Effect of scaling on class modeling with the simca method

The first step in multivariate analysis is almost always the scaling of the variables. The pattern recognition technique SIMCA provides the possibility of scaling the variables over all the objects of the training set (classical scaling), or only over the objects belonging to the same group (separate scaling). The former method of scaling is the more used. The effect of separate scaling on the classification of objects with SIMCA is investigated for a data set consisting of the percentage distribution of fatty acids in olive oils originating from two neighbouring regions in Italy. It is shown that separate scaling has a beneficial effect on the classification.     

Crosslinking Reaction of Type Aa Involving Intramolecular Cyclization

An alternative way is proposed to approach crosslinking reactions of Type A _a by taking into consideration intramolecular cyclization. The sol fraction for postgelation is investigated to deduce the equilibrium number distribution of n-mer, with the Flory-Stockmayer distribution as a criterion. Furthermore, scaling study leads directly to a generalized scaling law.     

Universal scaling laws of diffusion: application to liquid metals

This work focuses on the universal scaling laws, which relate scaled diffusivity to excess entropy in fluids and their mixtures. The derivation of the new scaling law for diffusivity proposed recently [A. Samanta, Sk. M. Ali, and S. K. Ghosh, Phys. Rev. Lett. 92, 145901 (2004)] is discussed in details highlighting the nature of approximations involved. Also the applicability of the scaling law is extended to a new class of liquids, viz., liquid metals. The results calculated based on the scaling laws are shown to be in very good agreement with the simulation results for liquid Rb and Cs metals along the liquid-vapor coexistence curve corresponding to a wide variation of temperature and density. The new universal scaling law discussed here is superior to the earlier empirically proposed scaling laws and provides a very simple route to calculate a dynamical quantity such as diffusivity from an equilibrium property such as the radial distribution function.     

Coil-globule transition of a single short polymer chain: an exact enumeration study

The authors present an exact enumeration study of short self-avoiding walks in two as well as in three dimensions that addresses the question, "what is the shortest walk for which the existence of all the three scaling regimes--coil, globule, and the theta--could be demonstrated." Even though they could easily demonstrate the coil and the globule phase from free energy considerations, they could demonstrate the existence of a theta temperature only by using a scaling form for the distribution of gyration radius. That even such short walks have a scaling behavior is an unexpected result of this work.     

Statistical theory for hydrogen bonding fluid system of AaDd type (Ⅱ): Properties of hydrogen bonding networks

Making use of the invariant property of the equilibrium size distribution of the hydrogen bonding clusters formed in hydrogen bonding system of AaDd type, the analytical expressions of the free energy in pregel and postgel regimes are obtained. Then the gel free energy and the scaling behavior of the number of hydrogen bonds in gel phase near the critical point are investigated to give the corresponding scaling exponents and scaling law. Meanwhile, some properties of intermolecular and intramolecular hydrogen bonds in the system, sol and gel phases are discussed. As a result, the explicit relationship between the number of intramolecular hydrogen bonds and hydrogen bonding degree is obtained.     

Statistical theory for hydrogen bonding fluid system of A_aD_d type(II):Properties of hydrogen bonding networks

Making use of the invariant property of the equilibrium size distribution of the hydrogen bonding clus- ters formed in hydrogen bonding system of AaDd type,the analytical expressions of the free energy in pregel and postgel regimes are obtained.Then the gel free energy and the scaling behavior of the number of hydrogen bonds in gel phase near the critical point are investigated to give the corre- sponding scaling exponents and scaling law.Meanwhile,some properties of intermolecular and in- tramolecular hydrogen bonds in the system,sol and gel phases are discussed.As a result,the explicit relationship between the number of intramolecular hydrogen bonds and hydrogen bonding degree is obtained.     

Aggregates of polymer-substituted fullerenes

We analyzed the self-organized supermolecular architectures observed in solutions of singly polymer-substituted fullerenes by light-scattering and fluorescence spectroscopy, as well as the surface pattern obtained from spraying the solution by atomic force microscopy. We found that the concentration dependence on aggregate size and fluorescence intensity can be explained quantitatively using a scaling argument, assuming that the aggregates in solution are self-emulsified micelles. Our results indicate that the core of the structure is unreacted fullerenes. Based on our scaling arguments, we predict that there is a critical molecular weight that allows for a narrow distribution of the self-assembled structures in solution.     

ON THE POLYCONDENSATION REACTION WITH MONOMER OF A_aB_b TYPE (Ⅲ)——THE SCALING STUDY

For the polycondensation reaction with monomer of A_aB_b type, the scaling problem asso-ciated with sol-gel transition is investigated to obtain an asymptotic form of the equilibriumnumber fraction distribution of n-mer near the gel point. As a direct result, a generalizedscaling law is obtained.     

布朗扩散系数对胶体聚集粒子簇大小动态标度影响的Monte Carlo模拟研究

通过对胶体扩散控制聚集机理的MonteCarlo模拟,表明粒子簇的布朗扩散系数与其大小和形状有关,粒子簇的大小分布可被标度.在微观或介观的层次上,揭示了表征粒子簇结构的几何形状因子在扩散控制聚集过程中对动态标度和粒子簇分布的影响.     

