Pulsed DC Plasma Research Articles

Evolution of hard carbon layers on Ti-45Al-2Nb-2Mn-1B by plasma enhanced chemical vapor deposition of hydrocarbons and hydrogen

Extension of pulsed DC plasma enhanced CVD (PECVD) of carbon from hydrocarbons resulted in thick hard carbon layers on Ti-45Al-2Nb-2Mn-1B. PECVD for less than 2 h led to the formation of surface interlayers of TiC and Ti2AlC compounds under the hard carbon layer. After 5 h until 20 h, the formation of carbons layers surpassed the growth of TiC and Ti2AlC layers and resulted in a thicker hard carbon layer. This can be explained with the deceleration of carbon diffusion into the base alloy that resulted in the accumulation of a hard structure of carbon on the outer surface. The thickness of the hard carbon layers reached up to ∼40 μm, which were revealed using SEM microcopy. The hardness on the outer surface of hard carbon layer was around 600–1500 HV0.5 (5.88–14.71 GPa). EDX analysis across the surface layers showed ∼50–100 at. % carbon on the outermost layers. Raman spectroscopy of carbon layers showed sp3 (D) and sp2 (G) peaks of carbon at ∼1330 and ∼1560 cm−1 and peaks of TiC at ∼200 cm−1 peaks.

Investigation of pulsed direct-current plasma jets in a turbulent boundary layer

Characteristics of the plasma jet produced by a pulsed direct-current (pulsed-DC) dielectric barrier discharge (DBD) and its interaction with a turbulent boundary layer (TBL) are investigated in detail using stereo particle imaging velocimetry. Quiescent-flow characterization results show that a positive starting vortex and a negative near-wall jet structure are induced by the pulsed-DC DBD plasma actuator. With increasing pulse width and discharge frequency, the jet velocity magnitude increases monotonously, as a direct result of the extension of fluid particle acceleration time. During the interaction with a cross-flow TBL, two streamwise vortices with opposite signs are observed at the two sides of the electrode junction, which essentially originate from the starting vortex and negative jet in quiescent air. The skin-friction drag variations are dominated by the cross-stream momentum transportation of streamwise vortices, with drag reduction in the vortex upwash zone and drag increase in the downwash zone. Compared with the conventional alternating-current DBD plasma actuators, the turbulent fluctuations produced by pulsed-DC DBD are much higher, which also affects the skin-friction drag. Further proper orthogonal decomposition (POD) analysis reveals that two distinctly different flow patterns are produced by pulsed-DC DBD working at small and large pulse widths. The dominant POD modes causing the most velocity fluctuation are the spanwise translation and deformation of plasma-induced streamwise vortices. These results provide insights into the basic phenomenon of pulsed-DC plasma jets in cross flow, which recently has demonstrated its promising applications in turbulent skin-friction reduction.

Dielectric exposed on DC-pulsed helium plasma jet: Hollow distribution of the surface electric field

In this study, the electric field distribution on the dielectric surface exposed to helium atmospheric pressure plasma jets with varying operating parameters was investigated. The study considered the effect of applied voltage (7–10 kV), pulse width (0.9–150 μs), and rate (400–2000 sccm), with the investigation based on the Pockels effect. The results suggest that the surface electric field arises from the charge transfer by ionization waves and the expansion of surface discharges. The evolution of the electric field can be segmented into three phases: establishment, maintenance, and dissipation. The gas flow rate determines the mixing “boundary” between helium and air, and consequently affects the shape of the surface electric field distribution. At 400 sccm, the field structure exhibits a solid, elongated shape. At 800–1200 sccm, the field takes on a hollow-ring appearance, and at 1600–2000 sccm, it is presented as two channels. As the applied voltage increases, the surface discharge can be extended further, even surpassing the designated boundary. At 10 kV, the maximum surface electric field and total deposited charge are 5.9 kV/cm and 4.3 nC, respectively. At a fixed frequency of 5 kHz, changing the pulse width affects charging time. The maximum value of the surface electric field and the total deposited charge exhibits an initial increase followed by a decrease. Short pulses (0.9–2 μs) result in incomplete charge transfer, while long pulses (100–150 μs) form a “dark channel” of predominantly negative charge at falling edge of the pulse, which neutralizes the positive charge in the subsequent pulse. A pulse width of 20 μs allows complete charge transfer with less influence from the dark channel.

