Table Of ContentReshaping elastic nanotubes via self-assembly of surface-adhesive nanoparticles
Josep C. P`amies and Angelo Cacciuto∗
Department of Chemistry, Columbia University, 3000 Broadway, New York, NY 10027
Elastic sheets with macroscopic dimensions are easy to deform by bending and stretching. Yet
shaping nanometric sheets by mechanical manipulation is hard. Here we show that nanoparticle
self-assembly could be used to this end. We demonstrate by Monte Carlo simulation that spherical
nanoparticlesadheringtotheoutersurfaceofanelasticnanotubecanself-assembleintolinearstruc-
turesasaresultofcurvature-mediatedinteractions. Wefindthatnanoparticlesarrangeintoringsor
helicesonstretchablenanotubes,andasaxialstringsonnanotubeswithhighrigiditytostretching.
These self-assembled structures are inextricably linked to a variety of deformed nanotube profiles,
which can be controlled by tuning the concentration of nanoparticles, the nanoparticle-nanotube
diameter ratio and the elastic properties of the nanotube. Our results open the possibility of de-
1 signingnanoparticle-ladentubularnanostructureswithtailoredshapes,forpotentialapplicationsin
1 materials science and nanomedicine.
0
2
Directmechanicalmanipulationcanmakemacroscopic logically richer — behavior emerges when nanoparticles
n
sheets conform to specific shapes. However, it is difficult aggregate on elastic surfaces rather than in fluid inter-
a
J to use mechanical means to reshape sheets with sizes in faces. Clearly,formultipleindenting(adhering)nanopar-
themicrometerrangeandsmaller. Analternativeatsub- ticles, one expects that the shape of the deformed sur-
5
micrometricscaleswouldbetheuseofadhesivenanopar- face is inextricably linked to the configuration of the ad-
] ticles that induce local deformation and may catalyze hered nanoparticles. Here, we use computer simulation
t
f global shape changes. Indeed, nanoparticles have been to demonstrate that nanoparticles adhering and locally
o
showntodrivethefoldingofgraphene[1],ofthinfilmsof deforming the surface of an elastic nanotube can self-
s
. silicon [2] and carbon nanotubes [3], the budding of fluid assemble into linear structures which themselves cause
t
a membranes by protein aggregation [4, 5], and the defor- the global reshape of the nanotube. Our results suggest
m mationofvesiclesviatheadhesionofnanoparticles[6,7]. a new, nanoparticle-based route to design the effective
elastic properties and the overall shape of a flexible sur-
- Whennanoparticlesadhereanddeformasurface,effec-
d face.
tive, curvature-mediated nanoparticle interactions arise
n
o asaresultofthetendencyofthesurfacetominimizethe To this aim, we considered a simple, coarse-grained
c deformationcausedbythenanoparticleimprints. Oneof modelofanelasticnanotubewithbendingandstretching
[ the first studies of curvature-mediated interactions was deformation modes, and of adhesive spherical nanopar-
that of Goulian and colleagues [8], who calculated such ticles (smaller than the tube radius) which adhere to
1
v effects in the context of protein aggregation in biolog- the nanotube via a generic short-range potential act-
7 ical membranes. The effective pair interaction in fluid ing between any nanoparticle and the surface. Monte
5 membranes is usually isotropic, has a Casimir-like func- Carlo simulations of this model show that, for a wide
8 tional dependence on particle separation [9] and a non- rangeofnanoparticle-nanotubediameterratios,nanopar-
0 trivial dependence on the protein shape. However, for ticles arrange into one-dimensional strings. Moreover,
1. elastic (tethered) surfaces, the overall effect of the forces the strings acquire a preferential orientation which de-
0 atplayismore complicated. Unlikefluidsurfaces, which pendsontheelasticrigidityofthenanotube: onstretch-
1 cannot withstand shear, elastic thin sheets can stretch able nanotubes, strings of nanoparticles arrange as rings
1 in response to the forces applied by strongly adhering or helices; on nanotubes with high rigidity to stretch-
v: nanoparticles. If the nanoparticles are able to diffuse on ing they form axial strings. We analyzed these arrange-
i thesheet,theyshouldbeabletoself-assembleinaconfig- mentsasafunctionofbendingrigidity,nanoparticlearea-
X
urationthatreducesthemechanicalcostofdeformingthe density(thatis,therelativeareaofthesurfacecoveredby
ar surface. However, the stretching energy associated with nanoparticles) and the average nanoparticle bound area.
