^{1}

## 1 Introduction

Microgels are soft, compressible colloidal particles, typically composed of cross-linked
polymer networks that, when dispersed in a solvent, can swell significantly in size and
can respond sensitively to environmental changes. Equilibrium particle sizes are
determined both by the elasticity of the gel network and by temperature, particle
concentration, and solvent quality.^{1, 2, 3, 4}
Ionic microgels, which acquire charge via dissociation of counterions, further respond to
changes in H and salt concentration. Tunable particle size results in unusual materials
properties, with practical applications to filtration, rheology, and drug delivery.^{5, 6, 7, 8}

Over the past two decades, numerous experimental and modeling studies have characterized
the elastic properties of single microgel particles^{9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}
and the equilibrium and dynamical behavior of bulk suspensions.^{25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}
Connections between single-particle properties, such as swelling ratio, and bulk properties,
such as osmotic pressure, thermodynamic phase behavior, pair structure, and viscosity, have been
probed experimentally by static and dynamic light scattering, small-angle neutron scattering,
confocal microscopy, and osmometry.^{37, 38, 39, 40, 41, 42, 43, 44, 45}
While the swelling/deswelling behavior of microgels has been extensively studied, the
full implications of elasticity and compressibility of these soft colloids for bulk
suspension properties remain only partially understood.

Previous modeling studies have examined the influence of elastic interparticle
interactions on structure and phase behavior. In an extensive simulation survey,
Pamiès et al. ^{46} mapped out the thermodynamic phase diagram of
a model of elastic, but incompressible, spheres interacting via a Hertz pair
potential.^{47} For a related model of ionic microgels,
one of us and coworkers^{48} recently applied
molecular dynamics simulation and thermodynamic perturbation theory to compute the
osmotic pressure and static structure factor of a suspension of Hertzian spheres that
interact also electrostatically, via an effective Yukawa pair potential.^{49}
At the single-particle level, numerous studies have established that the classic
Flory-Rehner theory of swelling of cross-linked polymer networks,^{50}
though originally developed for macroscopic gels, provides a reasonable description
also of the elastic properties of microgel particles.^{17, 18, 33, 34, 20, 16, 21, 19, 22, 14, 13, 12, 3, 15, 4, 51}
However, the implications of particle compressibility and intrinsic size polydispersity
for bulk properties of microgel suspensions have not been fully explored.
The purpose of this paper is to analyze the combined influences of particle elasticity,
compressibility, and associated size fluctuations on bulk thermal and structural properties
of microgel suspensions.

The remainder of the paper is organized as follows. In Sec. 2, we describe a model of microgels as compressible spheres, whose swelling is governed by a single-particle free energy derived from the Flory-Rehner theory of cross-linked polymer networks, and whose interparticle interactions are represented by a Hertz elastic pair potential. In Sec. 3, we outline Monte Carlo simulation methods by which we modeled bulk suspensions of compressible, size-fluctuating microgels. Section 4 presents numerical results for thermodynamic properties, specifically osmotic pressure and liquid-solid phase behavior, and structural properties, including particle volume fraction, radial distribution function, and static structure factor. Finally, in Sec. 5, we summarize and conclude.

## 2 Model

### 2.1 Swelling of Microgel Particles

We model a microgel as a spherical particle of dry (collapsed, unswollen) radius
and swollen radius , consisting of a cross-linked network of monomers, with
a uniform distribution of cross-linkers that divide the network into
distinct chains (see Fig. 1). Although idealized, this simple model
provides a reference that can be generalized to heterogeneous microgels with
nonuniform distribution of cross-linkers, such as core-shell^{52, 19, 17, 41, 18, 53} or
hollow^{54, 55, 56} microgels.
To highlight the interplay between intra- and inter-particle elasticity, without the
added complexity of electrostatic interactions, we consider here only nonionic microgels.

