Heterogeneities and Topological Defects in Two-Dimensional Pinned Liquids

1.4 MB·English

by  J. -X. Lin

Heterogeneities and Topological Defects in Two-Dimensional Pinned Liquids
J.-X. Lin1,2, C. Reichhardt1, Z. Nussinov1,3, Leonid P. Pryadko2, and C.J. Olson Reichhardt1
1Center for Nonlinear Studies and Theoretical Division,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545
2Department of Physics, University of California, Riverside, California 92521
3Department of Physics, Washington University, St. Louis, Missouri 63130
(Dated: February 6, 2008)
6 We simulate a model of repulsively interacting colloids on a commensurate two-dimensional tri-
0 angular pinning substrate where the amount of heterogeneous motion that appears at melting can
0
becontrolled systematically byturningoff a fraction of thepinningsites. Wecorrelate theamount
2
ofheterogeneous motion with theaveragetopological defect number,timedependentdefect fluctu-

n ations, colloid diffusion,and theform of thevanHovecorrelation function. Whenthepinningsites
a are all off or all on,the melting occurs in a single step. When afraction of thesites are turned off,
J the melting becomes considerably broadened and signatures of a two-step melting process appear.

5 The noise power associated with fluctuations in the number of topological defects reaches a maxi-
mum when half of the pinning sites are removed, and the noise spectrum has a pronounced 1/fα

] structure in the heterogeneous regime. We find that regions of high mobility are associated with
t
f regions of high dislocation densities.
o
s PACSnumbers: 82.70.Dd
.
t
a
m I. INTRODUCTION inthe formofstringlikejumps [13]. Thisresultsuggests
- thatthedynamicalheterogeneitiesaredirectlycorrelated
d
with the motion, creation, and annihilation of topolog-
n Glassy and liquid assemblies of particles in two and
ical defects. Further evidence that in 2D the topologi-
o three dimensions have been shown to exhibit dynami-
caldefectsareassociatedwithdynamicalheterogeneities
c
cal heterogeneities, where the motion of the particles is
[ has also been reported in the recent experiments of Dul-
notuniformbutoccursincorrelatedstringsincertainre-
lens and Kegel on 2D colloidal suspensions, where the
1 gions,whileotherregionsarelessmobile[1,2,3]. Numer-
non-sixfold-coordinated colloids were more mobile than
v ousnumericalsimulationshavefoundevidenceofdynam-
sixfold-coordinated colloids [14].
4
ical heterogeneities near the glass transition [4]. Direct
0 In a related class of systems, glassiness does not arise
observations of correlated regions of motion have been
1 solely from the particle interactions but instead occurs
1 obtained in imaging experiments on three-dimensional due to coupling with an underlying quenched substrate.
0 (3D) colloidal assemblies as the system approaches a
Ithasalreadybeenshownin2Dsystemsofclassicalelec-
6 glassy phase [5, 6], and nonuniform motion has been
trons in the presence of quenched disorder that the par-
0 found in polymer melts [7, 8, 9]. Heterogeneous motion
ticle motion occurs in string-like dynamical structures
/
t has also been observed directly for systems that do not whereachainofparticlesmovespastotherparticlesthat
a
form a glassy phase but that can have a dense liquid re-
m are pinned [15]. These motions are very similar to the
gion near crystallization, including 2D colloids [10] and
string-likedynamicalheterogeneitiesobservedinsystems
- dusty plasmas [11, 12].
d withoutquencheddisorder. Itisimportanttonote,how-
n In recent work on 2D systems of repulsive colloids or ever, that in Ref. [15], where the disorder was simulated
o vorticeswhichformatriangularlatticecrystallinephase, as a collection of randomly located pins, the quenched
c
itwasshownthatattemperaturesjustabovethemelting disorderhadatendencytocreatetopologicaldefectseven
:
v transition, topologicaldefects in the formof dislocations atverylow temperatures[16]. Therefore,the connection
Xi undergo correlated annihilation and creation, giving rise between the dynamical heterogeneities and the topologi-
to a 1/fα noise signal in the time dependent disloca- cal defects was not immediately apparent, and it would
r
a tion density [13]. The 1/f noise also coincides with the be desirable to identify a system in which the amount of
appearanceofdynamicalheterogeneities. Athighertem- heterogeneousmotioncouldbecontrolledsystematically.
