Surface energy and pressure of diamond and silicon nanocrystals

Based on the pair potential of interatomic interaction, we study the dependence of various properties of diamond and silicon nanocrystals with a free surface on size, surface shape, and temperature. A model nanocrystal has the form of a parallelepiped faceted by {100} planes with a square base. The number of atoms N in the nanocrystals is varied from 5 to infinity. The Debye temperature, Gruneisen parameter, specific surface energy, isochoric derivative of specific surface energy with respect to temperature, and surface pressure are calculated as a function of the size and shape of diamond and silicon nanocrystals at temperatures ranging from 20 K to the melting point. The surface pressure P _sf(N) ∼ N ^−1/3 is much lower than the pressure calculated by the Laplace formula for similar nanocrystals for given values of density, temperature, and number of atoms. As the temperature increases from 20 K to the melting point, the isotherm P _sf(N) lowers and changes the shape of the dependence on N; at high temperatures, it goes to the region of extension of small nanocrystals of diamond and silicon.     

Oxygen diffusion in uranium dioxide in the temperature range of phase transitions

The structure of and oxygen diffusion in UO₂ are studied by the molecular dynamics method in the range of transition to the superionic state (melting of the oxygen sublattice) and near the melting point of UO₂. The temperature dependence of the diffusion coefficient of a doubly charged oxygen ion in UO₂ is constructed. In the crystalline state at temperatures between 1800 and 2600 K, this dependence is described by an exponential dependence with a diffusion activation energy of 2.6±0.2 eV. In the superionic state (2600–3100 K), the activation energy of diffusion of an oxygen anion decreases to 1.88±0.13 eV. In melt (3100–3600 K), the exponential dependence of the diffusion coefficient of O^2- persists but the activation energy of diffusion decreases still further, to 0.8±0.2 eV. Our experimental results agree (within the limits of experimental error) with data on oxygen diffusion in the crystalline phase obtained by other researchers.     

Shape and size dependent thermophysical properties of nanocrystals

A theoretical model based on thermodynamic variables is employed in the present work to study the thermophysical properties of nanomaterials of different shapes and sizes. The model proposed by Qi and Wang [19] is applied to determine the cohesive energy of nanomaterial. The number of atoms on the surface to the total number of atoms in nanosolid is considered in terms of shape factor (α) and size of nanocrystal. The variation of cohesive energy?(E_cn?), melting temperature?(T_mN), Debye temperature (θ_DN), Specific heat capacity (C_pN), and Energy band gap (E_gN?) is studied for spherical, regular tetrahedral, regular hexahedral and regular octahedral nanocrystals. The cohesive energy, melting temperature and Debye temperature are found to decrease as the grain size is reduced. However, the energy band gap and specific heat capacity are found to increase with decrease of grain size of nanomaterial. The results achieved in the present study are compared with the available experimental and also with those calculated from other theoretical models. The consistency between the present calculated results and the results reported earlier confirms the validity of the present model theory to explain the shape and size dependence of thermophysical properties of nanomaterials.     

Dependence of the surface energy on the size and shape of a nanocrystal

An expression is derived for the surface energy σ as a function of the size and shape of a nanocrystal. It is shown that the wider the deviation of the shape parameter f from unity, the more pronounced the decrease in the surface energy σ with a decrease in the number N of atoms in the nanocrystal. The dependences of the average coordination number, the surface energy, and the melting temperature on the number N exhibit an oscillatory behavior with maxima at points corresponding to numbers of atoms forming a defect-free cube. The surface energy decreases with an increase in the temperature T. It is found that the smaller the nanocrystal size or the greater the deviation of the nanocrystal shape from the thermodynamically most stable shape (a cube), the larger the quantity-(dσ/dT). It is established that the nanocrystal undergoes melting when the surface energy decreases to a value at which it becomes independent of the nanocrystal size and shape. The conditions providing fragmentation and dendritization of the crystal are discussed. It is demonstrated that, at N>1000, the dependence σ(N) coincides, to a high accuracy, with the dependence of the surface tension of the nanocrystal on N. The inference is made that bimorphism is characteristic of nanocrystals. This implies that nanocrystals can have platelike and rodlike shapes with equal probability.     

