Square-well Potential Research Articles

Structural phases of colloids interacting via a flat-well potential

Using Langevin dynamics simulations we investigate the self-assembly of colloidal particles in two dimensions interacting via an isotropic potential, which comprises both a hard-core repulsion and an additional softened square-well potential of controllable width α. In dilute concentrations, the particles assemble in small clusters with a well-defined crystalline order. For small values of α the particles form triangular lattices. As α is increased, more particles can be captured by the potential well giving rise to different crystalline symmetries and the structural phase transitions between them. The main structures observed are triangular, square, and a mixture of square and triangular cells forming an Archimedean tiling. In the concentrated regime the particles form a single percolated cluster with essentially the same orderings at the same ranges of α values as observed in the dilute regime, thus showing that cluster boundary effects have a minor influence on the cluster crystal symmetry. By using energy analysis and geometry arguments we discuss how the different observed structures minimize the system energy at different values of α.

Vorticity of threshold strength in neutron drip-line nuclei

By adopting a finite square-well potential, the analytic expressions of vorticity strengths, which are related to transverse currents, are derived for some selected E1 and E2 transitions. From these analytic expressions it can be seen clearly that, like the threshold strengths, the vorticity strengths are also sharply distributed just above the threshold in nuclei near the drip-line and mainly come from the orbits with small binding energy.

Quantum effective force in an expanding infinite square-well potential and Bohmian perspective

The Schrödinger equation is solved for the case of a particle confined to a small region of a box with infinite walls. If the walls of the well are moved, then, due to an effective quantum nonlocal interaction with the boundary, even though the particle is nowhere near the walls, it will be affected. It is shown that this force apart from a minus sign is equal to the expectation value of the gradient of the quantum potential for vanishing at the walls boundary condition. The variation of this force with time is studied. A selection of Bohmian trajectories of the confined particle is also computed.

Pseudospin Symmetry in Single Particle Resonant States

The pseudospin symmetry (PSS) is a relativistic dynamical symmetry connected with the small component of the Dirac spinor. The origin of PSS in single particle bound states in atomic nuclei has been revealed and studied extensively. By examining the zeros of Jost functions corresponding to the small components of Dirac wave functions and phase shifts of continuum states, we show that the PSS in single particle resonant states in nuclei is conserved when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign. The exact conservation and the breaking of the PSS are illustrated for single particle resonances in spherical square-well and Woods-Saxon potentials.

Studying the general properties of potentials by means of dimensionless scaling variables

The general properties of the three-parametric Woods-Saxon potential and the two-parametric Hulthen and square-well potentials are studied by converting them to dimensionless scaling variables. The relations between the parameters of the potentials and such characteristics of a system as binding energies and the asymptotic normalization coefficients of wave functions are investigated.

Universal scaling behaviour of surface tension of molecular chains

We use and extend the universal relationship recently proposed by Galliero [G. Galliero, J. Chem. Phys. 133, 074705 (2010)], based on a combination of the corresponding-states principle of Guggenheim [E. A. Guggenheim, J. Chem. Phys. 13, 253 (1945)] and the parachor approach of Macleod [J. Macleod, Trans. Faraday Soc. 19, 38 (1923)], to predict the vapour-liquid surface tension of fully flexible chainlike Lennard-Jones molecules. In the original study of Galliero, the reduced surface tension of short-chain molecules formed by up to five monomers is expressed as a unique function of the difference between the liquid and vapour coexistence densities. In this work, we extend the applicability of the recipe and demonstrate that it is also valid for predicting the surface tension of two different chainlike molecular models, namely, linear tangent chains that interact through the Lennard-Jones intermolecular potential and fully flexible chains formed by spherical segments interacting through the square-well potential. Computer simulation data for vapour-liquid surface tension of fully flexible and rigid linear Lennard-Jones, and fluid flexible square-well chains is taken from our previous works. Our results indicate that the universal scaling relationship is able to correlate short- and long-chain molecules with different degrees of flexibility and interacting through different intermolecular potentials.

Free energy landscape of protein-like chains with discontinuous potentials

In this article the configurational space of two simple protein models consisting of polymers composed of a periodic sequence of four different kinds of monomers is studied as a function of temperature. In the protein models, hydrogen bond interactions, electrostatic repulsion, and covalent bond vibrations are modeled by discontinuous step, shoulder, and square-well potentials, respectively. The protein-like chains exhibit a secondary alpha helix structure in their folded states at low temperatures, and allow a natural definition of a configuration by considering which beads are bonded. Free energies and entropies of configurations are computed using the parallel tempering method in combination with hybrid Monte Carlo sampling of the canonical ensemble of the discontinuous potential system. The probability of observing the most common configuration is used to analyze the nature of the free energy landscape, and it is found that the model with the least number of possible bonds exhibits a funnel-like free energy landscape at low enough temperature for chains with fewer than 30 beads. For longer proteins, the free landscape consists of several minima, where the configuration with the lowest free energy changes significantly by lowering the temperature and the probability of observing the most common configuration never approaches one due to the degeneracy of the lowest accessible potential energy.

