Thomson Equation Research Articles

Solvents effects in the formation and viscoelasticity of DBS organogels

The gel behavior of dibenzylidene sorbitol (DBS) in different solvents was studied by rheology, optical microscopy and transmission electron microscopy. It was found that DBS molecules would self-assemble into fibrils with a helical structure, and the fibrillar diameter decreased with increasing solvent polarity. The gel dissolution temperature was extremely sensitive to the DBS concentration and solvent polarity, which could be explained by the Gibbs–Thomson equation through interfacial stress. The phase diagram, which classified the sol, the gel and the cluster regime in different solvents was determined by rheology and optical microscopy. The critical gel concentration was found to increase as the difference in the polar and hydrogen-bonding components of the solubility parameter (Δδph) between the gelator and solvents decreased. Meanwhile, the difference in the critical gel concentration in different solvents could explain the difference in the relaxation exponent and the gel strength at the gel point well. In the stable gel state, the plateau modulus depended on the gelator concentration according to a power-law scaling, GoN ∝ c2, which was consistent with entanglement theory and independent of the type of solvent. However, the gelator–solvent interaction and its temperature dependence were found to affect the stability of the gel substantially under large amplitude oscillatory shear. The critical strain would increase as Δδph decreased, and showed more evident temperature dependence on solvents with a smaller Δδph.

一类Thomson方程边值问题解的相似构造法

针对Thomson方程的一类边值问题的解式进行研究，发现其解式可由相似核函数和非齐次左边值条件中的系数进行组装得到，而相似核函数由引解函数以及齐次右边值条件中的系数来构造，由此获得了求解该类边值问题的一种新方法——相似构造法。 Aiming at the solution for a class of boundary value problem of Thomson equation, we found that its solution can be obtained by structuring similar kernel function with the coefficients of non-homogeneous left boundary condition. While the similar kernel function is constructed by both the function of guide solution and the coefficients of homogeneous right boundary condition. Thus, a new method of solving such kind of boundary value problem is obtained—Similar Constructive Method.

Numerical integration of the Gibbs–Thomson equation for multicomponent systems

The differential form of the Gibbs–Thomson equation is derived for non-stoichiometric, partially stoichiometric and fully stoichiometric precipitates in a multicomponent system. This form can be readily used in a numerical integration scheme based on separation of variables. The validity of the proposed approach has been demonstrated with binary (Al–Sc) and ternary (Al–Mn–Si) systems. Good agreement with other approaches (e.g. analytical or Thermo-Calc) has been shown. The proposed approach aims at bridging the gap between open thermodynamic databases and precipitation models.

Stepwise quasi-equilibrium crystallization of low-density polyethylene

Hydrostatic weighing of a polymer in an immersion liquid using low-density polyethylene as an example was used to show that a condition for the lack of recrystallization of partially crystalline polymers upon annealing is the attainment at this temperature of thermomechanical equilibrium that determines the ratio of the amounts of macromolecule elementary units existing in amorphous and crystalline regions during the crystallization of these polymers. A basis was found for the viewpoint that the Gibbs—Thomson equation is applicable only for estimating the dimensions of the last (not bonded to each other by connecting chains) crystallites disappearing at the true melting point.

Average Sequential Water Molecule Binding Enthalpies of M(H2O)19−1242+ (M = Co, Fe, Mn, and Cu) Measured with Ultraviolet Photodissociation at 193 and 248 nm

The average sequential water molecule binding enthalpies to large water clusters (between 19 and 124 water molecules) containing divalent ions were obtained by measuring the average number of water molecules lost upon absorption of an UV photon (193 or 248 nm) and using a statistical model to account for the energy released into translations, rotations, and vibrations of the products. These values agree well with the trend established by more conventional methods for obtaining sequential binding enthalpies to much smaller hydrated divalent ions. The average binding enthalpies decrease to a value of ~10.4 kcal/mol for n > ~40 and are insensitive to the ion identity at large cluster size. This value is close to that of the bulk heat of vaporization of water (10.6 kcal/mol) and indicates that the structure of water in these clusters may more closely resemble that of bulk liquid water than ice, owing either to a freezing point depression or rapid evaporative cooling and kinetic trapping of the initial liquid droplet. A discrete implementation of the Thomson equation using parameters for liquid water at 0 °C generally fits the trend in these data but provides values that are ~0.5 kcal/mol too low.

