Celebrating Soft Matter's 10th anniversary: Testing the foundations of classical entropy: colloid experiments

Assessment of Anharmonicities in Clusters: Developing and Validating a Minimum-Information Partition Function

Precise thermodynamic calculations are essential for understanding the dynamics of cluster systems and new particle formation. However, the widely employed harmonic statistical mechanical approach often falls short in terms of accuracy. In this study, we present an improved statistical model that incorporates vibrational anharmonicity via a novel partition function that requires only one additional system-specific input parameter. In addition to considering vibrational aspects, we also account for anharmonicity related to the configurational space. The role of anharmonicities is thoroughly examined in the case of general clusters, where the complete sets of conformers, mechanically stable spatial arrangements, are known up to clusters composed of 14 monomers. By performing consistent Monte Carlo simulations on these systems, we benchmark the statistical model's efficacy in reproducing key thermodynamic properties (formation free energy and potential energy) in the classical limit. The model exhibits exceptional alignment with simulations, accurately reproducing free energies within a precision of 2kBT and reliably capturing cluster melting temperatures. Furthermore, we demonstrate the significance and applicability of the model by reproducing thermodynamic barriers in homogeneous gas-phase nucleation of larger clusters. The transferability of our developed approach extends to more complex molecular systems and bears relevance for atmospheric multicomponent clusters, in particular.

Predicting Rubisco-Linker Condensation from Titration in the Dilute Phase

The condensation of Rubisco holoenzymes and linker proteins into "pyrenoids," a crucial supercharger of photosynthesis in algae, is qualitatively understood in terms of "sticker-and-spacer" theory. We derive semianalytical partition sums for small Rubisco-linker aggregates, which enable the calculation of both dilute-phase titration curves and dimerization diagrams. By fitting the titration curves to surface plasmon resonance and single-molecule fluorescence microscopy data, we extract the molecular properties needed to predict dimerization diagrams. We use these to estimate typical concentrations for condensation, and successfully compare these to microscopy observations.

A computational toolbox for the assembly yield of complex and heterogeneous structures

The self-assembly of complex structures from a set of non-identical building blocks is a hallmark of soft matter and biological systems, including protein complexes, colloidal clusters, and DNA-based assemblies. Predicting the dependence of the equilibrium assembly yield on the concentrations and interaction energies of building blocks is highly challenging, owing to the difficulty of computing the entropic contributions to the free energy of the many structures that compete with the ground state configuration. While these calculations yield well known results for spherically symmetric building blocks, they do not hold when the building blocks have internal rotational degrees of freedom. Here we present an approach for solving this problem that works with arbitrary building blocks, including proteins with known structure and complex colloidal building blocks. Our algorithm combines classical statistical mechanics with recently developed computational tools for automatic differentiation. Automatic differentiation allows efficient evaluation of equilibrium averages over configurations that would otherwise be intractable. We demonstrate the validity of our framework by comparison to molecular dynamics simulations of simple examples, and apply it to calculate the yield curves for known protein complexes and for the assembly of colloidal shells.

The "Real" Gibbs Paradox and a Composition-Based Resolution

There is no documented evidence to suggest that J. W. Gibbs did not recognize the indistinguishable nature of states involving the permutation of identical particles or that he did not know how to justify on a priori grounds that the mixing entropy of two identical substances must be zero. However, there is documented evidence to suggest that Gibbs was puzzled by one of his theoretical findings, namely that the entropy change per particle would amount to kBln2 when equal amounts of any two different substances are mixed, no matter how similar these substances may be, and would drop straight to zero as soon as they become exactly identical. The present paper is concerned with this latter version of the Gibbs paradox and, to this end, develops a theory characterising real finite-size mixtures as realisations sampled from a probability distribution over a measurable attribute of the constituents of the substances. In this view, two substances are identical, relative to this measurable attribute, if they have the same underlying probability distribution. This implies that two identical mixtures do not need to have identical finite-size realisations of their compositions. By averaging over composition realisations, it is found that (1) fixed composition mixtures behave as homogeneous single-component substances and (2) in the limit of a large system size, the entropy of mixing per particle shows a continuous variation from kBln2 to 0, as two different substances are made more similar, thereby resolving the "real" Gibbs paradox.

