Melting point depression
Encyclopedia
Melting-point depression is a term referring to the phenomenon of reduction of the melting point
of a material with reduction of its size. This phenomenon is very prominent in nanoscale materials
which melt at temperatures hundreds of degrees lower than bulk materials.
Melting-point depression is most evident in nanowire
s, nanotube
s and nanoparticle
s, which all melt at lower temperatures than bulk amounts of the same material. Changes in melting point occur because nanoscale materials have a much larger surface-to-volume ratio than bulk materials, drastically altering their thermodynamic and thermal properties.
This article focuses on nanoparticles because researchers have compiled a large amount of size-dependent melting data for near spherical nanoparticles. Nanoparticles are easiest to study due their ease of fabrication and simplified conditions for theoretical modeling. The melting temperature of a nanoparticle decreases sharply as the particle reaches critical diameter, usually < 50 nm for common engineering metals. Figure 1 shows the shape of a typical melting curve for a metal nanoparticle as a function of its diameter.
Melting point depression is a very important issue for applications involving nanoparticles, as it decreases the functional range of the solid phase. Nanoparticles are currently used or proposed for prominent roles in catalyst, sensor
, medicinal, optical, magnetic, thermal, electronic, and alternative energy applications. Nanoparticles must be in the solid state to function at elevated temperatures in several of these applications.
(TEM) can be used to melt nanoparticles. The melting temperature is estimated from the beam intensity, while changes in the diffraction conditions to indicate phase transition from solid to liquid. This method allows direct viewing of nanoparticles as they melt, making it possible to test and characterize samples with a wider distribution of particle sizes. The TEM limits the pressure range at which melting point depression can be tested.
More recently, researchers developed nanocalorimeter
s that directly measure the enthalpy
and melting temperature of nanoparticles. Nanocalorimeters provide the same data as bulk calorimeters, however additional calculations must account for the presence of the substrate supporting the particles. A narrow size distribution of nanoparticles is required since the procedure does not allow users to view the sample during the melting process. There is no way to characterize the exact size of melted particles during experiment.
s bind in the solid phase with less cohesive energy because they have fewer neighboring atoms in close proximity compared to atoms in the bulk of the solid. Each chemical bond
an atom shares with a neighboring atom provides cohesive energy, so atoms with fewer bonds and neighboring atoms have lower cohesive energy. The average cohesive energy per atom of a nanoparticle has been theoretically calculated as a function of particle size according to Equation 1.
Where: D=nanoparticle size
As Equation 1 shows, the effective cohesive energy of a nanoparticle approaches that of the bulk material as the material extends beyond atomic size range (D>>d).
Atoms located at or near the surface of the nanoparticle have reduced cohesive energy due to a reduced number of cohesive bonds. An atom experiences an attractive force with all nearby atoms according to the Lennard-Jones potential
. The Lennard-Jones pair-potential shown in Figure 2 models the cohesive energy between atoms as a function of separation distance.
The cohesive energy of an atom is directly related to the thermal energy required to free the atom from the solid. According to Lindemann’s criterion, the melting temperature of a material is proportional to its cohesive energy, av (TM=Cav). Since atoms near the surface have fewer bonds and reduced cohesive energy, they require less energy to free from the solid phase. Melting point depression of high surface to volume ratio materials results from this effect. For the same reason, surfaces of bulk materials can melt at lower temperatures than the bulk material.
The theoretical size-dependent melting point of a material can be calculated through classical thermodynamic analysis. The result is the Gibbs–Thomson equation
shown in Equation 2.
Where: TMB=Bulk Melting temperature
A normalized Gibbs–Thomson Equation for gold nanoparticles is plotted in Figure 1, and the shape of the curve is in general agreement with those obtained through experiment.
and covalently bonded nanoparticles may have a different dependence on particle size. The covalent character of the bonds changes the melting physics of these materials. Researchers have demonstrated that Equation 3 more accurately models melting point depression in covalently bonded materials.
Where: TMB=Bulk Melting temperature
Equation 3 indicates that melting point depression is less pronounced in covalent nanoparticles due to the quadratic nature of particle size dependence in the melting Equation.
