Table Of ContentPhase behavior of repulsive polymer-tethered colloids
∗
Behnaz Bozorgui, Maya Sen, William L. Miller, Josep C. Pa`mies and Angelo Cacciuto
Department of Chemistry,
Columbia University, 3000 Broadway,
New York, New York 10027
(Dated: January 21, 2010)
We report molecular dynamics simulations of a system of repulsive, polymer-tethered colloidal
particles. We use an explicit polymer model to explore how the length and the behavior of the
polymer(idealorself-avoiding) affect theabilityoftheparticlestoorganize intoorderedstructures
when thesystem is compressed to moderate volumefractions. Wefindavariety of different phases
0
whose origin can be explained in terms of the configurational entropy of polymers and colloids.
1
Finally, we discuss and compare our results to those obtained for similar systems using simplified
0
coarse-grained polymer models, and set thelimits of their applicability.
2
n
a I. INTRODUCTION plex networks between the particles via linker-mediated
J
dsDNA-dsDNAinteractions[19–21],noexplicitattractive
1 forces are introduced in our system. As a result, any
Understanding how colloidal particles spontaneously
2
phase described in this paper will be mostly driven by
organize into ordered macroscopic aggregates is a long-
a nontrivialbalance between the configurationalentropy
] standing challenge that has recently acquired an extra
ft degree of complexity. In fact, advances in particle syn- of the colloids and that of the chains.
o thesis [1–5] have opened the way to the production of In this paper we use molecular dynamics simulations
s to understand the phase behavior of a system of repul-
colloidalparticlesthatareanisotropicboth inshape and
.
t surface chemistry. This provides an unlimited number sive,polymer-tetheredcolloidalparticles. Specifically,we
a
consider a system in which each colloid is connected to
m of building blocks that can spontaneously assemble into
one of the end groups of a single polymer, and we study
an unprecedented variety of structures with potentially
-
how different structures emerge depending on the poly-
d novel functional, mechanical, and optical properties.
n The effect of the anisotropy of nanoparticles on their mer length. Furthermore, we explicitly analyze both the
o case of ideal and self-avoiding polymers.
macroscopic ordering can be addressed in terms of (a)
c Capone et al. [22] have recently analyzed the phase
the form of the inter-particle interaction, and (b) their
[
shape. Not surprisingly, for both cases there is ample behavior of model of di-block copolymers. Their study
2 evidence (see for example [6–15] and references therein) bears similarities with our work, as some of the self-
v assembledstructuresarecommontobothsystems. How-
of a strong correlation between the physical properties
3 ever, the two systems differ in the way the polymers
of the components and those of the resulting aggregates.
0
are modeled. Here we use an explicit beads-and-springs
This phenomenology must be thoroughly explored as it
1
model,whilesphereswithasoftpotentialwereemployed
4 mayleadthe wayto a rationaldesignofthe components
by Capone et al. Though computationally expensive,
. to target desired macroscopic structures.
9 our model enables us to gain a detailed understanding
Herewefocusontheroleofparticleshape. Specifically,
0
of the mechanisms behind the nontrivial phase behavior
9 we study the phase behaviorofa particularlyinteresting
emerging in this system, and this choice will turn out to
0 class of deformable particles that is obtained by graft-
: ing a single long chain to a each colloid. What makes be quite critical when considering the case of non-ideal
v polymers. To the best of our knowledge this is the first
this hybridcolloid intriguing is that, because of the flex-
i
X ibility of the polymer, the overall shape of the particle computational study that explicitly accounts for the in-
ternal degrees of freedom of the polymer for this partic-
r is not fixed, but canbe spontaneouslyaltereddepending
a ular system.
onthespecificthermodynamicstatesimposedonthesys-
tem. Afewexperimentalrealizationsofparticlescompat-
ible with our model have recently been synthesized [16–
II. MODEL
18]. The dual nature of these nanoparticles may open
the door to exotic self-assembled structures that are not
typically seen in systems of nanoparticles with intrinsic We model the polymer-tethered colloids as a polymer
(invariable) shape, and its key elements are very similar of N +1 monomers, with N monomers of diameter σ1,
to those of asymmetric diblock copolymers. and monomer N +1, representing the colloidal particle,
Unlike recent experimental and theoretical studies on of diameter σ2. See Fig. 1 for a depiction ofthe particle.
particles coated with dsDNA, which can form com- In this model the Nth monomer is not constrainedto be
at a specific location on the surface of the colloid, but
canfreelydiffuseonit;constrainingthismonomerwould
yield the same equilibrium properties.
