Memory in the relaxation of a polymer density modulation

Müller, Marcus

doi:10.1063/5.0084602

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, $\bar{N} \equiv {(ρ_{0} R_{e}^{3} / N)}^{2}$ (with ρ₀, 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.

Traditionally, one characterizes the one-component polymer system by the local segment density, 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,

ρ (r) = \sum_{j = 1}^{n} \sum_{i = 1}^{N} δ (r - r_{j i}) = \sum_{j = 1}^{n} N \int_{0}^{1} d s δ (r - r_{j} (s)),

(1)

where the sum runs over all polymers, j = 1, …, n, in a system of volume, V. Each polymer is discretized into N segments. The spatial position of the ith 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_{k} = \frac{1}{V} \int d r e^{i kr} ρ (r),

(2)

where ρ₀ = nN/V denotes the spatial average of the segment density.

Within SCFT, the free-energy landscape is quantified by a functional, F[ρ_k], of the local segment density. ΔF does not depend on the sign and complex phase of ρ_k. Thus, a Landau expansion [Random-Phase Approximation (RPA)] of ΔF begins with |ρ_k|². The thermal average of segment-density fluctuations defines the collective structure factor,

S_{k}^{coll} = ⟨ | V ρ_{k} |^{2} ⟩ / (n N)

⁠, that is experimentally accessible by scattering experiments. Thus, a shallow density fluctuation with wavevector k increases the free energy by an amount^6,15

Δ F [ρ_{k}] = \frac{k_{B} T V}{2 ρ_{0}} \sum_{k} \frac{| ρ_{q} |^{2}}{S_{k}^{coll}} + O (| ρ_{k} |^{3}),

(3)

where k_BT denotes the thermal energy. Since polymers in an ideal gas do not interact, individual molecules independently contribute to the collective structure factor, i.e.,

S_{k}^{coll} = S_{k}

⁠, 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 = {(k R_{e})}^{2} / 6

⁠,^6,15

g (x) = \frac{2}{{[{(k R_{e})}^{2} / 6]}^{2}} (e^{- \frac{{(k R_{e})}^{2}}{6}} - 1 + \frac{{(k R_{e})}^{2}}{6}) .

(4)

Since the segment number is a conserved quantity, its density obeys the continuity equation¹⁶^,

\frac{d ρ_{k}}{d t} = i k \cdot j_{k},

(5)

where j denotes the density current. For a shallow density fluctuation, ρ_k ≪ ρ₀, 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,

j_{k} = i k Λ_{k} \frac{μ_{k}}{k_{B} T} with μ_{k} = \frac{N ρ_{0}}{V} \frac{\partial F}{\partial ρ_{k}} \approx k_{B} T \frac{ρ_{k}}{S_{k} / N},

(6)

where Λ_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. ] in conjunction with this model-B dynamics¹⁶ predicts an exponential decay of a density fluctuation in time,

\frac{d ρ_{k}}{d t} = - k^{2} Λ_{k} \frac{ρ_{k}}{S_{k} / N},

(7)

with a wavevector-dependent relaxation rate,

\frac{1}{τ_{k}} = \frac{k^{2} Λ_{k}}{S_{k} / N}

⁠. Equation ignores thermal fluctuations, and therefore, ρ_k(t) corresponds to the average over different realizations of the stochastic relaxation process.

In the following, we create an initial, shallow density modulation with a specific wavevector, 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

\frac{Δ F}{k_{B} T} = N h \int d r \cos (qr) ρ_{h} (r)

(8)

to the excess free energy, with

ρ_{h} (r) = \sum_{j = 1}^{n} \int d s γ (s) δ (r - r_{j} (s))

⁠. A weak external preparation field gives rise to a spatial modulation of the density 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)]/ρ₀, for qR_e = 4π. This wavevector corresponds to a spatial period of L = R_e/2 ≈ 1.22R_g, where $R_{g} = R_{e} / \sqrt{6}$ 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). ${[\dots]}_{ρ_{q} (0)}$ 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).

FIG. 1.

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Relaxation of the amplitude of shallow density modulations in an ideal gas of homopolymers. Time, t, is measured in units of the Rouse time, τ_R. Symbols mark results of particle-based simulations. The dashed line corresponds to the predictions of D-SCFT [Eq. (7)] with an Onsager coefficient, Λ_k = D, where $D = \frac{R_{e}^{2}}{3 π^{2} τ_{R}}$ stands for the polymer’s self-diffusion coefficient. Solid lines show the predictions of LRT/D-RPA [Eq. (21)]. The black solid line coincides with the decay of the dynamic structure factor [Eq. (10)].

Since the description in terms of the collective segment density, ρ_q(t), involves a projection from the individual segment coordinates [cf. Eq. ], 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,

{[\dots]}_{ρ_{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. for a more detailed derivation within LRT]

\begin{align} S_{q}^{coll} (t) & = ⟨{[ρ_{q} (t)]}_{ρ_{q} (0)} ρ_{q}^{*} (0)⟩ = ⟨\frac{{[ρ_{q} (t)]}_{ρ_{q} (0)}}{ρ_{q} (0)} ρ_{q} (0) ρ_{q}^{*} (0)⟩ \\ = \frac{{[ρ_{q} (t)]}_{ρ_{q} (0)}}{ρ_{q} (0)} S_{q}^{coll} . \end{align}

(9)

Here, we have assumed that

{[ρ_{q} (t)]}_{ρ_{q} (0)} / ρ_{q} (0)

only is a function of 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,

S_{q}^{coll} (t) = S_{q} (t)

⁠, and an explicit expression is known within the Rouse model (see Appendix 4.III of Ref. 36),

\begin{align} S_{q} (t) & = N \int_{0}^{1} d s \int_{0}^{1} d s^{'} e^{- \frac{{(q R_{e})}^{2} | s - s^{'} |}{6} - \frac{{(q R_{e})}^{2} t}{3 π^{2} τ_{R}}} \\ \times e^{- \frac{2 {(q R_{e})}^{2}}{3 π^{2}} \sum_{p > 0} \frac{1 - e^{- p^{2} t / τ_{R}}}{p^{2}} \cos (π p s) \cos (π {p s}^{'})} . \end{align}

(10)

The Rouse time, τ_R, and the self-diffusion coefficient of the molecule’s center of mass are related by

D = R_{e}^{2} / (3 π^{2} τ_{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
$\frac{d ρ_{q} (t)}{d t} = - \int_{- \infty}^{t} d t^{'} q^{2} Λ_{q} (t - t^{'}) \frac{ρ_{q} (t^{'})}{S_{q} / N} .$
(11)
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, 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 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

Even if the weak external field acted at times, 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. ]^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. ] one can invert the relation³¹

\frac{d}{d t} S_{q} (t) = - \int_{0}^{t} d t^{'} q^{2} N Λ (t - t^{'}) \frac{S_{q} (t^{'})}{S_{q}}

(12)

to construct the time-dependent Onsager coefficient.

