Multiscale modeling of binary polymer mixtures: Scale bridging in the athermal and thermal regime.
ABSTRACT Obtaining a rigorous and reliable method for linking computer simulations of polymer blends and composites at different length scales of interest is a highly desirable goal in soft matter physics. In this paper a multiscale modeling procedure is presented for the efficient calculation of the static structural properties of binary homopolymer blends. The procedure combines computer simulations of polymer chains on two different length scales, using a united atom representation for the finer structure and a highly coarse-grained approach on the mesoscale, where chains are represented as soft colloidal particles interacting through an effective potential. A method for combining the structural information by inverse mapping is discussed, allowing for the efficient calculation of partial correlation functions, which are compared with results from full united atom simulations. The structure of several polymer mixtures is obtained in an efficient manner for several mixtures in the homogeneous region of the phase diagram. The method is then extended to incorporate thermal fluctuations through an effective χ parameter. Since the approach is analytical, it is fully transferable to numerous systems.
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ABSTRACT: A coarse-grained (CG) model of polyethylene glycol (PEG) was developed and implemented in CG molecular dynamics (MD) simulations of PEG chains with degree of polymerization (DP) 20 and 40. In the model, two repeat units of PEG are grouped as one CG bead. Atomistic MD simulation of PEG chains with DP = 20 was first conducted to obtain the bonded structural probability distribution functions (PDFs) and nonbonded pair correlation function (PCF) of the CG beads. The bonded CG potentials are obtained by simple inversion of the corresponding PDFs. The CG nonbonded potential is parameterized to the PCF using both an inversion procedure based on the Ornstein-Zernike equation with the Percus-Yevick approximation (OZPY(-1)) and a combination of OZPY(-1) with the iterative Boltzmann inversion (IBI) method (OZPY(-1)+IBI). As a simple one step method, the OZPY(-1) method possesses an advantage in computational efficiency. Using the potential from OZPY(-1) as an initial guess, the IBI method shows fast convergence. The coarse-grained molecular dynamics (CGMD) simulations of PEG chains with DP = 20 using potentials from both methods satisfactorily reproduce the structural properties from atomistic MD simulation of the same systems. The OZPY(-1)+IBI method yields better agreement than the OZPY(-1) method alone. The new CG model and CG potentials from OZPY(-1)+IBI method was further tested through CGMD simulation of PEG with DP = 40 system. No significant changes are observed in the comparison of PCFs from CGMD simulations of PEG with DP = 20 and 40 systems indicating that the potential is independent of chain length.The Journal of Chemical Physics 12/2011; 135(21):214903. · 3.12 Impact Factor -
Article: Effective potentials for representing polymers in melts as chains of interacting soft particles.
[Show abstract] [Hide abstract]
ABSTRACT: This paper outlines the derivation of an analytical pair potential in a coarse grained description of polymer melts where each chain is represented as a collection of soft spheres. Each particle is located at the center of mass of a polymer subchain, while the polymer is divided into an arbitrary number of identical chain subsections, each comprised of a large number of monomers. It is demonstrated that the soft effective pair potentials acting between these center-of-mass sites is described by a soft repulsive region at separation distances less than the average size of each coarse grained unit and a long repulsive tail, with a small attractive component. The attractive component is located at a length scale beyond the size of the coarse grained unit and its form varies with the level of interpenetration between the coarse-grained units. Consistent with numerically derived potentials, it is found that the short range features of the potential dominate the liquid structure, while the long-tail features dominate the virial-route thermodynamics of the system. It follows that the accurate determination of the effective potential in both short and large separation distances is relevant for ensuring structural and thermodynamic consistency in the coarse-grained description of the macromolecular liquid. It is further shown that due to the sensitivity of thermodynamic properties to the large-scale features of the potential, which are irrelevant to the reproducibility of structural correlations, the determination of thermodynamically accurate potentials by numerical optimization of structure alone is not a reliable strategy in the high-density regime for high levels of coarse-graining.The Journal of Chemical Physics 09/2013; 139(12):124906. · 3.12 Impact Factor - SourceAvailable from: Marina Guenza[Show abstract] [Hide abstract]
ABSTRACT: We present the equation of state for a coarse-grained model of polymer melts where each chain is represented as a soft colloidal particle centered on its center-of-mass. The formalism is based on the solution of the Ornstein−Zernike equation and is analytical, allowing for the formal investigation of the elements that ensure thermodynamic consistency in coarse-grained models of polymer melts. By comparing predictions from our expressions with those from computer simulations of the coarse-grained system and with atomistic polymer integral equation theory, we demonstrate that both structural and thermodynamic consistency with the atomistic level description is maintained during our coarse-graining procedures.
