Continuous Phase Research Articles

A new highly accurate and efficient pure visco‐acoustic wave equation for tilted transversely isotropic attenuating media

AbstractThe propagation of seismic waves in attenuating anisotropic media exhibits amplitude dissipation and phase dispersion. To describe its effects, the fractional Laplacian pure visco‐acoustic wave equations capable of producing stable and noise‐free wavefields have been derived. However, except for acoustic approximation, previous wave equations utilize the approximations with lower accuracy in simplifying the denominator of the approximate complex‐valued dispersion relation, resulting in reduced accuracy. To address this concern, we use a combination of complex stiffness coefficients to replace the denominator term of the approximate complex‐valued dispersion relation. This approximation effectively reduces the loss of accuracy caused by ignoring the influence of the velocity anisotropy parameter ε and the attenuation anisotropy parameter εQ in the denominator term, leading to a wave equation with high accuracy in media with large anisotropic parameters ε and δ. In addition, the new wave equation only contains two high‐order spatial partial derivatives and has high computational efficiency. Theoretical analysis and numerical examples demonstrate that the proposed pure visco‐acoustic tilted transversely isotropic wave equation outperforms the previous pure visco‐acoustic wave equation in terms of simulation accuracy. The newly developed wave equation is well suited for the application of Q‐compensated reverse time migration and full waveform inversion in attenuating anisotropic media.

Coercivity enhancement and microstructure optimization of Nd-Fe-Co-B magnets by Dy70Al10Cu20 grain boundary diffusion

The effect of grain boundary diffusion on the magnetic properties and microstructure of Nd-Fe-Co-B sintered magnets is investigated using Dy70Al10Cu20 alloy as the diffusion source. After the secondary annealing of the Nd-Fe-Co-B sintered magnet (Hcj = 7.94 kOe), its coercivity is substantially increased to 13.10 kOe. Following post-diffusion annealing, the coercivity of the sintered magnet significantly increased to 20.86 kOe. Microstructure analysis revealed that after the second annealing, the main phase grains in the magnet were connected but lacked thin intergranular phases. The distribution of Co was uneven, with an abundance of soft magnetic phases rich in Co, which was not conducive to improving coercivity. After diffusion, Dy formed a shell structure with high magnetic crystal anisotropy on the outer side of the main phase grains, leading to a significant increase in coercivity. Al and Cu entered the interior of the magnet, reducing the melting point of the grain boundary phase and promoting the uniform distribution of Co through the flow of the liquid phase. A large amount of thin continuous intergranular phase is generated inside the magnet, which helps reduce the exchange coupling between hard magnetic grains. The experimental results indicate that the Dy-Al-Cu alloy can effectively enhance the grain boundary structure of Nd-Fe-Co-B magnets, reduce the abundance of Co-rich phases, and improve the coercivity of the magnet.

A Numerical Study on Dust Control: Evaluating the Impact of Spray Angle and Airflow Speed in the Coalescence of Droplets and Dust

Spray dust reduction is one of the most economical and effective technologies for controlling coal dust in coal mining faces. We aimed to reproduce a spray dust reduction process in a simulation and investigate the mechanism by which the spray angle and airflow speed influence the dust reduction effect. Based on the DPM (discrete phase model) and the mixture model, we constructed a spray dust reduction evaluation model by considering two-way momentum coupling between the discrete phase and the continuous phase. The results showed that installing nozzles near the dust source (coal mining drum) significantly reduced the dust concentration at the coal mining face from 0.0005 kg/m3 to 0.0001 kg/m3. The increase in airflow speed and spray angle enhanced the horizontal transportation of droplets and dust, providing opportunities for the droplets to condense the dust; however, if the droplets have too large an angle, this will result in an insufficient concentration of droplets in the vicinity of the dust source. When the spray angle is 45°, increasing the airflow speed provides a better dust reduction effect. The nozzle position should also be set scientifically according to the airflow speed. Based on simulation results, a mathematical calculation model of spray dust reduction efficiency was constructed. These results can guide the key parameters of spray dust reduction systems, such as the installation position of the nozzle, the spray angle, and the airflow speed. This paper provides ideas for simulating spray dust reduction for other dust types.

