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, $N\u0304\u2261(\rho 0Re3/N)2$ (with *ρ*_{0}, *N*, and *R*_{e} 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.

*ρ*(

**r**), as a function of space,

**r**, that can be computed from the segment positions,

**r**

_{ji}, that completely specify the microscopic configuration of the system at a given time,

*j*= 1, …,

*n*, in a system of volume,

*V*. Each polymer is discretized into

*N*segments. The spatial position of the

*i*th segment along polymer

*j*is denoted by

**r**

_{ji}. In the latter expression, we have taken the continuum limit of fine discretization of the molecular contour,

*N*→ ∞, introduce a continuous contour variable, 0 ≤

*s*= (

*i*− 1/2)/

*N*≤ 1, and characterize the single-chain configurations by

**r**

_{j}(

*s*). The spatial Fourier transform of the collective segment density is defined by

*ρ*

_{0}=

*nN*/

*V*denotes the spatial average of the segment density.

*F*[

*ρ*

_{k}], of the local segment density. Δ

*F*does not depend on the sign and complex phase of

*ρ*

_{k}. Thus, a Landau expansion [Random-Phase Approximation (RPA)] of Δ

*F*begins with |

*ρ*

_{k}|

^{2}. The thermal average of segment-density fluctuations defines the collective structure factor, $Skcoll=\u27e8|V\rho k|2\u27e9/(nN)$, that is experimentally accessible by scattering experiments. Thus, a shallow density fluctuation with wavevector

**k**increases the free energy by an amount

^{6,15}

*k*

_{B}

*T*denotes the thermal energy. Since polymers in an ideal gas do not interact, individual molecules independently contribute to the collective structure factor, i.e., $Skcoll=Sk$, where

*S*

_{k}denotes the static, single-chain structure factor. For a continuous Gaussian chain, it is given by the Debye function,

*S*

_{k}=

*Ng*(

*x*), with $x=(kRe)2/6$,

^{6,15}

^{16}

^{,}

**j**denotes the density current. For a shallow density fluctuation,

*ρ*

_{k}≪

*ρ*

_{0}, Linear-Response Theory (LRT) gives rise to a linear relation between the current and the gradient of the chemical potential. In Dynamic Self-Consistent Field Theory (D-SCFT),

^{7,10,17–31}this relation is assumed to be local in time,

_{k}denotes the wavevector-dependent Onsager coefficient.

^{10,16,20,28,29,31–33}The wavevector dependence accounts for the spatially extended chain structure. Thus, the accurate free-energy functional [Eq. (3)] in conjunction with this model-B dynamics

^{16}predicts an exponential decay of a density fluctuation in time,

*ρ*

_{k}(

*t*) corresponds to the average over different realizations of the stochastic relaxation process.

**q**, by a weak external preparation field of strength,

*h*, for times

*t*< 0. Let

*γ*(

*s*) = 1 characterize all segments on which the initial external fields acts, whereas

*γ*(

*s*) = 0 for all other segments along the molecular contour. For

*t*< 0, the external preparation field contributes

*ρ*(

**r**) ∼

*ρ*

_{h}(

**r**) ∼ −

*h*cos(

**qr**) for

*t*≤ 0.

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*)]/*ρ*_{0}, for *qR*_{e} = 4*π*. This wavevector corresponds to a spatial period of *L* = *R*_{e}/2 ≈ 1.22*R*_{g}, where $Rg=Re/6$ 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). $[\u2026]\rho q(0)$ 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).

*ρ*

_{q}(

*t*), involves a projection from the individual segment coordinates [cf. Eq. (1)], one should expect on general grounds

^{34,35}that this projection imparts memory onto the time evolution of the coarse-grained order parameter,

*ρ*

_{q}, resulting in a nonexponential decay in time. Generally, a typical shallow density fluctuation,

*ρ*

_{q}(0), decays like the collective dynamic structure factor of density fluctuations. Distinguishing between the average, $[\u2026]\rho q(0)$, over realizations of the stochastic relaxation process, ensuing from configurations with an initial

*ρ*

_{q}(0), and the thermal average ⟨⋯⟩ over initial configurations, we obtain [see Eq. (21) for a more detailed derivation within LRT]

*t*and

**q**but independent from the microscopic configuration at

*t*= 0. For an ideal gas, the time-dependent collective and single-chain structure factors coincide, $Sqcoll(t)=Sq(t)$, and an explicit expression is known within the Rouse model (see Appendix 4.III of Ref. 36),

*τ*

_{R}, and the self-diffusion coefficient of the molecule’s center of mass are related by $D=Re2/(3\pi 2\tau R)$.

