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crystals

Article

Dislocation Structure and Mobility in Hcp Rare-Gas

Solids: Quantum versus Classical

Santiago Sempere 1, Anna Serra 2ID , Jordi Boronat 1and Claudio Cazorla 3,*ID

1

Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain;

ssemllag@gmail.com (S.S.); jordi.boronat@upc.edu (J.B.)

2Departament d’Enginyeria Civil i Ambiental, Universitat Politècnica de Catalunya, Campus Nord C2,

E-08034 Barcelona, Spain; a.serra@upc.edu

3School of Materials Science and Engineering, The University of New South Wales Australia,

Sydney 2052, Australia

*Correspondence: c.cazorla@unsw.edu.au

Received: 13 December 2017; Accepted: 20 January 2018; Published: 29 January 2018

Abstract:

We study the structural and mobility properties of edge dislocations in rare-gas crystals

with the hexagonal close-packed (hcp) structure by using classical simulation techniques. Our results

are discussed in the light of recent experimental and theoretical studies on hcp

4

He, an archetypal

quantum crystal. According to our simulations classical hcp rare-gas crystals present a strong tendency

towards dislocation dissociation into Shockley partials in the basal plane, similarly to what is observed

in solid helium. This is due to the presence of a low-energy metastable stacking fault, of the order

of 0.1 mJ/m

2

, that can get further reduced by quantum nuclear effects.

We compute

the minimum

shear stress that induces glide of dislocations within the hcp basal plane at zero temperature, namely,

the Peierls stress, and ﬁnd a characteristic value of the order of 1 MPa.

This threshold

value is similar

to the Peierls stress reported for metallic hcp solids (Zr and Cd) but orders of magnitude larger

than the one estimated for solid helium. We ﬁnd, however, that in contrast to classical hcp metals

but in analogy to solid helium, glide of edge dislocations can be thermally activated at very low

temperatures, T∼10 K, in the absence of any applied shear stress.

Keywords: dislocations; rare-gas solids; molecular dynamics; quantum nuclear effects

1. Introduction

Dislocations are line defects related to the accommodation of plastic deformation in crystals. They are

characterized by the Burgers vector that represents the magnitude and direction of the lattice distortion

along the dislocation line. Dislocations are ubiquitous in materials and can alter significantly their

physical properties. Due to their fundamental and technological interests,

the structure

and mobility of

dislocations in classical metals with the three elemental crystal structures, namely, face-centered cubic (fcc),

body-centered cubic (bcc), and hexagonal close-packed (hcp) (e.g., Al, Fe, and Zr), have been extensively

investigated both with theory and experiments (see [1–4] and references therein).

Quantum and classical solids are fundamentally different. In quantum crystals, typically

4

He

and H

2

, the kinetic energy per particle in the

T→

0 limit is much larger than

kBT

(where

kB

is the

Boltzmann constant) and the ﬂuctuations of the atoms around the equilibrium lattice sites are up

to

∼

10% of the distance to the neighboring lattice sites [

5

–

9

]. In classical crystals, on the contrary,

those quantities are practically negligible at low temperatures. Because of these important differences,

one might expect ﬁnding disparate dislocation phenomena in the two types of crystals. In fact,

Haziot et al.

analysed the plastic properties of hcp

4

He by means of direct stress-strain measurements

and found that the resistance to shear along directions contained in the basal plane nearly vanishes at

T≈

0.1 K due to the free glide of dislocations [

10

]. This intriguing effect, which has been termed as

Crystals 2018,8, 64; doi:10.3390/cryst8020064 www.mdpi.com/journal/crystals

Crystals 2018,8, 64 2 of 19

“giant plasticity”, disappears in the presence of

3

He impurities or when the temperature is raised [

10

].

Meanwhile, in a recent quantum Monte Carlo simulation study on hcp

4

He [

11

],

Landinez-Borda et al.

have shown that the Peierls stress for the ﬂow of dislocations in the basal plane is nominally zero,

essentially due to the zero-point motion of the atoms. The “giant plasticity” observed in solid helium,

therefore, appears to be a manifestation of its quantum character.

Quantum and classical solids, however, also present some similarities as regards dislocation

behavior. For instance, Landinez-Borda et al. have shown that in solid helium either screw or edge

dislocations with Burgers vectors contained in the hcp basal plane tend to dissociate into Shockley

partial dislocations separated by ribbons of fcc-like stacking fault [

11

]. The same behavior is observed

in classical metals with the hcp structure like, for instance, Zr [

2

,

12

,

13

]. Nevertheless, since the nature

of the atomic interactions in rare gases and metallic systems are so different, the physical origins of

such similarities (or the differences explained above) are not totally understood. Actually, studies

on the structure and mobility of dislocations in classical hcp rare-gas solids are to the best of our

knowledge absent in the literature (probably due to the lack of related applied interests). Consequently,

straightforward and physically insightful comparison between classical and quantum hcp rare-gas

crystals in terms of dislocation behavior is not possible.

In this article, we analyze the structure and mobility of edge dislocations in a model hcp rare-gas

crystal with classical simulation methods, and compare our results to those obtained in other hcp

crystals and solid helium. We focus on the atomic structure and glide of edge dislocations with

Burgers vector contained in the basal plane, as this type of line defect and dislocation motion are most

likely to occur in solid helium [10,11,14,15]. Our results reveal a strong tendency towards dislocation

dissociation into Shockley partials separated by wide regions of fcc-like stacking fault, in analogy to

what occurs in solid helium. We ﬁnd that the Peierls stress for the glide of edge dislocations in the

hcp basal plane amounts to

∼

1 MPa, which is very similar in magnitude to the values reported for

classical metals with the hcp structure (e.g., Zr and Cd) [

12

,

13

]. However, in contrast to other classical

solids but in analogy to solid helium, edge dislocations in hcp rare gases turn out to be extremely

mobile: they can diffuse with an approximate velocity of 50 m/s in the absence of any applied stress at

temperatures as low as 25 K (that is, well below the corresponding Debye temperature

ΘD∼

65 K [

16

]).

