Dynamic coarse-graining of polymer systems using mobility functions
DDynamic coarse-graining of polymer systems usingmobility functions
B Li , K Daoulas , F Schmid Institut f¨ur Physik , Johannes Gutenberg-Universit¨at Mainz, 55099 Mainz,Germany Max-Planck Institut f¨ur Polymerforschung, Ackermannweg 10, 55128 Mainz,Germany
Abstract.
We propose a dynamic coarse-graining (CG) scheme for mappingheterogeneous polymer fluids onto extremely CG models in a dynamically consistentmanner. The idea is to use as target function for the mapping a wave-vector dependentmobility function derived from the single-chain dynamic structure factor, which iscalculated in the microscopic reference system. In previous work, we have shown thatdynamic density functional calculations based on this mobility function can accuratelyreproduce the order/disorder kinetics in polymer melts, thus it is a suitable startingpoint for dynamic mapping. To enable the mapping over a range of relevant wavevectors, we propose to modify the CG dynamics by introducing internal frictionparameters that slow down the CG monomer dynamics on local scales, withoutaffecting the static equilibrium structure of the system. We illustrate and discussthe method using the example of infinitely long linear Rouse polymers mapped ontoultrashort CG chains. We show that our method can be used to construct dynamicallyconsistent CG models for homopolymers with CG chain length N = 4, whereas forcopolymers, longer CG chain lengths are necessary. Keywords : polymer simulations; coarse-graining; dynamics; friction; dynamic structurefactor; dynamic density functional theory; mobility
Submitted to:
J. Phys.: Condens. Matter a r X i v : . [ c ond - m a t . s o f t ] M a r ynamic coarse-graining of polymer systems using mobility functions
1. Introduction
Mixing polymers of different types is a simple and inexpensive way to create novelmaterials[1, 2]. However, chemically different polymers usually do not mix well.Polymeric composite materials therefore tend to be heterogeneous on local scalesand filled with internal interfaces, which largely determine the resulting materialproperties[3]. The morphology of the materials depend on the history, i.e., the way theyhave been processed. Understanding the dynamics of polymer kinetics in inhomogeneousmaterials is thus crucial if one wants to understand and predict the structure andproperties of the resulting materials.Computer simulations are a powerful tool to study soft matter systems. Due to thelarge size of the polymers and the even larger typical length scales of the inhomogeneities,simulations in full atomistic details are usually not possible, and using coarse-grained(CG) models instead has a long and successful history[4]. In CG polymer models,monomers or groups of monomers are lumped into one ”bead” of simpler structure.Generic models offer insight into universal features, and specific models with parametersadjusted to concrete molecules are used for quantitative studies. Designing such specificCG models requires the development of mapping procedures that allow to derive theparameters of the CG models from the microscopic static and dynamic features of thetarget systems [5, 6, 7, 8, 9, 10, 11].With respect to the static properties of equilibrium systems, such methods areby now well-established. Various protocols have been proposed to derive effectivepotentials of coarse-grained models from microscopic simulations by analyzing localcorrelations or force distributions[9, 10]. In addition, established mesoscopic conceptssuch as the Flory Huggins χ -parameter[2], the statistical segment length[12], or theMaier-Saupe parameter[13, 14, 15] are used to map microscopic models (or experimentaldata) on continuum models and then back to extremely CG particle-based polymermodels[16, 17, 18]. In the latter case, the target quantity in the CG parameteroptimization is often the static structure factor, and polymer theories like the randomphase approximation (RPA) or the self-consistent field theory (SCF) help to establishthe connection between fine-grained and CG models[19, 20, 21].Motivated by these successes, similar efforts are made to design mapping and CGmethods for polymer dynamics. In the earliest and still very popular approach[22, 23],the CG model is simulated by standard molecular dynamics and a single time scale– e.g., the time scale of diffusion – is used to mapped the CG system onto the fine-grained system. However, it has been long known through the work of Mori andZwanzig[24, 25, 26], that coarse-graining has a much more fundamental effect on thestructure of the equations of motion: Integrating out degrees of freedom invariably turnsa Hamiltonian system into a dissipative system with memory. Based on this insight,several recent efforts have been devoted to deriving generalized Langevin (GLE) modelsfor polymer melts and solutions, using as target quantities for the mapping the (absoluteor relative) velocity autocorrelation function of the center of mass[27, 28, 29, 30, 31] ynamic coarse-graining of polymer systems using mobility functions ? ]. However, it is not clearwhether this approach will also work for large molecules, where internal chain motionis a significant source of memory and friction. An alternative route that is closer tothe static coarse-graining strategies developed for mesoscopic scales would be to usethe dynamic structure