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, *S*_{q}(*t*), of density fluctuations at the wavevector, **q**, of the density modulation.

Both observations demonstrate that the collective density, *ρ*_{q}, at a given time *t* = 0, alone cannot appropriately characterize the initial state of the system. Therefore, by using *ρ*_{q}(*t*) as the only order parameter, traditional D-SCFT^{7,10,17–31} 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,^{31,33,40,43}This 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,(11)$d\rho q(t)dt=\u2212\u222b\u2212\u221etdt\u2032q2\Lambda q(t\u2212t\u2032)\rho q(t\u2032)Sq/N.$*τ*_{R},^{36}and accounts for the fact that the chain configurations need not to be in equilibrium with the instantaneous density distribution,*ρ*_{q}(*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,*ρ*_{q}(*t*) and*ρ*_{hq}(*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} Rather than projecting the degrees of freedom associated with the chain configurations onto the segment density,

*ρ*_{q}(*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 model^{64} 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

*t*< 0, on all segments, i.e.,

*γ*= 1 for all

*s*, traditional D-SCFT

^{7,10,17–31}with a wavevector-dependent Onsager coefficient fails to describe the short-time dynamics. Only within the generalized model-B [Eq. (11)]

^{31,33,40}with an Onsager coefficient that encodes the memory of the chain configurations, can we accurately describe the relaxation of a shallow density fluctuation. Since such a density fluctuation decays like the dynamic single-chain structure factor [Eq. (9)] one can invert the relation

^{31}

In the limit of long wavelengths, *qR*_{e} → 0, a density fluctuation decays by the diffusion of the chain’s center of mass, and *S*_{q}(*t*) → *N* exp(−**q**^{2}*Dt*).^{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 *qR*_{e}, we obtain an exponential shown by the dashed-red line in Fig. 1.

*vide infra*). Numerically, we split the Onsager coefficient into a local, ultra-short time contribution and a nonlocal contribution in time,

^{31}

^{,}

*K*(

*t*) according to Λ(

*t*) = d

*K*/d

*t*.

^{67}This integrated memory kernel can be iteratively constructed as $K(t)=\u2211n=0\u221eKn(t)$ with

*S*

_{q}(0)

*K*

_{0}(

*t*) = d

*S*

_{q}/d

*t*and $Kn+1(t)=\u2212\u222b0tdt\u2032Kn(t\u2032)K0(t\u2212t\u2032)$.

^{67}

Having numerically determined the memory kernel for various values of *qR*_{e}, we present the nonlocal contribution of the Onsager coefficient, Λ_{nlq}(*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, Λ_{nlq}, extends up to the Rouse time, *τ*_{R}, encompassing the entire spectrum of relaxation times of the Rouse chain. The data in Fig. 2 approach the limit of small length scales, *qR*_{e} ≫ 1, and short time scales, *t* ≪ *τ*_{R}, 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 Λ_{nlq}(*t*) is a function of a single scaling variable, $t*=(qRe)4t/(3\pi 2\tau R)=(qRe)2q2Dt$, for Rouse dynamics, extending up to *t* ∼ *τ*_{R} or $t*\u223c(qRe)4/(3\pi 2)$. 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 Λ_{nlq}(*t*) in this regime.

### B. External field acted on a portion of a chain

Since the Onsager coefficient in the generalized model-B dynamics^{31,33,40} 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, *ρ*_{h}(**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 *ρ*_{q}. If the field only acted on a portion of a chain, we employ two order-parameter fields, *ρ*_{hq}, and the density of segments, *ρ*_{nq} ≡ *ρ*_{q} − *ρ*_{hq}, on which the external field did not act for *t* < 0, i.e., we describe the system as a block copolymer.

^{15}

^{,}

^{31,33,40}

^{38–42}i.e., the LRT for an ideal gas of block copolymers subjected to a time-dependent external field, $V\u20d7q(t)$,

*t*< 0.

