Complete Scaling Theory Research Articles

Scaling theory of critical strain-stiffening in disordered elastic networks

Disordered elastic networks provide a framework for describing a wide variety of physical systems, ranging from amorphous solids, through polymeric fibrous materials to confluent cell tissues. In many cases, such networks feature two widely separated rigidity scales and are nearly floppy, yet they undergo a dramatic stiffening transition when driven to sufficiently large strains. We present a complete scaling theory of the critical strain-stiffened state in terms of the small ratio between the rigidity scales, which is conceptualized in the framework of a singular perturbation theory. The critical state features quartic anharmonicity, from which a set of nonlinear scaling relations is derived. Scaling predictions for the macroscopic elastic modulus beyond the critical state are derived as well, revealing a previously unidentified characteristic strain scale. The predictions are quantitatively compared to a broad range of available numerical data on biopolymer network models and future research questions are discussed.

Thermalized buckling of isotropically compressed thin sheets.

The buckling of thin elastic sheets is a classic mechanical instability that occurs over a wide range of scales. In the extreme limit of atomically thin membranes like graphene, thermal fluctuations can dramatically modify such mechanical instabilities. We investigate here the delicate interplay of boundary conditions, nonlinear mechanics, and thermal fluctuations in controlling buckling of confined thin sheets under isotropic compression. We identify two inequivalent mechanical ensembles based on the boundaries at constant strain (isometric) or at constant stress (isotensional) conditions. Remarkably, in the isometric ensemble, boundary conditions induce a novel long-ranged nonlinear interaction between the local tilt of the surface at distant points. This interaction combined with a spontaneously generated thermal tension leads to a renormalization group description of two distinct universality classes for thermalized buckling, realizing a mechanical variant of Fisher-renormalized critical exponents. We formulate a complete scaling theory of buckling as an unusual phase transition with a size-dependent critical point, and we discuss experimental ramifications for the mechanical manipulation of ultrathin nanomaterials.

Application of the renormalization group theory for critical asymmetry to surface criticality of one-component fluids

It is well-known that the Tolman length which characterizes the first order curvature corrections to surface tension is associated with the asymmetry in fluid phase coexistence. Recently, we have developed a global renormalization group (GRG) method for describing such asymmetry both near to and far from the critical point. In this work, we use the GRG method and an approximate thermodynamic relation to calculate Tolman lengths of one-component simple fluids in a wide temperature range including the critical vicinity where the density functional theory and the square-gradient theory is inapplicable. Far from the critical point, the accuracy of this work is comparable to the density functional theory and the square-gradient theory. Near the critical point, the GRG method yields the scaling behavior consistent with the prediction of complete scaling theory. We also obtain an accurate method for calculating the near-critical surface tensions.

Liquid–Liquid Phase Equilibria of Binary Solutions of {1-Butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide + 1-Butanol or 2-Methyl-1-propanol}

The liquid–liquid coexistence curves for the room temperature ionic liquid (RTIL) solutions of 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([C4mim][NTf2]) in 1-butanol and 2-methyl-1-propanol have been precisely measured. The values of the critical exponent corresponding to the coexistence curve were found to be consistent with theoretical predictions of the 3D-Ising universality. The complete scaling theory was used to analyze the asymmetric behavior of the coexistence curve. The reduced consolute temperatures and the reduced consolute densities of the binary solutions of [C4mim][NTf2] and alcohols were also calculated to discuss the interactions in the RTIL solutions driving the liquid–liquid phase separations.

Liquid-liquid equilibrium and heat capacity for the critical binary solution of {n-decane + benzonitrile}

The liquid-liquid coexistence curve in a temperature range of about 8 K from the upper consolute point and the specific isobaric heat capacity in a wide temperature range for the critical binary solution of {n-decane + benzonitrile} have been precisely measured. The upper consolute mole fraction xc of benzonitrile and the upper consolute temperature Tc were determined to be xc = 0.530, Tc = 286.930 K, respectively. From the data collected near the upper consolute point, the critical exponents corresponding to the heat capacity and the coexistence curve were deduced and found to be consistent with the theoretical predictions of the 3D-Ising universality. The chain-length dependences of the critical properties were investigated, and the asymmetric behavior of the coexistence curve was analyzed based on the complete scaling theory, which confirmed that the heat capacity related term was of importance.

