Signatures of Equilibration – A Design-Space Map
Mapping Mpemba-like effects in aqueous solutions
Version: 0.4 Stewardship: Ulrich Warring, University of Freiburg Status: Teaching and Research framework, not peer-reviewed. Use for experimental design and discussion.
What This Document Is
This is a structured guide for exploring an Mpemba–like effect—the counterintuitive observation that hot water can sometimes freeze faster than cold water. Rather than claiming to explain the effect, this document maps out:
What you can control in an experiment (parameters)
What mechanisms might dominate under different conditions
What claims are testable (falsifiable boundaries)
What remains unknown (open questions)
The framework helps you design experiments that can distinguish between competing explanations.
Background Physics
Why might hot water freeze faster?
At first glance, this seems impossible: hot water has more thermal energy to lose. But "freezing" involves two distinct processes:
Cooling — removing heat until the water approaches 0 °C
Nucleation — forming the first stable ice crystal
Water often supercools below 0 °C before freezing begins. The nucleation step is probabilistic: it depends on impurities, container surfaces, and random molecular fluctuations.
Key insight: If preheating changes the nucleation landscape (e.g., by modifying container surfaces or redistributing impurities), it might reduce supercooling and trigger earlier freezing—even though cooling takes longer.
Classical nucleation theory (the essential idea)
Ice doesn't form spontaneously in pure water at 0 °C. A small ice cluster must overcome an energy barrier to grow. This barrier depends on:
Temperature: Deeper supercooling lowers the barrier
Surface properties: Rough or chemically active surfaces provide "nucleation sites" where ice can form more easily
Impurities: Particles in the water can seed ice formation
The nucleation rate (how quickly ice appears) increases dramatically with supercooling depth and nucleation site density.
Established Physics (External Constraints)
These are well-tested principles we take as given:
Classical nucleation theory
Ice formation requires overcoming an energy barrier; surfaces and impurities lower this barrier
Newton's law of cooling
Cooling rate is proportional to temperature difference with surroundings
Thermodynamics
Energy is conserved; phase changes require latent heat removal
Stochastic nucleation
Freezing time = (deterministic cooling) + (random nucleation delay)
We don't re-derive these; we use them as constraints on any proposed explanation.
1. What You Can Control (Experimental Parameters)
Bulk impurities
Dissolved substances, particles, or gas bubbles in the water
Conductivity meter (μS/cm); particle counter; use ultrapure or deliberately seeded water
P_b
Wall nucleation state
How "rough" or chemically active the container surface is
Surface roughness Ra (μm) via profilometer; contact angle θ (°) via goniometer
P_w
Container material
Glass type, plastic, metal—affects surface chemistry and ion leaching
Categorical choice: borosilicate glass (best), soda-lime glass, HDPE, PTFE
M
Thermal history
Has the container been heated before? How many times?
Protocol: specify number of heat-cool cycles, maximum temperature, hold time
H
Cooling rate
How fast heat leaves the sample
Bath temperature; stirring speed; sample geometry
κ
Sample volume
Larger volumes have more thermal inertia and internal gradients
Direct measurement (mL); recommend 5–20 mL for statistical studies
V
Initial temperature
How hot is the "hot" sample?
Temperature difference from cooling bath: ΔT₀ (K)
ΔT₀
System openness
Can water evaporate?
Sealed (lid) vs. open; if open, weigh before and after
σ
Freeze criterion
What counts as "frozen"?
See below—this matters enormously
F
The freeze criterion problem
Many contradictory results in the literature come from different definitions of "freezing":
Nucleation onset
First ice crystal forms
Temperature spike (latent heat release); video observation
Surface ice
Visible ice layer on top
Visual/camera
Full solidification
Entire sample is solid
Temperature plateau ends; physical probing
Important: The nucleation-redistribution hypothesis (that preheating changes where ice starts forming) should show up most clearly at F = nucleation onset. It may not persist to F = full solidification, where total heat removal dominates.
2. Which Mechanism Dominates Where?
Different experimental conditions favour different mechanisms. This table shows likely primary drivers—not certainties.
Very pure water, smooth new container, sealed
Random convection patterns
Stochastic supercooling
High variability between trials; no consistent Mpemba-like effect
Controlled impurities, roughened container, thermal cycling, sealed
Wall nucleation changes
Reduced supercooling
Earlier nucleation in preheated samples; lower variance
Open container, large ΔT₀
Evaporative mass loss
Enhanced convection
Shorter total freeze time, but confounded—you've lost water mass
Large volume, slow cooling
Convection currents
Internal temperature gradients
Results depend on container shape; variable
Small volume, controlled nucleation, sealed
Nucleation timing
Minimal confounders
Cleanest test of nucleation hypothesis
How regimes connect (transition arrows)
Increasing volume (V ↑) while decreasing cooling rate (κ ↓) → shifts from nucleation-dominated to convection-dominated
Opening the system (σ: sealed → open) → introduces evaporation; mass loss confounds all other effects
Smoothing container walls (P_w ↓) → reduces nucleation site density; increases supercooling depth and variance
Switching from borosilicate to soda-lime glass (M change) → may introduce ion leaching into P_b
3. Testable Claims (Falsifiable Boundaries)
A good scientific framework makes claims that could be proven wrong. Here are four:
Boundary 1: Nucleation site threshold
Claim: If the container is very smooth (Ra < 0.05 μm) and the water is ultrapure, preheating won't produce a Mpemba-like effect at nucleation onset.
