Blob Theory Research Articles

Behaviors of a Polymer Chain in Channels: From Zimm to Rouse Dynamics

The effects of confinement and hydrodynamic interactions on single-chain diffusion behaviors are studied by using a combination of molecular dynamics and multiparticle collision dynamics simulations. For polymers in free space, the simulation results showed that the diffusion coefficient D∞ for long chains scales with the chain length N as D∞ ∼ N–ν (ν = 0.588), consistent with the Zimm dynamics, but it deviates from the Zimm dynamics for short chains. For polymers confined in channels with width H, the diffusion coefficient is found to follow two different scaling relations. The observed behaviors could be understood by introducing a microscopic hydrodynamic length ξh, below which the overall effect of hydrodynamic interactions becomes less important. For a confined chain, the diffusion behavior exhibits a crossover from Zimm (nondraining) to Rouse (free draining) dynamics as the channel size H decreases. When H is larger than ξh, the confinement experienced by the polymer chain is weak and the diffusion coefficient scales as D ∼ H0.7, in accordance with the prediction of blob theory; when H is smaller than ξh, D becomes independent of H, implying a free draining condition. A general analytical expression of D is derived by extending the partially permeable sphere model to the blob scale, which gives a quantitative description of the transition from Zimm to Rouse dynamics as H decreases.

How Implementation of Entropy in Driving Structural Ordering of Nanoparticles Relates to Assembly Kinetics: Insight into Reaction-Induced Interfacial Assembly of Janus Nanoparticles.

The ability to understand and exploit entropic contributions to ordering transition is of essential importance in the design of self-assembling systems with well-controlled structures. However, much less is known about the role of assembly kinetics in entropy-driven phase behaviors. Here, by combining computer simulations and theoretical analysis, we report that the implementation of entropy in driving phase transition significantly depends on the kinetic process in the reaction-induced self-assembly of newly designed nanoparticle systems. In particular, such systems comprise binary Janus nanoparticles at the fluid-fluid interface and undergo phase transition driven by entropy and controlled by the polymerization reaction initiated from the surfaces of just one component of nanoparticles. Our simulations demonstrate that the competition between the reaction rate and the diffusive dynamics of nanoparticles governs the implementation of entropy in driving the phase transition from randomly mixed phase to intercalated phase in these interfacial nanoparticle mixtures, which thereby results in diverse kinetic pathways. At low reaction rates, the transition exhibits abrupt jump in the mixing parameter, in a similar way to first-order, equilibrium phase transition. Increasing the reaction rate diminishes the jumps until the transitions become continuous, behaving as a second-order-like phase transition, where a critical exponent, characterizing the transition, can be identified. We finally develop an analytical model of the blob theory of polymer chains to complement the simulation results and reveal essential scaling laws of the entropy-driven phase behaviors. In effect, our results allow for further opportunities to amplify the entropic contributions to the materials design via kinetic control.

Scaling Behavior of Neutral and Charged Polymers in Nanochannel Confinement

We expand the blob theory for freely-jointed chains and perform molecular dynamics simulations to study the behavior of polymers confined in cylindrical channels. From weak to strong confinement, five scaling regimes, de Gennes, extended de Gennes, transition, backfolding, and Odijk regimes, are distinguished for neutral polymers.The size scalings in each regime are derived as a function of the channel width. The scaling exponents $-1$ and $-1/3$ are obtained for the transition and backfolding regimes, respectively, which result from the reduction of the excluded volume of the segments by restriction of the segment's orientation in the narrowed channels. For charged flexible chains, the de Gennes regime is split into Flory-de Gennes and electro-de Gennes regimes owing to strong Coulomb repulsion in electrostatic blobs. Nonetheless, the extended de Gennes and transition regimes are shrunken. The study of the fluctuations of the chain size shows consistent scaling demarcations for both the neutral and charged chain systems.

Evaluation of Blob Theory for the Diffusion of DNA in Nanochannels.

We have measured the diffusivity of λ-DNA molecules in approximately square nanochannels with effective sizes ranging from 117 nm to 260 nm at moderate ionic strength. The experimental results do not agree with the non-draining scaling predicted by blob theory. Rather, the data are consistent with the predictions of previous simulations of the Kirkwood diffusivity of a discrete wormlike chain model, without the need for any fitting parameters.

Optimal Reactivity and Improved Self-Healing Capability of Structurally Dynamic Polymers Grafted on Janus Nanoparticles Governed by Chain Stiffness and Spatial Organization.

