纳米晶体的尺寸和压力效应
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摘要
系统总结了根据Lindemann熔化准则和Mott方程推导出的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。进一步,结合该模型和尺寸依赖的熔化熵模型,预测了纳米晶体熔化焓和结合能的尺寸效应。根据建立的尺寸依赖的结合能模型,预测了尺寸和界面对铁磁、铁电和超导纳米晶体临界温度的影响以及半导体纳米固体能隙的尺寸效应。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,建立了半导体和分子晶体的温度-压力相图。以上各模型的预测结果均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 alteredfrom those of either the bulk or a single molecule. Among the above special properties ofnanocrystals, the size-dependent melting temperature of nanocrystals has received considerableattention since Takagi in 1954 experimentally demonstrated that ultrafine metallic nanocrystalsmelt below their corresponding bulk melting temperature. However, it has not been accompaniedby the necessary investigation of other size-dependent thermodynamic proerties of nanocrystals,such as the melting entropy, the melting enthalpy, and the cohesive energy. Such an investigationshould deepen our understanding of the nature of the thermal stability of nanocrystals. Moreover,size effects on other physical properties including critical temperatures of ferromagnetic,ferroelectric, and superconductive nanocrystals and the bandgap of semiconductor nanosolids etal. have been extensively investigated theoretically and experimentally due to their scienfic andindustrial importance.
It is obvious that an important variable?pressure is ignored in above investigations sincemost of them process under ambient pressure and the pressure effect is negligible. As the pressureincreases, the structural phase transition under pressure has been a subject of considerableexperimental and theoretical research activity in recent years, especially for semiconductors andmolecular crystals. The study of temperature-pressure phase diagram for both bulk material andnanocrystals may extend phase transition theory and possible industry applications and enrich theknowledge on the aspect of the structure.
Recently, a simple model without adjustable parameters for size-dependent meltingtemperature of nanocrystals under ambient pressure is established based on Lindemann′scriterion for the melting and Mott′s expression for the vibrational entropy at the meltingtemperature. In comparison with other model conventions, this model covers all the essentialconsiderations of early models and it has wider size range suitability. In this thesis, the abovemodel has been extended to predict the size dependences of the melting enthalpy and thecohesive energy of nanocrystals, the critical temperatures of ferromagnetic, ferroelectric, andsuperconductor nanocrystals, and the bandgap of semiconductor nanosolids. Moreover, thepressure effect on the phase transition temperatures is considered according to the Clapeyronequation, 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 summarizedsystematically. In terms of such model, the melting temperature of a free nanocrystal decreases asits size decreases while nanocrystals embedded in a matrix can melt below or above the meltingpoint of corresponding bulk crystals. If the bulk melting temperature of nanocrystals is lowerthan that of the matrix, the atomic diameter of nanocrystals is larger than that of the matrix, andthe interfaces between them are coherent or semi-coherent, an enhancement of the melting pointis present. Otherwise, there is a depression of the melting point. Our model can predict meltingtemperuatures of low-dimensional crystals with different chemical bonds, different dimensions,and different surface or interface conditions and the surface melting temperature of nanocrystals.As an example, an agreement is found between model predictions and experimental results for Innanocrystals. Based on the above model and the size-dependent melting entropy model, weestablisehed a model to predict the size dependence of the mleing enthalpy and further to predictthe size effect on the cohesive energy of nanocrystals. It is found that our model predictions areconsistent with available experimental results for the melting enthalpy of In nanocrystals and
cohesive energies of Mo and W nanoparticles.2. Based on the Ising premise and the bond order-length-strength correlation mechanism, thecritical temperatures of ferromagnetic, ferroelectric and superconductive nanocrystals aresupposed to be proportional to their cohesive energies. In tems of this premise and asize-dependent cohesive energy model, such size-dependent critical transition temperatures aremodeled in a unified form without any adjustable parameter. According to this model, the criticaltemperature of a nanocrystal progressively reduces with decreasing of its sizes. For freenanoparticles or nanowires and films having weak interactions with substrates compared to theinner interaction of the films, the dominant factor affecting the suppression of the criticaltemperatue seems to be the size while for films having strong