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Two-dimensional melting of dislocation vector systems
Surface Science 125 (1983) 285-290 North-Holland Publishing Company
TWO-DIMENSIONAL SYSTEMS
285
MELTING OF DISLOCATION
VECTOR
Y. SAITO Institut fiir Festkijrperjorschung Germany
der KertI/orschungsanloR
Received
for publication
13 May 1982; accepted
Jiiltch. D - JI 70 J&-h,
Fed. Rep. oj
IO June 1982
Dislocation vectorsystems with various dislocation core energies are simulated, and the nature and the mechanism of the melting phase transition there is determined by means of the energy, specific heat. dislocation density, renormalized coupling constant. shear modulus and orientational stiffness constant as well as microscopic configurations of dislocation vectors. For a system with a large core energy the melting transition is found lo be continuous, caused by the dislocation unbinding mechanism predicted by Kosterlitz-Thouless and Halperin-Nelson-Young. For a system with a small core energy, grain boundary loops are nucleated in the process of melting and the phase transition turns out to be first order. The latter agrees with most of the computer experiments on atomistic systems.
1. Introduction The mechanism of the melting transition in three dimensions is not yet known definitely. In two dimensions new mechanisms of melting transition have been proposed, namely dislocation unbinding [I] and dischnation unbinding [2]. These two mechanisms lead two consecutive continuous melting transitions, first to a hexatic phase and then to a true liquid phase. On the contrary most of the computer experiments on various atomistic systems indicate the melting transition to be of first order with discontinuity and hysteresis [3-7). Recently a first order melting transition is claimed to be possible if one takes into account a collective excitation of dislocations, namely a grain boundary [8]. In spite of all these investigations, however, the relation between theories and computer experiments is not clear, since experiments deal with atomistic systems with Lennard-Jones, hard core, soft core or Coulomb interactions, whereas theories deal with an elastic continuous medium with additional dislocations. Here I report results of simulations of dislocation vector systems directly [9] and compare the resulting phase transition with theories and atomistic computer experiments. The Burgers vectors of dislocations in a triangular lattice crystal have six 0039-6028/83/CtOOO-oooO/$03.00
0 1983 North-Holland
286
Y. Saito / 2D melting
of dislocakm
vector systems
orientations, ( If:a,, 0) and (+ ia,, + ffiao), where a, is the lattice parameter of the triangular lattice. According to the continuum elasticity theory [IO], the dislocations interact with each other with a Hamiltonian
(1) Here J is the coupling constant related to the Lame coefficients of the triangular lattice X and F by J = p(p + X)/n(2~ + X), and a and E, are the dislocation core radius and core energy respectively. The number and the distribution of dislocations are determined according to the grand canonical distribution ( CKe-‘K‘/T ) at a temperature T. Since an isolated dislocation costs energy proportional to the logarithm of system size, the following “neutrality” condition should be satisfied:
Cbi = 0.
(2)
Monte Carlo simulations are performed for a system with finite size under periodic boundary conditions. Therefore the following two modifications are applied on the system: (I) The position of a dislocation vector is restricted to a triangular mesh lattice with a lattice parameter 2a. (2) Accounting for a finite size effect and the symmetry of the triangular mesh lattice, the two-dimensional Green function In r is replaced by a triangular lattice Green function in the Hamiltonian (1). The orientation dependent interaction is also replaced by the corresponding one in a finite system. See details in ref. 19). The creation or annihilation of a nearest neighbour (nn) dislocation pair with the neutrality condition (2), as well as diffusion of a dislocation vector is permitted. The simulated sizes are N = 418 and 1672 mesh sites, and 2000 to 4000 Monte Carlo steps per nn bond are performed. I have chosen two systems with core energies EC = 0.82 J and 0.57 J, in order to find the change of nature and mechanism of the melting transition as a function of core energy.
2. Continuous melting for a system with a large core energy The energy E and the specific heat C per site for a system with a large core energy EC = 0.82 J behave smoothly, as shown in fig. la, indicating a continuous phase transition. On heating, the dislocation density n increases according to the Arrhenius law at low temperatures t = T/J pi 0.22, as shown in fig. lb. The activation energy corresponds to the formation of a nn dislocation pair. At high temperatures, on the other hand, n increases more than the Arrhenius law. On inspecting the microscopic configurations some dislocations are found
Y. Saito / 20 melting of dislocation vector systems
287
,_,0 !
I/ c
1-2
1(I- +
Fig. 1. Simulation results for a system with a large core energy, E, = 0.82 J. (a) The energy E and the specific heat C. (b) The dislocation density n versus the inverse temperature t- ’ = J/T. (c) The renormalized coupling constant K, and the singular part of the shear modulus cS_ (d) The q-dependence of the orientational stiffness KA( q).
