Penetrable Sphere Model Research Articles

Complexes formed between polyelectrolyte and acetylated cationic dendrimers

Gene therapies are receiving a revival, especially because of the potential for use in cancer and infectious disease vaccination. However, effectively condensing and maintaining deoxyribonucleic acid (DNA) carriers for gene therapy are used, such as dendrimers. The penetrable sphere model developed by Qamhieh et al., explain the interaction between linear polyelectrolyte (LPE) and an ion-penetrable sphere was employed to study the complexation of negatively charged DNA and acetylated dendrimer. Throughout the study, we emphasized the effect of acetylation on two types of complexations: single dendrimer-DNA chain complex and multiple dendrimers-DNA chain complex. The interaction between three different DNA lengths: L = 90 nm of 265; L = 184 nm of 541 bp; and L = 680 nm of 2000 bp individually with the poly (amido amine) (PAMAM) dendrimer of generation 5 was studied. For a single dendrimer-DNA complex, the number of condensed monomers around the dendrimer and the fraction of optimal length wrapping around the dendrimer was studied. It was found that by increasing acetylation, the number of condensed monomers and the fraction of optimal length decreased. For multiple dendrimers-DNA complex, the optimal length wrapping around the dendrimer and the linker that forms between complexes were studied. As a result, with increasing acetylation, the will shorten while the linker will increase significantly. The acetylation also affected the number of turns wrapping around the dendrimer and the net charge of the complex for single and multiple dendrimer-DNA complexes, and it was found that the longer the chain length, the more turns there are around the dendrimer. The net charge of all complexes was negative due to increased acetylation. In this study, it is revealed that the developed model is suitable for demonstrating the complexation between the dendrimer and the DNA.

Analytical study of Linker Formation in Complexation of Two Dendrimers and one Polyelectrolyte

The concept of gene therapy is to fix a genetic problem at its source by repairing defective genetic material. Dendrimers proved to be an effective carrier vectors for gene therapy that efficiently compact and protect DNA from degradation by serum nucleases in the blood when injected intravenously. The complexation between Poly(amido amine) (PAMAM) dendrimer and Deoxyribonucleic acid (DNA) has been studied using the penetrable sphere model describing the interaction between linear polyelectrolyte (LPE) chain and ion penetrable sphere (Porous sphere) representing the dendrimer. It was found that the wrapping degree of LPE around the dendrimer is increased by increasing the LPE length, and the Bjerrum length which represents the strength of the electrostatic interaction and is decreased by increasing the pH of the solution. As a result, the linker, represents the unwrapped part of the LPE chain, is decreased by increasing the Bjerrum length and increased by increasing the pH

Multifractal statistic of porous structures simulated through fully penetrable sphere models

Theoretical simulations of porous materials are important tools to establish correlations between physical properties and the pore structure. The final objective is to predict physical properties by determining microstructural parameters that can be directly measured in simple experiments. In this work, it is shown that the pore structure simulated through a fully-interpenetrable sphere model features a multifractal statistic. In essence, the study calculates the multifractal generalized dimensions and the singularity spectra for different simulated pore structures. The data indicate that the multifractal parameters depend on the porosity, a result in concordance with experimental findings where multifractal dimensions have been correlated with the logarithmic of the porosity. The simulations also evidence that porous structures with the same porosity show slightly different multifractal dimensions. This signalizes that multifractal parameters can be correlated to some extent with the heterogeneity level of the solid-phase, or in other words, that the granular structure can also influence the fractal geometry of the porous space. In summary, this report suggests that the multifractal statistic of the pore space can be explored to classify porous materials, and it can be used to correlate physical properties with the pore structure.

Simulation of the pyrolysis of charring polymers: influence of the porous media properties

