Transient nucleation driven by solvent evaporation.

We theoretically investigate homogeneous crystal nucleation in a solution containing a solute and a volatile solvent. The solvent evaporates from the solution, thereby continuously increasing the concentration of the solute. We view it as an idealized model for the far-out-of-equilibrium conditions present during the liquid-state manufacturing of organic electronic devices. Our model is based on classical nucleation theory, taking the solvent to be a source of the transient conditions in which the solute drops out of the solution. Other than that, the solvent is not directly involved in the nucleation process itself. We approximately solve the kinetic master equations using a combination of Laplace transforms and singular perturbation theory, providing an analytical expression for the nucleation flux. Our results predict that (i) the nucleation flux lags slightly behind a commonly used quasi-steady-state approximation. This effect is governed by two counteracting effects originating from solvent evaporation: while a faster evaporation rate results in an increasingly larger influence of the lag time on the nucleation flux, this lag time itself is found to decrease with increasing evaporation rate. Moreover, we find that (ii) the nucleation flux and the quasi-steady-state nucleation flux are never identical, except trivially in the stationary limit, and (iii) the initial induction period of the nucleation flux, which we characterize as a generalized induction time, decreases weakly with the evaporation rate. This indicates that the relevant time scale for nucleation also decreases with an increasing evaporation rate. Our analytical theory compares favorably with results from a numerical evaluation of the governing kinetic equations.

Preferential ordering of incommensurate-length guest particles in a smectic host.

Using density functional theory, we study the preferential ordering of rod-like guest particles immersed in a smectic host fluid. Within a model of perfectly aligned rods and assuming that the guest particles do not perturb the smectic host fluid, simple excluded-volume arguments explain that guest particles that are comparable in length to the host particles order in phase with the smectic host density layering, whereas guest particles that are considerably shorter or longer order in antiphase. The corresponding free-energy minima are separated by energetic barriers on the order of the thermal energy kBT, suggesting that guest particles undergo hopping-type diffusion between adjacent smectic layers. Upon introducing a slight orientational mismatch between the guest particles and the perfectly aligned smectic host, an additional, smaller free-energy barrier emerges for a range of intermediate guest-to-host length ratios, which splits the free-energy minimum into two. Guest particles in this range occupy positions intermediate between in-phase ordering and in-antiphase ordering. Finally, we use Kramers' theory to identify slow, fast, and intermediate diffusive regimes for the guest particles as a function of their length. Our model is in qualitative agreement with experiment and simulation and provides an alternative, complementary explanation in terms of a free-energy landscape for the intermediate diffusive regime, which was previously hypothesized to result from temporary caging effects [M. Chiappini, E. Grelet, and M. Dijkstra, Phys. Rev. Lett. 124, 087801 (2020)]. We argue that our simple model of aligned rods captures the salient features of incommensurate-length guest particles in a smectic host if a slight orientational mismatch is introduced.

Nematic to Cholesteric Transformation in the Cellulose Nanocrystal Droplet Phase.

Nucleation, growth, and transformation of chirality in nanomaterial systems is a growing research topic with broad interest in tunable and configurable chiroptical materials. Similar to other one-dimensional nanomaterials, cellulose nanocrystals (CNCs), which are nanorods of naturally abundant biopolymer cellulose, display chiral or cholesteric liquid crystal (LC) phases in the form of tactoids. However, the nucleation and growth of the cholesteric CNC tactoids to equilibrium chiral structures and their morphological transformations are yet to be critically assessed. We noticed that the onset of liquid crystal formation in CNC suspensions is characterized by the nucleation of a nematic tactoid that grows in volume and spontaneously transforms into a cholesteric tactoid. The cholesteric tactoids merge with the neighboring tactoids to form bulk cholesteric mesophases with various configurational palettes. We applied scaling laws from the energy functional theory and found suitable agreement with the morphological transformation of the tactoid droplets monitored for their fine structure and orientation by quantitative polarized light imaging.

Relaxational dynamics of the T-number conversion of virus capsids.

