Random sequential adsorption of straight rigid rods on a simple cubic lattice

0.25 MB·English

by  G. D. García

Random sequential adsorptionofstraightrigidrods ona simplecubic lattice
G. D. Garc´ıa,1 F. O. Sanchez-Varretti,1 P. M. Centres,2 and A. J. Ramirez-Pastor†2
1Universidad Tecnolo´gica Nacional, Facultad Regional San Rafael,
Gral J. J. De Urquiza 340, C.P. M5602GCH, San Rafael, Mendoza, Argentina
2Departamento deF´ısica,InstitutodeF´ısicaAplicada, UniversidadNacional deSanLuis-CONICET,
Eje´rcito de los Andes 950, D5700HHW, San Luis, Argentina
(Dated:January30,2015)
5
1 Randomsequentialadsorptionofstraightrigidrodsoflengthk(k-mers)onasimplecubiclatticehasbeen
0 studiedbynumericalsimulationsandfinite-sizescalinganalysis. Thecalculationswereperformedbyusinga
2 newtheoreticalscheme,whoseaccuracywasverifiedbycomparisonwithrigorousanalyticaldata.Theresults,
obtainedfor k rangingfrom2to64, revealedthat (i)inthecaseofdimers(k=2), thejammingcoverage is
an q j=0.918388(16). Ourestimateofq jdifferssignificantlyfromthepreviouslyreportedvalueofq j=0.799(2)
[Y.Y.TarasevichandV.A.Cherkasova,Eur. Phys. J.B60,97(2007)]; (ii)q exhibitsadecreasingfunction
J j
5 whenitisplottedintermsofthek-mersize,beingq j(¥ )=0.4045(19)thevalueofthelimitcoverageforlarge
k’s; and (iii) the ratiobetween percolation threshold and jamming coverage shows anon-universal behavior,
2
monotonicallydecreasingwithincreasingk.

]
h PACSnumbers:64.60.ah,64.60.De,68.35.Rh,05.10.Ln
c
e
m INTRODUCTION been studied: linear k-mers (particles occupying k adjacent
- latticesites)[9–13],flexiblepolymers[14,15],T-shapedob-
at Thestudyofsystemsofhardnon-sphericalparticlesisone jectsandcrosses[16],squares[17–20],disks[21],regularand
t star polygons[22], etc. Inallcases, thelimitingorjamming
s ofthecentralproblemsinstatisticalmechanicsandhasbeen
t. attracting a great deal of interest since long ago. In a sem- coverage, q j, strongly depends on the shape and size of the
a depositingparticles.
inal contribution to this subject, Onsager [1] predicted that
m
Inthecaseoflatticemodelsstraightrigidrods(orlineark-
rod-like particles interacting with only excluded-volume in-
- mers),whichisthetopicofthispaper,theRSA problemhas
d teractioncanleadtolong-rangeorientational(nematic)order.
been exactly solved for one-dimensional(1D) substrates. In
n Thisnematicphase,characterizedbyabigdomainofparallel
o molecules,isseparatedfromanisotropicstatebyaphasetran- thislimit,q (t)canbewrittenas[9],
c
sition occurringat a finite critical density. Numerousmodel
[ systemsofanisotropicobjects, rangingfrominorganictoor- q (t)=k texp −u−2k(cid:229)−1 1−e−ju du. (1)
1 ganicandalsobiologicalparticles,wereanalyzedbyusingthe Z0 " j=1(cid:18) j (cid:19)#
v
Onsager’stheory. Anextensiveoverviewofthisfield canbe
4 From Eq. (1), the dependence on k of the jamming cover-
foundintheexcellentreviewsofRefs. [2,3].
7 age can be obtained. Note that q (t) represents the fraction
1 The long-range orientational order disappears in the case oflatticesitescoveredattimet bythedepositedobjectsand,
6 of irreversible adsorption (no desorption), where the distri- consequently,q (t=¥ )=q .
