Field theoretic simulations of polymer nanocomposites.

Field theoretic simulations of polymer nanocomposites Jason Koski, Huikuan Chao, and Robert A. Riggleman Citation: The Journal of Chemical Physics 139, 244911 (2013); doi: 10.1063/1.4853755 View online: http://dx.doi.org/10.1063/1.4853755 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Structure and effective interactions of comb polymer nanocomposite melts J. Chem. Phys. 141, 204901 (2014); 10.1063/1.4902053 Integral equation theory study on the structure and effective interactions in star polymer nanocomposite melts J. Chem. Phys. 126, 014906 (2007); 10.1063/1.2426340 Structure and effective interactions in polymer nanocomposite melts: An integral equation theory study J. Chem. Phys. 124, 144913 (2006); 10.1063/1.2187489 Molecular dynamics simulations of polymer transport in nanocomposites J. Chem. Phys. 122, 134910 (2005); 10.1063/1.1874852 Structure, surface excess and effective interactions in polymer nanocomposite melts and concentrated solutions J. Chem. Phys. 121, 6986 (2004); 10.1063/1.1790831

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THE JOURNAL OF CHEMICAL PHYSICS 139, 244911 (2013)

Field theoretic simulations of polymer nanocomposites Jason Koski, Huikuan Chao, and Robert A. Rigglemana) Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

(Received 12 October 2013; accepted 27 November 2013; published online 31 December 2013) Polymer field theory has emerged as a powerful tool for describing the equilibrium phase behavior of complex polymer formulations, particularly when one is interested in the thermodynamics of dense polymer melts and solutions where the polymer chains can be accurately described using Gaussian models. However, there are many systems of interest where polymer field theory cannot be applied in such a straightforward manner, such as polymer nanocomposites. Current approaches for incorporating nanoparticles have been restricted to the mean-field level and often require approximations where it is unclear how to improve their accuracy. In this paper, we present a unified framework that enables the description of polymer nanocomposites using a field theoretic approach. This method enables straightforward simulations of the fully fluctuating field theory for polymer formulations containing spherical or anisotropic nanoparticles. We demonstrate our approach captures the correlations between particle positions, present results for spherical and cylindrical nanoparticles, and we explore the effect of the numerical parameters on the performance of our approach. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4853755] I. INTRODUCTION

Polymer nanocomposites have become an increasing topic of interest due to the enhanced properties achieved from the coexistence of organic and inorganic nanoparticles within a polymer matrix. The enhanced properties1–8 of these systems range from, but are not limited to, electrical, thermal, and mechanical, making polymer nanocomposites desirable for a wide array of applications such as microelectronics,4, 9–11 sensing applications,5, 12, 13 and the development of advanced materials.6–8 The enhanced properties found in these systems are due to the large surface-to-volume ratio of the nanoparticles, and the resulting polymer nanocomposite material frequently possesses properties which are not achievable with nanoparticles or polymers alone. While significant development has occurred in our understanding of the equilibrium properties of polymer nanocomposites, many applications would benefit greatly from a unified framework that can capture the fundamental aspects that dictate the behavior of these systems. The ability to control the spatial distribution of the nanoparticles within the polymer matrix is of particular interest as it is well-understood that the distribution plays a critical role in obtaining desirable material properties.5, 8, 14–21 As a result, it is important to develop analytical and simulation techniques capable of accurately and efficiently predicting equilibrium morphologies, which is particularly challenging when the nanoparticles have a chemically heterogeneous surface, such as Janus or patchy particles,22–24 or if the nanoparticles are functionalized with a block copolymer.25–28 Particle-based simulations of polymer nanocomposites have been employed using models whose resolution spans from fully atomistic29–33 to coarse-grained,34–49 and mesoscale approaches such as dissipative particle a) Electronic mail: [email protected]

0021-9606/2013/139(24)/244911/11/$30.00

dynamics.50, 51 While these approaches are ideal for providing a detailed, molecular picture of the structure and properties of the material in the immediate vicinity of the nanoparticle surface, for highly entangled polymers with large nanoparticles or nanoparticles grafted with polymer chains, equilibration using these approaches remains a tremendous challenge. For this reason molecular simulations of these systems with multiple nanoparticles remain uncommon. When the polymer matrix has its own microphase, such as a diblock copolymer, then the simultaneous equilibration of the matrix and the nanoparticle distribution becomes even more challenging. Integral equation theories, in particular the Polymer Reference Interaction Site Model (PRISM), have been used to deduce structural and thermodynamic detail of polymer nanocomposite systems.52 A set of nonlinear matrix integral equations based on input pair potentials can be used to calculate quantities of interest such as the structure factor, the potential of mean force between nanoparticles, and to provide insight on the correlations of the components in the system. As a strictly theoretical method, PRISM requires much less computational expense than simulation methods but lacks the ability to visually inspect and analyze the resulting morphologies of the system. Additionally, the majority of PRISM studies deal with ideal chain conformations and have primarily been geared towards studying bare nanoparticles.53, 54 Recently, hybrid PRISM-Monte Carlo (MC) techniques have been implemented to study grafted nanoparticles and as a way to incorporate non-ideal chain conformations.55, 56 However, in both PRISM and hybrid PRISM-MC techniques, studying inhomogenous or phase separated systems remains a challenge. Similarly, density functional theory (DFT) approaches have recently been employed to describe the interactions between small groups of spherical nanoparticles immersed in a polymer matrix57 or grafted nanorods in a polymer matrix.58 However, many-body effects between the nanoparticles

