Crystal Symmetry Properties

According to equation (3.16) the change of the elastic free energy∆Frelative to the reference con-figuration is for an elastically homogeneous and isothermal system of constant particle numberN and dimensionalitydin absence of external stress (σ =0)

∆F=F(T,ε)−F(T,ε=0) =1 2 Z

d^dR C_{i jkl}ε_{i j}ε_kl≡1 2 Z

d^dR C_αβε_αε_β. (3.41)

8Taking the second derivative of (3.7) results in:

ε_{i j|kl} = 1

2

u_i|jkl+u_j|ikl

ε_{kl|i j} = 1

2

u_{k|li j}+u_{l|ki j}

ε_il|jk = 1

2

u_{i|l jk}+u_{j|i jk}

ε_jk|il = 1

2

u_j|kil+u_k|jil

(Abbreviation used here: u(R)_i|j≡∂u_i/∂R_j, and so on for higher derivatives.) Due to the continuity of the dis-placements and their derivatives the order of derivatives can be interchanged. Finally one obtains by addition of all four relations the mathematical identity (3.38).

9The totally asymmetric tensor ∋

i jkhas the value+1 on even interchange of the indicesi, jandkbeing all different, the value−1 on odd interchange, and the value 0 otherwise.

3.3.1 Isotropic Systems

Isotropic systems have two independent elastic constants, which are the so-calledLam´e coefficients λ andµ [Lan91]. The elastic part of the free energy for a system with dimensiond has the form

∆F = ^Z d^dR where the bulk modulusBis connected to the Lam´e coefficients via

B=λ+2

dµ. (3.44)

The constant µ describes the pure shear, because all volume changes have been incorporated in the second summand by subtraction of the trace of the shear tensor in the second step. It is called theshear modulus.

3.3.2 Two-Dimensional Structures

The components of the elasticity tensor ofhexagonally ordered two-dimensional systemscan be written in pictorial form¹⁰

So we are left over with the two independent elastic constantsC11andC12. The free energy reads

∆F=1

It can be brought to the harmonic form

∆F = 1 Comparison of the coefficients of these two expressions for∆Fgives

C11 = K1+K2

C12 = K1−K2

4C₃₃ = K₃=2(C₁₁−C₁₂)

10In the pictorial form of the symmetric strain tensor only non-zero elements of the upper half tensor are marked. Solid circles connected via a solid line mark identical entries. The symbol×denotes the entry¹₂(C₁₁−C₁₂).

or

K₁ = 1

2(C₁₁+C₁₂)≡B K₂ = 1

2(C₁₁−C₁₂)≡µ₁ (3.49)

K3 = 2(C11−C12)≡µ2=4K2=4µ1.

In the paper of Senguptaet al.[Sen00b] the bulk modulusB and the shear modulus µ1=µ2/4 have been calculated for the 2D lattice by evaluation of the microscopic strain-strain fluctuations obtained from Metropolis Monte-Carlo simulations of hard discs arranged in a triangular lattice under zero external stress. For a 2D system, which is not isotropic or hexagonally ordered, it is C336=¹₂(C11−C12)resulting in two different shear moduliµ1andµ2(cf. Fig3.2).

For the 2D system the elements of the elasticity tensor are explicitly written in terms of the ele-ments of the compliance tensor of equation (3.33):

C₁₁ = S₁₁

(S₁₁−S₁₂)(S₁₁+S₁₂)

C₁₂ = −S₁₂

(S11−S12)(S11+S12) (3.50)

C33 = 1 S₃₃ and according to equation (3.49)

K₁=B = 1

2(S11+S12) ≡ 1 S++

K2=µ1 = 1

2(S₁₁−S₁₂) ≡ 1

S₋₋ (3.51)

K₃=4µ₁ = 1 4S₃₃.

The dimensionlessYoung’s modulus(modulus of extension)Edescribes the response of the system to applied uniaxial stress,i.e. E=σ11/ε11for dilatation inx-direction. For the 2D-system we get in terms of the 2D Lam´e constantsλ andµ and the bulk modulusB=λ+µ, [Cha00]

E= 4Bµ B+µ =

1 4

1 µ + 1

λ+µ ₋₁

. (3.52)

The strainε₂₂along the normal direction determines the dimensionlessPoisson ratio ν= −ε₂₂

ε11 =B−µ

B+µ. (3.53)

According to the KTHNY theory¹¹, two-dimensional solids melt via two successive continuous transitions involving the unbinding of dislocations and disclinations, respectively. Within this

11 Kosterlitz, Thouless, Halperin, Nelson, and Young (KTHNY) [Str88,Nel02] developed a theory based upon topo-logical order considerations which predicts a two-stage melting scenario for 2D systems (in contrast to a single first-order solid-liquid phase transition in 3D). In the first KTHNY transition step the crystalline system becomes

theory, which is usually formulated for triangular structures under zero external stress, the 2D Young’s modulus E depends on the fugacity e^(Ec/kBT) of the core dislocation energyE_c and the

