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synthetic surface and spm specific features

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# Lateral Force Simulation
Simulate lateral (friction) force signals from topography data, modeling how the local surface slope affects the cantilever torsion signal in contact-mode AFM. Equivalent to Gwyddion's `latsim.c` module.
## Inputs
| Name | Type | Required | Description |
|------|------|----------|-------------|
| field | DATA_FIELD | Yes | Input topography surface field |
## Outputs
| Name | Type | Description |
|------|------|-------------|
| forward | DATA_FIELD | Lateral force signal for the forward (+x) scan direction, in Newtons |
| reverse | DATA_FIELD | Lateral force signal for the reverse (-x) scan direction, in Newtons |
## Controls
| Name | Type | Default | Description |
|------|------|---------|-------------|
| direction | dropdown | forward | Scan direction to compute: forward, reverse, or both. When set to forward or reverse, both outputs carry the same single-direction result |
| friction_coefficient | FLOAT | 0.3 | Coulomb friction coefficient between tip and sample (0.0–10.0) |
| adhesion | FLOAT | 1e-9 | Tip-sample adhesion force in Newtons (0.0–1e-6) |
| load | FLOAT | 10e-9 | Applied normal load on the cantilever in Newtons (1e-12–1e-6) |
## Notes
- The lateral force is computed from a contact-mechanics model where the measured torsion signal depends on the local surface tilt angle. The x-gradient of the topography gives the slope, and the resulting lateral force combines the gravitational component along the slope with the friction force (proportional to the normal component of load plus adhesion): F_lateral = (F_load sin(theta) + mu (F_load cos(theta) + F_adhesion)) / (cos(theta) - mu sin(theta)).
- Forward and reverse scans produce different lateral force signals because friction opposes the scan direction. The forward scan (+x) adds the friction contribution to the slope component, while the reverse scan (-x) subtracts it, producing the characteristic "friction loop" seen in LFM experiments.
- Typical friction coefficients for common AFM sample materials: mica ~0.1–0.3, silicon ~0.2–0.5, polymers ~0.3–0.8, metals ~0.3–0.6. Use lower values for atomically smooth or lubricated surfaces.
- Output values represent the lateral force on the cantilever tip in Newtons. To convert to photodetector voltage, divide by the lateral sensitivity of the optical lever system.
- This node is the equivalent of Gwyddion's `latsim.c` lateral force simulation and uses the same contact-mechanics formulation for topography-induced friction artifacts.

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# MFM Current Simulation
Computes the magnetic stray field from an infinite current-carrying strip and the resulting force on an MFM tip. Useful for simulating the MFM response to current-carrying traces and interconnects.
## Inputs
| Name | Type | Required | Description |
|------|------|----------|-------------|
| field | DATA_FIELD | Yes | Topography used for grid dimensions (x/y size and resolution) |
## Outputs
| Name | Type | Description |
|------|------|-------------|
| hx | DATA_FIELD | In-plane (x) magnetic field component (A/m) |
| hz | DATA_FIELD | Out-of-plane (z) magnetic field component (A/m) |
| force | DATA_FIELD | Vertical force on the MFM tip (N) |
## Controls
| Name | Type | Default | Description |
|------|------|---------|-------------|
| height | FLOAT | 100e-9 | Tip-sample separation in metres |
| current | FLOAT | 1e-3 | Strip current in amperes |
| width | FLOAT | 100e-9 | Width of the current-carrying strip in metres |
| tip_magnetization | FLOAT | 1e5 | Effective tip magnetic moment per unit volume in A/m |
## Notes
- The current strip is infinite along y and centred at x = 0, so the field varies only in the x direction.
- Uses the Biot-Savart law for an infinite conducting strip of finite width to compute Hx and Hz.
- Hx is the in-plane field component; Hz is the out-of-plane component.
- Force is calculated with the point-dipole approximation: Fz = mu_0 * m_tip * dHz/dz.
- Useful for simulating MFM response to current-carrying traces/interconnects.
- Equivalent to Gwyddion's mfm_current.c.

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# MFM Domain Generation
Generate the stray field from a periodic pattern of parallel magnetic stripe domains with alternating up/down magnetization. Equivalent to Gwyddion's mfm_parallel.c.
