Lattice animals in diffusion limited binary colloidal system

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by  Zakiya Shireen

Lattice animals in diffusion limited binary colloidal system∗
Zakiya Shireen and Sujin B Babu†
Department of Physics, Indian Institute of Technology Delhi
Hauz Khas, New Delhi 110016,India.
(Dated: January 5, 2017)
In soft matter system controlling the structure of the amorphous materials have been a key
challenge. In this work we have modeled irreversible diffusion limited cluster aggregation of binary
colloids, which serves as a model for chemical gels. Irreversible aggregation of binary colloidal
particlesleadtotheformationofpercolatingclusterofonespeciesorbothspeciesalsocalledbigels.
Before the formation of the percolating cluster the system form self similar structure defined by a
fractaldimension. Foraonecomponentsystemwhenthevolumefractionisverysmalltheclusters
7
are far apart from each other and the system has a fractal dimension of 1.8. Contrary to this we
1
0 will show that for the binary system we observe the presence of lattice animals which has a fractal
2 dimension of 2 irrespective of the volume fraction. When the clusters start inter penetrating we
observe a fractal dimension of 2.5 same as in the case of one component system. We were also able
n
to predict the formation of bigels using a simple inequality relation. We have also shown that the
a
growth of clusters follows the kinetic equations introduced by Smoluchowski for diffusion limited
J
clusteraggregation. Furthermorewearealsoproposingauniversalparameterforirreversiblebinary
4
colloidal system, which follows the scaling laws proposed by percolation theory.

]
h
p I. INTRODUCTION a model of irreversible DLCA for binary colloidal parti-
- clesbothhavingthesamediameteranddifferonlyinthe
p
way particles interact. The particles of the same species
m Irreversible aggregation of colloidal particles like
form irreversible bonds on collision, while particle of dif-
proteins[1,2],DNAgraftednano-materials[3–5]etcleads
o ferent species have hard core repulsive interaction. De-
to the formation of percolating clusters or gels. These
c pending on the fraction of species in the system we ob-
amorphous materials are very important from a funda-
.
s servetheappearanceofpercolatedclusterofoneorboth
mental point as well as find a range of application in
c
species also called bigels.
i industry[6, 7]. The scaling laws associated with struc-
s ture and kinetics of amorphous materials are very well Here we report on the simulation study of DLCA bi-
y
h explained by the percolation theory[8]. Two limiting narycolloidalsystemwithshortrangeinteraction,where
p cases for irreversible aggregation are diffusion limited we have observed the appearance of bigels or one com-
[ cluster aggregation (DLCA) and reaction limited clus- ponent gel depending on the fraction of each species for

ter aggregation (RLCA) model both of which have been a particular volume fraction. We have proposed an in-
1
extensively studied [9–13]. The aggregation number m equality relation where by we were able to predict the
v
3 of the self similar clusters formed from irreversible ag- appearance of bigel for a particular volume fraction. We
1 gregation of particle is related to the radius of gyration have also shown that the aggregation kinetics of the sys-
9 R by m∝Rdf where d is the fractal dimension of the tem are very well described by the Smoluchowski rate
g g f
0 equation. In the flocculation region instead of the frac-
clusters[14]. AlsothenumberdensityoftheclusterN(m)
0 tal dimension of d = 1.8 we observe the appearance
scale with the aggregation number as N(m) ∝ m−τ [8]. f
.
1 In DLCA if the cluster are far apart from each other we of lattice animals which has a df = 2, which we have
0 also confirmed using the scaling between N(m) and m.
have the flocculation regime characterized by d = 1.8
7 f We have proposed a universal parameter for the species
and τ = 0, while when the clusters start to interpene-
1 whose knetics is aressted and can not form percolating
tratewehave thepercolationregime where d =2.5and
: f
v τ = 2.2. In reversible aggregation it has already been cluster. We have also shown that the universal parame-
i ter follows the scaling laws proposed by the percolation
X shown that the system undergoes a transition from con-
theory.
tinuous to directed percolation depending on strength of
r
a attraction [15].
