Brownian Relaxation Mechanism Research Articles

Magnetic-field dependence of the magnetic dynamics of barium hexaferrite nanoplatelet suspensions

Because of their unique structural and magnetic properties, ferrofluid suspensions consisting of barium hexaferrite nanoplatelets have been in the spotlight of intensive research. Analysis of the dynamic magnetic susceptibility spectra can elucidate both of these properties using the magnetic-field dependence of the Brownian relaxation mechanism. Applying a small probe field and uniform static bias fields in different configurations, the effect of single particles and collective behaviour is distinctly separated. The importance of considering magnetic interparticle interactions is illustrated by comparing models neglecting or considering such interactions, highlighting limitations considering particle concentration, interaction and excitation-field strengths.

A prediction model for magnetic particle imaging–based magnetic hyperthermia applied to a brain tumor model

Background and objectiveBrain tumor is a global health concern at the moment. Thus far, the only treatments available are radiotherapy and chemotherapy, which have several drawbacks such as low survival rates and low treatment efficacy due to obstruction of the blood-brain barrier. Magnetic hyperthermia (MH) using magnetic nanoparticles (MNPs) is a promising non-invasive approach that has the potential for tumor treatment in deep tissues. Due to the limitations of the current drug-targeting systems, only a small proportion of the injected MNPs can be delivered to the desired area and the rest are distributed throughout the body. Thus, the application of conventional MH can lead to damage to healthy tissues. MethodsMagnetic particle imaging (MPI)-guided treatment platform for MH is an emerging approach that can be used for spatial localization of MH to arbitrarily selected regions by using the MPI magnetic field gradient. Although the feasibility of this method has been demonstrated experimentally, a multidimensional prediction model, which is of crucial importance for treatment planning, has not yet been developed. Hence, in this study, the time dependent magnetization equation derived by Martsenyuk, Raikher, and Shliomis (which is a macroscopic equation of motion derived from the Fokker-Planck equation for particles with Brownian relaxation mechanism) and the bio-heat equations have been used to develop and investigate a three-dimensional model that predicts specific loss power (SLP), its spatio-thermal resolution (temperature distribution), and the fraction of damage in brain tumors. ResultsBased on the simulation results, the spatio-thermal resolution in focused heating depends, in a complex manner, on several parameters ranging from MNPs properties to magnetic fields characteristics, and coils configuration. However, to achieve a high performance in focused heating, the direction and the relative amplitude of the AC magnetic heating field with respect to the magnetic field gradient are among the most important parameters that need to be optimized. The temperature distribution and fraction of the damage in a simple brain model bearing a tumor were also obtained. ConclusionsThe complexity in the relationship between the MNPs properties and fields parameter imposes a trade-off between the heating efficiency of MNPs and the accuracy (resolution) of the focused heating. Therefore, the system configuration and field parameters should be chosen carefully for each specific treatment scenario. In future, the results of the model are expected to lead to the development of an MPI-guided MH treatment platform for brain tumor therapy. However, for more accurate quantitative results in such a platform, a magnetization dynamics model that takes into account coupled Néel-Brownian relaxation mechanism in the MNPs should be developed.

Fluid Hyperthermia and Ultrasonic Studies on Nickel Ferrite Magnetic Nanoparticles

Superparamagnetic nickel ferrite (NiFe2O4) nanoparticles were synthesized by low temperature hydrothermal method using EDTA as templating agent. The synthesized nanoparticles were characterized by FTIR, XRD, FESEM and TEM analysis. The XRD analysis shows the presence of cubic spinel phase with crystallite size of 26.42 nm. The morphology analyzed by FESEM and TEM techniques indicate the presence of cubical shape particles having average particle size of 28.44 nm with no agglomeration. The VSM analysis gives “S” shape curve with zero coercivity and saturation magnetization (Ms) of 30 emu/g indicating the presence of small magnetic particles exhibiting the superparamagnetic behaviour. Nickel ferrite nanofluids were prepared by mixing carrier fluid water with varying amounts of nickel ferrite nanopowder. Zeta potential measurements of dilute solution of nickel ferrite show the formation of stable nickel ferrite nanofluids. Hyperthermia and ultrasonic studies were carried out on various concentrations of nickel ferrite nanofluids. The SAR values and ultrasonic parameters were calculated. Less concentrated nickel ferrite nanofluids quickly attain threshold hyperthermia temperature of 43 °C when exposed to lower applied AC magnetic field of range 5 mT to 20 mT. Ultrasonic studies show at lower concentrations nickel nanoparticles interact with carrier fluid water through cohesive forces mainly hydrogen bonding and keep the particles in an isolated state there by continue to exhibit superparamagnetic characteristics. At higher concentrations, particle-particle interaction predominantly occurs leading to increase in particle size beyond 30 nm, so that the magnetic characteristics of nickel ferrite nanoparticles changes from superparamagnetic to ferromagnetic nature. At low concentrations induction heating occurs through Neel and Brownian relaxation mechanism whereas at high concentrations hysteresis loss mechanism operate to attain the threshold temperature of 43 ºC.

