The Search for Precursor and Aftershock Dynamics in Aqueous Foam

Michael Folkerts Sam Stanwyck, Oleg Shpyrko Department of Physics University of California San Diego APS March Meeting 2010

Aqueous Foam

Jammed System Laser Diffraction Heterogeneous Dynamics Millisecond Time Scale

Two-Time Correlation 〈 I t 1 ⋅I t 2 〉 q g 2 t 1 , t 2 = 〈 I t 1 〉 q 〈 I t 2 〉q

Illustrates Evolution of Dynamics Central Ridge  t = t 1 2  Broadening => Slowing Autocorrelation  Perpendicular to t = t 1 2 We expect similar plot... (Graphic: A Fluerasu 2004 Coherent X-Ray Studies of Non-Equilibrium Processes)

Two Time Correlation 〈 I t 1 ⋅I t 2 〉 q g 2 t 1 , t 2 = 〈 I t 1 〉 q 〈 I t 2 〉q

• • • • •

Unexpected result Broadening on short intervals. Large rearrangements dominate plot Huge intermittent component Trace parallel to ridge: – Fixed offset – Time Resolved Correlation

Time Resolved Correlation (TRC) 〈 I t ⋅I t〉 q c I t ,= 〈 I t 〉q 〈 I t 〉q • • • • •

(a) Holding tau fixed. Level of correlation between two frames offset by tau. TRC is averaged over to get one point on the autocorrelation plot. Key feature: Abrupt changes are downward spikes. Noisy Data – Skewed PDF.

L Cipelleti 2003 Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics Duri et al. PRE 72 051401

Temporal Contrast

I max − I min max  I min

T.C. t , =〈 I

〉q

I max =max {I t , I t 1 , ... , I  t−1  , I  t} I min=min{I  t  , I  t1 , ... , I  t−1 , I t }

Temporal Contrast and Time Resolved Correlation

Time (ms)

Rearrangement Correlation Random - 80 spikes/min

Rearrangement Profile

2 second windows centered on 17 rearrangements

Rearrangement Profile

19 random 2 second windows

Precursor/Aftershock

Poisson Distribution  Consider a random system  If the expected number of occurrences in this interval is λ, then the probability that there are exactly k occurrences is equal to: k −  e P k , = k! Our Data  6 events per minute on average  We see 2 per second  20 fold increase  Probability on order 10-7

Summary/Conclusions .- Temporally heterogeneous dynamics can be studied with Time Resolved Correlation and Temporal Contrast. - Temporal Contrast simplifies the data into large spikes atop a smooth baseline. - Foam is a rigid system with stress. It's not surprising that dynamics are coupled in space and time. - Rearrangements may set off chain reactions: sudden → continuous → sudden → continuous...

Thank You

Questions?

The Search for Precursor and Aftershock Dynamics in Aqueous Foam

Michael Folkerts Sam Stanwyck, Oleg Shpyrko Department of Physics University of California San Diego APS March Meeting 2010

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Hello. My name is Michal Folkerts. I am an undergraduate physics student from the Univ. of California San Diego. Today I will talk about precursor and aftershock dynamics in aqueous foam.

Aqueous Foam

Jammed System Laser Diffraction Heterogeneous Dynamics Millisecond Time Scale

2

*This talk is a continuation Sam's talk. *By correlating intensity values from a speckle pattern as a function of time. *We gain insight into the physical processes present in our aging foam sample. -----Transition----*We have already observed how the dynamics in our sample slow down with age. *One traditional approach is the two-time correlation function.

Two-Time Correlation g 2 t 1 , t 2 =

〈 I t 1 ⋅I t 2 〉 q 〈 I t 1 〉 q 〈 I t 2 〉q

Illustrates Evolution of Dynamics Central Ridge t = t 1 2  Broadening => Slowing Autocorrelation  Perpendicular to t = t 1 2 We expect similar plot... 3

(Graphic: A Fluerasu 2004 Coherent X-Ray Studies of Non-Equilibrium Processes)

*Here is an example of what a two time correlation plot can look like. *This 2 dimensional function correlates two intensities in time. *Lets take a moment to understand what this plot is showing. *The high central ridge corresponds to speckles correlating with them selves. *The drop off away from this ridge represents intensity correlations between frames separated by increasing time offsets. *This is precisely the second order correlation function, G2(τ), mentioned by Sam. *The broadening of this ridge represents increasing relaxation time with age, meaning the dynamics in this example are slowing down. _____ *We expected a similar plot when correlating our intensity data. However...

