Size and Shape Dependencies of Nanomaterial Properties: Thermodynamic Considerations
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Size and Shape Dependencies of Nanomaterial Properties: Thermodynamic Considerations Grégory Guisbiers Institute of Mechanics, Materials and Civil Engineering, Catholic University of Louvain, 2 Place Sainte-Barbe, Louvain-La-Neuve, 1348, Belgium. E-mail : [email protected] ABSTRACT A top-down approach using classical thermodynamics is presented in this paper to deduce size and shape dependencies of different material properties. Particular attention is focused on the thermal expansion coefficient. The theory developed here can also be used to deduce information on surface energies. INTRODUCTION Understanding how materials behave at tiny length scales is crucial for developing future nanotechnologies. The advances in nanomaterials modeling coupled with new characterization tools are the key to study new properties and capabilities and then to design devices with improved performance [1]. This study of size and shape effects on material properties has attracted enormous attention due to their scientific and industrial importance [2-4]. Nanomaterials have different properties from the bulk due to their high surface area to volume ratio and possible appearance of quantum effects at the nano-scale [5-7]. The determination of nanomaterials properties is still in its infancy and many materials properties are unknown or illcharacterized at the nano-scale [8, 9]. Therefore, thermodynamics can be particularly helpful at the nano-scale where traditional methods as molecular dynamics simulations and density functional theory (DFT) techniques are very computationally demanding and thus are size limited. Indeed, molecular dynamics generally consider less than 105 atoms in order to keep calculations time within reasonable values [10]. Predictions using DFT is generally limited to a few nanometers[11]. Generally, thermodynamics and computational techniques are not competing but complementary [11]. THEORY Thermodynamics implies that we are dealing with a large number of particles and therefore we require a size limit on the applicability of thermodynamics at the nano-scale. A first answer can be given by the fact that thermodynamics describes a material in thermodynamic equilibrium, defined as a volume where thermal fluctuations are small i.e. fluctuate by less than 1% and this occurs for sizes higher than ~4nm [12, 13]. A second answer can be given by the appearance of quantum effects at the nano-scale which signifies that classical thermodynamics is no more applicable when the discrete character of the energy levels appears. It occurs when the energy bandgap between two successive levels becomes larger than the thermal energy. Approximately, energy level spacings of about 1K can be found in particles with sizes around ~10nm. This size depends on the material and varies only within a range of about 50% [14].
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To describe a system thermodynamically, we have to use a thermodynamic potential, depending on the constraints imposed on the system. Let us note, as the number of atoms present in nanostructures is limited, we
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