Quadratic Behavior Research Articles

Effect of nitrogen rates applying controlled-release and conventional urea blend in maize

Blending polymer-sulfur coated urea (PSCU) and conventional urea (U) for maize (Zea mays L.) fertilization can supply nitrogen (N) during the crop cycle with a single application. Proper placement of PSCU + U (0.15 m below and 0.1 m to the side of seed row) in band application at sowing is necessary to reduce salt stress that can decrease dry weight (DU) and N uptake (NU) of maize plant compromising maize yield. It is not clear the proper N rate in the proper placement for band application of PSCU + U at maize sowing to avoid salt stress. In the current literature, reduction of N rates are being recommended using PSCU + U without consider the probably salt stress provided by high rates of PSCU + U. DW and NU in maize plant as well as soil pH and electrical conductivity (EC) were evaluated in a greenhouse pot trial. N treatments were equivalent to 0, 90, 180, 360 and 540 kg N ha−1 applied incorporated in band in two contrasting soils (Rhodic Eutrustox and Typic Haplustox) using 70%PSCU + 30%U. At V10 (vegetative leaf stage 10), DW and NU of maize aerial part had quadratic behavior in response to increase N rates in the Typic Haplustox soil. In the Rhodic Eutrustox was not observed known behavior for DW and NU in response to increase N rates. Soil pH and EC was higher in the fertilizer row than sowing row. A N rate above of 180 kg N ha−1 using 70%PSCU + 30%U incorporated in bands can reduce DW and NU in early maize plant growth associated with salt concentration of N fertilizer in a Typic Haplustox soil, which could compromise maize yield.

Nutritional and fermentative profile of forage sorghum irrigated with saline water

ABSTRACT This study aimed to evaluate the nutritional and fermentative potential of silage of the forage sorghum Ponta Negra irrigated with saline water. The bromatological characteristics, losses and fermentative profile of sorghum silages were observed. For the bromatological characteristics of the sorghum plant, a decreasing linear behavior was observed for the contents of dry matter and organic matter while a positive linear effect was verified for crude protein and mineral matter as the leaching fractions were increased (P < 0.05). A quadratic behavior was observed for non-fibrous carbohydrates, neutral detergent fiber, acid detergent fiber, and cellulose (P < 0.05). The pH values showed a quadratic effect (P < 0.05) with lower pH values observed at 15 days of silo opening for water depths of 10 and 15%. Regarding the silages, the leaching fractions provided differences (P < 0.05) for dry matter recovery, gas loss and effluent loss. A negative linear behavior for the contents of dry matter and organic matter and an increasing linear effect for mineral matter (P < 0.05). The highest concentrations of ammoniacal nitrogen were observed when irrigation was performed with a 10% saline water depth (P < 0.05).

Quality of leucaena, gliricidia, and pornunça silages with different old man saltbush levels

This study aimed to evaluate the quality of leucaena, gliricidia, and pornunça silages with different levels of old man saltbush inclusion in their composition. A completely randomized design was used with five levels of old man saltbush (Atriplex nummularia) inclusion (0, 25, 50, 75, and 100%) in leucaena (Leucaena leucocephala), gliricidia (Gliricidia sepium), and pornunça (Manihot sp.) silages, with 5 replications per treatment, totaling 75 experimental mini-silos maintained in a covered shed and free of opportunistic animals. Silos were opened after 100 days of ensilage, and the silage of the first 15 cm of their ends was discarded. The remaining material was homogenized and sampled for determining the bromatological composition, in vitro digestibility, and fermentation profile. Old man saltbush inclusion levels in leucaena silages promoted a negative linear effect for dry matter, organic matter, crude protein, ethereal extract, pH, and ammoniacal nitrogen, while a positive linear effect was found for mineral matter, in vitro dry matter digestibility, total carbohydrates, non-fibrous carbohydrates, and propionic acid (P &lt; 0.05). This silage also had a quadratic effect for lactic acid. Gliricidia silages showed a positive linear behavior for dry matter, mineral matter, and total carbohydrates, whereas organic matter, crude protein, and ethereal extract presented a negative linear behavior (P &lt; 0.05). The contents of organic lactic, acetic, and propionic acids showed a quadratic effect for gliricidia silage as old man saltbush levels increased (P &lt; 0.05). Regarding pornunça silage, dry matter, mineral matter, total carbohydrates, propionic acid, and butyric acid presented a linear increasing behavior, while organic matter, crude protein, ethereal extract, and non-fibrous carbohydrates presented a linear decreasing behavior (P &lt; 0.05). Lactic acid and acetic acid concentrations for pornunça silage showed a quadratic behavior according to the increasing old man saltbush inclusion levels (P &lt; 0.05). The highest contribution of old man saltbush in the silages promoted changes in the profile of organic acids, allowing an improvement in silage quality, favoring lactic fermentation rather than acetic fermentation. Butyric acid contents were low for all treatments and indicated the control of undesirable fermentation processes. These results indicate that old man saltbush inclusion in silage composition may contribute to bromatological composition and fermentation process of the studied silages.