tetheredsurfacesimposesglobalgeometricconstraintsto We also find that the cross-sectional profile of the nan-
nanoparticle arrangements, and this leads to nontrivial otube can be shaped into ellipsoidal, triangular, rectan-
many-body effects that extend across the surface. We gularandotherregularandirregularforms,dependingon
expect the effective nanoparticle interactions to depend thenumberandorientationoftheself-assembledstrings,
on the bending and stretching rigidities of the surface, thenanoparticle-nanotubediameterratioandtheelastic
itstopology, andtherelativelocationandspecificextent properties of the nanotube. Overall, our results suggest
ofthelocaldeformationsimprintedbythenanoparticles. that nanoparticle organization and surface deformation
As a result, a significantly different — and phenomeno- can be controlled by tuning the concentration of adhe-
sive nanoparticles and the mechanical properties of the
surface.
We modeled the elastic surface following a beads-and-
∗Electronicaddress: [email protected] string scheme [10]. The nanotube consists of a triangu-
2
bound to 15 beads approximately. The adhesion term is
a ramp well potential acting between each nanoparticle-
bead pair, with a tangent distance λ =5.5σ. For conve-
t
niencewechoseλ =6.5σ andλ =7.5σ. Wehaveseen
r m
that the specific shape of the adhesive term of the po-
tential does not affect qualitatively our results, provided
that the potential is short-ranged [14].
Clearly, both elastic constants and the adhesion en-
FIG. 1: Detail of the particle (left) and triangulated-surface ergy per bead affect the extent of the deformation. We
(right) representations of the nanoparticle-nanotube system. findthataconvenientgeometricparametertocharacter-
Nanotubes are made of hexagonally arranged beads of diam- ize deformation is the average nanoparticle area that is
eter σ. Nanoparticles have a diameter of 10σ and interact boundtothenanotube(seecaptionofFig.1foraprecise
withthebeadswithenergyperbead(cid:15)whenabeadiswithin definition). We simulate the nanotube with a periodic
a distance 2σ from the nanoparticle surface. The nanoparti-
boundaryinthedirectionparalleltoitslongitudinalaxis
cle bound area (bottom, shaded) is defined by the triangles
x, and use periodic boundaries in the three cartesian co-
whoseallthreeverticesfallwithintheannularvolumedefined
ordinatesforthenanoparticles. Theinitialconfiguration
bytheexcludedvolumeoftheparticlesandasphereofradius
consists of an unperturbed nanotube, and of nanopar-
6.5σ. We report the nanoparticle bound area as the average
value,A ,forallnanoparticlesboundtothenanotubeandas ticles randomly placed in the simulation box. Monte
b
a percentage of the nanoparticle surface. Carlo moves at constant number of particles and tem-
perature sample the configurational space by attempt-
ing changes in the position of vertices and nanoparti-
latedsurfaceofhardbeadsconnectedbyelasticlinksand cles. In order to allow for stretch-free configurations,
arranged in a hexagonal lattice with an overall cylindri- changes in the length of the box parallel to the nan-
cal shape, as sketched in Fig. 1. The triangulation of otube axis were also attempted, with constant pressure
thenetworkallowsforaneasycomputationofelasticen- P =0. Theresultspresentedinthisworkcorrespondto
x
ergies, with rigidities to bending and stretching, κ and simulations with nanoparticles of 10σ in diameter. Ad-
κF, respectively, characterizing the resistance to elastic ditionally, we have simulated systems with larger, 15σ-
deformation of the nanotube. The nanoparticles are de- nanoparticles at various points in the parameter space,
scribed as hard spheres, and interact with the surface and have also compared our results to a similar simula-
beads with a maximum energy per bead, (cid:15). Beads and tion model [15]. System-size and surface-discretization
nanoparticles do not interact with spheres of their own effectsdonotappeartobesignificant. Itisimportantto
type,besidessatisfyingexcludedvolumeconstraints. We point out that the nanoparticle area-density affects self-
define the total energy of the model as the sum the Hel- assembly and deformation. Indeed, we observe that a
frich bending elasticity, the stretching energy of links, nanoparticle area-density larger than ≈ 0.2 is necessary
and a nanoparticle-surface adhesive potential, which are for the nanoparticles to aggregate into linear structures.
written as As the area-density increases, the linear structures be-
(cid:16) (cid:17)2 comeprogressivelyinterconnected,approachingthelimit
(cid:80)
(cid:88)3 j|rijarccos(nij ·nji)| of homogeneous nanoparticle coverage. In this study we
H = κ +
(cid:80) havefocusedintheintermediatearea-densityregime,for
8 A
i j ij which considerable deformation and a larger degree of
+ (cid:88)−κFl2 ln(cid:0)1−[(r −l )/l ]2(cid:1)+ (1) nanotube shapes can be accessed.