We rationalize the origins of this effect in terms of the exceptionally weak interatomic interactions in

rare gases.

The organization of this article is as follows. In the next section, we provide the technical details

of our classical simulations and explain the methods that we have employed to analyze the structure

and mobility of dislocations. Then, we present our results and discuss them in the light of previous

classical and quantum simulation studies. Special consideration is put on the technical aspects of the

calculations as regards the impact of ﬁnite-size effects, relaxation of the simulation cell, and sampling

of different thermodynamic ensembles with molecular dynamics. Finally, we summarize our main

ﬁndings and conclusions in Section 4.

2. Methods Outline

2.1. Classical Simulations

All the geometry relaxations and molecular dynamics (MD) simulations were performed with

the LAMMPS code [

17

]. Our model hcp crystal consists of xenon (Xe) atoms interacting through a

pairwise Lennard–Jones (L–J, 6–12) potential with parameters

e=

0.01881 eV and

σ=

4.06 Å [

18

–

20

].

We note that the employed L–J pairwise potential, in spite of being analytically simple, was originally

devised to reproduce a considerable amount of experimental data measured in solid Xe, including the

elastic constants, sound velocities, and equation of state, among others. (The ground-state structure

of solid Xe is known to be cubic fcc; however, since our focus here is on classical hcp rare gases,

we chose the species with the largest possible atomic weight and most intense atomic forces as an

upper-bound). A particle–particle particle-mesh

k

-space solver was used to compute the long-range

Crystals 2018,8, 64 3 of 19

van der Waals interactions and forces beyond a cut-off distance of 20 Å at each relaxation and

MD step

.

The initial dislocation conﬁguration was generated by removing a

(

11

2

0

)

semi-plane of Xe atoms

from an orthorhombic simulation box containing a perfect hcp lattice; the generated Burgers vector

then was equal to

b= (a

, 0, 0

)

as expressed in Cartesian coordinates, where

a=

4.26 Å represents

the equilibrium in-plane lattice parameter. (It was checked that, upon full geometry optimization,

the relaxed system was identical to that obtained when starting from an initial conﬁguration generated

with the Osetsky and Bacon’s method [21].)

The geometry relaxations were performed with a conjugate gradient algorithm and convergence

was reached after the forces on the atoms and mechanical stresses were smaller than 10

−10

eV/Å and

10

−8

eV/Å

3

, respectively. Regarding the MD simulations, the pressure and/or temperature of the

system were kept ﬂuctuating around a set-point value by using thermostatting and/or barostatting

techniques in which some dynamic variables are coupled to the particle velocities and/or simulation

box dimensions. Large simulation boxes containing several thousands of atoms were employed

in the dynamical simulations, and periodic boundary conditions normally were applied along the

three Cartesian directions. Examples of the simulation-cell dimensions considered in this study

are

Lx=

614.64,

Ly=

61.27, and

Lz=

577.59 Å (344,544 atoms),

Lx=

104.38,

Ly=

107.20,

and Lz=101.13 Å (18,242 atoms)

, and

Lx=

43.00,

Ly=

45.75, and

Lz=

43.26 Å (1368 atoms).

Newton’s equations of motion were integrated by using the customary Verlet’s algorithm with a time

step of 10−3ps.

2.2. Analysis Methods

In order to identify with precision the structure and position of the edge dislocation in our model

crystal, we employed three different analysis methods that are brieﬂy explained next.

2.2.1. Differential Displacement Analysis (DD)

The presence of line defects makes the atoms contained in the slip plane of the dislocation to

displace. We can quantify the spread of such a disregistry through the distribution of partial Burgers

vector components,

[bx

,

by]

, within the glide plane [

12

,

22

]. These partial components can be expressed as:

bi(x) = d(∆ui)

dx , (1)

where

x

represents the direction perpendicular to the dislocation line contained in the hcp basal plane,

i=x,y, and ∆uiis the atomic disregistry. The latter quantity can be deﬁned as:

∆ui=uabove,i−ubelow,i, (2)

where

up,i=rdisl oc

p,i−rper f

p,i

,

rdisl oc

p,i

are the positions of the atoms above/below the glide plane in the

system containing the dislocation, and

rper f

p,i

the positions of those same atoms in the perfect-lattice

system. In a general case, the partial

x

components of the Burger vectors describe the edge part of the

dislocation, whereas the

y

components the screw. The components of the total Burger vector contained

in the simulation cell,

[bT

x

,

bT

y]

, then can be computed by integrating the respective partial components

along the glide direction as:

bT

i=ZLx

0bi(x)dx , (3)

where

Lx

represents the size of the simulation box along the

x

-direction. In our particular case,

bT

y

should be always equal to zero, as we are dealing exclusively with edge dislocations. The presented

differential displacement (DD) analysis is especially useful for detecting the presence of stacking fault

ribbons bounded by two partial dislocations, and for estimating the width of dislocation cores.

Crystals 2018,8, 64 4 of 19

2.2.2. Nearest Neighbor Analysis (NN)

In this method, the number of atoms within a certain radial distance from a selected atom,

nc

,

is computed

. The cut-off distance deﬁning such an interval normally is chosen to be a value between

the distances to the ﬁrst and second shells of atomic nearest neighbors. In the particular case of hcp

systems, a possible deﬁnition of the cut-off distance is [23]:

rhcp

c=1

2 1+r4+2x2

3!a, (4)

where

x= (c/a)/

1.633 and

c

and

a

represent the hcp lattice parameters. In the perfect hcp lattice,

the number of nearest neighbors is 12 for every atom; in the system containing the dislocation,

the atoms displaying

nc6=

12 values then can be identiﬁed with a highly distorted region of the crystal

like, for instance, the core of the dislocation. The nearest neighbor (NN) analysis turns out to be very

useful for locating dislocation cores and hence monitoring the motion of line defects.