Investigation of electric field distribution on dielectric exposed to DC-pulsed He plasma jet with shielding gas

Atmospheric pressure plasma jets (APPJs) produce reactive species and electric fields for biomedical applications. Gas shields control plasma plume-surrounding gas interactions, regulating reactive species generation and electric field strength. However, the surface electric field distribution is still unclear and needs urgent attention. Here, the electric field distribution on the surface exposed to a helium APPJ with shielding gas is investigated using the Pockels technique. This study considers the influence of the type of shielding gas (ambient air, dry air, nitrogen, oxygen, nitrogen-oxygen mixture) and the flow rate (2000–6000 sccm). The results show that the surface electric field develops in three phases: establishment, maintenance, and dissipation. Both flow rate and oxygen content of the shielding gas significantly influence surface discharge behavior and the maximum electric field value. The analysis suggests that the establishment phase of the electric field results from charge transfer by ionization waves to the dielectric, while the maintenance of the electric field depends on pulse duration. During the dissipation phase, the positive surface charge attracts negatively charged species to the surface (electrons and negative ions), which causes charge neutralization at the surface. The oxygen content in the shielding gas impacts the electric field establishment phase, with a low oxygen content leading to lower photo-ionization rates and, consequently, surface discharges with branching. Shielding gas flow rates affect the amount of shielding gas mixed into the helium channel. Mixing less oxygen into the APPJ increases the electric field strength, as the ionization potential is lower than nitrogen. Excessive oxygen mixing traps more free electrons due to electronegativity, causing fewer ionized collisions and more negative ions in APPJ, ultimately lowering the electric field strength. This study shows that shielding gas type and flow rates can adjust surface charging, aiding in optimizing biomedical APPJ.

Metrology of Ar-N2/O2 Mixture Atmospheric Pressure Pulsed DC Jet Plasma and its Application in Bio-Decontamination.

Atmospheric pressure plasma jets are gaining a lot of attention due to their widespread applications in the field of bio-decontamination, polymer modification, material processing, deposition of thin film, and nanoparticle fabrication. Herein, we are reporting the disinfection of Pseudomonas aeruginosa, Staphylococcus aureus, and Escherichia coli bacteria using plasma jet. In this regard, Ar-O2, Ar-N2, and Ar-O2-N2 mixture plasma is generated and characterized using optical and electrical characterization. Variation in plasma parameters like electron temperature, electron density, and reactive species production is monitored with discharge parameters such as applied voltage and feed gas concentration. Results show that the peak average power consumed in Ar-O2, Ar-N2, and Ar-O2-N2 mixture plasma is found to be 4.45, 2.93, and 4.35 W respectively, at 8 kV. Moreover, it is noted that by increasing applied voltage, the electron temperature, electron density, and reactive species production also increases. It is worth noting that electron temperature increases with increase in oxygen concentration in the mixture (, while it decreases with increase in nitrogen concentration in the mixture (Ar-N2). Similarly, a decreasing trend in electron temperature is noted for Ar-O2-N2 mixture plasma. On the other hand, a decreasing trend in electron density is noted for all the mixtures. Reduction in viable colonies of Pseudomonas aeruginosa, Staphylococcus Aureus, and Escherichia coli were confirmed by the serial dilution method. The inactivation efficiency of pulsed DC plasma generated, in the Ar-N2 mixture at 8 kV and 6 KHz, was evaluated against P. aeruginosa, S. aureus and E. coli bacteria by measuring the number of surviving cells versus plasma treatment time. Results showed that after 240 s of plasma treatment, the number of survival colonies of the mentioned bacteria was reduced to less than 30 CFU/mL.

