Difference between revisions of "First Order Phase Transition Material"
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== First Order Phase Transition == | == First Order Phase Transition == | ||
This exploratory [[Material Models|MPM material]] models a first order phase transition between two materials (it is currently only available in [[OSParticulas]]). It has no material | This exploratory [[Material Models|MPM material]] models a first order phase transition between two materials (it is currently only available in [[OSParticulas]]). It has no material response itself. Instead, it acts as a "parent" material to two "child" materials, where one child is a "solid phase" and the other is a "liquid" phase. The properties of this material control transition from the solid to liquid phase, where solid is the low-temperature phase, and the liquid is the high temperature phase. | ||
A first order phase transition occurs at a melting temperature, T<sub>melt</sub>. The meaning of a first order transition is that thermodynamic energy functions (e.g., Gibbs free energy) are continuous at T<sub>melt</sub>, | A first order phase transition occurs at a melting temperature, T<sub>melt</sub>. The meaning of a first order transition is that thermodynamic energy functions (''e.g.'', Gibbs free energy) are continuous at T<sub>melt</sub>, but the phase which is the lowest energy changes. Although free energy is constant, the first derivatives of free energy undergo a discrete change (hence the origin of the term first-order transition). Namely, there is a change in enthalpy (ΔH<sub>f</sub>), entropy (ΔS<sub>f</sub>), and volume (ΔV<sub>f</sub>), where subscript "f" means fusion. Because free energies of the two phases are equal at T<sub>melt</sub>, the first two are related by | ||
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<math>\Delta S_f = {\Delta H_f\over T_m}</math> | <math>\Delta G(T_m) = 0 = \Delta H_f - T_m \Delta S_f \qquad{\rm or}\qquad \Delta S_f = {\Delta H_f\over T_m}</math> | ||
=== Enthalpy and Volume Changes in the Transition === | === Enthalpy and Volume Changes in the Transition === | ||
Most materials do not have a sharp transition, but rather undergo a transition over a range of temperatures, such as from T<sub>i</sub> to T<sub>f</sub>. The enthalpy of | Most materials do not have a sharp transition, but rather undergo a transition over a range of temperatures, such as from T<sub>i</sub> to T<sub>f</sub>. The enthalpy of fusion is experimentally related to the heat capacity during this transition by: | ||
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<math>\Delta H_f = \int_{T_i}^{T_f} \Delta C\ dT</math> | <math>\Delta H_f = \int_{T_i}^{T_f} \Delta C\ dT</math> | ||
where ΔC is the excess heat capacity during the transition compared to the heat capacity of the material in the absence of a transition. For numerical modeling of heat of fusion, you enter the total heat of fusion and a transition temperature range ΔT, such that T<sub>i</sub> = T<sub>melt</sub>-ΔT and T<sub>f</sub> = T<sub>melt</sub>+ΔT. In the numerical implementation, the excess heat capacity | where ΔC is the excess heat capacity during the transition compared to the heat capacity of the material in the absence of a transition. For numerical modeling of heat of fusion, you enter the total heat of fusion and a transition temperature range ΔT, such that T<sub>i</sub> = T<sub>melt</sub>-ΔT and T<sub>f</sub> = T<sub>melt</sub>+ΔT. In the numerical implementation, the excess heat capacity is spread out over the temperature range as a hat function with the peak value equal to: | ||
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<math>\Delta C_{max}= {\Delta H_f\over\Delta T}</math> | <math>\Delta C_{max}= {\Delta H_f\over\Delta T}</math> | ||
Because implementation of heat of fusion is done through heat capacity, all simulations | Because implementation of heat of fusion is done through heat capacity, all simulations with phase transition materials should activate [[Thermal_Calculations#Conduction|coupled conduction calculations]] (conduction is also needed to see changes in temperature that can lead to phase changes). | ||
In general, the density of the liquid will differ from the density of the solid, which means there is a volume change of fusion. The volume change of fusion can be expressed from thermal expansion (or shrinkage) during the transition of | In general, the density of the liquid will differ from the density of the solid, which means there is a volume change of fusion. The volume change of fusion can be expressed from thermal expansion (or shrinkage) during the transition of | ||
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<math>\Delta V_f = \int_{T_i}^{T_f} \Delta \alpha\ dT</math> | <math>\Delta V_f = \int_{T_i}^{T_f} \Delta \alpha\ dT</math> | ||
where Δα is the excess thermal during the transition compared to the thermal expansion of the material in the absence of a transition. For numerical modeling, the total volume change is determined by the densities of the child, and like heat of fusion, it is spread out over T<sub>melt</sub>-ΔT to T<sub>melt</sub>+ΔT as a hat function with the peak value equal to: | where Δα is the excess thermal during the transition compared to the thermal expansion of the material in the absence of a transition. For numerical modeling, the total volume change is determined by the densities of the child solid and liquid materials, and like heat of fusion, it is spread out over T<sub>melt</sub>-ΔT to T<sub>melt</sub>+ΔT as a hat function with the peak value equal to: | ||
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<math>\Delta \alpha_{max}= {\rho_s/\rho_l\over\Delta T}</math> | <math>\Delta \alpha_{max}= {\ln(\rho_s/\rho_l)\over\Delta T}</math> | ||
=== Transition Temperature Range === | === Transition Temperature Range === | ||
The temperature range for which a material undergoes a transition is | The temperature range for which a material undergoes a transition is commonly caused by spatial heterogeneity in the object. For example, polymer materials (which have a particularly large temperature range for first order transitions) have wide variation in crystal perfection and therefore a wide temperature range for melting. The ΔT property mentioned above is one way to model a temperature range, but that parameter is more associated with numerical implementation of implementing an infinite heat capacity or infinite thermal expansion that is seen for a sharp transitions. A second approach to modeling a transition temperature range is to allow T<sub>melt</sub> to be a random variable representing heterogeneity in the crystal structures. This form of variation is modeled by entering the standard deviation for the melting temperature. It can be picked based on observations of the width of the melting transition. | ||
The current implementation randomly assigns a normal distribution of melting points to all particles with zero spatial correlation. In the future, spatial correlation will be added. | |||
== Material Properties == | == Material Properties == | ||
Revision as of 18:12, 21 January 2016
First Order Phase Transition
This exploratory MPM material models a first order phase transition between two materials (it is currently only available in OSParticulas). It has no material response itself. Instead, it acts as a "parent" material to two "child" materials, where one child is a "solid phase" and the other is a "liquid" phase. The properties of this material control transition from the solid to liquid phase, where solid is the low-temperature phase, and the liquid is the high temperature phase.