factor as a starting point for mapping. Such a strategy will beexplored in the present paper.We target systems containing polymers of large molecular weight, i.e., made ofthousands of monomers, and CG strategies that map these molecules onto muchshorter chains of soft blobs. The dynamics of such polymers is theoretically describedas an overdamped motion in a background medium created by the other polymers,e.g., Rouse, Zimm, or reptation dynamics[12]. Successful static CG strategies arebased on ”theoretically informed” soft potentials that are derived from static densityfunctionals[36, 37, 38, 39, 40, 41, 42, 43, 44, 17] and reproduce key quantities such as the χ parameter. Here we take a similar approach, but use as mesoscopic reference theorythe overdamped dynamic density functional theory (DDFT). The standard Ansatz ofsuch a DDFT equation for polymers has the form[45, 46, 47, 48, 49, 50, 51] ∂ρ α ( r , t ) ∂t = (cid:88) β ∇ r (cid:20)(cid:90) d r (cid:48) Λ αβ ( r , r (cid:48) ) ∇ r (cid:48) µ β ( r (cid:48) , t ) (cid:21) . (1)Here ρ α ( r , t ) is the local density of component α at the position r , the quantity ∇ r (cid:48) µ β ( r (cid:48) , t ) is the thermodynamic force acting on component β at position r (cid:48) , andΛ αβ ( r , r (cid:48) ) a non-local mobility function that accounts, e.g., for chain connectivityeffects. Obviously, Eq. (1) is Markovian and does not include memory effects. Moregeneral versions of (1) that includes a memory kernel K ( r , r (cid:48) , t ) have been proposed bySemenov[52] and more recently by M¨uller and coworkers[53, 54]. Eq. (1) represents aMarkovian approximation to the full GLE which accounts for different relaxation timeson different length scales in an effective manner. We will discuss this in more detail inthe next section.In previous work, we have devised a way to extract mobility functions in polymersystems in a bottom-up fashion from fine-grained simulations, using as input data thesingle chain dynamics structure factor, g ( q , t ) = N (cid:104) (cid:80) Nn,m =1 e i q · ( R n ( t ) − R m (0)) (cid:105) , where thesum n, m runs over all N monomers of the chain, and R n ( t ) is the position of monomer n at time t . Knowing g ( q , t ), one can calculate the rescaled single-chain mobility[55] inFourier space asˆΛ( q ) = 1 k B T N g ( q , G − ( q ) g ( q ,
0) with G ( q ) = q N (cid:90) ∞ d t g ( q , t )(2) ynamic coarse-graining of polymer systems using mobility functions α (length N α ) in the numberconcentration c α , the total mobility function is then given by[55]Λ αβ = δ αβ (cid:88) α c α N α ˆΛ ( α ) (3)The generalization to block copolymers is straightforward [55, 56]. In our previous work,we have shown that a DDFT (1) based on this approach can accurately reproduce thekinetic evolution of block copolymer melts after sudden changes of the χ -parameter,when compared to fine-grained reference simulations.These successes suggest that the mobility functions ˆΛ( q ) should be a suitable targetfor dynamic CG schemes that map fine-grained models to particle-based CG models.In the present paper, we will investigate this possibility. We will show that a naive”mapping” based on matching a single time scale fails to reproduce the kinetics onboth local and polymeric length scales. This can partly be remedied by modifying theinternal polymer dynamics in the CG model. We will present a simple approach to doso and discuss its limitations.The remainder of the paper is organized as follows: In the next section, we brieflydiscuss the background of the method. We first introduce the mobility function in somemore detail, and then discuss finite chain length effects and the ensuing problems withsimple time mapping. In Section 3, we propose a method to modify the CG dynamicswhithout affecting the static properties of the systems and show results for an extremelycoarse-grained polymer. We close with a brief summary in Section 4.
2. Background
To set the frame, we begin with a brief derivation of Eq. (2). It follows the spirit ofthe derivation presented in Ref. [55], but specifically highlights the relation between themobility function and the corresponding single-chain memory kernel. For simplicity, weagain consider homopolymers.We make two important assumptions. First, we assume that we can determinethe mobility function in a homogeneous reference system (i.e., it is transferable toinhomogeneous systems), and second we take a mean-field approach. We consider atagged polymer that moves in the average background potential provided by the otherchains of the reference system. Since the reference system is homogeneous, we can writea generalized DDFT equation for the monomers of the tagged polymer as follows: ∂ρ ( s ) ( r , t ) ∂t = ∇ r (cid:90) d r (cid:48) (cid:90) t −∞ d s K ( s ) ( r − r (cid:48) , t − s ) ∇ r (cid:48) µ ( s ) ( r (cid:48) , s ) , (4)which in Fourier space reads ∂ρ ( s ) ( q , t ) ∂t = − q (cid:90) t −∞ d s K ( s ) ( q , t − s ) µ ( s ) ( q , s ) , (5) ynamic coarse-graining of polymer systems using mobility functions f ( q ) = (cid:82) d r e i q · r f ( r ) for the Fourier transform), where µ ( s ) ( q ) = V δF ( s ) /δρ ( s ) ( − q ) is derived from the free energy F ( s ) of the single tagged chain system.Eqs. (4) account for memory effects via the single-chain memory kernel K ( s ) ( τ ). Theydo not include corresponding correlated stochastic currents, but these