Voronoi0_respres

7/31/2019 Voronoi0_respres

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Statistical geometric analysis of

hard-sphere microstructures

V. Senthil Kumar

Advisor: Prof. V. Kumaran

Department of Chemical Engineering

Indian Institute of Science


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Outline 1

Theme: Statistical geometry is useful for deriving structure-

property correlations, characterizing the microstructures

and locating structural transitions.

Introduction

- Hard-disks & Hard-spheres,

- Voronoi tessellation

Cell-volume distribution Configurational entropy

Equilibrium neighbor statistics EHSD of LJ fluid

Sheared neighbor statistics Non-equilibrium transitions

Bond-orientational analysis Martensitic transformation


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Characterization of microstructures 2

m = 9.44 m = 5.72

Aim: To develop measures of microstructural randomness

and utilize them to characterize the state and quantify theproperties.


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Hard-disks & Hard-spheres

- Introduction


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Hard-disk/sphere properties 4

v = specific volume

vp = volume of a particle

=vp

v, packing fraction

vc, c = volume, packing fraction at closed packing

y =

c

, normalized packing fraction

hard rod hard disk hard sphere is length diameter diametervp

4

2 6

3

vc 32 2 12

3

F - 0.691 0.494sF/k - 0.36 1.16M - 0.716 0.545LRP - 0.772 0.002 0.555 0.005DRP - 0.82 0.02 0.64 0.02c 1

23

32


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Voronoi tessellation - Introduction


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Poisson Voronoi Tessellation 5

Point = particle center, departmental store etc.

Convex space-filling polygons formed by the perpendicular

bisectors

Any point inside the Voronoi polygon is closer to its nuclei

than any other nuclei - definition of local volume

Geometric neighbors are the competing zones


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Hard-disk Voronoi tessellation 6

As approaches c Voronoi cells become regular

Useful framework to analyze all structures, random to regular

Geometric definition of local volume and neighbors


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Occurrence: Crystallization, plant cells 7

Source: www

All nuclei appear simultaneously and fixed during growth

Isotropic, linear growth rate

Growth ceases when a cell comes in contact with a neigh-boring cell


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Utility: Microstructural characterization 8

Applications

Microstructural analysis of packings, glass, foam, cel-

lular solids, proteins etc. Widely used in Meteorology,

Geology, Ecology, Metallography, Archeology.

Properties Typically Studied Zhu et al. (2001)

Distribution of cell-volume, number of neighbors, cell

surface area, vertex angles etc.

We show that Voronoi statistics are useful in character-

izing the microstructures and estimating the properties.


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Cell-volume distribution and

configurational entropy of

hard-spheres

J. Chem. Phys. 123, 114501 (2005).


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Configurational entropy 9

Partition Function, Q =

2mkTh2

D N2 Qconf

Config. Integral, Qconf =1

N! V V exp( U) dr1 drNF = kT log Q; Fconf = kT log Qconf

S = F

TN,V ; Sconf =

Fconf

T

N,V

S = Sconf +D

2N k log

2mkT

h2

+

D

2N k

Sconf contains all the thermodynamic information


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Free-volume theory 10

Kirkwood (1950) by spatial coarse-graining showed that

sconf = scell + scom.

() Cellular entropy scell single occupancy of regular

cells.() Communal entropy scom multiple occupancy of reg-

ular cells.

Cohen and Grest (1979) assumed the same partitioningfor the Voronoi irregular cells.

() Ansatz scell = k

f(v)log[f(v)]dv, in analogy with

mixing entropy of binary fluids.

() Communal entropy scom due to the existence of fluidclusters.

() Used in the theory of supercooled liquids.

Aim: To check the validity of Cohen-Grest ansatz for thehard sphere system. First, we will focus on f(v).


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Poisson cell-volume distributions 11

0 0.5 1 1.5 2 2.5 3 3.5

0

0.2

0.4

0.6

0.8

1

v

f(v)

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

v

f(v)

2D 3D

f(v) =m

(m)v(m1) ev; v =

m

D m Reference1 2 Exact result, Kiang(1966)2 3.61 Weaire et. al (1986)2 3.57 DiCenzo and Wertheim (1989)

2 3.578 Current work3 5.56 Andrade and Fortes (1988)3 5.562 Current work


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Hard-disk cell-volume distributions 12

0.85 0.9 0.95 1 1.05 1.10

2

4

6

8

10

12

14

16

18

v

f(v

)