288
Y. Saito / 2D melting
of dislocation vector systems
unbound and free, and the excess density corresponds to these free dislocations. According to ref. [2], when dislocation unbinding takes place, the renormalized coupling constant
should change discontinuously from 16~ to zero. Here X n and /_~n are the renormalized Lame coefficients. K, of the simulation shown in fig. lc is almost equal to the unrenormalized value K = 47r/t at low temperatures, takes the value about 16~7 at t - 0.22, and then decreases steeply and remains rather small at high temperatures. The shear modulus in fig. lc decreases also very rapidly near t - 0.22. In ref. [2], the high temperature phase with free dislocations is said not to be a true liquid, but a hexatic phase with enhanced orientational order. The orientational stiffness constant K, defined as the long wave-length limit of the q-dependent stiffness KA( q), where T/K,(q)=
((q.b(q))(q.b(-q)))/q’
is predicted to be finite in the hexatic phase, whereas the fluid and solid phases respectively. KA( q) calculated fig. Id indicates that in the long wave-length limit K, finite value at f - 0.23. All the findings in this section performs a Kosterlitz-Thouless-type continuous phase
it is zero and infinite in from the simulation in changes from zero to a indicate that the system transition.
3. First order melting of a system with a small core energy Since the theory [ 1,2] uses the “fugacity (- e-Ec/T)” expansion, it may be possible to have a new type of phase transition for a system with a small core energy. In fact one finds a large discontinuity and hysteresis in the energy E and specific heat C as is shown in fig. 2a for a system with a small core energy, E, = 0.57 J. The dislocation density n in fig, 2b also shows a large discontinuity. At low temperatures n follows the Arrhenius law rather well with the activation energy of a nn dislocation pair formation. At high temperatures n is almost saturated. Microscopic configurations such as in fig. 2c reveal that dislocation vectors align to form grain boundary loops. The entangled and folded structure of grain boundary loops can destroy the global orientational order and the high temperature phase is expected to be a true liquid. This is confirmed by the orientational stiffness K,(q) shown in fig. 2d. At high temperatures, T/K,(q) is very large, beyond the value of the instability condition r/72. It also seems to show an increasing tendency in the long wavelength limit (q -+ 0). All these facts indicate that the melting in this system is caused by nucleation of grain boundary loops, and it is of first order to the true liquid phase. This is in agreement with the previous computer experiments on atomistic systems, where the phase transition is of first order and similar loops of dislocations are observed [4].
289
Y. Saito / 213 melting of d~sl~at~a~ vector systems
h
C
1.0
)-
lo-
ozoo 0175 0150
/
/
I
0125
--
f
Iml 1
”
IO 0.5
lo-
) TiK,(ql
- t - i -. \- t/t-\’ ‘\‘-, / -\ --/ -‘- \\ \ _‘\ ,‘_ \ I\--\ \ I
I
I\’
-_
__-_(
--I
--’ _,‘- ;
’
I
,
--i
1\,
\
\
I
if\ -
I
\
\ ‘\\
Fig. 2. Simulation results for a system with a small core energy, E, = 0.57 5. (a) The energy E and the specific heat C. (b) The dislocation density n. (c) Microscopic configuration of dislocation vectors at a high temperature f = 0.15. (d) The orientational stiffness KA(q).
290
Y. S&o / 20 melting of dislocation vector systems
4. Conclusion The order and the mechanism of the melting phase transition is found to depend on the core energy. For a system with a large core energy the melting transition is caused by the dislocation unbinding mechanism and is of continuous order. For a system with a small core energy, melting is caused by the grain boundary loop formation, and is of first order.
References [I] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181; J.M. Kosterlitz, J. Phys. C7 (1974) 1046. [2] B.I. Halperin and D.R. Nelson, Phys. Rev. Letters 41 (1978) 121; D.R. Nelson and B.I. Halperin, Phys. Rev. B19 (1979) 2457. [3] F.F. Abraham, Phys. Rev. Letters 44 (1980) 463. [4] S. Toxvaerd, Phys. Rev. Letters 44 (1980) 1602; Phys. Rev. A24 (1981) 2735 (51 F. van Swol, L.V. Woodcock and J.N. Cape, J. Chem. Phys. 23 (1981) 913. [6] R.W. Hackney and T.R. Brown, J. Phys. C8 (1975) 1813. [7] R.H. Morf, Phys. Rev. Letters 43 (1979) 931. [8] ST. Chui, Phys. Rev. Letters 48 (1982) 933. [9] Y. Saito, Phys. Rev. Letters 48 (1982) 1114. [IO] F.R.N. Nabarro, Theory of Dislocations (Clarendon,
Oxford,
1967).
