Second-order Response Functions Research Articles

Nonlinear frequency response analysis of oxygen reduction reaction on silver in strong alkaline media

Understanding the kinetics of the oxygen reduction reaction on silver in highly concentrated alkaline solutions (e.g., 11 M NaOH) and elevated temperature is of great interest both in terms of technical applications of this reaction (e.g., in chloralkali electrolysis) and in fundamental terms. To gain a better understanding of this process, a nonlinear frequency response analysis was performed in combination with a rotating ring disk analysis. Firstly, the ORR was analyzed under steady-state conditions. The number of exchanged electrons was determined based on rotating ring disk experiments. Secondly, two different dynamic ORR kinetic models have been formulated. These models were used to derive the theoretical first- and second-order frequency response functions (FRFs). Furthermore, the first and second-order FRFs were determined experimentally for both 0.1 M NaOH and 11 M NaOH at different temperatures. In the next step, the kinetic model parameters were numerically estimated from the experimental frequency-domain data. Despite its simplicity, the so-called 1-step model was able to reproduce experimental first- and second-order FRFs in 0.1 M NaOH under different conditions (steady-state potentials and rotation rates). At the same time, this model was able to reproduce the first-order FRF in 11 M NaOH at 60 °C, but not the features observed in the second-order FRF spectra. The more complex 4-step model has shown a better capability in explaining experimental findings, like the first and second-order FRFs, but also experimentally observed change of the Tafel slope as well as the influence of mass transport on the number of exchanged electrons. However, it was also not able to reproduce all experimental features. Possible reasons for these discrepancies are discussed.

Modeling and Optimization of the Flocculation Process of Multi-Component Coal Waste Slurry Using Hybrid Polymer

ABSTRACT In the present study, the flocculation of multi-component coal waste slurry was investigated using an eco-friendly hybrid polymer naturally derived. The three significant operational parameters of the flocculation process, polymer dosage, suspension pH, and mixing time, were varied and the results were evaluated with the Central Composite Rotatable Design. In order to the flocculation process, mathematical model equations were derived by computer simulation programming applying the least-squares method using Minitab 15. Second-order response functions were produced for residual turbidity (RT), settleable solid speed (SSS), and settleable solid volume (SSV). The predictions were found to be in perfect agreement with the observed values (R2 values of between 0.82 and 0.86). The optimum results were obtained for the RT, SSS, and SSV as 3.59 NTU, 10 mm/min, and 143 ml, respectively, at polymer dosage – 350 ppm, suspension pH – 10, and mixing time – 1.0 min). Additionally, the predicted models have been presented as 3D (three-dimensional) response surface graphs to understand the results.

Gauge invariance and Ward identities in nonlinear response theory

We present a formal analysis of nonlinear response functions in terms of correlation functions in real- and imaginary-time domains. In particular, we show that causal nonlinear response functions, expressed in terms of nested commutators in real time, can be obtained from the analytic continuation of time-ordered response functions, which are more easily amenable to diagrammatic calculation. This generalizes the well-known result of linear response theory. We then use gauge invariance arguments to derive exact relations between second-order response functions in density and current channels. These identities, which are non-perturbative in the strength of inter-particle interactions, allow us to establish exact connections between nonlinear optics calculations done in different electromagnetic gauges.

Application of reflux classifier with closely spaced inclined channels in pre-concentrate process of fine antimony oxide particles

In this work, the reflux classifier with closely spaced inclined channels is used as the pre-concentration facility to improve the separation efficiency before the shaking table separation. Three operating parameters of reflux classifier (RC) to pre-concentrate fine(0.023–0.15 mm) tailings of antimony oxide were optimized by response surface methodology (RSM) using a three-level Box-Behnken design (BBD). The parameters studied for the optimization were feeding speed, underflow, and ascending water speed. Second-order response functions were produced for the Sb grade and recovery rate of the concentrate. Taking advantage of the quadratic programming, when the factors of feeding, underflow and ascending water are respectively 225, 30 and 133 cm3/min, a better result can be achieved for the concentrate grade of 2.31% and recovery rate of 83.17%. At the same time, 70.48% of the tailings with the grade of 0.20% were discarded out of the feeding. The results indicated that the reflux classifier has a good performance in dealing with fine tailings of antimony oxide. Moreover, second-order polynomial equations, ANOVA, and three-dimensional surface plots were developed to evaluate the effects of each parameter on Sb grade and recovery rate of the concentrate.