peratures, the dynamical heterogeneities disappear and In this work, we propose a model of repulsively in-
the noise spectrum of the fluctuating topological defect teracting particles on a substrate in which the disorder
densitybecomeswhite,indicatingthelossofcorrelations. potential is perfectly commensurate with the triangular
The same system has been studied in the case where the crystalandthereforedoesnotfavorthe creationoftopo-
particles are quenched from a high temperature liquid logical defects. Specifically, we study colloidal particles
phase where there is a high density of dislocations to interactingviaascreenedCoulombrepulsioninthepres-
a low temperature phase where the ground state is a ence of a triangular pinning substrate where the number
triangular lattice. In the low temperature regime the of colloids is commensurate with the number of pinning
dislocations created in the quench annihilate over time. sites. We introduce disorder by shutting off a specified
The particle motion in this annihilation process occurs fraction of randomly selected pinning sites. The crystal
2
1
0.8
6
P
0.6
0.4
(a)
0
-0.5
T
d
/ -1
6
P
d
-1.5
(b)
-2
0 0.5 1 1.5 2 2.5 3
T
FIG.2: (a)Thefractionofcolloidswithsix-foldcoordination
FIG. 1: Location of the pinning sites (circles) and colloids
(black dots). Large dark circles indicate pinning sites with a number P6 vs T for various np. From right to left, np =
1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, and 0. (b) The
finitefp whilesmall gray circles indicatepinsthathavebeen
shut off by setting fp = 0.0. In this image, the fraction of corresponding dP6/dT curves.
sites with finite fp is np =0.5.
1.8
phaseisstabilizedinregionsofthesystemthatcontaina
locallylargedensityofactivepins. Asthetemperatureis
1.6
increased, the melting occurs first in regions with higher
densities of nonactive pinning sites.
h
The system we consider can be realized experimen- widt
tally for colloids interacting with periodic arrays of op- g 1.4
n
tical traps [17, 18, 19, 20, 21, 22, 23]. The melting of elti
charged colloids interacting with triangular and square m
pinning arrays has already been studied experimentally
1.2
[18, 19] and in numerical simulations [20]. Experimen-
tal evidence for a coexistence of a liquid and a solid has
been obtained in a system where colloids located at pin-
ningsitesremainimmobilewhilecolloidsintheunpinned 1
0 0.2 0.4 0.6 0.8 1
interstitial regions are mobile [19]. Related systems that n
p
can be modeled as repulsive particles interacting with
a periodic substrate include vortices in superconductors
FIG. 3: The width of themelting curveversuspinningfrac-
with artificialpinning sites [24, 25] and vortices in Bose-
tion np obtained for thesystem in Fig. 2.
Einstein condensates interacting with optical traps [26].
the low volume fraction limit. A single colloid i obeys
II. NUMERICAL SIMULATION METHOD AND
PARAMETERS the overdamped equation of motion
dR
We model a 2D system of a monodisperse assembly of η i =Fcc+FT +Fs (1)
N colloids using a Brownian dynamics simulation with dt i i i
periodic boundary conditions in the x and y directions.
The equations of motion for the colloidal particles are Here η is the damping constant which is set to
overdamped and we neglect hydrodynamic interactions, unity. The colloid-colloid interaction force is Fcic =
N
whichisareasonableassumptionforchargedparticlesin −q ∇ V(r ), where the colloid-colloid interaction
i i6=j i ij
P
3
(a) (b)
FIG.4: (a)Colloidal trajectories (blacklines)andcolloidal positions(blackdots)overafixedperiod oftimeinasystemwith
np = 0.5 at T = 1.5. (b) Voronoi plot of the system in (a) for a single frame. Small black dots indicate sixfold-coordinated
particles while large black dots indicate particles with coordination numbersother than six.