could be addedeasily and would drop out in the next step of the derivation.The single chain structure factor is then given by g ( q , t ) = N (cid:104) ρ ( s ) ( q , t ) ρ ( s ) ( − q , (cid:105) ,where (cid:104)·(cid:105) denotes the thermal average over chain configurations. This results in thefollowing equation for g ( q , t ): ∂g ( q , t ) ∂t = − q N (cid:90) t −∞ d s K ( s ) ( q , t − s ) (cid:104) µ ( s ) ( q , s ) ρ ( s ) ( − q , (cid:105) . (6)To calculate µ ( s ) , we linearize the tagged chain free energy F ( s ) and expand it in powersof the tagged monomer density ρ ( s ) , ‡ F ( s ) = const. + k B T N (cid:88) q ρ ( s ) ( − q ) g − ( q , ρ ( s ) ( q ) + · · · (7)By truncating this equation at the second order, we implicitly assume that the chainconformations stay close to equilibrium and are not strongly distorted. Taking thederivative § , µ ( s ) ( q , t ) = k B T VN g − ( q , ρ ( s ) ( q , t ), and inserting it in Eq. (6), we obtain ∂g ( q , t ) ∂t = − q k B T VN (cid:90) t −∞ d s K ( s ) ( q , t − s ) g − ( q , g ( q , s ) . (8)Next we carry out a one-sided Fourier transform in the time domain iω ˜ g ( q , ω ) − g ( q ,
0) = − q k B T VN ˜ K ( s ) ( q , ω ) ˜ g − ( q ,
0) ˜ g ( q , ω ) , (9)which finally allows to calculate ˜ K ( s ) ( q , ω ) as˜ K ( s ) ( q , ω ) = Nk B T V q ( g ( q , − iω ˜ g ( q , ω )) ˜ g − ( q , ω ) g ( q , . (10)We emphasize that K ( s ) ( q , τ ) represents a single-chain memory kernel, which describesthe self-diffusion of the tagged chain. Wang et al[53] have recently calculated the collective memory kernel for incompressible block copolymer melts within the randomphase approximation and obtained a different expression, which is related to thecollective structure factor. For the purpose of dynamical mapping, it is more convenientto use the single chain structure factor as target quantity, since it can be accessedmore easily over the whole range of q vectors even from fine-grained simulations of verysmall systems. A second advantage is that the single-chain structure factor is muchless affected by dynamic slowdown close to phase transitions, which may occur due toslow collective critical or near-critical fluctuations [ ? ]. This makes it easier to justifythe Markovian approximation described below. ‡ In Ref. [55], the corresponding equation, Eq. (16), contains an additional erroneous factor 1 /V . § In Ref. [55] (before Eq. (17), the factor V is missing. ynamic coarse-graining of polymer systems using mobility functions K ( s ) ( q , τ ) by K ( s ) ( q , τ ) ≈ Λ ( s ) ( q )2 δ ( τ ), where the single-chain mobility is the integral over the memory kernelΛ ( s ) ( q ) = (cid:90) ∞ d τ K ( s ) ( q , τ ) = ˜ K ( s ) ( q ,
0) (11)Inserting Eq. (10), identifying G ( q ) = q N ˜ g ( q ,
0) and rescaling (cid:107) via ˆΛ = Λ ( s ) VN ,we recover Eq. (2). Within the Markovian approximation, g ( q , t ) decays exponentially(see Eq. (6)): The multiple relaxation times contributing to the memory kernel arereplaced by one effective relaxation time, which is, however, a function of q . Via this q -dependence, one still accounts, to some extent, for the spectrum of characteristicrelaxation modes in polymers. As we have seen in our previous work[55], this seems tobe sufficient to reproduce the ordering/disordering kinetics in melts at a quantitativelevel.The mobility function ˆΛ( q ) can thus be used to characterize the polymer dynamicsin a fine-grained system. Based on this insight, we propose to use it as target functionfor a dynamically consistent mapping of fine-grained systems onto CG systems. As weshall see in the next subsection, such a mapping is far from trivial. In a previous publication[56], we have derived an expression for the single-chain mobilityfunction of ideal infinitely long chains in the Rouse regime. The result was lengthy andshall not be repeated here. However, simple expressions were obtained for the limitingcases of very small or very large length scales. For homopolymers, we get qR G → ∞ : ˆΛ( q ) → D c k B T · .
279 (12) qR G → q ) ≈ D c k B T · (1 − ( qR G ) , (13)where R G is the radius of gyration, and D c the diffusion constant of the chain.In CG polymer models, one represents polymers by relatively short, possibly veryshort chains. This turns out to have a significant impact on the mobility function. Toinvestigate the chain length effects, we have carried computer simulations of spring-beadchains with harmonic bond potentials and different numbers of beads. Apart from beingconnected by bonds, monomers do not interact with each other. They move accordingto overdamped Brownian dynamics equations with a monomer friction constant ζ . Todetermine the mobility functions from the simulation data, we first determine the singlechain structure factor g ( q , t ) from the simulation trajectories and then evaluate theintegral G ( q ) and finally ˆΛ according to Eq. (2), applying an extrapolation procedureas described in Ref. [55] if necessary. The results for different chain lengths are presentedin Fig. 1. To normalize the data, the mobilities are divided by the respective polymer (cid:107) In the corresponding expressions in Ref. [55] (after (14) and before (19)) a factor V is missing. ynamic coarse-graining of polymer systems using mobility functions qR g Λ ^ [ D c / k B T ] N = 4N = 10N = 20N = 40Theory (Rouse chain)
10 20 qa
10 20
N = 4N = 10N = 20N = 40a) b)
Figure 1.