Asynchronous verifiable information dispersal protocol of achieving optimal communication

The evolution of distributed systems necessitates robust protocols to facilitate secure and efficient data storage and retrieval amid challenges like asynchronous communication and adversarial threats. However, achieving the theoretical lower bounds of communication cost in Asynchronous Verifiable Information Dispersal (AVID) protocols remains an unresolved issue, prompting the need for innovation in this domain.In this paper, we propose a new AVID protocol designed to optimize communication cost. In contrast to previous approaches, we leverage vector commitment (VC) technology combined with erasure codes, replacing conventional hash tree structures or error correction codes. This approach results in significant O(logn) communication savings. Additionally, our research reveals that while ensuring simultaneous output from all honest nodes is desirable to satisfy Totality, it is not necessary for all nodes to receive corresponding shares or proofs during the Dispersal phase. This insight allows us to circumvent the common O(n2κ) communication overhead.Our AVID protocol achieves an optimal dispersal cost of O(|M|+n2) compared to the prior best of O(|M|+κn2). Furthermore, each node in our protocol incurs a storage cost of O(|M|/n+κ) bits, while the retrieval cost per client is O(|M|+nκ).

Ecological risk assessment of organochlorine pesticide residues in sediment samples from Lake Tana and Hayqe in Northwest Ethiopia

In most developing countries, including Ethiopia, the excessive use of toxic and persistent pesticides for agricultural and malaria control purposes poses a significant challenge. This study focuses on evaluating the concentration and ecological risks associated with organochlorine pesticides (OCPs) in the sediments of Lake Tana and Hayqe. Employing a laboratory-based cross-sectional approach, the residues of OCPs in sediment samples were extracted using the Quick, Easy, Cheap, Effective, Rugged, and Safe (QuEChERS) method, followed by dispersive solid phase extraction cleanup (d-SPE) techniques and analyzed for 18 OCPs by gas chromatography coupled to tandem mass spectrometry (GC-MS). The risk Quotient (RQ) method was adopted for ecological risk assessments by using fish, daphnia, and algae as indicators. The concentration of OCPs ranged from 10.11 to 125.65 μg/kg and 10.96–45.17 μg/kg in Lake Tana and Hayqe, respectively. There was a statistically significant difference in the mean concentration of OCPs across the sampling sites in Lake Tana. Deltamethrin and lindane exhibit comparatively lower concentrations among the OCPs, whereas endrin, DDT, and their metabolites were the predominant pesticides in the study area, exceeding WHO and USEPA residue limits, except for endosulfan. Risk Quotient values ranged from 0.006 to 865.4 in Lake Tana and 0.005 to 729.9 in Lake Hayqe, indicating potential low to high ecological risks for aquatic organisms. High ecological risks are particularly evident in fish, daphnia, and algae species exposed to α and β-Endosulfan, DDT, and their metabolites, while specifically in daphnia, endrin poses high ecological risks. The ANOVA test revealed there were statistically significant differences (p-value <0.05) among sampling sites in Lake Tana and the mean values between Lake Tana and Lake Hayqe, underscoring the importance of addressing pesticide-related ecological concerns in the region.

The contribution of auditory imagery and visual rhythm perception to sensorimotor synchronization with external and imagined rhythm.

Sensorimotor synchronization (SMS) refers to the temporal coordination of an external stimulus with movement. Our previous work revealed that while SMS with visual flashing patterns was less consistent than with auditory or tactile patterns, it was still evident in a sample of nonmusicians. Although previous studies have speculated the potential role of auditory imagery, its contribution to visual SMS performance is not well quantified. Utilizing a synchronization-continuation finger-tapping task with a visual stimulus that included implied motion, we aimed to examine how participants' imagery ability, musicality, and rhythm perception affected SMS performance. We quantified participants' SMS consistency in synchronization (with visual cues) and continuation (without visual cues) phases. Participants also performed a perception task assessing their ability to detect temporal perturbations in the visual rhythm and completed musical ability and imagery questionnaires. Our linear regression model for SMS consistency included the trial phase, self-reported auditory imagery control and musicality, and visual rhythm perception as predictors. Significant effects of trial phase and auditory imagery scores on SMS consistency suggested that participants performed SMS more consistently while the guiding visual stimulus was present and that the higher one's self-reported auditory imagery ability, the better their SMS when continuing with unguided rhythm. One's visual rhythm perception accuracy significantly correlated with SMS consistency during the synchronization phase, and there was no correlation between rhythm perception and auditory imagery control. Overall, our results suggested relatively independent contributions of auditory imagery and visual rhythm perception to SMS with visual rhythm. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Synthesis of ZIF-8/PVA microspheres with the assistance of a microfluidic device and their controlled drug release properties