Structurally dynamic polymers are recognized as a key potential to revolutionize technologies ranging from design of self-healing materials to numerous biomedical applications. Despite intense research in this area, optimizing reactivity and thereby improving self-healing ability at the most fundamental level pose urgent issue for wider applications of such emerging materials. Here, the authors report the first mechanistic investigation of the fundamental principle for the dependence of reactivity and self-healing capabilities on the properties inherent to dynamic polymers by combining large-scale computer simulation, theoretical analysis, and experimental discussion. The results allow to reveal how chain stiffness and spatial organization regulate reactivity of dynamic polymers grafted on Janus nanoparticles and mechanically mediated reaction in their reverse chemistry, and, particularly, identify that semiflexible dynamic polymers possess the optimal reactivity and self-healing ability. The authors also develop an analytical model of blob theory of polymer chains to complement the simulation results and reveal essential scaling laws for optimal reactivity. The findings offer new insights into the physical mechanism in various systems involving reverse/dynamic chemistry. These studies highlight molecular engineering of polymer architecture and intrinsic property as a versatile strategy in control over the structural responses and functionalities of emerging materials with optimized self-healing capabilities.

Torque-Induced Rotational Dynamics in Polymers: Torsional Blobs and Thinning

By using the blob theory and computer simulations, we investigate the properties of a linear polymer performing a stationary rotational motion around a long impenetrable rod. In particular, in the simulations the rotation is induced by a torque applied to the end of the polymer that is tethered to the rod. Three different regimes are found, in close analogy with the case of polymers pulled by a constant force at one end. For low torques the polymer rotates maintaining its equilibrium conformation. At intermediate torques the polymer assumes a trumpet shape, being composed by blobs of increasing size. At even larger torques the polymer is partially wrapped around the rod. We derive several scaling relations between various quantities as angular velocity, elongation and torque. The analytical predictions match the simulation data well. Interestingly, we find a "thinning" regime where the torque has a very weak (logarithmic) dependence on the angular velocity. We discuss the origin of this behavior, which has no counterpart in polymers pulled by an applied force.

Matrix-free Brownian dynamics simulation technique for semidilute polymeric solutions.

Evaluating the concentration dependence of static and dynamic properties of macromolecules in semidilute polymer solutions requires accurate calculation of long-range hydrodynamic interactions (HI) and short range excluded volume (EV) forces. In conventional Brownian dynamics simulations (BDS), computation of HI necessitates construction of a dense diffusion tensor commonly performed via Ewald summation. Krylov subspace techniques allow efficient decomposition of this tensor [computational cost scales as O(N^{2}), where N is the total number of beads in bead-spring representation of macromolecules in a simulation box] and computation of Brownian displacements in the box. In this paper, a matrix-free approach for calculation of HI is implemented which leads to O(NlogN) scaling of computational expense. The fidelity of the algorithm is demonstrated by evaluating the asymptotic value of center-of-mass diffusivity of polymer molecules at very low concentrations and their radius of gyration scaling as a function of number of beads for dilute and semidilute solutions (with concentrations up to 5 times the overlap concentration). In turn, a favorable comparison between our results and the blob theory is shown.

Kirkwood diffusivity of long semiflexible chains in nanochannel confinement.

We compute the axial diffusivity of asymptotically long semiflexible polymers confined in square channels. Our calculations employ the Kirkwood approximation of the mobility tensor by combining computational fluid dynamics (CFD) calculations of the hydrodynamic tensor in channel confinement with pruned-enriched Rosenbluth method (PERM) simulations of a discrete wormlike chain model. Three key results emerge from our study. First, for the classic de Gennes regime, we confirm that Brochard and de Gennes' blob theory correctly predicts the scaling of the axial diffusivity, contrary to the conclusions of previous analyses. Second, for the extended de Gennes regime, we show that a modified blob theory, which has been used to incorporate the effect of local stiffness on DNA diffusion in nanoslits, explains the deviation from the prediction of classic blob theory for diffusion in nanochannels. Third, we provide a calculation similar to the modified blob theory to explain the relative insensitivity of the diffusivity to channel size for channels between the extended de Gennes regime and the Odijk regime, which is the most relevant regime for experiments and technological applications of DNA confinement in nanochannels. Our results are not only relevant to the dynamics of confined semiflexible polymers such as DNA, but also reveal interesting analogies between confinement in channels and slits.

Evaluation of the Kirkwood approximation for the diffusivity of channel-confined DNA chains in the de Gennes regime.

We use Brownian dynamics with hydrodynamic interactions to calculate both the Kirkwood (short-time) diffusivity and the long-time diffusivity of DNA chains from free solution down to channel confinement in the de Gennes regime. The Kirkwood diffusivity in confinement is always higher than the diffusivity obtained from the mean-squared displacement of the center-of-mass, as is the case in free solution. Moreover, the divergence of the local diffusion tensor, which is non-zero in confinement, makes a negligible contribution to the latter diffusivity in confinement. The maximum error in the Kirkwood approximation in our simulations is about 2% for experimentally relevant simulation times. The error decreases with increasing confinement, consistent with arguments from blob theory and the molecular-weight dependence of the error in free solution. In light of the typical experimental errors in measuring the properties of channel-confined DNA, our results suggest that the Kirkwood approximation is sufficiently accurate to model experimental data.

