Attractive Square Research Articles

Cooling-driven coagulation

ABSTRACT Astrophysical gases such as the interstellar-, circumgalactic-, or intracluster-medium are commonly multiphase, which poses the question of the structure of these systems. While there are many known processes leading to fragmentation of cold gas embedded in a (turbulent) hot medium, in this work, we focus on the reverse process: coagulation. This is often seen in wind-tunnel and shearing layer simulations, where cold gas fragments spontaneously coalesce. Using 2D and 3D hydrodynamical simulations, we find that sufficiently large (≫cstcool), perturbed cold gas clouds develop pulsations which ensure cold gas mass growth over an extended period of time (≫r/cs). This mass growth efficiently accelerates hot gas which in turn can entrain cold droplets, leading to coagulation. The attractive inverse square force between cold gas droplets has interesting parallels with gravity; the ‘monopole’ is surface area rather than mass. We develop a simple analytic model which reproduces our numerical findings.

Simulation study on the adsorption of polymer chains on checkerboard‐patterned surfaces

AbstractThe adsorption behavior of polymer chains on a checkerboard‐patterned surface consisting of attractive and neutral square blocks is simulated using dynamic Monte Carlo method. The polymer chain is adsorbed on the surface below the critical adsorption temperature Tc and finally form a recognizable state below the pattern recognition temperature Tr. Both Tc and Tr increase with increasing the block width L. The increase of Tc is because that the effect of neutral blocks reduces with increasing L, while the increase of Tr indicates that it is easy for the adsorbed polymer chain to recognize large pattern. Moreover, below Tc, the polymer chain is in a multi‐block adsorption state (MBAS) and multi‐block recognition state (MBRS) for small L and in a single‐block adsorption state (SBAS) and single‐block recognition state (SBRS) for large L. For moderate L, as the temperature decreases, the polymer changes from the SBAS to the MBAS or from SBRS to MBRS below the single‐block to multi‐block adsorption temperature Tsm. Tc and Tsm increase with increasing N, whereas Tr decreases with increasing N. Lastly, the intra‐polymer attraction helps the polymer chain to recognize in a SBRS.

The Interplay between the Theories of Mode Coupling and of Percolation Transition in Attractive Colloidal Systems.

In the recent years a considerable effort has been devoted to foster the understanding of the basic mechanisms underlying the dynamical arrest that is involved in glass forming in supercooled liquids and in the sol-gel transition. The elucidation of the nature of such processes represents one of the most challenging unsolved problems in the field of material science. In this context, two important theories have contributed significantly to the interpretation of these phenomena: the Mode-Coupling theory (MCT) and the Percolation theory (PT). These theories are rooted on the two pillars of statistical physics, universality and scale laws, and their original formulations have been subsequently modified to account for the fundamental concepts of Energy Landscape (EL) and of the universality of the fragile to strong dynamical crossover (FSC). In this review, we discuss experimental and theoretical results, including Molecular Dynamics (MD) simulations, reported in the literature for colloidal and polymer systems displaying both glass and sol-gel transitions. Special focus is dedicated to the analysis of the interferences between these transitions and on the possible interplay between MCT and PT. By reviewing recent theoretical developments, we show that such interplay between sol-gel and glass transitions may be interpreted in terms of the extended F13 MCT model that describes these processes based on the presence of a glass-glass transition line terminating in an A3 cusp-like singularity (near which the logarithmic decay of the density correlator is observed). This transition line originates from the presence of two different amorphous structures, one generated by the inter-particle attraction and the other by the pure repulsion characteristic of hard spheres. We show here, combining literature results with some new results, that such a situation can be generated, and therefore experimentally studied, by considering colloidal-like particles interacting via a hard core plus an attractive square well potential. In the final part of this review, scaling laws associated both to MCT and PT are applied to describe, by means of these two theories, the specific viscoelastic properties of some systems.