The modified Thomson equation in the theory of heterogeneous vapor nucleation on charged solid particles

Thermodynamic theory of heterogeneous vapor nucleation on charged nanosized solid particles is considered and formulas for the condensate chemical potential in the droplet and the nucleation activation barrier are derived. A modified Thomson equation for the pressure of vapor saturated over a liquid droplet formed around the spherical charged particle is proposed. This equation determines the saturated vapor pressure as a function of the particle size, charge magnitude and sign, and the droplet radius. The approach assumes that the droplet liquid film between the particle and the vapor phase is polarized in the central electric field of the charged particle and is thin, i.e., surface layers of the film at the boundaries with the particle and the vapor phase are overlapped and the arising disjoining pressure affects the droplet state. It is shown that the film disjoining pressure, the Maxwell stress of the film in the central electric field of the particle, the degree of the particle wetting, and the particle size interplay in determination of the threshold value of vapor supersaturation ratio at barrierless nucleation. The threshold values of the vapor supersaturation ratio were found for different model isotherms of the disjoining pressure and different relations between size and charge of the solid particles.

Use of the successive self-nucleation and annealing technique to characterize 60Co gamma irradiated HDPEs

The application of the Successive Self-nucleation and Annealing (SSA) thermal fractionation technique can yield detailed information of the structural changes induced in linear polyethylene by irradiation. The production of tertiary carbons during the crosslinking reactions can be equivalent to the structural heterogeneity present in branched polyethylenes since in both cases interruption of the linear crystallizable sequences occurs, and these are structural differences that can be easily detected by thermal fractionation. We demonstrate how correlations between melting point and short chain branching content employed for branched polymers can be useful to characterize the distribution of chain heterogeneity produced by crosslinking. As the radiation dose is increased and the crosslinking content also increases, the distribution of chain heterogeneity gets broader as detected by SSA. When the results are coupled with morphological observations made by transmission electron microscopy, valuable information on the morphological changes produced by crosslinking can also be ascertained, since the distribution of lamellar thicknesses substantially broadens with crosslinking. Such a broad distribution can also be predicted from SSA by simple calculations performed employing a modified version of the Gibbs–Thomson equation and is expected on the basis of random crosslinking reactions.

The modified Thomson equation in the theory of heterogeneous vapor nucleation on charged solid particles

Thermodynamic theory of heterogeneous vapor nucleation on charged nanosized solid particles is considered and formulas for the condensate chemical potential in the droplet and the nucleation activation barrier are derived. A modified Thomson equation for the pressure of vapor saturated over a liquid droplet formed around the spherical charged particle is proposed. This equation determines the saturated vapor pressure as a function of the particle size, charge magnitude and sign, and the droplet radius. The approach assumes that the droplet liquid film between the particle and the vapor phase is polarized in the central electric field of the charged particle and is thin, i.e., the surface layers of the film at the boundaries with the particle and the vapor phase are overlapped and the arising disjoining pressure affects the droplet state. It is shown that the film disjoining pressure, the Maxwell stress of the film in the central electric field of the particle, the degree of the particle wetting, and the particle size interplay in determination of the threshold value of vapor supersaturation ratio at barrierless nucleation. The threshold values of the vapor supersaturation ratio are found for different model isotherms of the disjoining pressure and different relations between size and charge of the solid particles.

