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The chemical gelation viewed through a percolation model simulation
D. Lairez, Denys Durand, J. Emery
To cite this version:
D. Lairez, Denys Durand, J. Emery. The chemical gelation viewed through a percolation model simulation. Journal de Physique II, EDP Sciences, 1991, 1 (8), pp.977-993. �10.1051/jp2:1991121�.
�jpa-00247569�
J. Phys II France 1 (1991) 977-993 AOOT1991, PAGE 977
ciassificanon Physics Abstracts
02 40 05 70 82 20 82 70
The chemical gelation viewed through a percolation model simulation
D Lairez i*), D Durand and J R. Emery
URA-CNRS n° 509 et 807, Unlversitd du Maine, 72017 Le Mans Cedex, France
(Received8 October J990, rev~ed 27 March J99J, accepted J9April J99J)
R4sumk. De nombreux articles ou revues prhsentent la th60rie de la percolation cornrne
pertinente pour le probldnle de la gkhfication chimique Mais la plupart de ces Etudes concement le comportement cntique du moddle de percolation norrnal de site ou de lien De tels moddles
ignorent totalement les aspects ch1mlques spkcifiques qui autonsent la grande vanktk de
structures et de propndtks des rkseaux macromolkculaires L'iddal serait d'aboutir I un moddle
capable de temr compte des particulantds de chaque systdnle ktudid Mais une question de fond se pose alors ces singulantks changent-elles le comportement umversel? Cette Etude tente de
rdpondre h cette question dans le cas particuher de la gkhfication obtenue par polymdnsation par dtape Une telle gdhfication peut dtre dvltke par l'introduction de monomdres monofonctionnels
tuant la croissance des arnas Ce cas qui est ktudik dans cet article, correspond I un probldme de
percolation de site-lien Le diagramme des phases est ktabh et di~krents chemlns traversant la
hgne critique de ce diagramme sont ktudiks Les rksultats souhgnent que les contramtes
topologiques I la connexion des monomdres peuvent conduire I un comportement cntique non
umversel La levke de ces contramtes en donnant une mobilitd fictive aux monomdres, redonne
au systdme un comportement cntique universel Ce travail justifie a posteriori les Etudes expknmentales du comportement antique de la gkhfication conduite sur des systdmes figks amvks h complete rkaction Il confirme dgalement l'appartenance de la gkhfication ch1mlque par
polymknsation par dtape I la mdme dasse d'umversahtd que le moddle de percolation normale et illustre la fa90n dont ce nlodele peut dtre rendu plus ikahste
Abstract. Many papers or reviews present the percolation theory as pertinent to the chemical
gelation problem But most of these studies are related to the cr1tlcal behaviour of standard bond
or site percolation models. Such approaches ignore totally the specific chemical features which allow the large variety of structures and properties exhibited by chemical networks The ideal would be to have a model able to mimic realistically the chemical gelation process by taking into
account the specificities of each chemical system investigated But then a basic question arises do
the singularities change the universal behaviour? Thls study aims to contnbute to answer this question in the particular case of the gelation made by stepwise polymenzation In such systems, gelation may be avoided by introducing monofunctlonal mononlers which are killing the cluster growth This case is examined in this paper and corresponds to a site-bond percolation problem.
(*) Present address Service de Physique de l'ttat Condensd, CE Saclay, 91191 Gif-sur-Yvette Cedex. France
978 JOURNAL DE PHYSIQUE II N 8 Phase diagram is established and the different ways to cross the cntical line of this diagram are
investigated The results outline the fact that topological constraints applied to the nlonomer
connection may prevent the systern from having a universal critical behavlour Deleting these
topological constraints, by giving a fictive mobility to the mononlers, allows us to find again a universal behavlour for the system Thls work justifies a posteriori expenmental studies on the critical behaviour of the chemical gelation run with quenched systems. It also confirms that chemical stepwise gelation and standard percolation belong to the same universality class and illustrates how this model may be modified to be more realistic
1. The question of universality in the chemical gelafion : what is universal and what is system dependent ?
In a wide vanety Of processes in physics, chemistry, biology, the union Of many separate small elements into clusters Of vanous sizes and the formation Of macroscopic phases Of connected
elements is an extremely important phenomenon. The polymenzation Of polyfunctional
organic molecules called monomers leading to macromolecular network or gel is one of these important processes. In fact, the long range of connectivity wh~ch confers the specific physical properties of these systems results from an increase of connectivity at a microscopic scale by
random links between monomers through specific chemical rules.