ZIF-8/PVA microspheres with an even size were synthesized with the assistance of a simple microfluidic device. The ZIF-8/PVA microdroplets were first generated in a simple co-flow microfluidic device using a PVA aqueous solution dispersed with ZIF-8 nanoparticles as the dispersed phase and fatty acid methyl ester (FAME) as the continuous phase. Subsequently, the ZIF-8/PVA microdroplets were further extracted to form ZIF-8/PVA microspheres. The influence of the type of continuous phase on the sphericity of microspheres was investigated and the effect of ZIF-8 content was studied. Further studies showed that decreasing the extraction rate and using additives of NaCl in the dispersed phase could improve the sphericity of the ZIF-8/PVA microspheres. Finally, the synthesized ZIF-8/PVA microspheres were used to study the loading and release of tetracycline. The results showed that the loading of ZIF-8/PVA microspheres was more than 5 times higher than that of pure PVA microspheres. The drug release time of ZIF-8/PVA microspheres was much longer than that of the PVA microspheres and the release rate was significantly affected by the pH of the environment.

Drag reduction assisted emulsification in a micro-channel

An experimental study of the turbulent dynamics of emulsification in a cross-slot microfluidic device is presented. The continuous phase contains a minute amount of an inelastic polymer (xanthan). The Reynolds numbers are sufficiently large (up to 16000) so the drag reduction phenomenon is observed during the emulsification process. The statistics of droplet sizes in the resulting emulsions are measured ex-situ by means of digital microscopy in a wide range of Reynolds numbers and polymer concentrations in the continuous phase. Integral measurements of the statistics of the pressure drops in the micro-channel allow one to systematically map the drag reduction states. Corresponding to each state, the space–time dynamics of the emulsification process are assessed by means of in-situ high speed imaging of the interface between the two fluids which further allows one to extract the characteristic time and space scales associated to the dynamics of the interface. Various dynamic regimes of the microscopic emulsification process are mapped in terms of the Reynolds number and shear thinning rheology of the continuous phase.

Application of magnetic nanoparticles modified with poly(8-hydroxyquinoline) as a nanosorbent for magnetic dispersive micro-solid phase extraction of copper in vegetable, water, and soil samples

In this study, a novel nanosorbent, namely F3O4@SiO2 modified with poly(8-hydroxyquinoline) (Fe3O4@SiO2@PHQ), was synthesized and employed for magnetic dispersive micro-solid phase extraction of Cu(II) followed by flame atomic absorption spectrometry detection in water, vegetables, and soil samples. This nanosorbent was characterized using FT-IR, XRD, VSM, FE-SEM, and EDS. To attain the best extraction recovery, effective parameters were studied and optimized. Under the optimal experimental conditions, the extraction recovery was 92%. The limits of detection and quantification were determined to be 0.64ngmL-1 and 2.13ngmL-1, respectively. The preconcentration factor was 92, and relative standard deviations for intra-day and inter-day measurements were 1.6%, and 2.5%, respectively. The linearity ranged from 2 to 200ngmL−1 with an R2 value greater than 0.998 and relative recoveries in the range of 90% to 96% in real sample analysis.

Preparation of a hydrogel sorbent based on agar-polyethylene glycol stabilized with iminodiacetic acid using electron irradiation method and its use for dispersive micro-solid phase extraction of Mn(II)

An agar/polyethylene glycol hydrogel was synthesized and modified with iminodiacetic acid (IDA-agar/PEG), and its ability as a biocompatible sorbent for extraction of manganese (II) ions from different environmental and food samples was investigated. The prepared IDA-agar/PEG hydrogel was characterized by scanning electron microscopy (SEM), Fourier transform infrared (FTIR) spectroscopy, energy dispersive X-ray (EDX) spectroscopy, thermogravimetric analysis (TGA), and X-ray diffraction (XRD). The crucial parameters of manganese (II) extraction before its determination by flow injection flame atomic absorption spectrometry were optimized. At optimized conditions, a detection limit of 0.12 μg L−1, a limit of quantification of 0.41 μg L−1, and the relative standard deviations of 3.5 % (intra-day) and 5.5 % (inter-day) for six replicate quantifications of Mn (5.0 μg L−1) were obtained. The linearity range (LR) and preconcentration factor (PF) were determined as 0.4–50.0 µg/L and 187.5, respectively. The IDA-agar/PEG hydrogel sorbent was successfully employed for the dispersive micro-solid phase extraction (DMSPE) of Mn(II) in waters, soil, vegetables, and nuts with good recoveries (97.2–101.0 %).

