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Incorporation of guest additives within inorganic single crystals offers a unique strategy for creating nanocomposites with tailored properties. While anionic additives have been widely used to control the properties of crystals, their effective incorporation remains a key challenge. Here, we show that cationic additives are an excellent alternative for the synthesis of nanocomposites, where they are shown to deliver exceptional levels of incorporation of up to 70 wt % of positively charged amino acids, polymer particles, gold nanoparticles, and silver nanoclusters within inorganic single crystals. This high additive loading endows the nanocomposites with new functional properties, including plasmon coupling, bright fluorescence, and surface-enhanced Raman scattering (SERS). Cationic additives are also shown to outperform their acidic counterparts, where they are highly active in a wider range of crystal systems, owing to their outstanding colloidal stability in the crystallization media and strong affinity for the crystal surfaces. This work demonstrates that although often overlooked, cationic additives can make valuable crystallization additives to create composite materials with tailored composition-structure-property relationships. This versatile and straightforward approach advances the field of single-crystal composites and provides exciting prospects for the design and fabrication of new hybrid materials with tunable functional properties.

The surface processes that control crystal growth from solution can be probed in real-time by

Calcite crystals grow by means of molecular steps that develop on {10.4} faces. These steps can arise stochastically via two-dimensional (2D) nucleation or emerge steadily from dislocations to form spiral hillocks. Here, we determine the kinetics of these two growth mechanisms as a function of supersaturation. We show that calcite crystals larger than ∼1 μm favor spiral growth over 2D nucleation, irrespective of the supersaturation. Spirals prevail beyond this length scale because slow boundary layer diffusion creates a low surface supersaturation that favors the spiral mechanism. Sub-micron crystals favor 2D nucleation at high supersaturations, although diffusion can still limit the growth of nanoscopic crystals. Additives can change the dominant mechanism by impeding spiral growth or by directly promoting 2D nucleation.

Organic additives play key roles in controlling the precipitation of calcium carbonate in the environment, industry, and biology, where they can direct polymorphism, alter crystal morphologies, and sometimes even become occluded, modifying bulk properties. However, significant questions remain regarding the pathway from adsorption on crystal surfaces to occlusion. Here, the optical properties of calcite crystals grown in the presence of the dye Congo red are used to characterize the organization of the additives within the crystal. Complemented by the analysis of surface adsorption through in situ atomic force microscopy (AFM), molecular simulations, and changes in crystal morphologies, we show that the occluded dye molecules are randomly oriented under fast growth conditions, but that slow growth engenders ordering of the dye within islands, whose orientation is determined by the dye/terrace interaction free energy. The islands are subsequently overgrown such that their internal structure is preserved. These results reveal that the occlusion of organic macromolecules into calcite can be understood by thermodynamics operating at the adsorption stage. This new insight will ultimately enable the design of additives to give specific material properties.

The classical model of crystal growth assumes that kinks grow via a sequence of independent adsorption events where each solute transitions from the solution directly to the crystal lattice site. Here, we challenge this view by showing that some calcite kinks grow via a multistep mechanism where the solute adsorbs to an intermediate site and only transitions to the lattice site upon the adsorption of a second solute. We compute the free energy curves for Ca and CO

The brittlestar Ophiocoma wendtii is theorised to employ a technique already used in metallurgy in order to optimise the mechanical properties of calcitic microlenses within their skeletons. These microlenses contain arrays of Mg-rich nanoprecipitates, which are proposed to inhibit crack propagation through the compression of the local host lattice. Here, we employ classical molecular dynamics in order to study the effects of Mg-rich nanoprecipitates on lattice strain, stress distributions and crack propagation in calcite. Our quantitative results on lattice strain and stress induced on the host matrix are compatible with empirical estimates. Simulations of crack propagation demonstrate that the inclusion of a Mg-rich region results in an increase in stress required to fracture the crystal, as well as higher residual stress in the fractured crystal. This is the result of an inhomogeneous stress distribution causing a more disordered fracture, as well as deflections of the crack away from the lowest energy (10.4) surface. The results agree with the proposal that the compression of the host lattice inhibits propagation, and offer insight into other mechanisms through which the nanoprecipitates affect crack propagation.

The rapid (< 1 ms) transport of biological material to and from the cell nucleus is regulated by the nuclear pore complex (NPC). At the core of the NPC is a permeability barrier consisting of intrinsically disordered Phe-Gly (FG) nucleoporins (FG Nups). Various types of nuclear transport receptors (NTRs) facilitate transport by partitioning in the FG Nup assembly, overcoming the barrier by their affinity to the FG Nups, and comprise a significant fraction of proteins in the NPC barrier. In previous work, we revealed that the experimental binding of the NTRs NTF2 and – the larger – Impβ to different planar assemblies of FG Nups follows a universal physical law defined by negative cooperativity, which was further validated by a minimal physical model that treated the FG Nups as flexible homopolymers and the NTRs as uniformly cohesive spheres Zahn et al. (2016). Here, we build upon our original study by first parametrizing our model to experimental data, and next to predict the effects of crowding by different types of NTRs. We show how varying the amounts of one type of NTR modulates how the other NTR penetrates the FG Nup assembly. Notably, at similar and physiologically relevant NTR concentrations, our model predicts demixed phases of NTF2 and Impβ within the FG Nup assembly. The functional implication of NTR phase separation is that NPCs may sustain separate transport pathways that are determined by inter-NTR competition.

