Bottom-up construction of dynamic density functional theories for inhomogeneous polymer systems from microscopic simulations
aa r X i v : . [ c ond - m a t . s o f t ] S e p Bottom-up construction of dynamic density functional theories for inhomogeneouspolymer systems from microscopic simulations
Sriteja Mantha, Shuanhu Qi, and Friederike Schmid ∗ Institut für Physik, Johannes Gutenberg Universität Mainz, Staudingerweg 9, 55128 Mainz, Germany Key Laboratory of Bio-inspired Smart Interfacial Science and Technology of Ministry of Education,School of Chemistry, Beihang University, Beijing 100191, China
We propose and compare different strategies to construct dynamic density functional theories(DDFTs) for inhomogeneous polymer systems close to equilibrium from microscopic simulationtrajectories. We focus on the systematic construction of the mobility coefficient, Λ( r , r ′ ) , whichrelates the thermodynamic driving force on monomers at position r ′ to the motion of monomersat position r . A first approach based on the Green-Kubo formalism turns out to be impracticalbecause of a severe plateau problem. Instead, we propose to extract the mobility coefficient froman effective characteristic relaxation time of the single chain dynamic structure factor. To test ourapproach, we study the kinetics of ordering and disordering in diblock copolymer melts. The DDFTresults are in very good agreement with the data from corresponding fine-grained simulations. I. INTRODUCTION
Inhomogeneous polymer systems assemble into or-dered morphologies due to incompatible interactions be-tween different constituents in the systems . Thesemorphologies have found applications as thermoplasticelastomers , materials for drug delivery and release , gascapture , water purification , energy conversion , andalso in soft lithography . Understanding the relation be-tween the molecular features of polymers and the orderedmorphologies formed by them has been a subject of activeinvestigation for a long time . An equally interestingtopic is the effect of polymer dynamics on the process ofself-assembly , e.g., on the kinetics of defect formationdepending on the way a nanostructured polymer mate-rial is processed . This has lead to experimental andtheoretical investigations to understand the polymer dy-namics in inhomogeneous systems and its effect on theformation of ordered morphologies.Different scattering and reflectometry techniques havebeen employed to study the kinetic pathways leadingto order-order and order-disorder transitions in blockcopolymer systems . The same techniques are used toinvestigate the adsorption dynamics and the formation ofinterfaces in an incompatible homo-polymer blend .However, the dynamics in inhomogeneous polymer sys-tems involves relaxation processes occurring over multi-ple length and time scales. For example, the molecularfeatures of polymers determine the local rearrangementsof chains. On the other hand, the mesoscopic orderingof polymer chains takes place on length and time scaleswhich are multiple orders of magnitude higher than themolecular length and time scales. As a result, findingan experimental technique that can capture the dynam-ics over the entire spectrum of length and time scales isan extremely involved task. Dynamic density functionaltheory (DDFT) or the dynamic self-consistent fieldtheory have been promoted as a theoretical alternative tostudy the polymer dynamics on the relevant mesoscopiclength and time scales. In a DDFT, the dynamics of an inhomogeneous poly-mer system is described by a diffusive equation in themonomer densities ∂ρ α ( r , t ) ∂t = X β ∇ r (cid:20)Z d r ′ Λ αβ ( r , r ′ ) ∇ r ′ µ β ( r ′ , t ) (cid:21) (1)Here, ρ α ( r , t ) is the density of monomers of type α , Λ αβ ( r , r ′ ) is the mobility matrix and ( −∇ r ′ µ β ( r ′ , t )) alocal thermodynamic force acting on monomers of type β .The matrix Λ ( r , r ′ ) relates the monomer density currentto the thermodynamic driving force and depends on themonomer-monomer correlations in the system. The field µ β ( r , t ) can be interpreted as a local chemical potentialfor unconnected monomers of type β and is derived from afree energy functional F , (i.e., µ β ( r ′ , t ) = δF/δρ β ( r ′ , t ) ),which is typically taken from self-consistent field (SCF)theory. Since ρ α ( r , t ) are coarse-grained quantities, theirdynamic evolution equations describe the kinetics in thesystem on mesoscopic scales. A typical SCF theory for polymers retains microscopic information on the chainarchitectures. This combination of mesoscopic and mi-croscopic aspects makes DDFT a promising techniquein the pursuit of studying polymer dynamics in an in-homogeneous system. DDFT has been used to explorethe kinetic pathways for micelle to vesicle transition inmicellar solutions , morphological transitions in di-block copolymer melts and also scaling laws forthe polymer inter-diffusion during interfacial broaden-ing in polymer blends . DDFT models have alsobeen extended to study the effects of hydrodynamics and reptation . Recent investigations have also usedDDFT in conjunction with the string method to deter-mine the mean free-energy path for pore formation andrupture in cell membranes .Although DDFT has significantly advanced our under-standing of polymer dynamics, it suffers from the prob-lem that DDFT models are typically constructed in anad hoc manner. The dynamics of polymers is well-knownto be governed by relaxation processes on multiple timescales . When projecting the dynamical equations formonomer coordinates onto a dynamical equation for den-sities such as Eq. (1) in a systematic manner, e.g., us-ing the Mori-Zwanzig formalism , this invariably re-sults in a generalized Langevin equation with a memorykernel . In DDFT, the memory kernel is replaced byone single, time independent (but nonlocal) effective mo-bility function. This greatly increases the computationalefficacy of the resulting coarse-grained model, however,the optimal way to choose such an effective mobility isnot clear.Currently, all approaches in the literature are based onheuristic assumptions. For chains in the Rouse regime,these approximate schemes can broadly be categorizedinto local and nonlocal approaches . In the local ap-proach, monomers are assumed to diffuse in the systemindependent of each other. In the nonlocal approaches,polymers are assumed to diffuse as a whole. These ap-proximations significantly reduce the complexity in han-dling the DDFT equation. However they come with theirown caveats. Most importantly, it was found that thechoice of DDFT approach may influence the pathwaysof self-assembly that are observed in DDFT calculations.One example is the dynamics of vesicle formation fromhomogeneous nucleation, where nonlocal DDFT calcu-lations predicted the existence of competing pathwaysof self-assembly (which was then confirmed both byexperiments and simulations ), whereas only onepathway was present in local DDFT simulations . More-over, local DDFT calculations greatly overestimate thefrequency of vesicle fusion events , which are largelysuppressed in nonlocal DDFT simulations consistentwith experiments . When comparing to particle-basedsimulations, local DDFT calculations generally tend tooverestimate the speed of structure formation, and non-local DDFT calculations tend to underestimate it .It should be noted that none of these approaches in-corporate knowledge on the microscopic dynamics in theunderlying polymer dynamics. In recent years, bottom-up coarse-graining techniques have become increasinglypopular in materials science, where coarse-grained mod-els are constructed from fine-grained simulations in a sys-tematic manner. Examples are techniques for derivingeffective potentials in coarse-grained models or effec-tive friction coefficients or even memory kernels indynamical equations. Since SCF models bridge betweenmicroscopic and the mesoscopic length scales, it shouldbe possible to apply similar ideas for the constructionof DDFT equations in order to improve their predictivecapabilities.In this article, we explore two physically motivatedbottom-up construction schemes for determining DDFTmobility functions Λ ( r , r ′ ) from microscopic simulations.In the first approach, we follow a classical approachto this type of problem and consider the Green-Kuborelation that relates Λ ( r , r ′ ) to an integral overan appropriate current-current time correlation function.Unfortunately, the result turns out to be not very useful,for reasons that we shall discuss below. In a second ap- proach, we therefore propose to extract Λ ( r , r ′ ) from thecharacteristic relaxation time of the dynamic structurefactor of single chains.To test our approach, we study two related problems:The first is the dynamics associated with the formationof the lamellar structure in diblock copolymer melts, thesecond is the relaxation of a lamellar structure into a ho-mogeneous state. We specifically choose these problemsbecause existing local and non-local DDFT schemes areknown to significantly under- or overestimate the timescales of (dis)ordering in comparison to fine grained sim-ulations of the same systems. We show that the bottom-up constructed DDFT models are able to capture boththe global dynamics and the relaxation due to local re-arrangements of the chain at the relevant length scales.This significantly improves the DDFT predictions for theabove listed problems.The rest of the manuscript is organized as follows: Inthe next section, we first introduce the general frame-work of DDFT theory and briefly describe the Ansätzefor mobility functions that have been proposed in the lit-erature. Then we present and discuss our two bottom-upapproaches. Finally, in the fourth section, we apply theapproach to the study of ordering and disordering in di-block copolymer melts. We conclude with a summaryand an outlook. II. GENERAL FRAMEWORK OF DDFT