2.2.3. Common Neighbor Analysis (CNA)

This method, which originally was introduced by Honeycutt and Andersen [

24

], consists of

creating a 4-index sequence for each pair of atoms,

α

and

β

. The ﬁrst index is equal to “1” if

α

and

β

are nearest neighbors, “2” otherwise (two atoms are nearest neighbors if the distance between them is

smaller than a certain cut-off value, e.g., see Section 2.2.2 ). The second index adopts a value that is

equal to the number of common nearest neighbors shared by

α

and

β

(e.g., “4” in a perfect hcp or fcc

lattice when the ﬁrst sequence index is equal to “1”). The third index indicates the number of bonds

between common neighbors. The fourth index is introduced to differentiate diagrams with same ﬁrst,

second and third indexes but with different types of bonds between common neighbors. For instance,

in a perfect fcc system, all the 4-index sequences ascribed to nearest neighbors are equal to “1421”; in a

perfect hcp system, half of the sequences ascribed to nearest neighbors are equal to “1421” while the

other half are equal to “1422”; in a perfect bcc system, we ﬁnd 4-index sequences describing nearest

neighbors that are equal to “1441” and “1661”. The common neighbor analysis (CNA) method turns

out to be very useful for locating dislocation cores and also stacking faults.

3. Results and Discussion

3.1. Edge Dislocation Structure

The shortest perfect Burgers vector in an hcp lattice is

b=1

3h

11

2

0

i

, and the most common

dislocation slip planes are the basal,

(

0001

)

, and prism,

{

10

1

0

}

, planes [

2

,

12

,

25

]. The preference of the

glide plane is determined by the energy and stability of a stacking fault. If a low-energy metastable

stacking fault with vector

1

3h

1

1

00

i

exists,

I2

, then the dislocation normally dissociates in the basal plane

into two Shockley partial dislocations bounding a ribbon of fcc-like stacking fault [

2

,

12

,

25

]. Such a

dissociation process is described in crystallographic notation as:

1

3[1120]→1

3[1010] + 1

3[0110]. (5)

The resulting system geometry generally consists of two partial dislocations lying on the basal plane

at

±

30

◦

of the initial Burgers vector

b

(or

±

60

◦

, if referred to the dislocation line) and with partial

Burgers vectors |bp|=|b|/√3.

In Figure 1a,b, we represent the ﬁnal relaxed conﬁguration of our model hcp Xe solid in which we

initially created an edge dislocation with its line oriented along the

y

-direction. The full relaxation was

performed via minimization of all the atomic forces,

Fi

, and mechanical stresses,

σij

(see Section 2.1).

By using the CNA analysis method (see Section 2.2.3), we are able to distinguish the atoms that belong

to the dislocation core (green) or to the fcc-like stacking fault (blue), and those that render the usual hcp

Crystals 2018,8, 64 5 of 19

ordering (yellow). The relaxed structure clearly shows two Shockley partial dislocations oriented as

+

30

◦

and

−

30

◦

with respect to the initial Burgers vector

b= (a

, 0, 0

)

, and a ribbon of fcc-like stacking

fault between them; we note that the same structural behavior is observed also in classical metallic

(e.g., Zr [2,12]) and quantum rare-gas (e.g., 4He [11]) hcp crystals.

Figure 1.

Sketch of a fully relaxed system containing an edge dislocation from different views. Green

spheres represent atoms belonging to the dislocation core, blue spheres atoms belonging to the

fcc-like stacking fault, and yellow spheres atoms with common hcp atomic coordination features.

Red arrows represent the Burgers vectors of the partial dislocations. (

a

,

b

) represent different views of

the simulation cell.

In order to provide a quantitative description of the relaxed dislocation conﬁguration,

we employed

the differential displacement (DD) analysis method (Section 2.2.1). In Figure 2a,b, we plot

the relative displacement of the atoms delimiting the glide plane, and, in Figure 2c,d,

the corresponding

partial Burgers vector components

[bx

,

by]

. It is shown that, as expected, integration of

bx

and

by

along

Crystals 2018,8, 64 6 of 19

the

x

-direction leads to non-zero edge and null screw total dislocation components, respectively.

The width of the resulting fcc-like stacking fault,

ωs f

, as deduced from the distance between the two

maxima in Figure 2c, is approximately equal to 50

a

. The width of the dislocation core, which can be

deﬁned as the region in which the atomic disregistry is greater than the half of its maximum, is found

to be

∼

12.5

a

. This latter quantity has an unusually large value, which indicates the presence of very

mobile dislocations (we will comment again in this point in Section 3.3).

Figure 2.

Relative displacement (

a

,

b

) and differential displacement (

c

,

d

) of the atoms above and

below the glide plane of the edge dislocation in the

x

- and

y

-directions. The simulation cell contains

n= 344,544 atoms and is fully relaxed.

Concerning the technical aspects involved in the simulation of dislocations, we have analyzed the

effects of reducing the size of the simulation cell on the determination of the ﬁnal

equilibrium state

.

This type of analysis is especially useful for interpreting the results obtained in quantum and

ﬁrst-principles simulations where, due to the high computational expense involved, one only can

handle systems made up of few hundreds or thousands of atoms [

11

,

26

]. Figures 3and 4show the

DD analysis performed in two simulation cells containing 18,424 and 1368 atoms, respectively. In the

n

= 18,424 case, we have also analyzed the effects of constraining the shape of the simulation cell to

orthorhombic, that is, of not relaxing it (hence

σij 6=

0). In Figure 3a–d (blue lines), it is appreciated that

ωs f

now is equal to 12

a

and the width of the dislocation core is

∼

5

a

. These values are signiﬁcantly

smaller than the results obtained in the simulation cell containing 344,544 atoms, which in principle

are not affected by ﬁnite-size errors. Nevertheless, integration of the corresponding

bx

and

by

partial

Burgers vector components along the

x

direction still provides non-zero edge and null screw total

dislocation components, and the two Shockley partial dislocations can be clearly differentiated in the

DD plots shown in Figure 3. Meanwhile, it is found that when the shear stresses on the simulation cell

are not minimized the separation between the two partial dislocations reduces to

approximately 10a

.