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remain a challenge, and the grafted chains were described as strongly adsorbed chains; the results showed very good agreement with associated self-consistent field theory calculations on a comparable system. Field-based techniques such as self-consistent field theory have emerged as an increasingly popular method to study inhomogenous polymer systems. Field theory decouples inter-particle interactions through Hubbard-Stratonovich transformations and makes it such that the model is no longer a function of particle coordinates but fields instead. As a result, field theory is particularly advantageous when studying polymer systems because the number of discretization points of the fields in field-based models is often much less than the number of particle coordinates in particle-based models. Polymer field theory also describes the thermodynamics of inhomogenous polymer systems using a small number of parameters (e.g., Flory-Huggins χ parameter) that can be obtained experimentally.59–61 Two distinct approaches have been developed for incorporating nanoparticles into a polymer field theory: the combined self-consistent field theory/density functional theory (SCFT/DFT) methods62–65 and the hybrid particle-field theory (HPF) approaches.66–70 In the SCFT/DFT approaches, an expression for the free energy of a system containing both polymers and nanoparticles is derived using standard particleto-field transformation techniques and the subsequent meanfield approximation. The resulting free energy expression is then augmented with DFT corrections that take into account the hard-sphere correlations in the nanoparticle positions. The SCFT/DFT approach may be limited by the accuracy of the DFT expressions used to augment the free energy expression, and there is not always a straightforward route for improving the approximations. Although fluctuations could formally be included into the SCFT/DFT methods to move beyond the mean-field approximation, this has not yet been demonstrated, and it is unclear whether the augmented DFT terms would effectively double count correlations in the particle positions that would arise through the field that maintains incompressibility. In the hybrid particle-field theory approach, upon invoking the particle-to-field transformation the positions of the nanoparticles are left as explicit degrees of freedom in the partition function, and the nanoparticles are given their shape through the use of a masking function centered at the particle positions.66–70 The masking function is used in combination with the incompressibility constraint to exclude polymers from the nanoparticle centers, and Flory-like interaction terms can control selective wetting of the nanoparticle surfaces. When the fields that govern the distribution of the polymer chains are described under the mean-field approximation, the HPF Hamiltonian is purely real, and nanoparticle positions can in principle be sampled using standard MC sampling techniques or Langevin dynamics sampling schemes. However, the complex nature of the field-theoretic Hamiltonian makes it less straightforward to relax the mean-field approximation for the HPF approach; MC approaches must be modified to circumvent the sign problem, and the complex Langevin technique60, 61, 71–73 would involve forces that are complex-valued. The complex-valued forces would push

J. Chem. Phys. 139, 244911 (2013)

the nanoparticles out of the real plane, and it is not clear how to treat aspects of the simulation such as periodic boundary conditions in this case. A coarse-grained Monte Carlo approach has also been used to study morphologies of polymer nanocomposite systems.74 In this approach, local densities are calculated either via a smoothing function or a particle-to-mesh technique. The interactions between the polymers are calculated based on a functional of these local densities, while the particleparticle and particle-polymer interactions are taken through an explicit pairwise interaction potential. The bonded and non-bonded energies are calculated based on these potentials and typical Monte Carlo schemes are used to equilibrate the system. A clear advantage of the coarse-grained approach is that it is trivial to incorporate advanced bonded and nonbonded potentials. However, it can be difficult to move the nanoparticles in these systems, particularly in a dense polymer system or in a system with large nanoparticles, and it is thus challenging to equilibrate without advanced Monte Carlo moves or long simulation times. Recently, for electrolyte solutions Wang has shown75, 76 that giving the small ions a finite size by distributing their charge over a small Gaussian rather than treating them as point charges lead to a field theory that could accurately capture Born solvation effects in systems with a heterogeneous dielectric. Another benefit of the smearing approach was that the resulting field theory was free of ultraviolet (UV) divergences, which occurs when properties calculating using the resulting field theory do not converge as the collocation grid is further refined.60, 77 More recently it was shown78 how the same idea can be used to distribute the mass of the segments over a small Gaussian volume in a complex coacervateforming system, which also leads to a theory that is free of UV divergences. In this paper, we demonstrate how these ideas can be applied to the cavity functions typically used in the hybrid particle-field theory simulations. The cavity functions can be traced through the particle-to-field transformation to yield a pure field theory containing both polymeric and nanoparticle components, and the resulting models are formally equivalent to the models used in previous HPF calculations. This enables straightforward implementation of complex Langevin field-theoretic simulations containing nanoparticles without any of difficulties for the SCFT/DFT or the HPF methods described above. We demonstrate that our approach is readily extended to anisotropic nanoparticles, and we also show how the Gaussian smearing used previously78 leads to other physically meaningful results, such as liquid-like oscillations in the density near hard walls. The structure of the fluid as measured by g(r) is demonstrated to agree identically with particle-based simulations. The goal of this paper is to detail the framework for incorporating finite-sized polymer segments and nanoparticles into a field theoretic simulation and to provide demonstrative results for systems containing bare nanoparticles and anisotropic nanoparticles. The remainder of this paper is organized as follows: In Sec. II, we provide a detailed derivation of our theory for spherical nanoparticles embedded in a diblock copolymer matrix containing discrete Gaussian chains


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before outlining how to incorporate anisotropic nanoparticles; Sec. III presents the numerical methods for both meanfield and complex Langevin implementations of our theories. Section IV provides demonstrative results for each system of interest, including a simple Gaussian fluid, a pure nanoparticle system, as well as a diblock copolymer containing bare or anistropic nanoparticles. Finally, in Sec. V we provide a brief discussion of our approach and comment on future extensions. II. THEORY A. Diblock copolymers with bare spherical nanoparticles

nD NA i

δ(r − ri,j )

(1)

δ(r − ri,j ),

(2)

j

and ρˆB,c (r) =

nD NB i

j

where the subscript c indicates the particle centers. The total mass density of each polymer component is given as ρÂ (r) =

nD NA i

ρˆB (r) =

i

h(r − ri,j ),

(3)

h(r − ri,j ),

(4)

j

nD NB j

where h(r) is given by h(r) =

1 2π b2

3/2

e−|r|

2

/2b2

.

