Simple Scaling Theory Research Articles

Molecular dynamics simulations of electrophoresis of polyelectrolytes in nano confining cylindrical geometries

We consider molecular dynamics simulations of the electrophoretic motion of charged polymers in straight cylinders and cylinders with a periodic variation (or modulation) of the diameter. The electric field is always orientated along the axis of the cylinders, and it does not change direction. The fluid is modeled explicitly, also the co-ions, counterions, and the charged polymer monomers. In straight geometries, we observe no separation effect for a wide range of applied electric fields. In modulated geometries, the charged polymer chains can be separated only at low fields. At large fields, we observe separation effects in modulated geometries only if the applied field is a dc pulse electric field. A simple scaling theory to explain the observed behavior is presented.

Designing Bio‐Inspired Adhesives for Shear Loading: From Simple Structures to Complex Patterns

AbstractThe gecko has inspired numerous synthetic adhesive structures, yet under shear loading conditions, general design criteria remains underdeveloped. To provide guidance for bio‐inspired adhesives under shear, a simple scaling theory is used to investigate the relevant geometric and material parameters. The total compliance of an elastic attachment feature is described over many orders of magnitude in aspect ratio through a single continuous function using the superposition of multiple deformation modes such as bending, shear deformation, and tensile elongation. This allows for force capacity predictions of common geometric control parameters such as thickness, aspect ratio, and contact area. This superposition principal is extended to develop criteria for patterned interfaces under shear loading. Importantly, the adhesive patterns under shear are controlled through the compliance in the direction of loading. These predictions are confirmed experimentally using macroscopic building blocks over an extensive range of aspect ratio and contact area. Over 25 simple and complex patterns with various contact geometries are examined, and the effect of geometry and material properties on the shear adhesion behavior is discussed. Furthermore, all of these various attachment features are described with a single scaling parameter, offering control over orders of magnitude in adhesive force capacity for a variety of applications.

Looking Beyond Fibrillar Features to Scale Gecko‐Like Adhesion

Hand-sized gecko-inspired adhesives with reversible force capacities as high as 2950 N (29.5 N cm(-2) ) are designed without the use of fibrillar features through a simple scaling theory. The scaling theory describes both natural and synthetic gecko-inspired adhesives, over 14 orders of magnitude in adhesive force capacity, from nanoscopic to macroscopic length scales.

Assessing of IDF curves for hydrological design by simple scaling of 1-day precipitation totals

Assessing of IDF curves for hydrological design by simple scaling of 1-day precipitation totalsIn this paper the scaling properties of short term extreme rainfall in Slovakia were investigated. The simple scaling theory was applied to the intensity-duration-frequency (IDF) characteristics of a short duration rainfall. This method allows for the estimation of the design values of rainfall of selected recurrence intervals and durations shorter than a day by using only the daily data. The scaling behavior of rainfall intensities was examined, and the possibility of using simple scaling in Slovakia was verified. The methodology for the simple scaling of rainfall is demonstrated using an example of the meteorological station in Ilava.

Self-propulsion and dispersion of reactive colloids due to entropic anisotropy

In this paper, self-motion of reactive colloids and their dispersion behaviour are theoretically examined. The motion is driven by an osmotic force imbalance arising from non-uniform atmospheres of reactive solutes around the colloids. The propulsion here is not limited to Janus-like particles. It can also occur to particles having ‘uniform’ reactivity due to the more universal mechanism – entropic anisotropy created by breaking in rotational symmetry. The idea is demonstrated by examining the motion of a reactive particle due to asymmetry in its shape or to the presence of an additional particle. In the two-particle problem, in particular, we find that sink (source) particles can self-migrate towards (apart from) each other at velocities varying asR−2, resembling Coulomb attraction (repulsion), whereRis the inter-particle distance. Because of this Coulomb-like nature, a suspension of sink particles could undergo collective flocculation due to unscreened osmotic attraction. The criterion for an occurrence of the flocculation is also established. It reveals that the flocculation can occur if the particle volume fraction is within a certain window in terms of the solute concentration and the particle reactivity. The stability of reactive suspensions is also discussed using the modified Derjaguin–Landau–Verwey–Overbeek (DLVO) theory that takes account of the competition between long-range reaction-induced osmotic forces and short-range colloidal forces. A more generalized view for the present self-driven particle motion is elucidated by a simple scaling theory, providing lucid accounts for the self-motion of two particles, composite bodies, and Janus particles – all are driven by dipolar distortions in potential energy. Comparison with phoretic self-swimmers is also discussed.

