Conformational Properties of a Polymer in an Ionic Liquid: Computer Simulations and Integral Equation Theory of a Coarse-Grained Model.

Subscriber access provided by UNIV OF YORK

Article

Conformational Properties of a Polymer in an Ionic Liquid: Computer Simulations and Integral Equation Theory of a Coarse-Grained Model Eunsong Choi, and Arun Yethiraj J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp508876q • Publication Date (Web): 13 Oct 2014 Downloaded from http://pubs.acs.org on October 18, 2014

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry B is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 20

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

The Journal of Physical Chemistry

Conformational Properties of a Polymer in an Ionic Liquid: Computer Simulations and Integral Equation Theory of a Coarse-Grained Model Eunsong Choi† and Arun Yethiraj⇤,‡ Department of Physics,University of Wisconsin, Madison, Wisconsin, 53706, and Department of Chemistry, University of Wisconsin, Madison, Wisconsin, 53706 E-mail: [email protected]

⇤ To

whom correspondence should be addressed of Physics,University of Wisconsin, Madison, Wisconsin, 53706 ‡ Department of Chemistry, University of Wisconsin, Madison, Wisconsin, 53706 † Department

1

ACS Paragon Plus Environment


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 study the conformational properties of polymers in room temperature ionic liquids using theory and simulations of a coarse-grained model. Atomistic simulations have shown that single poly(ethylene oxide) (PEO) molecules in the ionic liquid 1-Butyl 3-Methyl Imidazolium tetrafluoroborate ([BMIM][BF4 ]) are expanded at room temperature, i.e., the radius of gyration, Rg , scales with molecular weight, Mw , as Rg ⇠ Mw0.9 , instead of the expected self-avoiding walk behavior. The simulations were restricted to fairly short chains, however, which might not be in the true scaling regime. In this work we investigate a coarse-grained model for the behavior of PEO in [BMIM][BF4 ]. We use existing force fields for PEO and [BMIM][BF4 ]and Lorentz-Berthelot mixing rules for the cross interactions. The coarse-grained model predicts that PEO collapses in the ionic liquid. We also present an integral equation theory for the structure of the ionic liquid and the conformation properties of the polymer. The theory is in excellent agreement with the simulation results. We conclude that the properties of polymers in ionic liquids are unusually sensitive to the details of the intermolecular interactions. The integral equation theory is sufficiently accurate to be a useful guide to computational work.

KEYWORDS: Ionic liquid, polymers, coarse-grained models, integral equation theory

Introduction Room temperature ionic liquids are usually composed of a large organic cation and a small anion. They have become popular for the variety of potential applications 1,2 and interesting static and dynamic properties. 3 There has been recent interest in the behavior of mixtures of polymers and ionic liquids. 4,5 One can envisage applications where the polymer material forms a matrix (such as a membrane) which contains the ionic liquid as a working fluid. Although there have been a number of experiments on polymers in ionic liquids, 6–14 there has been little theoretical work. 15–17 In this paper we study the behavior of a single poly(ethylene oxide) (PEO) molecule in the ionic liquid 1-Butyl 3-Methyl Imidazolium tetrafluoroborate ([BMIM][BF4 ]) using computer simulation and integral equation theory of a coarse-grained model. 2


Page 2 of 20

Page 3 of 20

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


Atomistic simulations of PEO in [BMIM][BF4 ] show that the polymer is expanded at room temperature, in dilute solution. 14 Mondal et al. 14 performed atomistic simulations for degrees of polymerization, N, ranging from 9 to 40 and found that the chain size, characterized by the root mean square radius of gyration, Rg , scaled as Rg ⇠ N 0.9 . Such “polyelectrolyte" behavior is surprising because PEO chains are not charged, and one would therefore expect good solvent behavior, i.e., Rg ⇠ N 3/5 . The prediction was consistent, however, with experiments 13 that measured Rg ⇠ c

1/4

in the semi-dilute regime, which implies Rg ⇠ N in dilute solution. Standard molecular

dynamics simulations did not sample conformational space and we resorted to replica exchange methods, which are computationally intensive for large systems. This precluded the study of longer chains using atomistic simulations. Coarse-grained (CG) models are useful because they allow for the simulation of larger systems for longer times while maintaining some degree of chemical realism. When parameterized to reproduce experiments they are expected to provide a reasonable description of the system at a fraction of the computational coast. Recently Merlet et al. 18 have parameterized a coarse-grained forcefield for [BMIM][BF4 ] based on the force field of Roy and Maroncelli 19 (RM) for [BMIM][PF6 ]. RM performed extensive computer simulations and demonstrated that their model was in quantitative agreement with experiments for the molar volume, self-diffusion coefficient, viscosity, and conductivity over a wide temperature range (300K to 500K). The force field developed by Merlet et al. 18 was accurate (when compared to experiment and atomistic models) for the density, diffusion constant, viscosity, and surface tension at temperatures ranging from 373K to 500K, and in qualitative agreement with atomistic simulations for the pair correlation functions. In this work we investigate a coarse-grained model for PEO in [BMIM][BF4 ]. A CG model for PEO is available in the BMW-MARTINI framework 20 and we use Lorentz-Berthelot mixing rules for the cross interactions. We study the structure of the ionic liquid and the conformational properties of the PEO using MD simulations and integral equation theory. In contrast to the atomistic simulations, the coarse-grained model predicts that PEO collapses in the ionic liquid at 300K and 400K. The integral equation theory is in good agreement with the simulations for the pair

