We have used a realistic model for double stranded DNA and Monte Carlo simulations to compute the extension (mean span) of a DNA molecule confined in a nanochannel over the full range of confinement in a high ionic strength buffer. The simulation data for square nanochannels resolve the apparent contradiction between prior simulation studies and the predictions from Flory theory, demonstrating the existence of two transition regimes between weak confinement (the de Gennes regime) and strong confinement (the Odijk regime). The simulation data for rectangular nanochannels support the use of the geometric mean for mapping data obtained in rectangular channels onto models developed for cylinders. The comparison of our results with experimental data illuminates the challenges in applying models for confined, neutral polymers to polyelectrolytes. Using a Flory-type approach, we also provide an improved scaling result for the relaxation time in the transition regime close to that found in experiments.
Small-angle x-ray scattering experiments conducted with compositionally asymmetric low molar mass poly(isoprene)--poly(lactide) diblock copolymers reveal an extraordinary thermal history dependence. The development of distinct periodic crystalline or aperiodic quasicrystalline states depends on how specimens are cooled from the disordered state to temperatures below the order-disorder transition temperature. Whereas direct cooling leads to the formation of documented morphologies, rapidly quenched samples that are then heated from low temperature form the hexagonal C14 and cubic C15 Laves phases commonly found in metal alloys. Self-consistent mean-field theory calculations show that these, and other associated Frank-Kasper phases, have nearly degenerate free energies, suggesting that processing history drives the material into long-lived metastable states defined by self-assembled particles with discrete populations of volumes and polyhedral shapes.
SignificanceFormation of complex Frank–Kasper phases in soft matter systems confounds intuitive notions that equilibrium states achieve maximal symmetry, owing to an unavoidable conflict between shape and volume asymmetry in space-filling packings of spherical domains. Here we show the structure and thermodynamics of these complex phases can be understood from the generalization of two classic problems in discrete geometry: the Kelvin and Quantizer problems. We find that self-organized asymmetry of Frank–Kasper phases in diblock copolymers emerges from the optimal relaxation of cellular domains to unequal volumes to simultaneously minimize area and maximize compactness of cells, highlighting an important connection between crystal structures in condensed matter and optimal lattices in discrete geometry.
Cooling disordered compositionally asymmetric diblock copolymers leads to the formation of nearly spherical particles, each containing hundreds of molecules, which crystallize upon cooling below the order-disorder transition temperature (). Self-consistent field theory (SCFT) reveals that dispersity in the block degrees of polymerization stabilizes various Frank-Kasper phases, including the C14 and C15 Laves phases, which have been accessed experimentally in low-molar-mass poly(isoprene)--poly(lactide) (PI-PLA) diblock copolymers using thermal processing strategies. Heating and cooling a specimen containing 15% PLA above and below the from the body-centered cubic (BCC) or C14 states regenerates the same crystalline order established at lower temperatures. This memory effect is also demonstrated with a specimen containing 20% PLA, which recrystallizes to either C15 or hexagonally ordered cylinders (HEX) upon heating and cooling. The process-path-dependent formation of crystalline order shapes the number of particles per unit volume, /, which is retained in the highly structured disordered liquid as revealed by small-angle X-ray scattering (SAXS) experiments. We hypothesize that symmetry breaking during crystallization is governed by the particle number density imprinted in the liquid during ordering at lower temperature, and this metastable liquid is kinetically constrained from equilibrating due to prohibitively large free energy barriers for micelle fusion and fission. Ordering at fixed / is enabled by facile chain exchange, which redistributes mass as required to meet the multiple particle sizes and packing associated with specific low-symmetry Frank-Kasper phases. This discovery exposes universal concepts related to order and disorder in self-assembled soft materials.
Using pruned-enriched Rosenbluth method (PERM) simulations of a discrete wormlike chain model, we provide compelling evidence in support of Odijk's prediction of two distinct Odijk regimes for a long wormlike chain confined in a nanochannel. In both cases, the chain of persistence length l p is renormalized into a series of deflection segments of characteristic length D 2/3 l p 1/3 , where D is the channel size. In the first (classic) Odijk regime, these deflection segments are linearly ordered. In the second Odijk regime, thin, long wormlike chains can backfold at a length scale quantified by the global persistence length. We have measured this quantity by simulations and modified Odijk's global persistence length theory to account for thermal fluctuations. The global persistence length, which is defined to be independent of the effect of excluded volume, provides the requisite closure to Odijk's scaling theory for the second regime and thus allows us to resolve much of the confusion surrounding the so-called "transition" regime for DNA confined in a nanochannel. We show that Odijk's theory for the backfolded regime correctly describes both the average chain extension and the variance about this extension for wormlike chains in channel sizes between the classic Odijk regime and the de Gennes blob regimes, with our data spanning several decades in terms of Odijk's scaling parameter ξ. Although the backfolded Odijk regime occupies a very narrow range of D/l p , it is indeed a regime when viewed in terms of ξ and grows in size with increasing monomer anisotropy.
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