Hierarchical wrinkling patterns

This paper reports a simple and flexible method for generating hierarchical patterns from wrinkling instability. Complex features with gradually changing topographies are generated by using the spontaneous wrinkling of a rigid membrane (titanium) on a soft foundation (polystyrene) compressed via the diffusion of a solvent. We show that the morphology of these unreported wrinkled patterns is directly related to the rheological properties of the polymer layer and the geometry of the diffusion front. Based on these ingredients, we rationalize the mechanism for the formation of hierarchical wrinkling patterns and quantify our experimental findings with a simple scaling theory. Finally, we illustrate the relevance of our structuration method by studying the mechanosensitivity of fibroblasts.

On the use of the simple scaling of heavy rainfall in a regional estimation of IDF curves in Slovakia

The aim of this study was to test the applicability of a simple scaling methodology for a regional estimation of intensity-duration-frequency (IDF) curves in Slovakia. The analysis is based on the regionalization process of Gaal (2006), which focused on the delineation of homogeneous regions for a regional frequency analysis of precipitation maxima. In order to examine the regionally estimated IDF curves, a region covering the western parts of Slovakia was chosen. The selected region, which encompasses 19 raingauging stations, may be characterized by the dominant influence of Atlantic circulation patterns. Three of the 19 stations belonging to the target region were set aside and flagged as verification stations. The regional dimensionless growth curve of 1-day precipitation maxima in the warm season was derived for the region, and the local T-year quantiles were estimated by the index value method for the stations. At the same time, a regionally averaged scaling exponent was derived using all the stations except for the three verification ones. The local IDF curves at the verification stations were estimated by downscaling the Tyear quantiles of the 1-day precipitation maxima using the regionally averaged scaling exponent. Finally, the IDF curves for these stations were compared with those defined by Samaj, Valovic (1973). This study is the first step in assessing the applicability of a simple scaling theory for the regional estimation of IDF curves in Slovakia.

Kondo Resonance Narrowing ind- andf-Electron Systems

We develop a simple scaling theory for the effect of Hund's interactions on the Kondo effect, showing how an exponential narrowing of the Kondo resonance develops in magnetic ions with large Hund's interaction. Our theory accounts for the exponential reduction of the Kondo temperature with spin S of the Hund's coupled moment, first observed in d-electron alloys in the 1960s, and more recently encountered in numerical calculations on multiband Hubbard models. We discuss the consequences of Kondo resonance narrowing for the Mott transition in d-band materials, particularly iron pnictides, and the narrow ESR linewidth recently observed in ferromagnetically correlated f-electron materials.

Hygromorphs: from pine cones to biomimetic bilayers

We consider natural and artificial hygromorphs, objects that respond to environmental humidity by changing their shape. Using the pine cone as an example that opens when dried and closes when wet, we quantify the geometry, mechanics and dynamics of closure and opening at the cell, tissue and organ levels, building on our prior structural knowledge. A simple scaling theory allows us to quantify the hysteretic dynamics of opening and closing. We also show how simple bilayer hygromorphs of paper and polymer show similar behaviour that can be quantified via a theory which couples fluid transport in a porous medium and evaporative flux to mechanics and geometry. Our work unifies varied observations of natural hygromorphs and suggests interesting biomimetic analogues, which we illustrate using an artificial flower with a controllable blooming and closing response.