Mass Changes the Diffusion Coefficient of Particles with Ligand-Receptor Contacts in the Overdamped Limit

Inertia does not generally affect the long-time diffusion of passive overdamped particles in fluids. Yet a model starting from the Langevin equation predicts a surprising property of particles coated with ligands that bind reversibly to surface receptors: heavy particles diffuse more slowly than light ones of the same size. We show this by simulation and by deriving an analytic formula for the mass-dependent diffusion coefficient in the overdamped limit. We estimate the magnitude of this effect for a range of biophysical ligand-receptor systems, and find it is potentially observable for tailored micronscale DNA-coated colloids.

Geometric and topological entropies of sphere packing

We present a statistical mechanical description of randomly packed spherical particles, where the average coordination number is treated as a macroscopic thermodynamic variable. The overall packing entropy is shown to have two contributions: geometric, reflecting statistical weights of individual configurations, and topological, which corresponds to the number of topologically distinct states. Both of them are computed in the thermodynamic limit for isostatic and weakly underconstrained packings in 2D and 3D. The theory generalizes concepts of granular and glassy configurational entropies for the case of nonjammed systems. It is directly applicable to sticky colloids and predicts an asymptotic phase behavior of sticky spheres in the limit of strong binding.

Neglecting polydispersity degrades propensity measurements in supercooled liquids

We conduct molecular dynamics simulations of a bidisperse Kob-Andersen (KA) glass former, modified to add in additional polydispersity. The original KA system is known to exhibit dynamical heterogeneity. Prior work defined propensity, the mean motion of a particle averaged over simulations reconstructing the initial positions of all particles but with randomized velocities. The existence of propensity shows that structure and dynamics are connected. In this paper, we study systems which mimic problems that would be encountered in measuring propensity in a colloidal glass former, where particles are polydisperse (they have slight size variations). We mimic polydispersity by altering the bidisperse KA system into a quartet consisting of particles both slightly larger and slightly smaller than the parent particles in the original bidisperse system. We then introduce errors into the reconstruction of the initial positions that mimic mistakes one might make in a colloidal experiment. The mistakes degrade the propensity measurement, in some cases nearly completely; one no longer has an iso-configurational ensemble in any useful sense. Our results show that a polydisperse sample is suitable for propensity measurements provided one avoids reconstruction mistakes.

Reconfiguring confined magnetic colloids with tunable fluid transport behavior

Collective dynamics of confined colloids are crucial in diverse scenarios such as self-assembly and phase behavior in materials science, microrobot swarms for drug delivery and microfluidic control. Yet, fine-tuning the dynamics of colloids in microscale confined spaces is still a formidable task due to the complexity of the dynamics of colloidal suspension and to the lack of methodology to probe colloids in confinement. Here, we show that the collective dynamics of confined magnetic colloids can be finely tuned by external magnetic fields. In particular, the mechanical properties of the confined colloidal suspension can be probed in real time and this strategy can be also used to tune microscale fluid transport. Our experimental and theoretical investigations reveal that the collective configuration characterized by the colloidal entropy is controlled by the colloidal concentration, confining ratio and external field strength and direction. Indeed, our results show that mechanical properties of the colloidal suspension as well as the transport of the solvent in microfluidic devices can be controlled upon tuning the entropy of the colloidal suspension. Our approach opens new avenues for the design and application of drug delivery, microfluidic logic, dynamic fluid control, chemical reaction and beyond.