Page 1
Multiscale Modeling of Binary Polymer Mixtures: Scale Bridging in the Athermal
and Thermal Regime
J. McCarty and M.G. Guenza∗
Department of Chemistry and Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403
(Dated: August 6, 2010)
Obtaining a rigorous and reliable method for linking computer simulations of polymer blends and
composites at different length scales of interest is a highly desirable goal in soft matter physics.
In this paper a multiscale modeling procedure is presented for the efficient calculation of the static
structural properties of binary homopolymer blends. The procedure combines computer simulations
of polymer chains on two different length scales, using a united atom representation for the finer
structure and a highly coarse-grained approach on the meso-scale, where chains are represented
as soft colloidal particles interacting through an effective potential. A method for combining the
structural information by inverse mapping is discussed, allowing for the efficient calculation of
partial correlation functions, which are compared with results from full united atom simulations.
The structure of several polymer mixtures is obtained in an efficient manner for several mixtures
in the homogeneous region of the phase diagram. The method is then extended to incorporate
thermal fluctuations through an effective χ parameter. Since the approach is analytical, it is fully
transferable to numerous systems.
I.INTRODUCTION
Mixtures of polymers with different compositions are
of great interest for many technological and industrial ap-
plications. For example, multicomponent polymer mix-
tures facilitate the custom tailoring of desired physical
properties, leading to the creation of new materials with
enhanced performance.[1] However, since new materials
are typically processed in their liquid phase, the ability
to predict the mixing behavior of various polymer com-
posites is a desirable computational task. Because of the
underlying phase diagram, polymer mixtures can phase
separate at specific thermodynamic conditions of temper-
ature and polymer volume fraction. The structure and
dynamics of each component strongly depends on the re-
gion of the phase diagram that is explored, and different
length scales characterize the thermodynamic and dy-
namical behavior of the mixture, depending on how far
the system is from its phase separation.
While computer simulations are useful in elucidating
the structural and dynamic complexity of these systems,
they are limited in the extent of information that can
be collected by the precision of the calculation, which
degrades with the number of simulation steps, and by
the power of the available computational machines.[2] Al-
ready for polymer liquids it is not possible to collect all
the relevant information from a single simulation when
long polymers are involved, as information spreads over
many order of magnitude in time and length, going from
the monomer length scale, l, to the overall polymer di-
mension of the radius-of-gyration, Rg =
latter scales as the square root of the degree of poly-
merization, N, which can be fairly large for polymers of
√Nl/6. The
∗Author to whom correspondence should be addressed. Electronic
mail: mguenza@uoregon.edu
interest.[3] The added complexity of dealing with poly-
mer mixtures arises from the presence of a new length
scale of concentration fluctuations, ξφ, which diverges
as the system approaches its spinodal curve. The big
challenge for such systems becomes simulating the mix-
tures not only at different temperatures and composi-
tions, but also with a box size that increases with the
scale of concentration fluctuations.
equilibration step in molecular dynamics (MD) simula-
tions becomes extremely slow, as the time scale in which
polymer mixtures phase separate diverges as the temper-
ature approaches the critical point. Finally the timescale
of relaxation of the mixture depends on the proximity of
the system to the glass transition of its components. In
conclusion, classical MD simulations of polymer mixtures
have been limited so far to high temperatures close to the
athermal limit. In that region the thermodynamic be-
havior is mainly guided by entropy and local packing[3],
while enthalpic effects may be accounted for within a high
temperature perturbative framework.[4, 5]
Furthermore, the
In order to extend the ability of MD simulations to
make quantitative predictions about polymer blends at
temperatures approaching the spinodal, information is
needed on multiple length-scales of interest.