How does abundance scale with body size in coupled size‐structured food webs?

1. Widely observed macro-ecological patterns in log abundance vs. log body mass of organisms can be explained by simple scaling theory based on food (energy) availability across a spectrum of body sizes. The theory predicts that when food availability falls with body size (as in most aquatic food webs where larger predators eat smaller prey), the scaling between log N vs. log m is steeper than when organisms of different sizes compete for a shared unstructured resource (e.g. autotrophs, herbivores and detritivores; hereafter dubbed 'detritivores'). 2. In real communities, the mix of feeding characteristics gives rise to complex food webs. Such complexities make empirical tests of scaling predictions prone to error if: (i) the data are not disaggregated in accordance with the assumptions of the theory being tested, or (ii) the theory does not account for all of the trophic interactions within and across the communities sampled. 3. We disaggregated whole community data collected in the North Sea into predator and detritivore components and report slopes of log abundance vs. log body mass relationships. Observed slopes for fish and epifaunal predator communities (-1.2 to -2.25) were significantly steeper than those for infaunal detritivore communities (-0.56 to -0.87). 4. We present a model describing the dynamics of coupled size spectra, to explain how coupling of predator and detritivore communities affects the scaling of log N vs. log m. The model captures the trophic interactions and recycling of material that occur in many aquatic ecosystems. 5. Our simulations demonstrate that the biological processes underlying growth and mortality in the two distinct size spectra lead to patterns consistent with data. Slopes of log N vs. log m were steeper and growth rates faster for predators compared to detritivores. Size spectra were truncated when primary production was too low for predators and when detritivores experienced predation pressure. 6. The approach also allows us to assess the effects of external sources of mortality (e.g. harvesting). Removal of large predators resulted in steeper predator spectra and increases in their prey (small fish and detritivores). The model predictions are remarkably consistent with observed patterns of exploited ecosystems.

Viscous coalescence of expanding low-viscosity drops; the dueling drops experiment

A new experimental device was designed and developed for the systematic study of the interaction of two, very small low viscosity expanding drops that were pushed out of the ends of two aligned opposing capillaries into a polymeric liquid. The idea of this experiment is to mimic the interactions of two expanding bubbles during the formation of polymeric foams; for convenience, we call this device the dueling drops experiment. In the current setup, we are able to grow drops (50–200 μm maximum diameter) at a controlled flow rate (down to 0.1 μl/day). Optical observation of the pair of growing drops was achieved by the use of an optical zoom lens system for a side view, and a “long-range” microscope for a top view, thus ensuring complete alignment of the capillaries and partial observation of the thin-film. Measurements are reported for the drainage time between the apparent initial contact of the drops and film rupture. A simple scaling theory was developed based on the drainage of two approaching flat disk-like interfaces, the radius of which expands throughout the interaction as the volume of the drop increases.