We extend a recently proposed kinetic theory of virus capsid assembly based on Model A kinetics and study the dynamics of the interconversion of virus capsids of different sizes triggered by a quench, that is, by sudden changes in the solution conditions. The work is inspired by in vitro experiments on functionalized coat proteins of the plant virus cowpea chlorotic mottle virus, which undergo a reversible transition between two different shell sizes (T = 1 and T = 3) upon changing the acidity and salinity of the solution. We find that the relaxation dynamics are governed by two time scales that, in almost all cases, can be identified as two distinct processes. Initially, the monomers and one of the two types of capsids respond to the quench. Subsequently, the monomer concentration remains essentially constant, and the conversion between the two capsid species completes. In the intermediate stages, a long-lived metastable steady state may present itself, where the thermodynamically less stable species predominate. We conclude that a Model A based relaxational model can reasonably describe the early and intermediate stages of the conversion experiments. However, it fails to provide a good representation of the time evolution of the state of assembly of the coat proteins in the very late stages of equilibration when one of the two species disappears from the solution. It appears that explicitly incorporating the nucleation barriers to assembly and disassembly is crucial for an accurate description of the experimental findings, at least under conditions where these barriers are sufficiently large.

A kinetic model for the impact of packaging signal mimics on genome encapsulation.

Inspired by recent experiments on the spontaneous assembly of virus-like particles from a solution containing a synthetic coat protein and double-stranded DNA, we put forward a kinetic model that has as main ingredients a stochastic nucleation and a deterministic growth process. The efficiency and rate of DNA packaging strongly increase after tiling the DNA with CRISPR-Cas proteins at predesignated locations, mimicking assembly signals in viruses. Our model shows that treating these proteins as nucleation-inducing diffusion barriers is sufficient to explain the experimentally observed increase in encapsulation efficiency, but only if the nucleation rate is sufficiently high. We find an optimum in the encapsulation kinetics for conditions where the number of packaging signal mimics is equal to the number of nucleation events that can occur during the time required to fully encapsulate the DNA template, presuming that the nucleation events can only take place adjacent to a packaging signal. Our theory is in satisfactory agreement with the available experimental data.

The Dynamics of Viruslike Capsid Assembly and Disassembly.

Cowpea chlorotic mottle virus (CCMV) is a widely used model for virus replication studies. A major challenge lies in distinguishing between the roles of the interaction between coat proteins and that between the coat proteins and the viral RNA in assembly and disassembly processes. Here, we report on the spontaneous and reversible size conversion of the empty capsids of a CCMV capsid protein functionalized with a hydrophobic elastin-like polypeptide which occurs following a pH jump. We monitor the concentrations of T = 3 and T = 1 capsids as a function of time and show that the time evolution of the conversion from one T number to another is not symmetric: The conversion from T = 1 to T = 3 is a factor of 10 slower than that of T = 3 to T = 1. We explain our experimental findings using a simple model based on classical nucleation theory applied to virus capsids, in which we account for the change in the free protein concentration, as the different types of shells assemble and disassemble by shedding or absorbing single protein subunits. As far as we are aware, this is the first study confirming that both the assembly and disassembly of viruslike shells can be explained through classical nucleation theory, reproducing quantitatively results from time-resolved experiments.

Transient response and domain formation in electrically deforming liquid crystal networks.

Recently, three distinct, well-separated transient regimes were discovered in the dynamics of the volume expansion of shape-shifting liquid crystal network films in response to the switching on of an alternating electric field [Van der Kooij et al., Nat. Commun., 2019, 10, 1]. Employing a spatially resolved, time-dependent Landau theory that couples local volume generation to the degree of orientational order of mesogens that are part of a viscoelastic network, we are able to offer a physical explanation for the existence of three time scales. We find that the initial response is dominated by overcoming the impact of thermal noise, after which the top of the film expands, followed by a permeation of this response into the bulk region. An important signature of our predictions is a significant dependence of the three time scales on the film thickness, where we observe a clear thin-film-to-bulk transition. The point of transition coincides with the emergence of spatial inhomogeneities in the bulk of the film in the form of domains separated by regions of suppressed expansion. This ultimately gives rise to variations in the steady-state overall expansion of the film and may lead to uncontrolled patterning. According to our model, domain formation can be suppressed by (1) decreasing the thickness of the as-prepared film, (2) increasing the linear dimensions of the mesogens, or (3) their degree of orientational order when cross-linked into the network. Our findings provide a handle to achieve finer control over the actuation of smart liquid crystal network coatings.

Geometric percolation of hard-sphere dispersions in shear flow.