Test: Compare supercooling distributions in polished borosilicate with different thermal histories.
Current status: Supported in some careful experiments; results vary with protocol.
Boundary 2: Thermal history matters independently
Claim: The effect of preheating comes from changing the container surface (P_w), not just from starting hotter. Two samples at the same temperature but with different thermal histories should behave differently.
Test: Heat one sample slowly to 80 °C then cool to 40 °C; rapidly heat another directly to 40 °C. Compare their supercooling distributions.
How might thermal history change the surface?
Gas bubbles desorbing from surface cracks (timescale: minutes)
Chemical changes to surface oxides (timescale: minutes to hours)
Micro-crack formation or healing (timescale: hours)
Current status: Plausible but under-tested. Timescales are estimates.
Uncertainty note: If the timescales are wrong, the test protocols may not capture the relevant changes. Future work should systematically vary heating duration.
Boundary 3: Evaporation can be ruled out
Claim: In sealed containers with controlled purity and surface state, evaporation cannot explain observed nucleation-timing differences.
Test: Seal containers; weigh before and after; verify mass loss < 0.1%.
Current status: Established for sealed systems. Open systems remain confounded.
Boundary 4: Container material matters
Claim: Changing container material shifts which mechanism dominates, mainly through effects on surface nucleation (P_w) and ion leaching (P_b).
Test: Run identical protocols with borosilicate, soda-lime glass, and PTFE containers; compare supercooling statistics.
Current status: Expected from surface chemistry; systematic Mpemba-like studies limited.
4. Recommended Experimental Protocol
To test whether preheating affects nucleation timing:
Water purity (P_b)
Ultrapure (< 1 μS/cm) OR deliberately seeded with known nucleant (e.g., silver iodide)
Separates bulk from surface effects
Surface state (P_w)
Measure Ra and θ; compare polished vs. roughened containers
This is your main variable
Container (M)
Borosilicate glass; keep constant within experiment
Minimises ion leaching; stable surface
Thermal history (H)
Define protocol: e.g., "heat to 80 °C, hold 5 min, cool to start temperature"
Enables history-dependence test
Cooling rate (κ)
Stirred water bath at −10 °C
Fast enough to observe nucleation before deep supercooling
Volume (V)
5–20 mL; at least 30 replicates per condition
Statistical power; manageable gradients
System (σ)
Sealed; verify Δm < 0.1%
Eliminates evaporation
Freeze criterion (F)
Nucleation onset (temperature spike)
Direct test of nucleation hypothesis
What to measure:
Primary: Temperature at which nucleation occurs (T_nucleation) for each trial
Secondary: Variance across trials; correlation with surface roughness
5. What We Don't Know Yet
Can we predict nucleation from surface measurements? If we measure Ra and θ, can we calculate the expected supercooling distribution? What's the functional relationship?
How fast does thermal history act? If you heat a container for 1 minute vs. 10 minutes, does P_w change differently? At what point do returns diminish?
Does material choice matter beyond surface roughness? Does glass chemistry affect nucleation independently of roughness?
When does nucleation advantage persist to full freezing? Under what conditions does faster nucleation translate to faster complete solidification?
Can we model this quantitatively? Stochastic thermodynamics provides mathematical frameworks for random processes. Can these predict Mpemba-like statistics from measured parameters?
6. Key References
For further reading, organised by topic:
Nucleation theory (foundations):
Turnbull (1950) "Kinetics of heterogeneous nucleation" — the classic paper
Kelton & Greer (2010) Nucleation in Condensed Matter — comprehensive textbook
Mpemba-like effect experiments:
Burridge & Linden (2016) — careful study questioning reproducibility
Burridge & Hallstadius (2020) — protocol standardisation; addresses endpoint confusion
Theoretical approaches:
Lu & Raz (2017) — stochastic thermodynamics of Mpemba-like effects
Kumar & Bechhoefer (2020) — experimental Mpemba effect in colloidal system (not water)
Molecular simulations:
Huang et al. (2015) — MD simulations suggesting structural changes in preheated water
Document History
0.1
2026-01-23
Initial draft
0.2
2026-01-23
Added freeze criterion; split purity parameters
0.3
2026-01-23
Operationalised surface measurements; added material parameter; references
0.4
2026-01-23
Simplified language; added background physics; regime transition arrows; uncertainty propagation notes
How to Use This Document
For lab work: Use §4 (Recommended Protocol) as your starting point. Vary one parameter at a time.
For understanding the literature: Use §2 (Regime Map) to classify which conditions a paper used, then assess whether their conclusions apply to other regimes.
For discussion: Use §5 (Open Questions) to identify genuine unknowns vs. questions that careful experiments could resolve.
For writing reports: Cite specific boundaries (§3) when making claims; acknowledge which regime your experiment falls into.
This framework follows Open-Science Harbour conventions: established physics is referenced but not re-derived; claims are testable; uncertainties are flagged; versions are tracked.
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