Recently there have been many experimental and the-
oretical work on the aggregation of binary colloids using II. SIMULATION
patchyparticles[16],DNAgraftedontothecolloidalpar-
ticles [3–5] or using two different types of micelles[17], at
The simulation method used in the present work is
finitetemperatures. Inthepresentworkwehavestudied
called the Brownian cluster dynamics (BCD). BCD was
introduced primarily to study the kinetics and dynam-
ics of monomeric system. It has already been shown
thatthissimulationtechniquesagreewiththewellknown
∗ Afootnotetothearticletitle event driven molecular dynamics simulation[18]. In the
† [email protected] presentstudywehavemodifiedBCDtoaccommodatebi-
2
narysphericalparticles. WestartoursimulationwithN the1componentgelwehaveperformedsimulationsfor10
randomlydistributedspheresofunitsizeinacubicboxof differentconfigurationsinaboxsizeof50ataparticular
length L. The volume fraction of the system is given by combination of c and φ . If in more than 50% of the
A tot
φ = πN/L3. Werandomlypickafractionc =N /N trials only the B particles formed a percolating network
tot 6 A A
where N is the number of A particles and c =1−c and also the kinetics of A particles is arrested we have
A B A
is the fraction of B particles present in the system. In defined it as a 1 component gel. Likewise in 10 trials if
thepresentstudywewillbeworkingwithdifferentratios in more than 50% of the trials the configuration resulted
(c : c ) of A and B particle, and we have always kept in a percolating cluster for both A and B particles we
A B
c ≥ c . Inter species particles interact only via hard have defined it as a bigel. From Fig.1, we can observe
B A
core repulsion, while intra species particles interact via that when c < 0.15 we have a percolating cluster only
A
a very short range square well potential with an interac- fortheB particlesirrespectiveoftheφ . Theaggregat-
tot
tionrangeof0.1typicalofacolloid. Alltheparticlesare ingAclusterswillnotabletoformapercolatingcluster.
displaced in a random direction using a step size s, and The reasonbeing thegrowthof the A cluster ishindered
the time is incremented by ns2 where n is the number of by the presence of the B percolating cluster. As we go
simulation steps. It has already been demonstrated that to higher c = 0.18±0.01 for φ = 0.4, we have less
A tot
ifthestepsizesissufficientlysmallBCDisequivalentto particles in B as well as more A particles, which even-
Brownian dynamic simulation[19]. Thus t=1 is defined tually results in the appearance of A percolating cluster
as the time taken by a monomer to travel its own diam- as there is less hindrance from the B particles. As we go
eter, where the diffusion coefficient of monomer is given to lower φ , the critical value of c increases where we
tot A
by D = 1/6. During random diffusion as soon as same observe the formation of A percolating network.
0
kind of particles are within the range of the square well Thetransitionfroma1componentgeltobigelhappens
an irreversible bond is formed between them. All such for different φ at different c values as is obvious from
tot A
connected spheres together are called a cluster, where a Fig.1. Fromtheresultsofsimulationsforallthe φ , we
tot
monomer is considered as a cluster of aggregation num- were able to deduce an upper bound when the bigel will
ber 1. After the diffusion of individual monomers within appear in the system,
the cluster the center of mass of the cluster would have
2c
displaced in a random direction, on in other words we A ≥1 (1)
φ −φ
havefollowedRousedynamics[20]. Duringthemovement f tot
step if it leads to overlap with other sphere or leads to φ =0.74whichalsohappenstobethemaximumpack-
f
breakingofthebondwerejectthosemovementsteps. In ing fraction for spheres. cA is the free space avail-
additiontotheRousedynamicsthecenterofmassofthe able in a randomly distriφbfu−teφdtotspheres arrangement at
cluster is displaced in the same direction of the center of a particular φ for the A particles. When the free vol-
tot
masscalculatedfromtheRousedynamics. Thedisplace- ume available for the A particles is approximately equal
ment is now inversely proportional to its radius of the to half the volume of a sphere 2cA ≈ 1, we observe
cluster [18] and if it leads to overlap with other cluster φf−φtot
thatbigelstarttoappearinoursystem,asshownbythe
we reject the movement step mimicking Zimm dynamics
dashed line in Fig.1. When this fraction is less than one,
[21]. The box size was varied from L=50−100 and the
the A particles do not have enough space to diffuse and
resultsinthepresentstudyisnotinfluencedbyfinitesize
aggregateinsidetheporesofB percolatedclusterresult-
effects.