## Inputs
| Name | Type | Required | Description |
|------|------|----------|-------------|
| field | DATA_FIELD | Yes | Topography used for grid dimensions |
## Outputs
| Name | Type | Description |
|------|------|-------------|
| hz | DATA_FIELD | z-component of the stray field (A/m) |
| dhz_dz | DATA_FIELD | Vertical gradient of the stray field (A/m²) |
## Controls
| Name | Type | Default | Description |
|------|------|---------|-------------|
| height | FLOAT | 50e-9 | Tip-sample separation in metres |
| thickness | FLOAT | 20e-9 | Magnetic film thickness in metres |
| magnetization | FLOAT | 1e6 | Saturation magnetization in A/m |
| stripe_width_a | FLOAT | 200e-9 | Width of the A (+M) domain in metres |
| stripe_width_b | FLOAT | 200e-9 | Width of the B (-M) domain in metres |
| gap | FLOAT | 0.0 | Domain wall gap between stripes in metres |
## Notes
- Domains alternate between A (+M, magnetization up) and B (-M, magnetization down) with an optional gap of zero magnetization between them.
- The stray field is computed via FFT of the magnetization profile using an exponential decay transfer function.
- The field is uniform along y; stripes run parallel to the y-axis.
- dHz/dz is proportional to the MFM phase signal (force gradient), making it the quantity most directly comparable to raw MFM images.
- Useful for simulating MFM calibration samples and testing MFM analysis workflows.
- Typical stripe widths range from 100 nm to several microns.

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# PFM Analysis
Compute polarization vectors from Piezoresponse Force Microscopy (PFM) data by combining vertical (VPFM) and lateral (LPFM) amplitude and phase channels into polarization magnitude, azimuth, and inclination maps. Equivalent to Gwyddion's `pfm.c` module.
## Inputs
| Name | Type | Required | Description |
|------|------|----------|-------------|
| vpfm_amplitude | DATA_FIELD | Yes | Vertical PFM amplitude channel |
| lpfm_amplitude | DATA_FIELD | Yes | Lateral PFM amplitude channel |
| vpfm_phase | DATA_FIELD | Yes | Vertical PFM phase channel (radians) |
| lpfm_phase | DATA_FIELD | Yes | Lateral PFM phase channel (radians) |
## Outputs
| Name | Type | Description |
|------|------|-------------|
| magnitude | DATA_FIELD | Polarization vector magnitude |
| azimuth | DATA_FIELD | In-plane polarization angle |
| inclination | DATA_FIELD | Out-of-plane polarization angle (3D mode only) |
## Controls
| Name | Type | Default | Description |
|------|------|---------|-------------|
| mode | dropdown | 2d | Decomposition mode: 2d (in-plane only) or 3d (full polarization vector) |
| lateral_sensitivity | FLOAT | 1.0 | Scale factor applied to the lateral signal to compensate for differing VPFM/LPFM detector sensitivities |
## Notes
- PFM measures the local piezoelectric response of ferroelectric materials by detecting surface deformation under an applied AC bias via the AFM cantilever.
- In **2d** mode, only in-plane polarization is resolved from the LPFM channels; magnitude and azimuth are computed but inclination is not produced. In **3d** mode, both vertical and lateral channels are combined to reconstruct the full polarization vector, including inclination.
- The **lateral_sensitivity** parameter compensates for the fact that vertical and lateral deflection detectors typically have different sensitivities. Set this to the ratio of LPFM to VPFM detector sensitivity for accurate vector reconstruction.
- Phase inputs must be in radians. If your data is in degrees, convert before connecting (multiply by pi/180).
- Amplitude channels should be unsigned (absolute) piezoresponse magnitudes; the phase channels encode the sign (direction) of the polarization.

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# SEM Simulation
Simulate Scanning Electron Microscopy (SEM) secondary electron yield from topography data. Surface slopes and edges appear bright, while flat areas appear dark. Equivalent to Gwyddion's `semsim.c` module.
