Hexatic phase

The hexatic phase is a thermodynamic phase (physical state) between liquid and crystal in two-dimensional systems and represents an anisotropic liquid with a typically six-fold director field ( hexa = Greek six).

The hexatic phase is liquid because the shear modulus disappears due to the dissociation of dislocations . This phase is anisotropic because straight lines can be drawn along the nearest neighbors in six directions. The existence of this directional field means that there must be an elastic module against twisting or torsion in the plane, which is called the Frank constant, analogous to liquid crystals.

Only the dissociation of disclinations at a second phase transition at higher temperature or lower density creates an isotropic liquid in which the elasticity of orientation also disappears. The hexatic phase therefore contains dislocations, but not yet any disclinations.

Order parameters

The hexatic phase can be described by two order parameters , of which the translation order is short-range (exponential decrease) and the orientation order is quasi-long-range (algebraic decrease).

phase	Translation regulations	Orientation order	Defects
crystalline	quasi-long-range: ${\ displaystyle G _ {\ vec {G}} ({\ vec {R}}) \ propto R ^ {- \ eta _ {\ vec {G}}}}$	long-range: ${\ displaystyle \ lim _ {r \ to \ infty} G_ {6} ({\ vec {r}}) \ propto const.}$	defect free
hexatic (anisotropic liquid)	short-range: ${\ displaystyle G _ {\ vec {G}} ({\ vec {R}}) \ propto e ^ {- R / \ xi _ {\ vec {G}}}}$	quasi-long-range: ${\ displaystyle G_ {6} ({\ vec {r}}) \ propto r ^ {- \ eta _ {6}}}$	Dislocations
isotropic liquid	short-range: ${\ displaystyle G _ {\ vec {G}} ({\ vec {R}}) \ propto e ^ {- R / \ xi _ {\ vec {G}}}}$	short-range: ${\ displaystyle G_ {6} ({\ vec {r}}) \ propto e ^ {- r / \ xi _ {6}}}$	Dislocations and Disclinations

Translation regulations

Dislocation pairs destroy the discrete translation order (shift along the red arrows), but an orientation order can still be recognized, which is indicated in one direction by the black lines (above). Only disclinations destroy the orientation order (below).

If the positions of the atoms or particles are known, the translation order can be measured with the translation correlation function as a function of the distance between the lattice space at the location and the location , based on the two-dimensional density function in reciprocal space : ${\ displaystyle G _ {\ vec {G}} ({\ vec {R}})}$ ${\ displaystyle {\ vec {R}}}$ ${\ displaystyle {\ vec {0}}}$ ${\ displaystyle \ rho _ {\ vec {G}} ({\ vec {R}}) = e ^ {i {\ vec {G}} \ cdot [{\ vec {R}} + {\ vec {u }} ({\ vec {R}})]}}$

{\ displaystyle G _ {\ vec {G}} ({\ vec {R}}) = \ langle \ rho _ {\ vec {G}} ({\ vec {R}}) \ cdot \ rho _ {\ vec {G}} ^ {\ ast} ({\ vec {0}}) \ rangle}

The position vector here points to a lattice position in the crystal around which the atom can thermally fluctuate with the deflection , is a reciprocal lattice vector in the Fourier representation . The square brackets indicate the statistical averaging over all particle pairs at a distance R. ${\ displaystyle {\ vec {R}}}$ ${\ displaystyle {\ vec {u}} ({\ vec {R}})}$ ${\ displaystyle {\ vec {G}}}$

In the hexatic phase, the translational correlation function falls rapidly; H. exponentially. In the (two-dimensional) crystal, on the other hand, the translation order is quasi-long-range, the correlation function decays more slowly, i.e. H. algebraic; However, it is not perfectly long-range as in three dimensions, since the deflections due to the Mermin-Wagner theorem at temperatures above absolute zero grow logarithmically over all limits (diverge). ${\ displaystyle {\ vec {u}} ({\ vec {R}})}$

The disadvantage of the translation correlation function is that, strictly speaking, it is only defined in the crystal: at the latest in the isotropic liquid, when disclinations are present, the reciprocal lattice vector can no longer be determined.

