Cluster phases in nanocolloids:insights from computer simulations

Matt GlaserDepartment of Physics and LCMRCUniversity of Colorado, Boulder

University of Milan – March 24, 2009

Outline

I. MotivationII. Introduction to colloidsIII. Cluster phases in colloidal systems

A. Frustrated phase separationB. Soft shoulder clustering instability

IV. Colloidal assembly by designV. Conclusions

“There’s plenty of room at the bottom”“I would like to describe a field, in which little has

been done, but in which an enormous amount can be done in principle. This field is not quite the same as the others in that it will not tell us much of fundamental physics (in the sense of, “What are the strange particles?”) but it is more like solid-state physics in the sense that it might tell us much of great interest about the strange phenomena that occur in complex situations. Furthermore, a point that is most important is that it would have an enormous number of technical applications.

What I want to talk about is the problem of manipulating and controlling things on a small scale.”

Richard Feynman, APS Meeting, Caltech December 29th, 1959

Gold nanocolloids

• Vivid colors result from surface plasmon resonance

• Colloidal gold used to color glass since ancient timeshttp://www.ansci.wisc.edu/facstaff/Faculty/pages/

albrecht/albrecht_web/Programs/microscopy/colloid.html

“… known phenomena seemed to indicate that a mere variation in the size of [gold] particles gave rise to a variety of resultant colours.”

M. Faraday, “The Bakerian lecture: experimental relations of gold (and other metals) to light,” Philosophical

Transactions of the Royal Society of London 147, 145-181 (1847)

Damascus steel (circa 1700 AD)

multiwalled carbon nanotubes cementite nanowires

M. Reibold et al., Nature 444, 286 (2006)

• Unique combination of hardness and toughness

Small is different

• Effects of confined geometry– Plasmonic metal nanoparticles– Fluorescent semiconductor nanoparticles

• Surface/interfacial effects– Superadhesive and superhydrophobic surfaces (Gecko feet, Lotus

leaves)– Ultracapacitors– Nanostructured catalysts– Bulk heterojunction photovoltaics

• Emergence of novel effective properties in ‘metamaterials’– Nanostructured thermoelectrics– Giant magnetoresistance– Negative index materials

• Surface plasmon excitations of silver film yield negative dielectric permittivity

• This leads to enhancement of evanescent waves from the object• Sub-wavelength features of object are recovered in image plane

Science 308, 534 (2005)

• Subresolution image transfer over micron distances with stacked silver nanorod array

• Magnification achieved by splaying nanorod array• How can such structures be fabricated?

Nature Photonics 2, 438 (2008)

Motivation• Nanostructured materials have useful and unusual

properties• Various methods for fabricating nanostructures exist

– ‘Top-down’ approach• Optical and e-beam lithography

– ‘Bottom-up’ approach• Nanoparticle self-assembly

– Combination of top-down and bottom-up approaches• Self-assembly in lithographically defined confined geometries• Self-assembled nanoscale lithographic masks

• Better methods for controlling nanoparticle self-assembly are needed– Our approach: controlling self-assembly by engineering

interparticle interactions

What are colloids?• Dispersions of 1 nm - 1 m solid particles, fluid droplets,

macromolecules, or molecular assemblies in fluid media– Finely divided matter, with surface free energy comparable to bulk

free energy– Brownian motion is sufficient to inhibit sedimentation

• Stable suspensions can be obtained by polymeric or electrostatic stabilization– Examples: India ink and latex paint– Colloidal suspensions are often metastable

• The effective interactions between colloidal particles are highly complex, mediated by a large number of internal and solvent degrees of freedom

Colloids as artificial atoms

• Many phenomena characteristic of atomic systems (condensation, freezing, glass formation, etc.) are also observed in colloidal suspensions, at much larger (and more accessible) length- and timescales

• Techniques such as confocal microscopy and optical trapping allow real-time imaging and manipulation of individual colloidal ‘atoms’