Fall-to-the-centre as a symmetry breaking transition

The attractive inverse square potential arises in a number of physical problems such as a dipole interacting with a charged wire, the Efimov effect, the Calgero-Sutherland model, near-horizon black hole physics and the optics of Maxwell fisheye lenses. Proper formulation of the inverse-square problem requires specification of a boundary condition (regulator) at the origin representing short-range physics not included in the inverse square potential and this generically breaks the Hamiltonian’s continuous scale invariance in an elementary example of a quantum anomaly. The system’s spectrum qualitatively changes at a critical value of the inverse-square coupling, and we here point out that the transition at this critical potential strength can be regarded as an example of a symmetry breaking transition. In particular, we use point particle effective field theory (PPEFT), as developed by Burgess et al [1], to characterize the renormalization group (RG) evolution of the boundary coupling under rescalings. While many studies choose boundary conditions to ensure the system is unitary, these RG methods allow us to systematically handle the richer case of nonunitary physics describing a source or sink at the origin (such as is appropriate for the charged wire or black hole applications). From this point of view the RG flow changes character at the critical inverse-square coupling, transitioning from a sub-critical regime with evolution between two real, unitary fixed points ( symmetric phase) to a super-critical regime with imaginary, dissipative fixed points ( symmetry broken phase) that represent perfect-sink and perfect-source boundary conditions, around which the flow executes limit-cycle evolution.

Waiting-time dependent non-equilibrium phase diagram of simple glass- and gel-forming liquids.

Under numerous circumstances, many soft and hard materials are present in a puzzling wealth of non-equilibrium amorphous states, whose properties are not stationary and depend on preparation. They are often summarized in unconventional "phase diagrams" that exhibit new "phases" and/or "transitions" in which time, however, is an essential variable. This work proposes a solution to the problem of theoretically defining and predicting these non-equilibrium phases and their time-evolving phase diagrams, given the underlying molecular interactions. We demonstrate that these non-equilibrium phases and the corresponding non-stationary (i.e., aging) phase diagrams can indeed be defined and predicted using the kinetic perspective of a novel non-equilibrium statistical mechanical theory of irreversible processes. This is illustrated with the theoretical description of the transient process of dynamic arrest into non-equilibrium amorphous solid phases of an instantaneously quenched simple model fluid involving repulsive hard-sphere plus attractive square well pair interactions.

Repulsive, Densely Packed Ligand-Shells Mediate Interactions between PbS Nanocrystals in Solution

The surface capping ligands on nanocrystals (NCs) impart colloidal stability and determine the organization into ordered solids when cast from solution. Here, we use small-angle neutron scattering (SANS) to measure the solution structure factor in dispersions of fully passivated, oleate-capped PbS NCs up to concentrations of 16 v/v%. Fitting the data by assuming an attractive square well interaction potential between NCs yields a repulsive core diameter larger than the physical NC core diameter. This repulsive core stems from a densely packed region of the ligand-shell near the NC surface. The spatial extent of the repulsive core increases with temperature as ligand motion increases, becoming repulsive farther out in the ligand-shell, while the portion of ligand-shell outside of the repulsive region remains attractive with a strength of ∼1 kBT. SANS provides a molecular interpretation of the fundamental interparticle interactions in colloidal nanocrystal suspensions, ultimately relevant to understanding the self-organization of nanoscale materials.

Confinement of an ultra-cold-matter wave packet near the delocalization threshold by a waveguide bend with two or more contact impurities