βU0 =

nD N−1 3|ri,j − ri,j +1 |2 . 2b2 i j

(5)

Alternatively, we could write the mass density as a convolution of the Gaussian smearing function h(r) and the center distribution as ρˆK (r) = dr h(r − r ) ρˆK,c (r ) = (h ∗ ρˆK,c )(r), (6) where K refers to component A or B, and the last equality introduces our short-hand notation for a convolution integral.

(7)

In addition to the polymer chains, we also consider nP nanoparticles in our system, where each particle has a volume vP . Similar to the polymer segments above, the nanoparticle center distribution is given by a sum of delta functions, ρˆP ,c (r) =

Our model is similar to previous field-theoretic simulation59–61 and Monte Carlo approaches.39, 74, 79 We model our polymer chains as discrete Gaussian chains with N = NA + NB segments, and the composition of our polymer chains is given by f = NA /N. It should be noted that a discrete Gaussian chain model is used here in order to more directly compare pair distribution functions of polymer systems with pure particle systems but the methods outlined below would be equally valid for continuous Gaussian chain models. For the sake of simplicity we assume that each type of polymer segment has the same size b = bA = bB , and there are nD total diblock copolymers in our system. For the polymer species, we will assume that the mass of each polymer segment is Gaussian distributed about a central point. The distribution of particle centers is given by ρÂ,c (r) =

We note that each polymer segment has a unit mass and volume and that in the limit where b → 0 in the function h(r) we recover the point particle distribution. The polymer connectivity is described using Gaussian bonds,

nP

δ(r − ri ),

(8)

i

and the total nanoparticle volume is a convolution of the nanoparticle centers with the particle smearing function (r), ρˆP (r) = ( ∗ ρˆP ,c )(r).

(9)

We define (r) such that in the nanoparticle center, the density of the particles is equal to the total system density, ρ0 = (nD N + nP vP )/V , and the density of the nanoparticles goes smoothly from ρ 0 inside the nanoparticle core to 0 outside of the nanoparticle core. We choose to use a complementary error function for (r − r ), defined as |r − r | − RP ρ0 erfc , (10) (r − r ) = 2 ξ where RP is the nanoparticle radius and ξ controls the length over which the density changes from ρ 0to 0. The volume of the nanoparticles vP is defined as vP = dr (r). We assume that each component interacts with each other through a purely repulsive, Flory-like contact potential χI J (11) dr ρÎ (r)ρˆJ (r), βU1 = ρ0 I< J where I and J represent one of the chemical components, χ IJ is the Flory parameter governing the strength of the interaction between components I and J, and the sum is taken over distinct combinations of components A, B, or P. Finally, we assume a Helfand quadratic potential80 that penalizes deviations of the total density away from ρ 0 , κ (12) dr [ρˆ+ (r) − ρ0 ]2 , βU2 = 2ρ0 where ρˆ+ = ρÂ + ρˆB + ρˆP is the spatially varying total density and κ controls the strength of the density fluctuations; we note that the strictly incompressible model is recovered in the limit κ → ∞. The total partition function for the particle model can now be written as nP drnD N e−βU0 −βU1 −βU2 , (13) Z = z0 dr where z0 contains the numerical prefactors (e.g., the thermal de Broglie wavelengths) that will be unimportant for the calculations considered here. In order to proceed through the Hubbard-Stratonovich transformation from a particle model to a field theory,60, 81 it is first necessary to re-write the non-bonded potentials in


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Eq. (11) in a quadratic form. For a system containing more than two components this can be achieved by defining ρÎ(±) J (r) = ρÎ (r) ± ρˆJ (r) and re-writing Eq. (11) as the difference of two quadratic terms, χI J χI J (+) 2 2 dr [ρÎ J (r)] − dr [ρÎ(−) βU1 = J (r)] . 4ρ 4ρ 0 0 I =J (14) The result of the particle-to-field transformation is the fieldtheoretic partition function, −H [{w}] , (15) DwI(+) DwI(−) Z = z1 Dw+ J J e

The form of Eqs. (20) and (22) is the same as the terms that arise from the weighted density approximation82 used in the Tarazona DFT83 employed in several previous SCFT/DFT methods. The fields wA , wB , and wP are each given by (+) (+) (−) (−) wA = i w+ + wAB − wAB + wAP − wAP , (+) (+) (−) (−) wB = i w+ + wAB + wAB + wBP − wBP , (23) (+) (+) (−) (−) wP = i w+ + wAP + wAP + wBP + wBP . In our analysis below, two sets of density operators will be of importance. First, we have the distribution of polymer segment and nanoparticle centers, which are given by