• Colloids differ from atomic systems in that their effective interactions are tunable

W. B. Russel, D. A. Saville, and W. R. Showalter, Colloidal Dispersions (Cambridge University Press, 1990)

Van der Waals fluids

• Van der Waals interactions– Short-range repulsion– Long-range dispersion attraction

• Van der Waals equation of state

• Law of corresponding states– ‘Universal’ vapor-liquid coexistence

curve for van der Waals fluids• The van der Waals picture of

fluids– Structure determined primarily by

short-range excluded volume interactions

– Longer-range (‘soft’) interactions treated as perturbation

Lennard-Jones potential

r/σ

v(r)

/ε

The van der Waals picture

“According to the van der Waals picture, the average relative arrangements and motions of molecules in a liquid (that is, the intermolecular structure and correlations) are determined primarily by the local packing and steric effects produced by the short-ranged repulsive intermolecular forces. Attractive forces, dipole-dipole interactions, and other slowly varying interactions all play a minor role in the structure, and in the simplest approximation their effect can be treated in terms of a mean-field – a spatially uniform background potential – which exerts no intermolecular force and hence has no effect on the structure or dynamics, but merely provides the cohesive energy that makes the system stable at a particular density or pressure.”

D. Chandler, J. D. Weeks, and H. C. Andersen, Science 220, 787 (1983)

This suggests that spherical particles should assemble into simple, predominantly close-packed crystal structures

Dispersion interaction between mesoscopic colloidal particles

• Sum up dispersion interactions between molecules in two colloidal particles of radius a and center-to-center separation r:

• The total dispersion attraction is strong (>> kBT) and long-ranged (> a)

• As a result, colloidal dispersions are inherently unstable toward flocculation

• ‘Bare’ colloidal particles must be stabilized

Colloidal stabilization

• Charge stabilization– Charged colloidal particles + counterions form electrical

double layer– Effective interaction is of screened Coulomb (DLVO) form:

– Short-range dispersion attraction can still lead to instability• Steric stabilization

– Adsorbed or grafted polymer chains produce strong entropic repulsion at small separations

– Ancient technology (e.g., India Ink)

Depletion attraction

• Effective interaction induced by non-adsorbing polymer• Each particle excludes polymer chains from spherical

shell of thickness rg surrounding particle• Volume accessible to polymer chains increases when

particle surfaces are closer than 2rg

• Gives rise to short-range entropic attraction

Effective colloidal interactions• Van der Waals interactions – short-range dispersion attraction and excluded

volume interactions• Coulomb interactions - interactions between charged colloidal particles,

dependent on surface charge density, counterion and co-ion concentration• Depletion interactions - entropic attraction depending on the size and

concentration of ‘depleting’ particles or molecules• Texture-mediated interactions - effective interactions deriving from the

elasticity of a liquid crystalline medium• Pseudo-Casimir interactions - interactions due to solvent order parameter

fluctuations• Field-induced interactions - relatively weak external fields can produce large

changes in effective interparticle interactions• Structural forces - due to surface-induced changes in fluid structure• Hydrodynamic interactions - velocity-dependent, many-body interactions

mediated by the intervening fluid• Polymer-mediated interactions - soft entropic repulsion deriving from the

conformational entropy of grafted or adsorbed polymer chains• Interface-mediated interactions - interactions arising from capillary effects

in interfacial colloids• Marangoni forces – ‘chemical’ forces arising from gradients in surfactant

concentration

Nightmare, or opportunity?

