Counting Network Configurations in Frictional
Counting Network Configurations in Frictional Granular Materials Karen Daniels James Puckett, Ephraim Bililign, Jonathan Kollmer Department of Physics North Carolina State University
Force Chains in Particulate Materials Majmudar & Behringer Nature (2005)
Mueth, Jaeger, Nagel PRE (1998)
Even more ...
Brujic et al. Physica A (2003)
3D gel beads
3D emulsion
2D emulsion
Desmond & Weeks. Soft Matter (2013)
3D colloid
Lin, Bierbaum, Schall, Sethna, Cohen (2016)
Brodu, Dijksman, Behringer. Nat Comm. (2015)
Newton's Principia
Configurational Entropy & Statistical Ensembles
“ Sam Edwards
Edwards Entropy & Compactivity Edwards & Oakeshott 1989
S =ln Ω (V ) 1 ∂S = X ∂V
smallest system volume
V
larger volume
one valid configuration
Ω
more valid configurations
https://twitter.com/dal e_dixon/status/96439 4738278268928
Force Network Ensemble
●
●
count equations & constraints → # of degrees of freedom friction provides historydependence & changes the counting of valid states
Tighe, Snoeijer, Vlugt, van Hecke. Soft Matter (2010)
Ensembles of 2D packings James Puckett
Ephraim Bililign
Jonathan Kollmer
Measuring Interparticle Contact Forces birefringent disk
digital camera
right circular polarizer
light source left circular polarizer
Majmudar, Behringer, Nature (2005). Daniels, Kollmer, Puckett. Rev. Sci. Inst. (2017)
Which Particles are Reliable? Flaky?
“movie” of images taken at the same step
Compression → Forces → Fluctuations
steps
Kollmer, Daniels. Powders & Grains 2017
Configurations → Adjacency Matrix
Papadapoulous, Daniels, Porter, Bassett. JCN (2018) arXiv: 1708.08080
Network science metrics for different scales System
Global Efficiency
Efficiency of global signal transmission
2D Domain
Modularity
Local geographic domains
1D Curves
Geodesic Node Betweenness
Bottlenecks or centrality
0D Particles
Clustering Coefficient Local loop structures
Betweenness Centrality ●
sij = shortest path between particles i,j
●
can be either # of hops or weighted
●
●
b(n) i
b(n) = fraction of total # of shortest paths that go through particles n high b(n) ~ “airline hubs”
j Brain Connectivity Toolbox: https://sites.google.com/site/bctnet/measures/list
Betweenness centrality predicts forces 824 particles 80 cycles
Kollmer, Daniels (unpub)
Betweenness centrality predicts forces 824 particles 80 cycles
Kollmer, Daniels (unpub)
What sets P(f)? Wyart (2012)
= 0.42 g(r)
Probability
P(f) ~ f
r [d]
f = F/ Kollmer, Daniels (unpub)
= 0.42 > 0.32
Test the “Zeroth Law” for Compactivity high friction low friction
Is there temperature? equilibration Does Xbath = Xsubsys
? Puckett & Daniels. PRL (2013)
3 lighting schemes
white light particle positions
polarized light contact forces
fluorescence identify low-friction
Piston gravity
Piston
Local Voronoï Volumes sample Voronoï tessellation
Puckett & Daniels. PRL (2013)
3 example histograms (for subsystem only)
Compactivity Fails to Equilibrate red (low-friction system) and black (high-friction bath) do not have the same compactivity