The dynamic density functional theory is an extensionof the classical density functional theory, where the equi-librium free energy of a many-body system is expressedas a functional of coarse-grained field variables, the den-sity fields . A mathematical basis for this formal-ism is provided by the Hohenberg-Kohn theorem .Here we consider polymer systems with different typesof monomers α , hence our free energy functional de-pends on several fields, F ( { ρ α } ) . In practice, we will usethe functional provided by the self-consistent field (SCF)theory , which is a mean-field approach.The objective of the DDFT is to construct a physi-cally motivated scheme for the dynamical evolution ofthe microscopic densities, based on the given static func-tional. Such a scheme is expected to drive the systemalong a path of low free energy, with meaningful dynamicinformation, in order to reach the equilibrium state orat least a metastable minimum of F . Since the densityis a conserved field, its longest-wavelength Fourier com-ponents are slowly relaxing variables . This motivatesthe construction of a diffusive equation that involves thedynamic evolution of density fields only, resulting in so-called model B dynamics according to the classificationof Hohenberg and Halperin .A simple popular Ansatz is to assume the linear in-stantaneous form ∂ρ α ( r , t ) ∂t = ∇ r X β Z d r ′ Λ αβ ( r , r ′ ) ∇ r ′ µ β ( r ′ , t ) (2)with µ β ( r , t ) = δF/δρ β ( r , t ) . The mobility function Λ αβ ( r , r ′ ) relates the density current of the monomer α atposition r to the thermodynamic driving force ( −∇ µ β )on the monomer β at position r ′ . In the present pa-per, we will consider single-component homopolymer orcopolymer melts with average monomer density ρ , andassume that all chains have equal length N . Further-more, to simplify the notation, we will often use reducedquantities φ α = ρ α /ρ , ˆ µ β = Nρ δF/δφ β = N µ β , and ˆΛ = Λ /ρ N , which allows us to rewrite (2) as ∂φ α ( r , t ) ∂t = ∇ r X β Z d r ′ ˆΛ αβ ( r , r ′ ) ∇ r ′ ˆ µ β ( r ′ , t ) . (3)We note that the instantaneous assumption is ques-tionable in polymeric systems, which are known to ex-hibit memory effects , as already discussed in the intro-duction. In DDFT, one implicitly assumes that the mem-ory kernel can be replaced by a simple, time-independent(but not necessarily local) function. A second impor-tant approximation, which is typically made in polymericDDFT approaches and which we will also adopt here, isa mean-field approximation: In the spirit of the SCF the-ory which provides the static density functional F , poly-mers are assumed to move independently in an externalfield provided by the other polymers. This field may in-clude hydrodynamic flows and even entanglements, butonly in an averaged sense. Hence the mobility func-tion Λ describes the mobility of individual chains. Itincludes effects of intrachain monomer correlations, butnot those of interchain correlations. From Eq. (2), onecan thus extract a mobility function per chain , given by Λ ( s ) = Λ N/ρ = ˆΛ N .For melts in the Rouse regime (i.e., chains are non-entangled), three types of Ansatz for the mobility coeffi-cients have been proposed in the literature: (i) Local coupling scheme: In this approximation,monomer beads are assumed to diffuse independently ofeach other with the mobility D /k B T . This leads tothe following expression for ˆΛ αβ ( r , r ′ ) : ˆΛ Local αβ ( r , r ′ ) = D N k B T φ α ( r ) δ αβ δ ( r − r ′ ) (4) (ii) Chain coupling schemes: These approaches assumethat the internal structure of the polymer chain relaxeson a time scale much faster than the collective motionof the chain. As a consequence, the polymer chains areassumed to diffuse as a whole with the mobility D c = D /N . For this case, Maurits et al have derived theexpression ˆΛ Chain αβ ( r , r ′ , t ) = D c k B T P αβ ( r , r ′ , t ) /ρ N (5) where P αβ ( r , r ′ , t ) /ρ N is the pair correlation ofmonomers α , β on the same chain at position r and r ′ ,normalized to the integral one. Within the SCF theory,this quantity can be calculated exactly using a schemeproposed earlier by two of us . Further approximationshave been proposed, such as the external potential dy-namics (EPD) approximation (not discussed here) andthe Debye approximation, which approximates P αβ /ρ by the pair correlations of ideal Gaussian chains, i.e., theDebye correlation function ˆΛ Debye αβ ( r , r ′ ) = D c N k B T g αβ ( r − r ′ ) (6)Analytical expressions are available for the Fourier rep-resentation of g ( r − r ′ ) . For example, for diblock copoly-mers, one obtains g αα ( q ) = N f D ( h α , x ) (7) g AB = N { f D (1 , x ) − f D ( h A , x ) − f D ( h B , x ) } where x = q R g , h α is the fraction of block α , and f D ( h, x ) := x (cid:0) hx + e − hx − (cid:1) is the Debye function. (iii) Mixed coupling scheme: The predictions ofDDFTs based on local or non-local schemes have beencompared to simulations, and both were found to haveshortcomings . In a previous paper , two of us havetherefore proposed a mixed scheme where the dynam-ics is assumed to be governed by a local mobility func-tion on short wavelengths and a nonlocal one on largewavelengths. To this end, a filter function Γ ( r ) was in-troduced that filters out the long-wavelength part of thethermodynamic driving force via a convolution integral ˆ f Nonlocal α ( r ) = − Z d r ′ Γ ( | r − r ′ | ) ∇ ˆ µ α ( r ′ ) . (8)with Γ ( r ) = (cid:0) πσ (cid:1) − / exp {− r / σ } . (9)This ”coarsened” force is then taken to drive nonlocalchain diffusion, whereas the remaining part, ˆ f Local α ( r ) = −∇ ˆ µ α ( r ) − ˆ f Nonlocal α ( r ) (10)drives local rearrangements of the chain via a local mo-bility coefficient. The resulting interpolated scheme hasthe form ∂φ α ∂t = −∇ X β Z d r ′ h ˆΛ Nonlocal αβ ( r , r ′ ) ˆ f Nonlocal β ( r ′ )+ ˆΛ Local αβ ( r , r ′ ) ˆ f Local β ( r ′ ) i , (11)where ˆΛ Nonlocal can be any of the chain coupling schemesdiscussed above. The tunable parameter σ determinesthe length scale of crossover between the local and thenonlocal dynamics. When referring to mixed schemeDDFT calculations in the present paper, these are car-ried out by mixing local and Debye dynamics with thefilter parameter σ = 0 . R g , a value found to be optimalin our previous work . III. APPROACHES TO DETERMINE DDFTMOBILITY COEFFICIENTS FROMMICROSCOPIC SIMULATIONS
The expressions for the mobility coefficients discussedin the previous section were postulated more or lessheuristically, without much input on the underlying mi-croscopic dynamics. The only parameters that canbe used to match the microscopic and the DDFT dy-namics are the diffusion constant, and in case of themixed scheme, the tuning parameter σ . The purpose ofthe present work is to derive more informed bottom-upschemes, where the mobility coefficients are calculatedfrom simulations of a microscopic reference system. Wehave explored two such approaches which we will nowdiscuss below.In both cases, we will assume that our system is homo-geneous, hence Λ( r , r ′ ) is translationally invariant. Wecan then conveniently rewrite the DDFT equations inFourier representation as ∂ t ρ α ( q , t ) = − q X β Λ αβ ( q ) µ β ( q , t ) (12)with µ β ( q , t ) /V = δF/δρ β ( − q , t ) . Here and throughout,we define the Fourier transform via f ( q ) = Z d r e i q · r f ( r ) , f ( r ) = 1 V X q e − i q · r f ( q ) . A. Green-Kubo approach
The first approach is based on the Green-Kubo for-malism, which is a standard tool to determine transportcoefficients from simulations. Let us first recapitulatethe general formalism.
