms, which is the result of the fusion of pairs, usually dominate at low temperatures. The formation of single defects, disclinations (five or seven nearest neighbors), is also possible, but their concentration is usually much lower than the concentrations of dislocation pairs and quadruples because they create strong local elastic tension in the system at which their transformation to a dislocation is energetically favorable.

Figure 1 shows three phase states of the two-dimensional Yukawa system: (a) crystalline phase, (b) hexatic phase observed in the experiment with a colloidal plasma [18], and (c) equilibrium liquid near the melting curve. Defects are indicated in blue and red. Green color corresponds to crystal clusters with six nearest neighbors of the central particle. The inset of Fig. 1а demonstrates the images of dislocation pairs and quadruples, which are clusters of defects most widespread near the melting curve. The comparison of Figs. 1b and 1c indicates that it is difficult to visually distinguish the hexatic phase and melt, but the calculation of the two-dimensional structure factor (which is shown for each phase) allows one to do this easily because fine details of the angular distribution of the nearest neighbors disappear in the two-dimensional liquid and the structure factor becomes isotropic in the space of wave vectors (see, e.g., [7, 19, 20]).

Fig. 1.

(Color online) Three phases of the two-dimensional Yukawa system: (a) crystalline phase, (b) hexatic phase observed in the experiment with a colloidal plasma [18], and (c) equilibrium liquid near the melting curve. (a, c) Results of the numerical simulation and (b) experimental results. The number of nearest neighbors is indicated by the color of particles, which varies from blue to red with an increase in this number. Blue particles correspond to defects with five nearest neighbors and red particles correspond to defects with seven nearest neighbors. Green color corresponds to crystallites with six nearest neighbors. The inset in panel (а) demonstrates the images of dislocation pairs and quadruples, which are clusters of defects most widespread near the melting curve. The corresponding two-dimensional static structure factor, which clearly demonstrates structural differences between the indicated phases, is shown above each configuration.

The simplest characteristic of two-dimensional systems is the pair correlation function $g(r) = 1{\text{/}}N\left\langle {\sum\nolimits_{j \ne k} \delta (r - \;{\text{|}}{\mathbf{r}_{j}} - {\mathbf{r}_{k}}{\text{|}})} \right\rangle$. Its parameters such as the first peak g_max and the first nonzero minimum g_min can be used to quantitatively describe the phase state of the ensemble of particles, as in the three-dimensional case (e.g., [21, 22]). Figure 2 shows the dependences of the parameters g_max and g_min of the two-dimensional liquid near the melting curve on the screening parameter κ. The inset of Fig. 2a shows typical functions $g(r)$ for the Yukawa system near the crystal–liquid transition line at different dimensionless temperatures T* = T/T_m, where T_m is the melting temperature of the system, which are indicated by colors shown in the legend: blue and red lines correspond to the crystalline and liquid states, respectively. The breaking of the translational order (disappearance of long-range correlations) under the melting of the system is clearly seen.

Fig. 2.

(Color online) Parameters (a, red line) g_max and (b, green line) g_min in the two-dimensional Yukawa system near the melting curve versus the screening parameter. The inset in panel (a) shows the function $g(r)$ near the crystal–liquid transition line for κ = 3. The color of lines corresponds to the dimensionless temperature of the system T/T_m, where T_m is the melting temperature of the two-dimensional Yukawa system. Blue and red lines correspond to the crystal and liquid states of particles, respectively. It is clearly seen that the parameter g_min is quasi-universal in a wide κ range. A monotonic increase in g_max with the stiffness of the interaction (the parameter κ) and the quasi-universality of the parameter g_min allow noninvasive determination of the key interaction parameters in experiments with complex and colloidal plasmas. The number of particles in the system is N = 8 × 10⁴.