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(−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 qR_e, we obtain an exponential shown by the dashed-red line in Fig. 1.

Because of the convolution in time, in general, we can only provide an analytic expression for the Laplace transform of the Onsager coefficient (vide infra). Numerically, we split the Onsager coefficient into a local, ultra-short time contribution and a nonlocal contribution in time,³¹^,

Λ_{q} (t) = Λ_{0} (δ (t) - Λ_{nl q} (t) / τ_{R}),

(13)

with

Λ_{0} = - \frac{1}{q^{2} N} {\frac{d S_{q}}{d t}|}_{t = 0} = \frac{R_{e}^{2}}{3 π^{2} τ_{R}} = D,

(14)

and define an integrated memory kernel K(t) according to Λ(t) = dK/dt.⁶⁷ This integrated memory kernel can be iteratively constructed as

K (t) = \sum_{n = 0}^{\infty} K_{n} (t)

with S_q(0)K₀(t) = dS_q/dt and

K_{n + 1} (t) = - \int_{0}^{t} d t^{'} K_{n} (t^{'}) K_{0} (t - t^{'})

⁠.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, $t^{*} = {(q R_{e})}^{4} t / (3 π^{2} τ_{R}) = {(q R_{e})}^{2} q^{2} D t$ ⁠, for Rouse dynamics, extending up to t ∼ τ_R or $t^{*} \sim {(q R_{e})}^{4} / (3 π^{2})$ ⁠. 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.

FIG. 2.

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Dimensionless, nonlocal contribution, $Λ_{nl q} (t) = τ_{R} [δ (t) - Λ_{q} (t) / D]$ ⁠, to the Onsager coefficient, required for a generalized model-B description^31,33,40 [Eq. (11)] if the external preparation field acts on all segments, γ(s) = 1 for all s. The data for different wavevectors, q, are scaled according to the prediction of the Rouse model⁶⁴ in the short-wavelength limit, qR_e → ∞. Vertical lines indicate the Rouse time, τ_R.

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.

Combining these two order-parameter field into

{\vec{ρ}}_{q} = {(ρ_{h q}, ρ_{n q})}^{T}

⁠, we obtain a quadratic free-energy functional within RPA [in analogy to Eq. ],¹⁵^,

Δ F [{\vec{ρ}}_{q}] = \frac{k_{B} T V}{2 ρ_{0}} \sum_{q} {\vec{ρ}}_{q}^{T} S_{q}^{- 1} {\vec{ρ}}_{- q},

(15)

where

S_{q}

is the 2 × 2 matrix of partial, static, single-chain structure factors. The chemical potentials of the two segment species take the form

{\vec{μ}}_{q} = k_{B} T N S_{q}^{- 1} {\vec{ρ}}_{q}

⁠, resulting in the generalized model-B dynamics,^31,33,40

\begin{align} \frac{d {\vec{ρ}}_{q}}{d t} & = - q^{2} \int_{0}^{t} d t^{'} ⋀_{q} (t - t^{'}) \frac{{\vec{μ}}_{q} (t^{'})}{k_{B} T} \\ = - q^{2} N \int_{0}^{t} d t^{'} ⋀_{q} (t - t^{'}) S_{q}^{- 1} {\vec{ρ}}_{q} (t^{'}), \end{align}

(16)

where

⋀_{q} (t)

denotes the symmetric matrix of nonlocal Onsager coefficients. The Laplace transform (or one-sided temporal Fourier transform) of the above equation reads

i ω {\vec{\tilde{ρ}}}_{q} (ω) - {\vec{ρ}}_{q} (0) = - q^{2} N {\tilde{⋀}}_{q} (ω) S_{q}^{- 1} {\vec{\tilde{ρ}}}_{q} (ω),

(17)

with the transform

{\vec{\tilde{ρ}}}_{q} (ω) = \int_{0}^{\infty} d t e^{- i ω t} {\vec{ρ}}_{q} (t)

⁠.

These Onsager coefficients,

⋀_{q} (t)

⁠, can be obtained by comparing Eq. with the predictions of D-RPA,^38–42 i.e., the LRT for an ideal gas of block copolymers subjected to a time-dependent external field,

{\vec{V}}_{q} (t)

⁠,

\frac{{\vec{ρ}}_{q} (t)}{ρ_{0}} = - \int_{- \infty}^{t} d t^{'} x_{q} (t - t^{'}) {\vec{V}}_{q} (t^{'}),

(18)

where the matrix of linear-response functions,

x_{q}

⁠, is related to the matrix of partial dynamic structure factors,

S_{q} (t)

⁠, of the ideal-gas reference system by

x_{q} (t) = - \frac{d S_{q} (t)}{d t} for t \geq 0

(19)

and

x_{q} (t) = 0

for t < 0.

In the present case, the external field is constant for t < 0,

{\vec{V}}_{q} (t) \equiv {\vec{V}}_{-} = \frac{N h}{2} {(1, 0)}^{T}

⁠, and vanishes for t ≥ 0. Thus, Eq. can be simply integrated to

\frac{{\vec{ρ}}_{q} (t)}{ρ_{0}} = - S_{q} (t) {\vec{V}}_{-} .

(20)

The result for the segment density

ρ_{q} (t) = ρ_{h q} (t) + ρ_{n q} (t) = - [S_{h h q} (t) + S_{n h q} (t)] ρ_{0} N h / 2

takes the form

\begin{matrix} \frac{ρ_{\pm q} (t)}{ρ_{0}} & = & - \frac{N h}{2} \int_{0}^{1} d s^{'} \int_{0}^{1} d s γ (s) e^{- \frac{{(q R_{e})}^{2} | s - s^{'} |}{6} - \frac{{(q R_{e})}^{2} t}{3 π^{2} τ_{R}}} \\ \times e^{- \frac{2 {(q R_{e})}^{2}}{3 π^{2}} \sum_{p > 0} \frac{1 - e^{- p^{2} t / τ_{R}}}{p^{2}} \cos (π p s) \cos (π {p s}^{'})} . \end{matrix}

(21)

This prediction of LRT is plotted in Fig. 1 as colored solid lines for the different preparation processes described by γ(s), and it quantitatively agrees with the simulations of the particle-based model without an adjustable parameter.

Differentiating and eliminating

{\vec{V}}_{-}

⁠, we obtain

\frac{d {\vec{ρ}}_{q}}{d t} = - \frac{d S_{q} (t)}{d t} ρ_{0} {\vec{V}}_{-} = \frac{d S_{q} (t)}{d t} S_{q}^{- 1} (t) {\vec{ρ}}_{q} (t) .