Orientation order

The orientation order can be determined from the local directional field for a particle at the location by adding up the angles of the connecting axes to the nearest neighbors in the sixfold space and normalizing with the number of (on average six) neighbors: ${\ displaystyle {\ vec {r}} _ {i}}$ ${\ displaystyle \ theta _ {ij}}$ ${\ displaystyle N_ {j}}$

{\ displaystyle \ Psi ({\ vec {r}} _ {i}) = {\ frac {1} {N_ {j}}} \ sum _ {j = 1} ^ {N_ {j}} e ^ { i6 \ theta _ {ij}}}

${\ displaystyle \ Psi}$ is a complex number of magnitude and the phase indicates the direction of the six-digit director field; for a hexagonal crystal these are the directions of the crystal axes. For a particle with five or seven nearest neighbors, such as dislocations and disclinations, the local director field disappears , except for a very small contribution due to thermal fluctuations. The orientation correlation function can be determined for two particles i and k at a distance from the director field : ${\ displaystyle | \ Psi ({\ vec {r}}) | \ leq 1}$ ${\ displaystyle \ Psi \ sim 0}$ ${\ displaystyle {\ vec {r}} = {\ vec {r}} _ {i} - {\ vec {r}} _ {k}}$

{\ displaystyle G_ {6} ({\ vec {r}}) = \ langle \ Psi ({\ vec {r}} _ {i}) \ cdot \ Psi ^ {\ ast} ({\ vec {r} } _ {k}) \ rangle}

The square brackets again indicate the statistical averaging over all particle pairs. All three phases can be identified with the orientation correlation function. In the crystal, the correlation function does not decay, but rather approaches a constant value: the torsional stiffness is arbitrarily large, i. H. the Frank constant is infinite. In the hexatic phase, the orientation correlation function decays according to a power law (algebraic), in the double logarithmic representation this gives straight lines (shown in green in the figure). In the isotropic liquid, the correlation decays exponentially, resulting in curved curves in the log-log diagram (straight lines in the lin-log diagram). All curves are superimposed by the discrete structure of the atoms or the particles, which can be seen at the minima at half-integer mean particle distances . Particles that are poorly correlated in the positions are also poorly correlated in the local directional field. ${\ displaystyle a}$

The j nearest neighboring particles determine the sixfold directional field of the i-th particle

The orientation correlation of the local directional field as a function of the distance for the crystal (blue), the hexatic phase (green) and the isotropic liquid phase (red).

background

The theory of two-stage melting was developed by John Michael Kosterlitz and David J. Thouless and Bertrand Halperin , David R. Nelson and A. Peter Young in their work on melting in two dimensions. After the first letters of the surname of the authors, the name KTHNY theory has become established.

M. Kosterlitz and D. Thouless were awarded the Nobel Prize in Physics 2016 for the idea of melting through topological defects . The hexatic phase was predicted by D. Nelson and B. Halperin; it has no strict analogue in three-dimensional systems.

Two-digit director fields are known from liquid crystals ; there the anisotropy comes from the elongated (prolate) or flattened (oblate) shape of the individual molecules.

Web links

literature

JM Kosterlitz, DJ Thouless: Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory) . In: IOP Publishing (Ed.): Journal of Physics C: Solid State Physics . 5, No. 11, June 12, 1972, ISSN 0022-3719 , pp. L124-L126. doi : 10.1088 / 0022-3719 / 5/11/002 .
JM Kosterlitz, DJ Thouless: Ordering, metastability and phase transitions in two-dimensional systems . In: IOP Publishing (Ed.): Journal of Physics C: Solid State Physics . 6, No. 7, April 12, 1973, ISSN 0022-3719 , pp. 1181-1203. doi : 10.1088 / 0022-3719 / 6/7/010 .
JM Kosterlitz: The critical properties of the two-dimensional xy model . In: IOP Publishing (Ed.): Journal of Physics C: Solid State Physics . 7, No. 6, March 21, 1974, ISSN 0022-3719 , pp. 1046-1060. doi : 10.1088 / 0022-3719 / 7/6/005 .
David R. Nelson, JM Kosterlitz: Universal Jump in the Superfluid Density of Two-Dimensional Superfluids . In: American Physical Society (APS) (Ed.): Physical Review Letters . 39, No. 19, November 7, 1977, ISSN 0031-9007 , pp. 1201-1205. doi : 10.1103 / physrevlett.39.1201 .
BI Halperin, David R. Nelson: Theory of Two-Dimensional Melting . In: American Physical Society (APS) (Ed.): Physical Review Letters . 41, No. 2, July 10, 1978, ISSN 0031-9007 , pp. 121-124. doi : 10.1103 / physrevlett.41.121 .
David R. Nelson, BI Halperin: Dislocation-mediated melting in two dimensions . In: American Physical Society (APS) (ed.): Physical Review B . 19, No. 5, February 1, 1979, ISSN 0163-1829 , pp. 2457-2484. doi : 10.1103 / physrevb.19.2457 .
AP Young: Melting and the vector Coulomb gas in two dimensions . In: American Physical Society (APS) (ed.): Physical Review B . 19, No. 4, February 15, 1979, ISSN 0163-1829 , pp. 1855-1866. doi : 10.1103 / physrevb.19.1855 .
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M. Kosterlitz: Commentary on Ordering, metastability and phase transitions in two-dimensional systems . In: Journal of Physics C . 28, No. 48, 2016, p. 481001. doi : 10.1088 / 0953-8984 / 28/48/481001 .