benjamin p. vollmayr-lee bucknell university october 22,...
TRANSCRIPT
![Page 1: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/1.jpg)
Universality Classes in Coarsening
Benjamin P. Vollmayr-LeeBucknell University
October 22, 2009
![Page 2: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/2.jpg)
Universality Classes in Coarsening
Collaborators:
I Andrew Rutenberg, Dalhousie University
I Melinda Gildner, Bucknell → UPenn
I Will Rosenbaum, Reed → Indiana
I Fawntia Fowler, Reed → Stanford
I Sohei Yasuda, Bucknell → Purdue
Bucknell Physics REU
![Page 3: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/3.jpg)
Universality Classes in Coarsening
Coarsening Introduction
Theoretical Picture & Universality Conjecture
Asymptotic Defect Dynamics
Asymmetric Mobility — Numerical Test of Conjecture
Summary and Outlook
![Page 4: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/4.jpg)
Coarsening . . .
is a nonequilibrium relaxational process in which the characteristiclength scale grows with time.
Many examples in nature:
I binary alloys
I polycrystals
I magnetic domains
I binary fluids
I epitaxy
I salad dressing
I polymer blends
I soap froths
I colloids
I liquid crystals
I faceted surfaces
I and more . . .
![Page 5: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/5.jpg)
Phase Ordering Dynamics (binary alloys, polymer blends)
I Rapid quench into the forbiddenregion of a phase diagram
I system responds locally byequilibrating into one of the twophases
I leads to equilibrated domainsseparated by costly interface
I dissipative dynamics givescoarsening
φφeq1 φeq
2
Tf
T
![Page 6: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/6.jpg)
2D Dry Soap Froth
t = 1.95 h t = 21.5 h t = 166 h
Glazier, Gross, and Stavans, Phys. Rev. A 36, 306 (1987).
Self-similarity!
![Page 7: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/7.jpg)
3D Wet Soap Froth
Magnetic Resonance Imaging
Gonata et al., Phys. Rev. Lett.
75, 573 (1995).
![Page 8: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/8.jpg)
(a) Colloidal Suspension and (b) Polymer Solution
Tanaka, Nishikawa, and Koyama, J. Phys. Cond. Matt. 17, L143 (2005).
Universality!
![Page 9: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/9.jpg)
Homoepitaxial Islands
Cu on a Cu(100) surface
Pai et al., Phys. Rev. Lett. 79, 3210 (1997).
![Page 10: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/10.jpg)
Random Copolymers – PEH/PEB
Shimizu et al., Polymer 45, 7061 (2004).
![Page 11: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/11.jpg)
Why is coarsening so common?
Requirements:
I Excess free energy stored in stable, local defects (e.g., domainwalls): F − Feq ∝ ρdef
L
I Dissipation:dF
dt< 0 ⇒ dρdef
dt< 0
Result: growing characteristic length L(t)
![Page 12: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/12.jpg)
Basic Features of Coarsening
Sharp defects defect size ξ fixed, so for asymptotically latetimes L(t)� ξ ⇔ sharp-defect limit.
Self-similarity domain structure statistically invariant whenrescaled by L(t).
Implies correlation function scalingC(r, t) = f
(r/L(t)
)Power law growth characteristic scale L ∼ tα
Universality exponent α determined by only a few generalfeatures: conservation laws and nature oforder parameter
![Page 13: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/13.jpg)
Coarsening Models I: Kinetic Ising Models
Lattice of spins si = ±1, with hamiltonian H = −J∑〈ij〉
sisj
Spins initially random (Ti =∞). Quench at time t = 0 to T < Tc.
Glauber Dynamics
I spins flip with probability determined by energy ⇒nonconserved order parameter.
Kawasaki Dynamics
I neighboring spins exchanged ⇒ conserved OP.
I additional parameter ε = fraction of spins up
I appropriate for binary mixtures: ↑= Fe, ↓= Al.