Rescaled mobility functions Λ for polymer chains with different chain lengthas indicated. a) Logarithmic plot versus qR G . b) Linear plot versus qa , where a is thestatistical segment length. The black solid line shows the theoretical results from Ref.[56] diffusion constants D c = 1 /k B T ζN . In Fig. 1 a), we also show the theoretical result forinfinitely long Rouse polymers[56].The simulation data agree well with the theory for small q . At larger q , however,they deviate. Different from Rouse polymers, the mobility functions of finite chainsare nonmonotonic. They start from ˆΛ(0) = D c /k B T and first decay, initially closelyfollowing the theoretical curve, but then assume a minimum and grow again, untilthey reach the original value, ˆΛ( q ) = D c /k B T at q → ∞ . In the small q regime, thecurves for different chain lengths collapse onto each other if plotted against qR G ; in thelarge q regime, they collapse if plotted as a function of q only (made dimensionless bymultiplying with the the statistical segment length a ).In the DDFT (Eq. (1)), the asymptotic large q behavior of ˆΛ describes thatexpected for a fluid of monomers which move independently with the diffusion constant D = ˆΛ( ∞ ) N = 1 /k B T ζ [50, 57]. Hence, we observe a crossover from a collective”chain mobility” to a ”monomer mobility” in chains with finite length N . The crossoverpoint (the position of the minimum) scales roughly like ( qR G ) c ∼ N / as a function ofchain length. This seems to suggest that the crossover wavelength is determined by theaverage distance d of monomers in the coil, which is set by the local density, d ∼ ρ / with ρ = N/R G . In the limit of infinite chain length, the crossover point ( qR G ) c movesto infinity. However, the value of the bare wavevector at the crossover, ( qa ) c , moves tozero for infinite chain length.The reason why the mobility of the finite chain at large ( qR G ) differs from that ofthe infinite chain can be rationalized as follows: In the regime 1 (cid:28) ( qR G ) (cid:28) ( qR G ) c ,the local mobility is dominated by the collective motion of whole chain portions with alocally scale invariant conformations. The effective friction of such a ”wad” is reduced, ynamic coarse-graining of polymer systems using mobility functions qR G ) c (cid:28) ( qR G ),the effect of chain connectivity becomes negligible and monomers diffuse individually.The two regimes (”wad” diffusion and monomer diffusion) are well separated in realpolymer systems. However, in CG model systems of short chains, they move closer toeach other and overlap.When devising dynamic mapping schemes for such extremely CG polymers systemsthat cover kinetic processes, one is thus faced with a fundamental problem: It isimpossible to accurately represent dynamic processes on both global and local (”wad”)length scales with simple time scale matching. If one uses the time scale of chain diffusionfor time mapping, the time scales of local ordering, e.g., at interfaces, are overestimatedby a factor of roughly 3.6. On the other hand, if maps the time scale of local ordering,the global chain diffusion is underestimated.We should note that related finite chain length effects are also observed in the staticstructure factor, g ( q , /N g ( q ,
0) drops from 1 (at q = 0) to zero at large q R G → ∞ , whereas it levels off at1 /N for finite chains. In principle, this can be corrected by an appropriate backmappingprocedure [58], i.e., restoring structure in the coarse-grained beads in retrospect. In thecase of the dynamics, a different approach must be taken.