3



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

Page 4 of 20

correlation functions of the ionic liquid and, like the simulations, predicts that PEO is collapsed at 300K and 400K. The simulations and theory suggest that the molecular modeling of ionic liquids is subtle and depends on molecular details that are apparently lost in simple coarse-graining. The accuracy of the theory is promising, however, and suggests the theory maybe used to guide computational work.

Coarse-grained Model The CG model for the ionic liquid is shown in figure 1 where C1 , C2 , and C3 refer to the imidazolium, methyl, and butyl groups, and A refers to the anion. The geometry of the model for the

C3

C1

C2

A

Figure 1: Atomistic (left) and coarse grained (right) representation for the cation and anion, where C1 , C2 , and C3 are the imidazolium, methyl, and butyl groups, respectively, and A is the anion. cation can be found in references 18 or 19. For PEO we use a CG model recently developed by us within the BMW-MARTINI framework, where each site corresponds to three heavy atoms along the backbone, i.e., For simplicity we do not distinguish between end sites (CH3

O

[CH2

O

CH2 ] .

CH2 ) and repeat units (CH2

O

keep the bond lengths constant at 3.3 Å, and do not include dihedral or bond angle potentials. Note that we use the same identical model in both theory and simulations. Any discrepancies between theory and simulation therefore arise solely from the approximations inherent to the theory (and simulations). The interaction potential between sites i and j is given by, b ui j (r) = 4b ei j

⇣

si j ⌘12 r 4

⇣ s ⌘6 ij

r

+ lB


qi q j r

(1)

CH2 ),

Page 5 of 20

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


where b = 1/kB T , kB is Boltzmann’s constant, T is the temperature, si j = (si + s j )/2, and ei j = p ei e j . The Bjerrrum length, lB , is defined as lB ⌘ e2 /(4pe0 kT ), where e is the charge on an electron and e0 is the permittivity of free space. lB =167101.7/T Å i.e., lB =557 Å at T=300K. The interaction parameters for the sites are given in Table 1. Table 1: Interaction potential parameters for the coarse-grained model, taken from Merlet et al. 18 for the IL and Choi et al. 20 for PEO Type C1 C2 C3 A PEO

si (Å ) 4.38 3.41 5.04 5.06 4.30

ei (kJ/mol) q (e) 2.56 0.4374 0.36 0.1578 1.83 0.1848 3.24 -0.78 3.00 0

Integral equation theory We use the reference interaction site model (RISM) theory 21 to describe the pair correlation functions in the ionic liquid. Liquid state theories such as integral equations 22,23 and density functional theory 24,25 have been used for ionic liquids with considerable success. The RISM theory 21 has also been applied to polymers in ionic liquids 26–30 although the polymers were assumed to be Gaussian chains in that work. In this work, the polymer is at infinite dilution in the ionic liquid and therefore does not impact the pair correlation functions in the ionic liquid “solvent". We therefore first calculate the structure of the ionic liquid, use these correlation functions to calculate the effective solvent-induced interaction between polymer sites, and then calculate the properties of the polymer molecule. All four sites in the CG model have the same number density, r, which is the number of cations (or anions) per unit volume. The RISM equations for the ionic liquid are given (in reciprocal space) by ˆW ˆH ˆ =W ˆC ˆ + rW ˆC ˆ H

(2)

where the carets denote Fourier transforms, and H, C, and W are matrices of total, direct, and in5



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

Page 6 of 20

tramolecular correlation functions, respectively. The elements of H are given by Hi j (r) ⌘ gi j (r) 1 where gi j (r) is the pair correlation function between sites i and j, where i, j 2 {C1 ,C2 ,C3 , A}. The RISM equations may be considered a definition of the direct correlation functions Ci j (r). The molecules of the ionic liquid are rigid, and therefore for sites on the same molecule, the elements, ˆ are given by wˆ i j (k), of W wˆ i j (k) =

sin kli j kli j

(3)

where li j is the distance between sites i and j on the same (rigid) molecule, and k is the momentum transfer variable. When one of the sites is the anion wˆ i j (k)=0. The RISM equations must be supplemented by a closure relation between the direct and total correlation functions. For charged molecules, the hyper-netted chain (HNC) closure is expected to be the most appropriate. 31 The HNC closure is given by Ci j (r) = exp where gi j (r) ⌘ Hi j (r)