Coalescence of two equal-sized deformable drops in an axisymmetric flow

The coalescence of two equal-sized deformable drops in an axisymmetric flow is studied, using a boundary-integral method. An adaptive mesh refinement method is used to resolve the local small-scale dynamics in the gap and to retain a reasonable speed of computation. The thin film dynamics is successfully simulated, with sufficient stability and accuracy, up to a film thickness of O(10−4) times the undeformed drop radius, for a range of capillary numbers, Ca, from O(10−4–10−1) and viscosity ratios from O(0.1–10). The results are compared with experimental results from our earlier studies as well as the simple scaling theory for film drainage. The collisions for time-independent flow simulating head-on collisions in the experimental studies show two distinctively different regimes. At lower capillary numbers, the interfaces of the thin film between the colliding drops remain almost spherical up to the point of film rupture, and the dimensionless drainage time scales as tdG∼Ca. At higher capillary numbers, the film becomes dimpled at an early stage of the collision process, and the rate of the film drainage significantly slows down after the dimple is fully formed. In this case, the drainage time scales approximately as tdG∼Ca4∕3. The simulation, using a Hamaker constant with a fixed value calculated via Lifshitz theory, qualitatively agrees with the experimental results for the higher capillary numbers but not for the lower capillary numbers. The critical conditions for head-on collisions are also examined when the internal circulation within the drop, caused by the external flow, arrests the film drainage. Collisions in a time-dependent flow are also examined to simulate glancing collisions. Although the simulations predict many aspects of the experimental results, the results are quantitatively accurate, in comparison to the experimental data, only for the lowest viscosity ratio of 0.19. The interfaces of the thin film locally bulge outward when the drops are being pulled apart due to the suction pressure. This local deformation causes a local minimum in the film thickness. At the larger offsets, the coalescence angle continuously increases with Ca up to the separation angle (θ=55°–58°), for Ca<Cac. At smaller offsets, however, the local deformation for θ>45° cannot induce film rupture, even though coalescence is observed experimentally for the higher viscosity ratios.

Field-driven translocation of regular block copolymers through a selective liquid-liquid interface

We propose a simple scaling theory describing the variation of the mean first passage time (MFPT) τ(N,M) of a regular block copolymer of chain length N and block size M which is dragged through a selective liquid−liquid interface by an external field B. The theory predicts a non-Arrhenian τ vs B relationship which depends strongly on the size of the blocks, M, and rather weakly on the total polymer length, N. The overall behavior is strongly influenced by the degree of selectivity between the two solvents χ. The variation of τ(N,M) with N and M in the regimes of weak and strong selectivity of the interface is also studied by means of computer simulations using a dynamic Monte Carlo coarse-grained model. Good qualitative agreement with theoretical predictions is found. The MFPT distribution is found to be well described by a Γ-distribution. Transition dynamics of ring and telechelic polymers is also examined and compared to that of the linear chains. The strong sensitivity of the “capture” time τ(N,M) with...

Field-Driven Translocation of Regular Block Copolymers through a Selective Liquid−Liquid Interface

We propose a simple scaling theory describing the variation of the mean first passage time (MFPT) $\tau(N,M)$ of a regular block copolymer of chain length $N$ and block size $M$ which is dragged through a selective liquid-liquid interface by an external field $B$. The theory predicts a non-Arrhenian $\tau$ vs. $B$ relationship which depends strongly on the size of the blocks, $M$, and rather weakly on the total polymer length, $N$. The overall behavior is strongly influenced by the degree of selectivity between the two solvents $\chi$. The variation of $\tau(N,M)$ with $N$ and $M$ in the regimes of weak and strong selectivity of the interface is also studied by means of computer simulations using a dynamic Monte Carlo coarse-grained model. Good qualitative agreement with theoretical predictions is found. The MFPT distribution is found to be well described by a $\Gamma$ - distribution. Transition dynamics of ring- and telechelic polymers is also examined and compared to that of the linear chains. The strong sensitivity of the ``capture'' time $\tau(N,M)$ with respect to block length $M$ suggests a possible application as a new type of chromatography designed to separate and purify complex mixtures with different block sizes of the individual macromolecules.

Multiblock copolymers at selective liquid–liquid interfaces: Toward a block size chromatography

AbstractWe report some very recent studies on the localization of hydrophobic‐polar regular copolymers at a selective solvent–solvent interface with emphasis on the impact of block length M on their static properties and dynamics. A simple scaling theory is developed and its predictions are compared with extensive Monte Carlo simulations to provide an adequate description of copolymers behavior at such penetrable phase boundary. We have derived and verified the scaling of various quantities such as the components of the radius of gyration parallel and perpendicular to the interface, the lateral diffusion constant, as well as the characteristic relaxation times for the localization of the polymer coil in terms of chain length N and block size M for various strength of the interface. The copolymer dynamics is elucidated by a Rouse‐mode analysis, which reveals a specific coupling of the modes once a copolymer is adsorbed at the selective interface. For the case of copolymer chains being dragged by an external field through the selective interface, we provide results, which suggest a possible application as a new type of chromatography designed to separate and purify complex mixtures with respect to the chain length and the block size of the individual macromolecules. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2572–2588, 2006