Formulation of in situ chemically cross-linked hydrogel depots for protein release: from the blob model perspective.

The fast release rate and the undesirable covalent binding are two major problems often encountered in formulating in situ chemically cross-linked hydrogel as protein release depot, particularly when prolonged release over months is desirable. In this study, we applied the De Gennes' blob theory to analyze and tackle these two problems using a vinylsulfone-thiol (VS-SH) reaction based in situ hydrogel system. We showed that the simple scaling relation ξb ≈ Rg(c/c*)(-v/(3v-1)) is applicable to the in situ hydrogel and the mesh size estimated from the precursor polymer parameters is a reasonable match to experimental results. On the other hand, as predicted by the theory and confirmed by experiments, the drug diffusion within hydrogel depends mainly on polymer concentration but not the degree of modification (DM). The covalent binding was found to be caused by the mismatch of location between the reactive groups and the entanglement points. The mismatch and, thus, the protein binding were minimized by increasing the DM and concentration of the SH polymer relative to the VS polymer, as predicted by theory. Using these principles, an in situ hydrogel system for the controlled release of an antiangiogenic antibody therapeutics bevacizumab for 3 months was developed.

The Odijk Regime in Slits

De Gennes’ blob theory has been remarkably successful at describing weakly confined polymers in both slits and channels, and comparable results surround Odijk’s theory of deflection segments for strongly confined wormlike polymers in nanochannels. However, given the success of Odijk’s theory in channels, it is remarkable that there is no comprehensive theory for the simple case of a wormlike polymer strongly confined between two parallel plates. We propose such a theory by drawing inspiration from the existing literature on ideal wormlike chains in slits and Daoud and de Gennes’ idea of mapping a slit-confined chain to a two-dimensional chain. We postulate that the chain can be quantitatively described as a two-dimensional wormlike chain with a weak perturbation in the confining dimension due to deflection segments. By incorporating the effects of real chains, where the variable slit depth adds subtlety due to concomitant changes in the strength of excluded volume interactions, our theory predicts the existence of three distinct subregimes. We investigate the validity of our claims by performing Monte Carlo simulations of a slit-confined wormlike chain using an off-lattice implementation of the pruned–enriched Rosenbluth method. From these simulations, we find strong numerical evidence supporting our predictions, including the existence of subregimes within the Odijk regime.

Unperturbed dimensions of polymers revisited through the blob theory: the viscometric expansion factor

Abstract Plotting log([η]/M1/2) versus log(N/Nc) where [η] is the intrinsic viscosity of the macromolecular chain of molecular mass equal to M and N/Nc is the number of blobs of which this chain consists (N: the number of statistical segment of the chain and Nc : the number of statistical segment of one blob), we obtain the unperturbed dimensions parameter of the blob, Kθb. This graphical representation in this article is based on a modified, original equation of Han [6] based on the blob theory. The obtained value of Kθb for a given polymer, dissolved in a good solvent, is lower than the unperturbed dimensions parameter, Kθ, obtained for the same polymer in the Flory’s theta conditions. Having Kθb < Kθ we obtain for the viscometric expansion factors of a polymer, based on Kθb or Kθ, αηb > αη. With the obtained values of αηb we find αηb3/αs2αH ≈ 1, as predicted by Weill and des Cloizeauz [1], where using αη we obtain αη3/αs2αH ≈ 0.85 (αs and αH are the static and dynamical expansions factors).

Unperturbed dimensions of polymers revisited through the blob theory: the viscometric expansion factor

Unperturbed dimensions of polymers revisited through the blob theory: the viscometric expansion factor

Revisiting blob theory for DNA diffusivity in slitlike confinement.

Blob theory has been widely applied to describe polymer conformations and dynamics in nanoconfinement. In slit confinement, blob theory predicts a scaling exponent of 2/3 for polymer diffusivity as a function of slit height, yet a large body of experimental studies using DNA produce a scaling exponent significantly less than 2/3. In this work, we develop a theory that predicts that this discrepancy occurs because the segment correlation function for a semiflexible chain such as DNA does not follow the Flory exponent for length scales smaller than the persistence length. We show that these short length scale effects contribute significantly to the scaling for the DNA diffusivity, but do not appreciably affect the scalings for static properties. Our theory is fully supported by Monte Carlo simulations, quantitative agreement with DNA experiments, and the results reconcile this outstanding problem for confined polymers.