Recent large-scale experiments of cable tray fires have shown that the composition of the cable-sheath materials have a strong influence on the fire growth and spread. Thus, the correct prediction of the fire scenario requires the characterization of the sheath materials thermal and thermokinetic parameters. Yet, this task may be tricky for the materials whose pyrolysis generally leads to the formation of a porous residual. Whereas the physical constants related to the material degradation kinetics and to the virgin material properties can usually be determined using thermogravimetric analysis and differential scanning calorimetry, the porous residue properties are difficult to obtain and rarely available for pyrolysis models validation.The present paper deals with the simulation of reference cone calorimeter experiments carried out on PVC samples. An extension to porous media of the pyrolysis model of the CALIF3S-ISIS fire simulation software, developed at IRSN and validated on non-charring polymers, is used for the simulations. Special attention is paid to the modelling of the thermal diffusion effects in the porous residuals. On the one hand, based on the intrinsic conductivity of the different media (virgin material, intumescent intermediate state, char and gases), several effective conductivity models are tested to evaluate the relevant range for the global material conductivity. On the other hand, it is observed that the porous residuals of PVC generally contain large pores due to the material intumescence. That is why a radiative conductivity is included in the model to account for the radiative transfer in the pores. Without precise knowledge of the PVC residuals morphology, a Penetrable Sphere Model (PSM) is chosen to represent the porous medium geometry. This model is parametrized by a constant porosity deduced from the experimental data, and a variable pore size. It allows deducing a porous medium radiative conductivity which is added to the usual effective conductivity.It is shown that, in the particular case of PVC pyrolysis, the uncertainty on the purely conductive effects induces a very limited dispersion on the simulation of the Heat Release Rate (HRR), since all the considered models yield very similar results. Conversely, the sensitivity of the radiative conductivity and therefore of the pyrolysis kinetics to the PSM pore size is much larger. The optimal pore size with respect to the HRR temporal evolution is of the same order as the pore size observed in some PVC residuals. Finally, pyrolysis simulations are performed with this optimized parameter for different cone heater induced heat fluxes and sample sizes. A good overall agreement is found between the experimental and the numerical HRR evolution, except for the thinnest sample for which the numerical HRR is overestimated. More generally, this work shows that further effort on the morphological description of the porous residuals and on the modelling of the radiative transfer is required if one aims at performing predictive pyrolysis simulations.

Thermodynamics of Criticality: Percolation Loci, Mesophases and a Critical Dividing Line in Binary-Liquid and Liquid-Gas Equilibria

High-temperature and pressure boundaries of the liquid and gas states have not been defined thermodynamically. Standard liquid-state physics texts use either critical isotherms or isobars as ad hoc boundaries in phase diagrams. Here we report that percolation transition loci can define liquid and gas states, extending from super-critical temperatures or pressures to “ideal gas” states. Using computational methodology described previously we present results for the thermodynamic states at which clusters of excluded volume (VE) and pockets of available volume (VA), for a spherical molecule diameter σ, percolate the whole volume (V = VE + VA) of the ideal gas. The molecular-reduced temperature (T)/pressure(p) ratios ( ) for the percolation transitions are = 1.495 ± 0.015 and = 1.100 ± 0.015. Further MD computations of percolation loci, for the Widom-Rowlinson (W-R) model of a partially miscible binary liquid (A-B), show the connection between the ideal gas percolation transitions and the 1st-order phase-separation transition. A phase diagram for the penetrable cohesive sphere (PCS) model of a one-component liquid-gas is then obtained by analytic transcription of the W-R model thermodynamic properties. The PCS percolation loci extend from a critical coexistence of gas plus liquid to the low-density limit ideal gas. Extended percolation loci for argon, determined from literature equation-of-state measurements exhibit similar phenomena. When percolation loci define phase bounds, the liquid phase spans the whole density range, whereas the gas phase is confined by its percolation boundary within an area of low T and p on the density surface. This is contrary to a general perception and opens a debate on the definitions of gaseous and liquid states.

Percolation Transitions of the Ideal Gas and Supercritical Mesophase

High-temperature and pressure boundaries of the liquid and gaseous states have not been defined thermodynamically. Standard liquid-state physics texts use either critical isotherms or isobars as ad hoc boundaries in phase diagrams. Here we report that percolation transition loci can define liquid and gas states, extending from super-critical temperatures or pressures to “ideal gas” states. Using computational methodology described previously we present results for the thermodynamic states at which clusters of excluded volume (VE) and pockets of available volume (VA), for a spherical molecule diameter σ, percolate the whole volume (V = VE VA) of the ideal gas. The molecular-reduced temperature (T)/pressure (p) ratios (T* = kBT/pσ3) for the percolation transitions are T*PE = 1.495 ± 0.01 and T*PA = 1.100 ± 0.01. Further MD computations of percolation loci for the Widom-Rowlinson (W-R) model of a partially miscible binary liquid (A-B) show the connection between the ideal gas percolation transitions and the 1st-order phase-separation transition. A phase diagram for the penetrable cohesive sphere (PCS) model of a one-component liquid-gas is then obtained by analytic transcription of the W-R model thermodynamic properties. The PCS percolation loci extend from a critical coexistence of gas plus liquid to the low-density limit ideal gas. Extended percolation loci for argon, determined from literature equation-of-state measurements exhibit similar phenomena. When percolation loci define phase bounds, the liquid phase spans the whole density range, whereas the gas phase is confined by its percolation boundary within an area of low T and p on the density surface. This is contrary to a general perception, and reopens a debate of “what is liquid”. We append this contribution to the science of liquid-gas criticality and liquid-state bounds with further open debate.

