Configuration-dependent Properties Research Articles

Cluster expansions of multicomponent ionic materials: Formalism and methodology

The cluster expansion (CE) method has seen continuous and increasing use in the study of configuration-dependent properties of crystalline materials. The original development of the CE method along with the underlying mathematical formalism and assumptions was focused on the study of metallic alloys. Since then the methodology has been actively and successfully used in the study of ionic materials as well. In this work, we present a cohesive reformulation of the mathematical formalism underlying the CE method based on a synthesis of its original formulation, several additions and extensions that have been proposed since, and a revised representation of its constituent mathematical objects. We then proceed to describe some of the formal implications of using the methodology for charge-neutral configurations in ionic systems. In particular, we discuss the reduction of the size of configuration spaces and the resulting linear dependencies that arise among correlation functions that span the larger unconstrained configuration space. Additionally, we explore the effects of long-range electrostatic interactions. We also demonstrate how the previously proposed use of a point electrostatic term successfully accounts for the majority of the longer-range electrostatic interactions, and leaves the cluster expansion terms to capture mostly short-range interactions. Finally, we present and discuss a variety of recently developed methodologies, including training structure selection, oxidation state assignment, structure mapping, and regression algorithms, that are necessary to address these formal mathematical notions for a practical implementation of the CE method in the study of multicomponent ionic materials.

Machine Learning Force Field Aided Cluster Expansion Approach to Configurationally Disordered Materials: Critical Assessment of Training Set Selection and Size Convergence.

Cluster expansion (CE) is a powerful theoretical tool to study the configuration-dependent properties of substitutionally disordered systems. Typically, a CE model is built by fitting a few tens or hundreds of target quantities calculated by first-principles approaches. To validate the reliability of the model, a convergence test of the cross-validation (CV) score to the training set size is commonly conducted to verify the sufficiency of the training data. However, such a test only confirms the convergence of the predictive capability of the CE model within the training set, and it is unknown whether the convergence of the CV score would lead to robust thermodynamic simulation results such as order-disorder phase transition temperature Tc. In this work, using carbon defective MoC1-x as a model system and aided by the machine-learning force field technique, a training data pool with about 13000 configurations has been efficiently obtained and used to generate different training sets of the same size randomly. By conducting parallel Monte Carlo simulations with the CE models trained with different randomly selected training sets, the uncertainty in calculated Tc can be evaluated at different training set sizes. It is found that the training set size that is sufficient for the CV score to converge still leads to a significant uncertainty in the predicted Tc and that the latter can be considerably reduced by enlarging the training set to that of a few thousand configurations. This work highlights the importance of using a large training set to build the optimal CE model that can achieve robust statistical modeling results and the facility provided by the machine-learning force field approach to efficiently produce adequate training data.

A leap-frog algorithm-based efficient unified thermostat scheme for molecular dynamics

An important challenge for modern molecular simulations is to develop robust algorithms for molecular dynamics (MD), which employ a time interval (i.e., step size) of evolution as large as possible to achieve high efficiency while maintaining accuracy for sampling the distribution of configuration and momentum, e.g., for the canonical ensemble where the number of particles ( N ), the volume ( V ), and the temperature ( T ) are constant. Starting from the velocity-Verlet algorithm, in the recent paper we have proposed a unified scheme—the middle scheme for constructing efficient thermostatting integrators for configurational sampling of the canonical ensemble. In the present paper we extend the middle scheme to derive new thermostatting integrators based on the leap-frog algorithm. The leap-frog middle scheme is able to recover the exact marginal distribution of configuration and that of momentum in the harmonic limit, regardless of the time interval as long as the evolution of the MD trajectory is stable. It is also proved that the leap-frog middle scheme leads to the same marginal distribution of configuration as the velocity- Verlet middle scheme does even for general anharmonic systems. As illustrated in numerical examples (such as a 1-dimensional quartic potential, liquid water at room temperature with or without bond-length constraints, etc.), while both middle schemes perform better than conventional approaches (e.g., the side scheme and the default Langevin dynamics method in AMBER) and can calculate configuration-dependent properties with the same accuracy by using a larger time interval for from at least 4 times to even an order of magnitude, the leap-frog middle scheme produces a more accurate marginal distribution of momentum than that the velocity-Verlet middle scheme does. The sampling efficiency analysis by employing the characteristic correlation time step number (of the potential or of the Hamiltonian) suggests that both middle schemes share the same efficiency in sampling the configuration space and lead to similar efficiency in sampling the whole phase space (i.e., both configuration and momentum). When both only-configuration- dependent and only-momentum-dependent physical observables are considered, we suggest that the leap-frog middle scheme should in principle offer a better approach. When we study properties involving cross-terms of configuration and momentum, e.g., the fluctuation of the Hamiltonian (related to the heat capacity) and the energy current and its fluctuation (related to the thermal conductivity), analytic results in the harmonic limit and numerical data for anharmonic systems indicate that which of the two middle schemes performs better depends on the system and on the specific property. Note that the leap-frog middle scheme is of first-order accuracy for sampling the whole phase space while the velocity-Verlet middle scheme is of second-order accuracy. It is reasonable to choose the velocity-Verlet middle scheme to perform simulations for such properties for general molecular systems. Because either the leap-frog or the velocity-Verlet algorithm is the default integrator in molecular dynamics simulation software packages, the two unified middle schemes can offer novel, simple, accurate, and efficient thermostatting integrators for performing MD (or path integral MD (PIMD)) for the classical (or quantum) canonical ensemble.