The properties of confined quantum systems are of fundamental importance in the understanding of quantum behavior at extreme conditions. One of such quantum systems at extreme conditions is an ultra-cold quantum matter wave packet confined by a square well potential. The square well potential could be produced by a bend in a waveguide in an atom chip. Of further interest is the presence of contact impurities inside the bend produced by guide defects. In order to study such a system, its properties are well described by the nonlinear Schrodinger equation. In this work, we generalize our previous work (Mendez-Fragoso and Cabrera-Trujillo in Eur J Phys D 69:139, 2015) to study the ground state behavior of a quantum matter wave packet under confinement by a bend modeled by an attractive square well potential of strength $$V_0$$ and width 2R in the presence of several contact impurities represented by delta potentials with strength $$\beta $$ placed inside it at the threshold of delocalization, i.e., at the point where the well has trapped the maximum number of particles. We find that the maximum number of atoms held by the squared well potential only depends on the total strength of the contact impurities and not on their position. This implies that the maximum number of trapped particles is the same for one or more contact impurities as long as the combined strength is the same. For small $$R \sqrt{V_0}$$ values of the squared well potential, the number of particles trapped at the delocalization threshold is lower than when no contact impurities are present and decreases more as the contact impurity repulsive strength increases. This induces a threshold on the maximum number of particles trapped by an attractive squared well at $$2R V_0=\sum _i\beta _i$$ . The matter wave packet profile shows the effect of the contact impurity as a wedge shape at the contact impurity position.

Inverse Square Potential in a Space with Spin Noncommutativity of Coordinates

An attractive inverse square potential has been considered in a space with the spin noncommutativity of coordinates. The corresponding effective potential energy, as well as the total energy, is shown to be bounded from below. Using the variational method, the upper limit of the ground-state energy, which turns out to be negative for a sufficiently large coupling constant, is found. As a result, it is proved that the inverse square potential creates stationary levels in the space concerned, unlike the case of commutative space, where a particle falls to the center.

A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms

Abstract We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term ~x −1/2 with arbitrary strength and a repulsive centrifugal barrier core ~x −2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.

Non-Linear Stationary Solutions in Realistic Models for Analog Black-Hole Lasers

From both a theoretical and an experimental point of view, Bose–Einstein condensates are good candidates for studying gravitational analogues of black holes and black-hole lasers. In particular, a recent experiment has shown that a black-hole laser configuration can be created in the laboratory. However, the most considered theoretical models for analog black-hole lasers are quite difficult to implement experimentally. In order to fill this gap, we devote this work to present more realistic models for black-hole lasers. For that purpose, we first prove that, by symmetrically extending every black-hole configuration, one can obtain a black-hole laser configuration with an arbitrarily large supersonic region. Based on this result, we propose the use of an attractive square well and a double delta-barrier, which can be implemented using standard experimental tools, for studying black-hole lasers. We also compute the different stationary states of these setups, identifying the true ground state of the system and discussing the relation between the obtained solutions and the appearance of dynamical instabilities.

Trapping-desorption and direct-scattering of formaldehyde at Au(111).

Nonreactive surface scattering of atoms, molecules and clusters can be almost universally described by two mechanisms: trapping-desorption and direct-scattering. A hard cube model with an attractive square well provides a zeroth order description of the branching ratio between these two mechanisms as a function of the incidence energy. However, the trapping process is likely to be enhanced by excitation of internal degrees of freedom during the collision. In this molecular beam surface scattering study, we characterize formaldehyde/Au(111) scattering using angle resolved time-of-flight techniques. The two mechanisms are found to compete in the range of the investigated normal incidence energies between 0.1 and 1.3 eV. Whereas at low incidence energies trapping-desorption dominates, direct-scattering becomes more likely at incidence energies above 0.37 eV. This incidence energy is slightly higher than the desorption energy which is found to be 0.32 ± 0.03 eV by temperature programmed desorption techniques. A simple hard cube model underestimates the observed trapping probabilities indicating the importance of trapping induced by excitation of internal molecular degrees of freedom.

Non-equilibrium theory of arrested spinodal decomposition.