The gel charactenstics are very sensitive to the chemical reaction pathway used to build it up and it is well known that some changes m the expenmental conditions (catalyst or not, solvent or not) can affect the gelation threshold and lead to different final structures and
properties. In the same time, all these systems exhibit at the gelation transition, universal behaviours through the multiconnected gel cluster formed
One of the challenges for the theoretical approaches which aim to describe the network formation process is to be able to take into account the specific features of each chemical system and then to highlight which is universal and which is system dependent m the
investigated process [I]
As for any critical phenomenon, the universal behaviour is mainly manifested in the critical domain around the gel point, through the exponents which govem the evolution of physical quant1tles exhibiting singulanties at the gelation threshold. At the opposite, the location of the gel point and the properties of networks at the final stage are strongly dependent on the initial specific molecular parameters of the system Thus, it is worth to develop theoretical
approaches which can, at the same time, take into account the specific features of the considered system and correctly descnbe the system over the whole course of the gelation
process, including the sol-gel transition
2. Network modelling by computer sbnulafions.
The network modelhng by computer simulations seems to be at this moment the best route to include many of the complexities of particular systems and to describe the whole course of the network fornlation including the region in the near vicinity of the transition. But attention must be carefully focussed on the rules of bond fornlation chosen in computer models in order
to discuss their realistic character
Simulations of network fornlation on computers are mainly developed along three dominant lines
N 8 CHEMICAL GELATION AND PERCOLATION MODEL SIMULATION 979
1) The simulations off lattice [2-4] based on the Flory-Stockmayer model which do not take the excluded volume effect into account, neglect basically the cydisatlon (even if a certain
amount of finite size cycles can be involved) and consequently produce tree like cluster
structures m a space whlthout any dimensionahty. This classical approach can predict, from the parameters characterizing the initial components, the gel point and the evolution of
different molecular parameters (average molecular weights, gel fraction, cross link density, ...) except in the cntical region
ii) The simulations off lattice of kJnetic aggregation process [5-9] m which initially dispersed particules stick together under the action of given forces The particule-duster
aggregation called DLA (diffusion l~mlted aggregation) and the cluster-cluster aggregation
are the most famous models. If these models are realistic, they are too soph~sticated for being able to take into account the specifipities of the chemical gelation of polymers.
iii) The simulations on lattice based on percolation model [lo,11] which aim mainly to descnbe the cntical behaviour of systems m the transition domain. This approach takes into
account the excluded volume effects, allows the loop fornlation but gives a zero mobility to
monomers and clusters.
Simulation of network fornlation by computers can also provide an effective tool to exhibit wl~ich parameters (and more particularly which local rules for the connections) are relevant or not m respect to the unlversabty. As example, computer modelhng of the kJnetic of radical
initiated chamwise copolymenzation leading to gel, denved from a percolation model, has revealed that this process of gelation does not belong to the same universality class as the
standard percolation [12-15]
Thls paper is devoted to present a computer simulation of percolation based on Monte Carlo method which aims to m1mlc stepwise polycondensation leading to gel. It will be shown how easy such a model allows one to take into account specific features of a polycondensation
process and how valuable such an approach is to exhibit which are the relevant parameters to determine a universality class.
Two fundamental aspects which characterize any aggregation process will be considered here First, the kinetic aspect of the process wh~ch consists m a quantitative description of the
time evolution of the mean size and size distnbution of the clusters Secondly, the geometry aspect of the cluster which consists m a quantitative descnption of the structure of the clusters and involves mainly the concept of fractal
3. A computer modelbng of percolation fitting the reality of a polycondensafion process.
On a lattice of any cell, the standard percolation theory allows us to define two main types of percolation. The first is the « site percolation », where each site of the lattice is randomly occupied with probability p and empty with probab1llty (1 p) and clusters are groups of
neighbouring occupied sites The percolation threshold p~ is the value of p at which one cluster is extending from one side of the system to the opposite Tl~e counterpart of the site
percolation is called « bond percolation and is defined as follows. imagine each site of the lattice to be occupied and lines drawn between neighbounng lattice sites Then each line can
be effective bond with the probability p or ineffective bond with the probability
iI p A cluster is then a group of sites connected by effective bonds. As previously, there is
a cntical bond fraction p~ at which an infinite network appears m an infinite lattice These two
percolation processes are equivalent and have the same cntical behaviour which depends only
on the space dimensionahty.