For the uniform-sphere model, with particle size swelling ratio ,
the Flory-Rehner theory of polymer networks combines mixing entropy, polymer-solvent interactions,
and elastic free energy to predict the total Helmholtz free energy of the network:^{50}

(1) | |||||

where at temperature and is the polymer-solvent interaction
(solvency) parameter.
Swelling of a microgel particle results from stretching of polymer coils, subject to
the constraints of a cross-linked network.^{57}
Of the two terms in square brackets in Eq. (1), the first accounts for
the entropy of mixing of the microgel with solvent molecules, setting to zero
the number of individual polymer chains in the network structure. The second term
represents a mean-field approximation for the interaction between polymer monomers
and solvent molecules, which entirely neglects interparticle correlations.
The last term in Eq. (1) describes the elastic free energy associated with
stretching the microgel, assuming isotropic deformation, neglecting any change in
internal energy of the network, and modeling the polymer chains as Gaussian coils.
The assumption of a purely entropic elastic free energy ignores enthalpic contributions
stemming from structural changes in surrounding solvent upon stretching a polymer coil,
while the Gaussian coil approximation is valid only for swelling ratios not exceeding
the polymer contour length.^{56}
Given the Flory-Rehner free energy [Eq. (1)], the size of a single,
isolated microgel (i.e., in a dilute solution) fluctuates according to a
probability distribution,

(2) |

In thermal equilibrium, an isolated, swollen microgel in a dilute solution has a most probable radius , corresponding to the maximum of this distribution (i.e., minimum free energy). A suspension of particles thus has a fluctuating particle size distribution, i.e., dynamical size polydispersity, governed by Eq. (2).

### 2.2 Microgel Pair Interactions

While swelling of isolated microgels in dilute solutions is governed only by intraparticle
interactions, modeled by Eq. (1), swelling in concentrated solutions is
influenced also by interparticle interactions. To account for this additional influence,
we model the repulsion between a pair of microgel particles,
of instantaneous radii and at center-to-center separation ,
via a Hertz effective pair potential,^{47}

(3) |

The total internal energy associated with pair interactions is then given by

(4) |

where is the distance between the centers of particles and . For particles of average volume , with angular brackets denoting an ensemble average over configurations, the Hertz pair potential amplitude is

(5) |

which depends on the elastic properties of the gel through Young’s modulus
and the Poisson ratio .^{47} Scaling theory of
polymer gels in good solvents^{58} predicts that the bulk modulus
scales linearly with temperature and density of cross-linkers (or chain number):
. The denser the cross-links, the stiffer the gel.
It follows that the reduced amplitude, , is
essentially independent of temperature and particle volume, neglecting
dependence of on , and scales linearly with .

The elastic properties of bulk, water-swollen hydrogels have been measured using
scanning force microscopy.^{59} For poly(N-isopropylacrylamide)
(PNIPAM) hydrogel with 0.25 mol % cross-linker, Young’s modulus was determined
as 0.33 kPa in the fully swollen state at 10 C and 13.9 kPa in the
collapsed state at 40 C. In contrast, poly(acrylamide) (PA) hydrogels
with cross-linker between 1 and 5 mol % have Young’s moduli measured in the range
from 90 to 465 kPa at 20 C. Typical Poisson ratios for these gels are
. In Sec. 4, we appeal to such measurements to select
realistic pair potential parameters for our simulations.

### 2.3 Suspensions of Swollen Microgels

At constant , the equilibrium thermodynamic state of a suspension of
particles in a volume depends on the average number density, .
For a suspension of microgels, the dry volume fraction, ,
is defined as the fraction of the total volume occupied by the particles in their
dry state. Correspondingly, the swollen volume fraction is defined as the ratio
of the most probable volume of a swollen particle to the volume per particle:
. In suspensions of highly swollen
particles, can substantially exceed . For reference, we also define
the generalized volume fraction^{22, 18} as
the ratio of the volume of a particle of most probable size in the dilute limit
to the volume per particle: . Note that
(since ) and that and have no upper bounds.
In particular, in concentrated suspensions beyond close packing, it is possible that
and . At such high concentrations, crowded microgel particles
may not only compress in size, but may also distort in shape. The presently studied
model allows the former, but not the latter, response to crowding, although the
Hertz potential may be interpreted as allowing for faceted deformations. In a
more refined model, the microgels could be represented as
elastic spheres^{60, 61, 62}
or ellipsoids^{63, 64, 65, 66}
that can compress along three axes with an associated deformation energy.