⇥

⇤ b ui j (r) + gi j (r)

1

gi j (r)

(4)

Ci j (r). Unfortunately we are not able to get the HNC closure to converge

using our method of solving the RISM equations. We therefore use a combination of the HNC and partially linearized HNC (PLHNC) closures. In the PLHNC closure 32 the exponential in equation 4 is linearized if

b ui j (r) + gi j (r) > 0, i.e., Ci j (r) = b ui j (r)

if

b ui j (r) + gi j (r) > 0, and the HNC closure (equation 4) used if

(5) b ui j (r) + gi j (r)  0. We use

the HNC closure for all correlation functions involving the anion, and the PLHNC closure for the other correlation functions. The RISM equations are solved via a Picard iteration procedure using the fast Fourier transform. For computational convenience, i.e., to avoid numerical Fourier transforms of long-ranged

6


Page 7 of 20

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


functions, we separate the correlation functions into a short-ranged and long-ranged part 33 , i.e., Ci j (r) = Cisj (r) Hi j (r) = Hisj (r)

qi q j r qi q j lB e r

lB

kr

(6)

where k 2 = 4prlB Âi q2i . We solve (iteratively) for the functions Cisj (r) and Hisj (r), which are short-ranged. The Fourier transforms of the long-ranged components, i.e., the functions 1/r and exp( kr)/r are performed analytically. The correlation functions are discretized with 211 points with a spacing in r-space of 0.04 Å . We make an initial guess for the functions Cisj (r) and calculate Cîsj (k) using the fast Fourier transform. The direct correlation functions in Fourier space are given by Cî j (k) = Cîsj (k) 4plB qi q j /k2 , and the total correlation functions Hˆ i j (k) are calculated using the RISM equations (2). The short-ranged part of the total correlation functions is Hˆ isj (k) = Hˆ i j (k) + 4plB qi q j /(k + k)2 . The functions gˆs i j k) ⌘ Hˆ s i j (k)

Cˆs i j (k) are inverted, using the fast Fourier

transform, to give gisj (r). The next guess for Cisj (r) is obtained from the closure relations. For example, for the HNC closure Cisj (r) = exp

✓ gisj (r)

lB qi q j e r

kr

b usij (r)

◆

1

gisj (r) +

lB qi q j e r

kr

(7)

where usij (r) is the Lennard-Jones interaction in equation (1). The procedure is continued until convergence. In the self-consistent polymer RISM theory, 34 the system of polymer in the ionic liquid is replaced by a single polymer interacting via the bare polymer-polymer interaction plus a solventmediated interaction that is calculated approximately. This solvent mediated potential is assumed to be pair-wise decomposable, and is obtained by solving the RISM equations for two PEO sites at infinite dilution in the ionic liquid. In this limit, the RISM equations for the correlation functions between the polymer site and the sites of the ionic liquid are: ˆ ps ˆ ps = Sˆ C H 7


(8)


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

Page 8 of 20

where Sˆ is the matrix of partial structure factors of the ionic liquid, i.e., ˆ + rH ˆ Sˆ = W

(9)

and C ps and H ps are the vector of direct and total correlation functions between the polymer site and the sites of the ionic liquid. These functions are expected to be short-ranged because the PEO sites are not charged. We solve this integral equation as follows. We start with an initial guess for Cos,i (r), use the fast Fourier transform to calculate Cˆ ps,i (k), calculate the functions gˆps,i (k) ⌘ Hˆ os,i (k)

Cˆ ps,i (k) using equation 8, and g ps,i (r) using the inverse fast Fourier transform. The next

guess for C ps,i (r) is then obtained from the Percus-Yevick closure, 31 h C ps,i (r) = e

b u pi (r)

i 1 [1 + g ps,i (r)]

(10)

where u pi (r) is the interaction potential between a PEO site and site i on the ionic liquid. The iteration is continued until convergence. The solvent-mediated potential, Wsol (r), is obtained from an inverse Fourier transform of 35 b Wˆ sol (k) = r Â Cˆ ps,i (k)Sî, j (k)Cˆ ps, j (k).