Upper limit on shift current generation in extended systems

The response functions of a material characterize its behavior under external stimuli, such as electromagnetic radiation. Such responses may grow linearly with the amplitude of the incident radiation, as is the case of absorption, or may be nonlinear. The latter category includes a diverse set of phenomena such as second harmonic generation (SHG), shift current, sum frequency generation, and excited state absorption, among others. Despite decades of research into nonlinear response theory, and the occasional discovery of materials with large nonlinear responses, there has been no systematic investigation into the maximum amount of nonlinear optical response attainable in solid-state materials. In this work, we present an upper bound on the second-order response functions of materials, which controls the SHG and shift current responses. We show that this bound depends on the band gap, band width, and geometrical properties of the material in question. We find that Kuzyk's bound for the maximum SHG of isolated molecules can be exceeded by conjugation or condensation of molecules to form molecular solids, and that strongly coupled systems generally have larger responses than weakly coupled or isolated ones. As a proof of principle, we perform first-principles calculations of the response tensors of a wide variety of materials, finding that the materials in our database do not yet saturate the upper bound. This suggests that new large SHG and shift current materials will likely be discovered by future materials research guided by the factors mentioned in this work.

Applications of higher-order frequency response functions to the detection and damage assessment of general structural systems with breathing cracks

Two types of cracks are often encountered in engineering structural systems and these are the open cracks and the breathing cracks. Existence of open cracks often leads to the loss of physical stiffness, resulting in a mostly linear structure with reduced load bearing capacity and vibration frequencies. The development of breathing cracks not only reduces structural stiffness, but tends to render the otherwise linear structure to become nonlinear, due to their bilinear stiffness characteristics associated with open and closed states. The nonlinear structural vibration responses can then be investigated and potentially employed to detect and identify breathing cracks within a structural system. In the present study, breathing cracks are modeled based on fracture mechanics from which bilinear stiffness values are obtained. These values are then incorporated into finite element models to compute the first- and second-order frequency response functions (FRFs) based on a proposed correlation technique which is both very accurate and resilient against measurement uncertainties. The existence of well-defined second-order FRFs has been firmly established for the bilinear oscillator, as well as a cantilevered beam with breathing cracks. By expressing the bilinear restoring force as a polynomial series, analytical derivation of second-order FRFs of general structural systems such as the GARTEUR AG11 structure with breathing cracks has been established for the first time. Further, a method of identification has been developed to identify the physical parameters of breathing cracks using second-order FRFs. With these new developments that are presented in this paper, a solid foundation has been laid for the potential applications of higher-order FRFs to damage identification and assessment of general real structural systems.

Resonant-convergent second-order nonlinear response functions at the levels of Hartree-Fock and Kohn-Sham density functional theory.

The second-order nonlinear (or cubic) response function is derived from the Ehrenfest theorem with inclusion made of the finite lifetimes of the excited states, representing the extension of the derivation of the quadratic response function in the same framework [P. Norman et al., J. Chem. Phys. 123, 194103 (2005)]. The resulting damped response functions are physically sound and converging also in near-resonance and resonance regions of the spectrum. Being an accurate approximation for small complex frequencies (defined as the sum of an optical frequency and an imaginary damping parameter), the polynomial expansion of the complex cubic response function in terms of the said frequencies is presented and used to validate the program implementation. In terms of approximate state theory, the computationally tractable expressions of the damped cubic response function are derived and implemented at the levels of Hartree-Fock and Kohn-Sham density functional theory. Numerical examples are provided in terms of studies of the intensity-dependent refractive index of para-nitroaniline and the two-photon absorption cross section of neon. For the latter property, a numerical comparison is made against calculations of the square of two-photon matrix elements that are identified from a residue analysis of the resonance-divergent quadratic response function.

Modeling and Optimization of Vertical Pulsating High Gradient Magnetic Separator for Iron ore Slime Processing Using Response Surface Methodology

Due to the increasing demand in the high grade ores for the metallurgical operations and the stringent environmental conditions on the mining activity it is essential utilize the waste tailing pond slimes, recovery of iron values form these tailing ponds not only enhance the life of the existingoperating mines also finds the route to achieve the sustainable process. The present study aims to recover iron values from waste tailing ponds of Donimali area of Karnataka using vertical pulsating high gradient magnetic separator, a three-level Box–Behnken factorial design combined with response surface methodology (RSM) for modelling and optimizing of process parameters of Vertical Pulsating High Gradient Magnetic Separator (VPHGMS), namely Magnetic Intensity, matrix Pulsation and revolution of the Ring (RPM) for the separation of Fe (Hematite) from a deslimed iron ore slimy sample was studied. Second-order response functions were utilized for the grade and recovery of the Fe in the concentrate fraction. With the advantage of the optimization function in the statistical software MINTAB 14, optimized levels of the process variables have been determined to achieve the maximum grade of 65.6%, and recovery was 80.64% with combined desirability of 0.8 of Fe in the concentrate fraction was predicted. The influence of the process variables of the VPHGMS on grade and recovery of the Iron bearing minerals in the Magnetic fraction was presented as 3D response surface graphs.

