Structure and Fragmentation in Colloidal Artificial Molecules and Nuclei

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by  C. J. Olson Reichhardt

EPJ manuscript No.
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0 Structure and Fragmentation in Colloidal Artificial Molecules
0
2 and Nuclei

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a
C.J. Olson Reichhardt, C. Reichhardt and A.R. Bishop
J

1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
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the dateof receipt and acceptance should beinserted later
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o Abstract. Motivated by recent experiments on colloidal systems with competing attractive and repulsive
s interactions, we simulate a two-dimensional system of colloids with competing interactions that can un-
t. dergo fragmentation. In theabsence of any otherconfiningpotential, thecolloids can form stable clusters
a depending on the strength of the short range attractive term. By suddenly changing the strength of one
m
oftheinteraction termswefindarichvarietyoffragmentation behaviorwhichisaffectedbytheexistence
- of “magic” cluster numbers. Suchsoft matter systems can beused toconstruct artificial nuclei.
d
n
PACS. 82.70.Dd Colloids –45.70.Qj Patternformation –05.70.Ln Nonequilibriumandirreversible ther-
o
modynamics
c
[

2 The issue of pattern stability has attracted consider- types [15], and the details are sensitive to the nature of
v
able interest in a wide range of different areas. A com- the trapping potential.
6
monly encountered mechanism for pattern formation is There has recently been growing activity in studying
3
the competition betweenshort-rangeattractionandlong- colloidalsystemswithcompetinglongrangerepulsiveand
5
7 range repulsion, which occurs in many systems including short attractive interactions, where various types of clus-
0 magneticfilms,Langmuirmonolayers,diblockcopolymers ters are observed to form [16,17,18,19,20]. For instance,
6 [1], water-oil mixtures [2], and two-dimensional electron clusters and “artificial atoms” were created in an experi-
0 systems [3]. Similar types of pattern formation are ex- mental system of rotating disks with no confining poten-
t/ pected to occur in dense nuclear matter, where they have tial [21]. Other current experiments use combinations of
a been termed“pasta”phases.These phasesmay occurun- entropic attractive forces with additional repulsive inter-
m
der conditions of stellar collapse [4] or in the crust of a actions to form clusters and study their decay [22], and
- neutronstar[5].Modelsofclassicallyinteractingparticles long-rangerepulsionbetweencolloidshasbeenachievedin
d
with competing attractiveandrepulsiveinteractionshave nonpolarsolvents[23]andwater-oilemulsions[24].Analo-
n
been used to study a variety of topics in nuclear mat- gieshavebeenmadebetweencolloidalclustersandnuclear
o
c ter. For example, if two nuclei collide, they temporarily matter,leadingtoproposalsformimickingnuclearmatter
: formahighdensitystate whichthenexpandsrapidlyand phenomena with a colloidal system [20]. Although there
v
can fracture in a complex manner [6]. Nuclear fragmen- have been several colloidal studies of equilibrium cluster
i
X tation processes in both two-dimensional [7] and three- shapes that have identified magic numbers, to our knowl-
r dimensional[8]geometriescausethenucleustobedecom- edge the fragmentation of colloidal clusters has not been
a posedintoclustersofallsizesthroughmechanismsranging considered previously.
from evaporation to multifragmentation. Here,weformclustersoutofcolloidsthathaveacom-
peting short-range attraction and a long-range Coulomb
Models of small numbers of interacting particles, out- repulsion. We identify the stability line as a function of
side the thermodynamic limit, can be useful in under- the strength of the short-range attraction for clusters of
standingthestructureandstabilityofnaturallyoccurring different sizes. When the attraction falls above this line,
clusters such as nuclei. Repulsively interacting particles clusters exist in the absence of a confining potential. We
can be forced to form clusters by confining the particles thenstudydifferentmechanismsfordestabilizingtheclus-
in a trappotential. A wide variety of systems exhibit this ters,including thermalexcitationandasuddenquenchof
type of behavior, including electrons in quantum dots [9] the attractive term to a lower value. The latter process is
oronthesurfaceofliquidhelium[10],vorticesinsuperflu- inspiredbythe suddenexpansionofanenergizednucleus.
ids [11], colloidal particles in circular traps [12], confined The microscopic details of the decays can be accessed di-
ferromagneticparticles [13], andchargeddust particles in rectly through our simulations. We find numerous decay
plasmatraps[14].Theresultingstructureshavebeenstud- modesdependingontheclustersizeandthequenchdepth.