TWO-DIMENSIONAL SYSTEMS
285
MELTING OF DISLOCATION
VECTOR
Y. SAITO Institut fiir Festkijrperjorschung Germany
der KertI/orschungsanloR
Received
for publication
13 May 1982; accepted
Jiiltch. D - JI 70 J&-h,
Fed. Rep. oj
IO June 1982
Dislocation vectorsystems with various dislocation core energies are simulated, and the nature and the mechanism of the melting phase transition there is determined by means of the energy, specific heat. dislocation density, renormalized coupling constant. shear modulus and orientational stiffness constant as well as microscopic configurations of dislocation vectors. For a system with a large core energy the melting transition is found lo be continuous, caused by the dislocation unbinding mechanism predicted by Kosterlitz-Thouless and Halperin-Nelson-Young. For a system with a small core energy, grain boundary loops are nucleated in the process of melting and the phase transition turns out to be first order. The latter agrees with most of the computer experiments on atomistic systems.
1. Introduction The mechanism of the melting transition in three dimensions is not yet known definitely. In two dimensions new mechanisms of melting transition have been proposed, namely dislocation unbinding [I] and dischnation unbinding [2]. These two mechanisms lead two consecutive continuous melting transitions, first to a hexatic phase and then to a true liquid phase. On the contrary most of the computer experiments on various atomistic systems indicate the melting transition to be of first order with discontinuity and hysteresis [3-7). Recently a first order melting transition is claimed to be possible if one takes into account a collective excitation of dislocations, namely a grain boundary [8]. In spite of all these investigations, however, the relation between theories and computer experiments is not clear, since experiments deal with atomistic systems with Lennard-Jones, hard core, soft core or Coulomb interactions, whereas theories deal with an elastic continuous medium with additional dislocations. Here I report results of simulations of dislocation vector systems directly [9] and compare the resulting phase transition with theories and atomistic computer experiments. The Burgers vectors of dislocations in a triangular lattice crystal have six 0039-6028/83/CtOOO-oooO/$03.00
0 1983 North-Holland
286
Y. Saito / 2D melting
of dislocakm
vector systems
orientations, ( If:a,, 0) and (+ ia,, + ffiao), where a, is the lattice parameter of the triangular lattice. According to the continuum elasticity theory [IO], the dislocations interact with each other with a Hamiltonian
(1) Here J is the coupling constant related to the Lame coefficients of the triangular lattice X and F by J = p(p + X)/n(2~ + X), and a and E, are the dislocation core radius and core energy respectively. The number and the distribution of dislocations are determined according to the grand canonical distribution ( CKe-‘K‘/T ) at a temperature T. Since an isolated dislocation costs energy proportional to the logarithm of system size, the following “neutrality” condition should be satisfied:
Cbi = 0.
(2)
Monte Carlo simulations are performed for a system with finite size under periodic boundary conditions. Therefore the following two modifications are applied on the system: (I) The position of a dislocation vector is restricted to a triangular mesh lattice with a lattice parameter 2a. (2) Accounting for a finite size effect and the symmetry of the triangular mesh lattice, the two-dimensional Green function In r is replaced by a triangular lattice Green function in the Hamiltonian (1). The orientation dependent interaction is also replaced by the corresponding one in a finite system. See details in ref. 19). The creation or annihilation of a nearest neighbour (nn) dislocation pair with the neutrality condition (2), as well as diffusion of a dislocation vector is permitted. The simulated sizes are N = 418 and 1672 mesh sites, and 2000 to 4000 Monte Carlo steps per nn bond are performed. I have chosen two systems with core energies EC = 0.82 J and 0.57 J, in order to find the change of nature and mechanism of the melting transition as a function of core energy.
2. Continuous melting for a system with a large core energy The energy E and the specific heat C per site for a system with a large core energy EC = 0.82 J behave smoothly, as shown in fig. la, indicating a continuous phase transition. On heating, the dislocation density n increases according to the Arrhenius law at low temperatures t = T/J pi 0.22, as shown in fig. lb. The activation energy corresponds to the formation of a nn dislocation pair. At high temperatures, on the other hand, n increases more than the Arrhenius law. On inspecting the microscopic configurations some dislocations are found
Y. Saito / 20 melting of dislocation vector systems
287
,_,0 !
I/ c
1-2
1(I- +
Fig. 1. Simulation results for a system with a large core energy, E, = 0.82 J. (a) The energy E and the specific heat C. (b) The dislocation density n versus the inverse temperature t- ’ = J/T. (c) The renormalized coupling constant K, and the singular part of the shear modulus cS_ (d) The q-dependence of the orientational stiffness KA( q).