potential is a screened Coulomb interaction of the form atthesesites. AtT =0thecolloidsformatriangularlat-
tice that is commensurate with the substrate. To deter-
V(r )=(q /|r −r |)exp(−κ|r −r |). (2)
ij j i j i j mine the melting behavior as a function of temperature
Here q is the charge on particle j(i), κ is the inverse we measure the density of sixfold coordinated particles
j(i)
screening length which is set to 3/a, and ri(j) is the po- P6 using a Voronoi or Wigner-Seitz construction. For a
sition of particle i(j). Throughout this study the den- perfectlytriangularlattice,P6 =1.0. Topologicaldefects
sity of colloids is kept fixed at n = 1.0 which gives a such as dislocations produce 5− and 7−fold coordinated
c
colloid lattice constant of a = 1.0. We also fix the sys- particles. In Fig. 2(a) we plot P6 vs T for systems with
tem size to L = 24. Because the colloid-colloid inter- varied pinning fractions of np =0 (no pins active) to 1.0
action is screened, at long distances the force between (all pins active). When thermally induced defects begin
two colloids is negligible; thus, we place a cutoff on the to appear, P6 drops. As np increases, the drop in P6
interaction at 5a. For larger cutoffs we find no change shifts to higher temperatures. The sharpest drop in P6
in the results. The thermal force FT is modeled as occurs at np =0 where none of the pins are active. The
random Langevin kicks with the properties hFTi = 0 dropbecomessteeperagainasnp approaches1whereall
i
and hFT(t)FT(t′)i = 2πηk Tδ(t − t′). The pinning the pins are active. For intermediate fillings the drop in
B
comes from the substrate force Fsi. The pinning sites P6 is broadened. At high temperatures T > 2.5, all of
are modeled as parabolic traps of radius rp and maxi- the P6 curves come together near a value of P6 =0.4.
mum strength f = 2.0 which are placed in a triangular In Fig. 2(b) we plot the derivatives of the curves in
p
array with lattice constant a, commensurate with the Fig. 2(a). For intermediate filling 0.4<n <0.75, there
p
colloidal lattice. As an initial condition, each colloid is are two dips in dP /dT. The first dip at lower tempera-
6
placed inside a pinning site. We turn off some of the tures is associated with the onset of dislocations among
pins by setting f =0 at some sites, keeping f finite at thecolloidsintheunpinnedregions,whiletheseconddip
p p
a fraction n of randomly chosen “active” pinning sites. correspondstothe onsetofdislocationsinthe pinnedre-
p
We gradually increase the temperature up to 3.0 in in- gions. For n < 0.4, the first dip near T = 0.65 is the
p
crements of 0.01. The melting transition is identified by sharpest and the dip shifts to higher temperatures as n
p
examining the density of topologicaldefects and the dif- increases. The magnitude of the first dip decreases un-
fusion. A clean system with all the pinning turned off til n = 0.5, and for n > 0.5 the first dip is lost. At
p p
melts at T =0.6. n =0.5a smallshoulderstartstooccurindP /dT near
p 6
T = 1.75, indicating the onset of the second dip. The
second dip continues to grow in magnitude and shifts to
III. TOPOLOGICAL DEFECTS AND NOISE higher T with increasing n until at n = 1.0 the dip
p p
occurs at T = 2.0. For higher temperatures T > 2.5, all
A. Defect Density and Melting of the dP /dT curves merge.
6
ThedatainFig.2(b)indicatethattherearetwochar-
InFig.1weillustrateasystemwithn =0.5wherewe acteristic disordering regimes. The first coincides with
p
haveturnedoffhalfofthepinningsitesbysettingf =0 the temperature at which the particles located in the
p
4
(a) (b)
FIG. 5: Colloidal trajectories (black lines) and positions (black dots) for a system at fixed T = 1.5 for (a) np = 0.2 and (b)
np =0.8.
unpinned regions effectively melt, and the second corre- ture. In general,for high temperatures atall fillings, the
sponds to the temperature at which the particles in the motion is homogeneous and the defect density is high.