On the size dependence of surface tension in the temperature range from melting point to critical point

The problem of size dependence of surface tension was investigated in view of a more general problem of the applicability of Gibbs’ thermodynamics to nanosized objects. For the first time, the effective surface tension (coinciding with the specific excess free energy for an equimolecular dividing surface) was calculated within a wide temperature range, from the melting temperature to the critical point, using the thermodynamic perturbation theory. Calculations were carried out for Lennard-Jones and metallic nanosized droplets. It was found that the effective surface tension decreases both, with temperature and particle size.     

On the surface properties of a nanodiamond

The dependences of the specific surface energy, the surface tension, and the surface pressure on the size, the surface shape, and the temperature of a nanodiamond with a free surface have been investigated using the Mie-Lennard-Jones interatomic interaction potential. The nanocrystal has the form of a parallelepiped faceted by the (100) planes with a square base. The number of atoms N in the nanocrystal varies from 5 to ∞. The isothermal isomorphic dependences of the specific surface energy, its isochoric derivative with respect to the temperature, the surface tension, and the surface pressure on the nanodiamond size have been calculated at temperatures ranging from 20 to 4300 K. According to the results of the calculations, the surface energy under this conditions is positive, which indicates that the nanodiamond cannot be fragmented at temperatures up to 4300 K. The surface pressure for the nanodiamond P _sf (N) ∼ N ^−1/3 is considerably less than the Laplace pressure P _ls (N)^−1/3 ∼ N ^−1/3 for the same nanocrystal at the given values of the temperature, the density, and the number of atoms N. As the temperature increases from 20 to 4300 K, the lowering of the isotherm P _sf (N) is considerably more pronounced than that of the isotherm P _ls (N). At high temperatures, the isotherm P _sf (N) changes the shape of the size dependence and goes to the range of extension of small nanocrystals. It has been demonstrated that the lattice parameter of the nanodiamond can either decrease or increase with a decrease in the nanocrystal size. The most significant change in the lattice parameter of the nanodiamond is observed at temperatures below 1000 K.     

Acoustic studies of melting and crystallization of sodium nitrite nanocrystals in the pores of mesoporous silicate matrices

A conventional phase-pulse acoustic method was used to study melting and crystallization of sodium nitrite embedded in the pores of mesoporous silicate matrices. The pore diameter was 20, 37, and 52 Å. The measurements were performed at a frequency of 3–8 MHz in the temperature interval 290–560 K. The temperature dependence of ultrasonic velocity was found to exhibit anomalies corresponding to phase transitions of sodium nitrite. The transitions were smeared in temperature and shifted to lower temperatures from the melting point T _b of bulk sodium nitrite; the shift in crystallization temperature was greater than that of the melting temperature. The irreversible character of melting was revealed. The size dependence of the melting temperature of sodium nitrite was obtained. Phenomena observed in the experiments were discussed with the use of different size effect models.     

The temperature dependence of nanocrystal heat capacity

The temperature dependence of the specific (per atom) entropy and heat capacity of a nanocrystal is studied using a nanocrystal model in the form of a rectangular parallelepiped with variable surface shape. Accounting for the temperature dependence of the surface energy showed that the temperature dependence of the surface contribution to specific entropy is described by the same function that determines the temperature dependence of the isochoric heat capacity of a macrocrystal. Thus, at T → 0 K at T/Θ > 2 the surface contribution to the specific heat is zero. The maximal surface contribution to specific heat is reached at T/Θ = 0.2026 and is equal to c _st/k _B = 1.0115 (where k _B is the Boltzmann constant, Θ is the characteristic temperature depending both on the size and the shape of the nanocrystal). The applicability of the Grüneisen rule for a nanocrystal both at low and high temperatures is studied. It has been found that a case when the surface contribution to specific heat would be negative c(N) < c(∞), i.e. c _st(N) < 0 can occur for nanocrystals with a noncubic habitus.     

Influence of particle size,stoichiometry, and degree of long-range order on magnetic susceptibility of titanium monoxide

In situ measurements of the magnetic susceptibility of ordered and disordered titanium monoxides TiO_y in the temperature range from 300 to 1200 K have revealed that it depends on the size of crystals, their stoichiometry, and long-range order parameters. Analysis of the data for both the ordered and disordered TiO_y has demonstrated that the dependence of the Van Vleck paramagnetism on the nanocrystal size is inversely proportional due to the breaking of symmetry of the local environment of titanium and oxygen atoms near the surface of nanocrystals. It has been found that the Van Vleck contribution from the atomic vacancy disorder in monoxide nanocrystals of superstoichiometric composition, as well as in the crystalline stoichiometric monoxide, is proportional to the deviation of the degree of long-range order from the maximum value.     

