Percolation transition in fluids: scaling behavior of the spanning probability functions

Percolation transition and bimodal density distribution in hydrogen fluoride

Hydrogen-bond networks in associating fluids can be extremely robust and characterize the topological properties of the liquid phase, as in the case of water, over its whole domain of stability and beyond. Here, we report on molecular dynamics simulations of hydrogen fluoride (HF), one of the strongest hydrogen-bonding molecules. HF has more limited connectivity than water but can still create long, dynamic chains, setting it apart from most other small molecular liquids. Our simulation results provide robust evidence of a second-order percolation transition of HF's hydrogen bond network occurring below the critical point. This behavior is remarkable as it underlines the presence of two different cohesive mechanisms in liquid HF, one at low temperatures characterized by a spanning network of long, entangled hydrogen-bonded polymers, as opposed to short oligomers bound by the dispersion interaction above the percolation threshold. This second-order phase transition underlines the presence of marked structural heterogeneity in the fluid, which we found in the form of two liquid populations with distinct local densities.

Self-dual quasiperiodic percolation

How does the percolation transition behave in the absence of quenched randomness? To address this question, we study two nonrandom self-dual quasiperiodic models of square-lattice bond percolation. In both models, the critical point has emergent discrete scale invariance, but none of the additional emergent conformal symmetry of critical random percolation. From the discrete sequences of critical clusters, we find fractal dimensions of D_{f}=1.911943(1) and D_{f}=1.707234(40) for the two models, significantly different from D_{f}=91/48=1.89583... of random percolation. The critical exponents ν, determined through a numerical study of cluster sizes and wrapping probabilities on a torus, are also well below the ν=4/3 of random percolation. While these new models do not appear to belong to a universality class, they demonstrate how the removal of randomness can fundamentally change the critical behavior.

Geometric percolation of hard-sphere dispersions in shear flow

We combine a heuristic theory of geometric percolation and the Smoluchowski theory of colloid dynamics to predict the impact of shear flow on the percolation threshold of hard spherical colloidal particles, and verify our findings by means of molecular dynamics simulations. It appears that the impact of shear flow is subtle and highly non-trivial, even in the absence of hydrodynamic interactions between the particles. The presence of shear flow can both increase and decrease the percolation threshold, depending on the criterion used for determining whether or not two particles are connected and on the Péclet number. Our approach opens up a route to quantitatively predict the percolation threshold in nanocomposite materials that, as a rule, are produced under non-equilibrium conditions, making comparison with equilibrium percolation theory tenuous. Our theory can be adapted straightforwardly for application in other types of flow field, and particles of different shape or interacting via other than hard-core potentials.

Shape, geometric percolation, and electrical conductivity of clusters in suspensions of hard platelets

Using Monte Carlo simulations, we investigate how geometric percolation and electrical conductivity in suspensions of hard conducting platelets are affected by the addition of platelets and their degree of spontaneous alignment. In our simulation results for aspect ratios 10, 25, and 50, we consistently observe a monotonically decreasing percolation threshold as a function of volume fraction, i.e., the addition of particles always aids percolation. In the nematic phase, the distribution of particles inside the percolating clusters becomes less spherically symmetric and the aspect ratio of the clusters increases. However, the clusters are also anisotropically shaped in the isotropic phase, although their aspect ratio remains constant as a function of volume fraction and is only weakly dependent on the particle aspect ratio. Mapping the percolating clusters of platelets to linear resistor networks, and assigning unit conductance to all connections, we find a constant conductivity both across the isotropic-nematic transition and in the respective stable phases. This behavior is consistent with the other observed topological properties of the networks, namely, the average path length, average number of contacts per particle, and the Kirchhoff index, which all remain constant and unaffected by both the addition of particles and the degree of alignment of their suspension. In contrast, using an anisotropic conductance model that explicitly accounts for the relative orientation of the particles, the network conductivity decreases with increasing volume fraction in the isotropic, and further diminishes at the onset of the nematic while preserving the same trend deep in the nematic. Hence, our observations consistently suggest that, unlike for rodlike fillers, the network structures that arise from platelet suspensions are not very sensitive to the particle aspect ratio or to alignment. Hence platelets are not as versatile as fillers for dispersion in conductive composite materials as rods.

