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A Coarse-grained Molecular Dynamics Force-Field for Polyacrylamide in Infinite Dilution Derived from Iterative Boltzmann Inversion and MARTINI Force-Field Pallavi Banerjee, Sudip Roy, and Nitish Nair J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b09019 • Publication Date (Web): 26 Dec 2017 Downloaded from http://pubs.acs.org on December 30, 2017

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The Journal of Physical Chemistry

A Coarse-grained Molecular Dynamics Force-field for Polyacrylamide in Infinite Dilution Derived from Iterative Boltzmann Inversion and MARTINI Force-field Pallavi Banerjee,†,‡ Sudip Roy,† and Nitish Nair∗,† †Shell Technology Center, Bande Kodigehalli, Bengaluru, Karnataka 562149, India ‡Department of Chemistry, Indian Institute of Science Education and Research, Pune 411008, Maharashtra, India E-mail: [email protected] Phone: +91-7259027943

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ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract We present a mesoscale model of aqueous polyacrylamide in the infinitely dilute concentration regime, by combining an extant coarse-grained (CG) force-field, MARTINI and the Iterative Boltzmann Inversion protocol (IBI). MARTINI force-field was used to retain the thermodynamics of solvation of the polymer in water, whereas the structural properties and intra-polymer interactions were optimized by IBI. Atomistic molecular dynamics simulations of polymer in water were performed to benchmark the mesoscale simulations. Our results from the CG model show excellent agreement in structure with the atomistic system. We also studied the dynamical behavior of our CG system by computing the shear viscosity and compared it with the standard IBI model. The viscosity trends of our model were similar to the atomistic system, whereas the standard IBI model was highly dissimilar as expected. In summary, our hybrid CG model sufficiently mimics an infinitely dilute system, and is superior to both MARTINI and IBI in representing the structure and thermodynamics of the atomistic system respectively. Our hybrid coarse-graining strategy promises applicability in large-scale simulations of polymeric/biological systems where the structure needs to be replicated accurately while preserving the thermodynamics of a smoother surrounding.

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1

Introduction

Polyacrylamide, popularly known as PAM, is the polymer of choice in polymer-enhanced oil recovery, that is used to enhance the viscosity of water to match that of oil. 1–5 PAM is a water-soluble polymer synthesized by the free-radical polymerization of acrylamide. 6 The interaction of PAM with water molecules via hydrogen bonding depresses the mobility of water leading to an increased viscosity of the solution, a phenomenon that is widely employed in the process of enhanced oil recovery (EOR). 7,8 Based on the gel-like behavior of polyacrylamide, its other applications include those in biomaterials such as gel electrophoresis 9 , manufacture of soft contact lenses 10 , also soil conditioning. 11 Such a fascinating behavior of PAM in water has spiked considerable interest in experimentalists and computational scientists alike. Understanding the properties and dynamics of complex macromolecular systems, such as polymers, necessitates experimental and computational investigations to go in tandem. Computer simulations scrutinize the system at the atomistic/molecular level, thereby providing a deeper insight into the mechanisms underlying the behaviour of polymers in melts or solutions. Simulations allow more nuanced investigations into local effects such as polymer architecture, chemistry behind interactions in the system, organization and ordering of the polymeric chains and networks, and transport properties of the chains. With the aid of computer simulations, intelligent product design and optimization of methods can be realized, which in turn would lead to more targeted experiments. Simulations of polymers present formidable technical limitations because of the large number of degrees of freedom inherent in the system. Moreover, polymer dynamics sweep across a wide range of temporal (from 10−10 m to 1 m) and spatial (from 10−14 s to 107 s) scales. 12 There has been significant amount of research done on atomistic models of polymers 13–15 , however capturing macroscale phenomena with these models is impractical due to long simulation times involved. Polymers are high-molecular weight chains characterized by long chain relaxation times resulting in insufficient sampling of phase space. The typical molecular weight of PAM that is used in EOR is 18-20 million daltons comprising 104 − 105 monomers. The most intensive atomistic 3



