论文标题
二嵌段共聚物与非局部扩散的层状相位溶液
Lamellar phase solutions for diblock copolymers with nonlocal diffusions
论文作者
论文摘要
对于具有总链长$γ> 0 $和质量比$ m \ m \ in（-1,1）$的二嵌段共聚物，我们考虑了最小化双重非局部自由能$$ $$ \ MATHCAL {e} _ {e} _ {\ VAREPSILON} _ +\ frac {1} {\ varepsilon^{2S}}} \int_Ωw（u）\，dx +\ frac {1} {2} \int_Ω \ left |（-γ^{2}δ） 潜在的。这是在与非局部扩散的二嵌段共聚物的微相分离现象的研究中产生的。 在单位间隔中，我们将$γ$ limit确定为$ \ varepsilon \ to0^+$，并且还发现明确的隔离本地最小化器与情况相关的层状形态阶段$ m = 0 $，前提是链条足够短，或者非局限性的交互作用足够短，或者是非局限性的相互作用（即足够强大）（即$ s $ s $ s $ \ ys0^+bes ys0^+ys os $ s）。我们强调，这种额外的条件对于非局部情况是新的，并且不存在经典模型。虽然基本的证明需要仔细分析非局部积分。
For a diblock copolymer with total chain length $γ>0$ and mass ratio $m\in(-1,1)$, we consider the problem of minimizing the doubly nonlocal free energy $$ \mathcal{E}_{\varepsilon}(u) =\mathcal{H}(u) +\frac{1}{\varepsilon^{2s}} \int_ΩW(u)\,dx +\frac{1}{2}\int_Ω \left|(-γ^{2}Δ)^{-\frac{1}{2}}(u-m)\right|^2\,dx $$ in a domain $Ω$, where $\mathcal{H}(u)$ is a fractional $H^s$-norm with $s\in(0,\frac12)$, and $W$ is a double-well potential. This arises in the study of micro-phase separation phenomena for diblock copolymers with nonlocal diffusions. On the unit interval, we identify the $Γ$-limit as $\varepsilon\to0^+$, and also find explicit isolated local minimizers associated the lamellar morphology phase in the case $m=0$, provided that the chain is sufficiently short or the nonlocal interaction is sufficiently strong (i.e. as $s\to0^+$). We stress that such extra condition is new for the nonlocal case and is not present in the classical model. The proof, while elementary, requires a careful analysis of the nonlocal integrals.