*t*< 0, $V\u20d7q(t)\u2261V\u20d7\u2212=Nh2(1,0)T$, and vanishes for

*t*≥ 0. Thus, Eq. (18) can be simply integrated to

*γ*(

*s*), and it quantitatively agrees with the simulations of the particle-based model without an adjustable parameter.

^{33,43,68}

*ρ*

_{hq}and

*ρ*

_{nq}, are equivalent to Eq. (20), they quantitatively describe the short-time dynamics in Fig. 1 even if the initial, external field only acted on a chain portion.

This equivalence of the generalized model-B dynamics (with the nonlocal Onsager coefficients, $\u22c0q$, given by this combination of single-chain, linear-response functions, $xq(t)$ and static, single-chain structure factors, $Sq$) and LRT/D-RPA^{38–42} 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 $\u22c0q(t)$ in the time domain, however, is a nontrivial challenge, and the memory kernel, $\u22c0q(t)$, 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

*P*, that quantifies the probability of a single-chain configuration. A single-chain configuration can either be specified by the

*N*segment positions,

**r**

_{i}with

*i*= 1, …,

*N*or, equivalently, by the

*N*Rouse modes,

**X**

_{p}with

*p*= 0, …,

*N*− 1.

^{64}The transformation between segment positions and Rouse modes reads

**X**

_{0}=

*∫*d

*s*

**r**(

*s*). For

*p*≥ 1, Eq. (25) becomes

*p*> 0 characterize the distribution of bond vectors and do not depend on the absolute position of the macromolecule in space.

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, *τ*_{R}/*p*^{2}.

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 model^{64} is identical to a deterministic D-DFT^{65,66} 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, **X**_{0} and **X**_{1}, 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, **X**_{0}, is uniformly distributed in the volume, *V*. In this case, the Cartesian components of the higher Rouse modes, *X*_{p} with *p* > 0, are statistically independent and Gaussian distributed with average, ⟨*X*_{p}⟩ = 0, and variance, $\u27e8Xp2\u27e9\u2261\sigma p2\u2192Re2/(6\pi 2p2)$, for *N* → ∞.^{36}

*P*(

**X**) to an external preparation field of strength,

*k*

_{B}

*Th*, acting on a portion of a chain with

*γ*(

*s*) = 1. In equilibrium, the distribution of the Rouse modes in the canonical ensemble is given by

*s*in the external field. In the canonical ensemble, the free-energy functional

*F*[

*P*] does not depend on the normalization of the molecular density [

*vide infra*Eq. (A24)]. Thus, we choose

*∫*D

**X**

*P*(

**X**) = 1, where $\u222bDX=\u222bVdX0\u222b\u2212\u221e\u221e\u220fp=1\u221edXp$ integrates over all molecular configurations.

*ρ*(

**r**), is obtained by projection

_{P}standing for the average with respect to the molecular density,

*P*. The molecular density,

*P*, however, cannot be reconstructed from the segment density,

*ρ*(

**r**).

*Nhγ*(

*s*) ≪ 1, and approximate

*h*= 0, they become correlated even within this linear, weak-field approximation. Whereas, for

*p*> 0, the averages

*P*(

**X**) remain unaltered, a weak, external, spatially periodic field induces a spatial variation of the moments of the Rouse modes, i.e., cross-correlations of the form

*vide infra*Fig. 5).

*t*= 0. Other Fourier components,

*ρ*

_{k}, of the segmental density with

**k**≠ 0, ±

**q**vanish in linear order.

*γ*(

*s*) = 1 for all

*s*, we obtain the well-known RPA relation

*h*

_{e}, i.e.,

*Nhγ*(

*s*) =

*h*

_{e}

*δ*(

*s*), we find

*h*

_{m}, i.e., $Nh\gamma (s)=hm\delta (s\u22121/2)$, we obtain

**q**. These results coincide with the prediction of static RPA or D-RPA [Eq. (20)] evaluated at time

*t*= 0. They are also confirmed by the particle-based simulations (see Table I).