Unusual geometric percolation of hard nanorods in the uniaxial nematic liquid crystalline phase

We investigate by means of continuum percolation theory and Monte Carlo simulations how spontaneous uniaxial symmetry breaking affects geometric percolation in dispersions of hard rodlike particles. If the particle aspect ratio exceeds about 20, percolation in the nematic phase can be lost upon adding particles to the dispersion. This contrasts with percolation in the isotropic phase, where a minimum particle loading is always required to obtain system-spanning clusters. For sufficiently short rods, percolation in the uniaxial nematic mimics that of the isotropic phase, where the addition of particles always aids percolation. For aspect ratios between 20 and infinity, but not including infinity, we find reentrance behavior: percolation in the low-density nematic may be lost upon increasing the amount of nanofillers but can be regained by the addition of even more particles to the suspension. Our simulation results for aspect ratios of 5, 10, 20, 50, and 100 strongly support our theoretical predictions, with almost quantitative agreement. We show that a different closure of the connectedness Ornstein-Zernike equation, inspired by scaled particle theory, is as least as accurate in predicting the percolation threshold as the Parsons-Lee closure, which effectively describes the impact of many-body direct contacts.

Continuum percolation of polydisperse rods in quadrupole fields: Theory and simulations

We investigate percolation in mixtures of nanorods in the presence of external fields that align or disalign the particles with the field axis. Such conditions are found in the formulation and processing of nanocomposites, where the field may be electric, magnetic, or due to elongational flow. Our focus is on the effect of length polydispersity, which-in the absence of a field-is known to produce a percolation threshold that scales with the inverse weight average of the particle length. Using a model of non-interacting spherocylinders in conjunction with connectedness percolation theory, we show that a quadrupolar field always increases the percolation threshold and that the universal scaling with the inverse weight average no longer holds if the field couples to the particle length. Instead, the percolation threshold becomes a function of higher moments of the length distribution, where the order of the relevant moments crucially depends on the strength and type of field applied. The theoretical predictions compare well with the results of our Monte Carlo simulations, which eliminate finite size effects by exploiting the fact that the universal scaling of the wrapping probability function holds even in anisotropic systems. Theory and simulation demonstrate that the percolation threshold of a polydisperse mixture can be lower than that of the individual components, confirming recent work based on a mapping onto a Bethe lattice as well as earlier computer simulations involving dipole fields. Our work shows how the formulation of nanocomposites may be used to compensate for the adverse effects of aligning fields that are inevitable under practical manufacturing conditions.

A general method for determining molecular interfaces and layers

A general and direct computational scheme to locate the surface separating arbitrarily shaped domains made up of molecules (or any other particles) has been developed and is described and illustrated for several, both artificial and physical examples. The proposed scheme consists of two modules: (i) triangulation and (ii) assignment of simplices to domains. Three different triangulation methods are employed, viz., the Delaunay triangulation, regular triangulation, and quasi-triangulation. In the triangulated system, the assignment step is carried out in two different ways, one based on the characteristic metric of a particular triangulation procedure and the other on the concept of a touching sphere. Some of the combinations of the triangulation and assignment steps lead to methods already used by others to find interfacial or surface molecules, namely the alpha-shape-based method of Usabiaga nad Duque [Phys. Rev. E 79 (2009) 046709] and GITIM of Sega et al. [J. Chem. Phys. 138 (2013) 044110]. The resulting surface is defined not only as a discrete set of particles, but it is build up of facets of the triangulation forming a broken line in two dimensions or a polyhedral surface in three dimensions. Individual molecular layers are identified in a very straightforward manner, starting with the interfacial layer itself and proceeding into the interior of the phase. The proposed scheme is illustrated first by identifying border molecules of pre-sampled domains of several shapes in a plane and then applied to five physically meaningful examples: thin films, near critical water, liquid water slab in an electric field, liquid water at a solid wall, and water at condition of electric-field-induced jetting. Performance of the considered methods is critically assessed. Treatment of domains forming percolating clusters through periodic boundary conditions is also described along with the determination of their periodicity and dimensionality.

Range effect on percolation threshold and structural properties for short-range attractive spheres

Percolation or aggregation in colloidal system is important in many fields of science and technology. Using molecular dynamics simulations, we study the percolation behavior for systems consisting of spheres interacting with short-range square-well (SRSW) which mimic colloidal particles, with different interaction ranges. We specifically focus on how the interaction range affects the percolation thresholds in the supercritical region. We find that the contact percolation boundaries are strongly dependent on the interaction ranges of SRSW, especially away from the liquid-liquid critical point. However, varying the interaction ranges of SRSW does not affect much the structure along percolation boundaries especially for low packing fractions. For instance, along the percolation boundary, distributions of coordination number show convergence, and distributions of cluster size are universal for different interaction ranges considered. In addition, either the bond percolation boundaries or isolines of average bond coordination number collapse to those for Baxter sticky model on phase diagram, which confirms the extended law of corresponding states.