system approaches its phase transition and the largest
length scale of interest diverges, one might be tempted
to neglect the details of the local scale; however, such
a procedure risks losing pertinent information, since the
monomeric structure of a blend’s component largely de-
termines its glass transition temperature, as well as the
shape of its phase diagram. Nonetheless, capturing the
global structure is equally as important since an accurate
determination of the thermodynamics of the system re-
quires obtaining the total distribution of particles in the
ensemble. For example, while system-specific local infor-
mation is contained within the correlation hole region of
the pair distribution function, g(r), the global structure
is still required, as truncation of g(r) will prevent accu-
As the
arXiv:1008.1038v1 [cond-mat.soft] 5 Aug 2010
Page 2
2
rate Fourier transform to the structure factor where the
low k limit is important.[6] Thus, both local and global
scale information need to be accounted for to achieve a
complete thermodynamic and dynamic description of the
mixture.
One strategy to improve our ability to simulate mix-
tures of complex fluids and collect information on an in-
creasingly large range of length scales is to use a hier-
archical approach, such as the multiscale modeling pro-
cedure that we present in this paper. In such a multi-
scaling scheme the system of interest is represented at
various levels of coarse-graining and simulations are per-
formed for each of these coarse-grained representations to
enhance computational efficiency. In a second step, the
information obtained from those simulations is collected
and combined into a complete description, bridging in-
formation at all the length scales of interest.[7]
For any coarse-grained model to be capable of faith-
fully reproducing the desired characteristics of the sys-
tem, the coarse level of description needs to include the
essential features of the atomistic structure while aver-
aging out the remaining degrees of freedom.
coarse-graining procedures have been developed in the
literature at various levels of “coarseness”. At the lowest
level, the united atom (UA) description, which represents
groups of CHxunits, with x = 1,2, or 3 into an effective
UA along the chain, is a particularly useful represen-
tation, for which potentials have been optimized to re-
produce the structure for most polyolefins.[8] Higher lev-
els of structure-based approaches simplify interactions,
while remaining close to the chemical structure, facilitat-
ing scale hopping between the different levels[9]; however,
these methods remain computationally demanding and
do not adopt an extreme enough level of coarse-graining
to mitigate the limitations of fully atomistic simulations
in reaching the large length scales of phase separation.[10]
In order to simulate the increasingly large scale phenom-
ena of interest to material scientists, an even coarser level
of description is needed, while remaining formally related
to the finer structure.
The highest level of coarse-graining in our procedure,
represents each polymer chain as a soft colloidal particle,
centered at the polymer center-of-mass. Soft colloids in-
teract through a Gaussian-like repulsive potential of the
order of the polymer radius-of-gyration, Rg, which de-
fines the characteristic length scale of this description.
Large-scale behavior is easily generated in a “mesoscale”
simulation of the coarse-grained, mixture and informa-
tion is collected on a scale equal to or larger than the
polymer radius-of-gyration.
that our mesoscale simulations capture the global struc-
ture of the polymer mixture providing information over
much of the miscible region of the phase diagram.[11]
Much attention has been given recently to relating
mesoscopic[12] or field-theoretical model parameters[13]
to their more detailed atomistic representation. For ex-
ample, Groot and Warren were able to express the re-
pulsive soft potential parameters used in dissipative par-
Several
Recently, we have shown
ticle dynamics (DPD) in terms of the Flory-Huggins χ
parameter, providing a bridge between DPD simulations
and atomistic simulations, from which solubility parame-
ters can be directly calculated.[14] This method has been
successfully implemented in mesoscale simulations of bi-
ological membranes once the model is parameterized for
specific molecular structures.[15] In our approach, the ef-
fective χ parameter is included directly in the model as an
input parameter which determines the length scale of the
of concentration fluctuations. This χ parameter provides
a convenient link between an atomistic or united atom
level description and a mesoscale picture, similar to other
mesoscale modeling approaches;[13, 14, 16] however, our
approach has the advantage over other methods such as
DPD in that the soft-core potential depends directly on
the χ parameter and not on any phenomenological ex-
pression to bridge the two theories. In other words, once
the χ parameter is specified, our multiscale modeling pro-
cedure will produce the expected phase behavior[11], and
the task in practice is to determine this parameter for a
given model.