simulation conducted on PAM, thus far, incorporated 100 monomers. 16 Atomistic studies on polymers are computationally expensive. One way to bypass this difficulty is to eliminate the unnecessary degrees of freedom by reducing the number of interaction points. This is achieved by clumping groups of atoms to form beads or superatoms in the coarser description. By coarsening, i.e. lowering the resolution of the system, meso-/macroscale processes can be realized since the dynamics would evolve much faster owing to the derived soft potentials. Several coarse-graining strategies have been employed in the past to study polymer properties including high-coordination lattice simulations 17–19 , bond-fluctuation models involving Monte-Carlo simulations to represent Rouse dynamics 20–24 , Dissipative Particle Dyanamics (DPD) that is based on bead-spring models of polymers 25 , and simplex optimization schemes. 26,27 With the advent of efficient coarse-grained models, efforts have also been made to develop multiscale simulations that enable switching between resolutions upon requirement. 28,29 Coarse-graining exists over multiple scales, right till the extent where the particles are replaced by a continuum field, for example, implicit solvent models. 30 Among the existing literature about coarse-graining of PAM, a model has been designed that deals with DPD, which retains only a small amount of the original structure. 31 Another piece of work is based on the MARTINI force-field (FF) 32 , but it fails to represent the structure quantitively. 33 Our interest lies in the mesoscale regime where sufficient chemistry of the atomistic model is retained along with faster computation of properties. We employ Coarse-grained Molecular Dynamics (CGMD) simulations that aim to extend molecular dynamics to lower-resolutions while maintaining the chemical identity of the system by incorporating related potentials in the FF. Studies involving CGMD simulations include the Kremer-Grest model that consists of a bead-spring model for polymer that is defined by FENE bonds and the Weeks-ChandlerAnderson (WCA) potential 34 , the force-matching method where the average potential of mean force between the pseudoatoms is matched with the atomistic system 35–37 , and also shape-based coarse graining where the shape of the atomistic system is targeted using topol-

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ogy conserving maps. 38,39 Among the most popular CGMD approaches, MARTINI 32 parameterization has found a gamut of applications in the biomolecular domain like lipids 40 , proteins 41 , DNA 42 and is also being used for polymeric materials 43,44 and nanoparticles too. 45 The MARTINI force-field has been designed primarily to mimic the thermodynamics of the system of interest. 32 Yet another popular CG strategy is the Iterative Boltzmann Inversion (IBI) that targets the structural representation of the atomistic description. 46,47 IBI is a protocol of iterative optimization on the interaction potentials between coarse-grained sites so as to match the structural distributions (pair correlations) of the underlying atomistic model, such as the bond, angle, dihedral probability distributions and the radial distribution functions (RDFs) between the CG sites. IBI is mostly used for homogenous systems like polymer melts. 26,48–54 In this paper, we introduce a novel idea of coarse-graining aqueous polyacrylamide by integrating the MARTINI framework with the IBI methodology with the aim of reproducing both the structure and thermodynamics of the basal atomistic system. This was achieved by inclusion of MARTINI water beads 32 in the CG system while optimizing potentials of the polymer-polymer interactions alone. MARTINI water was chosen as it has been shown to represent the free energy of hydration of the underlying system, and also because it offers a high level of coarse-graining, with 4 molecules of water mapped to one CG water bead. 32 This work focuses on coarse-graining the solution in the infinitely dilute regime, where one polymeric chain does not interact with the other chains. Since the semi-dilute concentration regime is relevant in the case of EOR, we also tested the derived potentials in higher concentrations. We would like to get the inter-polymer interactions right so as to model the viscosity enhancement. This would need long term, more complex features to be added to the CG potential in order to describe interactions in the semi-dilute regime. The layout of the paper is as follows: In the next section we describe the coarse-graining methodology and the details of the simulations performed, both atomistic and coarse-grained. Next, we present the results obtained for both single chain and multi-chains systems, and a thorough discussion of the same. We have provided the summary and outlook in the end.

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2

Methods

All the MD simulations and computation of observables have been performed using the GROMACS-5.0.5 55,56 suite of programs. A coarse system was constructed upon an atomistic foundation, simulation details of which are in the following section.

