Glass Molding Simulation

The Physics of Glass Forming

A Material Whose Viscosity Moves Ten Decades

The lead role in a glass forming simulation belongs to the temperature dependence of viscosity. For soda-lime glass it is about that of syrup in the melting range (10² Pa·s), 10³–10⁷ Pa·s in the working range where forming happens, 10¹² Pa·s at the annealing point and 10¹³·⁵ Pa·s at the strain point — more than ten orders of magnitude across the process as a whole. That temperature dependence cannot be expressed in Arrhenius form; the VFT (Vogel-Fulcher-Tammann) equation is the standard.

$$ \log_{10} \eta = A + \frac{B}{T - T_0} $$

These three constants (measured for each composition) are the heart of the property data in a process analysis, and error in the viscosity curve becomes error in the wall-thickness distribution and in whether the part can be formed at all. Fitting the VFT constants to viscosity measured at the reference points (softening point = 10⁶·⁶ Pa·s, annealing point = 10¹² Pa·s and so on) is where practice starts.

Heat in a Semitransparent Medium — Internal Radiation

Hot glass is a semitransparent (participating) medium that transmits the near infrared internally, so cooling proceeds not only by conduction and convection from the surface but by radiation out of the volume. Where the material is optically thick (wall thickness sufficiently larger than the absorption length), the Rosseland approximation folds this into the conduction as a "radiative conductivity", which shows up as an effective thermal conductivity several times higher at high temperature. For thin sections or weak absorption that approximation breaks down and band radiative transfer (DOM and the like) is required — switching the treatment of radiation according to wall thickness and wavelength band is the first fork in the road of a glass thermal analysis.

The Glass Transition Is a Rate Process — Structural Relaxation

Numerical Methods

The Forming Stage — Non-Isothermal Free-Surface Flow at Extreme Viscosity

The filling stage of press and blow forming is solved as a non-isothermal Stokes flow at viscosities of 10³–10⁷ Pa·s (inertia is essentially negligible: Re≪1). The numerical essentials are ① tracking the free surface and the mould contact (ALE / interface tracking), ② the strong coupling between temperature and viscosity (the positive feedback in which a thinned region cools, stiffens and then stretches even less), and ③ contact heat transfer with the mould — the contact conductance depends on contact pressure, release agent and surface roughness, and is a calibration parameter spanning several hundred to several thousand W/m²K. For blow forming there are also implementations that use a shell approximation, treating the process as pressure-driven membrane stretching, where predicting the wall-thickness distribution is the main objective.

The Cooling and Stress Stage — Viscoelasticity and TNM, the Two Pillars

Residual stress from the cooling process is the sum of two mechanisms. ① Thermo-viscoelastic stress — the "quench" mechanism in which the surface solidifies first and the interior contracts afterwards (tempered glass exploits this deliberately). Computed with a stress relaxation function (Prony series) and a temperature shift (thermorheological simplicity). ② Structural relaxation stress — arising from density differences produced by the fictive temperature distribution of the TNM model. Birefringence and dimensional stability of optical elements are governed here. Designing an annealing schedule is the quantification of "pass slowly through the region around the annealing point", and solving both mechanisms predicts the annealing time together with the residual stress and birefringence of the lot.

Numerical Pressure Points

• Viscosity spanning decades — a system whose viscosity differs by several orders of magnitude between elements produces an ill-conditioned matrix. Clipping viscosity at an upper bound (switching solidified regions to an elastic body) and progressively solidifying regions are the standard moves
• Uncertainty in contact heat transfer — the mould h is a first-class calibration parameter. Do not skip fitting it against measured temperature (thermocouples embedded in the mould)
• Choice of radiation model — decide up front from the optical thickness whether the approximation applies. Using Rosseland on a thin part overestimates the cooling
• Separation of time scales — forming takes seconds, annealing takes hours to days. Split the analysis by stage and hand data across at the interface

Practical Workflow

The Standard Workflow

1. Assemble the properties — VFT viscosity curve, thermal properties, absorption spectrum (for radiation), relaxation function / TNM parameters. Change the composition and all of it has to be measured again
2. Forming analysis — gob temperature, mould temperature, press speed / blow pressure → filling, wall-thickness distribution, surface-defect risk
3. Cooling analysis — conveyance and cooling conditions → temperature history (internal radiation included)
4. Stress and quality analysis — residual stress, birefringence, fictive temperature distribution → design of the annealing conditions
5. Calibration and validation — calibrate the model against measured wall thickness, stress (polarimetry) and temperature. Then extend it to predicting the effect of condition changes

Wall-Thickness Distribution Is the First Battle

In blow forming of containers and tubing, prediction accuracy on the wall-thickness distribution accounts for most of the value of introducing simulation. What governs wall thickness is the initial temperature distribution of the gob and the timing of mould contact — an accumulation of "wherever it touched first cools and stops stretching". Calibrate the gob temperature distribution and the contact h against a measured thickness map and you can rank the effect of a mould-shape change or a gob-shape change before cutting any tooling. Even at ±10% absolute accuracy it is quite enough for ranking conditions against each other — the same working philosophy as other forming simulations such as welding.

Mapping Quality Defects to Analysis Quantities

Tool Options

Tool Comparison

What to Select On

① Is the main aim forming (thickness, filling) or quality (stress, birefringence, annealing) — the former belongs to Polyflow-class flow solvers, the latter to Abaqus-class viscoelasticity plus TNM. ② The accuracy demanded of internal radiation (band radiation support if you are doing precision cooling of optical glass). ③ Your capability to assemble property data — with any tool, VFT constants, relaxation functions and absorption coefficients are presumed to be measured on your own material, and no precise argument can be built on a built-in "generic soda glass".

Frontiers

Precision Glass Moulding (GMP)

Moulding of aspherical lenses demands sub-micrometre fidelity of shape transfer, and the focus of simulation shifts from "wall thickness" to "predicting how much to correct the mould shape". High-fidelity coupled analysis that includes cooling shrinkage, the refractive-index distribution from structural relaxation, and elastic deformation of the mould itself is at a practical stage, cutting the number of mould-correction iterations. High-accuracy identification of the TNM parameters (differential scanning calorimetry plus beam-bending experiments) is what now limits the accuracy.

Process-Chain Coupling From Melting Tank to Forming

Process-chain simulation is advancing that joins CFD of the melting tank (combustion, electric boosting, bubbles, homogenisation) to the forming analysis, supplying the temperature and composition history of the gob from upstream. The motivation is simultaneous improvement of energy consumption (melting is extremely energy-intensive) and quality (cord, bubbles), and large-scale coupling that takes in participating-media radiation from combustion, Joule heat from electric heating and batch dissolution is at the research and industrial frontier.

Machine Learning in the Loop

Searching mould designs with a surrogate from forming conditions to wall-thickness distribution, predicting quality from furnace operating data (soft sensors), and optimising the annealing schedule are the realistic applications, in a grey-box arrangement of a calibrated physical model plus a data-driven correction. Research combining measurement with Bayesian estimation for inverse identification of properties (VFT, TNM) is also taking shape as quantification of parameter uncertainty (Bayesian calibration).

Troubleshooting

Symptoms, Causes and Fixes

The Order in Which to Start Up

Related: index of industrial thermal process articles, the basics of radiative heat exchange, reflow oven simulation.