Dynamical arrest: interplay of the glass and of the gel transitions

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by  Nagi Khalil

Dynamical arrest: interplay of the glass and of the gel transitions
NagiKhalil,a‡,AntonioConigliob,AntoniodeCandiab,d,AnnalisaFierrob,andMassimoPicaCiamarrab,c
4ReceivedXthXXXXXXXXXX20XX,AcceptedXthXXXXXXXXX20XX
1
0FirstpublishedonthewebXthXXXXXXXXXX200X
2

nDOI:10.1039/b000000x
a
J
The structural arrest of a polymeric suspension might be driven by an increase of the cross–linker concentration, that drives
8
2thegeltransition,aswellasbyanincreaseofthepolymerdensity,thatinducesaglasstransition. Thesedynamicalcontinuous
(gel) and discontinuous(glass) transitionsmightinterfere, since the glass transition mightoccur within the gel phase, and the

]geltransitionmightbeinducedinapolymersuspensionwithglassyfeatures.Herewestudytheinterplayofthesetransitionsby
h
investigatingviaevent–drivenmoleculardynamicssimulationtherelaxationdynamicsofapolymericsuspensionasafunction
c
eof the cross–linkerconcentrationand the monomervolumefraction. We show that the slow dynamicswithin the gel phase is
mcharacterizedbyalongsub-diffusiveregime,whichisduebothtothecrowdingaswellastothepresenceofapercolatingcluster.
-Inthisregime,thetransitionofstructuralarrestisfoundtooccureitheralongthegeloralongtheglassline, dependingonthe
t
alength scale at which the dynamics is probed. Where the two line meet there is no apparent sign of higher order dynamical
tsingularity. Logarithmic behavior typical of A singularity appear inside the gel phase along the glass transition line. These
s 3
.findingsseemtoberelatedtotheresultsofthemodecouplingtheoryfortheF schematicmodel.
t 13
a
m
1 Introduction in lightinducedpolymerizationprocesses(e.g., dentalfilling
-
pastes) or by heat (e.g., cooking). On increasing the bond-
d
nA polymeric suspension might behave as a hard sphere sys- ing probability,onedrivesthe system acrossa geltransition,
otem, when no cross–linkers are added to it, as its dynamics where a spanning cluster of connected monomers emerges.
cslows down as the concentration increases. In this case, a This leads to a gel transition line f (p) in the p–f plane.
[ gel
transitionofstructuralarrest,theglasstransition,occurswhen Thisstructuraltransitionalso correspondsto a dynamicalar-
1 the volume fraction f overcomes a critical threshold, f glass. resttransition,asthedynamicsbecomesfrozenonthesmallest
vThistransitionmarksthearrestofthedynamicsatallrelevant wavevectork beingthespanningclusterunabletodiffuse.
min
52length scales, for wavevectors ranging from kmin =2p /L to Contrarytotheglasstransition,thisisacontinuoustransition,
2kmax =2p /s , with L and s sizes of the system and of the astheorderparametercontinuouslyincreasesasoneentersthe
7particles. The glass transition is a discontinuous dynamical gelphaseonincreasing porf .
.phase transition because its order parameter, known as non–
1 Theinterferenceofdifferenttransitionsofdynamicalarrest
ergodicityparameter and defined as the infinite time limit of
0 have been previously investigated, via model coupling the-
4relaxationfunctions,iszerobelowthetransitionbutacquires
ory, in systems of particles interacting with a hard core re-
1a finite value at the transition. Polymeric suspensions may
pulsion competing with a very short range attraction. These
v:also undergo a different kind of structural arrest transition, systems show an attractive and a repulsive glass transition,
ithegeltransition. Thisoccursatfixedvolumefractiononin-
X both of them discontinuous, and mode coupling theory pre-
creasingthecross–linkerconcentration,whichisrelatedtothe
dicts that their interference leads to an A high order singu-
rprobabilitythattwoclosemonomersarepermanentlybonded. 3
a larity characterized by logarithmic decay of the correlation
Thisconcentrationcanbetuned,forinstance,byradiationas
functions1–3. Thesepredictionswereverifiedindifferentcol-
loidalsystems4–9,bothexperimentallyandnumerically.Sim-
a DepartamentodeF´ısica,UniversidaddeExtremadura,E-06071Badajoz, ilarfindingswerealsoobservedinRef.10, wherethedynam-
Spain
icsofaphysicalgelwasnumericallystudiedathighdensities
bCNR–SPIN,DipartimentodiScienzeFisiche,UniversityofNapoliFederico
II,Italy wheregelationandglasstransitioninterfere.