DC pulsed plasma magnetron sputtering of CdO/Cu/CdO multilayers thin films for self-cleaning and optoelectronic applications

In this study, CdO/Cu/CdO multilayers thin films were organized on glass substrates with different Cu intermetallic layer thickness engaging DC plasma magnetron sputtering. The optoelectronic properties and structural characteristics of the multilayers at various Cu intermetallic layer thicknesses which were varied from 4 to 16 nm were explored. The calculated band gap was reduced from 2.66 eV to 2.48 eV as the Cu intermetallic layer thickness increased from 4 to 16 nm. The refractive index and coefficient of extinction of CdO/Cu/CdO multilayers increased with increasing the Cu intermetallic layer thickness. The resistivity is reduced from 1.8 × 10−2 Ω cm for CdO single layer to reach a value of 2.7 × 10−4 Ω cm for CdO/Cu (16 nm)/CdO multilayer. Further, the sheet resistance is decreased from 1000 to 13.8 Ω/sq. with the variation in Cu intermetallic layer thickness from 0 to 16 nm. CdO/Cu (4 nm)/CdO multilayer film recorded the best figure of merit (2.3 × 10−4 Ω−1). After sunlight illumination for the multilayers, the surface wettability was improved and the contact angle recorded lowest value of nearly 24° for CdO/Cu (8 nm)/CdO and CdO/Cu (12 nm)/CdO.

Micro-second range pulsed DC plasma electrolytic oxidation on Ti and Nb

The presented research reports on the formation of oxide coatings on pure Ti and Ni substrates by micro-second pulsed plasma electrolytic oxidation in sodium phosphate electrolyte. The formation process is monitored by real-time imaging and optical emission spectroscopy. At the beginning of the process, micro-discharges appear randomly over the sample's surface and migrate towards the edges with prolonged processing. Emission lines originating from Ti substrate are well resolved, while appearance of high-resolution emission lines originating from Nb substrate in the visible range is reported for the first time. Subsequent characterization of formed oxide coatings on Nb shows that its roughness increases with PEO processing time, contrasting the case of Ti. It is observed that longer processing time of Ti substrate results in the anatase-to-rutile phase transformation, while pseudohexagonal TT-Nb2O5 phase is present throughout the processing of Nb substrate. Photoluminescent and photocatalytic properties of obtained coatings are dependent on their morphology and phase composition i.e., on processing time.

Kinetic analysis of the reduction of hematite fines by cold hydrogen plasma

This study deals with the reduction of hematite fines by cold hydrogen plasma in the temperature range of 360–450 °C. It aims to examine the rate-controlling processes of fine hematite reduction by hydrogen plasma. The gaseous reduction was also performed for comparison. The reduction experiments were performed in a DC pulsed plasma reactor and the phases were characterized by X-ray diffraction. Reacted fraction curves were obtained by using the gravimetric technique. The kinetic analysis suggests that the controlling step of the hematite reduction process by cold hydrogen plasma was the nucleation and growth of iron on the magnetite substrate. The controlling step for the gaseous reduction process was found to be the chemical reaction at the magnetite–gas interface. The activation energy values obtained for cold plasma and gaseous hydrogen reduction processes were 68.5 and 87.2 kJ/mol.

Optical Properties of Anti-Reflective Coatings on P-N Silicon and their Response to Gamma-Irradiation

The dc-pulsed plasma magnetron sputtering was used for depositing TiO2, ZrO2, and Ti50-Zr50 anti-reflection coating with thickness of 40 nm on p-n junction wafers and glass substrates. Furthermore, the antireflective properties of silicon nitride deposited elsewhere using plasma enhanced chemical vapor deposition (PECVD) were compared with those of metal oxide films. X-ray structural analysis reveals amorphous nature of the prepared films. The optical transmittance and reflectance measurements were investigated at room temperature in the wavelength range from 250 to 2500 nm. At long wavelengths, ZrO2 has a slightly higher transparency than that of TiO2 and Ti50-Zr50 oxide. On the other hand, the transparency at short wavelengths has different behavior. At about 300nm, a sharp decrease in the transparency of the three coatings was found. On the other hand, reflectance measurements show a decrease in reflectivity at 600 nm for ZrO2 and Ti50-Zr50 oxide as compared toTiO2. Samples irradiated with gamma rays at a dose of 5 kGray show different behavior in reflectivity due to displacement damage in Si-based cells.

Plasma-assisted rapid sintering of nanotitania powders

In this work, Plasma-Assisted Rapid Sintering (PARS), a pulsed DC plasma system in a hollow cathode regime, is presented as a novel technology to sinter nanoceramics. Nano-TiO2 powders are used as proof of concept and submitted to thermal treatment using several PARS conditions and sintering schedules. PARS heating process induced solely by the hollow cathode effect is consistent and affordable, providing a homogeneous temperature distribution to the compact. Furthermore, the heating rate and the maximum temperature are easily tunable by the discharge current applied in the plasma source and can go safely from room to maximum temperature in a matter of seconds with heating speed comparable with other reported rapid sintering techniques. Using 1-min of non-isothermal PARS cycles up to 1000 °C, porous nanostructured samples were obtained; reaching 80 % relative density while the grains remained at the nanoscale. Adding dwell times, the relative density was increased up to 96 % using a temperature plateau for up to 10 min, but the TiO2 grains grow intensively when temperatures exceeded 1000 °C. The results indicate that the developed process is a promising new alternative technology for sintering nanoceramics.