For a given microscopicsystem with Hamiltonian H , we consider the linear re-sponse of a quantity ˙ A to a perturbation of H causedby a generalized field Z B that couples to a quantity B (i.e., H = H − Z B B ). According to the Green-Kuboformalism, the response is given by h ˙ A i = λ AB Z B with λ AB = − k B T R ∞ d t h ˙ A ( t ) ˙ B (0) i in classical systems.To apply this formalism to our DDFT problem, wechoose A = ρ α ( q , t ) and B = ρ β ( q , t ) , where ρ ζ ( q , t ) (with ζ = α, β ) is derived from the monomer coordi-nates R k ( t ) via ρ ζ ( q , t ) = P k e i q · R k ( t ) γ ( ζ ) k with γ ( ζ ) k = 1 if monomer k is of type ζ , and γ ( ζ ) k = 0 otherwise.This results in ˙ A = i q · j α ( q , t ) and ˙ B = − i q · j β ( q , t ) with j ζ ( q , t ) = P k e i q · R k ( t ) ˙ R k ( t ) γ ( ζ ) k . The conti-nuity equation for ρ α in Fourier representation reads ∂ t ρ α ( q , t ) = i q · j α ( q , t ) = ˙ A . From Eq. (12), we henceknow ˙ A = − q P β Λ αβ ( q ) µ β ( q ) , where ( − µ β ( q , t ) /V ) couples to B . Now, in the linear response regime, an ex-ternal field Z B coupling to B would contribute additivelyto ( − µ β ( q , t ) /V ) and generate the same response, hencewe can identify Λ αβ = λ AB /q V and the Green-Kubo formalism results in the following expression: Λ αβ ( q ) = 1 V k B T Z ∞ d t D j α ( q , t ) j β ( − q , E : ˆqˆq , (13)with ˆq = q /q and the tensor products jj and ˆqˆq .However, the numerical evaluation of this expressionand a theoretical analysis for the special case of Rousechains shows that Eq. (13) yields zero for all nonzero q . This is demonstrated in more detail in the appendix.Only at q = 0 do we recover the familiar Green-Kuboexpression for the diffusion constant.The reason becomes clear if we recall the premises un-derlying the Green-Kubo relations. They describe theresponse of stationary currents to generalized thermody-namic forces. In our case, at q = 0 , a stationary currentis not possible, since it would generate indefinitely grow-ing density fluctuations ρ ( q , t ) . Since ρ ( q , t ) must satu-rate eventually, the flows j ( q , t ) will average to zero atlate times, independent of the applied generalized forces.Therefore, the Green-Kubo transport coefficients mustvanish for any nonzero q . Stationary currents are onlypossible at q = 0 . Hence the Green-Kubo formalism isnot suitable for determining q -dependent mobility func-tions for DDFT models.In fact, this problem is not uncommon in applica-tions of Green-Kubo integrals . For example, con-finement can prevent stationary currents, which is whyGreen-Kubo integrals may vanish in confined systems,even if locally, a description in terms of a Markovian dy-namical equations with well-defined transport coefficientsis appropriate. The q -dependent Green-Kubo integralsconsidered here, which describe the response to a spa-tially varying field, vanish for a similar reason. One pop-ular solution to this problem has been to assume thatthe time scales of local Markovian dynamics and globalconstrained dynamics are well separated, and to searchfor a plateau in the running Green-Kubo integrals. Inour case, however, the running integrals do not exhibita well-defined plateau (data not shown). We will discussthis point further in Sec. V B. Relaxation time approach
In the present subsection, we describe an alternativeapproach to deriving DDFT mobility coefficients frommicroscopic trajectories: We propose to estimate themdirectly from the characteristic relaxation time of the sin-gle chain dynamic structure factor.To motivate our Ansatz, we begin with discussing someimplications of the DDFT equations. We consider thedynamics of a single tagged chain s with correspond-ing monomer density ρ ( s ) α . In the mean-field spirit, theDDFT equation for ρ ( s ) α in Fourier representation takesthe form ∂ t ρ ( s ) α ( q , t ) = − q X β Λ ( s ) αβ ( q ) µ ( s ) β ( q , t ) , (14)where Λ ( s ) = ˆΛ N is the mobility per chain, and µ ( s ) β ( q ) = V δF ( s ) /δρ ( s ) β ( − q ) is derived from the free en-ergy F ( s ) of a single chain that moves independently inthe averaged background provided by the other chains.Next we multiply both sides with ρ ( s ) γ ( − q , and aver-age over chain conformations. Identifying g αγ ( q , t ) = N h ρ ( s ) α ( q , t ) ρ ( s ) γ ( − q , i , we obtain ∂ t g α,γ ( q , t ) = − q N X β Λ ( s ) αβ ( q ) D µ ( s ) β ( q , t ) ρ ( s ) γ ( − q , E . (15)To proceed, we expand F ( s ) in powers of ρ ( s ) ( q ) , giving F ( s ) = const. + k B T N V X q ρ ( s ) ( − q ) g − ( q , ρ ( s ) ( q ) + · · · (16)Here and in the following, we use a matrix notationfor convenience, i.e. ρ , ( ρ α ) , Λ , (Λ αβ ) etc. Tak-ing the derivative with respect to ρ ( s ) β ( − q ) , we obtain µ ( s ) ( q ) ≈ k B T N g − ( q ) ρ ( s ) ( q ) . Inserting this in Eq. (15)yields ∂ t g ( q , t ) ≈ − k B T q N Λ ( s ) ( q ) g − ( q , g ( q , t ) , (17)which can be solved in matrix form, giving g ( q , t ) = exp (cid:18) − k B T q N Λ ( s ) ( q ) g − ( q , t (cid:19) g ( q , . (18)This equation approximates the relaxation of the sin-gle chain under three assumptions: (i) Memory effectswere neglected (the basis of the DDFT approach), (ii) amean-field approximation was made (in Eq. (14)), and(iii) density fluctuations were assumed to be small (inEq. (16)). Within these approximations, the relaxationof the chain is determined by a q -dependent ”relaxationtime matrix” T ( q ) , g ( q , t ) = exp( − t T − ( q )) g ( q , and,using Λ ( s ) = ˆΛ N , we can identify ˆΛ( q ) = 1 k B T q N T − ( q ) g ( q , . (19)We can further simplify this expression by assuming thatthe relaxation of the chain is governed by a single q -dependent time constant τ ( q ) , i.e., T ( q ) ≈ · τ ( q ) . ThenEq. (19) can be rewritten as ˆΛ( q ) = 1 k B T q N τ ( q ) g ( q , . (20)The considerations above suggest the following proce-dure to determine an effective mobility coefficient for theDDFT model: We first conduct fine-grained simulationsof the polymer melt in a homogeneous reference system(i.e., in the case of the diblock copolymer melt, below theorder-disorder transition (ODT)). From the simulation trajectory for the full g ( q , t ) , we compute the relaxationtime τ ( q ) and insert it in the expression (19) or (20).The question remains how to define the characteris-tic relaxation time. This question is non-trivial, becausethe actual behavior of g ( q , t ) is driven by a multitudeof time scales, corresponding to the different internalmodes of the chain. At late times, the slowest diffusivemode dominates, and g ( q , t ) has the limiting behavior lim t →∞ g ( q , t ) ∝ exp( − D c q t ) , giving τ = 1 /D c q . In-serting this in (20), we recover the Ansatz of nonlocalDebye dynamics, (see (6)) ˆΛ( q ) = D c Nk B T g ( q ) .However, by the time this limiting behavior sets in,much of the structuring has already taken place. It wouldbe more desirable to define τ ( q ) such that it captures thedominant time scales of structure formation on the scale q . In the present work, we test two prescriptions fordetermining τ and then calculate ˆΛ via Eq. (20): ˆΛ τ R : from τ R = 1 g ( q , Z ∞ d t g ( q , t ) , (21) ˆΛ τ e : from g ( q , t = τ e ) ! = g ( q , /e, (22)where e is the Euler number and g ( q , t ) is the full single-chain structure factor, g ( q , t ) = X α,β g αβ ( q , t ) . (23)In a third approach, we generalize (21) to extract a fullrelaxation time matrix, ˆΛ T : from T ( q ) = Z ∞ d t g ( q , t ) g − ( q , . (24)and use that to determine ˆΛ via Eq. (19). Calculating ˆΛ with this method involves matrix inversions and mul-tiplications for every value of q . However, in the case of symmetric A:B diblock copolymers with fully equivalent A and B blocks, the prescription can be simplified. Forsymmetry reasons, g , T and ˆΛ then have the same ma-trix structure ( M αβ ) with M AA = M BB , M AB = M BA and thus share the same Eigenvectors, (1 , and (1 , − .Using these to diagonalize g and T , we obtain ˆΛ AA ( q ) = 14 k B T q N (cid:18) g ( q , τ R + ∆( q , τ ∆ (cid:19) (25) ˆΛ AB ( q ) = 14 k B T q N (cid:18) g ( q , τ R − ∆( q , τ ∆ (cid:19) (26)with g ( q , t ) and τ R defined as above (Eqs. (23), (21)), ∆( q , t ) = g AA ( q , t ) + g BB ( q , t ) − g AB ( q , t ) − g BA ( q , t ) ,and τ ∆ = q , R ∞ d t ∆( q , t ) .In practice, determining the integrals (21) and (24) bynumerical integration of simulation data only is not pos-sible for small q , because the relaxation time divergesfor q → . Therefore, an extrapolation procedure must q [1/R g ] Λ ^ ( q ) [ D c / k B T ] Λ ^ τ R Λ ^ τ e Λ ^ τ R Λ ^ τ e Λ ^ Local Λ ^ Debye q [1/R g ] Λ ^ ( q ) [ D c / k B T ] Λ ^ Local Λ ^ Debye Brownian dynamics Inertial dynamics a) b)