Classical density functional theory for a two-dimensional isotropic ferrogel model with labeled particles

In this study, we formulate a density functional theory (DFT) for systems of labeled particles, considering a two-dimensional bead-spring lattice with a magnetic dipole on every bead as a model for ferrogels. On the one hand, DFT has been widely studied to investigate fluidlike states of materials, in which constituent particles are not labeled as they can exchange their positions without energy cost. On the other hand, in ferrogels consisting of magnetic particles embedded in elastic polymer matrices, the particles are labeled by their positions as their neighbors do not change over time. We resolve such an issue of particle labeling, introducing a mapping of the elastic interaction mediated by springs onto a pairwise additive interaction (pseudosprings) between unlabeled particles. We further investigate magnetostriction and changes in the elastic constants under altered magnetic interactions employing the pseudospring potential. It is revealed that there are two different response scenarios in the mechanical properties of the dipole-spring systems: While systems at low packing fractions are hardened as the magnetic moments increase in magnitude, at high packing fractions softening due to diminishing effects from the steric force, associated with increases in the volume, is observed. The validity of the theory is also verified by Monte Carlo simulations with both real springs and pseudosprings. We expect that our DFT approach may promote our understanding of materials with particle inclusions.

On the Logic of a Prior Based Statistical Mechanics of Polydisperse Systems: The Case of Binary Mixtures

Most undergraduate students who have followed a thermodynamics course would have been asked to evaluate the volume occupied by one mole of air under standard conditions of pressure and temperature. However, what is this task exactly referring to? If air is to be regarded as a mixture, under what circumstances can this mixture be considered as comprising only one component called “air” in classical statistical mechanics? Furthermore, following the paradigmatic Gibbs’ mixing thought experiment, if one mixes air from a container with air from another container, all other things being equal, should there be a change in entropy? The present paper addresses these questions by developing a prior-based statistical mechanics framework to characterise binary mixtures’ composition realisations and their effect on thermodynamic free energies and entropies. It is found that (a) there exist circumstances for which an ideal binary mixture is thermodynamically equivalent to a single component ideal gas and (b) even when mixing two substances identical in their underlying composition, entropy increase does occur for finite size systems. The nature of the contributions to this increase is then discussed.

Cluster Formation and Entanglement in the Rheology of Antibody Solutions

Antibody solutions deviate from the dynamical and rheological response expected for globular proteins, especially as the volume fraction is increased. Experimental evidence shows that antibodies can reversibly bind to each other via F_ab and F_c domains and form larger structures (clusters) of several antibodies. Here, we present a microscopic equilibrium model to account for the distribution of cluster sizes. Antibody clusters are modeled as polymers that can grow via reversible bonds either between two F_ab domains or between F_ab and F_c domain. We propose that the dynamical and rheological behavior is determined by molecular entanglements of the clusters. This entanglement does not occur at low concentrations where antibody-antibody binding contributes to the viscosity by increasing the effective size of the particles. The model explains the observed shear-thinning behavior of antibody solutions.

Morphometric Approach to Many-Body Correlations in Hard Spheres

We model the thermodynamics of local structures within the hard sphere liquid at arbitrary volume fractions through the morphometric calculation of n-body correlations. We calculate absolute free energies of local geometric motifs in excellent quantitative agreement with molecular dynamics simulations across the liquid and supercooled liquid regimes. We find a bimodality in the density library of states where fivefold symmetric structures appear lower in free energy than fourfold symmetric structures and from a single reaction path predict a dynamical barrier which scales linearly in the compressibility factor. The method provides a new route to assess changes in the free energy landscape at volume fractions dynamically inaccessible to conventional techniques.

The Gibbs Paradox

The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs' notion of 'generic phase'). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. 'Distinguishability', in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.

Probability, Entropy, and Gibbs' Paradox(es)

Two distinct puzzles, which are both known as Gibbs’ paradox, have interested physicists since they were first identified in the 1870s. They each have significance for the foundations of statistical mechanics and have led to lively discussions with a wide variety of suggested resolutions. Most proposed resolutions had involved quantum mechanics, although the original puzzles were entirely classical and were posed before quantum mechanics was invented. In this paper, I show that contrary to what has often been suggested, quantum mechanics is not essential for resolving the paradoxes. I present a resolution of the paradoxes that does not depend on quantum mechanics and includes the case of colloidal solutions, for which quantum mechanics is not relevant.

