percolation properties near the gel transition

Percolation properties near the gel transition

Joris Billen

December 6, 2011

Contents

1 Survival rate at all temperatures 2

2 Aging 3

3 Averaged survival rate data 63.1 Averaging procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 All data available for T=0.35-T=0.40(1x averaged) 7

5 Percolation in a system of surviving junctions between endgroups thatform aggregates 8

6 Percolation probability 96.1 Averaging procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.2 Percolation probability VS ∆t for # averaging N=50 . . . . . . . . . . . . . . 96.3 Percolation probability VS survival rate for # averaging N=50 . . . . . . . . 106.4 Characteristic time from 50% percolation . . . . . . . . . . . . . . . . . . . . 116.5 1/e lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Size distributions 137.1 Relation size distribution and percolation probability VS survival rate . . . . 137.2 Evolution of average aggregate size of surviving bonds . . . . . . . . . . . . . 157.3 Percolation probability VS average aggregate size . . . . . . . . . . . . . . . . 16

8 Distribution of surving endgroups in a percolating system 178.1 T=0.425 percolating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 T=0.425 no percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.3 Distribution of links accross the gap . . . . . . . . . . . . . . . . . . . . . . . 19

1

1 Survival rate at all temperatures

The survival rate at a certain time is defined as the percentage of surviving junctions that areboth present at time tinitial and tinitial + ∆t. This means that the junction could have beenlost (and if ∆t is large probably will have been lost) between those two times, but formedagain and is present at tinitial + ∆t.All the data available for all all temperatures of interest is shown below:

1 100 10000 1e+06time [τ]

0.001

0.01

0.1

1

Surv

ival

rat

e

T=0.35T=0.375T=0.40T=0.425T=0.45T=0.55

Survival ratesall T all data starting from first run axxb

Figure 1: All the data available.

2

2 Aging

We noticed that the rate at wich the survival rate drops, depends on the starting point(aging). We show here the data at several different starting points and will discard the datauntil the point in time where the survival rate does not depend on starting time anymore.Note that the datafiles available are labelled axxa-axxb-...axxz-bxxa-bxxb-...-bxxz-cxxa- etc.where xx is a number that corresponds to some temperature (e.g. 20=T0.35).

1 10 100 1000 10000 1e+05time [τ]

1

Surv

ival

rat

e

a70b-fa70h-va70q-b70eb70f-b70u

T=0.30effect of aging

b70f-b70h

0.9

1 10 100 1000 10000 1e+05 1e+06 1e+07time [τ]

1

Surv

ival

rat

e


0.9

Figure 2: T=0.30. Left: Data started at several starting points showing we should skip thefirst part of it. Right: a70h-m70s.

1 10 100 1000 10000 1e+05 1e+06time [τ]

0.1

1

Surv

ival

rat

e

a20b 108 filesb20a - 80 filesc20b 70 filesd20b 45 filese20b 24 files


1 100 10000 1e+06time [τ]

0.01

0.1

1

Surv

ival

rat

e

b40a-b40oc40a-c40oa40g-a40u


Figure 3: T=0.35-0.375. Data started at several starting points showing we should skip thefirst part of it.

3

1 10 100 1000 10000 1e+05time [τ]

0.1

1

Surv

ival

rat

e

a19b 50 filesb19a 25 filesb19g - 17 filesb19o 21 filesc19a 21 files


1 100 10000 1e+06time [τ]

0.1

1

Surv

ival

rat

e [%

]

started at e19l; dt=88 filesstarted at g19g dt=78 filesstarted at e19w-i19v;77 filesstarted at b19a-c19z; 52 filesstarted at c19a-z; 26 filese19b-j19z ALL DATA d19a-e19y

T=0.40effect of aging; skipped data untill plateau, different starting points

Figure 4: Left: T=0.40 old starting point (begin). Right: T=0.40 new starting point(after plateau).

1 10 100 1000 10000 1e+05time [τ]

0.01

0.1

1

Surv

ival

rat

e

a34b - 12 filesa34c - 11 filesa34d - 10 files


1 10 100 1000 10000 1e+05time [τ]

0.01

0.1

1

Surv

ival

rat

e

a31 4 filesa31 3 filesa31 2 files


Figure 5: T=0.425-0.45

4

T reasonable start point based on aging data data available

0.30 a70h(skip 6 files: a70b-a70g) a70h-m70s0.35 c20a (skip 51 files: a20b-b20z) c20a-t20z0.375 b40a (skip 25 files: a40b-b40z) b40a-o40x0.40 e19b (skip ?? files: a19b-e19a) e19b-k19s0.425 a34c (skip 1 file: a34b) a34c-a34n0.45 a31c (skip 1 file) a31c-a31k

Based on the aging data above, we skip for each temperature the first part of the data,until the survival rate does not depend anymore on the starting point, as shown in the tableabove.

