Nonadditivity Parameter Research Articles

Phase equilibria in polydisperse nonadditive hard-sphere systems

Colloidal particles naturally exhibit a size polydispersity that can greatly influence their phase behavior in solution. Nonadditive hard-sphere (NAHS) mixtures are simple and well-suited model systems to represent phase transitions in colloid systems. Here, we propose an analytical equation of state (EOS) for NAHS fluid mixtures, which can be straightforwardly applied to polydisperse systems. For positive values of the nonadditivity parameter Delta the model gives accurate predictions of the simulated fluid-fluid coexistence curves and compressibility factors. NPT Monte Carlo simulations of the mixing properties of the NAHS symmetric binary mixture with Delta>0 are reported. It is shown that the enthalpy of mixing is largely positive and overcomes the positive entropy of mixing when the pressure is increased, leading to a fluid-fluid phase transition with a lower critical solution pressure. Phase equilibria in polydisperse systems are predicted with the model by using the density moment formalism [P. Sollich, Adv. Chem. Phys. 116, 265 (2001)]. We present predictions of the cloud and shadow curves for polydisperse NAHS systems composed of monodisperse spheres and polydisperse colloid particles. A fixed nonadditivity parameter Delta > 0 is assumed between the monodisperse and polydisperse spheres, and a Schulz distribution is used to represent the size polydispersity. Polydispersity is found to increase the extent of the immiscibility region. The predicted cloud and shadow curves depend dramatically on the upper cutoff diameter sigmac of the Schulz distribution, and three-phase equilibria can occur for large values of sigmac.

Liquid gallium-lead mixture spinodal, binodal, and excess thermodynamic properties

A statistical mechanical based theory is developed for incorporating the effect of the long-range forces in Ga-Pb alloy. In particular, the simplified random-phase approximation is employed in conjunction with the Grosdidier et al. [Phys. Rev. B 72, 024207 (2005)] model for GaGa and PbPb interactions while a suitable nonadditive pair potential is introduced between unlike atoms. We present analytical expressions for the equation of state and for the concentration fluctuations ${S}_{CC}(0)$, and then the role of the nonadditivity parameter is established by consulting the empirical critical parameters of the spinodal curve. It becomes possible to deduce the behavior of different thermodynamic functions such as Gibbs energy of mixing, excess Gibbs energy, isobaric heat capacity, and ${S}_{CC}(0)$ in the vicinity of the liquid-liquid critical point and under extremely high pressure. The impact of temperature and pressure on segregation, and compound-forming tendencies is also investigated. Moreover, the immiscibility gap of the alloy is calculated and compared with the empirical results. The results suggest that: (i) the nonadditivity of the potential tails plays a dominant role in the determination of the spinodal curve, (ii) ${S}_{CC}(0)$ is a very sensitive function in triggering the chemical short-range order, and (iii) the segregation or phase separation in Ga-Pb alloy is an outcome of the temperature dependence of the energy mismatch parameter. In conclusion, the present equation of state is sufficiently accurate that it provides a binodal curve, an excess Gibbs energy of mixing and isobaric heat capacity in quantitatively good agreement with the empirical results.

Multifractal spectrum of phase space related to generalized thermostatistics

We consider a self-similar phase space with specific fractal dimension d being distributed with spectrum function f(d). Related thermostatistics is shown to be governed by the Tsallis formalism of the non-extensive statistics, where the non-additivity parameter equals to τ̄(q)≡1/τ(q)>1, and the multifractal function τ(q)=qdq−f(dq) is the specific heat determined with multifractal parameter q∈[1,∞]. At that, the equipartition law is shown to take place. Optimization of the multifractal spectrum function f(d) arrives at the relation between the statistical weight and the system complexity. It is shown that the statistical weight exponent τ(q) can be modeled by hyperbolic tangent deformed in accordance with both Tsallis and Kaniadakis exponential functions to describe arbitrary multifractal phase space explicitly. The spectrum function f(d) is proved to increase monotonically from minimum value f=−1 at d=0 to maximum one f=1 at d=1. At the same time, the number of monofractals increases with the growth of the phase-space volume at small dimensions d and falls down in the limit d→1.

