Directional depletion interactions in shaped particles
Entropic forces in colloidal suspensions and in polymer-colloid systems are of long-standing and continuing interest. Experiments show how entropic forces can be used to control the self-assembly of colloidal particles. Significant advances in colloidal synthesis made in the past two decades have en...
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irk-123456789-1535022019-06-15T01:27:34Z Directional depletion interactions in shaped particles Scala, A. De Sanctis Lucentini, P.G. Entropic forces in colloidal suspensions and in polymer-colloid systems are of long-standing and continuing interest. Experiments show how entropic forces can be used to control the self-assembly of colloidal particles. Significant advances in colloidal synthesis made in the past two decades have enabled the preparation of high quality nano-particles with well-controlled sizes, shapes, and compositions, indicating that such particles can be utilized as "artificial atoms" to build new materials. To elucidate the effects of the shape of particles upon the magnitude of entropic interaction, we analyse the entropic interactions of two cut-spheres. We show that the solvent induces a strong directional depletion attraction among flat faces of the cut-spheres. Such an effect highlights the possibility of using the shape of particles to control directionality and strength of interaction. Сили ентропiї в колоїдних суспензiях i в системах полiмер-колоїд привертають постiйну увагу вже впродовж багатьох рокiв. Експерименти показують як ентропiйнi сили можна використати для контролю самоскупчення колоїдних частинок. Значний прогрес, досягнутий в областi синтезу колоїдних частинок впродовж минулих двох десятирiч, уможливлює виготовлення наночастинок високої якостi з добре контрольованими розмiрами, формами i складами i свiдчить про те, що такi частинки можуть бути використанi як “штучнi атоми” для створення нових матерiалiв. Для того, щоб пояснити вплив форми частинок на величину ентропiйної взаємодiї, ми аналiзуємо ентропiйну взаємодiю двох розрiзаних сфер. Ми показуємо, що розчинник iндукує спрямоване збiднювальне притягання мiж плоскими поверхнями розрiзаних сфер. Такий вплив вказує на можливiсть використання форми частинок для контролю спрямованостi i сили взаємодiї. 2014 Article Directional depletion interactions in shaped particles / A. Scala, P.G. De Sanctis Lucentini // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33007:1-6. — Бібліогр.: 26 назв.— англ. 1607-324X DOI:10.5488/CMP.17.33007 PACS: 05.20.Jj, 05.10.Ln, 47.57.-s, 65.80.-g, 82.70.Dd arXiv:1410.4294 http://dspace.nbuv.gov.ua/handle/123456789/153502 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Entropic forces in colloidal suspensions and in polymer-colloid systems are of long-standing and continuing interest. Experiments show how entropic forces can be used to control the self-assembly of colloidal particles. Significant advances in colloidal synthesis made in the past two decades have enabled the preparation of high quality nano-particles with well-controlled sizes, shapes, and compositions, indicating that such particles can be utilized as "artificial atoms" to build new materials. To elucidate the effects of the shape of particles upon the magnitude of entropic interaction, we analyse the entropic interactions of two cut-spheres. We show that the solvent induces a strong directional depletion attraction among flat faces of the cut-spheres. Such an effect highlights the possibility of using the shape of particles to control directionality and strength of interaction. |
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Scala, A. De Sanctis Lucentini, P.G. |
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Scala, A. De Sanctis Lucentini, P.G. Directional depletion interactions in shaped particles Condensed Matter Physics |
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Scala, A. De Sanctis Lucentini, P.G. |
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Scala, A. |
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Directional depletion interactions in shaped particles |
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Directional depletion interactions in shaped particles |
| title_full |
Directional depletion interactions in shaped particles |
| title_fullStr |
Directional depletion interactions in shaped particles |
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Directional depletion interactions in shaped particles |
| title_sort |
directional depletion interactions in shaped particles |
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Інститут фізики конденсованих систем НАН України |
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2014 |
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| citation_txt |
Directional depletion interactions in shaped particles / A. Scala, P.G. De Sanctis Lucentini // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33007:1-6. — Бібліогр.: 26 назв.— англ. |
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Condensed Matter Physics |
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AT scalaa directionaldepletioninteractionsinshapedparticles AT desanctislucentinipg directionaldepletioninteractionsinshapedparticles |
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2025-07-14T04:38:20Z |
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2025-07-14T04:38:20Z |
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1837595781272436736 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 3, 33007: 1–6
DOI: 10.5488/CMP.17.33007
http://www.icmp.lviv.ua/journal
Directional depletion interactions in shaped particles
A. Scala1, P.G. De Sanctis Lucentini2
1 ISC-CNR Physics Department, University “La Sapienza”, Piazzale Moro 5, 00185 Roma, Italy
2 Gubkin Russian State University of Oil and Gas, 65 Leninsky prospekt, Moscow, Russia
Received April 30, 2014, in final form May 29, 2014
Entropic forces in colloidal suspensions and in polymer-colloid systems are of long-standing and continuing
interest. Experiments show how entropic forces can be used to control the self-assembly of colloidal particles.