Measurement of second-order response without perturbation

We study the second-order response functions of a colloidal particle being subjected to an anharmonic potential. Contrary to typical response measurements which require an external perturbation, here we experimentally demonstrate that the system's susceptibilities up to second order can be obtained from the particle's equilibrium fluctuations. The measured susceptibilities are in quantitative agreement with those obtained from the response to an external perturbation.

Purification with Falcon gravity concentrator processing of low-grade bentonites and modeling

ABSTRACTAs advances in technology have led to increased use of bentonites, higher quality bentonite has been sought. The volume of high-quality bentonites available is shrinking and use of bentonite reserves containing impurities is inevitable. The aim of this study was to apply central composite rotatable design (CCRD) to model and optimize some operational parameters of a Falcon gravity concentrator (FGC) to produce bentonite concentrate. The three significant operational parameters of FGC are gravity force, solid concentration, and water pressure, and these parameters were varied and the results evaluated using the CCRD. Second-order response functions were produced for the cation exchange capacity (CEC), swelling, smectite content, and yield of smectite in the bentonite concentrates. Predicted values were calculated to be in good agreement with the experimental values (R2 values 0.897, 0.980, 0.948, and 0.904 for CEC, swelling, smectite, and yield of smectite of bentonite concentrations, respectively). Although in natural states this bentonite is not suitable for industrial use, purification enhanced its CEC, swelling properties, and smectite content to values of 92 meq/100 g CEC, 32 mls/2 g swelling, and 97% smectite, respectively.

Simulating two-dimensional infrared-Raman and Raman spectroscopies for intermolecular and intramolecular modes of liquid water.

Full classical molecular dynamics (MD) simulations of two-dimensional (2D) infrared-Raman and 2D Raman spectroscopies of liquid water were carried out to elucidate a mode-mode coupling mechanism using a polarizable water model for intermolecular and intramolecular vibrational spectroscopy (POLI2VS). This model is capable of describing both infrared and Raman spectra. Second-order response functions, which consist of one molecular polarizability and two molecular dipole moments for 2D IR-Raman and three molecular polarizabilities for 2D Raman spectroscopies, were calculated using an equilibrium-non-equilibrium hybrid MD approach. The obtained signals were analyzed using a multi-mode Brownian oscillator (BO) model with nonlinear system-bath interactions representing the intramolecular OH stretching, intramolecular HOH bending, hydrogen bonded (HB)-intermolecular librational motion and HB-intermolecular vibrational (translational) motion of liquid water. This model was applied through use of hierarchal Fokker-Planck equations. The qualitative features of the peak profiles in the 2D spectra obtained from the MD simulations are accurately reproduced with the BO model. This indicates that this model captures the essential features of the intermolecular and intramolecular motion. We elucidate the mechanisms governing the 2D signal profiles involving anharmonic mode-mode coupling, the nonlinearities of the polarizability and dipole moment, and the vibrational dephasing processes of liquid water even in the case that the 2D spectral peaks obtained from the MD simulation overlap or are unclear. The mode coupling peaks caused by electrical anharmonic coupling (EAHC) and mechanical anharmonic coupling (MAHC) are observed in all of the 2D spectra. We find that the strength of the MAHC between the OH-stretching and HB-intermolecular vibrational modes is comparable to that between the OH-stretching and HOH bending modes. Moreover, we find that this OH-stretching and HB-intermolecular vibrational coupling should be observed as off-diagonal cross peaks in the 2D spectra.

Optimisation Of Process Parameters In High Energy Mixing As A Method Of Cohesive Powder Flowability Improvement

Abstract Flowability of fine, highly cohesive calcium carbonate powder was improved using high energy mixing (dry coating) method consisting in coating of CaCO3 particles with a small amount of Aerosil nanoparticles in a planetary ball mill. As measures of flowability the angle of repose and compressibility index were used. As process variables the mixing speed, mixing time, and the amount of Aerosil and amount of isopropanol were chosen. To obtain optimal values of the process variables, a Response Surface Methodology (RSM) based on Central Composite Rotatable Design (CCRD) was applied. To match the RSM requirements it was necessary to perform a total of 31 experimental tests needed to complete mathematical model equations. The equations that are second-order response functions representing the angle of repose and compressibility index were expressed as functions of all the process variables. Predicted values of the responses were found to be in a good agreement with experimental values. The models were presented as 3-D response surface plots from which the optimal values of the process variables could be correctly assigned. The proposed, mechanochemical method of powder treatment coupled with response surface methodology is a new, effective approach to flowability of cohesive powder improvement and powder processing optimisation.