Phase behavior of polymer mixtures with nonadditive hard-sphere potential

The phase behavior of a binary mixture of homopolymers in which macromolecules are composed of tangent hard spheres was studied. The interaction of unlike units is characterized by the contact distance (1/2)(σA + σB)(1 + Δ), where σ i is the diameter of the ith sphere (unit) and Δ is the nonadditivity parameter. The effect of nonadditivity was taken into account by means of the perturbation theory relative to the additive system (Δ = 0) considered earlier (Polymer Science, 47, 2146 (2005)) in terms of the Percus-Yevick approximation. The theoretical consideration presented is completely analytical. It was found that a polymer mixture experiences phase separation with an increase in pressure; the two-phase region extends with an increase in both the size ratio between the units α = σA/σB and the length of the chain per se. Closed phase diagrams were first predicted for athermal mixtures; such diagrams appear at Δ < 0 and certain values of α. It was shown that the thermodynamics of an incompressible mixture of hard-chain molecules at α = 1 follows the Flory-Huggins theory with the temperature-independent interaction parameter. Phase separation in polymer solutions with the nonadditive hard-sphere potential was also analyzed.

Phase separation of model adsorbates in random matrices

The liquid-liquid phase behavior of binary mixtures in random pores is investigated with non-additive hard spheres using both ROZ (Replica Ornstein-Zernike) integral equations and cavity biased grand canonical Monte Carlo simulations. The critical densities of the coexistence phase envelopes are determined as function of the non-additivity parameter Delta, varying from Delta = 0.2, 0.4, 0.6, up to 0.8. The matrix is made of quenched hard spheres. Its porosity is varied to ascertain the effects of confinement, with packing densities rho(0) ranging from 0.1, 0.3, to 0.5. To obtain fiduciary results from ROZ, we use the accurate ZSEP closure relation proposed earlier with and without thermodynamic consistency. The ZSEP closure is known to enforce the zero-separation theorems via special adjustable parameters in the bridge function. Two versions of this closure are used to assess their accuracies (vis-à-vis the Monte Carlo data): first ZSEP-T, namely, the ZSEP closure with added thermodynamic consistency (the Gibbs-Duhem relation); and second purely ZSEP without adding thermodynamic consistency. It is found that both closures give correct qualitative trend, with errors of ZSEP falling within 8-9%, while ZSEP-T, being more accurate, to within 1-2%. As non-additivity is increased, both versions become more accurate. The critical density rho(c) is found to decrease with decreasing porosity. In addition, rho(c) also decreases with increasing Delta, in a non-monotone fashion.

Quantitative genetics of speciation: additive and non-additive genetic differentiation between Drosophila madeirensis and Drosophila subobscura

The role of dominance and epistasis in population divergence has been an issue of much debate ever since the neoDarwinian synthesis. One of the best ways to dissect the several genetic components affecting the genetic architecture of populations is line cross analysis. Here we present a study comparing generation means of several life history-traits in two closely related Drosophila species: Drosophila subobscura, D. madeirensis as well as their F1 and F2 hybrids. This study aims to determine the relative contributions of additive and non-additive genetic parameters to the differentiation of life-history traits between these two species. The results indicate that both negative dominance and epistatic effects are very important in the differentiation of most traits. We end with considerations about the relevance of these findings for the understanding of the role of non-additive effects in speciation.