(22)

The comparison of the Laplace transform of this equation

\begin{align} i ω {\vec{\tilde{ρ}}}_{q} (ω) - {\vec{ρ}}_{q} (0) & = (i ω {\tilde{S}}_{q} (ω) - S_{q}) {\tilde{S}}_{q}^{- 1} (ω) {\vec{ρ}}_{q} (t) \\ = - {\tilde{x}}_{q} (ω) {\tilde{S}}_{q}^{- 1} (ω) {\vec{ρ}}_{q} (t) \end{align}

(23)

and Eq. identifies an explicit, analytic expression for the matrix of Onsager coefficients,^33,43,68

q^{2} N {\tilde{⋀}}_{q} (ω) = {\tilde{x}}_{q} (ω) {\tilde{S}}_{q}^{- 1} (ω) S_{q} = i ω {[{\tilde{x}}_{q}^{- 1} (ω) - S_{q}^{- 1}]}^{- 1} .

(24)

Since these Onsager coefficients in conjunction with the generalized model B [Eq. ] for both, ρ_hq and ρ_nq, are equivalent to Eq. , 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, $⋀_{q}$ ⁠, given by this combination of single-chain, linear-response functions, $x_{q} (t)$ and static, single-chain structure factors, $S_{q}$ ⁠) 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 $⋀_{q} (t)$ in the time domain, however, is a nontrivial challenge, and the memory kernel, $⋀_{q} (t)$ ⁠, 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

Whereas SCFT uses the segment density to specify the configuration of a homopolymer system, PDFT employs the molecular density, 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_{p} \equiv \frac{1}{N} \sum_{i = 1}^{N} r_{i} \cos (\frac{π p}{N} [i - \frac{1}{2}]),

(25)

r_{i} = X_{0} + 2 \sum_{p > 0}^{N - 1} X_{p} \cos (\frac{π p}{N} [i - \frac{1}{2}]) .

(26)

The zeroth Rouse mode denotes the center of mass of a polymer, and in the limit of fine chain discretization, we obtain X₀ = ∫dsr(s). For p ≥ 1, Eq. becomes

X_{p} = \int_{0}^{1} d s r (s) \cos (π p s) = - \int_{0}^{1} d s \frac{d r}{d s} \frac{\sin (π p s)}{π p},

(27)

where the latter expression shows that Rouse modes with 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/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-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, 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, X_p with p > 0, are statistically independent and Gaussian distributed with average, ⟨X_p⟩ = 0, and variance, $⟨ X_{p}^{2} ⟩ \equiv σ_{p}^{2} \to R_{e}^{2} / (6 π^{2} p^{2})$ ⁠, for N → ∞.³⁶

In the following, we consider the response of P(X) to an external preparation field of strength, k_BTh, 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

P (X) \sim e^{- \sum_{p = 1}^{\infty} \frac{3 π^{2} p^{2} X_{p}^{2}}{R_{e}^{2}} - N h \int_{0}^{1} d s γ (s) \cos [q X_{0} + 2 \sum_{p = 1}^{\infty} q X_{p} \cos (π p s)]} .

(28)

The first term in the Boltzmann factor represents the bonded energy,

H_{b}

[also see Eq. ], whereas the second term quantifies the potential energy of the segment with contour index 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. ]. Thus, we choose ∫DXP(X) = 1, where

\int D X = \int_{V} d X_{0} \int_{- \infty}^{\infty} \prod_{p = 1}^{\infty} d X_{p}

integrates over all molecular configurations.

The segment density, ρ(r), is obtained by projection

ρ (r) = n \int D X P (X) \sum_{i = 1}^{N} δ (r - r_{i}) = n {[\sum_{i = 1}^{N} δ (r - r_{i})]}_{P},

(29)

with [⋯]_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).

The spatial Fourier transform of the segment density takes the form

\frac{ρ_{k}}{ρ_{0}} = \int D X P (X) \int d s^{'} e^{i k [X_{0} + 2 \sum_{p > 0} X_{p} \cos (π p s^{'})]} .

(30)

To make progress, we assume that the external preparation field is weak, Nhγ(s) ≪ 1, and approximate

\begin{align} P (X) & \sim e^{- \sum_{p > 0} \frac{3 π^{2} p^{2} X_{p}^{2}}{R_{e}^{2}}} \{1 - N h \int_{0}^{1} d s γ (s) \\ \times \cos [q X_{0} + 2 \sum_{p > 0} q X_{p} \cos (π p s)] + O (h^{2})\} . \end{align}

(31)

While different Rouse modes are statistically independent in the absence of an external field, h = 0, they become correlated even within this linear, weak-field approximation. Whereas, for p > 0, the averages

{[X_{p}]}_{P} = 0 and {[X_{p}^{2}]}_{P} = \frac{R_{e}^{2}}{2 π^{2} p^{2}}

(32)

over 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

{[\hat{q} X_{p} \sin (q X_{0})]}_{P} and {[{(\hat{q} X_{p})}^{2} \cos (q X_{0})]}_{P},

(33)

where

\hat{q} = q / | q |

denotes the unit vector of the periodic, spatial modulation, may not vanish (vide infra Fig. 5).

Within this linear, weak-field approximation [Eq. ], we obtain

\begin{matrix} \frac{ρ_{\pm q}}{ρ_{0}} & \approx & - \frac{N h}{2} \int d s \int d s^{'} γ (s) \int \prod_{p = 1}^{\infty} d X_{p} \frac{e^{- \frac{3 π^{2} p^{2} X_{p}^{2}}{R_{e}^{2}}}}{{(R_{e}^{2} / 3 π p^{2})}^{3 / 2}} \\ \times e^{\pm 2 \sum_{p > 0} i q X_{p} \{\cos (π p s^{'}) - \cos (π p s)\}} + O (h^{2}), \end{matrix}

(34)

\begin{matrix} = & - \frac{N h}{2} \int d s \int d s^{'} γ (s) e^{- \frac{{(q R_{e})}^{2}}{6} 2 \sum_{p > 0} {\{\frac{\cos (π p s^{'}) - \cos (π p s)}{π p}\}}^{2}} \\ = & - \frac{N h}{2} \int d s \int d s^{'} γ (s) e^{- \frac{{(q R_{e})}^{2}}{6} | s - s^{'} |}, \end{matrix}

(35)

which agrees with the value of Eq. at t = 0. Other Fourier components, ρ_k, of the segmental density with k ≠ 0, ±q vanish in linear order.

If the preparation field uniformly acts on all segments, γ(s) = 1 for all s, we obtain the well-known RPA relation

\frac{ρ_{\pm q}}{ρ_{0}} \approx - \frac{N h}{2} g_{D} (q R_{e}) = - \frac{h}{2} S_{q}

(36)

or

\frac{ρ (r)}{ρ_{0}} \approx 1 - h S_{q} \cos (q r) .

(37)

If the field only acts on an end segment with strength, h_e, i.e., Nhγ(s) = h_eδ(s), we find

\frac{ρ_{\pm q}}{ρ_{0}} \approx - \frac{h_{e}}{2} \frac{6}{{(q R_{e})}^{2}} (1 - e^{- \frac{{(q R_{e})}^{2}}{6}}) .