I =J

where z1 is the re-defined numerical prefactor that differs from z0 in that it contains the normalization constants from the Gaussian functional integrals used to de-couple the particle interactions, H [{w}] is the effective Hamiltonian of the system, and {w} is the set of chemical potential fields for our (+) (−) (+) (−) (+) (−) , wAB , wAP , wAP , wBP , and wBP . We note system, w+ , wAB that all of the integrals in Eq. (15) are along the real axis. The effective Hamiltonian is given by ρ0 H [{w}] = dr w+ (r)2 − iρ0 dr w+ (r) 2κ ρ0 ρ0 2 2 + (r) + dr wI(+) dr wI(−) J J (r) χ χ IJ I =J I J −nD ln Q[ωA , ωB ] − nP ln Q[ωP ],

(16)

where QD and QP are the partition functions of a single diblock copolymer or nanoparticle, respectively. QD is calculated from the chain propagator, q(j, r), 1 QD [ωA , ωB ] = dr q(N, r), (17) V which is in turn constructed by iterating a ChapmanKolmogorov equation −ωK (r) dr (r − r ) q(j, r), (18) q(j + 1, r) = e where ωK is either ωA or ωB if segment j + 1 is an A or B segment, respectively, and (r) is the normalized bond transition probability. The initial condition is taken from the A end of the block copolymer and is given as q(1, r) = e−ωA (r) .

ρÃ (r) =

nD ρ˜B (r) = V QD

ωP (r) = ( ∗ wP )(r).

(22)

(26)

ρ˘K (r) = (h ∗ ρ˜K )(r)

(27)

ρ˘P (r) = ( ∗ ρ˜P )(r).

(28)

and

Below, we discuss our complex Langevin and mean-field implementation of this model, where it is necessary to have the derivatives of H with respect to each of the potential fields. These expressions are given by ρ0 δH = w+ (r) − iρ0 δw+ (r) κ

δH δH

where K refers to species A or B, and

nP −ωP (r) e , V QP

respectively. The complementary propagator, q † (j, r), is determined by iterating the same Chapman-Kolmogorov equation described above in Eq. (18) starting from the B end of the diblock copolymer. Second, we have the distribution of total particle volumes, which are given by the convolution of the center densities with that species’ respective smearing function as

In Eqs. (17)–(19) we have suppressed the functional dependence of q on ωA and ωB for notational simplicity. The partition function of the nanoparticles is given by 1 QP [ωP ] = (20) dr e−ωP (r) . V

(21)

q(j, r) eωB (r) q † (N − j, r), (25)

j =NA +1

ρ˜P (r) =

δwI(+) J (r)

ωK (r) = (h ∗ wK )(r),

N

(24)

and

(19)

The chemical potential fields ωA , ωB , and ωP , are the convoluted chemical potential fields given by

NA nD q(j, r) eωA (r) q † (N − j, r), V QD j =1

=

δwI(−) J (r)

+i [ρ˘A (r) + ρ˘B (r) + ρ˘P (r)] ,

(29)

2ρ0 (+) w (r) + i [ρ˘I (r) + ρ˘J (r)] , χI J I J

(30)

=

2ρ0 (−) w (r) + ρ˘J (r) − ρ˘I (r). χI J I J

(31)

B. Anisotropic nanoparticles

As a final example of how our approach can be extended to a wide range of polymer nanocomposites, we outline how the above model changes for anisotropic nanoparticles. Specifically, we focus on nanorods with a cylindrical shape; at the particle level, such a nanoparticle must be specified by both its position and its orientation so that the total rod


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center density is given by nP ρˆP ,c (r) = du δ(r − ri ) δ(u − ui ),

J. Chem. Phys. 139, 244911 (2013)

(32)

i

where the integral over u sums all possible rod orientations, ui . For these anisotropic particles, the density smearing function is also orientation-dependent, and the form that we adopt is given by

ρ0 |u · (r − r )| − L/2 (|r − r |, u) = erfc 4 ξ

|u × (r − r )| − R , (33) ×erfc ξ where L and R are the length and radius of the cylindrical nanoparticle, respectively. It should be noted that explicit orientational interactions are not incorporated into the model. The Hamiltonian for a diblock copolymer system containing bare cylindrical nanorods is identical to Eq. (17), only the form of the partition function associated with the nanorods QP becomes 1 QP [wP ] = dr du exp [−( ∗ wP )(r, u)] , 4π V (34) where we have explicitly written out the dependence of the convolution of with the field wP on both r and u to emphasize that this convolution must be performed for each nanorod orientation, u. Also, note that the factor of 4π is for a 3D calculation; this factor is 2π for a 2D system. The density operator for anisotropic particle centers with a given orientation u is given as nP e−(∗wP )(r,u) , (35) ρ˜P (r, u) = 4π V QP and the total nanorod density is given by accumulating the convoluted nanorod density from each possible orientation, (36) ρ˘P (r) = du ( ∗ ρ˜P )(r, u). The local ordering of the nanorods can also be quantified by first defining a tensor order parameter,

du uα uβ − 13 δα,β ρ˜P (r, u) , (37) Sα,β (r) = du ρ˜P (r, u), where α and β refer to the Cartesian directions and δ α, β is the Kronecker delta function. The largest eigenvalue then quantifies the extent of local ordering, while its associated eigenvector indicates the primary direction of alignment.84 III. NUMERICAL METHODS