Significant advances in colloidal synthesis made in the past two decades have enabled the preparation of high
quality nano-particles with well-controlled sizes, shapes, and compositions, indicating that such particles can
be utilized as “artificial atoms” to build new materials. To elucidate the effects of the shape of particles upon
the magnitude of entropic interaction, we analyse the entropic interactions of two cut-spheres. We show that
the solvent induces a strong directional depletion attraction among flat faces of the cut-spheres. Such an effect
highlights the possibility of using the shape of particles to control directionality and strength of interaction.
Key words: entropic interactions, self-assembly, Monte Carlo
PACS: 05.20.Jj, 05.10.Ln, 47.57.-s, 65.80.-g, 82.70.Dd
1. Introduction
Small particles S being added to a suspension of bigger particles B, induce a “depletion” attraction
among the bigger particle that arises from an unbalanced osmotic pressure due to the exclusion of small
particles from the region between big particles. Small particles are often referred to as the “depletant”.
The strength of this interaction can be easily explained by modelling big and small particles by hard-
spheres of radii R and r ; simple Asakura-Oosawa geometric arguments [1, 2] show that depletion interac-
tion is attractive and is controlled by the concentration of small particles and the depletant-to-solute ratio
R/r , while the range of attraction depends on the radius r of small particles. A classical experimental
example is the case of adding non-adsorbing polymers (with radius of gyration r ) to a stable colloidal
solution of silica spheres (with colloids of size R) [3]; at such scales, interactions can be considered to
appear only due to steric factors. While the original Asakura-Oosawa model considered only hard-core
interactions— hence depletion forces have necessarily got an entropic origin— in principle, there can be
additional thermodynamic contributions from the non-sharp inter-particle repulsion potentials, attrac-
tive terms and an explicit contribution of the solvent.
Depletion interactions induce an attractive force among particles that can be tuned both in the
strength and in the range; hence, it is possible to induce — often via phase transitions mechanisms —
a variety of micro-structures such as gels, complex liquids or exotic crystals with the final aim of achiev-
ing mechanical and rheological properties suitable for particular applications.
To understand the mechanism of depletion interaction, let us consider a system of two big particles
in a bath of depletant particles and assume that only hard interactions are at play. At low depletant
concentrations, the free energy of a system of hard particles will be as follows:
F =−kBTρVfree , (1.1)
where ρ is the number density of the depletant particles and Vfree is the volume available to depletant;
hence, the presence of big particles increases the free energy of the system by reducing the free volume.
In particular, if we model the depletant as hard particles of diameter σ, each big particle will “occupy”
© A. Scala, P.G. De Sanctis Lucentini, 2014 33007-1
http://dx.doi.org/10.5488/CMP.17.33007
http://www.icmp.lviv.ua/journal
A. Scala, P.G. De Sanctis Lucentini
an excluded volume Vexcl that extends up to a distance r = σ/2 from its surface. Nevertheless, when
big particles come close together (their surfaces are at a distance less than σ), their excluded volumes
overlap and decrease the free energy by a factor proportional to the volume Ω of the overlap region: all
this results in an effective attractive interaction of magnitude ∼ kBTρΩ and range ∼ σ. It is, therefore,
possible to tune the range and the strength of depletion interaction by varying the size σ and the packing
fraction φ=πρσ3/6 of the depletant.
