KineticsEngine (FELIX)

Real-time SciPy integration toolkit for optimizing the parameters of a simulation of gold nanoparticle nucleation and growth kinetics via a non-linear ODE system.

Numerical optimization engine for gold nanoparticle synthesis kinetics. SciPy-backed non-linear ODE parameter fitting.

Audience: Physical chemists and material scientists

Key Features

Non-linear parameter estimation via SciPy
JS fallback solver for browser-only mode
Experimental dataset upload and fitting

Use Cases

Extracting rate constants from kinetics data
Fitting ODE models to experimental datasets

OVERVIEW

KINETICS & SPECIES

COMPUTE DISPATCHER

RUNTIME & EMBEDDING

KineticsEngine (FELIX)

FELIX stands for "Fast Equation Logical Investigation eXperiment".
Developed during my doctoral thesis, its primary purpose was to describe experimentally measured reaction kinetics data using a robust rate-equation model. FELIX achieves exactly this: a single, unified mathematical model that simulates complex gold nanoparticle (AuNP) synthesis by adapting to varying precursor factors, absolute rates, and physical parameters.

🎯 Design Philosophy: The Workbench bridges theoretical chemistry and applied kinetic extraction. By combining a live-updating numerical solver with advanced optimization backends (SciPy/CuPy), researchers can immediately observe the impact of physical variables and securely dial in unknown reaction constants automatically against their uploaded datasets.

I. Core Kinetics Mechanism (ODE) The KineticsEngine evaluates complex nanocluster synthesis reactions by mathematically defining them as a continuous system of Ordinary Differential Equations (ODEs). Precursor particles (e.g., Au³⁺) undergo controlled reduction sequences leading to nucleation (Au⁰). Each step's rate is dictated by dynamic rate constants, which can be scaled by physical environment parameters like temperature and surface-area volume ratios.

II. SciPy Optimization (FELIX) The FELIX optimizer backend interfaces natively with the experimental evaluation pipeline. By projecting experimental yield checkpoints against the ODE trajectory bounds, we utilize advanced solver algorithms (such as Levenberg-Marquardt or L-BFGS-B) to minimize the residual sum of squares constraint. Active parameters are flagged via the UI's "Optimize" toggle, dynamically freeing their boundary conditions in the Jacobian matrix computation.

III. Dataset Pipeline & Projection Experimental observation matrices (Time vs. Concentration/Yield) serve as the absolute ground truth. The engine calculates an overlap integral of the simulated kinetic curves against these discrete datasets, visually rendering divergences and generating robust artifact profiles in the library cache asynchronously.

Core Capabilities

Frontend React 18 + Vite Glassmorphic CSS Ecosystem PWA Ready

Numerical Core Native JavaScript Fallback Solver Python / FastAPI Integrator Mathematical Modeling

Optimization Engines SciPy (scipy.optimize.least_squares) CuPy (Hardware Acceleration)

Deployment Context Standalone application mode Iframe/Embed compliant mode (Origin restrictions apply)

Reaction Kinetics & Species

The Finke-Watzky Mechanism

At the center of the engine lies the fundamental Finke-Watzky model, which abstracts nanoparticle formation into two interdependent, non-linear phases: slow continuous nucleation and rapid autocatalytic surface growth.

Slow Nucleation: A ⟶ B (rate constant k₁)

Autocatalytic Growth: A + B ⟶ 2B (rate constant k₂)

💡 The "Genius" of Surface Scaling: Normally, abstract ODE rate equations have no concept of physical dimensions or spatial volume — they only track theoretical dimensionless concentrations. The breakthrough of this specific kinetics engine is the introduction of a dynamic physical dimension: Particle Diameter (d).

Autocatalysis does not just happen abstractly; it occurs on the available surface area of the growing gold nanoparticles. By tracking the realistically evolving particle diameter (scaling smoothly from d_min to d_max) and calculating a live Surface-to-Volume ratio at every time step, physical size finally enters the system. The equations perfectly merge pure chemical kinetics with the spatial reality of nanocluster synthesis!

Chemical Ratios

RWG (Water to Gold Ratio): Within this framework, variables such as RWG mathematically scale the synthesis outcomes. An optimal amount of water in this organic gold nanoparticle synthesis is crucial — it dramatically increases the kinetics and overall yield.

Tracked Chemical Species

Precursor [Au³⁺] The Origin State: Represents the absolute pool of reducible gold salts (e.g., dissolved HAuCl₄) at the onset of the reaction. It serves as the primary mass reservoir, meticulously tracked as its concentration is depleted to feed the subsequent kinetic cascades.

Intermediate Pool [Au¹_Free] The Solvated Reservoir: A free, solvated Au(I) state. While thermodynamically available for further reduction, these ions are kinetically isolated. They exist outside the immediate reactive microenvironments of the clusters, acting as a buffered waiting room before they can undergo particle-forming reactions.

Pre-Nucleation Clusters [Au¹_PNC] The Active Matrix: Au(I) intermediates that have physically aggregated into pre-nucleation clusters. Unlike their free counterparts, PNC-bound ions are structurally integrated into the heart of the reactive pool. They are positioned at the precise sites required for rapid, barrierless reduction into metallic zero-valent gold.