HD, = 0.82

close packed cell-vol = vc

Voronoi cell-vol, v vcFree-volume: vf = v vc

f(vf) =m

(m)v(m

1)f e

vf

Specific volume condition: vf =m

= v

vc

Fitting for vf, approximately compensates the steric effects


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Regularity factor 13

0.8 0.9 1 1.1 1.2 1.3 1.40

5

10

15

20

25

30

v

f(v)

=0.65

=0.70

=0.75

=0.80

=0.85

=0.90

0.8 1 1.2 1.4 1.60

2

4

6

8

10

v

f(v)

=0.70

=0.75

=0.78

=0.80

=0.82

=0.84

TD Rnd

Relative spread=Std. dev

Mean =1m

As cells become regular Std.devMean

, m

Dense TD structures approach Dirac-delta distn,

, m

Dense Rnd structures approach exponential distn, , m


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HD/HS Regularity factors 14

0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

14

16

m

NVE

Rnd

0 0.2 0.4 0.60

10

20

30

40

50

m

NVE

Rnd

On freezing, due to the onset of order, m increases sharply.

TD structures: , m Low density Rnd structures identical to TD structures.

Regularity factor is a structural order parameter. It distinguish densethermodynamic and dense random structures.

Dense Rnd structures: , m . Can we explain this?


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Maximum-entropy formalism 15

Information entropy:sI =

k

vcf(v) log [f(v)]dv =

k

0

f(vf) log f(vf)dvf sI/k = log [v(1 y)] + (m), where

(m) = (m 1)(m) + l o g [(m)] log(m) + m.

sI is a functional of m. Setting

sI

m

= 0 gives m = 1.Exponential distn is maximally random.

Notion of order: equality of cell vols Max disorder: exponential distn.

Least information state: Close packing crystal unit cell.Moderate info state : Ideal gas Rnd number generator.Maximum info state : Dense Random Packing (or jammed states).

Excess Info Ent : sexI /k = log (1 y) + (m) (m0)the reference ideal gas is at same specific volume.


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Structural inhomogeneities near DRP 16

0.85 0.9 0.95 1 1.05 1.10

2

4

6

8

10

12

14

16

18

v

f(v)

0.9 1 1.1 1.20

2

4

6

8

10

v

f(v)

HD = 0.82, N=256, conf=104 HD = 0.82, N=256, conf=1000

Structural inhomogeneities due to DRP algorithm poor fit Binodal f(v) dense and lean regions. Large-volume tails Line defects(2D) or planar defects(3D).

Speculation: An ideal DRP algorithm producing homogeneous struc-tures might exactly realize the max-ent prediction.


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Cellular entropy 17

Cohen-Grest ansatz: scell = k

f(v)log[f(v)]dv

Info ent: sI = kvc

f(v) log [f(v)]dv = k0

f(vf) log

f(vf)

dvfsexI /k = log (1 y) + (m) (m0)

Modified ansatz: sexcell = sexIsexcell/k = [log (1 y) + (m) (m0)]

For the ansatz to be useful, should be a density-independent con-stant.

sconf = scell + scom.scom due to the existence of fluid clusters. As c, no fluid clusters. Hence scom 0 and sconf scell.

We use this asymptotic limit to estimate .

Alder et al. (1966), Z = D/a + c0 + c1a + , wherea = (v vc)/vc = (1 y)/y is the fractional excess free-volume.

Using Z expression in sexconf/k = y0Z1y

dy,

gives sexconf/k = D log (1 y) (c0 c1 1) log y + c1/y + .Comparing the dominant terms, we get = D.

This identification is independent of the gamma fit and is valid forany short-ranged force model.


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Cellular entropy 18

0 0.2 0.4 0.6 0.815

10

5

0

svE/

k

VorVirialComposite

0 0.2 0.4 0.615

10

5

0

svE/

k

VorVirialComposite

HD HS

With = D, the cellular entropies are nearly equal to the configura-tional entropies for densities above freezing.

s = sconf + D2 k log2mkTh2

+ D

2k

The last two terms arising from integration of momenta arefirst-order homogeneous is D.

Toy model: HS in a box, x,y,z, No. of states=2D.N independent HSs, = (2D)N = 2ND

S = k log = ND log 2.


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Summary: Cell-volume distributions 19

m is a structural order parameter, which distinguishes

dense thermodynamic and dense random structures.

If number of states 2D, then entropy is first-order ho-

mogeneous in D. We have found a factor of D correction

to Cohen-Grest Ansatz, scell = kD

f(v)log[f(v)]dv.This identification is independent of the gamma fit and

is valid for any short-ranged force model.