## 3 Computational Methods

To study the influence of particle compressibility and fluctuating size polydispersity
on thermal and structural properties of bulk microgel suspensions, we performed a series
of constant- Monte Carlo (MC) simulations of systems of particles modeled by the
Flory-Rehner size distribution [Eq. (2)] and the Hertz pair potential
[Eq. (3)], as described in Sec. 2. For particles in a cubic box
of fixed volume with periodic boundary conditions at fixed temperature ,
we made trial moves consisting of combined particle displacements and size changes.
Following the standard Metropolis algorithm,^{67, 68}
a trial displacement and change in swelling ratio, from to ,
was accepted with probability

(6) |

where is the change in internal energy [Eq. (4)] associated with interparticle interactions and is the change in free energy [Eq. (1)] associated with swelling. At equilibrium, the particles adopt a size distribution that depends on the dry particle volume fraction and minimizes the total free energy of the system. Since the particles modeled here are compressible and polydisperse, the generalized volume fraction can exceed close packing of monodisperse hard spheres (). Such high concentrations create a strong interplay between the Hertz elastic energy and the Flory-Rehner swelling free energy.

To guide analysis of phase behavior, we computed structural properties that probe interparticle pair correlations and thermodynamic properties that govern phase stability. First, we determined the radial distribution function by standard means, histogramming into radial bins the radial separation of particle pairs in each configuration and averaging over configurations. Second, we computed the static structure factor , proportional to the Fourier transform of and to the intensity of scattered radiation at scattered wave vector magnitude . For a system of particles, the orientationally averaged two-particle static structure factor is defined as

(7) |

where angular brackets again denote an ensemble average over particle configurations – the same configurations as used to compute . Finally, we computed the osmotic pressure of the system from the virial theorem

(8) |

where is the Hertz pair force.

## 4 Results and Discussion

From simulations of particles initialized on an fcc lattice, we analyzed the equilibrium particle swelling ratio, structural properties, and osmotic pressure over a wide range of densities. The results presented below represent statistical averages of particle coordinates and radii over 1000 independent configurations, separated by intervals of 100 MC steps (total of steps), following an initial equilibration phase of MC steps, where a single MC step is defined as one combined trial displacement and size change of every particle. To confirm that the system had equilibrated, we computed the total internal energy and pressure and checked that both quantities had reached stable plateaus. A typical snapshot of the system is shown in Fig. 2.

In the low-density (dilute) regime, the initial lattice structure rapidly fell apart,
as particles freely diffused through the system, implying a stable fluid phase.
In the high-density (concentrated) regime, strongly interacting particles remained
in the fcc lattice structure, consistent with a stable or metastable crystalline phase.
At intermediate densities, relative stability of fluid and solid phases depended on
the interparticle interactions. We emphasize that we did not attempt to determine
thermodynamic phase stability, which would require more extensive simulations and
thermodynamic integration to compute total free energies and perform coexistence analyses.
Nor did we investigate the glass transition,^{69} for which purpose
molecular dynamics simulation would be better suited.
Nevertheless, our results suggest that compressibility of microgels may significantly
alter the phase diagram of strictly Hertzian spheres.^{46}

To explore dependence of bulk properties on particle size and pair interactions,
we studied two systems distinguished by widely different particle sizes and interactions.
The first system is a suspension of relatively small microgels (nanogels) of
dry radius nm, comprising monomers – consistent
with monomers of radius 0.3 nm – and chains, corresponding
to 0.05 mol % of cross-linker (assuming two chains per cross-linker).
Focusing on polymers in a good solvent, e.g., PNIPAM or PAA in water, we took .
For these loosely cross-linked particles, the Flory-Rehner free energy in the
dilute limit [Eq. (1)] attains a minimum at swollen radius nm
(). For particles of this size with Young’s modulus kPa
and Poisson ratio ,^{59} Eq. (5) yields a
reduced Hertz pair interaction amplitude .
To explore sensitivity to particle elasticity, we also considered microgels
with .