(11)

i, j

Equation (11) corresponds to the HNC approximation for the solvation potential. The PercusYevick approximation is b Wˆ sol,PY (k) =

"

#

ln 1 + r Â Cˆ ps,i (k)Sî, j (k)Cˆ ps, j (k) . i, j

(12)

It is not clear a priori which of these approximations is more appropriate because the solvent is charged (for which the HNC is recommended) but the polymer sites are not (for which the PY is recommended). It turns out that for ionic liquids bWsol (r)=1 for some values r and this results in a divergence in the PY solvation potential of equation (12), which does not seem physical. We

8


Page 9 of 20

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


therefore employ the HNC approximation of equation (11) in this work. Integral equation theory predictions for the conformational properties of the polymer are obtained from Monte Carlo simulations of a single chain interacting via the bare interaction plus a solvent-mediated interaction. The initial conformation is a self-avoiding walk, and the pivot algorithm 36 is used to evolve the system. In this move, an interior site of the polymer is chosen at random which divides the chain into two parts. The shorter part is then rotated a random angle about a randomly chosen cartesian axis. The move is accepted according to the Metropolis criterion. In the simulations reported here the chains rapidly collapse, after which most moves are not accepted. We therefore report properties averaged over several (5-10) independent simulations. The integral equation theory calculations are performed at constant density, and we use the density of the ionic liquid measured in the simulations. For 1 bar, the number density of cations (or anions) is 0.0029343 molecules/Å3 at T=400K and 0.0031593 molecules/ Å3 at T=300K.

Computer simulation method Molecular dynamics simulations are performed for the neat ionic liquid and for a single PEO chain in the ionic liquid. For the neat ionic liquid we use 1000 ion pairs at 300K and 400K. The simulation system is prepared by first inserting the ions randomly in a cubic box with a periodic boundary conditions. A steepest descent energy minimization is performed to remove bad contacts between the sites. For each temperature, the system is equilibrated at 1 bar in the isobaric-isothermal ensemble for 100 ns using Nose-Hoover thermostat 37,38 and the Berendsen barostat. 39 Equilibrium properties are then monitored from production runs of 400 ns in the canonical ensemble, with temperature maintained using the Nose-Hoover thermostat. Simulations for PEO in the IL are performed for single PEO chains with N = 9, 18, 27, and 36 solvated in 1000 [BMIM][BF4 ] ion pairs at 300K and 400K. A single PEO chain is first inserted into the box and then solvated 1000 ion pairs. The system is equilibrated at each temperature at 1 bar for 50 ns in the NPT ensemble using Nose-Hoover thermostat 37,38 and Berendsen barostat 39

9



followed by a 300 ns production run in the NVT ensemble using Nose-Hoover 37,38 thermostat. All the simulations are performed using GROMACS 40 version 4.6.5 with a time step of 5fs. The PEO bond lengths are constrained using the LINCS algorithm. 41 The Lennard-Jones interaction is shifted to zero between 10A and 14A. The Particle-Mesh-Ewald (PME) summation 42,43 is used for long-ranged electrostatic interactions with a Fourier spacing of 2A and a real space cut-off distance 14 A. The radial distribution functions are computed from 400ns of trajectory where the positions of atoms are recorded every 20ps. In order to estimate statistical errors, we divide the trajectory into eight 50ns blocks and calculate uncertainties as the standard deviation of the mean. The percentile errors are less than 1%.

Results and discussion The predictions of the integral equation theory for the site-site pair correlation functions in the neat ionic liquid are in excellent agreement with molecular dynamics simulation results. Figure 2 compares theoretical predictions to molecular dynamics simulations results for the site-site pair 1.4

7

2.5

(a)

(b) Me-Me Me-Bu Bu-Bu Me-Me Me-Bu Bu-Bu

2.0

1.0 1.5 Im-Im Im-Me Im-Bu Im-Im Im-Me Im-Bu

0.6 0.4 0.2

g(r)

0.8

4

6

8

r/A

10

12

14

Im-A Me-A Bu-A A-A Im-A Me-A Bu-A A-A

5 4 3

T=300K

1.0

2 0.5

1

T=300K 0.0

(c)

6

g(r)

1.2

g(r)

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

Page 10 of 20

T=300K 0.0 2

4

6

8

r/A

10

12

14

0

4

6

8

10

12

r/A

Figure 2: Comparison of integral equation theory predictions (lines) for the pair correlation functions to molecular dynamics simulations results (symbols) for T=300K. correlation functions. In the figure, Im, Me, Bu, and A, refer to the imidazolium, methyl, butyl, and anion groups, respectively. The cation sites are not very strongly correlated with each other with the maximum in g(r) taking on values of 1-2 in figures 2 (a) and (b). As expected the cor10


Page 11 of 20

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


relations between the anion and cation sites are much stronger, and reflects the magnitude of the electrostatic interactions. In all cases, the theory is in quantitative agreement with the simulations. For correlation functions involving the Im groups (figure 2 (a)) the theory misses some features in g(r) that are present in the simulations, e.g., the shoulders at approximately 7 Å in the Im-Bu g(r), 6 Å in the Im-Me g(r), and 7.5 Å in the Im-Im g(r). This is a well known shortcoming of the RISM theory and similar discrepancies are seen in theoretical predictions for the pair correlation functions of hard triatomic fluids. 44 Overall, the accuracy of the theory is similar to what was seen in previous tests of the theory for other models of ionic liquids. 22,24 The structural heterogeneity seen in atomistic simulations of [BMIM][BF4 ] is absent in the coarse-grained model. Computer simulations of atomistic models of ionic liquids show structural heterogeneity with hydrocarbon domains. An example is in figure 3 which shows a snapshot from jp056006yf00005.jpeg 999×1,484 pixels