For dense liquids, configurational entropy is mostly cel-

lular. For athermal systems like an ensemble of similarly

prepared powders, communal entropy is identically zero.

Hence cellular entropy solely defines the state.


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Neighbor statistics and

effective hard-sphere diameter of

a Lennard-Jones fluid

J. Chem. Phys. 123, 074502 (2005).


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Voronoi partitioning of g(r) 20

0 2 4 6 80

0.5

1

1.5

2

2.5

3

3.5

r

g(r)

g1g2g3

g4gsum

Rahman (1966)

g(r) =(r)

=

N(r, r + r)

V(r, r + r) N(r, r + r) = N1(r, r + r) + N2(r, r + r) +

g(r) = g1(r) + g2(r) +


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Relation between fn(r) and gn(r) 21

gn(r) =n(r)

=

Nn(r, r + dr)

V(r, r + dr)

fn(r)dr =Nn(r, r + dr)

0Nn(r, r + dr)dr

= Nn(r, r + dr)

Cn

gn(r) =Cn

fn(r)V(r,r+dr)

dr

=Cn

fn(r)

S(r)

g(r) = 1 Sr

Cn fn(r)

Cn and fn(r) contain the structural information in g(r).

Cn are sensitive indicators of microstructural changes.

g() = g1() =C1

f1()

S()

Z =pv

kBT= 1 + B2 g() = 1 +

2DC1 f1().

C1 =

n Pn


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Hard-sphere Cn 22

0 0.2 0.4 0.614

14.2

14.4

14.6

14.8

15

15.2

15.4

15.6

C1

NVE

Rnd

DRP

0 0.2 0.4 0.650

55

60

65

70

C2

NVE

Rnd

Due to onset of order Cn decreases on freezing, F 0.494.

DRP 0.64 0.02, match with Finneys experimental data. For perfect FCC lattice Cn = 10 n2 + 2, Fuller(1975),

C1 = 12, C2 = 42, . . .

The thermodynamic solid Cn approaches 14 near the regular close

packing, instead of 12, due to the topological instability of FCClattice for infinitesimal perturbations, Troadec et al.(1998).


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Topological instability of FCC lattice 23

The Wigner-Seitz cell of perfect FCC lattice is the rhombic dodec-ahedron.

Vertex A formed by the intersection of 3 planes, 8 nos. (stable),Vertex B formed by the intersection of 4 planes, 6 nos. (unstable).

On slight perturbation, a type-B vertex gets destroyed and an addi-tional face can form instead.


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Topological instability 24

1

3

4

2

5

6

1 1.5 2 2.50

1

2

3

4

5

r/

gn

(r)

gg1g2

There are equal chances for an additional face to form between thepairs (1,6), (2,4) and (3,5). Thus

C1=12+No. of type-B vertices

Prob of forming an additional face at a type-B vertex=12+613

=14.

Few second neighbors get promoted as first neighbors.

Due to the topological instability, polyhedra with faces 12 to 18

coexist, with the mean at 14, even near regular close packed limit.


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Thermodynamic consistency 25

ZHS Nconf C1 f1() Zvor0.68 36.34 256 1000 14.0252 15.00 36.07500 512 14.0260 15.13 36.37

864 297 14.0255 15.09 36.271372 187 14.0249 15.20 36.53

0.72 108.05 256 1000 14.0083 46.23 108.94500

512 14.0080 46.15 108.74

864 297 14.0083 46.29 109.081372 187 14.0085 46.63 109.86

Young and Alder(1979):

ZHS= 3

/+2

.566+0

.55

1

.19

2+5.95

3,where = (v vc)/vc = (1 y)/y, is the dimensionless excess free-volume.

Zvor = 1 + 2D C1 f1().


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Hard-sphere Pn 26

0 0.2 0.4 0.610

2

101

100

Pn

n=12

n=13

n=14

n=15

0 0.2 0.4 0.610

5

104

103

102

101

10

0

Pn

n=11

n=16

n=17

n=18

n=19

Due to the topological instability, polyhedra with faces 12 to 18coexist, with the mean at 14, even near regular close packed limit.


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Effective hard-sphere diameter 27

0 0.2 0.4 0.614

14.2

14.4

14.6

14.8

15

15.2

15.4

15.6

C1

NVE

Rnd

DRP

Boltzmann: The distance of closest approach in particle trajectories.Radial distribution function or its Fourier transform of soft fluids canbe matched with that of HS at some diameter, Kirkwood (1942),Verlet(1968). The notion of EHSD is useful in correlating soft po-tential equilibrium and transport properties with the hard fluid.

