{ "node": { "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" ] } }, "edges": [ { "id": "edge:2047", "source": "term:ficks-second-law", "target": "topic:topic-transport", "type": "belongsTo", "properties": { "sourceFile": "terms.json" } }, { "id": "edge:2048", "source": "term:ficks-second-law", "target": "term:ficks-first-law", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:2049", "source": "term:ficks-second-law", "target": "term:solid-state-diffusion", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:2050", "source": "term:ficks-second-law", "target": "term:newman-model", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:2051", "source": "term:ficks-second-law", "target": "term:single-particle-model", "type": "relatedTo", "properties": { "sourceFile": "terms.json", "meaning": "Related context; not a causal claim" } }, { "id": "edge:mentions/ficks-second-law/diffusion", "source": "term:ficks-second-law", "target": "term:diffusion", "type": "mentions", "properties": { "sourceFile": "terms.json", "method": "build_glossary.mention_index + mentions", "field": "definition", "meaning": "Text mention; not a causal claim" } }, { "id": "edge:2053", "source": "term:ficks-second-law", "target": "lesson:transport/ficks-first-law/", "type": "explainedIn", "properties": { "sourceFile": "terms.json", "url": "https://lemonergy.com/learn/transport/ficks-first-law/#assumptions" } }, { "id": "edge:2054", "source": "term:ficks-second-law", "target": "visual:diffusion-transient", "type": "illustratedBy", "properties": { "sourceFile": "visual-pages.json" } }, { "id": "edge:2055", "source": "term:ficks-second-law", "target": "reference:iupac-green", "type": "cites", "properties": { "sourceFile": "wiki-references.json", "scope": "direct technical reference" } } ], "neighbors": [ { "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: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": [] } }, { "id": "term:solid-state-diffusion", "type": "Term", "label": "Solid-state diffusion", "properties": { "id": "solid-state-diffusion", "name": "Solid-state diffusion", "section": "topic-transport", "definition": "Lithium moving through the crystal lattice of an active-material particle, with diffusion coefficients of about 10⁻¹⁶ to 10⁻¹³ m²/s — three or more orders of magnitude slower than the 10⁻¹⁰ m²/s of the liquid. It is usually the rate-limiting step for thick electrodes and large particles.", "unit": "m²/s", "related": [ "diffusion-coefficient", "diffusion", "particle-size", "rate-capability", "gitt" ], "lesson": "transport/ficks-first-law/#in-a-cell", "url": "https://lemonergy.com/learn/glossary/solid-state-diffusion/", "mentions": [] } }, { "id": "term:newman-model", "type": "Term", "label": "Newman model", "properties": { "id": "newman-model", "name": "Newman model", "abbr": "P2D", "aka": [ "pseudo-two-dimensional model", "Doyle–Fuller–Newman", "DFN" ], "section": "topic-transport", "definition": "The standard physics-based lithium-ion cell model: porous-electrode theory across the cell thickness, with solid diffusion solved in a representative spherical particle at every point — hence pseudo-two-dimensional. It predicts voltage from first principles but needs dozens of parameters.", "related": [ "porous-electrode-theory", "single-particle-model", "butler-volmer", "ficks-second-law", "parameterisation", "pybamm" ], "url": "https://lemonergy.com/learn/glossary/newman-model/", "mentions": [ "diffusion" ] } }, { "id": "term:single-particle-model", "type": "Term", "label": "Single-particle model", "properties": { "id": "single-particle-model", "name": "Single-particle model", "abbr": "SPM", "section": "topic-transport", "definition": "A simplified physics model that represents each electrode as one spherical particle and neglects electrolyte gradients. It runs fast enough for a BMS and is accurate at low to moderate currents.", "related": [ "newman-model", "equivalent-circuit-model", "solid-state-diffusion", "bms" ], "url": "https://lemonergy.com/learn/glossary/single-particle-model/", "mentions": [ "electrode", "electrolyte" ] } }, { "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-transient", "type": "Visual", "label": "Diffusion smooths a concentration variation", "properties": { "title": "Diffusion smooths a concentration variation", "summary": "A cosine concentration variation decays analytically under zero-flux boundaries.", "readout": "The initial variation retains 13.89% of its amplitude. Mean concentration stays constant.", "formula": "c/c_mean = 1 + 0.5 cos(πx/L) exp(−π²Dt/L²)", "note": "Constant diffusivity in a one-dimensional slab; no-flux boundaries at x/L = 0 and 1; no source or reaction.", "software": { "status": "reference", "label": "Reference concept", "body": "battery-core evaluates Fick’s first-law flux only. This is a closed-form second-law example, not a PDE solver or a particle diffusion simulation.", "href": "../transport/ficks-first-law/" }, "html": "
A closed-form diffusion mode
A closed-form diffusion modePosition x/L versus Concentration c/c_mean. Initial concentration; At selected time. Interpretation follows the graphs.0.400.70.2510.51.30.751.61Position x/LConcentration c/c_mean
Initial concentrationAt selected time (dashed)
" } }, { "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" } } ] }