Study on the dimensional, configurational and optical evolution of palladium nanostructures on c-plane sapphire by the control of annealing temperature and duration

Metallic nanostructures can find various applications such as in optoelectronic devices, nanostructure synthesis and catalytic applications and their applicability vary depending on their size, density and configuration dependent properties. In this paper, the dimensional and configurational evolution of self-assembled palladium (Pd) nanostructures is systematically studied on c-plane sapphire with the control of annealing temperature (AT) and annealing duration with the initial Pd layers of various thicknesses. Depending on the AT, two distinct growth regimes are observed based on the concurrent effect of surface diffusion, surface energy minimization and Pd sublimation: i.e. (i) agglomeration of Pd nano-clusters from voids (500<AT≤650°C) and (ii) round nanoparticle (NP) fabrication (650°C<AT≤900°C). At 950°C, due to the sublimation of Pd atoms, substantial decrease in the NP dimension is witnessed and results in the ring patterns around the NPs. Additional investigation is performed at 950°C to reveal the annealing duration effect on the NP evolution. Due to the dual effect of the Oswald ripening and atom sublimation, initially the dimension of NPs is grown and then gradually decays along with the duration, resulting in an inverted ‘V’ pattern evolution in diameter and height. Moreover, the evolution of optical properties such as absorption band and average reflectance are studied with the corresponding reflectance spectra as a function of wavelength over UV, visible and NIR region. The Raman spectra analysis depicts the variation of lattice vibration peak intensity and position based on the surface morphology of the Pd nanostructures.

A generalized approach for computing the transmission index of parallel mechanisms

This paper presents a novel approach for computing the transmission index of parallel mechanisms. The approach is based on an extended concept to compute the maximal virtual coefficient, which is an important notion involved in the formulation of dimensionally homogeneous transmission indices for singularity analysis and dimensional optimization of parallel mechanisms. By exploiting the dual property of the virtual coefficient, two characteristic points instead of one as in the current state of the art are defined: one characteristic point – termed the transmission characteristic point – is located on the ‘floating’ axis of the transmission wrench, as in existing approaches, while a second one – termed the output characteristic point – is located on the floating axis of the output twist of the platform, which is a novel concept. This allows one to define two characteristic lengths, namely, the transmission and output characteristic lengths, respectively, of which the larger is then used for the measure of the “distance” between the transmission wrench screw and the output twist screw. As shown in this paper, this new measure makes it possible to discern more finely the configuration-dependent properties of kinematic performance of parallel mechanisms, thus making it more suitable for dimensional optimization. Confidence in this statement is demonstrated through the comparative study of two in-parallel mechanisms using the new method and previously existing ones.

Phase relationships and structures of inorganic crystals by a combination of the clusterexpansion method and first principles calculations

Properties of crystalline solutions are generally dependent not only on their chemicalcomposition but also on the configurations of solute atoms and/or point defects.Quantitative knowledge of the configuration-dependent properties is therefore essential formaterials design. The cluster expansion (CE) method has been widely used to describe theconfigurational properties. Increases in computational power and advances in numericaltechniques enable us to perform a large set of systematic first principles calculations basedon density functional theory (DFT) to be combined with CE calculations. In this paper,our procedure of CE with optimal selections of clusters and DFT structures is described.Two examples of such calculations are then shown. One is the cation arrangement in aseries of spinel oxides. The other is arrangement of the oxygen vacancy in a series of tinsub-dioxides.