Climate science employs a hierarchy of models, trading the tractability of simplified energy balance models (EBMs) against the detail of Global Circulation Models. Since the pioneering work of Hasselmann, stochastic EBMs have allowed treatment of climate fluctuations and noise. However, it has recently been claimed that observations motivate heavy-tailed temporal response functions in global mean temperature to perturbations. Our complementary approach exploits the correspondence between Hasselmann’s EBM and the original mean-reverting stochastic model in physics, Langevin’s equation of 1908. We propose mapping a model well known in statistical mechanics, the Mori-Kubo Generalised Langevin Equation (GLE) to generalise the Hasselmann EBM. If present, long range memory then simplifies the GLE to a fractional Langevin equation (FLE). We describe the corresponding EBMs that map to the GLE and FLE, briefly discuss their solutions, and relate them to Lovejoy’s new Fractional Energy Balance Model.

Since Hasselmann and Leith, stochastic Energy Balance Models (EBMs) have allowed treatment of climate fluctuations, and at least the possibility of fluctuation-dissipation relations. However, it has recently been argued that observations motivate heavy-tailed temporal response functions in global mean temperature. Our complementary approach (arXiv:2007.06464v2[cond-mat.stat-mech]) exploits the correspondence between Hasselmann’s EBM and Langevin’s equation (1908). We propose mapping the Mori-Kubo Generalised Langevin Equation (GLE) to generalise the Hasselmann EBM. If present, long range memory then simplifies the GLE to a fractional Langevin equation (FLE). We describe the EBMs that correspond to the GLE and FLE, and relate them to Lovejoy et al’s FEBE [NPG Discussions, 2019; QJRMS, to appear, 2021].

We present an application of the extended stochastic Liouville--von Neumann equation (ESLN) method introduced earlier [G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 95, 125124 (2017); G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 97, 224310 (2018)], which describes the dynamics of an exactly thermalized open quantum system reduced density matrix coupled to a non-Markovian harmonic environment. Critically, the combined system of the open system fully coupled to its environment is thermalized at finite temperature using an imaginary-time evolution procedure before the application of real-time evolution. This initializes the combined system in the correct canonical equilibrium state rather than being initially decoupled. Here we apply our theory to the spin-boson Hamiltonian and develop a number of competing ESLN variants designed to reduce the numerical divergence of the trace of the open-system density matrix. We find that a careful choice of the driving noises is essential for improving numerical stability. We also investigate the effect of applying higher-order numerical schemes for solving stochastic differential equations, such as the Stratonovich-Heun scheme, and conclude that stochastic sampling dominates convergence with the improvement associated with the numerical scheme being less important for short times but required for late times. To verify the method and its numerical implementation, we first consider evolution under a fixed Hamiltonian and show that the system either remains in, or approaches, the correct canonical equilibrium state at long times. Additionally, evolution of the open system under nonequilibrium Landau-Zener (LZ) driving is considered and the asymptotic convergence to the LZ limit was observed for vanishing system-environment coupling and temperature. When coupling and temperature are nonzero, initially thermalizing the combined system at a finite time in the past was found to be a better approximation of the true LZ initial state than starting in a pure state.

The Stochastic Liouville-von Neumann (SLN) equation describes the dynamics of an open quantum system reduced density matrix coupled to a non-Markovian harmonic environment. The interaction with the environment is represented by complex coloured noises which drive the system, and whose correlation functions are set by the properties of the environment. We present a number of schemes capable of generating coloured noises of this kind that are built on a noise amplitude reduction procedure [Imai et al, Chem. Phys. 446, 134 (2015)], including two analytically optimised schemes. In doing so, we pay close attention to the properties of the correlation functions in Fourier space, which we derive in full. For some schemes the method of Wiener filtering for deconvolutions leads to the realisation that weakening causality in one of the noise correlation functions improves numerical convergence considerably, allowing us to introduce a well controlled method for doing so. We compare the ability of these schemes, along with an alternative optimised scheme [Schmitz and Stockburger, Eur. Phys. J.: Spec. Top. 227, 1929 (2019)], to reduce the growth in the mean and variance of the trace of the reduced density matrix, and their ability to extend the region in which the dynamics is stable and well converged for a range of temperatures. By numerically optimising an additional noise scaling freedom, we identify the scheme which performs best for the parameters used, improving convergence by orders of magnitude and increasing the time accessible by simulation., 23 pages, 7 figures

Computing the stochastic entropy production associated with the evolution of a stochastic dynamical system is a well-established problem. In a small number of cases such as the Ornstein–Uhlenbeck process, of which we give a complete exposition, the distribution of entropy production can be obtained analytically. For a general potential it is much harder. A recent development in solving the Fokker–Planck equation, in which the solution is written as a product of positive functions, addresses any system governed by the condition of detailed balance, thereby permitting nonlinear potentials. Using examples in one and higher dimension, we demonstrate how such a framework is very convenient for the computation of stochastic entropy production in diffusion processes.