explanation? equiprobability of jammed states only holds at jamming Martiniani, Schrenk, Ramola, Chakraborty, Frenkel. NaturePuckett Physics& (2017) Daniels PRL 2013
Configurations of grains & forces particle n ⃗f mn
⃗ d mn
particle m
force-moment tensor
^ ∑⃗ Σ= d mn ⃗f mn m ,n
angoricity ∂S ^ α ij = S =ln Ω ( Σ) ∂ Σ ij Bi, Henkes, Daniels, Chakraborty. Ann. Rev. Cond. Matt. Phys. (2015)
Photoelastic Inversion (1)
Majmudar, Behringer, Nature (2005). Daniels, Kollmer, Puckett. Rev. Sci. Inst. (2017)
Photoelastic Inversion (2)
optimize fringe pattern & force/torque balance each disk
Majmudar, Behringer, Nature (2005). Daniels, Kollmer, Puckett. Rev. Sci. Inst. (2017)
Photo → Vector Forces → Pseudo-photo grain scale forcemoment tensor:
̂ ∑⃗ Σ= d mn ⃗f mn m ,n
^ σ=
∑
^ Σ
cluster
decompose into normal, deviatoric:
Daniels, Puckett, Kollmer. Rev. Sci. Inst (2017)
1 p= (σ 1+ σ 2) 2 1 τ= (σ1 −σ 2 ) 2 http://github.com/jekollmer/PEGS
Packings with different histories
sample ≿100 configurations analyze 8-particle clusters bin by (V, )
V min −V disks V= V min Γ=⟨|σ xx +σ yy|⟩ Bililign, Kollmer, Daniels: 1802.09641
Histograms are “Thermometers” ●
by analogy with Boltzmann, calculate the probability of observing a macroscopic stress σ:
Ω(σ) −α σ P (σ)= e Z (α )
●
where σ is either p or τ and α is the associated temperature-like quantity, angoricity
●
density of states (σ) → not generally not known, but independent of α
●
partition function Z(α) → not generally known
●
the RATIO of two histograms measures RELATIVE angoricity
Dean & Lefèvre, PRL (2003) McNamara, Richard, de Richter, Le Caër, Delannay. PRE (2009)
P (σ∣Γ i ) Z (α j ) σ ( α ℜ= = e P (σ∣Γ j ) Z (αi )
j
−αi )
Bililign, Kollmer, Daniels: 1802.09641
Normal (p) Deviatoric (τ)
Normal: p P (σ∣Γ i ) Z (α j ) σ ( α ℜ= = e P (σ∣Γ j ) Z (αi ) log ℜ=c+ σ ( α j−α i ) relative angoricity
p
j
−αi )
Deviatoric: P (σ∣Γ i ) Z (α j ) σ ( α ℜ= = e P (σ∣Γ j ) Z (αi ) log ℜ=c+ σ ( α j−α i ) relative angoricity
j
−αi )
−α p p
P( p)∝ e
Bililign, Kollmer, Daniels: 1802.09641
−α τ τ−β τ
P( τ)∝ e
2
Constraints on Interparticle Forces Constraint: each particle is force and torque balanced
Force Balance → Maxwell-CremonaTiles
Bi, Henkes, Daniels, Chakraborty. Ann. Rev. Cond. Matt. (2015).
Represent Whole Packing in Force Space
moving this point corresponds to adjusting the contact forces in a way that preserves force balance
Why is there a e
–2
term? Conservation of area of Maxwell-Cremona tiles
Tighe & Vlugt JSTAT 2010
Sarkar, Bi, Zhang, Ren, Behringer, Chakraborty. PRE 2016
→keramicity
Temperature → Equations of State? -angoricity s n te i a h sta h c g e a c u for vide thro P( τ)∝e o l e r l l p iab we s e r x l a i a t v M a n e o h t em Cr
p-angoricity
p = c1(process)
2
−α τ τ−β τ
keramicity
= c2(process)
= c3
Angoricity is a variable of process because friction matters
Schröter, Powders & Grains 2017 (EPJ Proceedings 140, 01008)
Conclusions ●
●
●
A simple network measure (betweenness) guides the ensemble of observed force networks Histograms of the force-moment provide a temperature-like variable (“keramicity”) ← Maxwell-Cremona tile area is a conserved quantity Keramicity exhibits a protocolindependent EOS, but angoricity is process-dependent