Evaluation of the performance of the Kenya Dual Purpose Goat composites: Additive and non-additive genetic parameters

Growth data from 6800 progeny of straightbreds, intermediate crosses and the resulting Kenya Dual Purpose Goat (KDPG) composites from 1982 to 2004 were used to estimate additive and non-additive genetic parameters. The KDPG was developed by crossing Toggenburg (T), Anglo-Nubian (N), Small East African (E) and Galla (G) breeds in equal proportions. All parameters were simultaneously estimated for early growth traits by fitting an animal model using Derivative-Free-Restricted Maximum Likelihood (DFREML) procedures (Meyer, 1998). The model included sex, type of birth, period and season of birth, and age of dam classes as fixed effects; crossbreeding parameters were fitted as covariates. Relative to Small East African breed, Anglo-Nubian showed the highest breed effects for weaning weight (+1.69 kg), yearling weight (+1.74 kg) and pre-weaning average daily gains (+13.28 g/day). Between the indigenous breeds, breed differences were in favour of Galla breed except for yearling weight. Individual heterosis had a positive effect on birth weight (+0.05 kg), yearling weight (+0.36 kg) and post-weaning average daily gains (+3.04 g/day) but negative in pre-weaning traits. Maternal heterosis was favourable in the traits influenced by pre-weaning maternal environment (weaning weight and pre-weaning average daily gains). The effect of recombination was negative and, therefore, reduced performance in all traits. Heritability estimates were low with very narrow range between traits. These estimates were 0.13 ± 0.03, 0.16 ± 0.01 0.16 ± 0.04, 0.24 ± 0.01 and 0.10 ± 0.02 for birth weight, weaning weight, yearling weight, pre-weaning and post-weaning average daily gains, respectively. The study shows that the developed KDPG composites have not optimized on the positive dominance effects; an effect due to retained recombination loss caused by lack of selection during breed development. It was concluded therefore, that the KDPG composites are still segregating and have not stabilized into a new breed as was the aim of the breeding programme.

Demixing can occur in binary hard-sphere mixtures with negative nonadditivity

A binary fluid mixture of nonadditive hard spheres characterized by a size ratio gamma = sigma(2)/sigma(1) < 1 and a nonadditivity parameter Delta = 2 sigma(12)/(sigma(1) + sigma(2)) - 1 is considered in infinitely many dimensions. From the equation of state in the second virial approximation (which is exact in the limit d--> infinity) a demixing transition with a critical consolute point at a packing fraction scaling as eta approximately d2(-d) is found, even for slightly negative nonadditivity, if Delta >-1/8 (ln gamma)(2). Arguments concerning the stability of the demixing with respect to freezing are provided.

Estimating the Extent of Parameter Bias in the Classical Twin Design: A Comparison of Parameter Estimates From Extended Twin-Family and Classical Twin Designs

AbstractTheclassical twin design(CTD) circumvents parameter indeterminacy by assuming (1) negligible higher-order epistasis; and (2) either nonadditive genetic or common environmental effects are nonexistent, creating two potential sources of bias (Eaves et al., 1978; Grayson, 1989). Because the extended twin-family design (ETFD) uses many more unique covariance observations to estimate parameters, common environmental and nonadditive genetic parameters can be simultaneously estimated. The ETFD thereby corrects for what is likely to be the largest of the two sources of bias in CTD parameter estimates (Keller &amp; Coventry, 2005). In the current paper, we assess the extent of this and other potential sources of bias in the CTD by comparing all published ETFD parameter estimates to CTD parameter estimates derived from the same data. CTD estimates of the common environment were lower than ETFD estimates of the common environment for some phenotypes, but for other phenotypes (e.g., stature in females and certain social attitudes), what appeared as the common environment was resolved to be assortative mating in the ETFD. On average, CTD estimates of nonadditive genetic factors were 43% lower, and additive genetic factors 63% higher, than ETFD estimates. However, broad-sense heritability estimates from the CTD were only 18% higher than ETFD estimates, highlighting that the CTD is useful for estimating broad-sense but not narrow-sense heritability. These results suggest that CTD estimates can be misleading when interpreted literally, but useful, albeit coarse, when interpreted properly.