2 m ij 0 m
ij The average nanoparticle bound area controls the de-
(cid:15) λ ≤r <λ gree of deformation of the nanotube. For 20 (cid:46) κ (cid:46)
+ (cid:88)(cid:15)(λ2m−ri2k)/(λ2m−λ2r) λtr ≤riikk <λrm , 80kBT and Ab (cid:46) 5%, the elastic energy associated with
0 λ ≤r the nanoparticle imprints is of the order of the thermal
ik m ik
energy of the beads, and the nanoparticles can explore
Thediscretizedbendingtermhasbeenusedbefore[4,11], the entire surface on the nanotube. As A increases,
b
andisthesumofthebendingenergiesofalltrianglepairs nanoparticles bound to the surface become less mobile.
of the surface. r is the length of the link connecting When 4% (cid:46) A (cid:46) 10%, we observed that nanoparti-
ij b
vertices i and j; n and n are the unit normal vectors cles spontaneously organize into linear structures. For
ij ji
to the triangles sharing that link; and (cid:80) A the sum A (cid:38) 10%, the adhesive energy needed to reach such
j ij b
of the areas of the triangles sharing vertex i. We de- boundareabecomeslargerthan10k T pernanoparticle,
B
scribe the stretching contribution with a Finitely Exten- and this makes the nanoparticle mobility on the surface
sibleNonlinearElastic(FENE)potential[12],wherel is too slow for the simulations to reach equilibrium within
0
thezero-energylengthofalinkandl =3σitsmaximum oursimulationtimeframe. Thisisschematicallydepicted
m
length. We have chosen l = 1.229σ, which corresponds in Fig. 2, where we also describe the self-assembled lin-
0
toanareafractionforthebeadsof60%[13]. Thismeans ear structures as a function of stretching rigidity. For
that a nanoparticle of 10σ in diameter with A = 5% is fairly stretchable nanotubes, nanoparticles self-assemble
b
3
a
h
ngt Large deformation and low nanoparticle mobility
e
e str ➚Stretching energy ≈ 1 / bead
v
si
e
h
d
a
e
b
otu Rings / helices
n
na Coexistence of b c 7
— radial and axial Axial strings
e fragments 6
noparticl Reversible self-assembly e energy54
Na siv3
e
dh2
A
1
Small deformation and high nanoparticle mobility
Stretching rigidity 0 0.2 0.4 0.6 0.8 1 1.2
Stretching energy
FIG.2: Nanoparticlesself-assembleintoorderedlinearstruc- FIG. 3: a,b, cross-sectional cuts of deformed nanotubes.
tures at moderate adhesive strength, switching orientation Symbols locate the corresponding energies in c. Nanopar-
from rings and helices to axial strings when the stretching ticles have not been drawn for clarity. c, adhesion-stretching
rigidity is such that the stretching energy is approximately energies of configurations with axial strings (squares). One
1k T per bead. The boundaries of the simulated nanotubes point corresponding to a helix (circle) is shown for compar-
B
match up to avoid boundary effects. ison. Energies are given in units of k T per bead. The
B
straight line is a fit to simulation data of axial strings with
an(undeformed)nanotube-nanoparticlediameterratiointhe
into rings and helices, which can coexist on the same interval 1.017−2.033, with 15−55 nanoparticles, Ab in the
5−8% range, and bending and stretching rigidities within
nanotube. For slightly less stretchable tubes, for which
20−100k T and 120−960k T/σ2.
the stretching energy per bead is below ≈ 1k T, linear B B
B
aggregates are shorter and their orientation can span all
possible angles, although we have seen a preference for
radial and axial orientations. Above that energy, we see whelmsthatofthebending,effectivelyimposingaglobal
preferentially axial arrangements. constraint on the possible deformations of the nanotube.