pinnedregionsmelt. Fromthe curvesinFig.2thewidth In Fig. 5(a) we show the trajectories for the case of
ofeachmelting transitioncanbe determined by measur- n = 0.2 at T = 1.5 where the system is mostly in the
p
ingthedistancefromthebeginningofthe dipindP /dT liquid state with a small number of pinned colloids. In
6
to the temperature where dP /dT begins to saturate. In Fig. 5(b) we plot the trajectories for the same tempera-
6
Fig.3 we plotthe melting width vs n , showingthat the tureatn =0.8,wheremostofthesystemispinnedand
p p
width reaches a maximum value at n =0.5. The width a small number of colloids show extra motion at their
p
is smaller for n = 1.0 than for the unpinned case of sites but do not change neighbors.
p
n =0.
p
The system at n = 0.5 and T = 1.5 can be regarded
p
as a mixture of a solid pinned phase and a liquid phase, B. Fluctuations in the Defect Density and Noise
and thus the motion and diffusion of particles is highly
heterogeneous. At np = 0 and T = 1.5 in Fig. 2(a), We next consider how the filling fraction affects the
P6 = 0.5, indicating that at this temperature the sys- timedependentfluctuationsoftopologicaldefectdensity.
temisinastronglydisorderedliquidstate. Forthesame InFig.6 we plotthe time seriesofthe density ofsix-fold
T =1.5atnp =1.0,P6 =1.0,indicatingthatthesystem coordinated particles, P6(t), in a system with np = 0.5
isacompletely triangularsolid. Incontrast,fornp =0.4 forT =1.5(uppercurve)andT =4.0(lowercurve). For
atT =1.5,P6 =0.67. In this case,it wouldbe expected T = 4.0, the system is strongly disordered with hP6i =
that the colloids located at pinning sites that have been 0.36,whileatT =1.5thesystemis inthe heterogeneous
turned off should have a liquid like behavior, while the phase and P varies from 0.86 to 0.68. The fluctuations
6
colloids at the active pinning sites should behave like a in the defect density are much more rapid for T = 4.0
solid. Thedislocationsandfluctuationsinthedislocation than at T = 1.5. For much longer time series at T =
density shouldthen be associatedwith the liquidlike un- 1.5, there are additional long time fluctuations with P
6
pinned regions. risingashighas0.9onoccasion. Inordertoquantifythe
InFig.4(a)weplotthecolloidtrajectories(blacklines) fluctuations in the defect density, we compute the power
and colloidal positions (black dots) for a fixed period of spectrum of the time series,
time for a system with n = 0.5 and T = 1.5, showing
p
a highly heterogeneous motion of particles in correlated 2
S(f)= P (t)e−2πiftdt . (3)
groups where the pinning is deactivated. In Fig. 4(b) (cid:12)Z 6 (cid:12)
(cid:12) (cid:12)
we illustrate the corresponding Voronoi construction for (cid:12) (cid:12)
a single frame indicating the locations of the topological The noise power S i(cid:12)s the value of S(f(cid:12)) averaged over
0
defects as nonsixfold coordinated particles. In general, a specific frequency octave. Fig. 7(a) shows S(f) for
the dislocations are located in the same regions where T =1.5whereaclear1/fαsignalappearswithα=1.45.
the correlated particle motions are occurring. InFig.7(b)weplotS(f)forthe samesystematT =4.0
Theamountofheterogeneousmotionthatappearsde- wherethenoisespectrumisclosertowhitewithα=0.1.
pends on both the filling fraction n and the tempera- Forintermediatetemperaturesαgraduallychangesfrom
p
5
-4
0.9 10
0.8 10-5
0.7 S(f)10-6
0.6 -7
10
P6 (a) (b)
0.5 -8
10
0.001 0.01 0.1 0.001 0.01 0.1
f f
0.4
0.3 FIG. 7: Power spectrum S(f) for the same system in Fig. 6
at (a) T =1.5 and (b) T =4.0.