ing in the formation of 1 component gel. For very low
φ thisrelationmaynotbeexactlyfollowedandwemay
tot
observe the appearance of bigel for 2cA < 1. This is
III. RESULTS φf−φtot
because the A clusters can diffuse through the system as
well as aggregate much more freely than for the higher
A. Phase diagram φ , thereby equation 1 only giving us an upper bound
tot
for the appearance of bigel.
Irreversible aggregation of 1 component colloidal sys- InFig.2wehaveshownsnapshotsofbinarysystemfor
temalwaysleadstotheformationofpercolatingclusters 3differentc asindicatedontheleftsideofthefigurefor
A
or gels [9], which in the present work we have defined φ = 0.1. The red colored spheres represent A particle
tot
as a cluster which extends between the opposite end of and green colored spheres represent B particles. All the
the simulation box. In the binary colloidal system as snapshothavebeentakenatatimewhentheaggregation
we change the fraction of A particles 0 < c < 0.5, we kinetics of the system have stopped evolving. In Fig.2a
A
always observe a percolating cluster of the B particles wehaveshownbothAandB particleasitappearinthe
irrespective of the volume fraction, while the A particles system for c =0.1, evident from the fewer red particles
A
formspercolatingclusterdependingonthefractionofc . seen in the system. In Fig.2b we have kept only the
A
In Fig.1 we have plotted φ as a function of c , where B particle , where we observe the presence of a single
tot A
wehaveidentifiedregionsastheonecomponentgelwhen system spanning cluster, with thick tenuous branch. In
only B particles percolate and 2 component gel or bigel Fig.2c we have shown only the A particle for the same
[3–5] when both the particles percolate. For identifying systemwhichareformingfractallikeclustersdistributed
3
0.18 0.2 0.22 0.24 0.26 0.28 0.3
0.4 0.4 cA AB B A
a b c
0.35
0.35 0.1
0.3
0.3
φtot 1Component gel Bigel d e f
0.25
0.25 0.2
0.2
0.2
g h i
0.5
0.15
0.15
0.18 0.2 0.22 0.24 0.26 0.28 0.3
c
A
FIG.2. Snapshotofthebinarysystemofparticleswherered
FIG. 1. The volume fraction φ is plotted against the frac-
tot represents A particles and green represents B particles for
tion of A particles c . Below the curve shown in green we
A φ = 0.1 in a box size L = 50 for a range of increasing c
tot A
have only the 1 component gel and above we have gels of
valuesasindicatedbythearrow. (a)Snapshotatc =0.1of
A
both the particles or bigel. The error bar signifies two cases
bothAandBparticles,(b)fortheBparticlesafterremoving
where we have observed the appearance of 1 component gel
the A particles where we have one single percolating cluster,
andbigelsmorethan50%inallthetrials. Thedashedlineis
(c) for the A particles after removing the B particles where
given by c = 1(φ −φ ).
