First-principle results for the radial pair distribution function in strongly coupled one-component plasmas

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1 Contributions to Plasma Phys., No. 2/3, xxx-xxx (2015) / DOI /ctpp First-principle results for the radial pair distribution function in strongly coupled one-component plasmas Torben Ott 1 and M. Bonitz 1 1 Institute for Theoretical Physics and Astrophysics, University of Kiel, Leibnizstr. 15, Kiel, Germany Received 2 September 2014, revised 30 October 2014, accepted 30 October 2014 Key words One-Component-Plasma, Pair Distribution Function The radial pair distribution function (RPDF) is the most simple way to characterize the structure of a system. In this work, we provide a comprehensive overview over the dependence of the RPDF on the coupling parameter and screening length in Coulomb and Yukawa One-Component plasmas. These data allows for a precise assessment of the coupling strength of experiments and simulations via a structural measurement and give a benchmark for analytical models. 1 Introduction Strongly coupled plasmas in which the interaction energy exceeds the thermal energy are often modeled as one-component plasmas (OCP, e.g., Ref. [1]) which include pure Coulomb systems and screened Yukawa systems. These models are relevant for dusty or complex plasmas [2, 3], ultracold neutral plasmas [4], ions in traps [5, 6], warm dense matter [7, 8], and colloidal suspensions [3]. One of the primary appeals of the OCP is its simplicity. Only one parameter [besides the screening length in the case of Yukawa systems], = Q 2 /(ak B T ) (where a is the Wigner-Seitz radius and T the temperature), suffices to characterize the system completely and determines all thermodynamic properties. The coupling strength has a straightforward interpretation as the ratio of the nearest-neighbor Coulomb interaction energy Q 2 /a to the typical thermal energy k B T of one particle. Many fields are interested in the liquid strong-coupling regime, roughly defined by 1 c, where the crystallization point c is on the order of 100 and depends on the dimensionality and the screening [9, 10]. Knowledge of is, therefore, of paramount importance for a reliable modeling of a given experimental system. It can, however, be difficult to assess experimentally, since it requires separate measurements of the temperature T, the charge state Q, and the number density n = [(4π/3)a 3 ] 1 (n = [πa 2 ] 1 for 2D). We have, therefore, recently presented a very general method to assess the coupling strength of a threedimensional Coulomb and Yukawa system [11], extending an analogous previous work for two-dimensional systems [12]. In this method, the coupling strength is extracted from structural information alone, without the need to measure the charge state or the temperature. 1 It is based on the observation that the height g max of the first peak of the radial pair distribution function (RPDF) uniquely corresponds to the coupling strength of a system if the interaction potential is known. In the case of one-component plasmas, this requires knowledge of the inverse screening length κ, which then, together with known from g max, uniquely defines all thermodynamic properties of the system. 2 In this contribution, we provide additional numerical data on seven characteristics of the RPDF (see inset in Fig. 1). This substantially extends our previous works [11, 12] that were based on the nearest-neighbor distribution alone, and allows for a more detailed comparison with experimental results for the RPDF that are accessible, e.g., in dusty plasmas or colloidal suspensions. This potentially increases the accuracy with which can be obtained, in particular for systems with polydisperse (e.g., dust particles of varying size) Corresponding author: ott@theo-physik.uni-kiel.de 1 A similar approach has been used in Ref. [13]. 2 Unlike structural properties, dynamic properties such as diffusion can also be influenced by dissipation provided, e.g., by the neutral gas component in dusty plasmas. In this case, three parameters,, κ, and the normalized friction coefficient ν, are needed to define the system. However, since structural quantities are not affected by ν, g max can still be used to obtain.

2 2 T. Ott and M. Bonitz: Radial pair distribution function in strongly coupled one-component plasmas g(p 1) g(p 2) g(d 1) g(r) g= r(cv) r(p1) r(d1) r(p2) g(r) r/a r/a 0 Fig. 1 Radial pair distribution function of Coulomb systems. Left: 2D for = 1, 2, 5, 10, Right: 3D for = 1, 2, 5, 10, The inset shows the definition of the RPDF characteristics. or fluctuating parameters (e.g., fluctuating ion charge states in dense plasmas), where g max may be fluctuating as well. 2 Radial Pair Distribution Function (RPDF) from Langevin Dynamics We obtain the RPDF from first-principle Langevin Dynamics simulation, solving numerically the equations of motion for each particle, m r i = F i m ν v i + y i, i = 1... N (1) where ν is the friction coefficient, F i is the force on particle i due to all other particles and y i (t) is a Gaussian white noise with zero mean and the standard deviation y α,i (t 0 )y β,j (t 0 + t) = 2k B T m νδ ij δ αβ δ(t) (α and β are the Cartesian coordinates). A friction coefficient of ν = 0.1ω p is used for all simulations, where ω p = [DQ 2 /(ma 3 )] 1/2 is the nominal (Coulomb) plasma frequency; D = 2, 3 is the dimensionality of the system. The particles interact via the standard Coulomb or Yukawa potential, V (r) = Q r exp ( κr/a), where κ = a/λ is the dimensionless inverse of the screening length λ and the limit κ 0 recovers the Coulomb potential. We use particle numbers N = 4080 (D = 2) and N = 8192 (D = 3) and employ standard Ewald summation for Coulomb systems [18]. Each simulation spans a time of ωp 1. We consider homogeneous and isotropic N-particle systems. Given a particle at the origin, r = 0, the average density at a distance r from the origin is given by g(r)n, where n is the (areal) average number density and g(r) is the radial pair distribution function the probability of finding a pair of particles separated by r relative to the probability of finding such a pair in an ideal, non-interacting system. Formally, g(r) is obtained by angular integration of the pair distribution function g( r), g( r) = 1 N δ( r r ij ) Nn i,j=1 i j, g(r) = M 1 M k=1 N k(r, r). (2) N(N 1)V (r, r) The second formula shows the expression that is numerically evaluated based on the particle positions [19], where N k (r, r) is the number of particles with pair separation between r and r + r at time step k, V (r, r) is the volume (or area) of a spherical shell with radius r of thickness r in 3D (2D), and M is the number of time steps during measurement. 1 2