Limit Behaviour of a Singular Perturbation Problem for the Biharmonic Operator

We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in Dipierro et al. ( arXiv:1808.07696 , 2018), and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small values of the perturbation parameter. Some counterexamples to the uniform regularity are also provided if one does not impose some structural assumptions on the forcing term.

Response of mechanically-created neurites to extension

We use micromanipulation techniques and real-time particle tracking to develop an approach to study specific attributes of neuron mechanics. We use a mechanical probe composed of a hollow micropipette with its tip fixed to a functionalized bead to induce the formation of a neurite in a sample of rat hippocampal neurons. We then move the sample relative to the pipette tip, elongating the neurite while simultaneously measuring its tension by optically tracking the deflection of the beaded tip. By calibrating the spring constant of the pipette, we can convert this deflection to a force. We use this technique to obtain uniaxial strain measurements of induced neurites and investigate the dependence of the force-extension relationship on mechanical pull speed. We show that in the range of pull speeds studied (0.05–1.8 μm/s), the variation in the work to extend a neurite 10 μm is consistent across pull speeds. We do not observe statistically significant rate-dependent effects in the force-extension profiles; instead we find the same quadratic behaviour (with parameters drawn from the same distributions) at each pull speed.

Effects of foliar spraying with new zinc sources on rice seed enrichment, nutrition, and productivity

ABSTRACTNew Zn sources have emerged recently, which may boost the beneficial effects of leaf sprays on rice cropping. Thus, this study aimed to compare the effects of foliar sprays using Zn chelate against those with new Zn sources such as sorbitol-stabilised Zn nitrate and Zn oxide nanoparticles on rice seed enrichment and production. Treatments were arranged in a completely randomised design, comprising of three Zn sources and four Zn concentrations with four replicates. Rice plants were sprayed three times: at the beginning of tillering, maximum tillering, and booting stage. Experimental unity consisted of 7-dm³ pots filled with 5-dm³ medium textured sand. Each pot had 10 plants, of which 6 randomly harvested at the flowering stage and the remaining at the maturity stage. Plant dry matter, content and accumulation of Zn in the shoot, seed Zn content, and seed production at the grain-filling stage were also estimated. Seed production and Zn accumulation showed a quadratic behaviour with the use of novel sources. Overall the new Zn sources were more effective than the traditional EDTA chelated Zn in terms of rice seed production and quality, especially the sorbitol-stabilised Zn nitrate.