Нарушение закона броуновской релаксации для частиц феррита кобальта в воде

In the temperature range 0–90°C, the dynamic susceptibility of three samples of a magnetic fluid based on cobalt ferrite particles stabilized in water with a double layer of a surfactant was meas-ured. The first layer of the stabilizer was formed of auric acid molecules. The second layer consisted of a mixture of lauric acid and sodium dodecyl sulfate. The liquid samples had a small (about 10%) volume fraction of particles in order to reduce the influence of interparticle interactions and eliminate non-Newtonian rheological properties. The main parameter in which the samples differed was viscosity. It was changed by adding a small amount of polyvinyl alcohol. For all samples, in the studied temperature range the frequency dependence of the dynamic susceptibility had a quasi-Debye form with a pronounced maximum for the imaginary part. The value of the magnetization relaxation time was determined from the frequency of the maximum on the imaginary part of the dynamic susceptibility. As a result, the dependences of themagnetization relaxation time on tem-perature were obtained. To analyze the results obtained, viscosities of the samples were measured. All samples exhibited the properties of Newtonian fluids –the value of dynamic viscosity did not depend on the shear rate. The obtained results on the dynamic susceptibility and viscosity were compared by plotting relaxation times as a function of the ratio ofdynamic viscosity to absolute temperature. The resulting dependences do not fit on one universal straight line, as it is supposed to be according to the theory. The actual graph looks like a family of three different curves that di-verge as viscosity increases. In order to assess the influence of Néel relaxation on the resulting sus-ceptibility, the dynamic susceptibility of a frozen sample of the magnetic fluid was measured. The resulting susceptibility value was about 3% of the initial susceptibility of the liquid sample. Thus, despite the insignificant contribution of the Néel mechanism of magnetization relaxation to the dy-namic susceptibility, it is impossible to explain the obtained measurement results on the basis of the Brownian relaxation mechanism alone.

Preparation and thermogenic performance of monodisperse ferromagnetic Fe/SiO2 nanoparticles for magnetic hyperthermia and thermal ablation*

Ferromagnetic nanoparticles have great potential to be used in the field of magnetic hyperthermia and thermal ablation due to its high saturation magnetization (Ms) and high heating efficiency. In this paper, monodisperse spherical α-Fe2O3 nanoparticles were prepared by hydrothermal method using FeCl3, NaHCO3 and Na2SO4 as reactants. Then monodisperse Fe/SiO2 nanoparticles with core/shell structure were obtained by coating SiO2 on the surface of α-Fe2O3 nanoparticles and reducing at 500℃ in hydrogen atmosphere. The diameter of Fe core is about 230 nm and the thickness of SiO2 coating is about 75 nm. Magnetic measurements show that Fe/SiO2 nanoparticles exhibit typical ferromagnetic properties with Ms about 132.5 emu/g at 25 ℃. As-obtained Fe/SiO2 nanoparticles have higher heating efficiency than superparamagnetic ferrite nanoparticles. The high heating ability of the Fe/SiO2 nanoparticles is estimated to be related to the hysteresis loss and Brownian relaxation mechanism during alternating magnetization. The results show that Fe/SiO2 nanoparticles are promising for the applications of magnetic hyperthermia and thermal ablation.