![Page 14: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/14.jpg)
Coarsening Models I: Kinetic Ising Models
Lattice of spins si = ±1, with hamiltonian H = −J∑〈ij〉
sisj
Spins initially random (Ti =∞). Quench at time t = 0 to T < Tc.
Glauber Dynamics
I spins flip with probability determined by energy ⇒nonconserved order parameter.
Kawasaki Dynamics
I neighboring spins exchanged ⇒ conserved OP.
I additional parameter ε = fraction of spins up
I appropriate for binary mixtures: ↑= Fe, ↓= Al.
![Page 15: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/15.jpg)
Coarsening Models I: Kinetic Ising Models
Lattice of spins si = ±1, with hamiltonian H = −J∑〈ij〉
sisj
Spins initially random (Ti =∞). Quench at time t = 0 to T < Tc.
Glauber Dynamics
I spins flip with probability determined by energy ⇒nonconserved order parameter.
Kawasaki Dynamics
I neighboring spins exchanged ⇒ conserved OP.
I additional parameter ε = fraction of spins up
I appropriate for binary mixtures: ↑= Fe, ↓= Al.
![Page 16: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/16.jpg)
Kinetic Ising Models
Glauber (spin flip): nonconserved OP → L ∼ t1/2
Kawasaki (spin exchange): conserved OP → L ∼ t1/3
![Page 17: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/17.jpg)
Coarsening Models II: Phase Field Models
Field φ(x, t) describes local concentration. Free energy functional:
F [φ] =∫ddx{
12(∇φ)2 + V (φ)
}φeq
2φeq1
V
φ
Allen-Cahn equation
Nonconserved OP:∂φ
∂t= −δF
δφ⇒ ∂φ
∂t= ∇2φ− V ′(φ)
Cahn-Hilliard equation
Conserved OP:∂φ
∂t= −∇ · J and J = −∇δF
δφ
⇒ ∂φ
∂t= −∇2[∇2φ− V ′(φ)]
![Page 18: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/18.jpg)
Coarsening Models II: Phase Field Models
Field φ(x, t) describes local concentration. Free energy functional:
F [φ] =∫ddx{
12(∇φ)2 + V (φ)
}φeq
2φeq1
V
φ
Allen-Cahn equation
Nonconserved OP:∂φ
∂t= −δF
δφ⇒ ∂φ
∂t= ∇2φ− V ′(φ)
Cahn-Hilliard equation
Conserved OP:∂φ
∂t= −∇ · J and J = −∇δF
δφ
⇒ ∂φ
∂t= −∇2[∇2φ− V ′(φ)]
![Page 19: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/19.jpg)
Coarsening Models II: Phase Field Models
Field φ(x, t) describes local concentration. Free energy functional:
F [φ] =∫ddx{
12(∇φ)2 + V (φ)
}φeq
2φeq1
V
φ
Allen-Cahn equation
Nonconserved OP:∂φ
∂t= −δF
δφ⇒ ∂φ
∂t= ∇2φ− V ′(φ)
Cahn-Hilliard equation
Conserved OP:∂φ
∂t= −∇ · J and J = −∇δF
δφ
⇒ ∂φ
∂t= −∇2[∇2φ− V ′(φ)]
![Page 20: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/20.jpg)
Phase Field Models
Allen-Cahn eq: nonconserved OP → L ∼ t1/2
Cahn-Hilliard eq: conserved OP → L ∼ t1/3
![Page 21: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/21.jpg)
Universality?
Glauber
Allen-Cahn
L ∼ t1/2
Kawasaki
Cahn-Hilliard
L ∼ t1/3
![Page 22: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/22.jpg)
Universality Classes in Coarsening
Coarsening Introduction
Theoretical Picture & Universality Conjecture
Asymptotic Defect Dynamics
Asymmetric Mobility — Numerical Test of Conjecture
Summary and Outlook
![Page 23: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/23.jpg)
Theoretical Challenge
I Self-similar scaling state with universal power law growthgeneric. Demands explanation!