Estimating the Extent of Parameter Bias in the Classical Twin Design: A Comparison of Parameter Estimates From Extended Twin-Family and Classical Twin Designs

The classical twin design (CTD) circumvents parameter indeterminacy by assuming (1) negligible higher-order epistasis; and (2) either nonadditive genetic or common environmental effects are nonexistent, creating two potential sources of bias (Eaves et al., 1978; Grayson, 1989). Because the extended twin-family design (ETFD) uses many more unique covariance observations to estimate parameters, common environmental and nonadditive genetic parameters can be simultaneously estimated. The ETFD thereby corrects for what is likely to be the largest of the two sources of bias in CTD parameter estimates (Keller & Coventry, 2005). In the current paper, we assess the extent of this and other potential sources of bias in the CTD by comparing all published ETFD parameter estimates to CTD parameter estimates derived from the same data. CTD estimates of the common environment were lower than ETFD estimates of the common environment for some phenotypes, but for other phenotypes (e.g., stature in females and certain social attitudes), what appeared as the common environment was resolved to be assortative mating in the ETFD. On average, CTD estimates of nonadditive genetic factors were 43% lower, and additive genetic factors 63% higher, than ETFD estimates. However, broad-sense heritability estimates from the CTD were only 18% higher than ETFD estimates, highlighting that the CTD is useful for estimating broad-sense but not narrow-sense heritability. These results suggest that CTD estimates can be misleading when interpreted literally, but useful, albeit coarse, when interpreted properly.

Fluid–fluid and fluid–solid phase separation in nonadditive asymmetric binary hard-spheremixtures

Very asymmetric mixtures of hard spheres naturally arise in the modellization of colloidaldispersions. Effective potentials have emerged as a powerful tool for describing thesesystems and have often been employed to extract the phase diagram in both theadditive and nonadditive cases. However, most theoretical investigations havebeen carried out by means of mean-field-like approaches, so their quantitativeaccuracy remains to be assessed. Here we employ previously determined effectivepotentials for nonadditive hard-sphere mixtures to study the fluid–fluid phasetransition by the hierarchical reference theory (HRT), which is designed to takerealistically into account the effects of long-range fluctuations on phase separation.Fluid–solid equilibrium is addressed by supplementing HRT with thermodynamicperturbation theory for the solid phase. We apply this approach both to a potential withadjustable nonadditivity parameter (Louis et al 2000 Phys. Rev. E 61 R1028) and to theAsakura–Oosawa (AO) potential, which represents an extreme case of nonadditivity. Ourresults for the phase diagram, including modified hypernetted chain (MHNC)calculations, are compared to those of other liquid-state theories and are found toagree nicely with available simulation data. Unlike commonly adopted liquid-statetheories, HRT is capable both of getting arbitrarily close to the fluid–fluid criticalpoint, and of giving nontrivial critical exponents. In particular, the fluid–fluidcoexistence curve is much flatter than that obtained via perturbation theory, inagreement with a recent finite-size scaling Monte Carlo analysis of the AO model.

Estimation of Melting Points of Organic Compounds

A combination of additive group contributions and nonadditive molecular parameters is employed to estimate the normal melting points of 1215 organic compounds. The melting points are calculated from the ratio of the total phase change enthalpy and entropy of melting. The total phase change enthalpy of melting is calculated from the enthalpic group contributions, whereas the total phase change entropy of melting is estimated using a semiempirical equation based on only two nonadditive molecular parameters. The average absolute error in estimating the melting points of these organic compounds is 33.2 K. This is a relatively low value considering the wide range of pharmaceutically and environmentally relevant organic compounds included in this data set.

Structural and energetic consequences of expanding a highly cooperative stable DNA hairpin loop.

Many hairpin loops are expanded versions of smaller, stable ones. Herein we investigate the extent to which the energetics and structure of d(cGNAg) hairpin loops will tolerate sequence variation. Changing the closing base pair from CG to GC was found to completely eliminate loop-loop interactions; in contrast, expanding the loop at the 3'-end resulted in similar energetics and nonadditivity parameters as the parent loop, suggesting that loop-loop interactions remain intact and highly coupled upon expansion. Together, these data suggest that the CG closing base pair forms an essential platform upon which a stable d(GNA) hairpin loop can fold and that this loop can undergo 3'-expansion with little effect to its structure or energetics.