Axialstringsareessentiallystretch-freeconfigurations. Nextwediscusstheshapesofthedeformednanotubes
Indeed,ananotubecompletelyrigidtostretchingcannot mediated by the linear aggregation of the nanoparticles
accommodatenanoparticleringsorhelicesbecausethese bound to it, and also the connection between nanoparti-
configurations necessarily involve double-curvature im- cle self-assembly and the shape of the nanotube. Fig. 3a
prints, which have a stretching penalty. But the rigid showstwonanotubeprofiles: onewithahelicallyshaped
nanotube can deform around axial strings by curving string of nanoparticles and another with four, almost-
only in the radial direction. This is why axial strings parallelaxialnanoparticlestrings. Ringandhelicalstruc-
only appear above 1k T in stretching energy per bead. turesinduceascrew-liketypeofdeformationonthenan-
B
Indeed, there is a moderately sharp transition from the otube. Axial strings, however, induce a larger variety of
bending-dominated regime — for which rings and he- shapes, which can also show constrictions along the nan-
lices are the bending-minimizing configurations — to otube axis if the strings are not fully parallel. Fig. 3b
thestretching-dominatedregime,inwhichtheminimiza- shows four different shapes corresponding to two (ellip-
tion of stretching energy becomes dominant. We be- soidalprofile), three(triangularprofile)andfour(square
lieve that the fact that the transition between the two and bow-tie profiles) parallel axial strings. The more
regimes takes place within a relatively small range of bent the nanotube profile is, the larger the adhesive en-
stretching energies has its origin in the high relative ergy per bead of the configuration, as can be seen in
cost of stretching with respect to bending. Indeed, the Fig. 3c. This figure also shows that stretching the sur-
stretching-to-bending energy ratio due to an indentation face above 1k T per bead comes at a very steep cost in
B
of depth h in a thin shell of thickness t can be written as adhesion. This is largely due to the fact that the sur-
E /E ∼(h/t)2 ∼κ /κh2[16]. Thisisalsoapplicableto face wrapping an axial string does not form an entirely
s b F
a nanotube with radius R as long as h(cid:28)R. Clearly, for straight channel, but follows the curvature of the indi-
sufficientlythinsurfaces,thestretchingenergycostover- vidual nanoparticles forming the string.
4
regime, the shape of the nanotube can be controlled
by the area-density of nanoparticles (or analogously, the
8 number of axial strings) and the nanotube-nanoparticle
diameter ratio. Fig. 4 illustrates this by showing snap-
7 shotsoftheprofilesobtainedvaryingthesetwogeometric
parameters at constant bound area and elastic rigidities.
s6 We see that the profiles are fairly symmetrical but for
g
n the higher number of axial strings, for which the shape
al stri5 canpresumablygettrappedinmetastableconfigurations.
xi Also,asufficientlyhighnumberofaxialstringscancause
a
of a nanotube to collapse.
er 4
b By altering the profile of the nanotube, one has
m
Nu3 direct access to the effective bending rigidity of the
nanoparticle-nanotube composite in the axial direction,
as such rigidity depends linearly on the tube’s cross-
2
sectional moment of inertia. Furthermore, it should be
possible to reversibly switch nanoparticle self-assembly
1
on and off in experiments — for instance by tuning tem-
perature,orbyexploitingelectrostaticordepletioninter-
1 2 3 4
actions. Controlling nanotube shape may be relevant to
(Nanotube — nanoparticle diameter ratio) x 0.738
applications of tubular nanomaterials [17–19], of bioac-
tivenanotubes[20], andinmicrofluidics[21]. Webelieve
FIG. 4: Snapshots of nanotube profiles with high rigidity that experimental systems in which the here described
tostretching,forwhichnanoparticlesself-assembleintoaxial
coupling between nanoparticle self-assembly and nan-
strings. Thestringshavebeenconstrainedtospanthelength
otube deformation occurs can be readily realized. Also,
of the tube and to contain the same number of nanoparticles
our approach may inspire alternative routes to manipu-
each. The profiles thus have a uniform shape in the axial
lating the folding of thin films of silicon for photovoltaic
direction. Profiles in the leftmost column have been drawn
applications [2].
twiceasbigforclarity. Forallprofilesthenanoparticleradius
is 10σ, κ=40k T, κ =500k T/σ2 and A =10%. WethankAnđelaSˇari´candWilliamL.Millerforstim-
B F B b
ulating discussions. This work was supported by the
National Science Foundation under Career Grant No.
Fig. 3 suggests that, for the stretching-dominated DMR-0846426.
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