A 2 f tot the small cluster are not able to grow due to the hindrance
of B percolating cluster. (d) Snapshot at c = 0.2 of the
A
systemofAandB particles(e)B particlealonewheremore
free space is available and (f) A particle alone where we ob-
serve the presence of one big cluster and few smaller clusters
over the entire box. These small clusters are not able
distributedovertheentirebox. (g)Snapshotatc =0.5for
A
to aggregate further as they are stuck in the pores of B
both A and B particles, (h) showing only the B percolating
percolating cluster. In Fig.2d we are showing both the cluster,whicharemoreopencomparedtotheearlier2cases,
particle for cA = 0.2, where we are closer to the critical (i) showing only the A particle which has aggregated into a
pointofAparticles,whenitstartstopercolate. InFig.2e single percolating cluster.
wehaveshownonlytheBparticle,whereweobservethat
the gel of the B looks much more open than in the case
of Fig.2b. As the number of B particles is smaller than
B. Kinetics of the A particles
the previous system, the percolating cluster formed is at
a lower volume fraction of B particles. In the case of A
particles for the same c we observe the presence of one For studying the kinetics of aggregation we have cal-
A
very large cluster as well as smaller clusters distributed culated the mass average aggregation number m =
w
over the entire box. As we are close to the critical point (cid:80)∞ m2N(m)/(cid:80)∞ mN(m) of the A particles as a
m=1 m=1
weareabletoobserveclusterswiththintenuousbranches function of time for a range of c at φ =0.3 see Fig.3.
A tot
forthecaseofAparticles. Thesmallerclusterswillnever For c > 0.25 we observe that m diverges, indicating
A w
beabletoformpartofthepercolatingclusterastheyare the formation of percolating cluster. For higher fraction
stuck inside the B cluster. For the case of c = 0.5 we m divergesfasterastherearemoreparticlesofAspecies
A w
are having the same number of A and B particle and present in the system as well as B cluster becomes much
both are forming independent system spanning clusters moreopen,therebyAspecieswillpercolatefaster. When
whichareinterpenetratingamongeachotherseeFig.2g. c < 0.25 we observe that m grows for some time and
A w
In Fig.2h the B percolating cluster are much more open thenitstagnates,thereasonbeingthattheAclustersare
thantheprevious2casesasthepercolatingclusterisfor getting stuck inside the B percolating cluster. We know
a smaller number of B particles compared to the earlier that in the flocculation limit the kinetics of DLCA type
cases. The A particles for c = 0.5 are able to form aggregationisverywellexplainedbytheSmoluchowski’s
A
percolating cluster with thicker strands and all the A rate equation where m ∝ t for the monomeric system
w
particles are now part of one single percolating cluster [22, 23]. The solid line in Fig.3 has a slope of 1, which
see Fig.2i. is followed by all the fraction of c in the flocculation
A
4
0.0001 0.001 0.01 0.1 1 10 100 1,000 0.0001 0.001 0.01 0.1 1 10 100 1,000
103
0.50 1,000
monomer
0.40
binary
0.30
0.25
102
0.22 100
0.20 102
100
0.15
0.10
w
m w
m
101
10
101
10
100
1 100
1
10−4 10−3 10−2 10−1 100 101 102 103 10−4 10−3 10−2 10−1 100 101 102 103
t t
FIG. 3. Mass average aggregation number (m ) of only the FIG. 4. m is plotted as a function of time for the A parti-
w w
A particles is plotted as a function of time for range of c cles c = 0.2 for the binary system (diamond) at φ = 0.3
A A tot
at φ = 0.3 as indicated in the figure. The straight line and monomeric system (circles) with volume fraction 0.06.
tot
representsslopeofunityinaccordancewiththeSmoluchowski The solid line has a slope of 1 according to the prediction of
rate equation Smoluchowski rate equation [22, 23]
limit. For c > 0.22 we observe that the m diverges havedisplacedonlyapproximatelyitsowndiameter. For
A w
from the slope of 1, a signature of the formation of per- t>1 we observe that the kinetics of aggregation for the
colating cluster for the A particles. For c < 0.22 we binary system falls below a slope of 1. While for the vol-
A
observe that m deviates to a smaller slope and at a ume fraction 0.06 the aggregation goes on with the slope
w
later time approaches a stationary value indicating that 1, which is expected from Smoluchowski approach. For
AparticlesareaggregatinginsidethecagesoftheB clus- themonomericcasem deviatefromaslopeofunityfor
w
ters. As c decreases the stationary value of m also t > 10 indicating the system have started to interpene-
A w
decreases as there is less space available for the A parti- trate and later percolates at t=59. For the binary case
cles to aggregate because of the increased fraction of B m approaches a stationary value see Fig.4. For all the
w
particles. Although Smoluchowski approach is not valid volumefractionthemonomericcasewillformapercolat-
in the percolation regime, as the clusters interpenetrate ingnetworkwhileforthebinarycasetheAparticleswill
in this limit[24], we observe that for the present work it form a percolating network only above a critical fraction
is valid up to the critical point. As A clusters are caged for a particular φ .