Figure 1. Normalized mobility functions of homopolymerswith length N = 40 in a melt, as obtained via the relaxationtime method (Eq. (20)) with data from Brownian dynamics(a) and inertial dynamics (b) simulations. Two prescriptionsfor determining the single chain relaxation time are tested, τ R (green, Eq. (21)) and τ e (blue, Eq. (22)). Also shownfor comparison are the results from the Debye and the localapproximation (( Λ Debye ( q ) , red) and ( Λ Local ( q ) , black)). be devised. At late times, g αβ ( q , t ) is known to de-cay exponentially according to g ( q , t ) ∼ exp( − q D c t ) .Hence we make the Ansatz g αβ ( q, t ) = g αβ ( q, t i )) exp (cid:0) − q D eff ( t − t i ) (cid:1) , (27)for large t, t i with t > t i . Specifically, we fit the datafor g αβ ( q, t ) to Eq. (27) in time windows t ∈ [ t i , t f ] , us-ing the weighted least squares fit module in the Matlabsuite , and then choose those values of t i,f which yieldthe value of D eff that is closest to the theoretical value, D c = D /N . The integrals over t in (21) and (24) arethen evaluated by first numerically integrating the dataup to t = t i , and then using the extrapolation (27) in theintegral from t = t i to infinity. Typical values for t i , t f are t i ≈ t and t f ≈ t , where t is the simulationtime unit, see below.Fig. 1 shows results for the q -dependent mobility func-tions of homopolymers in a homopolymer melt. Theywere extracted from Brownian dynamics simulations(massless monomers, Fig. 1a) and molecular dynamicssimulations (massive monomers, Fig. 1b) of melts ofGaussian chains with length N = 40 , using the prescrip-tions (21) and (22). We note that in the case of ho-mopolymers, the prescription (24) is equivalent to (21).For comparison, we also show the mobility functions cor-responding to the local and the Debye approximation. Inthe local scheme, the mobility is constant, in the Debyescheme, it is proportional to the static structure factor.The results from the relaxation schemes are intermediatebetween the local and the Debye scheme. At small q ,they follow the Debye scheme. At larger q , the mobilityis enhanced, hence small wavelength modes relax faster.The effect is more pronounced for Brownian dynamicsthan for inertial dynamics, most likely because the iner-tial time scale contributes to the total relaxation time atsmall wavelengths (see also Fig. 10 b).Thus we find that the mobility functions obtained withthe relaxation time approach interpolate between the nonlocal mobility function (at small q ) and the local mo-bility function (at larger q ). This seems promising, sinceour previous studies have suggested that such an interpo-lation may be necessary to capture the kinetics of struc-ture formation in copolymer systems . We will now testour DDFT approach by performing a systematic compar-ison of fine-grained simulations and DDFT predictionsfor the ordering/disordering kinetics in block copolymermelts. IV. APPLICATION TO DIBLOCKCOPOLYMER MELTS
We consider melts of n c block copolymers containing N A beads of type A and N B beads of type B , in a box ofvolume V = L x × L y × L z with dimension L i in i directionand periodic boundary conditions. The average monomerdensity is thus ρ = n c N/V . Polymers are modelled asGaussian chains, i.e., chains of ”monomer beads” con-nected by harmonic springs. The non-bonded monomerinteractions are characterized in terms of a Flory Hugginsparameter χ , which controls the incompatibility between A and B monomers, and a Helfand parameter κ , whichcontrols the compressibility.We carry out fine-grained simulations of order/disorderprocesses in such systems and compare them with DDFTcalculations, using the SCF free energy functional andmobility functions that are extracted from fine-grainedsimulations at χ = 0 .Throughout this paper, lengths will be represented inunits of the radius of gyration R g of an ideal chain oflength N = N A + N B , energies in units of the thermalenergy, k B T , and time in units of t = R g /D , where D is the monomer diffusivity. A. Model and methods
1. Fine-grained model and simulation method
Since we focus on a comparison of dynamical propertiesof particle-based and field-based models here, we use asfine-grained model a particle-based implementation of anEdwards model , where the non-bonded monomerinteractions are described by the same Hamiltonian thanthat underlying the SCF free energy functional. At suf-ficiently high polymer density and sufficiently far fromcritical points, the static properties of such models areknown to be well represented by SCF functionals with-out much parameter adjustment .Non-bonded interactions are thus expressed as a func-tional of the local monomer densities . Let R m,j denotethe position of the j th monomer on the m th chain. TheHamiltonian H describing the monomer interactions isthen expressed as H/k B T = N R g n c X m =1 N X j =1 ( R m,j − R m,j − ) + ρ χ Z d r ˆ φ A ( r ) ˆ φ B ( r )+ ρ κ Z d r (cid:16) ˆ φ A ( r ) + ˆ φ B ( r ) − (cid:17) , (28)where the first term represents the bonded interactionsin the polymer, and the last two terms correspondto non-bonded interactions. The quantities ˆ φ α ( r ) arethe normalized microscopic densities of α -type beads( α = A or B ) at position r , defined as, ˆ φ α ( r ) = ρ P mj δ ( r − R mj ) δ α,τ mj , where τ mj = A or B char-acterizes the monomer sequence on chain m .In practice, the local densities are evaluated on a gridwith grid size ∆ x = ∆ y = ∆ z = 0 . R g , using a firstorder cloud in the cell (CIC) scheme . The grid sizeis an important ingredient of the model definition, as itsets the range of non-bonded interactions. In the simu-lations, we consider systems with average monomer den-sity ρ = · /R g , i.e., roughly 50 monomers per gridcell. For this choice of densities and grid parameters,grid artefacts are negligible, and the renormalizedvalues of χ and κ in the SCF theory are practicallyidentical to the corresponding ”bare” parameters in Eq.(28) ). Furthermore, fluctuation effects are small. Thestrength of thermal fluctuations can be characterized bythe Ginzburg parameter , C − = V /n c R g . In oursystem, this parameter is C − = 0 . or less.Monomers ( m, j ) with mass M m,j evolve in time ac-cording to a Langevin equation, M m,j ˙ v m,j ( t ) = − ∂H∂ R m,j − Γ v m,j + p k B T f m,j ( t ) . (29)The first term on the right hand side describes the con-servative interaction forces, the second term correspondsto a friction force (with v = d R /dt and the monomer fric-tion Γ = 1 /D ), and the last term to a stochastic forcerepresenting the effect of thermal fluctuations, where f mj ( t ) is a Gaussian distributed random noise with zeromean and variance h f mj ( t ) f nk ( t ′ ) i = δ mn δ jk δ ( t − t ′ ) .Hydrodynamic interactions are thus neglected, and sincethe interaction potentials defined by Eq. (28) are soft, en-tanglement effects are not included as well. We considerthe two cases M m,j ≡ k B T t /R g (inertial dynamics),and M m,j → (overdamped dynamics). In the secondcase, Eq. (29) is replaced by d R m,j dt = − D ∂H∂ R m,j + p D k B T f m,j ( t ) . (30)The equations of motion are integrated using theVelocity-Verlet scheme in the case of inertial dy-namics (Eq. (29)), and the Euler-Maruyama algorithmin the case of overdamped dynamics (Eq. (30)) with thetime step δt = 0 . t . Specifically, we consider copolymer melts in a simula-tion box of size R g × R g × R g . Unless stated otherwise,we consider symmetric copolymers, i.e., N A = N B , withtotal length N = 40 . For comparison, we also studycopolymers with length N = 20 or N = 100 , and varythe A:B fraction. In all cases the monomer density iskept fixed at ρ = · /R g . The Helfand parameter isset to κN = 100 . The systems are initially prepared bygrowing polymers at randomly picked points in the sim-ulation box. In three independent runs, configurationsare then equilibrated for 300000 time steps each. Datafor g ( q , t ) are subsequently collected over 200000 timesteps and used to extract the mobility functions. In aset of additional simulations, we monitor the formationof lamellar structure in the melt after a step change from χN = 0 to a finite χN above the ODT, and the decayof the lamellar structure after a step change from finite χN to χN = 0 . The systems are equilibrated as de-scribed above and the time evolution is then monitoredover 100000 time steps in 10 independent runs.