• The tunability of colloidal interactions raises the possibility of directed design of interaction potentials to promote specific modes of self-assembly

• In this respect, colloidal assemblies differ fundamentally from their atomic and molecular counterparts

• Although the majority of colloidal systems exhibit relatively simple phase behavior (isotropic fluid, hexagonal, FCC, and BCC crystal phases), there are other interesting possibilities

• Control of colloidal interactions allows us to escape the tyranny of the van der Waals picture

• 4-6 nm silver nanoparticles passivated with octanethiol, imaged with TEM• Clustering thought to be driven by competition between short-range

dispersion attraction and long-range dipolar repulsion

simulation

experiment

model potential

• Sterically stabilized (uncharged?) PMMA particles in density- and index-matched solvent, diameter ≈ 400 nm

• Short-range depletion attraction induced by nonadsorbing PS polymer

• Studied by light scattering and DIC microscopy

• Gelation of clusters resembles repulsive glass transition of hard spheres

2.5 μm

• Small-angle scattering and confocal microscopy studies of protein solutions and colloid-polymer mixtures

• Cluster formation observed in both systems, driven by competition between long-range repulsion and short-range attraction

Nature 432, 492 (2004)

• Sterically stabilized PMMA spheres, diameter ≈ 780 nm, Debye screening length ≈ 1 μm

• Short-range depletion attraction induced by nonadsorbing PS polymer, rg ≈ 92 nm

• Stable chainlike clusters consisting of face-sharing tetrahedra (‘Bernal spirals’)

• Chains associate into a 3D network at gelation point

Frustrated phase separation• Competition between short-range attraction and long-range

repulsion leads to incomplete phase separation• Binding energy of a finite cluster containing N particles:

– Cohesive energy of a growing cluster is negative and proportional to N– Surface tension contribution is proportional to N2/3 (the cluster surface area)– Long-range repulsion contributes a positive ‘bulk’ energy proportional to N2

– There is an optimal cluster size beyond which further growth of the cluster decreases its stability

• Similar mechanisms lead to clustering phenomena (‘emulsion phases’) in a variety of types of dense matter– Langmuir monolayers– Cuprate superconductors (‘Coulomb-frustrated phase separation’)– Magnetic materials– Biomolecular assemblies– Nuclear matter

D. Wu et al., J. Phys. Chem. 96, 4077 (1992)J. Groenewold and W. K. Kegel, J. Phys. Chem. B 105, 11702 (2001)

Nuclei as nucleon cluster phases

• Remarkably successful in predicting nuclear stability line• Coulomb term prevents macroscopic ‘condensation’ of nuclear matter• Competition between Pauli and Coulomb terms (last two terms) leads to

proton/neutron asymmetry

Semi-empirical mass formula for nucleus of atomic number Z and mass number A = Z + N (Weizsäcker, 1935):

colloidal cluster nucleus

J. Groenwold and W. Kegel, J. Phys.: Condens. Matter 16, S4877 (2004)

where

arXiv:nucl-th/0512020v1

sphere phase cylinder phase slab phase

cylindrical hole phase spherical hole phase

• Liquid crystal phases (‘nuclear pasta’) in the interior of neutron stars?

• Due to competition between long-range Coulomb repulsion between protons and short-range strong nuclear attraction

MD simulation of collection of particles with pair potential:

where

A hard core/soft shoulder potential:the WCA/generalized exponential model

• Particles start out on a 2d hexagonal lattice at very low temperature (kBT/ε = 0.001) and relatively high density

• Is the hexagonal lattice stable?

Soft repulsion can mimic effective attraction!

The ‘primitive’ hard core/soft shoulder model

• A simple model of soft colloids• Isotropic, repulsive pair interactions• We have investigated this model

(and related models) via Monte Carlo and molecular dynamics simulation

2d finite-temperature phase sequence ( = 5, t = 0.5)

• Complex sequence of inhomogeneous phases, reminiscent of lyotropic liquid crystals or block copolymers

• Liquid crystal phases from isotropic, repulsive pair interactions!