2.1

Atomistic Simulations

A united-atom (UA) force-field, GROMOS-53A6 57 was used to parameterize PAM of 30 monomers with charges of atoms derived from Wang, et al. 33 The polymer chain was chosen as atactic, since PAM is synthesized by free radical polymerization, a process that typically results in atactic polymers, as is also seen in synthetic HPAM (hydrolyzed PAM). 58 The rationale behind using PAM of N=30 and validity of the force field have been explained in Sec.S1 of the supporting information (SI). GROMOS has been used previously for modelling polymers by Oldiges, et al 16 to model PAM, Qimeng, et al 59 to model Polyethyleneglycol, and Alaa S., et al 60 to model terperidyls. SPC-E 61 was used as the force-field for water. A geometry-optimized PAM chain was solvated in a cubic box of edge 5.63 nm with 5698 water molecules. The system was energy-minimized using the steepest-descent algorithm. 62 It was then subjected to an NVT equilibration for 1 ns to allow the water to relax around the polymer. An NPT equilibration for 5 ns followed by an NPT production run for 300 ns were performed. Trajectories were analyzed from the last 200 ns of the simulation, saving the trajectory every 2 ps. The equations of motion were integrated with the leap-frog integrator 63 using a time step of 2 fs. The temperature was maintained at 300 K with the Nose-Hoover 64,65 thermostat and pressure was constant at 1 bar with the Parrinello-Rahman 66 barostat with coupling constants of 0.4 ps and 1.5 ps respectively. Electrostatics was employed using Particle Mesh Ewald (PME) method 67 with a real space cut-off of 1.2 nm and PME order of 4. With a cut-off radius of 1.2 nm, the neighbour list was updated at every 10th step. Lennard-Jones interactions were also truncated at 1.2 nm. The Linear Constraint Solver

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(LINCS) 68 algorithm applied constraints on all bonds.

2.2

Coarse-graining procedure

The mapping scheme to group atoms into beads for PAM was also derived from the work of Wang, et al 33 , who in turn had derived it from Rossi, et al’s 44 work in which two types of mapping for Polystyrene were compared, namely Mapping A and Mapping B. Mapping A describes a linear chain of polymer beads and Mapping B includes pendant groups. It was found that A fared better than B in representing experimental properties, and thereby we chose the better scheme, that is, Mapping A. Two types of beads, A and B, were used to describe the chemical specificity of the PAM chain. The mapping scheme is shown in Fig.1 Three heavy atoms of the pendant amide group are mapped to bead A; half each of the flanking carbon atoms with the central carbon are mapped to the backbone bead B. We maintained the same concentration of 2.03 weight% of PAM in the atomistic and coarse systems. The coarse-graining tool-kit VOTCA 69 has been employed in this work to perform IBI. This package is a unified scaffolding implementing three kinds of CG techniques: Inverse Monte Carlo, Force Matching, and Iterative Boltzmann Inversion, and presents a versatile platform for further development of CG methods. The coarse-grained PAM chain was solvated in 1424 MARTINI water beads. Each MARTINI water bead represents a clump of 4 water molecules, of type P4 (high polarity) as per MARTINI FF’s denomination. 32 The centre of mass of each group of atoms is the coordinate of the mapped bead. Our system of coarse-graining is a hybrid approach where the polymer-polymer interactions are optimized by the IBI scheme while the polymer-water and water-water interactions are kept fixed by using the respective MARTINI bead parameters. Abiding by MARTINI’s classification of bead types, bead A was parametrized as type SNda (intermediate polarity with hydrogen-donor and acceptor groups) and bead B as type SC1 (low polarity). Hence, non-bonded interaction pair potentials A-W (SNda-P4), B-W (SC1-P4), and W-W (P4-P4) were directly picked from MARTINI FF, as has been provided in Sec.S3 of the SI. IBI was performed for the re7



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maining bonded and non-bonded potentials. The reference bonded potentials were obtained after mapping the 300 ns-long atomistic trajectory to the coarse system, which include bond AB, angles ABA and BAB, and dihedral angle BABA (see Fig.1). The target sets for the non-bonded interactions were the radial distribution functions (RDFs) corresponding to pairs A-A, B-B and A-B. Boltzmann inversion over each of these distributions, Pq (q), where q is the concerned degree of freedom, served as guess potentials to initiate the iterative process (as shown in Eqn.4). Pq (q) stands for probability distribution for the bonded terms, and is the RDF, g(r), for the non-bonded interactions. In order to assess the advantages of our hybrid method of coarse-graining, we also coarse-grained our system by the standard IBI approach, wherein each atomistic SPC-E water molecule was mapped to a bead. The details of the procedure can be found in Sec.S3 of the SI. SC1