0 j
. bution of adsorbed objects is different from that obtained at In two and three dimensions, the inherent complexity of
1 equilibrium. Thus, at high coverage, the equilibrium state the system still represents a major difficulty to the develop-
0
correspondstoanematicphasewithlong-rangecorrelations, ment of accurate analytical solutions, and computer simula-
5
1 whereasthefinalstategeneratedbyirreversibleadsorptionis tionsappearasaveryimportanttoolforstudyingthissubject.
: adisorderedstate(knownasjammingstate),inwhichnomore Inthisdirection,Bonnieretal. investigatedthedepositionof
v
objects can be deposited due to the absence of free space of linear k-mers (with k ranging between 2 and 512 on a two-
i
X appropriate size and shape. The jamming state has infinite dimensional(2D)squarelattice[10]. Theauthorsfoundthat
r memory of the process and the orientational order is purely thejammingconcentrationmonotonicallydecreasesandtends
a
local[4–6]. to0.660(2)asthelengthoftherodsincreases.
Thephenomenonofjammingplaysanimportantroleinnu- Vandewalleetal. studiedthepercolationandjammingphe-
meroussystemswherethedepositionofobjectsisirreversible nomenaforstraightrigidrodsofsizek on2Dsquarelattices
overtimescalesofphysicalinterest[7]. Therandomsequen- [11] and found that, for values of k between 2 and 10, the
tialadsorption(RSA),introducedbyFeder[8],hasservedasa ratioq /q (beingq thepercolationthreshold)remainscon-
p j p
paradigmformodelingirreversibledepositionprocesses.The stant q /q ≈0.62, regardless of the length of the particle.
p j
main features of the RSA model are: (1) the objects are put Basedonthisfinding,theauthorssuggestedthatbothcritical
on randomly chosen sites, (2) the adsorption is irreversible phenomena(percolationandjamming)areintimatelyrelated.
and (3) at any time only one object is being adsorbed, so Kondrat and Pe¸kalski [12] extended the study of Ref. [11]
that the process takes place sequentially. In this theoretical tolargerlattices(latticesizeL=30,100,300,1000and2500)
frame,anisotropicparticlesofdifferentshapesandsizeshave and longer objects (1≤k≤2000). The results obtained re-
2
vealedthat:(1)asreportedinRef.[10],thejammingcoverage MODEL,BASICDEFINITIONSANDSIMULATION
decreasesmonotonicallyapproachingtheasymptoticvalueof SCHEME
q =0.66(1)forlargevaluesofk;(2)thepercolationthresh-
j
oldisanonmonotonicfunctionofthesizek,showingamin- The followingscheme is usually called standardmodelof
imum aroundk=13; and(3) the ratio of the two thresholds depositionorRSA.Letusconsiderthesubstrateisrepresented
q p/q j is nonmonotonictoo: after initial growth, it stabilizes byathree-dimensionalsimplecubiclatticeofM=L×L×L
betweenk=3andk=7,andthenitgrowslogarithmically. sitesandperiodicboundaryconditions. Inthefillingprocess,
straightrigidrodsoflengthk (withk≥2)aredepositedran-
domly,sequentially,andirreversiblyonaninitiallyemptylat-
Recently, Lebovka et al. studied an anisotropic RSA of
tice. Theprocedureofdepositionisasfollows: (i)oneofthe
straightrigid rodson 2D square lattices [13]. In this model,
threepossiblelatticedirections(x,y,z) andastartingsite are
the vertical and horizontal orientations occur with different
randomly chosen; (ii) if, beginning at the chosen site, there
probabilities,and the degreeof anisotropyofthe system can
are k consecutive empty sites, then a k-mer is deposited on
becharacterizedbyanorderparametermeasuringthediffer-
thosesites. Otherwise,theattemptisrejected. WhenN rods
ence (normalized)between the numberof line segmentsori-
aredeposited,theconcentrationisq =kN/M.
