Existence theorems for a crystal surface model involving the p-Laplace operator

The manufacturing of crystal films lies at the heart of modern nanotechnology. How to accurately predict the motion of a crystal surface is of fundamental importance. Many continuum models have been developed for this purpose, including a number of PDE models, which are often obtained as the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. In this paper we offer an analytical perspective into some of these models. To be specific, we study the existence of a weak solution to the boundary value problem for the equation - \Delta e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right)}+au=f, where $p>1, a>0$ are given numbers and $f$ is a given function. This problem is derived from a crystal surface model proposed by J.L.~Marzuola and J.~Weare (2013 Physical Review, E 88, 032403). The mathematical challenge is due to the fact that the principal term in our equation is an exponential function of a p-Laplacian. Existence of a suitably-defined weak solution is established under the assumptions that $p\in(1,2], \ N\leq 4$, and $f\in W^{1,p}$. Our investigations reveal that the key to our existence assertion is how to control the set where -\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right) is $\pm\infty$

Titanium Complexes of Tridentate Aminebiphenolate Ligands Containing Distinct <i>N</i>‑Alkyls: Profound N‑Substituent Effect on Ring-Opening Polymerization Catalysis

The synthesis, structural characterization, and reactivity studies of titanium complexes supported by tridentate amine biphenolate ligands of the type [RN­(CH₂-2-O-3,5-C₆H₂(<i>t</i>Bu)₂)₂]^2– {[R-ONO]^2–; R = <i>t</i>Bu (<b>1a</b>), <i>i</i>Pr (<b>1b</b>), <i>n</i>Pr (<b>1c</b>)} are described. Alcoholysis of Ti­(O<i>i</i>Pr)₄ with H₂[<b>1a</b>–<b>1c</b>] in diethyl ether solutions at 25 °C generates quantitatively the corresponding [R-ONO]­Ti­(O<i>i</i>Pr)₂ (<b>2a</b>–<b>2c</b>) as a yellow crystalline solid. X-ray diffraction studies of <b>2b</b> and <b>2c</b> showed them to be five-coordinate, trigonal-bipyramidal species. Ring-opening polymerization of ε-caprolactone (ε-CL) catalyzed by <b>2b</b> and <b>2c</b> proved to be living, as evidenced by the narrow molecular weight distributions of the derived polymers and the linear dependence of number-averaged molecular weights on the monomer-to-catalyst ratios or polymerization time. Kinetic studies revealed that the polymerization rates are first-order in the concentration of ε-CL and first-order in that of <b>2b</b> and <b>2c</b>. The propagation rate of <b>2c</b> is ca. 15 times faster than that of <b>2b</b>, highlighting a profound substituent effect of primary versus secondary <i>N</i>-alkyls. In sharp contrast, reactions employing catalytic <b>2a</b> produce either low-molecular-weight oligomers or polymers characteristic of somewhat wider molecular weight distributions, depending on the polymerization temperatures

Zirconium and Hafnium Complexes Containing <i>N</i>‑Alkyl-Substituted Amine Biphenolate Ligands: Unexpected Ligand Degradation and Divergent Complex Constitutions Governed by <i>N</i>‑Alkyls