In addition, the orientation of the two partial dislocations changes from

+

30

◦

and

−

30

◦

to

−

30

◦

and

+

30

◦

,

as compared

to the minimum-energy case

σij =

0. Nevertheless, it may be reasonably concluded

that, in the particular case of simulating edge dislocations, the inaccuracies deriving from the use of

relatively small orthorhombic boxes containing up to

∼

10

4

atoms are not critical. In the

n=

1368 case

(see Figure 4), by contrast, it is found that the edge dislocation hardly can get dissociated owing to

the limited size of the simulation cell, which artiﬁcially prevents the appearance of any stacking fault

(that is, only one diffuse maximum is appreciated in Figure 4c). Moreover, integration of the

by

partial

Burgers vector component along the

x

-direction neither provides an exact null value for the total screw

Crystals 2018,8, 64 7 of 19

dislocation component (see Figure 4d). In view of the results enclosed in Figures 2–4,

we may

conclude

that the use of small simulation cells containing just up to

∼

1000 atoms is likely to produce unrealistic

dislocation conﬁgurations (see, for instance, Ref. [26]).

Figure 3.

Relative displacement (

a

,

b

) and differential displacement (

c

,

d

) of the atoms above and

below the glide plane of the edge dislocation in the

x

and

y

directions. The simulation cell contains

n= 18,424 atoms

. Green and blue lines represent the results obtained in a non-relaxed (

σij 6=

0) and a

fully relaxed (σij =0) simulation cell, respectively.

Figure 4.

Relative displacement (

a

,

b

) and differential displacement (

c

,

d

) of the atoms above and

below the glide plane of the edge dislocation in the

x

- and

y

-directions. The simulation cell contains

n=1368 atoms and is fully relaxed.

In order to get quantitative insight into the metastable stacking fault that induces the dissociation

of the edge dislocation into Shockley partials within the basal plane, we have computed the stacking

Crystals 2018,8, 64 8 of 19

fault energy in our model hcp crystal as a function of the fault plane displacement,

γ(f)

[

12

].

For this

calculation, ﬁrst we rigidly displace one half of the crystal with respect to the other over a grid of 10

4f

points spanning all possible faults within the

x

–

y

plane. Subsequently, at each

f

point,

the atoms

are

allowed to relax perpendicular to the fault plane, which is along the

z

-direction, by potential energy

minimization (i.e., zero-temperature conditions are assumed). Our simulation cell contains a total

of 8100 Xe atoms, and we apply periodic boundary conditions over the

x

–

y

fault plane and rigid

boundary conditions along

z

. Our stacking fault energy results are represented in Figure 5, for which a

spline-based interpolation has been used in order to provide smooth iso-

γ

contours. We actually ﬁnd

a metastable stacking fault at

f=1

3h

1

1

00

i

, which is

I2

, similarly to what has been reported by other

authors for classical hcp metals [

12

]. According to our calculations, the energy of the

I2

stacking fault,

γs f

, is equal to 0.094 mJ/m

2

. (It is worth noting that the numerical accuracy in our

γs f

estimations is

below 0.001 mJ/m

2

.) We also calculated the energy of the metastable stacking fault associated to an

edge dislocation with its line laying on the hcp prism plane (see, for instance, Figure 2b in Ref. [

12

]).

In that case, we obtained a

γs f

value of 15 mJ/m

2

, which is about three orders of magnitude larger

than the value calculated for the basal plane. This result shows a major tendency towards dislocation

dissociation into Shockley partials in the basal plane.

0.0 0.2 0.4 0.6 0.8 1.0

x/a

0.0

0.2

0.4

0.6

0.8

1.0

y/a

0

5

10

15

20

25

30

35

40

45

γ (mJ/m2)

Figure 5.

The

γ

-surface of the analyzed classical hcp rare-gas crystal. Perfect hcp stacking positions

correspond to the four corners of the plot while large white spheres indicate metastable fault positions.

Iso-γcurves are represented with solid black lines at 5 and 10 mJ/m2intervals.

As expected, the

γs f

values calculated in Xe turn out to be extremely small as compared

to those obtained in other hcp crystals where the interactions between atoms are much stronger

(

e.g., γs f ∼100 mJ/m2

in Zr [

12

,

27

,

28

]). We note that Keyse and Venables already measured more

than 30 years ago the stacking fault energy in fcc Xe at low temperatures by means of transmission

electron microscopy techniques [

29

]. In particular, they found a

γs f

value of 1.96

±

0.65 mJ/m

2

at

a temperature of 25

±

5 K, which is about two (one) orders of magnitude larger (smaller) than the

stacking fault energy that we have determined for the basal (prism) plane in the hcp phase. The reasons

behind these discrepancies may be possibly understood in terms of the different crystal structure

considered in our calculations and also of likely inaccuracies present in the employed interaction

pairwise potential.

Crystals 2018,8, 64 9 of 19

Once the metastable stacking fault energy is known, we can estimate from elastic theory the

expected equilibrium distance between the Shockley partial dislocations, which is the width of the

resulting fcc-like stacking fault, with the formula [1,2]:

ωelas

s f =Gb2

p

4πγs f

, (6)

where likely elastic anisotropic effects have been disregarded,

G

represents the shear modulus of the

system (which we estimate here to be 200 MPa), and

bp

the modulus of the corresponding partial

Burgers vector. By performing the necessary numerical substitutions, we ﬁnd that in the basal plane

ωelas

s f

is equal to 24

a

. We recall that, in the larger simulation cell considered in this study, we have

found that

ωs f

approximately amounts to 50

a

(see Figure 2c,d), which turns out to be of the same order

of magnitude and larger than

ωelas

s f

. Consequently, the results obtained in the 344,544-atoms system

may be considered to be virtually free of ﬁnite-size bias.