Investigation of axial forces on dynamic properties of a flexible 3-PRR planar parallel manipulator moving with high speed

SUMMARYThe effect of axial forces on the dynamic properties is formulated and investigated for a 3-PRR planar parallel manipulator with three flexible intermediate links. A dynamic model of the manipulator system is developed based on the assumed mode method with the consideration of the effect of longitudinal forces on lateral stiffness is included. The flexible intermediate links are modeled as Euler–Bernoulli beams with pinned-pinned boundary conditions, which are verified by experimental modal tests. Natural frequencies of bending vibration of the intermediate links are derived as the functions of axial force and rigid-body motion of the manipulator. Dynamic behavior including the effect of axial forces on lateral deformation is investigated, and configuration-dependant frequencies are analyzed. Numerical simulations of configuration-dependent frequency properties and axial forces are performed to illustrate the effect of axial forces on the dynamic behaviors of the flexible parallel manipulator. Simulation results of mode amplitudes, deformations, axial forces, inertial, and coupling forces are presented, and further validate the theoretical derivations. These analyses and results provide a new and valuable insight to the design and control of the parallel manipulators with flexible intermediate links.

Efficient Monte Carlo method for free energy evaluation of polymer chains

We have extended the Continuous Configuration Boltzmann Biased (CCBB) Monte Carlo method to predicting the free energy and properties of multiple-chain condensed amorphous polymers. We illustrate the method by considering two independent polyethylene (PE) chains per cell (each with 100 monomers) using periodic boundary conditions (PBC). In CCBB-MC the free energy is calculated directly and simultaneously with pressure and other configuration dependent properties. The phase transition from polymer vapor to liquid is demonstrated by the density dependence of pressure and free energy. At low temperature, increasing the density causes the chains to change from globular to interpenetrating due to increased screening. At high density, increasing the temperature causes each chain to shrink since it becomes more flexible due to the screened interchain interactions. This leads to a dramatically increased population of short lengths in the end-to-end distance distribution.

Analysis of robot dynamics and compensation using classical and computed torque techniques

A classical analysis of the dynamics of robot manipulators is presented. It is shown that these systems have configuration-dependent properties and can be open-loop unstable. Due to this fact, present day linear controllers are inefficient. On the other hand, nonlinear hardware and software compensation methods also are shown to have some limitations. Controllers based on the direct design algorithm and the computed torque method have superior performances. These algorithms have nonlinear loops, yet it is shown that a linear analysis is still feasible. Therefore, classical design tools can be adopted in order to develop practical implementations. >

The Model Compounds and the Conformational Analysis of Poly(thiomethylene)

Abstract The conformational energies of model compounds of poly(thiomethylene), CH3(SCH2)nSCH3, (n=1–6) were calculated by the molecular-mechanics method. Based on these conformational energy differences, the configuration-dependent properties of poly(thiomethylene) were estimated under the rotational isomeric-state approximations. The first-order interaction energy of the (Remark: Graphics omitted.) bond, obtained from the energy differences among the conformers of CH3(SCH2)3SCH3, is coincident with that estimated from the same bond sequence for poly(1,3-dithiocane). Based on this coincidence, the monomeric sequential length of the model compound, which is representative of poly(thiomethylene), was decided. The propriety of the method of coupling the molecular-mechanics with the statistical-mechanics was also demonstrated.

The Model Compounds and Conformational Analysis of Poly(thiotrimethylene) Using Molecular Mechanics Method

Abstract Several model compounds of poly(thiotrimethylene), including one to four thiotrimethylene units, were studied by means of a molecular mechanics method. The geometric and statistical weight parameters were directly derived from the results of molecular mechanics calculations concerning several conformers of these model compounds. Although the first-order interactions of poly(thiotrimethylene) were significantly larger than those of poly(oxyalkylenes), the second-order interactions of this polymer were much smaller. The “gauche effect” by sulfur atoms seemed to be mainly caused by small second-order interactions. By means of the model, compounds including more than three repeating units, the calculated configuration-dependent properties were in better agreement with the experimental ones than that calculated earlier. The better agreement was obtained by varying some of the geometric parameters. Compounds including more than three repeating units of the polymer were available as model compounds. Since this was also true in poly(oxyalkylenes), it was found that a combination of the methods of molecular and statistical mechanics was sufficiently effective to investigate the configurational properties of poly(thiotrimethylene).

