Source-linked AI summary
Understanding shape entropy through local dense packing
Greg van Anders, Daphne Klotsa, N. Khalid Ahmed, Michael Engel, Sharon C. Glotzer
TL;DR
Shape entropy contributes to phase behavior through directional entropic forces, but their microscopic origin and strength had not been rigorously established. The paper defines and computes these forces by integrating out surrounding particles and analyzing pair PMFTs, finding few-kBT forces near crystallization and attributing them to entropy-maximizing local dense packing.
Problem
The origin and strength of directional entropic forces, and the role of shape entropy in laboratory systems, were not clearly established.
Method
The paper defines directional entropic forces for arbitrary systems, computes pair PMFTs, and integrates out surrounding sea particles using simulation-based statistical methods.
Results
A few kBT just below crystallization, directional entropic forces emerge in example hard-particle systems and arise from particles adopting local dense-packing configurations.
Takeaways & Limitations
Shape entropy can be compared directly with intrinsic particle forces and contributes to phase behavior when those forces are a few kBT or less.
Abstract
from arXiv · showhide
Entropy drives the phase behavior of colloids ranging from dense suspensions of hard spheres or rods to dilute suspensions of hard spheres and depletants. Entropic ordering of anisotropic shapes into complex crystals, liquid crystals, and even quasicrystals has been demonstrated recently in computer simulations and experiments. The ordering of shapes appears to arise from the emergence of directional entropic forces (DEFs) that align neighboring particles, but these forces have been neither rigorously defined nor quantified in generic systems. Here, we show quantitatively that shape drives the phase behavior of systems of anisotropic particles upon crowding through DEFs. We define DEFs in generic systems, and compute them for several hard particle systems. We show that they are on the order of a few kT at the onset of ordering, placing DEFs on par with traditional depletion, van der Waals, and other intrinsic interactions. In experimental systems with these other interactions, we provide direct quantitative evidence that entropic effects of shape also contribute to self-assembly. We use DEFs to draw a distinction between self-assembly and packing behavior. We show that the mechanism that generates directional entropic forces is the maximization of entropy by optimizing local particle packing. We show that this mechanism occurs in a wide class of systems, and we treat, in a unified way, the entropy-driven phase behavior of arbitrary shapes incorporating the well-known works of Kirkwood, Onsager, and Asakura and Oosawa.
I. INTRODUCTION
Shape influences structure through both intrinsic properties and an emergent geometric effect termed shape entropy. The paper investigates how shape entropy produces directional entropic forces and contributes to ordering, especially as crowded packing begins to dominate intrinsic interactions.
- I. INTRODUCTION: Computer simulations show that shape entropy becomes important at moderate density in sterically interacting model particles.Laboratory systems provide less control for isolating this effect, but it is expected to appear when packing dominates intrinsic interactions.
- I. INTRODUCTION: Entropy-driven behavior spans hard-sphere crystallization, hard-rod nematic ordering, and colloid-polymer depletion, but microscopic-to-macroscopic links remain difficult to establish.Global packing arguments explain some mixtures and polygons but not many other shapes, including tetrahedra and related particles.
- I. INTRODUCTION: Directional entropic forces were proposed to explain face-to-face alignment in convex polyhedral crystals, but their origin and strength were unclear.The paper addresses this gap by constructing a quantitative framework for arbitrary systems.
- I. INTRODUCTION: The paper quantifies pairwise directional entropic forces, connects them to local packing, and places monodisperse hard particles and colloid-depletant systems within one framework.It also distinguishes self-assembly from packing behavior and considers when shape entropy competes with intrinsic forces.
II. METHODS
The paper defines effective pair interactions by integrating out surrounding particles and evaluates them with Monte Carlo simulations and the potential of mean force and torque. For excluded-volume systems, the formulation incorporates steric constraints, sea-particle free volume, and a generalization of depletion theory to arbitrary shapes.
- II. METHODS: Monte Carlo simulations evaluate hard-particle systems with translations and rotations of 1000 particles at fixed volume, while depletant free volume is computed by Monte Carlo integration.Polyhedra overlaps are checked with the GJK algorithm.
- II. METHODS: Directional entropic forces are quantified at arbitrary density using the potential of mean force and torque.The PMFT provides an effective interaction for particle pairs after accounting for surrounding particles.