The second system studied contains much larger particles, of dry radius nm,
comprising monomers, more closely matching many experimentally
studied materials. To again model loosely cross-linked particles in a good solvent,
we chose (0.05 mol % cross-linker) and .
The Flory-Rehner free energy [Eq. (1)] is now a minimum at swollen radius
nm (dilute limit). Young’s moduli in the range kPa correspond
to reduced Hertz pair interaction amplitudes .
We note that the densities and temperatures ( values) studied here
correspond to thermodynamic states that fall well within the range of fcc crystal
stability in the known phase diagram of Hertzian spheres.^{46} Indeed,
we never observed structural reassembly from fcc into bcc or any other crystalline structure.

Normalized probability distributions for the equilibrium swelling ratio are shown in
Figs. 3 and 4 over a range of concentrations. Below close packing
(), steric (Hertzian) interparticle interactions are sufficiently weak that
the particles are not significantly compressed, as reflected by the distributions differing negligibly
from the intrinsic dilute-limit distribution [Eq. (2)]. With increasing ,
as close packing is approached and exceeded, the distributions not only shift toward smaller ,
as particles become compressed, but also broaden, i.e., become more polydisperse.
The smaller particles (Fig. 3) have a polydispersity (ratio of standard deviation
to mean) that increases from roughly 2% to 4% as increases from 0 to 2.
In contrast, the larger particles (Fig. 4) have negligible polydispersity
when uncrowded, but fluctuate significantly in size for . For both
small and large particles, with increasing Hertz pair potential amplitude, the
degree of compression and the breadth of polydispersity increase.
The reason why the polydispersity distribution broadens with increasing concentration
is that compression forces particles into a range of sizes in which the curvature
of the Flory-Rehner free energy [Eq. (1)] with respect to is
weaker than for uncompressed particles.
Measurements of particle size in suspensions of microgels whose dry radii have a static
polydisperse^{70} or bidisperse^{71} distribution show that
swollen particles have an equilibrium polydispersity that decreases with
increasing particle compression. Conversely, our study demonstrates that microgels
whose dry radius is monodisperse have swollen polydispersity – associated purely with
fluctuations in swelling ratio – that increases with particle compression.

Particle compressibility is illustrated further in Fig. 5, which plots the volume fraction vs. generalized volume fraction . Below close packing, where particle compression is negligible, . With increasing density, however, interparticle interactions become stronger and so energetically costly, relative to the free energy cost of compression, that the particles more readily contract to minimize interactions. As a result, the volume fraction is lower than the generalized volume fraction () for , the density at which nearest-neighbor particles in the fcc crystal would begin to overlap. For these systems, significant particle compression occurs only in the solid phase. Moreover, the higher the Hertz pair potential amplitude – for given values of , , and in the Flory-Rehner free energy – the more easily the particles yield to compression.

To explore the evolution of structure as a function of density and interparticle interactions, and to aid our diagnosis of the liquid-solid phase transition, we computed radial distribution functions and static structure factors, as described in Sec. 3. Figure 6 shows our results for over a range of densities that span the freezing transition. The emergence of distinct peaks signals the onset of crystallization at for nm and for nm (black curves in Fig. 6). For both particle sizes, the freezing density is insensitive to over the range of values considered. Suspensions of larger microgel particles thus crystallize at the same volume fraction as hard-sphere fluid, while suspensions of smaller nanogel particles remain fluid up to significantly higher volume fractions. This interesting difference in phase stability can be attributed to the relatively softer Hertz pair repulsion between the smaller particles.

Figure 7 shows corresponding results for , computed using Eq. (7)
by averaging over the same configurations used to compute in Fig. 6.
With increasing density, the peaks progressively sharpen and shift toward larger .
At the densities at which distinct peaks in indicate crystallization,
the main peak of attains a height in the range .
This freezing threshold is consistent with the Hansen-Verlet criterion for fluids
interacting via a Lennard-Jones pair potential,^{72} according to
which at freezing. In passing, we note that a different
freezing criterion proposed specifically for ultrasoft pair-potential
fluids^{73} does not apply here, since the Hertz pair potential,
although bounded, has amplitudes in our system that far exceed the thermal energy .
As a consistency check, we also computed the Lindemann parameter , defined as
the ratio of the root-mean-square displacement of particles from their equilibrium
lattice sites to the lattice constant, and confirmed that and remains
constant in the solid phase.