Unforced translocation of a polymer chain through a nanopore: the solvent effect

The authors have performed the Langevin dynamics simulation to investigate the unforced polymer translocation through a narrow nanopore in an impermeable membrane. The effects of solvent quality controlled by the attraction strength lambda of the Lennard-Jones cosine potential between polymer beads and beads on two sides of the membrane on the translocation processes are extensively examined. For polymer translocation under the same solvent quality on both sides of the membrane, the two-dimensional and three-dimensional simulations confirm the scaling law of tautrans approximately N1+2upsilon for the translocation in the good solvent, where tautrans is the translocation time, N is the chain length, and upsilon is the Flory exponent. For the three-dimensional polymer translocation under different solvent qualities on two sides of the membrane, the translocation efficiency may be notably improved. The scaling law between tautrans and N varies from tautrans approximately N1+2upsilon to tautrans approximately N with the increase of the difference of solvent qualities, and the crossover occurs at the theta temperature point, where a scaling law of tautrans approximately N1.27 is found. The simulation results here also show that the translocation time changes from a wide and asymmetric distribution with a long tail to a narrow and symmetric distribution with the increase of the difference of the solvent qualities.     

Dependence on chain length of NMR relaxation times in mixtures of alkanes

Many naturally occurring fluids, such as crude oils, consist of a very large number of components. It is often of interest to determine the composition of the fluids in situ. Diffusion coefficients and nuclear magnetic resonance (NMR) relaxation times can be measured in situ and depend on the size of the molecules. It has been shown [D. E. Freed et al., Phys. Rev. Lett. 94, 067602 (2005)] that the diffusion coefficient of each component in a mixture of alkanes follows a scaling law in the chain length of that molecule and in the mean chain length of the mixture, and these relations were used to determine the chain length distribution of crude oils from NMR diffusion measurements. In this paper, the behavior of NMR relaxation times in mixtures of chain molecules is addressed. The author explains why one would expect scaling laws for the transverse and longitudinal relaxation times of mixtures of short chain molecules and mixtures of alkanes, in particular. It is shown how the power law dependence on the chain length can be calculated from the scaling laws for the translational diffusion coefficients. The author fits the literature data for NMR relaxation in binary mixtures of alkanes and finds that its dependence on chain length agrees with the theory. Lastly, it is shown how the scaling laws in the chain length and the mean chain length can be used to determine the chain length distribution in crude oils that are high in saturates. A good fit is obtained between the NMR-derived chain length distributions and the ones from gas chromatography.     

Effect of emulsion drop-size distribution upon coalescence in simple shear flow: a population balance study

A population balance is used to examine the effect of the shape of the initial drop-size distribution of an emulsion upon its short and long-time evolution in simple shear flow. Initial distributions that are monodisperse, multidisperse, lognormal, bimodal, multimodal, and step functions are considered. At short times, it is shown that the rate of coalescence decreases by up to 25% for step distributions and up to 75% for lognormal distributions as the width of the distribution increases. Bimodal, multidisperse and multimodal distributions show intermediate decreases in the rate of coalescence, between these two values, with increases in the distribution width. Furthermore, it is found that the initial rate of coalescence is strongly dependent upon the presence of large drops. As the number fraction of large droplets within the distribution increases, the rate of coalescence also increases. At long times, all distributions move toward an asymptotic distribution shape in which the frequency of drops decreases algebraically with drop diameter at small drop diameters, and decreases exponentially with drop diameter at large drop diameters. Though portions of each distribution showed the expected asymptotic scaling behavior at long times, each asymptotic distribution nevertheless retains 'fingerprints' of the respective initial distribution. Overall, the rate of coalescence for a system is bounded by the initial rate, which is a function of the initial distribution shape, and the asymptotic rate, which is dependent upon the long-time scaling behavior. Finally, it is shown that the resolution with which the drop-size distribution of an emulsion is experimentally measured can have a significant effect upon predicted rates of coalescence.     

Critical point estimation of the Lennard-Jones pure fluid and binary mixtures

The apparent critical point of the pure fluid and binary mixtures interacting with the Lennard-Jones potential has been calculated using Monte Carlo histogram reweighting techniques combined with either a fourth order cumulant calculation (Binder parameter) or a mixed-field study. By extrapolating these finite system size results through a finite size scaling analysis we estimate the infinite system size critical point. Excellent agreement is found between all methodologies as well as previous works, both for the pure fluid and the binary mixture studied. The combination of the proposed cumulant method with the use of finite size scaling is found to present advantages with respect to the mixed-field analysis since no matching to the Ising universal distribution is required while maintaining the same statistical efficiency. In addition, the accurate estimation of the finite critical point becomes straightforward while the scaling of density and composition is also possible and allows for the estimation of the line of critical points for a Lennard-Jones mixture.     