http://danielslab.physics.ncsu.edu
R2 τ (r )=S 2 +basal friction r τ (r ) μ (r )= P
(P , τ) Zhu Tang
laser-cut leaf springs
γ˙ (r ) d I (r )= √ P /ρ
vwall
S R = 15 cm
v (r ) ∂v γ= ˙ ∂r
43
Local Granular Rheology
●
●
γ˙ d inertial number: ratio I = between √ P/ ρ –
micro timescale (T) to squeeze a particle into a hole
–
macro timescale of deformation
–
large I corresponds to rapid flow
density diameter d
P
stress ratio: ratio μ= τ between P –
shear stress
–
normal pressure
τ
v wall
∂v γ= ˙ ∂r
r
fixed
44
Local vs. Nonlocal Rheology Local ●
the local shear rate is determined by only the local shear stress
●
resistance to flow is a function of only the local shear rate Nonlocal
●
●
particle rearrangements in one part of a flow trigger rearrangements elsewhere resistance to flow is a function of both the local shear rate and these nonlocal events 45
Kamrin & Koval (PRL 2012)
Nonlocal Rheology
● ●
granular fluidity field
μ≡ τ P
length scale diverges at s fit parameters: A, b, s should be a property of the particles only (not geometry or driving)
– Kamrin, Koval, Hennan - - Bouzid, Claudin
outer edge
inner edge
Growing cooperative length ξ
Tang, Brzinski, Shearer, Daniels (submitted to Soft Matter)
Determination of s (1) upper limit of slowest (I) curve: s > 0.26 (2) maximum of (): s ~ 0.26
… but shouldn't it have to do with forces?
48
Force fluctuations change near μs
Tang, Brzinski, Shearer, Daniels (submitted to Soft Matter)
50
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: ⃗h∗=⃗h+ ⃗f lm
Ball & Blumenfeld PRL (2002) DeGiuli & McElwaine PRE (2011) Henkes & Chakraborty PRL (2005) PRE (2009)
Relationship to Continuum Mechanics ●
●
forces are locally balanced → ̂ is conserved Σ=V σ̂ Cauchy stress tensor can be calculated from the height field: ⃗ ×⃗h ̂ ∇ σ= ̂ Σ=V σ̂
Force Chains in Particulate Materials Majmudar & Behringer Nature (2005)
Mueth, Jaeger, Nagel PRE (1998)
Even more ...
Brujic et al. Physica A (2003)
3D gel beads
3D emulsion
2D emulsion
Desmond & Weeks. Soft Matter (2013)
3D colloid
Lin, Bierbaum, Schall, Sethna, Cohen (2016)
Brodu, Dijksman, Behringer. Nat Comm. (2015)
Newton's Principia
Configurational Entropy & Statistical Ensembles
“ Sam Edwards
Edwards Entropy & Compactivity Edwards & Oakeshott 1989
S =ln Ω (V ) 1 ∂S = X ∂V
smallest system volume
V
larger volume
one valid configuration
Ω
more valid configurations
https://twitter.com/dal e_dixon/status/96439 4738278268928
Force Network Ensemble
●
●
count equations & constraints → # of degrees of freedom friction provides historydependence & changes the counting of valid states
Tighe, Snoeijer, Vlugt, van Hecke. Soft Matter (2010)
Ensembles of 2D packings James Puckett
Ephraim Bililign
Jonathan Kollmer
Measuring Interparticle Contact Forces birefringent disk
digital camera
right circular polarizer
light source left circular polarizer
Majmudar, Behringer, Nature (2005). Daniels, Kollmer, Puckett. Rev. Sci. Inst. (2017)
Which Particles are Reliable? Flaky?