## Inputs
| Name | Type | Required | Description |
|------|------|----------|-------------|
| field | DATA_FIELD | Yes | Input topography field |
## Outputs
| Name | Type | Description |
|------|------|-------------|
| result | DATA_FIELD | Simulated SEM image (dimensionless intensity) |
## Controls
| Name | Type | Default | Description |
|------|------|---------|-------------|
| method | dropdown | integration | Simulation method: integration (fast, deterministic) or monte_carlo (stochastic) |
| sigma | FLOAT | 3.0 | Beam interaction distance in pixels, controlling the spatial resolution of the simulated interaction volume |
| n_samples | INT | 100 | Number of Monte Carlo samples per pixel (only used by the monte_carlo method) |
## Notes
- SEM imaging contrast arises because secondary electron yield depends on the local surface orientation relative to the incident beam. Tilted surfaces and sharp edges emit more secondary electrons and therefore appear brighter.
- The integration method evaluates yield analytically from local slopes, making it faster and fully deterministic. The Monte Carlo method samples random electron trajectories, introducing realistic statistical variation into the result.
- Sigma controls the effective size of the beam interaction volume. Larger values blur fine detail and simulate a broader excitation region; smaller values preserve sharp features.
- n_samples only affects the monte_carlo method. Higher values produce smoother, more converged images at the cost of longer computation time.
- This node is equivalent to Gwyddion's `semsim.c` data processing module.

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# SMM Analysis
Scanning Microwave Microscopy analysis: perform 3-point calibration and de-embedding to convert raw S11 reflection coefficient measurements into quantitative tip-sample capacitance and impedance maps. Equivalent to Gwyddion's smm.c and smm_apply.c modules.
## Inputs
| Name | Type | Required | Description |
|------|------|----------|-------------|
| s11_amplitude | DATA_FIELD | Yes | Measured S11 reflection coefficient amplitude |
| s11_phase | DATA_FIELD | Yes | Measured S11 reflection coefficient phase (radians) |
## Outputs
| Name | Type | Description |
|------|------|-------------|
| capacitance | DATA_FIELD | Calibrated tip-sample capacitance map (unit: F) |
| impedance_real | DATA_FIELD | Real part of the de-embedded tip-sample impedance (unit: Ohm) |
## Controls
| Name | Type | Default | Description |
|------|------|---------|-------------|
| frequency | FLOAT | 1e9 | Microwave excitation frequency in Hz |
| ref_impedance | FLOAT | 50.0 | Reference impedance of the transmission line in Ohm |
| cal_c1 | FLOAT | — | First calibration capacitance standard (F) |
| cal_c2 | FLOAT | — | Second calibration capacitance standard (F) |
| cal_c3 | FLOAT | — | Third calibration capacitance standard (F) |
| cal_s11_1 | FLOAT | — | Measured S11 (complex magnitude) at the first calibration standard |
| cal_s11_2 | FLOAT | — | Measured S11 (complex magnitude) at the second calibration standard |
| cal_s11_3 | FLOAT | — | Measured S11 (complex magnitude) at the third calibration standard |
## Notes
- SMM measures local microwave impedance and capacitance by recording the S11 reflection coefficient of a scanning probe coupled to a vector network analyser (VNA) operating at GHz frequencies.
- The 3-point calibration procedure uses three known capacitance standards to solve for the VNA error terms and correct systematic measurement errors, mapping raw S11 values to the true tip-sample impedance.
- The error model decomposes VNA systematics into three terms: e00 (directivity), e01 (tracking), and e11 (source match). These are determined from the three calibration measurements and then used to de-embed every pixel.
- Calibration capacitances should span the expected measurement range; widely spaced standards yield a better-conditioned error model and more accurate results.
- The phase input must be in radians. If your data is in degrees, convert before connecting to this node.

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# Synthetic Surface
Generate synthetic test surfaces for development, calibration, and algorithm testing. Equivalent to Gwyddion's *_synth.c modules.
Generate synthetic test surfaces for development, calibration, and algorithm testing. 28 patterns covering noise, geometry, growth simulations, phase separation, reaction-diffusion, and tiling. Equivalent to Gwyddion's *_synth.c modules.