The non-equilibrium self-consistent generalized Langevin equation theory of irreversible relaxation [P. E. Ramŕez-González and M. Medina-Noyola, Phys. Rev. E 82, 061503 (2010); 82, 061504 (2010)] is applied to the description of the non-equilibrium processes involved in the spinodal decomposition of suddenly and deeply quenched simple liquids. For model liquids with hard-sphere plus attractive (Yukawa or square well) pair potential, the theory predicts that the spinodal curve, besides being the threshold of the thermodynamic stability of homogeneous states, is also the borderline between the regions of ergodic and non-ergodic homogeneous states. It also predicts that the high-density liquid-glass transition line, whose high-temperature limit corresponds to the well-known hard-sphere glass transition, at lower temperature intersects the spinodal curve and continues inside the spinodal region as a glass-glass transition line. Within the region bounded from below by this low-temperature glass-glass transition and from above by the spinodal dynamic arrest line, we can recognize two distinct domains with qualitatively different temperature dependence of various physical properties. We interpret these two domains as corresponding to full gas-liquid phase separation conditions and to the formation of physical gels by arrested spinodal decomposition. The resulting theoretical scenario is consistent with the corresponding experimental observations in a specific colloidal model system.

Generalized Efimov Effect in One Dimension.

We study a one-dimensional quantum problem of two particles interacting with a third one via a scale-invariant subcritically attractive inverse square potential, which can be realized, for example, in a mixture of dipoles and charges confined to one dimension. We find that above a critical mass ratio, this version of the Calogero problem exhibits the generalized Efimov effect, the emergence of discrete scale invariance manifested by a geometric series of three-body bound states with an accumulation point at zero energy.

Effective interactions between fluid membranes.

A self-consistent theory is proposed for the general problem of interacting undulating fluid membranes subject to the constraint that they do not interpenetrate. We implement the steric constraint via an exact functional integral representation and, through the use of a saddle-point approximation, transform it into a novel effective steric potential. The steric potential is found to consist of two contributions: one generated by zero-mode fluctuations of the membranes and the other by thermal bending fluctuations. For membranes of cross-sectional area S, we find that the bending fluctuation part scales with the intermembrane separation d as d-2 for d≪√S but crosses over to d-4 scaling for d≫√S, whereas the zero-mode part of the steric potential always scales as d-2. For membranes interacting exclusively via the steric potential, we obtain closed-form expressions for the effective interaction potential and for the rms undulation amplitude σ, which becomes small at low temperatures T and/or large bending stiffnesses κ. Moreover, σ scales as d for d≪√S but saturates at √kBTS/κ for d≫√S. In addition, using variational Gaussian theory, we apply our self-consistent treatment to study intermembrane interactions subject to different types of potentials: (i) the Moreira-Netz potential for a pair of strongly charged membranes with an intervening solution of multivalent counterions, (ii) an attractive square well, (iii) the Morse potential, and (iv) a combination of hydration and van der Waals interactions.

Cluster formation and phase separation in heteronuclear Janus dumbbells

We have recently investigated the phase behavior of model colloidal dumbbells constituted by two identical tangent hard spheres, with the first being surrounded by an attractive square–well interaction (Janus dumbbells, Munaó et al 2014 Soft Matter 10 5269). Here we extend our previous analysis by introducing in the model the size asymmetry of the hard-core diameters and study the enriched phase scenario thereby obtained. By employing standard Monte Carlo simulations we show that in such ‘heteronuclear Janus dumbbells’ a larger hard-sphere site promotes the formation of clusters, whereas in the opposite condition a gas–liquid phase separation takes place, with a narrow interval of intermediate asymmetries wherein the two phase behaviors may compete. In addition, some peculiar geometrical arrangements, such as lamellæ, are observed only around the perfectly symmetric case. A qualitative agreement is found with recent experimental results, where it is shown that the roughness of molecular surfaces in heterogeneous dimers leads to the formation of colloidal micelles.

Inverse melting in a two-dimensional off-lattice model.

We carry out computer simulations of a simple, two-dimensional off-lattice model that exhibits inverse melting. The monodisperse system comprises core-softened disks interacting through a repulsive square shoulder located inside an attractive square well. By systematically varying the potential parameters, we increase the pressure range over which the liquid freezes to a crystal upon isobaric heating. The effect is largely controlled by the extent of the shoulder. Despite occurring in two dimensions, the melting transition is first order and to a liquid, rather than to a hexatic or quasicrystal phase. We also provide comment on a commonly employed correlation function used to determine the degree of translational ordering in a system.

