Using analytical considerations and particle-based simulations of a coarse-grained model, we study the relaxation of a density modulation in a polymer system without nonbonded interactions. We demonstrate that shallow density modulations with identical amplitudes and wavevectors that have been prepared by different processes exhibit different nonexponential decay behaviors. Thus, in contrast to the popular assumption of dynamic self-consistent field theory, the density alone does not suffice to characterize the configuration of the polymer system. We provide an analytic description within Linear-Response Theory (LRT) and the Rouse model that quantitatively agree with the results of the particle-based simulations. LRT is equivalent to a generalized model-B dynamics with an Onsager coefficient that is nonlocal in space and time. Alternatively, the Rouse description can be cast into a dynamic density-functional theory that uses the full probability distribution of single-chain configurations as a dynamic variable and yields a memory-free description of the dynamics that quantitatively accounts for the dependence on the preparation process. An approximate scheme that only considers the joint distribution of the first two Rouse modes—the ellipsoid model—is also explored.
I. INTRODUCTION AND BACKGROUND
The free-energy landscape of dense polymer systems can be accurately described by Self-Consistent Field Theory (SCFT).1–12 In the limit of a large invariant degree of polymerization, (with ρ0, N, and Re being the segment number density, the number of segments per polymer, and the polymer’s root mean squared end-to-end distance, respectively), one chain molecule interacts with many neighbors, and the mean-field theory provides a quantitatively accurate description of thermal equilibrium. This provides an excellent starting point for studying the kinetics of structure formation.13,14
In this paper, we study a deceptively simple problem—the temporal decay of a shallow density modulation in an ideal gas of polymers. Understanding this basic phenomenon is a prerequisite for studying more complex dynamic phenomena, such as phase separation in homopolymer blends, self-assembly of block copolymers, or polymer crystallization.
At t ≥ 0, the field is switched off, and the density modulation relaxes. In Fig. 1, we present the time evolution of the normalized amplitude, a(t) ≡ 2[ρ±q(t)]/ρ0, for qRe = 4π. This wavevector corresponds to a spatial period of L = Re/2 ≈ 1.22Rg, where denotes the radius of gyration of the Gaussian polymer. The relaxation of such a density modulation with a length scale that is comparable to or smaller than the molecular extension does not only involve the diffusion of the molecule’s center of mass but, additionally, the dynamics of (internal) molecular configurations. The data are obtained by particle-based simulations of an ideal gas of Gaussian polymer chains (cf. Appendix B for a description of the model and simulation technique). denotes the average over independent realizations of the stochastic relaxation process. One can clearly appreciate that the amplitude does not exponentially decay in time in marked contrast to Eq. (7).
Indeed, Fig. 1 shows that if the shallow density modulation has been prepared by an external spatially varying field that acts on all segments uniformly, γ = 1 for all s, the time evolution observed in the simulations (filled black circles) quantitatively agrees with the decay of a density fluctuation [Eqs. (9) and (10)] (black line) in equilibrium without an adjustable parameter. Note that the normalized amplitude of the density modulation is only 0.02; thus, LRT is accurate.
Figure 1, however, also reveals that if the density modulation has been prepared by a different process, its time evolution differs. If we use an external field for t < 0 that only acted on an end of a macromolecule, γ(s) = δ(s)/N (blue squares), we observe that the density modulation decays faster than predicted by Eq. (9) after the field has been switched off at t = 0. If the initial preparation field only acted on the segments in the middle of a chain, i = N/2 and N/2 + 1 [or γ(s) = δ(s − 1/2)/N, green triangles], the density modulation decays slower than predicted by the collective structure factor, Sq(t), of density fluctuations at the wavevector, q, of the density modulation.
