Dynamics and Conformation of Semiflexible Polymers in Strong Quasi

Publication Date (Web): January 22, 2014 ... The chain relaxation time (trelax) exhibits strong power-law dependence in H < P nanochannels ..... imple...
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Dynamics and Conformation of Semiflexible Polymers in Strong Quasi-1D and -2D Confinement Yeng-Long Chen,*,†,‡,§ Yu-Hui Lin,† Jen-Fang Chang,† and Po-keng Lin† †

Institute of Physics, Academia Sinica, Taipei, Taiwan Department of Physics, National Taiwan University, Taipei, Taiwan § Department of Chemical Engineering, National Tsing-Hua University, Hsinchu, Taiwan ‡

ABSTRACT: We investigate the conformation and relaxation dynamics of single DNA molecules in strong confinement (smaller than persistence length) with coarse-grained semiflexible chain (SFC) models using overdamped Langevin dynamics simulations. DNA properties in nanochannels and nanoslits are studied in confinement with height (H) ranging from the DNA radius of gyration (Rg) to smaller than the persistence length (P). Qualitatively different dependences of chain conformation and relaxation time on H in moderate (P < H < Rg) and strong (H < P) confinement are observed for very stiff SFC in the nanochannel but not in the nanoslit. The chain relaxation time (trelax) exhibits strong power-law dependence in H < P nanochannels, verified with and without including hydrodynamic interactions (HI). The inclusion of hydrodynamic interactions affects chain relaxation dynamics even in strong confinement, indicating the intersegmental hydrodynamic interactions affect dominant segmental relaxation mechanisms of strongly confined polymers.

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INTRODUCTION The physical properties of polymers under strong confining constraint have been of great interest in nanoengineering developments such as photoresists, thin-film nanocomposites, and drug delivery capsules. In the studies of molecular biology, the confining environment of macro-biomolecules such as DNA and proteins also affects molecular functions and reaction kinetics.1−3 Recently, there have been several theoretical attempts to understand how confinement changes the physical properties of large, micrometer-sized molecules in a crowded biological environment.4−6 However, it remains difficult to investigate the changes in the molecular structure and dynamics inside a cell or cell nucleus. It is more common to examine biological macromolecules in well-controlled confined systems such as in a fabricated microor nanochannel. Properties of single DNA molecules have been well-studied in microchannels with dimensions of 1 μm and smaller.7−23 Micro- and nanofluidic devices have well-defined geometries ideal for studying how strong confinement affects DNA conformations and dynamic properties. On a length scale much larger than a helical turn (3.4 nm), double-stranded DNA molecules are good models for semiflexible chains (SFC)24−26 for their molecular monodispersity, large persistence length P (≈50 nm), and radius of gyration Rg.27,28 In moderate confinement with the channel height P < H < Rg, the accessible phase space of polymer conformation becomes restricted and the chain configuration entropy decreases. In strong confinement with H < P, chain stiffness and relaxation are strongly © XXXX American Chemical Society

affected by confinement. In a quasi-one-dimensional (Q1D) nanochannel, strong confinement hinders chain motion and effectively filters long wavelength collective relaxation mechanisms such as coordinated chain rotation,20,29−31 restricting the dominant relaxation process to the confinement length scale. In quasi-two-dimensional (Q2D) nanoslits, stronger confinement leads to symmetry breaking in the segmental correlation length in the confined and unconfined dimensions.32−34 Theoretical analyses of the chain free energy and dynamics in moderate and strong confinement regimes have predicted the chain properties characterized by Rg and the relaxation time (trelax) on spatial confinement.35−43 A DNA molecule modeled as a SFC can be characterized by the length scales of the monomer size σm, P, Rg, and the contour length L = Nσm.39 Measured on a length scale smaller than P, the physical behavior of a SFC is qualitatively similar to a rod-like, rigid segment. On a length scale larger than Rg, the chain physical properties are coil-like. For fluorescently labeled λ-phage DNA molecules (48.5 kbps, Rg,bulk/P ≈ 13), the chain diffusivity has been found to follow self-avoiding walk statistics (SAW) in free solution (D ∼ D0N−0.6, D0 = monomer diffusivity).44 Recent theoretical analyses and MC simulations have questioned whether unlabeled λ-DNA molecules should be characterized as ideal random walk (RW) or SAW statistics Received: September 17, 2013 Revised: December 2, 2013

