Search for: All records

Let \(\Sigma\) be a closed subset of \(\mathbb{R}^{n+1}\) which is parabolic Ahlfors-David regular and assume that \(\Sigma\) satisfies a 2-sided corkscrew condition. Assume, in addition, that \(\Sigma\) is either time-forwards Ahlfors-David regular, time-backwards Ahlfors-David regular, or parabolic uniform rectifiable. We then first prove that \(\Sigma\) satisfies a weak synchronized two cube condition. Based on this we are able to revisit the argument of Nyström and Strömqvist (2009) and prove that \(\Sigma\) contain suniform big pieces of Lip(1,1/2) graphs. When \(\Sigma\) is parabolic uniformly rectifiable the construction can be refined and in this case we prove that \(\Sigma\) contains uniform big pieces of regular parabolic Lip(1,1/2) graphs. Similar results hold if \(\Omega\subset\mathbb{R}^{n+1}\) is a connected component of \(\mathbb{R}^{n+1}\setminus\Sigma\) and in this context we also give a parabolic counterpart of the main result of Azzam et al. (2017) by proving that if \(\Omega\) is a one-sided parabolic chord arc domain, and if \(\Sigma\) is parabolic uniformly rectifiable, then \(\Omega\) is in fact a parabolic chord arc domain. Our results give a flexible parabolic version of the classical (elliptic) result of David and Jerison (1990) concerning the existence of uniform big pieces of Lipschitz graphs for sets satisfying a two disc condition. 

As wastewater reclamation and reuse technologies become more critical to meeting the growing demand for water, a need has emerged for separation platforms that can be tailored to accommodate the highly varied feed water compositions and treatment demands of these technologies. Nanofiltration (NF) membranes based on copolymer materials are a promising platform in this regard because they can be engineered at the molecular scale to address an array of separation process needs. Here, for example, a resilient NF membrane is developed through the design of a poly(trifluoroethyl methacrylate- co -oligo(ethylene glycol) methyl ether methacrylate- co -glycidyl methacrylate) [P(TFEMA-OEGMA-GMA)] copolymer that can be dip-coated onto hollow fiber supports. By exploiting the microphase separation of the oligomeric ethylene glycol side chains from the copolymer backbone and by elucidating the processing–structure–property relationships for the dip-coating process, membranes with pores 2 nm-in-diameter that exhibit a hydraulic permeability of 15.6 L m −2 h −1 bar −1 were generated. The GMA repeat units were functionalized post-coating with hexamethylene diamine to incorporate positively-charged moieties along the pore walls. This functionality resulted in membranes that rejected 98% of the MgCl 2 from a 1 mM feed solution. Moreover, the reaction with the diamine crosslinked the copolymer such that the membranes operated stably in ethanol, an organic solvent that damaged the unreacted parent membranes irreparably. Finally, the stability of the crosslinked P(TFEMA-OEGMA-GMA) copolymer resulted in membranes that could operate continuously for a 24 hour period in aqueous solutions containing 500 ppm chlorine without exhibiting signs of structural degradation as evidenced by consistent rejection of neutral probe solutes. These results demonstrate how resilient, charge-selective NF membranes can be fabricated from microphase separated copolymers by engineering each of the constituent repeat units for a directed purpose.