All segments . | End segment . | Two middle segments . |
---|---|---|

$2\rho \xb1q\rho 0=0.0207$ | 0.0202 | 0.0205 |

$h=0.285128$ | $he=64128=0.5$ | $hm=17.564=0.273$ |

All segments . | End segment . | Two middle segments . |
---|---|---|

$2\rho \xb1q\rho 0=0.0207$ | 0.0202 | 0.0205 |

$h=0.285128$ | $he=64128=0.5$ | $hm=17.564=0.273$ |

*s*

_{h}= 0 or 1/2, respectively, it does not distort the internal distances, i.e., the joint distribution, $P\u0303(r,d)$, of segment

*s*

_{h}being at location

**r**and segments

*s*and

*s*′ being separated by a distance vector

**d**factorizes

*p*> 0 are uncorrelated or are distributed like in the absence of the external preparation field [cf. Eq. (31)]. Therefore, also a simple factorization of the relaxation of a density modulation into a time-dependent dynamics of the distribution,

*ρ*

_{h}(

**r**,

*t*), of the segment,

*s*

_{h}, on which the external preparation field has acted, and an equilibrium (i.e., time-independent) density distribution of all segments around this segment,

*s*

_{h}, is not exact because the dynamics of segment,

*s*

_{h}, and the internal distances of the chain molecule are coupled.

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, **X**_{p}, 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-SCFT^{7,10,17–31} 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, **X**_{p}, 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

*P*(

**X**,

*t*), at the time

*t*= 0, when the weak external field is switched off, is nontrivial, the ensuing dynamics of the Rouse modes in the absence of the external field is simple. Each Rouse mode independently relaxes toward their equilibrium distribution according to an Ornstein–Uhlenbeck process.

^{36,64}The

*p*th Rouse mode relaxes according to the Langevin equation of the Rouse model,

*W*

_{α}(with

*α*=

*x*,

*y*,

*z*denoting the Cartesian components) are the increments of the Wiener process, i.e., ⟨d

*W*

_{α}⟩ = 0 and ⟨d

*W*

_{α}(

*t*)d

*W*

_{β}(

*t*′)⟩ =

*δ*

_{α,β}

*δ*(

*t*−

*t*′)d

*t*. The factor 1 +

*δ*

_{p,0}could be easily eliminated by redefining the

*p*th Rouse mode as $Xp\u2032=2Xp$ for

*p*> 0.

*P*(

**X**,

*t*), evolves according to the corresponding Fokker–Planck equation,

*∫*D

**X**

*P*(

**X**,

*t*) = 1 also for

*t*> 0.

**X**, at time,

*t*, given the molecular configuration

**X**(0) at time

*t*= 0, is given by Green’s function of the Fokker–Planck equation (43). For an ideal gas (Rouse model), the Green’s function factorizes

*P*(

**X**, 0), given by Eq. (31), yields the molecular density at time

*t*> 0,

*P*(

**X**,

*t*) by projection [cf. Eq. (29)],

**X**can be performed and the prediction [Eq. (21)] of LRT/D-RPA is obtained, as expected. In fact, the dynamics of the Rouse modes is typically used to derive this explicit expression for the partial, dynamic, single-chain structure factors that are employed in LRT/D-RPA.

^{36}

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, *qR*_{e} ≳ 2*π*, or smaller, however, the preparation process matters because the initial distribution of the internal Rouse modes, **X**_{p} 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 *qR*_{e}. In the ultimate long-time limit, both amplitudes decay like $exp(\u2212q2Dt)=exp[\u2212(qRe)2t/(3\pi 2\tau R)]$ 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

*P*(

**X**,

*t*), in an ideal gas of homopolymers follows the Fokker–Planck Eq. (43). This dynamic equation can be exactly rewritten in form of a deterministic D-DFT equation as

^{65,66}

*F*[

*P*] takes the form of the canonical free-energy functional of PDFT [

*vide infra*Eq. (A24)],

^{65,66}[Eq. (50)] with the free-energy functional of equilibrium PDFT in the absence of nonbonded interactions. Thus, by using the molecular density,

*P*(

**X**), instead of the segment density,

*ρ*(

**r**), for the macromolecular liquid, we obtain a simple, memory-free time evolution.

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, $\Delta F$, 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}

^{69}Grzetic, Wickham, and Shi,

^{70}and Fredrickson and Orland

^{71}employed the Martin–Siggia–Rose formalism

^{72}to derive a Fokker–Planck equation for the molecular density as an intermediate step toward dynamic mean-field theory (DMFT),

*ζ*stands for the segmental friction and $K=Kb+Knb\u2208R3N$ denotes the sum of bonded and nonbonded forces that act on all segments. The nonbonded forces,

**K**

_{nb}, are self-consistently calculated from the molecular density,

*vide infra*. Here, we also demonstrate that this Fokker–Planck equation for the interacting system can be cast in the form of the deterministic D-DFT [Eq. (50)], with the canonical free-energy functional,

*F*[

*P*], of equilibrium PDFT for the interacting system according to Eq. (A24) and a simple mobility

^{65,66}For the three examples considered, we have solved the Fokker–Planck equation and obtained an explicit, analytic result, Eq. (47). Moreover, the Fokker–Planck equation also serves as basis for systematically deriving approximate schemes, like the ellipsoid model in Subsection III D.