Both observations demonstrate that the collective density, ρq, at a given time t = 0, alone cannot appropriately characterize the initial state of the system. Therefore, by using ρq(t) as the only order parameter, traditional D-SCFT7,10,17–31 fails to accurately describe the short-time dynamics of the segment density even in an ideal gas of homopolymers.37
In this paper, we explore two avenues to improve the description of the short-time dynamics:
- Exactly rewriting LRT with an ideal gas as the reference system, i.e., D-RPA,38–42 in Sec. II, we generalize the model-B dynamics [Eq. (7)] via an Onsager coefficient that is nonlocal in space and time,31,33,40,43This spatiotemporal nonlocality results from the extended shape of the macromolecules and their spectrum of relaxation times. The nonlocality in time quantifies memory effects that extend up to the Rouse time, τR,36 and accounts for the fact that the chain configurations need not to be in equilibrium with the instantaneous density distribution, ρq(t). In the case that the density modulation has been prepared by an external field that acted only on a portion of the chain contour (e.g., a chain end or middle segment), it turns out to be necessary to describe the time evolution of both, ρq(t) and ρhq(t). We obtain an explicit expression of the Onsager coefficients with memory (or rather their Laplace transforms) in terms of dynamic single-chain structure factors in accord with prior work on incompressible, multicomponent polymer systems.31,33(11)
Rather than projecting the degrees of freedom associated with the chain configurations onto the segment density, ρq(t), subsequently in Sec. III, we use the probability distribution, P, of single-chain configurations—the molecular density—as an order parameter that characterizes the system. Such a more detailed description is routinely employed in classical Density-Functional Theory (DFT) of molecules with orientational degrees of freedom or other internal degrees of freedom that characterize extended molecular configurations.44–47 The equilibrium properties of the molecular density, P, of polymer systems are described by Polymer Density-Functional Theory (PDFT),9,48–63 which has attracted abiding interest to accurately predict liquid-like packing effects of polymer fluids. In Appendix A, we show that static SCFT and PDFT yield the same description of equilibrium properties for continuous Gaussian chains.
Exactly rewriting the Rouse model64 of an ideal gas of chain molecules, we obtain a Dynamic Density-Functional Theory (D-DFT)65,66 that uses the free-energy functional of PDFT. This D-DFT provides a memory-free description of the dynamics and accurately accounts for the dependence on the preparation process observed in Fig. 1. A simplified but approximate approach—the ellipsoid model—that characterizes a molecular configuration only via the chain’s center of mass and overall shape of the segment distribution is also briefly discussed.
This paper concludes with a summary and a brief outlook.
II. GENERALIZED MODEL-B DESCRIPTION OF THE RELAXATION OF A DENSITY MODULATION IN AN IDEAL GAS
A. External field acted on all segments
In the limit of long wavelengths, qRe → 0, a density fluctuation decays by the diffusion of the chain’s center of mass, and Sq(t) → N exp(−q2Dt).32 In this limit, the above equation yields an Onsager coefficient that is local in time, Λ(t) → Dδ(t). Applying this expression at finite qRe, we obtain an exponential shown by the dashed-red line in Fig. 1.
Having numerically determined the memory kernel for various values of qRe, we present the nonlocal contribution of the Onsager coefficient, Λnlq(t), in Fig. 2. It is similar to the memory kernel of the generalized model-B dynamics for incompressible, symmetric diblock copolymer melts.31 We observe that the memory, Λnlq, extends up to the Rouse time, τR, encompassing the entire spectrum of relaxation times of the Rouse chain. The data in Fig. 2 approach the limit of small length scales, qRe ≫ 1, and short time scales, t ≪ τR, characteristic for the universal scale-invariant dynamics inside a Gaussian polymer coil.64 In this limit, we expect that wavevector- and time-dependent, nonlocal contribution Λnlq(t) is a function of a single scaling variable, , for Rouse dynamics, extending up to t ∼ τR or . This expectation is confirmed by the data in Fig. 2. We hypothesize that the scale invariance of the dynamics inside a Gaussian polymer coil gives rise to a power-law behavior of Λnlq(t) in this regime.
B. External field acted on a portion of a chain
Since the Onsager coefficient in the generalized model-B dynamics31,33,40 accounts for memory effects, the dynamics does require information of the system at times t < 0 when there is a distinction between segments that are subjected to the external field and others that are not. Thus, we characterize the system configuration at a given time by both, the total segment density, ρ(r), and the density of segments, ρh(r), on which the external field has been applied for t < 0, in order to incorporate the dependence on the preparation process. If the field was applied to all segments, the generalized model-B description only requires ρq. If the field only acted on a portion of a chain, we employ two order-parameter fields, ρhq, and the density of segments, ρnq ≡ ρq − ρhq, on which the external field did not act for t < 0, i.e., we describe the system as a block copolymer.
This equivalence of the generalized model-B dynamics (with the nonlocal Onsager coefficients, , given by this combination of single-chain, linear-response functions, and static, single-chain structure factors, ) and LRT/D-RPA38–42 is not restricted to an ideal gas but generally holds for interacting systems, such as incompressible, multicomponent polymer systems (within linear response).33
Formally, Eq. (24) provides an explicit, analytic expression for the Laplace transform of the Onsager coefficient. The accurate determination of in the time domain, however, is a nontrivial challenge, and the memory kernel, , in Eq. (16) increases the numerical complexity of integrating the generalized model-B equation. Therefore, it is worth to explore an approach that utilizes a more microscopic description of the system configuration but results in a simpler memory-free dynamics.