Measurements of the liquid-liquid coexistence curves of {(1 − x) dimethyl carbonate + x pentadecane} and {(1 − x) dimethyl carbonate + x heptadecane} in the critical region

The liquid-liquid coexistence curves in a temperature range of about 10 K from the upper consolute critical temperature for binary solutions of {dimethyl carbonate + pentadecane} and {dimethyl carbonate + heptadecane} have been precisely measured. The upper consolute mole fraction xc of n-alkane and the upper consolute temperature Tc were determined to be xc = 0.210, Tc = 312.234 K for {dimethyl carbonate + pentadecane} and xc = 0.183, Tc = 320.496 K for {dimethyl carbonate + heptadecane}, respectively. From the data collected in the near critical point, the critical exponent β was deduced and shown to be consistent well with its theoretical value for the 3-D Ising universality class. The dependences of critical properties on the chain length of n-alkane for a series of {dimethyl carbonate + n-alkane} binary solutions were investigated. Furthermore, it was found that the asymmetric behaviors of the coexistence curve could be well described by the complete scaling theory. The contribution of the heat capacity related term to the asymmetric behaviors of the coexistence curves was discussed.

Liquid–liquid equilibrium, heat capacity and turbidity of binary solution nitrobenzene + n -pentadecane

Abstract The liquid–liquid coexistence curve, the isobaric heat capacity per unit volume and the turbidity for the binary solution of nitrobenzene + n-pentadecane have been measured at various temperatures. The upper consolute mole fraction xc of nitrobenzene and the upper consolute temperature Tc of the binary solution were determined to be xc = 0.696 and Tc = 307.298 K, respectively. The values of the critical exponents related to the width of the coexistence curve, the heat capacity, the osmotic compressibility and the correlation length were deduced and found to be in good agreements with that of the 3-D Ising universality. With the heat capacity data the asymmetric behavior of the diameter of the coexistence curve was studied in terms of the complete scaling theory. The critical amplitudes related to the above four quantities were used to calculate the universal scaling ratios and to confirm their theoretical predictions.

Liquid-liquid equilibrium and heat capacity measurements of the binary solution {ethanol + 1-butyl-3-methylimidazolium hexafluorophosphate}

The liquid-liquid coexistence curves in a temperature range of about 10K from the upper consolute point and the specific isobaric heat capacities in a wide temperature range for the critical binary solution of {ethanol+1-butyl-3-methylimidazolium hexafluorophosphate} have been reported. The critical exponents corresponding to the heat capacity and the coexistence curve were deduced and found to be consistent with theoretical predictions. The complete scaling theory was used to analyze the asymmetric behavior of the diameter of the coexistence curve, which confirmed that the heat capacity related term was of importance. The crossover behavior from the 3D-Ising to the mean field was examined and shown to be monotonic and more pronounced than that of simple fluids. It was also found that the system-dependent critical amplitude A+ for the heat capacity had a comparative large value, and the reduced critical parameters calculated from the restricted primitive model (RPM) were significantly larger than the characteristic RPM values, indicating the nature of solvophobic criticality.

Universal fractality of morphological transitions in stochastic growth processes

Stochastic growth processes give rise to diverse and intricate structures everywhere in nature, often referred to as fractals. In general, these complex structures reflect the non-trivial competition among the interactions that generate them. In particular, the paradigmatic Laplacian-growth model exhibits a characteristic fractal to non-fractal morphological transition as the non-linear effects of its growth dynamics increase. So far, a complete scaling theory for this type of transitions, as well as a general analytical description for their fractal dimensions have been lacking. In this work, we show that despite the enormous variety of shapes, these morphological transitions have clear universal scaling characteristics. Using a statistical approach to fundamental particle-cluster aggregation, we introduce two non-trivial fractal to non-fractal transitions that capture all the main features of fractal growth. By analyzing the respective clusters, in addition to constructing a dynamical model for their fractal dimension, we show that they are well described by a general dimensionality function regardless of their space symmetry-breaking mechanism, including the Laplacian case itself. Moreover, under the appropriate variable transformation this description is universal, i.e., independent of the transition dynamics, the initial cluster configuration, and the embedding Euclidean space.