2d phase diagram (κ = 5)

• A large number of distinct stripe, cluster, and inverse phases• Phase behavior becomes increasingly complex with decreasing temperature

incr

easi

ng p

ress

ure

increasing temperature

2d zero-temperature phase sequence ( = 5)

• Obtained by simulated annealing, exact thermodynamic analysis• Large number (> 30!) of complex crystal structures• Cluster and stripe morphologies predominate

enthalpy vs. pressure

penetrable sphere model

Anisotropic melting in 2d ( = 2, t = 0.125)

• A realization of the anisotropic melting scenario of Ostlund and Halperin [Phys. Rev. B 23, 335 (1981)]

3d phase sequence ( = 4, t = 1.4)

2d 12-fold quasicrystal phase

• Entropy-stabilized random tiling quasicrystal

• Stable over a broad range of temperatures and pressures

What accounts for this complicated behavior?

• The complex behavior of the soft shoulder system derives from an underlying ‘soft core’ clustering instability

• Two complementary viewpoints:– Clustering instability of ‘homogeneous’ crystalline ground state– Clustering instability of homogeneous isotropic fluid state

• Competition between clustering behavior and excluded volume constraints (‘radial frustration’) leads to highly complex structures and phase behavior

Why do ‘soft’ particles cluster?

Repulsive Step

Impenetrable Core

Clustering lowers the potential energy of the system!

How general is this phenomenon?• Clustering behavior of hard core/soft shoulder systems derives from

clustering instability of bounded soft core component of potential• For bounded potentials, instability of an isotropic fluid toward clustering is

predicted to occur if the Fourier transform of the pair potential is negative for any finite wavevector q0

– W. Klein et al., Physica A 207, 738 (1994)– C. N. Likos et al., Phys. Rev. E 63, 031206 (2001)

• The Likos criterion predicts clustering for a wide variety of isotropic, repulsive pair potentials

• These systems freeze into simple crystal lattices (hexagonal in 2d, FCC or BCC in 3d), with a site occupancy that increases with increasing density

step linear ramp generalized exponential

If v(q) is negative for some wavevector q0, then S(q) diverges along a line in the density-temperature plane (λ line) defined by:

Likos et al. identify the divergence in the response function S(q) as an instability toward the formation of clusters of characteristic size

Within a mean field approximation, the thermodynamic properties depend only on the ratio of density to temperature, not on density and temperature separately

The Likos criterionC. N. Likos et al., Phys. Rev. E 63, 031206 (2001)

• At the level of mean field theory, the penetrable sphere fluid is unstable toward formation of a density wave with wavevector q0

• This spinodal instability leads to the formation of particle clusters, with characteristic cluster size 0 = 2 / q0 ~

• The Likos criterion is a useful guide for identifying colloidal systems with ‘clustering’ effective interactions

• However, to predict the morphology of cluster phases and compute the equilibrium phase diagram, we need other tools

Density functional theory of mean-field fluidsIn general, the Helmholtz free energy can be written as a functional of the single-particle density ρ(r):

Fid[ρ] is the free energy of an inhomogeneous ideal gas:

Fid[ρ] is the excess free energy (resulting from interactions). If correlations are neglected (mean-field approximation), then:

This is an excellent approximation for bounded potentials such as the repulsive step potential and the Gaussian core model

By minimizing F[ρ] w.r.t. variations of ρ(r) for each T and ρ, phase diagrams of bounded potentials can be mapped out

γ = 2 corresponds to the Gaussian core model; cluster phases occur for γ > 2

Example: polymorphic cluster phases of the generalized exponential model

Consider particles interacting via a generalized exponential potential:

Mladek et al., Phys. Rev. Lett. 96, 045701 (2006)

The sum ranges over a set of Bravais lattice vectors (e.g., FCC or BCC), and nc is the number of particles per cluster (lattice site occupancy)

The particle density in the crystal phases is modeled as a sum of Gaussians centered on lattice sites:

With this ansatz, the free energy per particle becomes a function of nc and α, F/N = f(nc,α), which can be minimized w.r.t. these parameters for each T and ρComparing the free energies of the various competing states (isotropic fluid, FCC, BCC, …) yields the phase diagram in the T-ρ plane

• Phase diagram determined from MC simulation and density functional theory• Freezing into BCC or FCC cluster crystal preempts clustering instability (λ line)• Phase behavior depends primarily on ratio of density and temperature

Phase diagram of the 3d generalized exponential model

Mladek et al., Phys. Rev. Lett. 96, 045701 (2006)

isotropic fluid FCC crystal

λ line

How do we incorporate excluded volume interactions into density functional theory?