tot
inside the B percolating cluster, the inter penetration of
Aclusterswillbeataminimumwhichcouldbeonepos-
sible reason why the A clusters follow the Smoluchowski D. Fractal dimension of binary system
equation even close to the percolation limit.
For monomeric DLCA type aggregation we know that
there is a transition from flocculation regime to perco-
C. Kinetics of monomeric and binary system lation regime, characterized by the change in fractal di-
mension from 1.8 to 2.5 [24]. In Fig.5a we plot the ag-
It has already been shown that for monomeric system gregation number m with the radius of gyration (R ) of
g
we will always form percolating cluster irrespective of the A clusters when m attains a stationary value. The
w
thevolumefraction[24]. InFig.4wehaveshowntheevo- dotted line in Fig.5a has a slope of 2, showing that the
lution of m for only A particles when φ = 0.3 at aggregation of A cluster inside the cage of B percolating
w tot
c =0.2 and for the monomeric system the volume frac- cluster are self similar in nature with a d = 2. In the
A f
tion is 0.06 the same as that for only A particles in the insetofFig.5awehaveshownmandR ofbothAandB
g
system. For the initial time t < 1 we observe the ag- particles for c = 0.2(triangle) and c = 0.5(square for
A A
gregation kinetics is following each other closely for the A particles and cross for B particles) before the system
monomeric and binary case, as the monomers involved percolates. At c = 0.2 B particles are forming clusters
A
5
A 1 10 100 1,000 B 0.1 1 10 100
103 0.c5A0→A 103 1,000
0.20→B
0.50→B
102
102 100
101
102 100
100
m m
10−1 100 101
101 10 101 10
100 101 102 103 10−1 100 101 102
Rg Rg
FIG. 5. (a) The aggregation number m is plotted as function of R is plotted for the A particles for c = 0.2 at φ = 0.2.
g A tot
Here the m attains a stationary value as A clusters are stuck inside the percolated B cluster. The dotted line has a slope of
w
2, showing that the A cluster is self similar and has a d =2. The inset shows the m and R for the A particles at c =0.5
f g A
(square) and for B species at c = 0.5 (cross) as well as c = 0.2 (triangle) just before the percolated cluster appears. The
A A
solid line has a slope of 2.5 showing that the system has already crossed over to percolation regime. (b) Here m and R is
g
plotted for φ =0.15 at c =0.5 for both A (square) and B (cross) during the early stage of aggregation. The dashed line
tot A
has a slope of 2 showing the clusters are still in the dilute regime with d =2.