entedintheverticaldirectionandthenumberoflinesegments
Duetotheblockingofthelatticebythealreadyrandomly
orientedinthehorizontaldirection. Theauthorsinvestigated
deposited dimers, the limiting or jamming coverage, q ≡
the effect of k-mer alignmenton the jamming properties. In j
q (t =¥ ) is less than that corresponding to the close pack-
thecaseofisotropicsystems(orderparameterequaltozero),
ing(q <1). Notethatq (t)representsthe fractionoflattice
theresultsobtainedbyLebovkaetal. areinexcellentagree- j
sitescoveredattimetbythedepositedobjects.Consequently,
mentwithprevioussimulationdatainRef. [12].
q ranges from 0 to q for objects occupying more than one
j
site,andthejammingcoveragedependsonthesizeofthede-
TheRSAproblembecomesmoredifficulttosolvewhenthe positedobject[4,9–13].
objectsaredepositedonthree-dimensional(3D)lattices, and Itiswellknownthatitisaquitedifficultmattertoanalyti-
onlyverymoderateprogresshasbeenreportedso far. Inthe callydeterminethevalueofthejammingcoverageforagiven
lineofpresentwork,TarasevichandCherkasovaexaminedthe lattice. Forsomespecialtypesoflattices,geometricalconsid-
percolation and jamming properties of dimers (straight rigid erationsenabletoderivetheirjammingthresholdsexactly[9].
rods with k=2) on 3D simple cubic lattices and found that For systems which do not present such a topologicaladvan-
(1)q =0.2555(1)andq =0.799(2),and(2)incontrastwith tage,jammingpropertieshavetobeestimatednumericallyby
p j
percolationandjammingonsquarelattices,q /q ≈0.32for meansofcomputersimulations.
p j
cubiclattices[23]. Inordertocalculatethejammingthresholds,weintroduce
theprobabilityW (q )thatalatticecomposedofL×L×Lele-
L
mentsreachesacoverageq . Inthesimulations,theprocedure
In the present paper, the study of Tarasevich and
to determine W (q ) consists of the following steps: (a) the
Cherkasova is extended to larger k-mers (2≤k≤64). Us- L
constructionoftheM=L×L×Llattice(initiallyempty)and
ing computationalsimulationsanda new methodintroduced
(b)thedepositionofparticlesonthelatticeuptothejamming
here,thejammingpropertiesofstraightrigidrodson3Dsim-
limitq . Inthelatestep,thequantitym(q )iscalculatedas
j i
ple cubic lattices are studied. The method is based on the
ocaflLcu×laLtio×nLoeflethmeepnrtosbreaabcilhiteysWacLo(qve)rathgaetqa.lAatstiictewciollmbpeodsiesd- mi(q )= 10 ffoorr qq >≤qq j. (2)
j
cussedindetailsinthenextsection, theintersectionpointof (cid:26)
thecurvesW (q )fordifferentvaluesofLgivesanaccuratees- n runsof such two steps (a)-(b)are carriedoutforobtaining
L
timationofthejammingthresholdintheinfinitesystem. The thenumberm(q )ofthemforwhichalatticereachesacover-
main objectivesof the paperare (1) to evaluatethe accuracy ageq ,
and applicability of the new technique to estimate jamming
n
properties, (2) to determine the dependence of the jamming m(q )=(cid:229) m(q ). (3)
i
coverageonthesizeofthedepositedk-mers,and(3)tocom- i=1
pareourresultswiththecorrespondingonesinRef. [23].The
Then, W (q )= m(q )/n is defined and the procedure is re-
workisanaturalextensionofourpreviousresearchinthearea L
peatedfordifferentvaluesoflatticesizes. Forinfinitesystems
ofpercolationandjammingoflineark-mers.
(L→¥ ),W (q )isa step function,being1 forq ≤q and0
L j
forq >q . ForfinitevaluesofL,W (q )variescontinuously
j L
Theoutlineofthepaperisasfollows.InSec.IIwedescribe between1and0, with a sharpfallaroundq . In conclusion,
j
the depositionmodel, along with the simulation scheme and W (q ) appearsas a goodparameterevidencingthe jamming
L
theprocedureusedtocalculatethejammingcoverage.InSec. thresholdanditsdefinitioniscomputationallyconvenient.