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in free solution.45,46 The argument against considering λ-DNA as SAW is due to the relatively small number of Kuhn segments, and intersegmental self-avoiding interaction is smaller than thermal energy.47 The critical number of Kuhn segments for chain swelling is Nc ∼ (2P/σm)2 ≈ 400 (with P = 50 nm and the effective DNA width including Debye screening length σm = 5 nm), whereas the λ-DNA consists of approximately 160 Kuhn segments. However, the scale argument only offers a rough order-of-magnitude estimate, and the properties of λ-DNA should likely be characterized as between RW and SAW. For fluorescently labeled λ-DNA, we are only aware of experimental evidence that found the molecular diffusivity follows the scaling predicted with the SAW model.44,48 For SFC under Q2D confinement, Monte Carlo (MC) studies of the chain conformation have shown that there is a moderate confinement regime where the statistical “blob” of segments is pancake-shaped and the segmental conformation change from SAW to RW in which the intrachain segmental interactions within a blob is smaller than the thermal energy.34 However, for strongly confined DNA molecules such those adsorbed on a surface, λ-DNA follows SAW statistics.49,50 We use the coarse-grained self-avoiding chain to model fluorescently labeled DNA molecules and discuss how the physical properties depend on confinement. For a polymer chain of Nseg = L/P persistence-length segments, scaling theory35,39 predicts that

R g,bulk ∼ PNseg v

The scaling exponent predictions of R∥ and trelax are illustrated in Figures 1a and 1b, in comparison to recent

(1)

The scaling exponent is v ≈ 3/5 (SAW) and 1/2 (RW). In twodimensional confinement, v = 3/4 (SAW) and 1/2 (RW). For a chain confined in a nanochannel, the transverse segmental correlation length l⊥ is proportional to the channel height H and v = 1 for real polymers. The H dependence can be found from the scaling argument R∥* = R∥/R∥,bulk ∼ (R∥/H)δ, where R∥ is the projected radius of gyration in the unconfined dimensions. Similarly, the polymer extension X is measured for DNA in nanochannels. The exponent δ = 1/4 and 2/3 for moderately confined chains in Q2D and Q1D, respectively.35 Recent single molecule experiments have attempted to verify these predictions with fluorescence microscopy.7−13,15,20−22,51 Polymer relaxation dynamics can also be directly measured with microscopy. The longest chain relaxation mode is the coordinated rotation and extension, with trelax = Rg2/Dchain = Rg2(ζchain/kBT), where ζchain is the chain friction coefficient. Without accounting for the intrachain hydrodynamic interactions (HI),27,52 the Rouse model predicts ζchain = Nζ, where ζ is the bead friction coefficient. With eq 1, the Rouse relaxation time trelax,R ∼ N11/5 for free DNA. In contrast, in the Zimm model that includes HI,27,53 Dchain ∼ kBT/ζRg ∼ D0N−3/5. With eq 1, the Zimm relaxation time trelax,Z ∼ N9/5 for free DNA. The dependence predicted by Zimm model has been verified for DNA diffusivity in free solution.44 Under moderate confinement in which polymer segments have “blob”-like conformation, the influence of HI for coordinated chain relaxation with segmental length scale larger than the blob size (or confinement length) is reduced.15,34,54 Thus, for a two-dimensional polymer, trelax(kBT/ζ) ∼ Rg2N ∼ N5/2 (SAW) and ∼N2(RW). For a real polymer in onedimensional confinement (channel or tube), Rg ∼ N and trelax ∼ N3. Following the scaling argument trelax* = trelax/trelax,bulk ∼ (R∥/ H)ε, one finds ε = 7/6 and 2 for moderately confined polymers in slits and channels, respectively.\

Figure 1. Illustration of the scaling dependences on H over 1 decade of (a) chain size and (b) relaxation time in nanochannels (dashed lines) and nanoslits (solid lines). Experimental data of Lin et al.7,15 (red circles), Tang et al.21 (green squares), Strychalski et al.22 (blue stars), and Reisner et al.20 (triangles) are shown. The solid lines are drawn to show scaling theory predictions, and the blue lines with circles show the best fits for the experimental data.