Zero-Valent [Au⁰] The Metallic Core: The final, solid nanoparticle product. As Au⁰ accumulates, it fundamentally alters the physics of the system by generating expanding surface areas, which in turn violently accelerates the autocatalytic rate cycles.

Kinetic Parameters Explained

The parameter matrix governs the transition rules determining the flux between species curves.

Nucleation (k₁ & k_N) The base probability of precursor atoms collapsing into stable seeds. k₁ covers continuous F-W style primary nucleation, while k_N allows for localized or singular-burst seeding sequences.

Autocatalysis (k_AC & k_AC2) The surface-assisted reduction rate. Modifies the speed at which free precursor ions are catalyzed into valid metallic gold upon striking existing cluster surfaces.

Particle Dimensions (d_min, d_max, d_cut) The spatial bounds. d_max governs the final nanocluster size. Crucially, d_cut defines the minimum activation diameter where Autocatalysis formally bridges over from standard nucleation.

Time Structure (t) & Topology (n) Physical runtime duration t drives the ODE forward, while the topological exponent n determines the fractal scaling factor for autocatalytic growth propagation.

System Diagnostics Explained

Particle Diameter (d) Spatial Topology: The calculated physical dimension of the growing gold nanoparticles, expressed in nanometers. Unlike purely abstract ODEs, our Finke-Watzky extended model continuously derives changing physical dimensions. Tracking this geometry allows the engine to accurately derive the real-world surface area available for catalytic reactions at any given millisecond.

Surface Ratio The Catalytic Boundary: Defines the active reactive perimeter exposed to the surrounding solution. As nucleation transitions into growth, the surface area expands exponentially. A high surface ratio dictates an aggressive amplification of autocatalysis, pulling precursor ions out of the liquid phase at massively accelerated rates.

Particle Concentration (nM) Population Density: A theoretical projection of the molar concentration of discrete, fully-formed nanoparticles in the system. This metric is derived mathematically by cross-referencing the total mass of the reduced Au⁰ pool against the current volumetric dimensions of an average particle. Note: Because nanoparticle formation is inherently stochastic, achieving a perfect empirical match with lab data here is notoriously challenging.

AC/N Ratio The Growth Signature: A highly sensitive diagnostic metric that quantifies the kinetic dominance between Autocatalysis (surface condensation) and pure Nucleation (the birth of new seeds). An AC/N ratio of 112x, for example, proves that the reaction is overwhelmingly driven by the swelling of existing clusters rather than the continued spontaneous generation of new ones.

Compute Dispatcher

The Fit Data Pipeline

Instead of manually tuning sliders to guess coefficients, the dataset dispatcher delegates problem solving to heavy-duty optimization backends. The user queues a target CSV/JSON file tracking empirical gold growth, and the orchestrator handles the rest.

1. User drops Target CSV (Time vs Yield)
2. Selects Free Parameters (e.g. k1, k2) & Sets Initial Boundaries
3. Fits Dispatched to Queue (Local or Remote Compute)
4. Optimizer evaluates residuals recursively to minimize error
5. Result persists into the 'Saved Records Library' as a Markdown artifact

Optimizer Architectures

CPU Fast-Math (SciPy) Utilizes vectorized evaluation for highly non-linear constraints. Best for standard time-series mappings.

Interactive Baseline Fallback solver operating locally within the browser context. Limits recursive depth to preserve main thread framerates.

CUDA / CuPy Execution Offloads batched parameter sweeps to GPU. Requires robust backend handshake and job queuing timeouts.

⚠️ Execution Safety: Bound limits (Min/Max) are highly recommended. Unrestricted parameter matrices can result in non-physical solutions or mathematically unsolvable divergences crashing the solver constraint chain.

Runtime & Embedding Constraints

The Bourier.biz Embed Contract

The Kinetics Engine is a standalone product that actively queries its mounting hierarchy to determine behavior parameters via window message events.

System Mode: The webapp switches into "Iframe Protocol" mode when receiving a trusted parentOrigin variable. Upon detecting this, the native Topbar is dynamically rewritten into a minimalist embedded toolbar relying entirely on the parent's thematic synchronization.

Event Handshake Payload

// Child (Workbench) requests sync:
{
  "type": "child:theme-request",
  "payload": { "app": "gold-kinetics-webapp" }
}

// Parent (Bourier.biz) transmits authority state:
{
  "type": "theme:init",
  "payload": { "theme": "dark" }
}

Origin Restrictions

Standalone Security Internal components ignore external CORS origins.

Authorized Parent Chains If a valid domain (like bourier.biz) initializes the embed context, the app mounts the message event listener. Valid parents must be specifically allowed within the backend node.

✅ Dynamic Overlay Parity: The documentation view you are reading now is symmetrically synchronized between the public documentation directory and the WebApp's internal React DocumentationOverlay.jsx component, ensuring 1:1 fidelity regardless of access context.