The chemical gelation in which clusters are fornled by chemical bonds between monomers
seems relevant to the bond percolation process m three dimensions, But the classical Monte
Carlo method used for simulating bond percolation has to take into account the specificities of
980 JOURNAL DE PHYSIQUE II N 8 the chemical gelation, which are mainly the functionality of the monomers and the
hmltatlons of the gelation process
Functionality ofmonomers in the standard bond percolation, the nearest neighbours of any given site can potentially be bound and the number z of these nearest neighbours depends
only on the lattice coordmance. If now chemical monomers having a functionality
f are put on the sites of a lattice of coordmance z, any given monomer cannot link more than
f monomers among the
z nearest neighbours. This corresponds to the l1mlted bond
percolation problem previously discussed by Gaunt et al. and Kersetsz et al. [16, 17] (all computation details are available on request)
Limitation of the gelation process there are three main ways to abort the gelation
process by killing the cluster growth
i) The cluster growth can be limited by introducing monofunctional monomers iA,
monomers) among the f-functional monomers iA~ monomers with f m 2). Above a certain fraction of monofunctional monomers p~, even at complete reaction ~p = I ), the size of the
largest cluster is always finite (in an infinite lattice) and the system can never become a gel.
ii) If we consider, for example, a chemical system composed of f-functional monomers iA~ monomers) able to react only with g-functional monomers iB~ monomers) above a
certain limit stoich~ometric ration r~ ii-e number of functions A/number of functions B), even
at complete consumption of the reactive groups having the lowest concentration, the cluster
size is alway limited and a gel does not appear
iii) The third mean to abort gelation process consists to carry out the reaction with solvent.
The clusters are totally surrounded by solvent molecules and even after complete reaction, only finite clusters, but strongly multiconnected by intramolecular links, can be formed and
gelation does not occur.
In these three cases, a sol-gel diagram can be established as a function of the reaction extent and a second parameter. molar fraction of monofunctional monomers ii), stoichiometnc ratio iii), solvent concentration iii).
From the point of mew of percolation, at complete reaction, the gelatlon process can be identified with a site percolation problem A reliable theoretical approach of these systems should m~x bond and site percolation models in a non-correlated manner. Simulations have
already been developed to describe solvent effect (iii) [see Ref 18] The computer simulation program developed in th~s study is able to take into account both problems : functionality and limitation of gelation process, in the scope of three dimensional percolation
4. Physical quantifies derived from the simulation data.
From the simulation data, many physical quantities charactenzing the kinetic aspect of the clusters growth and the size distribution of the clusters can be denved Figure I displays the kinetic aspect of the clusters growth for a percolation of hexafunctional monomers on a cubic lattice of size L
= 50a, where a is the lattice spacing and which has a number of sites
lLla)~
=
50~.
A cluster being defined as an assembly of connected sites, the mass of a cluster will be charactenzed by the number s of sites (monomers) belonging to the cluster (monomers will be considered as unit mass cluster) If N~ is the number of clusters of size s then,
N
= £N~ ii)
is the total number of clusters. At the beginning of the process, the major part of bonds is
N 8 CHEMICAL GELATION AND PERCOLATION MODEL SIMULATION 981
~w
SlZe
(x10~)
a 75 isa 3aa
number of bonds (xio~)
Fig I -Physical quant1tles denved from a simulation on a 503 lattice
mterclusters. Then each bond decreases of one unit the number of clusters, therefore the
slope of N versus the number of bond is I
The mean degree of connectivity can be charactenzed by a weight average mass N~ defined as the ratio of the second to first moment of the cluster mass distnbution
Nw= ([Ns.S~)/([Ns.S), 12)
Before the percolation threshold, N~ can be considered, from the chemical point of view, as the weight average degree of polymerization But, above the threshold, because of the
finiteness of the lattice, N~ has no physical meaning
The mass N * of the largest cluster behaves as the z-average mass of the clusters [18] which
can be defined by the relation
N*~N~= (£N~.s~)I(£N~.s~). 13)
~ ~
loco
£coo
NW lox ~
4000
fin
loco
Sax
~~~
40x ioox
o
P P
Fig 2 Examples of solvent effect simulations of AJAO percolation on a 20~ lattice with di~erent
monomer concentration
982 JOURNAL DE PHYSIQUE II M 8
oooo
»=&
&coo
Nw
<oar
2080
fmz4
a
P
Fig 3 Effect of the average functionality simulation of A~/Ag percolation on a 203 lattice
After the gelation threshold, N * can be identified with the gel fraction G. Tl~e scaling laws of these quantities as a function of the distance to the gelation threshold will be examined
The solvent effect can be mlmlcked by m1xlng randomly hexafunctional sites (monomers)
with zero functional sites (solvent) on the lattice nodes. Figures 2a and b shows qualitatively
that a decrease of the relative number of monomers decreases obviously the size of the clusters, but also delays the gelation threshold because the extent of intramolecular reactions at the gel point increases with the solvent concentration Figure 3 shows the effect of killing
the cluster growth by adding monofunctional monomers The characteristics of the initial
mixture of monomers are defined through number and weight functionalities of the
monomers
In
" £~i°fi /£~i (~)
,
fw " £ ~i f~ £ ~i Ii (5)