## Outputs
@@ -10,28 +10,202 @@ Generate synthetic test surfaces for development, calibration, and algorithm tes
## Controls
### Required
| Name | Type | Default | Description |
|------|------|---------|-------------|
| pattern | dropdown | fbm | Pattern type: fbm, white_noise, lattice, steps, particles, flat |
| xres | INT | 256 | Horizontal resolution in pixels (16–2048) |
| yres | INT | 256 | Vertical resolution in pixels (16–2048) |
| xreal | FLOAT | 1e-6 | Physical width in metres |
| yreal | FLOAT | 1e-6 | Physical height in metres |
| amplitude | FLOAT | 1e-9 | Peak-to-peak amplitude in metres |
| seed | INT | 42 | Random seed for reproducibility (0–999999) |
| hurst_exponent | FLOAT | 0.7 | FBM roughness exponent: 0 = rough, 1 = smooth (fbm only) |
| lattice_spacing | FLOAT | 100e-9 | Lattice period in metres (lattice only) |
| lattice_angle | FLOAT | 90.0 | Angle between lattice vectors in degrees (lattice only) |
| n_steps | INT | 5 | Number of step terraces (steps only) |
| n_particles | INT | 20 | Number of particles (particles only) |
| particle_radius_px | INT | 10 | Particle radius in pixels (particles only) |
| pattern | dropdown | fbm | Synthesis pattern (28 options, see Notes below) |
| xres | INT | 256 | Horizontal resolution in pixels (16--2048) |
| yres | INT | 256 | Vertical resolution in pixels (16--2048) |
| xreal | FLOAT | 1e-6 | Physical width in metres (1e-9--1.0) |
| yreal | FLOAT | 1e-6 | Physical height in metres (1e-9--1.0) |
| amplitude | FLOAT | 1e-9 | Peak-to-peak amplitude in metres (0--1e-3) |
| seed | INT | 42 | Random seed for reproducibility (0--999999) |
### Optional (pattern-specific)
| Name | Type | Default | Range | Used by |
|------|------|---------|-------|---------|
| hurst_exponent | FLOAT | 0.7 | 0.0--1.0 | fbm |
| lattice_spacing | FLOAT | 100e-9 | 1e-9--1e-3 | lattice |
| lattice_angle | FLOAT | 90.0 | 0--180 | lattice |
| n_steps | INT | 5 | 1--100 | steps |
| n_particles | INT | 20 | 1--500 | particles, objects, discs, plateaus, pileups, residues, fibres, rods, columnar, deposition, waves, wfr, voronoi |
| particle_radius_px | INT | 10 | 2--100 | particles, objects, discs, plateaus, pileups, residues, columnar, deposition, fibres (width), rods (width) |
| n_iterations | INT | 200 | 10--5000 | domains, ballistic, annealing, spinodal, pde, dla |
| direction_deg | FLOAT | 0.0 | 0--360 | dunes |
| feature_length_px | INT | 40 | 2--500 | fibres, rods |
| object_shape | dropdown | sphere | sphere, pyramid, box, cylinder, cone | objects |
| noise_type | dropdown | gaussian | gaussian, poisson, exponential, uniform, salt_pepper | noise |
| periodic_type | dropdown | checker | checker, hex, stripe, diamond, staircase, rings | periodic |
| spectral_exponent | FLOAT | 2.0 | 0.5--5.0 | spectral |
| frequency | FLOAT | 5.0 | 0.5--50.0 | waves, dunes, periodic, wfr |
## Notes
- **fbm**: Fractional Brownian motion via spectral synthesis. Hurst exponent controls roughness.
- **white_noise**: Gaussian random noise.
- **lattice**: Two-axis sinusoidal grid with configurable spacing and angle.
- **steps**: Terraced step structure with equal step heights.
- **particles**: Random spherical particles on a flat background.
- **flat**: Zero surface (useful as a baseline).
- All patterns are normalised to the specified amplitude range.
All 28 patterns are normalised to the specified amplitude range after generation and use the seed for reproducibility. Changing the seed produces a different random realisation of the same pattern type. Optional parameters that do not apply to the selected pattern are silently ignored.
---
### Random & Spectral
**fbm** -- Fractional Brownian motion via spectral synthesis. Generates a 2D FFT field whose power spectrum follows a power law controlled by the Hurst exponent. An exponent of 0 produces very rough, jagged surfaces; an exponent of 1 produces smooth, gently undulating terrain. This models naturally rough surfaces such as thin films, etched or polished substrates, and geological terrain. Use it whenever you need a statistically self-affine rough surface for testing roughness analysis or levelling algorithms.
- Controls: `hurst_exponent` (0.0--1.0, default 0.7)
**white_noise** -- Uncorrelated Gaussian random noise. Every pixel is drawn independently from a standard normal distribution. This models detector noise, thermal noise floors, or provides a baseline noise field for testing denoising algorithms and signal-to-noise ratio calculations. No additional parameters beyond the common required controls.