Thermodynamic behavior of {ethanol + butylpyridinium tetrafluoroborate} binary solution in the critical region

The liquid-liquid coexistence curve, heat capacity, and turbidity of binary solution {ethanol+butylpyridinium tetrafluoroborate]} have been precisely measured. The critical exponents and critical amplitudes corresponding to the heat capacity, width of coexistence curve, osmotic compressibility, and correlation length were obtained. The critical exponents and critical amplitude ratios showed well agreements with the theoretical values of the 3D-Ising universality class. The asymmetric behavior of the coexistence curve diameter was found to be well described by the complete scaling theory with the consideration of the heat capacity related term.

Liquid-liquid phase equilibrium and heat capacity of binary solution {2-propanol + 1-octyl-3-methylimidazolium hexafluorophosphate}

The liquid-liquid coexistence curve and the heat capacity for binary solution of {2-propanol+1-octyl-3-methylimidazolium hexafluorophosphate} have been precisely measured. The values of the critical exponents α and β, characterizing the critical anomalies of the heat capacity and the coexistence curve respectively, were deduced and found to be consistent with theoretical predictions. The asymmetric behavior of the diameter of the coexistence curve was studied in the frame of the complete scaling theory, demonstrating that the heat capacity related term is of importance. Furthermore, the restricted primitive model (RPM) was used to calculate the reduced upper consolute temperature and density, which together with a comparative larger value of A+ indicated a character of solvophobic criticality.

Critical phenomena of {1-butanol + 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide} binary solution

The liquid-liquid equilibrium data and the specific heat capacities for the binary room temperature ionic solution {1-butanol +1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide} have been reported. The corresponding critical exponents β and α were obtained and found to be well coincided with the 3D-Ising criticality. The critical complete scaling theory was applied to successfully describe the asymmetric behavior of the diameter of the coexistence curve with the consideration of the heat capacity related term. The large values of the reduced critical density, the reduced critical temperature and the critical amplitude related to the heat capacity were deduced, which indicated a solvophobic criticality of the studied system.

Asymmetric criticality of ionic solution containing 1-hexyl-3-methylimidazolium tetrafluoroborate and 2-propanol

The liquid–liquid coexistence curve for binary solution {2-propanol+1-hexyl-3-methylimidazolium tetrafluoroborate ([C6mim][BF4])} has been measured. The isobaric heat capacities per unit volume were obtained in both critical and non-critical regions. The critical exponents α and β were deduced and showed good agreement with those predicted for the 3D-Ising universality class. The asymmetric behaviour of the diameter of the coexistence curve was analysed based on the complete scaling theory, which indicated that the heat capacity related term plays an important role in describing the asymmetric behaviour of the coexistence curve. Furthermore, the RPM (restricted primitive model)-rescaled critical parameters were calculated and suggested the solvophobic criticality for this system.

Critical phenomena in {bromobenzene + 1-hexyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide} binary solution

The liquid-liquid coexistence curve for binary solution {bromobenzene + 1-hexyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([C6mim][NTf2])} has been measured in both critical and non-critical regions. The isobaric heat capacities per unit volume were measured in one-phase and two-phase regions. Moreover, the turbidity measurements were also performed in the near-critical one-phase region. From the liquid-liquid equilibrium, heat capacity and turbidity data collected in the critical region, the critical exponents α, β, γ and ν were obtained and shown to be consistent with those predicted for the 3D-Ising universality class. The universal critical amplitude ratios RB and X have been calculated and shown to reasonably coincide with the theoretical predicted values. The asymmetric behavior of the diameter of the coexistence curve was well described by the complete scaling theory, provided that the heat capacity related term was considered. Moreover, the coulombic criticality of the studied system was indicated by the small values of RPM reduced critical density and critical temperature.

(Liquid + liquid) phase equilibria of binary mixtures 1-butyl-3-methylimidazolium tetrafluoroborate and tert-butanol or 1,2-butanediol

Precise (liquid+liquid) coexistence curves of binary mixtures {1-butyl-3-methylimidazolium tetrafluoroborate ([C4mim][BF4])+tert-butanol or RS-1,2-butanediol} have been measured. Critical exponent β and critical amplitude B̂0Z were deduced from the (liquid+liquid) equilibria data in the critical regions, and it was found that the critical exponent β coincided well with the 3D-Ising value. The complete scaling theory was applied to describe the asymmetry behavior of the diameter of the coexistence curves, which suggested that the heat capacity related term showed significant importance. The reduced critical temperature and density were calculated by the restricted primitive model (RPM), which were used to discuss the driven force of phase separation. Moreover, the dependences of the critical temperature, the critical mole fraction, the reduced critical temperature and the reduced critical density on the permittivity of alcohol for a series of binary solutions of RTIL in alcohols are discussed.