The integral equation approaches due to Barker and Henderson (1976),Chandler, Weeks and Andersen (1983) and its modification by Lado(1984), integrate the repulsive part of the LJ potential with differ-ent criteria yielding different expressions for the EHSDs, results inBen-Amotz and Herschbach (1990).

By the equality of a soft potential C1 with that of the hard-sphereC1 we estimate the EHSD.


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EHSD of LJ fluid 28

LJ state values from theoretical models Voronoi analysisT BH WCA LWCA C1

0.8230 0.8010 1.0226 1.0193 1.0169 14.55 1.02951.0649 0.7000 1.0134 1.0113 1.0090 14.77 1.0246

2.7584 0.7195 0.9742 0.9718 0.9686 14.91 0.97353.2617 0.9200 0.9666 0.9600 0.9558 14.70 0.9540

LJ(r) = 4 LJ[(LJ/r)12 (LJ/r)6],T = kB T /LJ, = 3LJ,

= /LJ,

= (6 //)1/3.

EHSD predicted from C1 show less than 2% deviation from the BH,

WCA and LWCA models. This shows the validity of EHSD conceptand our computational procedure.

While Boltzmanns notion was based on trajectories, ours is basedon neighbor distribution.


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Summary: Neighbor statistics 29

We have shown that the information contained in g(r) is

partitioned into the sets Cn and fn(r), and C1 = nPn.We report the Cn and Pn for thermodynamic and randomhard-sphere structures. They are sensitive microstruc-

tural indicators.

Due to the topological instability of FCC lattice, even

near regular close packing C1 does not approach 12, but

approaches 14.

Using the C1 of Lennard-Jones fluid, we are able to esti-

mate its EHSD within 2% deviation from the theoretical

predictions.


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Sheared neighbor statistics and

non-equilibrium transitions

Phys. Rev. E 73, 051305 (2006).


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Lees-Edwards boundary conditions 30

+U

U

+U/2

U/2

Ly

Inelastic hard-particles are the simplest model rapid granular matter.

In central box: vx = U

(yly/2)ly

; = Uly

Top crossing: vx vx UBottom crossing: vx vx + U

Top/bottom crossings inputs energy into the central boxand inelastic collisions dissipate the energy.

= 1 for elastic collisions, < 1 for inelastic collisions.


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Voronoi partitioning of g(r) 31

g(r, ) = 1

n=1

N

nr,

Vr,=

n=1

gn(r, )

gn(r, ) =Cn

fn(r, )Vr,

dr d

=Cn

rfn(r, )

g(r, ) =1

r

n=1

Cn fn(r, ) (2D)

g(r,,) =1

r2

n=1Cn fn(r,,) (3D)

We show that the sheared structure Cn are useful scalar mea-

sures, which are sensitive to non-equilibrium structural tran-

sitions.


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FCC layers 32

Source: www.mines.edu

In FCC solids undergoing plastic deformation the (111) planes slidepast each other.

Colloidal crystals in plane Poiseuille flow have the (111) plane par-allel to the velocity vorticity plane, Kanai et al.(2002), and have atwinned-faulted FCC structure.


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Shear hard-sphere Cn 33

0 0.2 0.4 0.614

14.2

14.4

14.6

14.8

15

15.2

15.4

15.6

C1

NVE

=0.999

=0.90

=0.80

=0.70

x/

z

/

2 1 0 1 22

1.5

1

0.5

0

0.5

1

1.5

2

For = 0.999 0.80 shear ordering is in the range 0.52 - 0.53. For = 0.70 shear ordering is near 0.55. The shear ordering packingfraction in the preferred orientation shear varies in a narrow range.

Note the suppression of nucleation by homogeneous shear.

Close packed layers of hard-spheres move past each other in a zig-zagpath, causing a corresponding haze in the g(x, z) plot.

Close-packed layers sliding past each other offer lesser resistance toshear than a disordered structure. Such a layer-wise flow causesShear-thinning in colloids and dense suspensions.


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Possibility of shear-thickening 34

0 0.2 0.4 0.614

14.2

14.4

14.6

14.8

15

15.2

15.4

15.6

C1

NVE

=0.999

=0.90

=0.80

=0.70

For = 0.999 0.80 shear ordering is in the range 0.52 - 0.53. For = 0.70 shear ordering is near 0.55.

With a constant coefficient of restitution, due to the lack on an in-trinsic energy scale, the structures are shear-rate independent. Henceshear-thickening can be observed only with a velocity-dependent co-efficient of restitution.