3. Adapting the CG polymer dynamics on multiple length scales
We will now propose a way to adjust the CG dynamics of in a CG polymer system suchthat it has the same mobility function than the target system of large polymers overthe whole range of q vectors up to ( qR G ) c . The idea is slow down the internal modes,such that the CG monomers effectively have the mobility of a ”wad”, without changingthe diffusion constant of the whole chains and the static structure of the chains. Tothis end, we have to introduce different friction constants for internal modes and globaldiffusion. We consider linear Gaussian chains of length N with global chain friction γ t . Our goal isto devise a modified dynamical model that allows for different internal friction constantswhile not affecting the static behavior of the chain. The diffusion constant of the wholechain will be kept fixed.The monomer coordinates are given by R i ( t ), and the total potential is given by U [ { R i } ]. Thus the force acting on monomer i is given by f i = −∇ R i U . The centerof mass of the chain is given by R t ( t ) = N (cid:80) i R i and the total force acting on allmonomers is f t ( t ) = (cid:80) i f i ( t ). ynamic coarse-graining of polymer systems using mobility functions The simplestAnsatz is to introduce two friction constants, one for the center of mass motion ofthe chain and one for the relative motion with respect to the center of mass. Wewill illustrate this approach using the example of a overdamped Brownian dynamics.We introduce alternative coordinates R t (center of mass) and r i = R i − R t (internalcoordinates), i.e., (cid:80) i r i = 0. Rewriting the potential energy as a function of thesecoordinates, we obtain a new potential function˜ U [ R t , { r i } ] = U [ { R t + r i } ] . (14)To reproduce the identical static averages, the generalized forces ˜ f i acting on coordinates R t , r i are derived from ˜ U with an additional Lagrange multiplier λ (a vector) thataccounts for the constraint (cid:80) i r i ≡ f t = − ∇ R t ˜ U = (cid:88) i ( −∇ R i U ) ∂ R i ∂ R t = (cid:88) i f i = f t (15)˜ f i = − ∇ r i ( ˜ U + λ · (cid:88) i r i ) = f i − λ (16)The constraint forces must be chosen such that the constraint is fulfilled at all times.The dynamical equations are overdamped Langevin equations˙ R t = γ t ˜ f t + ξ t = γ t f t + ξ t (17)˙ r i = γ m ˜ f i + ξ i = γ m ( f i − λ ) + ξ i (18)with inverse friction constants γ t and γ m . The value of γ t is chosen such that thechain has the desired diffusion constant. The value of γ m can be used for mappingthe dynamics on short scales. The variables ξ t , ξ i describe uncorrelated Gaussiannoise with mean zero ( (cid:104) ξ α (cid:105) = 0) which satisfy the fluctuation-dissipation relation, i.e., (cid:104) ξ t ( t ) ξ t ( t (cid:48) ) (cid:105) = 2 k B T γ t δ ( t − t (cid:48) ), (cid:104) ξ i ( t ) ξ i ( t (cid:48) ) (cid:105) = 2 k B T γ m δ ( t − t (cid:48) ), and (cid:104) ξ α ( t ) ξ β ( t (cid:48) ) (cid:105) = 0for α (cid:54) = β . From the constraint (cid:80) i r i ≡
0, we derive (cid:80) i ˙ r i ≡
0, which allows to express λ as λ = N ( f t + γ m (cid:80) i ξ i ), hence Eq. (18) reads˙ r i = γ m ( f i − N f t ) + ξ i − N (cid:88) j ξ j . (19)This finally yields the modified equations of motion for monomers R i :˙ R i = γ m f i + γ t, eff f t + η i with γ t, eff = γ t − N γ m (20)where η i = ξ i + ( ξ t − N (cid:80) j ξ j ). Note that η i is again a correlated Gaussian distributionnoise with correlation matrix (cid:104) η i ( t ) η j ( t (cid:48) ) (cid:105) = 2 k B T δ ( t − t (cid:48) )[ γ m δ ij + γ t, eff ]. We recover theregular equations for linear Rouse polymers in the case γ t = N γ m . The above modifications can also be applied to regularLangevin dynamics (with inertia). For beads of mass m , we obtain the modified equationof motion m ¨ R i = f i − ζ m ˙ R i − ζ t, eff ˙ R t + f Ri ( t ) (21) ynamic coarse-graining of polymer systems using mobility functions ζ m = γ − m , ζ t, eff = N γ − t − γ − m , and f Ri ( t ) is a Gaussian distributed stochasticforce with correlation matrix (cid:104) f Ri ( t ) f Rj ( t ) (cid:105) = 2 k B T δ ( t − t (cid:48) )[ ζ m δ ij + N ζ t, eff]. As in Eq.(19), it can be implemented as a linear combination of uncorrelated random forces f Ri = θ i + N ( θ t − (cid:80) j θ j ) with (cid:104) θ t ( t ) θ t ( t (cid:48) ) (cid:105) = 2 k B T γ − t δ ( t − t (cid:48) ) and (cid:104) θ i ( t ) θ j ( t (cid:48) ) (cid:105) =2 k B T ζ m δ ( t − t (cid:48) ).Extensions to modified dynamical models with more than one internal frictionconstants are straightforward. For future reference, we briefly describe the resultingequations for a hierarchical model with three friction constants. We separate thepolymer into two blocks of equal length, A and B, such that the block A comprisesmonomers R i with i ∈ { , ..., N/ } := S A and the block B monomers R i with i ∈ { N/ , ..., N } =: S B . We distinguish between the forces f i acting on monomers i ,the force f t = (cid:80) i f i acting on the whole chain, and the forces f A,B = (cid:80) i ∈ S A,B f i actingon the individual blocks A and B. As generalized coordinates, we choose the center ofmass R t of the full chain, the center of masses r A,B of the blocks relative to R t , andthe coordinates r i of monomers relative to r A,B . The motion of R t , r A,B , and r i areassociated with separate inverse friction constants γ t , γ b , and γ m . Following the sameprogram as in the previous subsection, we obtain the following dynamical equations formonomers i belonging to the