“movie” of images taken at the same step
Compression → Forces → Fluctuations
steps
Kollmer, Daniels. Powders & Grains 2017
Configurations → Adjacency Matrix
Papadapoulous, Daniels, Porter, Bassett. JCN (2018) arXiv: 1708.08080
Network science metrics for different scales System
Global Efficiency
Efficiency of global signal transmission
2D Domain
Modularity
Local geographic domains
1D Curves
Geodesic Node Betweenness
Bottlenecks or centrality
0D Particles
Clustering Coefficient Local loop structures
Betweenness Centrality ●
sij = shortest path between particles i,j
●
can be either # of hops or weighted
●
●
b(n) i
b(n) = fraction of total # of shortest paths that go through particles n high b(n) ~ “airline hubs”
j Brain Connectivity Toolbox: https://sites.google.com/site/bctnet/measures/list
Betweenness centrality predicts forces 824 particles 80 cycles
Kollmer, Daniels (unpub)
Betweenness centrality predicts forces 824 particles 80 cycles
Kollmer, Daniels (unpub)
What sets P(f)? Wyart (2012)
= 0.42 g(r)
Probability
P(f) ~ f
r [d]
f = F/ Kollmer, Daniels (unpub)
= 0.42 > 0.32
Test the “Zeroth Law” for Compactivity high friction low friction
Is there temperature? equilibration Does Xbath = Xsubsys
? Puckett & Daniels. PRL (2013)
3 lighting schemes
white light particle positions
polarized light contact forces
fluorescence identify low-friction
Piston gravity
Piston
Local Voronoï Volumes sample Voronoï tessellation
Puckett & Daniels. PRL (2013)
3 example histograms (for subsystem only)
Compactivity Fails to Equilibrate red (low-friction system) and black (high-friction bath) do not have the same compactivity
explanation? equiprobability of jammed states only holds at jamming Martiniani, Schrenk, Ramola, Chakraborty, Frenkel. NaturePuckett Physics& (2017) Daniels PRL 2013
Configurations of grains & forces particle n ⃗f mn
⃗ d mn
particle m
force-moment tensor
^ ∑⃗ Σ= d mn ⃗f mn m ,n
angoricity ∂S ^ α ij = S =ln Ω ( Σ) ∂ Σ ij Bi, Henkes, Daniels, Chakraborty. Ann. Rev. Cond. Matt. Phys. (2015)
Photoelastic Inversion (1)
Majmudar, Behringer, Nature (2005). Daniels, Kollmer, Puckett. Rev. Sci. Inst. (2017)
Photoelastic Inversion (2)
optimize fringe pattern & force/torque balance each disk
Majmudar, Behringer, Nature (2005). Daniels, Kollmer, Puckett. Rev. Sci. Inst. (2017)
Photo → Vector Forces → Pseudo-photo grain scale forcemoment tensor:
̂ ∑⃗ Σ= d mn ⃗f mn m ,n
^ σ=
∑
^ Σ
cluster
decompose into normal, deviatoric:
Daniels, Puckett, Kollmer. Rev. Sci. Inst (2017)
1 p= (σ 1+ σ 2) 2 1 τ= (σ1 −σ 2 ) 2 http://github.com/jekollmer/PEGS
Packings with different histories
sample ≿100 configurations analyze 8-particle clusters bin by (V, )
V min −V disks V= V min Γ=⟨|σ xx +σ yy|⟩ Bililign, Kollmer, Daniels: 1802.09641
Histograms are “Thermometers” ●
by analogy with Boltzmann, calculate the probability of observing a macroscopic stress σ:
Ω(σ) −α σ P (σ)= e Z (α )
●
where σ is either p or τ and α is the associated temperature-like quantity, angoricity
●
density of states (σ) → not generally not known, but independent of α
●
partition function Z(α) → not generally known
●
the RATIO of two histograms measures RELATIVE angoricity
Dean & Lefèvre, PRL (2003) McNamara, Richard, de Richter, Le Caër, Delannay. PRE (2009)
P (σ∣Γ i ) Z (α j ) σ ( α ℜ= = e P (σ∣Γ j ) Z (αi )
j
−αi )
Bililign, Kollmer, Daniels: 1802.09641
Normal (p) Deviatoric (τ)
Normal: p P (σ∣Γ i ) Z (α j ) σ ( α ℜ= = e P (σ∣Γ j ) Z (αi ) log ℜ=c+ σ ( α j−α i ) relative angoricity
p
j
−αi )
Deviatoric: P (σ∣Γ i ) Z (α j ) σ ( α ℜ= = e P (σ∣Γ j ) Z (αi ) log ℜ=c+ σ ( α j−α i ) relative angoricity
j
−αi )
−α p p
P( p)∝ e
Bililign, Kollmer, Daniels: 1802.09641
−α τ τ−β τ
P( τ)∝ e
2
Constraints on Interparticle Forces Constraint: each particle is force and torque balanced
Force Balance → Maxwell-CremonaTiles
Bi, Henkes, Daniels, Chakraborty. Ann. Rev. Cond. Matt. (2015).