Several studies have involved a combination of heat and gravitational energy exchanges to create novel heat engines. A common theoretical framework is developed here to describe thermogravitational cycles which have the same efficiencies as the Carnot, Rankine, or Brayton cycles. Considering a working fluid enclosed in a balloon inside a column filled with a transporting fluid, a cycle is composed of four steps. Starting from the top of the column, the balloon goes down by gravity, receives heat from a hot source at the bottom, then rises and delivers heat to a cold source at the top. Unlike classic power cycles which need external work to operate the compressor, thermogravitational cycles can operate as a &ldquo, pure power cycle&rdquo, where no external work is needed to drive the cycle. To illustrate this concept, the prototype of a thermogravitational electrical generator is presented. It uses a hot source of average temperature near 57 &deg, C and relies on the gravitational energy exchanges of an organic fluorinated fluid inside a balloon attached to a magnetic marble to produce an electromotive force of 50 mV peak to peak by the use of a linear alternator. This heat engine is well suited to be operated using renewable energy sources such as geothermal gradients or focused sunlight.

We present a procedure for simulating epitaxial growth based on the phase-field method. We consider a basic model in which growth is initiated by a flux of atoms onto a heated surface. The deposited atoms diffuse in the presence of this flux and eventually collide to form islands which grow and decay by the attachment and detachment of migrating atoms at their edges. Our implementation of the phase-field method for this model includes uniform deposition, isotropic surface diffusion, and stochastic nucleation (in both space and time), which creates islands whose boundaries evolve as the surface atoms "condense" into and "evaporate" from the islands. Computations using this model in the submonolayer regime, prior to any appreciable coalescence of islands, agree with the results of kinetic Monte Carlo (KMC) simulations for the coverage-dependence of adatom and island densities and island-size distributions, for both reversible and irreversible growth. The scaling of the island density, as obtained from homogeneous rate equations, agrees with KMC simulations for irreversible growth and for reversible growth for varying deposition flux at constant temperature. For reversible growth with varying temperature but constant flux, agreement relies on an estimate of the formation energy of the critical cluster. Taken together, our results provide a comprehensive analysis of the phase-field method in the submonolayer regime of epitaxial growth, including the verification of the main scaling laws for adatoms and island densities and the scaling functions for island-size distributions, and point to the areas where the method can be extended and improved.

We present a two-state empirical valence bond (EVB) potential describing interactions between sulphuric acid and water molecules and designed to model proton transfer between them within a classical dynamical framework. The potential has been developed in order to study the properties of molecular clusters of these species, which are thought to be relevant to atmospheric aerosol nucleation. The particle swarm optimisation method has been used to fit the parameters of the EVB model to density functional theory (DFT) calculations. Features of the parametrised model and DFT data are compared and found to be in satisfactory agreement. In particular, it is found that a single sulphuric acid molecule will donate a proton when clustered with four water molecules at 300 K and that this threshold is temperature dependent.

Effects of acoustic waves on a phase transformation in a metastable phase were investigated in our previous work [S. R. Haqshenas, I. J. Ford, and N. Saffari, “Modelling the effect of acoustic waves on nucleation,” J. Chem. Phys. 145, 024315 (2016)]. We developed a non-equimolar dividing surface cluster model and employed it to determine the thermodynamics and kinetics of crystallisation induced by an acoustic field in a mass-conserved system. In the present work, we developed a master equation based on a hybrid Szilard-Fokker-Planck model, which accounts for mass transportation due to acoustic waves. This model can determine the kinetics of nucleation and the early stage of growth of clusters including the Ostwald ripening phenomenon. It was solved numerically to calculate the kinetics of an isothermal sonocrystallisation process in a system with mass transportation. The simulation results show that the effect of mass transportation for different excitations depends on the waveform as well as the imposed boundary conditions and tends to be noticeable in the case of shock waves. The derivations are generic and can be used with any acoustic source and waveform.

It is widely accepted that the nucleation of graphene on transition metals is related to the formation of carbon clusters of various sizes and shapes on the surface. Assuming a low concentration of carbon atoms on a crystal surface, we derive a thermodynamic expression for the grand potential of the cluster of N carbon atoms, relative to a single carbon atom on the surface (the cluster work of formation). This is derived taking into account both the energetic and entropic contributions, including structural and rotational components, and is explicitly dependent on the temperature. Then, using ab initio density functional theory, we calculate the work of formation of carbon clusters CN on the Ir(111) surface as a function of temperature considering clusters with up to N = 16 C atoms. We consider five types of clusters (chains, rings, arches, top-hollow, and domes), and find, in agreement with previous zero temperature studies, that at elevated temperatures the structure most favoured depends on N, with chains and arches being the most likely at N10. Our calculations reveal the work of formation to have a much more complex character as a function of the cluster size than one would expect from classical nucleation theory: for typical conditions, the work of formation displays not one but two nucleation barriers, at around N = 4-5 and N = 9-11. This suggests, in agreement with existing LEEM data, that five atom carbon clusters, along with C monomers, must play a pivotal role in the nucleation and growth of graphene sheets, whereby the formation of large clusters is achieved from the coalescence of smaller clusters (Smoluchowski ripening). Although the main emphasis of our study is on thermodynamic aspects of nucleation, the pivotal role of kinetics of transitions between different cluster types during the nucleation process is also discussed for a few cases as illustrative examples.