III. DYNAMIC DENSITY FUNCTIONAL THEORY OF THE RELAXATION OF A DENSITY MODULATION IN AN IDEAL GAS
In the following, we use the Rouse modes, X, instead of the segment positions, R, to specify a single-chain configuration because, within the framework of the Rouse model,64 the dynamics of the different Rouse modes are independent and are simply described by Ornstein–Uhlenbeck processes with relaxation times, τR/p2.
First, we study the equilibrium distribution of Rouse modes in the presence of a weak external preparation field that acts on a portion of the chain. This description is the analog of static RPA and provides information about the initial P(X, t) at the time, t = 0, when the external field that has prepared the density modulation is switched off. Second, we study the dynamics of P(X, t) in the absence of the external field, t > 0, and project onto the time evolution of the segment density. This strategy reproduces the predictions of LRT/D-RPA.38–42 Third, we demonstrate that the time evolution of P(X, t) according to the Rouse model64 is identical to a deterministic D-DFT65,66 using the free-energy functional of PDFT and a simple local mobility. Fourth, we explore an approximate description in terms of the first two Rouse modes, X0 and X1, only—the ellipsoid model.
A. Equilibrium distribution of Rouse modes of a single chain subjected to a weak external field
In the absence of an external field or nonbonded interactions, the zeroth Rouse mode, X0, is uniformly distributed in the volume, V. In this case, the Cartesian components of the higher Rouse modes, Xp with p > 0, are statistically independent and Gaussian distributed with average, ⟨Xp⟩ = 0, and variance, , for N → ∞.36
|All segments .||End segment .||Two middle segments .|
|All segments .||End segment .||Two middle segments .|
These static considerations within the Rouse model show that even if the variation of the segment density, ρ(r), is identical, the probability distribution of the Rouse modes, Xp, given by Eq. (28) or Eq. (31), and the density profile of specific segments, ρ(r, s), at position s along the molecular contour do depend on the process by which the density modulation has been prepared. This observation explicitly illustrates the failure of the basic assumption of D-SCFT7,10,17–31 that the segment density, ρ(r), is the only slow variable and that all other characteristics of the system, such as the probability distribution of the Rouse modes, Xp, or the density profile of specific segments, ρ(r, s), adopt unique values that minimize the free energy given the constrained segment density, ρ(r).
B. Relaxation of Rouse modes and segment density
We use this analytic prediction to calculate the ratio of time-dependent amplitudes, a*(t) = ρ±q(t)/ρ±q(0), with * = a denoting a field that acts on all segments and * = e corresponding to an external field that only acts on one chain end. The results for different wavevectors are presented in Fig. 3. In the limit of small wavevectors, q → 0, the density modulation decays by the diffusion of the center of mass, and it becomes independent from the process by which the density modulation has been prepared. If the density variation occurs on the length scale of the molecular extension, qRe ≳ 2π, or smaller, however, the preparation process matters because the initial distribution of the internal Rouse modes, Xp with p > 0, differs. Density modulations that have been prepared by an external field on one chain end decay faster than those, which have been fabricated by a field on all segments, and this discrepancy increases with qRe. In the ultimate long-time limit, both amplitudes decay like and the ratio of amplitudes tends to a wavevector-dependent constant.
C. Equivalence of Rouse dynamics and D-DFT for an ideal gas of macromolecules
The simple form of Eq. (50) suggests that it can be generalized to interacting multichain systems by additionally including nonbonded interactions in E[P] of Eq. (51). The free-energy functional including nonbonded interactions, , and its variation with respect to P are given in Eqs. (A24) and (A28), respectively. This D-DFT form of the polymer dynamics also allows us to establish connections to the wealth of knowledge about the dynamics of simple and complex fluids.65,66
Grzetic et al.70 did not have a strategy for tackling the high-dimensional Fokker–Planck equation71 and applied the scheme to monomers, where the segment density and the molecular density are identical, ρ(r) = P(R). In the monomer case, (i) the distinction between D-SCFT, using a free-energy functional of the local segment density, and D-DFT, as a functional of the molecular density, as well as (ii) the memory effects that stem from the spectrum of relaxation times in a single molecule are irrelevant.