**noise** -- Various noise distributions beyond Gaussian. Selecting a distribution changes the statistical character of the noise field:
- *gaussian* (default): standard normal distribution, symmetric tails.
- *poisson*: integer-valued noise with lambda = 5; models photon counting statistics in optical or X-ray detection.
- *exponential*: one-sided distribution; models rare-event intervals or surface feature waiting times.
- *uniform*: flat distribution over [0, 1]; every value equally likely.
- *salt_pepper*: impulse noise where ~5% of pixels are set to +1 or -1, rest zero; models dead pixels or spike artefacts in detector arrays.
Use this pattern to test how processing algorithms handle non-Gaussian noise statistics.
- Controls: `noise_type`
**spectral** -- FFT-based surface with a power-law spectrum P(k) proportional to k^(-alpha). This is a generalisation of FBM where the spectral exponent directly sets how power is distributed across spatial frequencies. A low exponent (near 0.5) concentrates energy at high spatial frequencies, producing rough, fine-grained textures. A high exponent (near 5.0) concentrates energy at low frequencies, producing smooth surfaces with long-range correlations. Use it to create test surfaces with precisely controlled spatial frequency content for filter testing or PSDF validation.
- Controls: `spectral_exponent` (0.5--5.0, default 2.0)
---
### Geometric Features
**particles** -- Random spherical particles on a flat background. Each particle is a hemispherical cap whose height is determined by the radius and position. Later particles overlay earlier ones (maximum height rule). This models nanoparticle deposits on substrates as seen in AFM imaging of colloidal or catalytic nanoparticle samples. Use it to test grain detection, particle counting, and size distribution algorithms.
- Controls: `n_particles` (count), `particle_radius_px` (radius in pixels)
**objects** -- Various 3D geometric shapes placed at random positions. Five shapes are available:
- *sphere*: hemispherical bumps (same geometry as particles).
- *pyramid*: four-sided pyramid with linear slopes meeting at a point.
- *box*: flat-topped rectangular blocks with vertical walls.
- *cylinder*: flat-topped circular pillars.
- *cone*: circular cone with linear profile tapering to a point.
Heights are randomised per object and the maximum-height rule applies. This models calibration gratings, nanofabricated structure arrays, and MEMS features. Use it to test shape recognition, tip characterisation, and volume measurement algorithms.
- Controls: `n_particles` (count), `particle_radius_px` (size), `object_shape`
**discs** -- Flat-topped circular disc features at random positions with random heights. The disc boundary is a sharp step edge. This models thin film islands, lithographic dot arrays, and etched circular features. Use it to test edge detection, step height measurement, and area coverage analysis.
- Controls: `n_particles` (count), `particle_radius_px` (radius)
**plateaus** -- Flat-topped circular features with smooth (tanh-profile) edge transitions instead of sharp steps. The edge transition width is 20% of the radius. This models mesa structures, etched plateaus, and features where the imaging tip or physical process rounds the edges. Useful for testing algorithms that need to distinguish between sharp and gradual step edges.
- Controls: `n_particles` (count), `particle_radius_px` (radius)
**pileups** -- Rounded rectangle structures with random aspect ratio (0.5--2.0) and random orientation. The profile uses a superelliptic (quartic) distance function, producing smoothly rounded rectangular footprints. This models rectangular nanostructures, MEMS features, and lithographic pads with rounded corners. Use it for testing grain shape analysis on non-circular features.
- Controls: `n_particles` (count), `particle_radius_px` (size)
**residues** -- Irregular elliptical deposits with random orientation and random aspect ratio (0.3--3.0). Each deposit has a Gaussian height profile, producing smooth, blob-like features. This models contamination residues, biological particles (cells, proteins), and other irregular deposits commonly encountered in AFM imaging of soft matter. Useful for testing feature detection on non-uniform, non-circular objects.
- Controls: `n_particles` (count), `particle_radius_px` (size)
---
### Linear Features
**fibres** -- Randomly oriented rectangular line features (constant cross-section, flat top). Each fibre has random position, orientation, and height. This models polymer fibres, nanowires, surface scratches, and elongated crystallites. Use it to test line detection, orientation analysis, and fibre network characterisation. The `particle_radius_px` parameter controls fibre width (half-width in pixels) while `feature_length_px` controls length.