I Characterizing scaling state a starting point for analysis of realsystems. Need universality classes!
Renormalization Group Scenario
I Critical-like behavior ⇒ dynamical RG fixed point controllingthe asymptotic dynamics.
I Not (yet) tractable — a strong-coupling fixed point.
How do we proceed?
![Page 24: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/24.jpg)
Theoretical Challenge
I Self-similar scaling state with universal power law growthgeneric. Demands explanation!
I Characterizing scaling state a starting point for analysis of realsystems. Need universality classes!
Renormalization Group Scenario
I Critical-like behavior ⇒ dynamical RG fixed point controllingthe asymptotic dynamics.
I Not (yet) tractable — a strong-coupling fixed point.
How do we proceed?
![Page 25: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/25.jpg)
Two Routes to Progress
Exact Solution — Lifshitz-Slyozov Theory (’58)
I Conserved OP coarsening in dilute ε→ 0 limit ⇒ isolateddroplets
I Derives scaling state, demonstrates its universality.
I Original prediction of L ∼ t1/3 exponent.
Assume Scaling, Derive Consequences
I Huse (’86) argued COP L ∼ t1/3 extends to all ε.
I Bray’s RG scenario (’89) also gives L ∼ t1/3.
I Bray-Rutenberg energy scaling approach (’94) ⇒ growthexponents. Explains universality classes!
![Page 26: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/26.jpg)
Two Routes to Progress
Exact Solution — Lifshitz-Slyozov Theory (’58)
I Conserved OP coarsening in dilute ε→ 0 limit ⇒ isolateddroplets
I Derives scaling state, demonstrates its universality.
I Original prediction of L ∼ t1/3 exponent.
Assume Scaling, Derive Consequences
I Huse (’86) argued COP L ∼ t1/3 extends to all ε.
I Bray’s RG scenario (’89) also gives L ∼ t1/3.
I Bray-Rutenberg energy scaling approach (’94) ⇒ growthexponents.
Explains universality classes!
![Page 27: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/27.jpg)
Two Routes to Progress
Exact Solution — Lifshitz-Slyozov Theory (’58)
I Conserved OP coarsening in dilute ε→ 0 limit ⇒ isolateddroplets
I Derives scaling state, demonstrates its universality.
I Original prediction of L ∼ t1/3 exponent.
Assume Scaling, Derive Consequences
I Huse (’86) argued COP L ∼ t1/3 extends to all ε.
I Bray’s RG scenario (’89) also gives L ∼ t1/3.
I Bray-Rutenberg energy scaling approach (’94) ⇒ growthexponents. Explains universality classes!
![Page 28: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/28.jpg)
So why am I here talking about universality classes?
Bray-Rutenberg ⇒ growth exponent (L ∼ tα) universality classes:
I α depends only on conservation law and nature of orderparameter
I does not depend on spatial dimension d, volume fraction ε, ormicroscopic details
But which quantities are universal?
Conventional wisdom: correlation function C(r, t) or structurefactor S(k, t) has same universality as the growth exponent.
Not true!
![Page 29: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/29.jpg)
So why am I here talking about universality classes?
Bray-Rutenberg ⇒ growth exponent (L ∼ tα) universality classes:
I α depends only on conservation law and nature of orderparameter
I does not depend on spatial dimension d, volume fraction ε, ormicroscopic details
But which quantities are universal?
Conventional wisdom: correlation function C(r, t) or structurefactor S(k, t) has same universality as the growth exponent.
Not true!
![Page 30: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/30.jpg)
So why am I here talking about universality classes?
Bray-Rutenberg ⇒ growth exponent (L ∼ tα) universality classes:
I α depends only on conservation law and nature of orderparameter
I does not depend on spatial dimension d, volume fraction ε, ormicroscopic details
But which quantities are universal?