block α ( i ) ( α = A, B ) (overdamped regime):˙ R i = γ m f i + γ b, eff f α ( i ) + γ t, eff f t + η i (22)with γ b, eff = γ b − N γ m , γ t, eff = γ t = γ b , where the Gaussian noise is correlated accordingto (cid:104) η i ( t ) η j ( t (cid:48) ) (cid:105) = 2 k B T δ ( t − t (cid:48) )[ γ m δ ij + γ b, eff δ α ( i ) ,α ( j ) + γ t, eff ] . (23)As in the previous examples, it can again be conveniently calculated as a sumover uncorrelated Gaussian noise terms. Similar equations can be derived for otherdistributions of friction constants. To evaluate our proposed approach, we consider an extreme test case and attemptto map a Rouse polymer onto very short discrete Gaussian chains (length N ). Asa preliminary remark, we note that the asymptotic value of the mobility functionˆΛ( q ) in the limit q → ∞ is bounded from below by the corresponding value forchains with frozen conformations, where the chains only move as a whole, i.e.[59, 57]ˆΛ( q ) frozen = D c k B T N g ( q, q →∞ → D c k B T N − . In order to be able to implement the limitingbehavior of ˆΛ( q ) for Rouse polymers given by Eq. (12), CG chains must thus have aminimum length of N = 4. Hence we will evaluate CG systems of Gaussian tetramers.We employ modified dynamics with two friction constants as described in the previoussection, Section 3.1.We first verify that the static behavior of the chain is not changed by the modifieddynamics model. We characterize the static properties by the static structure factor, i.e., g ( q, t ) at t = 0. Fig. 2a) shows the static structure factors g ( q,
0) for the polymer chains ynamic coarse-graining of polymer systems using mobility functions qR g g ( q , ) / g ( , ) qR g Λ ^ [ D c / k B T ] a) b) Figure 2. a) Normalized static structure factors g ( q,
0) for chains with length N = 4.Black dashed line shows the results from traditional overdamped Brownian dynamicssimulations, the symbols those from modified dynamics approaches with two inversefriction constants γ m = N γ t , N γ t /
16 and three inverse friction constants with two sets: ( γ b = N γ t / , γ m = N γ t ) and ( γ b = N γ t / , γ m = N γ t / q ) for the same chains from modified dynamics with two friction constants.For comparison the thin dashed line shows the results from traditional simulations,the thick solid line the theoretical results. For modified dynamics, the inverse relativemonomer friction γ m is decreased from top to bottom: γ m = N γ t , N γ t / N γ t / N γ t / N γ t / N γ t / moving according to the modified dynamics approaches with two friction constants( γ m = N γ t , N γ t /
16) and three friction constants (two sets: γ b = N γ t / , γ m = N γ t and γ b = N γ t / , γ m = N γ t / γ m = N γ t , N γ t / , N γ t / , N γ t / , N γ t / N γ t / γ t , which sets the diffusion constant of the whole chain, iskept fixed. Additionally shown is the result from traditional simulations (black dashedline), and the theoretical results for Rouse polymers from our previous paper, Ref. [56](black solid line).At small qR G ( qR G < qR G . At intermediateand large qR G , the internal relaxation becomes important. If one decreases γ m , themobility function decreases. At γ m ≈ N γ t /
16, the data for the CG chain match thoseof the Rouse polymer. Hence, we can indeed obtain the target mobility function in ynamic coarse-graining of polymer systems using mobility functions qR g g AA ( q , ) / g AA ( , ) qR g Λ ^ AA [ D c / k B T ] a) b) Figure 3.
Same as Fig. 2, but now for the first half block (block A) of the chain.a) Normalized static structure factors g ( q, γ m = N γ t , N γ t / N γ t / N γ t / N γ t / N γ t / modified dynamics simulations by tuning γ m . This is the central message of the presentpaper.However, the approach also has limitations. This becomes apparent when lookingat the partial mobility functions for chain blocks, which is important for dynamicalstudies of block copolymer ordering and disordering[55, 56]. To illustrate this, we splitour ultrashort chain ( N = 4) in two symmetric blocks A and B of length N = 2 (seeSec. 3.1.2) and evaluate separately their mobility functions ˆΛ AA ( q ) , ˆΛ BB ( q ) as well asthe cross-mobility ˆΛ AB ( q ). The same quantities can be calculated semi-analytically forRouse polymers using the expressions given in our previous work, Ref. [56].The results are shown in Fig. 3. Note that ˆΛ AA ( q ) = ˆΛ BB ( q ) due to symmetry andwe also have ˆΛ AB ( q ) = ˆΛ BA ( q ) and ˆΛ( q ) = (cid:80) αβ ˆΛ αβ ( q ). Since ˆΛ( q ) is known from Fig. 2,it suffices to plot the data for ˆΛ AA ( q ) here. The same holds for the static structure factor g αβ ( q ). In Fig. 3a), we verify that the latter is not affected by the modified dynamicsas expected. The data for the block mobility functions are given in Fig. 3b). At large qR G , if one decreases γ m , the block mobility function decreases, and the target value(the value for Rouse polymers) can be matched for γ m = N γ t /
16. Different from thetotal mobility function ˆΛ, however, the block mobility function ˆΛ AA is also affected by γ m . In regular dynamics ( γ m = γ t N ), the behavior at small qR G → γ m , it becomes smaller and deviates fromthe target. Hence it is not possible to match the kinetics of chain blocks on both shortand long length scales in a CG model with such ultrashort chains, if one uses modified ynamic coarse-graining of polymer systems using mobility functions t g ( q , t ) / g ( q , ) t ∆ ( q , t ) / ∆ ( q , ) γ m = N γ t γ m = N γ t /2 γ m = N γ t /4 γ m = N γ t /8 γ m = N γ t /16 γ m = N γ t /32 a) b) Figure 4.