The dependences of the parameters g_max and g_min on the screening parameter κ demonstrate an interesting effect for Yukawa melts: the maximum of the pair correlation function g_max increases with the stiffness of the interparticle interaction (i.e., with the screening parameter κ), whereas the parameter g_min hardly changes on the melting curve in the considered screening parameter range and at g_min ≈ 0.27 (for κ ≈ 0–8). We note that such a behavior of the parameters g_max and g_min as functions of the stiffness of the pair interaction was also observed for three-dimensional melts of the system with the inverse power-law repulsion [22]. Therefore, the parameters g_max and g_min are important characteristics of the two-dimensional liquid; the parameter g_min can be used as a quasi-universal indicator of melting for a wide class of systems, at least, for two-dimensional Yukawa and Coulomb systems.

In addition, the parameters g_max and g_min make it possible to easily and noninvasively determine the key parameters of the pair interaction in two-dimensional experiments with melts of complex and colloidal plasmas: the parameter g_min indicates that the liquid under study is a melt, and the stiffness of the interaction (the screening parameter κ) is determined from the parameter g_max. A single configuration with N ~ 10³–10⁴ particles in the system is sufficient for such a determination. These N values are easily achievable in experiments with two-dimensional colloidal (see, e.g., [11]) and complex plasmas (e.g., [23, 24]). It can be shown that the parameter g_min for the two-dimensional Yukawa liquid at T/T_m > 1 depends only on the dimensionless temperature T*. These dependences g_min(T*) for several screening parameters κ are shown in Fig. 3. It is clearly seen that g_min(T*) hardly depends on κ in a wide range κ ≈ 0–8 and T* ≈ 1–2.5. For this reason, the parameter g_min is a simple and convenient indicator, which (together with g_max) allows one to determine the location of the studied two-dimensional liquid on the phase diagram even if this liquid is far from the melting curve. Such a universality for three-dimensional Yukawa systems was revealed in [25].

Fig. 3.

(Color online) Parameter g_min in the two-dimensional Yukawa system near the melting curve versus the dimensionless temperature T* = T/T_m for some screening parameters κ. The color of lines corresponds to the parameter κ and varies from blue (κ = 1) to red (κ = 8), covering a wide range of stiffness of the interparticle interaction. It is clearly seen that all lines almost coincide after the “crystal–liquid” transition (for T* > 1); i.e., g_min depends only on T*, whereas the dependence on the screening parameter almost disappears at least up to temperatures T ≈ 2.5T_m.

We note that the integral characteristics of the pair correlation function, such as the pair entropy s₂ [26] and the cumulative pair entropy C_s2, can also be used to characterize the two-dimensional liquid [27–30]. The functions s₂ and C_s2 are defined as

$${{s}_{2}} = - \frac{n}{2}\int\limits_0^\infty {\left[ {g(r)\ln g(r) - g(r) + 1} \right]} dr,$$

(2)

$${{C}_{{s2}}}(r) = - \pi \int\limits_0^r {\left[ {g(x)\ln g(x) - g(x) + 1} \right]} xdx,$$

(3)

where n is the density of particles. It can be shown [27] that s₂ for two-dimensional systems diverges for crystalline and hexatic phases and converges for the liquid phase. The cumulative function C_s2 is a characteristic of this convergence (because ${{C}_{{s2}}}(\infty ) \equiv {{s}_{2}}$) and allows one to easily distinguish the liquid phase from the other aforementioned phases of the two-dimensional material. Figure 4 shows the dependence of s₂ on the dimensionless temperature of the system T* for two values of the parameter κ. The inset of Fig. 4 illustrates the behavior of the cumulative pair entropy C_s2(r) for various T* values and various phase states of matter and, in particular, its divergence for the solid phase. It is seen that s₂ is defined only for the liquid and only a part of the line with T* > 1 in Fig. 4 has a physical meaning. The quasi-universality of s₂ for the liquid near the melting curve is demonstrated for two κ values, but the parameter g_min is favorable in practice.

Fig. 4.

(Color online) Pair entropy s₂ of the two-dimensional Yukawa system near the melting curve versus the dimensionless temperature T* = T/T_m for two screening parameters κ. A sharp decrease in s₂ with increasing temperature T* corresponds to the melting of the system. The inset shows the behavior of the cumulative pair entropy C_s2(r) for various temperatures of the system, covering the solid and liquid phases. The function C_s2(r) converges only for the liquid phase; consequently, the dependence s₂(T*) has physical meaning only at T* > 1.