f
and for c = 0.5 both A and B species are forming self of 2, which confirms the fact that for both A and B par-
A
similar structure and have crossed over to the percola- ticlesinflocculationregimehavead =2. Inthepresent
f
tion regimes as shown in the inset of Fig.5a by the solid casewewereabletoshowthatintheflocculationregime
line which has a slope of 2.5. Usually d =2 is reported we have a d = 2, which will cross over to d = 2.5, as
f f f
in reversible aggregation of colloids as well as during the predicted by the percolation theory irrespective of type
formation of lattice animals [25, 26]. According to the of particle. While in the case of A particles when m
w
percolation theory lattice animals appear far away from reaches a stationary value, it will always remain as lat-
the percolation transition. Due to the presence of the tice animals irrespective of φ
tot
2 species of particles there are many collisions between
A and B particles, which do not lead to bond formation
as they interact only through hard core repulsion. The
E. Scaling relation for binary system
clusters so formed are able to densify before a collision
between same species happens leading to bond forma-
tion. This also leads to the conclusion that if 2 species ConventionalDLCAischaracterizedbyτ =0ford =
f
are undergoing irreversible aggregation, it leads to self 1.8intheflocculationregime,forlatticeanimalswehave
similar clusters where all the configuration for a given τ =1.5andd =2,whileinthepercolationlimitτ =2.2
f
aggregation number is equi-probable for both species of for d = 2.5 as N(m) ∝ m−τ where the exponent τ are
f
particles [27]in the flocculation regime. If the structure supposed to be exact [28]. In Fig.6a, we have plotted
oftheAparticlesareinfluencedbythehindrancecaused N(m) as a function m for φ = 0.2 at c = 0.2 for
tot A
bytheB particles,weshouldbeabletoobservethesame theAclusters,wherem attainsastationaryvalueorin
w
effectfortheB particles. Thiseffectwillbemostevident otherwordsAparticlescannotpercolate. Thedottedline
when we have c = 0.5, as we have same number of A isgivenby aslopeof 1.5, aspredicted forlattice animals
A
andB particles. InFig.5bwehaveplottedmandR for by the percolation theory. While in the inset we have
g
the case of c = 0.5 φ = 0.15 for the A and B in the plotted N(m) with m for B particles when c = 0.2,
A tot A
initial stages of aggregation, where our system will still also for both A and B particles at c = 0.5 also see
A
be in the flocculation regime. The solid line has a slope inset Fig.5a. The solid line which has a slope of 2.2,
which agrees with the fact that this system is already
a cA
6
A 1 10 100 1,000 B 1 10 100 1,000
10−4 0.50→A 10−5 1e−05
0.20→B
10−5 10−6 0.50→B 1e−05
10−6
1e−06
10−8
10−7
10−10 1e−07
10−6
m) 1e−06 m)
N( 100 101 102 103 N( 10−8
1e−08
10−9
10−7 1e−09
1e−07
10−10
1e−10
10−8 10−11
1e−08 1e−11
100 101 102 103 100 101 102 103
m m
FIG.6. (a)ThenumberdensityofclusterN(m)isplottedasafunctionoftheaggregationnumbermforφ =0.2atc =0.2
tot A
for the A clusters when m attains a stationary value. The dashed line has a slope of 1.5 as expected for lattice animals. The
w
inset of figure we have plotted the N(m) with aggregation number m before the appearance of percolated cluster for the A
clusters at c =0.5 (square) and for the B particles at c =0.5 (cross) and c =0.2 (triangle) for φ =0.2. The solid line
A A A tot
intheinsethasaslopeof2.2showingthatthesystemisinpercolationregime. (b)N(m)isplottedwithmforasystemwith
c = 0.5 at φ = 0.15 during the initial stages of the aggregation process for the A (square) and B (cross) clusters. The
A tot
dashed line has a slope of 1.5 as predicted for lattice animals.
in the percolation regime. Also it shows that we have the critical parameter for irreversible aggregation of bi-
a cross over for the binary system from lattice animals nary system. We know that percolation theory predicts
to percolation regime. In Fig.6b we have plotted N(m) that the correlation length ξ scale with (p−p )ν were
c
with m for φ = 0.15 at c = 0.5 both for A and B ν =0.88 below the percolation threshold. It has already
tot A
pTahretidcoletsteddulriinneghtahseainsiltoipael sotfa1g.e5scoofnafigrgmrienggattihoenfpacrotctehsast. been shown that Rgz ∝ξ, where Rgz =(cid:113)(cid:80)(cid:80)∞m∞m==11mm22NR(gm2)
[8]. In Fig.7b we have plotted Rg with c /φ and
both A and B have a fractal dimension of 2, contrary to z A tot
the solid line is given by the slope of 0.88, which is in
thefactthatintheflocculationregimeforDLCAwehave
agreement with the percolation theory. In Fig.7 com-
d =1.8. Thiswebelieveisadirectevidencethatweare
f pressibility and correlation length are showing universal
observing lattice animals in our system in the case when
behavior when plotted as function of c /φ with the
clusters are in the flocculation regime. A tot
scaling law predicted by percolation theory, where by we
In Fig.7 we have plotted the stationary value of m can confirm that irreversible binary system comes under
w
for the A particles as a function of c /φ for a range the universality class of percolation.