Finally, we turn to the osmotic pressure, which we computed from our simulations using
Eq. (8). Figure 8 presents plots of osmotic pressure vs. generalized
volume fraction (equation of state) for the two microgel systems studied. In the dilute regime,
the osmotic pressure is very close to that of the hard-sphere fluid, which is consistent
with the swelling ratio results (Fig. 5). At volume fractions above 20%,
deviations emerge and steadily grow with increasing volume fraction. The compressible
microgels have osmotic pressures consistently lower than for the hard-sphere fluid.
At volume fractions approaching close packing, the microgel osmotic pressure rapidly rises,
yet remains finite beyond close packing, in sharp contrast to the hard-sphere fluid,
whose pressure diverges at close-packing. Furthermore, as is clear from Fig. 8a,
the osmotic pressure rises more gradually with concentration the smaller and softer
the particles. This equilibrium trend may have implications for rheological properties,
such as shear viscosity. In fact, recent measurements^{74} of
the zero-shear viscosity of aqueous suspensions of monodisperse, highly branched,
phytoglycogen nanoparticles^{75} are consistent with
significant compression of the particles.

At the apparent freezing density of the microgel suspensions, as diagnosed by the main peak height of the static structure factor, the osmotic pressure plateaus, consistent with a thermodynamic phase transition between fluid and solid. Thus, the freezing behavior of the compressible microgels modeled here seem to be accurately described by the Hansen-Verlet criterion. Our methods are not suited, however, to identifying a glass transition, which could preempt crystallization.

Finally, to probe the dependence of system properties on solvent quality, we have repeated several of the calculations for , modeling a theta solvent. As increases, the particles become progressively compressed, as would be expected with worsening solvent quality, and exert lower osmotic pressure. However, with varying concentration, the systems display similar qualitative trends in polydispersity, volume fraction, structure, and osmotic pressure.

## 5 Conclusions

In summary, we designed and implemented Monte Carlo computer simulations of model suspensions of compressible, soft, spherical, colloidal particles that fluctuate in size and interact via an elastic Hertz pair potential. For this purpose, we introduced a novel trial move that allows particles to expand and contract, according to the Flory-Rehner free energy of cross-linked polymer networks, in response to interactions with neighboring particles. From a series of simulations over ranges of particle parameters chosen to be consistent with experimental systems, we analyzed the dependence of interparticle correlations and thermodynamic behavior on particle size (dry radius) and softness (elastic modulus) by computing a variety of thermal and structural properties.

Radial distribution functions, static structure factors, and osmotic pressures all display behavior similar to that of a hard-sphere fluid at low densities – volume fractions below about 0.3 – but reveal the particles’ intrinsically soft nature at higher densities. Swelling ratios confirm that, with increasing density above close packing, the particles become progressively compressed and polydisperse, compared with their dilute (uncrowded) states. While suspensions of relatively large, stiff particles (microgels of dry radius 100 nm) crystallize at the same volume fraction as a fluid of hard spheres, internal degrees of freedom associated with particle compressibility and size polydispersity enable suspensions of smaller, softer particles (nanogels of dry radius 10 nm) to remain in a stable fluid phase up to densities significantly beyond freezing of the hard-sphere fluid. For both nanogels and microgels, the freezing transition is accurately predicted by the Hansen-Verlet criterion.

Our calculations of equilibrium swelling ratios and bulk osmotic pressures are qualitatively consistent
with observations of significant compression of deformable particles, albeit ionic microgels,
only at volume fractions approaching and exceeding close packing.^{17, 24}
Similarly, our finding that particle softness lowers the osmotic pressure is consistent
with experimental measurements of reduced zero-shear viscosity in suspensions of soft
nanoparticles.^{74} Furthermore, our calculations may guide the choice
of system parameters in future experiments and molecular-scale simulations and could be extended
to map out equilibrium phase diagrams.