Retraction: Enhanced heat transfer properties of the magnetite nanofluids due to Neel and Brownian relaxation mechanisms

Retraction: Enhanced heat transfer properties of the magnetite nanofluids due to Neel and Brownian relaxation mechanisms

Magnetic particle imaging of particle dynamics in complex matrix systems

Abstract Magnetic particle imaging (MPI) is a young imaging modality for biomedical applications. It uses magnetic nanoparticles as a tracer material to produce three-dimensional images of the spatial tracer distribution in the field-of-view. Since the tracer magnetization dynamics are tied to the hydrodynamic mobility via the Brownian relaxation mechanism, MPI is also capable of mapping the local environment during the imaging process. Since the influence of viscosity or temperature on the harmonic spectrum is very complicated, we used magnetic particle spectroscopy (MPS) as an integral measurement technique to investigate the relationships. We studied MPS spectra as function of both viscosity and temperature on model particle systems. With multispectral MPS, we also developed an empirical tool for treating more complex scenarios via a calibration approach. We demonstrate that MPS/MPI are powerful methods for studying particle-matrix interactions in complex media.

Различие между броуновским и неелевским механизмами релаксации в подмагничивающем поле

Measurements of the dynamic susceptibility of a magnetic fluid based on cobalt ferrite particles stabilized in water by a double surfactant layer have been carried out. Cobalt ferrite, in comparison with magnetite, has a significantly higher energy of magnetic anisotropy. Therefore, for particles of cobalt ferrite, the Brownian mechanism of relaxation of magnetic moments is characteristic. The Debye (with a finite relaxation time) contribution to the dynamic susceptibility and the high-frequency (dispersionless) contribution are distinguished by constructing Cole-Cole diagrams. It was found that with an increase in the magnetizing field, the Debye contribution to the dynamic susceptibility decreases, while the high-frequency one (having a zero relaxation time) remains unchanged. The indicated property of the dynamic susceptibility of a fluid with a Brownian relaxation mechanism is radically different from the properties of the susceptibility of a fluid with Néel particles. Previously, measurements were made of the susceptibility of a fluid based on magnetite particles stabilized with oleic acid in kerosene. The magnetite particles have significantly lower anisotropy energy and are characterized by the predominance of the Néel relaxation mechanism. Turning on the magnetizing field caused a decrease in both the Debye part of the susceptibility and the high-frequency part of the susceptibility of magnetite particles.

Computational predictions of enhanced magnetic particle imaging performance by magnetic nanoparticle chains

The magnetic particle imaging (MPI) performance of collections of chains of magnetic nanoparticles with Néel and Brownian relaxation mechanisms was studied by carrying out simulations based on the Landau–Lifshitz–Gilbert equation and rotational Brownian dynamics, respectively. The effect of magnetic dipole–dipole interactions within chains on the time-domain average magnetic dipole moment and corresponding dynamic hysteresis loops, harmonic spectra, and point spread functions (PSFs) of the particle chains was evaluated. The results show that interactions within chains lead to ‘square-like’ dynamic hysteresis loops and enhanced MPI performance, compared to chains of non-interacting nanoparticles. For nanoparticles with the Brownian relaxation mechanism, subjected to a superimposed alternating and ramping magnetic field mimicking the magnetic field in MPI applications, we studied the dependence of x-space MPI performance of particle chains on parameters such as the amplitude of the alternating magnetic field, surface-to-surface separation between nanoparticles, solvent viscosity, and the number of nanoparticles in a chain. The results illustrate that magnetic dipole-dipole interactions within a chain contribute to enhanced MPI performance, and also suggest that there exist optimal values of the above parameters that lead to the best x-space MPI performance, i.e. maximum peak signal intensity and smallest full-width-at-half-maximum in PSFs.

Magnetically induced heating by iron oxide nanoparticles dispersed in liquids of different viscosities

