| Περίληψη: | Membranes find extensive applications today in numerous processes ranging from gas purification techniques to the treatment of industrial wastewater and the production of clean water because of their potential for better energy utilization and reduced production and equipment costs. A typical example is seawater desalination, where the use of advanced membrane technologies based on nanoporous, semipermeable materials with well controlled pore architectures would be favored over reverse osmosis due to lower operating cost and minimal environmental impact. But for membranes to achieve the desired levels of purification efficiency and effectiveness (they are also often susceptible to fouling and tend to exhibit low chemical resistance) they must possess an array of desired and novel properties such as high tensile strength and a well-defined nanoscale porous structure; the latter could allow the selective transport of (e.g.) water while simultaneously blocking undesired compounds (e.g., organic molecules). A typical such membrane operation is nanofiltration (NF), driven by applying a pressure difference between the two sides of the membrane.
In the last decade, a large number of experimental studies have identified carbon nanotubes (CNTs) as a very attractive new class of nanoporous materials for designing nanostructured polymeric membranes characterized by exceptionally selective and permeable nanopores. Unfortunately, contradicting experimental results have often been reported as far as the magnitude of flow enhancement is concerned during water transport through nanometer-wide CNTs embedded in micrometer thick membranes. For example, Holt et al. [Nano Letters, 2004] reported an enhancement factor of 4 to 5 orders of magnitude higher while Majumder et al. [Nature, 2005] found water flows that are 2 to 4 orders of magnitude larger than the predicted ones by macroscopic continuum models. More recent experimental results [Qin et al., Nano Letters, 2011] on individual ultra-long (several micrometers) CNTs with diameter in the range 0.81-1.59 nm reported flow enhancement rates below 1000, thus contradicting for the same diameter the results of the two previous studies. A thorough review of the existing literature [Kannam et al., JCP, 2013] has shown that data for the slip length (which characterizes the flow rate of water in CNTs) are scattered over 5 orders of magnitude for nanotubes of diameter 0.81–10 nm.
To help clarify some of the above observations, in this Master’s thesis, we have developed and implemented a coarse-grained method for simulating diffusion of a small molecule (water) within a glassy PMMA membrane containing CNTs which has allowed us to probe significantly longer times than what is possible today by atomistic molecular dynamics (MD) simulations. The method is known as kinetic Monte Carlo, is realized on a lattice, and uses as input data only the transition rates for a water molecule to hop from one lattice site to another. To take into account the nanostructure of the polymeric membrane and the fact that water diffuses much faster within a CNT than within a glassy polymer, lattice sites belonging to PMMA regions of the membrane have been assigned a different rate constant than lattice sites belonging to the interior of a CNT. The two constants have been computed by borrowing data for water diffusivity in the PMMA matrix and in a CNT either from experimental measurements or from independent simulation studies. At T=300K and for CNTs with a diameter D larger than about 2 nm, the rates are equal to 1.3x108 s-1 for PMMA and 2.3x1011 s-1 for CNT. That is, CNT sites correspond to “fast-diffusing” regions while PMMA ones to “slow-diffusing” regions, for a given water molecule.
The simulations begin by distributing a large number of ghost water molecules on the sites of the lattice and letting them hop from site to site by using the above predetermined transition rates. In the simulations, hopping from a PMMA site towards a CNT interior site and backwards is forbidden; the only possible way for a walker to enter-exit a CNT is via the CNT entrance region. From the KMC method we compute the mean square displacement (msd) of all walkers as a function of time and then we apply Einstein’s equation to extract the corresponding effective diffusivity Deff quantifying water transport in the entire polymeric membrane given that the diffusive motion of the penetrants is Fickian. We conducted several such KMC runs both for randomly placed and perfectly aligned CNTs in the matrix, and we calculated the dependence of Deff on the size of CNTs (their diameter D and length L) and their concentration C (% vol.) in the PMMA matrix. Our simulation results indicate that CNT orientation does not significantly affect the water effective diffusivity. We also found that Deff varies practically linearly with both the CNT aspect ratio and CNT concentration. This allowed us to come up with a simple linear expression for Deff as a function of C and L/D describing the mobility of water molecules in the membrane. The predictions of this analytical equation are in excellent agreement with the simulation findings.
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