Prediction of the degree of hydration at initial setting time of cement paste with particle agglomeration

This paper presents a computer simulation method for predicting the degree of hydration at the initial setting time of cement paste with particle agglomeration. Agglomeration is controlled by using a square-well potential for the interaction between two cement particles, and the degree of hydration at the initial setting time is determined using the burning algorithm. It is verified that, when the critical surface-to-surface distance for attachment is taken as 0.05μm, the simulation predictions are in good agreement with experimental results. For cement particles in water at the initial stage of hydration, the number of clusters obeys the power law with an average Fisher exponent of 1.9. Finally, the effects of key factors affecting the degree of hydration at the initial setting time are quantitatively evaluated.

Rational-function approximation for fluids interacting via piece-wise constant potentials

The structural properties of fluids whose molecules interact via potentials with a hard-core plus n piece-wise constant sections of different widths and heights are derived using a (semi-analytical) rational-function approximation method. The results are illustrated for the cases of a square-shoulder plus square-well potential and a shifted square-well potential and compared both with simulation data and with those that follow from the (numerical) solutions of the Percus-Yevick integral equation.

Semiclassical approach to model quantum fluids using the statistical associating fluid theory for systems with potentials of variable range

Thermodynamic properties of quantum fluids are described using an extended version of the statistical associating fluid theory for potentials of variable range (SAFT-VR) that takes into account quantum corrections to the Helmholtz free energy A, based on the Wentzel-Kramers-Brillouin approximation. We present the theoretical background of this approach (SAFT-VRQ), considering two different cases depending on the continuous or discontinuous nature of the particles pair interaction. For the case of continuous potentials, we demonstrate that the standard Wigner-Kirkwood theory for quantum fluids can be derived from the de Broglie-Bohm formalism for quantum mechanics that can be incorporated within the Barker and Henderson perturbation theory for liquids in a straightforward way. When the particles interact via a discontinuous pair potential, the SAFT-VR method can be combined with the perturbation theory developed by Singh and Sinha [J. Chem. Phys. 67, 3645 (1977); and ibid. 68, 562 (1978)]. We present an analytical expression for the first-order quantum perturbation term for a square-well potential, and the theory is applied to model thermodynamic properties of hydrogen, deuterium, neon, and helium-4. Vapor-liquid equilibrium, liquid and vapor densities, isochoric and isobaric heat capacities, Joule-Thomson coefficients and inversion curves are predicted accurately with respect to experimental data. We find that quantum corrections are important for the global behavior of properties of these fluids and not only for the low-temperature regime. Predictions obtained for hydrogen compare very favorably with respect to cubic equations of state.

Diffuse versus square-well confining potentials in modelling A@C60 atoms

A perceived advantage for the replacement of a discontinuous square-well pseudo-potential, which is often used by various researchers as an approximation to the actual C60 cage potential in calculations of endohedral atoms A@C60, by a more realistic diffuse potential is explored. The photoionization of endohedral H@C60 and Xe@C60 is chosen as the case study. The diffuse potential is modelled by a combination of two Woods–Saxon potentials. It is demonstrated that photoionization spectra of A@C60 atoms are largely insensitive to the degree η of diffuseness of the potential borders, in a reasonably broad range of ηs. These spectra are found to be insensitive to discontinuity of the square-well potential as well. Both potentials result in practically identical calculated spectra. New numerical values for the set of square-well parameters, which lead to a better agreement between experimental and theoretical data for A@C60 spectra, are recommended for future studies.

Fluid-fluid and fluid-solid transitions in the Kern-Frenkel model from Barker-Henderson thermodynamic perturbation theory

We study the Kern-Frenkel model for patchy colloids using Barker-Henderson second-order thermodynamic perturbation theory. The model describes a fluid where hard sphere particles are decorated with one patch, so that they interact via a square-well potential if they are sufficiently close one another, and if patches on each particle are properly aligned. Both the gas-liquid and fluid-solid phase coexistences are computed and contrasted against corresponding Monte Carlo simulations results. We find that the perturbation theory describes rather accurately numerical simulations all the way from a fully covered square-well potential down to the Janus limit (half coverage). In the region where numerical data are not available (from Janus to hard-spheres), the method provides estimates of the location of the critical lines that could serve as a guideline for further efficient numerical work at these low coverages. A comparison with other techniques, such as integral equation theory, highlights the important aspect of this methodology in the present context.