B

B

SC1

SNda SNda

A

A

Figure 1: Atomistic to coarse-grained mapping scheme; on the atomistic representation (left), grey = Carbon, red = Oxygen, blue = Nitrogen, white = Hydrogen; corresponding coarse-grained representation lies on the right It is assumed that the total potential energy of the system can be partitioned into bonded and non-bonded parts:

U=

∑

Ubond +

∑

Unon−bonded

(1)

This assumption implies that the bonded and non-bonded interaction functions can be optimized separately. It is further assumed that the probability distributions for the bonded interactions are uncorrelated and can be factorized into individual components:

P (r, θ, ϕ) = Pr (r)Pθ (θ)Pϕ (ϕ)

8


(2)

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where r, θ, ϕ are bond, angle, dihedral angle between beads in spherical coordinates. Histograms, Hq (q), of the probabilities were rescaled to obtain volume-normalized distribution functions, Pq (q), where q is r, θ, or ϕ 69 :

Pr (r) =

Hθ (θ) Hr (r) , P (θ) = , Pϕ (ϕ) = Hϕ (ϕ) θ 4πr2 sinθ

(3)

Boltzmann inversion on the normalized distributions gives us the respective potentials:

Uq (q) = −kB T lnPq (q)

(4)

where q is the degree of freedom in question, kB is the Boltzmann constant, and T is the temperature at which the simulation was run. This directly inverted potential serves as a reasonable first guess for the iterative scheme. Further refinements on these potentials are carried out by the following relation 47,70 : ( Uqi+1 (q)

=

Uqi (q)

+ kB T ln

Pqi (q) Pqtarget (q)

) (5)

Potentials were updated in the decreasing order of relative strengths - Ubond → Uangle → Unon−bonded → Udihedral . The standard protocol for updating potentials in IBI is to optimize solely one potential until convergence before moving on to the next in the order of updates, and then subsequently update the other potentials while keeping the converged potentials fixed. In the standard way, Ubond is optimized until it is converged before moving on to Uangle and similarly down the list. We followed a more fine-tuned process of updating potentials. The order of updates was the same, but we went over repeated cycles of the update order by optimizing one interaction potential in one iteration until convergence was achieved in all. Ubond is optimized in iteration 1 followed by Uangle in iteration 2 and similarly down the list in cycles that are repeated until convergence is reached by all of them. The cycle of updates was continued till the Pqi (q) converged to match Pqtarget (q) for all the interaction potentials.

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The extent of convergence is determined by the following error function: ∫ fmerit =

w(q)(Pqi (q) − Pqtarget (q))2 dq

(6)

where w(q) = exp(−q), a weighting function to avoid strong deviations at small distances. 71 We set fmerit = 0.1 to arrive at an acceptable convergence.

2.3

Coarse-grained Simulations

The coarse-grained system was prepared in a cubic box with the same concentration as the atomistic system. The trajectory was updated by imposing Langevin dynamics with a time-step of 4 fs. The simulations were carried out in an NVT ensemble with the Langevin thermostat maintaining the temperature at 300 K with the friction constant, γ = 0.5τ −1 where τ = 0.2 ps is the time constant. The neighbour list was updated every 10 time-steps. Following the usual MARTINI methodology, non-bonded potentials were made to smoothly approach zero at the cutoff value, 1.2 nm, by applying the force-switch algorithm 72 between 0.9 nm and 1.2 nm. Each iteration was 10 ns long. The final CG simulation was run for 210 ns, with the last 200 ns reserved for analysis.