Application of nmr spectroscopy to determine the thermodynamic characteristics of water bound to OX-50 nanosilica

We have used low-temperature 1H NMR spectroscopy to determine the thermodynamic characteristics of water bound to OX-50 nanosilica (SBET ≈ 50 m2/g) in different media: aqueous, air, chloroform medium, and gaseous methane. We demonstrate the difference between the hydration parameters of silica OX-50 on going from an aqueous suspension to a hydrated powder. We present the water cluster size distributions in the studied systems, calculated from the Gibbs–Thomson equation. We found that the average water cluster size in suspension is considerably larger than the cluster sizes in hydrated powders.

Thermodynamic description of condensed phases

Phases may be smaller than visible to the human eye. In order to characterize a microphase, a phase smaller than 1 μm, one must consider surface area and free energy in addition to the standard thermodynamic functions. As one approaches nanometer sizes, one also needs to know the changing thermodynamic functions within the phases. The Gibbs–Thomson equation can be used to characterize microphases, but not nanophases. For the latter, the glass transition is needed to assess the properties in the interior. In order to classify condensed phases as liquid, solid, mesophase, or crystal, one needs to consider the molecular motion in addition to the molecular structure. Most important are large-amplitude displacements in form of translation, rotation, and conformational motion. An operational definition based on experiments and an updated classification of the phases is given. The surprising result is the observation that crystals, earlier assumed prime examples of solids, can have order–disorder transitions to more mobile mesophases, as well as a glass transition without change in crystal structure, i.e., under certain condition, they cannot be identified as a solid. To these observations, one has to add the fact that large-amplitude motion may start gradually to a more mobile phase without abrupt changes in structure. These observations limit the usefulness of the 80-year-old classification of transitions as being of first or second order. Quantitative thermal analysis is shown to be an important tool to identify the possible total of 57 different condensed states in terms of their macroscopic properties as well as molecular structure and motion.

Homogeneous water nucleation in a laminar flow diffusion chamber

Homogeneous nucleation rates of water at temperatures between 240 and 270 K were measured in a laminar flow diffusion chamber at ambient pressure and helium as carrier gas. Being in the range of 10(2)-10(6) cm(-3) s(-1), the experimental results extend the nucleation rate data from literature consistently and fill a pre-existing gap. Using the macroscopic vapor pressure, density, and surface tension for water we calculate the nucleation rates predicted by classic nucleation theory (CNT) and by the empirical correction function of CNT by Wolk and Strey [J. Phys. Chem. B 105, 11683 (2001)]. As in the case of other systems (e.g., alcohols), CNT predicts a stronger temperature dependence than experimentally observed, whereas the agreement with the empirical correction function is good for all data sets. Furthermore, the isothermal nucleation rate curves allow us to determine the experimental critical cluster sizes by use of the nucleation theorem. A comparison with the critical cluster sizes calculated by use of the Gibbs-Thomson equation is remarkably good for small cluster sizes, for bigger ones the Gibbs-Thomson equation overestimates the cluster sizes.

Measurement and simulation of dendritic growth of ice in cement paste

We re-examine experiments by Helmuth (1962) from which he concluded that ice grows in the pores of cement paste under heat-flow control, and that the internal temperature rises to the melting point given by the Gibbs–Thomson equation. We find that his conclusions are correct, but the growth rates he reports are misleading. Using experimental and computational methods, we show that the ice grows in the form of dendrites, which allows a constant growth rate under heat-flow control. A modification of the experimental procedure permits accurate measurement of the growth rate of ice in the pores.