A detailed review of the literature for the last 5–10 years on epitaxial growth of graphene is presented. Both experimental and theoretical aspects related to growth on transition metals and on silicon carbide are thoroughly reviewed. Thermodynamic and kinetic aspects of growth on all these materials, where possible, are discussed. To make this text useful for a wider audience, a range of important experimental techniques that have been used over the last decade to grow (e.g. CVD, TPG and segregation) and characterize (STM, LEEM, etc.) graphene are reviewed, and a critical survey of most important theoretical techniques is given. Finally, we critically discuss various unsolved problems related to growth and its mechanism which we believe require proper attention in future research.

We investigate some unusual behaviour observed while performing molecular dynamics simulations of small molecular clusters using a constrained Langevin thermostat. Atoms appear to be thermalised to different temperatures that depend on their mass and on the total number of particles in the system. The deviation from the zeroth law of thermodynamics can be considerable for small systems of heavy and light particles. We trace this behaviour to the absence of thermal noise acting on the centre of mass of the system. This is demonstrated by solving the stochastic dynamics for the constrained thermostat and comparing the results with simulation data. By removing the constraint, the Langevin thermostat may be restored to its intended behaviour. We also investigate a Langevin thermostat constrained to have zero total force acting on its centre of mass, and find similar deficiencies.

The complete mechanism behind the thermal decomposition of ethylene (C2H4) on Ir(111), which is the first step of graphene growth, is established for the first time employing a combination of experimental and theoretical methods. High-resolution X-ray photoelectron spectroscopy was employed, along with calculations of core level binding-energies, to identify the surface species and their evolution as the surface temperature is increased. To understand the experimental results, we have developed a reaction sequence between the various CnHm species, from ethylene to C monomers and dimers, based on ab initio density functional calculations of all the energy barriers and the Arrhenius prefactors for the most important processes. The resulting temperature evolution of all species obtained from the simulated kinetics of ethylene decomposition agrees with photoemission measurements. The molecular dissociation mechanism begins with the dehydrogenation of ethylene to vinylidene (CH2C), which is then converted to acetylene (CHCH) by the removal and addition of an H atom. The C–C bond is then broken to form methylidyne (CH), and in the same temperature range a small amount of ethylidyne (CH3C) is produced. Finally methylidyne dehydrogenates to produce C monomers that are available for the early stage nucleation of the graphene islands.

An in-depth discussion of nuclear pore complexes illustrates the importance of macromolecular confinement in biological systems., Physically interesting behaviour can arise when soft matter is confined to nanoscale dimensions. A highly relevant biological example of such a phenomenon is the Nuclear Pore Complex (NPC) found perforating the nuclear envelope of eukaryotic cells. In the central conduit of the NPC, of ∼30–60 nm diameter, a disordered network of proteins regulates all macromolecular transport between the nucleus and the cytoplasm. In spite of a wealth of experimental data, the selectivity barrier of the NPC has yet to be explained fully. Experimental and theoretical approaches are complicated by the disordered and heterogeneous nature of the NPC conduit. Modelling approaches have focused on the behaviour of the partially unfolded protein domains in the confined geometry of the NPC conduit, and have demonstrated that within the range of parameters thought relevant for the NPC, widely varying behaviour can be observed. In this review, we summarise recent efforts to physically model the NPC barrier and function. We illustrate how attempts to understand NPC barrier function have employed many different modelling techniques, each of which have contributed to our understanding of the NPC.

The uctuation relations have received considerable attention since their emergence and development in the 1990s. As an aid to newcomers to the eld, we present a summary of the main results and suggest ways to interpret this material. Starting with a consideration of the under-determined time evolution of a simple open system, formulated using continuous Markovian stochastic dynamics, an expression for the entropy generated with time is proposed in terms of the probability of observing a trajectory associated with a prescribed driving protocol, and the probability of its time-reverse. This forms the basis for a more complete theoretical description of non-equilibrium thermodynamic processes. Having established a connection between entropy production and an inequivalence in probability for forward and time-reversed events, we proceed in the manner of Sekimoto and Seifert, in particular, to derive results in stochastic thermodynamics: a description of the evolution of a system between equilibrium states that ties in with well-established thermodynamic expectations. We derive uctuation relations, state conditions for their validity, and illustrate their operation in some simple cases, thereby providing some introductory insight into the various celebrated symmetry relations that have emerged in this eld.