50000 100000 150000 200000
Time 1.0×10-4
FIG. 6: Time series of the fraction of colloids with six-fold
coordination, P6, for a system with np = 0.5. Top curve,
T =1.5; lower curve,T =4.0. 7.5×10-5
1.45 to 0.1 or a white spectrum, indicating that there is S05.0×10-5
little or no correlation in the defect fluctuations at the
higher temperatures. We note that for the case where
there is no pinning, previous studies found dynamical 2.5×10-5
heterogeneities occurring just above the melting transi-
tion [13]. In these studies, similar 1/fα dislocation den-
sity noise appeared in this regime; however, the value of 0.0
0 1 2 3 4
α had a maximum of 1.0. The larger value of α that we T
observe here implies that for the pinned system, there
arestrongercorrelationsinthe annihilationandcreation FIG. 8: Noise power S0 vs T for a system with np = 0.0
ofthe dislocationscomparedto the unpinned system. In (squares) and np =0.5 (circles).
Fig. 8 we plot the noise power S vs T for systems with
0
n = 0 (squares) and n = 0.5 (circles). The clean sys-
p p
tem shows a maximum noise power at T = 0.75 and peratureswherethemotionisuniform,thecorrelationin
then a slow drop in noise power for higher temperatures the noise spectrum is lost.
as observed in previous simulations [13]. For the case of
n =0.5,thenoisepowerbeginstoincreasenearT =1.0
p
andreachesamuchhighermaximumnearT =1.8before C. Effect of Substrate Strength
decreasing at higher temperatures. The noise power for
thedifferentfillingsbecomesequalnearT =2.5,whichis We next consider the effects of varying the pinning
also the temperature at which the dP6/dT curves merge strength of the substrate. In the previous analysis the
in Fig. 2(b). For higher filling fractions, the peak in the pinning strength was fixed at f = 2.0 so that there
p
noise power shifts to higher T and there is a slight de- wasacleardistinctionbetweenthemeltingofthepinned
crease in the maximum value of the noise power. Al- species and of the unpinned species. When f is var-
p
though the width of the noise power peak is broadened ied, the width of the heterogeneous regime as a function
forthe np =0.5casecomparedtothe np =0case,wedo of T can be increased or decreased. We perform a se-
not observe two peaks, which would be indicative of two ries of simulations with fixed n = 0.5 and varied f .
p p
melting transitions. In Fig. 10 we plot the temperature at which the sys-
In Fig. 9 we show how α evolves as a function of tem- tem enters the liquid phase (squares) vs f . This tran-
p
perature for a system with n = 0.5 as obtained from sition is defined as the point where P saturates to a
p 6
the power spectrum. The peak value of α occurs at a value near 0.4. We also plot the temperature at which
temperature of T = 1.5 which also corresponds to the the first topological defects appear (circles) vs f . For
p
peak in the noise power S in Fig. 8. As T increases, f > 0.4, the temperature at which defects first appear
0 p
there is a slow fall off to a white noise spectrum with saturates at T ≈ 1.14, while the transition into the liq-
α ≈ 0 at T = 4.0. This result shows that in the regime uid phase shifts to higher temperatures with increasing
wherethemotionishighlyheterogeneous,thenoisespec- f . The saturation of the dislocation onset line delin-
p
trumisbroad,indicatingcorrelationsinthecreationand eates the crossoverto the strong pinning limit, and indi-
destruction of the topological defects. At the high tem- catesthateventhoughthepinningstrengthisincreasing,
6
1.5 4
3.5
3
Liquid
2.5
1
2
α T1.5 Heterogeneous
0.5
1
Solid
0
0 1 2 3 4
0.01 0.1 1
T
f
p
FIG.9: Theexponentα,obtained from thepowerspectrum,
vsT for thesame system in Fig. 6 with np =0.5. FIG. 10: Diagram of the different regimes for T vs pinning
strength fp at fixed np = 0.5. Circles indicate the onset of
the first topological defects. Squares show the temperature
the particles located in the non-pinned regions melt at a at which P6 saturates to a valueof 0.4.
