Percolation with Multiple Giant Clusters

0.18 MB·English

by  E. Ben-Naim

Percolation with Multiple Giant Clusters
E. Ben-Naim1,∗ and P. L. Krapivsky2,†
1Theoretical Division and Center for Nonlinear Studies,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545
5 2Center for Polymer Studies and Department of Physics,
0 Boston University, Boston, Massachusetts 02215
0
We study the evolution of percolation with freezing. Specifically, we consider cluster formation
2
via two competing processes: irreversible aggregation and freezing. Wefind that when thefreezing
n rateexceedsacertain threshold, thepercolation transition is suppressed. Below thisthreshold, the
a
system undergoes a series of percolation transitions with multiple giant clusters (“gels”) formed.
J
Giantclustersarenot self-averaging astheirtotalnumberandtheirsizes fluctuatefrom realization
0 to realization. The size distribution Fk, of frozen clusters of size k, has a universal tail, Fk ∼k−3.
1
Wepropose freezing as a practical mechanism for controlling thegel size.

]
h PACSnumbers: 82.70.Gg, 02.50.Ey,05.40.-a
c
e
m Percolationwas originally discoveredin the context of satisfies
polymerizationandgelation[1,2]. Percolationhasfound
- dc 1
t numerous applications in physics [3, 4], geophysics [5], k = (ic )(jc )−mkc −αc , (1)
a i j k k
dt 2
t chemistry [6], andbiology[7]. It plays animportantrole i+j=k
s X
. in a vastarrayofnaturalandartificialprocessesranging
at fromflowinporousmedia[8]andcloudformation[9,10] with m(t) the mass density of active clusters. The first
m toevolutionofrandomgraphs[11,12],combinatorialop- twotermsontheright-handsidedescribehowthecluster
- timization [13], algorithmic complexity [14], amorphous sizedistributionchangesduetoaggregation,andthelast
d computing,andDNAcomputingusingself-assembly[15]. termaccountsforlossduetofreezing. Thequantitym(t)
n
is subtle. Generally, it equals the mass density of all
o Westudytheevolutionofpercolationusingtheframe-
activeclustersincludingpossiblygiantclusters,butwhen
c
work of aggregation. An aggregation process typi-
[ there are no giant clusters, i.e., all clusters are finite in
cally begins with a huge number of molecular units
1 (“monomers”) that join irreversibly to form clusters size, then m(t) ≡ M1(t) with the moments defined via
M (t) ≡ hkni = knc (t). We are interested in the
v (“polymers”). At some time, a giant cluster (“gel”)con- n k≥1 k
8 evolution starting with finite clusters only.
taining a finite fraction of the monomers in the system P
1
The gelation transition. Initially, all clusters are finite
2 is born, and it grows to engulf the entire system. In this
in size, so m = M . The moments M provide a useful
1 classic percolation picture only a single gel forms, but in 1 n
0 many natural and artificial processes the system freezes probeofthedynamics. Fromthegoverningequation(1),
5 into a non-trivial final state with multiple gels or even the secondmomentofthe sizedistributionM2 obeysthe
0 closed equation dM /dt = M (M −α). For arbitrary
micro-gels[16,17]. InthisLetter,weshowthataggrega- 2 2 2
/
t tionwithfreezingnaturallyleadtoformationofmultiple initial condition,
a
m gelsandthatfreezingis alsoa convenientmechanismfor −1
α
- controlling the gel size. M2(t)=α −1 eαt+1 . (2)
d (cid:20)(cid:18)αc (cid:19) (cid:21)
We analyze the simplest aggregation with freezing
n
o process where there are two types of clusters: active There is a critical freezing rate αc = M2(0). For fast
c and frozen. Active clusters join by binary aggrega- freezing, α ≥ α , the second moment is always finite
c
:
v tion into larger active clusters. The aggregation rate indicating that clusters remain finite at all times. In
i is proportional to the product of the two cluster sizes this case, there is no gelation. For slow freezing, α<α ,
X c
[9, 18, 19]; this is equivalent to the gelation model of thereisafinitetimesingularityindicatingthataninfinite
r
a Flory and Stockmayer where a chemical bond between cluster,thegel,emergesinafinitetime[20]. Thegelation
two monomers joins their respective polymers [1, 2]. In time is
parallel, active clusters may become frozen at a size-
1 α
independent constant rate α. These frozen clusters are t =− ln 1− . (3)
g
α α
passive,thatistheydonotinteractwithother(activeor (cid:18) c(cid:19)
passive) clusters.