Mixed-partial-wave scattering with spin-orbit coupling and validity of pseudopotentials

We present exact solutions of a two-body problem for spin-$1/2$ fermions with isotropic spin-orbit (SO) coupling and interacting with an arbitrary short-range potential. We find that in each partial-wave scattering channel, the parametrization of a two-body wave function at short interparticle distance depends on the scattering amplitudes of all channels. This reveals the mixed-partial-wave scattering induced by SO couplings. By comparing with results from a square-well potential, we investigate the validity of original pseudopotential models in the presence of SO coupling. We find the $s$-wave pseudopotential provides a good approximation for low-energy solutions near $s$-wave resonances, given the length scale of SO coupling much longer than the potential range. However, near $p$-wave resonance the $p$-wave pseudopotential gives low-energy solutions that are qualitatively different from exact ones, based on which we conclude that the $p$-wave model can not be applied to the fermion system if the SO coupling strength is larger or comparable to the Fermi momentum.

Dynamical Arrest, Percolation, Gelation, and Glass Formation in Model Nanoparticle Dispersions with Thermoreversible Adhesive Interactions

We report an experimental study of the dynamical arrest transition for a model system consisting of octadecyl coated silica suspended in n-tetradecane from dilute to concentrated conditions spanning the state diagram. The dispersion's interparticle potential is tuned by temperature affecting the brush conformation leading to a thermoreversible model system. The critical temperature for dynamical arrest, T*, is determined as a function of dispersion volume fraction by small-amplitude dynamic oscillatory shear rheology. We corroborate this transition temperature by measuring a power-law decay of the autocorrelation function and a loss of ergodicity via fiber-optic quasi-elastic light scattering. The structure at T* is measured using small-angle neutron scattering. The scattering intensity is fit to extract the interparticle pair-potential using the Ornstein-Zernike equation with the Percus-Yevick closure approximation, assuming a square-well interaction potential with a short-range interaction (1% of particle diameter). (1) The strength of attraction is characterized using the Baxter temperature (2) and mapped onto the adhesive hard sphere state diagram. The experiments show a continuous dynamical arrest transition line that follows the predicted dynamical percolation line until ϕ ≈ 0.41 where it subtends the predictions toward the mode coupling theory attractive-driven glass line. An alternative analysis of the phase transition through the reduced second virial coefficient B(2)* shows a change in the functional dependence of B(2)* on particle concentration around ϕ ≈ 0.36. We propose this signifies the location of a gel-to-glass transition. The results presented herein differ from those observed for depletion flocculated dispersion of micrometer-sized particles in polymer solutions, where dynamical arrest is a consequence of multicomponent phase separation, suggesting dynamical arrest is sensitive to the physical mechanism of attraction.

Adsorption versus aggregation. Particles and surface of the same material

A model for the adsorption of colloidal particles on a planar surface is analyzed by using a thermodynamic chemical equilibrium model and Monte Carlo simulations. Central to this investigation are that (i) particles and surface are considered to be of the same material and (ii) the particle–surface and particle–particle interactions are related using the Derjaguin approximation using a surface–surface square-well potential as a basis. Thereby, all interactions within the system are characterized by the same parameters, and hence the difference between particle adsorption on the surface and particle aggregation in bulk is solely due to geometrical effects. Equilibrium constants for the different binary associations are calculated from the interaction potentials enabling a direct comparison between predictions based on a chemical equilibrium model and on computer simulations with no adjustable parameters. As the interaction gradually is made more attractive for a given particle concentration, we find the following sequence of events: (A) a weak particle adsorption onto the surface, (B) particle association on the surface forming a denser single adsorbed layer, (C) formation of a second adsorbed layer on the surface, (D) multiple adsorbed layers on the surface, and (E) bulk phase separation. There is a semi-quantitative agreement between the predictions of the equilibrium model and the results of the simulations. The equilibrium model calculations facilitate a conceptual understanding of the competition between association on a surface and in bulk. Our study is relevant both for understanding processes where colloidal particle adsorption is used to modify surface properties and also for the understanding of heterogeneous versus homogeneous nucleation.

Self-Assembly of Bifunctional Patchy Particles with Anisotropic Shape into Polymers Chains: Theory, Simulations, and Experiments

Concentrated solutions of short blunt-ended DNA duplexes, as short as 6 base pairs, are known to order into the nematic liquid crystal phases. This self-assembly is due to the stacking interactions between duplex terminals that promotes their aggregation into polydisperse chains with a significant persistence length. Experiments show that liquid crystal phases form above a critical volume fraction depending on the duplex length. We introduce and investigate via numerical simulations, a coarse-grained model of DNA double-helical duplexes. Each duplex is represented as an hard quasi-cylinder whose bases are decorated with two identical reactive sites. The stacking interaction between terminal sites is modeled via a short-range square-well potential. We compare the numerical results with predictions based on a free energy functional and find satisfactory quantitative matching of the isotropic–nematic phase boundary and of the system structure. Comparison of numerical and theoretical results with experimental findings confirm that the DNA duplex self-assembly can be properly modeled via equilibrium polymerization of cylindrical particles. This insight enables us to estimate the stacking energy.