Real-time and imaginary-time quantum hierarchal Fokker-Planck equations.

We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.

Exploring the effects of pulsed electric field processing parameters on polyacetylene extraction from carrot slices.

The effects of various pulsed electric field (PEF) parameters on the extraction of polyacetylenes from carrot slices were investigated. Optimised conditions with regard to electric field strength (1–4 kV/cm), number of pulses (100–1500), pulse frequency (10–200 Hz) and pulse width (10–30 μs) were identified using response surface methodology (RSM) to maximise the extraction of falcarinol (FaOH), falcarindiol (FaDOH) and falcarindiol-3-acetate (FaDOAc) from carrot slices. Data obtained from RSM and experiments fitted significantly (p < 0.0001) the proposed second-order response functions with high regression coefficients (R2) ranging from 0.82 to 0.75. Maximal FaOH (188%), FaDOH (164.9%) and FaDOAc (166.8%) levels relative to untreated samples were obtained from carrot slices after applying PEF treatments at 4 kV/cm with 100 number of pulses of 10 μs at 10 Hz. The predicted values from the developed quadratic polynomial equation were in close agreement with the actual experimental values with low average mean deviations (E%) ranging from 0.68% to 3.58%.

Modeling and Optimizations of Phosphate Removal from Aqueous Solutions Using Synthetic Zeolite Na-A

Synthetic zeolite Na-A was prepared from Egyptian kaolinite by hydrothermal treatment to be used as an adsorbent for removal of phosphate from aqueous solutions. The present work deals with the application of response surface methodology (RSM) and central composite rotatable design (CCRD) for modeling and optimization of the effect of four operating variables on the removal of phosphate from aqueous solution using zeolite Na-A. The parameters were contact time (0.5 - 6 h), phosphate anion concentrations (10 - 30 mg/L), adsorbent dosage (0.05 - 0.1 g), and solution pH (2 - 7). A total of 26 tests were conducted using the synthetic zeolite Na-A according to the conditions predicted by the statistical design. In order to optimize removal of phosphate by synthetic zeolite Na-A, mathematical equations of quadratic polynomial model were derived from Design Expert Software (Version 6.0.5). Such equations are second-order response functions which represent the amount of phosphate adsorbed (mg/g) and the removal efficiency (%) and are expressed as functions of the selected operating parameters. Predicted values were found to be in good agreement and correlation with experimental results (R2 values of 0.918 and 0.905 for amount of phosphate adsorbed and removal efficiency of it, respectively). To understand the effect of the four variables for optimal removal of phosphate using zeolite Na-A, the models were presented as cube and 3-D response surface graphs. RSM and CCRD could efficiently be applied for the modeling of removing of phosphate from aqueous solution using zeolite Na-A and it is efficient way for obtaining information in a short time and with the fewer number of experiments.

Use of Almond Shell Powder in Modification of the Physical and Mechanical Properties of Medium Density Fiberboard

This study evaluated the effects of almond shell powder content as an extender, as well as the effects of varying paraffin contents, temperature, and press time on the properties of medium-density fiberboard. Response surface methodology (RSM) based on a five-level, four-variable central composite rotatable design was applied to evaluate the effects of independent variables on the modulus of rupture (MOR), internal bonding (IB), and thickness swelling (TS) of medium-density fiberboard. Mathematical model equations were derived from computer simulation programming to optimize the properties of the panels. These equations, which are second-order response functions representing MOR, IB, and TS, were expressed as functions of four operating parameters of panel properties. Predicted values were found to be in agreement with experimental values (R2 values of 0.93, 0.90, and 0.90 for MOR, IB, and TS, respectively). The study showed that RSM can be efficiently applied in modeling fiberboard properties. It was found that almond shell powder maintained the MOR, IB, and TS at desirable levels up to 10.6% as an extender in the resin. Using 1.7% paraffin as a sizing agent, a press temperature of 158 °C, and a duration of 6.43 min had the highest impact on the improvement of the studied properties of medium-density fiberboard.