Two Time Correlation 〈 I t 1 ⋅I t 2 〉 q g 2 t 1 , t 2 = 〈 I t 1 〉 q 〈 I t 2 〉q

• • • • •

Unexpected result Broadening on short intervals. Large rearrangements dominate plot Huge intermittent component Trace parallel to ridge: – Fixed offset – Time Resolved Correlation 4

*This is the two-time correlation plot of our foam data. *This result was unexpected. *We do see some broadening of the ridge but only over relatively short intervals. *Large intermittent rearrangements in the sample reset the correlation to zero. *It would be interesting to couple to these large rearrangements and study them in more detail. *It turns out that one method for studying these large intermittent rearrangements is to take a trace parallel to the central ridge, along a line of fixed time offset. *This approach is called Time Resolved Correlation. _____ Lets take a little time to see how it works...

Time Resolved Correlation (TRC) c I t ,= • • • • •

〈 I t ⋅I t〉 q 〈 I t〉q 〈 I t 〉q

(a) Holding tau fixed. Level of correlation between two frames offset by tau. TRC is averaged over to get one point on the autocorrelation plot. Key feature: Abrupt changes are downward spikes. Noisy Data – Skewed PDF.

5 L Cipelleti 2003 Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics Duri et al. PRE 72 051401

*TRC is a function of time (not offset). It correlates intensities at a fixed time offset: tau. *When a sample is continuously changing, the TRC plot fluctuates about some mean value [ namely g2(tau) ]. *TRC is useful because it couples to intermittent rearrangements which show up as a downward spike in correlation. *Heterogeneity in dynamics shows up as skewness in the probability distribution for the TRC plot. *Where as Temporally homogeneous dynamics form symmetric Gaussian distribution. _____ *TRC works fine, but a much cleaner representation of heterogeneous dynamics is found with Temporal Contrast...

Temporal Contrast

I max − I min max  I min

T.C. t , =〈 I

〉q

I max =max {I t , I t 1 , ... , I  t−1  , I  t} I min=min {I  t  , I  t1 , ... , I t−1 , I t }

6

(Quick explanation!) *The "temporal contrast" is just the contrast ratio, but in the time dimension.  *When there are intermittent dynamics there will be a higher contrast, with global rearrangements corresponding to spikes in the contrast. _____ *If we compare this now to Time Resolved Correlation...

Temporal Contrast and Time Resolved Correlation

Time (ms)

7

*You can see we get the same type of information, but Temporal Contrast is nice and “clean.” *We now can measure magnitude and duration of each rearrangement. --(Pause: set up what is next)-*One question we could ask is: are these rearrangements are random or clustered? *We can see some hint here in this plot. They seem to come in groups. *Is this evidence of some collective dynamics in our sample? _____ *To answer this question we auto-correlate this time sequence...

Rearrangement Correlation Random - 80 spikes/min

8

*On the left we have the second order correlation function g_2(tau) sampling temporal contrast at 32 min aging time. *The initial relaxation corresponds to half the rearrangement duration. *But similar to pair correlation: we have a high prob of finding event(s) close by. *On the right is the same function plotted for random simulated rearrangements. As you can see there is a relatively smooth relaxation. __ *This gives us good evidence of precursor and/or aftershocks, but the direction of time is ambiguous. *Another approach is to look at the Temporal Contrast data alone...

Rearrangement Profile

2 second windows centered on 17 rearrangements 9

Another way to look: direction of time? *Select spikes above threshold *View +- 1 sec *Large events on BOTH sides *Close spikes: 2 per second

Rearrangement Profile

19 random 2 second windows 10

*There are a couple spikes but few and low magnitude. *Lets take a statistical approach.

Precursor/Aftershock

Poisson Distribution  Consider a random system  If the expected number of occurrences in this interval is λ, then the probability that there are exactly k occurrences is equal to:  k e− P k , = k! Our Data  6 events per minute on average  We see 2 per second  20 fold increase  Probability on order 10-7

11

*On ave expect .1 (6/60) over large window (based on num. pops) *But in window we see 2 per sec (20x) large increase *Assuming randomness (Poisson): prob would be 10^-7 *There IS clustering from a statistical standpoint. _____ **Events not completely random.

Summary/Conclusions .- Temporally heterogeneous dynamics can be studied with Time Resolved Correlation and Temporal Contrast. - Temporal Contrast simplifies the data into large spikes atop a smooth baseline. - Foam is a rigid system with stress. It's not surprising that dynamics are coupled in space and time. - Rearrangements may set off chain reactions: sudden → continuous → sudden → continuous...

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*Perhaps it's not too surprising. This foam is a rigid system with stress... coupling in space and time.

*Not one big (avalanche) also continuous dynamics: *Conclude avalanche – continuous - avalanche

Thank You

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Questions?

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