Recently, Landinez-Borda et al. have estimated an almost vanishing

γs f

value of 0.002 mJ/m

2

in solid

4

He at ultralow temperatures by using quantum Monte Carlo simulation techniques [

11

].

In an attempt to quantify the importance of quantum nuclear effects on the stacking fault energy of

solid helium, we have performed analogous classical

γ(f)

calculations to those described for Xe but

considering the same volume conditions, interatomic interactions, and atomic mass than in Ref. [

11

].

Our classical calculations in solid 4He render a stacking fault energy of 0.003 mJ/m2, which is orders

of magnitude smaller than the value estimated in solid Xe. By comparing this result to the stacking

fault energy calculated by Landinez-Borda et al., we may conclude that quantum nuclear effects are

responsible for a

γs f

reduction of the

∼

30% . Consequently, quantum nuclear ﬂuctuations further

contribute to the dissociation of edge dislocations into partials in solid

4

He and probably also in any

other quantum crystal (e.g., H2, Ne and LiH [5,30,31]).

3.2. The Peierls Stress

The Peierls stress,

τP

, is key to quantifying the resistance of a crystal to the motion of dislocations.

τPnormally

is referred to the critical stress that induces glide of dislocations in the absence of thermal

excitations. Here, we use two different methods to evaluate the Peierls stress in our model crystal as

concerns the motion of edge dislocations in the basal plane. We note that, since the glide of dislocations

involves the breaking and formation of atomic bonds, the value of

τP

in principle is expected to depend

strongly on the crystal structure and strength of the interatomic forces.

3.2.1. Method A: Fixed Boundary Conditions

We ﬁrst employ the usual method found in classical simulation studies based on force ﬁelds

(see, for instance, Refs. [

12

,

21

]), which is brieﬂy described next. The simulation cell is divided into three

main parts: “U”, the upper region containing frozen atoms, “L”, the lower region containing frozen

atoms, and “M”, the rest of the simulation cell containing mobile atoms (see Figure 6). Regions U and

L consist of several layers of atoms that are displaced together as a block. Essentially, a shear strain

deformation is applied on the simulation cell perpendicular to the dislocation line and the resulting

stresses are monitored. In our particular case, the dislocation line is parallel to the

y

-axis; hence,

we ﬁrst displace the U slab a small distance along the

x

-direction,

∆u

, and then proceed to minimize

the potential energy of the atoms in M while keeping the L slab ﬁxed. Periodic boundary conditions

are applied just along the

x

- and

y

-directions. The applied mechanical strain is straightforwardly

calculated as

∆η=∆u/Lz

, where

Lz

is the length of the simulation along the

z

-direction, and the

accompanying shear stress is

σxz =Fx/LxLy

, where

Fx

is the sum of all the forces along the

x

-direction

exerted on the atoms in region U. By iteratively repeating this procedure, we can reproduce with detail

the dependence of the shear stress on

η

. For small cell distortions,

σxz

is expected to increase almost

linearly, as it follows from elastic theory; however, when

η

is large enough so that it induces the glide

Crystals 2018,8, 64 10 of 19

of dislocations, the shear stress should decrease sharply. The maximum value of

σxz

just before that

sudden drop can be identiﬁed with the Peierls stress.

Figure 6.

Sketch of the system used to estimate the Peierls stress with Method

A

. Three main parts

are differentiated: the upper part “U”, the lower part “L”, and the region with mobile atoms “M”.

“P” indicates application of periodic boundary conditions and the dashed line the orientation of the

edge dislocation.

In Figure 7, we show the

σxz(η)

results obtained in a large simulation cell containing 344,544 atoms

by adopting two different

∆η

increments; the thickness of the upper region U,

dU

, was safely ﬁxed

to 5

c

in both cases [

21

]. As can be appreciated in Figure 7a (case

∆η

= 8

×

10

−4

), a regular pattern

emerges that follows elastic theory at small cell deformations (that is,

σxz ∝η

) and which allows for an

estimation of the Peierls stress (as identiﬁed with the

σxz

maximum) of

τP=

6.0

±

0.1 MPa.

When the

employed

∆η

is signiﬁcantly reduced (case

∆η=

1

×

10

−4

, see Figure 7b), the obtained

σxz(η)

curve

shows more irregularities and the expected elastic behavior is reproduced at conditions

η>

0.001.

This outcome reﬂects the intricacies found in the relaxation of such a large simulation cell, which at

small

η

values may easily end up on metastable conﬁgurations. In this latter case, we estimate a Peierls

stress (as identiﬁed with the σxz maximum) of τP=7.4 ±0.1 MPa.

Figure 7.

Evolution of the shear stress expressed as a function of strain for a simulation cell containing

n

= 344,544 atoms in which the thickness of the “U” region is taken to be

dU=

5

c

. Results obtained

with ∆η=8×10−4and 1 ×10−4are shown in (a,b), respectively (see text).

Crystals 2018,8, 64 11 of 19

In order to assess the effects of ﬁnite-size bias on the estimation of

τP

, we repeated the same

calculations in a smaller simulation cell containing

n

= 18,424 atoms. The stress proﬁles that we

obtained in this case (see Figure 8a) are intricate and do not allow for a clear estimation of the Peierls

stress (that is, an unambiguous maximum appearing periodically is missing). Such a ﬁnite-size effect is

related to the fact that the width of the fcc-like stacking fault is already of the same order of magnitude

than the characteristic size of the simulation cell (that is,

∼

100 Å). In order to somehow determine

τP

from the results shown in Figure 8a, we monitored the position of the two partial dislocations

with the differential displacement (DD) and nearest neighbor (NN) analysis methods (see Section 2.2

and Figure 8b), and averaged the value of the shear stress over the set of strain points at which the

partial dislocations change their position. (In the NN case, for the sake of simplicity, we have averaged

the position of all the atoms exhibiting a nearest neighbor number different from 12; consequently,

we obtain the center position of the stacking fault.) By proceeding like this, we obtained a Peierls stress

of

τP=

1.3

±

0.2 MPa, which is about 5 times smaller than the one estimated in the

n

= 344,544 atoms

simulation cell. Our results, therefore, show that ﬁnite-size errors affect critically the estimation of

τP

when using Method A(in agreement with previous conclusions by Osetsky and Bacon [21]).