The reactivity and thermal stability of zirconium and hafnium complexes containing the <i>N</i>-alkyl-substituted amine biphenolate ligands of the type [RN­(CH₂-2-O-3,5-C₆H₂(<i>t</i>Bu)₂)₂]^2– ([R-ONO]^2–; R = <i>t</i>Bu (<b>1a</b>), <i>i</i>Pr (<b>1b</b>), or <i>n</i>Pr (<b>1c</b>)) were investigated. The reactions of either [<b>1a</b>]­M­(O<i>i</i>Pr)₂ (M = Zr or Hf) with equimolar H₂[<b>1a</b>] or M­(O<i>i</i>Pr)₄(HO<i>i</i>Pr) (M = Zr or Hf) with 2 equiv of H₂[<b>1a</b>] at 25 °C in diethyl ether or 80 °C in toluene afford moderate yields of colorless crystals of M­[<b>1a</b>]­(O<i>i</i>Pr)­(<i>i</i>PrOCH₂-2-O-3,5-C₆H₂(<i>t</i>Bu)₂) (M = Zr (<b>4a</b>) or Hf (<b>5a</b>)). Controlled experiments revealed that the production of <b>4a</b> and <b>5a</b> proceeds via unexpected thermal degradation of H₂[<b>1a</b>] that produces a highly reactive, transient <i>ortho</i>-quinone methide intermediate. Similar reactions employing H₂[<b>1b</b>] and H₂[<b>1c</b>], however, led to the formation of homoleptic bis-ligand complexes Zr­[<b>1b</b>]₂ (<b>8b</b>) and M­[<b>1c</b>]₂ (M = Zr (<b>8c</b>) or Hf (<b>9c</b>)) as colorless crystals. Decisive factors governing these divergent reaction pathways and complex constitutions are discussed. The X-ray structures of <b>4a</b>, <b>5a</b>, <b>8b</b>, <b>8c</b>, and <b>9c</b> are presented

Zirconium and Hafnium Complexes Containing <i>N</i>‑Alkyl-Substituted Amine Biphenolate Ligands: Unexpected Ligand Degradation and Divergent Complex Constitutions Governed by <i>N</i>‑Alkyls

The reactivity and thermal stability of zirconium and hafnium complexes containing the <i>N</i>-alkyl-substituted amine biphenolate ligands of the type [RN­(CH₂-2-O-3,5-C₆H₂(<i>t</i>Bu)₂)₂]^2– ([R-ONO]^2–; R = <i>t</i>Bu (<b>1a</b>), <i>i</i>Pr (<b>1b</b>), or <i>n</i>Pr (<b>1c</b>)) were investigated. The reactions of either [<b>1a</b>]­M­(O<i>i</i>Pr)₂ (M = Zr or Hf) with equimolar H₂[<b>1a</b>] or M­(O<i>i</i>Pr)₄(HO<i>i</i>Pr) (M = Zr or Hf) with 2 equiv of H₂[<b>1a</b>] at 25 °C in diethyl ether or 80 °C in toluene afford moderate yields of colorless crystals of M­[<b>1a</b>]­(O<i>i</i>Pr)­(<i>i</i>PrOCH₂-2-O-3,5-C₆H₂(<i>t</i>Bu)₂) (M = Zr (<b>4a</b>) or Hf (<b>5a</b>)). Controlled experiments revealed that the production of <b>4a</b> and <b>5a</b> proceeds via unexpected thermal degradation of H₂[<b>1a</b>] that produces a highly reactive, transient <i>ortho</i>-quinone methide intermediate. Similar reactions employing H₂[<b>1b</b>] and H₂[<b>1c</b>], however, led to the formation of homoleptic bis-ligand complexes Zr­[<b>1b</b>]₂ (<b>8b</b>) and M­[<b>1c</b>]₂ (M = Zr (<b>8c</b>) or Hf (<b>9c</b>)) as colorless crystals. Decisive factors governing these divergent reaction pathways and complex constitutions are discussed. The X-ray structures of <b>4a</b>, <b>5a</b>, <b>8b</b>, <b>8c</b>, and <b>9c</b> are presented

Nickel(II) Complexes Containing Bidentate Diarylamido Phosphine Chelates: Kumada Couplings Kinetically Preferred to β‑Hydrogen Elimination