experiments.7,20,21,51 Best fits of the experimental observations of the DNA size found exponents of −0.16 and −0.85 in nanoslits and in nanochannels, respectively. The deviation from the predicted exponents in nanoslits could be due to the transition into the deflection regime, leading to much slower increase of R∥* as H* = H/Rg decreases. It may also be due to stronger surface-to-DNA interactions in very small nanoslits. In the range of nanoslit confinement examined, a clear transition between the blob and the deflection regimes was not observed. This may be attributed to the small range of R∥*/H* that can be probed. Most experiments are performed in nanoslits and nanochannels with height greater than 50 nm, whereas the deflection regime can only be clearly observed for H/P < 1. One may consider whether the theoretical exponents may be adjusted by considering the polymer as an ideal random walk in free solution. The ensuing analysis for DNA in nanoslits shows that R∥* ∼ H*(−1/2), which further deviates from the experimental observations. Future experiments to probe the deflection regime with DNA would need thinner nanoslits. Although some experiments agree very well with the predictions, measurement over at least 1 decade of variation in R∥ and trelax is required to verify the predicted exponents. This is a challenge for both experiments and simulations due to the large separation in length and time scales. Particularly for R∥ B

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for polymers in slits, measurement needs to be made over 4 decades of change in H to verify that R∥/R∥,bulk ∼ (H/ Rg,bulk)−1/4. Under strong confinement, the chain properties are strongly affected by the walls and chain conformation becomes “deflected”. Odijk’s analysis gives the correlation length as the deflection length l = (H2 P)1/3 for a SFC confined in a cylindrical tube.41 The chain extension may be considered as the projection of (L/l) rods of length l stretched along the tube axis. For small deflection angle (θ in Figure 1a), the maximum chain extension X is given by X = L[1 − A(H/P)2/3], where A = 0.17 for square channels,36,55 a qualitatively different dependence on H. Although the longest relaxation for strongly confined SFC is the reptation time, trep ∼ L3 (ζ/kBT), the dominant chain fluctuation relaxation processes in strong confinement are due to segmental motion on the length scale of H, and trelax ∼ H2/Dseg ∼ H3 or ∼H3/ln(H/σm) if the segments are considered as chain of connected rods. Qualitative changes in the chain conformation and relaxation mark the transition between the moderated confined “blob chain” regime to the strongly confined “deflected chain” regime. Recent experiments have observed the transition from the blob scaling regime to the deflection length regime for DNA molecules confined in nanochannels,20 where trelax increases as H decreases for Rg >H > 2P and trelax decreases as H decreases for H < 2P. The experimental data of Reisner et al.,20 shown in Figure 1b, found that the dominant observable DNA relaxation mechanism in nanochannels is the relaxation of segments on the length scale of H. Using micro- and nanochannels to probe single DNA properties,7−15,20,21 these studies are limited by the available DNA and channel sizes and the resolution of fluorescence microscopy. In addition, only the projected fluorescently labeled DNA images can be observed. Simulations of a coarse-grained model DNA can complementarily probe DNA properties. Recent coarse-grained simulations studies have found good agreement with the predictions for blob-todeflection transition and the scaling law dependences in nanochannels.30,31 MC studies have examined the conformation and the free energy of DNA in strong confinement.33,34,37,56 In Q1D nanochannels, a transition in the chain relaxation dynamics is also found in experiments and coarse-grained simulations of DNA.20,30,31 In Q2D nanoslits, it is less clear whether a qualitative change of chain properties occurs, as slit confinement does not inhibit long wavelength collective chain motion. Prior simulation studies have probed the chain property dependences for slit heights larger than the Kuhn length,54,57−59 but the coarse-grained flexible polymer model for DNA prevented investigations in the H < P regime. To better understand the conformation and dynamics of confined DNA molecules in nanoslits, coarse-grained dynamics simulations are performed to study how R∥ and trelax depend on H in nanoslits.