Performance of Holstein x Gyr dairy cows fed hydrolyzed sugarcane-based diets

Twelve multiparous 7/8 Holstein x Gyr dairy cows were fed diets based on chopped sugarcane supplied in natura (control) or after being hydrolyzed (fresh matter basis) for 24 h with 0.5%, 1% or 2% calcium oxide (CaO). The voluntary consumption and digestibility of dietary nutrients, milk production and composition, and feeding behavior of cows were evaluated. A switchback design with three periods, four treatments and three blocks was used. The linear and quadratic effects of the treatments were analyzed using orthogonal contrasts, and the effects were considered significant when P ? 0.05. Compared to the that in the control treatment, the Ca concentration increased by 297%, 570% and 1,083% in the diets including sugarcane hydrolyzed with 0.5%, 1% and 2% CaO, respectively, while the respective Ca:P ratios were 6.58:1, 10.70:1 and 19.77:1 (control Ca:P ratio: 1.55:1). The hydrolysis of sugarcane with increasing amounts of CaO promoted a linear reduction (P &lt; 0.01) in the intake of dry matter (DM), nonfibrous carbohydrates, crude protein (CP), organic matter (OM), neutral detergent fiber corrected for ash and protein (NDFap) and ether extract. The intake of total digestible nutrients, acid detergent fiber (ADF), lignin, hemicelluloses and cellulose presented quadratic behavior (P &lt; 0.01) in response to increasing CaO concentrations used for sugarcane hydrolysis. Increases in the amount of CaO used for sugarcane hydrolysis had no effect on the digestibility of NDFap (P &gt; 0.05), but there were linear reductions (P &lt; 0.05) in the digestibility of DM, OM and ADF; the production of milk corrected or not corrected to 4% fat; and the content of protein, lactose and total solids. Linear reductions (P &lt; 0.05) were also observed in the production of all milk components. Only the milk fat content was not altered (P &gt; 0.05) in response to the inclusion of hydrolyzed sugarcane in the diets. There was no effect (P &gt; 0.05) of sugarcane hydrolysis with increasing amounts of CaO on the time the cows spent idle, ruminating and feeding, but there was a linear increase (P &lt; 0.05) in the minutes spent ruminating 1 kg of DM and NDF. Chopped sugarcane hydrolyzed for 24 h with 0.5% to 2.0% CaO should not be used as exclusive roughage in diets for Holstein x Gyr dairy cows with an average milk production of 10 kg day?1.

Effect of including different levels of crude glycerin in feed on plasma triglyceride concentrations in pigs, and the composition and qualitative attributes of pork

This study was conducted to evaluate the effects of including two types of crude glycerin (CG) in feed on plasma parameters, quantitative and qualitative carcass characteristics, and the fatty acid profile of the Longissimus dorsi pig muscle. Sixty-three commercial line pigs were used, with an average weight of 30.53 ± 1.59 to 59.55 ± 4.27 kg in the growing phase and 60.35 ± 2.25 to 89.91 ± 5.23 kg in the finishing phase. The pigs’ diet included two sources of CG (vegetable and mixed) at four inclusion levels (3%, 6%, 9% and 12%). One CG was derived from soybean oil (VCG) and the other from 80% animal fat + 20% soybean oil (MCG). The control diet contained 0% glycerin. Blood samples were collected at the beginning, middle and final phases of the trial. In addition, at the end of the finishing phase, the animals were fasted for 12 h and then sacrificed to measure the quantitative and qualitative carcass characteristics and the fatty acid profile of the Longissimus dorsi. The results indicate that in both phases there was a quadratic effect on plasma triglycerides. There was no effect on carcass parameters. A quadratic behaviour for pH24 h was obtained, with the lowest value attained at 1.05% CG inclusion. There was an interaction between the type of glycerin and the inclusion level in the diets for polyunsaturated fatty acids. VCG and MCG can be used at up to 12% inclusion because they cause no negative effects on carcass characteristics and pig meat quality.

TiO₂ Nanostructures for Photoelectrocatalytic Degradation of Acetaminophen.

Advanced oxidation processes driven by renewable energy sources are gaining attention in degrading organic pollutants in waste waters in an efficient and sustainable way. The present work is focused on a study of TiO2 nanotubes as photocatalysts for photoelectrocatalytic (PEC) degradation of acetaminophen (AMP) at different pH (3, 7, and 9). In particular, different TiO2 photocatalysts were synthetized by stirring the electrode at different Reynolds numbers (Res) during electrochemical anodization. The morphology of the photocatalysts and their crystalline structure were evaluated by field emission scanning electron microscopy (FESEM) and Raman confocal laser microscopy (RCLM). These analyses revealed that anatase TiO2 nanotubes were obtained after anodization. In addition, photocurrent densities versus potential curves were performed in order to characterize the electrochemical properties of the photocatalysts. These results showed that increasing the Re during anodization led to an enhancement in the obtained photocurrents, since under hydrodynamic conditions part of the initiation layer formed over the tubes was removed. PEC degradation of acetaminophen was followed by ultraviolet-visible absorbance measurements and chemical oxygen demand tests. As drug mineralization was the most important issue, total organic carbon measurements were also carried out. The statistical significance analysis established that acetaminophen PEC degradation improved as hydrodynamic conditions linearly increased in the studied range (Re from 0 to 600). Additionally, acetaminophen conversion had a quadratic behavior with respect to the reaction pH, where the maximum conversion value was reached at pH 3. However, in this case, the diversity of the byproducts increased due to a different PEC degradation mechanism.