It is nowadays possible to engineer shaped building blocks at nanometer andmicrometer scales. New
pioneering techniques permit unprecedented synthesis and fabrication of shaped nano-particles and col-
loids; shape only can induce exotic phases [4] as it is well known from liquid crystal science [5]. The
final goal of producing self-assembling materials requires the capability of controlling and designing the
interactions among the building blocks. While depletion interactions in colloidal and nano systems are
of long-standing and continuing interest, entropic forces among anisotropic particles started to be sys-
tematically investigated only in the recent decades [6–9]. In this paper, we intend to further develop the
investigation of entropic forces among anisotropic particles.
2. Model and simulations
To elucidate the effects of the particle shape upon the magnitude of entropic interaction, we simulate
two cut-spheres in a bath of smaller hard spheres of diameter σ (the depletant). A cut-sphere (CS) is a
geometrical object obtained by chopping off two symmetrical caps from a hard sphere of diameter Σ at a
distance (thickness) L ÉΣ (figure 1). We use σ as the unit length and choose L/σ= 8 and Σ/σ= 10.
ΣL
Figure 1. A cut-sphere is a geometrical object obtained by chopping off two symmetrical caps from a hard
sphere of diameter Σ at a distance (thickness) L ÉΣ.
To study the system, we employ Monte Carlo simulations for hard bodies. Since hard bodies have
either a null interaction when they are separate or an infinite repulsion when they overlap, in a hard-
body Monte Carlo all the moves are accepted unless they lead to the overlapping of particles. Hence, to
implement the simulation code, we need three overlap-checking procedures: one to detect the overlaps
among two hard spheres (interaction among the depletant particles), one to detect the overlaps among a
hard sphere and a CS (depletant-CS interaction) and one to detect the overlaps among two CSs [10, 11].
We perform Monte Carlo simulations at a depletant packing fraction φ = 0.00, 0.05, 0.10, 0.15 with
104 depletant spheres for the highest φ. For each φ, we perform from 10 to 20 independent simulations
to reduce the errors in the estimates for our observables.
In each Monte Carlo step, we perform one attempt of random displacement for each depletant parti-
cle, 10 attempts of random displacements for each CS and 10 attempts of random rotations for each CS.
To perform random rotations, we employ the algorithm of [12] for linear molecules. The attempted dis-
placement and rotation for the CSs are tuned to reach an acceptance ratio of∼ 40% during thermalization
phase.
To ensure thermalization for simulated systems, we check for two conditions to apply: (i) the mean
square displacement
〈
∆r 2
〉
of the CS’s is of the order of the simulation box length squared and (ii) the
distribution of cosθα is flat, where θα is the angle between the axis of a CS and the α-axis (α= x, y, z).
After thermalising the system, we sample the observables over 1011 Monte Carlo steps per run; within
such time-scales, conditions (i) and (ii) are largely verified, hence ensuring a complete exploration of the
configuration space for the two CSs.
33007-2
Directional depletion interactions in shaped colloidal particles
Notice that our simulations involve only two CSs at a time; hence, we explore only two-body interac-
tions of CSs’ systems. Such an approach corresponds to the investigation of the infinite dilution limit for
CSs (not for the depletant!), where three-body and higher interactions among CSs can be disregarded.
3. Results
We first study the radial distribution function g (r ) of the center to center distance r between two CSs
g (r )= ρ−1 p(r )
4πr 2
, (3.1)
where p(r ) is the probability that the two CSs are at a distance r and ρ is the number density. To cal-
culate g (r ), we evaluate via histograms p(r ) and then apply the discretised version of identity p(r )dr =
ρ g (r )4πr 2dr .
