The conformations of semiflexible polymers in two dimensions confined in a strip of width D are studied by computer simulations, investigating two different models for the mechanism by which chain stiffness is realized. One model (studied by molecular dynamics) is a bead-spring model in the continuum, where stiffness is controlled by a bond angle potential allowing for arbitrary bond angles. The other model (studied by Monte Carlo) is a self-avoiding walk chain on the square lattice, where only discrete bond angles (0° and ±90°) are possible, and the bond angle potential then controls the density of kinks along the chain contour. The first model is a crude description of DNA-like biopolymers, while the second model (roughly) describes synthetic polymers like alkane chains. It is first demonstrated that in the bulk the crossover from rods to self-avoiding walks for both models is very similar, when one studies average chain linear dimensions, transverse fluctuations, etc., despite their differences in local conformations. However, in quasi-one-dimensional confinement two significant differences between both models occur: (i) The persistence length (extracted from the average cosine of the bond angle) gets renormalized for the lattice model when D gets less than the bulk persistence length, while in the continuum model it stays unchanged. (ii) The monomer density near the repulsive walls for semiflexible polymers is compatible with a power law predicted for the Kratky-Porod model in the case of the bead-spring model, while for the lattice case it tends to a nonzero constant across the strip. However, for the density of chain ends, such a constant behavior seems to occur for both models, unlike the power law observed for flexible polymers. In the regime where the bulk persistence length ℓp is comparable to D, hairpin conformations are detected, and the chain linear dimensions are discussed in terms of a crossover from the Daoud/De Gennes "string of blobs"-picture to the flexible rod picture when D decreases and/or the chain stiffness increases. Introducing a suitable further coarse-graining of the chain contours of the continuum model, direct estimates for the deflection length and its distribution could be obtained.
We present a unified scaling description for the dynamics of monomers of a semiflexible chain under good solvent condition in the free draining limit. We consider both the cases where the contour length L is comparable to the persistence length ℓ(p) and the case L ≫ ℓ(p). Our theory captures the early time monomer dynamics of a stiff chain characterized by t(3/4) dependence for the mean square displacement of the monomers, but predicts a first crossover to the Rouse regime of t(2ν/1 + 2ν) for τ¹ ~ ℓ(p)³, and a second crossover to the purely diffusive dynamics for the entire chain at τ2 ∼ L(5/2). We confirm the predictions of this scaling description by studying monomer dynamics of dilute solution of semi-flexible chains under good solvent conditions obtained from our Brownian dynamics (BD) simulation studies for a large choice of chain lengths with number of monomers per chain N = 16-2048 and persistence length ℓ(p) = 1-500 Lennard-Jones units. These BD simulation results further confirm the absence of Gaussian regime for a two-dimensional (2D) swollen chain from the slope of the plot of ⟨R(N)²⟩/2Lℓ(p) ~ L/ℓ(p) which around L/ℓ(p) ∼ 1 changes suddenly from (L/ℓ(p)) → (L/ℓ(p))(0.5), also manifested in the power law decay for the bond autocorrelation function disproving the validity of the worm-like-chain in 2D. We further observe that the normalized transverse fluctuations of the semiflexible chains for different stiffness √(⟨l(⊥)²⟩)/L as a function of renormalized contour length L/ℓ(p) collapse on the same master plot and exhibits power law scaling √(⟨l(⊥)²⟩)/L ~ (L/ℓ(p))(n) at extreme limits, where η = 0.5 for extremely stiff chains (L/ℓ(p) ≫ 1), and η = -0.25 for fully flexible chains. Finally, we compare the radial distribution functions obtained from our simulation studies with those obtained analytically.
Summary The linear stability of immiscible, two-phase-flow displacement processes in porous media is examined. Multiphase-flow characteristics are processes in porous media is examined. Multiphase-flow characteristics are included in the stability description through relative-permeability and capillary-pressure functions. A linear-stability analysis of the steady-state saturation and pressure distributions is carried out in terms of normal modes. The pressure distributions is carried out in terms of normal modes. The resulting linearized eigenvalue problems describing the early evolution of unstable modes show a certain similarity in the respective cases of negligible and non-negligible capillary effects to the Rayleigh and Orr-Sommerfeld equations governing the stability of unbounded shear flowThe stability of noncapillary displacement is first examined. Growth rates of the unstable modes as a function of the wavelength of instability are explicitly obtained for specific classes of initial total-mobility profiles. The Saffman-Taylor instability and layer instability follow profiles. The Saffman-Taylor instability and layer instability follow directly as limiting cases of the initial-mobility profiles. The effect of capillarity on flow stability is examined next. Stability curves are obtained for step-saturation initial profiles. Both capillary pressure and a smooth initial-mobility profile exert a stabilizing influence on the flow displacement. The linear-stability- analysis predictions are compared to the results of Chuoke et al., and an estimate for the effective interfacial tension (IFT) is derived. The results find application in prediction of the onset of instability and description of the early stages of unstable growth in immiscible displacement. Introduction Several important processes involve displacement of a fluid in a porous medium by continuous injection of another immiscible fluid. Examples of current interest are processes for the recovery of oil from reservoirs by the processes for the recovery of oil from reservoirs by the injection of fluids of suitable chemical composition or thermal energy content. Because of the inherent heterogeneity of porous-medium properties, flow stability is an essential requirement for the development of an efficient displacement. The frequently observed significant reduction in process efficiency that results from unstable displacement (viscous fingering ) has spurred intensive research to obtain a fundamental understanding of the instability mechanisms. The majority of past investigations on the stability of immiscible flow in