The concentration of defects n_d and the geometric properties of clusters of defects are very important characteristics of two-dimensional systems [31]. Defects for the triangular lattice are easily determined by the Voronoi diagram of the region occupied by particles (e.g., [31–33]). In this diagram, each particle of the system corresponds to a convex polygon with the number of sides equal to the number of nearest neighbors n_nn. Particles with n_nn ≠ 6 are treated as defects and their number in the system is n_d. A cluster of defects is defined as an assembly of defect particles that are nearest neighbors (for details, see [31]).

Here, we consider the dependences of the concentration of defects n_d only near the melting curve on the dimensionless temperature T* for several values of the screening parameter κ, which are shown in Fig. 5. A sharp increase in n_d at T* ≈ 1 corresponds to the melting of the system. The further behavior of the function n_d(T*) at T* > 1 is quasi-universal and is almost independent of the screening parameter κ at least to temperatures T/T_m ≈ 10. An important feature of the two-dimensional liquid is that the concentration of defects increases slowly (logarithmically) with the temperature; e.g., only half of all atoms are defects at T/T_m ≈ 10. This is illustrated in the inset of Fig. 5, where the distribution of defects in the two-dimensional Yukawa liquid at T/T_m ≈ 5 is shown. Particles with five, six, and seven nearest neighbors are shown in blue, green, and red, respectively. A significant number of crystalline clusters are seen. On the melting curve, n_d ≈ 0.25 for all considered κ values, which means, in particular, that the two-dimensional Yukawa melt consists mainly of crystallites (i.e., clusters of particles with six nearest neighbors). This property fundamentally distinguishes two-dimensional systems from three-dimensional ones (where the melt includes almost no crystalline clusters).

Fig. 5.

(Color online) Relative concentration of defects n_d in the two-dimensional Yukawa system near the melting curve versus the dimensionless temperature T* = T/T_m for some screening parameters κ (indicated on the plot). A sharp increase in n_d at T* ≈ 1 corresponds to the melting of the system. The further behavior of the function n_d(T*) at T* > 1 is quasi-universal and is almost independent of the screening parameter κ. It is noteworthy that n_d ≈ 0.25 on the melting curve, which means that the two-dimensional Yukawa melt consists mainly of crystallites (clusters of particles with six nearest neighbors). The inset shows the distribution of defects in the two-dimensional liquid at T/T_m ≈ 5. Particles with five, six, and seven nearest neighbors are shown in blue, green, and red, respectively.

In this work, the structural properties of a two-dimensional liquid have been considered by example of a Yukawa system. It has been shown that simple parameters of the pair correlation function $g(r)$ such as its first maximum g_max and the first nonzero minimum g_min, are important parameters characterizing two-dimensional systems. In particular, it has been found that the parameter g_min for two-dimensional Yukawa melts is quasi-universal in a wide range of stiffness of the interparticle interaction (i.e., screening parameter κ), which makes the parameter g_min a simple indicator of two-dimensional melting and allows one to easily identify the melt of such a system. The indicated quasi-universality of the parameter g_min apparently reflects the universality of melts of all two-dimensional close-packed systems such as Yukawa systems, Coulomb systems, and soft spheres (systems with inverse power-law repulsion).

A monotonic increase in the parameter g_max with the parameter κ and the quasi-universality of the indicator g_min allow one to easily and noninvasively determine the key parameters of the interparticle interaction in experiments with melts of complex (dusty) and colloidal plasmas, which previously required significant efforts. It has been established that the relative concentration of defects n_d in two-dimensional Yukawa liquid is universal: in a wide κ range, it depends only on the dimensionless temperature T * = T/T_m. Finally, two-dimensional melts consist primarily (75%) of crystallites, which fundamentally distinguishes two-dimensional systems from three-dimensional ones.