The Swelling Behavior of Charged Hydrogels

AbstractSummary: We describe recent advances obtained in understanding the microscopic interplay of a charged hydrogel with its confined mobile counterions. Results of extensive computer simulations are compared to simple scaling descriptions for various solvent conditions. We will briefly discuss our simulation package ESPResSo which was used to perform the simulations. We demonstrate why simple scaling theories suffice to adequately describe the swelling behavior of these systems despite their enormous complexity. We will conclude with a presentation of ongoing results for polyelectrolyte networks under poor solvent conditions.

On the validity of boundary layer analysis for flow and mass transfer caused by rotation during the solution growth of large, single crystals

The rate of growth of single crystals from liquid solutions depends on coupled kinetic and transport phenomena. However, when continuum transport is limiting, the maximum crystal growth rate is determined by the rate of solute transport through the liquid phase to the growing crystal. We examine the validity of simple scaling and boundary layer theories to assess the behavior of a model solution crystal growth system, namely the growth of potassium titanyl phosphate (KTP) from high-temperature solutions. The approximation of the transport to a crystal rotating steadily in a supersaturated solution is based on the classical analytical solution for flow driven by a semi-infinite rotating disk and associated mass transfer. Our results indicate that this boundary layer analysis is reasonable as long as the solution container is very large, the geometry of the system is nearly axisymmetric, and the imposed flows are predominantly steady. For many practical systems, these conditions do not hold and such boundary layer analyses are expected to be in considerable error.

Nested self-similar wrinkling patterns in skins

Stiff thin films on soft substrates are both ancient and commonplace in nature; for instance, animal skin comprises a stiff epidermis attached to a soft dermis. Although more recent and rare, artificial skins are increasingly used in a broad range of applications, including flexible electronics, tunable diffraction gratings, force spectroscopy in cells, modern metrology methods, and other devices. Here we show that model elastomeric artificial skins wrinkle in a hierarchical pattern consisting of self-similar buckles extending over five orders of magnitude in length scale, ranging from a few nanometres to a few millimetres. We provide a mechanism for the formation of this hierarchical wrinkling pattern, and quantify our experimental findings with both computations and a simple scaling theory. This allows us to harness the substrates for applications. In particular, we show how to use the multigeneration-wrinkled substrate for separating particles based on their size, while simultaneously forming linear chains of monodisperse particles.

Fast Directed Motion of “Fakir” Droplets

In this Letter, we report on the motion of water droplets on surfaces decorated with molecular gradients comprising semifluorinated (SF) organosilanes. SF molecular gradients deposited on flat silica substrates facilitate faster motion of water droplets relative to the specimens covered with an analogous hydrocarbon gradient. Further increase in the drop speed is achieved by advancing it along porous substrates coated with the SF wettability gradients. The results of our experiments are in quantitative agreement with a simple scaling theory that describes the faster liquid motion in terms of reduced friction at the liquid/substrate interface.

Lattice gas simulations of dynamical geometry in one dimension.

We present numerical results obtained using a lattice gas model with dynamical geometry. The (irreversible) macroscopic behaviour of the geometry (size) of the lattice is discussed in terms of a simple scaling theory and obtained numerically. The emergence of irreversible behaviour from the reversible microscopic lattice gas rules is discussed in terms of the constraint that the macroscopic evolution be reproducible. The average size of the lattice exhibits power-law growth with exponent at late times. The deviation of the macroscopic behaviour from reproducibility for particular initial conditions ('rogue states') is investigated as a function of system size. The number of such 'rogue states' is observed to decrease with increasing system size. Two mean-field analyses of the macroscopic behaviour are also presented.