Figure 2 shows that increasing φ, the depletion interaction attracts CSs nearby. In fact, we notice
that increasing φ, the radial distribution function develops peaks at r ∼ 10 (i.e., r ∼ Σ), at r ∼ 9/2 (i.e.
r ∼ (L+Σ)/2) and at r ∼ 8 (i.e. r ∼ L).
By first-principle calculations of the work to separate two particles at an infinite distance, it can be
shown [13] that the radial distribution function is related to the two-particle potential of the mean force
Veff by g (r ) = exp
[
−βVeff (r )
]
; such potential Veff accounts for integrating out all the other degrees of
freedom (particles). In our case, the effective interaction among the CSs is purely due to entropic effects
since we are considering hard-bodies. A simple way of estimating the potential among two particles in
the dilute limit is to calculate the work W to bring such particles from distance r to infinity: g (r ) =
exp
[
−βW (r )
]
[13]. Hence, to evaluate the magnitude of the entropic force while explicitly factoring out
the r > L hard-body constrain, we calculate Veff from the formula:
g (r )= gHS
0 (r ) exp
[
−βVeff (r )
]
, (3.2)
where to avoid the divergent contribution of the hard-core, we have factored out the radial distribution
function of hard spheres of diameter L in the infinite dilute limit gHS
0 .
In figure 3 we show Veff among two CSs at different packing fractions. Notice that the evolution of Veff
withφ shows competition among two terms: the attractive depletion interaction due to small spheres and
Figure 2. (Color online) Radial distribution function g (r ) for the center-to-center distance r between two
CSs at different values of the packing fraction φ. For φ = 0, only geometrical constrains matter (i.e., CSs
cannot overlap); hence, g (r ) decreases when the two CSs approach the minimum possible distance r = L.
For increasing φ, depletion interactions make CSs attract and peaks develop in the region L < r <Σ.
33007-3
A. Scala, P.G. De Sanctis Lucentini
Figure 3. (Color online) Effective interaction Veff(r ) among two CSs at different values of the packing
fractionφ. Forφ= 0, only geometrical constrainsmatter and Veff(r ) increases when the two CSs approach
theminimum possible distance r = L; in fact, the number of possible non-overlapping orientations — and
hence the entropy — of two CSs decreases with a decreasing distance. Attractive depletion interactions
develop local minima at r ∼Σ, at r ∼ (L +Σ)/2 and at r ∼ L for increasing φ’s.
the repulsive entropic interaction due to the geometry of the two CSs. In particular, at φ= 0, no depletion
interaction is present and Veff reflects the entropic contribution due to the fact that when two CSs are at
a distance r < Σ not all the orientations are possible; in particular, the number of possible orientations
vanishes for r = L where the φ= 0 potential has a repulsive maximum.
On the other hand, when the two CSs approach, the depletion interaction is influenced by the lo-
cal curvature of the surfaces of the particles coming at contact: in particular, it is maximum when flat
surfaces come together and minimum when curved surfaces come together. Hence, for high enough φ,
the depletion interaction wins over the entropic repulsion and CSs have an overall attractive potential;
we sketch in figure 4 the behaviour of the depletion potential based on these geometric arguments and
confirmed by visual inspection of the CSs configurations during the Monte Carlo runs.
c
a
b
Vβ
c
r
a
b
Figure 4. (Color online) Sketch of the attractive part of the effective potential induced by a depletion
interaction: since the attraction is proportional to the overlap volume, for a couple of cut-spheres there
is maximum when flat faces are at contact (case (a), distance ∼ L), minimum when curved faces are at
contact (case (c), distance ∼ Σ), intermediate when a flat face is at contact with a curved face (case (b),
distance ∼ (L +Σ)/2). Such configurations are observed when conducting visual inspection of the CSs
configurations during the Monte Carlo runs at non-zero φ’s.