porous media are based on the description of flow in terms of two macroscopically distinct, single-phase flow regions separated by an abrupt macroscopic interface. Guided by the premise that, as in a Hele-Shaw cell, the potential of incompressible single-phase flow in porous media satisfies the Laplace equation, Chuoke et al. extended the Saffman-Taylor stability analysis for Hele-Shaw flows. On the basis of the abrupt interface approximation, their results show that in the case of low IFT, the displacement is stable when the mobility of the displaced phase is higher than that of the displacing phase and unstable otherwise. To assess the contribution of capillary effects, Chuoke et al. proceeded further by assigning to the macroscopic interface an effective IFT, gamma e, that allows for the pressures in the two regions across the interface to be related pressures in the two regions across the interface to be related to the curvature of the interface. The resulting stability condition showed that capillarity exerts a stabilizing influence at small wavelengths with the onset of instability occurring at wavelengths larger than a critical value X,. With the abrupt interface approximation, several previous investigators examined related aspects of immiscible previous investigators examined related aspects of immiscible displacement, such as the growth and shape of unstable macroscopic fingers, the linear and weakly nonlinear stability of immiscible layers, and the stability of bounded flows. In a study that considers the two-phase-flow transition zone, Hagoort used an energy method to assess the contribution to stability of viscous and capillary effects. The objective of this and the companion paper is to initiate an investigation of the linear stability of immiscible, two-phase flows in porous media that includes a representation of the effects of simultaneous flow and capillarity. With commonly accepted principles for multiphase flow in porous media, a linear-stability analysis in terms of normal modes is implemented. The resulting eigenvalue problems are similar to the equations describing the flow stability of unbounded shear layers. For instance, noncapillary flow stability is the analog to the porous-medium Rayleigh problem for an unbounded porous-medium Rayleigh problem for an unbounded non-viscous shear layer, while capillary flow stability is the analog of the Orr-Sommerfeld problem for an unbounded viscous shear layer. Such an analogy allows the implementation in the present context of several of the methods developed for the study of the hydrodynamic stability of shear layers. SPERE P. 378
The problem of nonlinear rarefied Couette flow with heat transfer has been studied for both monatomic and diatomic gases using the Boltzmann equation with the Bhatnagar-Gross-Krook type models as the governing equation and the method of discrete ordinates as a tool. The calculated results have been compared with the existing experimental data in order to test the accuracy and the applicability of the statistical models for this one-dimensional problem. The calculated density results are found to be in good agreement with available experimental data; the calculated heat flux solution for the linear case are found to always be lower than the experimental data of Teagan and Springer. There seems to be insufficient published experimental data available to draw rigid conclusions. However, the comparisons made here indicate that the statistical models are indeed reasonably accurate so that their use is justified in the type of problems investigated.
-Semiflexible polymers characterized by the contour length L and persistent length ℓp confined in a spatial region D have been described as a series of "spherical blobs" and "deflecting lines" by de Gennes and Odjik for ℓp < D and ℓp ≫ D respectively. Recently new intermediate regimes (extended de Gennes and Gauss-de Gennes) have been investigated by Tree et al. [Phys. Rev. Lett. 110, 208103 (2013)]. In this letter we derive scaling relations to characterize these transitions in terms of universal scaled fluctuations in d-dimension as a function of L, ℓp, and D, and show that the Gauss-de Gennes regime is absent and extended de Gennes regime is vanishingly small for polymers confined in a 2D strip. We validate our claim by extensive Brownian dynamics (BD) simulation which also reveals that the prefactor A used to describe the chain extension in the Odjik limit is independent of physical dimension d and is the same as previously found by Conformations and dynamics of DNA inside a nanochannel have attracted considerable attention among various disciplines of science and engineering [1]. Important biomolecules, such as, chromosomal DNAs, or proteins whose functionalities are crucially dependent on the exact sequence of the nucleotides or amino acids usually exist in highly compact conformations. By straightening these molecules on a two dimensional sheet [2][3][4] or inside a nanochannel [5][6][7][8][9][10] it is possible to obtain the structural details of these molecules. It is believed that a complete characterization of the DNA sequence for each individual and a proper understanding the role of genetic variations will lead to personalized medicine for diseases, such as, cancer [11]. DNA confined and stretched inside a nanochannel offers significant promise towards this goal. Unlike, traditional sequencing using Sangers method [12] which requires fragmentation and replication, analysis of a single DNA will be free from statistical errors and sequence gaps while reconstruction [11]. Naturally quests for efficient but low cost techniques have attracted considerable attention. Along with optical maps [2,7], recently DNA melting characteristics inside a nanochannel have been studied showing further promises [9]. These recent experiments have generated renewed interests in theoretical and computational studies of confined polymers [13]- [24].Confined DNAs inside nanochannels often studied in high salt concentrations [1] where the charges of the individual nucleotides are heavily screened [5,16]. Besides, the resolution of optical studies set by the diffraction limit is typically of the order of 100 base pairs. Under these conditions a double-stranded DNA is often described as a worm-like chain (WLC) [25] whose end-to-end distanceHowever, for a very long chain eventually the excluded volume (EV) effect becomes important [26,27], and for L ≫ ℓ p the end-to-end distance in d dimensions should be characterized by the bulk conformation of a swollen semiflexible chain [28,29] where a is the effective width of the...
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