- II. METHODS: The framework fixes a particle pair, integrates over the degrees of freedom of the remaining sea particles, and uses relative position-orientation coordinates.These coordinates are invariant under translating and rotating the pair; generic three-dimensional pairs have six scalar degrees of freedom.
- II. METHODS: For excluded-volume pairs, a Heaviside factor makes the effective potential infinite for overlapping configurations using the minimum pair separation as the steric criterion.The minimum separation is negative for overlap and positive otherwise.
- II. METHODS: In the penetrable hard-sphere limit, the sea contribution is evaluated from the free volume available to ideal-gas depletants.The resulting formulation generalizes the Asakura–Oosawa depletion result from spherical particles to arbitrary shapes.
A. Entropic Forces in Monodisperse Hard Systems
In dense monodisperse hard-particle fluids and crystals, PMFT measurements reveal shape-dependent directional attractions concentrated around facet centers. Increasing density strengthens these effects, supporting local dense packing as their microscopic origin and explaining enhanced face-to-face alignment.
- A. Entropic Forces in Monodisperse Hard Systems: Face-to-face configurations are expected to correspond to the deepest effective-attraction wells for polyhedral particle pairs.The PMFT is used to identify and quantify these directional preferences.
- A. Entropic Forces in Monodisperse Hard Systems: As density increases, the first PMFT minimum deepens and approaches contact for hard tetrahedra, faceted spheres, and cubes.The stronger alignment is concentrated near facet centers rather than arising solely from reduced average particle separation.
- A. Entropic Forces in Monodisperse Hard Systems: Entropic patches lie at facet centers and represent geometric features that facilitate local dense packing.For polyhedra, their coordination follows the vertices of the dual polyhedron; the sea-particle contribution drives cubes together despite the pair contribution favoring separation.
- A. Entropic Forces in Monodisperse Hard Systems: Particles sharing point-group symmetry can still have different PMFT shapes and strengths, with a facet-to-truncated-vertex difference exceeding 1 kBT at φ = 0.4.This indicates that small shape changes can strongly affect dense-fluid structural coordination.
- A. Entropic Forces in Monodisperse Hard Systems: Increasing packing fraction from φ = 0.5 to φ = 0.6 strengthens PMFT anisotropy in nearly cubic faceted spheres, with comparable-point differences increasing by more than 1.5 kBT.These particles self-assemble a simple cubic lattice at sufficiently high packing fractions.
B. Entropic Forces with Penetrable Hard Sphere Depletants
The paper shows that modifying particle shape creates entropic patch sites whose interactions can tune binding specificity and alignment in systems with weak depletants. Increasing faceting or elongation strengthens patch-mediated attraction and angular dependence.
- Reduced-curvature regions on faceted particles and elongated spherocylinders act as entropic patch sites that promote locally dense packing.The two model systems continuously vary faceting or elongation while maintaining weakly interacting, small depletants.
- Increasing shape alteration strengthens attraction between the corresponding entropic patches.Patch adjacency is used to quantify this stronger attraction as shape alteration increases.
- Patch size and depletant pressure tune binding specificity, distinguishing patch-to-patch, patch-to-non-patch, and non-patch-to-non-patch binding.Increasing patch size corresponds to greater faceting in the reported curves.
- For ideal depletants, the isolated patch-torque expression is independent of depletant pressure under fixed near-contact separation and orientation conditions.The analysis fixes separation at 1% above the minimum and sets both orientations normal to the separation vector.
- For spherocylinders, larger entropic patches produce a more pronounced PMFT dependence on the angle between particle symmetry axes.This angular dependence indicates patch coordination together with a torque favoring alignment.
A. PMFT as an Effective Potential at Finite Density
The PMFT provides a restricted effective-potential description of anisotropic particle pairs at finite density, separating bare interactions, pair configurational entropy, and sea-particle free energy. Its entropic contribution favors local dense packing, producing directional alignment and distinguishing assembly from global densest packing.
- PMFT as an effective potential: The PMFT is a restricted effective potential for a reference pair, with the remaining particles integrated out as an implicit solvent.It describes two-particle behavior in the presence of the surrounding sea, not the bare interaction potential of the entire system.
- PMFT as an effective potential: The PMFT contains bare pair interactions, a Jacobian counting relative configurations, and the free energy of sea particles held around the pair.For hard systems, the third term is the entropy of sea-particle microstates available to each pair configuration.
- Directional entropic forces: At finite density, sea-particle entropy drives pairs toward denser local packing configurations, with strength scaling on the order of Pσ3/kBT.The effect grows with density because pair rearrangements alter the local stress experienced by the surrounding particles.