9/1/14, 12:14 PM

an atomistic simulation of [BMIM][PF6 ] where the hydrocarbon tails are coloured green and the imidazolium groups are coloued red. These domains are not seen in the coarse-grained simulations (right panel of figure 3). This is not surprising, because the butyl group is only a sphere, and not that much larger than the imidazolium group. The absence of such structural heterogeneity is a potentially significant difference between the atomistic and coarse-grained models that has not been widely recognized.

Figure 3: Snapshots from atomistic 45 (left) and coarse-grained (right) simulations. The hydrocarbon tails in the atomistic simulations and C3 site in the CG simulations are colored green and the imidazolium groups are colored red. The atomistic simulations are for [BMIM][PF6 ], but similar results are expected for [BMIM][BF4 ]. http://pubs.acs.org.ezproxy.library.wisc.edu/appl/literatum/publisher/…006.110.issue-7/jp056006y/production/images/large/jp056006yf00005.jpeg

Page 1 of 2

The absence of structural domains in the CG model is also manifested in the static structure 11



factor. Figure 4 compares theoretical predictions for Sˆ11 (k), Sˆ22 (k), and Sˆ33 (k) to computer simulation results at 300K. The agreement between theory and simulation is excellent, as might be expected from the accuracy of the theory for the pair correlation functions. In atomistic simulations there is a prominent peak, at k ⇠ 0.4 Å

1,

in the Bu-Bu and Me-Me structure factors. In the

CG model this peak is absent in the Bu-Bu structure factor and only a slight bump is seen in the Me-Me structure factor, consistent with the snapshots in figure 3. 2.0

1.5

S(k)

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

1.0

Im-Im Me-Me Bu-Bu Im-Im Me-Me Bu-Bu

0.5

0.0 0.0

0.5

1.0

k/A

-1

1.5

2.0

Figure 4: Comparison of theoretical predictions (lines) for the partial structure factors to simulation results (symbols) at 300K. The solvent-mediated potential between PEO sites is repulsive at short distances and attractive at longer distances. The short-ranged repulsion almost cancels the bare interaction between PEO sites and the result is an interaction that has a minimum at roughly 9 Å (see figure 5). This leads to a minimum energy configuration that is not at closest approach of the sites but rather closer to a solvent-separated pair distance. As a consequence, when the chain collapses, it is not into a compact globule, but a more open structure. The simulations and self-consistent integral equation theory both predict that the PEO is collapsed in [BMIM][BF4 ] at 300K and 400K. Figure 6 compares simulation results and theoretical predictions for the root mean square radius of gyration, Rg as a function of N for two temperatures. In all cases, the scaling exponent is consistent with collapsed chains with an exponent n where Rg ⇠ N n in the range 0.28 to 0.34, consistent with the value n=0.33 for collapsed chains. The collapse of the chains is readily confirmed from snapshots from the simulations. Figure 7 depicts a final snapshot from simulations and theory for N=37. The end beads are in a different color. The 12


Page 12 of 20

4 3

Energy/kT

2

βW(r) β(U(r)+W(r))

1 0 -1

βU(r) -2

4

6

8

10

12 14 r(A)

16

18

20

Figure 5: The bare interaction, U(r), solvent-induced interaction, W(r), and the total effective interaction between polymer sites for 300K.

10 MD 300 MD 400 PRISM 300 PRISM 400

9 8 7

g

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


R (A)

Page 13 of 20

6 5

4

8 9 10

20

30

40

N

Figure 6: Radius of gyration as a function of degree of polymerization. In all cases, the results are consistent with the scaling for collapsed chains.

13



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

chain is collapsed and the nearest non-bonded beads are roughly a solvent-separated distance from each other.

Figure 7: Snapshot from theory (left) and simulation (right) for the final conformation of a 37-mer at 300K. The chain is collapsed with non-bonded nearest neighbor sites roughly a solvent separated distance apart.

