TL;DR

Thermal energy can be stored by sensible heating (raise temperature), latent heat (phase change, e.g., melting/solidifying), or thermochemical reactions. Latent‑heat storage uses phase‑change materials (PCMs) that absorb/release lots of energy near a constant temperature. Melting is a dance of molecules: order fades and the material flows. We model it with heat‑transfer theory (the Stefan problem) and watch it using molecular dynamics (MD).

Learning goals

• Tell the difference between sensible, latent, and thermochemical storage.
• Define latent heat of fusion and describe a melting curve.
• Sketch heat flow during melting and identify the Stefan number.
• Describe what molecular dynamics does and how to spot melting (RDF, MSD, diffusion).

1) What is Thermal Energy Storage?

When energy demand and supply don’t line up, storage helps. TES smooths the bumps by parking heat now and using it later.

1. Sensible heat — store energy by changing temperature (hot‑water tanks, rocks).
   Q = m c_p ΔT
2. Latent heat (phase change) — store energy by changing state (solid ↔ liquid).
   Q = m L_f (dominant near melting point)
3. Thermochemical — store energy via reversible reactions (e.g., salt hydration).

2) Thermophysical: The Melting Process

2.1 Phase‑Change Materials (PCMs)

Common PCMs: paraffins, fatty acids, salt hydrates, eutectic alloys. Selection tips: target melting temperature, high L_f, adequate thermal conductivity (can be boosted with fins/graphite/metal foams), cycling stability, low supercooling, non‑toxic, affordable.

2.4 Classic melting (Stefan problem)

Consider a semi‑infinite solid at T_m. At t = 0, the surface jumps to T_s > T_m. A moving interface separates solid and liquid.

• Heat conduction in each region follows the heat equation.
• At the interface, incoming heat equals latent heat consumed.

Interface position often scales like x_i ∝ √(α t), where α = k/(ρ c_p) is thermal diffusivity.

2.5 Real‑world wrinkles

• Supercooling: liquid can remain below T_m before freezing starts.
• Convection in the melt accelerates heat transfer beyond pure conduction.
• Volume change on melting can stress containers.
• Encapsulation (micro/macro) prevents leakage and improves heat transfer.

2.6 Quick numerical example

1 kg paraffin with L_f = 200 kJ/kg and c_p = 2.1 kJ/(kg·K) from 20°C to 55°C:

Q ≈ (1)(2.1)(55 − 20) + (1)(200) = 73.5 + 200 = 273.5 kJ. During phase change, temperature is nearly flat at 55°C.

3) Theory: Equations for Melting

3.1 Enthalpy method

Define enthalpy H(T) to include latent effects; the transient conduction equation becomes ρ (∂H/∂t) = ∇·(k ∇T). Within the mushy zone, H increases rapidly while T changes little.

3.2 Stefan condition (moving interface)

ρ L_f (dx_i/dt) = k_s ∇T_s·n − k_l ∇T_l·n. The interface sits where incoming heat just supplies latent heat.

3.3 Melting curve & pressure

Clausius–Clapeyron: dP/dT = (ΔS/ΔV) = (L_f/T)/ΔV. Most substances expand on melting (ΔV > 0), so dP/dT > 0.

4) Molecular Simulations: Seeing Melting Atom‑by‑Atom

4.1 What is Molecular Dynamics (MD)?

MD integrates Newton’s equations using an interatomic potential (e.g., Lennard‑Jones; EAM for metals; OPLS/CHARMM for organics).

4.2 Typical MD workflow

1. Build a crystal (e.g., FCC) and replicate (≈10⁴–10⁶ atoms).
2. Minimize energy and equilibrate below T_m (NVT/NPT).
3. Heat gradually or place solid–liquid in contact to form an interface.
4. Run dynamics; monitor melting indicators.
5. Analyze and visualize.

4.3 How to tell it’s melting

• Radial distribution function g(r): sharp periodic peaks (solid) → smoother (liquid).
• Mean square displacement (MSD): plateaus in solids; grows linearly in liquids.
• Diffusion coefficient D: from MSD slope (Einstein relation).
• Bond‑orientational order (e.g., q₆) drops at melting.
• Potential energy: often shows a jump around T_m.

4.4 Minimal LAMMPS‑style pseudocode

# build fcc, apply potential
units metal
atom_style atomic
lattice fcc 3.6
region box block 0 20 0 20 0 20
create_box 1 box
create_atoms 1 box
pair_style eam
pair_coeff * * some_metal.eam.alloy Fe

# equilibrate solid
fix 1 all npt temp 300 300 0.1 iso 1 1 1.0
run 20000

# heat towards melting and monitor
fix 1 all npt temp 300 1600 0.1 iso 1 1 1.0
compute msd all msd
compute rdf all rdf 100
thermo_style custom step temp pe c_msd[4]
run 200000

4.5 Limits & tips

• Time/length scales: ns and nm; real systems can be much larger.
• Potentials matter: choose validated force fields.
• Finite‑size effects: use larger cells or finite‑size scaling.
• Hysteresis: melting on heating vs. freezing on cooling may differ.

5) Try‑it ideas (classroom‑friendly)

• Ice‑salt demo: show freezing point depression; record temperature vs. time.
• PCM cup: melt a small paraffin sample with warm water; plot temperature and note the plateau at T_m.
• Spreadsheet model: 1‑D slab using an enthalpy method; track the moving melt front.
• Sim sandbox: use an online MD toy to watch order fade as T rises.

6) Design checklist for a PCM TES

• Target operating temperature range
• Energy to store (kWh) and allowed volume/weight
• Charge/discharge time; need for heat‑transfer enhancement (fins, foams, graphite)
• Cycling stability & supercooling mitigation
• Containment & compatibility (corrosion, leakage)

7) Glossary

• Latent heat (L_f): energy absorbed/released during phase change at constant temperature.
• Mushy zone: region containing both solid and liquid.
• Stefan number (Ste): c_p(T_s − T_m)/L_f.
• Thermal diffusivity (α): how quickly temperature spreads, α = k/(ρ c_p).
• MD (Molecular Dynamics): simulation that tracks atom motions using interatomic forces.
• g(r): probability of finding a neighbor at distance r; reveals structural order.

8) Build this page in WordPress (quick guide)

• Use a single‑column template; place the TOC in a sidebar block or at the top.
• Recreate sections with Heading, Paragraph, List, and Code blocks.
• Add a Callout block for Stefan number or key equations.
• Optional: add a Table block for PCM properties later.
• Accessibility: write alt‑text; keep color contrast high; explain jargon in the Glossary.

Credits: student‑led STEM learning series. Reuse allowed with attribution.