,
where n~ is the number of monomers having a functionality f~
S. Sol-Gel diagrams of a mixture of A~ and A, monomers.
Figure 4 represents a typical sol-gel diagram m which the continuous line gives the fraction of effective bonds at the gelation threshold p~ as a function of the molar fraction p of the
A6 monomers The p~ values are determined as the p values from wh~ch a cluster connects one
side to the opposite side of the lattice Two limiting ways of crossing the cntical line can, be considered :
I) In a system composed only of A~ monomers, the only increase of the number of bonds allows one to get percolation ThJs process corresponds to the classical bond percolation.
ii) If all the systems at maximal bond extent ~p = I are considered, the crossing of the
gelation transition might be considered as a site percolation and the cntical behaviour should be identical to the previous case.
M 8 CHEMICAL GELATION AND PERCOLATION MODEL SIMULATION 983
gel
P
sol
a
P
Fig. 4 Phase diagram of a mixture of A6/Aj monomers on 503 lattice possible ways to cross the cntical line
In addition to these two limiting ways, there are many other possibilities for crossing the
sol-gel transition at a given p value (vertical way) or at a given p value lhorizontal way) The
main open question is the following is the cntical behavlour of the system dependent or not
on the way of crossing th~s cntlcal line ?
Before exanumng th~s problem, let us charactenze the cluster mass distribution along the cntical line
gel
b P
o.1
~~~
o
P
Fig. 5 Phase diagram of a mixture of A~/Aj monomers on 503 lattice investigated points for the clusters mass distnbution
6. Cluster mass distribution along the critical line of a sol-gel diagram and fractal dimension of the clusters.
Figure 5 indicates the ip, p ) coordinates associated with the points where the cluster mass
distributions have been investigated In percolation model, the cluster mass distnbution which gives the evolution of N~ as a function of s follows a power law.
N~
~ s~ ~. (6)
IOURN AL DE PHYSIQUEII T I w 8 AOOT I%1 45
984 JOURNAL DE PHYSIQUE II M 4,soo
~
~,soo
store=2.25
2.700
iogt%s>
i,coo
o.900
n
n n
n o.coo
1. .~
logts)
Fig. 6a Cluster mass dlstnbutlon at the critical threshold with 100 fb of A~ (p = I (A-procedure)
4,soo
i~
~.GBO
slooe~2.25
z.70e
iogt«s>
i ,coo
o.900
-n
o.oo.
i ,s
lagts)
Fig 6b Cluster mass distribution at the cntical threshold with 50 9b of A6 (P = 0 5 (A-procedure)
N 8 CHEMICAL GELATION AND PERCOLATION MODEL SIMULATION 985
«,soo
C
3.SOD
~
slope=2.25
2?OO
log(%s)
i,coo
a
o.900
a°
a w
a aaa
I.coo
,s
loots)
Fig. 6c. Cluster mass distnbution at the cntical threshold with 20 9b of A6 (P
= 0 2) (A-procedure)
The values of the exponent T can be considered as identical whatever the cntical point (Figs. 6a, b and c) and the value of 2.25 found is very closed to the theoretical value of 2 20
predicted for the classical percolation problem [18]
The size f of the largest cluster which measures the spatial extension of connectivity and called the correlation length is linked to the mass N* of the largest cluster by the fractal
dimension di:
~ * ~di ~~~
argest
luster is
G
= N*ii d (8)
where d is the space dimensionahty.
By using this equation and the scaling laws (10 and II) presented m the following
paragraph, equation (9) called « hyperscalling relation » is found
d/di = r (9)
This relation indicates that, at the percolation threshold, the density of the largest cluster
(N*/f~) is equal to one of the giant cluster (G/L~). It also expresses the fact that the cluster
distribution reflects the fractal structure of the clusters Therefore, it can be concluded from the simulation data that percolation process, developed from different1nltial composition
mixtures, builds clusters exhibiting the same fractal structure as predicted [19].
