纳米晶体熔化温度的尺寸和压力依赖性
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摘要
本文根据Lindemann熔化准则和Mott方程推导出尺寸依赖的熔化温度模型,该模型简单且无自由参数,能够预测不同化学键类型纳米晶体的熔化,既能解释具有自由表面的纳米晶体的过冷现象,又能解释镶嵌于基体的纳米晶体的过热现象,还能应用于不同维数的低维晶体。并通过纳米晶体熔化模型和扩散公式推导出了尺寸依赖的扩散模型。该模型能够很好的解释纳米材料的扩散比大块材料要更加迅速,并且随着纳米材料的尺寸减少,扩散系数逐步增加。并通过模型分析了影响纳米材料扩散的各方面因素。此外,本文还根据表面应力模型和上述尺寸依赖的熔化温度模型,应用Clapeyron方程计算了相变温度的压力效应,建立了氧化物的温度-压力相图。以上各模型的预测结果与实验以及其它理论结果具有良好的一致性。
As the size of low-dimensional materials decreases to nanometer size range, electronic, magnetic, optic, catalytic and thermodynamic properties of the materials are significantly altered from those of either the bulk or a single molecule. Takagi in 1954 demonstrated for the first time that ultrafine metallic particles melt below their corresponding bulk melting temperature. It is now known that the melting temperature of all low dimensional crystals, including metal, semiconductor and organic crystals, depends on their sizes. Although there are relatively extensive investigations on the size-dependent melting of nanocrystals, it has not been accompanied by the necessary investigation of the size-dependent thermodynamics of nanocrystals. Such an investigation should deepen our understanding of the size effect of melting. In particular, a complete understanding of the melting transition in nanocrystals cannot be obtained without a clear understanding of the enthalpy of melting, which is an important property of melting. Recently, a simple model without adjustable parameters for size-dependent melting temperature of nanocrystals under ambient pressure is established based on Lindemann′s criterion for the melting and Mott′s expression for the vibrational entropy at the melting temperature. In comparison with other model conventions, this model covers all the essential considerations of early models and it has wider size range suitability.
With further research of nanomaterial, its diffusion problem gradually becomes one of interesting domains of scientists. Experience results show nanomaterial diffuses much faster than bulk, namely, diffusion of materials is size-dependent. Combining the thermodynamic size-dependent melting temperature model that is based on the Lindemann criterion on melting and Arrhenius diffusion equation, one can obtain size-dependent diffusion equation. The article gives free parameter model with a universal model ,which predicts diffusion coefficient of different nanocyrstal.
As the pressure increases, the structural phase transition under pressure has been a subject of considerable experimental and theoretical research activity in recent years, especially for oxides crystals. The study of temperature-pressure phase diagram for both bulk material and nanocrystals may extend phase transition theory and possible industry applications and enrich the knowledge on the aspect of the structure. Moreover, the pressure effect on the phase transition temperatures is considered according to the Clapeyron equation, and thus, bulk and size-dependent temperature-pressure phase diagrams are calculated. The concrete contents are listed as follows:
1. The size-dependent melting temperature model for nanocrystals is summarized systematically. In terms of such model, the melting temperature of a free nanocrystal decreases as its size decreases while nanocrystals embedded in a matrix can melt below or above the melting point of corresponding bulk crystals. If the bulk melting temperature of nanocrystals is lower than that of the matrix, the atomic diameter of nanocrystals is larger than that of the matrix, and the interfaces between them are coherent or semi-coherent, an enhancement of the melting point is present. Otherwise, there is a depression of the melting point.