Two-dimensional percolation at the free water surface and its relation with the surface tension anomaly of water

The percolation temperature of the lateral hydrogen bonding network of the molecules at the free water surface is determined by means of molecular dynamics computer simulation and identification of the truly interfacial molecules analysis for six different water models, including three, four, and five site ones. The results reveal that the lateral percolation temperature coincides with the point where the temperature derivative of the surface tension has a minimum. Hence, the anomalous temperature dependence of the water surface tension is explained by this percolation transition. It is also found that the hydrogen bonding structure of the water surface is largely model-independent at the percolation threshold; the molecules have, on average, 1.90 ± 0.07 hydrogen bonded surface neighbors. The distribution of the molecules according to the number of their hydrogen bonded neighbors at the percolation threshold also agrees very well for all the water models considered. Hydrogen bonding at the water surface can be well described in terms of the random bond percolation model, namely, by the assumptions that (i) every surface water molecule can form up to 3 hydrogen bonds with its lateral neighbors and (ii) the formation of these hydrogen bonds occurs independently from each other.

Microscopic origin of the surface tension anomaly of water

We investigate the hydrogen bonding percolation threshold of water molecules at the surface of the liquid-vapor interface. We show that the percolation temperature agrees within statistical accuracy with the high-temperature inflection point of the water surface tension. We associate the origin of this surface tension anomaly of water with the sudden breakup of the hydrogen-bonding network in the interfacial molecular layer.

Ising-like phase transition of an n-component Eulerian face-cubic model

By means of Monte Carlo simulations and a finite-size scaling analysis, we find a critical line of an n-component Eulerian face-cubic model on the square lattice and the simple cubic lattice in the region v>1, where v is the bond weight. The phase transition belongs to the Ising universality class independent of n. The critical properties of the phase transition can also be captured by the percolation of the complement of the Eulerian graph.

Thermal stability of the hydrogen-bonded water network in the hydration shell of islet amyloid polypeptide

The effect of temperature on the connectivity of hydrogen bonds in the hydration shells of the islet amyloid polypeptides (IAPPs) is studied by means of computer simulations. The hydrogen-bonded network of hydration water homogeneously envelopes a peptide at low temperature and breaks into an ensemble of small clusters upon heating. This thermal break occurs via a percolation transition, which is not found to be sensitive to the chemical modifications of IAPP (IAPP with and without a disulfide bridge, human and rat IAPP). The radius of gyration of IAPP starts to increase when the hydration water network breaks upon heating. The fluctuations of the number of intra-peptide hydrogen bonds show negative correlation with the fraction of molecules in the largest cluster of hydration water. The thermal stability of the network of hydration water is enhanced upon increasing number of intra-peptide hydrogen bonds, which makes the peptide surface more hydrophobic. The thermal stabilities of the hydrogen-bonded water networks in the hydration shells of IAPPs and of several other biomolecules are found to be rather similar: the network breaks between 300 and 330 K, i.e., in the temperature interval where the biological activity of living organisms is maximal.

Percolation Threshold of Water in Ideal Binary Mixture

Percolation transitions of water and solute molecules in an ideal completely miscible aqueous solution are studied by computer simulations. Three concentration ranges with different kinds of water and solute clustering can be distinguished. An infinite water network, present in pure liquid water, breaks at some solute concentration. Accordingly, the infinite network of solute molecules, present in pure liquid solute, breaks upon addition of water. At ambient and high temperatures, there is a concentration range were both components are below their respective percolation thresholds and there are no water or solute infinite networks. The existence of such concentration range is a characteristic of a completely miscible binary mixture. Upon supercooling, the threshold concentrations decrease and simultaneous existence of the infinite networks of water and of solute becomes possible. The presence of two inter-penetrating infinite networks of molecules may be one of the conditions required for the liquid-liquid transition of one-component isotropic fluid.

Percolation threshold parameters of fluids

Extensive Monte Carlo simulations on three qualitatively different model supercritical fluids (square-well fluid, Lennard-Jonesium, and primitive water) have been performed to examine percolation threshold parameters for continuum (correlated) models and their relation to general results valid for random lattice models; random-site percolation simple-cubic lattice has therefore been considered as well. Two different bond criteria, the configurational and self-bound ones, defining a cluster have been used. In addition to the percolation threshold occupation probability pc and the percolation threshold fluid density rhoc, the correlation length exponent nu and the wrapping probability at the percolation threshold Rw,c have also been evaluated. It is found that parameters nu and Rw,c exhibit not only strong temperature dependence but also, unlike the case of lattice systems, dependence on the nature of the system considered and the employed definition of the cluster.