(38)

If the field only acts on the middle segment with strength, h_m, i.e.,

N h γ (s) = h_{m} δ (s - 1 / 2)

⁠, we obtain

\frac{ρ_{\pm q}}{ρ_{0}} \approx - \frac{h_{m}}{2} \frac{12}{{(q R_{e})}^{2}} (1 - e^{- \frac{{(q R_{e})}^{2}}{12}}) .

(39)

Equations , , and quantify the strengths of the external fields necessary to bring about a segment-density variation of a given amplitude and wavevector, q. These results coincide with the prediction of static RPA or D-RPA [Eq. ] evaluated at time t = 0. They are also confirmed by the particle-based simulations (see Table I).

TABLE I.

Amplitude of the initial density modulation and strength of the external preparation field in units of the thermal energy k_BT used in the particle-based simulation with SOMA.

All segments	End segment	Two middle segments
$2 \frac{ρ_{\pm q}}{ρ_{0}} = 0.0207$	0.0202	0.0205
$h = \frac{0.285}{128}$	$h_{e} = \frac{64}{128} = 0.5$	$h_{m} = \frac{17.5}{64} = 0.273$

Equations and have a simple interpretation: Since the external field only acts on a single segment, s_h = 0 or 1/2, respectively, it does not distort the internal distances, i.e., the joint distribution,

\tilde{P} (r, d)

⁠, of segment s_h being at location r and segments s and s′ being separated by a distance vector d factorizes

\begin{align} \tilde{P} (r, d) & = \int D X P (X) δ [r - X_{0} - 2 \sum_{p > 0} X_{p} \cos (π p s_{h})] \\ \times δ [d - 2 \sum_{p > 0} X_{p} (\cos (π p s) - \cos (π p s^{'}))] \end{align}

(40)

≃ \frac{1 - h \cos (qr)}{V} {(\frac{3}{2 π R_{e}^{2} | s - s^{'} |})}^{3 / 2} e^{- \frac{3 d^{2}}{2 R_{e}^{2} | s - s^{'} |}} .

(41)

Unfortunately, however, the fact that the internal-distance distribution adopts its field-free equilibrium form, independently from the position of the segment on which the external preparation field is applied, does not imply that the Rouse modes with p > 0 are uncorrelated or are distributed like in the absence of the external preparation field [cf. Eq. ]. 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

Whereas the initial distribution, 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 pth Rouse mode relaxes according to the Langevin equation of the Rouse model,

d X_{p} = - \frac{p^{2}}{τ_{R}} X_{p} d t + \sqrt{2 \frac{1 + δ_{p, 0}}{6 π^{2} τ_{R}} R_{e}^{2}} d W,

(42)

where dW_α (with α = x, y, z denoting the Cartesian components) are the increments of the Wiener process, i.e., ⟨dW_α⟩ = 0 and ⟨dW_α(t)dW_β(t′)⟩ = δ_α,βδ(t − t′)dt. The factor 1 + δ_p,0 could be easily eliminated by redefining the pth Rouse mode as

X_{p}^{'} = \sqrt{2} X_{p}

for p > 0.

The molecular density, P(X, t), evolves according to the corresponding Fokker–Planck equation,

\frac{\partial P (X, t)}{\partial t} = \nabla_{X} \cdot (\frac{p^{2}}{τ_{R}} X P + \frac{1 + δ_{p, 0}}{6 π^{2} τ_{R}} R_{e}^{2} \nabla_{X} P) .

(43)

This time evolution conserves the normalization of the single-chain probability distribution, i.e., ∫DX P(X, t) = 1 also for t > 0.

The conditional probability of observing the molecular configuration, 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 . For an ideal gas (Rouse model), the Green’s function factorizes

G (X, t | X (0)) = G_{0} (X_{0}, t | X_{0} (0)) \prod_{p > 0} G_{p} (X_{p}, t | X_{p} (0))

(44)

\begin{matrix} = & {(\frac{4 R_{e}^{2} t}{3 π τ_{R}})}^{- 3 / 2} \exp (- \frac{3 π^{2} τ_{R}}{4 R_{e}^{2} t} {\{X_{0} - X_{0} (0)\}}^{2}) \\ \times \prod_{p > 0} {(\frac{R_{e}^{2}}{3 π p^{2}} (1 - e^{- 2 p^{2} t / τ_{R}}))}^{- 3 / 2} \\ \times \exp (- \frac{3 π^{2} p^{2}}{R_{e}^{2}} \frac{{\{X_{p} - X_{p} {(0)}^{- p^{2} t / τ_{R}}\}}^{2}}{1 - e^{- 2 p^{2} t / τ_{R}}}) . \end{matrix}

(45)

The convolution with the initial distribution, P(X, 0), given by Eq. , yields the molecular density at time t > 0,

P (X, t) = \int D X (0) G (X, t | X (0)) P (X (0), 0)

(46)

\begin{matrix} = & P_{0} (X) \{1 - N h \int d s γ (s) \\ \times e^{- \frac{{(q R_{e})}^{2} t}{3 π^{2} τ_{R}} - \frac{{(q R_{e})}^{2}}{3 π^{2}} \sum_{p > 0} (1 - e^{- 2 p^{2} t / τ_{R}}) \frac{\cos^{2} (π p s)}{p^{2}}} \\ \times \cos [q X_{0} + 2 \sum_{p > 0} q X_{p} e^{- p^{2} t / τ_{R}} \cos (π p s)]\}, \end{matrix}

(47)

where

P_{0} (X) = \frac{1}{V} \prod_{p > 0} {(\frac{R_{e}^{2}}{3 π p^{2}})}^{- 3 / 2} \exp (- \frac{3 π^{2} p^{2}}{R_{e}^{2}} X_{p}^{2})

(48)

denotes the equilibrium distribution of Rouse modes in the absence of an external field. The time evolution of the segment density, in turn, is obtained from P(X, t) by projection [cf. Eq. ],

\frac{ρ_{\pm q} (t)}{ρ_{0}} = \int D X P (X, t) \int d s^{'} e^{\pm i q [X_{0} + 2 \sum_{p = 1}^{\infty} X_{p} \cos (π p s^{'})]} .

(49)

After straightforward calculation, the integrals over X can be performed and the prediction [Eq. ] 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 ≳ 2π, 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 $\exp (- q^{2} D t) = \exp [- {(q R_{e})}^{2} t / (3 π^{2} τ_{R})]$ and the ratio of amplitudes tends to a wavevector-dependent constant.

FIG. 3.

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Ratio of the time-dependent amplitudes, a_a(t) and a_e(t), of decaying density modulations that have been prepared by a weak spatially modulated external field with wavevector q, acting on all segments or only one end segment, respectively. The arrows on top mark τ_R for qR_e = π and 2π. For t ≫ τ_R, the ratio tends toward a constant.