Surface tension (γLV), surface energy (γSV) and crystal-melt interfacial energy (γSL) of metals

The results of temperature-dependent surface tension calculations of pure liquids aluminium (933-1200 K) and iron (1811-2500 K), in the framework of the theoretical considerations suggested by Eyring, are presented. It is observed that the surface tension decreases linearly with temperature. The calculated surface tension data are fitted as γ = 985-0.275(T − Tm) and γ = 1560-0.387(T − Tm) for Al and Fe, respectively. Moreover, the surface tension (γ_LV) at melting point, surface energy (γ_SV) and crystal-melt interfacial energy (γ_SL) are calculated for many metals. The agreement between the calculated and the reported measured values is reasonable.     

On viscous mechanism for surface diffusion at high temperatures (T/Tm > 0.75) due to formation of a 2D dense fluid on metallic surfaces

Experimental determinations of temperature dependence of surface self-diffusion coefficient of several metals exhibit a strong increase in D_s values and in activation energy for temperatures near the melting point T_m. This variation is illustrated by a bending of the Arrhenius plot of surface self-diffusion coefficients of tungsten, which are obtained experimentally by tip profile variation technique. For T/T_m < 0.75 the apparent activation energy for W is 2.85 eV and the pre-exponential term is equal to 0.24 cm²/s, while for T/T_m > 0.75 we have respectively 5.57 eV and 1.08 × 10⁴ cm²/s. To account for these unexpected variations in the activation energy and diffusivities, the hypothesis that the surface mass transport mechanism changes from individual atomic jumps at low temperatures towards a cooperative motion at temperatures near the bulk melting point, namely a viscous mechanism, is proposed. This model is based on the postulation of the formation of a 2D dense fluid on the metallic surfaces about 75% of the bulk melting temperature. Discussions of existing models on surface diffusion proposed by Rhead, by Bonzel, or by Tsong are given, and a technique to characterize surface viscosity of a 2D dense fluid is suggested.     

Temperature dependence of trap luminescence of CdS doped glasses

The temperature dependence of the optical absorption edge and integrated luminescence of CdS semiconductor doped glasses, having similar characteristics but manufactured by different manufactures like Corning 373 and Schott 1743 has been investigated. Transmission Electron Microscope (TEM) analysis is presented for these samples to understand the contribution of the inhomogeneous distribution of nanocrystal size. The energy gap shrinkage coefficient (α) and effective phonon energy (θ) have been derived from the temperature dependence of the energy gap. The activation energy of thermal photoluminescence (PL) quenching has been derived from the temperature dependence of luminescence intensity. An anomalous behaviour of the luminescence intensity and of bandwidth in the temperature range 50-10 K has been discussed using configuration co-ordinate model. The difference in the behaviour of different samples in the range 50-10 K has been suggested to arise from the inhomogeneous size distribution, which is supported by TEM analysis.     

On the pressure dependence of the effective ionic charge of cesium halides

The importance of temperature effects in the use of the second Szigeti relation to calculate the volume dependence of the effective ionic charge of cesium halide crystals is studied. The effect obtained is larger than the one for alkali halide crystals of the NaCl structure. Positive values of (? ln s/? In v) are obtained using low temperature experimental data with the Grüineisen parameter γ_t(T → 0K) calculated using the generalized first Szigeti relation. Values of γ_t are also obtained with a rigid ion model and they differ little from the previous ones.     

Elastic properties of diamond,silicon, and germanium nanocrystals as functions of their size and shape

The dependence of elastic modulus B on the size (number of atoms (N)) and the shape of a nanocrystal of a simple monoatomic substance is studied when the nanocrystal is considered as a rectangular parallelepiped with a varied surface shape. The elastic modulus is shown to decrease during an isomorphic-isothermal decrease of the nanocrystal size. At low temperatures, the B(N) dependence is less pronounced, and the case where the B(N) function increases during an isomorphic-isothermal decrease of the nanocrystal size is possible here. The size dependences of the elastic modulus, Poisson ratio μ, Young’s modulus Y, shear modulus G, and lattice parameter compression are calculated for diamond, Si, and Ge. It is shown that B, Y, and G decrease and μ increases during an isomorphic-isothermal decrease of the nanocrystal size. The surface pressure compresses a nanocrystal at low temperatures and expands it at high temperatures. The larger the deviation of the nanocrystal shape from the most energetically favorable shape, the more pronounced the changes of these functions during an isothermal decrease of the nanocrystal size.     