∗Electronicaddress: [email protected] Excluded volume interactions between any two par-
2
N+1 dimensionless ∆P was typically set to 0.01, and was re-
fined or extended depending on the distance from the
1 2 3 N−1 N
transition point. This procedure ensures that the chains
have the time to fully equilibrate. The relatively short
chain lengths considered in this study, and the relative
σ1/2 σ1 low densities at which most of the phases occur, result
in a system that does not have a pathological dynami-
σ
2 cal behavior. Therefore, the very slow pressure anneal-
ing described above, together with the monitoring of all
FIG. 1: Schematic representation of our model for a hybrid,
the observables considered in this study, including the
polymer-grafted colloid. The first N monomers of diameter
σ1 represent the chain, whereas the N +1st monomer of di- average cluster size and its distribution where sufficient
ameterσ2 representsthecolloidalparticle. Bothparticlesare to establish equilibrium. Moreover, we checked that we
assumed to haveequal mass. could reproduce all the phases starting from completely
different initial configurations.
In our study we considered tethers with a minimum
ticles in the system are enforced via a purely repulsive of N = 5 and a maximum of N = 300 monomers, and
shifted-truncated Lennard-Jones potential colloidsofdiameterrangingfromσ =2σ toσ =18σ .
2 1 2 1
All of our simulations are carried out using a total of
512hybridcolloidsatroomtemperature,andthelongest
12 6
σ σ 1
UE(r)=ǫ i,j − i,j + , ∀ r ≤21/6σsimulations took about six months of computer time on
i,j i,j r r 4 i,j i,j
"(cid:18) i,j(cid:19) (cid:18) i,j(cid:19) # an Intel Xeon X5355 2.66GHz processor. In all simula-
(1) tions we set the time step to ∆t = 0.015τ . For each
0
Theindicesi,j ∈{1,2}indicatetheidentityofthepar- pressure annealing step, ∆P, simulations were run for a
ticle(polymerorcolloid,respectively.) σ ≡(σ +σ )/2. minimum of 106 (for the small chains) to a maximum of
i,j 1 2
r isthedistancebetweenthecentersofmassofanytwo 108 timesteps (for the long chains.)
i,j
particles. Finally, ǫi,j = 10kBT ∀ i,j when considering Everyobservablereportedinthispaperisexpressedin
self-avoiding polymers, and we set ǫ = 0k T for the dimensionless units.
1,1 B
case of ideal polymers.
In each hybrid colloid, particles are linearly connected
via the harmonic spring potential III. RESULTS
US =k (r −r0 )2 (2)
i,i+1 s i,i+1 i,i+1 Apart from the harmonic potential, which serves a
where k = 150k T is the spring constant, and r0 = purely structural purpose by enforcing connectivity be-
s B i,i+1
σ +σ /2 is the equilibrium distance. tween the different components of our hybrid colloid,
i,i+1 1
We performNPT moleculardynamics simulationsus- there are no attractive interactions in our system. As
ing the LAMMPS (Large-scale Atomic/Molecular Mas- a consequence, the free energy is dominated, at the low
sively Parallel Simulator) package [23]. Pressure and concentrations considered in our study, by the configu-
temperature are kept constant by means of a Nos´e- rational entropy of its components. Although the con-
Hooverthermostat[24]andbarostat[25]withadditional figurationalentropyis,strictly speaking,associatedwith
drag terms, with coefficients ξT = 1τ0−1 and ξP = 1τ0−1 hardpotentials,wehavechosenalargevalueforks toen-
respectively (τ is the reduced time unit), to damp the surethatbondsareveryclosetotheirequilibriumlength,
0
dynamics and suppress large temperature and pressure and have also run a few simulations with a stricter ex-
oscillations. cluded volume constraint by setting ǫi,j = 500kBT. We
The simulation box is a cuboid with periodic bound- findnodiscernibledifferencebetweenthetwocasesunder
aryconditionsand,forpressurecontrol,weusedecoupled several thermodynamic conditions.