III we present the results and discussion of the simulations. Usingasystemoffullyalignedrodsdepositedon3Dcubic
Finally,thegeneralconclusionsaregiveninSec. IV. latticesasanexampleofapplication[25],weshowinthefol-
3

1.0 1.0
Aligned dimers
k = 5 k = 2
0.8 0.8
0.6 j = 0.864656(18) 0.6 * j = 0.918388(16)
WL 0.51
WL WL
0.4 0.4 j = 0.736061(12)
L/k
L = 72
18
L = 96
24
0.2 L = 120 0.2 30
L = 144
36
L = 168
42
0.0 0.0
0.862 0.864 0.866 0.868 0.735 0.736 0.918 0.920
Fig. 2: G. Garcia et al.
Fig. 1: G. Garcia et al.
FIG.1: CurvesofW vsq forfullyaligneddimersonsimplecubic FIG.2:CurvesofW asafunctionofthedensityq fortwovaluesof
L L
latticesofdifferent sizesasindicated. Fromtheirintersectionsone k-mersize(k=2,curvesontheright,andk=5,curvesontheleft)
obtainedq j.Inthiscase,q j=0.864656(18). andlatticesizesrangingbetweenL/k=18andL/k=42.Horizontal
dashed line shows theW∗ point. Vertical dashed lines denote the
L
jammingthresholdsinthethermodynamiclimit.
lowing how to determine the correspondingjamming cover-
agefromtheprobabilitiesW (q ). Thismodeloffullyaligned
L
rodscanbeexactlysolved[9],whichallowsusnotonlytoil-
lustratetheuseofthemethodintroducedhere,butalsototest
itsvalidityandaccuracybycomparingoursimulationresults
with thoseofRef. [9]. Fivek-mersizeswere chosenforthe locatedataverywelldefinedvalueintheq -axesdetermining
comparisonk(=2,3,4,5,6).Then,asetofn=105runswere
thejammingcoverageforeachk.TheprocedureinFig.1was
carriedoutforeachpairkandL/k(=18,24,30,36,42).
repeatedfork(=3,4,5,6)(datadonotshownhereforreasons
TableI:Simulationandtheoreticalvaluesofq (asindicated ofspace)andtheresultsarecollectedinthesecondcolumnof
j
inthetext)forfullyalignedrodsonasimplecubiclatticeand TableI.
krangingfrom2to6.
Thehighaccuracyofthesimulationdatacanbeassertedby
Alignedrods comparisonwithexactresultsforq presentedinRef. [9]. In
j
k q [Eq.(1)] q [W ′s] fact,forafullyalignedsystem,asstudiedhere,thek-mersare
j j L
deposited only along an axis of the lattice, and the jamming
2 0.864665 0.864656(18)
problemreducestothe1Dcase. Inthislimit,q canbecalcu-
3 0.823653 0.823646(16) j
latedfromEq. (1)[9]. Theresultsareshowninthefirstcol-
4 0.803893 0.803892(12)
umnofTableI.Aremarkableagreementisobtainedbetween
5 0.792276 0.792284(15) theoreticalandsimulationdata,validatingtheapplicabilitythe
6 0.784630 0.784637(16) methodintroducedheretocalculatejammingthresholds.
In Fig. 1, the probabilitiesW (q ) are presentedfor a sys-
L
temofaligneddimers(k=2)andseveralvaluesofLasindi- Inthenextsection, wewillanalyzethedependenceofthe
cated. As it can be observedfrom the figure, curvesfor dif- jamming coverage on the size of the deposited objects for a
ferentlatticesizescrosseachotherinauniquepoint,whichis systemofstraightrigidrodsonasimplecubiclattice.
