---
title: "Patterns on a growing domain Interactive Lab | Sandbox Physics"
description: "What happens to a chemical pattern when its domain expands? Growth stretches the available wavelengths and dilutes each substance over a larger area."
canonical_url: "https://sandboxphysics.com/labs/growing-domain-patterns/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/growing-domain-patterns/"
last_modified: "2026-10-28"
---

FR21 · Growth and dilution

# Patterns on a growing domain

**What happens to a chemical pattern when its domain expands?** Growth stretches the available wavelengths and dilutes each substance over a larger area. Even when the total amount stays constant, concentration falls, which changes the reaction and the pattern.

Expand the same patterned square and watch old peaks stretch and new ones emerge. Compare concentration with total amount, then turn reactions off to isolate dilution.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Uniform exponential expansion stretches a periodic material square. Diffusion uses the current physical spacing and both concentrations dilute with area. Growth stops at 1.7 times the initial side; tissue mechanics and cell division are omitted.

Growth time $10$

Mean concentration $1$

Total activator amount $256$

Physical side length $18.6$

SIMULATION RECORD AND ANALYTIC REFERENCE

## Concentration falls as the area increases

Cyan: mean activator concentration relative to the prepared state. Gold: total activator amount relative to that state. Reactions can change total amount; switch to Conserved amount to remove both source and sink.

The field uses one fixed color scale: dark blue at zero, cyan near one and amber at 2.5 and above. Opposite edges touch.

[Read the derivation and worked examples](https://sandboxphysics.com/stories/morphogenesis/)

Physics tutorial

## Follow material, then count molecules

**Background** When the patterned material expands, neighboring material points move apart. The same chemical amount occupies more area, so its concentration falls unless reactions replace it.

**Why it matters** Changing the picture size alone misses this dilution. Following the material lets us keep track of diffusion, chemical production and the growing area together.

Start with the essentials

- **Focus question**: How can concentration fall even when no chemical is lost?
- **One-sentence intuition**: Concentration is amount per area; the area changes during growth.

### Core mathematical model

#### Coordinates attached to the material

$$
x=L(t)\xi,\qquad y=L(t)\eta,\qquad g=\frac{\dot L}{L}
$$

The material coordinates remain fixed as the physical square grows. Length and time are dimensionless, and the growth rate is the fractional increase of side length per unit time.

#### Reaction, diffusion and area dilution

$$
\left.\partial_t u\right|_{\xi,\eta}=\frac{D_u}{L(t)^2}\nabla_{\xi,\eta}^2u+f(u,v)-2gu
$$

The factor of two comes from growth in two directions. The substrate has the same dilution term, with its own diffusivity and reaction rate.

#### Total amount on the square

$$
M_u=L(t)^2\langle u\rangle,\qquad \frac{dM_u}{dt}=L(t)^2\langle f(u,v)\rangle
$$

Periodic diffusion transfers material internally. With all reactions off, the total amount remains constant even though concentration changes.

### Common difficulties

#### Concentration and amount answer different questions

**Typical misconception** A falling concentration means the chemical has disappeared.

**Better mental model** An expanding area can lower concentration without removing any molecules. Multiply mean concentration by the current area to calculate total amount.

#### The old equilibrium need not remain an equilibrium

**Typical misconception** The same reaction parameters should preserve the original mean concentration.

**Better mental model** Growth continuously dilutes both species. Their reaction rates respond to the new concentrations, and the combined balance can move away from the fixed-domain equilibrium.

### Run the experiment

1. 01

   #### Compare the two quantities

   Start with Growing pattern and compare the cyan and gold curves.

   **What to observe:** Concentration and total amount follow different histories because the area is changing and reactions continue.

2. 02

   #### Remove chemical production and loss

   Select Conserved amount.

   **What to observe:** The gold curve stays at one. The cyan curve decreases as the square expands; all of this decrease comes from dilution.

3. 03

   #### Hold the domain still

   Select Fixed domain, then compare with Fast expansion.

   **What to observe:** The fixed domain retains its original physical spacing. Growth stretches the old pattern and changes which physical wavelengths fit inside the square.

Physical conclusion

To describe a growing pattern, evolve chemical amount and geometry consistently.

## Related experiments and articles

Morphogenesis and reaction–diffusion · Reading path

### Reaction, diffusion, growth and active stress

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why concentration falls as a tissue grows**](https://sandboxphysics.com/stories/morphogenesis/growing-domains/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)