Conventional wisdom: correlation function C(r, t) or structurefactor S(k, t) has same universality as the growth exponent.
Not true!
![Page 31: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/31.jpg)
Distinct Universality (for conserved scalar OP)
Quantities that affect the correlation function but not the growthexponent:
Trivial
I volume fraction ε
I spatial dimension d
. . . everyone knew that already. ε = 1/2 ε < 1/2
Less Trivial
I anisotropic surface tension σ(n̂) (e.g. Ising model)
σ(n̂) exact Lifshitz-Slyozov solution fordilute coarsening
[BVL & Rutenberg ‘99; Gildner,
Rosenbaum, Fowler, and BVL ’09]
![Page 32: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/32.jpg)
Distinct Universality (for conserved scalar OP)
Quantities that affect the correlation function but not the growthexponent:
Trivial
I volume fraction ε
I spatial dimension d
. . . everyone knew that already. ε = 1/2 ε < 1/2
Less Trivial
I anisotropic surface tension σ(n̂) (e.g. Ising model)
σ(n̂) exact Lifshitz-Slyozov solution fordilute coarsening
[BVL & Rutenberg ‘99; Gildner,
Rosenbaum, Fowler, and BVL ’09]
![Page 33: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/33.jpg)
Questions
I Does the scaled correlation function have any universality?
I If so, what are its universality classes?
I And, which quantities belong to exponent universality classes,versus correlation function universality classes?
[Higher order correlation functions, curvature distribution,autocorrelation exponents, persistence exponents, growth lawamplitudes, . . . ]
![Page 34: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/34.jpg)
Answer?
Conjecture:
I Growth exponents are a special case. Superuniversal due toconstraints
I Correlation function universality reflects domain morphologyuniversality [Ockham, circa 1300], so
focus on the domain morphology!
I Domain morphology universality reflects defect (domain wall)dynamics universality, so
focus on the defect dynamics!
![Page 35: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/35.jpg)
Answer?
Conjecture:
I Growth exponents are a special case. Superuniversal due toconstraints
I Correlation function universality reflects domain morphologyuniversality [Ockham, circa 1300], so
focus on the domain morphology!
I Domain morphology universality reflects defect (domain wall)dynamics universality, so
focus on the defect dynamics!
![Page 36: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/36.jpg)
Universality Classes in Coarsening
Coarsening Introduction
Theoretical Picture & Universality Conjecture
Asymptotic Defect Dynamics
Asymmetric Mobility — Numerical Test of Conjecture
Summary and Outlook
![Page 37: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/37.jpg)
Asymptotic Defect Dynamics
What are the dynamical rules for the interfaces?
For a given domain configuration, e.g.
how will it evolve? What is the sequence of future domainconfigurations?
Use late-time asymptotia to reduce to simpler sharp defectdynamics.
![Page 38: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/38.jpg)
Example: conserved scalar OP with isotropic σ
Step 1. Surface Tension
Consider a flat interface atx = 0 with b.c. as shown:
φ2eq
x
φ
φ1eq
Equilibrium concentration profile given by
0 = µ(x) =δF
δφ(x)= V ′(φ)− c∇2φ+ . . .
Solution φint(x) gives free energy per unit interface:
Surface Tension: σ ≡ F [φint(x)]/A
For curved interfaces, σ(κ) = σ +O(κ)
![Page 39: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/39.jpg)
Example: conserved scalar OP with isotropic σ
Step 2. Bulk Mobility
I In bulk φ ≈ φeq1 , so local chemical potential proportional tothe supersaturation:
µ(x) ∼ V ′′(φeq1 )(φ(x)− φeq1
)I Asymptotic current:
J = −M(φ)∇µ ∼ −M(1)V ′′(1)∇φ
I Gives diffusion equation in bulk:
∂φ
∂t= −∇ · J ∼ D∇2φ
I φ and µ equilibrate to ∇2µ in time teq ∼ L2
![Page 40: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/40.jpg)
Example: conserved scalar OP with isotropic σ
Step 3. Gibbs-Thomson at interfaces:
µ(x) =σ
∆φeqκ(x) +O(κ2)
Step 4. Quasistatic in bulk: ∇2µ = 0since diffusion field equilibrates fasterthan interfaces move.
