A singularly perturbed phase field model used to model lithium-ion batteries including chemical and elastic effects is considered. The underlying energy is given by

where f is a double well potential, \({\mathbb {C}}\) is a symmetric positive definite fourth order tensor, c is the normalized lithium-ion density, and u is the material displacement. The integrand contains elements close to those in energy functionals arising in both the theory of fluid-fluid and solid-solid phase transitions. For a strictly star-shaped, Lipschitz domain \(\varOmega \subset {\mathbb {R}}^2,\) it is proven that \(\varGamma - \lim _{\varepsilon \rightarrow 0} I_\varepsilon = I_0,\) where \(I_0\) is finite only for pairs (u, c) such that \(f(c) = 0\) and the symmetrized gradient \(e(u) = ce_0\) almost everywhere. Furthermore, \(I_0\) is characterized as the integral of an anisotropic interfacial energy density over sharp interfaces given by the jumpset of c.

考虑用于模拟锂离子电池的奇摄动相场模型，包括化学和弹性效应。潜在能量由下式给出

其中f是双阱势，\（{\ mathbb {C}} \）是对称正定四阶张量，c是归一化的锂离子密度，u是材料位移。被积物包含与流体-流体相变和固-固相变理论中出现的能量功能近似的元素。对于严格的星形Lipschitz域\（\ varOmega \ subset {\ mathbb {R}} ^ 2，\），证明了\（\ varGamma-\ lim _ {\ varepsilon \ rightarrow 0} I_ \ varepsilon = I_0，\）其中\（I_0 \）仅对（u，c ）是有限的， 使得\（f（c）= 0 \）对称的梯度\（e（u）= ce_0 \）几乎无处不在。此外，\（I_0 \）的特征是由c的跳跃集给出的尖锐界面上的各向异性界面能密度的积分。