The gelation point marks two phases. Prior to the
This process is conveniently studied using the rate gelation point, the system contains only finite clusters
equation approach. The density c (t) of active clusters that undergo cluster-cluster aggregation (“coagulation
k
of mass k at time t (that is, made up from k monomers) phase”). Past the gelation point, the gel grows via
2
cluster-gel aggregation (“gelation phase”). We analyze 1
these two phases in order. m(τ), total mass
Coagulation phase. Coagulation occurs for α≥αc at all 0.8 s(τ), sol mass
times or for α < α when t < t . From (1), the mass g(τ), gel mass
c g
density of active clusters satisfies dm/dt = −αm, and
thus, ordinary exponential decay occurs, 0.6
m(t)=m(0)e−αt. (4)
0.4
For concreteness, we consider the monodisperse initial
conditions c (0) = δ . In this case M (0) = 1 and
k k,1 n
consequently,α =1. The cluster size distribution is ob- 0.2
c
tained using the transformed distribution, c =e−αtC ,
k k
and the modified time variable
0
τ = tdt′e−αt′ = 1−e−αt (5) 0 0.5 τ1 1.5 2
α
Z0
FIG.1: Thetotalmass,thesolmass,andthegelmassversus
that increases monotonically with the physical time and
modified time τ for α = 1/2. These curves hold as long as
reachesτ →1/αas t→∞. With these transformations,
thegel remains active.
Eq. (1) reduces to the no-freezing case
dCk = 1 (iCi)(jCj)−kCk. (6) where G(z)= k≥1 kkk−!1 ekz is the “tree” function [23].
dτ 2 During the gelation phase, active clusters consist of
i+j=k
X P
finite clusters, the “sol”, with mass s, and the gel with
From the well-known solution of this equation [21, 22], mass g. The total mass density of clusters, m = s+g,
the cluster-size distribution is decays according to
ck(t)= kk−2 τk−1e−kτ−αt. (7) dm =−αs. (10)
k! dt
Generally, the size distribution decays exponentially at The sol mass decays according to ds/dt = −gM −αs
2
largesizesand the typicalcluster size is finite. The gela- obtained from (1) using s=M . The first two moments
1
tion time (3) is simply τg = 1. Of course, no gelation follow from the generating function, M1 = c(0) and
occurs when α > 1 because τ < 1/α < 1. Otherwise, M = c′(0), and using the identity G′(z)=G/(1−G)
2
as the gelation point is approached, t → tg, the charac- [24], yields M2 =s/(1−sτeαt). The evolution equation
teristic cluster size diverges k ∼(tg−t)−2. The gelation for the sol mass becomes explicit,
pointismarkedbyanalgebraicdivergenceofthesizedis-
ds s(m−s)
tributionck ∼(1−α)k−5/2 forlargek. Wenotethatthe =− −αs. (11)
dt 1−sτeαt
mass density decreases linearly with the modified time,
m = 1−ατ, and that at the gelation point, the mass is Equations (10) and (11) are subject to the initial condi-
simply m(τg)=1−α. tions m(tg)=s(tg)=1−α. Once the masses are found,
Gelationphase. Pastthegelationtransition,agiantclus- the formal solution (8) becomes explicit. Results of nu-
ter containing a finite fraction of the mass in the system merical integration of Eqs. (10) and (11) are shown on
forms. Inadditiontocluster-clusteraggregation,cluster- Fig. 1. In the vicinity of the gelation transition, the gel
gelaggregationtakesplacewiththegiantclustergrowing mass grows linearly with time:
at the expense of finite clusters. In parallel, all clusters
g(t)≃2(1−α)2(t−t ) (12)
may undergofreezing andparticularly,the gelitself may g
freeze. as t↓t . The quadratic dependence on the freezing rate
g
Formally, the size distribution (7) generalizes to implies that the emerging gel is very small when α↑α .
c
Thus, micro-gels, that may be practically indistinguish-
kk−2
c (t)= τk−1e−ku−αt (8) able fromlargeclusters, emerge. Moreover,the maximal
k
k!
gel size must be smaller than 1−α. We conclude that
with u(t) = tdt′m(t′). Statistical properties of the freezing can be used to controlthe gel size, as gels of ar-
0 bitrarily small size can be produced using freezing rates
size distribution are derived from the generating func-
tion c(z,t)= R kc (t)ekz that equals just below criticality.
k≥1 k
Multiple giant clusters. At any time during the gelation
c(Pz,t)=τ−1e−αtG(z+lnτ −u) (9) phase, the gel itself may freeze. This freezing process
3
1 0
10
-1 simulation
10
0.8 -3
(1/2)k
-2
10
0.6
-3
m 10
F
k
0.4 10-4
-5
10
0.2
-6
10
0
0 2 4 6 8 10 -7
τ 10 0 1 2
10 10 10
k
FIG.2: Themassdensitymversustimeτ. Shownareresults
of a Monte Carlo simulation with system size N = 105 and
FIG. 3: The size distribution of the frozen clusters for α =
freezing rate α=0.1. The system alternates between theco- 1/2. The simulation results represent an average over 102
agulation phase and the gelation phase. In the former phase independentrealizations in a system of size N =106.
the mass decreases linearly according to (4) such that deple-
tion occurs at time τ = 1/α. In the latter phase, the active
massdecreases slowerthanlinearaccordingto(10)and(11).
tionofthegelationphaseisrandom,thesizeofthegiant
The gelation phase endswhen thegel freezes.
clusters, and their number are both random variables.