We evaluate the grand potential of a cluster of two molecular species, equivalent to its free energy of formation from a binary vapour phase, using a nonequilibrium molecular dynamics technique where guide particles, each tethered to a molecule by a harmonic force, move apart to disassemble a cluster into its components. The mechanical work performed in an ensemble of trajectories is analysed using the Jarzynski equality to obtain a free energy of disassembly, a contribution to the cluster grand potential. We study clusters of sulphuric acid and water at 300 K, using a classical interaction scheme, and contrast two modes of guided disassembly. In one, the cluster is broken apart through simple pulling by the guide particles, but we find the trajectories tend to be mechanically irreversible. In the second approach, the guide motion and strength of tethering are modified in a way that prises the cluster apart, a procedure that seems more reversible. We construct a surface representing the cluster grand potential, and identify a critical cluster for droplet nucleation under given vapour conditions. We compare the equilibrium populations of clusters with calculations reported by Henschel et al. [J. Phys. Chem. A 118, 2599 (2014)] based on optimised quantum chemical structures.

Atmospheric aerosols play a vital role in affecting climate by influencing the properties and lifetimes of clouds and precipitation. Understanding the underlying microscopic mechanisms involved in the nucleation of aerosol droplets from the vapour phase is therefore of great interest. One key thermodynamic quantity in nucleation is the excess free energy of cluster formation relative to that of the saturated vapour. In our current study, the excess free energy is extracted for clusters of pure water modelled with the TIP4P/2005 intermolecular potential using a method based on nonequilibrium molecular dynamics and the Jarzynski relation. The change in free energy associated with the "mitosis" or division of a cluster of N water molecules into two N/2 sub-clusters is evaluated. This methodology is an extension of the disassembly procedure used recently to calculate the excess free energy of argon clusters [H. Y. Tang and I. J. Ford, Phys. Rev. E 91, 023308 (2015)]. Our findings are compared to the corresponding excess free energies obtained from classical nucleation theory (CNT) as well as internally consistent classical theory (ICCT). The values of the excess free energy that we obtain with the mitosis method are consistent with CNT for large cluster sizes but for the smallest clusters, the results tend towards ICCT; for intermediate sized clusters, we obtain values between the ICCT and CNT predictions. Furthermore, the curvature-dependent surface tension which can be obtained by regarding the clusters as spherical droplets of bulk density is found to be a monotonically increasing function of cluster size for the studied range. The data are compared to other values reported in the literature, agreeing qualitatively with some but disagreeing with the values determined by Joswiak et al. [J. Phys. Chem. Lett. 4, 4267 (2013)] using a biased mitosis approach; an assessment of the differences is the main motivation for our current study.

A phase transformation in a metastable phase can be affected when it is subjected to a high intensity ultrasound wave. In this study we determined the effect of oscillation in pressure and temperature on a phase transformation using the Gibbs droplet model in a generic format. The developed model is valid for both equilibrium and non-equilibrium clusters formed through a stationary or non-stationary process. We validated the underlying model by comparing the predicted kinetics of water droplet formation from the gas phase against experimental data in the absence of ultrasound. Our results demonstrated better agreement with experimental data in comparison with classical nucleation theory. Then, we determined the thermodynamics and kinetics of nucleation and the early stage of growth of clusters in an isothermal sonocrystallisation process. This new contribution shows that the effect of pressure on the kinetics of nucleation is cluster size-dependent in contrast to classical nucleation theory.

It is nearly 150 years since Maxwell challenged the validity of the second law of thermodynamics by imagining a tiny creature who could sort the molecules of a gas in such a way that would decrease entropy without exerting any work. The demon has been discussed largely using thought experiments, but it has recently become possible to exert control over nanoscale systems, just as Maxwell imagined, and the status of the second law has become a more practical matter, raising the issue of how measurements manage our ignorance in a way that can be exploited. The framework of stochastic thermodynamics extends macroscopic concepts such as heat, work, entropy and irreversibility to small systems and allows us explore the matter. Some arguments against a successful demon imply a second law that can be suspended indefinitely until we dissipate energy in order to remove the records of his operations. In contrast, under stochastic thermodynamics the demon fails because on average more work is performed upfront in making a measurement than is to be extracted by exploiting the outcome. This requires us to exclude systems and a demon that evolve under what might be termed self-sorting dynamics, and we reflect on the constraints on control that this implies while still working within a thermodynamic framework.

We compute statistical properties of the stochastic entropy production associated with the nonstationary transport of heat through a system coupled to a time dependent nonisothermal heat bath. We study the one-dimensional stochastic evolution of a bound particle in such an environment by solving the appropriate Langevin equation numerically, and by using an approximate analytic solution to the Kramers equation to determine the behavior of an ensemble of systems. We express the total stochastic entropy production in terms of a relaxational or nonadiabatic part together with two components of housekeeping entropy production and determine the distributions for each, demonstrating the importance of all three contributions for this system. We compare the results with an approximate analytic model of the mean behavior and we further demonstrate that the total entropy production and the relaxational component approximately satisfy detailed fluctuation relations for certain time intervals. Finally, we comment on the resemblance between the procedure for solving the Kramers equation and a constrained extremization, with respect to the probability density function, of the spatial density of the mean rate of production of stochastic entropy.

