The study of defects or vacancies helps to understand the transport mechanism that occurs electrically and thermally in materials. It also deals with the mechanical properties of materials. In the present study, the extension of qualitative size-dependent model proposed by Jiang for cohesive energy of nanomaterials is done and size dependence of vacancy formation energy, vibrational frequency and vacancy concentration in nanomaterials is studied. The variation in vacancy concentration in nanomaterials is studied using the extended Jiang model for different dimensional nanomaterials with size and temperature. The model calculations depict the increase in vacancy concentration in free-standing nanomaterials as size of the nanomaterial is reduced and temperature is increased. The behaviour of concentration variation is found just opposite in embedded nanomaterials. The present predicted results of variation in concentration of vacancies follow the same trend as explained in previous studies. The study validates that defects help in the displacement of atoms. The variation in melting temperature and vibrational frequency of nanomaterials with size-dependent vacancy concentration is also discussed.

对缺陷或空位的研究有助于了解材料中电和热发生的传输机制。它还涉及材料的机械性能。在本研究中，对蒋提出的纳米材料内聚能定性尺寸相关模型进行了扩展，并研究了纳米材料中空位形成能、振动频率和空位浓度的尺寸依赖性。使用扩展的蒋模型研究了纳米材料中空位浓度的变化，适用于不同尺寸和温度的不同维度纳米材料。模型计算描述了随着纳米材料尺寸的减小和温度的升高，自支撑纳米材料中空位浓度的增加。在嵌入的纳米材料中发现浓度变化的行为正好相反。目前空缺浓度变化的预测结果与先前研究中解释的趋势相同。该研究证实了缺陷有助于原子的位移。还讨论了纳米材料的熔化温度和振动频率随尺寸相关的空位浓度的变化。