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Jamming Meets Experiments(Day 2: Theoretical Frameworks)
Karen DanielsDept. of Physics, NC State University
http://nile.physics.ncsu.edu @karenedaniels [email protected]
Boulder School 2017: Frustrated and Disordered Systems
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2https://xkcd.com/1867/
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quasi-2D experiments
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Photoelastic Inversion
optimize fringe pattern & force/torquebalance each disk
Daniels, Kollmer, Puckett. Rev. Sci. Inst. (2017)
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Network Science
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Writing Data as an Adjacency Matrix
Papadopoulos, Daniels, Porter, Bassett (on arXiv soon!)
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System 2D Domain 1D Curves
Efficiency of global signal transmission
Local geographic domains
Bottlenecks or centrality
Global Efficiency ModularityGeodesic Node
Betweenness
0D Particles
Local loop structures
Clustering Coefficient
Network science metrics for different scales
Bassett, Owens, Daniels, Porter. PRE (2012)
http://netwiki.amath.unc.edu/
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Force Network Ensemble
● count equations & constraints # of degrees of →freedom
● friction – provides history-dependence
– changes the counting of valid states
Tighe, Snoeijer, Vlugt, van Hecke. Soft Matter (2010)
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Experiment Version of FNE
Kollmer & Daniels, Powders & Grains 2017
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i (1- std[
i])
std[i
]
i
Bi
Jonathan Kollmer
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Null Model Matters
Bassett, Owens, Porter, Manning, Daniels. Soft Matter (2015)
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Effect of Resolution
Parameter
Bassett, Owens, Porter, Manning, Daniels. Soft Matter (2015)
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Gap Factor Hull Ratio
Bassett, Owens, Porter, Manning, Daniels. Soft Matter (2015)
Huang & Daniels. Granular Matter. (2015)
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Network Measures Distinguish Exp/Sim
Bassett, Owens, Porter, Manning, Daniels. Soft Matter (2015)
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Multilayer Networks
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Black = Higher Friction BathRed = Lower Friction Subsystem
Puckett & Daniels PRL (2013)Papadoploulos, Puckett, Daniels, Bassett. PRE (2016)
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Papa
dopl
oulo
s, Pu
cket
t, D
anie
ls, B
asse
tt. P
RE
(201
6)
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21Papadoploulos, Puckett, Daniels, Bassett. PRE (2016)
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Configurational Entropy & Statistical Ensembles
“
”
Sam Edwards
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Edwards' Central Idea
smallest system volume
only one validconfiguration
larger volume
more validconfigurations
S=lnΩ(V )1X
=∂ S∂V
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Test the “Zeroth Law”Zeroth law
requirestemperatureequilibration
Does Xbath = Xsubsys
?low
friction
highfriction
Puckett & Daniels. PRL (2013)
James Puckett
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3 lighting schemes
white light
particle positions
polarized light
contact forces
fluorescence
identify low-friction
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Pist
on
Pistongravity
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Local Voronoï Volumes
Puckett & Daniels. PRL (2013)
3 example histograms (for subsystem only)
sample Voronoï tessellation
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Plot the Overlapping Histograms
Puckett & Daniels. PRL (2013)
log
slope difference in compactivity→
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Compactivity Fails to Equilibratered (low-friction system) and black (high-friction bath) do not
have the same compactivity
Puckett & Daniels PRL 2013
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Compactivity Fails to Equilibratered (low-friction system) and black (high-friction bath) do not
have the same compactivity
Puckett & Daniels PRL 2013
explanation? equiprobability of jammed states only holds at jamming
Martiniani, Schrenk, Ramola, Chakraborty, Frenkel. Nature Physics (2017)
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Quantifying Interparticle Forces
d⃗ mn
f⃗ mn
particle m
particle n
Σ̂=∑m ,n
d⃗ mn f⃗ mn
force-moment tensor
Σ̂=V σ̂ Γ=Tr Σ̂
pressurestress tensor
Bi, Henkes, Daniels, Chakraborty. Ann. Rev. Cond. Matt. (2015)
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Photo Vector Forces Pseudo-photo→ →
Daniels, Puckett, Kollmer. Rev. Sci. Inst (2017) http://github.com/jekollmer/PEGS
Σ̂=∑m ,n
d⃗ mn f⃗ mnp=12(σ1+σ2)
τ=12(σ1−σ2)
grain scale force- moment tensor:
σ̂= ∑cluster
Σ̂
decompose into normal, deviatoric:
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Constraints on Interparticle Forces
What do you know about the 5 black arrows?
Bi, Henkes, Daniels, Chakraborty. Ann. Rev. Cond. Matt. (2015).
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Force Balance Tiles→
Bi, Henkes, Daniels, Chakraborty. Ann. Rev. Cond. Matt. (2015).
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Conservation: Maxwell-Cremona tile area
Tighe & Vlugt JSTAT 2010.
Sarkar, Bi, Zhang, Ren, Behringer, Chakraborty. PRE 2016
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Represent Whole Packing in Force Space
moving this point corresponds to adjusting the contact forces
in a way that preserves force balance
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Forces Field Theory→
● define a vector gauge field h(x, y) on the dual space of voids (, )
● going counterclockwise around a grain, increment the height field by the contact force between the two voids:
Ball & Blumenfeld PRL (2002) DeGiuli & McElwaine PRE (2011) Henkes & Chakraborty PRL (2005) PRE (2009)
h⃗∗= h⃗+ f⃗ lm
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Relationship to Continuum Mechanics
● forces are locally balanced is conserved→● Cauchy stress tensor can be calculated from the
height field: σ̂=∇⃗× h⃗
Σ̂=V σ̂
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Caveat: friction can cause non-convexity
Sarkar, Bi, Zhang, Behringer, Chakraborty. PRL (2013)