box lengths in each of the three Cartesian coordinates. Whatfollowsarethephasediagramsforidealandself-
This allows box aspect ratios to vary slightly to accom- avoiding tethers as a function of the volume fraction of
modate crystalline structures. colloids, φ ≡ πσ23Nc/(6V), and the effective polymer-
The system initial configurations are prepared by per- colloid size ratio, which we define as α = 2Rg/σ2.
forming NVT simulations in the gas phase. Once the Nc = 512 is the number of colloids, V the volume of
system is equilibrated, and the initial pressure P0 is ex- the simulation box, and Rg is the radius of gyration of
tracted from the thermalized configurations, we slowly a polymer tether, which scales as Rg ∼ (N)1/2 for ideal
ramp the pressure to the desired value P starting from polymers and as R ∼ N3/5 for self-avoiding ones. The
1 g
P (all pressures referred in this paper are rescaled with calculated bulk prefactors for our models are 0.57 and
0
respect to the colloidal interaction energy ε and the 0.60 for ideal and self-avoiding chains respectively. Ex-
22
colloidal diameter σ ). Each subsequent simulation per- perimentally, one can easily control the behavior of the
2
formed at a constant pressure P starts from the ther- polymer by altering the properties of the solvent. For
i
malized configurationat pressurePi−1 (Pi−1 <Pi). The instance, the polymer will behave ideally at the solvent
3
θ point, and as a SAW at larger temperatures. ments [26],
2
R
g
∆f ∝n , (3)
R
Ideal chains (cid:18) m(cid:19)
where R is the radius of the micelle, from which we
m
Figure 2, obtained using several combinations of col- estimatethatthepressureofthesystemshouldscalewith
loidalradii andchain lengths, shows the different phases micellar radius as
arising from the organization of the particles in the sys-
tem as a function of volume fraction for different val- Rg2
P ∝N n , (4)
ues of α, and presents several interesting features. The m R5
m
lines area guide to the eye,and are found by identifying
thethresholddensitiesandsizeratiosabovewhichphase where Nm is the number of micelles forming the crystal.
change occurred. Figure3showshowalldatacollectedfordifferentcom-
For α sufficiently small, α . 1, the presence of the binationsofcolloidalradiiandpolymerlengthsinthemi-
tethers does not alter the ability of the colloids to crys- cellarcrystalphasecanindeedbecollapsedintothesame
tallize into a macroscopic FCC crystal once the system mastercurvewhenproperlynormalized. A powerlaw fit
is compressedaboveathresholdvolumefraction. This is to the data, i.e. the pressure P imposed in our simu-
exactly how tether-free colloids crystallize under analo- lationsandthe correspondingmeasuredaveragemicellar
gous conditions, and is achieved in our system by chain radius,yieldsapressuredependanceonRm,P ∝Rm5.4(2),
localizationinto either the interstitial space between the which is consistent with the eq. 4 for large values of R .
m
colloids (for very small α) or into crystal vacancies as Clearly, our theory breaks down at very large densities,
depicted in Fig. 2A. This is only possible as long as the i.e. small micellar radii, where long tethers begin to
chains are short enough to fit within a vacancy without radiate out of the micellar cores. This happens when
exerting a significant amount of pressure arising from themainmechanismofmicellarshrinkageinvolvesexclu-
chain confinement. The formation of crystal vacancies sivelycolloidalexpulsionfromthemicellarsurface,caus-
is the first hint of colloidal/polymer segregation. This ing significant thickening and layering of colloids in the
phase is preceded by a fluid phase of small micelles at inter-micellar regions.
a lower volume fraction (Fig. 2 region D). These de- Unfortunatelyextendingthevalidityofourscalingbe-
form and freeze as the system pressure is increased into haviortolargesizeratioswouldrequireperformingsimu-
structurally FCC-compatible cages: the vacancies in the lationswithverylongpolymers,andthisbecomesquickly
colloidal crystal lattice. Each vacancy is typically filled untreatable. Nevertheless,thefactthatallthedatagath-
by the polymer chains of all colloids surrounding it, and ered in our simulations for different size ratios collapse
theirlocationspresentnoobvioustranslationalorder. In onto the same master curve is strongly suggestive that
fact, we find a non-negligible number of vacancy pairs the dominant contribution to the system pressure comes
distributed across the colloidal crystal. indeedfromthefreeenergypenaltyassociatedwithchain
Interestingly, for 1 . α . 1.3, the colloidal crystal confinement.