interactions with substrates, thesubstrate effect also becomes evident and the suppression trend is weaker than the former. Themodel predictions correspond to the experimental and other theoretical results in the full sizerange. Moreover, it is found that the characteristics of the Co-Ni alloys can be roughly expressedas an algebraic sum of elements of which the alloys consist and the evaporation entropy equalingthirteen times the ideal gas constant has been proven as a good approximation for compounds.3. With miniaturization of semiconductor solids down to the nanometer scale, quantum andinterfacial effects become dominant, which have led to significant changes of their optical andelectrical properties from the corresponding bulk. One of the characteristics of thenanometer-sized semiconductor is the increase of the valence-conduction bandgap withdecreasing of its size. According to the nearly-free-electon approach, the bandgap is supposed tobe proportional to the cohesive energy. Based on this approach and a model for size-dependentcohesive energy, the size-dependent bandgap of semiconductor nanocrystals are modeled withoutany adjustable parameter. The model predictions are supported by the available experimental andother theoretical results in the full size range. Our model can not predict the bandgap of clustersaccurately since the long-range periodicity in clusters is absent and the bond structures of clusters
differ form the corresponding crystals. Moreover, based on the size and temperature-dependentconductivity, we have proven that the upper-limt value of the bandgap is twice as many as itsbulk value.4. Although the updated temperature-pressure phase diagrams of semiconductors andmolecular crystals have been developed experimentally, further theoretical works are still neededdue to the limited measuring accuracy of high-temperature and high-pressure experiments. Theclassic Clapeyron equation governing all first-order phase transitions of pure substances may beuseful to determine the temperature-pressure curve theoretically. However, in order to know therelation between the equilibrium values of the pressure and the temperature, only approximatemethods can be used in carrying out the integration of the Clapeyron since both the transitionenthalpy and the transition volume are functions of pressure and temperature. To find a solutionof the Clapeyron equation, as a first-order approximation, two reasonable simplifications that thetransition enthalpy is a weak function of the pressure while the transition volume is a weakfunction of the temperature are assumed. Based on the above consideration, thetemperature-pressure phase diagrams for bulk seimiconductors Si and Ge and the melting curvesof Si nanocrystal and bulk CO2 crystal are calculated thorugh the Clapeyron equation where thepressure-dependent volume difference is modeled and the corresponding thermodynamic amountof solid transition enthalpy is calculated by introducing the effect of surface stree inducedpressure and reasonable initial points for integrations are selected. The model predictions arefound to be consistent with the present experimental and other theoretical results. Since theClapeyron equation may govern all first-order phase transitions, the Clapeyron equation suppliesa 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 alteredfrom those of either the bulk or a single molecule. Among the above special properties ofnanocrystals, the size-dependent melting temperature of nanocrystals has received considerableattention since Takagi in 1954 experimentally demonstrated that ultrafine metallic nanocrystalsmelt below their corresponding bulk melting temperature. However, it has not been accompaniedby the necessary investigation of other size-dependent thermodynamic proerties of nanocrystals,such as the melting entropy, the melting enthalpy, and the cohesive energy. Such an investigationshould deepen our understanding of the nature of the thermal stability of nanocrystals. Moreover,size effects on other physical properties including critical temperatures of ferromagnetic,ferroelectric, and superconductive nanocrystals and the bandgap of semiconductor nanosolids etal. have been extensively investigated theoretically and experimentally due to their scienfic andindustrial importance.
It is obvious that an important variable?pressure is ignored in above investigations sincemost of them process under ambient pressure and the pressure effect is negligible. As the pressureincreases, the structural phase transition under pressure has been a subject of considerableexperimental and theoretical research activity in recent years, especially for semiconductors andmolecular crystals. The study of temperature-pressure phase diagram for both bulk material andnanocrystals may extend phase transition theory and possible industry applications and enrich theknowledge on the aspect of the structure.