The spatial-temporal evolution process of flow field generated by a pulsed-DC plasma actuator in quiescent air

Flow control technology using the pulsed-DC plasma actuators has been demonstrated as an effective way of reducing the turbulent boundary-layer drag, which has attracted great interest from the aeronautical industry. However, the detail of the flow field induced by the pulsed-DC plasma actuator in quiescent air has rarely been addressed. In order to further reveal the controlling mechanism of the pulsed-DC plasma actuator, the experimental investigation on the flow field created by a pulsed-DC plasma actuator has been carried in quiescent air by using Schlieren visualization technique and Particle Image Velocimetry (PIV) measurement. In addition to the pressure wave induced by the pulsed-DC plasma actuator at the initiation stage, it is of great interest that two bi-direction vortices (vortex A and vortex B) and one corner vortex (vortex C) created by the pulsed-DC plasma actuator are observed and the tangential and vertical positions of the vortex core of the vortex A which is moving along the lower electrode are observed to scale with t1/5 as it develops at 24.5 degrees to the wall. Meanwhile, the formation mechanisms of the three vortices are discussed. Based on the aforementioned results, the whole spatial-temporal evolution process of the induced flow field is summarized for affording a comprehensive detail of the pulsed-DC plasma actuator. The results lay a foundation for understanding the controlling mechanism and advancing the numerical simulation model of the pulsed-DC plasma actuator.

An investigation of adhesion mechanisms between plasma-treated PMMA support and aluminum thin films deposited by PVD

The metallization of polymers has been the subject of several scientific studies Polymethylmethacrylate PMMA is known to have very bad adhesion to plasma deposited aluminum thin films. This study focuses on understanding the phenomena of adhesion between PMMA and metallic aluminum films deposited by magnetron sputtering. Before deposition, in situ pulsed DC plasma pre-treatment was performed in various atmospheres. We showed that oxygen-rich and short duration plasma pre-treatment considerably increase PMMA’s adhesives properties. Contact angle measurements showed a significant increase in the wettability of plasma pre-treated samples. XPS analysis revealed that the interaction between plasma species and polymer surface results in two simultaneous phenomena: surface functionalization by the introduction of oxygenated chemical functions such as carbonyl groups (CO) and surface degradation by the splitting of carbon chains by plasma UV radiation. Moreover, AFM analyses showed that the slight increase in surface roughness leads to best adhesion between PMMA and aluminum.

Adhesion Studies of CrC/a-C:H Coatings Deposited with Anode Assisted Reactive Magnetron Sputtering Combined with DC-Pulsed Plasma Enhanced Chemical Vapor Deposition.

We studied the effect of CrC interlayers with different carbon contents on the adhesion of CrC/a-C:H coatings prepared by anode assisted reactive magnetron sputtering combined with DC-pulsed plasma enhanced chemical vapor deposition. The adhesion of the coating was measured by indentation and scratching. The coatings were characterized by Raman, XPS, SEM and Nanoindentation. The adhesion of the CrC/a-C:H coating is best when the carbon content in the interlayer of CrC is 44.5%, the scratch adhesion is 74 N, and the indentation adhesion is HF1. In this case, the elastic modulus of the interlayer CrC (284 GPa) is closest to that of the a-C:H layer (274 GPa). In conclusion, when there is no graphitization in the CrC interlayer, and the elastic modulus of the CrC interlayer is close to that of the a-C:H layer, the CrC/a-C:H coatings show the best adhesion.