Each Fraenkel spring has the potential energy Uspring,i =

2σm

2

(| ri ⃗ − ri +⃗ 1| − σm)2

(3)

where kv = 400, kBT is the unit energy, and σm is the unit length corresponding to a DNA segment length. ri⃗ is the position of the ith bead of the chain. The beads repel each other with the repulsive Morse potential with a cutoff distance of 1.5σm Uev,i = εmkBT ∑ exp[−αm(| ri ⃗ − rj⃗| − σm)] j

(4)

εm = 0.2 and αm = 6 are the repulsion strength and range. The chain rigidity is introduced by a bending potential energy of the angle between consecutive segments, given by ⎛ v ⃗ ·v ⃗ ⎞ Ubend,i = k bendkBT ⎜1 − i − 1 i ⎟ |vi⃗ − 1||vi⃗| ⎠ ⎝

(5)

where vi⃗ = ri⃗ − ri⃗ +1. The bending modulus kbend is directly related to P, which is determined from the segmental correlation function C(j) = ⟨vi⃗ ·vi⃗ +j⟩i ∼exp(−|i − j|/P). In free solution, a N = 1600, P = 20 polymer with these polymer interaction parameters is found to follow SAW statistics (Rg ∼ N0.6). Chains with P = 20 and 1 (kbend = 20 and 0) for N = 1600 (Rg,bulk/P ≈ 4.7 and 36.8) are compared to examine how chain flexibility affect chain conformation in confinement. P = 40 (kbend = 40), N = 400 and 800 chains are also examined in order to probe the H/P ≪ 1 regime. By fully accounting for HI, how HI influences chain relaxation can be isolated. This is done only with P = 4 (kbend = 4), N = 160 (Rg,bulk/P ≈ 3.6) chains due to computational cost limitations. SFC with P = 40, 20, 1, and 4 correspond to σm = 1.25, 2.5, 50, and 12.5 nm, respectively, when matched to P = 50 nm for double-stranded DNA.25 The variation in the model DNA width may be considered as corresponding to a change in the electrostatic Debye layer thickness, which is controlled by the buffer solution ionic strength. Both the DNA width and persistence length change with the ionic strength corresponding, and a careful calculation is needed to find the solution ionic strength to match the persistence length-to-width ratio [D. Stigter, Biopolymers 16, 1435 (1977)]. The effective polymer width is a constraint on the minimal confinement length. To avoid significant finite size artifacts in chain property measurements, the minimum slit/ channel height H′ is chosen such that H′/σm = 4. The beads are repelled from the walls with f wall based on a short-ranged cubic repulsive potential.62 The distance of closest approach between a bead and a wall is 1.1σm, and the reported effective slit height is H = (H′ − 2.2σm) as was used in prior studies.33,34 fR is a Gaussian random force with zero mean and variance ⟨f xRf xR⟩ = 2kBTζ/dt that satisfies the fluctuation− dissipation theorem. ffric = −ζ(um − uf) is the friction force on a bead, with bead velocity um and fluid velocity uf. HI is included by coupling polymer motion to a lattice Boltzmann fluid (LB), which is a well-known methodology.63,64 In LB, the fluid velocity distribution function ni(x,t) is solved on a discretized cubic lattice with coordinates x, lattice spacing Δx = σm, and 19 discrete velocities ci. The discrete Boltzmann equation is solved by stepping

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METHOD Overdamped Langevin dynamics simulations with a coarsegrained DNA modeled as a bead−spring semiflexible chain with N beads are performed. Segmental dynamics is tracked with the velocity-Verlet method.60 The forces on each bead i is determined from the interaction potential61 ∂U fi = − = fspring + fbend + fev + f R + ffric + fwall ∂ri

k vkBT

ni(x + ci dt , t + dt ) = ni(x , t ) + Lij[ni(x , t ) − nieq (x , t )] + next(x , t )

(2) C

(6)