ied extensively for a range of interaction types and trap This system can serve as a model for understanding the
2 C.J. Olson Reichhardt,C. Reichhardt,A.R. Bishop: Structureand Fragmentation...
0.2
(a) (b)
6 0.15
N
d0.1
Bc5.5 B/c
(a,5) (b,6) (c,7) (d,8) (e,9) d
0.05
5
0
4.5
0 20 40 60 80 0 10 20 30 40 50 60 70 80
N N
(f,10) (g,11) (h,12) (i,13) (j,14)
Fig.2. (a)Dots:CriticalvalueBc forstableclusterformation
versuscluster size N.Dotted line: Fit toBc =4.2+0.5ln(N).
(b) dBc/dN for the same data. Negative values of dBc/dN
indicateclustersthataremorestablethanneighboringcluster
(k,15) (l,16) (m,17) (n,18) (o,19) sizes.
most lattice-like are also the most stable. These “magic”
clusters include N = 7, 12, and 19. Note that the magic
(p,20) (q,22) (r,24) (s,26) (t,30)
cluster N = 12 forms a nonhexagonal segment of a tri-
Fig. 1. Structure of stable clusters for N=(a) 5, (b) 6, (c) 7, angular lattice, so that it is highly stable even though
(d) 8, (e) 9, (f) 10, (g) 11, (h) 12, (i) 13, (j) 14, (k) 15, (l) 16, it does not meet the proposed magic cluster criterion of
(m) 17, (n) 18, (o) 19, (p) 20, (q) 22, (r) 24, (s) 26, (t) 30. N =1+3p(p−1) for integer p [26]. The strong tendency
towardtriangularorderingisincontrasttosystemsofcon-
fined repulsively interacting particles, which tend to form
complexinteractionsbetweenchargedfragmentsinaclus-
ringlike structures instead. The smaller clusters shown in
ter that is breaking apart.
Fig. 1 resemble structures obtained in the cluster phase
We simulate N interacting colloids in a system with
of either a periodic system [27] or one confined to a trap
open boundary conditions. No confining force is applied
[28]. In these systems, interactions with neighboring clus-
tothecolloids,whicheitherremaininaclusterduetothe
ters alter the shapes of largerclusters and can distortthe
colloid-colloidinteractionswhenplacedina stableconfig-
triangular ordering. Thus, the configurations we observe
uration, or scatter under Coulomb repulsion when placed
here for larger unconfined clumps differ from those found
inanunstableconfiguration.Theoverdampedequationof
in confined or periodic systems.
motionforcolloidiisηdr /dt=f =PN f +fT,where
i i j6=i ij To measure the stability of the configurations at dif-
ηisaphenomenologicaldampingterm.Thecolloid-colloid
ferent values of N, we perform a series of simulations in
interaction f = −∇U(r )ˆr for colloids separated by
ij ij ij whichB isinitiallysettoavaluewhichgivesastablecon-
r = r −r consists of a long-range Coulomb repulsion
ij i j figuration, and is then abruptly lowered to a new value.
and a short-range exponential attraction:
We can identify when B < B by observing the cluster
c
break into two or more pieces. In Fig. 2(a) we plot B
U(r)=1/r−Bexp(−κr). (1) c
versus N through N = 80. Configurations with B ≥ B
c
At small and large r the repulsive Coulomb term domi- are stable, while those with B <Bc break apart. We find
nates. The attractive interaction can be varied using the that Bc increases approximately logarithmically with N,
inverse screening length κ and the parameter B. Here we but that there are noticeable fluctuations which are par-
fix κ = 1. As we will demonstrate, clusters are stabilized ticularlypronouncedforsmallN.Thisismoreeasilyseen
above a critical value Bc which is a function of N. The in a plot of dBc/dN, shown in Fig. 2(b). Magic clusters
temperature is modeled as Langevin random kicks with appear as pronounced dips in dBc/dN. This general be-
thepropertieshfT(t)i=0andhfT(t)fT(t′)i=2ηk Tδ(t− haviorissimilartothatobservedinartificialatoms,where
B
t′). To initialize the system, we set B ≥B and allow the a confining potential is present [29].