Generically, the system exhibits a series of percolation
is random: the gel lifetime T is a random variable that transitions, each producing a frozen gel, so that overall,
is exponentially distributed, P(T) = αe−αT. Until the multiplegelsareproduced. The randomfreezingprocess
gel freezes, the system evolves deterministically, so the governs the number of percolation transitions as well as
mass ofthe frozengelis g(t +T). When the gelfreezes, the size of the frozen gels.
g
the total active mass m(t) is discontinuous: it exhibits The magnitude of the second moment when the gel
a downward jump (Fig. 2). Given that the duration of freezes determines whether a successive gelation occurs.
the gelation phase is governed by a random process, the Because the second moment diverges at the gelation
mass of the frozen gel is also random. It fluctuates from point, there is a time window past the gelation time
realization to realization, i.e., it is not a self-averaging where the second moment exceeds the freezing rate. If
quantity. the gel freezes during this window, another percolation
Whenthegelfreezes,thesystemre-entersthecoagula- transition is bound to occur. Therefore, the maximal
tion phase because all remaining clusters are finite. The number of frozen gels is unbounded.
initialconditionsaredictatedbythedurationofthepre- Monte Carlo simulations confirm this picture (Fig. 2).
ceding gelation phase and are therefore also stochastic. In the simulations, we keep track the total aggrega-
Nevertheless,oncetheinitialstateisset,theevolutionin tion rate R = N(M2 − M )/2 and the total freezing
a 1 2
the coagulation phase is deterministic. The gel is frozen rate R = αNM . Aggregation occurs with probability
f 0
so it no longer affects the evolution and only cluster- R /(R +R ), and freezing occurswith the complemen-
a a f
cluster aggregation occurs. Let us assume that the gel taryprobability. Aclusterischosenforaggregationwith
freezes at time tf. Reseting time to zero, the first and probability proportional to its size. Time is augmented
the second moments are simply given by Eqs. (4) and by ∆t = 1/(R +R ) after each aggregation or freezing
a f
(2), respectively, with Mn(0) replaced by Mn(tf). Note event.
that M (t ) contains contributions from finite clusters
n f
Frozen clusters. We now turn to the frozen clusters.
only. Asecondgelationoccursifthefreezingrateissuffi-
TheirdensityisfoundfromdF /dt=αc . Whenα≥α ,
cientlysmall,α<M (t ). Otherwise,thesystemforever k k c
2 f
thedensityofactiveclustersovertheentiretimerangeis
remains in the coagulationphase.
given by Eq. (7), and therefore F (t) is found by simple
k
Cyclic dynamics. The general picture is now clear: the
integration. In particular, the final density is
process starts and ends in coagulation and throughout
theevolution,the systemalternatesbetweencoagulation α
F (∞)= γ(k,k/α) (13)
and gelation. Once the initial conditions are set, the be- k k2·k!
haviorthroughoutthecoagulationphaseandthroughout
the gelation phase are both deterministic. Eachgelation whereγ(n,x)= xdyyn−1e−y istheincompletegamma
0
phaseendswithfreezingoftheactivegel. Sincethedura- function. At large sizes, the size distribution of frozen
R
4
clusters decays according to We acknowledge US DOE grant W-7405-ENG-36
(EBN) for support of this work.
1 ·k−3 α=1
F (∞)≃ 2 (14)
k (A(α)k−7/2exp[−B(α)k] α>1
∗ Electronic address: [email protected]
where A=(2π)−1/2α2/(α−1) and B =α−1+lnα−1. † Electronic address: [email protected]
Similarly, we may obtain the density of frozen clus- [1] P. J. Flory, J. Amer. Chem. Soc. 63, 3083–3090 (1941).
ters formed in the first coagulation case. Integrating (7) [2] W. H. Stockmayer,J. Chem. Phys. 11, 45–55 (1943).
givesF (t )= α γ(k,k). Thissizedistributiondecays [3] D. Stauffer and A. Aharony, Introduction to Percolation
algebrakicaglly,Fk2(·tk!)≃ αk−3. However,enumerating the Theory (Taylor & Francis, London, 1992).
k g 2 [4] A.BundeandS.Havlin(editors),PercolationandDisor-
frozenclusters that are bornpastthe first gelationpoint deredSystems: TheoryandApplications,PhysicaA366,
is more complicated since the deterministic patches are
(1998).
punctuated by random jumps. In fact, even the total [5] M. Sahimi, Flow and Transport in Porous Media and
mass of frozen clusters becomes a random quantity be- Fractured Rock (VCH,Boston, 1995).
cause it is a complementary quantity to the final mass [6] P. J. Flory, Principles of Polymer Chemistry (Cornell
density of frozen gels. Interestingly, numerical simula- University Press, Ithaca, 1953).