We develop a set of microscopic rate equations describing the growth and decay of molecular clusters adsorbed onto a seed particle. Such Becker–Doring equations are fundamental to descriptions of homogeneous nucleation, but do not appear to have been developed for the rather more complicated case of heterogeneous nucleation. We show that the familiar Fletcher theory of heterogeneous nucleation emerges from such a Becker–Doring description, but only if the concentration of adsorbed single molecules on the surface is estimated in a rather rudimentary manner. For small seed particles, this approach fails and one needs a proper Becker–Doring approach to provide a better estimate. The change in predicted nucleation rate can be several orders of magnitude for nanometre-size seed particles. We go on to include into the Becker–Doring treatment the processes of growth and decay of clusters by monomer surface diffusion on the seed. We use recent high quality experimental data to show that the latter process can make only a small contribution to the nucleation current for nanoparticle seeds. We also use the data to demonstrate that the traditional Fletcher theory fails to account for critical sizes and nucleation currents correctly, and that a modification to the implied underlying cluster properties is necessary.

The possibility of magnesium oxide being the first species to nucleate in the cooling outflows around M stars has been investigated. By treating the formation of the seed nuclei as a homogeneous nucleation problem and using molecular dynamics data obtained with the 'compressible ion potential' for MgO, free energy calculations are performed to obtain an estimate of the population densities of MgO clusters of various sizes. It is found that a free energy barrier of at least hundreds of k B T would need to be climbed in order for MgO to nucleate in significant amount in typical circumstellar shells, hence ruling out MgO as a realistic candidate for the primary nucleating dust species. This is in agreement with a similar conclusion reached in earlier studies, although the present calculations are based on a much more robust potential model for MgO.

We derive analogues of the Jarzynski equality and Crooks relation to characterise the nonequilibrium work associated with changes in the spring constant of an overdamped oscillator in a quadratically varying spatial temperature profile. The stationary state of such an oscillator is described by Tsallis statistics, and the work relations for certain processes may be expressed in terms of q-exponentials. We suggest that these identities might be a feature of nonequilibrium processes in circumstances where Tsallis distributions are found.

The selection of an equilibrium state by maximizing the entropy of a system, subject to certain constraints, is often powerfully motivated as an exercise in logical inference, a procedure where conclusions are reached on the basis of incomplete information. But such a framework can be more compelling if it is underpinned by dynamical arguments, and we show how this can be provided by stochastic thermodynamics, where an explicit link is made between the production of entropy and the stochastic dynamics of a system coupled to an environment. The separation of entropy production into three components allows us to select a stationary state by maximizing the change, averaged over all realizations of the motion, in the principal relaxational or nonadiabatic component, equivalent to requiring that this contribution to the entropy production should become time independent for all realizations. We show that this recovers the usual equilibrium probability density function (pdf) for a conservative system in an isothermal environment, as well as the stationary nonequilibrium pdf for a particle confined to a potential under nonisothermal conditions, and a particle subject to a constant nonconservative force under isothermal conditions. The two remaining components of entropy production account for a recently discussed thermodynamic anomaly between over- and underdamped treatments of the dynamics in the nonisothermal stationary state.

A conceptual framework to predict microphysical and optical properties of contrail particles within a wingspan behind the source aircraft is developed. Results from two decades of contrail observations and numerical simulations are reviewed forming the basis of theoretical model development. The model utilizes cloud theory applied to the dynamics and thermodynamics of jet aircraft exhaust plumes in upper tropospheric conditions. Droplet nuclei include soot particles emitted from aircraft engines and atmospheric particles entrained into the plume. These precursor particles activate into copious homogeneously freezing water droplets as the plume relative humidity rises beyond liquid water saturation. A unimodal size spectrum of ice particles develops wherein ice particles grow to micrometer mean sizes. Contrail particle formation is analyzed over a wide range of soot emissions relating to conventional jet fuels as well as to alternative aviation fuels producing much less soot and volatile particle emissions. For current aviation fuels and propulsion technology, the number of contrail ice particles scales roughly in proportion to the number of emitted soot particles that act as water condensation nuclei despite their poor hygroscopicity. Close to the contrail formation threshold, only few plume particles can be water activated and freeze. Implications for effects of alternative fuels on contrails, an arena for future scientific exploration, are outlined.

The excess free energy of a molecular cluster is a key quantity in models of the nucleation of droplets from a metastable vapour phase; it is often viewed as the free energy arising from the presence of an interface between the two phases. We show how this quantity can be extracted from simulations of the mechanical disassembly of a cluster using guide particles in molecular dynamics. We disassemble clusters ranging in size from 5 to 27 argon-like Lennard-Jones atoms, thermalised at 60 K, and obtain excess free energies, by means of the Jarzynski equality, that are consistent with previous studies. We only simulate the cluster of interest, in contrast to approaches that require a series of comparisons to be made between clusters differing in size by one molecule. We discuss the advantages and disadvantages of the scheme and how it might be applied to more complex systems.