Replica Ornstein-Zernike self-consistent theory for mixtures in random pores.

We present a self-consistent integral equation theory for a binary liquid in equilibrium with a disordered medium, based on the formalism of the replica Ornstein-Zernike (ROZ) equations. Specifically, we derive direct formulas for the chemical potentials and the zero-separation theorems (the latter provide a connection between the chemical potentials and the fluid cavity distribution functions). Next we solve a modified-Verlet closure to ROZ equations, which has built-in parameters that can be adjusted to satisfy the zero-separation theorems. The degree of thermodynamic consistency of the theory is also kept under control. We model the binary fluid in random pores as a symmetrical binary mixture of nonadditive hard spheres in a disordered hard-sphere matrix and consider two different values of the nonadditivity parameter and of the quenched matrix packing fraction, at different mixture concentrations. We compare the theoretical structural properties as obtained through the present approach with Percus-Yevick and Martinov-Sarkisov integral equation theories, and assess both structural and thermodynamic properties by performing canonical standard and biased grand canonical Monte Carlo simulations. Our theory appears superior to the other integral equation schemes here examined and provides reliable estimates of the chemical potentials. This feature should be useful in studying the fluid phase behavior of model adsorbates in random pores in general.

The one-dimensional fully non-additive binary hard rod mixture: exact thermophysical properties

The non-additive binary hard particle mixture is characterized by a collision parameter between unlike hard particles that is not the arithmetic mean of the two pure component collision diameters. Instead, σ12=(σ11+σ22)(1+Δ)/2, where σ12 is the collision diameter between species 1 and 2 and Δ is a non-additivity parameter in which Δ≥−1 (Δ=0 corresponds to the additive mixture). For mixtures in which repulsive forces are mostly responsible for the observed behavior, the binary non-additive hard particle mixture provides a useful model for examining the thermophysical properties of a wide class of real fluid mixtures. To extend further the range of applicability of these mixtures, another layer of complexity can be considered by including potential interactions that exist past contact which themselves may also be defined as non-additive. For example, the well depth of the interaction of unlike species can be defined as ε12=δ∣ε11ε22∣1/2, where δ is another non-additivity parameter which may take on any real value (note that Δ=0 and δ=1 correspond to the Lorentz–Berthelot mixing rules).To gain insight into these complicated mixtures in higher dimensions, we consider the one-dimensional non-additive binary hard rod mixture with various attractive interactions. We generate the exact thermophysical properties of these mixtures, including a general form of the equation of state. Similarly, we also determine the radial distribution functions, g ij(r), using the exactly known nearest-neighbor probability density distributions. Such relations enable one to determine the effects of the various non-additive parameters on the properties of the mixtures.

Investigation of the Microstructure of Micelles Formed by Hard-Sphere Chains Interacting via Size Nonadditivity by Discontinuous Molecular Dynamics Simulation

Micelle formation by short nonadditive hard surfactant chains was investigated at different size ratios, reduced densities, and nonadditivity parameters using molecular dynamics simulation. It was found that spherical, cylindrical, lamellar, and reverse micelles can form in systems with different head, tail, and solvent characteristics. Hard-core surfactant chains composed of a head segment and three tail segments were simulated in a solvent of hard spheres. The formation of micelles was found to be a strong function of the packing fraction and nonadditivity parameter. Micelles were more stable at higher densities and larger nonadditivity parameters. At lower densities, micelles tended to break into small, dynamic globules.

Influence of monomer sequence on microstructure of nonadditive hard chain copolymers: Simulation and equation of state

Influence of copolymer structure on self-assembly of the copolymer was investigated by introducing nonadditive size interactions among different type segments in copolymers composed of touching hard spheres. The copolymer structures investigated are diblock, alternating, multiblock, and random. The copolymers were simulated using discontinuous molecular dynamics simulation. Equations of state of the different model copolymers were developed using the TPT1 theory. Diblock copolymers were found to self assemble readily at high values of the nonadditivity parameter Δ and at high densities. Multiblock and random copolymers self assembled to a smaller extent and the alternating copolymer self-assembled only at very high Δ and densities. The TPT1 equation of state provided good prediction of the compressibilities of the different copolymer systems at negative and small positive values of Δ. Deviations between the model and simulation results were observed at high Δ and high densities. Deviations were associated with self-assembly of the copolymers.