Figure 8.

(

a

) evolution of the shear stress expressed as a function of strain for a simulation cell

containing

n

= 18,424 atoms in which the thickness of the “U” part is equal to

dU=

2.5

c

. Several

mechanical strain steps,

∆e

, are considered; (

b

) position of the corresponding dissociated edge

dislocation expressed as a function of strain. “DD” and “NN” stand for the analysis methods of

differential displacement and nearest neighbors, respectively.

3.2.2. Method B: Periodic Boundary Conditions

As we have just shown in the previous section, Method

A

requires of very large simulation

cells (i.e.,

N∼

10

5

–10

6

atoms) in order to remove all possible ﬁnite-size bias affecting the estimation

of

τP

. This technical aspect suggests that accurate calculation of the Peierls stress with Method

A

and quantum atomistic or electronic ﬁrst-principles simulation techniques (in which typically

Crystals 2018,8, 64 12 of 19

N∼102–103

atoms [

11

,

32

]) is hardly achievable in practice. Moreover, by construction customary

electronic ﬁrst-principles techniques (e.g., plane-wave density functional theory [

5

,

33

]) generally

demand the application of periodic boundary conditions in all directions in order to ensure the

periodicity and continuity of the electrostatic potential in space. Therefore, it is desirable to work out

reliable τPcomputational methods in which all boundaries of the simulation cell are treated equally.

Here, we present a method in which a particular tilt is introduced in the simulation cell containing

the edge dislocation and the accompanying change in the total energy is monitored upon constrained

relaxation of the system; the relaxation is performed by applying periodic boundary conditions

along the three Cartesian directions and optimizing all degrees of freedom of the system except the

initial tilt

. The resulting stresses then can be calculated numerically with the well-known expressions

from elastic theory. (A similar approach has been employed by Wang et al. to investigate the

dynamics of screw dislocations in bcc tantalum [

34

]; in our case, however, the simulations are

strictly performed at zero-temperature conditions.) Speciﬁcally, the lattice vectors describing our

simulation cell are

a1= (Lx

, 0, 0

)

,

a2= (x2

,

Ly

, 0

)

, and

a3= (x3

,

y3

,

Lz)

, where

x3

represents the

introduced tilt. The corresponding

shear strain is

ηxz =x3/Lz

, and the resulting shear stress can be

estimated as [35–37]:

σxz =1

V

∂E

∂ηxz , (7)

where Vrepresents the volume of the system and Ethe corresponding total energy.

We enclose the total energy and shear stress results obtained in a simulation cell containing

18,424 atoms in Figure 9a,b. In both cases, the proﬁles that we obtain as a function of applied shear

strain are periodic, in contrast to what we found with Method

A

when using a simulation cell of

the same dimensions (see Figure 8), and elastic behavior is observed for small system deformations.

By identifying the global maximum in the

σxz

curve with the Peierls stress, we obtain a value of

3.40

±

0.01 MPa (see Figure 9b). This value is roughly two times smaller than the free-of-bias result

obtained with Method

A

in the simulation cell containing

n

= 344,544 atoms. Therefore, we may

conclude that, as compared to Method

A

, ﬁnite-size bias appear to affect less critically the calculation

of

τP

with Method

B

(we recall that, with Method

A

, we obtained a Peierls stress of 1.3

±

0.2 MPa in a

same simulation cell containing 18,424 atoms (see previous Section 3.2.1)).

In spite of this favorable outcome, we should acknowledge that the use of periodic boundary

conditions in systems containing dislocations is not exempt of important limitations. For instance,

periodic boundary conditions are inconsistent with the existence of a net Burgers vector in the

simulation cell; consequently, a dipole or quadrupole of dislocations needs to be introduced in

the system [

38

,

39

]. In order to further incorporate this technical aspect on the calculation of

τP

with

Method B

, we constructed a large simulation cell of

n

= 247,680 atoms containing a dislocation

quadrupole. Upon introduction of a moderate tilt and by proceeding to relax the system, however,

we found that all the dislocations merged into a big stacking fault and were annihilated (we note that

a similar behavior has been observed also by Wang et al. in bcc tantalum [

34

]). This outcome suggests

that, unfortunately, even larger simulation cells are required to correct for the inaccuracies associated

to Method B.

The main conclusions emerging from Sections 3.2.1 and 3.2.2 is that the Peierls stress in our model

hcp crystal is of the order of 1 MPa. This result is orders of magnitude smaller than the

τP

values

reported for archetypal crystals with cubic symmetry (e.g., Fe and Mo) [

13

]; however, to our surprise,

it is very similar in magnitude to the Peierls stresses found in classical metals with the hcp structure

(e.g., Zr, Cd, and Mg) [

12

,

13

]. Very recently, Landinez-Borda et al. have shown in solid helium that

τP

nominally amounts to zero, that is, dislocations can move freely throughout the crystal in the absence

of thermal excitations and shear stresses [

11

]. The authors of that study have argued that such an

effect is quantum in nature as is essentially originated by zero-point ﬂuctuations. Our Peierls stress

results obtained in classical rare-gas hcp solids come to corroborate Landinez-Borda et al.’s conclusion,

Crystals 2018,8, 64 13 of 19

as we have demonstrated that weak interparticle interactions alone cannot render practically vanishing

τPvalues.

Figure 9.