2. SCF free energy functional
As discussed earlier, we use the SCF theory to con-struct the free energy functional in our DDFT equations.The SCF theory is one of the most powerful equilibriumtheories for inhomogeneous polymer systems and hasbeen well documented elsewhere . Here, we justbriefly summarize the main equations, adjusted to oursystem. We model the copolymers as continuous Gaus-sian chains , and parameterize the contour lengthby a continuous variable s ∈ [0 : 1] . The free energyfunctional F [ { φ α ( r ) } ] of our block copolymer system isexpressed as F/k B T = ρ N (cid:26) Z d r h χN φ A ( r ) φ B ( r ) (31) + κN ( φ A ( r ) + φ B ( r ) − i − X α = A,B Z d r φ α ( r ) ω α ( r ) − V ln Q (cid:27) , where φ α is the normalized density field of monomers oftype α , ω α the corresponding conjugate field, and Q isthe single chain partition function in the external field ω α . The conjugate fields are determined implicitly bythe requirement φ A ( r ) = VQ Z N A /N d s q f ( r , s ) q b ( r , − s ) ,φ B ( r ) = VQ Z N B /N d s q b ( r , s ) q f ( r , − s ) . (32)Here q f ( r , s ) and q b ( r , s ) are the end-integrated forwardand backward chain propagators, respectively, which canbe obtained from solving the following differential equa-tion: ∂q ( r , s ) ∂s = R g ∇ q ( r , s ) − ω ( r ) q ( r , s ) (33)with initial condition q f,b ( r ,
0) = 1 and ω ( r ) = ω A ( r ) or ω B ( r ) , depending on s : q f ( r , s ) is obtained by setting ω ( r ) = ω A ( r ) for s < N A /N and ω ( r ) = ω B ( r ) other-wise, and q b ( r , s ) by setting ω ( r ) = ω B ( r ) for s < N B /N and ω ( r ) = ω A ( r ) otherwise. Knowing q f or q b , one cancalculate the single chain partition function Q via Q = 1 V Z d r q f ( r ,
1) = 1 V Z d r q b ( r , (34)At equilibrium, F [ { φ α ( r ) } ] assumes a minimum withrespect to φ α ( r ) , leading to a second set of conditionsfor the values of the conjugate fields, ω α : ω SCF A ( r ) = χN φ B + 2 κN ( φ A + φ B − ω SCF B ( r ) = χN φ A + 2 κN ( φ A + φ B − (35)However, in DDFT calculations, these conditions arenot imposed. Instead, the system is dynamically driventowards the equilibrium state via the diffusive dynamicalequation (3) with ˆ µ α ( r ) = ( ω SCF α ( r ) − ω α ( r )) .The SCF and DDFT calculations in the present workare effectively one-dimensional, i.e., we assume that den-sities vary only in the z direction. Space is discretizedwith grid size ∆ z = 0 . R g The propagator equation,Eq. (33) is solved using the pseudo spectral scheme with discretization ∆ s = 0 . . As in our earlier work ,the time step in the DDFT calculations depends on theDDFT scheme: We use ∆ t = 10 − t N for DDFT cal-culations based on Debye dynamics or any of the otherpre-determined mobility functions ˆΛ( r − r ′ ) discussed inSec. III B, ∆ t = 10 − t N for full chain dynamics, Eq.(5), and ∆ t = 10 − t N for local dynamics (4) or mixeddynamics (11). B. Mobility functions
Based on simulations of the fine-grained model dis-cussed above, mobility functions were extracted from thesimulation data using the different variants of the relax-ation time approaches discussed in Section III B. In thefollowing, we will consider melts of symmetric A:B di-block copolymer melts.Fig. 2 shows the results for the full-chain mobilityfunction, ˆΛ( q ) = P αβ ˆΛ αβ ( q ) for different chain lengths( N = 20 , , ) at fixed χ = 0 , and for different valuesof χN ( χN = 0 , , ) at fixed chain length N = 40 .These values were chosen such that ( χN ) is still be-low the value ( χN ) ODT ≈ . where the order-disorder transition sets in for symmetric diblock copoly-mers, hence the melt is disordered and isotropic. The be-havior of ˆΛ( q ) at q → reflects the translational diffusionof chains and takes the asymptotic value ˆΛ = D c . There-fore, the curves are rescaled by the chain diffusion con-stant D c , which has been calculated independently from q [1/R g ] Λ ^ τ R ( q ) [ D c / k B T ] N=20N=40N=100Theory q [1/R g ] Λ ^ τ R ( q ) [ D c / k B T ] N=20N=40N=100 q [1/R g ] Λ ^ τ R ( q ) [ D c / k B T ] χ N=0 χ N=5 χ N=10 q [1/R g ] Λ ^ τ R ( q ) [ D c / k B T ] χ N=0 χ N=5 χ N=10Brownian DynamicsBrownian Dynamics Inertial DynamicsInertial Dynamics χ N=0 χ N=0N=40 N=40 a) b)c) d)
Figure 2. Normalized full-chain mobility function of symmet-ric A:B copolymers in a melt, as obtained via the relaxationtime method (Eq. (21)) from Brownian dynamics simulations(a,c) and inertial dynamics simulation data (b,d) for differentchain lengths N or interaction parameter χ as indicated. Solidline in (a) shows theoretical prediction for N = 40 obtainedby inserting Eq. (A7) into Eq. (21). the mean-square displacement of the chain. For example,for N = 40 , we obtain D BDc = (0 . ± . R g /t inBrownian dynamics simulations, and D IDc = (0 . ± . R g /t in inertial dynamics simulations, which isclose to the value for free Rouse chains, D c = 0 . R g /t .Since the interactions between monomers are very soft inthe particle-based model, they do not affect the diffusionconstant significantly in the disordered phase.The full-chain mobility function is found to dependweakly on the chain length N (Fig. 2a,b), the effectsbeing most pronounced in the regime of high q : If oneincreases N , the mobility function for high q decreasesin the Brownian dynamics case and increases in the in-ertial dynamics case, such that both mobility functionsapproach each other. In contrast, the Flory Huggins pa-rameter χ has practically no influence on the chain mo-bility function in the disordered regime (Fig. 2c,d)). Mo-tivated by this finding, we will use the mobility functionsobtained at χ = 0 in all DDFT calculations below.Next we turn to the discussion of the monomer-speciesresolved mobility functions ˆΛ αβ . The results extractedfrom Brownian dynamics simulation trajectories for sym-metric diblock copolymers of length N = 40 are shownin Fig. 3 for the different relaxation time approaches dis-cussed in Sec. III B. Since ˆΛ AA ( q ) = ˆΛ BB ( q ) for symmet-ric systems, and ˆΛ AB ( q ) = ˆΛ BA ( q ) , only the results for ˆΛ AA ( q ) and ˆΛ AB ( q ) are shown.If one assumes that the mobility matrix ˆΛ is governedby a single relaxation time τ ( q ) (Eqs. (21) or (22)), theresulting mobility curves are qualitatively similar to thecurves obtained from the Debye approximation (6), ex-cept that ˆΛ αβ ( q ) is enhanced at high q values like the q [1/R g ] Λ ^ AA ( q ) [ D c / k B T ] Λ ^ Local Λ ^ Debye Λ ^ AA ( q ) [ D c / k B T ] Λ ^ τ R Λ ^ τ e Λ ^ T q [1/R g ] Λ ^ A B ( q ) [ D c / k B T ] Λ ^ Local Λ ^ Debye Λ ^ τ R Λ ^ τ e Λ ^ T a) b) Figure 3. Normalized mobility function ˆΛ αβ of symmet-ric A:B diblock copolymers (length N = 40 ), obtained fromBrownian dynamics simulations at χ = 0 , using different vari-ants of the relaxation time method: Eqs. (21) (light greenline), (22) (dark green line), and (24) (blue line). Also shownfor comparision is the result from the Debye approximation(red line) and the local approximation in a homogeneous melt(black). full-chain mobility function. However, if one derives ˆΛ from a full relaxation time matrix which is calculated ac-cording to Eq. (24), the mobility functions change quali-tatively. The intra-block mobility ˆΛ TAA ( q ) becomes muchlarger than in the other nonlocal schemes, especially atsmall q . Hence monomer rearrangements inside blocksare faster than anticipated in the Debye approximation.Nevertheless, ˆΛ TAA ( q ) never reaches the level of the localcoupling scheme, where monomers are taken to move in-dependently ( ˆΛ Local AA ( q ) = ˆΛ Local BB ( q ) ≡ . D c /k B T for sym-metric A:B copolymers in homogeneous melts accordingto Eq. (4)).In contrast, the inter-block mobility ˆΛ TAB ( q ) is muchsmaller than in the other nonlocal schemes already at q = 0 . It then decreases further with increasing q andeven becomes slightly negative, until it rises again andreaches zero at large q . We note that the slightly negativevalues of ˆΛ AB ( q ) do not destabilize the system, since theEigenvalues of Λ( q ) are still positive. The inter-blockmobility is practically zero for q values above qR g ≈ .The same is obtained with a local approximation, wherethe motion of A and B monomers is also uncorrelated.An important consequence is that the values of ˆΛ AA ( q ) and ˆΛ AB ( q ) at q → differ from each other in the relax-ation time matrix scheme ˆΛ T (Eq. (24), whereas they areequal in the other nonlocal schemes. This influences theprediction for the relaxation of composition fluctuations m ( t ) = (Φ A ( t ) − Φ B ( t )) / . From Eq. (3), one can derive ∂ t m ( q , t ) = − q (cid:0) ˆΛ AA ( q ) − ˆΛ AB ( q ) (cid:1) ˆ µ ( q , t ) , (36)where ˆ µ = (ˆ µ A − ˆ µ B ) is conjugate to m . If m ( t ) issmall, one can apply the random phase approximation(RPA) and approximate ˆ µ ( q , t ) ≈ Γ ( q ) m ( q , t ) ,where the RPA-coefficient Γ ( q ) can be identified withthe inverse of the collective structure factor of the copoly-mer melt. Expanding Γ ( q ) in powers of q and neglect-ing compressibility effects, one obtains to leading order z [R g ] φ A , M a x t = 0 t t = 2 t t = 4 t t = 10 t t = 20 t z [R g ] φ A , M a x t = 0 t t = 2 t t = 4 t t = 10 t t = 20 t DDFT: Λ ^ T BD Simulation a) b) ( χ N) init =15( χ N) init =15N=40 N=40 Figure 4. Evolution of density profile of A-monomers after asudden change from ( χN ) init = 15 to χN = 0 at t = 0 , ac-cording to (a) Brownian dynamics simulations and (b) DDFTcalculations based on the relaxation time method, Eq. (24). Γ ( q ) ≈ k B T /q R g for symmetric diblock copolymers.At small q , Eq. (36) thus takes the limiting form ∂ t m ( q , t ) ≈ − k B TR g (cid:0) ˆΛ AA ( q ) − ˆΛ AB ( q ) (cid:1) m ( q , t ) . (37)Since (cid:0) ˆΛ AA (0) − ˆΛ AB (0) (cid:1) > in the relaxation time ma-trix scheme, composition fluctuations are predicted to de-cay with a finite relaxation time in the limit q → . In theother nonlocal schemes, one has (cid:0) ˆΛ AA (0) − ˆΛ AB (0) (cid:1) = 0 at q → , i.e., the relaxation time for long-wavelengthcompositional fluctuations is predicted to diverge. Insimulation studies , the relaxation time is found tobe finite and of order (2 /π ) R g /D c (the Rouse time ofthe chain), implying (ˆΛ AA (0) − ˆΛ AB (0)) ≈ . D c /k B T .This is consistent with the data in Fig. 3 obtained withthe relaxation time matrix method. C. Comparison of DDFT calculations withsimulations
In order to evaluate the mobility functions discussedin the previous section, we have compared DDFT calcu-lations with fine-grained simulations for different situa-tions of dynamical ordering/disordering in block copoly-mer melts. In the following, we report the results forBrownian dynamics simulations. The results for inertialdynamics simulations are similar.