5

3 Averaged survival rate data

For the available data, leaving out the first part as indicated before, we average over 5 setsof a certain time interval ∆t (as long as we have data available) to get a smooth curve forthe survival rate at different temperatures.

3.1 Averaging procedure

# averaging: N (e.g. here N=3)

Averaging for survival rate3 parameters:

time the begin point will shift : shift beginlength over which to monitor survival rate:

ttotal

t∆t∆

∆ t

∆t

Begin points: ( i−1)*shift begin, i=1..N

3.2 Results

1 100 10000 1e+06time [τ]

0.001

0.01

0.1

1

Sur

viva

l rat

e

T=0.60; start a16c;shift begin=1ktau; dt=4995 tauT=0.55; start:a25c; shift begin=5ktau;dt=49995tauT=0.45; start:a31c; shift begin=2.5ktau; dt=32495tauT=0.425; start:a34c; shift begin=1k tau dt=34995 tauT=0.40; start e19b; shift begin= ; dt=50ktau;dt=455ktauT=0.375; start b40a; shift begin=50k tau; dt=1255kfatuT=0.35

Figure 6: Average survival rates

6

4 All data available for T=0.35-T=0.40(1x averaged)

.

1 100 10000 1e+06time [τ]

0.01

0.1

1S

urvi

val r

ate

T=0.40 after plateauT=0.40T=0.35 c20a-g20zl20b-m20zl20ah20a-k20zn20a-n20rdata ran againo20a-t20zT=0.35 error j20m-j20o j20n missing

Full data - begin point(aging)T=0.35-0.375-0.4

Figure 7: T=0.35 and T=0.40 data

7

5 Percolation in a system of surviving junctions between end-groups that form aggregates

The picture below shows how the network changes in a certain time interval dt. The sur-vivalrate in this case is 70 % and the system that is initially percolating is not percolatinganymore after some vital junctions have been broken. Note that:

Figure 8: Illustration of how the junctions formed between endgroups lead to the creation ofaggregates.

• The polymer chain consists of 8 atoms, but only the 2 endgroups (in red) are shownsince only endgroups can form junctions amongst each other. The 6 atoms in betweenare represented by the blue line.

• The endgroups connected by a thick line have formed a junction.

• All the endgroups that can be reached, starting from a certain endgroup followingjunctions, are part of the same aggregate. For each aggregate the size is given in thepicture. Hence in an aggregate, not every endgroup will necessarily have a junctionwith every other endgroup in the aggregate.

• The thick green lines are the junctions that will not survive over the given time interval.

• Some chains are grafted to the walls: 5 % of the chains.

• We talk about percolation when there exists a percolating path that starts at an ag-gregate grafted to the top wall and extends to an aggregate grafted to the bottom wall.The aggregates are linked to each other by the polymer chains. Therefore our problemis one of percolation of aggregates of endgroups, and not merely percolation of atoms.

8

6 Percolation probability

6.1 Averaging procedure

length of 1 percolation run: t ∆

Averaging for percolation data2 parameters:

# averaging: N (e.g. here N=3)

ttotal

t∆t∆t∆

total ∆Ν

Begin points: (t − t) * (i−1), i=1..N

6.2 Percolation probability VS ∆t for # averaging N=50

We select several sets of junctions with different starting points tinitial that survive over ∆t.Note again that in this case it is possible that the junction was broken at some point andwas formed again later. For higher temperature, the system still percolates at longer times.For T=0.40 there is still percolation at all timescales available.

1 100 10000 1e+06

∆ t[τ]0

20

40

60

80

100

% p

erco

latio

n

T=0.8T=0.70T=0.60T=0.55T=0.50T=0.45T=0.425T=0.40T=0.375

Figure 9: Percolation probability VS time

9

1∆ t[τ]

0

20

40

60

80

100

% p

erco

latio

n

T=0.8T=0.70T=0.60T=0.55T=0.50T=0.45T=0.425T=0.40

Figure 10: Normalized by dividing by 50% percolation time point

6.3 Percolation probability VS survival rate for # averaging N=50

We display the percolation probability as a function of the survival rate for several temper-atures. The survival rate at which percolation still occurs is dependent of the temperature:at lower temperature there is still percolation at lower survival rate and the transition froma percolating system to a non-percolating one is less sharp.