cDivisionofPhysicsandAppliedPhysics,SchoolofPhysicalandMathemat-
Inthispaperwestudytheinterferenceofacontinuousand
icalSciences,NanyangTechnologicalUniversity,Singapore
dINFN,SezionediNapoli,ComplessoUniversitariodiMonteS.Angelo,Via ofadiscontinuoustransitionofstructuralarrestfocusingona
Cintia,Edificio6,80126Naples,Italy system with permanentbondswhere, as illustrated in Fig. 1,
1–7 |1
102 1
Gel line
0.6 Glass line 101 0.8
lity, p Gel & Glass 2r(t)1100-01 F(k,t)s00..46 k = k
bi0.4 10-2 0.2 k = 1max
a
b Gel 10-3 0
pro 10120-210-1100 101 102 103 104 105 110-210-1100 101 102 103 104 105
d monomers k = kmax
on0.2 Sol 101 percolating 0.8 k = 1
b Glass 2r(t) 100 F(k,t)s00..46
0 10-1 0.2
0.2 0.3 0.4 0.5 0.6
volume fraction, f 10-2 0
10-2 10-1 100 101 102 103 104 10-210-1100 101 102 103 104 105
time time
Fig. 1 Structuralarrestdiagramasafunctionofthevolume
fraction,f ,andofthebondingprobability, p,illustratingthe Fig. 2 Meansquaredisplacement(left)andsISF(right).Toppanel:
interplayofthegelandtheglasstransitionlines.Thegellineis onapproachingtheglasstransitionatp=0,withvolumefraction
determinedviapercolativeanalysis,afterintroducingbondswith fromf =0.52tof =0.59(fromlefttoright).Bottompanel:on
probabilitypinanequilibrium(fullsymbols)orinan enteringthegelphaseatp=0.3(f gel≈0.4),withvolumefraction
out–of–equilibrium(opensymbols)monomersuspension. Theglass fromf =0.4tof =0.51(fromlefttoright).
lineisdefinedasthatwheretheextrapolateddiffusioncoefficient
vanishes.Onlypointsuptop=0.5f =0.54areshown,wherethe
selfIntermediateScatteringFunction(sISF)exhibitsalogarithmic (monomers) of mass m and diameters s and 1.4s , in a box
behavior,whichweassociatetoanA3singularityaspredictedbythe ofsizeLwithperiodicboundaryconditions. Thechosensize
F13modecouplingschematicmodel(seetextandFig.8).Solid ratioisknowntoeffectivelypreventcrystallization13–16. The
linesareguidestotheeye. volume fraction f =Nv/L3, where v is the average particle
volume,istunedbychangingthesizeLofthebox. Themass
m, the diameter s of the smaller particles, and the tempera-
thegelf (p)andtheglasstransitionlinef (p)intersect.
gel glass tureT,fixourmass,lengthandenergyscales,whilethetime
Weaddresssomequestionsraisedbythepresenceofthesetwo unit is pms 2/T. After thermal equilibration at the desired
transition lines. First, we note that the gel transition corre-
volume fraction, permanentbonds are introducedwith prob-
spondstostructuralarrestoccurringatthelargest(k )scale,
min ability pbetweenanypairofparticlesseparatedbyless than
whileallscales,includingthesmallestone(kmax),arerelevant 1.5s . Abondcorrespondstoaninfinitesquarewellpotential,
in the structural arrest corresponding to the glass transition. extendingfroms to1.5s . Theprocedureweusetoinsertthe
Therefore, we consider the length scale dependence of the
bondsmimicsalight–inducedpolymerizationprocess,asthe
transitionof structuralarrest. Second, we note thatthe glass numberofbondsdependsonboth pandf .
transitionlineentersinthegelphaseatapointwherebothcon-
Usingthepercolationapproach,weidentifythegelphaseas
tinuousanddiscontinuoustransitionoccur. Henceweexpect
thestateinwhichapercolatingclusterispresent17,18,andthe
that the model system here studied might have some analo-
gelation transition as the percolationline. A standardfinite–
gieswiththemodecouplingtheoryforschematicF model2,
13 size scaling analysis of the mean cluster size19 is applied to
whichgivestwoarrestedlines: acontinuoustransition,which
identifythepercolationline.Theresultinglineisillustratedin
meetsadiscontinuousone.Thisdiscontinuoustransitionends
Fig.1. Thegeltransitionoccursatacriticalbondprobability
onahighordercriticalpoint(A3singularity)characterizedby p (f )decreasingwithf .
logarithmdecay1–3 oftherelaxationfunctions. Topologically gel
the phase diagram is similar to the one found here (Fig. 1).