A tot
of φ for three different c as indicated in the figure.
tot A
It seems that c /φ is the parameter that determines
A tot
the variation of stationary value of m once A particles IV. DISCUSSION
w
are stuck inside the pores of B particles. In the Fig.3
we observe that as we go towards the percolation tran- Hechtet. alhaveanalyzedthefractaldimensionofbi-
sition the stationary value of m or the compressibility nary aggregating colloidal particle, where they have re-
w
of the system keeps on increasing till we have an infinite ported a d = 1.8, which is predicted for 1 component
f
spanning network or in other words m diverges. We DLCA [3]. With the experimental data of m and R ,
w g
also observe that m scales with c /φ with a slope of they have reported it is very difficult to differentiate a
w A tot
1.8 see solid line in Fig.7a. It is a well established fact slope of 1.8 and 2. Unfortunately they have not calcu-
thatm ∝(p−p )γ belowthecriticalvaluep according latedN(m)asafunctionofm,whichwebelievewillgive
w c c
to percolation theory. This gives us a clear indication much more precise value of d , as shown in the present
f
that if we consider (p−p )∝c /φ , may be c /φ is study. Inthepresentworkboththeparticlesstartaggre-
c A tot A tot
7
A B
0.1 1 10 0.1 1 10
101
101
102
100
z
w g
m R
0.10
0.15
0.20
101
10
10−1 100 101 10−1 100 101
c /φ c /φ
A tot A tot
FIG. 7. (a) The stationary value of m for the A particles is plotted as a function of c /φ at different φ for the values
w A tot tot
of c as indicated in the figure. The solid line has a slope of 1.8 consistent with the prediction of percolation theory. (b) The
A
stationaryvalueofz averageradiusofgyrationisplottedagainstc /φ fordifferentc asindicatedinthefigureatdifferent
A tot A
φ for the A particles when it cannot percolate. The solid line has a slope of 0.88 .
tot
gatingsimultaneously,whileintheexperimentalcasethe gregation [30]. For smaller attraction the system under-
aggregationofAandB particlestartsatdifferenttimes. goes spinodal decomposition for the B particles, while
It has already been shown that compared to a random the fraction of A densifies which will require a higher
distribution an aggregated system have more free space fraction in order to percolate.
[29]. Thus we believe that the diffusion and the kinetics
Inthepresentworkwehaveshownthatbycontrolling
of the system may be different, but there should not be
φ and c we are able to produce self similar clusters
tot A
much difference in the final structure of these systems.
of particular size for the A particles there by controlling
The appearance of bigel have also been reported by
the pore size of B . This result can help in creation
Varrato et. al, in reversible aggregation of binary col-
of new functional materials of a particular size simply
loidal system. They have reported 3 distinct region,
by controlling the pores size created by the B particles
where both A and B percolated, only B percolated and
[31]. Itwillbeinterestingtofurtherextendthisstudyto
region where neither A or B percolated. In our case we
includeasymmetricparticles,andstudyhowtheporosity
onlyhave2regionsbigelor1componentgel,asexpected
of the corresponding gel varies.
from irreversible aggregation. For φ > 0.2 our results
tot
are similar with the results of Varrato et. al, but for
φ <0.2 they have reported c much higher than what
tot A
wehaveobserved inoursystem. Inthecaseofreversible
aggregation it was shown that for low volume fraction ACKNOWLEDGMENTS
onlyhighattractionstrengthcouldmimicirreversibleag-
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