In the current paper, we focus on bridging the length
scales of binary homopolymer mixtures of different chem-
ical architecture to provide a complete description of the
static structure of the blend under a given set of ther-
modynamic conditions. Blends of polyolefins represent a
good test case for our procedure since united atom pa-
rameters have been well optimized for these systems, and
they provide a stringent test of the method, as subtle
differences in chemical architecture, such as the extent of
branching, greatly affects the global structure and mis-
cibility. One advantage of our coarse-graining and mul-
tiscale modeling approaches is that since they are based
on the formalism of the Ornstein-Zernike equation, the
rescaling of the structure to recover the monomer level
of description is straightforward. Furthermore, we have
shown that the correct dynamics can be obtained di-
rectly from simulations of the coarse-grained structures
by proper rescaling.[17] Thus, by performing simulations
of mesoscopic particles and subsequently reinserting the
relevant chemical details afterwards, we are able to ob-
tain quantitative information about large-scale proper-
ties.
Large scale information, obtained form mesoscale MD
simulations, is later combined with local-scale united-
atom simulations of the same system, using a multiscale
formalism obtained from the solution of the Ornstein-
Zernike equation, presented in this paper. In a spirit
similar to the united atom simulations to which we test
our approach, we initially take the high temperature limit
as a reference state, and set χ = 0 in order to calculate
monomer level total correlation functions. The model is
then extended to incorporate thermal effects on the con-
centration fluctuation through the interaction parameter,
χ, which serves to bridge the gap between mesoscale and
atomistic models.
This multiscale modeling approach, is an extension
to polymer mixtures of our coarse-graining and multi-
Page 3
3
scale modeling procedure for polymer melts.[18] By com-
parison with united atom simulations of polymer mix-
tures, our approach shows good agreement between the
two descriptions. Furthermore, our procedure has the
advantage of being analytically solved, so that thermo-
dynamic parameters defining the free energy potential
appear explicitly in the formalism, making the whole
procedure general and transferible to other systems in
different thermodynamic conditions.
tage of our method lies in the fact that it provides a
quantitative picture of how the local details such as the
specific chemical architecture and thermodynamic con-
ditions are connected to the observed, global properties.
While atomistic and united atom simulations are limited
in the maximum length scale that can be simulated, our
approach easily provides information on a wide range of
length scales, once mesoscale simulations are combined
with atomistic or monomer level simulations. The num-
ber of test systems presented in the paper is determined
by the available united-atom simulations against which
our procedure is tested; however, our procedure is di-
rectly applicable to other mixtures as well.
While the focus here is on the equilibrium structural
properties of simple polyolefins, it is worth noting that
many interesting systems could be investigated through
our method and important problems approached. For
example, the proposed multiscale procedure could be ap-
plied to study various aspects of binary blends, including
critical behavior, mechanical properties and viscosity, the
effects of non-random mixing, stiffness disparity, shape
disparity, and different architectures, as well as the dy-
namics of demixing at equilibrium and under shear, in-
terdiffusion in equilibrium and during phase separation,
and the different propensity for crystallization in the dy-
namics of miscible blends approaching the phase transi-
tion. As one of the polymers in the mixture crystalizes
it is important to investigate, for example, the kinetics
of interface formation and structure. Additionally the
procedure could be extended to encompass mixtures in-
cluding block and homo-copolymers, blends close to sur-
faces and in thin films, and mixtures of colloids and poly-
mers. Although the dynamics of coarse-grained systems
is unrealistically accelerated, we have recently proposed
a new first-principle method to rescale the dynamics of
the coarse-grained system to its atomistic value, enabling
dynamical properties of these systems to be investigated
with quantitative precision.[17] This method can be use-
ful in evaluating long-time dynamics directly from the
mesoscale simulation.