1. Relaxation of an initially lamellar symmetric diblockcopolymer melt into the homogeneous state
In the first example, we study the relaxation of an ini-tially lamellar block copolymer melt into a homogeneousstate. Diblock copolymer melts were prepared in a lamel-lar state by equilibrating them above the order-disordertransition, i.e., at ( χN ) init > ( χN ) ODT . Then, startingfrom such a configuration, χ was turned off (to χ = 0 )at time t = 0 and the evolution of the profiles was mon-itored. Fig. 4 shows an example of a series of resulting0 t [t N] φ A , M a x t [t N] φ A , M a x Local Dyn.Full ChainMixed Dyn. Λ ^ Debye Λ ^ τ R Λ ^ τ e Λ ^ T Simulation ( χ N) init =15 ( χ N) init =20 a) b) Figure 5. Relaxation of the maximum in the density profile ofA-monomers for configurations that were initially equilibratedin an ordered phase at ( χN ) init = 15 (a) and ( χN ) init = 20 (b), after a sudden change to χ = 0 at t = 0 , for differentDDFT schemes as indicated, and compared to Brownian dy-namics simulations at N = 40 . The initial value φ A, Max ( t = 0) is the same in all calculations. density profiles for A monomers at different times, asmeasured in a Brownian dynamics simulation run (Fig.4 a)), and the corresponding results from DDFT calcu-lations based on the relaxation time matrix (Fig. 4 b)).The DDFT calculations are in excellent agreement withthe simulations.To further quantify the comparison, we plot in Fig.5 the maximum value of the profile Φ A ( z ) versus timefor systems that were initially prepared at ( χN ) init = 15 (Fig. 5 a)) and ( χN ) init = 20 (Fig. 5 b)). Symbols showthe simulation results, averaged over ten independentruns, and, lines the results from different DDFT calcula-tions. We find that DDFT calculations based on a chaincoupling assumption (i.e., full chain dynamics, Eq. (5) orDebye dynamics ˆΛ Debye , Eq. (6)), consistently underesti-mate the speed of the relaxation process. DDFT schemeswith mobility functions ˆΛ τ that were extracted assuminga single relaxation time τ ( q ) (i.e., Eqs. (21) and (22))perform better, but the dynamics is still too slow. Thecurves calculated with the ”mixed coupling” scheme ,Eq. (11), are close by and also too slow. DDFT cal-culations based on a local coupling assumption overesti-mate the relaxation speed. In contrast, the predictions ofDDFT calculations based on the relaxation time matrix,i.e., on ˆΛ T (Eq. (24)), are in excellent agreement withthe simulation data.
2. Ordering kinetics in a symmetric diblock copolymer melt
In our second example, we study the dynamics of struc-ture formation in the block copolymer melt after a suddenquench from χN = 0 to some value ( χN ) > ( χN ) ODT .An example for the time evolution of an A-density profileobtained from a Brownian dynamics simulation run andcompared to DDFT calculations based on the relaxationtime matrix is shown in Fig. 6. In both cases, the initialdensity profile is exactly the same, i.e., small density fluc-tuations in the simulation profile were also transferred tothe initial configuration in the DDFT calculation. Nev- z [R g ] φ A , M a x t = 0 t t = 10 t t = 20 t t = 40 t t = 80 t t = 90 t z [R g ] φ A , M a x t = 0 t t = 10 t t = 20 t t = 40 t t = 80 t t = 90 t DDFT: Λ ^ T Simulation a) b) χ N=15 N=40 χ N=15 N=40
Figure 6. Evolution of density profile of A-monomers froman initially disordered conformation after the monomer in-teraction is suddenly raised from χN = 0 to χN = 15 at t = 0 , according to (a) Brownian dynamics simulations and(b) DDFT calculations with mobility function based on Eq.(24). ertheless, the agreement between simulations and DDFTcalculations is less impressive than in the relaxation case,Fig. 4. First, the location of the density maxima differs.This can be explained from the fact that the maximaemerge spontaneously at random positions in both cases.Second, the melt seems to order faster in the simulationsthan in the DDFT simulations. At the time t = 40 t after the quench, the amplitude of the oscillation in theA-density profile has almost saturated in the simulations,whereas it has only reached about one fourth of the finalvalue in the DDFT calculations.On the other hand, looking at the simulations, onenotices that the ordering time also differs between differ-ent simulation runs. Fig. 7 shows results for the maxi-mum value of the A-monomer density profile as a functionof time for ten different independent simulations, whichall started from exactly the same initial configuration at t = 0 . In every run, the lamellar ordering sets in at a dif-ferent time (Fig. 7 a)). However, if one aligns the curves,i.e., adds a time offset such that they coincide at halfmaximum, their slopes fall largely on top of each other:The statistical spread of the onset of the ordering is much t [t N] φ A , M a x t [t N] φ A , M a x a) b) χ N=15.0 χ N=15.0
Figure 7. (a) Original and (b) aligned curves for the time evo-lution of the maximum in the spatial density of A-monomerafter a sudden quench from χN = 0 to χN = 15 at t = 0 from ten different Brownian dynamics simulation runs. -0,5 0 0,5 1 1,5 2 ∆ t [t N] φ A , M a x Local Dyn.Full ChainMixed Dyn. Λ ^ Debye Λ ^ τ R Λ ^ τ e Λ ^ T t [t N] φ A , M a x Simulation ∆ t [t N] φ A , M a x Local Dyn.Full ChainMixed Dyn. Λ ^ Debye Λ ^ τ R Λ ^ τ e Λ ^ T t [t N] φ A , M a x Simulation χ N=15a) b)c) d) χ N=20
Figure 8. (a,c): Time evolution of the maximum density ofA-monomers after a sudden quench from χN = 0 to when χN = 15 . (a) and χN = 20 . (c), according to Browniandynamics simulations (symbols) and different DDFT schemes(lines) as indicated. The initial density profile in z directionis the same in all calculations. Grey shades indicate spreadof simulation curves (see Fig. 7). (b,d): Same curves, alignedin time t . larger than the statistical noise after the ordering has setin. In the following, we therefore not only compare thekinetics of ordering on an absolute time scale, but alsothe shape of the curves after they have been aligned.Fig. 8 shows the corresponding results for quenches to χN = 15 (Fig. 8 a,b), and to χN = 20 (Fig. 8 c,d), com-pared to a DDFT predictions from the different schemesdiscussed above. As reported in our earlier work , andconsistent with our observations for the relaxation kinet-ics, Fig. 5, DDFT calculations based on local dynamics(Eq. (4, black line) underestimate the ordering time, andDDFT calculations based on global chain dynamics (fullchain dynamics (5) or Debye dynamics (6, red lines) over-estimate it. Using DDFT mobilities that were extractedfrom bulk simulations assuming a single relaxation time,(Eqs. (21) or (22), green lines), the predicted ordering isfaster than in the case of Debye dynamics, but still tooslow.The best results are again obtained with the DDFTscheme ˆΛ T based on the relaxation time matrix, Eq. (24).The ordering in the DDFT calculations sets in later thanin the simulations, but once started, the dynamics ofordering is comparable. The delayed onset may be ex-plained by the role of thermal fluctuations in initiatingthe ordering process. The DDFT calculations are purelydeterministic and do not include fluctuations. Since theinitial configurations are chosen identical to the simu-lated configurations, they include some noise, and thatnoise has the correct amplitude. As we have shown inearlier work , the ordering would have been further de- t [t N] φ A , M a x q [1/R g ] Λ ^ T ( q ) [ D c / k B T ] Λ ^ TAA Λ ^ TAB -0.5 0 0.5 ∆ t [t N] φ A , M a x SimulationLocal Dyn. Λ ^ Debye Λ ^ T t [t N] φ A , M a x SimulationLocal Dyn. Λ ^ Debye Λ ^ T q [1/R g ] Λ ^ T ( q ) [ D c / k B T ] Λ ^ TBA Λ ^ TBB χ N=20 a) b)c) d) ( χ N) init =20f A =0.6 f A =0.6 Figure 9. (a) Normalized mobility function ˆΛ Tαβ of asymmet-ric A:B diblock copolymers with block ratio 6:4 and length N = 40 obtained from simulations at χN = 0 with therelaxation time matrix approach (Eq. (24)). (b-d) Corre-sponding time evolution of the maximum spatial density ofA-monomers (b) after suddenly switching from χN = 20 to χN = 0 (c,d) and from χN = 0 to χN = 20 at t = 0 in abso-lute time (c) and aligned in time (d). Symbols correspond toBrownian Dynamics simulations with chain length N = 40 ,lines to results from DDFT calculations as indicated. Greyshades indicates spread of simulation curves from 10 indepen-dent configurations with identical starting configuration. layed in all DDFT schemes if the initial noise level hadbeen chosen lower. Nevertheless, adding noise to the ini-tial configuration of a deterministic DDFT calculation isapparently not sufficient, if one wishes to faithfully re-produce the onset of ordering. To improve on this, onewould have to include thermal noise in the DDFT equa-tions (see Sec. V). Once initiated, the ordering proceedsin a deterministic manner and is very well captured bythe DDFT calculations based on ˆΛ T (Fig. 8 b,d, blueline).The results from "mixed dynamics" calculations (Eq.