Normalized single chain dynamic structure factor (a) and ∆( q, t ) (b) ofpolymer chain with length N = 4 obtained from modified dynamics at qR g = 0 . γ m is varied as N γ t , N γ t / N γ t / N γ t / N γ t / N γ t / dynamics with two friction constants.To analyze this in more detail, we inspect the structure of the block mobilityfunctions. In Ref. [55], we have derived the following general expressions for ˆΛ αβ ( q ):ˆΛ AA ( q ) = 14 k B T q N (cid:18) g ( q, τ R + ∆( q, τ ∆ (cid:19) (24)ˆΛ AB ( q ) = 14 k B T q N (cid:18) g ( q, τ R − ∆( q, τ ∆ (cid:19) (25)where τ R = g ( q, (cid:82) ∞ dtg ( q, t ), ∆( q, t ) = g AA ( q, t ) + g BB ( q, t ) − g AB ( q, t ) − g BA ( q, t ), and τ ∆ = q, (cid:82) ∞ dt ∆( q, t ). Since g ( q,
0) and ∆( q,
0) are not affected by γ m (shown inFig. 2a) and Fig. 3a)), the dependence of τ R and τ ∆ on γ m will determine the behaviorof Λ AA and Λ AB . The time scale τ R characterizes the dynamics of the whole chain, and∆ characterizes the relaxation dynamics of blocks with respect to each other.Here we focus on the small qR G regime. Fig. 4 shows the normalized single chaindynamic structure factor g ( q, t ) (a) and the quantity ∆( q, t ) (b) of the CG chains asobtained from modified dynamics as a function of the simulation time t at qR G = 0 . γ m has practically no effect on the behavior of the singlechain dynamic structure factor, hence the relaxation time τ R does not change. For∆( q, t ), however, the relaxation slows down with decreasing γ m , which results in anincrease of τ ∆ . Combined with the equations above, we conclude that decreasing γ m will lead to a decrease in Λ AA and a increase in Λ AB . The individual blocks relax moreslowly and the two blocks move more cooperatively at small qR G if the relative monomerfriction is increased.We have tested whether it is possible to decouple the motion of blocks at small qR G by using a more versatile modified dynamics scheme with three friction constants. To ynamic coarse-graining of polymer systems using mobility functions qR g Λ ^ [ D c / k B T ] qR g Λ ^ AA [ D c / k B T ] a) b) Figure 5.
Mobility functions Λ (a) and ˆΛ AA (b) for polymer chains with length N = 4 from modified dynamics with three friction constants. The black solid lineshows the target function, the mobility function for Rouse polymers. For modifieddynamics, the inverse total friction constant γ t and the inverse relative monomerfriction γ m = N γ t /
16 are kept fixed, and the inverse block friction parameter γ b decreases from top to bottom: N γ t / N γ t / N γ t / N γ t / N γ t / this end, we have adopted the hierarchical model described in Sec. 3.1.2 and introducedan additional inverse block friction constant γ b . Some representative results are shownin Fig. 5. The black line shows again the target mobility functions. In this example,we fix the inverse monomer friction parameter at a large value, γ m = N γ t /
16, such thatrelative monomer motions are largely suppressed, and vary the inverse block frictionconstant γ b is varied. As can be seen from Fig. 5, introducing the hierarchical schemewith three friction constants does not improve the quality of the mapping. At the levelof the block mobilities, the problems persist, and even the mapping of the total mobilityfunction (Fig. 5a)) is not as good as in the system with two friction constants (Fig. 3).We have explored all possible parameter combinations of γ b and γ m and did not obtainany better results. Hence we conclude that dynamic mapping of block copolymers ontotetramers is not possible, and longer CG chains must be used to model such systems.Given that the chain length N = 4 is the minimum chain length for homopolymermapping as explained at the beginning of this section, it is perhaps not surprising thatit is too small to map individual blocks.