Grzetic *et al.*^{70} did not have a strategy for tackling the high-dimensional Fokker–Planck equation^{71} 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 Orland^{71} 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 simulations^{75,76} in the mean-field limit, $N\u0304\u2192\u221e$. 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, $Hb$. SCMF simulations represent the weak nonbonded interactions of a segment with its surrounding by quasi-instantaneous fields, $\omega (r)=\delta \Delta F/\delta \rho $ 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 $N\u0304\u2192\u221e$ 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 $N\u0304\u2192\u221e$.

There is, however, one important difference between SCMF simulations for finite $N\u0304<\u221e$ and DMFT/SCMF simulations for $N\u0304\u2192\u221e$. 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 theory^{15} 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 $N\u0304$ capture these fluctuation and correlation effects,^{76} neither D-DFT nor DMFT/SCMF simulations for $N\u0304\u2192\u221e$ are expected to describe these effects for they vanish in the limit, $N\u0304\u2192\u221e$.

### 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-SCFT^{7,10,17–31} or *ρ* and *ρ*_{h} 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, $X12$, of the first Rouse modes.^{82,83} This approach has the advantage to reproduce the accurate thermodynamic description of SCFT^{1–12} while retaining some additional information about nonequilibrium molecular conformations. Here, we take a related but different approach by using **X**_{0} and **X**_{1} to characterize the system.

**X**

_{0}, is the slowest diffusing degree of freedom and related to the conserved number of polymer chains. The other Rouse modes with

*p*> 0 relax with different

*p*-dependent time scales,

*τ*

_{p}=

*τ*

_{R}/

*p*

^{2}, according to Eq. (44). The first Rouse mode,

**X**

_{1}, is the next slowest degree of freedom that characterizes the overall shape of a molecule around its center of mass. Progressively higher Rouse modes describe smaller-scale details of the internal molecular configurations and relax faster. Although there is no sharp time-scale separation between the Rouse modes, we try to approximate the molecular density by assuming that all Rouse modes with

*p*> 1 instantaneously relax to their (constraint) equilibrium values,

*P*

_{eq}(

**X**

_{2},

**X**

_{3}, …|

**X**

_{0},

**X**

_{1}), given

**X**

_{0}and

**X**

_{1}. Thus, we approximate

*P*

_{ell}(

**X**

_{0},

**X**

_{1},

*t*). We refer to this approximate description as the ellipsoid model because

*P*

_{ell}(

**X**

_{0},

**X**

_{1},

*t*) describes the system by soft particles. Each particle represents an entire polymer molecule and is characterized by its center-of-mass position,

**X**

_{0}, and the anisotropic, overall shape,

**X**

_{1}. In a similar vein, such a description has previously been devised by Murat and Kremer

^{84}and others

^{85–89}for highly coarse-grained computer simulations of polymer solutions, blends, and copolymers.

*h*= 0,

*P*

_{eq}, adopts the particularly simple form

**X**

_{0}and

**X**

_{1}. In this case, the molecular density of the ellipsoid model coincides with the marginal of

*P*(

**X**,

*t*),

^{90}

*P*

_{ell}(

**X**

_{0},

**X**

_{1},

*t*) can be either obtained as marginal of

*P*(

**X**,

*t*), given in Eq. (47), or as the limit of Eq. (47) when the relaxation times of the higher internal Rouse modes vanish,

*τ*

_{p}→ 0 for

*p*> 1.

