{
  "abstract": "Silicon-like electrode materials expand several-fold on lithiation, so solid-phase diffusion must be solved in a sphere whose radius, volume and host-site concentration change with the state of lithiation. The reduced-order solutions developed for fixed particles (polynomial profiles, eigenfunction Galerkin collocation and, most recently, Galerkin weak forms with symmetric polynomials for constant and concentration-dependent diffusivity) cannot be applied to such a particle directly, because the governing equation contains a lattice velocity, a dilution term and a moving boundary. We show that, for a homogeneous isotropic expansion of the particle, the lithium balance of a swelling particle reduces exactly to the standard problem of spherical diffusion in a fixed sphere when it is written for the scaled moles X = R_a³ C_a / (R_a0³ C^H_a0) (the moles per unit initial volume, that is, the referential concentration of finite-deformation diffusion theory, which is also the local occupancy of the host sites) in a coordinate that moves with the lattice. In this variable the time derivative is multiplied by τZ², where Z = R_a/R_a0 is the stretch of the particle, the surface flux is Z²τQ/3 and the average of X is the state of lithiation exactly. For a prescribed stretch history this holds for any diffusivity law; with the closure Z³ = f(X_avg) the radius and its rate of change, which appear explicitly in the concentration equation, are eliminated through the average of X, and for a concentration-dependent diffusivity the argument of the diffusivity carries the time-dependent stretch. Because the swelling problem takes the standard form, the weak-form equations and tables of the symmetric-polynomial formulation apply directly. We give them for constant diffusivity, for the moment form and for concentration-dependent diffusivity, in non-dimensional and in dimensional form, together with the relations needed to couple them to a pseudo-two-dimensional electrode model. For a volume law that reaches 3.75 at full capacity, charged to 2.47 times the initial volume, the weak form with three to six unknowns reaches errors of 10⁻⁴ to 10⁻⁸ in the surface occupancy after the first 100 s for constant diffusivity and 10⁻⁴ to 10⁻⁷ for a diffusivity D_ref·e^(6X), where a cell-centered finite-difference scheme needs 32 to more than 128 cells. Over the whole history, including the first seconds and the instants after a change of current, the low-order weak forms are less accurate than a fine mesh. The moles are conserved to round-off by all schemes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "name": "Venkat R. Subramanian"
    },
    {
      "name": "Puneet Jindal"
    },
    {
      "name": "Taylor R. Garrick"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/frgjdvgegsrn5.bib",
  "cited_by": 0,
  "doi": "10.1149/osf.io/g8svr_v1",
  "fulltext_url": null,
  "id": "frgjdvgegsrn5",
  "issns": [],
  "landing_url": "https://osf.io/preprints/ecsarxiv/g8svr_v1/",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "bepress",
      "Engineering"
    ],
    [
      "openalex",
      "2208|Advancements in Battery Materials"
    ],
    [
      "osf-provider",
      "chemistry/electrochemistry-energy"
    ]
  ],
  "openalex": "W7224112595",
  "pdf_url": "https://osf.io/g8svr/download",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/frgjdvgegsrn5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:52Z",
      "native_id": "g8svr",
      "source": "osf",
      "url": "https://osf.io/preprints/ecsarxiv/g8svr_v1/"
    }
  ],
  "subjects": [
    {
      "field": "chemistry",
      "origin": "osf-provider:chemistry/electrochemistry-energy",
      "subfield": "electrochemistry-energy"
    }
  ],
  "title": "Efficient formulation and simulation of solid-phase diffusion in silicon-like volume-expanding materials",
  "title_text": "Efficient formulation and simulation of solid-phase diffusion in silicon-like volume-expanding materials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:52Z",
  "url": "https://research.draconitic.com/paper/frgjdvgegsrn5",
  "venue": "ECSarXiv",
  "versions": []
}