The LDM represents the binding energy of a nanoparticles as function of the free energies of the volume and surface. Equation 4 gives the normalized, size dependent melting temperature of a material according the liquid-drop model.
Where: σsv=solid-vapor interface energy
The model calculates melting conditions as a function of two competing order parameters using Landau potentials. One order parameter represents a solid nanoparticle, while the other represents the liquid phase. Each of the order parameters is a function of particle radius.
The parabolic Landau potentials for the liquid and solid phases are calculated at a given temperature, with the lesser Landau potential assumed to be the equilibrium state at any point in the particle. In the temperature range of surface melting, the results show that the Landau curve of the ordered state is favored near the center of the particle while the Landau curve of the disordered state is smaller near the surface of the particle.
The Landau curves intersect at a specific radius from the center of the particle. The distinct intersection of the potentials means the LSN predicts a sharp, unmoving interface between the solid and liquid phases at a given temperature. The exact thickness of the liquid layer at a given temperature is the equilibrium point between the competing Landau potentials.
Equation 5 gives the condition at which an entire nanoparticle melts according to the LSN model.
Where: d0=atomic diameter
The BOLS mechanism states that if one bond breaks the remaining neighbouring ones become shorter and stronger. The cohesive energy, or the sum of bond energy, of the less coordinated atoms determines the thermal stability including melting, evaporating and other phase transition. The lowered CN changes the equilibrium bond length between atoms near the surface of the nanoparticle. The bonds relax towards equilibrium lengths, increasing the cohesive energy per bond between atoms, independent of the exact form of the specific interatomic potential. However, the integrated cohesive energy for surface atoms is much lower than bulk atoms due to the reduced coordination number and overall decrease in cohesive energy.
Using a core–shell configuration, the melting point depression of nanoparticles is dominated by the outermost two atomic layers yet atoms in the core interior remain their bulk nature.
The BOLS model and the core–shell structure have been applied to other size dependency of nanostructures such as the mechanical strength, chemical and thermal stability, lattice dynamics (optical and acoustic phonons), Photon emission and absorption, electronic colevel shift and work function modulation, magnetism at various temperatures, and dielectrics due to electron polarization etc. Reproduction of experimental observations in the above mentioned size dependency has been realized. Quantitative information such as the energy level of an isolated atom and the vibration frequency of individual dimer has been obtained by matching the BOLS predictions to the measured size dependency.
Where: c=materials constant
The shape parameter is 1 for sphere and 3/2 for a very long wire, indicating that melting-point depression is suppressed in nanowires compared to nanoparticles. Past experimental data show that nanoscale tin platelets melt within a narrow range of 10 oC of the bulk melting temperature. The melting point depression of these platelets was suppressed compared to spherical tin nanoparticles.
Reduced solubility can affect the catalytic properties of nanoparticles. In fact it has been shown that size-induced instability of Fe-C mixtures represents the thermodynamic limit for the thinnest nanotube that can be grown from Fe nanocatalysts.
Melting point
The melting point of a solid is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depends on pressure and is usually specified at standard atmospheric pressure...
of a material with reduction of its size. This phenomenon is very prominent in nanoscale materials
Nanotechnology
Nanotechnology is the study of manipulating matter on an atomic and molecular scale. Generally, nanotechnology deals with developing materials, devices, or other structures possessing at least one dimension sized from 1 to 100 nanometres...
which melt at temperatures hundreds of degrees lower than bulk materials.
Introduction
The melting temperature of a bulk material is not dependent on its size. However as the dimensions of a material decrease towards the atomic scale, the melting temperature scales with the material dimensions. The decrease in melting temperature can be on the order of tens to hundreds of degrees for metals with nanometer dimensions.Melting-point depression is most evident in nanowire
Nanowire
A nanowire is a nanostructure, with the diameter of the order of a nanometer . Alternatively, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less and an unconstrained length. At these scales, quantum mechanical effects are important —...
s, nanotube
Carbon nanotube
Carbon nanotubes are allotropes of carbon with a cylindrical nanostructure. Nanotubes have been constructed with length-to-diameter ratio of up to 132,000,000:1, significantly larger than for any other material...
s and nanoparticle
Nanoparticle
In nanotechnology, a particle is defined as a small object that behaves as a whole unit in terms of its transport and properties. Particles are further classified according to size : in terms of diameter, coarse particles cover a range between 10,000 and 2,500 nanometers. Fine particles are sized...
s, which all melt at lower temperatures than bulk amounts of the same material. Changes in melting point occur because nanoscale materials have a much larger surface-to-volume ratio than bulk materials, drastically altering their thermodynamic and thermal properties.