Temperature Dependence of the Electrical Resistivity of Amorphous Co80−xErxB20 Alloys: Localization and Electron–Electron Interaction Effects

The amorphous Co80−xErxB20 alloys with x = 0, 3.9, 5.5 and 8.6% have been prepared by melt spinning in pure argon atmosphere. The electrical resistivity has been measured over the temperature range from 4.2 to 300 K. All investigated amorphous alloys exhibit a logarithmic resistivity versus temperature below the temperature of the resistivity minimum Tmin. In addition, the resistivity shows a quadratic temperature behavior in the temperature range Tmin < T < 77 K. At high temperature, the electrical resistivity was discussed by the extended Ziman theory. For the studied alloys, the composition dependence of the temperature coefficient of electrical resistivity α shows a change in sign related to a structural short-range that is occurring in the composition range 8–9%.

A comparison between rad-hard float zone silicon diodes as gamma dosimeter in radiation processing

In this work, we report on the results obtained with rad-hard Standard Float Zone (STFZ) and Diffused Oxygenated Float Zone (DOFZ) silicon diodes in radiation processing dosimetry. The dosimetric probes were designed to operate in the direct current mode, as on-line radiation dosimeter. The irradiation of the samples was performed using a 60Co source with a dose rate of almost 2.4 kGy/h. The current response of each diode was measured as a function of the exposure time in steps from 5 kGy up to 50 kGy to achieve a total absorbed dose of 275 kGy. In this dose range it is observed a significant decrease in the photocurrent generated in both devices due to gamma radiation defects produced in their active volumes. To mitigate this effect, the samples were pre-irradiated with Co-60 gamma rays at 700 kGy. Despite of being less sensitive, these devices presented stable and reproducible current signals with a relative sensitivity decrease of about 19% within the whole range of dose studied. The dose-response curves of the pre-irradiated diodes showed quadratic behavior with correlation coefficient higher than 0.9999 for total absorbed dose up to 275 kGy. The comparison of the FZ and DOFZ responses evidenced that the latter was slightly superior to the first. However, it is important to note that all pre-irradiated diodes can be used as gamma dosimeters in radiation processing applications.

Modified approach to modeling barrier inhomogeneity in Schottky diodes

The conventional model describing barrier inhomogeneities in Schottky diodes with a single series resistance leads to many anomalies including crossing over effect in current–voltage–temperature (I–V–T) plots. A new model is therefore proposed here which entails a parallel combination of several diodes, each with its own series resistance. Further, the barrier heights follow a Gaussian distribution function ρ(φ) with a mean barrier height and standard deviation σ. The series resistance is believed to vary inversely with the ρ(φ) value of the concerned barrier height and hence the area occupied. The occurrence of anomalies with the conventional model is examined in depth, and their effective elimination with the proposed model is discussed with the undertaken current simulations. In addition to predicting the increase in apparent barrier height with increase in temperature, decrease in ideality factor with increase in temperature, and elimination of modeling anomalies observed in the conventional model, the model is successful in predicting that apparent barrier height as a function of inverse thermal energy is a quadratic behavior even with a single distribution. This previously required unlikely scenarios involving multiple mean barrier heights and standard deviations with the conventional model. Finally, the description also explains the observed increase in the apparent barrier height with decrease in temperature of a Schottky diode when the values are deduced from the C–V–T data under reverse bias.

Radial symmetry of non-maximal entire solutions of a bi-harmonic equation with exponential nonlinearity

AbstractWe study radial symmetry of entire solutions of the equation0.1$$\Delta ^2u = 8(N-2)(N-4)e^u\quad {\rm in}\;R^N\;\;(N \ges 5).$$It is known that (0.1) admits infinitely many radially symmetric entire solutions. These solutions may have either a (negative) logarithmic behaviour or a (negative) quadratic behaviour at infinity. Up to translations, we know that there is only one radial entire solution with the former behaviour, which is called ‘maximal radial entire solution’, and infinitely many radial entire solutions with the latter behaviour, which are called ‘non-maximal radial entire solutions’. The necessary and sufficient conditions for an entire solutionuof (0.1) to be the maximal radial entire solution are presented in [7] recently. In this paper, we will give the necessary and sufficient conditions for an entire solutionuof (0.1) to be a non-maximal radial entire solution.