(11), cyan line) are also in very good agreement with thesimulation data. However, it should be noted that thisscheme has been postulated heuristically, without anymicroscopic justification, and it has one free parameter(the parameter σ in Eq. (9)) which has been optimizedfor this specific ordering situation in our earlier work .In contrast, the mobility functions in the relaxation timescheme were determined from independent bulk simula-tions without any adjustable parameter. Also, from apractical point of view, mixed dynamics calculations havethe disadvantage that they require smaller time steps.2
3. Asymmetric diblock copolymer melt
So far, we have evaluated our different DDFT schemesby examining systems of symmetric diblock copolymermelts. To test whether the results depend on the sym-metry of the system, we have repeated the analysis fora different A:B block fraction. The results are shown inFig. 9. We consider the same two situations as above:One where an initially lamellar morphology (set up inthe ordered phase at ( χN ) init = 20 ) relaxes into a ho-mogeneous structure after turning χ off, and one wherean initially disordered melt develops lamellar order afterperforming a quench into the ordered phase at χN = 20 .The results are essentially the same as in the symmet-ric case: When using DDFT with ”local dynamics”, thedynamics is too fast, when using global chain dynamics(Debye dynamics), it is too slow. When using the re-laxation time matrix approach, the onset of ordering isslightly delayed in the DDFT calculations compared tosimulations, but the actual ordering kinetics (the shapeof the curves) is in very good agreement with the simu-lation data. V. DISCUSSION AND SUMMARY
The purpose of the present work was to develop sys-tematic bottom-up coarse-graining strategies for con-structing nonlocal mobility functions ˆΛ( q ) in DDFTmodels for polymeric systems. The goal was to extractthese mobility functions from trajectories of fine-grained,microscopic simulations. We have explored two physi-cally motivated approaches.The first was based on the Green-Kubo formalism.However, the Green-Kubo integrals were found to alwaysvanish except at q = 0 , due to the fact that the cor-responding stationary current cannot exist at q = 0 . Itwas not even possible to identify a well-defined plateau inthe running Green-Kubo integrals. Español et al haverecently discussed such ”plateau problems” and proposedan alternative approach to evaluating Green-Kubo trans-port coefficients: They suggested to analyze the late-timebehavior of quantities − (cid:16) dd t C ( t ) (cid:17) C − ( t ) , where C ( t ) isthe time-dependent correlation function of the quantitiesof interest. In our case, the relevant correlation func-tion is the single chain structure factor, g ( q , t ) . Insert-ing Eq. (17) yields ˆΛ( q ) ∝ − q ( ∂ t g ( q , t )) g − ( q , t ) g ( q , .Since the long-time behavior of g ( q , t ) is dominated bythe diffusive behavior of whole chains, one has g ( q , t ) ∝ exp( − D c q t ) at t → ∞ and hence gets Λ( q ) ∝ D c g ( q , ,which corresponds to Debye dynamics. Thus the re-sulting DDFT model is a ”chain coupling” model wherechains move as a whole.In practice, however, we are interested in local order-ing processes with characteristic time scales that are typ-ically smaller than the diffusive time. Therefore, we have explored a second scheme, where a characteristic relax-ation time matrix is first determined independently foreach q -vector from fine-grained simulations, and this isthen used to derive a q -dependent mobility matrix. Asone can see from Figs. 1 and 3, the resulting mobilityfunctions are intermediate between ”chain coupling dy-namics” (chains move as a whole) and ”local couplingdynamics” (monomers move independently). We havetested the approach by examining two kinetic processes inblock copolymer melts: The process of disordering froman initially lamellar phase and the process of orderingafter a quench into the lamellar phase. Comparing theDDFT calculations with the simulation results, we con-clude that our new scheme is capable of describing theordering/disordering kinetics at a quantitative level. Al-though we applied our model to study the order/disorderkinetics of lamellar structures only, the method can beapplied to other morphologies as well (e.g. spheres, cylin-ders etc.).We should note that, although the kinetics of orderingand disordering are well-captured by the DDFT model,the onset of ordering is later than it should be, comparedto simulations. We attribute this to the effect of thermalfluctuations, which are omitted in our DDFT calcula-tions. They could be included by adding thermal noiseto the density currents , i.e., replace Eq. (2) by ∂ t ρ α = ∇ r n X β Z d r ′ Λ αβ ( r , r ′ ) ∇ r ′ µ β + j α (cid:9) , (38)where the stochastic current j ζ ( r , t ) is to a Gaussian ran-dom vector field with zero mean ( h j ζ ( r , t ) i = 0 ) and corre-lations according to the fluctuation-dissipation theorem: h j I α ( r , t ) j J β ( r ′ , t ′ ) i = 2 k B T δ ( t − t ′ ) Λ αβ ( r , r ′ ) δ IJ ( I, J are cartesian coordinates).It is worth recapitulating some of the approximationsand assumptions that are entering our coarse-grainingscheme.First, we have assumed that the dynamics of inhomo-geneous polymer systems can be described by an effectiveMarkovian model. To account for the multitude of differ-ent relaxation times in polymer systems, we have treatedthe mobility as an adjustable q -dependent function; how-ever, explicit memory effects were neglected. Wang etal have recently devised a dynamic RPA theory forpolymer systems with a frequency dependent Onsagercoefficient and showed that it successfully describes thedecay of composition fluctuations in diblock copolymermelts (similar to Fig. 4 here) and the onset of spinodaldecomposition in homopolymer mixtures. Their Ansatzcan easily be generalized to a dynamic SCF theory witha time-dependent memory kernel. It has the advantagethat it includes memory explicitly, and does not requiread hoc adjustments of ”effective” mobility functions. Onthe other hand, effective Markovian models are compu-tationally more efficient in many cases.Second, in Eq. (2), the mobility matrix describing thetime evolution of density fluctuations should really be3derived from the collective density correlations. Here,we have replaced them by a sum over intrachain densitycorrelations, in the spirit of a mean-field theory. Re-cently, Ghasimakbari and Morse have used the collec-tive structure factor to analyze the effective q -dependentdiffusive relaxation of compositional fluctuations in sym-metric diblock copolymer melts. They fitted the decayof the dynamic collective structure factor by a single ex-ponential. Their results in the regime ( χN ) < . arecomparable to ours in Fig. 2.Third, when deriving our final expression for Λ( q ) inEq. (16), we have linearized the free energy density func-tional and thus assumed that density variations are small.We determine the mobility function ˆΛ( q ) from simula-tions of a homogeneous bulk melt at χN = 0 , but thenuse them in DDFT calculations for inhomogeneous, or-dered systems. This is partly motivated by the find-ing that ˆΛ( q ) hardly depends on χ in the disorderedregime of a block copolymer melt. Nevertheless, at high χ and/or in strongly inhomogeneous systems, correctionsmust probably be applied.We have formulated our approach for diblock copoly-mer melts, but it can easily be generalized to mixtures.Starting from Eq. (2), one can simply replace the mo-bility function Λ αβ ≈ Λ ( s ) αβ ρ /N , by a sum over chainmobilities, i.e. Λ αβ ( r , r ′ ) = X γ N γ ¯ ρ ( γ ) ( r , r ′ ) Λ ( s,γ ) αβ ( r , r ′ ) (39)where the sum γ runs over chain types, N γ is the lengthof chains of type γ , ¯ ρ ( γ ) ( r , r ′ ) the locally averaged den-sity of monomers from chains of type γ (hence ¯ ρ γ /N γ is achain density), and Λ ( s,γ ) αβ the corresponding single chainmobility function. Note that the prescription for deter-mining the local average ¯ ρ ( γ ) ( r , r ′ ) must be symmetricwith respect to r and r ′ (e.g., ¯ ρ ( γ ) ( r , r ′ ) = ρ ( γ ) ( r + r ′ ) .In mixtures, the diffusion of chains of different typeadds another slow time scale to the dynamics of the sys-tem. In our previous work , we have compared the dy-namics of interdiffusion at A/B homopolymer interfacesfrom different DDFT calculations with simulations. Wefound that the results obtained with local and nonlocalDDFT coupling schemes were very similar, and all in verygood agreement with the simulations. We conclude thatstudies of homopolymer interdiffusion do not seem to bea very sensitive test of the quality of a DDFT model, andtherefore expect that the new schemes proposed here willalso perform well.Our bottom-up approach for constructing mobility ma-trices has been tested for Rouse chains, but it is not re-stricted to that. It only requires as input the single chaindynamic structure factors from simulations of the targetmicroscopic systems. In future work, we plan to studypolymer mixtures and melts in other dynamical regimes,e.g., entangled melts, or systems where hydrodynamicsare important.The DDFT theory relies on the assumption that the polymer system under consideration is only weakly dis-turbed from equilibrium. It assumes that the polymerconformations are close to local equilibrium at all timesand that the dynamic process under consideration is stillsuitably described in terms of a free energy landscapepicture. Therefore, it cannot be applied in situationsfar from equilibrium where the distribution of polymerconformations is distorted, such as, e.g., polymers undershear at high Weissenberg numbers which are stretchedout. Studying such systems with DDFT models requiresnovel approaches where not only the mobility functions,but also the density functionals themselves have to bereconsidered . However, DDFT theories that wereconstructed as proposed in the present paper can beused to study ordering processes and spontaneous self-assembly in inhomogeneous polymer mixtures, and thusto evaluate the role of processing and pathways for thefinal structures. Acknowledgements
We thank Marcus Müller for a critical reading of themanuscript and many useful comments. This researchwas supported by the German Science Foundation (DFG)via SFB TRR 146 (Grant number 233630050, projectC1). S.Q. acknowledges research support from the Na-tional Natural Science Foundation of China under theGrant NSFC-21873010. The simulations were carried outon the high performance computing center MOGON atJGU Mainz.
Appendix A: Evaluation of the Green-Kubo integral
In this appendix, we discuss the results from the eval-uation of the integral (13). In the spirit of mean-fieldtheory, we will assume that the mobility can be derivedfrom a single chain mobility,
Λ = ρ N Λ ( s ) , which is derivedfrom the current-current correlations of a single chain,i.e., the quantity I αβ ( q , t ) = N X k,j =1 e i q · ( R k ( t ) − R j (0)) ˙ R k ( t ) ˙ R j (0) γ ( α ) k γ ( β ) j . (A1)If interchain correlations can be neglected, one has h j α ( q , t ) j β ( − q , t ) i = n c I αβ ( q , t ) , where n c = V ρ N is thenumber of polymers in the system, and hence Λ ( s ) , GK αβ ( q ) = 1 k B T Z ∞ d t I αβ ( q , t ) : ˆqˆq . (A2)The full chain mobility (all monomers) is given by thesum Λ ( s ) ( q ) = P αβ Λ ( s ) αβ ( q ) .We first discuss the full chain mobility at q = 0 .Eq. (A1) then reduces to I (0 , t ) = P αβ I αβ (0 , t ) = P kj h ˙ R k ( t ) ˙ R j (0) i . After evaluating the average of ˆqˆq with respect to all possible directions ˆq , we recover thewell-known relation between the chain mobility and thevelocity autocorrelation function of the center of mass ofthe chain ( V ( t ) = N P k ˙ R k ( t ) ): Λ ( s ) , GK (0) = N k B T Z ∞ d t h V ( t ) V (0) i = D c N k B T . (A3)Here D c is the diffusion constant of the whole chain, andthe factor N accounts for the fact that Λ ( s ) describesthe response of monomer current (scaling with the num-ber N of monomers) to a thermodynamic force acting onmonomers (i.e., the total force again scales with N ).For q = 0 and t > , I αβ ( q , t ) can be derived from thesingle chain dynamic structure factor, defined as g αβ ( q , t ) = 1 N * N X k,j =1 e i q · ( R k ( t ) − R j (0)) γ ( α ) k γ ( β ) j + (A4)by taking the second derivative with respect to t : I αβ ( q , t ) : qq = − N d dt g αβ ( q , t ) . (A5)Putting everything together, we finally obtain the follow-ing Green-Kubo relation between the mobility functionand the single chain dynamic structure factor, Λ ( s ) , GK αβ ( q ) = Nk B T (cid:18) q lim t → (cid:20) ddt g αβ ( q, t ) (cid:21) (A6) + lim ǫ → Z ǫ d t I αβ ( q ) : ˆqˆq (cid:19) . This quantity can be measured in microscopic simula-tions. The second term in Eq. (A6) has to be addedexplicitly if the microscopic model evolves according tooverdamped Brownian dynamics, to account for the con-tribution of the delta-correlated stochastic white noise at t = 0 to Eq. (13).Fig. 10 shows simulation results for single chains in ahomogeneous melt from Brownian dynamics and inertialdynamics simulations (see Sec. IV A 1 for a detailed de-scription of the simulation models). Fig. 10 a,b) show re-sults for g ( q , t ) for qR g = 1 (a) and qR g = 4 (b) and com-pares them with an analytic result for ideal free Rouse chains , which is exact in the limit N → ∞ : g ( q, t ) = 1 N X ij exp (cid:20) − q D c t − | i − j | ( qR g ) N − qR g ) π N X p =1 p cos (cid:18) pπiN (cid:19) cos (cid:18) pπjN (cid:19)(cid:26) − exp (cid:18) − D c tp π R g (cid:19)(cid:27)(cid:21) . (A7)Here, the index p represents the p th Rouse mode, and theindices i, j represent the i th and j th beads on the poly-mer chain. The agreement with the Brownian dynamicssimulaton data is very good. Fig. 10 c,d) shows the cor-responding Green-Kubo mobility functions. Somewhatdisappointingly, they are found to be zero within the sta-tistical and systematic error. Deviations from zero canbe traced back to discretization artefacts when taking thederivative ddt g ( q, t ) numerically.In the case of overdamped Rouse homopolymers, wecan evaluate (A6) exactly, using the relation ddt ln g ( q, t ) = − g ( q, k B TN (A8) X kj h H kj exp( i q · ( R k − R j )) i : qq with the Rouse mobility matrix H kj = D δ kj . Thefirst term in (A6) yields q ddt g ( q, t ) (cid:12)(cid:12) t → = − D N k B T .The noise term contributes with k B T D N R ǫ d tδ ( t ) = D N k B T . Since these two terms cancel, the resultingGreen-Kubo mobility is zero, as suggested by the simu-lations. t [t ] g ( q , t ) / g ( q , ) t [t ] g ( q , t ) / g ( q , ) Eq. (20)Brownian DynamicsInertial Dynamics q [1/R g ] Λ ( s ) , GK ( q ) [ D c N / k B T ] Theory . ∆ t=0.08 t ∆ t=0.04 t ∆ t=0.02 t ∆ t=0.01 t q [1/R g ] Λ ( s ) , GK ( q ) [ D c N / k B T ] Theory . ∆ t=0.08 t ∆ t=0.04 t ∆ t=0.02 t ∆ t=0.01 t a)b) c)d) Inertial DynamicsBrownian Dynamics qR g =1qR g =4 Figure 10. 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