phase ceases to form, and is replaced by a disordered
micellar phase (see Fig. 2B). This is clearly due to the
increased free energy cost associated with chain con- Self-avoiding chains
finement into a vacancy which grows quadratically with
α,[26] ∆F ∼ n(2Rg/σ2)2 = nα2, where n is the number Figure 4 shows the phase behavior as a function of
of chains in the same vacancy. To mitigate this effect, particle volume fraction for different values of particle-
the typical cage sizes become larger and the geometrical to-polymer size ratio when self-avoiding chains are con-
rearrangementintoanFCC-cellbecomesexpensive. The nectedtothecolloids,andpresentsaquitedifferentland-
presence of these unstructured micelles at large volume scape. We still find that for sufficiently small α, col-
fraction frustrates and disrupts the formation of a high loidscrystallizeinto anFCCcrystalbyfitting the chains
density colloidal ordered phase. in the colloidal interstitial spaces (Fig. 4A). However,
Above α∼ 1.3, the system assembles into low-density chains never mix to form vacancies, and as the length
micellarcrystals(seeFig.2C,withthecolloidalparticles of the polymer increases, the colloidal crystal becomes
freely diffusing at their surfaces. This phase is analo- frustratedandeventuallyceasestoform. Unlikethecase
gousto that observedusing a coarse-grained,soft-sphere ofidealpolymers,weseenoevidenceofamicellarphase.
model for the polymers [22]. Webelievethisisduetothelargeentropicbarrierassoci-
We argue that the dominant contribution to the sys- atedwithoverlappingmultipleconfinedchains. Thiscan
tem pressure in the micellar regime comes from the free beestimatedbycomputingtheconfinementfreeenergyof
energypenalty associatedwithchainconfinementwithin a polymer of length equal to the sum of all chains in the
eachmicelle. The free energy cost per micelle associated cavity, which would grow as ∆F ∝ (Rgeff/Rm)3/(3ν−1),
with it can be readily evaluated by simple scaling argu- where Reff is the radius of gyration of a chain of length
g
4
FIG.2: (Coloronline) Phasediagram ofcolloids with idealtethersasafunction ofthepolymer-colloid sizeratioαandcolloid
volumefraction φ. Snapshotsofthephasesinregion (A),(B),and(C),depictingthecolloidal crystal,thedisorderedmicellar,
and the micellar crystal phase, respectively, are also shown. For the sake of clarity, in snapshot (B) and (C), the colloidal
particlesaredepictedusingalight,low-densitypixelrepresentation,whilethedarkregionsshowwherethepolymerchainsare
located.
nN and ν ≃ 3/5. Clearly, the free energy dependence effective lateralpacking of the chains. This phase is pre-
onboth the number ofchains n andsize ofthe cageR , ceded by the formation of small colloidal clusters driven
m
∆F ∝n9/4(R /R )3.75, is much stronger than what ob- together by a combination of depletion interactions and
g m
tainedforidealchains[27]. Asaresult,assoonaschains chain-chain repulsions (Fig. 4B). The colloid-rich region
becomeconfined,anysignificantamountofpolymerover- presents,in both cases,a significantdegree ofcrystalline
lap is highly unfavorable. order.
As the polymer size increases, for 0.5≤ α ≤ 1.75, the The cluster phase is stable within a relatively narrow
dense phase presents no colloidal order. Chains do not range of volume fractions, and is promptly transformed
mix with each other and occupy the interstitial spaces into the bicontinuous phase as soon as φ is sufficiently
in between colloids. The overall shape of the chains is large for the clusters to merge. Fig. 5 shows how the
elongated,asthisgeometryisentropicallymorefavorable size of the largest colloidal cluster in the system, nor-
than a spherical one [28–30].. malized by the total number of colloids, grows with the
As soon as α becomes larger than 1.75, the micellar system volume fraction. It is worthmentioning that col-
phase found for ideal chains is replaced by a disordered loidalclusterscangrowquite thick, andthis canonly be
bicontinuous phase (Fig. 4C), which allows for a more attained at the expense of the entropy of the polymers
5
-1
ofacolloidalparticle. Wefindavarietyofself-assembled
α = 1.4 structures as a function of polymer-colloidsize ratio and
2)) αα == 11..67 volume fraction. The structures are driven by compress-
Rg -2 α = 1.8 ing the disordered low-density states and can be under-
nm α = 2.0 stood in terms of the entropy of both tethers and col-
N α = 2.1
P/( loids. We have identified chain confinement as the key
g( -3 parameter to sort out the physical mechanisms driving
o
l self-assembly in this system.