Analytic and numerical calculations are performed on the production of sulphuric acid aerosol in conditions of a very large nucleation event observed in the upper troposphere. The numerical results feature a growing peak in the size distribution whose magnitude is reproduced well analytically, and are consistent with the observed particle number concentration at sizes greater than 25 nm (measured dry diameter), but suggest that most of the aerosol was at unobserved smaller sizes. Because of growth and coagulation, number concentrations of the aerosol rapidly become independent of the number initially nucleated, so that conclusions as to the nucleation process, either homogeneous or ion-induced nucleation, cannot easily be drawn from existing atmospheric observations. The final concentration is very insensitive to the magnitude of the SO2 source, but, if condensation on, and coagulation with, a remnant background aerosol occurs, such nucleation events will be cut off for source magnitudes less than a specific value. Anthropogenic emissions of SO2 which exceed this value can produce higher aerosol number concentrations in the atmosphere with consequences for the indirect effect of aerosols on the climate.

Mean field density functional theory (MFDFT) has been employed to calculate the free energy of a pair of attractive hard rods on a ring. The results for homogeneous and optimal inhomogeneous density profiles have been compared with the exact free energy as a test of the approach. We discuss the problems in applying MFDFT to small systems and suggest modifications which allow a reasonably accurate treatment of this particular, rather extreme, case.

The principles of statistical mechanics have been used to develop a theory of the nucleation of a phase transition, but a number of subtle questions remain and are highlighted in this review. A central issue is the cluster definition, the mathematical scheme which distinguishes a molecular cluster from a collection of separate molecules. There is also the question of whether thermodynamic transition state theory is suitable to describe the process, or whether the development of kinetic rate equations describing cluster growth and decay is a better approach. The classical theory of nucleation is flawed but appears to provide useful estimates in some cases, including water. Phenomenological extensions of the classical theory can improve matters. However, improvements in the theory from microscopic considerations are not simple to apply, require major computational effort and suffer from uncertainties due to lack of knowledge of the fundamental intermolecular interactions. Calculations of the nucleation rate of water droplets are especially difficult since this substance is notoriously difficult to model. Nevertheless, as capabilities improve, accurate calculations should come within reach, which will offer better understanding of the process for practical applications such as the transition from dry to wet steam.

Fusion reactors create extreme conditions for structures close to the plasma. It seems unlikely that materials currently being considered can meet all performance requirements under such conditions. We explore the possibility of separating functionality in composite structures to overcome this barrier. To this end, several suggestions of directions are made for the search for such materials. In particular, we note some of the new materials that have become available only in the last two decades. Those discussed include the use of diamondlike carbon coatings, nano-structured materials, layered structures, stacked structures, and viscous coatings, including more complex carbon composite materials. Materials modelling will be an important component in the search for viable materials. However, the extreme conditions and the nature of the radiation damage demand extensions both to molecular dynamics and to the much-used Norgett–Robinson–Torrens model. We identify some of the relevant condensed matter challenges for modellin ga nd materials testing in the fusion context, including the relevance of spallation source neutron testing to fusion materials evaluation.

Charged particle colloids typically consist of particles with negative surface charge suspended in an aqueous electrolyte solution. A layer of ions forms around each particle, and previous research has shown that the dynamics of this charge distribution, induced by an external field, lead to effective and potentially useful dielectric properties (Lyklema J and Van Leeuwen H P 1999 Adv. Colloid and Interface Sci. 83 33, O’Konski C T 1960 J. Phys. Chem. 64 605). Charged nanoparticle colloids are potentially of great benefit because they are extremely stable and their small dimensions make them highly versatile. Possible uses include electromagnetic shielding devices and medical applications. This research focuses on the dielectric properties of charged polystyrene nanoparticle colloids, which are investigated across a range of radio and microwave frequencies. The results show two distinct regions in the permittivity behaviour. The first is the dipolar relaxation of water at microwave frequencies. The second is a high amplitude, low frequency relaxation due to the diffusive motion of ions in the bulk electrolyte beyond the double layer. Effective control of the resulting dielectric response is demonstrated with particle size ranging from 20 to 220 nm. It is shown that the small nanoparticle sizes lead to faster relaxation times than those expected from conventional micron particle sizes, such that the loss extends to microwave frequencies. The results are fitted to a model where the zeta potential, ξ , and double layer thickness, κ −1 , are used as fitting parameters.

We consider chemical reactions taking place in aerosol droplets when the mean number of reactant molecules present is less than unity. Situations of this kind are possible for ultrafine droplets in an atmosphere containing trace amounts of gas phase reactants. We describe an analytic description of such ‘stochastic’ chemistry, and compare the results with the predictions of traditional chemical kinetics. We establish criteria to indicate when the traditional approach can overpredict a reaction rate considerably. We determine that it is rare for such conditions to occur in the atmosphere.