2.4

Multiple chains in water

To test the transferability of our CG force-field, we used the potentials derived from the single-chain-in-water case to simulate aqueous PAM at higher concentrations, i.e., of more than one chain of PAM in water. The concentrations tested were: 3.93 wt%, 5.87 wt%, 9.69 wt%, and 19.08 wt%. These systems were built by solvating 2, 3, 5, and 10 chains of PAM30 (PAM chain of 30 monomers) respectively, in water. The same atomistic or coarse-grained forcefield parameters were used in the infinitely dilute and concentrated cases to check the range of transferability of the coarse-grained potential. Each of these CG simulations was run for 210 ns, collecting data from the last 200 ns. 10


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2.5

Viscosity

In order to study the rheological behavior of the resulting CG system, we computed the shear viscosity of the aqueous polymer solutions with the Transverse Current Autocorrelation Function (TCAF) method. 73 This method involves calculating transport coefficients by an analysis of the momentum gradients in the system. The microscopic momentum field, u⊥ is described by:

u⊥ (k, t) =

N ∑

ˆ ⊥ · pj (t)sin[k · rj (t)] k

(7)

ˆ ⊥ · pj (t)cos[k · rj (t)] k

(8)

j=1

u⊥ (k, t) =

N ∑ j=1

ˆ ⊥ is where pj is the momentum, and rj is the center of mass of molecule j. The vector k the unit vector normal to k. 73 The k values are:

k=

2π (n1, n2, n3) L

(9)

where n1, n2, n3 represent crystallographic indices. The TCAF, C⊥ (k, t) decays with the following function 74–76 :

[ ] [ ] [ ] [ ] 1 1 1 (1 + ω)t 1 (1 − ω)t C⊥ (k, t) = 1− exp − + 1+ exp − 2 ω 2τ 2 ω 2τ where ω =

√

1 − 4τ (µ/ρ)k 2 .

µ is an even function of k, and to order k 2 , the relation becomes 73 :

µ(k) = µ∞ + ak 2

(10)

where µ∞ is the infinite limit of µ obtained by extrapolating the relation to k → 0 limit. 11



The value, µ∞ , is the bulk viscosity of the system. Three trials each were conducted for both the atomistic and CG systems, for polymer concentrations 0 wt% (only water) to 3.43 wt% (2 chains). NPT equilibration for 2 ns followed by 5 ns long NPT production run were simulated at 300 K and 1 bar. The trajectory was saved every 10 fs. The decay constant was acquired by an exponential fit to the decaying TCAF, and further converted into the shear viscosity of the system.

3

Results and Discussion

3.1

Single chain of PAM in water

The iterative process was run for 60 steps at the end of which sufficient convergence for all the interactions in the coarse-grained system had been achieved. Incorporating MARTINI water in the CG system provided the benefit of a considerably high level of coarse-graining because of the 4-molecules-to-one-bead mapping and led to much faster evolution of the system. Apart from speedy dynamics, our method of coarse-graining entailed no pressure correction, a situation that emerges in cases of IBI 46,47 and requires further optimization of potentials. The inclusion of MARTINI water guarantees that the thermodynamics of the system is maintained. This is because MARTINI FF was parameterized based on thermodynamic data such as partitioning free energy. 32 Fig.2 shows that all the bonded distributions in the CG system (red) - bond, both the angles, and the dihedral angle - have matched the atomistic targets (black) quite well. This suggests that the bonded degrees of freedom of the CG chain are well in place. The three non-bonded pairs of interactions that comprise the polymer-polymer interactions in the system are: A-A, B-B, and A-B. Fig.3 shows the RDF profiles of these pairs of superatoms. As is evident from the pair-correlation functions here, there was an excellent overlap between the results from the CG simulations (red) and those from the underlying atomistic profiles (black). These results prove that the local structure of the atomistic system 12


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has been well-reproduced by our hybrid approach of coarse-graining. For an atom ’i’, consecutively bonded neighbours i+1, i+2 and i+3 were not considered in the intra-molecular RDF calculation. Hence, both the intra- and inter-molecular arrangements have been replicated by our coarse-grained model.

Figure 2: Comparison of bonded distributions between atomistic (black) and CG (red) models; Probability distributions of bond AB, angle ABA, angle BAB, dihedral BABA

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(a)

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(b)

(c)

Figure 3: Reproduction of the atomistic RDFs (black) by the CG model (red) for the interaction pairs (a) A-A, (b) B-B, (c) A-B

Table 1: Comparison of structural properties (Rg and Ree ) between atomistic and CG simulations Atomistic Coarse-grained

Rg (nm) Ree (nm) 0.724±0.028 1.061±0.390 0.719±0.036 1.156±0.417

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(a)

(b)

Figure 4: Histograms of (a) Rg and (b) Ree of the CG system (red) against atomistic profiles (brown)