Consistentlywithmodecouplingtheory,wefindlogarithmic 3 Standard glassandgel behaviours
behavioratapointwellinthegelphasealongtheglassline.
Since we are interested in the interplay of the gel and of the
glass transition lines, we start by shortly reviewing the main
2 Numerical model features of the slowing down of the dynamics when these
transitions do not interfere. First, we consider the transition
Weperformeventdrivenmoleculardynamicssimulations11,12 at zero bonding probability, where our system reduces to a
of a 50:50 binary mixture of N = 103 hard spheres hardspheresuspensionandundergoesaglasstransitionasthe
2| 1–7
volume fraction increases. This crowding induced dynami- tothefluctuationsoftheclusterstructure,andthethirdtothe
caltransitionischaracterizedbywellknownsignaturesinthe diffusionofitscenterofmass.Ofthese,onlythethirdreaches
meansquaredisplacement,r2(t), andintheselfintermediate thediffusiveregimeatlongtimes. Atthegeltransition,when
scatteringfunction(sISF),F(k,t),definedrespectivelyby: the diffusionof the center of mass of the largestcluster van-
s
ishes, the mean square displacement remains subdiffusive at
hr2(t)i= 2 N(cid:229) /2r2(t), (1) alltimes,whileinsidethegeliteventuallyreachesaplateau.
N i ThebehaviourofthesISFcritically dependsonthelength
i=1
scaleatwhichitisprobed.Atsmallscales,thatisfork=k ,
max
and
it does not exhibit any particular feature on approachingthe
F(k,t)= 2 N(cid:229) /2eik(ri(t)−ri(0)), (2) gel transition. On the other hand, at larger length scales,
s
N k = k , the relaxation time diverges when f approaches
i=1 min
wherethesumsextendonlytothelargerparticles. Theseare f gel(p). At the transition, the correlator decays in time as
a power law, while inside the gel asymptotically tends to a
reviewedinthetoprowofFig.2. Athighdensities,particles
plateau, f , the non–ergodicityparameter, that is zero at the
rattle in the cages formed by their neighbours before enter- k
ing the diffusive regime. Accordingly, the mean square dis- the transition, and continuously grows on entering the gel
placementdevelopsaplateauatintermediatetimes,knownas phase. Thegrowthiscontinuousasthevalueof fk is related
tothedensityoftheparticlesinthepercolatingcluster,andof
Debye–Wellerfactor.Thisplateaubecomeslongerandlonger
asthedensityincreases. ThediffusivityD=limt→¥ r2(t)/6t themonomersthataretrappedinthepercolatingcluster. The
decreases on increasing f , and vanishes at the glass transi- geltransitionthereforeinducesacontinuousdynamicalphase
tion. Consistently with the behaviourof r2(t), the relaxation transition. Thesepropertiesarefoundingellingsystemsboth
functionatlargekdevelopsatwostepdecay,thefirstone(b experimentally20–22andnumerically23–26. InRef.s27,28,deep
relaxation)associatedtotherattlingwithinthecages,thesec- analogiesofgelationandspinglasstransitionwassuggested.
ondone(a )totheonsetofdiffusivemotion.Theb relaxation
becomeslessandlessvisibleaskdecreases,andlengthscales
4 Dynamicsinthe gel phase
largerthan those associated to the typicalvibrationalmotion
areprobed. Atthe glass transition, onlythe b relaxationoc-
We now consider how the features of the relaxation process
curs,andF(k,t)asymptoticallyreachesafinitevalue,known
s
changesasthevolumefractionincreases, andthesystemen-
asnon–ergodicityparameter f . Theglasstransitionisadis-
k
tersthegelphaseandapproachestheglasstransition. Thisis
continuousphasetransitionas f variesdiscontinuouslyacross
k
identifiedasthevolumefractionwheretheextrapolateddiffu-
thetransition.