*qR*

_{e}= 4

*π*. The strength of the preparation field is identical for the Rouse and the ellipsoid model. The instantaneous relaxation of the higher internal Rouse modes,

*p*> 1, at time

*t*= 0

^{+}, reduces the amplitude of the density modulation at

*t*= 0

^{+}. Only after the relaxation time of the second Rouse mode,

*t*>

*τ*

_{2}=

*τ*

_{R}/2

^{2}, do the ellipsoid model and the Rouse model start to agree. This observation suggests that the description at early times can be systematically improved by retaining the time evolution of higher Rouse modes,

**X**

_{3},

**X**

_{4}, …, instead of assuming that they instantaneously relax to their equilibrium values. In turn, if we only accounted for the zeroth Rouse mode, i.e., the center of mass, and assumed all internal Rouse modes to equilibrate instantaneously, we would obtain a single-exponential decay with rate 1/

*τ*

_{q}=

**q**

^{2}

*D*. Indeed, this relaxation due to center-of-mass diffusion is compatible with the ultimate long-time decay,

*t*≫

*τ*

_{R}.

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*(**X**_{0}, **X**_{1}, *t*) = *P*_{ell}(**X**_{0}, **X**_{1}, *t*)/*P*_{ell}(**X**_{0}, **X**_{1}, ∞) − 1 ≈ ln *P*_{ell}(**X**_{0}, **X**_{1}, *t*)/*P*_{ell}(**X**_{0}, **X**_{1}, ∞) of the joint distribution function of **X**_{0} and **X**_{1} from its equilibrium value as a function of the first two Rouse modes, $X0=q\u0302X0$ and $X1=q\u0302X1$, 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, *X*_{0}, and its overall orientation and extension, *X*_{1}, greatly differ although all three preparation processes result in a periodic density modulation with a single characteristic wavevector, *qR*_{e} = 4*π*.

If the external preparation field acted on all segments, molecules with a center of mass, *X*_{0}, in a high density region, *qX*_{0}/2*π* = 1/2, are characterized by a narrower distribution of *X*_{1}, 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 *X*_{1} is largely independent for the molecule’s center-of-mass position. In these two cases, the probability distribution is symmetric for **X**_{1} ↔ −**X**_{1}. 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*/*τ*_{R} = 0.05 and 0.1. Whereas the amplitude of the variations, *R*(**X**_{0}, **X**_{1}, *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 *X*_{0} − *X*_{1} 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, **X**_{0}, of a molecule diffuses, whereas the internal degrees of freedom, quantified by higher Rouse modes, **X**_{p} with *p* > 0, exhibit a spectrum of relaxation times up to the Rouse time, *τ*_{R}.

We prepare the density modulation with length scale ≲*R*_{e} 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-SCFT^{7,10,17–31} 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 model^{64} 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-RPA^{38–42} 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, *ρ*_{h}, on which the external preparation field acted at *t* < 0, and the density of other segments, *ρ*_{n} = *ρ* − *ρ*_{h}. LRT can be exactly rewritten in terms of a generalized model-B dynamics^{31,33,40} [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* = *ρ*_{h}, 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 $\Lambda \u0303(\omega )$ by its zero-frequency limit, i.e., $\Lambda k(t)\u2248\Lambda \u0304k\delta (t)$, with $\Lambda \u0304k=\u222b0\u221edt\Lambda k(t)=Sk/(Nk2\tau k)$, where *τ*_{k} 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, **X**_{p} 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, **X**_{0} and **X**_{1}, 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, **X**_{0}, and its overall orientation and extension, **X**_{1}. Such an approximate description of the molecular density turns out to be fairly accurate for the studied example (at times larger than *τ*_{2} = *τ*_{R}/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 approaches^{70,71} 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-link^{78,91–93} and slip-spring models^{94–96} 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 interfaces^{27,102} in polymer blends, or process-directed self-assembly^{14} after a sudden quench of thermodynamic condition (e.g., pressure^{103}), mechanical deformation,^{104,105} or a chemical reaction^{106} 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, $\u223cRe$, of a single unit cell of the periodically modulated structure, the relaxation toward the metastable state occurs on the time scale, *τ*_{R}. 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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflict of interests to declare.

