Week 12 Module: Inverse Material Design

Research question#

Every previous module ran forward: given a material and a process, predict what happens. A product developer’s actual question usually runs the other way — “design a cracker that stays crisp for at least three days once opened, at typical kitchen humidity” — and asks which formulation and process choices would achieve that target, not what a given choice produces. Can the eleven forward models built across this course be inverted into a design procedure?

Specimen capstone cracker, target shelf-life at fixed humidity

Where this sits#

Closing module of Stage 4, Prediction and Material Design, and of the course. This module does not introduce new physics; it is explicitly a synthesis exercise that requires selecting the right forward models from Stages 1–3 and running them backward, using the fitting discipline from Module 10 to stay honest about how much confidence an inverted prediction actually deserves. It is the direct precursor to the Inverse Material Design Project assessment.

The model: inversion as a design procedure#

An inverse design problem has three parts, and confusing them is the most common way such a problem goes wrong:

1. A target outcome, stated as a measurable quantity with a tolerance — here, “water activity below the specimen’s critical a_w for at least 3 days at 60% relative humidity” (Module 11’s framing), not “stays crisp,” which is not by itself measurable.
2. A forward model relating process/formulation variables to that outcome — here, two models from Module 11 used together, not either one alone: the Gordon-Taylor T_g(moisture) curve sets the moisture content at which the cracker leaves the glassy region, and Module 11’s moisture-uptake equation, w(t) = w_eq + (w₀ − w_eq)e^−t/τ, predicts how many days it takes to reach that moisture content at the stated humidity. Gordon-Taylor alone gives a boundary, not a shelf life; the uptake equation is what turns the boundary into a day count.
3. The free variables the model actually lets you act on — for a cracker, plausibly starch:water ratio at baking (shifts w₁, w₂ in Gordon-Taylor), bake time/temperature (shifts final moisture content and, via Module 8’s Arrhenius kinetics, how much Maillard browning accompanies it), and packaging barrier properties (shifts the rate of equilibration Module 11 flagged as outside the state diagram itself).

Inverting the model means solving for the free variables that place the target outcome inside the model’s predicted glassy/stable region — but because every forward model in this course carries explicit assumptions and illustrative rather than measured parameters (see every module’s provenance note), an inverted answer is a hypothesis about which direction to move a process variable, to be tested, not a guaranteed specification. Module 10’s residual-plot discipline is exactly the tool for checking, once real data exists, whether the model that was inverted was the right one to trust.

Figure — forward prediction and inverse design, side by side

Design steps

3 (target, model, variables)

Inversion gives

a hypothesis, not a spec

Stays honest via

Module 10's residuals

What you measure and analyse#

Given a stated target (critical water activity, ambient humidity, and a shelf-life target in days) and Gordon-Taylor and moisture-uptake parameters for a candidate starch formulation, you solve for the maximum initial moisture content the baked cracker can have and still meet the target, and identify which one process variable (bake time, at fixed temperature) would need to change to hit that moisture content, referencing Module 7’s transport model for why longer baking removes moisture.

Activity — the course’s synthesis problem#

Choose one Chinese food system from any earlier module (not necessarily the one it was originally anchored to) and pose your own inverse design target for it — a specific, measurable outcome with a tolerance, not a vague texture word (Module 1’s opening argument, now closing the loop). State which two governing models from this course you would need to invert to reach it, and what free variables you would act on. This is deliberately open — the point is to practise the three-part structure above on a system of your choosing, and it is the shape the Inverse Material Design Project assessment expects your submission to take.

References#

• Roos, Y.H. Phase Transitions in Foods, 2nd ed. Academic Press.
• Gordon, M. & Taylor, J.S. (1952). “Ideal Copolymers and the Second-Order Transitions of Synthetic Rubbers. I. Non-Crystalline Copolymers.” Journal of Applied Chemistry 2(9), 493–500.

The spec

What you bring to the session. Written so a reader can tell whether it has been met without asking you.

• state the difference between a forward model (predict outcome from process) and an inverse problem (choose process from target outcome)
• identify which two prior modules' models are needed to solve a stated crispness-shelf-life design target
• propose a formulation or process change and justify it by tracing back through the specific equation it acts on

Teaching team

• Idris Fenn — tutor
• Marisol Quaye — convenor