First Order Phase Transition Material

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, T_melt. The meaning of a first order transition is that thermodynamic energy functions (e.g., Gibbs free energy) are continuous at T_melt, 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_f), entropy (ΔS_f), and volume (ΔV_f), where subscript "f" means fusion. Because free energies of the two phases are equal at T_melt, 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 T_i to T_f. 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 T_i = T_melt-ΔT and T_f = T_melt+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 T_melt-ΔT to T_melt+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]

Finally, because both ΔC_max and Δα_max involve dividing by ΔT, this value must be greater than zero. For numerically stability, ΔT should probably be several times larger than the expected temperature change per time step in the transition range such that the excess quantities are spread out over multiple steps. When you are unsure of the expected temperature change rates, a few simulations with various values of ΔT should indicate the minimum value needed to sable results.

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 T_melt 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
SolidPhase	Enter the solid phase material by ID (scripted files only) or by number	none	none
LiquidPhase	Enter the liquid phase material by ID (scripted files only) or by number	none	none
Tmelt	The transition temperature (K). The solid phase is stable below Tmelt and liquid phase above it	K	none
Tsigma	The standard deviation for the transition temperature (K). The temperature range will be limited to +/- three standard deviations	K	0
DeltaT	A temperature range for application of increased heat capacity and increased thermal expansion (must be > 0)	K	10
Hfusion	The heat of fusion for the transition	heat of fusion units	0

Note that the materials that are capably of being phase materials are limited to those that have been revised to work with the phase transition material. The currently allowed materials are Mooney Material, Neo-Hookean Material, Isotropic, Hyperelastic-Plastic Material, Isotropic, Hyperelastic-Plastic Mie-Grüneisen Material, Clamped Neo-Hookean Material, and Tait Liquid Material.

History Variables

This material tracks two history variables:

1. The currently active phase with 0 for solid phase and 1 for liquid phase.
2. The melting point of the particle. It is constant throughout the analysis, but can be visualized as random variable if Tsigma>0.

For history variable n>2, the output is history variable n-2 of the solid phase material. The liquid phase may also have history variables, but currently those variables can not be archived.