Role of elasticity on polymeric droplet generation and morphology in microfluidic cross-junctions

Recently, our direct numerical simulations [Duan et al., Phys. Fluids 36, 033112 (2024)] showed that fluid elasticity affects the extension length and pinch-off time of the droplet formation process, thus changing the flow pattern. However, the effect of fluid elasticity on the morphology and properties of polymeric droplets is not yet fully understood. In this work, by analyzing the stretched state of the polymer macromolecule and the velocity distribution of the flow process, we find that the increase in fluid elasticity (characterized by the relaxation time) inhibits the contraction of the dispersed phase during droplet pinching and resists the effect of surface tension after droplet generation, which significantly affects the droplet geometry, volume, and generation frequency. The results demonstrate that the length and volume of polymeric droplets increase with the relaxation time of the polymer fluid, while the generation frequency decreases. Meanwhile, the effects of polymer viscosity and the superficial velocity ratio of the continuous to the dispersed phase on the droplets' morphology are investigated. The semi-empirical models for the length, volume, and generation frequency of polymeric droplets are developed for the first time by considering the elastic interaction. The purpose of our work is to provide a better understanding and experimental guidance for controlling the parameters of polymeric droplets with viscoelasticity of different shapes and sizes.

Microdroplet formation of water and alumina nanofluid in a T-junction microchannel

A scarcity of studies about nanofluids’ utilization in droplet formation inside microdevices currently hovers in the literature although potential applications of nanoparticles in a microfluidic environment are foreseen. For this purpose, experimentally assessing both nanofluid and microdroplet characteristics is fundamental. This work reports a series of experimental tests on the microdroplet formation of distilled water (DIW) and DIW-based aluminum oxide (Al2O3) nanofluid in a microfluidic T-junction. While water and nanofluid are used as the dispersed phase, mineral oil is used as the continuous phase. Microdroplet formation in the squeezing, transitional, and dripping regimes is characterized and scaling laws for the non-dimensional droplet volumes are presented. The effects of flow rate, capillary number, microchannel aspect ratio, and nanoparticle concentration are investigated. The addition of Al2O3 nanoparticles to the water is observed to have a major impact in the transitional regime (up to 40% increase), whereas in the dripping regime its influence is lower, with less than 10% difference. This was attributed to the nanofluid's enhanced interfacial tension and viscosity compared to the DIW, as well as possible adsorption at the surface.

Study of droplet formation in parallel flow focusing microchannel under electrostatic field control

To achieve efficient preparation of emulsion droplet, this paper uses a two-dimensional mathematical model coupled with a phase field model and an electrostatic field model to calculate the droplet formation in parallel flow focusing multichannel, and analyzes the effect of electric field strength on the droplet formation process. In addition, the effects of different capillary numbers and dispersed phase viscosity on droplet formation were also analyzed. The results show that the emulsion in the channel always generates droplet by dripping regime, and the droplet have the best mono-dispersity. Applying appropriate electric field strength can realize droplet formation frequency and size regulation. Increasing the dispersed phase viscosity increases the breakup thread of the dispersed phase fluid, and the droplet size obtained gradually decreases, the difference in droplet size can be slowed down by applying the electric field. Increasing the electrostatic force leads to a higher droplet deformation rate. The droplet shape classification is plotted using the capillary number (Ca) and electrostatic number (CaE). It is found that the droplet flow pattern in the channel remains a circle only when Ca and CaE are sufficiently small.

