Non-equilibrium materials – maximum strength under

extreme conditions

Alfred Hübler

Center for Complex Systems ResearchUniversity of Illinois at Urbana-Champaign

Research supported in part by the National Science Foundation

(DMS-03725939 ITR)

We study:-The strength of materials with large heat flow-Typical structure of materials with a large flow: Structure of materials with high-voltage currents

We find:-Materials have their maximum strength in a large heat flow if they are produced in a large heat flow.

-Materials produced in a high-voltage current develop fractal structures which maximize the conductivity for the applied current. These fractal structures can be predicted with graph-theoretical models.

Strength of materials with large heat flow

α = Linear expansion coefficient:, c1=Tensile stiffness, c2 = Nonlinear stiffness, k=conductivityProduction temperatures: T1,p, T2,p <=> heat flow Qp= k (T1,p- T2,p)Application temperatures: T1, T2 <=> heat flow Q = k (T1- T2)Equilibrium length at production temperatures: xp = x1,0 (1+ α T1,p) = x2,0 (1+ α T2,p) Equilibrium lengths at application temperatures: x1 = x1,0 (1+ α T1), x2 = x2,0 (1+ α T2)Tensile stresses: F1 = c1 (x-x1)-c2 (x-x1)2 , if x-x1 < c1/(2c2) F2 = c1 (x-x2) -c2 (x-x2)2, if x-x2 < c1/(2c2)

↓heat flow

Strength of materials with a large heat flow

Net tensile stress: Fn(x) =F1(x)+F2(x)

Strength: F = max(Fn(x))Stength F depends on production

temperatures and application temperatures.

Result: The material has maximum application strength if

the production temperatures match the application

temperatures, i.e.Production heat flow Qp

= Application heat flow Q

Figure 1. The strength versus the production temperature T2,p , where T1,p=10.

The application temperatures are T1=10, and T2=20, for α=1, c1=0.95, c2=1

Experimental Study of Structural Changes in Materials due to High-voltage Currents:

Growth of Fractal Transportation Networks

20 kV

needle electrode sprays charge over oil surface

air gap between needle electrode and oil surface approx. 5 cm

ring electrode forms boundary of dish

has a radius of 12 cm

oil height is approximately 3 mm, enough to cover the particles

castor oil is used: high viscosity, low ohmic heating, biodegradable

particles are non-magnetic stainless steel, diameter D=1.6 mm

particles sit on the bottom of the dish

Phenomenology Overview

12 cm

t=0s 10s 5m 13s 14m 7s

14m 14s 14m 41s 15m 28s 77m 27s

stage I:strand

formation

stage II:boundary

connection stage III: geometric expansion

stationary state

Adjacency defines topological species of each particle

Termini = particles touching only one other particle

Branching points = particles touching three or more other particles

Trunks = particles touching only two other particles

Particles become termini or three-fold branch points in stage III. In addition there are a few loners (less than 1%). Loners are not connected to any other particle. There are no closed

loops in stage III.

Relative number of each species is robust

Graphs show how the number of termini, T, and branching points, B, scale with the total number of particles in the tree.

Qualitative effects of initial distribution

N = 752T = 131B = 85

N = 720T = 122B = 106

N = 785T = 200B = 187

N = 752T = 149B = 146

(N = Number of Particles, T = Number of Termini, B=Number of Branch Points)

Can we predict the structure of the emerging transportation network?

?

Predicting the Fractal Transporatation Network

Left: Initial condition, Right: Emergent transporation network

Predictions of structural changes in materials due to a high voltage current: Predicting fractal network growth

Task: Digitize stage II structure and predict stage III transporation network.

1) Determine neighbors, since particles can only connect to their neighbors. All the links shown on the left are potential connections for the final tree.

2) Use a graph-theoretical algorithms to connect particles, until all available particles connect into a tree.

Some particles will not connect to any others (loners). They commonly appear in experiments.

We test three growth algorithms:

1) Random Growth: Randomly select two neighboring particles & connect them, unless a closed loop is formed (RAN)

2) Minimum Spanning Tree Model: Randomly select pair of very close neighbors & connect them, unless a closed loop is formed (MST)

3) Propagating Front Model: Randomly select pair of neighbors, where one of them is already connected & connect them, unless a closed loop is formed (PFM)

loner

Random Growth Model: Randomly select two neighboring particles

Typical connection structure from RAN algorithm.

Distribution of termini produced from 105 permutations run on a single

experiment.

Number of termini produced for all experiments, plotted as a function of

N.

Minimum Spanning Tree Model: Randomly select pair of very close neighbors

Typical connection structure from MST algorithm.

Distribution of termini produced from 105 permutations run on a single

experiment.

Number of termini produced for all experiments, plotted as a function of

N.

Propagation Front Model: Randomly select connected pair of neighbors

Typical connection structure from PFM algorithm.

Distribution of termini produced from 105 permutations run on a single

experiment.

Number of termini produced for all experiments, plotted as a function of

N.

Comparison of all models to experiments

Main Result: The Minimum Spanning Tree (MST) growth model is the best predictor of the emerging fractal transportation network

Structural changes of materials in high voltage current

Experiment: J. Jun, A. Hubler, PNAS 102, 536 (2005)1) Three growth stages: strand formation, boundary connection, and

geometric expansion;2) Networks are open loop;3) Statistically robust features: number of termini, number of branch

points, resistance, initial condition matters somewhat;4) Minimum spanning tree growth model predicts emerging pattern.5) To do: random initial condition, predict other observables, control network

growth, study fractal structures in systems with a large heat flowApplications: Hardware implementation of neural nets, absorbers, batteriesM. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the

Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165 (1999)

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random initial distribution compact initial distribution