Some applications of graph theory to the study of polymer configuration

The spatial configuration of polymer molecules in an Euclidean space depends on the adjacency of their units. The configuration dependent properties of the so called Gaussian polymer molecules of different structures can be expressed in terms of graph-theoretical categories such as the number of spanning trees, path lengths, generalized inverses and spectra of the Kirchhoff matrices of their graphs. We review here the methods and results concerning the distribution function of the square radius of gyration and its first moment, the mean square radius of gyration, for various types of polymer molecules.

Applicability of the configurational analysis to some problems of interest in polymer science

AbstractComparison between theoretical and experimental values of configuration dependent properties of polymers is used to estimate the values of the conformational energies associated with the rotational states of molecular chains. It is shown that the critical interpretation of the configurational properties of the polymers obtained by cationic polymerization of heterocycles, may be an useful tool to elucidate the ring‐opening mechanisms. The influence of the molecular flexibility in both the glass transition temperature and melting temperature of polymers is discussed in terms of the configurational entropy.

Rotational Isomeric State Analysis of Poly(oxyethylene). Conformational Energies and the Random-Coil Configuration

Configuration-dependent properties of the poly(oxyethylene) (POE) chain were calculated within the framework of the rotational isomeric state (RIS) approximation. Conformational energies of the chain have been determined by the 1H and 13C NMR studies on 1,2-dimethoxyethane (DME) and POE as described in our recent paper as well as in the present work. The unperturbed chain dimension C(=‹r2›0/nl2), the dipole moment ‹μ2›/nm2, and the molar Kerr constant ‹mK›/x were calculated as a function of RIS parameters such as bond angles, rotational angles, and conformational energies. The value of C(=5.2) determined in organic solutions by Beech and Booth was found to be favorably reproduced by these calculations. The SANS result (C=6.9) recently reported by Kugler et al. on the melt exceeds the limit of a probable range estimated from uncertainties involved in each of the RIS parameters. Within the limit mentioned above, a set of RIS parameters was chosen so as to give an optimum agreement with observed values of the configurational properties of POE. Conformational energies thus estimated are as follows: Eσ=−0.5 kcal mol−1 for the C–C bond and Eρ=0.8 kcal mol−1 for the C–O bond, both representing the energy of the gauche form expressed relative to the trans state. Use of the conformational energies Eσ=−1.2 and Eρ=1.1 kcal mol−1 determined in D2O leads to a value of C=3.2, suggesting that the end-to-end dimension of the chain for the state unperturbed by longrange interactions may be considerably small in pure water.

Conformational analysis of polyoxyethylene using empirical force field methods

AbstractConformanational energies of various oligo (oxyethylene) isomers were calculated by the empirical force field method, and statistical mechanics calculations of the chain dimensions and the dipole moments were carried out. The results are seriously different from those obtained for poly (oxymethylene). Although the first‐order steric interactions in POE are of the same order of magnitude as those in n‐alkane chains, the second‐order interactions are significantly different. The second‐order interactions arising from rotations around two consecutive bonds of OCH2CH2OCH2 in g± g∓ conformation have greatly reduced steric repulsions. It was found that the reduced second‐order interaction plays an important role in the configurationdependent properties of POE. The agreement of the calculated values for the unperturbed dimensions and the temperature coefficients obtained from the force field method with experimentally observed ones is fairly good.

Dynamics of helical wormlike chains. I. Dynamic model and diffusion equation

A model, called the discrete helical wormlike chain, is constructed to be suitable for the study of the dynamics of polymer chains, both flexible and stiff. It is a chain of N rigid sub-bodies such that, apart from its location, its configuration may be specified by N sets of Euler angles, each associated with one sub-body, and its equilibrium conformational behavior is almost identical with that of the continuous helical wormlike chain. It is shown that the model can simulate rather well several types of elementary motion of real chains, by evaluating the change in the potential energy caused by each motion. The configurational diffusion equation is then formulated. Following the procedure of Fixman, rigid constraints are imposed on the lengths of sub-bodies (and hence on the total chain length) by setting the components of the flux associated with the constrained coordinates equal to zero through constraining forces, so that the diffusion equation and all configuration-dependent properties may be written in terms of only the unconstrained coordinates, i.e., Euler angles. For the present model, all dynamic properties in the regime of linear response may be expressed in terms of time-correlation functions of the Wigner 𝒟 functions of Euler angles. Thus, formal expressions for them are given along with the formal solution for the Green’s function of the linearized diffusion equation. The eigenvalue problem which arises is also discussed.