• Observations:– Intra-cluster structure and thermodynamics governed by hard-core interactions– Inter-cluster structure and thermodynamics determined by soft-shoulder

interactions

• Approach:– Treat soft cluster-cluster interactions within mean-field approximation (as before)– Treat intra-cluster thermodynamics using a local density approximation based on

the hard sphere reference system

… but mainly P. Ziherl and A. Kosmrlj

M. A. Glaser et al., Europhys. Lett. 78, 46004 (2007)

Density functional theory of hard core/soft shoulder systems

The contribution due to ‘soft’ interactions is computed at the mean-field level:

Hard-core interactions are treated within a local density approximation:

The total free energy is written as the sum of two contributions:

Here, f (ρ) is the Helmholtz free energy per particle of a uniform system of hard spheres with bulk density ρ. For fluid clusters we can use a simple empirical equation of state such as the Carnahan-Starling approximation:

Here, vsoft(r) is the soft part of the potential (excluding hard core interactions)

Here, η is the packing fraction. For crystalline clusters, a cell theory estimate of f (ρ) for the hard-disk crystal is used

Theoretical phase diagram of the hard core/soft shoulder system

• Reasonable semi-quantitative agreement between theory and simulation• Density functional theory reproduces both fluid and crystalline cluster phases• Modulation period of cluster phases is nearly independent of pressure, but

has a systematic dependence on cluster morphology

phase diagram of modulated phases modulation period

• Can we design spherical colloids that self-assemble into a diamond lattice (e.g., to create photonic bandgap materials)?

Zero-temperature phase diagram of the 2d HCSS system

http://phycomp.technion.ac.il/~nika/diamond_structure.html

Tuning the Potential

1.08 1.43 1.84 4.34 5.83 6.57 10.2 31.9P =

3D Zero-T Phase Sequence, κ = 1.63

1.08 1.43 1.84 4.34 5.83 6.57 10.2 31.9P =

3D Zero-T Phase Sequence

Phonon spectrum of diamond lattice

• Softened version of HCSS potential (repulsive power-law + GEM)• Thermally and mechanically stable diamond lattice by self-assembly

of isotropic spheres!

• Can the soft shoulder clustering mechanism be realized experimentally? Are there observed phenomena that can be explained on this basis?

• Voids form on long timescales in suspensions of charged colloidal particles at low salt concentrations

• Suggests a long-range attractive interaction, contrary to DLVO theory• These observations have led to a re-examination of the theory of charged

colloids

‘Core-softened’ paramagnetic spheres in a magnetic field

N. Osterman et al., Phys. Rev. Lett. 99, 248301 (2007)

experiment

simulation

• 2D system of paramagnetic particles confined between glass surfaces, with a magnetic field perpendicular to the surfaces

• Short-range dipolar repulsion is ‘softened’ by allowing particles to move over one another slightly

• Amphiphilic dendrimers with solvophobic core and solvophilic shell• Simulations suggest generalized-exponential-like effective interactions

schematic of effective potentialmeasured potential and Fourier transform

Conclusions

• There are at least two distinct ‘colloidal’ routes to the formation of cluster phases– Frustrated phase separation

• Competition between short-range attraction and long-range repulsion

– Soft shoulder clustering instability• Soft repulsion mimics effective attraction• Excluded volume interactions compete with soft core clustering behavior

• These mechanisms produce rich crystalline, liquid crystalline, and quasicrystalline polymorphism

• This is largely unexplored territory – so there are many interesting opportunities for controlling colloidal assembly

Zach Smith, Robert Blackwell, Steve Kadlec, Julia Santos, Johannes Hausinger, Paul Beale, Noel Clark (Colorado) Randy Kamien (Penn)

Chris Santangelo, Greg Grason (U. Mass.)

