the phases of hard sphere systems...outline: • order by disorder • constant temperature &...
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The Phases of Hard Sphere Systems
Tutorial for Spring 2015 ICERM workshop “Crystals, Quasicrystals and Random Networks”
Veit Elser Cornell Department of Physics
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outline:• order by disorder • constant temperature & pressure ensemble • phases and coexistence • ordered-phase nucleation • infinite pressure limit • what is known in low dimensions • the wilds of higher dimensions
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eliminating time: ergodic hypothesis
sphere centers: x1, x2, . . . , xN
arbitrary property:✓(x1, x2, . . . , xN )
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eliminating time: ergodic hypothesis
sphere centers: x1, x2, . . . , xN
arbitrary property:✓(x1, x2, . . . , xN )
✓ =1
T
Z T
0dt ✓(x1(t), . . . , xN (t))time average:
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eliminating time: ergodic hypothesis
sphere centers: x1, x2, . . . , xN
arbitrary property:✓(x1, x2, . . . , xN )
✓ =1
T
Z T
0dt ✓(x1(t), . . . , xN (t))time average:
h✓i =Z
dµ ✓(x1, . . . , xN )configuration average:
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eliminating time: ergodic hypothesis
sphere centers: x1, x2, . . . , xN
arbitrary property:✓(x1, x2, . . . , xN )
✓ =1
T
Z T
0dt ✓(x1(t), . . . , xN (t))time average:
h✓i =Z
dµ ✓(x1, . . . , xN )configuration average:
✓ = h✓iergodic hypothesis:
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Gibbs measure: dµ = d
Dx1 · · · dDxN e
��H0(x1,...,xN )
(x1, . . . , xN ) 2 A
NA ⇢ RDdomain: “box”
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Gibbs measure: dµ = d
Dx1 · · · dDxN e
��H0(x1,...,xN )
(x1, . . . , xN ) 2 A
NA ⇢ RDdomain: “box”
1/� = kBTabsolute temperature:
asphere diameter:
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Gibbs measure: dµ = d
Dx1 · · · dDxN e
��H0(x1,...,xN )
(x1, . . . , xN ) 2 A
NA ⇢ RDdomain: “box”
H0(x1, . . . , xN ) =
⇢0 if 8 i 6= j kxi � xjk > a1 otherwise
Hamiltonian:
1/� = kBTabsolute temperature:
asphere diameter:
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constant temperature & pressure ensemble
environment
T, p
V
pV = work performed on constant pressure environment in expanding box to volume V
H(x1, . . . , xN ;V ) = H0(x1, . . . , xN ) + pV
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dimensionless variables & parameters
xi = a xi
V = aD V
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dimensionless variables & parameters
xi = a xi
V = aD V
� p V = p V
p = (kBT/aD) p
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rescaled constant T & p ensemble
dµ(p) = d
Dx1 · · · dDxN dV e
�H0�pV
measure:
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rescaled constant T & p ensemble
dµ(p) = d
Dx1 · · · dDxN dV e
�H0�pV
measure:
configurations:
(x1, . . . , xN ) 2 A(V )N
8 i 6= j kxi � xjk > 1
A(V ) = RD/(V 1/DZ)D (periodic box)
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rescaled constant T & p ensemble
dµ(p) = d
Dx1 · · · dDxN dV e
�H0�pV
measure:
configurations:
(x1, . . . , xN ) 2 A(V )N
8 i 6= j kxi � xjk > 1
A(V ) = RD/(V 1/DZ)D (periodic box)
pdimensionless pressure (parameter):specific volume (property): v = hV i/N
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p = 2.00 v = 1.73
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p = 2.00 v = 1.73
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p = 20.00 v = 1.02
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p = 20.00 v = 1.02
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108 spheres (D = 3)
v = 1/p2
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� =|BD(1/2)|
vpacking fraction:
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T
p gas crystal
T
pcrystal
gas
T
pcrystal
gas
quiz: hard sphere phase diagram
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T
p gas crystal
T
pcrystal
gas
T
pcrystal
gas
quiz: hard sphere phase diagram
p = p (kB/a3)T a ⇡ 4Ap ⇡ 10 (krypton)
p (kB/a3) ⇡ 20 atm/K
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thermal equilibriumHere we are, trapped in the amber of the moment.
— Kurt Vonnegut
• configuration sampling via Markov chains, small transitions
• Markov sampling is not unlike time evolution
• diffusion process: slow for large systems
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approaching equilibrium
�p
�t= p > 0 � > 0
108 spheresp = 10�5
10�6
10�7
10�8
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approaching equilibrium
�p
�t= p > 0 � > 0
108 spheresp = 10�5
10�6
10�7
10�8
p⇤�1 �2
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quiz: between phases
Suppose we sample hard spheres (3D) at fixed volume, corresponding to an intermediate packing fraction (0.52), what should we expect to see in a typical configuration?