Fredrickson and Orland71 proposed to solve the equations numerically by explicitly evolving chain replica configurations in time, i.e., the molecular density is represented by an infinitely large ensemble of independent single-chain configurations that obey Langevin equations, involving the mean force, K. This scheme was adopted in later applications.73,74 The resulting particle-based simulations are essentially identical to SCMF simulations75,76 in the mean-field limit, . SCMF simulations use force-biased MC simulations,77,78 whereas DMFT uses a Langevin equation for the individual segment positions.71,73,74 Both schemes explicitly account for the strong, bonded interactions, . SCMF simulations represent the weak nonbonded interactions of a segment with its surrounding by quasi-instantaneous fields, according to Eq. (A29) 75,76 whereas DMFT uses mean-forces, −∇ω(r).71,73,74 Both quantities are calculated from the segment densities of the ensemble of chain replica configurations on a collocation lattice.
Therefore, D-DFT and DMFT/SCMF simulations for will give rise to essentially the same results, albeit by solving a deterministic partial differential equation for the molecular density, P(R), in the case of D-DFT or an infinite set of ordinary stochastic differential equations for the explicit segment positions in case of DMFT/SCMF simulations for .
There is, however, one important difference between SCMF simulations for finite and DMFT/SCMF simulations for . In the case of a finite ensemble of chain replica configurations, the quasi-instantaneous fields fluctuate and these fluctuations are physically meaningful.75,76 Using an appropriate discretization of space and molecular contour and frequently updating the quasi-instantaneous fields,76 SCMF simulations capture fluctuation effects, such as the shift of the transition of a symmetric diblock copolymer from second-order as predicted by Leibler’s mean-field theory15 to a fluctuation-induced first-order transition,79 capillary waves at interfaces,80 or the power-law decay of the bond–bond autocorrelation of chains in a nearly incompressible melt.81 Whereas SCMF simulations at finite capture these fluctuation and correlation effects,76 neither D-DFT nor DMFT/SCMF simulations for are expected to describe these effects for they vanish in the limit, .
D. Ellipsoid model
Qualitatively, the detailed molecular density, P(X, t), employed in D-DFT, results in a memory-free description of the dynamics because all characteristics of a single molecule are explicitly considered. P, however, is a high-dimensional probability distribution that contains much more detailed information than the segment density, ρ, used in D-SCFT7,10,17–31 or ρ and ρh employed in the generalized model-B dynamics. Moreover, the concomitant free-energy functional, F[P], may be difficult to obtain for systems with interacting polymers. Therefore, it is worth to explore simpler yet approximate descriptions.
Recently, we have made efforts to obtain the free energy as a functional of the segment density, ρ, and the variance, , of the first Rouse modes.82,83 This approach has the advantage to reproduce the accurate thermodynamic description of SCFT1–12 while retaining some additional information about nonequilibrium molecular conformations. Here, we take a related but different approach by using X0 and X1 to characterize the system.
The reduction of the ellipsoid model and the observation that its probability distribution corresponds to the marginal of the molecular density for an ideal gas also help us to visualize the time evolution of the molecular density. In Fig. 5, we plot the relative deviation R(X0, X1, t) = Pell(X0, X1, t)/Pell(X0, X1, ∞) − 1 ≈ ln Pell(X0, X1, t)/Pell(X0, X1, ∞) of the joint distribution function of X0 and X1 from its equilibrium value as a function of the first two Rouse modes, and , along the spatial modulation. The top row displays the initial molecular distribution at time t = 0+ for the three different preparation processes. One can appreciate that the joint distribution of the molecule’s center-of mass-position along the density modulation, X0, and its overall orientation and extension, X1, greatly differ although all three preparation processes result in a periodic density modulation with a single characteristic wavevector, qRe = 4π.
If the external preparation field acted on all segments, molecules with a center of mass, X0, in a high density region, qX0/2π = 1/2, are characterized by a narrower distribution of X1, indicating that the chains are less extended than those, whose center of mass is localized in a low density region. If the external preparation field acted only on the middle segment of a polymer, the distribution of X1 is largely independent for the molecule’s center-of-mass position. In these two cases, the probability distribution is symmetric for X1 ↔ −X1. This symmetry, however, is broken if the external preparation field, γ(s) = δ(s)/N, acted only on one chain end. Given these pronounced differences in the initial molecular density, the significant influence of the preparation process on the dynamics even in this simple system can be rationalized.