Primoz Ziherl, Andrej Kosmrlj (Ljubljana)

Acknowledgments

Work supported by the NSF MRSEC program

• Can we predict the phase behavior and materials properties of liquid crystals?

• What physical mechanisms lead to liquid crystalline order?

Paths to liquid crystalline order

Are there other paths to liquid crystallinity?

nanophasesegregation

shapeanisotropy

blockcopolymers

chromoniclyotropics

amphiphiliclyotropics thermotropics

colloids &polyelectrolytes

fd virus

‘topology-frustrated phase separation’ ‘phase space frustration’

‘Radial frustration’: a novel route to liquid crystalline order

• Systems of particles with soft, repulsive, isotropic pair potentials can exhibit clustering instabilities (soft repulsion = effective attraction)

• Competition between clustering instabilities and excluded volume interactions leads to extraordinarily rich phase behavior– Liquid crystal phases– Wide variety of unusual crystal structures– Quasicrystals

• Similar clustering behavior can arise due to competition between short-range attraction and long-range repulsion– Frustrated phase separation leads to liquid crystalline ordering

• Spherically symmetric particles can form liquid crystal phases!

M. A. Glaser et al., Europhys. Lett. 78, 46004 (2007)

Cluster phases• Cluster phases tend to appear when bulk phase

separation is frustrated– Competition between long-range repulsion and short-range

attraction– Coulomb-frustrated phase separation

• But soft shoulder mechanism is different?– Cluster phases are ground state for soft shoulder systems, due

to clustering instability– But introducing frustration (e.g., via excluded volume

interactions) leads to extraordinary diverse phase behavior• Multiple clustering mechanisms can be present in a given

system

“Without standards for the publication of small-angle scattering data, and especially for structural results in the protein field, the broader nonexpert community is bound to exercise great caution.”

Void structure in colloidal dispersions

K. Ito, H. Yoshida, and N. Ise, Science 263, 66 (1994)

• Confocal laser scanning microscopy

• Dispersions of charged latex particles (D = 0.96 μm, σ = 12.4 μC/cm2) in density-matched solvent

• Voids form after several hours

• Suggests long-ranged attractive interaction between like-charged macroions

Origin of heterogeneity and dynamical arrest in attractive colloids

• Traditional viewpoint:- characteristic size of aggregates determined by kinetic

factors (‘dynamical arrest’) [M. E. Cates et al., J. Phys.: Condens. Matter 16 S4861–S4875 (2004)]

• Alternative viewpoint:- characteristic size of aggregates determined by

equilibrium physics (emergent clustering lengthscale)- subsequent gelation of equilibrium clusters leads to

‘cluster glass transition’ [F. Sciortino et al., Phys. Rev. Lett. 93, 055701 (2004); K. Kroy et al., Phys. Rev. Lett. 92, 148302 (2004)]

• PMMA particles in density- and index-matched solvent, Debye screening length ≈ 12 nm

• Short-range depletion attraction induced by nonadsorbing PS polymer• Suggests long-ranged repulsive interaction of unknown origin

“When the going gets tough, the tough get going.”

Joseph P. Kennedy

“When the going gets weird, the weird turn pro.”