288
Y. Saito / 2D melting
of dislocation vector systems
unbound and free, and the excess density corresponds to these free dislocations. According to ref. [2], when dislocation unbinding takes place, the renormalized coupling constant
should change discontinuously from 16~ to zero. Here X n and /_~n are the renormalized Lame coefficients. K, of the simulation shown in fig. lc is almost equal to the unrenormalized value K = 47r/t at low temperatures, takes the value about 16~7 at t - 0.22, and then decreases steeply and remains rather small at high temperatures. The shear modulus in fig. lc decreases also very rapidly near t - 0.22. In ref. [2], the high temperature phase with free dislocations is said not to be a true liquid, but a hexatic phase with enhanced orientational order. The orientational stiffness constant K, defined as the long wave-length limit of the q-dependent stiffness KA( q), where T/K,(q)=
((q.b(q))(q.b(-q)))/q’
is predicted to be finite in the hexatic phase, whereas the fluid and solid phases respectively. KA( q) calculated fig. Id indicates that in the long wave-length limit K, finite value at f - 0.23. All the findings in this section performs a Kosterlitz-Thouless-type continuous phase
it is zero and infinite in from the simulation in changes from zero to a indicate that the system transition.
3. First order melting of a system with a small core energy Since the theory [ 1,2] uses the “fugacity (- e-Ec/T)” expansion, it may be possible to have a new type of phase transition for a system with a small core energy. In fact one finds a large discontinuity and hysteresis in the energy E and specific heat C as is shown in fig. 2a for a system with a small core energy, E, = 0.57 J. The dislocation density n in fig, 2b also shows a large discontinuity. At low temperatures n follows the Arrhenius law rather well with the activation energy of a nn dislocation pair formation. At high temperatures n is almost saturated. Microscopic configurations such as in fig. 2c reveal that dislocation vectors align to form grain boundary loops. The entangled and folded structure of grain boundary loops can destroy the global orientational order and the high temperature phase is expected to be a true liquid. This is confirmed by the orientational stiffness K,(q) shown in fig. 2d. At high temperatures, T/K,(q) is very large, beyond the value of the instability condition r/72. It also seems to show an increasing tendency in the long wavelength limit (q -+ 0). All these facts indicate that the melting in this system is caused by nucleation of grain boundary loops, and it is of first order to the true liquid phase. This is in agreement with the previous computer experiments on atomistic systems, where the phase transition is of first order and similar loops of dislocations are observed [4].
289
Y. Saito / 213 melting of d~sl~at~a~ vector systems
h
C
1.0
)-
lo-
ozoo 0175 0150
/
/
I
0125
--
f
Iml 1
”
IO 0.5
lo-
) TiK,(ql
- t - i -. \- t/t-\’ ‘\‘-, / -\ --/ -‘- \\ \ _‘\ ,‘_ \ I\--\ \ I
I
I\’
-_
__-_(
--I
--’ _,‘- ;
’
I
,
--i
1\,
\
\
I
if\ -
I
\
\ ‘\\
Fig. 2. Simulation results for a system with a small core energy, E, = 0.57 5. (a) The energy E and the specific heat C. (b) The dislocation density n. (c) Microscopic configuration of dislocation vectors at a high temperature f = 0.15. (d) The orientational stiffness KA(q).
290
Y. S&o / 20 melting of dislocation vector systems
4. Conclusion The order and the mechanism of the melting phase transition is found to depend on the core energy. For a system with a large core energy the melting transition is caused by the dislocation unbinding mechanism and is of continuous order. For a system with a small core energy, melting is caused by the grain boundary loop formation, and is of first order.
References [I] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181; J.M. Kosterlitz, J. Phys. C7 (1974) 1046. [2] B.I. Halperin and D.R. Nelson, Phys. Rev. Letters 41 (1978) 121; D.R. Nelson and B.I. Halperin, Phys. Rev. B19 (1979) 2457. [3] F.F. Abraham, Phys. Rev. Letters 44 (1980) 463. [4] S. Toxvaerd, Phys. Rev. Letters 44 (1980) 1602; Phys. Rev. A24 (1981) 2735 (51 F. van Swol, L.V. Woodcock and J.N. Cape, J. Chem. Phys. 23 (1981) 913. [6] R.W. Hackney and T.R. Brown, J. Phys. C8 (1975) 1813. [7] R.H. Morf, Phys. Rev. Letters 43 (1979) 931. [8] ST. Chui, Phys. Rev. Letters 48 (1982) 933. [9] Y. Saito, Phys. Rev. Letters 48 (1982) 1114. [IO] F.R.N. Nabarro, Theory of Dislocations (Clarendon,
Oxford,
1967).