Energy (

a

) and shear stress (

b

) expressed as a function of shear strain for a simulation

cell containing

n

= 18,424 atoms in which periodic boundary conditions are applied along the three

Cartesian directions.

3.3. Dislocation Mobility: Finite-T Simulations

We have estimated the basal mobility of an edge dislocation in our model hcp rare-gas solid at

T6=

0 conditions by performing molecular dynamics (MD) simulations in a simulation cell containing

n

= 18,424 atoms. Fully periodic boundary conditions are employed and a

z

-vacuum slab is introduced

in order to avoid the presence of additional dislocations in the upper and lower edges of the simulation

cell (which otherwise would interact with the principal dislocation). No external stresses are considered

in our MD simulations, which are performed in the canonical,

(N

,

V

,

T)

, microcanonical,

(N

,

V

,

E)

,

and isothermal-isobaric,

(N

,

P

,

T)

ensembles. For the cases in which the volume is ﬁxed, we use the

simulation cell obtained through full geometry relaxation of the system. Likewise, the pressure is set to

zero in the

(N

,

P

,

T)

calculations [

17

]. All simulations are performed with a time step of 10

−3

ps and last

for a total of 800 ps. The position of the (dissociated) edge dislocation is monitored with three different

methods: the differential displacement (DD, Section 2.2.1), the nearest neighbor (NN, Section 2.2.2),

and the common neighbor (CNA, Section 2.2.3). In the DD case, due to the high sensitivity of this

method to thermal ﬂuctuations, we have averaged the positions of the atoms over ﬁve consecutive

time steps. In the NN case, for the sake of simplicity, we have averaged the position of the atoms

with nearest neighbor number different from 12, hence we have determined the center position of the

stacking fault; we note that the NN method provides inaccurate results in the

(N

,

P

,

T)

simulations

owing to the ﬂuctuations of the simulation cell, thus that particular case must be disregarded in what

follows (shown here just for completitude).

In Figures 10 and 11, we represent the position of the (dissociated) edge dislocation expressed as

a function of time at a temperature of 25 and 50 K, respectively. It is appreciated that, in spite of the

absence of shear stresses, the dislocation moves at temperatures as low as 25 K, which are well below

the corresponding Debye temperature (

ΘD∼

65 K [

16

]). (Cautiously, we have monitored the size of

the ﬂuctuating stresses in our MD simulations, which are null in average, and checked that in fact

Crystals 2018,8, 64 14 of 19

they are not responsible for the observed dislocation glide (e.g.,

σ

ﬂuctuations

τP

in the

(N

,

P

,

T)

runs). The partial dislocations move either to the left or to the right along the

x

-direction with equal

probability, which is fully consistent with the absence of applied stresses. In our MD simulations,

the partial

dislocations practically remain rigid along the

y

-direction as no kinks or jogs are observed

along their dislocation lines (although for a more detailed analysis of the structural properties of the

mobile dislocations the dimensions of our simulation cell should probably need to be increased). It is

worth noting that, in the

(N

,

V

,

E)

simulations, we have selected

E=

0 and assigned initial velocities

to the atoms reproducing the temperatures chosen in the

(N

,

V

,

T)

and

(N

,

P

,

T)

runs; consequently,

after initializing the

(N

,

V

,

E)

simulations, half of the total kinetic energy is transformed into potential

and the effective temperature of the system is halved (i.e.,

T=

12.5 and 25 K, respectively). In spite of

such a reduction in temperature, one still can observe in Figure 9b that the dislocation remains mobile

in the

(N

,

V

,

E)

simulations. These results clearly make evident a very low resistance of the rare-gas

lattice to

dislocation glide

. From Figures 10 and 11 we can also estimate the width of the fcc-like

stacking fault in our MD simulations,

ωs f

, which corresponds to the position difference between

the two

dislocation cores

.

We enclose

our averaged

ωs f

results in Table 1, expressed as a function of

temperature and simulation ensemble. It is appreciated that all three ensembles provide consistent

results and that, as expected, thermal excitations tend to increase the ωs f ﬂuctuations.

Figure 10.

Position of the dissociated edge dislocation expressed as a function of time for a simulation

cell containing

n

= 18,424 atoms. Molecular dynamics simulations have been performed in the three

thermodynamic ensembles (

a

)

NVT

, (

b

)

NV E

, and (

c

)

NPT

; the temperature has been ﬁxed to

25 K, the pressure to zero, and the volume to the equilibrium one. In the

NV E

case,

an equilibrium

temperature of 12.5 K is reached. “DD”, “NN”, and “CNA” stand for the analysis methods of differential

displacement, nearest neighbors, and common neighbor, respectively.

Crystals 2018,8, 64 15 of 19

Figure 11.

Position of the edge dissociated dislocation expressed as a function of time for a simulation

cell containing

N=

18424 atoms. Molecular dynamics simulations have been performed in the

three thermodynamic ensembles (

a

)

NVT

, (

b

)

NV E

, and (

c

)

NPT

; the temperature has been ﬁxed to

50 K, the pressure to zero, and the volume to the equilibrium one. In the

NV E

case, an equilibrium

temperature of 25 K is reached. “DD”, “NN”, and “CNA” stand for the analysis methods of differential

displacement, nearest neighbors, and common neighbor, respectively.

In Figure 12, we show the time-accumulated average displacement of the dissociated edge

dislocations,

∆x

, expressed as a function of time, temperature, and simulation ensemble. In particular,

we calculate this quantity with the formula:

∆x(t) = |x(t)−x(t−δt)|+∆x(t−δt), (8)

where

x(t)

corresponds to the average position of the atoms belonging to the dislocations along the

x

direction at time

t

(that is, as shown in Figures 10 and 11) and

δt

is equal to a time step in our molecular

dynamic simulations. A series of kinks appear in the ﬁgure that are a consequence of the dislocation

cores passing through the boundaries of the simulation cell (that is, due to use of periodic boundary

conditions during the crossing of an edge, some of the atoms belonging to the same dislocation core are

located in one extreme of the box, whereas the rest remain in the opposite boundary).