The subsequent rows depict the molecular density of the ellipsoid model at times, t/τR = 0.05 and 0.1. Whereas the amplitude of the variations, R(X0, X1, t), decreases in time, the pattern does not qualitatively change with time. Only for the middle column, γ(s) = δ(s)/N, the pattern rotates clockwise in the X0 − X1 plane. Again, one can clearly observe that the structure that has been prepared by an external field acting on an end segment (middle column) decays much faster than that prepared by an external field on the middle segment.
IV. SUMMARY AND OUTLOOK
We have studied the temporal decay of a density modulation with wavevector q in an ideal gas of homopolymers. The thermodynamics of a shallow density modulation can be accurately described by SCFT or PDFT that result in identical predictions for noninteracting, continuous Gaussian chains. The dynamics of the molecules is described by the Rouse model.64 The center of mass, X0, of a molecule diffuses, whereas the internal degrees of freedom, quantified by higher Rouse modes, Xp with p > 0, exhibit a spectrum of relaxation times up to the Rouse time, τR.
We prepare the density modulation with length scale ≲Re by an external field that acts on a portion, γ(s), of a polymer chain for t < 0. The initial distribution of the center of mass and the internal Rouse modes (Fig. 5) depends on the preparation process, specified by γ(s), even if the projected segment density, ρ(r) is identical.
Therefore, the basic assumption of D-SCFT7,10,17–31 that the segment density is the only slow order parameter and all other degrees of freedom are in instantaneous equilibrium with this segment density, ρ(r, t), is overtly inaccurate. Whereas D-SCFT with a wavevector-dependent Onsager coefficient predicts an exponential decay of the amplitude of the density modulation in time, a qualitatively different short-time evolution is predicted by exact analytic calculations within the Rouse model64 and observed in particle-based simulations. Thus, while SCFT has proven extremely successful in providing an exquisite level of insights into the structure and thermodynamics of multicomponent polymer systems, its generalization to the dynamics remains a challenge.
We have used LRT/D-RPA38–42 to predict the decay of a density modulation for t > 0. Within this framework, we have to describe the time evolution of both, the density of segments, ρh, on which the external preparation field acted at t < 0, and the density of other segments, ρn = ρ − ρh. LRT can be exactly rewritten in terms of a generalized model-B dynamics31,33,40 [Eq. (16)] with Onsager coefficients that are nonlocal in space and time. Equation (24) relates the Laplace transform of these Onsager coefficients to dynamic, single-chain properties and agrees with previous studies of incompressible, multicomponent systems.33 In the special case that the external preparation field acted on all segments, ρ = ρh, only a single Onsager coefficient, Λ(t), is required, and its time dependence is presented in Fig. 2—this memory effect accounts for the spectrum of relaxation times of the internal Rouse modes. One can recover the D-SCFT-form by approximating the Laplace transform by its zero-frequency limit, i.e., , with , where τk defines a single wavevector-dependent relaxation time.28,29,31,33
In analogy to equilibrium PDFT, we can alternatively utilize the entire molecular density, P, of single-chain configurations as order parameter. Here, we use the Rouse modes, Xp with p = 0, …, N − 1, to completely specify a single-chain configuration because their dynamics in an ideal gas decouple and are described by independent Ornstein–Uhlenbeck processes according to the Rouse model. The corresponding Fokker–Planck equation (43) is equivalent to a deterministic D-DFT [Eq. (50)]65,66 with the free-energy functional of PDFT and a local mobility [Eq. (52)].
Additionally, we have explored the ellipsoid model that only considers the joint probability distribution of the first two Rouse modes, X0 and X1, but assumes that all higher internal Rouse modes instantaneously relax to their equilibrium distribution. Thereby, we describe the polymer configuration by its center of mass, X0, and its overall orientation and extension, X1. Such an approximate description of the molecular density turns out to be fairly accurate for the studied example (at times larger than τ2 = τR/4) and provides some intuition for the time evolution of the molecular density.