[7] P. Grassberger, Math. Biosci. 63, 157 (1983).
tions suggest that the tail behavior
[8] S.R.BroadbentandJ.M.Hammersley,Proc.Cambridge
F ≃Dk−3 (15) Phil. Soc. 53, 629 (1957).
k [9] R. L. Drake, in Topics in Current Aerosol Researches,
eds. G.M. Hidyand J. R.Brock (Pergamon, New York,
is universal (Fig. 3). Even the prefactor D appears to
1972), pp.201.
dependonlyweaklyuponthefreezingrateα. Thisshows [10] J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry
thatthefinaldistributionoffrozenclustersis dominated and Physics (John Wiley & Sons, New York,1998).
by the coagulation phase. Indeed, in each such phase, [11] S. Janson, T. L uczak, and A. Rucinski, Random Graphs
the tailbehaviorisk−3,andfurthermore,inthe gelation (John Wiley & Sons, New York,2000).
[12] G. Grimmett, Percolation, (Springer, Berlin, 1999).
phase, large clusters are more likely to be consumed by
[13] S. Cocco, L. Ein-Dor, and R. Monasson, in New Opti-
the gel. These results suggest that freezing leads to an
mization Algorithms in Physics, eds. A. K. Hartmann
additional non-trivial critical exponent γ =3.
and H.Rieger (Wiley–VCH, Weinheim, 2004).
Inconclusion,wehavefoundthatfreezingleadstomul- [14] B. Bollob´as, C. Borgs, J. T. Chayes, J. H. Kim, and
tiple percolation transitions. The system evolves in a D. B. Wilson, Rand.Struct. Alg. 18, 201 (2001).
cyclic fashion alternating between coagulation and gela- [15] E.Winfree,F.R.Liu,L.A.Wenzler,andN.C.Seeman,
tion. Depending on the freezing rate, the system may Nature 394, 539–544 (1998).
form no gels, a single gel, or multiple gels. Most statis- [16] M. J. Murray and M. J. Snowden, Adv. Coll. Int. Sci.
54, 73 (1995).
tical characteristics are non-self-averaging because they
[17] B. R. Saunders and B. Vincent, Adv. Coll. Int. Sci. 80,
are controlled by the random freezing of gels.
1 (1999).
Freezingprovidesapracticalmechanismforcontrolling [18] F. Leyvraz,Phys. Rep. 383, 95 (2003).
gelation. Itmaybeusedtoengineermicro-gelsofdesired [19] A. A.Lushnikov,Phys. Rev.Lett. 93, 198302 (2004).
sizebyimplementingvariablecoolingrates. Slowfreezing [20] In a finite system, the gels size equals a finite a fraction
followed by rapid freezing can be used to produce gels of thesystem size.
[21] R.M.Ziff,E.M.Hendriks,andM.H.Ernst,Phys.Rev.
of prescribed size, while near-critical freezing produces
Lett.49, 593 (1982); R.M.Ziff, M.H.Ernst, andE.M.
micro-gels of arbitrarily small size.
Hendriks, J. Phys. A:Math. Gen. 16, 2293 (1983).
Thereareanumberofinterestingpotentialgeneraliza- [22] E. Ben-Naim and P. L. Krapivsky, J. Phys. A 37, L189
tionsofthepresentwork. Themostnaturalispercolation (2004); cond-mat/0408620.
infinite dimensions. Percolationis analyticallytractable [23] S.Janson,D.E.Knuth,T.L uczak,andB.Pittel,Rand.
only in two dimensions, though even in that situation it Struct. Alg. 3, 233 (1993).
is not clear how to derive the exponent characterizing [24] H. S. Wilf, generatingfunctionology (Academic Press,
Boston, 1990).
the tail of the final distribution of frozen clusters. One
[25] P. L. Krapivsky and E. Ben-Naim, J. Phys. A 33, 5465
mayalsoconsidersituationswithdifferentfreezingmech-
(2000).
anisms [25], particularly for finite and infinite clusters.
[26] When finite clusters remain active while infiniteclusters
We also solved for the case where infinite clusters freeze instantaneously freeze, this version was essentially sug-
immediately [26]. In this case, there is no breakdown gested by Stockmayer [2] and was subsequently studied
of self-averaging and the final density of frozen clusters e.g. by Ziff and co-workers [21] and by D. J. Aldous,
mimics the critical behavior of active clusters, γ =5/2. Math. Proc. Camb. Phil. Soc. 128, 465–477 (2000).