The global structure of a polymeric chain is best described by static structural properties, such as radius of gyration, Rg , and end-to-end distance, Ree . Table.1 reports the averaged values of the Rg and Ree from both the coarse-grained and the atomistic simulations. When the distribution of these structural properties match, the success of the potential also gets confirmed. Fig.4 shows the comparison between the distribution of the structural features of the atomistic foundation (brown) and the derived CG model (red). The CG distributions exhibit good overlap with their atomistic counterparts. There exists a marginal left-shift in the Rg of the CG system. The reason could be understood on examining Fig.1. The distance between the center of mass of a peripheral bead (bead A/B) and the CG chain’s center is always less than the distance between the center of mass of a peripheral atom (Carbon/Hydrogen) and the atomistic chain’s center.

3.2

Higher concentrations of aqueous PAM

As described in Sec.2.4, the applicability of the CG potentials designed for a single chain of PAM in water (infinitely dilute) was tested for higher concentrations. To estimate the transferrability of the constructed force-field, global structural properties, Rg and Ree of

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the concerned systems were computed. The same exercise was performed using the singlechain potentials derived from IBI (only) on all the interactions, as described in Sec.S3 of the SI. Fig.5 and Fig.6 show the comparison between the two methods of coarse-graining, IBI+MARTINI and IBI, by placing the results from both against the atomistic target. The plots indicate that for the lowest concentration, 3.93 wt%, IBI-MARTINI models the system of target better than the all-IBI methodology. Looking at the concentration of 3.93 wt%, we see that red curves (all-IBI) are much broader than the target ones, whereas the green (IBI-MARTINI) curves are well close. On going up on the concentration ladder though, the distributions from both the models stray away from the target, intimating towards generation of potentials differently for the higher concentrations. From the results of IBI+MARTINI here, it is evident that the CG force-field that works perfectly for one chain of PAM in water, i.e. infinitely dilute concentration, works only qualitatively till 5.87 wt%. On going beyond, CG distributions deviate from the atomistic ones. This is because the CG potentials were built for only a single chain of polymer in water and is hence, limited to an infinitely dilute solution. The results from the IBI+MARTINI work better than those from the allIBI methodology in the lowest concentration regime. Incorporating water molecules in IBI is non-trivial, therefore, IBI works well when the solvent effect is built in the potentials inclusively. Therefore, for the IBI+MARTINI method, re-optimization of potentials is necessary to move on to higher concentrations of the solution, a fact which serves as a limitation of our method. However, the protocol itself can be used to construct CG versions of polymers of different architectures (e.g., combs, stars), chemistries, and monomer arrangements. Once the CG versions are built, their effects on viscosity can be determined even before synthesizing the molecule in the lab.

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3.93 wt%

800

Count

Count

400

0.8

1

1.2

Rg (nm)

500

0 0.6

1.4

0.8

Atomistic IBI IBI+MARTINI

1.2

1.4

Rg (nm)

1.6


800

Count

600

1

19.08 wt%

9.69 wt%

800

400 200 0


1000

200 0 0.6

5.87wt%

1500


600

Count

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


600 400 200

0.8

1.2

1.6

Rg (nm)

0

2

0.8

1

1.2

1.4

Rg (nm)

1.6

1.8

2

Figure 5: Comparison of Rg between atomistic (black) and CG models, pure IBI (red) and IBI+MARTINI (green) at different concentrations of PAM

Figure 6: Comparison of Ree between atomistic (black) and CG models, pure IBI (red) and IBI+MARTINI (green) at different concentrations of PAM

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3.3

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Viscosity

Shear viscosity values were computed for polymer solutions of concentrations 0 wt% (plain water) and 3.43 weight% (2 PAM chains in water), in both atomistic and CG levels, using the potentials derived from single-chain-in-water system. As can be inferred from Tab.2, the viscosity of water from the hybrid model of IBI+MARTINI is quite close to that of the atomistic SPC-E water model. The CG MARTINI water model is closer to the experimental viscosity reported for water, 0.89 cP 77 , than the atomistic SPCE water model. Due to this slight upper shift in viscosity of water, the vertical offset is maintained in the slightly higher concentration of 3.43wt% as well. Table 2: Comparison of viscosities of CG models against atomistic Water 2PAM in water