Atomic clusters with addressable complexity

A general formulation for constructing addressable atomic clusters is introduced, based on one or more reference structures. By modifying the well depths in a given interatomic potential in favour of nearest-neighbour interactions that are defined in the reference(s), the potential energy landscape can be biased to make a particular permutational isomer the global minimum. The magnitude of the bias changes the resulting potential energy landscape systematically, providing a framework to produce clusters that should self-organise efficiently into the target structure. These features are illustrated for small systems, where all the relevant local minima and transition states can be identified, and for the low-energy regions of the landscape for larger clusters. For a 55-particle cluster, it is possible to design a target structure from a transition state of the original potential and to retain this structure in a doubly addressable landscape. Disconnectivity graphs based on local minima that have no direct connections to a lower minimum provide a helpful way to visualise the larger databases. These minima correspond to the termini of monotonic sequences, which always proceed downhill in terms of potential energy, and we identify them as a class of biminimum. Multiple copies of the target cluster are treated by adding a repulsive term between particles with the same address to maintain distinguishable targets upon aggregation. By tuning the magnitude of this term, it is possible to create assemblies of the target cluster corresponding to a variety of structures, including rings and chains.

Gibbs Paradox Revisited from the Fluctuation Theorem with Absolute Irreversibility

The inclusion of the factor ln(1/N!) in the thermodynamic entropy proposed by Gibbs is shown to be equivalent to the validity of the fluctuation theorem with absolute irreversibility for gas mixing.

Free energy of singular sticky-sphere clusters

Networks of particles connected by springs model many condensed-matter systems, from colloids interacting with a short-range potential and complex fluids near jamming, to self-assembled lattices and various metamaterials. Under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero-frequency vibrational modes, yet such singular modes are at the epicenter of many interesting behaviors in the systems above. We consider a system of N spherical particles, and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged. Although the contribution to the partition function from singular clusters diverges in the limit, its asymptotic value can be calculated and depends on only two parameters, characterizing the depth and range of the potential. The result holds for systems that are second-order rigid, a geometric characterization that describes all known ground-state (rigid) sticky clusters. To illustrate the applications of our theory we address the question of emergence: how does crystalline order arise in large systems when it is strongly disfavored in small ones? We calculate the partition functions of all known rigid clusters up to N≤21 and show the cluster landscape is dominated by hyperstatic clusters (those with more than 3N-6 contacts); singular and isostatic clusters are far less frequent, despite their extra vibrational and configurational entropies. Since the most hyperstatic clusters are close to fragments of a close-packed lattice, this underlies the emergence of order in sticky-sphere systems, even those as small as N=10.

Segregation of "isotope" particles within colloidal molecules

Clusters of spherical particles are called "colloidal molecules" because they adopt structures that resemble those of true molecules. In this analogy, the particles are the atoms, the attractive interactions between them are bonds, and the different structures that appear in equilibrium are isomers. We take this analogy a step further by doping colloidal molecules with colloidal "isotopes," particles that have the same size but different bonding energies from the other particles in the system. Our molecules are two-dimensional clusters consisting of polystyrene and silica microspheres held together by depletion interactions. Using a combination of optical microscopy and particle tracking, we examine an ensemble of 4- and 5-particle molecules at different isotope ratios. We find that the isotopes tend to segregate to particular positions in the various isomers. We explain these findings using a statistical mechanical model that accounts for the rotational entropy of the isomers and the different interaction potentials between the different types of particles. The model shows how to optimize the yield of any particular isomer, so as to put the isotopes in desired locations. Our experiments and models show that even in systems of particles with isotropic interactions, the structures of self-assembled molecules can in principle be controlled to a surprisingly high extent.