Critical-point and coexistence curve properties of a symmetric mixture of nonadditive hard spheres: A finite size scaling study

The behavior of binary mixtures of nonadditive hard spheres with positive nonadditivity parameter is investigated. The coexistence curves are calculated with a high accuracy in Monte Carlo simulations. Finite size scaling analysis is used to determine precisely the location of the critical point. The calculations strongly support the hypothesis that mixtures of nonadditive hard spheres belong to the Ising universality class. The dependence of the critical packing fraction on the nonadditivity parameter is investigated.

Stability and interfacial properties of confined nonadditive hard-sphere binary mixture

Monte-Carlo simulations are performed to investigate fluid–fluid phase equilibria and interfacial properties of nonadditive hard-sphere binary mixtures adsorbed in a slit pore. The behavior of critical density with wall separation and a nonadditive parameter is analyzed using mean field theory and simulations. The mixtures exhibit phase separation with the critical density ρ c larger than its bulk counterpart. It is found that ρ c decreases by increasing both the pore width H and the nonadditive parameter. The mean-field-like perturbation theory predicts ρ c − ρ bulk c ∼ H −1; for large H and small ρ c , good agreement with simulation is found. The phase diagram, density profile, and interfacial tension are also obtained using different simulation methods. Interfacial tension, γ, increases with the nonadditive parameter for a given total density and H. The effect of confining is to increase γ for a given nonadditive parameter. A coupling between pore size and nonadditivity is discussed in connection with a possible spatially anisotropic separation.

Is Crossbreeding the Answer to Questions of Dairy Breed Utilization?

The current interest in crossbreeding in the commercial dairy industry, even though it is quite limited, raises questions of breed utilization. Fewer than 5% of US dairy cattle are other than purebred or grade Holsteins. The large advantage of Holsteins for additive genetic merit for lactation milk yield is apparently responsible for this trend. Why, then, this interest in crossbreeding? The economic importance of traits such as reproduction, health, and survival in dairy production systems is likely the basis for the interest in crossbreeding, even though these traits are secondary to milk yield. Several US studies and a Canadian study confirmed that while several crossbred groups were equivalent to Holsteins for lactation milk yield, none were superior. Two crossbred groups in the Canadian study had lifetime yields, milk value, and net returns equivalent to Holsteins. In the New Zealand study, Friesian-Jersey reciprocal crossbreds were predicted to exceed Friesians in first-lactation fat yield. Crossbred performance is dictated by a combination of additive and nonadditive genetic effects. Evidence exists for direct, maternal, heterosis, and cytoplasmic maternal effects. Heterosis of 15 to 20% for lifetime traits was found in two studies. Results from previous crossbreeding studies have something to recommend for inclusion of Holstein, Ayrshire, Brown Swiss, and Jersey breeds in a crossbreeding scheme. However, multiple-generation lifetime performance on an array of purebreds and crossbreds under US condition does not exist. Full unique identification of individual animals, including breed composition, would permit the use of DHIA data to estimate additive and nonadditive genetic parameters for the traits recorded therein. Survival data from birth and health data would need to be fully recorded to provide complete data on lifetime performance. Self-propagation of crossbred replacements is mandatory if any crossbreeding system is to be successful. Based on current empirical data, a two-breed rotational crossing system appears to be the most viable system to maximize economic merit. The theoretical advantages of a three-breed rotational crossing system are clear, but the data to recommend the third breed and this system in practice are limited. Full-scale long-term breeding experiments or analysis of field data paired with a comprehensive modeling of alternative breed utilization strategies for US conditions are recommended.