C. Equivalence of Rouse dynamics and D-DFT for an ideal gas of macromolecules

The time evolution of the molecular density, P(X, t), in an ideal gas of homopolymers follows the Fokker–Planck Eq. . This dynamic equation can be exactly rewritten in form of a deterministic D-DFT equation as^65,66

\begin{gathered} \frac{\partial P (X, t)}{\partial t} = \frac{1 + δ_{p, 0}}{6 π^{2} τ_{R}} R_{e}^{2} \nabla_{X} \cdot \{P (\frac{X}{σ_{p}^{2}} + \nabla_{X} \ln P)\} \\ \frac{\partial P (X, t)}{\partial t} = Γ \nabla_{X} \cdot \{P \nabla_{X} \frac{δ}{δ P} \frac{F [P]}{n k_{B} T}\} . \end{gathered}

(50)

F[P] takes the form of the canonical free-energy functional of PDFT [vide infra Eq. ],

\frac{F [P]}{k_{B} T} = \ln n! + \frac{E [P]}{k_{B} T} + n \int D X P (X) \ln P (X),

(51)

where

E [P] = n \int D X P (X) H_{b} (X) = k_{B} T \frac{ρ_{0} V}{N} \int D X P (X) \sum_{p > 0} \frac{X_{p}^{2}}{2 σ_{p}^{2}}

denotes the average, bonded energy in the case of an ideal gas of chain molecules. Comparing Eqs. and , we identify the mobility, Γ,

Γ = \frac{1 + δ_{p, 0}}{6 π^{2} τ_{R}} R_{e}^{2} .

(52)

The key observation is that the Fokker–Planck equation for the Rouse modes has the form of a deterministic D-DFT^65,66 [Eq. ] 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, $Δ F$ ⁠, 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

Starting from the Langevin equation of interacting polymer segments, Fredrickson and Helfand,⁶⁹ Grzetic, Wickham, and Shi,⁷⁰ and Fredrickson and Orland⁷¹ employed the Martin–Siggia–Rose formalism⁷² to derive a Fokker–Planck equation for the molecular density as an intermediate step toward dynamic mean-field theory (DMFT),

\frac{\partial P (R, t)}{\partial t} = \frac{k_{B} T}{ζ} \nabla_{R}^{2} P - \frac{1}{ζ} \nabla_{R} \cdot (K, P)

(53)

where ζ stands for the segmental friction and

K = K_{b} + K_{nb} \in R^{3 N}

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. ], with the canonical free-energy functional, F[P], of equilibrium PDFT for the interacting system according to Eq. and a simple mobility

\frac{\partial P (R, t)}{\partial t} = \frac{k_{B} T}{ζ} \nabla_{R} \cdot \{P (\nabla_{R} \ln P - \frac{K}{k_{B} T})\}

(54)

= \frac{R_{e}^{2} N}{3 π^{2} τ_{R}} \nabla_{R} \cdot \{P \nabla_{R} \frac{δ}{δ P} \frac{\overset{= F [P]}{\overset{︷}{k_{B} T n \ln n + k_{B} T n \int DR P (\ln P - 1) + E [P]}}}{n k_{B} T}\} .

(55)

Thus, we identify the mean force with

K = - \frac{1}{n} \nabla_{R} \frac{δ E}{δ P}

⁠, where

E [P] = n \int DR P H_{b} (R) + Δ F [ρ [P]]

is given by Eq. .

Δ F [ρ]

is the excess free energy of nonbonded interactions. This relation immediately clarifies the equilibrium statistical mechanics.^65,66 For the three examples considered, we have solved the Fokker–Planck equation and obtained an explicit, analytic result, Eq. . 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 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 simulations^75,76 in the mean-field limit, $\bar{N} \to \infty$ ⁠. 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, $H_{b}$ ⁠. SCMF simulations represent the weak nonbonded interactions of a segment with its surrounding by quasi-instantaneous fields, $ω (r) = δ Δ F / δ ρ$ 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 $\bar{N} \to \infty$ 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 $\bar{N} \to \infty$ ⁠.

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

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, $X_{1}^{2}$ ⁠, 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 X₀ and X₁ to characterize the system.

The molecule’s center of mass, X₀, 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/p², according to Eq. . The first Rouse mode, X₁, 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(X₂, X₃, …|X₀, X₁), given X₀ and X₁. Thus, we approximate

P (X, t) \approx P_{ell} (X_{0}, X_{1}, t) P_{eq} (X_{2}, X_{3}, \dots | X_{0}, X_{1})

(56)

and only follow the dynamics of P_ell(X₀, X₁, t). We refer to this approximate description as the ellipsoid model because P_ell(X₀, X₁, t) describes the system by soft particles. Each particle represents an entire polymer molecule and is characterized by its center-of-mass position, X₀, and the anisotropic, overall shape, X₁. In a similar vein, such a description has previously been devised by Murat and Kremer⁸⁴ and others^85–89 for highly coarse-grained computer simulations of polymer solutions, blends, and copolymers.

In the case of an ideal gas of macromolecules without an external field, h = 0, P_eq, adopts the particularly simple form

P_{eq} (X_{2}, X_{3}, \dots | X_{0}, X_{1}) = \prod_{p > 1} {(\frac{R_{e}^{2}}{3 π p^{2}})}^{- 3 / 2} e^{- \frac{3 π^{2} p^{2}}{R_{e}^{2}} X_{p}^{2}},

(57)

which is independent from X₀ and X₁. In this case, the molecular density of the ellipsoid model coincides with the marginal of P(X, t),

\begin{align} P_{ell} (X_{0}, X_{1}, t) & = \frac{\int \prod_{p > 1} d X_{p} P (X, t)}{\int \prod_{p > 1} d X_{p} P_{eq} (X_{2}, X_{3}, \dots | X_{0}, X_{1})} \\ = \int \prod_{p > 1} d X_{p} P (X, t) . \end{align}

(58)

Inserting the approximate molecular density [Eq. ] into the Fokker–Planck equation , we obtain the reduced Fokker–Planck equation of the ellipsoid model,

τ_{R} \frac{\partial P_{ell}}{\partial t} = \nabla_{X_{0}} \cdot (\frac{R_{e}^{2}}{3 π^{2}} \nabla_{X_{0}} P_{ell}) + \nabla_{X_{1}} \cdot (X_{1} P_{ell} + \frac{R_{e}^{2}}{6 π^{2}} \nabla_{X_{1}} P_{ell}),

(59)

with the concomitant Green’s function

\begin{align} G_{ell} (X_{0}, X_{1}, t | X_{0} (0), X_{1} (0)) \\ = {(\frac{3 π τ_{R}}{4 R_{e}^{2} t})}^{3 / 2} e^{- \frac{3 π^{2} τ_{R}}{4 R_{e}^{2} t} {\{X_{0} - X_{0} (0)\}}^{2}} \\ \times {(\frac{3 π}{R_{e}^{2} (1 - e^{- 2 t / τ_{R}})})}^{3 / 2} e^{- \frac{3 π^{2}}{R_{e}^{2}} \frac{{\{X_{1} - X_{1} {(0)}^{- t / τ_{R}}\}}^{2}}{1 - e^{- 2 t / τ_{R}}}} . \end{align}

