Continuum Mechanics and Thermodynamics

, Volume 29, Issue 6, pp 1181–1194 | Cite as

Thermodynamic coupling between gradient elasticity and a Cahn–Hilliard type of diffusion: size-dependent spinodal gaps

Original Article


In the electrode materials of lithium ion batteries, the large variations of Li concentration during the charge and discharge processes are often accompanied by phase separations to lithium-rich and lithium-poor states. In particular, when the composition of the material moves into the spinodal region (linearly unstable uniform compositions) or into the miscibility gap (metastable uniform compositions), it tends to decompose spontaneously under composition fluctuations. If the lattice mismatch of the two phases is not negligible, coherency strains arise affecting the decomposition process. Furthermore, when the dimensions of a specimen or a grain reduce down to the nanometer level, the phase transition mechanisms are also substantially influenced by the domain size. This size effect is interpreted in the present article by developing a thermodynamically consistent model of gradient elastodiffusion. The proposed formulation is based on the coupling of the standard Cahn–Hilliard type of diffusion and a simple gradient elasticity model that includes the gradient of volumetric strain in the expression of the Helmholtz free energy density. An initial boundary value problem is derived in terms of concentration and displacement fields, and linear stability analysis is employed to determine the contribution of concentration and strain gradient terms on the instability leading to spinodal decomposition. It is shown that the theoretical predictions are in accordance with the experimental trends, i.e., the spinodal concentration range shrinks (i.e., the tendency for phase separation is reduced) as the crystal size decreases. Moreover, depending on the interplay between the strain and the concentration gradient coefficients, the spinodal region can be completely suppressed below a critical crystal size. Spinodal characteristic length and time are also evaluated by considering the dominant instability mode during the primary stages of the decomposition process, and it is found that they are increasing functions of the strain gradient coefficient.


Gradient elasticity Spinodal decomposition Size effects Diffusion Coherency strains Stability analysis 


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Financial support by the General Secretariat of Research and Technology (GSRT) of Greece and by the European Union under the projects “Shift of the Phase Equilibria in Nanograined Materials” (ERA-NET scheme, no.88839) and “Internal Length Gradient Mechanics across Scales and Materials: Theory, Experiments and Applications” (No. 88257) is acknowledged. The support of the Ministry of Education and Science of Russian Federation under Mega Grant Project “Fabrication and Study of Advanced Multi-Functional Metallic Materials with Extremely High Density of Defects” (No. 14.Z50.31.0039) to Togliatti State University is also gratefully acknowledged.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Aristotle University of ThessalonikiThessalonikiGreece
  2. 2.Michigan Technological UniversityHoughtonUSA
  3. 3.Beijing University of Civil Engineering and ArchitectureBeijingChina
  4. 4.ITMO UniversitySt. PetersburgRussia
  5. 5.Togliatti State UniversityTogliattiRussia

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