constant temperature. The melting of the non-pinned
speciesoccurswhenthethermalmotionisstrongenough
ing temperature. Once the first dislocations appear, the
toovercometherepulsivecolloid-colloidinteractionforce.
pinning in this regime is not strong enoughto create the
Sincethecolloidinteractionstrengthisnotchangingasa
two step melting found above f =0.4.
functionoff ,the unpinnedcolloidmelting temperature p
p
saturates. The colloids located at active pinning sites
can only melt when the thermal fluctuations are strong
IV. DIFFUSION MEASURES
enough to enable the particles to hop out of the pinning
sites. As f is increased, the activated hopping temper-
p
ature also increases. For f < 0.4, in the weak pinning A. Average Diffusion Coefficient
p
regime, the melting does not occur in a two step fashion
but instead occurs in a single step, similar to the case We next consider diffusive measures. We calculate
of f = 0.0. At f = 0.0 there is still a finite window of the distance traveled by the colloids during a fixed pe-
p p
temperaturefallingbetweentheonsetofdislocationsand riod of time δt = t − t . The diffusion is given by
1 0
the saturationofP . Inthis case,the motioncanstillbe D =(r(t )−r(t ))2/δt for fixed δt. Based on the initial
6 1 0
heterogeneous, as has been previously studied; however, colloidpositions,wecandistinguishbetweentheinitially
the dynamical heterogeneities for f =0 are not located pinned and initially unpinned colloids, and we measure
p
at specified regions but are moving over time so that all the diffusion of the two species separately. In Fig. 11(a)
theparticlestakepartinthemotionoverlongtimes[13]. we plot D vs T for a systemwith f =2.0 and variedn
p p
This is in contrast with the finite f case where, due to forthecolloidsthatwereinitiallyplacedinsitesthathad
p
the existence of pinning sites, only certain particles take f = 0. In Fig. 11(b) we plot D for the same system in
p
part in the motion. Fig.11(a)fortheparticlesthatwereinitiallyplacedatac-
An interesting effect indicated by Fig. 10 is that, up tivepinningsites. FortheunpinnedcolloidsinFig.11(a),
to f = 0.4, pinning can effectively increase the melting for n =0.0 at low temperatures up to T =0.6, there is
p p
temperatureoftheentirelatticebyupto∆T =0.3,while aninitial increaseof D with increasingT. This is due to
for f > 0.4, the two step melting scenario occurs. This the factthat the thermalforcescausethe colloidsto rat-
p
increase in melting temperature for weak pinning occurs tleinsidethecagingpotentialcreatedbytheinteractions
since even at n = 0.5, the triangular colloidal lattice is with the other colloids; however, the system maintains
p
still commensurate with the triangular substrate. The triangular ordering and the particles do not hop out of
thermal fluctuations that the colloids experience origi- their initial locations. There is a subsequent sharper in-
nate both from the applied thermal Langevin force and creaseinD forT >0.6whenthesystemmeltsandtopo-
also from the disordered motion of the surrounding col- logical defects proliferate. Above the melting transition,
loids. Since the colloids that are at the pinning sites are D continues to increase with T as the particle can move
more constrained, they fluctuate less than unpinned col- a greater distance during δt at the higher temperatures.
loids. This leads to an overall reduced fluctuating force For higher values of n in Fig. 11(a),the initial values of
p
on all the colloids and results in an increase of the melt- DforT <0.6aremonotonicallyshifteddown,indicating
7
100 (a) (b)
1.6
10
1 1.4
D
0.1
1.2
0.01
T
1
0.001
1 2 3 1 2 3
0.8
T T
FIG. 11: (a) Diffusion D of the initially unpinned particles 0.6
vs T for a system with fp = 2.0 and np = (left) 0, 0.1, 0.2,
0 0.2 0.4 0.6 0.8 1
0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 (right). (b) D vs T for the n
same system in (a) for the initially pinned colloids at np = p
(left) 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 (right).
FIG. 12: Open circles: Temperature at which the diffusion
noticeablyincreasesfortheunpinnedcolloidsfromFig.11(a).