- Assembly versus packing: Self-assembly follows local dense-packing optimization, whereas infinite-pressure global packing removes the pair-entropy contribution and remains an all-body problem.Thus, assembled structures and densest packing structures can coincide, but need not do so.
- Directional entropic forces: Increasing spherocylinder length makes the PMFT more angle-dependent, indicating stronger entropic torques that align the particles’ patches.Short cylinders show weak angular dependence, whereas larger entropic patches produce more pronounced dependence on χ.
B. Unification
The paper unifies monodisperse hard-particle systems and depletion systems as entropy-driven systems governed by a preference for local dense packing. This framework extends across particle shape, sea-particle softness, and relative size, while recognizing boundaries in how depletants are modeled.
- Shared entropic mechanism: De Boer’s potential-of-mean-force picture likewise describes preferred pair distances as a balance between particle repulsion and sea-particle packing pressure.This links isotropic entropic interactions to the broader local-packing mechanism.
- Scope and boundaries: Depletants may be large, hard, or aspherical, but the boundary at which they cease to act as depletants remains an open modeling question.Experimental and accurate-model depletants are not freely interpenetrable, and polymer depletants are not intrinsically spherical.
- Shared entropic mechanism: Monodisperse hard shapes and colloid-depletant systems share an entropy-driven mechanism based on maximizing local dense packing.The same framework connects hard spheres, rods, tetrahedra, and colloid-depletant systems.
- Parameter-space unification: The unified family spans colloid-pair shape, sea-particle softness, and inverse sea-particle size, with monodisperse systems and penetrable depletants at limiting locations.Additional axes, including sea-particle shape and system density, are omitted for simplicity.
C. DEFs in Experimental Systems
DEFs can matter in experimental systems because their magnitude near crystallization is comparable to ordinary colloidal interactions. Particle trajectories with both positions and orientations can provide a route to measuring them directly.
- Experimental relevance: DEFs are on the order of a few kBT around the onset of crystallization, placing them in the interaction range relevant to experiments.This estimate comes from systems in which shape entropy is the only factor determining behavior.
- Experimental measurement: Confocal microscopy can extract PMFTs from experimental trajectories when anisotropic colloid positions and orientations are both measured.These measurements make direct comparison with the computed directional entropic forces possible.
V. CONCLUSIONS
The paper concludes that shape entropy drives anisotropic-particle phase behavior through directional entropic forces emerging from local dense packing. These forces become comparable to intrinsic interactions at experimentally relevant energy scales and apply broadly across soft-matter systems.
- Shape entropy drives anisotropic-particle phase behavior through emergent directional entropic forces.The conclusions identify directional entropic forces as the route by which shape entropy affects phase behavior.
- A few kBT below crystallization onset, directional entropic forces in example hard-particle systems are comparable to intrinsic particle forces.The paper places these emergent forces on the same footing as intrinsic interactions relevant to self-assembly.
- Shape entropy contributes to phase behavior when intrinsic particle forces are on the order of a few kBT or less.This threshold guides how strongly intrinsic interactions must be controlled in experiments to observe shape-entropy effects.
- Maximizing shape entropy drives particles to adopt local dense-packing configurations, generating directional entropic forces.The proposed mechanism connects local packing optimization with the emergence of effective directional forces.
- The same local-packing mechanism generates directional entropic forces across a wide class of soft-matter systems with entropy-driven phase behavior.The authors present this as a unified mechanism spanning the systems considered in the paper.
Appendix A: Supplementary Methods
The appendix develops coordinate and computational methods for representing particle configurations, calculating the PMFT, and extracting forces and torques. It uses invariant descriptions of relative position and orientation, with Cartesian facet coordinates for practical calculations.
- General Coordinate System for PMFT: Six scalar invariants describe a general pair’s relative position and orientation in three dimensions.The listed invariants include separation magnitude, orientation-derived quantities, and dot products involving the relative direction.
- General Coordinate System for PMFT: Particle orientations are represented with SU(2), the double cover of SO(3), to simplify calculations involving three-dimensional rotations.The orientations are written as rotation operators using conventions based on quantum mechanics and Pauli matrices.
- General Coordinate System for PMFT: Combining two orientations yields two scalars, three vectors, and one tensor through a Clebsch–Gordan decomposition.These components are then used to construct scalar invariants together with the relative particle position.