Effect of BN nanodomains on the mechanical properties of graphene monolayers

In the present contribution first-principles calculation was applied to investigate the modifications on the mechanical and electronic properties of a graphene monolayer induced by the inclusion of a boron nitride nanodomain. The calculation results indicate a significant reduction in Young's modulus. Such nanodomain can be seen as an extended defect, which can be pointed out as the crack seed for the layer disruption. The strain energy shows a quadratic behavior, which is typical for structures submitted to tensile strain. A significant increment in the band gap energy was found.

Características fermentativas, composição química e fracionamento da proteína da silagem de gliricídia submetida a diferentes períodos de fermentação

The objective of this study was to evaluate the effect of the fermentation time on the qualitative and chemical characteristics of Gliricidia sepium silage grown in the sergipe agreste. A completely randomized experimental design with five treatments and six replicates was used. The treatments were the different opening periods of the silos (7, 14, 30, 42 and 60 days after the closing date). PVC minisilars with 10 cm diameter and 30 cm length were used. The production of lactic acid was influenced (P = 0.01) of the fermentation time, presenting a quadratic positive behavior. The buffer capacity showed a linear behavior (P = 0.02) along the fermentation time. Negative quadratic behavior (P = 0.01) was observed for pH values, as a function of fermentation time. Positive quadratic behavior (P 0.05) of the fermentation time for the other chemical variables with mean values ​​of 230.2 g / kg, 66.4; 514.8; 407.6; 66.7; 107.2; 447.4 and 213.4 (g / kg DM) for MS, MM, NDF, ADF, Lig, Hem, Cel and NFC respectively. No effect (P> 0.05) of the fermentation time was observed for the crude protein (CP) content and the fractionation protein values ​​with mean values ​​of 204.2 g / kg, 356.2, 128.8, 353, 162.6, for PB, fraction A, fraction B1 + B2, fraction B3 and fraction C, respectively. Gliricidia demonstrated potential as a food source in the form of silage, as the fermentation retained its bromatological characteristics and is suitable for consumption of the animals at 60 days of fermentation.

Pinning and disorder relevance for the lattice Gaussian Free Field II: The two dimensional case

This paper continues a study initiated in [34], on the localization transition of a lattice free field on $\mathbb Z^d$ interacting with a quenched disordered substrate that acts on the interface when its height is close to zero. The substrate has the tendency to localize or repel the interface at different sites. A transition takes place when the average pinning potential $h$ goes past a threshold $h_c$: from a delocalized phase $h h_c$ where the field sticks to the substrate. Our goal is to investigate the effect of the presence of disorder on this phase transition. We focus on the two dimensional case $(d=2)$ for which we had obtained so far only limited results. We prove that the value of $h_c(\beta)$ is the same as for the annealed model, for all values of $\beta$ and that in a neighborhood of $h_c$. Moreover we prove that in contrast with the case $d\ge 3$ where the free energy has a quadratic behavior near the critical point, the phase transition is of infinite order $$\lim_{u\to 0+} \frac{ \log \mathrm{F}(\beta,h_c(\beta)+u)}{(\log u)}= \infty.$$

A study of the nonlinearity of a building thermal behavior based on metamodeling

In this work, we present a study of the nonlinearity of building thermal behavior based on a metamodel for cooling energy needs. We studied the nonlinearity of the thermal behavior of an office. The building quadratic behavior and interactions between its components were analyzed based on the metamodel coefficients. The metamodel was fitted with a reduced number of dynamic simulations. The nonlinearity was first assessed as function of the mean outdoor air temperature in fifteen typical European climates and then as function of the internal heat gains for the coldest and hottest climates. The metamodel provided highly accurate results with fast calculation time. However, a higher accuracy was generally obtained for hot climates, high internal heat gains and lightweight thermal mass. Conversely, the nonlinearity of thermal behavior was accentuated in cold climates and with low internal heat gains. Moreover, the interactions between the building components were found to be more influential on cooling energy needs than quadratic behavior. We propose a classification of thermal behavior into three regimes: Highly nonlinear when the energy needs are close to zero; intermediate with decreasing nonlinearities that can be expressed by power functions; and finally, a quasi-linear regime with almost-steady nonlinearities.