Conformational transition of polyelectrolyte chains extending over the de Gennes regime in slitlike nanochannels

The confinement-induced conformational transitions of the polyelectrolyte chain are characterized with the coarse-grained Brownian dynamics simulations and the blob theory. Submicron-sized biopolymer xanthan is chosen as a model polyelectrolyte taking into account both flexible and semiflexible chains for comparison. Confined flexible and semiflexible chains exhibit a nonmonotonic variation in size in weak confinements, where the relative radius of gyration shows a dip and then increases when decreasing the channel width. The rigid chain, realized at low screening, exhibits a sigmoidal transition without minima in size. Major attention should be on the dependence of scaling law exponents on the screening effect of the solution in the moderate confinement of the de Gennes regime. Our findings are expected to provide useful information and new insight into the confined polyelectrolytes when relevant micro/nanochannels are designed and fabricated.

Convective transport by intermittent blob-filaments: Comparison of theory and experiment

A blob-filament (or simply “blob”) is a magnetic-field-aligned plasma structure which is considerably denser than the surrounding background plasma and highly localized in the directions perpendicular to the equilibrium magnetic field B. In experiments and simulations, these intermittent filaments are often formed near the boundary between open and closed field lines, and seem to arise in theory from the saturation process for the dominant edge instabilities and turbulence. Blobs become charge-polarized under the action of an external force which causes unequal drifts on ions and electrons; the resulting polarization-induced E × B drift moves the blobs radially outwards across the scrape-off-layer (SOL). Since confined plasmas generally are subject to radial or outwards expansion forces (e.g., curvature and ∇B forces in toroidal plasmas), blob transport is a general phenomenon occurring in nearly all plasmas. This paper reviews the relationship between the experimental and theoretical results on blob formation, dynamics and transport and assesses the degree to which blob theory and simulations can be compared and validated against experiments.

DNA Collisions with a Large, Conducting Post

We theoretically treat the problem of DNA collisions with large, ideally conducting posts during electrophoresis. We exploit a separation of time and length scales that allows the problem to be broken into two connected problems: (i) compression of a molecule against a flat wall by an applied transverse field and (ii) transport of a molecule across the post surface due to both tangential diffusion and convection. We address the former using a combination of statistical mechanics and blob theory and the latter utilizing a Fokker−Planck approach. The theoretical predictions are then compared with the results of Brownian dynamics simulations.

Crossover behavior in static and dynamic properties of a single DNA molecule from three to quasi-two dimensions

We studied the conformation and dynamics of a single DNA molecule in a thin slit by a fluorescent microscope. In a slit thinner than the Flory radius in three dimensions, the length of the major axis, the translational self-diffusion coefficient and the rotational relaxation time in a dilute solution show the apparent dependence on the thickness of the slit. The observed dependence is in agreement with that predicted by blob theory, despite the number of blobs is very small. The radial distribution of the segments around the center of mass of a single molecule was also studied and compared with that calculated for a Gaussian and an excluded volume chain. The influence of the polymer concentration on the geometrical confinement by slits was also studied in a semidilute solution near the overlap concentration c∗. The confinement effect is found to be not so serious near c∗ and is only significant in the so-called "two-dimensional pancake" region.

Revisiting of dimensional scaling of linear chains in dilute solutions

The dimensions of linear polymer chains are scaled to their molar mass ( M) as R = kM α with α = 1/2 and 3/5 in a theta and an athermal solvent, respectively. In a good solvent, both k and α are a function of the solvent quality and chain length range. A high-temperature laser light-scattering spectrometer was used to measure the average radius of gyration (〈 R g〉) and hydrodynamic radius (〈 R h〉) of a set of narrowly distributed linear polystyrene chains in decalin over a wide temperature range. k and α in the scaling experimentally varying with T over a chain length range was analyzed. The results reveal that for 〈 R g〉, α = 0.59 − 0.09exp(− τ/0.066) and k = 0.60 τ 2 α−1 , reasonably agreeing with the thermal blob theory. For 〈 R h〉, α = 0.59 − 0.09exp(− τ/0.106), but k deviates from the relationship of k ∝ τ 2 α−1 , reflecting that the hydrodynamic interaction and chain draining are not considered in the thermal blob theory.

Chain Extension of DNA Confined in Channels

The mechanism of DNA elongation in nanochannels was explored by Monte Carlo simulations as a function of the channel dimension D, DNA length, and stiffness. Simulations were based on the bead-spring model, representing double-stranded DNA chains of moderate length at a high salt concentration. As a rule, the channel-induced elongation profiles of R( parallel) vs D from the simulations were in qualitative agreement with those from microfluidic measurements of DNA. The longitudinal chain elongation in narrow channels was found to be correctly predicted by the Odijk relation for the deflection regime. The scaling relation of R( parallel) vs D(-1), based on the statistics of ideal-chain blobs, was used to explain the simulation data at the intermediate channel widths. Contrary to the blob-theory presumption, the nonlinear dependence of DNA elongation R( parallel) on the chain length N was observed in simulations at moderate confinement. It was suggested that discrepancies found between the simulations and the blob theory arose from the formation of various DNA hairpin structures within channels.