Effects of fullerene confining potential on the ionization of the hydrogen atom by a strong femtosecond VUV pulse

Ionization of atoms trapped inside a fullerene cage by femtosecond VUV pulses is studied beyond the perturbative regime. When solving the time-dependent Schrödinger equation, the cage surrounding the hydrogen atom is modelled by a spherical square-well potential with the parameters corresponding to the fullerenes C60 and C36. The influence of the cage on the yields and angular distributions of electrons in the main and first ATI lines is revealed for ionization from the ground 1s and metastable 2s states.

Simple analytic equations of state for the generalized hard-core Mie(α, β) and Mie(α, β) fluids from perturbation theory

New, simple and analytic perturbation theory equations of state for generalized hard-core Mie HCMie(α, β) and Mie(α, β) fluids are proposed. They are based on the second-order Barker–Henderson perturbation theory in the macroscopic compressibility approximation and the new analytical expression of the radial distribution function of hard spheres, g HS(r), developed by Sun in terms of a polynomial expansion of base functions adapted to the square-well and Sutherland potentials [Can. J. Phys. 83 (2005) 55], the combination of which yields the HCMie(α, β) and Mie(α, β) functions. The compressibility factors, the residual internal energies and the radial distribution function at contact with the hard core are then obtained from this equation of state for the HCLJ(12, 6) potential, which is a particular case of the HCMie(α, β) potentials with α = 12 and β = 6. The results are in good agreement with the existing Monte Carlo (MC) simulation data, and compare favorably with those obtained from five other equations of state, three of which contain numerical coefficients fitted to the Monte Carlo results. For the Mie(α, 6) (α = 8, 10, 12), fluids, the present equation of state is a good representation of recent molecular dynamics (MD) simulations of the pressure and internal energy. It is more accurate than the statistical associating fluid theory of variable range (SAFT-VR Mie(n, 6)) theory for n = 8, and 10, while for n = 12 the SAFT-VR theory is best. For the Mie(14, 7) fluid, which is outside the range of application of the SAFT-VR theory, the results for the pressure are in good agreement with the analytical equation of state obtained from the MC simulation data.

On the chemical and the physical approaches to ion association

For a system of particles interacting by a square-well potential in one dimension it is shown that the chemical approach of describing molecular association by the mass action law is the maximum term of the complete partition function. Different definitions of the partition functions of the associates are given and the consequences for the equilibrium constant, the distribution of bounded and unbounded particles as function of temperature and pressure are analysed. Although the thermodynamic is independent of the chosen definition for the associated state, the equilibrium constant and the particle distribution depend on the chosen definition. In this spirit different definitions of ion-pairs for the system of charged hard spheres in a dielectric continuum are discussed. Various criteria for the stability range of an ion-pair are considered including a new criterion based on the requirement of mechanical stability. It turned out that at low temperatures in the vicinity of the critical point the considered definitions of ion pairs are numerically equivalent but have different consequences for treating the ion–ion interactions, which become important at lower reduced Coulomb energies or higher reduced temperatures.

Thermodynamic modeling of hydrogen sulfide solubility in ionic liquids using modified SAFT-VR and PC-SAFT equations of state

Equations of state based on the statistical associating fluid theory for potentials of variable range (SAFT-VR) and the perturbed chain statistical associating fluid theory (PC-SAFT) have been used to model the PVT behavior of ionic liquids and the solubility of H 2S in six imidazolium-based ionic liquids. The studied systems included [bmim][PF6], [hmim][PF6], [bmim][BF4], [hmim][BF4], [bmim][NTF2] and [hmim][NTF2] at various temperatures and pressures. For pure components, parameters of the models have been obtained by fitting the models to experimental data on liquid densities; the average relative deviation between the calculated and experimental densities for ionic liquids is less than 2.42% in the PC-SAFT model and 5.44% in the SAFT-VR approach, the latter which incorporates the square-well potential for short-range interactions. In both models an additional term has been added to account for dipole–dipole interactions between solute molecules resulting from the permanent charges on the chain molecules of the solvents. The model parameters have also been correlated as functions of the molecular weight of the solvents. For binary mixtures of ionic liquids and H 2S, the association interactions between H 2S molecules and between the ionic liquids and H 2S molecules have also been taken into account in both approaches, using binary interaction coefficients. The results show an average deviation of less than 5% in the calculation of the mole fraction of H 2S in the ionic liquids. The effect of inclusion of the polar term has been studied for binary systems in both models.