Stochastic dynamics of penetrable rods in one dimension: Entangled dynamics and transport properties.

The dynamical properties of a system of soft rods governed by stochastic hard collisions (SHCs) have been determined over a varying range of softness using molecular dynamics simulations in one dimension and analytic theory. The SHC model allows for interpenetration of the system's constituent particles in the simulations, generating overlapping clustering behavior analogous to the spatial structures observed in systems governed by deterministic bounded potentials. Through variation of an assigned softness parameter δ, the limiting ranges of intermolecular softness are bridged, connecting the limiting ensemble behavior from hard to ideal (completely soft). Various dynamical and structural observables are measured from simulation and compared to developed theoretical values. The spatial properties are found to be well predicted by theories developed for the deterministic penetrable-sphere model with a transformation from energetic to probabilistic arguments. While the overlapping spatial structures are complex, the dynamical properties can be adequately approximated through a theory built on impulsive interactions with Enskog corrections. Our theory suggests that as the softness of interaction is varied toward the ideal limit, correlated collision processes are less important to the energy transfer mechanism, and Markovian processes dominate the evolution of the configuration space ensemble. For interaction softness close to hard limit, collision processes are highly correlated and overlapping spatial configurations give rise to entanglement of single-particle trajectories.

Geometric Models for Isotropic Random Porous Media: A Review

Models for random porous media are considered. The models are isotropic both from the local and the macroscopic point of view; that is, the pores have spherical shape or their surface shows piecewise spherical curvature, and there is no macroscopic gradient of any geometrical feature. Both closed-pore and open-pore systems are discussed. The Poisson grain model, the model of hard spheres packing, and the penetrable sphere model are used; variable size distribution of the pores is included. A parameter is introduced which controls the degree of open-porosity. Besides systems built up by a single solid phase, models for porous media with the internal surface coated by a second phase are treated. Volume fraction, surface area, and correlation functions are given explicitly where applicable; otherwise numerical methods for determination are described. Effective medium theory is applied to calculate physical properties for the models such as isotropic elastic moduli, thermal and electrical conductivity, and static dielectric constant. The methods presented are exemplified by applications: small-angle scattering of systems showing fractal-like behavior in limited ranges of linear dimension, optimization of nanoporous insulating materials, and improvement of properties of open-pore systems by atomic layer deposition of a second phase on the internal surface.

Hybrid effective medium approximations for random elastic composites

Several popular effective medium approximations for elastic constants of random composites are reformulated in terms of a pair of canonical functions and their transform variables. This choice of reformulation enables easier comparisons of the results of all these methods with rigorous bounds. Furthermore, insight into the various methods gained by taking this point of view suggests a number of new effective medium approximations that, in some cases, are natural variants and/or combinations (i.e., hybrids) of the existing ones, and in other cases are new ones based in part on the bounds themselves. Numerical comparisons are given for several standard inclusion models — including spherical, needle, and penny-shaped inclusions — as well as the penetrable sphere model. Of the various alternatives considered, a new method called the split-step differential (SSD) scheme is one of the more useful ones, as it simplifies the differential scheme by replacing half of this scheme’s integration routines with a simple update formula for the bulk modulus.

Surface area and volume fraction of random open-pore systems

For the first time, explicit approximate formulas are presented for the volume fraction and specific surface area of random open-pore systems with poly-disperse pore size distributions. It is shown that the formulas are valid for broad classes of models for porous media characterized by tunable pore size distributions and a variable degree of inter-penetrability of pores. The formulas for the poly-disperse case are based on expressions derived previously for mono-disperse penetrable-sphere models. The results are obtained by analysis of a series of open-pore models, which are prepared by computer simulation of systems of randomly packed partially penetrable spheres with various poly-disperse size distributions such as gamma, lognormal, and Gaussian. The formulas are applied in a study of atomic layer deposition processes on open-pore systems, and the effective Young's modulus and the effective thermal conductivity of Al2O3 coated porous polypropylene electrodes for lithium ion batteries are predicted.