“coarse-graining” length scale. The solid-hexatic dislocation unbinding transition is predicted to occur when the 2D Young’s modulus of equation (3.52) equals the constant 16/π. However, the two-stage melting process of the KTHNY theory may be preempted by a single first-order transition between the solid and isotropic liquid, because only the point at which the solid becomes unstable to the generation of free dislocations is predicted. In simulations both scenarios have been found, whereas in experiment the hexatic phase has been identified [Zah99]. Simulations of 2D hard disc systems in the vicinity of melting have been preformed for example in [Sen00a,Bat00].

The self-written simulation code for 2D-systems of triangular lattices was tested with respect to the results of Senguptaet al.[Sen00b]. Further studies on 2D systems have been done in the group of Prof. P. Nielaba at the University of Konstanz by K. Franzrahe [Fra08].

3.3.3 Cubic Crystals

Forcubic systems(simple cubic (SC), face-centered cubic (FCC), or body-centered cubic (BCC) crystals) only three elastic constantsC11,C12, andC44 are independent [Lan91] and the tensor of elastic constants in Voigt notation has the following pictorial form

C_αβ =

After evaluation of the Einstein summation convention the free energy for cubic crystals is given as

unstable and topological defects calleddislocationsare formed from dissociation of dislocation pairs. Proliferation of free dislocations produces the so-calledhexaticphase which is a liquid crystalline phase with quasi-long ranged orientational order but short ranged positional order. A second KTHNY transition destroys the quasi-long ranged orientational order by breaking up the dislocations into so-called disclinations(scalar charges). This brings the hexatic to the isotropic liquid phase. The KTHNY theory makes several interesting predictions, like for example for the value of the critical temperatureT_cand the behavior of the order parameter correlation length and susceptibility nearT_c, where they form essential singularities (∝e^bt−ν witht≡T/T_c−1). Predictions from KTHNY theory in principle can be checked by simulations. For superparamagnetic colloids confined to the planar liquid-gas interface of a hanging droplet Zahn and coworkers [Zah99] demonstrated experimentally that the melting process follows the two-stage scenario described by the KTHNY theory.

In the second step the conditioni6= jof the last two sums has been removed, and in the last step all contributing terms have been written down explicitly.

For central potentials additionally the so-called Cauchy relation holds, which is the symme-try [Bor68]

C₁₂=C₄₄. (3.58)

Using the following definitions of linear combinations of the Lagrangian strains e1 ≡ ε11+ε22+ε33

e₂ ≡ ε₁₁−ε₂₂ e₃ ≡ ε₁₁−ε₃₃

e₄ ≡ ε₂₂−ε₃₃ (3.59)

e₅ ≡ ε₂₃ e6 ≡ ε13

e7 ≡ ε12

the free energy (3.57) can be brought to the harmonic form

∆F=1 2 Z

d³R

Ae²₁+B(e²₂+e²₃+e²₄) +4C₄₄(e²₅+e²₆+e²₇)

(3.60) Again comparison of the coefficients gives

C11 = A+2B

C₁₂ = A−B (3.61)

or equivalently

A = 1

3(C₁₁+2C₁₂)

B = 1

3(C₁₁−C₁₂). (3.62)

For cubic symmetry the relations between the elastic constantsC_αβ and the compliance constants S_αβ ≡C_αβ⁻¹ (Eq. (3.33)) are explicitly written as

C₄₄ = 1 S₄₄

C11 = S11+S12

(S₁₁−S₁₂)(S₁₁+2S₁₂) (3.63)

C₁₂ = −S₁₂

(S₁₁−S₁₂)(S₁₁+2S₁₂). Insertion of the equations (3.63) into the equations (3.62) gives

A = 1

3(S11+2S12)≡ 1 SAA

B = 1

3(S11−S12) ≡ 1

SBB. (3.64)

The identification of the compliance constants SAA andSBB in the relations (3.64) can be done, because the free energy in harmonic form (3.60) does not have any cross relations between the strains (3.59).

Stability: The cubic crystal (in absence of external stress) is stable against homogeneous defor-mations when the coefficients of the free energy of equation (3.60) are all positive definite,i.e.

A=C11+2C12>0, B=C11−C12>0, C44>0. (3.65) Forcubic crystals under isotropic pressureone has, according to equation (3.36), the following three independent stress-strain coefficients

B11=C11−P

B12=C12+P (3.66)

B₄₄=C₄₄−P. ObviouslyC12=C44does not implyB12=B44.

Physical meaning of the elements of the stress-strain tensor:

y

x

(a) (b) (c)

Figure 3.2: Three different types of body deformation: (a)expansion or dilatation,(b)shear 1,(c) shear 2. Shown are sketches for the cuts in thex-yplane through the center of a 3D body.