Determines µ(x) everywhere!
Step 5. Interface velocity determined by bulk flux to interface:
∆φeqv(x) = n̂ · (J+ − J−) ⇒ v(x) =M1 n̂ · ∇µ1 −M2 n̂ · ∇µ2
∆φeq
Huse: v ∼ L̇, ∇µ ∼ 1/L2 ⇒ L̇ ∼ 1/L2 ⇒ L ∼ t1/3
![Page 41: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/41.jpg)
Example: conserved scalar OP with isotropic σ
Step 3. Gibbs-Thomson at interfaces:
µ(x) =σ
∆φeqκ(x) +O(κ2)
Step 4. Quasistatic in bulk: ∇2µ = 0since diffusion field equilibrates fasterthan interfaces move.
Determines µ(x) everywhere!
Step 5. Interface velocity determined by bulk flux to interface:
∆φeqv(x) = n̂ · (J+ − J−) ⇒ v(x) =M1 n̂ · ∇µ1 −M2 n̂ · ∇µ2
∆φeq
Huse: v ∼ L̇, ∇µ ∼ 1/L2 ⇒ L̇ ∼ 1/L2 ⇒ L ∼ t1/3
![Page 42: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/42.jpg)
Example: conserved scalar OP with isotropic σ
Step 3. Gibbs-Thomson at interfaces:
µ(x) =σ
∆φeqκ(x) +O(κ2)
Step 4. Quasistatic in bulk: ∇2µ = 0since diffusion field equilibrates fasterthan interfaces move.
Determines µ(x) everywhere!
Step 5. Interface velocity determined by bulk flux to interface:
∆φeqv(x) = n̂ · (J+ − J−) ⇒ v(x) =M1 n̂ · ∇µ1 −M2 n̂ · ∇µ2
∆φeq
Huse: v ∼ L̇, ∇µ ∼ 1/L2
⇒ L̇ ∼ 1/L2 ⇒ L ∼ t1/3
![Page 43: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/43.jpg)
Example: conserved scalar OP with isotropic σ
Step 3. Gibbs-Thomson at interfaces:
µ(x) =σ
∆φeqκ(x) +O(κ2)
Step 4. Quasistatic in bulk: ∇2µ = 0since diffusion field equilibrates fasterthan interfaces move.
Determines µ(x) everywhere!
Step 5. Interface velocity determined by bulk flux to interface:
∆φeqv(x) = n̂ · (J+ − J−) ⇒ v(x) =M1 n̂ · ∇µ1 −M2 n̂ · ∇µ2
∆φeq
Huse: v ∼ L̇, ∇µ ∼ 1/L2 ⇒ L̇ ∼ 1/L2 ⇒ L ∼ t1/3
![Page 44: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/44.jpg)
Example: conserved scalar OP with isotropic σ
Take case of equal bulkmobilities: M1 = M2 = M .
I For all such systems v(x) same at each point along theinterface, up to an overall factor Mσ/(∆φeq)2.
I All systems will evolve through the same sequence ofconfiguration: they have the same defect trajectories.
I In rescaled time τ = Mσ(∆φeq)2
t, all systems evolve identically!
I If M1 6= M2, the above still hold for all systems with the sameratio M1/M2.
![Page 45: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/45.jpg)
Domain Morphology Universality
Conjecture: Domain morphology has same universality as thedefect trajectories.
Wrong if
I different trajectories can lead to the same morphology(superuniversal)
I different morphologies possible from same trajectories(history dependent)
Corollary: in rescaled time, growth law L ∼ Aτα is determined bythe morphology ⇒ the growth law amplitude should have thesame universality as the correlation function.