Comparison of thermal properties of a hot target magnetron operated in DC and long HIPIMS modes

Thermal properties of the magnetron discharge with uncooled copper and chromium targets were studied experimentally and theoretically for DC and long HiPIMS (L-HiPIMS) operation modes. A set of thermal fluxes was considered to build a numerical model of the hot target exposed to DC and high-power pulsed plasma. The modeling results were tested in the experiments. The temperature of targets was measured directly in course of magnetron discharge operation with an elaborated contact thermocouple method. The measurements were made in two modes. At first the temporal evolution of temperature was recorded for a fixed applied discharge power. The results were found to well agree with temperature values expected from calculations. At 2 kW power, it takes ~50 s to reach the melting point of copper. The steady-state temperature values were also measured for a number of discharge power levels. The obtained dependence clearly demonstrated that the main mechanisms of the target heat balance at high temperatures are surface radiation and heat transfer due to thermal conductivity of the heat insulation supports between the target and water-cooled cathode. The parameters of DC hot target magnetron were compared to the high-power pulsed regime with the pulse-on time 20 ms. A promising method of a hot target magnetron discharge operation was considered that involves applying long (>20 ms) high-power pulses to the target pre-heated in the DC mode during the pulse-off period.

Development of thermal control coatings on AA7075 by plasma electrolytic oxidation (PEO) process

The thermal control coating was produced on AA7075 by pulsed DC plasma electrolytic oxidation (PEO) process in an aqueous electrolyte containing 3 g/l sodium silicate (Na2SiO3) and 6 g/l potassium hydroxide (KOH) along with 4 g/l nanoparticles of Cupric oxide (CuO) or Titanium oxide (TiO2) or CuO and TiO2 with equal mass. PEO coatings were fabricated with different electrolyte compositions with a current density of 150 mA/cm2 and a frequency of 1000 Hz for 6 min. The phase composition, surface morphology, and thickness of the coating were assessed using X-ray diffraction (XRD), scanning electron microscope (SEM), and eddy current thickness gauge, respectively. Thermal diffusivity of the coated sample was measured by the laser flash apparatus and the practical adhesive strength between the substrate and coating layer was evaluated by the scratch test. It was perceived that all the coated samples exhibited superior thermal control properties compared to the bare aluminium and the CuO particles incorporated PEO coating showed the highest thermal diffusivity (80.18 mm2/sec at 30 °C) compared to other PEO coated samples. The nano TiO2 and nano CuO incorporated PEO coatings shown the least porosity compared to the PEO coating without nanoparticles and the scratch testing critical loads are 13.58 N and 9.53 N, respectively.

Spatio-temporal characterization of a pulsed DC atmospheric pressure plasma jet interacting with substrates

Atmospheric pressure plasmas generate a variety of chemically active species in open air, thus providing the unique ability to treat a variety of materials that do not require or are not compatible with vacuum systems. Producing the plasma-surface interaction that leads to a desired change in the substrate is complicated by the codependency between the plasma and the substrate: while the plasma will modify the surface, the surface will also influence the plasma properties. In this work, a pulsed-DC plasma jet produced in helium and impinging upon glass and metal substrates is studied over a range of applied voltage pulse widths extending from 1 to 10 µs. Current–voltage measurements, high speed images, and time-resolved optical emission from three important He and nitrogen excited species are used to examine the evolution of the plasma and its interaction with the surface. At ignition, a streamer is ejected into the open air from the jet exit and eventually collides with the substrate. For a glass substrate, the streamer will hit the surface and form a short-lived plasma across it. This surface plasma is almost completely unaffected by changes in the voltage pulse width. In contrast, when the streamer hits a metal substrate, a surface discharge will form that will last the entirety of the voltage pulse. If the pulse is long enough, a ‘reflected discharge’ will slowly develop that extends from the substrate back towards the outlet of the plasma jet. The emission intensity of the surface discharge closely matches that of the initial streamer, but not the reflected discharge, which suggests different electron kinetics between the two features. The addition of capacitors or resistors between the metal substrate and ground show how differences in substrate electrical properties can account for some of these behaviors. Emission line ratios are used to examine the evolution of electron temperature and the relative importance of Penning processes during the different plasma phases.