A. gas phase
B. crystal phase
C. quantum superposition of gas and crystal
D. none of the above
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quiz: between phases
Suppose we sample hard spheres (3D) at fixed volume, corresponding to an intermediate packing fraction (0.52), what should we expect to see in a typical configuration?
A. gas phase
B. crystal phase
C. quantum superposition of gas and crystal
D. none of the above
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phase coexistenceAt the coexistence pressure, gas and crystal subsystems are in equilibrium with each other.
We can accommodate a range of volumes by changing the relative fractions of the two phases.
crystal
crystalgas
gas �1 = 0.494
�2 = 0.545
�1 < � < �2{p⇤ = 11.564
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thermodynamics
Z(p) =
Zdµ(p) =
Zdµ(0)e�pVnumber of microstates:
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thermodynamics
Z(p) =
Zdµ(p) =
Zdµ(0)e�pVnumber of microstates:
s(p) =kBN
logZ(p)entropy per sphere:
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thermodynamics
Z(p) =
Zdµ(p) =
Zdµ(0)e�pVnumber of microstates:
s(p) =kBN
logZ(p)entropy per sphere:
F (p) = �Ts(p)free energy per sphere:
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thermodynamics
Z(p) =
Zdµ(p) =
Zdµ(0)e�pVnumber of microstates:
s(p) =kBN
logZ(p)entropy per sphere:
F (p) = �Ts(p)free energy per sphere:
v =
1
NhV i = � 1
N
d
dplogZ(p) = � 1
kB
ds
dp
identity:
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p⇤
vcryst
vgas
v = �k�1B
ds
dp�v
p⇤
sgas
scrysts(p)
sgas(p⇤) = scryst(p
⇤)
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… suppose, young man, that one Marine had with him a tiny capsule containing a seed of ice-nine, a new way for the atoms of water to stack and lock, to freeze. If that Marine threw that seed into the nearest puddle … ?
—Kurt Vonnegut (Cat’s Cradle)
phase equilibrium: the ice-9 problem
• Phase coexistence implies there is an entropic penalty, a negative contribution from the interface, proportional to its area.
• The interfacial penalty severely inhibits the spontaneous formation of the ordered phase, especially near the coexistence pressure.
“critical nucleus”
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size of critical nucleus
Assume interface-surface entropy is isotropic, so the critical nucleus is spherical:
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size of critical nucleus
Assume interface-surface entropy is isotropic, so the critical nucleus is spherical:
snuc = (scryst � sgas)Vnuc
v� �Anuc
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size of critical nucleus
Assume interface-surface entropy is isotropic, so the critical nucleus is spherical:
snuc = (scryst � sgas)Vnuc
v� �Anuc
scryst(p⇤ +�p)� sgas(p
⇤ +�p) = kB(�vcryst + vgas)�p
= kB(��v)�p > 0
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size of critical nucleus
Assume interface-surface entropy is isotropic, so the critical nucleus is spherical:
snuc = (scryst � sgas)Vnuc
v� �Anuc
Vnuc = (4⇡/3)R3 Anuc = 4⇡R2
scryst(p⇤ +�p)� sgas(p
⇤ +�p) = kB(�vcryst + vgas)�p
= kB(��v)�p > 0
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size of critical nucleus
Assume interface-surface entropy is isotropic, so the critical nucleus is spherical:
snuc = (scryst � sgas)Vnuc
v� �Anuc
Vnuc = (4⇡/3)R3 Anuc = 4⇡R2
scryst(p⇤ +�p)� sgas(p
⇤ +�p) = kB(�vcryst + vgas)�p
= kB(��v)�p > 0
snucRc
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snuc(Rc) = 0 ) Rc = 3(�/kB)(v/�v)1
�p
exponentially small nucleation rate: e�Vnuc/v0
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snuc(Rc) = 0 ) Rc = 3(�/kB)(v/�v)1
�p
exponentially small nucleation rate: e�Vnuc/v0
The critical nucleus would in any case have to be at least as large to contain most of a kissing sphere configuration, thus making the spontaneous nucleation rate decay exponentially with dimension.
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infinite pressure limit
limp!1
✓�k�1
B
ds
dp
◆= vmin
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infinite pressure limit
limp!1
✓�k�1
B
ds
dp
◆= vmin
s(p)/kB ⇠ �vmin p+ C
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infinite pressure limit
limp!1
✓�k�1
B
ds
dp
◆= vmin
s(p)/kB ⇠ �vmin p+ C
Z(p) ⇠ e�N(vmin p+C)
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infinite pressure limit
limp!1
✓�k�1
B
ds
dp
◆= vmin
s(p)/kB ⇠ �vmin p+ C
Z(p) ⇠ e�N(vmin p+C)
In three dimensions we know there are many sphere packings — stacking sequences of hexagonal layers — with exactly the same 𝑣min.
The constant C depends on the stacking sequence.
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free volume: disks
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free volume: disks
Allowed positions of central disk when surrounding disks are held fixed.