(60)

Again, the Green’s function of the Rouse model [Eq. ] and that of the ellipsoid model are simply related by

G_{ell} (X_{0}, X_{1}, t | X_{0} (0), X_{1} (0)) = \int \prod_{p > 1} d X_{p} G (X, t | X (0)),

(61)

and we obtain consistently

P_{ell} (X_{0}, X_{1}, t) = \int \prod_{p > 1} d X_{p} \int D X (0) G (X, t | X (0)) P (X (0), 0)

(62)

\begin{matrix} = & \int d X_{0} (0) d X_{1} (0) G_{ell} (X_{0}, X_{1}, t | X_{0} (0), X_{1} (0)) \\ \times \int \prod_{p > 1} d X_{p} (0) P (X (0), 0) . \end{matrix}

(63)

Performing the integrals, we derive⁹⁰

\begin{matrix} P_{ell} (X_{0}, X_{1}, t) & = & \frac{1}{V} \frac{{(3 π)}^{3 / 2}}{R_{e}^{3}} e^{- \frac{3 π^{2}}{R_{e}^{2}} X_{1}^{2}} \{1 - N h \int_{0}^{1} d s γ (s) \\ \times \cos [q X_{0} + 2 q X_{1} e^{- t / τ_{R}} \cos (π s)] \\ \times e^{- \frac{{(q R_{e})}^{2} t}{3 π^{2} τ_{R}} - \frac{{(q R_{e})}^{2}}{3 π^{2}} [(1 - e^{- 2 t / τ_{R}}) \cos^{2} (π s) - \sum_{p > 1} \frac{1}{p^{2}} \cos^{2} (π p s)]} \} . \end{matrix}

(64)

Importantly, for the dynamics in an ideal gas without an external field, P_ell(X₀, X₁, t) can be either obtained as marginal of P(X, t), given in Eq. , or as the limit of Eq. when the relaxation times of the higher internal Rouse modes vanish, τ_p → 0 for p > 1.

The projection onto the segment density results in

\begin{matrix} \frac{ρ_{ell \pm q} (t)}{ρ_{0}} & = & - \frac{N h}{2} \int_{0}^{1} d s^{'} \int_{0}^{1} d s γ (s) e^{- \frac{{(q R_{e})}^{2} | s - s^{'} |}{6} - \frac{{(q R_{e})}^{2} t}{3 π^{2} τ_{R}}} \\ \times e^{- \frac{2 {(q R_{e})}^{2}}{3 π^{2}} [(1 - e^{- 2 t / τ_{R}}) \cos (π s) \cos (π s^{'}) + \sum_{p > 1} \frac{1}{p^{2}} \cos (π p s) \cos (π {p s}^{'})]} . \end{matrix}

(65)

In Fig. 4, we compare this result of the ellipsoid model to the exact Rouse result [Eq. ] for the decay of a density modulation with wavevector qR_e = 4π. 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 > τ₂ = τ_R/2², 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, X₃, X₄, …, 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 = q²D. Indeed, this relaxation due to center-of-mass diffusion is compatible with the ultimate long-time decay, t ≫ τ_R.

FIG. 4.

Comparison of the temporal decay of a shallow density modulation with qRe = 4π, obtained by the Rouse model64 [solid lines, Eq. (21)] and the ellipsoid model [dashed lines with symbols, Eq. (65)], which assumes that higher internal Rouse modes with p > 1 instantaneously equilibrate. Three preparation processes are indicated in the key. The red dashed-dotted vertical line marks the relaxation time of the second Rouse mode, τ2 = τR/4—the slowest mode assumed to be instantaneous relaxed. The red dashed line indicates an exponential decay with rate 1/τq=q2⁡D=(qR)23π2τR.

View large Download slide

Comparison of the temporal decay of a shallow density modulation with qR_e = 4π, obtained by the Rouse model⁶⁴ [solid lines, Eq. (21)] and the ellipsoid model [dashed lines with symbols, Eq. (65)], which assumes that higher internal Rouse modes with p > 1 instantaneously equilibrate. Three preparation processes are indicated in the key. The red dashed-dotted vertical line marks the relaxation time of the second Rouse mode, τ₂ = τ_R/4—the slowest mode assumed to be instantaneous relaxed. The red dashed line indicates an exponential decay with rate $1 / τ_{q} = q^{2} D = \frac{{(q R)}^{2}}{3 π^{2} τ_{R}}$ ⁠.

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(X₀, X₁, t) = P_ell(X₀, X₁, t)/P_ell(X₀, X₁, ∞) − 1 ≈ ln P_ell(X₀, X₁, t)/P_ell(X₀, X₁, ∞) of the joint distribution function of X₀ and X₁ from its equilibrium value as a function of the first two Rouse modes, $X_{0} = \hat{q} X_{0}$ and $X_{1} = \hat{q} X_{1}$ ⁠, 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, qR_e = 4π.

FIG. 5.

Relative deviation R(X0, X1, t) = Pell(X0, X1, t)/Pell(X0, X1, ∞) − 1 of the time-dependent joint distribution of qX0/2π (horizontal axis) and 6π2X1/Re (vertical axis) along the direction, q̂, of the density modulation. qX0/2π ≈ 0 and 1 correspond to low density regions, whereas the density is maximal at qX0/2π = 1/2. The three columns represent the cases that the external preparation field that created a density modulation with the wavevector qRe = 4π for t < 0 acted on all segments (left column), on one end chain (middle column), or on the middle segment of the polymer (right column). In the first case, we plot R/(hN), whereas in the latter two cases, when the field only acts on one segment, the graph displays R/h. The three rows show the results for times t = 0 (top row), t = 0.05τR (center row), and t = 0.1τR (bottom row).

View large Download slide

Relative deviation R(X₀, X₁, t) = P_ell(X₀, X₁, t)/P_ell(X₀, X₁, ∞) − 1 of the time-dependent joint distribution of qX₀/2π (horizontal axis) and $\sqrt{6 π^{2}} X_{1} / R_{e}$ (vertical axis) along the direction, $\hat{q}$ ⁠, of the density modulation. qX₀/2π ≈ 0 and 1 correspond to low density regions, whereas the density is maximal at qX₀/2π = 1/2. The three columns represent the cases that the external preparation field that created a density modulation with the wavevector qR_e = 4π for t < 0 acted on all segments (left column), on one end chain (middle column), or on the middle segment of the polymer (right column). In the first case, we plot R/(hN), whereas in the latter two cases, when the field only acts on one segment, the graph displays R/h. The three rows show the results for times t = 0 (top row), t = 0.05τ_R (center row), and t = 0.1τ_R (bottom row).