- Controls: `n_particles` (count), `feature_length_px` (length), `particle_radius_px` (width)
**rods** -- Like fibres but with a semicircular cross-section profile instead of a flat top. This produces a more physically realistic rounded wire shape. Models nanowires, carbon nanotubes, and other cylindrical structures imaged by AFM where the cross-section appears as a half-circle due to tip convolution. Use it for testing tip deconvolution and cross-section fitting algorithms.
- Controls: `n_particles` (count), `feature_length_px` (length), `particle_radius_px` (width)
---
### Periodic & Tiling
**lattice** -- Two-axis sinusoidal grid. Two cosine waves are superimposed along directions separated by the specified angle, both with the same spatial period. At 90 degrees this produces a square grid; at 60 degrees a hexagonal-like pattern. This models crystal surfaces, calibration gratings, and periodic nanostructure arrays. Use it for testing Fourier analysis, lattice measurement, and periodicity detection.
- Controls: `lattice_spacing` (period in metres), `lattice_angle` (angle between lattice vectors in degrees, 0--180)
**steps** -- Terraced step structure with equal step heights across the x-axis. The surface is a staircase function with the specified number of terraces. This models vicinal crystal surfaces and atomic step trains as seen in STM/AFM imaging of semiconductor or metal surfaces. Use it for testing step detection, terrace fitting, and step height measurement algorithms.
- Controls: `n_steps` (number of terraces, 1--100)
**periodic** -- Repeating tiling patterns. Six tiling types are available:
- *checker* (default): alternating black/white squares (checkerboard).
- *hex*: hexagonal array of circular dots.
- *stripe*: vertical stripes (binary, based on sine threshold).
- *diamond*: diamond/rhombus tiling pattern.
- *staircase*: stepped ramp repeating across x.
- *rings*: concentric ring pattern centred on the field.
This models lithographic patterns, photonic crystal layouts, and periodic test structures. Use it for testing spatial frequency analysis, periodicity detection, and pattern recognition. The `frequency` parameter controls the spatial frequency (number of pattern repeats across the field).
- Controls: `frequency` (spatial frequency, 0.5--50.0), `periodic_type`
**flat** -- Zero surface (all pixel values are zero before amplitude scaling; the output is identically zero regardless of amplitude). This serves as a baseline for testing, a null input for algorithm validation, or a starting point to be combined with other operations via Field Arithmetic. No additional parameters.
---
### Wave Patterns
**waves** -- Superposition of decaying circular waves from random source points. Each source emits a cosine wave that decays exponentially with distance (decay constant of 3.0 in normalised coordinates). Source amplitudes are randomised. This models surface acoustic wave interference, ripple patterns from localised impacts, and capillary wave superposition. Use it for testing wavelet analysis and source localisation algorithms.
- Controls: `n_particles` (number of wave sources), `frequency` (spatial frequency)
**dunes** -- Asymmetric sawtooth-like ripples along a specified direction. The profile ramps gradually over 70% of the period and drops sharply over the remaining 30%, mimicking wind-blown dune asymmetry. A small amount of Gaussian noise (3% amplitude) is added for realism. This models aeolian dune patterns, directional ion beam sputtering ripples, and other asymmetric periodic surface textures. Use it for testing directional analysis and asymmetry quantification.
- Controls: `frequency` (ripple frequency), `direction_deg` (propagation direction in degrees, 0--360)
**wfr** -- Concentric wavefronts from random sources without amplitude decay -- pure interference. Unlike `waves`, there is no exponential fall-off, so all sources contribute equally everywhere, producing complex moire-like interference patterns. The result is normalised by the number of sources. This models wave interference experiments, moire patterns, and standing wave fields. Use it for testing interference fringe analysis and phase extraction.
- Controls: `n_particles` (number of sources), `frequency` (spatial frequency)
---
### Growth & Deposition Simulations
**columnar** -- Gaussian pillar growth at random positions. Each column is a 2D Gaussian bump whose width is set by the radius parameter and whose height is randomised. Columns are summed (additive stacking), so overlapping columns produce taller features. This models columnar thin film growth as seen in PVD (physical vapour deposition) and CVD (chemical vapour deposition) processes. Use it for testing grain analysis on columnar microstructures.