33007-4
Directional depletion interactions in shaped colloidal particles
4. Discussion
We have investigated, via Monte Carlo simulations, the depletion interaction among cut-spheres. Cut-
spheres are a simple model system of shaped particles with only two curvatures. We show that even
in this case, a complex, directional potential arises from the geometry of a system, hence indicating the
possibility of controlling depletion-driven assembly by engineering the shape of the particles, and, in
particular, the local curvature of their surfaces.
By varying the number of flat regions (cuts) in the spheres, interactions can be made selective and
directional, hence implementing a variant of the so-called patchy colloidal particles. In patchy particles,
inter-particle interactions depend on the specific regions of the particles that come close to contact, lead-
ing to assembled structures with substantially fewer defects than via conventional self-assembly meth-
ods [14, 15] and enriching both the equilibrium phase diagrams and the possible out-of-equilibrium be-
haviours of the system [16]. Finally, at a low packing fraction, the presence of short-range repulsive poten-
tials makes the system akin to core-softened potentials systems [17], whose phase behaviour is definitely
unusual for one-component systems, since they can show iso-structural solid-solid and/or liquid-liquid
phase transitions [18].
Our simulation approach is in the Asakura-Oosawa spirit and permits to capture the phenomenology
of the shape induced depletion interaction; nevertheless, an important advance would be to develop ana-
lytical approaches similar to the ones already introduced for the depletion interaction of spherical parti-
cles, such as density functional theory [19], virial expansions [20], the Derjaguin approximation [21], the
integral equation formulation of liquid state theory [22] or Laplace transforms of the radial distribution
functions [23]. In particular, it would be important to take into account the oscillatory behaviour due
to excluded volume correlation effects following the entropic attraction arising at short distances and
predicted by accurate numerical [24, 25] and theoretical [19, 26] calculations of the effective potential
between hard-sphere colloids and depletants.
Acknowledgements
AS thanks D. Blaak for his help in setting up and testing the algorithm for checking the overlap among
cut-spheres.
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Спрямованi збiднювальнi взаємодiї у сформованих
частинках
А. Скаля1, П.Г. Де Санктiс Лучентiнi2
1 Фiзичний факультет, Унiверситет “Ла Сап’єнца”, 00185 Рим, Iталiя
2 Росiйський державний Унiверситет нафти i газу iм. Губкiна, Москва, РФ
Сили ентропiї в колоїдних суспензiях i в системах полiмер-колоїд привертають постiйну увагу вже впро-
довж багатьох рокiв. Експерименти показують як ентропiйнi сили можна використати для контролю само-
скупчення колоїдних частинок. Значний прогрес, досягнутий в областi синтезу колоїдних частинок впро-
довж минулих двох десятирiч, уможливлює виготовлення наночастинок високої якостi з добре контро-
льованими розмiрами, формами i складами i свiдчить про те, що такi частинки можуть бути використанi
як “штучнi атоми” для створення нових матерiалiв. Для того, щоб пояснити вплив форми частинок на
величину ентропiйної взаємодiї, ми аналiзуємо ентропiйну взаємодiю двох розрiзаних сфер. Ми показу-
ємо, що розчинник iндукує спрямоване збiднювальне притягання мiж плоскими поверхнями розрiзаних
сфер. Такий вплив вказує на можливiсть використання форми частинок для контролю спрямованостi i
сили взаємодiї.
Ключовi слова: ентропiйнi взаємодiї, самоскупчення, Монте Карло
33007-6
http://dx.doi.org/10.1016/S0378-4371(01)00566-0
http://dx.doi.org/10.1023/A:1018631426614
http://dx.doi.org/10.1103/PhysRevE.57.6785
http://dx.doi.org/10.1016/0378-4371(95)00206-5
http://dx.doi.org/10.1016/S0378-4371(02)00991-3
http://dx.doi.org/10.1016/0021-9797(86)90250-X
http://dx.doi.org/10.1021/la010047d
http://dx.doi.org/10.1063/1.474367
http://dx.doi.org/10.1103/PhysRevE.59.5744
http://dx.doi.org/10.1063/1.458556
Introduction
Model and simulations
Results
Discussion
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