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/τ_R = 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, 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 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-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 $\tilde{Λ} (ω)$ by its zero-frequency limit, i.e., $Λ_{k} (t) \approx {\bar{Λ}}_{k} δ (t)$ ⁠, with ${\bar{Λ}}_{k} = \int_{0}^{\infty} d t Λ_{k} (t) = S_{k} / (N k^{2} τ_{k})$ ⁠, 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, 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 = 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-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, $\sim R_{e}$ ⁠, 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

Within the grand canonical ensemble, we explicitly verify that PDFT and SCFT yield the same equilibrium distribution.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 = {r₁, …, r_N} or, alternatively but equivalently, by the N Rouse modes, {X₀, …, 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,

ρ (r) = \int DR z P (R) \sum_{i = 1}^{N} δ (r - r_{i}) .

(A1)

Integrating Eq. over space, we obtain the normalization of the molecular density, ∫DRzP = ⟨n⟩. The starting point of PDFT is the grand canonical partition function,

Z_{gc} \sim \sum_{n = 0}^{\infty} \frac{z^{n}}{n!} \int \prod_{j = 1}^{n} D R_{j} e^{- \frac{\sum_{j = 1}^{N} H_{b} (R_{j}) + Δ F [\hat{ρ}]}{k_{B} T}},

(A2)

where

H_{b}

stands for the bonded interactions that are additive for the different chains. For the standard Gaussian chain model, bonded interactions along a single macromolecule can be expressed in terms of the segment positions or the Rouse modes,^6,36

\frac{H_{b}}{k_{B} T} = \sum_{i = 1}^{N - 1} \frac{3 (N - 1)}{2 R_{e}^{2}} {(r_{i + 1} - r_{i})}^{2} = \sum_{p = 1}^{N - 1} \frac{X_{p}^{2}}{2 σ_{p}^{2}},

(A3)

with

σ_{p}^{2} \to R_{e}^{2} / (6 π^{2} p^{2})

for N → ∞. The expression in terms of the Rouse modes is more convenient because

H_{b} (X)

is also additive for the different Rouse modes, X_jp of a given polymer j.

Δ F [ρ]

denotes the free energy of nonbonded interactions that depends on the segment coordinates only via the microscopic segment density,

\hat{ρ} (r)

⁠, that is calculated from the microscopic segment coordinates [cf. Eq. ]. The hat highlights that this quantity is calculated from the segment coordinates. For an ideals gas,

Δ F = 0

⁠, whereas for a system with the zero-ranged, soft pairwise repulsion potential,

Φ = k_{B} T \frac{κ}{⟨ n ⟩ N / V} δ (r - r^{'})

⁠, between two segments at positions, r and r′, the nonbonded interaction free energy takes the form

Δ F [ρ] = \frac{k_{B} T κ}{2 \frac{⟨ n ⟩ N}{V}} \int d r ρ^{2} (r),

(A4)

where the model parameter κN is related to the inverse, isothermal compressibility.⁷⁶

Analog to

\hat{ρ} (r)

⁠, we define a microscopic molecular density

z \hat{P} (R) = \sum_{j = 1}^{n} \prod_{i = 1}^{N} δ (r_{i} - r_{j i})

⁠, where the hat again indicates the dependence on all segment coordinates. With this definition, we can rewrite the microscopic segment density in a form that is the analog of Eq. ,

\hat{ρ} (r) = \int DR z \hat{P} (R) \sum_{i = 1}^{N} δ (r - r_{i}) .

(A5)

Using the standard procedure of SCFT, we introduce the auxiliary field variable, W(R), that depends on the 3N degrees of freedom, R, of a single polymer,

Z_{gc} \sim \sum_{n = 0}^{\infty} \frac{z^{n}}{n!} \int D W D P \prod_{j = 1}^{n} D R_{j} e^{- \frac{\sum_{j = 1}^{n} H_{b} (R_{j}) + Δ F [\hat{ρ}]}{k_{B} T}} e^{\int DR \frac{W}{k_{B} T} z (P - \hat{P})},

(A6)

\sim \sum_{n = 0}^{\infty} \frac{z^{n}}{n!} \int D W D P e^{- \frac{Δ F [ρ]}{k_{B} T} + z \int D R P \frac{W}{k_{B} T}} Q^{n},

(A7)

with

Q [W] = \int D R e^{- \frac{H_{b} (R) + W (R)}{k_{B} T}}

(A8)

Z_{gc} \sim \int D W D P e^{- \frac{G [W, P]}{k_{B} T}},

(A9)

with

G = Δ F [ρ] - z \int D R P W - k_{B} T z Q .

(A10)

Approximating the functional integral over the auxiliary field by the saddlepoint method, we obtain

Z_{gc} \sim \int D P e^{- \frac{Ω [P]}{k_{B} T}} with Ω [P] = G [W^{*}, P],

(A11)

and the saddlepoint condition reads

0 = {\frac{δ G [W, P]}{δ W (R)}|}_{W^{*}} = - z P (R) - k_{B} T z {\frac{δ Q [W]}{δ W (R)}|}_{W^{*}}

(A12)

or

W^{*} (R) = - H_{b} (R) - k_{B} T \ln P (R) .

(A13)

This relation allows us to rewrite Eq. in the form of an equation of state that relates the activity, z, to the average number of molecules, ⟨n⟩,

Q [W^{*}] = \int D R P (R) = \frac{⟨ n ⟩}{z} .

(A14)

Within this mean-field approximation, the grand potential, Ω[P], as a functional of the molecular density, P(R), takes the form

\frac{Ω [P]}{k_{B} T} = \frac{E [P]}{k_{B} T} + z \int D R P (\ln P - 1)

(A15)

with the nonbonded and bonded interaction (free) energy

E [P] = Δ F [ρ [P]] + z \int D R P H_{b} (R)

(A16)

= Δ F [ρ [P]] + ⟨ n ⟩ \underset{\equiv {[H_{b}]}_{P}}{\underset{︸}{\frac{\int D R P H_{b} (R)}{\int D R P}}},

(A17)

where we have introduced the average, [⋯]_P, of a single-chain quantity with respect to the molecular distribution, P.

Minimizing this density functional, Ω[P] with respect to the molecular density, P(R), we obtain the second saddlepoint condition

0 = {\frac{δ Ω [P]}{δ P (R)}|}_{P^{*}} = {\frac{δ E [P]}{δ P (R)}|}_{P^{*}} + k_{B} T z \ln P^{*} .

(A18)

This condition yields

P^{*} (R) = e^{- \frac{H_{b} (R)}{k_{B} T} - \frac{1}{k_{B} T} \int d r \frac{δ Δ F}{δ ρ (r)} \sum_{i = 1}^{N} δ (r - r_{i})}

(A19)

= \exp (- \frac{H_{b} (R) + \sum_{i = 1}^{N} ω (r_{i})}{k_{B} T}),

(A20)

with

ω (r) = \frac{δ Δ F [ρ]}{δ ρ (r)}

⁠. At this saddlepoint value of P, the auxiliary field takes the form

W^{*} (R) = \sum_{i = 1}^{N} ω (r_{i})

⁠. Inserting this molecular distribution in equilibrium into the density functional, we calculate the equilibrium grand potential, Ω* = Ω[P*],

Ω^{*} = - k_{B} T ⟨ n ⟩ + Δ F [ρ [P^{*}]] - z \int D R P^{*} \sum_{i = 1}^{N} ω (r_{i}) .