c
colloid positions to relax into a stationary state in the The structure of our clusters is predominantly trian-
absence of temperature. gular,but we candefine the equivalent ofouter and inner
Due to the attractive component of the colloid-colloid shellsofcolloidsintheclusters.Unlikesystemsofparticles
interaction, stable clusters of colloids form even in the confined in traps, our shells are not circular. The evolu-
absence of a confining potential. We illustrate the stable tionofthe shellstructureis showninFig.3(a),wherethe
clusters obtained for representative values of N ranging number of colloids N in each of the first four shells is
shell
from N = 5 to N = 30 in Fig. 1. In general, the colloids plotted as a function of cluster size N. The second shell
tend to form a small portion of a triangular lattice when- first appears at N = 7, while the third shell arises for
everpossible,notunlikethehypotheticaltriangularstruc- N ≥ 18 and the fourth shell for N ≥ 29. There is some
ture assumed in the nuclear lattice model [25]. Since the tendencyfortheshellstofavoranoccupancyofN =6,
shell
underlyingsubstrateisflat,geometricallynecessarydislo- asseenbythe presenceofastepinN atN =6in
shell shell
cations do not arise.Certain configurationswhich are the Fig.3(a) for allthree ofthe shells.These results resemble
C.J. Olson Reichhardt,C. Reichhardt,A.R. Bishop: Structureand Fragmentation... 3
1 1
15 (a) 10 (b) (a) (b)
0.8 0.8
10 0.6 0.6
Nshell Tm 5 P0.4 P0.4
5 0.2 0.2
0 0
0 0 4.7 4.8 B*4.9 5 4.8 B*5 5.2
0 10 20 30 5 6 7 8 1
N B (c) 15 (d)
0.8
sFhiegl.l(3•.),(sae)coTnhdeshneulml(b(cid:3)e)r,othficrdollsohiedlsl(N(cid:7)s)h,eallndinfotuhrethoushteerllm(o△st) P0.6 M>10
as a function of N. (b) Disintegration temperature Tm as a 0.4 < 5
function of B for B > Bc and five different values of N. •: 0.2
N =3. (cid:3): N =6. (cid:7): N =9. △: N =12. ×: N =15.
0 0
4.8 5 5.2 5.4 4.6 4.8 5 5.2 5.4
* *
B B
measurements in dusty plasma systems [14]. We expect
Fig. 4. (a) Decay modes for N = 5. P is the probability
the shell structure shown here to be generally applicable of observing the mode, and B∗ is the quench depth. •: (15);
tothetypeofcolloid-colloidinteractionwehaveassumed, (cid:3): (13,21); N: (11,22); ⋄: (21,31); (cid:13): (12,31); ×: (11,41). (b)
and not to be parameter dependent. Decay modes for N =12. •: (112); (cid:3): (111,21); ×: (110,22); (cid:7):
We next consider how to destabilize our cluster struc- (16,23); △: (22,31,51); +: (14,21,32); (cid:4): (15,22,31); N: (13,33);
turessothattheyfragment.Firstwestudythermaldesta- (cid:13): (12,32,41); ⋄: (11,31,42). (c) Major decay modes for N =
bilization. Here, we fix B at a stable value B ≥ Bc, and 15. •: (17,24); (cid:3): (16,23,31); N : (15,22,32); ⋄: (14,21,33); ◭:
simulate the system for a fixed number of time steps at (13,21,32,41); ▽: (12,33,41); +: (12,21,32,51); (cid:7): (11,33,51); ×:
each value of T, increasing T until the thermal fluctua- (21,31,52);(cid:13):(11,31,51,61);(cid:4):(12,21,51,61);△:(31,51,71);H:
tions break the cluster apart.We perform a series of runs (11,21,51,71);∗: (12,61,71); ⊳:(11,72);◮:(21,61,71);⊲:(71,81).