4
SIMULATIONRESULTS:RANDOMSEQUENTIAL
ADSORPTIONOFSTRAIGHTRIGIDRODSONASIMPLE
CUBICLATTICE
The proceduredescribed in Section II was applied to cal- 0.4
culate the jamming coverage corresponding to a system of 16
straightrigidrodsonasimplecubiclattice. k-mersoflength
k = 2...64 were considered, and the finite-size study was 12
performed for lattices ranging between (36×36×36) and
L
(1152×1152×1152) sites. In addition, n=105 runs were 8
carriedoutforeachpairkandL. Thesevaluesoftheparam-
etersallowedustoobtainveryaccuratemeasurementsofthe 4
jammingthresholds,witherrorslessthan0.008%inallcases.
0
In Fig. 2, the probabilitiesW (q ) are shown for two dif-
L (a)
ferentvalues of k (k=2 and k=5 as indicated)and L/k(=
Fig. 3(a): G. Garcia et al.
18,24,30,36,42).Fromasimpleinspectionofthefigure(and
from data do not shown here for a sake of clarity) it is ob-
served that: (a) for each k, the curves cross each other in a
uniquepointW∗;(b)thosepointsdonotmodifytheirnumer-
L
ical value for the different k used (ranged between k=2 to
k=64). InthecasesofFig. 2,W∗=0.509(9)fork=2,and
L
W∗=0.505(9)fork=5;(c)thosepointsarelocatedatvery
L
well defined values in the q -axes determining the jamming
thresholdforeachk,beingq =0.918388(16)fork=2,and
j
q =0.736061(12)fork=5;and(d)q decreasesforincreas-
j j 0.8
ingk-mersizes.
16
Inthecaseofdimers(k=2),thevalueofq obtainedinFig.
j
2 differssignificantlyfrom the value q =0.799(2)reported
j 12
byTarasevichandCherkasova[23]. Duetothemethodology
used in this contribution, our estimate of q j is expected to L8
be more accurate than that reported previously. In order to
reinforce the validity of our result, typical configurations of 4
dimers adsorbed on a simple cubic lattice are shown in Fig.
3: (a) low-density state (q ≈ 0.4), (b) intermediate-density 0
state(q ≈0.8),and(c)jammingstate(q ≈0.918). Different (b)
colorsrepresentdifferentdirections. Clearly, the valueof q Fig. 3(b): G. Garcia et al.
j
informedinRef.[23]correspondstoanintermediatestateand
isfarfromthelimitdensityofthesystem.
TheprocedureofFig. 2wasrepeatedforkrangingbetween
2and64,andtheresultsareshowninFig.4(solidcircles)and
secondcolumnofTableII.Thepointscorrespondingtok=32,
48 and 64 were calculated for three relatively small values
of L/k (12, 18, 24). The jamming coverage decreases upon
increasingk. Atthebeginning,forsmallvaluesofk,thecurve
rapidlydecreases. However,itflattenoutforlargervaluesof
0.918
kandfinallyasymptoticallyconvergestowardadefinitevalue
ask→¥ . 16
Following a similar scheme to that proposed in Ref. [10]
for straight rigid rods on 2D square lattices, the curve of q 12
j
L
vs. kwasfittedtoafunction
8
2.63(5) 5.2(2)
q (k)=0.4045(19)+ − (k≥6), (4) 4
j k k2
beingq (¥ )=0.4045(19)theresultforthelimitcoverageof 0
j
(c)
asimplecubiclatticebyinfinitelylongk-mers.
Fig. 3(c): G. Garcia et al.
Figure 4 includes also the behavior of the percolation
FIG.3: Typicalconfigurationsofdimersadsorbedona(16×16×
5
TableII:Valuesofq ,q andq /q (asindicatedinthetext)
j p p j
forisotropicrodsonasimplecubiclatticeandkranging
from2to64.