In the models presented above, we have adopted the discrete Gaussian chain description of our polymers, so to calculate the relevant propagators for our system we have to iterate the Chapman-Kolmogorov equation above in Eq. (18). Since Eq. (18) is in the form of a series of convolutions, these iterations can be carried out efficiently using a series of Fourier transform/inverse Fourier transform pairs, which we perform

using a parallel implementation of the Fastest Fourier Transform in the West (FFTW) library.85 For anisotropic particles, we discretize the unit sphere using a combination of spherical coordinates and Gaussian-Legendre weights to calculate the spatial coordinates of the aforementioned orientation vector, u. The spherical angles, θ and φ, are discretized into Nu and 2Nu points, respectively, so that the total number of points on the unit sphere is 2Nu 2 . Sampling the fully fluctuating field theory is not trivial due to the complex nature of the effective Hamiltonian equation (17) that arises from the particle-to-field transformation, which precludes the use of standard sampling techniques such as MC. For polymer field theories, complex Langevin (CL) sampling60, 61, 71–73 has emerged as a powerful tool for avoiding the so-called “sign problem” and efficiently sampling the complex fields appearing in Eqs. (15) and (17). To implement a CL simulation, the integrals in the partition function Eq. (15) are extended over the entire complex plane, and the fields are each evolved according to δH ∂w(r) = −λ + η(r, t), (38) ∂t δw(r) where η(t) is a Gaussian white noise term that obeys the statistics η(r, t) = 0,

(39)

η(r, t)η(r , t ) = 2λδ(r − r )δ(t − t ).

(40)

We emphasize that the t variable in Eqs. (38)–(40) does not represent physical time, and evolving the fields using Eq. (38) does not lead to a realistic evolution of the dynamics of our systems. Averages calculated over a trajectory evolved according to Eq. (38) where w(r) is free to explore the entire complex plane correspond to averages taken in the original equilibrium system, where w(r) should in principle be integrated along the real axis.60, 72, 73 Our implementation uses a discretized form of Eq. (38) that is numerically integrated using a first-order operator splitting technique.71 The mean-field calculations presented below numerically integrate Eq. (38) with the noise term set to zero. IV. RESULTS A. Gaussian fluid

The first goal of this work is to demonstrate that the particle smearing approach leads to correlations in the particle centers in a manner that is expected from a particle simulation with a finite-ranged potential. To do so, it is instructive to begin by analysing some results for a monomeric Gaussian fluid, which can be realized by applying the above polymer model with N = NA = 1 and nP = 0, and comparing the structure predicted by the field-theoretic simulations with that of a pure particle-based simulation. For a field theoretic simulation containing only monomers of the A component, all of the potentials involving χ are neglected, and the only potential that contributes to the Hamiltonian is the compressibility term Eq. (12), which can be simplified to



κ 2ρ0

=

κ 2ρ0

dr [ρÂ (r) − ρ0 ]2

(41)

nκ , 2

(42)

dr ρÂ (r)2 −

0.9

where the nκ term is an irrelevant constant shift in the potential energy. The potential energy calculated in Eq. (42) is between the smeared particle densities. To perform an equivalent particle-based simulation we require the effective potential between the particle centers, which can be derived as κ dr ρÂ (r)2 2ρ0 κ = 2ρ0 =

1 2

dr

dr

dr ρÂ (r) δ(r − r ) ρÂ (r )

(43)

dr ρÂ,c (r) v(r − r ) ρÂ,c (r ),

(44)

with κ −|r−r |2 /4b2 v(r − r ) = e . 4πρ0

1

(45)

A Monte Carlo particle-based simulation using Eq. (45) should be exactly equal to a field-theoretic simulation of the above model with N = NA = 1 in the absence of nanoparticles. Next, we require a means to calculate the pair correlation function from the field-theoretic simulation that is equivalent to the particle-based approach, which is ρÂ,c (r)ρÂ,c (r )/ρ02 . In the field-theoretic approach, expressions for the pair correlation function are typically derived using a double functional integration by parts to express an operator for the pair correlation function as a correlation of the field variables.60, 61 However, the integration by parts approach fails for the smeared particle densities due to the presence of h(r) in each of the ln Q terms, which implies that the segment center density cannot be isolated in a manner which allows the integration by parts. To circumvent this issue for these demonstrative calculations, when we perform the particle-to-field transformation above, one “tagged” particle is left in the particle representation fixed at the center of the simulation box, and its degrees of freedom are not integrated out. This is equivalent to the hybrid particle-field approach of Sides et al.,66 only the segment that retains its explicit coordinates in the partition function is identical to the segments that are represented as fields. We can then calculate the average distribution of the field-based segment centers around the explicit particle to obtain a pair correlation function that can be directly compared to the particle-based Monte Carlo simulation. Figure 1 shows the pair distribution function g(r) calculated from a two-dimensional particle-based Monte Carlo simulation, a complex Langevin field-theoretic simulation, and a mean-field calculation for the Gaussian fluid interacting with the potentials given by Eqs. (42) and (45) for the particleand field-based calculations, respectively, at κ = 40 and 200. The field-theoretic approach is able to capture the liquid-like

1.04

g(r)

βU2 =

J. Chem. Phys. 139, 244911 (2013)

g(r)

244911-6

0.8

1.02 1 0.98

0.7

0.96 1.5

0.6 0

2

3

2.5

3.5

r [b] 1

2

3

4

5

6

7

8

r [b] FIG. 1. Comparison of the pair distribution function calculated from a particle-based Monte Carlo simulation (solid lines), a field-theoretic simulation using CL (red squares), and a mean-field calculation (blue diamonds). The solid lines represent calculations with κ = 40 and the dashed line κ = 200. The error bars on the CL calculations are taken as the standard error after averaging over three independent CL trajectories. The inset shows a close-up of the first peak in g(r). In the main figure, only every third point from the field-based calculations is shown for clarity.