Measurements of heat capacities and turbidities for binary mixtures {water + 2,6-dimethylpyridine} and {water, or heavy water + 2,6-dimethylpiperidine} in the critical regions

The isobaric heat capacities per unit volume in both one-phase and two-phase regions and the turbidities in the one-phase region for the significantly asymmetric binary mixtures of {water + 2,6-dimethylpyridine} and {water, or heavy water + 2,6-dimethylpiperidine} were carefully measured, from which the corresponding critical exponents were obtained and found to be consistent with the 3D-Ising values. The values of the system-dependent critical amplitudes for the heat capacity, the correlation length and the osmotic compressibility were deduced, which together with the values of the critical amplitudes related to the coexistence curves reported previously were used to test some universal critical amplitude ratios. Moreover, the importance of the heat capacity related term in the formalism of the complete scaling theory was confirmed. It was also found that the complete scaling formalism possibly becomes somewhat inapplicable for the significantly asymmetric systems.

(Liquid + liquid) phase equilibrium and critical behavior of binary solution {heavy water + 2,6-dimethylpyridine}

The (liquid+liquid) coexistence curves, the isobaric heat capacities per unit volume and the turbidities for the binary solution of {heavy water+2,6-dimethylpyridine} have been precisely measured. The values of the critical exponents were obtained, which confirmed the 3D-Ising universality. It was found that the critical temperature dropped by 5.9K and the critical amplitude of the coexistence curve significantly increased as compared to the binary solution of {water+2,6-dimethylpyridine}. The complete scaling theory was applied to well describe the asymmetric behavior of the diameter of the coexistence curve as the heat capacity contribution was considered. Moreover, the values of the critical amplitudes of the correlation length and the osmotic compressibility were deduced, which together with the critical amplitudes of the coexistence curve and the heat capacity to test universal amplitude ratios.

Critical asymmetry of liquid–liquid coexistence curve of binary mixture nitromethane + 3-pentanol

The liquid–liquid coexistence curves and the isobaric heat capacities per unit volume of the binary solution {nitromethane+3-pentanol} have been measured in a wide temperature range. The values of the critical exponents were derived and found to be in good agreements with the theoretical ones of the 3D-Ising universality. It was found that the asymmetric behavior of the diameter of the coexistence curve was well described by the complete scaling theory with the consideration of the heat capacity related term. Moreover, analysis of the asymmetric behavior of the diameters of coexistence curves showed an unusual large asymmetric coefficient indicating the compressibility of this solution was not negligible, which was significantly different from that of {polar+nonpolar} binary solutions we investigated previously.

Measurements of the liquid–liquid phase equilibria of binary solutions of 1-methyl-3-octylimidazolium tetrafluoroborate with 1-heptanol or 2-pentanol

Liquid–liquid phase equilibria of binary solutions of the room temperature ionic liquid (RTIL) 1-methyl-3-octylimidazolium tetrafluoroborate ([C8mim][BF4]) with 1-heptanol or 2-pentanol were measured. The critical exponents β obtained from the liquid–liquid equilibrium data in the critical region were consistent with the 3D-Ising value, which confirmed the 3D-Ising criticality of the investigated binary ionic solutions. The asymmetry of the diameters of the coexistence curves was analyzed by the complete scaling theory, indicating a significant contribution of the heat capacity related term. The RPM (restricted primitive model)-rescaled critical parameters were calculated, which showed dual characters of both solvophobic and coulombic criticalities of the studied binary ionic solutions.

Liquid–Liquid Phase Equilibrium and Heat Capacity of Binary Mixture 1-Ethyl-3-methylimidazolium Bis(trifluoromethylsulfonyl)imide + 1-Propanol

The measurements of the liquid–liquid coexistence curve and the heat capacity for binary mixture {1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([C2mim][NTf2]) + 1-propanol} have been precisely performed. The values of the critical exponents α and β in the critical region were obtained and coincided well with the 3D-Ising ones. The complete scaling theory was applied to well represent the asymmetric behavior of the diameter of the coexistence curve, indicating an important role of the heat capacity related term in the complete scaling formulizm. A comparison of the reduced critical parameters to those predicted by the restricted primitive model clearly showed the solvophobic criticality of the studied system.