Conclusions We investigate the conformational properties of polymers in ionic liquids using a coarse-grained model for the polymer, PEO, and the ionic liquid [BMIM][BF4 ]. The simulations show that the polymer is collapsed at 300K and 400K. This is in contrast to atomistic simulations, which show that the polymer is extended at room temperature. We also investigate an integral equation theory for the structure of the ionic liquid and the conformational properties of a single chain in the ionic liquid. The theory is in excellent agreement with the simulations for the liquid structure, and in good agreement with the simulations for the size of the polymer molecule. The theory also predicts that the chains are collapsed at 300K and 400K. The atomistic and CG simulations suggest that the conformational properties of the polymer are sensitive to details of the modeling of the ionic liquid. The CG model does not show structural heterogeneity that is predicted by atomistic models. If the polymer molecule segregates to the tail regions in a heterogeneous solvent, the conformations might be sensitive to the fractal nature of the heterogeneity, which is absent in the CG model for the liquid. If this were the case, then the CG 14


Page 14 of 20

Page 15 of 20

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


model misses an important structural characteristic. This implies that the level of coarse-graining employed here is too severe and it might be important to keep the chain-like nature of the alkyl tail as well as the local stiffness of the polymer on short length-scales. A model intermediate to the atomistic and CG models is the united atom model where the hydrogen atoms are not treated explicitly, but all heavy atoms are. The challenge, of course, is that a united atom model might not provide the required computational savings over the atomistic model. The accuracy of the theory suggests that it might be useful as a predictive tool to screen ionic liquids for desired structural properties. The theory might be used to screen interaction potentials as well as degree of coarse-graining to provide insight into regions in parameter space where more computationally intensive simulations might be performed. For example, it would be interesting to investigate intermediate levels of coarse-graining, such as united atom models, to determine what level of chemical detail is necessary for a faithful description of the structure and conformational properties of polymers in ionic liquids.

Acknowledgments This material is based upon work supported by the National Science Foundation under Grant No. CHE-1111835.

References (1) Wassercheid, P. Volatile Times for Ionic Liquids. Nature 2006, 439, 797–797. (2) Plechkova, N.; Seddon, K. Applications of Ionic Liquids in the Chemical Industry. Chem. Soc. Rev. 2008, 37, 123–150. (3) Zhu, F.; Liang, Y.; Liu, W. Ionic Liquid Lubricants: Designed Chemistry for Engineering Applications. Chem.Soc.Rev. 2009, 38, 2590–2599.

15



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

(4) Zhao, H.; Xia, S.; Ma, P. Use of ionic liquids as ’green’ solvents for extractions. J.Chem.Tech.Biotech. 2005, 80, 1089–1096. (5) Lodge, T. A Unique Platform for Materials Design. Science 2008, 321, 50–51. (6) Rodriguez, H.; Rogers, R. Liquid Mixtures of Ionic Liquids and Polymers as Solvent Systems. Fluid Phase Equilib. 2010, 294, 7–14. (7) Harner, J.; Hoagland, D. Thermoreversible Gelation of an Ionic Liquid by Crystallization of a Dissolved Polymer. J.Phys.Chem.B 2010, 114, 3411–3418. (8) Rodriguez, H.; Francisco, M.; Rahman, M.; Sun, N.; Rogers, R. Biphasic Liquid Mixtures of Ionic Liquids and Polyethylene Glycols. Phys.Chem.Chem.Phys. 2009, 11, 10916–10922. (9) Sarkar, A.; Trivedi, S.; Pandey, S. Polymer Molecular Weight-Dependent Unusual Fluorescence Probe Behavior within 1-Butyl-3-methylimidazolium Hexafluorophosphate plus Poly(ethylene glycol). J. Phys.Chem. B 2009, 113, 7606–7614. (10) Sarkar, A.; Trivedi, S.; Baker, G.; Pandey, S. Multiprobe Spectroscopic Evidence for "Hyperpolarity" within 1-Butyl-3-methylimidazolium Hexafluorophosphate Mixtures with Tetraethylene Glycol. J.Phys.Chem. B 2008, 112, 14927–14936. (11) Tsuda, R.; Kodama, K.; Ueki, T.; Kokubo, H.; Imabayashi, S.; Watanabe, M. Lcst-Type Liquid-Liquid Phase Separation Behaviour of Poly(Ethylene Oxide) Derivatives in an Ionic Liquid. Chem. Comm. 2008, 40, 4939–4941. (12) Ueki, T.; Karino, T.; Kobayashi, Y.; Shibayama, M.; Watanabe, M. Difference in Lower Critical Solution Temperature Behavior between Random Copolymers and a Homopolymer Having Solvatophilic and Solvatophobic Structures in an Ionic Liquid. J.Phys.Chem. B 2007, 111, 4750–4754. (13) Triolo, A.; Russina, O.; Keiderling, U.; Kohlbrecher, J. Morphology of Poly(ethylene oxide)