sivityofthemonomersvanishes(seeFig.1).Inthissectionwe
Wenowconsiderthetransitionathighp,whereonincreas-
ingf thesystemundergoesageltransitionata volumefrac- focusonahighenoughvalueofthebondprobability,p=0.3,
tionf ≪f , so thatthetwotransitionsdonotinterfere. forwhichf gel≈0.4issensiblesmallerthanf glass≈0.58.
gel glass
Thetypicalbehaviourofr2(t)andofF(k,t)onapproaching
s
thistransitionisillustratedinFig.2(bottomrow). Duetothe
4.1 Meansquaredisplacement
presenceofbonds,inthesystemthereareclusterswithadif-
ferentsizes. Inthelongtimelimit,eachclusterdiffuseswith Inthe gelphase, themeansquaredisplacementatintermedi-
adiffusivitydecreasingwiths. Accordingly,atlongtimesthe atetimesshowsasubdiffusiveregime,similartothatobserved
mean square displacement is essentially fixed by that of the intheslowdynamicsofcolloidalparticlesdiffusinginama-
fastest clusters, the monomers. Since the volume fraction is trix of disordered hard sphere obstacles30. We have charac-
small,thediffusivityofthemonomersisonlyslightlyaffected terizeditbyinvestigatingthetimedependenceofthesubdif-
bythevolumefractionandhasnotspecialfeaturesatthegel fusiveexponentq,definedbyq=dlogr2(t)/dlogt. Dataare
transition,abovewhichthemonomersdiffusewithintheper- illustratedinFig.3. Thisexponentexhibitsacrossoverfrom
colatingpolymernetwork.Onthecontrarythelargestcluster, theballisticshorttimevalueq=2,tothediffusivelongtime
whosemeansquaredisplacementisalso illustratedinFig.2, value,q=1. Atintermediatetimes,asubdiffusivebehaviour
slowsdownonapproachingthe geltransition, and indeedits isfound,q<1,whichbecomesprominentasthevolumefrac-
diffusivityvanishesatthetransition(wherethelargestcluster tion increases. Further increasing the volume fraction, very
percolates). We note that, on increasing f , the mean square peculiarfeaturesappear,astheexponentqmightalsoexhibit
displacement of the largest cluster does not reach the diffu- two minima. This behaviour is rationalized considering that
sive regime on the time scale of our simulations. Indeed, it themeansquaredisplacement,Eq.(1),hascontributionsfrom
hasthreecontributions:thefirstisduetothevibrationsofthe particlesbelongingtodifferentfiniteclustersofsizes,aswell
particlesinthecagesformedbynearestneighbors,thesecond as from the spanning cluster (s=¥ , in the thermodynamic
1–7 |3
3 2
10 10
2
f=0.44 60
q f=0.48 t
2 f=0.52
10 f=0.56
1 101 n 40 s
101 s20 Ær2æ
0 s = ¥
(t)100 10-210-1100101t102103104 (t)100 0
2 2 0 1 2 3 4 5
r r s
Gel effect
-1
10
-1
10
-2
10
Cage effect
-3 -2
10 10
-2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
time time
Fig. 3 Meansquaredisplacementinthegelphase,for p=0.3and Fig. 4 Mainpanel:meansquaredisplacementaveragedoverallthe
f from0.42to0.56(fromtoptobottom).Intheinset,the particles,hr2i,overparticlesbelongingtofiniteclustersofsize
subdiffusiveexponentq=dlogr2(t)/dlogt,asfunctionoftimefor s=1,2,3,andoverparticlesbelongingtothepercolatingclusters,
p=0.3andf =0.44, 0.48, 0.52, 0.56. for p=0.3andf =0.54.Inset:sizedistributionoffiniteclusters
forthesamevaluesof pandf .Thereisalsoapercolatingcluster,
withs≈930particles.