Recently, a simple model without adjustable parameters for size-dependent meltingtemperature of nanocrystals under ambient pressure is established based on Lindemann′scriterion for the melting and Mott′s expression for the vibrational entropy at the meltingtemperature. In comparison with other model conventions, this model covers all the essentialconsiderations of early models and it has wider size range suitability. In this thesis, the abovemodel has been extended to predict the size dependences of the melting enthalpy and thecohesive energy of nanocrystals, the critical temperatures of ferromagnetic, ferroelectric, andsuperconductor nanocrystals, and the bandgap of semiconductor nanosolids. Moreover, thepressure effect on the phase transition temperatures is considered according to the Clapeyronequation, 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 summarizedsystematically. In terms of such model, the melting temperature of a free nanocrystal decreases asits size decreases while nanocrystals embedded in a matrix can melt below or above the meltingpoint of corresponding bulk crystals. If the bulk melting temperature of nanocrystals is lowerthan that of the matrix, the atomic diameter of nanocrystals is larger than that of the matrix, andthe interfaces between them are coherent or semi-coherent, an enhancement of the melting pointis present. Otherwise, there is a depression of the melting point. Our model can predict meltingtemperuatures of low-dimensional crystals with different chemical bonds, different dimensions,and different surface or interface conditions and the surface melting temperature of nanocrystals.As an example, an agreement is found between model predictions and experimental results for Innanocrystals. Based on the above model and the size-dependent melting entropy model, weestablisehed a model to predict the size dependence of the mleing enthalpy and further to predictthe size effect on the cohesive energy of nanocrystals. It is found that our model predictions areconsistent with available experimental results for the melting enthalpy of In nanocrystals and
cohesive energies of Mo and W nanoparticles.2. Based on the Ising premise and the bond order-length-strength correlation mechanism, thecritical temperatures of ferromagnetic, ferroelectric and superconductive nanocrystals aresupposed to be proportional to their cohesive energies. In tems of this premise and asize-dependent cohesive energy model, such size-dependent critical transition temperatures aremodeled in a unified form without any adjustable parameter. According to this model, the criticaltemperature of a nanocrystal progressively reduces with decreasing of its sizes. For freenanoparticles or nanowires and films having weak interactions with substrates compared to theinner interaction of the films, the dominant factor affecting the suppression of the criticaltemperatue seems to be the size while for films having strong interactions with substrates, thesubstrate effect also becomes evident and the suppression trend is weaker than the former. Themodel predictions correspond to the experimental and other theoretical results in the full sizerange. Moreover, it is found that the characteristics of the Co-Ni alloys can be roughly expressedas an algebraic sum of elements of which the alloys consist and the evaporation entropy equalingthirteen times the ideal gas constant has been proven as a good approximation for compounds.3. With miniaturization of semiconductor solids down to the nanometer scale, quantum andinterfacial effects become dominant, which have led to significant changes of their optical andelectrical properties from the corresponding bulk. One of the characteristics of thenanometer-sized semiconductor is the increase of the valence-conduction bandgap withdecreasing of its size. According to the nearly-free-electon approach, the bandgap is supposed tobe proportional to the cohesive energy. Based on this approach and a model for size-dependentcohesive energy, the size-dependent bandgap of semiconductor nanocrystals are modeled withoutany adjustable parameter. The model predictions are supported by the available experimental andother theoretical results in the full size range. Our model can not predict the bandgap of clustersaccurately since the long-range periodicity in clusters is absent and the bond structures of clusters
differ form the corresponding crystals. Moreover, based on the size and temperature-dependentconductivity, we have proven that the upper-limt value of the bandgap is twice as many as itsbulk value.4. Although the updated temperature-pressure phase diagrams of semiconductors andmolecular crystals have been developed experimentally, further theoretical works are still neededdue to the limited measuring accuracy of high-temperature and high-pressure experiments. Theclassic Clapeyron equation governing all first-order phase transitions of pure substances may beuseful to determine the temperature-pressure curve theoretically. However, in order to know therelation between the equilibrium values of the pressure and the temperature, only approximatemethods can be used in carrying out the integration of the Clapeyron since both the transitionenthalpy and the transition volume are functions of pressure and temperature. To find a solutionof the Clapeyron equation, as a first-order approximation, two reasonable simplifications that thetransition enthalpy is a weak function of the pressure while the transition volume is a weakfunction of the temperature are assumed. Based on the above consideration, thetemperature-pressure phase diagrams for bulk seimiconductors Si and Ge and the melting curvesof Si nanocrystal and bulk CO2 crystal are calculated thorugh the Clapeyron equation where thepressure-dependent volume difference is modeled and the corresponding thermodynamic amountof solid transition enthalpy is calculated by introducing the effect of surface stree inducedpressure and reasonable initial points for integrations are selected. The model predictions arefound to be consistent with the present experimental and other theoretical results. Since theClapeyron equation may govern all first-order phase transitions, the Clapeyron equation suppliesa new way to determine the temperature-pressure phase diagram of materials.
引文
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