Say at = 0.545, the shear rate is such that

0.80, the system is

ordered. At a higher shear rate say 0.70, then the system getsdisordered.


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vim-dependent coefficient of restitution 35

vim vY, plastic deformation, 1.18 v1/4 , where v = vim/vY,Johnson (1985).

= A

(A4 + v)1/4

,

A 0.2 to 0.4, Spahn et al. (1997).

vim

vY, viscoelastic deformation, = 1

1 v

1/5im + 2 v

2/5im

i v

i/5im

Schwager and Poschel(1998)

McNamara and Falcon (2005):

= 1 (1 Y) v1/5 if vim < vY;

Y v1/4

if vim > vY.

,

Current work:

=A + B exp(C v1/5 )

[(A + B)4 + v

]1/4

,

with A = 1.18, and B = 0.12, C = 0.44 for Nylon data of Labous etal. (1996).


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Shear amorphization at = 0.545, = 20 mm 36

x/

z/

2 1 0 1 2

2

1.5

1

0.5

0

0.5

1

1.5

2

x/

z/

2 1 0 1 2

2

1.5

1

0.5

0

0.5

1

1.5

2

= 2400 s1 = 2500 s1

Sh hi ti t 0 545


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Shear amorphization at = 0.545. 37

mm s1 C1 C210 4800 14.0959 53.1667

4900 14.0978 53.1682

5000 14.5478 57.10505200 14.5705 57.2308

20 2300 14.1024 53.16172400 14.0959 53.1667

2500 14.5592 57.23283000 14.5627 57.1953

Savage and Jeffreys(1980) predict c 1 for granular suspensions,quoted in Barnes(1989).

At a given packing fraction, c/vY is the only material dimensionlessgroup. Hence at shear amorphization c = constant.

For charge stabilized colloidal suspension, c 2, since the gov-erning dimensionless group is c2/(020).

S Sh d i hb t ti ti


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Summary: Sheared neighbor statistics 38

The information contained in g(r) is partitioned into the

sets Cn and fn(r). We report Cn for homogeneously

sheared inelastic hard-spheres. They are sensitive indi-

cators of shear-ordering and shear-amorphization transi-tions.

The suppression of nucleation by homogeneous shear is

evident in these statistics. The near-elastic sheared struc-

tures are identical to the thermodynamic structures, be-

low F.

Using velocity-dependent coefficient of restitution, we

have demonstrated a non-hydrodynamic pathway to re-

alise shear-amorphization or shear-thickening. We find

that c is a constant, for a given vY and packing frac-tion.


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Bond-orientational analysis of

sheared microstructures

J. Chem. Phys. 124, 204508 (2006).

Global bond orientational analysis


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Global bond-orientational analysis 39

Angular orientation of each bond Ylm(, ), where 0 and0 < 2. Rotationally invariant combinations are used to charac-terize the microstructures, Steinhardt et al (1983).

Ql =

4

2 l + 1

lm=l

|Ylm|21/2

Q6 = 0 for random structures andQ6 = 0 for common crystalline structures.Hence, Q6 is used to locate any kind of crystallization.

0.3 0.4 0.5 0.6 0.70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Q6

TD

Rnd

=0.60

=0.70

=0.80

=0.90

=0.99

Local bond orientational analysis


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Local bond-orientational analysis 40

Ql1 l2 l3 =

m1+m2+m3=0

l1 l2 l3

m1 m2 m3

Ql1 m1 Ql2 m2 Ql3 m3,

Q6 = 0 for all crystalline structures. Third order invariants are usedfor structure discrimination

Mitus et al.(1995) showed that Q446 is useful for FCC structuredetermination.For an FCC cluster: Q446 > Qc446, Q

c446 = 0.7

103.

0 0.5 1 1.5 2 2.5 3

3

0

2500

5000

Q446

f(Q

446

)

=0.49

=0.50

=0.55

=0.60

=0.65

0 0.5 1 1.5 2 2.5 3

3

0

2000

4000

Q446

f(Q

446

)

=0.52

=0.55

=0.60

=0.64

TD Shear, = 0.90

Fraction of FCC clusters 41


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Fraction of FCC clusters 41

0.3 0.4 0.5 0.6 0.70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Q6

TD

Rnd

=0.60

=0.70

=0.80

=0.90

=0.99

0.35 0.4 0.45 0.5 0.55 0.6 0.650

0.2

0.4

0.6

0.8

1

f

TD

Rnd

=0.60

=0.70

=0.80

=0.90

=0.99

Q6 is high for dense sheared structures, so the system is crystallized. But even in the dense, near-elastic limit there is only 40% FCC

clusters. This is a clear evidence of the coexistence of multiplecrystalline orders.