Represent Whole Packing in Force Space
moving this point corresponds to adjusting the contact forces in a way that preserves force balance
Why is there a e
–2
term? Conservation of area of Maxwell-Cremona tiles
Tighe & Vlugt JSTAT 2010
Sarkar, Bi, Zhang, Ren, Behringer, Chakraborty. PRE 2016
→keramicity
Temperature → Equations of State? -angoricity s n te i a h sta h c g e a c u for vide thro P( τ)∝e o l e r l l p iab we s e r x l a i a t v M a n e o h t em Cr
p-angoricity
p = c1(process)
2
−α τ τ−β τ
keramicity
= c2(process)
= c3
Angoricity is a variable of process because friction matters
Schröter, Powders & Grains 2017 (EPJ Proceedings 140, 01008)
Conclusions ●
●
●
A simple network measure (betweenness) guides the ensemble of observed force networks Histograms of the force-moment provide a temperature-like variable (“keramicity”) ← Maxwell-Cremona tile area is a conserved quantity Keramicity exhibits a protocolindependent EOS, but angoricity is process-dependent
http://danielslab.physics.ncsu.edu
R2 τ (r )=S 2 +basal friction r τ (r ) μ (r )= P
(P , τ) Zhu Tang
laser-cut leaf springs
γ˙ (r ) d I (r )= √ P /ρ
vwall
S R = 15 cm
v (r ) ∂v γ= ˙ ∂r
43
Local Granular Rheology
●
●
γ˙ d inertial number: ratio I = between √ P/ ρ –
micro timescale (T) to squeeze a particle into a hole
–
macro timescale of deformation
–
large I corresponds to rapid flow
density diameter d
P
stress ratio: ratio μ= τ between P –
shear stress
–
normal pressure
τ
v wall
∂v γ= ˙ ∂r
r
fixed
44
Local vs. Nonlocal Rheology Local ●
the local shear rate is determined by only the local shear stress
●
resistance to flow is a function of only the local shear rate Nonlocal
●
●
particle rearrangements in one part of a flow trigger rearrangements elsewhere resistance to flow is a function of both the local shear rate and these nonlocal events 45
Kamrin & Koval (PRL 2012)
Nonlocal Rheology
● ●
granular fluidity field
μ≡ τ P
length scale diverges at s fit parameters: A, b, s should be a property of the particles only (not geometry or driving)
– Kamrin, Koval, Hennan - - Bouzid, Claudin
outer edge
inner edge
Growing cooperative length ξ
Tang, Brzinski, Shearer, Daniels (submitted to Soft Matter)
Determination of s (1) upper limit of slowest (I) curve: s > 0.26 (2) maximum of (): s ~ 0.26
… but shouldn't it have to do with forces?
48
Force fluctuations change near μs
Tang, Brzinski, Shearer, Daniels (submitted to Soft Matter)
50
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: ⃗h∗=⃗h+ ⃗f lm
Ball & Blumenfeld PRL (2002) DeGiuli & McElwaine PRE (2011) Henkes & Chakraborty PRL (2005) PRE (2009)
Relationship to Continuum Mechanics ●
●
forces are locally balanced → ̂ is conserved Σ=V σ̂ Cauchy stress tensor can be calculated from the height field: ⃗ ×⃗h ̂ ∇ σ= ̂ Σ=V σ̂