limit),as
hr2(t)i= 1 (cid:229) sn hr2(t)i, (3) 103
N s s
s 2
10
wherethesumrunsovertheclustersizes,hr2(t)iisthemean
s 1
square displacementof clusters of size s, and n is the num- 10
s
berofclusterwithsizes. Theclustersizedistributionns and )100
themeansquaredisplacementfordifferentsareillustratedin (t
2
Fig.4,for p=0.3andf =0.54. Thefigureclarifiesthattwo r10-1
phenomenalead to the subdiffusivebehaviour. The first one
-2
isthecrowdinginducedcageeffecttypicalofglasses, which 10
involvesbothparticlesbelongingtofiniteclusters, aswellas -3
10
particlesbelongingtothepercolatingcluster. Thesecondone
only involves the particles of the percolating cluster, whose 10-4
-2 -1 0 1 2 3 4
meansquaredisplacementreachesaplateauatlongtimes,as 10 10 10 10 10 10 10
time
thepercolatingclusterisunabletodiffuse.Thecoexistenceof
thesetwoeffects,whoserelevantimportanceisfixedbyf and Fig. 5 Themeansquaredisplacementofthemonomersinthegel
p, leadsto the unusualfeaturesof the meansquaredisplace- (opensymbols)iscomparedwiththatobtainedwhenthespanning
mentplottedinFig.3. clusterisfrozen(fullsymbols).Squares: p=0.2,f =0.52;Circles:
As we have observed above, the diffusive problem in the p=0.25,f =0.55.
gelphasesharessomesimilaritieswiththemuchinvestigated
problemofsingle–particlediffusioninamatrixofdisordered
hardsphereobstacles29–31,wheresingleparticlesareconsid- completelyarrestedbyfreezingthepercolatingcluster,show-
ingthattherelaxationofthesystem iscompletelyduetothe
ered to move in a frozen environment. Conversely, in the
fluctuationsofthepercolatingcluster.Thisresultisrelevantto
present case, monomers and finite clusters diffuse in a fluc-
modeldiffusioninbiologicalsystems,e.g. toinvestigatedrug
tuating environment,as the percolatingclusters may vibrate.
deliveryprocesses.
These fluctuationsaffectthe diffusionof free particlesas we
seeinFig.5,wherethemeansquaredisplacementobtainedin
theactualgeliscomparedwith thatobtainedbyfreezingthe
4.2 Correlationfunctions
percolatingcluster. Thefigureclarifiesthatthefluctuationsof
thepercolatingclusterstronglyinfluencethediffusivityofthe Inthissection,wedescribetherelaxationofthesysteminthe
particles. For instance, at p=0.25, f =0.55, the system is gelphasethroughthestudyofthesISF,Eq. (2). Convention-
4| 1–7
ally, the relaxationtime t ofthe system is definedrequiring
k f
that F(k,t )≡1/e, or a similar small value, and the transi- glass
s k
tionofstructuralarrestisconsideredtooccurwhent reaches 0.55
k
ahighvaluet ,relatedtheexperimentaltimescale. Here,
arrest k = 0.1
wefixt ≡105,andexplicitlyconsiderthedependenceon k = 1
arrest 0.5 k = 3
the threshold used to define the relaxation time. Therefore, est
wederiveak andxdependentstructuralarrestlinef (p), arr
arrest
definedasthevolumefractionatwhichF(k,t )≡x. f 0.45
s arrest
Fig. 6 (top panel) illustrates the dependenceof f on x
arrest
forthreedifferentvaluesofk,at p=0.3. Thefigureclarifies 0.4 f
gel
theexistenceoftwocharacteristicvaluesforf ,f ≈0.4
arrest gel
and f ≈ 0.58. Indeed, regardless of x, f ≈ f at
glass arrest gel 0.35
small k, and f arrest ≈f glass at large k. At intermediate val- 0 0.2 0.4 x 0.6 0.8 1
uesofk,f exhibitsacrossoverfromf tof asxin-
arrest gel glass
creases. Thedecayofthecorrelationfunctionatwavevectors
k <k <k is therefore influenced by both the gel and 0.9 0.58
min max
the glass transitions. The bottom panel of Fig. 6 illustrates 0.8 0.56
the same crossover for different wavevectors, that influence 0.7 0.54
thevalueofxatwhichthecrossoveroccurs. Theglassytime 0.6 0.52
scale affectsthe initialdecayof thecorrelationfunction,and 0.5
x 0.5
it is thus observed for large x, while the gel relaxation cor- 0.48
0.4
responds to the final relaxation, and it is observed for small 0.46
0.3
x. 0.44
DatashowninFig.7furtherclarifythesefindings. Asalso 0.2 0.42
found in Ref.10, we can recognize three different relaxation 0.1 0.4
time scales: tb , due to the rattling of particlesin the nearest 0.1 1
neighbourcage,doesnotdivergeatall;ta ,duetotheopening k
of the cage, divergingat the glass transition line; and finally
t , dueto the relaxationofthe largestcluster, divergingat
perc
thegeltransitionline(dataplottedinFig.7refertof >f , Fig. 6 Weconsiderthetransitionofstructuralarresttooccurwhen
gel
thenFs(k,t)doesnotrelaxtozero,andreachesatlongtimea Fs(k,tarrest)≡x,withtarrest≡105.Thetoppanelillustratesthe
finitevalue). dependenceoff arrestonx,forthreevaluesofk,andp=0.3.The
bottompanelisacolourmapshowingthedependenceoff arrestonk,
for p=0.3.Thefiguresclarifythatatsmallk,f arrest≈f gel,atlarge
5 Interference ofstructural arrestlines k,f arrest≈f glass,whileatintermediatekf arrestexhibitsacrossover
betweenthesetwolimitingvaluesonincreasingx.