The paper begins with a brief overview of the the-
ory for coarse-graining of polymer blends, from which
we calculate the effective potential, v(r), input to the
mesocale molecular dynamic simulations of the coarse-
grained system.Mesoscale simulations of the coarse-
grained polymer mixture, where chains are represented
as soft colloidal particles, are presented in the following
section. Those simulations provide the large-scale infor-
mation, which is combined with local-scale united-atom
The key advan-
simulations by means of our multiscale modeling proce-
dure. Next, we present our results in the form of the
total correlation function, hα,β(r) with α,β ∈ A,B for
each pair of components, which is predicted and com-
pared with data of full united-atom simulations. First,
we present correlation functions of the mixture in the
limit where χ = 0 to demonstrate the tenor of our scale
bridging approach. This is followed by an extension of
the model to include finite temperature effects, where we
present the change in the concentration structure factor
upon inclusion of realistic values for the χ parameter. A
brief discussion concludes the paper.
II.METHODOLOGY
A.Effective Pair Potentials for Polymer Mixtures
An analytical coarse graining procedure for binary poly-
mer blends was derived by Yatsenko, et al.[19], to for-
mally map the blend onto a mixture of soft-colloidal
particles. Here, we briefly review the results of that
work. The polymer species (A and B) in the blend are
characterized by a radius of gyration, RgA and RgB,
a total density of monomers, ρ, and a segment length
σA=?6/NARgAand σB=?6/NBRgB, where NAand
of the A component is given by φ and the ratio of seg-
ment lengths is given by γ = σA/σB. The main physical
quantity of interest in the study of complex liquids, is the
pair distribution function, or the related total correlation
function, as from this quantity any thermodynamic prop-
erty of interest can be directly calculated.[20] Here the
total correlation function has contributions from three
terms: hAA(k), hBB(k), hAB(k), representing the three
types on interaction of a mixture of particles A and B.
Analytical expressions for the total correlation functions
at the coarse grained level are obtained by solving a gen-
eralized Ornstein-Zernike (OZ) matrix relation
NBare the number of chain units. The volume fraction
H(k) = Ω(k)C(k)[Ω(k) + H(k)] ,(1)
where H(k) is the matrix of total intermolecular partial
correlation functions (pcfs), C(k) is the matrix of direct
pcfs, and Ω(k) represents the matrix of intramolecular
pcfs. Solving Equation 1 gives the relation for a generic
pair[21] {α,β ∈ (A,B)}
?
ωmm
hcc
αβ(k) =
ωcm
αα(k)ωcm
αα(k)ωmm
ββ(k)
ββ(k)
?
hmm
αβ(k),(2)
where ω(k) for each type of chain is the intramolecular
form factor, and the superscripts, cc, cm, or mm, de-
notes center of mass-center of mass, monomer-center of
mass, or monomer-mononomer interactions, respectively.
The derivation of Equation 2 is an extension of the pro-
cedure outlined by Krakoviack, et al.[22] for homopoly-
Page 4
4
mers and formally connects the center of mass (cm) dis-
tribution functions to monomer-monomer (mm) distri-
bution functions. Both the intramolecular form factors
in Equation 2, namely the monomer-monomer ωmm(k)
and the monomer-cm ωcm(k), are expressed in analyt-
ical forms.[23, 24] Yatsenko, et al.
analytical form for the monomer total correlation func-
tions, hmm
αβby extending the PRISM-blend thread model
of Tang and Schweizer[25] to include asymmetries in local
chemical structure and flexibility.[19]
Following this procedure, analytical forms of hcc
coarse-grained at the cm level, can be calculated from
Equation 2. In real space they are given by:
introduced a new
αβ,
hAA(r) =
1 − φ
φ
φ
1 − φIφ
AB(r) + Iφ
Iφ
AA(r) + γ2Iρ
AA(r) ,
hBB(r) =
BB(r) +
1
γ2Iρ
BB(r) ,(3)
hAB(r) = −Iφ
AB(r) ,
with
Iλ
αβ(r) =3
4
?
ξ?
rϑαβ2
?Rgαβ
3
π
ξ?
ρ
Rgαβϑαβ1
?
√3+
2Rgαβ
?
cαβ
ξ2
λ
1 −
ξ2
cαβ
ξ2
λ
?
e−3r2/(4R2
gαβ)
−1
2
ρ
1 −
r√3
ξ2
?2
eR2
gαβ/(3ξ2
?Rgαβ
λ)
(4)
??