In general, any deformation of a body can be divided into three different parts: ε=ε^B+ε^S1+ε^S2 consisting of a bulk expansion ε^B and two different shear modes ε^S1 and ε^S2 which keep the volume unchanged (cf. Fig:3.2). The stress-strain coefficients related to these deformations will now be derived.

• The tensor ε^B describes the dilatation of a body as being sketched in Fig. 3.2(a). It is diagonal and has the components ε_{i j}^B = ¹₃ε_kkδi j, where the trace ε_kk= ^∆V_V0 is the relative change of the body volume (cf. equation (3.10)). Due to the generalized Hooke’s law the stress tensor is also diagonal. For isotropic external pressure ∆P we have τ_{i j} =−∆Pδ_{i j}. Insertion of these stress and strain tensors into equation (3.30) results in

−∆Pδi j=B_{i jkl}ε_kl^B= 1

3B_{i jkl}εmmδ_kl=1

3B_{i jkk}εmm.

It is clear from reflection symmetry for a cubic crystal thatB_{i jkk}=δ_{i j}B_nnkkand

−∆Pδ_{i j}= 1

3B_nnkkε_mmδ_{i j}=1

3(B₁₁+2B₁₂) ∆V V₀ δ_{i j}.

According to the definition of the bulk modulusB(cf. Eq. (3.23)) one gets

• The strain tensorε^S1describing the shear S1 of Fig.3.2(b) is diagonal and traceless with

ε^S1=

The corresponding stress tensorτ^S1is also diagonal, with the non-zero elements τ₁₁ = 2

• The shear S2 of Fig.3.2(c) is described by the strain tensor

ε^S2=

This results in the non-zero elements of the stress tensorτ^S1:

τ23=τ13=τ12=B44(ε23+ε13+ε12). (3.71) So, the value the elementB44=C44−Pis responsible for the magnitude of the shear S2.

Thedirection dependency of Young’s modulusEin direction of the unit vectorn= (n_x,n_y,n_z) of a cubic crystal is [Lan91]

1 cubic crystal. On the edges Young’s modulus has the value

E= σ

∆l/l0 =(B₁₁+2B₁₂)(B₁₁−B₁₂)

B11+B12 , (3.73)

which is identical to the value of isotropic crystals, where B₄₄= ¹₂(B₁₁−B₁₂). In absence of external stress/ pressure, which will be the case for the systems under consideration, theB_αβ can be replaced by theC_αβ.

Another interesting physical constant is the so-calledPoisson ratioν, which is a measure of the cross contraction∆d/d₀in directionmwhen the body is stretched by∆l/l₀in the directionn per-pendicular tom. Analogous to Young’s modulus described above we get the following direction dependency: case this reduces to the extremal value

ν= B₁₂ B₁₁+B₁₂

(3.66)= C₁₂+P

C₁₁+C₁₂. (3.75)

Isotropic limit: It is helpful to rewrite the free energy (3.56) in terms of Fourier transforms of the displacement field using equation (3.29). This gives

∆F = 1

In this expression only the first summand is not invariant under rotations, because all other sum-mands contain vector dot products. Therefore the coefficient of the first summand must vanish in the isotropic limit:

C₁₁−C₁₂−2C₄₄=0. (3.77)

This condition can be used as a measure of the isotropy of the crystalline system. Inserting con-dition (3.77) into (3.56) and comparing the free energy with the free energy for the isotropic crystal (3.43) returns the following connection of the elastic constants to the Lam´e coefficients

λ =C12 and µ =C44. (3.78)

If we insert the equations (3.77) and (3.78) into the cubic expression for the bulk modulus (3.67), we find agreement with the bulk modulus for an isotropic system of equation (3.44) (in absence of external pressure)

B= 1

3(C₁₁+2C₁₂)^(3.77)= 1

3(3C₁₂+2C₄₄)^(3.78)= λ+2

3µ. (3.79)

Analogous calculations give the isotropic expressions for Young’s modulus E= 9Bµ

3.3.4 Elasticity Tensors for Some Lower Symmetries

Finally, we state the elasticity tensor and the free energy for two other important crystal symme-tries:

• Crystals withhexagonal symmetry(symmetry class:C₆):

C_αβ =

This elasticity tensor implies the free energy [Lan91,Lax66]

F = 1

• Crystals withrhombohedral symmetry(symmetry classes:C3v,D3,D3d):

C_αβ =

Open circles denote the negative value of full circles, and the symbol^u%denotes a value two times the value of the solid circle. Consequently, the free energy is [Lan91,Lax66]

F = 1

Im Dokument Computer-Simulationen zu Strukturen und Phasenumwandlungen in Modell-Kolloiden (Seite 34-42)