(A21)

Starting from the grand potential, Ω[P], in Eq. , 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. ],

\frac{F [P]}{k_{B} T} = \frac{Ω [P]}{k_{B} T} + ⟨ n ⟩ \ln z

(A22)

= n \ln \frac{n}{e} + E [P] + n (\frac{\int D R P \ln \frac{P}{\int D R P}}{\int D R P})

(A23)

= \ln n! + \frac{E [P]}{k_{B} T} + n {[\ln \frac{P}{\int D R P}]}_{P} .

(A24)

Note that the free-energy functional, F[P], of the canonical ensemble does not depend on the normalization ∫DRP(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 ∫DRP(R) = 1 in the main text.

The variation of the free-energy functional, F[P], yields the molecular chemical potential

\frac{δ F [P]}{δ P (R)} = \frac{δ E [P]}{δ P (R)} + k_{B} T n (\frac{\ln P}{\int D R P} - \frac{\int D R P \ln P}{{(\int D R P)}^{2}})

(A25)

\begin{align} = & \int d r ω (r) \frac{δ ρ (r)}{δ P (R)} + n (\frac{H_{b}}{\int D R P} - \frac{\int D R P H_{b}}{{(\int D R P)}^{2}}) \\ + k_{B} T \frac{n}{\int D R P} (\ln P - {[\ln P]}_{P}) \end{align}

(A26)

\begin{align} = & \int d r ω (r) \frac{1}{\int D R P} (n \sum_{i = 1}^{N} δ (r - r_{i}) - ρ (r)) \\ + \frac{n}{\int D R P} (H_{b} - {[H_{b}]}_{P} + k_{B} T (\ln P - {[\ln P]}_{P})) \end{align}

(A27)

\begin{align} = & \frac{n}{\int D R P} (\sum_{i = 1}^{N} ω (r_{i}) + H_{b} + k_{B} T \ln P \\ - \underset{independent from R}{\underset{︸}{{[\sum_{i = 1}^{N} ω (r_{i}) + H_{b} + k_{B} T \ln P]}_{p})}} . \end{align}

(A28)

In equilibrium, the molecular chemical potential is independent from R,

{\frac{δ F [P]}{δ P (R)}|}_{P^{*}} =

const, resulting in

P^{*} \sim \exp (- \frac{H_{b} + \sum_{i = 1}^{N} ω (r_{i})}{k_{B} T})

⁠, in accord with Eq. . The saddlepoint of the auxiliary field is given by

ω (r) = {\frac{δ Δ F}{δ ρ (r)}|}_{ρ^{*} (r)},

(A29)

and the equilibrium density is given by Eq. . Equations and coincide with the self-consistent set of equations that describe equilibrium in SCFT.

Inserting this equilibrium molecular density into the canonical free-energy functional [Eq. ], we obtain the canonical free energy, F* = F[P*],

\frac{F^{*}}{k_{B} T} = \frac{E [P^{*}]}{k_{B} T} + n \ln \frac{n}{e} + n {[\ln P^{*}]}_{P^{*}} - n \ln \int D R P^{*}

(A30)

\begin{matrix} = & n \ln \frac{n}{e} + \frac{E [P^{*}]}{k_{B} T} - n {[\frac{H_{b} + \sum_{i = 1}^{N} ω (r_{i})}{k_{B} T}]}_{P^{*}} \\ - n \ln \int D R \exp (- \frac{H_{b} + \sum_{i = 1}^{N} ω (r_{i})}{k_{B} T}) \end{matrix}

(A31)

\begin{matrix} = & n \ln \frac{n}{V e} + \frac{Δ F [ρ^{*}]}{k_{B} T} - \frac{n}{k_{B} T} {[\sum_{i = 1}^{N} ω (r_{i})]}_{P^{*}} \\ - n \ln \frac{1}{V} \int D R \exp (- \frac{H_{b} + \sum_{i = 1}^{N} ω (r_{i})}{k_{B} T}), \end{matrix}

(A32)

which agrees with the standard expression of the free energy in SCFT.

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_BTh 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, $\bar{N}$ ⁠. 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 $D = 9.48 \cdot 1 0^{- 6} R_{e}^{2}$ /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, $τ_{R} = \frac{R_{e}^{2}}{3 π^{2} D} = 3563$ 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.

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Published open access through an agreement with University of Göttingen Faculty of Physics

Memory in the relaxation of a polymer density modulation

I. INTRODUCTION AND BACKGROUND

II. GENERALIZED MODEL-B DESCRIPTION OF THE RELAXATION OF A DENSITY MODULATION IN AN IDEAL GAS

A. External field acted on all segments

B. External field acted on a portion of a chain

III. DYNAMIC DENSITY FUNCTIONAL THEORY OF THE RELAXATION OF A DENSITY MODULATION IN AN IDEAL GAS

A. Equilibrium distribution of Rouse modes of a single chain subjected to a weak external field

B. Relaxation of Rouse modes and segment density

C. Equivalence of Rouse dynamics and D-DFT for an ideal gas of macromolecules

D. Ellipsoid model

IV. SUMMARY AND OUTLOOK

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

APPENDIX A: SCFT-LIKE DERIVATION OF POLYMER DENSITY-FUNCTIONAL THEORY

APPENDIX B: COARSE-GRAINED PARTICLE MODEL AND SIMULATION TECHNIQUES

REFERENCES

Citing articles via

Resources

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pubs.aip.org

Connect with AIP Publishing

Memory in the relaxation of a polymer density modulation

I. INTRODUCTION AND BACKGROUND

II. GENERALIZED MODEL-B DESCRIPTION OF THE RELAXATION OF A DENSITY MODULATION IN AN IDEAL GAS

A. External field acted on all segments

B. External field acted on a portion of a chain

III. DYNAMIC DENSITY FUNCTIONAL THEORY OF THE RELAXATION OF A DENSITY MODULATION IN AN IDEAL GAS

A. Equilibrium distribution of Rouse modes of a single chain subjected to a weak external field

B. Relaxation of Rouse modes and segment density

C. Equivalence of Rouse dynamics and D-DFT for an ideal gas of macromolecules

D. Ellipsoid model

IV. SUMMARY AND OUTLOOK

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

APPENDIX A: SCFT-LIKE DERIVATION OF POLYMER DENSITY-FUNCTIONAL THEORY

APPENDIX B: COARSE-GRAINED PARTICLE MODEL AND SIMULATION TECHNIQUES

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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