Comparison through a CFD approach of static mixers in an emulsification process

Emulsions, characterized by droplet dispersions of immiscible liquids in a continuous phase, are often found in many diverse fields of the chemical industry. The energy needed to break these droplets can be sourced from mechanical agitators in stirred tanks, active mixers like pulse-flow mixers, and ultrasonic mixers. However, static mixers serve as a convenient alternative due to their typically lower maintenance costs and increased resistance to failures. Static mixers are available in diverse shapes and dimensions, with two widely used types being the Kenics Static Mixers and the Sulzer Static Mixers. While these two static mixers have been extensively studied in the literature in terms of pressure drop and mixing efficiency, to the best of the authors’ knowledge there is no study comparing the two in emulsification processes. This work aims to compare the performance of these static mixers in the production of emulsions by employing a Computational Fluid Dynamics approach. Results showed that the Sulzer Static Mixers allow to obtain higher values of turbulent energy dissipation, which in turn leads to smaller droplet diameters.

Aromatic poly (amino acids) as an effective low-temperature demulsifier for treating crude oil-in-water emulsions

Amphiphilic aromatic poly (amino acids) polymers were designed as biodegradability demulsifiers with higher aromaticity, stronger polarity, and side chain-like combs. The effects of demulsifier dosage, structural characteristics and emulsion properties such as pH, salinity, and oil content on the demulsification efficiency were investigated. The results show that the poly (L-glutamic-benzyl ester)-block-poly (L-phenylalanine) (PBLG15-b-PPA15) as the demulsifier can remove more than 99.97% of the oil in a 5.0 wt% oil-in-water (O/W) emulsion at room temperature within 2 min. The poly (L-tyrosine)-block-poly (L-phenylalanine) (PTyr15-b-PPA15) with environmental durability demonstrates high effectiveness, universality, and demulsification speed. It achieves a remarkable demulsification efficiency of up to 99.99% for a 20.0 wt% O/W emulsion at room temperature. The demulsification mechanism indicates that demulsifiers have sufficient interfacial activity can quickly migrate to the oil-water interface after being added to the emulsions. Additionally, when demulsifiers are present in a continuous phase in the molecular form, their "teeth" side chains are beneficial for increasing coalescence and flocculation capacities. Furthermore, according to the Density Functional Theory (DFT) calculations, enhancing the intermolecular interactions between demulsifiers and the primary native surfactants that form an oil-water interfacial film is a more efficient approach to reducing demulsification temperature and improving demulsification efficiency and rate.

Constructing high-density dislocations by primary (Nb,Ti)(C,N) to induce massive secondary precipitations in austenitic heat-resistant cast steel

The microstructural evolution and the high-temperature tensile properties of a novel Nb-Ti-N alloyed austenitic heat-resistant cast steel were investigated during the isothermal aging at 900 °C for up to 1000 h. The mechanisms of enhancing strength-ductility synergy and the high-temperature tensile fracture behaviors after the aging process were analyzed in detail through tensile tests at 900 °C. Obtained results show that the high-density dislocation regions and the Cr segregation, formed around the intragranular primary (Nb,Ti)(C,N), provided numerous nucleation sites for submicron-sized secondary M23C6 precipitates. The density of secondary M23C6 significantly increased, while the coarsening rate was notably reduced during the aging process. Simultaneously, the uniform dispersed precipitation of nano-sized secondary (Nb,Ti)C was promoted by massive dislocations, which exhibited high resistance to coarsening during long-term aging. The improvement of high-temperature strength was attributed to the precipitation strengthening effects induced by the submicron-sized M23C6 and nano-sized (Nb,Ti)C particles. Additionally, the enhancement of ductility during aging was mainly due to the more effective stress transmission and the ameliorated interface relationship between primary (Nb,Ti)(C,N) and austenitic matrix, which was facilitated by the abundant precipitation of secondary phases. The current findings indicate that high-density dislocations constructed by the intragranular primary (Nb,Ti)(C,N) induce the extensive dispersion of secondary strengthening phases, which enhanced both ultimate tensile strength of 158 MPa and elongation of 45.3% at 900 °C after isothermal aging for 1000 h.