Cryopreservation of semen from goats fed a diet supplemented with flaxseed

ABSTRACT The objective of this study was to evaluate the effects of the inclusion of flaxseed in the diet of male goats on the resistance of semen to the cryopreservation process. Sixteen males were distributed in four groups and fed a diet supplemented with 0, 4, 8, and 12% of flaxseed for a period of 60 days. Only the ejaculates that presented motility and vigor above 70% and 3, respectively, were sent through the cryopreservation process. After thawing, the semen was evaluated through thermo resistance, hypoosmotic, and acrossomal integrity tests. The data were submitted to analysis of variance and regression at 5% of significance. There was positive quadratic behavior for motility after 60, 120, and 180 min in the thermoresistance test (TTR), and positive quadratic behavior for sperm vigor after thawing after 120 and 180 min in the TTR (P &lt; 0.05). However, negative quadratic behavior was obtained for plasma membrane integrity according to the hypoosmotic test (P &lt; 0.05) and there was a difference in the acrosomal integrity test, presenting an optimum maximum level of 3.25% of flaxseed for acrosomal integrity of 65.83% (P &lt; 0.05). The results obtained demonstrated that the addition of as much as 12% of flaxseed to the diet of goat breeders improved post-thawing sperm quality.

Levels of inclusion of spineless cactus (Nopalea cochenillifera Salm Dyck) in the diet of lambs

The objective of this study was to evaluate the intake of nutrients and water, digestibility, nitrogen balance, microbial protein synthesis, performance and blood parameters of lambs fed increasing levels (0, 150, 300 and 450 g/kg of DM basis) of fresh spineless cactus cv. Miúda (Nopalea cochenillifera Salm Dyck) included in the diet. Forty male (noncastrated) crossbred Santa Inês lambs, averaging 18.6 ± 2.8 kg body weight (BW) at the beginning of the study, were used in this trial. A completely randomized design was used, with ten replicates per treatment and initial BW as a covariate. Quadratic behaviour was observed for dry matter (DM), organic matter (OM), crude protein (CP), ether extract (EE), total carbohydrates (TC) and metabolizable energy (ME) intake, with maximum values estimated with 242.9, 236.4, 229.2, 218.2, 238.2 and 284.5 g/kg of spineless cactus inclusion, respectively. Non-fibrous carbohydrates (NFC) and neutral detergent fibre (NDF) intakes presented higher and lower values respectively in lambs fed 450 g/kg of spineless cactus. Voluntary water intake decreased with increasing quantities of spineless cactus, however, water intake from feed and total water intake increased. The digestibility of DM, OM, NFC and TC presented higher values with 450 g/kg of spineless cactus. Total weight gain and daily weight gain showed an increasing linear effect, while feed conversion decreased. Urine volume and microbial protein were higher for lambs fed 450 g/kg of spineless cactus, while urea excretion was lower. There was a quadratic effect on serum glucose and calcium levels with maximum values for inclusion of spineless cactus of 368.9 and 150 g/kg respectively, whereas ß-hydroxybutyrate and blood urea decreased linearly; phosphorus and magnesium concentrations increased, with lower and higher values in animals fed the highest levels of spineless cactus, respectively. The inclusion of spineless cactus by up to 450 g/kg of DM improves the microbial efficiency, nutrient utilization and the performance of the animals, without causing adverse effects on blood parameters.

Effect of Thermal Vacancy on Thermodynamic Behaviors in BCC W Close to Melting Point: A Thermodynamic Study.

As temperature increases, the thermal vacancy concentration in pure metals dramatically increases and causes some strongly non-linear thermodynamic behaviors in pure metals when close to their melting points. In this paper, we chose body-centered cubic (bcc) W as the target and presented a thermodynamic model to account for its Gibbs energy of pure bcc W from 0 K to melting point by including the contribution of thermal vacancy. A new formula for interaction part was proposed for describing the quadratic temperature behavior of vacancy formation energy. Based on the experimental/first-principles computed thermodynamic properties, all the parameters in the Gibbs energy function were assessed by following the proposed two-step optimization strategy. The thermodynamic behaviors, i.e., the strong nonlinear increase for temperature dependence of heat capacities at high temperatures and a nonlinear Arrhenius plot of vacancy concentration, in bcc W can be well reproduced by the obtained Gibbs energy. The successful description of thermal vacancy on such strongly non-linear thermodynamic behaviors in bcc W indicates that the presently proposed thermodynamic model and optimization strategy should be universal ones and are applicable to all other metals.