×
?
er/ξλerfc
ξλ
?
− er/ξλerfc
ξλ
√3−
r√3
2Rgαβ
and
ϑαβ1 =
?
1 −
?
1 −ξ2
ξ2
cααξ2
ξ2
cββ
cαβξ2
ξ2
cαβ
ξ2
λ
λ
?
?
1 −
,(5)
ϑαβ2 =
?
cαα
ξ2
λ
?
??
ξ2
1 −
ξ2
cββ
ξ2
λ
?
1 −
cαβ
ξ2
λ
?2
,(6)
where ξcαβ is the average correlation hole length scale,
and Rgαβ ≡
radius-of-gyration.Here, ξλ ∈ {ξρ,ξφ} identifies the
length scale for the density and concentration fluctuation
correlations. The concentration fluctuation of a polymer
blend is defined as
σAB
?
where we introduce the miscibility parameter χ for which
the quantity χ/ρ is analogous to the Flory-Huggins in-
teraction parameter[21] such that χ ∝ 1/T. The con-
centration fluctuation, ξφ, diverges at the spinodal tem-
perature where χ = χs. The average segment length is
σ2
A, while the density fluctuation, ξρ,
is defined as ξ−1
Although for compressible blends, there is, in princi-
ple, more than one χ parameter resulting from the dis-
tinct partial osmotic compressibilities of a two compo-
nent blend[26], in this work we adopt a single interaction
?
(R2
gα+ R2
gβ)/2 = ξcαβ
√2 is an average
ξφ=
24χs(1 −
χ
χs)φ(1 − φ)
,(7)
AB= φσ2
B+(1−φ)σ2
ραβ= πρσ2
αβ/3+ξ−1
cαβand ξ?
ρ= 3/(πρσ2
AB).
parameter which determines the length scale of concen-
tration fluctuations. The choice of a single adjustable
parameter is appropriate in this case, since the aim is to
obtain quantitative agreement with experiment, and the
adoption of a single “effective” χ maintains a straight-
forward connection to SANS experiments. The notion of
extracting a single effective χ parameter was also invoked
Dudowicz et al. in a general lattice cluster theory anal-
ysis of compressible blends,[27] and by Schweizer in con-
necting the PRISM blend approach with methods used
in experimental SANS analysis.[28] While we initially set
χ = 0, the extension to more realistic models is achieved
by using the experimental χ parameter within this gen-
eral framework, which has been shown to represent well
both upper and lower critical phase diagrams.[21] Results
presented in Section III B of this paper are examples of
both kinds of systems. In the athermal regime, however,
the theory is presently being implemented to reproduce
quantitatively and in a self-consistent way, the experi-
mental values of χath[29].
Finally, it should be noted that here we are only inter-
ested in capturing the global structure and thus the use
of the simple analytic PRISM thread result is adequate
as it accurately captures the structure of the polymer at
distances greater the Rgand correctly predicts the cor-
relation hole. While the PRISM thread result does not
capture local effects such as solvation shells, such detail is
not relevant at the coarse-grained level since at this level,
local interactions are averaged out. Local packing in the
structural g(r) is introduced later using the multiscale
modeling procedure discussed later, where local scale in-
formation from UA MD simulations is paired with global
information obtained from course-grained simulations.