The Empirical Enskog-Like Approximation of Shear Viscosity in the Penetrable-Sphere Model

Key Laboratory for Advanced Materials, Department of Chemistry, East China University of Science and Technology, Shanghai 200237, P. R. ChinaReceived February 7, 2012, Accepted March 2, 2012Key Words : Molecular dynamics simulation, Empirical Enskog-like approximation, Shear viscosity, Penetra-ble-sphere modelThe so-called penetrable-sphere (PS) interaction potentialis defined as,(1)where σ is the diameter penetrable sphere, and isthe strength of the repulsive energy barrier between twooverlapping spheres when they penetrate each other. The PSsystem simply reduces to the hard-sphere (HS) model in thezero-temperature limit, i.e., T

N]pT Monte Carlo simulations of the cluster-crystal-forming penetrable sphere model

Certain models with purely repulsive pair interactions can form cluster crystals with multiply-occupied lattice sites. Simulating these models' equilibrium properties is, however, quite challenging. Here, we develop an expanded isothermal-isobaric [N]pT ensemble that surmounts this problem by allowing both particle number and lattice spacing to fluctuate. It is particularly efficient at high T, where particle insertion is facile. Using this expanded ensemble and thermodynamic integration, we solve the phase diagram of a prototypical cluster-crystal former, the penetrable sphere model, and compare the results with earlier theoretical predictions. At high temperatures and densities, the equilibrium occupancy n(c)(eq) of face-centered cubic crystal increases linearly. At low temperatures, although n(c)(eq) plateaus at integer values, the crystal behavior changes continuously with density. The previously ambiguous crossover around T ~ 0.1 is resolved.

침투성 구형 모델에 관한 분자 전산 연구 : II. 충돌 특성

침투 가능한 구형 모델 유체의 충돌 특성을 고찰하고자 분자 동력학 방법을 이용한 전산 모사를 수행하였다. 이로부터 다양한 범위의 입자 충전 분율 φ 및 척력적 에너지 ε 조건에 대한 충돌 빈도수, 평균 자유 행로, 강체형 반 사 충돌각 및 연체형 침투 충돌각의 분포도, 유효 충전 분율 등을 계산하였다. 낮은 척력적 에너지 조건에서는 연체 형 충돌이 주된 특성이나, 반면 높은 척력적 에너지 조건에서는 강체형 충돌이 주된 요인이었다. 매우 흥미롭게도, 전체 충돌 빈도수에 대한 연체형 충돌비(또는, 강체형 충돌비)는 정준 앙상블의 몬테카를로 전산 모사에서 받아들임 확률(또는, 되돌림 확률)로 표시되는 볼쯔만 인자와 직접적으로 관계되었다. 이와 같은 거동 입자들의 동적 충돌 특 성들은 ε 3.0 및 φ 0.7 범위의 높은 척력적 에너지 및 높은 충전 분율 조건에서 제한적이었으며, 이는 침투성 구형 모델에서 나타나는 클러스터 형성 구조를 함축하고 있다.

침투성 구형 모델에 관한 분자 전산 연구 : 1. 상태 방정식

침투 가능한 구형 모델 유체의 상태 방정식을 고찰하고자 다양한 범위의 입자 충전 분율 φ 및 척력적 에너지 상수 ε* 에 대하여 분자 동력학 방법을 이용한 전산 모사를 수행하였다. 전산 모사로부터 얻어진 결과는 문헌에 보고된 고침투 근사식 및 저침투 근사식으로 알려진 두 가지의 한계적 이론식들과 직접 비교하였다. 낮은 척력적 에너지를 갖는 ε*<3.0의 경우 전산 모사 결과는 이들 두 이론식들과 일치하였으나, 반면 입자간 상호 포텐셜 에너지가 입자 자체 평균 운동 에너지의 두 배 이상 높은 척력적 에너지를 갖는 ε*≥3.0의 경우 이들 이론식들 모두 전산 모사 결과를 재현하지 못하였다. 이는 특히 높은 입자 밀도와 높은 척력적 에너지를 갖는 φ≥0.7 및 ε*＝6.0의 경우 입자들의 클러스터 형성 및 자체 입자 배제 부피에 따른 비연속적 크기 효과에 기인되었다.

Ground State at High Density

Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global properties of high-density ground state configurations in bounded domains and in infinite space. Our main result is a theorem stating that for interactions having a strictly positive Fourier transform the distribution of particles tends to be uniform as the density increases, while high-density ground states show some pattern if the Fourier transform is partially negative. The latter confirms the conclusion of earlier studies by Vlasov (in J. Phys. (USSR) IX:25–40, 1945), Kirzhnits and Nepomnyashchii (in Sov. Phys. JETP 32:1191–1197, 1971), and Likos et al. (in J. Chem. Phys. 126:224502, 2007). Other results include the proof that there is no Bravais lattice among high-density ground states of interactions whose Fourier transform has a negative part and the potential diverges or has a cusp at zero. We also show that in the ground state configurations of the penetrable sphere model particles are superimposed on the sites of a close-packed lattice.

