1

Of oranges and mayonnaise:O o a ges a d ayo a seThe mathematics of packing spheres

Rob Farr & Rob Groot

September 2011p

2

1611

?74048.018/ ≈π ?

3

12/π9069.0≈

4

In 2 dimensions it’s clear:

Any packing of equal circles can be split up into trianglesp p g

The triangle with the highest packing is this one with 91%this one, with 91% coverage:

Magically this triangle canMagically, this triangle can be used to tile the plane:

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In 3 dimensions there is a fly in the ointment:

The best packed polyhedron (at 78%)...

... can’t be fitted together to tile all of space.

So the globally best packed state is not the locally best packed statepacked state

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Different kinds of packing:1 C t lli ki f di h1. Crystalline packing of monodisperse spheres

Hi tHistory:

1611 Kepler conjecture for1611 Kepler conjecture for monodisperse sphere packing:π/181/2≈0.7405 from face centred cubic.

1831 Gauss proved this for regular1831 Gauss proved this for regular packings

1997 Thomas Hales provided a full proof... Face Centred Cubic

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Crystalline packing of monodisperse spheres

Seen in colloidal crystals with enough time to equilibrate:

e

ratu

re

Gas FCC crystalence

of

d fc

c

ossi

ble

Tem

per y

Coe

xist

ega

s an

d

Not

po

T

0 494 0 545 0 7404

Volume fraction0.494 0.545 0.7404

Note: Because interaction is hard sphere, no T dependence

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Random close packing of monodisperse spheres

At high volume fractions, it takes a long time to form crystalsN 413 694 (2001)

d

Nature 413, 694 (2001)ut

e

ntra

ted

Dil

Con

cen

C

0 494 0 545Liq. Coexistence f.c.c ??

The system can get trapped in a disordered glassy state,

0.494 0.545

y g pp g yeven at a volume fraction where it should crystallize.

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Random close packing of monodisperse spheres

If the system is compressed further in the glassy state, there is a maximum packing fraction which is achievable for this random, disordered state, which is

64.0max ≈φ

First studied in detail by Bernal in 1956, & appears to be a y , ppwell defined state (still controversial – see Torquato et al.).

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Why do we care?y

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The emulsion clockMost foods are emulsions. Both water and oil phases are usually structured.

Margarine Mayonnaise

SDressing

Low FatMargarine

Sauce

SoupW/O O/W

HIGH OIL HIGH OIL

Li id M iNon-DairyCreamVLF Dairy

HIGH WATERHIGH WATER

Liquid MargarineIce Cream

CreamVLF Dairy Spread Fresh Cheese

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The importance of random close packing

In fluid systems of hard particles, viscosity is related to how

Viscosity of (hard) particle suspensions

y p , yfar below RCP you are (e.g. Krieger-Dougherty equation)

[ ]φ[ ] max1

φη

φφη

−

⎟⎠⎞⎜

⎝⎛ −=r Rheology of hard spheres

J Chem Phys 83(9) 4717 (1985)maxφ ⎠⎝ J. Chem Phys. 83(9), 4717 (1985)

At low shear ratesc

osity

At low shear rate, the particles will be randomly arranged

Vis

At high shear rate may get aligned into string

Volume fraction

get aligned into string phases (higher φmax)

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The importance of random close packing

You can go above RCP if the spheres are Yield stress of emulsions & elastic particle gels

deformable, and then you get a solid. The elastic and yield properties of emulsions (e.g. full fat mayo.) are related to how far above RCP they areare related to how far above RCP they are.

ld s

tress

Yiel

J. Coll. Int. Sci. 79 (1996) 439CSLM of dense emulsion

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The importance of random close packing

D’Arcy flow through packed beds (oil through sandstone, Specific surface area and permeability

coffee through grounds) is related to average pore sizes, specific surface area and so indirectly to packing fraction

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Packing of different size distributionsWhat if you mix two powders and tap them down to random close packing?

How much space will they occupy?

Or two different emulsions – what will the viscosity be?

You can run computer simulations of packing, but it is not easy and each simulationnot easy, and each simulation takes hours or days – even with the latest code [Rob Groot].the latest code [Rob Groot].

The only remaining option is to think harder...

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What does polydispersity do to R.C.P.?Think of packing spheres with two different sizes:

(a) (b)

In general, this increases the close packing vol. fraction:

• You can sometimes think of packing small spheres inYou can sometimes think of packing small spheres in the gaps between big ones,

• Or grouping small spheres to form large ones which• Or grouping small spheres to form large ones, which frees up space.