Experimental Measurement and Thermodynamic Modeling of the Mixed CH4 + C3H8 Clathrate Hydrate Equilibria in Silica Gel Pores: Effects of Pore Size and Salinity

We measured hydrate phase equilibria for the ternary CH(4) (90%) + C(3)H(8) (10%) + water mixtures in silica gel pores with nominal diameters of 6.0, 15.0, 30.0, and 100.0 nm and for the quaternary CH(4) (90%) + C(3)H(8) (10%) + NaCl + water mixtures of two different NaCl concentrations (3 and 10 wt %) in silica gel pores with nominal diameters of 6.0, 15.0, and 30.0 nm. The CH(4) (90%) + C(3)H(8) (10%) hydrate-water interfacial tension (sigma(HW)) of 42 +/- 3 mJ/m(2) was obtained through the Gibbs-Thomson equation for dissociation within cylindrical pores. With this value, the experimental results were in good agreement with the calculated ones based on the van der Waals and Platteeuw model. A correction term for the capillary effect and a Pitzer model for electrolyte solutions were adopted to calculate the activity of water in the aqueous electrolyte solutions within silica gel pores. At a specified temperature, three-phase H-L(W)-V equilibrium curves of pore hydrates were shifted to higher-pressure regions depending on pore sizes and NaCl concentrations. From the cage-dependent (13)C NMR chemical shifts of enclathrated guest molecules, the mixed CH(4) (90%) + C(3)H(8) (10%) gas hydrate was confirmed to be structure II.

1H NMR cryoporosimetry of swollen hypercrosslinked polymers

The porous structure of two laboratory samples of hypercrosslinked polystyrene networks with a crosslinking degree of 200%, the commercial hypercrosslinked sorbent MN-270 (Purolite Int.), and macroporous polystyrene sorbents Amberlite XAD-4 and Amberlite XAD-1600 were investigated by means of 1H NMR cryoporosimetry. It was determined that the signal intensity of benzol, dioxane, dichloroethane and water protons previously frozen in polymer pores increases during solvent melt. Dependencies of the integral proton signal intensities on the sample temperature obtained in various solvents are discussed in terms of sample micropore accessibility and solvent—polystyrene interaction energy. Pore dimensions are estimated using the melting temperature value of ice according to the Gibbs—Thomson equation. We conclude that the obtained distribution curves reveal all hypercrosslinked polymers to be mainly microporous with diameter distribution maxima around 10 A. It is shown that along with pores 120 A in diameter, XAD-4 has a considerable amount of micropores. In contrast, XAD-1600 is ranged as a mesoporous sorbent.

Synthesis and DSC study on Sn3.5Ag alloy nanoparticles used for lower melting temperature solder

The traditional Sn–Pb eutectic solder alloys are being phased out from the electronics industry due to the toxicity of lead (Pb), leading to the development and implementation of lead-free solders. Sn3.5Ag lead-free solder alloy, considered to be one of the promising alternatives to replace the traditionally used Sn–Pb solder, however, still has some weaknesses, such as its higher melting temperature than that of the Sn–Pb solder alloy. A possible way to decrease the melting temperature of a solder alloy is to decrease the alloy particle size to the nanometer range. Sn3.5Ag nanoparticles with different size distribution were synthesized using chemical reduction method by applying NaBH4 as reduction agent. The melting properties of these synthesized nanoparticles were studied by differential scanning calorimetry (DSC), and size-dependent melting temperature depression of these nanoparticles has been observed. Gibbs–Thomson equation was used to analyze the size-dependent melting temperature property, giving a good prediction of the melting temperature depression for the Sn-based lead-free solder alloy nanoparticles.

Thermodynamic properties of HFO-1234yf (2,3,3,3-tetrafluoropropene)

Measurements of the thermodynamic properties of HFO-1234yf were conducted. The critical temperature T C, critical density ρ C, and critical pressure P C, were measured by the visual observation of the meniscus disappearance, and were determined to be 367.85 ± 0.01 K, 478 ± 3 kg/m 3, and 3382 ± 3 kPa, respectively. Vapor pressures were measured by a batch-type calorimeter with a metal-bellows in the temperature range from 310 K to 360 K and correlated by the Wagner-type equation. Surface tensions were measured by the differential capillary-rise method in the temperature range from 273 K to 340 K and correlated by the van der-Waals type equation. The acentric factor was determined to be 0.280 with the vapor pressure correlation. Based on the critical parameters and acentric factor, saturated vapor and liquid densities were estimated by the Peng–Robinson equation and the Hankinson–Thomson equation, respectively. The heat of vaporization was also calculated from the Clausius–Clapeyron equation.