By adopting the hypernetted-chain (HNC) closure re-
lation, the effective pair interaction potential, vcc
connected to the pcfs by the relationship
αβ(r), is
(kBT)−1vcc
αβ(r) = hcc
αβ(r)−ln[1+hcc
αβ(r)]−ccc
αβ(r) , (8)
where ccc
particles, defined in reciprocal space as
αβis the direct pcf for a mixture of soft colloidal
ccc
αα(k) =
1
ρc,α
−
Scc
ββ(k)
(ρc,α+ ρc,β)|Scc(k)|,
Scc
αβ(k)
(ρc,α+ ρc,β)|Scc(k)|.ccc
αβ(k) = (9)
The chain density, ρc,α, of chain type, A, is given as
ρc,A = φρ/NA, and for chain type, B, as ρc,B = (1 −
φ)ρ/NB. For a binary mixture the static structure fac-
tors, Scc
αβ, are given by
Scc
Scc
Scc
AB(k) = φ(1 − φ)ρchhAB(k) ,
AA(k) = φ + φ2ρchhAA(k) ,
BB(k) = 1 − φ + (1 − φ)2ρchhBB(k) ,
(10)
Page 5
5
where the total chain density ρch= ρ/N, and |Scc(k)| =
Scc
mesoscopic static structure factor matrix. Whereas at
the monomer level of description, molecular closures
are required to ensure the correct scaling of the χ
parameter[30], the use of the site level HNC is appropri-
ate here because at this level of coarse-graining, the poly-
mers are treated as a simple liquid of soft-colloids.[31]
By inserting Equations 3 and 9 into equation 8, the
effective pair potentials are calculated, which are input
to the mesocale simulations of the coarse grained binary
mixture. The potential so obtained explicitly depends
on the structural parameters of the polymer, such as ρ,
N, φ, and Rg. Since these parameters enter into the
UA description, they do not represent additional pa-
rameters needed in moving to a higher level of coarse-
graining. In other words, the potential is state depen-
dent, being a free energy obtained from the monomer
frame of reference; however, it is fully transferable to
different systems since it is analytically derived in a well-
defined manner. In addition to these structural param-
eters, there is the additional parameter, χ, that enters
into the mesoscale description through Equation 7, and
describes the monomer-specific interactions that drive
phase separation.Once these parameters are defined,
mesoscale simulations may be performed and structural
properties calculated for any system of interest.
AA(k)Scc
BB(k) − [Scc
AB(k)]2is the determinant of the
B. Mesoscale Simulations of Coarse-Grained
Polymer Mixtures
In continuing our multi-scaling approach, mesoscale
simulations (MS) of various binary polymer mixtures
with χ = 0 were performed using the effective poten-
tial calculated in the previous section. These simulations
provide a reference system representative of the case in
which the blend is well mixed and fluctuations are small.
Systems investigated were blends of polyethylene (PE),
polyisobutylene (PIB), and polypropylenes in their head-
to-head (hhPP), isotactic (iPP), and syndiotactic (sPP)
forms. Table I shows the simulation parameters used in
both UA and MS descriptions for each polyolefin blend
studied.
TABLE I: Simulation Parameters for Polyolefin Blends (NA =
NB = 96)
Blend [A/B]
hhPP/sPP 0.50
hhPP/PE
PIB/PE
PIB/sPP
iPP/PE
hhPP/PIB 0.50
φρ [sites/˚ A] RgA [˚ A] RgB [˚ A]
0.033212.18
0.033212.32
0.03439.76
0.03439.76
0.0328 11.33
0.034312.41
γ
13.87
16.48
16.38
13.78
16.69
9.69
1.14
1.34
1.68
1.41
1.48
1.28
0.50
0.50
0.50
0.75
Our MS MD simulations were performed on systems
of point particles evolving in the microcanonical (N, V ,
E), ensemble. Initially all particles were placed on a cu-
bic lattice with periodic boundary conditions, where the
type of particle (A/B) occupying a particular lattice site
was determined at random. The potential and its cor-
responding derivative were entered as tabulated inputs
to the program, and linear interpolation was used to de-
termine function values not supplied as the algorithm
proceeds. Each site was given an initial velocity pooled
from a Mersenne Twister random number generator,[32]
and the system subsequently was evolved using a veloc-
ity Verlet integrator. We used reduced units such that all
the units of length were scaled by RgAB(r∗= r/RgAB)
and energies were scaled by kBT.
Equilibrium was induced by a quenching procedure,
in which velocities were rescaled at regular intervals to
maintain the desired temperature. Proper equilibration
was verified by observing a Maxwell-Boltzmann distri-
bution of velocities, a steady temperature, a stabilized
Boltzmann H-theorem function, and a decayed transla-
tional order parameter. Once velocity rescaling was dis-
continued during the production stage, we monitored the
fluctuating steady temperature to ensure the system re-
mains in equilibrium throughout the duration of the sim-
ulation.