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Random Close Packing of bidisperse systemsThe bidisperse case (mixture of two different sizes) has been long studied – but even this simplest case is not trivial

tTwo parameters:• Ratio R of sphere sizes, and

s pr

esen • Relative amounts of the two

types (w)

parti

cles

f the

se p

Mas

s of

Sphere diameter

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Random Close Packing of bidisperse systems

mass fraction in larger spheres

(Data from R. Groot)

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Predicting the volume fraction of polydisperse hard sphere packingshard sphere packings

Difficulty 1: Small spheres may “rattle around” between large spheres. However, if th i il i th i ht j t i t fthey are similar size, they might just interfere with the packing of the large spheres.

Difficulty 2: It is not just one question: we t t k th i ki f tiwant to know the maximum packing fraction

of any distribution (mixtures of 2, 3, ... 1000) different spheres1000) different spheres.

What are we are trying to approximate?

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The mathematical zoo

3 NumbersFunction

Numbers

Operator

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g(x)=12J0(x)Functions

OperatorFunctions

FunctionsFunctional

Numbers7.2

Functions Numbers

? (Exercise f th d )

Numbersfor the reader)

Functionsg(x)=cosh(x)

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There are other animals, like functionalals and operatorators, but not many have been successfully domesticated.

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What we are looking for

)( φaDPFWe want to approximate the functional F, where

max3 )(: φaDP DF

P3D(D) is the size Pdistribution of the spheres (could be anything)

P3D

anything) size D

This is much more difficult than approximating a function

How can we simplify the problem?

22Mapping the 3D problem onto 1DS t t ith h di t ib ti d tSuppose we start with our sphere distribution, and create a collection of rods in the following way:

∫∞

∝ DD dDDPLLP )()( 31This gives: ∫L DD )()( 31gIf the 3D arrangement of spheres were true RCP, then the length fraction of the rods would be exactly φRCP

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Mapping the 3D problem onto 1D

How do we pack this collection of rods in 1D, in such a way that it represents a section through a 3D RCP state?

There must exist a many-body potential

y p g

),...,,( 21 NxxxVwhich is minimized for the correct positions of the rods.

V will be very complex, reflecting topological properties of 3D space (interactions of next nearest neighbours etc.).

However, the potential V has some basic properties:

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Basic properties of the potential V(1) It should lead to a maximum packing fraction that i h d if ll th d (is unchanged if all the rods (or spheres) are magnified by an equal amount

(2) The potential should be "hard“: it is either 0 or ∞.equal amount.

(3) The interaction between large rods should “reach through” small rods so very small rods can “rattle around” in the gapssmall rods, so very small rods can rattle around in the gaps between the large rods, while the large rods carry the load.

(4) Interaction range for very unequal rods should be determined by size of the smallest rod, so small rods can form a random packed system in between the large rods.

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What is the simplest 2-body potential with these properties?Suppose we have rods on a line:pp

d i t d t h d ti l i t ti... and we introduce a strange hard particle interaction:The rods bump into each other before they actually touch:

L LL1 L2

minimum separation is ),min( 21 LLf

Free to move

This gives ourcannot get closer

This gives our approximation to V

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Results for bidisperse spheres

But, how accurate is this approximation to the potential V?

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Results for log-normal distribution

⎥⎦

⎤⎢⎣

⎡−∝ 2

20

3 2)]/[ln(exp

21)(

σπσDD

DDP D

⎦⎣ 22 σπσD

σ=0 6σ=0.6

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Results for log-normal distribution

⎥⎦

⎤⎢⎣

⎡−∝ 2

20

3 2)]/[ln(exp

21)(

σπσDD

DDP D

⎦⎣ 22 σπσD

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Results for tridisperse spheres

1 : 3 : 9

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ConclusionsConclusions

It is a long story ... but:

We now have a fast and jolly accurate method for predicting RCP for any given sphere size distributionpredicting RCP for any given sphere size distribution

This could be useful for design of systems which areThis could be useful for design of systems which are approximately hard sphere.

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Quick advertisementQuick advertisement

An free and open-source program called “ h k1d” ill b il bl th “S“spherepack1d” will soon be available on the “Source Forge” website:http://sourceforge net/projects/spherepack1dhttp://sourceforge.net/projects/spherepack1d

This will make predictions for arbitrary mixtures of p ysphere sizes, including mixtures of lognormal distributions.