Effect of the Alumina Shell on the Melting Temperature Depression for Aluminum Nanoparticles

The dependence of aluminum (Al) melting temperature on particle size was studied using a differential scanning calorimeter and thermogravitmetric analyzer for particles encapsulated in an oxide shell. Pressure generation within the Al core leads to an increase in melting temperature in comparison with traditional melting temperature depression calculated using the Gibbs−Thomson equation. On the basis of elasticity theory, the pressure in the Al core at the onset of melting is caused mainly by surface tension at the alumina−air and Al−alumina interfaces. This implies that pressure due to the difference in thermal expansion of aluminum and alumina relaxes. A possible relaxation mechanism is discussed. The static strength of the alumina shell and the maximum static generated pressure in aluminum were evaluated. Mechanically damaging the oxide shell was shown to reduce the melting temperature due to a decrease in generated pressure within the Al core. Thus, reduction in melting temperature can be used as a qu...

Nanoparticles of SnAgCu lead-free solder alloy with an equivalent melting temperature of SnPb solder alloy

Nanoparticles were prepared with a consumable-electrode direct current arc (CDCA) technique. The results showed that the calorimetric melting onset temperature of the nanoparticles of SnAgCu solder alloy could be as low as 179 °C, equivalent to that of the traditionally used SnPb eutectic alloy (183 °C). Moreover, the homogenous melting model (HMM) and Gibbs–Thomson equation were employed to theoretically estimate the size-dependent melting temperature of the as-prepared nanoparticles. The structure and morphology of the nanoparticles were analyzed with a high-resolution transmission electron microscopy (HRTEM). The CDCA technique showed promising prospect in manufacturing large amounts of nanoparticles with controlled shape, small size, narrow particle size distribution and nearly oxide-free composition. This undoubtedly puts forward a novel feasible approach to manufacture high quality lead-free solders for electronic products.

Determination of crystal nucleus size of potassium chloride from ethanol solution caused by ultrasonic irradiation

The solubilities of electrolyte salts, such as sodium chloride (NaCl) and potassium chloride (KCl), are small in alcohol solution. As an application of such dilute solubility, nucleation phenomena were investigated by the dynamic laser scattering (DLS) method at different supersaturation degrees, prepared by different solubilities of KCl in aqueous ethanol solutions. The supersaturated solution of KCl was very stable, and therefore ultrasonic irradiation was utilized as a trigger of nucleation. The time-change of the sizes of fine KCl crystals was measured, and the nucleus size was obtained by extrapolating the size of the fine crystals to zero-time. The nucleus size of KCl was correlated by the modified Gibbs–Thomson equation as a function of supersaturation. It was found that the nucleus size tended to increase when supersaturation decreased, but three different curves for three different temperatures could be obtained. The modified Gibbs–Thomson equation represented the nucleus size at three different temperatures. The correlated “minimum supersaturation energy” parameters were converged to zero at the saturated temperature. The parameter “minimum supersaturation energy” could be interpreted as “ultrasonic effective energy” that could be defined as the received energy in the solution from ultrasonic irradiation.

A simple classical model for predicting onset crystallization temperatures on curved substrates and its implications for phase transitions in confined volumes

For small confinement volumes, phase transition temperatures are determined by the scarcity of the crystallizing material, rather than the magnitude of the energy barrier, as the supply of molecules undergoing the phase transition can be depleted before a stable nucleus is attained. We show this for the case of crystallization from the melt and from the solution by using a simple model based on an extended classical nucleation theory. This has important implications because it enables a simple and direct measurement of the critical nucleus size in crystallization. It also highlights that predicting the observable melting points of nanoparticles by using the Gibbs-Thomson equation can lead to substantial errors.