Here,weconsidertheinterferenceoftheglassandofthegel
transitionsinamodelofachemicalgel,wherebondsareper-
manent. As we have noticed, the glass transition is a dis-
continuous transition and the gel transition is a continuous
one. Hence, we expect that the model system here studied Inordertoappreciatehowthislogarithmicdecayemerges,we
might have some analogies with the mode coupling theory followtheevolutionofthesISFonincreasingpalongtheglass
for schematic F model2, which gives two arrested lines: a line, in the gel phase. Fig. 9 shows that on increasing p the
13
continuoustransition,whichmeetsa discontinuousone. The valueoftheplateauassociatedtocagemotionoftheparticles
modelpredictthatthediscontinuoustransitionendsonahigh decreases, and the plateaubecomesshorter. On increasing p
ordercritical point(A singularity), where the plateau of the wealsoobservethevalueoftheplateauassociatedtothegel
3
twotransitionscoincide,characterizedbylogarithmdecay1–3 transitiontogrow,andthetimeneededtoreachittoincrease.
ofthe relaxationfunctions. Topologicallythe phase diagram Accordingly,thehigherordercriticalpointwherealogarithm
issimilartotheonefoundhere(Fig. 1). decay is found seems to be associated to the state where the
In our model, due to long relaxation time involved, it is twoplateaucoincide,inagreementwiththeA singularityof
3
rather difficult to localize such singularity. However we do the F model. We stress thatthe highordersingularitymay
13
findevidenceofalogarithmicdecayinaregioninsidethegel occurathighervaluesof pincludingpossibly p=1. Further
phase, close to the glass transition line, as shown in Fig. 8. investigationisnecessary.
1–7 |5
1.0 1.0 1.0
t
b
0.8 0.8 0.8
(k,t)s0.6 t a F(k,t)s00..46 00..46
F 0.4 0.153
0.2
0.3
0.2 0.4 0.2
0.5
0.2
t
perc
0.0 0.0
-2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5
0.0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
10-2 10-1 100 101 102 103 104 105 time time
time
Fig. 9 ThesISF,Fs(k,t),atdifferentvaluesof pasindicated,
Fig. 7 ThesISF,Fs(k,t),fork=3, p=0.3,andf =0.54,larger movingalongtheglassline,fork=kmax(leftpanel)andfork=3
thanf gel≈0.4.Thethreerelaxationprocessesareclearly (rightpanel).Theplateauassociatedwiththegelandtheglassare
distinguishable. indicatedbyanarrow.
1.0
Animportantopenquestionaheadregardstherelevanceof
themodecouplingtheoryforschematicF model2 forthese
13
0.8
systems. Ourresultssupportthisscenario,aswedonotonly
recoverthe interferenceof a continuousand of a discontinu-
) 0.6 ous transition, but we also find a point where the relaxation
t
k, functionseemstodecaylogarithmically,whichshouldbeas-
(
Fs0.4 sociatedtotheA3singularity.However,moreworkisneeded
inthisdirection,particularlytobetterlocalizethishighorder
criticalsingularity.
0.2
Acknowledgement
0.0 -2 -1 0 1 2 3 4 5 The authors would like to thank Dr. M. Sellitto for having
10 10 10 10 10 10 10 10
time drawntheirattentiononthepropertiesoftheF13model.
Fig. 8 ThesISF,Fs(k,t),for p=0.5,f =0.52,andwavevector References
k=1,2,3,4,5,6.28(fromtoptobottom).Continuouslinesare
logarithmicfunctions.
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