{ "node": { "id": "term:ficks-first-law", "type": "Term", "label": "Fick’s first law", "properties": { "id": "ficks-first-law", "name": "Fick’s first law", "section": "topic-transport", "definition": "J = −D dc/dx: diffusive flux is proportional to the concentration gradient and points down it. It is a constitutive law, valid at one instant; Fick’s second law follows from it when mass is conserved.", "formula": "J = −D · dc/dx", "unit": "mol/(m²·s)", "related": [ "diffusion-coefficient", "concentration-gradient", "ficks-second-law", "diffusion" ], "lesson": "transport/ficks-first-law/", "example": { "title": "Read the direction of a flux", "text": "With D = 1×10⁻¹⁰ m²/s and dc/dx = 1×10⁶ mol/m⁴, J = −1×10⁻⁴ mol/(m²·s). The negative sign means diffusion points toward decreasing concentration." }, "url": "https://lemonergy.com/learn/glossary/ficks-first-law/", "mentions": [] } }, "edges": [ { "id": "edge:26", "source": "term:ficks-first-law", "target": "workflow:fick-flux", "type": "usedIn", "properties": { "sourceFile": "wiki-blocks.json" } }, { "id": "edge:2039", "source": "term:ficks-first-law", "target": "topic:topic-transport", "type": "belongsTo", "properties": { "sourceFile": "terms.json" } }, { "id": "edge:2040", "source": "term:ficks-first-law", "target": "term:diffusion-coefficient", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:2041", "source": "term:ficks-first-law", "target": "term:concentration-gradient", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:2042", "source": "term:ficks-first-law", "target": "term:ficks-second-law", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:2043", "source": "term:ficks-first-law", "target": "term:diffusion", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:2044", "source": "term:ficks-first-law", "target": "lesson:transport/ficks-first-law/", "type": "explainedIn", "properties": { "sourceFile": "terms.json", "url": "https://lemonergy.com/learn/transport/ficks-first-law/" } }, { "id": "edge:2045", "source": "term:ficks-first-law", "target": "visual:diffusion", "type": "illustratedBy", "properties": { "sourceFile": "visual-pages.json" } }, { "id": "edge:2046", "source": "term:ficks-first-law", "target": "reference:iupac-green", "type": "cites", "properties": { "sourceFile": "wiki-references.json", "scope": "direct technical reference" } } ], "neighbors": [ { "id": "workflow:fick-flux", "type": "Workflow", "label": "Calculate diffusion flux", "properties": { "id": "fick-flux", "title": "Calculate diffusion flux", "kind": "core", "summary": "Lemonergy Core evaluates this relation.", "steps": [ "Open the lesson and inspect the equation.", "Change one input while holding the others fixed." ], "example": "For D = 1 × 10⁻¹⁰ m²/s and dc/dx = 2 × 10⁶ mol/m⁴, J = −2 × 10⁻⁴ mol/(m²·s).", "observe": "Compare the result with the equation and units.", "limits": "This does not solve second-law time evolution or specify boundary conditions.", "url": "https://lemonergy.com/learn/transport/ficks-first-law/", "label": "Open the worked lesson", "evidence": [ "src/battery_core/diffusion.py" ] } }, { "id": "topic:topic-transport", "type": "Topic", "label": "Transport and modelling", "properties": { "id": "topic-transport", "name": "Transport and modelling", "lede": "How ions move, and the equations and models that describe a cell.", "context": "Specify phase, direction and boundaries.", "example": "The same concentration difference across 10 μm gives ten times the gradient across 100 μm; at equal D, first-law flux magnitude is ten times greater.", "lesson_url": "https://lemonergy.com/learn/transport/ficks-first-law/", "lesson_label": "Open diffusion" } }, { "id": "term:diffusion-coefficient", "type": "Term", "label": "Diffusion coefficient", "properties": { "id": "diffusion-coefficient", "name": "Diffusion coefficient", "abbr": "D", "aka": [ "diffusivity" ], "section": "topic-transport", "definition": "The proportionality constant D in Fick’s first law, in m²/s. It sets how fast a species spreads for a given concentration gradient.", "unit": "m²/s", "related": [ "ficks-first-law", "solid-state-diffusion", "gitt", "diffusion" ], "lesson": "transport/ficks-first-law/#law", "url": "https://lemonergy.com/learn/glossary/diffusion-coefficient/", "mentions": [ "concentration-gradient" ] } }, { "id": "term:concentration-gradient", "type": "Term", "label": "Concentration gradient", "properties": { "id": "concentration-gradient", "name": "Concentration gradient", "section": "topic-transport", "definition": "The rate at which concentration changes with distance, dc/dx, in mol/m⁴. It is the driving force in Fick’s law; current creates one across the electrolyte and inside each particle, and it relaxes away when current stops.", "unit": "mol/m⁴", "related": [ "ficks-first-law", "concentration-polarization", "relaxation", "diffusion" ], "lesson": "transport/ficks-first-law/#sign", "url": "https://lemonergy.com/learn/glossary/concentration-gradient/", "mentions": [ "electrolyte" ] } }, { "id": "term:ficks-second-law", "type": "Term", "label": "Fick’s second law", "properties": { "id": "ficks-second-law", "name": "Fick’s second law", "section": "topic-transport", "definition": "The time-dependent diffusion equation, ∂c/∂t = D ∂²c/∂x², obtained by combining Fick’s first law with conservation of mass. It is what a cell model solves inside each particle to follow lithium concentration over time.", "formula": "∂c/∂t = D · ∂²c/∂x²", "related": [ "ficks-first-law", "solid-state-diffusion", "newman-model", "single-particle-model" ], "lesson": "transport/ficks-first-law/#assumptions", "example": { "title": "Compare diffusion time scales", "text": "For diffusion over an illustrative length of 10 μm with D = 10⁻¹⁴ m²/s, the scale L²/D is 10,000 s, about 2.8 h. Halving the length reduces this scale to one quarter; exact solutions also depend on geometry and boundaries." }, "url": "https://lemonergy.com/learn/glossary/ficks-second-law/", "mentions": [ "diffusion" ] } }, { "id": "term:diffusion", "type": "Term", "label": "Diffusion", "properties": { "id": "diffusion", "name": "Diffusion", "section": "topic-transport", "definition": "The net movement of a species from where it is concentrated to where it is scarce, driven by random thermal motion. In a cell lithium diffuses through the electrolyte in the pores and, far more slowly, through the solid particles themselves.", "related": [ "ficks-first-law", "diffusion-coefficient", "solid-state-diffusion", "concentration-gradient", "migration" ], "lesson": "transport/ficks-first-law/#in-a-cell", "url": "https://lemonergy.com/learn/glossary/diffusion/", "mentions": [ "electrolyte" ] } }, { "id": "lesson:transport/ficks-first-law/", "type": "Lesson", "label": "Fick’s first law", "properties": { "url": "https://lemonergy.com/learn/transport/ficks-first-law/" } }, { "id": "visual:diffusion", "type": "Visual", "label": "Flux follows the concentration slope", "properties": { "title": "Flux follows the concentration slope", "summary": "The minus sign in Fick’s law sends material down its concentration gradient.", "readout": "Flux J = 500 μmol/(m²·s), toward increasing position.", "formula": "J = −D dc/dx", "note": "The profile is imposed and static. Time evolution requires Fick’s second law and boundary conditions.", "software": { "status": "implemented", "label": "Implemented in battery-core", "body": "ficks_first_law_flux evaluates J = −D dc/dx. It does not solve a diffusion PDE, estimate the gradient or apply boundary conditions.", "href": "../transport/ficks-first-law/" }, "html": "
Imposed concentration profile
Imposed concentration profilePosition (μm) versus Concentration (mol/m³). Concentration. Interpretation follows the graphs.00500251,000501,500752,000100Position (μm)Concentration (mol/m³)
Concentration
" } }, { "id": "reference:iupac-green", "type": "Reference", "label": "Quantities, Units and Symbols in Physical Chemistry, 3rd edition, 2nd printing", "properties": { "title": "Quantities, Units and Symbols in Physical Chemistry, 3rd edition, 2nd printing", "publisher": "IUPAC", "url": "https://media.iupac.org/publications/books/gbook/IUPAC-GB3-2ndPrinting-Online-22apr2011.pdf", "type": "Technical reference", "scope": "Physical-chemistry notation, thermodynamics, electrochemistry and transport quantities.", "year": 2011, "accessed": "2026-10-27" } } ] }