Photoelastic Studies of Some Polybutadiene Networks

Two butadiene polymers were used in this investigation, one with 98.5% cis-1,4 units and the other with an approximately equibinary mixture of cis and trans units. Elastomeric networks prepared from these polymers were studied in elongation, in both the swollen and unswollen states over the temperature range −30 to 95°C. There is evidence for crystallization in these networks, particularly as manifested by marked increases in birefringence at relatively low elongations and at temperatures as high as 40°C. As expected, the birefringence and related quantities were found to be more sensitive to crystallization than the force, with the optical-configuration parameter and the stress-optical coefficient showing the greatest sensitivity. In the case of the cis–trans copolymer, the crystallization involves trans sequences, which are of relatively high melting point, and thus occurs at a temperature higher than for the lower melting cis sequences in the high-cis networks. The results which were free from the effects of network crystallization were used to calculate values of the temperature coefficient of the unperturbed dimensions of the chains, and values of the optical-configuration parameter. These configuration-dependent properties were found to be in satisfactory agreement with previously published theoretical results based on a rotational isomeric state model of these chain molecules.

Pressure dependence of polymer chain conformational parameters

Configurational statistics of chain molecules relates the average spatial configuration and the other configuration-dependent properties of a chain to the relative free energy AF of rotational isomers for each rotatable skeletal bond. The statistical weights uij = exp[ Mij/(RT)] are assigned to the pairs of conformations about adjoining bonds and are arranged as elements of statistical weight matrix’). The comparison of calculated and experimental values of configuration-dependent properties is the usual way for the assignment of statistical weights uij . Alternatively, uij can be determined from the conformational energy calculations for the compounds. Customarily, only the dependence of uij on temperature is assumed in conformational studies. For low molecular weight compounds there are many examples of the striking environmental (especially solvent) influence on conformational equilibria2). In configurational statistics this effect can be accounted for by introducing the solventdependent statistical weights uij . For example, the solvent effect on mean chain dimensions is simulated by a variation of some uij values’). It is common belief that the environmental effect on torsional potentials about single bonds and on the associated conformational free energies AF is important in polar systems only. However, there are also indications of the importance of the environmental effect on the conformational equilibria for nonpolar e.g. hydrocarbon molecule^^^^). One of the possible reasons for it is the modification of torsional potential in the condensed phase due to the different volumes of trans and gauche conformers’). In other words, the statistical weights for hydrocarbon chain molecules in the condensed phase are affected by changes of internal or external pressure or by changes of sample volume. In this paper the pressure variation of statistical weights of the polymethylene chain is estimated as a representative case of non-polar polymers.

Configurational properties of aromatic polyesters with sulfur atoms in their structure

The dipole moments of poly (thiodiethylene glycol terephthalate) chains were determined as a function of temperature by means of dielectric constant measurements in dioxane. The experimental results were found to be in fair agreement with theoretical results based on a rotational isomeric state model in which the required conformational energies were obtained from previous configurational analysis on poly(ethylene terephthalate), poly(diethylene glycol terephthalate) and poly(thiodiethylene glycol). Since poly(thiodiethylene glycol terephthalate) can be considered an alternating copolymer of ethylene terephthalate and thioethylene units, its configuration-dependent properties were compared with those of poly(ethylene terephthalate) and poly(ethylene sulfide). It was found the flexibility of the copolymer, as expressed by the partition function, intermediate to that of its parent homopolymers. The theoretical results also indicate that the dimensions of poly(thiodiethylene glycol) are similar to those of poly(ethylene terephthalate) while its dipole moment ratio resembles that of poly(ethylene sulfide).

Molecular structure, conformation and properties of macromolecules

The spacio—geotnetric characteristics of macromolecular chains are discussed from the point of view of the essential connections they provide between chemical constitution and properties. Quantities such as molecular weight, radius of gyration, degree of branching, etc., that characterize the molecule as a whole do not suffice for an understanding of the virtually endless variations in the properties that distinguish one polymer from another. For this purpose it is necessary to examine the chemical structures of the constituent units, taking into account the geometric parameters pertaining to chemical bonds, the conformations accessible to the chain skeleton and the resulting spatial configurations. The physical principles underlying the statistical mechanical treatment of conformations of long chain molecules and of their configuration—dependent properties are summarized. Correlations between bonds of the chain and the range of such correlations provide insights into the spatial characteristics of polymer chains. Persistences expressing the configurational averages of vectors spanning sequences of chain units are particularly useful in this connection. Experimental methods that provide quantitative information on conformation and the correlations between units of the chain are briefly discussed.