This article focuses on nanoparticles because researchers have compiled a large amount of size-dependent melting data for near spherical nanoparticles. Nanoparticles are easiest to study due their ease of fabrication and simplified conditions for theoretical modeling. The melting temperature of a nanoparticle decreases sharply as the particle reaches critical diameter, usually < 50 nm for common engineering metals. Figure 1 shows the shape of a typical melting curve for a metal nanoparticle as a function of its diameter.
Sensor
A sensor is a device that measures a physical quantity and converts it into a signal which can be read by an observer or by an instrument. For example, a mercury-in-glass thermometer converts the measured temperature into expansion and contraction of a liquid which can be read on a calibrated...
, medicinal, optical, magnetic, thermal, electronic, and alternative energy applications. Nanoparticles must be in the solid state to function at elevated temperatures in several of these applications.
Measurement techniques
Two techniques allow measurement of the melting point of nanoparticle. As described above, the electron beam of the Transmission electron microscopeTransmission electron microscopy
Transmission electron microscopy is a microscopy technique whereby a beam of electrons is transmitted through an ultra thin specimen, interacting with the specimen as it passes through...
(TEM) can be used to melt nanoparticles. The melting temperature is estimated from the beam intensity, while changes in the diffraction conditions to indicate phase transition from solid to liquid. This method allows direct viewing of nanoparticles as they melt, making it possible to test and characterize samples with a wider distribution of particle sizes. The TEM limits the pressure range at which melting point depression can be tested.
More recently, researchers developed nanocalorimeter
Calorimeter
A calorimeter is a device used for calorimetry, the science of measuring the heat of chemical reactions or physical changes as well as heat capacity. Differential scanning calorimeters, isothermal microcalorimeters, titration calorimeters and accelerated rate calorimeters are among the most common...
s that directly measure the enthalpy
Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
and melting temperature of nanoparticles. Nanocalorimeters provide the same data as bulk calorimeters, however additional calculations must account for the presence of the substrate supporting the particles. A narrow size distribution of nanoparticles is required since the procedure does not allow users to view the sample during the melting process. There is no way to characterize the exact size of melted particles during experiment.
History
Takagi first observed melting point depression of several types of metal nanoparticles in 1954. A variable intensity electron beam from a transmission electron microscope melted metal nanoparticles in early experiments. Diffraction patterns changed from characteristic crystalline patterns to liquid patterns as the small particles melted, allowing Takagi to estimate the melting temperature from the electron beam energy.Physics
Nanoparticles have a much greater surface to volume ratio than bulk materials. The increased surface to volume ratio means surface atoms have a much greater effect on chemical and physical properties of a nanoparticle. Surface atomAtom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s bind in the solid phase with less cohesive energy because they have fewer neighboring atoms in close proximity compared to atoms in the bulk of the solid. Each chemical bond
Chemical bond
A chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electromagnetic force attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction...
an atom shares with a neighboring atom provides cohesive energy, so atoms with fewer bonds and neighboring atoms have lower cohesive energy. The average cohesive energy per atom of a nanoparticle has been theoretically calculated as a function of particle size according to Equation 1.
Where: D=nanoparticle size
-
- d=atomic size
- Eb=cohesive energy of bulk
As Equation 1 shows, the effective cohesive energy of a nanoparticle approaches that of the bulk material as the material extends beyond atomic size range (D>>d).
Atoms located at or near the surface of the nanoparticle have reduced cohesive energy due to a reduced number of cohesive bonds. An atom experiences an attractive force with all nearby atoms according to the Lennard-Jones potential
Lennard-Jones potential
The Lennard-Jones potential is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules. A form of the potential was first proposed in 1924 by John Lennard-Jones...
