Note: Melting criterion for soft particle systems in two dimensions
aa r X i v : . [ c ond - m a t . s o f t ] A p r Note: Melting criterion for soft particle systems in two dimensions
Sergey Khrapak
1, 2, 3 Institut f¨ur Materialphysik im Weltraum, Deutsches Zentrum f¨ur Luft- und Raumfahrt (DLR), 82234 Weßling,Germany Aix Marseille University, CNRS, Laboratoire PIIM, 13397 Marseille, France Joint Institute for High Temperatures, Russian Academy of Sciences, 125412 Moscow,Russia (Dated: 9 November 2018)
According to the Berezinskii-Kosterlitz-Thouless-Halperin-Nelson-Young (BKTHNY) theory, melting intwo dimensions (2D) is a two-stage process. The crys-tal first melts by dislocation unbinding to an anisotropichexatic fluid and then undergoes a continuous transitioninto isotropic fluid. The dislocation unbinding occurswhen the Young’s modulus reaches the universal value of16 π , 4 µ ( µ + λ )2 µ + λ b k B T = 16 π, (1)where µ , λ are the Lam´e coefficients of the 2D solid, b isthe lattice constant, and k B T is the thermal energy. TheLam´e coefficients to be substituted in Eq. (1) should beevaluated taking into account (i) thermal softening and(ii) renormalization due to dislocation-induced softeningof the crystal. Simplistic theoretical estimates usingthe elastic constants of an ideal crystalline lattice at T =0 yield melting temperatures overestimated by a factorbetween ≃ . ≃ BKTHNY scenario has been confirmed experimentally,in particular for systems with dipole-like interactions.
More recently, it has been reported that 2D melting sce-nario depends critically on the potential softness. Onlyfor sufficiently soft long-range interactions does meltingproceed via the BKTHNY scenario. For steeper inter-actions the hard-disk melting scenario with first orderhexatic-liquid transition holds.
The focus of this Note is on 2D soft particle systems.It is demonstrated that a melting criterion can be intro-duced, which states that the melting occurs when theratio of the transverse sound velocity of an ideal crys-talline lattice to the thermal velocity reaches a certainquasi-universal value.The Lam´e coefficients of an ideal 2D lattice can beexpressed in terms of the longitudinal ( C L ) and trans-verse ( C T ) sound velocities as µ = mρC and λ = mρ ( C − C ), where m and ρ are the particle massand number density. Then the condition (1) can berewritten as 2 π √ v = C (cid:0) − C /C (cid:1) , (2)where v T = p k B T /m is the thermal velocity. For soft re-pulsive potentials, independently of space dimensionality,the following strong inequality, C /C ≫
1, holds.
This implies that Eq. (2) can be further simplified to C T /v T ≃ const (3) TABLE I. Selected properties of 2D one-component plasmawith logarithmic (OCP log), Coulomb (OCP 1 /r ) interac-tions, and of the 2D system with the dipole-like interaction.Here C T is the transverse sound velocity of an ideal triangu-lar lattice, v T is the thermal velocity, and Γ m is the couplingparameter at melting.System f ( x ) C T /v T a Γ m b C T /v T | Γ m OCP log − ln x p Γ / ≃ ÷ ≃ . ÷ . /r /x . √ Γ ≃ ÷ ≃ . ÷ . /x . √ Γ ≃ ÷ ≃ . ÷ . a See e.g. Ref. 17 for OCP log, Ref. 5 for OCP 1 /r , and Ref. 18for the dipole system. b See Refs. 19 and 20 for OCP log; Refs. 21 and 22 for OCP 1 /r ,and Refs. 7 and 18 for the dipole system. at melting. The value of the constant that follows fromEq. (2) is ≃ .
30. However, this does not take intoaccount thermal and dislocation induced softening. Aworking hypothesis to be verified is that a simple renor-malization of the constant in Eq. (3) can account for theseeffects. In this case, Eq. (3) would be identified as a sim-ple 2D universal melting rule for soft particle systems.Let us verify whether the ratio C T /v T does assumea universal value at melting. We consider three exem-plary 2D systems with soft long-ranged repulsive inter-actions: one-component plasmas with logarithmic poten-tial (OCP log),
2D electron system with Coulomb ∝ /r potential (OCP 1 /r ), , and dipole-like systemwith ∝ /r interaction. The pair-wise interactionpotential φ ( r ) can be written in a general form as φ ( r ) /k B T = Γ f ( r/a ) , where Γ is the coupling parameter and a = 1 / √ πρ is the2D Wigner-Seitz radius. The system is usually referredto as strongly coupled when Γ ≫
1. The fluid-solid phasetransition is characterized by a system-dependent criti-cal coupling parameter Γ m (the subscript “m” refers tomelting). All systems considered here form hexagonallattices in the crystalline phase (more complicated inter-actions and lattices should be considered separately).The discussed soft-particle systems have been exten-sively studied in the literature and some relevant infor-mation is summarized in Table I. In particular, the lastcolumn lists the ratios C T /v T at melting. The valuespresented indicate that as the potential steepness growssome weak increase of the ratio C T /v T at melting is likely. MD, Ref [23] Eq. (3), Present work Eq. (1), Ref. [5] m FIG. 1. Melting curve of a 2D Yukawa crystal in the ( κ ,Γ) plane. The solid curve corresponds to the condition C T =4 . v T . The symbols correspond to the results of the numericalmelting “experiment”. The dotted line corresponds to thesolution of Eq. (1) with the asymptotic T = 0 values of elasticconstants. At the same time, all the values are scattered in a rel-atively narrow range, 4 . ± .
3. This can justify usingEq. (3) as an approximate one-phase criterion of meltingof 2D crystals with soft long-ranged interactions.As an example of the application of the proposed cri-terion, the melting curve of a 2D Yukawa crystal hasbeen calculated. The Yukawa potential is characterizedby f ( x ) = exp( − κx ) /x , where κ is the screening pa-rameter (ratio of the mean interparticle separation a tothe screening length). This potential is used as a reason-able first approximation to describe actual interactions incolloidal suspensions and complex (dusty) plasmas. In the latter case, the screening is normally weak, κ .
1, which corresponds to the soft interaction limit.Thermodynamics and dynamics of 2D Yukawa systemsare well understood, simple practical expressionfor thermodynamic functions and sound velocities have been derived. In particular, the transverse soundvelocity of an ideal Yukawa lattice can be expressed us-ing the Madelung constant M ( κ ) as C = v (cid:18) κ ∂ M∂κ + κ ∂M∂κ − M (cid:19) , where the product M Γ defines the lattice energy per par-ticle in units of temperature (reduced lattice sum). Us-ing the condition (3) with an “average” const = 4 .
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