A first order phase transition occurs at a melting temperature, Tmelt. The meaning of a first order transition is that thermodynamic energy functions (e.g., Gibbs free energy) are continuous at Tmelt, but the phase which is the lowest energy changes. Although free energy is constant, the first derivatives of free energy undergo a discrete change (hence the origin of the term first-order transition). Namely, there is a change in enthalpy (ΔHf), entropy (ΔSf), and volume (ΔVf), where subscript "f" means fusion. Because free energies of the two phases are equal at Tmelt, the first two are related by
[math]\displaystyle{ \Delta G(T_m) = 0 = \Delta H_f - T_m \Delta S_f \qquad{\rm or}\qquad \Delta S_f = {\Delta H_f\over T_m} }[/math]
Enthalpy and Volume Changes in the Transition
Most materials do not have a sharp transition, but rather undergo a transition over a range of temperatures, such as from Ti to Tf. The enthalpy of fusion is experimentally related to the heat capacity during this transition by:
[math]\displaystyle{ \Delta H_f = \int_{T_i}^{T_f} \Delta C\ dT }[/math]
where ΔC is the excess heat capacity during the transition compared to the heat capacity of the material in the absence of a transition. For numerical modeling of heat of fusion, you enter the total heat of fusion and a transition temperature range ΔT, such that Ti = Tmelt-ΔT and Tf = Tmelt+ΔT. In the numerical implementation, the excess heat capacity is spread out over the temperature range as a hat function with the peak value equal to:
[math]\displaystyle{ \Delta C_{max}= {\Delta H_f\over\Delta T} }[/math]
Because implementation of heat of fusion is done through heat capacity, all simulations with phase transition materials should activate coupled conduction calculations (conduction is also needed to see changes in temperature that can lead to phase changes).
In general, the density of the liquid will differ from the density of the solid, which means there is a volume change of fusion. The volume change of fusion can be expressed from thermal expansion (or shrinkage) during the transition of
[math]\displaystyle{ \Delta V_f = \int_{T_i}^{T_f} \Delta \alpha\ dT }[/math]
where Δα is the excess thermal during the transition compared to the thermal expansion of the material in the absence of a transition. For numerical modeling, the total volume change is determined by the densities of the child solid and liquid materials, and like heat of fusion, it is spread out over Tmelt-ΔT to Tmelt+ΔT as a hat function with the peak value equal to:
[math]\displaystyle{ \Delta \alpha_{max}= {\ln(\rho_s/\rho_l)\over\Delta T} }[/math]
Transition Temperature Range
The temperature range for which a material undergoes a transition is commonly caused by spatial heterogeneity in the object. For example, polymer materials (which have a particularly large temperature range for first order transitions) have wide variation in crystal perfection and therefore a wide temperature range for melting. The ΔT property mentioned above is one way to model a temperature range, but that parameter is more associated with numerical implementation of implementing an infinite heat capacity or infinite thermal expansion that is seen for a sharp transitions. A second approach to modeling a transition temperature range is to allow Tmelt to be a random variable representing heterogeneity in the crystal structures. This form of variation is modeled by entering the standard deviation for the melting temperature. It can be picked based on observations of the width of the melting transition.
The current implementation randomly assigns a normal distribution of melting points to all particles with zero spatial correlation. In the future, spatial correlation will be added.
Material Properties
The unusual task for this material is to use multiple Gk and tauk properties (all with the same property name) to enter a material with multiple relaxation times.
| Property | Description | Units | Default |
|---|---|---|---|
| pressureLaw | Picks the constitutive law use for time independent pressure. The options are 0 to linear elastic law and 1 to use MGEOS equation of state. | none | 0 |
The total number of Gk and tauk properies must be equal. In XML files, that total number must match the supplied ntaus property.
History Variables
This material tracks internal history variables (one for each relaxation time and each component of stress) for implementation of linear viscoelastic properties, but currently none of these internal variables are available for archiving.
This material also tracks J (total relative volume change) and Jres (volume change of free expansion state) as history variables 1 and 2. Note that Jres is only needed, and therefore only tracked, when using MGEOS for pressure constitutive law (when pressureLaw is 1). If not tracked, it is always 1.