. The Lennard-Jones pair-potential shown in Figure 2 models the cohesive energy between atoms as a function of separation distance.
The theoretical size-dependent melting point of a material can be calculated through classical thermodynamic analysis. The result is the Gibbs–Thomson equation
Gibbs–Thomson equation
The Gibbs–Thomson effect in Physics refers to the observation that small crystals of a liquid melt at a lower temperature than the bulk liquid.The melting point depression is inversely proportional to the pore size, as given by the Gibbs–Thomson equation....
shown in Equation 2.
Where: TMB=Bulk Melting temperature
-
- σsl=solid–liquid interface energy
- Hf=Bulk heat of fusion
- ρs=density of solid
- d=particle diameter
A normalized Gibbs–Thomson Equation for gold nanoparticles is plotted in Figure 1, and the shape of the curve is in general agreement with those obtained through experiment.
Semiconductor/covalent nanoparticles
Equation 2 gives the general relation between the melting point of a metal nanoparticle and its diameter. However, recent work indicates the melting point of semiconductorSemiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
and covalently bonded nanoparticles may have a different dependence on particle size. The covalent character of the bonds changes the melting physics of these materials. Researchers have demonstrated that Equation 3 more accurately models melting point depression in covalently bonded materials.
Where: TMB=Bulk Melting temperature
-
- c=materials constant
- d=particle diameter
Equation 3 indicates that melting point depression is less pronounced in covalent nanoparticles due to the quadratic nature of particle size dependence in the melting Equation.
Proposed mechanisms
The specific melting process for nanoparticles is currently unknown. The scientific community currently accepts several mechanisms as possible models of nanoparticle melting. Each of the corresponding models effectively match experimental data for melting of nanoparticles. Three of the four models detailed below derive the melting temperature in a similar form using different approaches based on classical thermodynamics.Liquid drop model
The liquid drop model (LDM) assumes that an entire nanoparticle transitions from solid to liquid at a single temperature. This feature distinguishes the model, as the other models predict melting of the nanoparticle surface prior to the bulk atoms. If the LDM is true, a solid nanoparticle should function over a greater temperature range than other models predict. The LDM assumes that the surface atoms of a nanoparticle dominate the properties of all atoms in the particle. The cohesive energy of the particle is identical for all atoms in the nanoparticle.The LDM represents the binding energy of a nanoparticles as function of the free energies of the volume and surface. Equation 4 gives the normalized, size dependent melting temperature of a material according the liquid-drop model.
Where: σsv=solid-vapor interface energy
-
- σlv=liquid-vapor interface energy
- Hf=Bulk heat of fusion
- ρs=density of solid
- ρl=density of liquid
- d=diameter of nanoparticle
Liquid shell nucleation model
The liquid shell nucleation model (LSN) predicts that a surface layer of atoms melts prior to the bulk of the particle. The melting temperature of a nanoparticle is a function of its radius of curvature according to the LSN. Large nanoparticles melt at greater temperatures as a result of their larger radius of curvature.The model calculates melting conditions as a function of two competing order parameters using Landau potentials. One order parameter represents a solid nanoparticle, while the other represents the liquid phase. Each of the order parameters is a function of particle radius.
The parabolic Landau potentials for the liquid and solid phases are calculated at a given temperature, with the lesser Landau potential assumed to be the equilibrium state at any point in the particle. In the temperature range of surface melting, the results show that the Landau curve of the ordered state is favored near the center of the particle while the Landau curve of the disordered state is smaller near the surface of the particle.
The Landau curves intersect at a specific radius from the center of the particle. The distinct intersection of the potentials means the LSN predicts a sharp, unmoving interface between the solid and liquid phases at a given temperature. The exact thickness of the liquid layer at a given temperature is the equilibrium point between the competing Landau potentials.
Equation 5 gives the condition at which an entire nanoparticle melts according to the LSN model.