- Controls: `n_particles` (number of columns), `particle_radius_px` (column width)
**deposition** -- Spherical particles deposited sequentially with gravity-like stacking. Each particle is a hemisphere; when it lands on existing material, it sits on top of the highest point within its footprint. This produces layered, stacked structures where later particles rest on earlier ones. Models sequential particle deposition, powder coatings, and colloidal stacking. Use it for testing height profile analysis of layered structures.
- Controls: `n_particles` (number of deposited particles), `particle_radius_px` (particle radius)
**ballistic** -- Ballistic deposition with neighbour adhesion. In each iteration, ~30% of randomly chosen sites receive a particle. The deposited height is one plus the maximum of the site's own height and its four nearest neighbours, producing lateral correlation and rough, porous growth fronts. This is a vectorised simulation of the classic ballistic deposition model. Models thin film growth with surface tension effects and produces surfaces with characteristic scaling exponents. Use it for testing roughness scaling analysis and growth exponent estimation.
- Controls: `n_iterations` (10--5000)
**dla** -- Diffusion-limited aggregation via iterative boundary growth from a central seed. At each iteration the cluster boundary is dilated, and a fraction (~12.5%) of boundary pixels are randomly added to the aggregate with random heights. This produces fractal, dendritic structures characteristic of DLA. Models electrodeposition, frost formation, mineral dendrites, and crystallisation from solution. Use it for testing fractal dimension analysis and morphological characterisation.
- Controls: `n_iterations` (10--5000)
---
### Phase Separation & PDE
**domains** -- Phase-separated domain patterns via a 2D Ising model using the checkerboard Metropolis algorithm. The surface is initialised with random +/-1 spins and evolved at inverse temperature beta = 0.55 (above the critical temperature for ordering). More iterations produce larger, more coarsened domains. This models magnetic domain patterns, block copolymer self-assembly, and binary alloy phase separation. The output is a binary +/-1 field that is normalised to the amplitude range. Use it for testing domain size analysis, boundary detection, and correlation length measurement.
- Controls: `n_iterations` (10--5000)
**spinodal** -- Spinodal decomposition via the Cahn-Hilliard equation solved with an FFT-based semi-implicit scheme. The initial condition is a near-uniform concentration (0.5) with small random perturbations. The system phase-separates into interconnected, bicontinuous domain structures whose characteristic length scale grows with iteration count. This models spinodal decomposition in alloys, polymer blends, and lipid membranes. Use it for testing structure factor analysis, domain coarsening kinetics, and bicontinuous morphology characterisation.
- Controls: `n_iterations` (10--5000)
**pde** -- Gray-Scott reaction-diffusion system producing Turing patterns. Two chemical species (u, v) react and diffuse with parameters F = 0.035, k = 0.065, which lies in the spots/stripes regime. The reaction is seeded in a central square region and grows outward. With few iterations (< 200) the pattern is sparse; with many iterations (500--2000) the pattern fills most of the field with well-developed spots and labyrinthine stripes. This models biological pattern formation (animal coat patterns, morphogenesis), chemical self-organisation (Belousov-Zhabotinsky reaction), and provides complex test patterns for segmentation algorithms.
- Controls: `n_iterations` (10--5000; recommend 500--2000 for fully developed patterns)
**annealing** -- Simulated annealing surface relaxation. Starts with a random Gaussian noise surface and progressively smooths it by averaging with nearest neighbours while adding decreasing amounts of thermal noise. The temperature decreases linearly from 1.0 to 0.01 over the iteration count. This models thermal relaxation, surface diffusion at elevated temperature, and produces surfaces with controlled correlation lengths. More iterations yield smoother surfaces. Use it for testing smoothness quantification and correlation analysis.
- Controls: `n_iterations` (10--5000)
---
### Tessellation
**voronoi** -- Voronoi tessellation with random heights per cell. Random seed points are placed in the field and each pixel is assigned to its nearest seed point. Every cell receives a uniform random height, producing a mosaic of flat-topped polygonal grains separated by sharp boundaries. This models grain boundaries in polycrystalline thin films, biological cell boundaries, and territorial partitioning. Use it for testing grain boundary detection, grain size distribution analysis, and watershed segmentation algorithms.
- Controls: `n_particles` (number of Voronoi sites)