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where Lij is the collision operator and nieq is the equilibrium fluid velocity distribution function. Bead friction and fluctuations are coupled to the fluid through next and propagated through the fluid through the immersed boundary method.63−67 We adopted nonslip fluid boundary condition, implemented through the bounce back boundary condition in LB. The integration time step is dt = 10−2tD, where tD = ζσm2/ kBT is the bead diffusion time. Each trial is performed for 2 × 107tD to capture chain relaxation to equilibrium, with the longest chain relaxation time found to be approximately 2 × 106tD. Ensemble properties are averaged over more than 200 uncorrelated samples for each trial (40 trials and over 5 chain relaxation times). The error bar for each data point is less than 8% and within the symbol size reported. Discretization errors due to the choice of fluid lattice spacing and bond length were previously characterized64 and found to be negligible for the choice of parameters in this study. Furthermore, finite-size effects on chain diffusivity measurements were corrected for the periodic boundary conditions and found to be negligible for box size chosen to be greater than 6 chain radius of gyration.63,65 The current approach for studying polymer dynamics has been recently tested for polymers in free solution with comparisons to multiparticle collision dynamics, molecular dynamics, Brownian dynamics, and a different implementation of lattice Boltzmann fluid−polymer coupling and found to be in excellent agreement.54,66,67 In addition, the methodology for confined polymers has previously been compared with experimentally measured DNA diffusivity in nanoslits and also a different implementation of LB polymer−fluid coupling, and the results for the scaling law dependences were found to agree.15 For different computational studies of polymers using short-ranged repulsive potentials, we expect that the properly rescaled results would collapse for different choices of L, P, and σ m in the regime where the effective bead diameter (characterized by the parametrization) is much smaller than all the relevant physical length scales (σm ≪ H, σm ≪ P ≪ Rg).

Figure 2. (a) R∥* as a function of the scaled slit height for N = 1600, P = 20 (empty circles) and the MC results from Table 1, index 5 of Dai et al.34 (+, R∥,bulk/P ≈ 6.8), Cifra33 (red stars, R∥,bulk/P ≈ 3.5). (b) Comparison with experimental results from Lin et al.7 (red triangles), Tang et al.21(green down triangles), and Strychalski et al.51 (blue right triangles). The inset shows Rg* = Rg/Rg,bulk as a function of H* = H/ R//,bulk and the comparison for N = 1600, P = 20 and 1 (solid circles). The dashed lines show power law scaling with an exponent of −0.25.

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RESULTS AND DISCUSSION The dependence of the projected polymer radius of gyration on slit confinement, which has been carefully measured in recent studies,12−14,22,34,68 is first verified. Figure 2a shows that under moderate confinement R∥ increases monotonically with H−1/4 for N = 1600, P = 20, as predicted by scaling theory for SAW. For H* > 0.1, good agreement is found with MC. However, a transition from the blob to deflection regime is not observed in the current simulations. This may be attributed to the smaller chain stiffness and the lower limit of nanoslit heights studied, which only probes into the range where MC begins to observe the transition near H* = 0.1,34 where H is much smaller than P. In contrast to the projected radius of gyration R∥*, Figure 2a inset shows that Rg varies nonmonotonically for slit height around H* = 1. As H further decreases, Rg increases and follows the H−1/4 scaling. For polymer confined in a nanoslit, this reflects the competing effects of decreasing chain size in the confined dimension and increasing chain size in the unconfined dimensions.69 This phenomenon has also been observed for polymers near a large colloid.70 The coarse-grained simulation results are also in good agreement with several experimental measurements of the projected DNA radius of gyration in nanoslits, as shown in Figure 2b. It is notable that the experiments from different research groups are in good agreement with each other, despite

differences in the chosen reagents and buffer solution. In coarse-grained simulations of semiflexible polymers, the chosen chain stiffness and chain length could affect results for chain conformation, particularly under strong confinement. The effect of segmental flexbility on chain properties are examined in Figure 2b inset. It is observed that for a flexible polymer (P = 1) the measured Rg* is consistently higher than the stiffer chain under the same confinment conditions relative to Rbulk while exhibiting the same nonmonotonic dip near H* = 1 and the same power-law dependence for H* < 1. The quantitative difference between N = 1600, P = 20, and P = 1 is due to the larger number of Kuhn steps for the more flexible chain, which allows the chain to expand and swell more under confinement. In addition, a finite length effect for a confined SFC with small number of Kuhn segments is observed. In a previous MC study by Cifra,33 R∥* appeared to exhibit a qualitative change that indicates the blob-deflection transition for H/P < 1, as shown in Figure 3. This is contrast with a more recent study by Dai et al.34 and also the current study. This difference may be due to the relatively shorter polymers examined. A similar case is examined here with shorter, stiff chains with P = 40, N = 400, and 800 (Rg,bulk/P ≈ 1.5 and 2.1), as shown in Figure 3. Simulation results with shorter, stiffer chains are qualitatively D

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As shown in Figure 4, coarse-grained simulation results for trelax show that it increases monotonically as H decreases in the

Figure 3. R∥* as a function of H/P for the same data in Figure 2, and P = 40, N = 400 (green squares) and 800 (blue down triangles). The inset shows the projected radius of gyration scaled by the rod radius of gyration.