The coarse-grained model of nonionic, spherical microgels studied here may be
refined by incorporating a more realistic swelling free energy^{56}
that goes beyond the limitations of the Flory-Rehner theory noted in Sec. 2.1,
and may be further developed in several directions.
For example, the model may be extended to ionic microgels, whose properties
emerge from an interplay between elastic and electrostatic interparticle interactions.
Such an extension would enable analyzing the coupled influences of fluctuating particle size
and counterion distribution on swelling, structure, and phase behavior of concentrated
suspensions of soft, charged colloids. By incorporating shape fluctuations, our simulation
method also may be generalized to model dynamically deformable particles, as in recent studies
of elastic colloids,^{60, 61, 62} and
extended to study the influence of added hard crowders, as in our previous studies
of polymer-nanoparticle mixtures.^{64, 65, 66}
Finally, rheological properties, such as diffusion coefficients, viscosity, elastic moduli,
jamming and glass transition densities^{69} could be explored by wedding
our model of soft, compressible, size-fluctuating colloids to appropriate simulation methods,
such as Brownian dynamics, lattice Boltzmann, or molecular dynamics of
micromechanical models.^{61, 62}

Acknowledgments

This work was partially supported by the National Science Foundation (Grant No. DMR-1106331).

### Footnotes

- footnotetext: Department of Physics, North Dakota State University, Fargo, ND 58108-6050, USA. E-mail: alan.denton@ndsu.edu