The AC magnetically induced heating characteristics of uncoated and silica-coated magnetic iron oxide nanoparticles dispersed in liquids of different viscosities were investigated. The aim was to synthesize and prepare uncoated and silica-coated nanoparticles of maghemite, and evaluate their ability to hyperthermically dissipate heat under an applied AC magnetic field when they are dispersed in liquids of different viscosities. A conceptual approach on the relative contributions of the Néel and the Brownian relaxation mechanisms to the hyperthermic heating of these suspensions is proposed. The microstructure, the physical and chemical properties of the uncoated and silica-coated nanoparticles were assessed by transmission electron microscopy; powder X-ray diffraction; 57Fe Mössbauer spectroscopy; Fourier transform infrared spectroscopy; zeta potential measurements; and conventional chemical analysis. Results of the Rietveld refinement of the XRD patterns and analysis of the collected Mössbauer data are well consistent with maghemite as being the only iron oxide phase. The mean diameters of the uncoated nanoparticles increased from ~6 to 7 nm, to 35 and 78 nm, for added silica coating amounts varying from 1 to 6-fold, respectively. Zeta potential measurements confirmed the efficiency of the nanoparticles sol-gel coating method. When dispersed in low viscosity media (water, triethanolamine, ethylene glycol), uncoated nanoparticles efficiently release heat via both Néel and Brown relaxation mechanisms under an applied oscillating magnetic field, achieving a temperature raise of approximately 21 °C. The silica layers tend to inhibit the Brownian motion of coated particles, making heat dissipation to be preferentially governed by the Néel relaxation of the magnetization vector. This leads to modest rises in temperature of 9 °C when they are dispersed in water or in the PVC (polyvinyl chloride). SAR tests indicate that the maghemite-silica core-shell systems might be useful for advanced technologies in medical practices based on local hyperthermia, particularly in oncology.

Multi-spectral Magnetic Particle Spectroscopy for the investigation of particle mixtures

Magnetic Particle Spectroscopy (MPS) records the non-linear response of magnetic nanoparticles (MNP) to large ac magnetic fields. It captures a harmonic spectrum for the characterization of MNP samples, especially in the context of Magnetic Particle Imaging (MPI). MPS is sensitive to minute variations in magnetization response, thus different particle types or batches are easily distinguished. Also, since the dynamic magnetization of the particles is driven by both the Néel and the Brownian relaxation mechanisms, the harmonic spectra in MPS scale with ambient parameters, such as temperature, viscosity or binding state of the particles. Previous studies mostly observed low-frequency harmonic ratios (e.g. 5f0/3f0) to distinguish particles or to obtain ambient parameters. Here, we propose a multi-spectral analysis technique, using the full available spectrum, to decompose the collective response of different particles from an integral measurement on an MNP sample. This work focuses on the description of the multi-spectral method and its application to quantify binary and ternary mixtures of different particles.

Determination of dominating relaxation mechanisms from temperature-dependent Magnetic Particle Spectroscopy measurements

Abstract Magnetic Particle Spectroscopy (MPS) is a characterization method for investigating the nonlinear properties of magnetic nanoparticles (MNP) using magnetic field strengths in the order of a few tens of millitesla. Its exploitation for particle characterization is of high significance for biomedical applications such as Magnetic Particle Imaging (MPI) and magnetic hyperthermia. Since the dynamic characteristics of MNP are influenced by both the Neel and the Brownian relaxation mechanism, harmonic spectra in MPS measurements are directly linked to ambient influences like temperature or viscosity of the surrounding medium. Experimental data of multiparametric measurements helps one to evaluate and validate mathematical models of dynamic particle magnetization. This contribution deals with the investigation of temperature-dependent harmonic spectra of different commercially available single-core and multi-core particle systems. It is shown, that dominating relaxation mechanisms can be determined from temperature-dependent MPS measurements.

Heating efficiency of magnetic nanoparticles decreases with gradual immobilization in hydrogels

Abstract Magnetic nanoparticles (MNP) applied in magnetic fluid hyperthermia cancer treatment interact with cancerous tissue in various ways. The impact of these interactions on MNP heating efficiency is hard to quantify and strongly depends on the MNP mobility inside the cancerous environment. There, this MNP mobility is inhibited by cell attachment and internalization. In this study, we model this impact and analyze the MNP heating under gradual immobilization of MNP in acrylamide hydrogels with tailored mesh size. Our results confirm a clear particle heating dependency on the state of immobilization of MNP. This state is related to the mean mesh size of the respective hydrogel. From this, the contributions of Brownian and Neel relaxation mechanisms to the overall particle heating are estimated. In fact, the heating efficiency decreases by up to 35% for the highest immobilization state of MNP. This result is discussed in the context of the field-dependent Brownian and Neel relaxation time, showing that the former significantly contributes to the heating efficiency even for small particles under the field parameters employed in this study.