Critical behaviour of the anchoring energy of the director at the free surface of a nematic liquid crystal

A new structural transition occurs at the free surface of some nematic liquid crystals when the temperature reaches a critical value T₀. In this work we study the temperature dependence of the anchoring energy of the director at the free surface close to the critical point. We find that the anchoring energy tends to zero with the critical exponent δ = 1 when the temperature approaches the critical value T₀. The experimental results are interpreted in terms of the Parsons and Mada theories.     

Analysis of dispersion of long-wavelength optical phonons in zinc with the help of light scattering

The temperature dependences (5–300 K) of the Raman spectra of E _2g phonons and optical constants in zinc single crystals are measured in the excitation energy range 1.4–2.54 eV. It is found that phonon damping decreases upon an increase in the wavelength of exciting radiation. The obtained results are compared with the dependence of the phonon width on the excitation energy (the probed wave vector of the excitations under investigation), which are presented for the first time for the transition metal osmium, as well as with the calculated electron-phonon renormalization of damping, taking into account the actual distribution of wave vectors.     

Diamond-graphite phase transition in ultradisperse-diamond clusters

A systematic study of the diamond-graphite structural phase transition in ultradisperse-diamond clusters obtained by the detonation technique is reported. Samples of two types, differing in the kinetics of detonation-product cooling, were investigated. The phase transition was achieved under heating in an inert atmosphere in the temperature range 720–1400 K. The transition was identified by Raman scattering and x-ray diffraction data. Raman and x-ray characterization showed the ultradisperse diamond, irrespective of the cooling rate used, to be cluster material possessing diamond structure with a characteristic nanocrystal size of 43 Å. The diamond-graphite phase transition in ultradisperse diamond is shown to start from the cluster surface inwards at T _pt≈1200 K, i.e. at substantially lower temperatures than is the case with bulk diamond single crystals.     

Study of surface cell Madelung constant and surface free energy of nanosized crystal grain

Surface cell Madelung constant is firstly defined for calculating the surface free energy of nanosized crystal grains, which explains the physical performance of small crystals and may be greatly beneficial to the analysis of surface states and the study of the dynamics of crystal nucleation and growth. A new approximative expression of the surface energy and relevant thermodynamic data are used in this calculation. New formula and computing method for calculating the Madelung constant α of any complex crystals are proposed, and the surface free energies and surface electrostatic energies of nanosized crystal grains and the Madelung constant of some complex crystals are theoretically calculated in this paper. The surface free energy of nanosized-crystal-grain TiO₂ and the surface electrostatic energy(absolute value) of nanosized-crystal-grain α -Al₂O₃ are found to be the biggest among all the crystal grains including those of other species.     

On the superconducting transition temperature for metallic nanocrystals

A new approach is proposed for calculating the Debye temperature of a nanocrystal in the form of an n-dimensional rectangular parallelepiped with an arbitrary microstructure and the number of atoms N ranging from 2ⁿ to infinity. The geometric shape of the system is determined by the lateral-to-basal edge ratio of the parallelepiped. The size dependences of the Debye and melting temperatures for a number of materials are calculated using the derived relationship. The theoretical curves thus obtained agree well with the experimental data. The calculated dependences of the superconducting transition temperature T _c on the size d of aluminum, indium, and lead nanocrystals are also in reasonable agreement with the experimental estimates of T _c (d). It is demonstrated that, as the nanocrystal size d decreases, the greater the deviation of the nanocrystal shape from an equilibrium shape (in our case, a cube), the higher the temperature of the superconducting transition T _c (d). The superconducting transition temperature is calculated as a function of the thickness (diameter) of a plate (rod) with an arbitrary length. It is found that a decrease in the thickness (diameter) of the plate (rod) leads to an increase in the temperature T _c (z): the looser the microstructure of the metallic nanocrystal, the higher the temperature T _c (z).     

Thermal conductivity of liquid rubidium in the interval of 312–873 K

The thermal conductivity λ and the thermal diffusivity a of liquid rubidium were measured by the laser flash method in the temperature interval from the melting point up to 873 K. The measurement error was 4–6%. The data of this paper were compared with the results of other authors. Approximation equations and the table of reference values for the temperature dependence of λ and a have been obtained. The dependence of the Lorentz number on temperature has been calculated.