The regulatory role of cysteine in active phase construction of polyoxometalate-based hydrodesulfurization catalysts: Reduction, chelation, and dispersion

Polyoxometalate-based hydrodesulfurization (HDS) catalysts were modified using cysteine to construct efficient CoMoS active phase. Cysteine, characterized by chelating (–COOH, –NH2) and reducing (−SH) groups, assumed a triple role in catalyst synthesis, including reduction, chelation, and dispersion. Cysteine pre-reduced polyoxometalates to heteropoly blue, significantly increasing the catalyst sulfidation. The increased nucleophilicity of heteropoly blue enhanced their attraction to Co2+ ions, intensifying the Co-Mo bimetallic synergy. The formation of Co-chelate species delayed Co sulfidation, allowing for the concurrent sulfidation of both Co and Mo species and promoting the generation of the CoMoS active phase. Carbonaceous deposits from the high-temperature carbonization of cysteine weakened the metal-support interaction, facilitating the dispersion of the active phase. The S-CoPMo12-Cys/Al2O3 catalyst demonstrated high activity in the HDS reaction of dibenzothiophene, showcasing their potential for advanced HDS reaction.

Ion-irradiation effect on swelling and microstructure of BN particle dispersed liquid phase sintering SiC

The boron nitride (BN) particle dispersed silicon carbide (SiC) materials were fabricated by liquid phase sintering (LPS) with Al2O3 and Y2O3 sintering additives as the reference matrix material for SiC composites. They were irradiated to 0.1/1/3/10 dpa by 5.1 MeV Si2+ ions at 800 °C. Raman spectra indicated that the pure h-BN structure was partially destroyed following irradiation, while yttrium aluminum garnet (YAG) structure at grain boundaries formed from the sintering additives wasn’t destroyed. TEM images and diffraction patterns indicated that the mechanism of microstructure evolution of SiC after irradiation up to high fluence at 800 °C was defect clusters induced by irradiation.

MOF@N–doped carbon dots–based dispersive solid phase extraction combined dispersive liquid–liquid microextraction of several aflatoxins from raw cow’s milk samples prior to HPLC–FLD

A dispersive solid-phase extraction method based on metal–organic framework and nitrogen–doped carbon dot nanocomposite combined with magnetic ionic liquid-based dispersive liquid–liquid microextraction was established for the extraction of aflatoxins from raw milk samples. For this purpose, the proteins in the sample were precipitated using a zinc sulfate solution. After vortexing (for 5 min) and centrifuging, 35 mg of the sorbent were added into the aqueous phase obtained from previous step, and the mixture was vortexed for five minutes. In the following, the sorbent particles were separated and eluted with methanol in the presence of vortex agitation (4 min). After centrifuging, the eluent phase was mixed with magnetic ionic liquid (1- octyl-3-methylimidazolium tetrachloroferrate) and rapidly injected into deionized water to perform the dispersive liquid–liquid microextraction procedure. Finally, the magnetic solvent was separated in the presence of a magnet and injected into a high-performance liquid chromatography-fluorescence detector after diluting with the mobile phase. Referring to the results, the limits of detection and quantification of the studied aflatoxins were in the range of 0.69–1.09 and 2.31–3.63 ng/L, respectively. Extraction recoveries were in the range of 61 to 73 %. At the end, the suggested approach was employed in the analysis of the aflatoxins in the raw cow’s milk samples collected from different regions of East Azarbaijan and AFM1 was found in several samples (34 out of 60 samples). Also, AFB1 was found in three samples.

Magnetic dispersive solid-phase extraction of clarithromycin and azithromycin by Fe3O4@ionic salt g-C3N4 adsorbent and HPLC-UV

In this study, Fe3O4@ionic salt g-C3N4 was synthesized as a new adsorbent for magnetic dispersive solid phase extraction (MDSPE) of two antibiotics (clarithromycin and azithromycin) in biological samples followed by HPLC-UV. The new nanocomposite was characterized by Fourier-transform infrared (FT-IR), field emission-scanning electronic microscope (FE-SEM), transmission electron microscope (TEM), nitrogen adsorption–desorption analysis, X-ray powder diffraction (XRD), and vibrating sample magnetometer (VSM). Box-Behnken design (BBD) and response surface methodology (RSM) were used for the optimization and design of experiments. The method had a suitable linear range (1–300 µg/L), a high coefficient of determination (R2; 0.9988 and 0.9993), and a low detection limit (0.33 and 0.42 µg/L) for clarithromycin and azithromycin, respectively. The applicability of the method was evaluated by estimating the recovery of analytes from urine and blood serum samples. Recovery of more than 90.0 % and RSDs of less than 5.2 % show the acceptable accuracy and precision of the proposed method.