Figure 4. Normalized chain relaxation time for N = 160, P = 4 with and without HI (solid and empty circles). Data from Lin et al.7 (solid triangles) and Tang et al.21 (solid squares) are also shown for comparison. The solid, dotted, and dashed lines show H−7/6, H−1, and H−1/2 dependences, respectively. Inset shows the same data as a function of H/P.

similar to the transition observed in the Cifra study, where R∥ increases more weakly as H decreases starting around H/P = 1. In contrast, the same is not observed for the longer N = 1600 polymer with L/P = 80 and longer coarse-grained DNA model used in the Dai et al. study.34 For the shorter, stiffer chains, a comparison to the conformation of a rod can be made by rescaling with respect to the radius of gyration of a rod of length L, Rg,rod = L/√12. Figure 3 inset shows that for H/P < 1 the projected polymer radius of gyration is comparable to the rod radius of gyration for N = 400, P = 40, indicating that the polymer conformation becomes rod-like for short, stiff polymers under strong confinement. It is interesting to note that the results for N = 1600, P = 20 overlap with that of Dai et al.,34 and the results for N = 400, P = 40 agree with that of Cifra.33 Because of the smaller number of Kuhn segments for the stiffer polymer, the chain swells and becomes rod-like conformation due to strong confinement, and the radius of gyration as a function of the slit height plateaus in small slits. This behavior is expected for DNA molecules with contour lengths shorter than 500 nm (L/P = 10), and hopefully future experiments could verify this observation. As shown in Figure 1b, for DNA confined in nanochannels, prior experiments have found that the chain relaxation time increases as the channel height decreases for H > 140 nm, and the relaxation time decreases as the channel decreases for H < 140 nm.20 For DNA in nanoslits, DNA relaxation time has been found to monotonically increase in smaller slits,7,21 but one previous study has suggested a nonmonotonic behavior similar to in nanochannels.9 If DNA relaxation in nanoslits exhibited nonmonotonic dependence on confinement, it suggests the blob-to-deflection transition could be easier to observe than the conformational change. The dependence of trelax on H is examined as follows. The relaxation dynamics of DNA in nanoslits is directly measured from the time correlation function CR(t) = ⟨δR∥(0) δR∥(t)⟩ ∼ exp(−t/trelax), δR∥(t) = R∥(t) − ⟨R∥⟩. The relaxation time is extracted from an exponential fit over a variation of 1/e of CR(t), with the regression coefficient greater than 0.99. In nanochannels, trelax is similarly determined from ⟨δX(0) δX(t)⟩.30

nanoslit, whether intrachain hydrodynamic interactions are included or not. However, the qualitative dependences are different with and without HI for H < Rg,bulk. Without accounting for HI, the measured trelax is found to increase as H−1/2. With HI, the measured trelax is found to increase with H−7/6 as predicted. Slight deviation from the predicted scaling is found for H/P = 0.45, which could be attributed to finite size effects for σm/H′ = 4. The simulation results are in agreement with the scaling arguments. Results with HI also agree well with recent experimental measurements,7,21 although the best fit through the experimental data is H−1. This may also be due to the larger statistical errors in the experimental measurements in smaller nanoslits, which could lead to a lower estimate of the DNA relaxation time. The role of including hydrodynamic interactions is further examined with simulation results that account for HI. In a slit or channel with H > Rg, the chain relaxes slower without HI, as expected. As the confinement length decreases, intrachain HI is expected to become screened due to exponential decay of HI in the confined dimensions, particularly for coordinated chain dynamics on the length scale of the polymer radius of gyration. As found in prior studies,27,71 the influence of HI on segmental relaxation of strongly confined polymers becomes dependent on both segmental rearrrangment due to confinement and the range of HI in quasi-2D and -1D geometries. Figure 5 shows that in the nanoslit trelax measured with and without HI approach the same value for H/P ≈ 1 and H/P < 1, which indicates that the effects of intrachain hydrodynamic interactions becomes negligible only in very small slits. It also confirms observations of monotonic increase of DNA relaxation time in smaller nanoslits.7,21 In comparison, for the N = 160, P = 4 SFC in nanochannels, trelax increases as H decreases for H > 3P, and trelax decreases as H decreases for H < 3P. Unlike for SFC in slits, the value of trelax for SFC in nanochannels with and without HI approach the same value as the channel height decreases to H ≈ Rg,bulk. For H < Rg,bulk, HI appears to enhance chain relaxation (faster trelax). Although the qualitative dependE