Blacksquares: Temperatureatwhichthediffusionnoticeably
thatthe pinnedcolloidsareexertingastrongerconfining increasesforthepinnedcolloidsfromFig.11(b). Dashedlines
force on the unpinned colloids. As np is increased, the are linear fits which both havethesame slope.
sharpincreaseinDdenotingthemeltingoftheunpinned
particles is shifted to higher T, consistent with the mea-
surements of P from Fig. 2. This jump broadens as
6 same value, as was the case for Fig. 11(a).
n increases. For fillings of n = 0.3 to 0.7, D shows
p p
a plateau like feature followed by an additional smaller In Fig. 12 we plot the temperature at which the diffu-
increase at a higher temperature which corresponds to sion noticeably increases for the pinned particles (black
the temperature at which the particles in the pinning squares)and the unpinned particles (open circles)vs np.
sites melt. The derivative of dD/dT shows a single peak The temperature at which the unpinned colloids begin
at n = 0.0 and 0.9 and two peak like features around to diffuse significantly is lower than the temperature at
p
n = 0.5. The plateaulike feature in D at intermediate whichthepinnedparticlesbegintodiffuse. Thediffusion
p
fillings occurs because even though the unpinned parti- onsettemperaturefortheunpinnedcolloidsincreaseslin-
cles have melted, their motion is confined to the regions early with np up to np = 0.75. For np > 0.75, the dif-
where there is no pinning. This limits the magnitude of fusion onset temperature jumps to a value close to the
the long time diffusion. Once the colloids at the pinning diffusiononsettemperature for the pinned colloids. This
sites also melt, the unpinned colloids are no longer con- is due to the fact that for np >0.75 most of the colloids
fined and can diffuse freely. At higher temperatures all at unpinned sites are completely surrounded by pinned
of the diffusion curves come together. colloids, and the time scale for an unpinned colloid to
hop to another site becomes much longer.
Figure 11(b) illustrates D vs T for the colloids that
were initially placed in active pinning sites. There is an For the pinned particles, the diffusion onset temper-
initial slow increase in D for T < 1.0 which is similar ature also increases linearly with n for n > 0.4. For
p p
to the slow increase seen in Fig. 11(a). In this case D n < 0.4 the diffusion onset temperature remains fixed
p
for the pinned colloids has a much smaller value than D at a constant value. We note that for the lower pin fill-
for the unpinned colloids. This is due to the confining ings, the pinned colloids are mostly surrounded by un-
forceexertedbythepinningwellsonthe pinnedcolloids. pinned colloids. The region in which a clearly defined
For n =0.1, the colloids at the pinning sites only begin two-step melting transition occurs is 0.4 < n < 0.75,
p p
to jump out of the wells at T > 1.0, in contrast to the which is consistent with the values of n at which two
p
caseinFig.11(a)wherethejump indiffusionfortheun- peaks occur in dP /dT and dD/dT.
6
pinned particles occurs at much lower temperatures. As
n increases, the temperature at which the diffusion of
p
pinned colloidsbegins alsoincreases. This indicates that
the motion of the unpinned particles affects the melting
B. van Hove correlation function
of the pinned particles by providing an extra effective
thermalnoise. Wenotethatpinnedcolloidslocatednear
regions of unpinned colloids do not experience the same Another measureusedto identify andanalyze dynam-
fluctuating interaction force as pinned colloids that are ical heterogeneities is the self part of the van Hove cor-
surroundedmostly by pinned colloids. At high tempera- relation function G. This measure gives the probability
tures,allofthecurvesinFig.11(b)cometogethertothe distributionthataparticlehasmovedadistancerduring
8
100 that corresponds to the colloids in the liquidlike regions
(a) (b)
which can travel a much larger distance. We plot G for
-1
10 T = 1.0 and n = 0.1 in Fig. 13(c). There is a much
p
G10-2 smaller narrow peak at x = 0, consistent with the fact
thatamuchsmallerfractionofthecolloidsispinnedand
10-3 thus fewer colloids have low mobility. In Fig. 13(d) we
show the case for n = 0.9 at the same temperature,
p
-4
10 where the peak at x = 0.0 is high and the wide part
(c) (d)
of the distribution is smaller. For all n at T > 3.0,
-1 p
10
the distribution function appears very similar to the one
G10-2 shown in Fig. 13(a), with increased width.