### References

- W. O. Baker, Ind. Eng. Chem., 1949, 41, 511–520.
- R. H. Pelton and P. Chibante, Colloids Surf., 1986, 20, 247–256.
- R. H. Pelton, Adv. Colloid Interface Sci., 2000, 85, 1–33.
- B. R. Saunders, N. Laajam, E. Daly, S. Teow, X. Hu and R. Stepto, Adv. Colloid Interface Sci., 2009, 147, 251.
- Hydrogel Micro and Nanoparticles, ed. L. A. Lyon and M. J. Serpe, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2012.
- Microgel Suspensions: Fundamentals and Applications, ed. A. Fernández-Nieves, H. Wyss, J. Mattsson and D. A. Weitz, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011.
- L. A. Lyon and A. Fernández-Nieves, Annu. Rev. Phys. Chem., 2012, 63, 25–43.
- P. J. Yunker, K. Chen, D. Gratale, M. A. Lohr, T. Still and A. G. Yodh, Rep. Prog. Phys., 2014, 77, 056601–1–29.
- R. Borrega, M. Cloitre, I. Betremieux, B. Ernst and L. Leibler, Euro. Phys. Lett., 1999, 47, 729–735.
- M. Cloitre, R. Borrega, F. Monti and L. Leibler, C. R. Physique, 2003, 4, 221–230.
- B. H. Tan, K. C. Tam, Y. C. Lam and C. B. Tan, J. Rheol., 2004, 48, 915–926.
- A. Fernández-Nieves, A. Fernández-Barbero, B. Vincent and F. J. de las Nieves, Macromol., 2000, 33, 2114–2118.
- A. Fernández-Nieves, A. Fernández-Barbero, B. Vincent and F. J. de las Nieves, J. Chem. Phys., 2003, 119, 10383–10388.
- J. J. Liétor-Santos, B. Sierra-Martín, R. Vavrin, Z. Hu, U. Gasser and A. Fernández-Nieves, Macromol., 2009, 42, 6225–6230.
- Y. Hertle, M. Zeiser, C. Hasenöhrl, P. Busch and T. Hellweg, Colloid Polym. Sci., 2010, 288, 1047.
- P. Menut, S. Seiffert, J. Sprakel and D. A. Weitz, Soft Matter, 2012, 8, 156–164.
- G. Romeo, L. Imperiali, J.-W. Kim, A. Fernández-Nieves and D. A. Weitz, J. Chem. Phys., 2012, 136, 124905–1–9.
- G. Romeo and M. P. Ciamarra, Soft Matter, 2013, 9, 5401–5406.
- J. J. Liétor-Santos, B. Sierra-Martín, U. Gasser and A. Fernández-Nieves, Soft Matter, 2011, 7, 6370–6374.
- B. Sierra-Martín and A. Fernández-Nieves, Soft Matter, 2012, 8, 4141–4150.
- J. J. Liétor-Santos, B. Sierra-Martín and A. Fernández-Nieves, Phys. Rev. E, 2011, 84, 060402(R)–1–4.
- B. Sierra-Martín, Y. Laporte, A. B. South, L. A. Lyon and A. Fernández-Nieves, Phys. Rev. E, 2011, 84, 011406–1–4.
- S. M. Hashmi and E. R. Dufresne, Soft Matter, 2009, 5, 3682–3688.
- M. Pelaez-Fernandez, A. Souslov, L. A. Lyon, P. M. Goldbart and A. Fernández-Nieves, Phys. Rev. Lett. , 2015, 114, 098303–1–5.
- T. G. Mason, J. Bibette and D. A. Weitz, Phys. Rev. Lett. , 1995, 75, 2051–2054.
- F. Gröhn and M. Antonietti, Macromol., 2000, 33, 5938–5949.
- Y. Levin, Phys. Rev. E, 2002, 65, 036143–1–6.
- A. Fernández-Nieves and M. Márquez, J. Chem. Phys., 2005, 122, 084702–1–6.
- S. P. Singh, D. A. Fedosov, A. Chatterji, R. G. Winkler and G. Gompper, J. Phys.: Condens. Matter, 2012, 24, 464103.
- R. G. Winkler, D. A. Fedosov and G. Gompper, Curr. Opin. Colloid Interface Sci., 2014, 19, 594 – 610.
- X. Li, L. E. Sánchez-Diáz, B. Wu, W. A. Hamilton, P. Falus, L. Porcar, Y. Liu, C. Do, A. Faraone, G. S. Smith, T. Egami and W.-R. Chen, ACS Macro Lett., 2014, 3, 1271–1275.
- S. A. Egorov, J. Paturej, C. N. Likos and A. Milchev, Macromol., 2013, 46, 3648–3653.
- T. Colla, C. N. Likos and Y. Levin, J. Chem. Phys., 2014, 141, 234902–1–11.
- T. Colla and C. N. Likos, Mol. Phys., 2015, 113, 2496–2510.
- S. Gupta, M. Camargo, J. Stellbrink, J. Allgaier, A. Radulescu, P. Lindner, E. Zaccarelli, C. N. Likos and D. Richter, Nanoscale, 2015, 7, 13924–13934.
- S. Gupta, J. Stellbrink, E. Zaccarelli, C. N. Likos, M. Camargo, P. Holmqvist, J. Allgaier, L. Willner and D. Richter, Phys. Rev. Lett., 2015, 115, 128302–1–5.