It would be interesting to test whether, for self-
-4 avoiding polymers, an ordered bicontinuous phase and
a crystal phase of inverted micelles can indeed be ob-
tained, and to study how the phase behavior presented
in this manuscript changes as a function of the number
-5
1.2 1.4 1.6 1.8
of grafted polymers.
log(R )
m We wish to stress that both disordered and ordered
micellar phases were observed by Capone and collabo-
FIG. 3: (Color online) Data collapse of the rescaled pressure
rators [22] while studying a system of diblock copoly-
P/(NmnRg2) as a function of micellar size Rm for different mersmodeledasanidealandself-avoidingpolymerwith
values of α.
a density-dependent effective soft-sphere potential. This
seemstosuggestthat(a)thenatureofthemicellarphase
connected to the particles at the core of each cluster, as for ideal tethers is not too sensitive to the details of the
they need to be partially unwrapped. We believe that interaction, and (b) for α sufficiently large, ideal chains
this free energycostmayactually limitthe overallthick- areindeedwell-characterizedbyanadditiveeffectivepair
ness of the clusters and incentivize linear, rather than potential. Theproblembecomesmorecomplicatedwhen
isotropic, cluster growth. dealingwithself-avoidingpolymers. Whenmultiplepoly-
The overall phenomenology in this region can again mersareconfinedwithinthesameregiontheirinteraction
be understood in terms of chain confinement. It is well energydoesnotscalelinearlywiththe numberofchains,
known [31] that the free energy cost to completely over- but as n9/4, and up to n3 for even larger densities [27].
laptwounconfinedchainsisabout2k T,independentof This is clearly not pairwise additive. Some preliminary
B
thepolymerlength. Asaresult,atlowvolumefractions, results obtained using an effective soft-spherical poten-
thereisn’tasignificantdrivingforceforself-organization. tial to describe the polymer (to be published elsewhere)
However, as φ increases and the chain sizes become indicate quite different phase behavior,including several
smallerthanR ,∆F acquires,asdiscussedabove,anon- ordered phases which are not found in our simulations
g
trivial dependence of the number of chains, n, sharing with explicit polymers. This seems to suggest that a
the samevolume. Thisleadstochainreorganizationand more sophisticated coarse-graining of self-avoiding poly-
subsequentcolloidalclustering. Theseclusterspresentno mers is required to obtain the correct phenomenological
translationalorderor size monodispersity,andarestabi- behavior of this system.
lizedbytheirmutualeffectiverepulsions,whichextendto
a surface-to-surface range that is typically smaller than
Acknowledgments
R . As soon as φ is sufficiently large, clusters merge to
g
further minimize chain-chaininteractions and the bicon-
tinuous phasediscussedaboveis formed. Forevenlarger This work was supported by the National Science
volume fractions we observe significant ordering of the Foundation under CAREER Grant No. DMR-0846426.
overallstructure ofthe bicontinuous phase; however,the
system sizes considered in this study are too small to
make any conclusive claim in this regard.
Thecolloidal-clusterphasecanbeinterpretedasadis-
ordered inverted micellar phase. We cannot a priori ex-
clude the existence of an inverted micellar crystal phase
for evenlargerpolymerlengths thanthe onesconsidered
in this study, but such an analysis is out of the reach of
our computational resources.
IV. CONCLUSIONS AND DISCUSSION
Wereportthephasebehaviorofasystemofhybridcol-
loids formed by grafting a single polymer on the surface
6
FIG.4: (Color online) Phasebehaviorfor thecaseof self-avoiding chains. Inthegraph in thetop-left cornerof thefigure,the
verticalaxisindicatesthepolymer-colloidsizeratioαandthehorizontalaxisisthecolloidalvolumefractionφ. Snapshotsofthe
phasesinregion (A),(B) and(C),depictingthecolloidal crystal,thecolloidal cluster,andthebicontinuousphaserespectively
are also shown. For the sake of clarity, in snapshots (B) and (C), the polymers are depicted using a light, low density pixel
representation.
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