- Forces and Torques: For facet-resolved calculations, Cartesian coordinates adapted to particle facets are used after integrating over selected angular degrees of freedom.The appendix also explains that forces and torques follow from the negative gradient of the PMFT.
- Monodisperse Systems: The PMFT is measured by histogramming relative pair configurations from Monte Carlo trajectories and taking the logarithm of their frequencies.Relative orientations are integrated over for the main-text results, while configurations are recorded on a discrete position grid.
- Monodisperse Systems: The PMFT approximation fails where it varies rapidly or where a grid bin is partially forbidden, especially near hard-overlap boundaries.These regions are difficult to sample and require care because the effective potential is relatively hard there.
b. Penetrable Hard Sphere Systems
The appendix treats penetrable hard-sphere depletants by computing pair PMFTs from depletant free volume and relating them to explicit-depletant Monte Carlo moves. It also defines entropic binding probabilities through integration over configuration-space basins.
- Penetrable Hard Sphere Systems: The PMFT for hard colloids in penetrable hard-sphere depletants is computed from the free volume available to the depletants.The calculation isolates the free-volume contribution for fixed colloid configurations and combines it with the colloid pair potential and Jacobian.
- Penetrable Hard Sphere Systems: Free-volume changes are estimated by Monte Carlo integration over the intersection region of spheres enclosing the two colloids.Random depletants are sampled uniformly in the intersection region to estimate the fraction intersecting both colloids.
- Penetrable Hard Sphere Systems: Specific entropic binding is defined as the probability of configurations within the basin of attraction of perfect alignment at entropic patch sites.The probability is obtained by integrating the Boltzmann weight over the relevant configuration set.
- Penetrable Hard Sphere Systems: The same integration framework extends to semi-specific and non-specific binding between patch and non-patch sites.These cases are distinguished by whether one or both interacting sites are non-patch regions.
- Penetrable Hard Sphere Systems: At η = 0.4, tetrahedra, tetrahedrally facetted spheres, and cubes all prefer face-to-face alignment, but force strength and shape depend on particle shape.The DEFs are obtained as finite-difference approximations to the negative PMFT gradient in Cartesian coordinates.
- Comparison of Methods for Penetrable Hard Sphere Depletants: Free-volume calculations precisely match explicit ideal-depletant Monte Carlo simulations because the PMFT controls average colloid-move acceptance rates.The equivalence follows by comparing forward and reverse move probabilities in the small-move limit.
Appendix B: Supplementary Results
The supplementary results quantify directional entropic forces through PMFT calculations, showing how density, particle shape, and sea-particle stress produce alignment and effective attraction. They also establish scope limits for pair-potential descriptions and connect shape entropy with depletion-like interactions.
- Entropic Forces In Monodisperse Hard Systems: PMFT-derived forces in tetrahedra, tetrahedrally faceted spheres, and cubes at η = 0.4 are on the order of kBT/σ.Finite-difference gradients of the PMFT show repulsion near tetrahedron vertices and attraction at face centers; removing vertices removes the repulsion but preserves face attraction.
- Supplementary Results: A few kBT of angular PMFT differences at 50% density generate entropic torques that strongly favor alignment in hard hemispheres.The reported potential differences occur at fixed separation distance and produce alignment even when angular differences are small.
- Supplementary Results: Shape enhances face-to-face contact beyond the density-driven effective attraction expected along a cube-face axis.The vertex-crossing profile in Fig. 10 shows a less pronounced density-dependent attraction than the face-centered profile in Fig. 2c.
- Penetrable Hard Sphere Limit: Sea particles generate effective forces through boundary stress on the pair, with force scale Pσ3/kBT when the stress tensor is isotropic.Moving the pair changes the effective box available to sea particles, whose stress tensor acts on the steric boundary.
- Penetrable Hard Sphere Limit: Correlations among sea particles make locally preferred dense packing differ from the globally densest pair packing, while smaller soft depletants further strengthen monodisperse directional entropic forces.Experimental depletants commonly induce interactions of ∼4 kBT to ∼8 kBT, with estimates reaching hundreds of kBT at very high concentrations.
- Many-Body Interactions: Pair PMFTs cannot generally capture higher-density many-body effects, and accurate n-body extensions become increasingly difficult as n increases.Octahedra provide an example where coincident face-to-face alignment is not preferred; the paper attributes such behavior to many-body effects captured by a many-body PMFT.