- P. S. Mohanty and W. Richtering, J. Phys. Chem. B, 2008, 112, 14692–14697.
- T. Eckert and W. Richter, J. Chem. Phys., 2008, 129, 124902–1–6.
- A. N. St. John, V. Breedveld and L. A. Lyon, J. Phys. Chem. B, 2007, 111, 7796–7801.
- M. Muluneh and D. A. Weitz, Phys. Rev. E, 2012, 85, 021405–1–6.
- J. Riest, P. Mohanty, P. Schurtenberger and C. N. Likos, Z. Phys. Chem., 2012, 226, 711–735.
- P. S. Mohanty, A. Yethiraj and P. Schurtenberger, Soft Matter, 2012, 8, 10819–10822.
- D. Paloli, P. S. Mohanty, J. J. Crassous, E. Zaccarelli and P. Schurtenberger, Soft Matter, 2013, 9, 3000–3004.
- U. Gasser, J.-J. Liétor-Santos, A. Scotti, O. Bunk, A. Menzel and A. Fernández-Nieves, Phys. Rev. E, 2013, 88, 052308–1–8.
- P. S. Mohanty, D. Paloli, J. J. Crassous, E. Zaccarelli and P. Schurtenberger, J. Chem. Phys., 2014, 140, 094901–1–9.
- J. C. Pàmies, A. Cacciuto and D. Frenkel, J. Chem. Phys., 2009, 131, 044514–1–8.
- L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Elsevier, Amsterdam, 3rd edn., 1986.
- M. M. Hedrick, J. K. Chung and A. R. Denton, J. Chem. Phys., 2015, 142, 034904–1–12.
- A. R. Denton, Phys. Rev. E, 2000, 62, 3855–3864.
- P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, 1953.
- A. Moncho-Jordá and J. Dzubiella, Phys. Chem. Chem. Phys., 2016, 18, 5372–5385.
- M. Stieger, W. Richtering, J. S. Pedersen and P. Lindner, J. Chem. Phys., 2004, 120, 6197–6206.
- A. Moncho-Jordá, J. A. Anta and J. Callejas-FernÃ¡ndez, J. Chem. Phys., 2013, 138, 134902.
- M. Quesada-Pérez and A. Martín-Molina, Soft Matter, 2013, 9, 7086–7094.
- I. Adroher-Benítez, S. Ahualli, A. Martín-Molina, M. Quesada-Pérez and A. Moncho-Jordá, Macromol., 2015, 48, 4645–4656.
- A. M. Rumyantsev, A. A. Rudov and I. I. Potemkin, J. Chem. Phys., 2015, 142, 171105–1–5.
- C. N. Likos, Phys. Rep., 2001, 348, 267–439.
- P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell, Ithaca, 1979.
- T. R. Matzelle, G. Geuskens and N. Kruse, Macromol., 2003, 36, 2926–2931.
- J. Riest, L. Athanasopoulou, S. A. Egorov, C. N. Likos and P. Ziherl, Sci. Rep., 2015, 5, 15854–1–11.
- M. Cloitre and R. T. Bonnecaze, in High Solid Dispersions, ed. M. Cloitre, Springer, Heidelberg, 2010, pp. 117–161.
- J. R. Seth, L. Mohan, C. Locatelli-Champagne, M. Cloitre and R. T. Bonnecaze, Nature Mat., 2011, 10, 838–843.
- V. M. O. Batista and M. Miller, Phys. Rev. Lett., 2010, 105, 088305–1–4.
- W. K. Lim and A. R. Denton, J. Chem. Phys., 2014, 141, 114909–1–10.
- W. K. Lim and A. R. Denton, J. Chem. Phys., 2016, 144, 024904–1–10.
- W. K. Lim and A. R. Denton, Soft Matter, 2016, 12, 2247–2252.
- D. Frenkel and B. Smit, Understanding Molecular Simulation, Academic, London, 2nd edn., 2001.
- K. Binder and D. W. Heermann, Monte Carlo Simulation in Statistical Physics: An Introduction, Springer, Berlin, 5th edn., 2010.
- X. Di, X. Peng and G. B. McKenna, J. Chem. Phys., 2014, 140, 054903–1–9.
- D. K. Gupta, B. V. R. Tata, J. Brijitta and R. G. Joshi, AIP Conf. Proc., 2011, 1391, 266–268.
- A. Scotti, U. Gasser, E. S. Herman, M. Pelaez-Fernandez, J. Han, A. Menzel, L. A. Lyon and A. Fernández-Nieves, PNAS, 2016, 113, 5576–5581.
- J. P. Hansen and L. Verlet, Phys. Rev., 1969, 184, 151.
- C. N. Likos, A. Lang, M. Watzlawek and H. Löwen, Phys. Rev. E, 2001, 63, 031206–1–9.
- E. Papp-Szabo, C. Miki, H. Shamana, J. Atkinson and J. R. Dutcher, preprint, 2016.
- J. D. Nickels, J. Atkinson, E. Papp-Szabo, C. Stanley, S. O. Diallo, S. Perticaroli, B. Baylis, P. Mahon, G. Ehlers, J. Katsaras and J. R. Dutcher, Biomacromol., 2016, 17, 735–743.