A Modified Enskog-Like Equation of Self-Diffusion Coefficients for Penetrable-Sphere Model Fluids

Key Laboratory for Advanced Materials, Department of Chemistry, East China University of Science and Technology, Shanghai 200237, P. R. ChinaReceived December 31, 2010, Accepted March 3, 2011Molecular dynamics simulations have been performed to investigate the transport properties of self-diffusioncoefficients in the penetrable-sphere model system. The resulting simulation data for the product of the packingfraction and the self-diffusion coefficient exhibit a transition from an increasing function of density in lowerrepulsive systems, where the soft-type collisions are dominant, to a decreasing function in higher repulsivesystems, where most particle collisions are the hard-type reflections due to the low-penetrability effects. Amodified Enskog-like equation implemented by the effective packing fraction with the mean-field energycorrection is also proposed, and this heuristic approximation yields a reasonably good result even in systems ofhigh densities and high repulsive energy barriers. Key Words : Molecular dynamics simulation, Self-diffusion, Modified Enskog-like approximationIntroductionDuring the last couple of decades much progress has beenmade in our understanding of the static and dynamicproperties of soft-condensed matter including colloids,polymers, foams, gels, granular materials, and a number ofbiological materials. The effective model potentials can beof various nature,

Reentrant and Isostructural Transitions in a Cluster-Crystal Former

We study the low-temperature behavior of a simple cluster-crystal forming system through simulation. We find the phase diagram to be hybrid between the Gaussian core model and the penetrable sphere model. The system additionally exhibits S-shaped doubly reentrant phase sequences as well as critical isostructural transitions between crystals of different average lattice site occupancy. Because of the possible annihilation of lattice sites and accompanying clustering, the system moreover shows an unusual softening upon compression.

Nonadditivity of orientational interactions and perturbation theory for dipole hard spheres

The procedure of approximate summation is applied to a series of many-body perturbation theory for the internal energy of a dipolar hard sphere (DHS) fluid to produce an expression similar to the mean field approximation in the Widom-Rowlinson penetrable sphere model, which is well consistent with the experiment at moderate or high densities. Similar results are obtained from the hindered rotation model generalized for arbitrary density. The critical parameters ρc, Pc and Tc of both models are consistent with the data of machine experiments and are close to the parameters of the percolation transition in the DHS system.

Molecular dynamics simulation study of self-diffusion for penetrable-sphere model fluids

Molecular dynamics simulations are carried out to investigate the diffusion behavior of penetrable-sphere model fluids characterized by a finite energy barrier ϵ. The self-diffusion coefficient is evaluated from the time-dependent velocity autocorrelation function and mean-square displacement. Detailed insights into the cluster formation for penetrable spheres are gained from the Enskog factor, the effective particle volume fraction, the mean free path, and the collision frequency for both the soft-type penetrable and the hard-type reflective collisions. The simulation data are compared to theoretical predictions from the Boltzmann kinetic equation and from a simple extension to finite ϵ of the Enskog prediction for impenetrable hard spheres (ϵ→∞). A reasonable agreement between theoretical and simulation results is found in the cases of ϵ∗ ≡ ϵ/k(B)T=0.2, 0.5, and 1.0. However, for dense systems (packing fraction ϕ>0.6) with a highly repulsive energy barrier (ϵ∗ = 3.0), a poorer agreement was observed due to metastable static effects of clustering formation and dynamic effects of correlated collision processes among these cluster-forming particles.

Simple relationship between the virial-route hypernetted-chain and the compressibility-route Percus–Yevick values of the fourth virial coefficient

As is well known, approximate integral equations for liquids, such as the hypernetted chain (HNC) and Percus-Yevick (PY) theories, are in general thermodynamically inconsistent in the sense that the macroscopic properties obtained from the spatial correlation functions depend on the route followed. In particular, the values of the fourth virial coefficient B(4) predicted by the HNC and PY approximations via the virial route differ from those obtained via the compressibility route. Despite this, it is shown in this paper that the value of B(4) obtained from the virial route in the HNC theory is exactly three halves the value obtained from the compressibility route in the PY theory, irrespective of the interaction potential (whether isotropic or not), the number of components, and the dimensionality of the system. This simple relationship is confirmed in one-component systems by analytical results for the one-dimensional penetrable-square-well model and the three-dimensional penetrable-sphere model, as well as by numerical results for the one-dimensional Lennard-Jones model, the one-dimensional Gaussian core model, and the three-dimensional square-well model.