From the

slope of

the linear ﬁts to the

∆x

data points, which are not affected by such a periodic boundary artifact, we can

deduce the module of the average diffusion velocity of the dislocations,

vd

, as a function of temperature

and simulated thermodynamic ensemble. The results enclosed in Table 1show that simulations

performed both in the

(N

,

V

,

T)

and

(N

,

P

,

T)

ensembles render very similar

vd

values;

by contrast

,

Crystals 2018,8, 64 16 of 19

simulations performed in the

(N

,

V

,

E)

ensemble systematically provide smaller dislocation diffusion

velocities due to the effective reduction in the temperature of the system (

see preceding paragraph

).

Interestingly, our MD results suggest a square root-like dependence of

vd

on temperature,

vd∝√T

.

For instance

, in the

(N

,

V

,

T)

and

(N

,

P

,

T)

cases, we realize that

vd(2T)'√2vd(T)

,

and, when

comparing those results with the values obtained in the

(N

,

V

,

E)

ensemble, we consistently ﬁnd

that

vNVT

d(T)'√2vNV E

d(T)

. This behavior appears to depart signiﬁcantly from the Arrhenius-like

relation that is expected for thermally activated dislocations, namely

vd∝exp [−∆G/kBT]

, and which

has been observed in other materials at high temperatures [2,40,41].

0 100 200 300 400 500 600 700

Time

(

ps

)

0

100

200

300

400

500

600

700

800

∆

x

(

Å

)

NVT (25K)

NPT (25K)

NVE (25K)

NVT (50K)

NPT (50K)

NVE (50K)

Figure 12.

Time-accumulated average displacement of the dissociated edge dislocations expressed as

a function of time, temperature, and simulated thermodynamic ensemble (

n= 18,424 atoms

). Solid

lines represent the actual dislocation positions and dashed lines are linear ﬁts performed on regions in

which the dislocation motion is not disturbed by the simulation cell boundaries. Dislocation diffusion

velocities are deduced directly from the slope of the linear ﬁts.

Table 1.

Average width of the fcc-like stacking fault,

ωs f

, and edge dislocation diffusion velocity,

vd

,

expressed as a function of temperature, and simulated thermodynamic ensemble (

n

= 18,424 atoms).

ωs f

results are expressed in units of lattice parameter

a

and the ﬁgures within parentheses indicate the

corresponding statistical uncertainty.

T=25K

vd(m/s)ωs f (a)

NVT 57.2 (0.5)12.0 (0.3)

NVE 37.0 (0.5)12.1 (0.6)

NPT 53.1 (0.5)12.1 (0.6)

T=50K

vd(m/s)ωs f (a)

NVT 79.6 (0.5)11.9 (3.3)

NVE 56.2 (0.5)11.9 (3.3)

NPT 76.1. (0.5)11.9 (3.3)

It is physically insightful to compare the

vd

values obtained in our model rare-gas solid with

those reported for other materials with the hcp structure. In the case of Zr, Khater and Bacon have

estimated edge dislocation velocities within the basal plane of about 100 ms

−1

at room temperature

and practically vanishing applied shear stresses (see Figure 7a in Ref. [

12

]). In the present case, similar

vd

values are obtained already at a much lower temperature of 50 K and nominally zero mechanical

stress. This comparison comes to show that edge dislocations in classical hcp rare-gas solids are much

more mobile than in structurally analogous metals. The origins of such differences may reside on the

Crystals 2018,8, 64 17 of 19

interatomic interactions (mind that the atomic masses of Zr and Xe atoms are roughly comparable),

which in the case of rare gases are extremely weak [33].

4. Conclusions

We have presented a comprehensive computational study on the structural and mobility

properties of edge dislocations in classical rare-gas solid with the hcp structure. We have shown

that dissociation of edge dislocations into Shockley partials, as induced by the presence of a low-energy

metastable stacking fault, is a common process in hcp rare-gas crystals. On the other hand, we have

inferred that quantum nuclear effects further enhance the dissociation of edge dislocations into

partials as they tend to decrease the energy of the actual stacking fault. A dislocation-related quantity

that indirectly appears to be drastically affected by quantum nuclear effects is the Peierls stress,

τP

.

While we have calculated a Peierls stress value of the order of 1 MPa in our model classical rare-gas

crystal, other researchers have estimated a practically vanishing

τP

in archetypal quantum crystal

hcp

4

He. Meanwhile, the mobility of edge dislocations in rare-gas solids in general is very large,

owing to the characteristic weak interactions between atoms. In the present case, we have found that

glide of dislocations can be activated at temperatures as low as

∼

10 K in the absence of any applied

shear stress, achieving large dislocation diffusion velocities of the order of 10 ms

−1

. Furthermore,

our molecular dynamics results suggest that the diffusion velocity of edge dislocations depends on

temperature as the square root, namely

vd∝√T

, in contrast to what has been observed in other

crystals.

The conclusions

presented in this study provide valuable new insights into the structure

and mobility of edge dislocations in rare-gas solids, and allow for a quantitative assessment of the

importance of quantum nuclear effects in solid 4He dislocation behavior.

Acknowledgments:

This research was supported by the Australian Research Council under the Future Fellowship

funding scheme (Grant No. FT140100135). J.B. and A.S. acknowledge ﬁnancial support from MINECO (Spain)

Grants No. FIS2014-56257-C2-1-P and No. FIS2015-69017-P. Computational resources and technical assistance

were provided by the Australian Government and the Government of Western Australia through Magnus under

the National Computational Merit Allocation Scheme and The Pawsey Supercomputing Centre.

Author Contributions:

S.S., C.C., and J.B. conceived and designed the study; S.S. and C.C. performed the

calculations; S.S., C.C., A.S. and J.B. analyzed the results and wrote the paper.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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