Brownian Dynamics Simulations of Magnetic Nanoparticles Captured in Strong Magnetic Field Gradients

The behavior of spherical single-domain magnetic nanoparticles in strong inhomogeneous magnetic fields is investigated through Brownian dynamics simulations, taking into account magnetic dipole–dipole interactions, repulsive hard-core Yukawa potential, hydrodynamic particle-wall interactions, and the mechanism of magnetic dipole rotation in the presence of a magnetic field. The magnetic capture process of nanoparticles in prototypical magnetic field gradients generated by a sudden reversal in perpendicular magnetization of a flat substrate (defining a “capture line”) is studied as a function of strength of the magnetic field and volume fraction of the magnetic nanoparticles. Capture curves show a regime where capture follows a power law model and suggest that particles with the Brownian relaxation mechanism are captured at a slightly faster rate than particles with the Néel relaxation mechanism under similar conditions of the field gradient. Additionally, evaluation of the shape of the aggregates of captured particles suggests that greater dipole–dipole interactions result in aggregate structures that are flatter/wider than in the case of negligible dipole–dipole interactions. These results can help guide the design of systems for magnetically directed assembly of nanoparticles into complex shapes at a substrate.

Enhanced Heat Transfer Properties of Magnetite Nanofluids due to Neel and Brownian Relaxation Mechanisms

Fe3O4 nanoparticles were prepared through solvo-thermal method for further heat transfer applications. TEM, XRD, TGA, and VSM were applied to characterize the obtained nanoparticles. XRD pattern confirmed that nanoparticles were composed of 6-nm crystallites; however, TEM images showed the formation of ca. 75-nm highly dispersed magnetite clusters, made up of about 6-nm nanoparticles. Since, VSM analysis confirmed the superparamagnetic characteristics of Fe3O4 nanoclusters, heat transfer properties of the resulting nanofluids were studied to investigate the influence of the magnetic field on the behavior of the magnetite-based nanofluids. The findings indicated that the convective heat transfer coefficient increased up to 48% and 15%, respectively, for nanofluids containing 0.005 wt% magnetite particles dispersed in water and EG, when the frequency of the alternating magnetic field was changed from 50 Hz to 1 MHz. According to the results, compared to the water-based nanofluids, at higher field amplitudes, the h enhancements of EG-based ones were more pronounced, for instance, at H 0 = 36,000 A/m, the h measurements are augmented by about 74% and 109%, respectively, compared to the water and EG as the base fluids. These findings could be explained by the use of specific lost powers of the nanofluids in the exposure of an external alternating magnetic field.

Magnetic response of gelatin ferrogels across the sol-gel transition: the influence of high energy crosslinking on thermal stability.

As emerging responsive materials, ferrogels have demonstrated significant potential for applications in areas of engineering to regenerative medicine. Promising techniques to study the behavior of magnetic nanoparticles (MNPs) in such matrices include magnetic particle spectroscopy (MPS) and magnetorelaxometry (MRX). This work investigated the magnetic response of gelatin-based ferrogels with increasing temperatures, before and after high energy crosslinking. The particle response was characterized by the nonlinear magnetization using MPS and quasistatic magnetization measurements as well as MRX to discriminate between Néel and Brownian relaxation mechanisms. The effective magnetic response of MNPs in gelatin was suppressed, indicating that the magnetization of the ferrogels was strongly influenced by competing dipole-dipole interactions. Significant changes in the magnetic behavior were observed across the gelatin sol-gel transition, as influenced by the matrix viscosity. These relaxation processes were modeled by Fourier transformation of the Langevin function, combined with a Debye term for the nonlinear magnetic response, for single core MNPs embedded in matrices of changing viscosities. Using high energy electron irradiation as a crosslinking method, modified ferrogels exhibited thermal stability on a range of timescales. However, MRX relaxation times revealed a slight softening around the gelatin sol-gel transition felt by the smallest particles, demonstrating a high sensitivity to observe local changes in the viscoelasticity. Overall, MPS and MRX functioned as non-contact methods to observe changes in the nanorheology around the native sol-gel transition and in crosslinked ferrogels, as well as provided an understanding of how MNPs were integrated into and influenced by the surrounding matrix.