oscillations in the pair distribution function that are present in the Monte Carlo simulation, and all of the points agree with the particle-based calculation within the resolution of our calculations. Interestingly, the mean-field calculation captures the correlations in the segment positions, and it is numerically indistinguishable from the fluctuating field theory. This implies that the particle smearing approach does indeed capture correlations in the particle positions that are not detectable in a theory with point particles. Another common situation where these oscillations are expected to be prevalent is in the layering of the particle density near a hard wall such as a hard substrate or the surface of a nanoparticle; it is known that such layering is missed by typical implementations of polymer field theories using point particles.60 We include a hard wall in the center of our simulation box such that the middle of the wall sits at y = 0. We use a masking function that defines a wall density and is included in ρˆ+ (r) above in Eq. (12). The masking function then imparts a repulsion onto the polymer segments whose strength is governed by κ. A 2D mean-field calculation is conducted where periodic boundary conditions are incorporated in all 4 directions. From those calculations, a density profile averaged over the x-direction is shown in Figure 2 for a homopolymer melt with N = NA = 15, where the liquid-like oscillations in the segment density are clearly visible. In order to emphasize the liquid-like oscillations, only the positive y-values are shown as it represents an exact mirror image of the negative y-values. B. Block copolymers with spherical nanoparticles

Having demonstrated that giving the segments a finite size leads to correlations in the particle positions, we next turn our attention to a system consisting solely of nanoparticles. First, we wish to emphasize that the nanoparticles included in this model are not strictly hard spheres except in the incompressible limit where κ → ∞. For two chemically identical nanoparticles positioned at r1 and r2 whose smearing


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J. Chem. Phys. 139, 244911 (2013)

κ=

2

1 10 30 100 300 1000 3000

g(r) + C

ρ(y)

1.5

1

0.5

0

10

5

20

15

FIG. 2. Distribution of segment centers ρÃ (y) (solid black line), total segment density ρ˘A (y) (dashed black line), and the surface density (solid red line) for a polymer melt confined by a repulsive wall. These results were obtained from a 2D mean-field calculation with N = NA = 15, κ = 50, and a simulation box of size Lx = 10b by Ly = 40b resolved in a grid with dimension Nx = 63 by Ny = 245.

function is given by Eq. (10), the effective pair potential between the particle centers is given by the overlap of the two smearing functions, κ βu(|r1 − r2 |) = dr (r − r1 ) (r − r2 ). (46) 2ρ0 This function changes smoothly from 0 to κvP /2 as the particle centers approach, as shown in Figure 3. It is important to note that the nanoparticle smearing functions (r) and hence the volume vP depend on the segment density, ρ 0 . This implies that at higher polymer concentrations where the segment density increases, such as those employed in the results presented below, the repulsion represented by Eq. (46) quickly becomes many factors of kB T. For example, if our system is parameterized such that the polymer concentration is C ≈ 17 for polymer chains that are discretized into N = 40 sites, then the corresponding segment density is ρ 0 = 40. Using κ = 1.5, the pair potential shown in Figure 3 for particles with Rp = 2 reaches ∼10kB T when the nanoparticles are separated by a distance r ≈ 3.65, which demonstrates that the repulsion between nanoparticle centers quickly becomes very strong. To more clearly demonstrate that the field-theoretic treatment prevents overlap of the nanoparticles for a sufficiently

2 u(|r1 - r2|) / κ vP

1 0.8 0.6 0.4 0.2 0 0

1

2

3

0

0.5

1

1.5

2

2.5

r [2Rp]

y [b]

4

5

6

|r1 - r2| [b] FIG. 3. Effective pair potential between two nanoparticle centers given by Eq. (46) for two nanoparticles with Rp = 2, ξ = 0.1.

FIG. 4. Pair distribution function g(r) calculated between particle centers from a mean-field calculation containing only nanoparticles with a radius of Rp = 1 and an interfacial width parameter ξ = 0.1 for different values of κ and the value of ρ 0 was fixed at ρ 0 = 1. The particles only interact through the repulsive potential given by Eq. (46). The particle volume fraction is φ P ≈ 0.57, and the each curve is shifted vertically by a constant factor C for clarity.

large κ, we have performed a series of mean-field calculations comparable to those performed above for the Gaussian fluid. We fixed a single nanoparticle in the center of a twodimensional simulation box, and then calculated the density of the nanoparticles surrounding the central particle as a function of the strength of the repulsion, κ. Figure 4 plots the pair distribution function, g(r), as we increase the values of κ. We observe a transition from soft, slightly overlapping nanoparticles at κ = 1 to liquid-like layering for κ = 10–100. For values of κ ≈ 300 and larger, we calculate a structure closer to crystalline order where the pair distribution function becomes approximately zero after the nearest-neighbor peak. In addition, we observe√the emergence of a shoulder on the second peak close to 2 3Rp , corresponding to the second-nearest neighbor of a triangular crystalline lattice, which would be the expected crystalline lattice for 2D hard spheres. After exhibiting the ability to control the strength of repulsion between nanoparticles within our model’s framework, we now focus on a system containing both block copolymers and nanoparticles. Figure 5 shows an image of a 3D CL simulation averaged over approximately 38 000 CL time steps. We observe both spherical domains that are of the order of the size of a single particle as well as larger domains that represent clusters of particles. Under the mean-field approximation, the nanoparticle density does not exhibit the correlations observed in Figure 5. When we calculate the particle density profiles, the distribution of particle centers is dualpeaked in both the mean-field and the CL simulations (Figure 6(a)), but the field fluctuations cause the peaks to separate further and broaden the distribution. The distribution of the total particle density ρ˘P (r) is also dual-peaked in the fluctuating field theory (Figure 6(b)). In contrast, the mean-field calculations show a single peak in the total nanoparticle density because the smearing functions cause the two particle center peaks to overlap. After taking into account the difference in dimension (2D vs. 3D) as well as the different values of ρ 0 used to generate Figure 4 against Figures 5 and 6, the calculations shown in Figures 5 and 6 would correspond to κ ≈ 67 in Figure 4; this value is more than sufficient to exclude the