We combine a heuristic theory of geometric percolation and the Smoluchowski theory of colloid dynamics to predict the impact of shear flow on the percolation threshold of hard spherical colloidal particles, and verify our findings by means of molecular dynamics simulations. It appears that the impact of shear flow is subtle and highly non-trivial, even in the absence of hydrodynamic interactions between the particles. The presence of shear flow can both increase and decrease the percolation threshold, depending on the criterion used for determining whether or not two particles are connected and on the Péclet number. Our approach opens up a route to quantitatively predict the percolation threshold in nanocomposite materials that, as a rule, are produced under non-equilibrium conditions, making comparison with equilibrium percolation theory tenuous. Our theory can be adapted straightforwardly for application in other types of flow field, and particles of different shape or interacting via other than hard-core potentials.

Structure of nematic tactoids of hard rods.

We study by means of Monte Carlo simulations the internal structure of nematic droplets or tactoids formed by hard, rod-like particles in a gas of spherical ghost particles that act as depletion agents for the rods. We find that the shape and internal structure of tactoids are strongly affected by the size of the droplets. The monotonically increasing degree of nematic order with increasing particle density that characterizes the bulk nematic phase is locally violated and more so the smaller the tactoid. We also investigate the impact of an external quadrupolar alignment field on tactoids and find that this tends to make the director field more uniform, but not to very significantly increase the tactoid's aspect ratio. This agrees with recent theoretical predictions yet is at variance with experimental observations and dynamical simulations. We explain this discrepancy in terms of competing relaxation times.

Continuum percolation in colloidal dispersions of hard nanorods in external axial and planar fields.

We present a theoretical study on continuum percolation of rod-like colloidal particles in the presence of axial and planar quadrupole fields. Our work is based on a self-consistent numerical treatment of the connectedness Ornstein-Zernike equation, in conjunction with the Onsager equation that describes the orientational distribution function of particles interacting via a hard-core repulsive potential. Our results show that axial and planar quadrupole fields both in principle increase the percolation threshold. By how much depends on a combination of the field strength, the concentration, the aspect ratio of the particles, and percolation criterion. We find that the percolated state can form and break down multiple times with increasing concentration, i.e., it exhibits re-entrance behaviour. Finally, we show that planar fields may induce a high degree of triaxiality in the shape of particle clusters that in actual materials may give rise to highly anisotropic conductivity properties.

Enhanced ordering in length-polydisperse carbon nanotube solutions at high concentrations as revealed by small angle X-ray scattering.

Carbon nanotubes (CNTs) are stiff, all-carbon macromolecules with diameters as small as one nanometer and few microns long. Solutions of CNTs in chlorosulfonic acid (CSA) follow the phase behavior of rigid rod polymers interacting via a repulsive potential and display a liquid crystalline phase at sufficiently high concentration. Here, we show that small-angle X-ray scattering and polarized light microscopy data can be combined to characterize quantitatively the morphology of liquid crystalline phases formed in CNT solutions at concentrations from 3 to 6.5% by volume. We find that upon increasing their concentration, CNTs self-assemble into a liquid crystalline phase with a pleated texture and with a large inter-particle spacing that could be indicative of a transition to higher-order liquid crystalline phases. We explain how thermal undulations of CNTs can enhance their electrostatic repulsion and increase their effective diameter by an order of magnitude. By calculating the critical concentration, where the mean amplitude of undulation of an unconstrained rod becomes comparable to the rod spacing, we find that thermal undulations start to affect steric forces at concentrations as low as the isotropic cloud point in CNT solutions.

Combined Force-Torque Spectroscopy of Proteins by Means of Multiscale Molecular Simulation.

Assessing the structural properties of large proteins is important to gain an understanding of their function in, e.g., biological systems or biomedical applications. We propose a method to examine the mechanical properties of proteins subject to applied forces by means of multiscale simulation. Both stretching and torsional forces are considered, and these may be applied independently of each other. As a proof of principle, we apply torsional forces to a coarse-grained continuum model of the antibody protein immunoglobulin G using fluctuating finite element analysis and use it to identify the area of strongest deformation. This region is essential to the torsional properties of the molecule as a whole because it represents the softest, most deformable domain. Zooming in, this part of the molecule is subjected to torques and stretching forces using molecular dynamics simulations on an atomistically resolved level to investigate its torsional properties. We calculate the torsional resistance as a function of the rotation of the domain while subjecting it to various stretching forces. From this, we assess how the measured twist-torque profiles develop with increasing stretching force and show that they exhibit torsion stiffening, in qualitative agreement with experimental findings. We argue that combining the twist-torque profiles for various stretching forces effectively results in a combined force-torque spectroscopy analysis, which may serve as a mechanical signature for a biological macromolecule.