0 20 40 60 80 100survival rate [%]

0

20

40

60

80

100

% p

erco

latio

n

T=0.8T=0.7T=0.60T=0.55T=0.45T=0.425T=0.40T=0.375T=0.5

Figure 11: Percolation probability VS survival rate

10

6.4 Characteristic time from 50% percolation

1 1.5 2 2.5 31/T

1

100

10000

1e+06

1e+08

∆ t 5

0% p

erco

latio

n

Updated dataPrevious fit: dt=a*exp(b/(T-c)); a=0.28054; b=1.48605; c=0.29375New fit updated data: a = 0.29572; b = 1.46715; c = 0.29498Interpolated dataFit of interpolated data: a=0.31877; b=1.43458; c=0.29714

Figure 12: Time interval at which 50% of the sets show a percolating path.

11

6.5 1/e lifetimes

1 1.5 2 2.5 31/T

1

100

10000

1e+06

1/e

lifet

ime

[τ]

a*exp(b/(T-c)); a=3.464;b=0.596;c=0.343

Figure 13: Time interval at which 1/e of the junctions survive, taken from the figure 6

12

7 Size distributions

7.1 Relation size distribution and percolation probability VS survival rate

The percolation probability is different with similar survival rate at different T. Possiblythis can be explained by the different size distribution (shown below): at higher T thesystem consists of more smaller aggregates (at T=0.45 the average size is about 14) that areconnected to each other and span accross the gap and if then a couple are missing the systemwill stop percolating. For lower T, the system consists of a few very large aggregates that areconnected, so it will only stop percolating if the whole aggregate is broken up. This is shownin the picture on the next page where in two different situations with the same survival rateof 50% the system at low T would still percolate, while the system at higher T would not.

A

AAAAAAAAAAA

AAA

AAA

A

A

AA

A

A

A

AAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

0 10 20 30 40 50 60Size

0

0.1

P(s

ize)

T=0.375A A

T=0.35T=0.40T=0.425T=0.45T=0.50T=0.60T=0.70T=0.80T=0.90T=1.00

Grafted removed wall

0 20 40Size

0

0.05

0.1

0.15

0.2

P(s

ize)

T=0.35T=0.375 T=0.40T=0.50T=0.35 high shear

Grafted removed wall

Figure 14: (left)Aggregate size distributions at different T.(right) Same data, only low T.

13

low T high T

Survival rate=50%

tinitial tinitialtinitial + dt tinitial + dt

Figure 15: Illustration of two systems at different T with different survival rate. The one atlow T has larger aggregates and after time dt at survival rate of 50 % there is still percolation.The one at high T has smaller aggregates and gets broken up after the same time and survivalrate.

14

7.2 Evolution of average aggregate size of surviving bonds

In this section we are interested in how the size distribution of the network created by thesurviving junction evolves. The longer the time interval, the more junctions will break upand smaller aggregates will be formed. As a result it is expected that the average aggregatesize of the network < s > decreases with time. < s > can be calculated as follows:

< s >=2000

Nagg + Nones

(1)

with Nagg the number of aggregates of size 2 and up in the network created by the survivingjunctions and Nones all the endgroups that are not present in the Nagg aggregates.

0 2000 4000time [τ]

0

2

4

6

8

10

<s>

of s

urvi

ving

junc

tions

T=0.45T=0.425T=0.40

0 50000 1e+05 1.5e+05 2e+05time [τ]

0

5

10

15

<s>

of s

urvi

ving

junc

tions

T=0.45T=0.425T=0.40

Figure 16: (left)Evolution of aggregate size for different T .(right): Same data zoomed in.

15

7.3 Percolation probability VS average aggregate size

1∆ t[τ]

0

20

40

60

80

100

% p

erco

latio

n

T=0.8T=0.70T=0.60T=0.55T=0.50T=0.45T=0.425T=0.40

Figure 17: Percolation probability vs average aggregate size of surviving system.

16

8 Distribution of surving endgroups in a percolating system

As a first step to determine the heterogeneity of the system, some snapshots are shown forthe system close to the percolation treshold at different T. Some info on the pictures:

• The color of the endgroup is indicative for the size of the aggregate the endgroup is in:green for size < 10, copper for size between 10-15, blue for size 15-20, and white forsize 20-30.

• If an aggregate was split in 2 because of periodic boundary conditions, a correction ismade and some endgroups are displaced so the whole aggregate is close to each other.

• The links between the aggregates are shown too.

8.1 T=0.425 percolating

Figure 18: The surviving bonds after ∆t = 7500τ (survival rate 22 % and percolation % 98)at T=0.425. This system is percolating.

17

8.2 T=0.425 no percolation

Figure 19: The surviving bonds after ∆t = 15000τ (survival rate 10 %) at T=0.425. Thissystem is not percolating.

18

8.3 Distribution of links accross the gap

0 5 10 15 20 25 30distance from bottom wall [σ]

0

5

10

15

20

# of

link

s in

that

slic

enon percolatingpercolating set

Distribution of links across the gap

Figure 20: Number of links as a function of the distance from bottom wall

19

percolation properties near the gel transition

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