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mal number of points in which to discretize the unit sphere, (Nu ), in an effort to optimize computation time with accuracy. We plot the nanorods’ excess chemical potential and the average orientation angle of nanorods (θ ) with respect to the z axis as a function of Nu in Figures 7(a) and 7(b), respectively. We observe that the excess chemical potential of the nanorods quickly converges when Nu is increased from 4 to 14. Similarly, we can see that the average orientation angle of the nanorods quickly saturates for Nu values greater than 8. Based on these results, we fix Nu at 12 for the ensuing calculations. The computational expense of calculating both the nanorod partition function QP and the density operators for the nanorods scale as 2Nu 2 Mlog (M), and so for Nu = 12, the rods are only slightly more expensive to incorporate in the simulation than the diblocks. We next focus on the spatial distribution and orientation of the nanorods in the block copolymer system. In Figures 8(a) and 8(b), we present a visual representation of the nanorods’ spatial distribution and orientation vector, respectively, within a diblock copolymer melt. Similar to the corresponding results for the spherical particles, we observe that in short CL simulations, the nanorods form clusters inside the A layer of the diblock. The nanorods’ center distribution averaged over the xy plane is shown in Figure 9(a) and the average angle between the long axis of the rods with the unit normal in Figure 9(b) as a function of the z position for both CL and mean-field calculations. Figure 9(a) shows that field fluctuations lead to a broader distribution in the particle center density than the mean-field calculations where the nanorod centers are primarily concentrated in the middle of the A layer. Interestingly, the average orientation shows that in the CL simulations the nanorods closer to the interface are more aligned with the interfacial plane, while the nanorods in the middle of the A layer have room to rotate and sample a wide distribution of angles. However, mean-field calculations show that essentially all of the nanorods lie in the plane of the lamellae and are tightly pinned to the center of the domain.

FIG. 5. Visualization of the local volume fraction of the A-block of a block copolymer containing spherical nanoparticles averaged over a short complex Langevin simulation with C = 21.52, χ AB N = χ PB N = 20, χ AP N = 0, RP = 0.387Rg , fA = 0.375 and a nanoparticle volume fraction of 5%. The isosurfaces enclose regions where the A density is given by the colors on the colorbar, while the solid black surfaces enclose regions where the nanoparticle volume fraction is greater than 0.3. The remaining parameters of these simulations were κN = 40, δx = 0.237Rg , and the chains were discretized into N = 40 sites; the units of the axes are Rg .

nanoparticle cores from each other. We note that the simulation box length normal to the lamellae was held fixed at Lz ≈ 8.29Rg ; the optimal box length for the CL simulations may differ from the mean-field calculations. While schemes exist that enable variation of the simulation box size and shape for field-theoretic simulations,86, 87 they are not implemented in the present work. C. Anisotropic nanoparticles

V. SUMMARY AND DISCUSSION

Finally we provide representative calculations examining the distribution of anisotropic nanorods within a lamellae forming block copolymer melt. First, we investigate the opti-

In this paper, we have presented a technique to incorporate finite sized particles into field theoretic simulations. We

0.05 0.15

φP(z)

ρP,c(z)

0.04 0.03

0.1

0.02 0.05 0.01 0 0

2

4

z [Rg]

6

8

0

2

4

z [Rg]

6

8

FIG. 6. Comparison of the distribution of nanoparticle centers ρ˜P (r) (a) and the total nanoparticle volume fraction φP (z) = ρ˘P (z)/ρ0 (b) when calculated using CL (solid black line) compared to a SCFT calculation (dashed red line). The CL results are taken from the simulations used to generate Figure 5, while the SCFT calculations used identical parameters but with a grid spacing of δx = 0.1315Rg .


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J. Chem. Phys. 139, 244911 (2013)

8 6 65 4

μex,P - μex,P( Nu = 14)

70

60

2 55 0 50 -2 2

4

6

8

Nu

10

12

4

14

6

8

Nu

10

12

14

FIG. 7. Convergence properties of CL simulations with discretization of the unit sphere. The excess chemical potential of the nanorods μex, P = − ln Qp (shifted by the value at θ u = 14) (a) and averaged angle θ between the z axis and the nanorods (b) are plotted as a functions of Nu .These results are generated from 3D CL simulations of a block copolymer system with χ AB N = χ BP N = 32, κN = 52, N = 40, fA = 0.375, and C = 21.52. The simulation box is of size Lx = Ly = 4.678Rg , Lz = 4.376Rg with δx = 0.234. The radius of nanorods is Rp = 0.387Rg , their aspect ratio is 3, and the volume fraction of nanorods is fixed at 3%.