Geometric percolation of hard nanorods: The interplay of spontaneous and externally induced uniaxial particle alignment.

We present a numerical study on geometric percolation in liquid dispersions of hard slender colloidal particles subject to an external orienting field. In the formulation and liquid-state processing of nanocomposite materials, particle alignment by external fields such as electric, magnetic, or flow fields is practically inevitable and often works against the emergence of large nanoparticle networks. Using continuum percolation theory in conjunction with Onsager theory, we investigate how the interplay between externally induced alignment and the spontaneous symmetry breaking of the uniaxial nematic phase affects cluster formation in nanoparticle dispersions. It is known that particle alignment by means of a density increase or by an external field may result in a breakdown of an already percolating network. As a result, percolation can be limited to a small region of the phase diagram only. Here, we demonstrate that the existence and shape of such a "percolation island" in the phase diagram crucially depends on the connectivity length-a critical distance defining direct connections between neighboring particles. For some values of the connectivity range, we observe unusual re-entrance effects, in which a system-spanning network forms and breaks down multiple times with increasing particle density.

Real-Time Assembly of Viruslike Nucleocapsids Elucidated at the Single-Particle Level.

While the structure of a multitude of viral particles has been resolved to atomistic detail, their assembly pathways remain largely elusive. Key unresolved issues are particle nucleation, particle growth, and the mode of genome compaction. These issues are difficult to address in bulk approaches and are effectively only accessible by the real-time tracking of assembly dynamics of individual particles. This we do here by studying the assembly into rod-shaped viruslike particles (VLPs) of artificial capsid polypeptides. Using fluorescence optical tweezers, we establish that small oligomers perform one-dimensional diffusion along the DNA. Larger oligomers are immobile and nucleate VLP growth. A multiplexed acoustic force spectroscopy approach reveals that DNA is compacted in regular steps, suggesting packaging via helical wrapping into a nucleocapsid. By reporting how real-time assembly tracking elucidates viral nucleation and growth principles, our work opens the door to a fundamental understanding of the complex assembly pathways of both VLPs and naturally evolved viruses.

DNA partitions into triplets under tension in the presence of organic cations, with sequence evolutionary age predicting the stability of the triplet phase.

Using atomistic simulations, we show the formation of stable triplet structure when particular GC-rich DNA duplexes are extended in solution over a timescale of hundreds of nanoseconds, in the presence of organic salt. We present planar-stacked triplet disproportionated DNA (Σ DNA) as a possible solution phase of the double helix under tension, subject to sequence and the presence of stabilising co-factors. Considering the partitioning of the duplexes into triplets of base pairs as the first step of operation of recombinase enzymes like RecA, we emphasise the structure-function relationship in Σ DNA. We supplement atomistic calculations with thermodynamic arguments to show that codons for 'phase 1' amino acids (those appearing early in evolution) are more likely than a lower entropy GC-rich sequence to form triplets under tension. We further observe that the four amino acids supposed (in the 'GADV world' hypothesis) to constitute the minimal set to produce functional globular proteins have the strongest triplet-forming propensity within the phase 1 set, showing a series of decreasing triplet propensity with evolutionary newness. The weak form of our observation provides a physical mechanism to minimise read frame and recombination alignment errors in the early evolution of the genetic code.

Connectivity, Not Density, Dictates Percolation in Nematic Liquid Crystals of Slender Nanoparticles.

We show by means of continuum theory and simulations that geometric percolation in uniaxial nematics of hard slender particles is fundamentally different from that in isotropic dispersions. In the nematic, percolation depends only very weakly on the density and is, in essence, determined by a distance criterion that defines connectivity. This unexpected finding has its roots in the nontrivial coupling between the density and the degree of orientational order that dictate the mean number of particle contacts. Clusters in the nematic are much longer than wide, suggesting the use of nematics for nanocomposites with strongly anisotropic transport properties.

Directing Liquid Crystalline Self-Organization of Rodlike Particles through Tunable Attractive Single Tips.