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studies over a significantly larger range of H/Rg and Rg/P. Indeed, much stronger dependences of the chain relaxation time on the confinement length suggest that measuring trelax is a better indicator for probing the blob−deflection transition over a limited variable range. Current experimental measurements for the coil size cover only a very small range in H/P, and experiments and simulations with larger separation of length scales between Rg, P, and σm are needed to further test the scaling theory predictions. In particular, for the scaling relation R∥* ∼ (H/Rg,bulk)−1/4 for a two-dimensionally confined polymer, measurements over 1 order of magnitude in R∥*, and correspondingly 4 orders of magnitude in (H/Rg,bulk), are needed. Such measurements would be challenging to perform experimentally or computationally. The strong dependence of chain relaxation time to chain length (trelax ∼ Rg2/D ∼ N1.8 in free solution), combined with the computation time dependence on N, leads to a total time to collect equilibrated data that grows with N2.8. The difficulty of studying longer polymers will be computationally expensive but may be overcome with accelerated algorithms in the near future. Finally, the simulation results show that hydrodynamic interactions can strongly affect chain relaxtion in nanochannels and should not be neglected even under strong confinement due to the relation between the observed dominant relaxtion process and the confinement length scale.

Figure 5. Chain relaxation times with (solid symbols) and without hydrodynamic interactions (empty symbols) for N = 160, P = 4 in nanoslits (squares) and in nanochannels (red circles). The solid, dotdashed, and dashed lines show H 3, H 3 /ln(H/σm ), and H−7/6 dependences, respectively.

ences on the channel height are similar for trelax with and without HI, it is observed that HI enhances chain relaxation even in very small nanochannels. This is consistent with the argument that the chain dynamics is coordinated-motiondominant in large channels and segmental-motion-dominant under strong confinement. Furthermore, the chain relaxation time appears to follows the H3/ln(H/σm) scaling as predicted by scaling theory, albeit only for a limited regime a the chain relaxation time is bounded by the bead relaxation time (tD). This confirms that the dominant chain relaxation mechanism measured from the chain stretch relaxation is determined by segments of length comparable to the channel height in subpersistence length channels. In moderate confinement channels, coordinated chain relaxation is dominant and trelax increases as H decreases. At the same time, the effects of intrachain HI become screened over the length scale of the entire chain is reduced. For H < P, chain relaxation dynamics is dominated by segmental fluctuations of segmental lengths comparable to H, and the dynamics of the shorter segments is the dominant relaxtion mode. On the length scale relevant to segmental motion that is shorter than the channel height, HI strongly influences segment motion.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (Y.-L.C.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project is funded by NSC (ROC) 98-2112-M-001-004MY3, 101-2112-M-001-003-MY3, NCTS Taiwan and AS CDA100-M01. Y.-L. thanks Chih-Chen Hsieh, Murugappan Muthukumar, Peter Cifra, Bela Mulder, Elizabeth Strychalski, Dai Liang, Patrick S. Doyle, and Chia-Fu Chou for many insightful and helpful discussions.

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REFERENCES

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CONCLUSION With coarse-grained dynamics simulations that includes hydrodynamic interactions, we have investigated the dominant relaxation processes of polymer molecules confined in nanoslits and nanochannels smaller and comparable to the persistence length. Our results show that the chain size and relaxation time follow the predicted scaling exponents in the H < P and P < H < Rg regimes. In nanochannels, the “mode filtering effect” of chain relaxation mechanisms leads to an interesting dynamic where the influence of HI waxes and wanes depending on the dominant relaxation processes. It indicates that quantitative predictions of chain dynamic properties requires the inclusion of HI even under very strong confinement because the length scale of the dominant dynamics is also reduced by confinement. However, accurate measurements of the scaling exponents between theory, simulation, and experiment are limited by the small accessible probing range of Rg/P and P/σm for DNA, which only vary by 1 decade. The weak dependences of Rg on H for polymers in moderate and strong confinement demand F

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