![Page 46: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/46.jpg)
Predicted Universality Classes — conserved scalar OP
I anisotropic σ(n̂) modifies µ(x) at interface, so morphologydepends on σ(n̂, T ).
I field-dependent mobility M(φ), specifically the ratioM(φeq1 )/M(φeq2 ).
M1=M2 M1/M2
M
φ1 φ2
so same UC UCdifferent
so same UCsame
I volume fraction ε and spatial dimension d.
Morphology universality determines correlation function, growthlaw amplitude, persistence exponents, . . . .
![Page 47: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/47.jpg)
Universality Classes in Coarsening
Coarsening Introduction
Theoretical Picture & Universality Conjecture
Asymptotic Defect Dynamics
Asymmetric Mobility — Numerical Test of Conjecture
Summary and Outlook
![Page 48: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/48.jpg)
Asymmetric Cahn-Hilliard Equation
Field φ(x, t) describes local concentration. Free energy functional:
F [φ] =∫ddx{
12(∇φ)2 + V (φ)
}µ(x) =
δF
δφ(x)= −∇2φ+ V ′(φ) φeq
2φeq1
V
φ
Conservation:∂φ
∂t= −∇ · J and J = −M(φ)∇δF
δφ
where M(φ) = 1 +mφ
⇒ asymmetric Cahn-Hilliard Eq.
φ−1 1
M
Define R ≡ M(1)M(−1)
![Page 49: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/49.jpg)
Power Law Growth of Domain
10
100
1 10 100 1000 10000 100000 1e+06
L
t
t1/3
t1/4
R=1248
![Page 50: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/50.jpg)
Structure Factor — Scaling Collapse
S(k, t) = t2/3 g(kt1/3)
1e-05
0.0001
0.001
0.01
0.1
1
10
100
0.1 1 10 100
St-2
/3
kt1/3
R=1, t=1*103
3*103
1*104
3*104
R=4, t=1*103
3*103
1*104
3*104
![Page 51: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/51.jpg)
Structure Factor — Different R
0.1
1
10
100
1000
0.01 0.1 1
S
k
R=1248
Little R dependence, if there is any!
![Page 52: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/52.jpg)
Number of Domains
1
10
100
1000
100 1000 10000 100000
t-2/3
R=1248
Faster phase has more domains
![Page 53: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/53.jpg)
Ratio of Number of Domains at t = 104
0
2
4
6
8
10
12
0 1 2 3 4 5 6 7 8 9
N1/
N-1
R
![Page 54: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/54.jpg)
Universality Classes in Coarsening
Coarsening Introduction
Theoretical Picture & Universality Conjecture
Asymptotic Defect Dynamics
Asymmetric Mobility — Numerical Test of Conjecture
Summary and Outlook
![Page 55: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/55.jpg)
Conclusions
I The growth law exponent and the correlation function do nothave the same universality.
I The growth law amplitude and the correlation function dohave the same universality, determined by the morphology.
I These universality classes apply to the complete asymptoticscaling state, and might be determined defect dynamics.
I Numerical tests of the asymmetric Cahn-Hilliard equationoffer preliminary confirmation.
I Structure factor is not a sensitive measure — need to look atdomain number
![Page 56: Benjamin P. Vollmayr-Lee Bucknell University October 22, 2009bvollmay/talks/2009_10_22_William... · 2010. 6. 4. · I soap froths I colloids I liquid crystals I faceted surfaces](https://reader033.vdocument.in/reader033/viewer/2022060709/6075b8d2a010e00d507ff773/html5/thumbnails/56.jpg)
Future Work
I Generalize defect trajectory analysis (vector order parameter,liquid crystals, hydrodynamics, facets, froths, . . . ). WithSteven Watson.
I For numerical tests, we need larger system sizes to push runsto later times.
I We’ll investigate the Cahn-Hilliard equation with asymmetricpotential.