Specific Volume (pvT) Fitting

Specific Volume is the inverse of Specific Gravity, and its dependence on temperature and pressure forms an equation of state from which several physical quantities can be derived. The Specific Volume Fitter is a sophisticated tool designed to fit specific volume test data of a polymer or composite to various mathematical models beginning with the Tait model, and it is currently executed from the Material Modeler - Polymer application or via the Compose REPL. This function is compatible with data from various testing machines.

To ensure accurate fitting, the data set must include at least 10 temperatures for each pressure set, with 3 points at or above the transition temperature for the given pressure. Also, there must be a minimum of two pressure curves. The fitter can tolerate the first curve having one element fewer than the others without negatively impacting the results. This can happen when a data set includes the transition temperature for the second and subsequent curves, but not the first.

The data imported for the Tait model fitter should be in the following units: Temperature: [°C], Pressure: [MPa], and Specific Volume: [cm³/kg].

The initial fitter output provides diagnostic information about the fitting process. Reviewing the initial Tait Parameters and subsequent refinements can be useful. The log can help enter the initial guess before refining the solution. The parameters from each step are reported in the screen log. After the Tait model, key values are reused to fit other supported models.

The fitting process involves the following steps:

  1. Morphology Categories Defined:
    • Amorphous: No melting point but a distinct glass transition temperature. The material softens until it can flow over a broad temperature range. There is no exponential volumetric expansion near the transition temperature.
    • Semi-crystalline: A distinct melt transition temperature and an exponential volumetric expansion with heating, or contraction with cooling, is observed in the data.
    • Hybrid: This category looks much like the amorphous polymer except that there is no distinct transition temperature, there is a non-trivial exponential volumetric expansion near the transition temperature and the curve is continuous on its derivatives.
    • Polyetherimides & Polymer Alloys: PEI polymers and several polymer alloys exhibit a change in the shape of their curves as pressure increases, revealing two transition temperatures and a volume loss between them as pressure and temperature increase below the melt transition. This category frequently does not fit well using the Tait or Schmidt-type pvT models. The current Tait fitter will reject this morphology if it detects this behavior. The curve can be fit manually in Excel and the parameters can be imported for use in Material Modeler.
  2. Obtain Initial Key Values:
    • Hybrid Morphology Test
    • Main Morphology Routine (bypassed if strong hybrid behavior is detected)
    • Determine inflection point(s) and linear fits for amorphous and hybrid polymers
    • Determine transition temperature and polynomial fits for semi crystalline polymers
  3. Compute Initial Tait Parameters from the previously identified key points.
  4. Refine the P=0 curve, given b5 (parameters b1 & b2 for molten and solid states, b7, & b8).
  5. Further refine parameters b6, b3 & b4 for molten and solid states, and b9, based on the solution of the P=0 curve.