16


Page 16 of 20

Page 17 of 20

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


Dissolved in a Room Temperature Ionic Liquid: A Small Angle Neutron Scattering Study. J.Phys.Chem. B 2006, 110, 1513–1515. (14) Mondal, J.; Choi, E.; Yethiraj, A. Atomistic Simulations of Poly(ethylene oxide) in Water and an Ionic Liquid at Room Temperature. Macromolecules 2014, 47, 438–446. (15) Kirchner, B. Ionic Liquids; 2009; Vol. 290; pp 213–262. (16) Costa, L.; Ribeiro, M. Molecular Dynamics Simulation of Polymer Electrolytes Based on Poly(Ethylene Oxide) and Ionic Liquids. Ii. Dynamical Properties. J.Chem.Phys. 2007, 127, 164901. (17) Costa, L.; Ribeiro, M. Molecular Dynamics Simulation of Polymer Electrolytes Based on Poly(Ethylene Oxide) and Ionic Liquids. I. Structural Properties. J.Chem.Phys. 2006, 124, 184902. (18) Merlet, C.; Salanne, M.; Rotenberg, B. New Coarse-Grained Models of Imidazolium Ionic Liquids for Bulk and Interfacial Molecular Simulations. J.Phys. Chem. C 2012, 116, 7687– 7693. (19) Roy, D.; Maroncelli, M. Simulations of Solvation and Solvation Dynamics in an Idealized Ionic Liquid Model. J.Phys. Chem. B 2012, 116, 5951–5970. (20) Choi, E.; Mondal, J.; Yethiraj, A. Coarse-Grained Models for Aqueous Polyethylene Glycol Solutions. J.Phys. Chem. B 2014, 118, 323–329. (21) Chandler, D.; Andersen, H. C. Optimized Cluster Expansions for Classical Fluids 2. Theory of Molecular Liquids. J. Chem. Phys. 1972, 57, 1930–1937. (22) Bruzzone, S.; Malvaldi, M.; Chiappe, C. A RISM Approach to the Liquid Structure and Solvation Properties of Ionic Liquids. Phys. Chem. Chem. Phys. 2007, 9, 5576–5581. (23) Bruzzone, S.; Malvaldi, M.; Chiappe, C. Solvation Thermodynamics of Alkali and Halide Ions in Ionic Liquids through Integral Equations. J. Chem. Phys. 2008, 129. 17



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

(24) Forsman, J.; Woodward, C. E.; Trulsson, M. A Classical Density Functional Theory of Ionic Liquids. J. Phys. Chem. B 2011, 115, 4606–4612. (25) Turesson, M.; Szparaga, R.; Ma, K.; Woodward, C. E.; Forsman, J. Classical Density Functional Theory & Simulations on a Coarse-Grained Model of Aromatic Ionic Liquids. Soft Matter 2014, 10, 3229–3237. (26) Zherenkova, L. V.; Komarov, P. V.; Pavlov, A. S. Long-Range Correlations in PolymerContaining Ionic Liquids: The Case of Good Solubility. J. Phys. Chem. Lett. 2010, 1, 1186– 1190. (27) Zherenkova, L. V.; Komarov, P. V.; Pavlov, A. S. A Polymer in an Ionic Liquid: Structural Properties of a System during Attraction between the Polymer and Cations of the Ionic Liquid. Polym. Sci. Ser. A 2010, 52, 864–871. (28) Zherenkova, L. V.; Komarov, P. V. The Study of Structure Formation in a Polymer-Containing Ionic Liquid in Terms of the Integral Equation Theory. Polym. Sci. Ser. A 2011, 53, 252–260. (29) Zherenkova, L. V.; Komarov, P. V.; Belov, A. N.; Pavlov, A. S. A Polymer in an Ionic Liquid: Effect of the Length of the Cationic Nonpolar Tail on the Character of Interchain Correlations. Polym. Sci. Ser. A 2012, 54, 147–154. (30) Zherenkova, L. V.; Komarov, P. V.; Zubkov, V. V. Spatial Correlations of a Flexible-Chain Oligomer in an Ionic Liquid. Polym. Sci. Ser. A 2013, 55, 577–585. (31) Hansen, J. P.; McDonald, I. R. Theory of Simple Lquids; Elsevier, 1990. (32) Kovalenko, A.; Hirata, F. Potential of Mean Force Between Two Molecular Ions in a Polar Molecular Solvent: A Study by the Three-Dimensional Reference Interaction Site Model. J. Phys. Chem. B 1999, 103, 7942–7957. (33) Shew, C. Y.; Yethiraj, A. Integral Equation Theory of Solutions of Rigid Polyelectrolytes. J. Chem. Phys. 1997, 106, 5706–5719. 18