3 Radial pair distribution function in strongly coupled one-component plasmas 3 The RPDF determines many thermodynamic properties of a system, including short- and long-range order, compressibility, energy, and pressure [14]. It is also the central input quantity for semi-analytical theories such as the Quasi-Localized Charge Approximation [15, 16]. For an ideal system, g(r) 1. A repulsive interaction lowers g at small distances, giving rise to the so-called correlation void. With increasing coupling, the RPDF develops a series of peaks and dips which indicate an increased particle ordering and shells of locally increased or decreased density around each particle. Upon freezing the peaks exhibit a sudden transformation. The phase transition is associated with a critical height of the first peak which can be regarded as the real-space analogue of the Hansen-Verlet criterion for solidification [17, 11, 12]. The shape of g(r) can therefore serve as a unique indicator of the degree of non-ideality in a system. By modeling the type of the interaction potential operative in a given experimental system, one can uniquely map g(r) to the proper physical coupling strength of the system. 3 Results and discussion Figure 1 shows the variation of the RPDF of a Coulomb system with in two and three dimensions. The growth of the correlation void proceeds rapidly at small values of and slows down at larger, where, instead, the peak structure increases in prominence. The formation of the first (second) peak occurs around = 3 ( = 6). One also observes that the pair distances at which g(r) = 1 are almost completely independent of. The general shape of the RPDF is comparable between 2D and 3D systems, although the former shows a more pronounced peak structure, which reflects the higher packing density of 2D systems. In the following, we consider seven key properties of the RPDF (see inset of Fig. 1): the heights and positions of the first two peaks, the height and position of the first dip, and as a measure of the correlation void the distance at which g(r) first reaches the value 1/2. All lengths are given in units of a, while heights are dimensionless. These data constitute the main result of our paper as they allow one to find the most similar RPDF for a given experiment, allowing for a precise determination of the coupling parameter without recourse to the kinetic temperature or ionic charge state. A position [a] κ = p 2 d 1 p 1 cv B peak position ratio κ = p 2/p 1 d 1/p C peak height κ = p 1 p 2 d 1 D peak height ratio κ = p 1/d 1 p 1 1 d 1 1 p 2/p Fig. 2 2D: Peak heights (A), positions (C), and their ratios (B,D) for Coulomb and Yukawa (κ = ) systems. The labels refer to first peak ( p 1 ), first dip ( d 1 ), second peak ( p 2 ) and correlation void ( cv ).

4 4 T. Ott and M. Bonitz: Radial pair distribution function in strongly coupled one-component plasmas Figures 2 and 3 show these characteristics for the examples of a Coulomb and a Yukawa system at κ =. Subfigures A show the absolute positions of the first two peaks, the first trough and the correlation void. The peak and trough positions are only weakly dependent on and the pair interaction. The correlation hole grows rapidly at small [11] whereas, at larger, this growth slows down being due mainly to the narrowing of the first peak. Even though the peak positions are largely independent of the coupling, their relative positions vary more strongly with (see Subfigures B). The absolute heights of the peaks and troughs (Subfigures C) are a more sensitive measure of the structural properties of the system. The increase in the first peak height proceeds rapidly with, as does the depth of the first dip and, to a lesser extent, the second peak height. The peak height ratio (Subfigures D) increases almost linearly with. For a meaningful comparison of the heights of the first maximum and the first minimum, we also compare their deviation from unity. Their ratio depends in a non-monotonic manner on, with larger ratios at small and large coupling strengths and a shallow minimum in-between. This behavior is more prominent for two-dimensional systems. Tables 1-6 (see Supplementary Material) give more detailed data for the parameters of Figs. 2 and 3 and also include data for additional screening lengths. A position [a] κ = p 2 d 1 p 1 cv B position ratio κ = p 2/p 1 d 1/p C κ = p 1 D κ = peak height p 2 height ratio p 1 1 d 1 1 p 1/d 1 d 1 p 1/p Fig. 3 3D: Same as Fig. 2 but for 3D. To summarize, in this paper, we have presented first-principle data of the radial pair distribution function for 2D and 3D plasmas where the interaction is either unscreened or screened. These data extend those reported in our earlier papers [11, 12] where we have developed methods for obtaining the coupling parameter without knowledge of the kinetic temperature or the charge state and have presented analytical expressions for an effective coupling parameter of screened systems. With the additional data presented here, a more fine-tuned matching between the structure [i.e., g(r)] and the coupling parameter is possible. For experimental systems composed of polydisperse particles (e.g., dusty plasmas) or with fluctuating parameters, a particularly useful quantity to determine the physical coupling parameter was found to be the amplitude ratio of the first peak to the first minimum, p 1 /d 1.