2. F. G. Shi theoretically presented a model for size-dependent melting temperature based on Lindemann criterion. The model can perfectly interpret melting behaviour of nanocrystals, it is applicable both for melting depression and for superheating. For free-standing nanocrystals, the fraction of surface atoms increases with size decreasing and the thermal vibration of surface atoms is larger than that of internal atoms, mean square displacement increases with size decreasing, thus the melting temperature decreases with size decreasing. For nanocrystals embedded in a matrix, the melting temperature will decrease when the interface between nanocrystals and matrix is incoherent. When the interface is coherent or semi-coherent, the amptute of surface atoms is less than that of internal atoms leading to mean square displacement of nanocrystals decrease, the melting temperature increases with size decreasing. But this model has drawback for it including a undefined parameter which is obtained through experiment.
3. Whether diffusion coefficient of bulk and that of corresponding nanomaterial essentially is equal at melting temperature, namely, size-independent. According to the assumption, combining the thermodynamic size-dependent melting temperature model that is based on the Lindemann criterion on melting and Arrhenius diffusion equation, one can obtain size-dependent diffusion equation. In our model, we know diffusion coefficient of nanocrystals increases, when its size decreases, the model predicts an exponent relation between the diffusion coefficient and reciprocal radius of nanocrystals. Diffusion coefficient is related to atom size, vibration entropy and dimension of material. We deeply realize the important reason that the reduce of activation energy induces increase of diffusion coefficient is that the excess surface free energies of nanocrystals with the large surface/volume ratio make atoms easily break away and migrate. The predictions of the model are in agreement with the experimental results of Ag diffusing into Au nanoparticles and N diffusing into Fe particle. The article gives free parameter model with a universal model ,which predicts diffusion coefficient of different nanocyrstal.
4. Although the updated temperature-pressure phase diagrams of oxides crystals have been developed experimentally, further theoretical works are still needed due to the limited measuring accuracy of high-temperature and high-pressure experiments. The classic Clapeyron equation governing all first-order phase transitions of pure substances may be useful to determine the temperature-pressure curve theoretically. However, in order to know the relation between the equilibrium values of the pressure and the temperature, only approximate methods can be used in carrying out the integration of the Clapeyron since both the transition enthalpy and the transition volume are functions of pressure and temperature. To find a solution of the Clapeyron equation, as a first-order approximation, two reasonable simplifications that the transition enthalpy is a weak function of the pressure while the transition volume is a weak function of the temperature are assumed. Moreover, the pressure effect on the phase transition temperatures is considered according to the Clapeyron equation, and thus, bulk and size-dependent temperature-pressure phase diagrams are calculated. The model predictions are found to be consistent with the present experimental and other theoretical results. Since the Clapeyron equation may govern all first-order phase transitions, the Clapeyron equation supplies a new way to determine the temperature-pressure phase diagram of materials.
As the size of low-dimensional materials decreases to nanometer size range, electronic, magnetic, optic, catalytic and thermodynamic properties of the materials are significantly altered from those of either the bulk or a single molecule. Takagi in 1954 demonstrated for the first time that ultrafine metallic particles melt below their corresponding bulk melting temperature. It is now known that the melting temperature of all low dimensional crystals, including metal, semiconductor and organic crystals, depends on their sizes. Although there are relatively extensive investigations on the size-dependent melting of nanocrystals, it has not been accompanied by the necessary investigation of the size-dependent thermodynamics of nanocrystals. Such an investigation should deepen our understanding of the size effect of melting. In particular, a complete understanding of the melting transition in nanocrystals cannot be obtained without a clear understanding of the enthalpy of melting, which is an important property of melting. Recently, a simple model without adjustable parameters for size-dependent melting temperature of nanocrystals under ambient pressure is established based on Lindemann′s criterion for the melting and Mott′s expression for the vibrational entropy at the melting temperature. In comparison with other model conventions, this model covers all the essential considerations of early models and it has wider size range suitability.
With further research of nanomaterial, its diffusion problem gradually becomes one of interesting domains of scientists. Experience results show nanomaterial diffuses much faster than bulk, namely, diffusion of materials is size-dependent. Combining the thermodynamic size-dependent melting temperature model that is based on the Lindemann criterion on melting and Arrhenius diffusion equation, one can obtain size-dependent diffusion equation. The article gives free parameter model with a universal model ,which predicts diffusion coefficient of different nanocyrstal.
