Geometric memory in incomplete phase transitions across dimensions

We model a solid-state phase transition through a nucleation-and-growth process in which plates have simple, regular shapes—squares, cubes, or square-faced lamellae—and grow homothetically (self-similarly) until they either reach a randomly assigned maximum size or are stopped by impingement with previously formed plates. The transformation is represented by the preferential disappearance of smaller plates, while larger plates are retained during an incomplete reversion. A subsequent direct transformation therefore produces a modified plate-size distribution—a that forms the main focus of this study. Building upon an earlier two-dimensional ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mn>2</a:mn> <a:mi mathvariant="normal">D</a:mi> </a:mrow> </a:math> ) formulation, we extend the model to ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mn>3</c:mn> <c:mi mathvariant="normal">D</c:mi> </c:mrow> </c:math> ) and to ( <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mn>3</e:mn> <e:msub> <e:mi mathvariant="normal">D</e:mi> <e:mi>L</e:mi> </e:msub> </e:mrow> </e:math> ) in order to examine how dimensionality affects transformation memory. We introduce a quantitative descriptor of memory, the and find that memory is robust in all geometries but overall stronger in <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mn>2</g:mn> <g:mi mathvariant="normal">D</g:mi> </g:mrow> </g:math> than in <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mn>3</i:mn> <i:mi mathvariant="normal">D</i:mi> </i:mrow> </i:math> or <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mn>3</k:mn> <k:msub> <k:mi mathvariant="normal">D</k:mi> <k:mi>L</k:mi> </k:msub> </k:mrow> </k:math> . We provide growth snapshots, arrest-regrowth cycles, size distributions, and differential scanning calorimetry simulations, and we compute the Shannon size-entropy to quantify configurational diversity. Although motivated by the thermal memory effect in shape-memory alloys, the model more generally identifies a purely geometric mechanism for memory in first-order solid-solid transformations, highlighting the role of dimensionality and geometric blocking in controlling the strength of transformation memory.