z-cone model for the energy of a foam s. hutzler, r. murtagh, d. whyte, s. tobin and d. weaire...
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Z-Cone Model for the Energy of a FoamS. Hutzler, R. Murtagh, D. Whyte, S. Tobin and D. Weaire
School of Physics, Trinity College Dublin.Foams and Complex Systems Group www.tcd.ie/physics/foams
©E.Finch
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Aim: understanding foam structure in static equilibrium
Total energy proportional to surface area (for gas and liquid incompressible)
Z- Cone Model provides alternative to numerical Surface Evolver description of foam structure.
It offers new insight into interaction potential between barely touching bubbles.
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Related theoretical studies
Morse and Witten, 1993 (droplet on surface)
Lacasse, Grest and Levine, 1996 (droplet between two plates)
Durian, 1995 (soft disk model - 2d)harmonic potential
3D: Energy-force relationship for small compression: ε ~ - f2 [log(f)-c]
large compression: approx. harmonic
Remaining questions (3D):Dependence on number of contacts Z?Energy-compression relationship?
3D 3D 2D
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The Z-cone approximation
Volume of bubble is divided into Z cones Total solid angle is conserved: opening angle of cone:
θ = arcos(1-2/Z)
Ziman 1961, Fermi Surface of Copper
2θ
area of constant mean curvature
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The geometry of the Z-cone model
boundary conditions:
0atcot
at
zr
hzr
z
z
)1()(0
Rhhc
ξ: dimensionless compression parameter
z
R0: radius of undeformed spherical sector
body of revolution
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Minimisation of cone surface area
under constraint of constant volume
using Euler-Lagrange equation
with Lagrange multiplier λ
Mathematics of minimal surfaces
dzdz
drzrZA
h
0
2
2 1)(2/
Cr
rz
z
LL
)(1)(2/ 2
22
zrdz
drzr
hZ
L
)(3
)0()(/
3
0
2
tg
rdzzrZV
h
0atcot
at
zr
hzr
z
z
boundary conditions:
z
)0(/ r
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Exact equations for energy of cone model
Dimensionless excess energy
Deformation
Definite elliptic integrals
2/1
222222
2
1 2
1 222
1 22
)()1()1(4
),,(with
),,(),(
),,()(),(
),,()(),(
Z
ZZf
dZfZK
dZfZJ
dZfZI
1
),(61
2
),()1(1),( 3/2
3/1
22
ZJZ
ZZ
zKZ
Z
Z
),(
12
2
),(312
2/4
1),(
3/1
ZIZ
Z
ZJZ
ZZ
Z
1)(
),(0
ZA
AZ
)0(/ r (varies between 0 and 1)
A0: area of curved surface of undeformed cap
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Evolution of cone under compression
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Cone model: good approximation of energy variation at low values of deformation
wet limit
for Z=12
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Clear deviations from harmonic potential at small and large deformations
Deformation
Energy / (Deformation2)
Roughly harmonic(more pronounced for lower values of Z;pre-factor ~ Z2)
Logarithmic asymptote
for Z=12
wet limit
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Wet limit: logarithmic asymptotic
ln2
),(2Z
Z
Simple asymptote
Deformation
Energy / (Deformation2)
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Application: variation of energy with liquid fraction (for Z=12)
Dry limit described by ε(φ)=e0 -e1 φ 1/2
Wet limit
)ln(
)(
)1(18)(
2/1
2
c
c
c
Z
1
/43
1
11
3/1
Z
Zc
c
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Application: computation of osmotic pressure
Hoehler: empirical formula for experimental data
gVV
E
)(
2/12)(3.7)//( c
R
log-term
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Conclusions and outlook• Cone model agrees well with SE data in the wet limit and has
right asymptotic form also in the dry limit
• Interaction potential is proportional to Z for each cone in harmonic regime BUT constant in limit of touching bubbles
• Possible extension of energy dependence ε(φ) to random foams, where Z varies with liquid fraction φ
• Wet foam often considered as frictionless granular packing: but note deviations from harmonic potential