A fast and remote magnetonanothermometry for a liquid environment

This study reports on a new approach for remote nanothermometry with short response time (milliseconds) aiming to operate in liquid media using AC susceptibility components of a suspended magnetic nanoparticle subjected to the Brownian relaxation mechanism. A simple, low cost, and accurate system was designed to measure AC susceptibility using an AC magnetic field at small amplitude (6 Oe) and frequency range (5 kHz) superimposed on a weak DC magnetic field (up to 30 Oe). A model based on the AC susceptibility of magnetic nanoparticles (30 nm average diameter) was constructed to describe the temperature measurement sensitivity of the dominated Brownian relaxation time. A new approach for remote nanothermometry was achieved with measured AC susceptibility by the designed system and the proposed model. Our experimental results show that our magnetonanothermometer allows temperature errors lower than 0.3 K with standard deviations lower than 0.1 K in the temperature range from 310 to 320 K.

Heating of polyacrylamide ferrogel by alternating magnetic field

Ferrogel based on polacryamide network with embedded maghemite nanoparticles with mean number average particle diameter 12nm was synthesized by radical polymerization in water-based ferrofluid. The network structure of ferrogel was characterized by Flory–Rehner theory and it was shown that the embedded particles were substantially larger than the mesh size. It prevented the translational movement of particles in the ferrogel. The immobilization of particles was confirmed by dynamic light scattering. The adhesion of macromolecular chains to the particles was determined by calorimetry using thermochemical cycle. The enthalpy of interfacial adhesion was found several orders of magnitude higher than the energy of dipoles in typically applied magnetic fields. Despite the differenve in the mobility of particles in ferrofluid and ferrogel the comparative study of their heating in alternating magnetic field, however, revealed their close similarity. In both cases it was goverened by superposing of Neel and Brownian relaxation mechanisms.

Ferrofluid droplet vaporization under very large magnetic power: Effects of pressure and effective thermal conductivity of liquid

The aim of current analysis is to quantify the influence of the effective thermodynamic and transport coefficients and of the transient process of mass and energy accumulation in the gas phase (pressure effect) on the heating and vaporization of a single ferrofluid droplet. Ferrofluids under external alternating magnetic field heat up themselves due to the magnetic Brownian relaxation mechanism. Under the condition of very large magnetic power compared to the thermal power provided by heat transfer from the gas phase, the magnetic heat source together with the heat transfer from the gas phase impose a thermal boundary layer adjacent to the droplet surface in the liquid side and the temperature presents a maximum inside the droplet, not at the surface. Since the transport coefficient increases significantly with a dispersion of a small quantity of nanoparticles, the heat transfer from the thermal boundary layer to the droplet core increases. Then the temperature of that region increases faster comparing to the case without nanoparticle dispersion. The temperature inside the thermal boundary layer increases slower because of the heat transfer to the droplet core as well as to the droplet surface. Therefore, the boiling condition which is found inside the thermal boundary layer is reached later when considering effective thermal conductivity. The droplet vaporization rate is augmented by the heat transfer from the thermal boundary layer to the droplet surface. In addiction, the strong dependence of the magnetic relaxation mechanism on temperature imposes a dependence of the vaporization rate on the initial condition of the problem.

Ferrofluid droplet heating and vaporization under very large magnetic power: A thermal boundary layer model

In this work, heating and vaporization of a liquid droplet with dispersed magnetic nanoparticles (ferrofluid) are analyzed. The ferrofluid droplet is in a quiescent inert gas phase with a temperature which is set down equal to, higher and lower than the liquid boiling temperature. Under these conditions, an alternating magnetic field is applied and, as a result, the magnetic nanoparticles generate heat by the Brownian relaxation mechanism. In this mechanism, the magnetic dipoles present a random orientation due to collisions between the fluid molecules and nanoparticles. The magnetic dipoles tend to align to the magnetic field causing rotation of the nanoparticles. Consequently the temperature increases due to the energy dissipated by the friction between the resting fluid and the rotating nanoparticles. Assuming a very large magnetic power and a uniform distribution of nanoparticles, the droplet core is uniformly heated. A thermal boundary layer is established in the liquid-phase adjacent to the droplet surface due to heat flux from the ambient atmosphere. The temperature profile inside the thermal boundary layer is obtained in appropriate time and length scales. In the present model, the ferrofluid droplet is heated up to its boiling temperature in a very short time. In addition, the combination of the heat generated by magnetic nanoparticles and heat conduction from gas phase results in a higher vaporization rate. Under specific conditions, the boiling temperature is achieved not at the surface but inside the thermal boundary layer. Moreover, the results point out that the thermal boundary layer depends directly on the vapor Lewis number but the vaporization rate reciprocally on it.