As the pressure increases, the structural phase transition under pressure has been a subject of considerable experimental and theoretical research activity in recent years, especially for oxides crystals. The study of temperature-pressure phase diagram for both bulk material and nanocrystals may extend phase transition theory and possible industry applications and enrich the knowledge on the aspect of the structure. Moreover, the pressure effect on the phase transition temperatures is considered according to the Clapeyron equation, and thus, bulk and size-dependent temperature-pressure phase diagrams are calculated. The concrete contents are listed as follows:
1. The size-dependent melting temperature model for nanocrystals is summarized systematically. In terms of such model, the melting temperature of a free nanocrystal decreases as its size decreases while nanocrystals embedded in a matrix can melt below or above the melting point of corresponding bulk crystals. If the bulk melting temperature of nanocrystals is lower than that of the matrix, the atomic diameter of nanocrystals is larger than that of the matrix, and the interfaces between them are coherent or semi-coherent, an enhancement of the melting point is present. Otherwise, there is a depression of the melting point.
2. F. G. Shi theoretically presented a model for size-dependent melting temperature based on Lindemann criterion. The model can perfectly interpret melting behaviour of nanocrystals, it is applicable both for melting depression and for superheating. For free-standing nanocrystals, the fraction of surface atoms increases with size decreasing and the thermal vibration of surface atoms is larger than that of internal atoms, mean square displacement increases with size decreasing, thus the melting temperature decreases with size decreasing. For nanocrystals embedded in a matrix, the melting temperature will decrease when the interface between nanocrystals and matrix is incoherent. When the interface is coherent or semi-coherent, the amptute of surface atoms is less than that of internal atoms leading to mean square displacement of nanocrystals decrease, the melting temperature increases with size decreasing. But this model has drawback for it including a undefined parameter which is obtained through experiment.
3. Whether diffusion coefficient of bulk and that of corresponding nanomaterial essentially is equal at melting temperature, namely, size-independent. According to the assumption, combining the thermodynamic size-dependent melting temperature model that is based on the Lindemann criterion on melting and Arrhenius diffusion equation, one can obtain size-dependent diffusion equation. In our model, we know diffusion coefficient of nanocrystals increases, when its size decreases, the model predicts an exponent relation between the diffusion coefficient and reciprocal radius of nanocrystals. Diffusion coefficient is related to atom size, vibration entropy and dimension of material. We deeply realize the important reason that the reduce of activation energy induces increase of diffusion coefficient is that the excess surface free energies of nanocrystals with the large surface/volume ratio make atoms easily break away and migrate. The predictions of the model are in agreement with the experimental results of Ag diffusing into Au nanoparticles and N diffusing into Fe particle. The article gives free parameter model with a universal model ,which predicts diffusion coefficient of different nanocyrstal.
4. Although the updated temperature-pressure phase diagrams of oxides crystals have been developed experimentally, further theoretical works are still needed due to the limited measuring accuracy of high-temperature and high-pressure experiments. The classic Clapeyron equation governing all first-order phase transitions of pure substances may be useful to determine the temperature-pressure curve theoretically. However, in order to know the relation between the equilibrium values of the pressure and the temperature, only approximate methods can be used in carrying out the integration of the Clapeyron since both the transition enthalpy and the transition volume are functions of pressure and temperature. To find a solution of the Clapeyron equation, as a first-order approximation, two reasonable simplifications that the transition enthalpy is a weak function of the pressure while the transition volume is a weak function of the temperature are assumed. Moreover, the pressure effect on the phase transition temperatures is considered according to the Clapeyron equation, and thus, bulk and size-dependent temperature-pressure phase diagrams are calculated. The model predictions are found to be consistent with the present experimental and other theoretical results. Since the Clapeyron equation may govern all first-order phase transitions, the Clapeyron equation supplies a new way to determine the temperature-pressure phase diagram of materials.
引文
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