Separation analysis of dry high intensity induced roll magnetic separator for concentration of hematite fines

In this study, a three level Box–Behnken factorial design combined with response surface methodology was used for modeling and optimization of process parameters of induced roll dry high intensity magnetic separator. The influence of magnetic field intensity, roll speed and feed rate was analyzed for the separation of hematite minerals. Second-order response functions were produced for the grade and recovery of the hematite minerals in the magnetic fraction. Taking advantage of the quadratic programming, optimized levels of the process variables have been determined as optimum levels to achieve the maximum grade of 51.2% Fe, and maximum level of recovery was 97.8% Fe whereas minimum grade of 23.9% SiO2 in the magnetic product was predicted. Further optimization was carried out by targeting the Fe (%) content and recovery of the magnetic product of above 50%. The influence of the process variables of the induced roll high intensity magnetic separator on grade and recovery of the hematite mineral in the magnetic fraction were presented as 3D response surface graphs.

The lamination influence on properties of agro-based particleboard

Response surface methodology (RSM) based on a three-level, three-variable central composite rotatable design was applied to evaluate the effects of the parameters such as ratio of rice straw (Oryza sativa)/poplar (Populus deltoids) wood particle bonded with urea-formaldehyde resin in panels, species of wood veneer coating these panels and amount of adhesive in glue line on the modulus of rupture (MOR), internal bonding (IB), and thickness swelling (TS) of panels. Mathematical model equations were derived by computer simulation programming to optimize the properties of the particleboard. These equations that are second-order response functions representing MOR, IB, and TS were expressed as functions of three operating parameters of panel properties. Predicted values were found to be in a good agreement with experimental values (R 2 values of 0.96, 0.98 and 0.98 for MOR, IB, and TS, respectively). This study has shown that the RSM could efficiently be applied for modeling panel properties. It was found that the variables affected the properties of panels. Straw usage up to 30% in the mixture did not cause a significant decrease in MOR, IB, and TS. Using beech veneer and 190 g/m2 glue line had the highest MOR and lowest TS.

Cement-bonded particleboard with a mixture of wheat straw and poplar wood

We investigated the hydration behavior and some physical/mechanical properties of cement-bonded particleboard (CBPB) containing particles of wheat straw and poplar wood at various usage ratios and bonded with Portland cement mixed with different levels of inorganic additives. We determined the setting time and compression strength of cement pastes containing different additives and particles, and studied the effects of these additives and particles on thickness swelling, internal bond strength and modulus of rupture of CBPB by using RSM (Response Surface Methodology). The mathematical model equations (second-order response functions) were derived to optimize properties of CBPB by computer simulation programming. Predicted values were in agreement with experimental values (R2 values of 0.93, 0.96 and 0.96 for TS, IB and MOR, respectively). RSM can be efficiently applied to model panel properties. The variables can affect the properties of panels. The cement composites with bending strength > 12.5 MPa and internal bond strength > 0.28 MPa can be made by using wheat straw as a reinforcing material. Straw particle usage up to 11.5% in the mixture satisfies the minimum requirements of International Standard, EN 312 (2003) for IB and MOR. The dose of 4.95% calcium chloride, by weight of cement, can improve mechanical properties of the panels at the minimum requirement of EN 312. By increasing straw content from 0 to 30%, TS was reduced by increasing straw particle usage up to 1.5% and with 5.54% calcium chloride in the mixture, TS satisfied the EN 312 standard.

Modeling and optimization of spiral concentrator for separation of ultrafine chromite

In this study, a three-level Box–Behnken factorial design combined with Response Surface Methodology (RSM) for modeling of process parameters of spiral concentrator for the separation of ultrafine chromite has been developed. The three significant operational parameters of spiral concentrator, such as feed rate (m3/h.), feed pulp density (% solids by wt.), splitter position (radial distance from the central axis in cm) were considered for the investigation. The obtained results were evaluated with the Box–Behnken factorial design, RSM and also Quadratic Programming (QP). Second-order response functions were produced for Cr2O3 in terms of grade (%) and recovery (%) of the concentrate fraction. Taking advantage of the QP, it was observed that maximum grade of 51.05% Cr2O3 achieved at the feed rate of 1.2m3/h, feed pulp density of 14.81% solids by wt. and splitter position of 12cm. Similarly, maximum Cr2O3 recovery of 46.73% can be achieved at the same feed rate, feed pulp density of 10% solids by wt. and 16cm of splitter position. Predicted values of responses obtained using model equations were in good agreement with the experimental values. The influence of the process variables of the spiral concentrator on grade and recovery of concentrate fraction was presented and discussed.