Body-centered tetragonal structure 42


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Body-centered tetragonal structure 42

ideal ideal shear

Martensitic transformation: rapidly quenched metals FCC-BCTtransition. When close-packed layers of spheres slide past each other,

BCT structures are formed. Close-packed layers of spheres are stacked such that adjacent layer

spheres are above the mid-points of neighboring layer spheres. Lat-tice ratios: {a,a,c}, 10 neighbors at a distance c = , 4 neighbors ata distance a = 3/2, a/c = 3/2.

BCC-like polyhedra (8 hexagons and 6 squares) signature polyhe-dra, useful in identifying local structures.

Shear hard-sphere Pn 43


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Shear hard sphere Pn 43

0 0.2 0.4 0.6

55

60

65

70

C2

NVE

=0.999

=0.90

=0.80

=0.70

0 0.2 0.4 0.6 0.810

3

102

101

100

Pn

NVE n=12

NVE n=14

LEBC n=12

LEBC n=14

The sheared solid Cn are lower than the thermodynamic solid Cn, thisis not to be interpreted that the sheared solid structures are compactthan the thermodynamic solid structures.

Note that P12 drops sharply in the sheared solid structures, whileP14 rises. These trends are unlike those in the thermodynamic solidstructures.

P14 increases on shear, while Pn for n = 12, 13, 15-18 decreases, dueto the formation of BCT structure.

Distribution of lattice ratios - BCT structures 44


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Distribution of lattice ratios BCT structures 44

a/c

a/b

1 1.1 1.2 1.3 1.4 1.51

1.1

1.2

1.3

1.4

1.5

a/c

a/b

1 1.1 1.2 1.3 1.4 1.51

1.1

1.2

1.3

1.4

1.5

Shear = 0.59 = 0.64

a/c = 3/2 1.225.

Maximum shearable limit: max 0.64.

For ord < < max, the signature polyhedra are BCT type.For > max, the signature polyhedra are FCC type.

Dense sheared structures show Martensitic signatures.

Distribution of lattice ratios - FCC structures 45


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Distribution of lattice ratios FCC structures 45

a/c

a/b

1 1.1 1.2 1.3 1.4 1.51

1.1

1.2

1.3

1.4

1.5

a/c

a/b

1 1.1 1.2 1.3 1.4 1.51

1.1

1.2

1.3

1.4

1.5

TD = 0.55 = 0.65

14-faceted polyhedra are formed by additional faces at a pair ofoppositely placed B vertices.

For > F, the signature polyhedra have a/c =

2 1.414.

Summary: Bond-orientational analysis 46


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Summary: Bond orientational analysis 46

Global analysis: Q6 is useful to locate any kind of crystallization.

Local analysis: Q446 is useful to locate FCC clusters. Dense random structures show negligible crystallization.

Even in the dense, near-elastic limit there are only about 40% FCCclusters in sheared structures, even though Q6 is high. This is a clearevidence of the existence of multiple crystalline orders.

FCC: ABC, HCP: ABA. When close-packed layers slide past eachother random stacking faults are generated, FCC-to-HCP transition.We showed the presence of HCP structure using a pair analysis (not

presented).

On shearing hard-spheres show FCC-to-BCT or the Martensitic tran-sition. On shearing P14 increases, with a decrease in Pn for n = 12,13, 15 - 18, due to the formation of BCT structure. The presence ofBCT structure is also shown using the signature polyhedra analysis.

In a nut-shell 47


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Statistical geometry is the appropriate starting point to

develop structure-property correlations: e.g.

scell = k D

f(v)log[f(v)]dv,

Z = 1 + 2D C1 f1(), C1 =

n Pn,

effective hard-sphere diameter.

Statistical geometrical measures are useful in character-

izing microstructures, e.g. m, Cn, Pn, Q6, Q446 etc.

These measures are useful in locating structural transi-

tions: e.g. freezing, shear-ordering, shear-amorphization

and Martensitic transitions.

Acknowledgments 48


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g

Prof. Kumaran, for his course Physics of Fluids, andallowing me pursue a statistical-geometric approach.

Prof. Binny Cherayil for introducing Callen.

Prof. Anindya Chatterjee for his course on nonlinear os-

cillations.

JRD Tata Library and SERC.

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