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free volume: spheres
� = 0.45
cubic hexagonal AB …
hexagonal AC …
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hexagonal stacking
A
BC
“hexagonal close packing” (hcp) : … ABAB …“face-centered cubic” (fcc) : … ABCABC …
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limp!1
�s(p)/kB = Cfcc � Chcp ⇡ 0.001164
N = 1000 :
prob(fcc)
prob(hcp)
⇡ 3.2
densest & most probable sphere packing
fcc and hcp stackings are the extreme cases; other stackings have intermediate probabilities.
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hard sphere phases in other dimensions
• The statistical ensemble, through phase transitions, detects any kind of order.
• Good packings assert themselves at lower packing fractions.
• Different dimensions have qualitatively different phase behaviors.
what we know so far …
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D = 1
• only one phase (gas)
• exercise: calculate s(p) in your head
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D = 2
* Brown Faculty
• If you had to devote your life to one dimension, this would be the one.
• three phases: gas, hexatic, crystal
• gas:
• gas-hexatic coexistence:
• hexatic:
• crystal:
• hexatic-crystal transition is Kosterlitz*-Thouless
• logarithmic fluctuations in crystal (Mermin-Wagner)
• gas-hexatic coexistence resolved only recently (Michael Engel, et al.)
0.720 < �
0.714 < � < 0.720
0.702 < � < 0.714, p⇤ = 9.185
0 < � < 0.702
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2 < D < 7
• As in three dimensions, two phases: gas and crystal.
• Symmetry of crystal phase consistent with densest lattice.
• Ice-9 problem avoided by introducing tethers to crystal sites whose strengths are eventually reduced to zero (“Einstein-crystal method”). Possible ordered phases limited to chosen crystal types.
• Interesting behavior of coexistence pressure:
D �gas � �cryst �max
p⇤ type3 0.494 - 0.545 0.741 11.56 D3
4 0.288 - 0.337 0.617 9.15 D4
5 0.174 - 0.206 0.465 10.2 D5
6 0.105 - 0.138 0.373 13.3 E6
source: J.A. van Meel, PhD thesis
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hard spheres in higher dimensions:
three possible universes
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Leech’s universe
• Every dimension has a unique and efficiently constructible densest packing of spheres.
• At low density the gas phase maximizes the entropy.
• There is only one other phase, at higher density, where spheres fluctuate about the centers of the densest sphere packing (appropriately scaled).
• Classic order by disorder!
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The Stillinger-Torquato* universe• It is unreasonable to expect to find lattices optimized for packing
spheres in high dimensions.
• Argument: For a lattice in dimension D the number of parameters in its specification grows as a polynomial in D. But the number of spheres that might impinge on any given sphere is of order the kissing number and grows exponentially with D.
• Conjecture: In high dimensions and large volumes V, there is a proliferation of sphere packings in V very close in density to the densest (much greater in number than defected crystal packings). Because of this proliferation, the densest packings are actually disordered.
• Since the lack of order in the densest limit is qualitatively no different from the lack of order in the gas, there is only one phase!
* S. Torquato and F. H. Stillinger, Experimental Mathematics, Vol. 15 (2006)
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Conway’s universe
• Mathematical invention knows no bounds, and natural phenomena never fail to surprise.
• Perhaps two dimensions was not an anomaly: multiple phases of increasing degrees of order may lurk in higher dimensions.
• Testimonial: unbiased kissing configuration search in ten dimensions.
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• The densest known sphere packing in 10 dimensions, P10c, has center density δ = 5/128. For comparison, D10 has center density δ = 2/128.
• P10c, a non-lattice, has maximum kissing number 372.
• Output of kissing number search with constraint satisfaction algorithm:
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• The densest known sphere packing in 10 dimensions, P10c, has center density δ = 5/128. For comparison, D10 has center density δ = 2/128.
• P10c, a non-lattice, has maximum kissing number 372.
• Output of kissing number search with constraint satisfaction algorithm:
372 spheres
0.0 0.1 0.2 0.3 0.4 0.5
cos!Θ"
374 spheres
0.0 0.1 0.2 0.3 0.4 0.5
cos!Θ"disordered ordered
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• The densest known sphere packing in 10 dimensions, P10c, has center density δ = 5/128. For comparison, D10 has center density δ = 2/128.
• P10c, a non-lattice, has maximum kissing number 372.
• Output of kissing number search with constraint satisfaction algorithm:
372 spheres
0.0 0.1 0.2 0.3 0.4 0.5
cos!Θ"
374 spheres
0.0 0.1 0.2 0.3 0.4 0.5
cos!Θ"disordered ordered
cos ✓ 2 {0, (3±p3)/12, 1/2}
(378 spheres in complete configuration)
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The new kissing configuration extends to a quasicrystal in 10 dimensions with center density δ = 4/128 !
Sphere centers projected into quasiperiodic plane; five kinds of contacting spheres.
Quasiperiodic tiling of squares, triangles and rhombi.
V. Elser and S. Gravel, Discrete Comput. Geom. 43 363-374 (2010)