0.025

100

0.02

80

0.015

60

ρp,c (z)

FIG. 8. Visualization of the local volume fraction of the block copolymer A segments with nanorods (a) and orientation vectors of nanorods (b) averaged over 50 000 CL time steps. The system is identical to that depicted in Figure 7, except Lz is expanded to ∼8.30 Rg and Nu is fixed at 12.

0.01

40

0.005

20

0 0

2

4

z [Rg]

6

8

0 0

2

4

z [Rg]

6

8

FIG. 9. Comparison of the distribution of nanoparticle centers ρ˜P ,c (r) (a) and the average angle θ between the long axis of the rods and the z axis (b) when calculated using CL (solid black line) compared to a mean-field calculation (dashed red line). These systems are identical to those shown above in Figure 8, only the results are taken from an average over 400 000 CL time steps.


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have shown that a generic smearing function can be traced through the particle-to-field transformation such that finite sized particles can be incorporated in an exclusively fieldbased method. The mathematical outline in this approach merits that any smearing function may be used as long as the chosen smearing function is positive semi-definite and can be efficiently numerically evaluated. We have provided simple systems to assess the validity of this technique and illustrate the potential for which our approach has to offer: (a) a Gaussian fluid; (b) pure nanoparticles; (c) block copolymers with spherical nanoparticles; and (d) block copolymers with anistropic particles. The Gaussian fluid provides a platform to perform an exact computational comparison between a particle and field based simulations, and it demonstrates that the structures of the fluid predicted by both the particle and field based simulations are quantitatively identical. The pure nanoparticle system shows how the strength of repulsion can easily be controlled such that it may be possible to approach the hard sphere limit in a field theory context. Finally, the distribution of spherical and rod like nanoparticles in diblock copolymer matrices is qualitatively consistent with expectations based on previous simulation studies and experiments. The method introduced here is formally identical to the hybrid particle-field method when no additional potentials are introduced between the nanoparticles. The utility of this approach is embedded in the ability to model finite sized particles while retaining the inherent advantages of field theory. Namely, our approach circumvents the need to augment the free-energy expression in a manner similar to the hybrid SCFT/DFT approach, the complexities associated with maintaining the particle coordinates in the HPF Hamiltonian, as well as computational expense or equilibration problems that are involved with particle-based simulations. Although incorporating fluctuation effects in a field theory requires advanced sampling techniques such as complex Langevin sampling, recent advances in the numerical algorithms used for polymer field theory make these simulations computationally tractable.71, 88–90 Additionally, it is trivial to parallelize a field based simulation,91 which is a significant advantage over Monte Carlo approaches. Many of the approximations for our description of the nanoparticles are controlled by numerical parameters, such as the interfacial width ξ or the effective repulsion between the nanoparticle cores that is controlled by the compressibility potential and κ; this enables a systematic investigation of the effect of the numerical parameters on the resulting properties. Finally, we note that this approach can easily be modified to incorporate charged nanoparticles, and we are currently working to expand this technique to both grafted and Janus nanoparticles, which will greatly extend the range of applications of the method. While our technique provides several advantages outlined above, it has some inherent limitations. Specifically, the nanoparticles in our system are soft for finite values of kappa; while we demonstrated that we can model hard nanoparticles for large kappa values, large kappa values can lead to CL trajectories that are more prone to numerical instabilities. Additionally it remains a challenge to incorporate sophisticated particle-particle interactions which are easily implemented in

J. Chem. Phys. 139, 244911 (2013)

particle-based or HPF simulations.39, 92 In the same regard, the polymer-particle interactions are currently governed by a chi parameter as a crude approximation and it remains unclear how to implement more advanced interaction potentials. 1 A.

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Efficient field-theoretic simulation of polymer solutions.

Synthesis of Polymer-Mesoporous Silica Nanocomposites.

Polymer conformations in polymer nanocomposites containing spherical nanoparticles.

Surface dynamics of miscible polymer blend nanocomposites.

Self-consistent field model simulations for statistics of amorphous polymer chains in crystalline lamellar structures.

Molecular dynamics simulations of the structural, mechanical and visco-elastic properties of polymer nanocomposites filled with grafted nanoparticles.

Polyhedral Oligomeric Silsesquioxane (POSS)-Containing Polymer Nanocomposites.

Biomimetic Reversible Heat-Stiffening Polymer Nanocomposites.

Lattice Monte Carlo simulations of polymer melts.

Rouse mode analysis of chain relaxation in polymer nanocomposites.

Green aqueous surface modification of polypropylene for novel polymer nanocomposites.

Interfacial and bulk nanostructure of liquid polymer nanocomposites.

polymer nanocomposites.

Clay Nanocomposites.

Biodegradable polymer iron oxide nanocomposites: the future of biocompatible magnetism.

Flexible high-temperature dielectric materials from polymer nanocomposites.

Highly electrically conductive nanocomposites based on polymer-infused graphene sponges.

Novel kojic acid-polymer-based magnetic nanocomposites for medical applications.

Polymer-Ag nanocomposites with enhanced antimicrobial activity against bacterial infection.

Microscopic Chain Motion in Polymer Nanocomposites with Dynamically Asymmetric Interphases.

Stimuli-Responsive Polymer-Clay Nanocomposites under Electric Fields.

Block-copolymer-mediated nanoparticle dispersion and assembly in polymer nanocomposites.

Piezoresistance in polymer nanocomposites with high aspect ratio particles.

Carbon Nanotube Nanocomposites and Their Electrorheological Response.