Page 18 of 20

Page 19 of 20

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


(34) S., S. K.; Honnell, K. G.; Curro, J. G. Reference Interaction Site Model-Theory of Polymeric Liquids - Self-consistent Formulation and Nonideality Effects in Dense Solutions and Melts. J. Chem. Phys. 1992, 96, 3211–3225. (35) Chandler, D.; Singh, Y.; Richardson, D. M. Excess Electrons in Simple Fluids. 1. General Equilibrium Theory for Classical Hard-sphere Solvents. J.Phys. Chem. B 1984, 81, 1975– 1982. (36) Madras, N.; Sokal, A. D. The Pivot Algorithm - A Highly Efficient Monte-Carlo Method For The Self-Avoiding Walk. J. Stat. Phys. 1988, 50, 109–186. (37) Nosé, S. A Molecular Dynamics Method for Simulations in the Canonical Ensemble. Mol.Phys. 1984, 52, 255–268. (38) Hoover, W. Canonical Dynamics: Equilibrium Phase-Space Distributions. Phys.Rev. A 1985, 31, 1695–1697. (39) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. Molecular Dynamics with Coupling to an External Bath. J. Chem. Phys. 1984, 81, 3684–3690. (40) Hess, B.; Kutzner, C.; Van der Spoel, D.; Lindahl, E. GROMACS 4: Algorithms for Highly Efficient, Load-Balanced, and Scalable Molecular Simulation. J. Chem. Theory Comput. 2008, 4, 435–447. (41) Hess, B.; Bekker, H.; Berendsen, H.; Fraaije, J. G. E. LINCS: A Linear Constraint Solver for Molecular Simulations. J. Comput.Chem. 1997, 18, 1463–1472. (42) Darden, T.; York, D.; Pederson, L. G. Particle Mesh Ewald: An Nlog(N) Method for Ewald Sums in Large Systems. J. Chem. Phys. 1993, 98, 10089. (43) Essman, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pederson, L. G. A Smooth Particle Mesh Ewald Method. J. Chem. Phys. 1995, 103, 8577.

19



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

(44) Phan, S.; Kierlik, E.; Rosinberg, M. L.; Yethiraj, A.; Dickman, R. Perturbation DensityFunctional Theory And Monte-Carlo Simulations For The Structure Of Hard Triatomic Fluids In Slitlike Pores. J. Chem. Phys. 1995, 102, 2141–2150. (45) Lopes, J. N. A. C.; Padua, A. A. H. Nanostructural Organization in Ionic Liquids. J.Phys. Chem. B 2006, 110, 3330–3335.

atomistic

PEO in [BMIM][BF4] coarse-grained Figure 8: Table of content graphic: Atomistic models predict the chain is expanded but coarsegrained models predict it is collapsed.

20


Page 20 of 20

Properties of a soft-core model of methanol: an integral equation theory and computer simulation study.

Calculation of the intrinsic solvation free energy profile of an ionic penetrant across a liquid-liquid interface with computer simulations.

Classical density functional theory & simulations on a coarse-grained model of aromatic ionic liquids.

Mass and Charge Transport in the Polymer-Ionic-Liquid System PEO-EMImI: From Ionic-Liquid-in-Polymer to Polymer-in-Ionic-Liquid Electrolytes.

Computer simulations of N-methyl-D-aspartate receptor-induced membrane properties in a neuron model.

Dissipative particle dynamics with an effective pair potential from integral equation theory of molecular liquids.

Force spectroscopy of polymer desorption: theory and molecular dynamics simulations.

An integral equation theory for solvation effects on the molecular structural fluctuation.

Properties of the Lennard-Jones dimeric fluid in two dimensions: an integral equation study.

A conformational comparison of two stereoisomeric cyclic dermorphin analogues employing NMR and computer simulations.

Integral equation model for warm and hot dense mixtures.

Density-functional theory and Monte Carlo simulations of the phase behavior of a simple model liquid crystal.

Influence of the Ionic Liquid Type on the Gel Polymer Electrolytes Properties.

Glassy dynamics of athermal self-propelled particles: Computer simulations and a nonequilibrium microscopic theory.

Coarse-grained simulations of an ionic liquid-based capacitor: I. Density, ion size, and valency effects.

Universal conformational properties of polymers in ionic nanogels.

Molecular dynamics simulations of the protonated G4 PAMAM dendrimer in an ionic liquid system.

Rigidity and soft percolation in the glass transition of an atomistic model of ionic liquid, 1-ethyl-3-methyl imidazolium nitrate, from molecular dynamics simulations--Existence of infinite overlapping networks in a fragile ionic liquid.

Photoreversible gelation of a triblock copolymer in an ionic liquid.

Double-stimuli-responsive spherical polymer brushes with a poly(ionic liquid) core and a thermoresponsive shell.

Solvation thermodynamics of organic molecules by the molecular integral equation theory: approaching chemical accuracy.

The application of the integral equation theory to study the hydrophobic interaction.

Computer simulations of activation in an anatomically based model of the human ventricular conduction system.

A quasi-solid-state rechargeable lithium-oxygen battery based on a gel polymer electrolyte with an ionic liquid.