In the present paper, we have only focused on the simplest case of short-time dynamics in an ideal gas of homopolymers. The more relevant cases that the polymers are subjected to an external field or mutually interact remain to be explored. Given related approaches70,71 discussed in Sec. III C, we expect that the D-DFT description also can be transferred to systems of interacting chains. Additionally, the noncrossability of the molecular backbones results in entangled dynamics for long, mutually interacting polymers. Whereas these effects can be captured by slip-link78,91–93 and slip-spring models94–96 in a particle-based description, such as DMFT and SCMF simulations, the generalization of D-DFT toward entangled dynamics remains an important challenge.97–99
Generally, the kinetics of structure formation and transformation in polymer systems is important because these complex macromolecular fluids rarely reach full thermodynamic equilibrium and, often, the experimentally observed state depends on the process history.100,101 Understanding the short-time dynamics is important, for instance, to predict the early stages of spinodal decomposition,20,32 and the formation or broadening of interfaces27,102 in polymer blends, or process-directed self-assembly14 after a sudden quench of thermodynamic condition (e.g., pressure103), mechanical deformation,104,105 or a chemical reaction106 in a periodically modulated phase of copolymers. After the quench, the copolymer system spontaneously relaxes on the free-energy landscape into one of a multitude of metastable, periodic mesostructures that have no analog in the equilibrium phase diagram. Since the structural changes often only involve the scale, , of a single unit cell of the periodically modulated structure, the relaxation toward the metastable state occurs on the time scale, τR. More generally, the short-time kinetics of structure formation may template the final morphology.
Another potential application that requires the quantitative prediction of the short-time dynamics is the Heterogeneous Multiscale Method (HMM)107–110 or alternate projective integration methods where the dynamics of slow variables, e.g., the segment density, is inferred from the observation of the short-time dynamics. These techniques are promising strategies for extending the simulation time scale of these fascinating complex fluids.
M.M. acknowledges stimulating discussions with Kostas Ch. Daoulas, Jörg Rottler, David Steffen, and De-Wen Sun. Financial support by the Deutsche Forschungsgemeinschaft (DFG) Mu 1674/15-2 is gratefully acknowledged. We thank the von Neumann Institute for Computing (NIC) for access to the supercomputer JUWELS at the Jülich Supercomputing Centre (JSC). Additional computational resources at the HLRN Berlin/Göttingen were also made available to the project.
Conflict of Interest
The author has no conflict of interests to declare.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
APPENDIX A: SCFT-LIKE DERIVATION OF POLYMER DENSITY-FUNCTIONAL THEORY
APPENDIX B: COARSE-GRAINED PARTICLE MODEL AND SIMULATION TECHNIQUES
To verify the predictions of LRT/D-RPA38–42 and the Rouse model,64 we use a coarse-grained particle model75,76,111,112 in conjunction with the Graphics Processing Unit (GPU)-accelerated simulation code SOft coarse grained Monte carlo Acceleration (SOMA).112 We discretize the molecular contour into N = 128 segments. The interactions are given by Eq. (A3), and we use a spatially periodic, external field of strength kBTh per monomer to prepare the initial density modulation. This coarse-grained model will also allow us to extend the study to cases with soft, nonbonded interactions for which no analytic solution is available.
We study a system of spatial extension, Re, and apply periodic boundary conditions along all Cartesian directions. The simulation box contains n = 65 536 polymers, corresponding to a high invariant degree of polymerization, . In the absence of nonbonded interactions, however, the chains are decoupled and no mean-field approximation needs to be invoked in SCFT. In the present study, the high density of the ideal gas merely allows for efficient parallel simulations. We use Smart Monte Carlo (SMC) moves that use the bonded forces to propose trial displacements of the segments. This Monte Carlo (MC) dynamics closely mimics the Rouse model.93 The self-diffusion coefficient is /MCS, where each segment is attempted to be once displaced in one Monte Carlo Step (MCS). Length scales are measured in units of Re, and time scales are reported in units of the Rouse time, SMC.
The amplitude of the initial density modulation and the strength of the external preparation field are reported in Table I. The relation between the field strength, h, and the amplitude of the initial density modulation is in quantitative agreement with the analytical predictions for the continuous Gaussian chain in the weak-field limit [Eqs. (36) and (39)] if the external preparation field acts on all segments or the two middle segments. In the case of the field acting on a single end segment, there is a small deviation of 6% that stems from the fact that (i) he = 0.5 is not small and (ii) chain discretization effects are particularly pronounced if the field acts on a single end segment.
Simulation data in Fig. 1 have been averaged over more than 400 stochastic realizations of the relaxation process. The density profiles at a given time, t, are averaged over the realizations and, subsequently, the amplitude, ρ±q(t), is extracted from the averaged profiles.
Short-time dynamics refers to the universal dynamics of the polymers on the scale of a fraction of the coil size, Re. This dynamics is dictated by the self-similar structure and dynamics of Gaussian polymer chains64 and not by the dynamics on the scale of a monomeric repeating units that is determined by the specific chemistry of the monomers.