Dispersions of rodlike colloidal particles exhibit a plethora of liquid crystalline states, including nematic, smectic A, smectic B, and columnar phases. This phase behavior can be explained by presuming the predominance of hard-core volume exclusion between the particles. We show here how the self-organization of rodlike colloids can be controlled by introducing a weak and highly localized directional attractive interaction between one of the ends of the particles. This has been performed by functionalizing the tips of filamentous viruses by means of regioselectively grafting fluorescent dyes onto them, resulting in a hydrophobic patch whose attraction can be tuned by varying the number of bound dye molecules. We show, in agreement with our computer simulations, that increasing the single tip attraction stabilizes the smectic phase at the expense of the nematic phase, leaving all other liquid crystalline phases invariant. For a sufficiently strong tip attraction, the nematic state may be suppressed completely to get a direct isotropic liquid-to-smectic phase transition. Our findings provide insights into the rational design of building blocks for functional structures formed at low densities.

Supramolecular copolymers predominated by alternating order: Theory and application.

We investigate the copolymerization behavior of a two-component system into quasilinear self-assemblies under conditions that interspecies binding is favored over identical species binding. The theoretical framework is based on a coarse-grained self-assembled Ising model with nearest neighbor interactions. In Ising language, such conditions correspond to the antiferromagnetic case giving rise to copolymers with predominantly alternating configurations. In the strong coupling limit, we show that the maximum fraction of polymerized material and the average length of strictly alternating copolymers depend on the stoichiometric ratio and the activation free energy of the more abundant species. They are substantially reduced when the stoichiometric ratio noticeably differs from unity. Moreover, for stoichiometric ratios close to unity, the copolymerization critical concentration is remarkably lower than the homopolymerization critical concentration of either species. We further analyze the polymerization behavior for a finite and negative coupling constant and characterize the composition of supramolecular copolymers. Our theoretical insights rationalize experimental results of supramolecular polymerization of oppositely charged monomeric species in aqueous solutions.

Experimental and Theoretical Determination of the pH inside the Confinement of a Virus-Like Particle.

In biology, a variety of highly ordered nanometer-size protein cages is found. Such structures find increasing application in, for example, vaccination, drug delivery, and catalysis. Understanding the physiochemical properties, particularly inside the confinement of a protein cage, helps to predict the behavior and properties of new materials based on such particles. Here, the relation between the bulk solution pH and the local pH inside a model protein cage, based on virus-like particles (VLPs) built from the coat proteins of the cowpea chlorotic mottle virus, is investigated. The pH is a crucial parameter in a variety of processes and is potentially significantly influenced by the high concentration of charges residing on the interior of the VLPs. The data show a systematic more acidic pH of 0.5 unit inside the VLP compared to that of the bulk solution for pH values above pH 6, which is explained using a theoretical model based on a Donnan equilibrium. The model agrees with the experimental data over almost two orders of magnitude, while below pH 6 the experimental data point to a buffering capacity of the VLP. These results are a first step in a better understanding of the physiochemical conditions inside a protein cage.

Nanoscale insight into silk-like protein self-assembly: effect of design and number of repeat units.

By means of replica exchange molecular dynamics simulations we investigate how the length of a silk-like, alternating diblock oligopeptide influences its secondary and quaternary structure. We carry out simulations for two protein sizes consisting of three and five blocks, and study the stability of a single protein, a dimer, a trimer and a tetramer. Initial configurations of our simulations are ß-roll and ß-sheet structures. We find that for the triblock the secondary and quaternary structures upto and including the tetramer are unstable: the proteins melt into random coil structures and the aggregates disassemble either completely or partially. We attribute this to the competition between conformational entropy of the proteins and the formation of hydrogen bonds and hydrophobic interactions between proteins. This is confirmed by our simulations on the pentablock proteins, where we find that, as the number of monomers in the aggregate increases, individual monomers form more hydrogen bonds whereas their solvent accessible surface area decreases. For the pentablock ß-sheet protein, the monomer and the dimer melt as well, although for the ß-roll protein only the monomer melts. For both trimers and tetramers remain stable. Apparently, for these the entropy loss of forming ß-rolls and ß-sheets is compensated for in the free-energy gain due to the hydrogen-bonding and hydrophobic interactions. We also find that the middle monomers in the trimers and tetramers are conformationally much more stable than the ones on the top and the bottom. Interestingly, the latter are more stable on the tetramer than on the trimer, suggesting that as the number of monomers increases protein-protein interactions cooperatively stabilize the assembly. According to our simulations, the ß-roll and ß-sheet aggregates must be approximately equally stable.