Mathematical modeling of the multi-walled carbon nanotube growth by the pyrolysis of a ferrocene solution in benzene

The possibility of scaling up the growth of multi-walled carbon nanotubes (MWCNTs) synthesized by the pyrolysis of aerosol of a ferrocene solution in benzene was experimentally and theoretically studied. The morphology and yields of the reaction products and the influence of the volume of the introduced solution and ferrocene concentration on the duration of the process were investigated. A quantitative mathematical model of the nanotube growth process was proposed, and the possibility of using the model in scaling up the synthesis was analyzed. The experimental and model curves of the distribution of sample weights over the reactor length were compared and showed that the proposed model is correct.     

Statistical theory for hydrogen bonding fluid system of AaDd type (Ⅱ): Properties of hydrogen bonding networks

Making use of the invariant property of the equilibrium size distribution of the hydrogen bonding clusters formed in hydrogen bonding system of A_aD_d type, the analytical expressions of the free energy in pregel and postgel regimes are obtained. Then the gel free energy and the scaling behavior of the number of hydrogen bonds in gel phase near the critical point are investigated to give the corresponding scaling exponents and scaling law. Meanwhile, some properties of intermolecular and intramolecular hydrogen bonds in the system, sol and gel phases are discussed. As a result, the explicit relationship between the number of intramolecular hydrogen bonds and hydrogen bonding degree is obtained. Supported by the National Natural Science Foundation of China (Grant Nos. 20303006 and 20574016), the Natural Science Foundation of Hebei Province (Grant Nos. B2006000959 and B2004000093) and the Natural Science Foundation of Education Committee of Hebei Province (Grant No. 2003101)     

The influence of polymer molecular-weight distributions on pulsed field gradient nuclear magnetic resonance self-diffusion experiments

 The influence of polymer molecular-weight distributions on the outcome of pulsed field gradient (PFG) NMR self-diffusion experiments has been considered. The self-diffusion coefficient, D, of monodisperse poly(ethylene oxide) (PEO) polymers has been determined in order to accurately determine the scaling behavior of D both with molecular weight and concentration. In order to investigate the influence of polydispersity on the PFG NMR signal, a model system consisting of ten reasonably monodisperse PEO polymers was made, and the PFG NMR signal intensities were recorded at a low total concentration. The data were analyzed using both inverse Laplace transformation and nonlinear least-squares fitting to a prescribed distribution function of D. Finally, the molecular-weight distribution was obtained by use of the values of the scaling parameters. We also present some model calculations used to investigate the sensitivity of the degree of polydispersity on the NMR signal decays. Received: 27 May 1999 Accepted in revised form: 19 October 1999     

Random copolymers at a selective interface: saturation effects

Combining scaling arguments and Monte Carlo simulations using the bond fluctuation method we have studied concentration effects for the adsorption of symmetric AB-random copolymers at selective, symmetric interfaces. For the scaling analysis we consider a hierarchy of two length scales given by the excess (adsorption) blobs and by two dimensional thermal blobs in the semidilute surface regime. When both length scales match, a densely packed array of adsorption blobs is formed (saturation). We show that for random copolymer adsorption the interface concentration can be further increased (oversaturation) due to reorganization of excess blobs. Crossing over this threshold results in a qualitative change in the behavior of the adsorption layer which involves a change in the average shape of the adsorbed chains towards a hairpinlike form. We have analyzed the distribution of loops and tails of adsorbed chains in the various concentration regimes as well as the chain order parameter, concentration profiles, and the exchange rate of individual chains. We emphasized the role of saturation scaling which dominates the behavior of static and dynamic quantities at higher surface concentration.     

Polymer chain conformation of copolymers with different monomer size: C NMR spectroscopy and MALS study

Copolymer composition, distribution and molecular size of the comonomer influence the final properties of polymer materials. Such influence can be followed from the effect on the chain conformation in solution determined from the scaling law between radius of gyration and molecular weight.     

Rheology of gelling polymers in the Zimm model

In order to study rheological properties of gelling systems in dilute solution, we investigate the viscosity and the normal stresses in the Zimm model [B. H. Zimm, J. Chem. Phys. 24, 269 (1956)]. for randomly cross-linked monomers. The distribution of cluster topologies and sizes is assumed to be given either by Erdos-Renyi random graphs or three-dimensional bond percolation. Within this model the critical behavior of the viscosity and of the first normal stress coefficient is determined by the power-law scaling of their averages over clusters of a given size n with n. We investigate these scaling relations numerically and conclude that the scaling exponents are independent of the hydrodynamic interaction strength. The numerically determined exponents agree well with experimental data for branched polymers. However, we show that this traditional model of polymer physics is not able to yield a critical divergence at the gel point of the viscosity for a polydisperse dilute solution of gelation clusters. A generally accepted scaling relation for the Zimm exponent of the viscosity is thereby disproved.     

On the time evolution of light scattering intensity from binary liquids quenched near and above the critical point

A method is presented for calculating the time evolution and the angular distribution or the intensity intensity scattered by a system which is quenched rapidly above and near its critical point. If the system is quenched exactly to the critical temperature, the two-point, equal-time correlation function has a universal scaling form, as a function of time and wave- number