Hunter S. Thompson

The self-assembly problem• How are condensed phase properties encoded in the

chemical structure of molecular constituents?– A formidably difficult statistical mechanics problem– Self-assembly can be governed by kinetics or equilibrium

thermodynamics– Frustration plays a key role in soft materials

• Goals:– Interpret experiments– Predict materials properties– Facilitate materials design– Develop intuition

• Methods:– Ab initio quantum mechanics– Atomistic and coarse-grained molecular simulation– Liquid state theory, density functional theory– Phenomenological theory– Continuum modeling

A brief overview of liquid state theory

Much of liquid state theory deals with the pair distribution function g(r), which is defined in terms of the density-density correlation function, G(r),

through the relation:

Here, the number density of point particles at point r is:

g(r) measures the probability of finding a second particle at a distance r from a given particle, and approaches 1 for large r

Fluid structure can be characterized in reciprocal space via the structure factor S(q),

where

are the Fourier components of the (instantaneous) density (r)

S(q) can be expressed in terms of the Fourier transform of g(r):

S(q) can also be interpreted as a response function; the linear response (q) of a homogeneous fluid to a weak external field (q) is:

which serves to define the susceptibility (q)

Much of liquid state theory devolves from the Ornstein-Zernike (OZ) relation,

which relates the pair correlation function h(r) = g(r) - 1 to the direct correlation function c(r), and which defines c(r) in terms of h(r); the OZ relation cannot be solved without an additional closure relation, and much of the considerable apparatus of liquid state theory is devoted to developing approximations for c(r)

Qualitatively, the OZ relation tells us that the total correlation between a pair of particles can expressed as the sum of a direct correlation, due to the direct interaction between a pair of particles (assumed to be short ranged) and the indirect correlation, mediated by interactions with other particles

In reciprocal space, the OZ relation becomes:

where c(q) is the Fourier transform of c(r)

Workable approximation schemes for c(r) can be constructed starting from its thermodynamic definition as a functional derivative of the excess Helmholtz free energy:

where the total free energy has been split into ideal gas and excess terms, both of which are functionals of the single particle density (r):

The ideal part of the free energy is

where is the thermal de Broglie wavelength

Distinct approximations for the excess contribution to the free energy give rise to distinct approximations for c(r)

Mean-field approximationBounded potentials at high densities and temperatures are accurately described by a simple mean-field approximation for the excess free energy:

In this approximation, the direct correlation function is given by

and the Ornstein-Zernike relation becomes:

where v(q) is the Fourier transform of the pair potential

Why is this interesting?

• This is a novel (and largely unexplored) route to the the creation of exotic self-assembling materials:– Novel colloidal liquid crystal phases– Photonic materials– Nanostructured materials/metamaterials– Active materials, sensors

• The soft shoulder clustering mechanism may help explain a variety of observations of clustering behavior in colloidal and biomolecular systems

Charged colloids in the presence of non-adsorbing polymer

• Soft shoulder-type effective pair potentials can be produced by tuning polymer concentration

Effective potential between charged colloidal particles:

where

and

Derjaguin-Landau-Overbeek-Verwey potential

Asakura-Oosawa depletion potential

• Hard core/soft shoulder model displays solid-solid phase transitions and melting curve minima and maxima, similar to behavior of Cerium and Cesium

κ = 1.2 κ = 1.5

• Isotropic fluid phase unstable to the formation of ‘clumps’ at high densities and low temperatures

• Cluster crystal competes with a large number of metastable amorphous cluster configurations at low temperatures (analogous to mean-field spin glass)

repulsive step or ‘penetrable sphere’ potential

• Hard core/soft shoulder systems display a wide variety of unusual crystalline ground states, including ‘open’ structures such as the honeycomb lattice

• Both ordered (liquid crystalline?) and disordered (fingerprint) stripe phases are observed for σ1/σ0 = 2.5

Nature Materials 2, 97 (2003)

• Softened soft-shoulder potential gives rise to rich phase behavior, including Kagome lattice and stripe (liquid crystal?) phases

• Direct measurement of entropic interactions between colloidal particles in suspensions of fd bacteriophage

• A ramp-like effective potential is measured at high salt concentrations