6 research outputs found
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 are given numbers and 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
, and . 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
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 [RNÂ(CH<sub>2</sub>-2-O-3,5-C<sub>6</sub>H<sub>2</sub>(<i>t</i>Bu)<sub>2</sub>)<sub>2</sub>]<sup>2–</sup> {[R-ONO]<sup>2–</sup>; 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 TiÂ(O<i>i</i>Pr)<sub>4</sub> with H<sub>2</sub>[<b>1a</b>–<b>1c</b>] in diethyl ether solutions at 25 °C generates quantitatively
the corresponding [R-ONO]ÂTiÂ(O<i>i</i>Pr)<sub>2</sub> (<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 [RNÂ(CH<sub>2</sub>-2-O-3,5-C<sub>6</sub>H<sub>2</sub>(<i>t</i>Bu)<sub>2</sub>)<sub>2</sub>]<sup>2–</sup> ([R-ONO]<sup>2–</sup>; 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>]ÂMÂ(O<i>i</i>Pr)<sub>2</sub> (M = Zr or Hf) with
equimolar H<sub>2</sub>[<b>1a</b>] or MÂ(O<i>i</i>Pr)<sub>4</sub>(HO<i>i</i>Pr) (M = Zr or Hf) with 2 equiv of H<sub>2</sub>[<b>1a</b>] at 25 °C in diethyl ether or 80 °C
in toluene afford moderate yields of colorless crystals of MÂ[<b>1a</b>]Â(O<i>i</i>Pr)Â(<i>i</i>PrOCH<sub>2</sub>-2-O-3,5-C<sub>6</sub>H<sub>2</sub>(<i>t</i>Bu)<sub>2</sub>) (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 H<sub>2</sub>[<b>1a</b>] that produces a highly reactive, transient <i>ortho</i>-quinone methide intermediate. Similar reactions employing
H<sub>2</sub>[<b>1b</b>] and H<sub>2</sub>[<b>1c</b>],
however, led to the formation of homoleptic bis-ligand complexes ZrÂ[<b>1b</b>]<sub>2</sub> (<b>8b</b>) and MÂ[<b>1c</b>]<sub>2</sub> (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 [RNÂ(CH<sub>2</sub>-2-O-3,5-C<sub>6</sub>H<sub>2</sub>(<i>t</i>Bu)<sub>2</sub>)<sub>2</sub>]<sup>2–</sup> ([R-ONO]<sup>2–</sup>; 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>]ÂMÂ(O<i>i</i>Pr)<sub>2</sub> (M = Zr or Hf) with
equimolar H<sub>2</sub>[<b>1a</b>] or MÂ(O<i>i</i>Pr)<sub>4</sub>(HO<i>i</i>Pr) (M = Zr or Hf) with 2 equiv of H<sub>2</sub>[<b>1a</b>] at 25 °C in diethyl ether or 80 °C
in toluene afford moderate yields of colorless crystals of MÂ[<b>1a</b>]Â(O<i>i</i>Pr)Â(<i>i</i>PrOCH<sub>2</sub>-2-O-3,5-C<sub>6</sub>H<sub>2</sub>(<i>t</i>Bu)<sub>2</sub>) (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 H<sub>2</sub>[<b>1a</b>] that produces a highly reactive, transient <i>ortho</i>-quinone methide intermediate. Similar reactions employing
H<sub>2</sub>[<b>1b</b>] and H<sub>2</sub>[<b>1c</b>],
however, led to the formation of homoleptic bis-ligand complexes ZrÂ[<b>1b</b>]<sub>2</sub> (<b>8b</b>) and MÂ[<b>1c</b>]<sub>2</sub> (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>-ArNC<sub>6</sub>H<sub>4</sub>PR<sub>2</sub>)<sup>−</sup> (<b>1a</b>, Ar = 2,6-C<sub>6</sub>H<sub>3</sub>Me<sub>2</sub>, R =
Ph; <b>1b</b>, Ar = 2,6-C<sub>6</sub>H<sub>3</sub><i>i</i>Pr<sub>2</sub>, R = Ph; <b>1c</b>, Ar = 2,6-C<sub>6</sub>H<sub>3</sub>Me<sub>2</sub>, R = <i>i</i>Pr; <b>1d</b>,
Ar = 2,6-C<sub>6</sub>H<sub>3</sub><i>i</i>Pr<sub>2</sub>, R = <i>i</i>Pr) have been prepared and structurally characterized.
The dimeric nickel chloride derivatives {[<b>1b</b>–<b>d</b>]ÂNiÂ(μ-Cl)}<sub>2</sub> (<b>2b</b>–<b>d</b>) were isolated as brick red microcrystals in high yields
from the reactions of NiCl<sub>2</sub>(DME) with either LiÂ[<b>1b</b>–<b>d</b>]Â(solv)<sub><i>x</i></sub> or HÂ[<b>1b</b>–<b>d</b>] in the presence of NEt<sub>3</sub>. Similar reactions employing [<b>1a</b>]<sup>−</sup>, however, generated homoleptic NiÂ[<b>1a</b>]<sub>2</sub> (<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>]ÂNiClÂ(PMe<sub>3</sub>) (<b>4b</b>,<b>c</b>). Interestingly, solution NMR spectroscopic and X-ray crystallographic
data of these PMe<sub>3</sub> 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, CH<sub>2</sub>SiMe<sub>3</sub>, Ph) in THF at −35 °C produced high yields of red or
brownish red crystalline [<b>1b</b>,<b>c</b>]ÂNiRÂ(PMe<sub>3</sub>) (R = Me (<b>5b</b>,<b>c</b>), CH<sub>2</sub>SiMe<sub>3</sub> (<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>]ÂNiHÂ(PMe<sub>3</sub>) (<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 trisÂ(3,5-di-<i>tert</i>-butyl-2-hydroxyphenyl)Âphosphine (H<sub>3</sub>[O<sub>3</sub>P])
with 3 equiv of <i>n</i>BuLi in diethyl ether at −35
°C affords hexanuclear Li<sub>6</sub>[O<sub>3</sub>P]<sub>2</sub>(OEt<sub>2</sub>)<sub>2</sub> (<b>1</b>) as colorless crystals.
In situ lithiation of H<sub>3</sub>[O<sub>3</sub>P] with 3 equiv of <i>n</i>BuLi in THF at −35 °C followed by metathetical
reactions with MnCl<sub>2</sub> or NiCl<sub>2</sub>(DME) gives crystals
of forest green pentanuclear MnLi<sub>4</sub>[O<sub>3</sub>P]<sub>2</sub>(THF)<sub>3</sub> (<b>2</b>) or dark brown tetranuclear
Ni<sub>2</sub>Li<sub>2</sub>[O<sub>3</sub>P]<sub>2</sub>(THF)<sub>2</sub> (<b>3</b>), respectively. Alkane elimination of ZnR<sub>2</sub> (R = Me, Et) with H<sub>3</sub>[O<sub>3</sub>P] in THF at
25 °C generates high yields of colorless crystalline trinuclear
Zn<sub>3</sub>[O<sub>3</sub>P]<sub>2</sub>(THF)<sub>2</sub> (<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 MnÂ(II) (<i>S</i> = 5/2) as indicated by solution and solid state magnetic
susceptibility measurements