5 Radial pair distribution function in strongly coupled one-component plasmas 5 4 Supplementary Material r(cv)/a g(p 1 ) r(p 1 )/a g(d 1 ) r(d 1 )/a g(p 2 ) r(p 2 )/a Table 1 2D, : Properties of the RPDF, see definition in Fig. 1. Missing values indicate non-existing or shallow extrema.

6 6 T. Ott and M. Bonitz: Radial pair distribution function in strongly coupled one-component plasmas r(cv)/a g(p 1 ) r(p 1 )/a g(d 1 ) r(d 1 )/a g(p 2 ) r(p 2 )/a Table 2 2D, κ =

7 Radial pair distribution function in strongly coupled one-component plasmas 7 r(cv)/a g(p 1 ) r(p 1 )/a g(d 1 ) r(d 1 )/a g(p 2 ) r(p 2 )/a Table 3 2D, κ =

8 8 T. Ott and M. Bonitz: Radial pair distribution function in strongly coupled one-component plasmas r(cv)/a g(p 1 ) r(p 1 )/a g(d 1 ) r(d 1 )/a g(p 2 ) r(p 2 )/a Table 4 3D,

9 Radial pair distribution function in strongly coupled one-component plasmas 9 r(cv)/a g(p 1 ) r(p 1 )/a g(d 1 ) r(d 1 )/a g(p 2 ) r(p 2 )/a Table 5 3D, κ =

10 10 T. Ott and M. Bonitz: Radial pair distribution function in strongly coupled one-component plasmas r(cv)/a g(p 1 ) r(p 1 )/a g(d 1 ) r(d 1 )/a g(p 2 ) r(p 2 )/a Table 6 3D, κ =

11 References 11 Acknowledgements This work was supported by the DFG via SFB TR-24, project A7, and the HLRN via project shp References [1] S. Ichimaru, H. Iyetomi, and S. Tanaka, Phys. Rep. 149, (1987). [2] M. Bonitz, C. Henning, and D. Block, Rep. Prog. Phys. 73, (2010). [3] A. Ivlev, H. Löwen, G. Morfill, and P. Royall, Complex Plasmas and Colloidal Dispersions: Particle-resolved Studies of Classical Liquids and Solids, (World Scientific Publishing Company, Incorporated, 2012). [4] T. Killian, T. Pattard, T. Pohl, and J. Rost, Phys. Rep. 449, (2007). [5] A. Dantan, J. P. Marler, M. Albert, D. Guénot, and M. Drewsen, Phys. Rev. Lett. 105, (2010). [6] J. Wrighton et al., Contrib. Plasma Phys. 52, 45 (2012) [7] M. Koenig, et al., Plasma Phys. Controlled Fusion 47, B441 (2005). [8] D. O. Gericke, K. Wünsch, A. Grinenko, and J. Vorberger, J. Phys. Conf. Ser. 220, (2010). [9] S. Hamaguchi, R. Farouki, and D. Dubin, Phys. Rev. E 56, (1997). [10] P. Hartmann, G. J. Kalman, Z. Donkó, and K. Kutasi, Phys. Rev. E 72, (2005). [11] T. Ott, M. Bonitz, L. G. Stanton, and M. S. Murillo, Phys. Plasmas 21, (2014) [12] T. Ott, M. Stanley, and M. Bonitz, Phys. Plasmas 18, (2011). [13] J. Clérouin, G. Robert, P. Arnault, J. D. Kress, and L. A. Collins, Phys. Rev. E 87, (2013). [14] J. Hansen and I. McDonald, Theory of Simple Liquids (Academic Press, London, 2006). [15] G. Kalman and K. I. Golden, Phys. Rev. A 41, (1990). [16] T. Ott, H. Kählert, A. Reynolds, and M. Bonitz, Phys. Rev. Lett. 108, (2012). [17] J. P. Hansen and L. Verlet, Phys. Rev. 184, (1969). [18] M. Deserno and C. Holm, J. Chem. Phys 109, (1998). [19] J. Haile, Molecular Dynamics Simulation: Elementary Methods (Wiley, New York, 1976).

arxiv: v1 [cond-mat.stat-mech] 1 Oct 2009

arxiv: v1 [cond-mat.stat-mech] 1 Oct 2009 Shell Structure of Confined Charges at Strong Coupling J. Wrighton, J. Dufty, M. Bonitz, 2 and H. Kählert 2 Department of Physics, University of Florida, Gainesville, FL 326 arxiv:9.76v [cond-mat.stat-mech]