A series of divalent nickel complexes containing diarylamido phosphine ligands of the type (<i>o</i>-ArNC₆H₄PR₂)⁻ (<b>1a</b>, Ar = 2,6-C₆H₃Me₂, R = Ph; <b>1b</b>, Ar = 2,6-C₆H₃<i>i</i>Pr₂, R = Ph; <b>1c</b>, Ar = 2,6-C₆H₃Me₂, R = <i>i</i>Pr; <b>1d</b>, Ar = 2,6-C₆H₃<i>i</i>Pr₂, R = <i>i</i>Pr) have been prepared and structurally characterized. The dimeric nickel chloride derivatives {[<b>1b</b>–<b>d</b>]­Ni­(μ-Cl)}₂ (<b>2b</b>–<b>d</b>) were isolated as brick red microcrystals in high yields from the reactions of NiCl₂(DME) with either Li­[<b>1b</b>–<b>d</b>]­(solv)_<i>x</i> or H­[<b>1b</b>–<b>d</b>] in the presence of NEt₃. Similar reactions employing [<b>1a</b>]⁻, however, generated homoleptic Ni­[<b>1a</b>]₂ (<b>3a</b>) as paramagnetic, dark red prisms in high yield. Addition of trimethylphosphine to red solutions of <b>2b</b>,<b>c</b> in THF at room temperature afforded emerald crystals of [<b>1b</b>,<b>c</b>]­NiCl­(PMe₃) (<b>4b</b>,<b>c</b>). Interestingly, solution NMR spectroscopic and X-ray crystallographic data of these PMe₃ adducts reveal the exclusive formation of <i>cis</i>-<b>4b</b> and <i>trans</i>-<b>4c</b>, as defined by the mutual orientation of the two phosphorus donors incorporated. Metathetical reactions of <b>4b</b>,<b>c</b> with RMgCl (R = Me, CH₂SiMe₃, Ph) in THF at −35 °C produced high yields of red or brownish red crystalline [<b>1b</b>,<b>c</b>]­NiR­(PMe₃) (R = Me (<b>5b</b>,<b>c</b>), CH₂SiMe₃ (<b>6b</b>,<b>c</b>), Ph (<b>7b</b>,<b>c</b>)). Analogous reactions of <b>4c</b> with EtMgCl or <i>n</i>BuMgCl, however, led instead to the isolation of the hydrido species [<b>1c</b>]­NiH­(PMe₃) (<b>8c</b>) in quantitative yield. Solution NMR data of the methyl complexes <b>5b</b>,<b>c</b> indicate the presence of both cis and trans isomers; the major component of <b>5b</b> is cis whereas that of <b>5c</b> is trans. In contrast, complexes <b>6b</b>,<b>c</b>, <b>7b</b>,<b>c</b>, and <b>8c</b> all exist exclusively in the trans form. The chloro complexes <b>2b</b>–<b>d</b> are all active catalyst precursors for Kumada couplings under mild conditions. In particular, this catalysis is compatible with alkyl nucleophiles that contain β-hydrogen atoms, even in reactions with chlorobenzene. The X-ray structures of <b>2d</b>, <b>3a</b>, <b>4c</b>, <b>5c</b>, <b>6b</b>,<b>c</b>, <b>7c</b>, and <b>8c</b> are presented

Homo- and Heteropolynuclear Clusters of Phosphine Triphenolates

The synthesis and structural characterization of a series of homo- and heteropolynuclear clusters constructed with a potentially tetradentate phosphine triphenolate ligand are presented. Treatment of tris­(3,5-di-<i>tert</i>-butyl-2-hydroxyphenyl)­phosphine (H₃[O₃P]) with 3 equiv of <i>n</i>BuLi in diethyl ether at −35 °C affords hexanuclear Li₆[O₃P]₂(OEt₂)₂ (<b>1</b>) as colorless crystals. In situ lithiation of H₃[O₃P] with 3 equiv of <i>n</i>BuLi in THF at −35 °C followed by metathetical reactions with MnCl₂ or NiCl₂(DME) gives crystals of forest green pentanuclear MnLi₄[O₃P]₂(THF)₃ (<b>2</b>) or dark brown tetranuclear Ni₂Li₂[O₃P]₂(THF)₂ (<b>3</b>), respectively. Alkane elimination of ZnR₂ (R = Me, Et) with H₃[O₃P] in THF at 25 °C generates high yields of colorless crystalline trinuclear Zn₃[O₃P]₂(THF)₂ (<b>4</b>). The cluster structures of <b>1</b>–<b>4</b> were all determined by single crystal X-ray diffraction studies. These molecules represent the first examples of metal complexes supported by phosphine triphenolate derivatives. The cluster <b>2</b> contains a paramagnetic core of high spin Mn­(II) (<i>S</i> = 5/2) as indicated by solution and solid state magnetic susceptibility measurements