Deposition and characterization of tungsten oxide (WO3) nanoparticles incorporated diamond-like carbon coatings using pulsed-DC PECVD

In this study a novel approach using pulsed-DC plasma enhanced chemical vapour deposition (DC-PECVD) to incorporate tungsten oxide nanoparticles into diamond-like carbon coatings (DLC) thin film deposited on tappet valve (metallic substrate) was successfully achieved. The DLC-W film obtained by incorporating WO3 nanoparticles into DLC was characterized using scanning electron microscope/energy dispersive spectroscopy (SEM/EDS), X-ray diffraction (XRD), Raman spectroscopy, scratch test, Rockwell C indention test and nano-hardness measurement. The thickness, hardness and elastic modulus were measured as 789.11 ± 16 nm, 18.64 ± 0.11 GPa and 152.03 ± 2.1 GPa respectively. The ID/IG was 0.95, D and G peaks positioned at 1341.66 and 1562.17 cm−1 respectively and W2C was observed from the XRD. The coating had a low apparent COF of 0.075 and excellent adhesion properties.

Deposition and characterization of molybdenum oxide (MoO3) nanoparticles incorporated diamond-like carbon coatings using pulsed-DC PECVD

In this study a novel approach for the deposition of diamond-like carbon coating incorporated with molybdenum nanoparticles (DLC-Mo) thin film on tappet valve (metallic substrate) by incorporating MoO3 nanoparticles into DLC using pulsed-DC plasma enhanced chemical vapour deposition (PECVD) was successfully achieved. The as-deposited DLC-Mo was further characterized using scanning electron microscope/energy dispersive spectroscopy (SEM/EDS), X-ray diffraction (XRD), Raman spectroscopy, scratch test, Rockwell C indention test and nano-hardness measurement. The thickness was found to be 899.7 ± 16 nm. Hardness and elastic modulus 18.72 ± 0.1 and 150.03 ± 1.1 GPa respectively. The ID/IG was 0.98, D and G peaks positioned at 1350.92 and 1569.26 cm−1 respectively and Mo2C was observed from the XRD. The adhesion properties from the scratch and Rockwell C test were excellent, with low COF.

Time-resolved imaging and spectroscopy of atmospheric pressure plasma bullet propagation and RONS production

An atmospheric pressure plasma jet visually appears as a continuous plasma, but is actually comprised of a series of fast moving ionization waves, commonly called plasma bullets. Plasma bullets are the visible front of ionization streamers. Studying their spatiotemporal formation and propagation can help understand the plasma dynamics. In this work, the formation and lifetime of a high voltage pulsed dc plasma jet and the subsequent bullets were observed using time-resolved synchronized measurements with an intensified charge-coupled device (ICCD) camera and an ICCD spectrometer. The operating voltage, helium flow rate, and pulse width were varied to observe changes in bullet behavior. The measured velocities of the plasma bullets largely depended on voltage but showed that flow rate can have an effect. The formation of OH and excited N2 (2+) was measured at the nanosecond timescale. It was seen that the formation of these species begins at the visible bullet, but the peak emission and decay occur ∼50–100 ns after bullet passage, indicating a finite time of reaction, excitation, and emission. Observing the temporal formation of plasma bullets and reactive species provides insight into some of the reactions occurring in the discharge and how to tailor the plasma operating conditions.

MD-Simulation of Duty Cycle and TaN Interlayer Effects on the Surface Properties of Ta Coatings Deposited by Pulsed-DC Plasma Assisted Chemical Vapor Deposition

In this work, molecular dynamics (MD) simulations were employed to investigate the effects of duty cycle changes and utilization of tantalum nitride interlayer on the surface roughness and adhesion of Ta coating deposited by pulsed-DC plasma assisted chemical vapor deposition. To examine the simulation results, some selected deposition conditions were experimentally implemented and characterized through scanning electron microscopy, atomic force microscopy (AFM) and microscratch tests. The Ta and Ta/TaN coatings were deposited on AISI316L stainless steel substrate in a plasma atmosphere consist of Ar, H2, N2 and TaCl5 vapor at 350 °C. The results showed that at the same duty cycles the surface roughness of Ta/TaN coating is at least 40% less than that of the single layer Ta coating. By increasing the duty cycle from 17 to 40%, the surface roughness significantly decreases about 80%. This is attributed to the further exposure of surface against high energy ions bombardment at higher duty cycles. The presence of TaN interlayer due to its lower lattice mismatch with Ta (under 2%), contributes to the nucleation of Ta grains which consequently leads to reducing the surface roughness. The enhanced adhesion of the Ta coatings on TaN is discussed in view of improving the interfacial stresses as shown by the MD simulations. The MD simulations results were shown to be in good agreement with the experimental data.