1.0 Isotropicrods
0.3
k q [W ′s] q Ref. [24] q /q
j L p p j
0.2 / 2 0.918388(16) 0.2555(1) [23] 0.2777
0.8 p j

j 3 0.838860(14) 0.2129(1) 0.2538
0.1
4 0.780344(16) 0.1800(1) 0.2307
0.6 0.0
0 20 40 60 5 0.736061(12) 0.1555(1) 0.2113
k
6 0.701346(13) 0.1364(1) 0.1945
0.4
7 0.673355(14) 0.1218(1) 0.1809
j( ) = 0.4045(19) 8 0.650282(11) 0.1089(1) 0.1675
0.2 9 0.630901(12) 0.0990(1) 0.1569
p
10 0.614384(15) 0.0901(1) 0.1467
0.0 11 0.600130(10) 0.0831(1) 0.1385
0 20 40 60
12 0.587696(11) 0.0772(1) 0.1314
k Fig. 4: G. Garcia et al.
13 0.576780(14) 0.0714(1) 0.1238
14 0.567044(11) 0.0661(1) 0.1164
15 0.558360(13) 0.0632(1) 0.1128
20 0.525676(18) 0.0478(1) 0.0909
24 0.507750(18) 0.0411(1) 0.0807
FIG.4: Thethresholdsq j (solidcircles)andq p (opencircles)asa
functionofk. Thesolidlinecorresponds ofthebestfitofthejam- 32 0.483360(34) 0.0299(1) 0.0620
mingvaluesasindicatedinthetext.Inset:Ratioq p/q jasafunction 48 0.456071(32) 0.0191(1) 0.0417
ofk.
64 0.440655(34) 0.0143(1) 0.0318
threshold q as a function of k for the system studied here On the other hand, the jamming coverage dependence on
p
thek-mersizehasbeenreported.Inthecaseofdimers(k=2),
(opencircles). Thecorrespondingnumericalvalues,obtained
q =0.918388(16)was obtained. This value differs signifi-
from Ref. [24], are tabulated in the third column of Table j
cantlyfromthevalueq =0.799(2)calculatedbyTarasevich
II. The percolation threshold is a decreasing function of k. j
Combiningtheresultsofq andq , the ratioq /q wascal- and Cherkasova [23]. Due to the methodology used in this
j p p j contribution,ourestimateofq isexpectedtobemoreaccu-
culated (see inset in Fig. 4 and fourth column of Table II). j
ratethanthatreportedpreviously.
As for 2D square lattices, the ratio between percolation and
On the basis of the behavior of q vs. k in the range
jammingthresholdsexhibitsanon-universalbehavior. Inthis j
case,q /q decreasesinallrangeofk. 2 ≤ k ≤ 64, and the best fit to this curve, the expression
p j q (k)=0.4045+2.63/k−5.2/k2wasfound,beingq (¥ )=
j j
0.4045(19)theresultforthelimitcoverageofasimplecubic
latticebyinfinitelylongk-mers.
CONCLUSIONS
Finally, and based on previous findings from our group
[24], the possible relationship betweenpercolationthreshold
Randomsequentialadsorptionof straightrigid k-mersde- and jammingcoveragewas examined. The results showeda
positedona simplecubiclattice hasbeenstudiedbynumer- non-universalbehavior for the ratio q /q , which decreases
p j
icalsimulationsand finite-size analysis. Severalconclusions uponincreasingk.
canbedrawnfromthepresentwork.
On the one hand, a new theoretical scheme to determine
jamming thresholdswas introducedhere. The method relies ACKNOWLEDGMENTS
onthe definitionofthe probabilityW (q ) thata lattice com-
L
posedofL×L×Lelementsreachesacoverageq . Thevalue ThisworkwassupportedinpartbyCONICET(Argentina)
of q can be obtained from the crossing point of the curves under project number PIP 112-201101-00615; Universidad
j
ofW for different lattice sizes. Comparisons with rigorous Nacional de San Luis (Argentina) under project 322000;
L
analyticalresultsdemonstratedthemethod’saccuracy. Universidad Tecnolo´gica Nacional, Facultad Regional San
6
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