-3
10
-4
10
-1 0 1 -1 0 1 V. CONCLUSION
x x
We have studied the dynamic and topological hetero-
FIG. 13: The self part of the van Hove correlation distribu-
tion function for the x direction for fp = 2.0, T = 1.0, and geneitiesina2Dsystemofinteractingparticleswithpin-
(a) np =0.0, (b) np =0.5, (c) np =0.1, and (d) np =0.9. ning. The number of colloids is fixed and is commensu-
rate with a triangular pinning substrate. By shutting
off a fraction of the pinning sites randomly, we can con-
a fixed time interval t: troltheamountofheterogeneousmotion. Forsufficiently
strong pinning, there can be a two step melting process
N
in which the colloids in the unpinned regions melt first
G(r,t)=N−1 δ r−r (0)+r (t) . (4)
(cid:28) i i (cid:29) followed by the colloids in the pinned regions. The two
Xi=1 (cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1) step melting appears as a double peak in the derivative
Systems with dynamical heterogeneity have non- of the density of six-fold coordinated particles. The mo-
Gaussian average velocity distributions. Experimental tionattemperaturesbetweenthetwomeltingtransitions
studieshaveshownthatinaliquidstatewithoutdynam- is heterogeneous, and the topological defects are associ-
icalheterogeneities,thismeasureproducesasingleGaus- ated with the more mobile unpinned regions. The cre-
sianfit. Intheheterogeneousphase,thereisextraweight ation and annihilation of the topological defects in the
atlargerdistancesandadoubleGaussianfitcanbeused mixed liquid-solid phase shows a prominent 1/fα power
[14]. The Gaussian distribution for the short distances spectrum. The noise power reaches a much higher value
canbeinterpretedascorrespondingtoslowparticlesthat in the mixed phase than in a system with no pinning.
arelocatedinregionswithlowmobility,whilethesecond For weaker disorder, the two step melting is lost. Signa-
wider Gaussiandistributionthat fits the largerdistances tures of the heterogeneous motion can also be observed
corresponds to faster particles located in regions with in the van Hove correlation function, which is composed
higher mobility. of two overlapping Gaussian distributions. In the high
In Fig. 13(a) we plot the self part of the van Hove temperature homogeneous phase, only a single Gaussian
correlationfunctionGcomputedalongthexdirectionfor distribution appears. Our results suggest that the dy-
asystemwithn =0.0andT =1.0. Thissystemisinthe namicallyheterogeneousregionsinwhichthecolloidsare
p
liquid state, since the initial disordering temperature for more mobile may be associated with regions that con-
n =0occursatT =0.6. Here,wefindasingleGaussian tainalargernumberoftopologicaldefectsandwhichare
p
distribution,indicatingisotropictransport. InFig.13(b) locally molten. We also predict that the temporal fluc-
we show G for T = 1.0 and n = 0.5, where the system tuations of the density of defects in regions that show
p
is in the heterogeneous regime. Two features appear. dynamical heterogeneities will have a 1/fα noise signa-
There is a peak near x = 0.0 which corresponds to a ture, and that at higher temperatures where the hetero-
group of particles that are much less mobile during the geneitiesarelost,thesystemwillhaveawhitenoisespec-
time frame of the measurement. These are the colloids trum. Thisisthefirststudyofdynamicalheterogeneities
located at the pinning sites, which undergo only a small in a system with quenched disorder.
amount of motion within the pinning sites, producing We thank M. Dykman for helpful discussions. This
a peak in G with very narrow width. Superimposed on work was supported by the U.S. DoE under Contract
thisnarrowpeakisasecond,widerGaussiandistribution No. W-7405-ENG-36.
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