Atomistic (cP) 0.695±0.004 0.743±0.006

IBI (cP) IBI+MARTINI (cP) 0.151±0.004 0.741±0.006 0.165±0.003 0.827±0.004

We also conducted the same experiment on the CG system obtained from all-IBI methodology (refer Sec S3 of the SI). As can be seen in Tab.2, the viscosity of the CG water from the all-IBI model is much lower than the atomistic model. This big difference is attributed to the loss of many degrees of freedom in the CG system and is well known in literature. In the process of reproducing the structure of the system, and the coarse-graining of groups of atoms (less degrees of freedom), improper representation of dynamics of the solvent result in artificial dynamical behavior of the whole system. Hence, it is justified that the IBI+MARTINI model should be chosen over IBI to represent the dynamical behavior for solvated systems. As our FF could not replicate the structures of the atomistic systems of higher concentrations, we limited our viscosity calculations only to the systems shown in Tab.2, since viscosity of a polymeric solution depends on the radius of gyration of the polymer 78 . Further work is required to re-parameterize the force-field for different concentration regimes. 18


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4

Conclusions

In this work, we present a hybrid approach to coarse-grain polyacrylamide in an aqueous environment, which conjoins the MARTINI force-field and the procedure of IBI. We successfully modeled the coarse-grained description of a single chain of PAM in water, by the bottom-up approach of building on the information gathered from an atomistic trajectory of aqueous PAM. Molecular dynamics simulations have been conducted in this study for both levels of resolution. The aim of the study was to develop a CG force-field for aqueous PAM that would be in agreement with the structural and dynamical observables of the basal atomistic description. A united-atom model of aqueous PAM had been picked from literature upon which our CG model is founded. We have shown that the local structure of the polymer chain is perfectly matched in the CG model, and the global structure is only very marginally shifted which can be explained by end effects. Overall, the structure of the atomistic system is very well mimicked by our CG model. We investigated the applicability of the derived potentials in higher concentration regimes. It was observed that the interaction functions derived from single-chain-in-water system worked qualitatively till ≈6wt%, whereas they failed upon going higher. We compared our CG model with the CG system derived from the standard IBI method where there was no inclusion of MARTINI parameters. The standard IBI model also reproduces the structure fairly well, but on taking the concentration up a notch, the structural distributions become worse. We also explored the dynamical aspects of the system by computing the shear viscosity using the TCAF method. MARTINI water gives a viscosity value much closer to the experimentally obtained value for water than does the atomistic SPC-E water model. The viscosity of the IBI model of water was a value too low to be acceptable. This is because in the process of iterating based on structural distributions, the thermodynamic quantities are ignored, while MARTINI FF has been developed keeping the thermodynamics of solvation into consideration. Studies of viscosity of solutions of concentrations higher than 4 wt% require reparameterization of potentials as the derived CG FF could not mimic their atomistic counterparts. The IBI+MARTINI model 19



was seen to be 4 times faster than the standard IBI model. Hence, in more than one way, IBI+MARTINI wins over the pure IBI case. We conclude that the CG model that we derived by integrating MARTINI and IBI schemes can be used to study infinitely dilute systems of aqueous PAM. It performs better than MARTINI in terms of structure and better than IBI in terms of thermodynamics. One could also use this model to study the effect of inclusion of other chemicals on the structural behavior of PAM when infinitely diluted, such as surfactants or monovalent or divalent salts, that could be crucial in the subject of oil recovery. The integrated model can also be extended to study polymers of different architectures like copolymers or branched polymers. Further challenges include devising a strategy to generate potentials that satisfy larger ranges of concentration, and also modelling poly-electrolytes such as Hydrolyzed-PAM (HPAM) as they are more extensively employed in oil fields than neutral polymers.

Supporting Information Available The Supporting Information is available free of charge. • Atomistic PAM melt: strategy of choosing the chain length of PAM • Coarse-graining by all IBI: Procedure and results • Autocorrelation

Acknowledgment We thank Shell for granting us the permission to publish this work and equipping us with immense computational power and resources, and DST for the INSPIRE Fellowship to facilitate the project. Special mention goes to Foram Thakkar from Shell and Dr. Arnab Mukherjee from IISER Pune to have participated in the discussions to provide their invaluable suggestions. 20


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