Where: d0=atomic diameter
Liquid nucleation and growth model
The liquid nucleation and growth model (LNG) treats nanoparticle melting as a surface initiated process. The surface melts initially, and the liquid-solid interface quickly advances through the entire nanoparticle. The LNG defines melting conditions through the Gibbs-Duhem relations, yielding a melting temperature function dependent on the interfacial energies between the solid and liquid phases, volumes and surface areas of each phase, and size of the nanoparticle. The model calculations show that the liquid phase forms at lower temperatures for smaller nanoparticles. Once the liquid phase forms, the free energy conditions quickly change and favor melting. Equation 6 gives the melting conditions for a spherical nanoparticle according to the LNG model.Bond-order-length-strength (BOLS) model
The bond-order-length-strength (BOLS) model employs an atomistic approach to explain melting point depression. The model focuses on the cohesive energy of individual atoms rather than a classical thermodynamic approach. The BOLS model calculates the melting temperature for individual atoms from the sum of their cohesive bonds. As a result, the BOLS predicts the surface layers of a nanoparticle melt at lower temperatures than the bulk of the nanoparticle.The BOLS mechanism states that if one bond breaks the remaining neighbouring ones become shorter and stronger. The cohesive energy, or the sum of bond energy, of the less coordinated atoms determines the thermal stability including melting, evaporating and other phase transition. The lowered CN changes the equilibrium bond length between atoms near the surface of the nanoparticle. The bonds relax towards equilibrium lengths, increasing the cohesive energy per bond between atoms, independent of the exact form of the specific interatomic potential. However, the integrated cohesive energy for surface atoms is much lower than bulk atoms due to the reduced coordination number and overall decrease in cohesive energy.
Using a core–shell configuration, the melting point depression of nanoparticles is dominated by the outermost two atomic layers yet atoms in the core interior remain their bulk nature.
The BOLS model and the core–shell structure have been applied to other size dependency of nanostructures such as the mechanical strength, chemical and thermal stability, lattice dynamics (optical and acoustic phonons), Photon emission and absorption, electronic colevel shift and work function modulation, magnetism at various temperatures, and dielectrics due to electron polarization etc. Reproduction of experimental observations in the above mentioned size dependency has been realized. Quantitative information such as the energy level of an isolated atom and the vibration frequency of individual dimer has been obtained by matching the BOLS predictions to the measured size dependency.
Particle shape
Nanoparticle shape impacts the melting point of a nanoparticle. Facets, edges and deviations from a perfect sphere all change the magnitude of melting point depression. These shape changes affect the surface to volume ratio, which affects the cohesive energy and thermal properties of a nanostructure. Equation 7 gives a general shape corrected formula for the theoretical melting point of a nanoparticle based on its size and shape.Where: c=materials constant
-
- z=shape parameter of particle
The shape parameter is 1 for sphere and 3/2 for a very long wire, indicating that melting-point depression is suppressed in nanowires compared to nanoparticles. Past experimental data show that nanoscale tin platelets melt within a narrow range of 10 oC of the bulk melting temperature. The melting point depression of these platelets was suppressed compared to spherical tin nanoparticles.
Substrate
Several nanoparticle melting simulations theorize that the supporting substrate affects the extent of melting-point depression of a nanoparticle. These models account for energetic interactions between the substrate materials. A free nanoparticle, as many theoretical models assume, has a different melting temperature (usually lower) than a supported particle due to the absence cohesive energy between the nanoparticle and substrate. However, measurement of the properties of a freestanding nanoparticle remains impossible, so the extent of the interactions cannot be verified through experiment. Ultimately, substrates currently support nanoparticles for all nanoparticle applications, so substrate/nanoparticle interactions are always present and must impact melting point depression.Solubility
Within the size–pressure approximation, which considers the stress induced by the surface tension and the curvature of the particle, it was shown that the size of the particle affects the composition and temperature of a eutectic point (Fe-C ) the solubility of C in Fe and Fe:Mo nanoclusters.Reduced solubility can affect the catalytic properties of nanoparticles. In fact it has been shown that size-induced instability of Fe-C mixtures represents the thermodynamic limit for the thinnest nanotube that can be grown from Fe nanocatalysts.