∗
at each value of B to obtain an average melting temper- (d)Average multiplicity hMi of thedecay as afunction of B
ature T as a function of B. The results for five different for N =5 (•), 12 ((cid:3)), and 15 (×).
m
cluster sizes are shown in Fig. 3(b). In each case, T in-
m
creaseswithB sincetheattractiveenergyofthe clusteris
increasing. The rate of increase steepens as N increases. numberofclustersoftypei,andtherearemdifferentclus-
The destabilization is triggered by the escape of at least tertypes. We alsomeasurethe averagemultiplicity ofthe
one of the particles from the cluster, and can thus be re- decay at a given value of B, hM(B)i = hPm n (B)i.
i=1 jm
gardedasafirstpassageprocessthatbecomesmorelikely We illustrate the probability P of each decay mode for
asT increases.Inouroverdampedsystem,itisdifficultfor N = 5 as a function of B∗ in Fig. 4(a). Here we observe
a single particle to accumulate enough energy to escape all six possible decay modes, and find two dominant de-
the cluster at low temperatures, unlike the evaporation cay modes: a breakup into one 1-particle cluster and two
processthat canoccurin a massivesystem.This tends to 2-particle clusters, denoted (11,22), or complete disinte-
stabilize the clusters at low temperatures. gration into five 1-particle clusters, denoted (15). For the
The structures can also be destabilized by starting magicclusterN =7(notshown),wefindonlyasinglede-
fromaconfigurationwithB >Bc,andthensuddenlylow- caymodeofcompletedisintegration,(17),unlessB∗ isex-
eringB toavalueB∗ <Bc.Thistypeofprocesscouldoc- tremelyclosetoBc.TheN =7statehassuchstrongsym-
cur after two nuclei collide and form a highly compressed metry that the particles are not able to form subclumps
but unstable structure, which subsequently fragments. In asthey breakawayfromthe mainclump. Fig.4(b) shows
a colloidalsystem, sudden adjustments to the formof the the decaymodes for the magic clusterN =12,illustrated
interactionpotentialmightbe achievedthroughmagnetic in Fig. 5(a). The two dominant decay modes are both
interactions, as was considered for purely repulsive par- symmetric: a breakup into three 1-particle clusters and
ticles in Ref. [30], or through electric interactions, as in three 3-particle clusters (13,33), illustrated in Fig. 5(b),
Ref. [31]. We begin with configurations prepared slightly andcomplete disintegration,(112), shownin Fig. 5(c). At
above Bc and perform a series of quenches at small but B∗ values falling in the transition region between these
finiteT todifferentvaluesofB∗.Wemeasure20quenches tworegimesforthecaseofa clusterheldatfinite temper-
for each value of B∗ in order to accumulate statistics on ature, numerous decay modes occur with no single mode
the resulting fragment configurations. The clusters break dominating.Inthe caseofanasymmetric,nonmagicclus-
into smaller clusters which are stable at B∗. As Bc−B∗ ter such as N = 15, plotted in Fig. 4(c), a series of de-
increases, the number of fragment clusters increases and cay modes appears. Images of representative decays are
the average size of a fragment cluster drops. showninFig.5(d-f).ForB∗ closetoB ,theclusterbreaks
c
Differentclustersshowdifferenttypesofdecaymodes. into two large asymmetric pieces, such as (61,91) shown
Wenotatethesemodesusing(jnj1,jnj2,...jnjm),wherej in Fig. 5(d). As B∗ is lowered, smaller pieces begin to
1 2 m i
is the number of colloids in clusters of type i, n is the appear, as in the case of (21,61,71) in Fig. 5(e). For B∗
ji
4 C.J. Olson Reichhardt,C. Reichhardt,A.R. Bishop: Structureand Fragmentation...
confiningpotential.Wehaveshownthatsomeclustersizes
aremagic,andcorrespondtoadefect-freesegmentofatri-
angularlattice.Thestrengthoftheshort-rangeattractive
term required to stabilize the clusters increases logarith-
mically with the cluster size. Clusters can be melted by
increasing the temperature, or destabilized by suddenly
quenching the attractive portion of the interaction to a
(a) (b) (c) lower value. We find a variety of decay modes in the lat-
ter case, with a structure that depends on the symmetry
ofthe originalcluster.Theseresultssuggestthatcolloidal
systemsofself-confinedclusterswheretheinteractionscan
be controlledmayoffer insightsinto the effects ofinterac-
tions between charged fragments during decay processes
such as multifragmentation for a larger class of systems
including molecular clusters and nuclei.
(d) (e) (f)
We thank W.K. Kegel for useful comments and bringing to
ourattentionsomeoftherecentexperimentalworkoncolloidal
systemswithcompetinginteractions.Thisworkwassupported
bytheU.S.DepartmentofEnergyunderContractNo.W-7405-
ENG-36.
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