4. Summary and Conclusion
To summarize, in this paper, we have presented a dynamic coarse-graining scheme forpolymer systems with the goal of mapping the time scales of local kinetic processes overa large range of relevant length scales. The scheme builds on the single-chain mobilitymatrix, a wave-vector dependent integrated quantity that is derived from the single- ynamic coarse-graining of polymer systems using mobility functions q . The reason is that in short chains, the motion of different monomersdecouples for large qR G , whereas the dynamics remains cooperative in Rouse polymers.As a remedy, we have proposed a class of modified CG dynamics schemes where therelative motion of monomers is artificially slowed down, and shown that this can greatlyimprove the quality of dynamic mapping of homopolymers, even if the length of the CGchains is as short as N = 4.We have also investigated the limitations of the method. For homopolymers, wehave established by analytical considerations that N = 4 is the minimum CG chainlength where consistent dynamic mapping is possible. In the case of block copolymers,this still seems too short and dynamic mapping of symmetric diblock copolymers ontotetramers was not possible. We found that slowing down the monomers increases thedynamic correlation between the different blocks in an undesired way, and it was notpossible to find dynamical parameters that reproduce the mobility matrix functionˆΛ αβ ( q ) in a satisfactory manner over the whole range of q vectors for CG chains with N = 4. We conclude that dynamically consistent ”extreme” coarse-graining of blockcopolymers onto CG requires either further modifications or less extreme coarse-graining(i.e., larger N ). When respecting these limitations, we believe that our scheme can havea wide range of interesting applications. We have tested it on linear Rouse polymers, butit can also be applied to polymers in other dynamic regimes, e.g., entangled polymers,and to other polymer architectures.Our dynamic coarse-graining scheme is motivated by a Markovian approximationto the dynamics (Eq. (1)) that does not explicitly account for memory effects inpolymer dynamics. Mapping strategies that target the full frequency dependent mobilitymatrix of the GLE, e.g., Eq. (10), should be even more accurate. However, it willlikely not be possible to implement them without introducing frequency dependentmobility coefficients at the level of the CG model as well[34], which would greatlyreduce the efficiency of CG simulations. On the other hand, CG simulations based onmodified dynamics, e.g., Eqs. (20) or (21), are not much more expensive than regularCG simulations, as they neither require additional force evaluations, nor extra efforts(storage of data, auxiliary variables) to account for memory kernels[29]. The approachcan additionally be motivated by the observation that polymer DDFTs based on theMarkovian approximation – when using wave-vector dependent (i.e., nonlocal) mobilityfunctions as in Eq. (1) – were found to reproduce kinetic processes in inhomogeneouspolymer systems fairly accurately on time scales well below the Rouse time[55]. Wehave studied this for chains in the Rouse regime, corresponding investigations of otherdynamical regimes are currently under way.A large number of different internal friction constants can be introduced following ynamic coarse-graining of polymer systems using mobility functions Acknowledgments
We thank Shuanhu Qi for valuable discussions. This work was done within theCollaborative Research Center SFB TRR 146; corresponding financial support wasgranted by the Deutsche Forschungsgemeinschaft (DFG) via Grant 233530050.
References [1] Utracki L A and Wilkie C A (eds) 2014
Polymer Blends Handbook
Handbook of Multiphase Polymer Systems (Wiley)[3] Stamm M (ed) 2008
Polymer Surfaces and Interfaces
J. Chem. Phys. Adv. Polymer Science:Viscoelasticity, atomistic models, statistical chemistry ( Advances in Polymer Science vol 152)ed Abe, A (Springer) pp 41–156[6] M¨uller-Plathe F 2002
ChemPhysChem Soft Matter Faraday Discuss. (0) 9–24[9] Brini E, Algaer E A, Ganguly P, Li C, Rodr´ıguez-Ropero F and van der Vegt N F A 2013
SoftMatter (7) 2108–2119[10] Noid W G 2013 J. Chem. Phys.
J. Chem. Phys.
The Theory of Polymer Dynamics (Oxford University Press)[13] Maier W and Saupe A 1959
Z. Naturforsch.
Z. Naturforsch.
The Physics of Liquid Crystals (Oxford University Press)[16] Olsen B D, Shah M, Ganesan V and Segalman R A 2008
Macromolecules J. Chem. Phys.
Chem. Mater. Macromolecules Mol. Syst. Design & Eng. J. Chem. Phys.
Acta Polymerica Acta Polymerica Physical Review ynamic coarse-graining of polymer systems using mobility functions [25] Mori H 1965 Progress of Theoretical Physics Nonequilibrium statistical mechanics (New York: Oxford University Press)[27] Li Z, Bian X, Li X and Karniadakis G E 2015
J. Chem. Phys.
J. Chem. Phys.
J. Chem. Phys.
Soft matter Soft Matter (36) 8330–8344[32] Chen M, Li X and Liu C 2014 J. Chem. Phys.
J. Chem. Phys.
J. Chem. Phys.
J. Chem. Phys.
Physical Review E Europhys. Lett. J. Chem. Phys.
J. Chem. Phys.
J. Chem. Phys.
J. Chem. Phys.
J. Stat. Phys.
J. Phys.-Condens. Mat. Soft Matter Physica A
349 – 413[46] Kawasaki K and Sekimoto K 1988
Physica A
361 – 413 ISSN 0378-4371[47] Fraaije J G E M 1993
J.Chem.Phys. J.Chem.Phys.
Int. J. Mod. Phys. C Incorporating Fluctuations and Dynamics in Self-Consistent FieldTheories for Polymer Blends (Berlin, Heidelberg: Springer Berlin Heidelberg) pp 1–58[51] te Vrugt M, L¨owen H and Wittkowski R 2020
Adv. in Physics JETP Macromolecules ACS Nano (10) 13986–13994[55] Mantha S, Qi S and Schmid F 2020 Macromolecules Polymers Macromolecules ACS Macro Lett. J. Chem. Phys.
J. Chem. Theory Comp. Soft matter EPL (Europhysics Letters)128