Molecular configurations which count as snapshots of a quasibound cluster are identified through a retrospective dynamical definition. The trajectory of a molecular cluster is followed, and a clear evaporation event is considered to have occurred when a molecule moves a very long distance away from the others. The cluster is judged to have broken before this condition is satisfied, however, at the instant that the energy of the departing molecule in the center of mass frame becomes positive. The decay of a cluster is therefore defined dynamically as the production of a molecule with positive energy on a separating trajectory. Not all positive energy molecules created by the system follow such a trajectory, hence the need to examine the subsequent behavior in molecular dynamics. We simulate a sequence of decays by repairing broken clusters as they occur. This approach enables us to calculate mean decay rates of isolated Lennard-Jones clusters in what promises to be a physically realistic fashion.

The process of nucleation is normally described using rate equations for the mean populations of molecular clusters. This approach can be justified for cases where these mean populations are large. However, it may be unsuitable in the case of heterogeneous nucleation on small particles if the mean populations are of the order of unity or less. In such a case, considering the average populations might be erroneous since the statistical fluctuations in the molecular populations should be taken into account. Here a stochastic treatment of heterogeneous nucleation kinetics is presented that is described by a set of master equations, and a modified expression for the nucleation rate has been deduced. Furthermore, a numerical method for solving the stochastic system has been examined, and the results show that the rate of nucleation can differ greatly from that obtained with the traditional kinetics.

Properties of the critical clusters that are instrumental in the nucleation of a first-order phase transition can be obtained through the analysis of nucleation rate data. The theoretical tools for doing so are known as nucleation theorems. We consider solid-phase critical clusters forming from a liquid phase, and it is shown that their size and binding energies can be extracted from data gathered at a range of temperatures and pressures. Ice crystal nucleation rates described recently are used to study ice clusters containing several hundred water molecules.

Cloud chamber data for the nucleation of droplets from supersaturated n- and i-octane vapors are analyzed using nucleation theorems. We obtain the excess energies of pure and mixed component molecular clusters with sizes ranging from 19 to 58 molecules. We plot this information in the form of an excess energy surface for a range of cluster compositions. Since the two species are similar we also combine the data into a plot of excess energy against the total number of molecules in the cluster. We show that the capillarity approximation fails to predict the critical cluster composition, though it does provide a rough estimate of the excess energy of a specified cluster.

The nuclear pore complex (NPC) is the gate for transport between the cell nucleus and the cytoplasm. Small molecules cross the NPC by passive diffusion, but molecules larger than ~5 nm must bind to nuclear transport receptors to overcome a selective barrier within the NPC1. Whilst the structure and shape of the cytoplasmic ring of the NPC are relatively well characterized2-5, the selective barrier is situated deep within the central channel of the NPC and depends critically on unstructured nuclear pore proteins5,6, and is therefore not well understood. Here, we show that stiffness topography7 with sharp atomic force microscopy tips can generate nanoscale cross sections of the NPC. The cross sections reveal two distinct structures, a cytoplasmic ring and a central plug structure, which are consistent with the three-dimensional NPC structure derived from electron microscopy2-5. The central plug persists after reactivation of the transport cycle and resultant cargo release, indicating that the plug is an intrinsic part of the NPC barrier. Added nuclear transport receptors accumulate on the intact transport barrier and lead to a homogenization of the barrier stiffness. The observed nanomechanical properties in the NPC indicate the presence of a cohesive barrier to transport, and are quantitatively consistent with the presence of a central condensate of nuclear pore proteins in the NPC channel.

A laminar flow tube reactor was designed and constructed to provide an accurate, quantitative measurement of a nucleation rate as a function of supersaturation and temperature. Measurements of nucleation of a supersaturated vapor of dibutylphthalate have been made for the temperature range from −30.3 to +19.1 °C. A thorough analysis of the possible sources of experimental uncertainties (such as defining the correct value of the initial vapor concentration, temperature boundary conditions on the reactor walls, accuracy of the calculations of the thermodynamic parameters of the nucleation zone, and particle concentration measurement) is given. Both isothermal and the isobaric nucleation rates were measured. The experimental data obtained were compared with the measurements of other experimental groups and with theoretical predictions made on the basis of the self-consistency correction nucleation theory. Theoretical analysis, based on the first and the second nucleation theorems, is also presented. The crit...

We generalize the second nucleation theorem to multicomponent systems. Nucleation theorems are used to extract the molecular composition and excess internal energy of the critical cluster from experimental nucleation rates in a water–ethanol mixture. The excess internal energy is found to depend only weakly on temperature and to be almost solely a function of the molecular numbers of water and ethanol in the cluster. We estimate the contribution of the kinetic pre-factor to our analysis, and find that it is small in the case of the first theorem, but significant for the second theorem. We find that capillarity approximation fails to predict the experimental critical size and excess energy in this highly nonideal system.