Attractive Inverse Square Potential,U(1)Gauge, and Winding Transitions

The inverse square potential arises in a variety of different quantum phenomena, yet notoriously it must be handled with care: it suffers from pathologies rooted in the mathematical foundations of quantum mechanics. We show that its recently studied conformality breaking corresponds to an infinitely smooth winding-unwinding topological transition for the classical statistical mechanics of a one-dimensional system: this describes the tangling or untangling of floppy polymers under a biasing torque. When the ratio between torque and temperature exceeds a critical value the polymer undergoes tangled oscillations, with an extensive winding number. At lower torque or higher temperature the winding number per unit length is zero. Approaching criticality, the correlation length of the order parameter-the extensive winding number-follows a Kosterlitz-Thouless-type law. The model is described by the Wilson line of a (0+1) U(1) gauge theory, and applies to the tangling or untangling of floppy polymers and to the winding or diffusing kinetics in diffusion-convection reactions.

Viscosity of a neutron gas

Abstract The viscosity of nucleonic matter and in particular a neutron gas is studied both classically and in a quantum mechanical description. The collisions between particles are modeled as hard sphere scattering as a baseline for comparison and as scattering from an attractive square well potential. Properties associated with the unitary limit are developed which are shown to be approximately realized for a system of neutrons. The issue of near perfect fluid behavior of neutron matter is remarked on. Using some results from hard sphere molecular dynamics studies near perfect fluid behavior is discussed further.

Tunneling conductivity in composites of attractive colloids

In conductor-insulator nanocomposites in which conducting fillers are dispersed in an insulating matrix, the electrical connectedness is established by inter-particle tunneling or hopping processes. These systems are intrinsically non-percolative and a coherent description of the functional dependence of the conductivity σ on the filler properties, and in particular of the conductor-insulator transition, requires going beyond the usual continuum percolation approach by relaxing the constraint of a fixed connectivity distance. In this article, we consider dispersions of conducting spherical particles which are connected to all others by tunneling conductances and which are subjected to an effective attractive square-well potential. We show that the conductor-insulator transition at low contents φ of the conducting fillers does not determine the behavior of σ at larger concentrations, in striking contrast to what is predicted by percolation theory. In particular, we find that at low φ the conductivity is governed almost entirely by the stickiness of the attraction, while at larger φ values σ depends mainly on the depth of the potential well. As a consequence, by varying the range and depth of the potential while keeping the stickiness fixed, composites with similar conductor-insulator transitions may display conductivity variations of several orders of magnitude at intermediate and large φ values. By using a recently developed effective medium theory and the critical path approximation, we explain this behavior in terms of dominant tunneling processes which involve inter-particle distances spanning different regions of the square-well fluid structure as φ is varied. Our predictions could be tested in experiments by changing the potential profile with different depletants in polymer nanocomposites.

Resonant states and transmission coefficient oscillations for potential wells and barriers

The oscillatory behavior of the transmission coefficient T as a function of energy is examined for an attractive square well and a rectangular barrier. We calculate T using resonant state boundary conditions and demonstrate that the maxima in T are correlated with the broad resonances generated by these potentials. For barrier potentials the maxima signify resonances occurring at energies above the barrier height. It is shown that the resonance position and width can also be generated from the complex poles of the amplitude of the transmitted plane wave. We also explain the relation between the positions of the resonances generated by the square well and the rectangular barrier to the energy eigenvalues of the corresponding rigid box with the same range. We show for a potential with an attractive well and a repulsive barrier that T exhibits oscillations when the particle energy is below the barrier, implying that in many cases the simple WKB type barrier penetration expression for T is not adequate. These features of T are likely to hold for most attractive potentials and flat repulsive barriers. We also discuss the attractive modified Poschl–Teller type potential for which T does not show oscillations as a function of energy.