## DATA AVAILABILITY

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

^{9}We consider a system with polymer activity,

*z*, volume,

*V*, and temperature,

*T*. Whereas the basic degree of freedom of the free-energy functional,

*F*[

*ρ*], of SCFT is the local density,

*ρ*(

**r**), PDFT provides a description in terms of the molecular density, i.e., the unnormalized probability distribution,

*P*, of single-chain configurations. A single-chain configuration is specified either by the

*N*segment positions,

**R**= {

**r**

_{1}, …,

**r**

_{N}} or, alternatively but equivalently, by the

*N*Rouse modes, {

**X**

_{0}, …,

**X**

_{N−1}}. Since the molecular density,

*P*(

**R**), provides more information than the segment density,

*ρ*(

**r**), one can obtain the segment density by a projection of the molecular density,

*∫*D

**R**

*zP*= ⟨

*n*⟩. The starting point of PDFT is the grand canonical partition function,

^{6,36}

*N*→ ∞. The expression in terms of the Rouse modes is more convenient because $Hb(X)$ is also additive for the different Rouse modes,

**X**

_{jp}of a given polymer

*j*.

**r**and

**r**′, the nonbonded interaction free energy takes the form

*κN*is related to the inverse, isothermal compressibility.

^{76}

*W*(

**R**), that depends on the 3

*N*degrees of freedom,

**R**, of a single polymer,

*z*, to the average number of molecules, ⟨

*n*⟩,

*P*], as a functional of the molecular density,

*P*(

**R**), takes the form

_{P}, of a single-chain quantity with respect to the molecular distribution,

*P*.

*P*] with respect to the molecular density,

*P*(

**R**), we obtain the second saddlepoint condition

*P*, the auxiliary field takes the form $W*(R)=\u2211i=1N\omega (ri)$. Inserting this molecular distribution in equilibrium into the density functional, we calculate the equilibrium grand potential, Ω* = Ω[

*P**],

*P*], in Eq. (A15), we obtain the canonical free-energy functional,

*F*[

*P*], at given

*n*= ⟨

*n*⟩,

*V*, and

*T*by Legendre transformation in conjunction with the equation of state [Eq. (A14)],

*F*[

*P*], of the canonical ensemble does not depend on the normalization

*∫*D

**R**

*P*(

**R**) of the molecular density. The change of the normalization is equivalent to multiplying a constant factor to the activity,

*z*, or adding a constant to the chemical potential, respectively. For the considerations in the canonical ensemble, we choose

*∫*D

**R**

*P*(

**R**) = 1 in the main text.

*F*[

*P*], yields the molecular chemical potential

**R**, $\delta F[P]\delta P(R)P*=$ const, resulting in $P*\u223cexp\u2212Hb+\u2211i=1N\omega (ri)kBT$, in accord with Eq. (A20). The saddlepoint of the auxiliary field is given by

*F** =

*F*[

*P**],

### APPENDIX B: COARSE-GRAINED PARTICLE MODEL AND SIMULATION TECHNIQUES

To verify the predictions of LRT/D-RPA^{38–42} and the Rouse model,^{64} we use a coarse-grained particle model^{75,76,111,112} 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 *k*_{B}*Th* 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, *R*_{e}, 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, $N\u0304$. 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 $D=9.48\u22c510\u22126Re2$/MCS, where each segment is attempted to be once displaced in one Monte Carlo Step (MCS). Length scales are measured in units of *R*_{e}, and time scales are reported in units of the Rouse time, $\tau R=Re23\pi 2D=3563$ 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) *h*_{e} = 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.

## REFERENCES

*The Equilibrium Theory of Inhomogeneous Polymers*

*Soft Matter*

*Fddd*network in triblock and diblock copolymer melts

*The Theory of Polymer Dynamics*

Short-time dynamics refers to the universal dynamics of the polymers on the scale of a fraction of the coil size, *R*_{e}. This dynamics is dictated by the self-similar structure and dynamics of Gaussian polymer chains^{64} and not by the dynamics on the scale of a monomeric repeating units that is determined by the specific chemistry of the monomers.

Taking the zero-frequency limit of Eq. (24), we find $q2N\Lambda \u0303q(\omega )\u2192q2N\Lambda \u0303q(0)=SqSq\u22121(0)Sq$ and obtain an approximation for a wavevector-dependent Onsager coefficient $\Lambda q(t)\u2248\delta (t)\u222b0\u221edt\u2032\Lambda q(t\u2032)=\delta (t)\Lambda \u0303q(0)=\delta (t)Sq/(Nq2\tau q)$,^{29,31} with a single relaxation time, *τ*_{q}.

*s*th segment and the center of mass in an unperturbed Gaussian chain along a Cartesian direction,