During the production stage, trajectories were col-
lected over a traversal of ∼ 8Rg.
simulation included ∼ 4000 particles, evolving for 30,000
computational steps. Several simulations were run using
the LONI TeraGrid system[33] to facilitate performing
numerous simulations at a time. As a benchmark, a typ-
ical run on a single CPU workstation took ∼ 5 h of wall
clock time; however, since our codes are not yet subject
to any parallelization process, we stress that this repre-
sents an underestimate of the potential gain in compu-
tational time as opposed to running full atomistic or UA
simulations.
A typical MS MD
TABLE II: Mesoscale Simulation (MS-MD) Particle Number
and Box Dimension Compared to UA Box Dimension. All
UA simulations are for 1600 chains.
Blend [A/B] Particles LMS [˚ A] LUA [˚ A]
hhPP/sPP2744
hhPP/PE5324
PIB/PE4096
PIB/sPP 5488
iPP/PE1728
hhPP/PIB3456
199.07
246.21
218.66
245.19
168.57
230.73
166.61
166.61
164.91
164.91
167.27
164.91
Mesoscale simulations provide the center-of-mass to-
tal correlation functions that describe the polymer mix-
tures on the large scale and are readily calculated from
the simulation coordinates. As an exmple we show in
Figure 1(a) the plot of hcc
AA(k) for a 50:50 mixture of
Page 6
6
hhPP/sPP (χ = 0). Data from mesoscale simulations
and theoretical predictions are compared against united
atom simulations for the center-of-mass total correlation
functions and show an excellent agreement. Analytical
theory, mesoscale simulations, and united atom simula-
tions are all consistent in depicting the structure of the
fluid on the length scale of the polymer radius of gyration
and larger. Although all of the structural information is
already contained in the analytic expression, the MS-MD
simulations are useful as they can provide further infor-
mation, for example, on the dynamics of the system close
or far from equilibrium, both of which could be in prin-
ciple investigated with our coarse-grained systems.
0.1
0.20.30.4
k (Å-1)
-1
-0.8
-0.6
-0.4
-0.2
0
!chhcc
AA(k)
(a)
0.1
0.15
k (Å-1)
0.2
0.25
-80
-60
-40
-20
!hmm
AA(k)
(b)
FIG. 1: (a) Plot of hcc
FIG. 1: (a) Plot of hcc
mesoscale simulation (open red circles).
UA data (filled circles) and theoretical predictions (solid line)
shows quantitative agreement. (b) Plot of hmm
using the inverse mapping procedure, Equation 11, (open red
circles) compared to data from the full UA MD simulation
(solid circle).
AA(k) for hhPP/sPP obtained from mesoscale simulation (open red circles).
Comparison with UA data (filled circles) and theoretical predictions (solid line) shows quantitative
agreement. (b) Plot of hmm
AA(k) calculated using the inverse mapping procedure, Equation 11, (open
red circles) compared to data from the full UA MD simulation (solid circle).
23
AA(k) for hhPP/sPP obtained from
Comparison with
AA(k) calculated
However, while hcc(k) describes the center of mass cor-
relations between particles, when we are concerned with
the structure of the liquid on the local scale, it is nec-
essary to look at the monomer total correlation func-
tions. Although the large scale information is completely
provided by the mesoscale information, the information
on the local-scale structure is averaged out during the
coarse-graining. To take advantage of the gain in com-
putational time given by mesoscale simulation, it is con-
venient to combine the large scale information from the
latter with a short united atom simulation that describes
the local structure. The two are merged together through
a multiscale modeling procedure as presented in the next
section. The advantage in performing this two-steps pro-
cedure is that the simulation with atomistic resolution
can now be limited to short simulation runs and to a
small ensemble of molecules, as the needed large-scale,
long-time information comes from the fast mesoscale sim-
ulations.
C.Recovering the Monomer Level Description:
Implementation of the Multiscale Modeling
Procedure for Polymer Blends
In our multiscale modeling we combine the mesoscale
simulation, which correctly captures the large scale,
global structure, with UA simulations, needed to deter-
mine the local properties of the polymer. By this pro-
cedure we aim at bridging the different length scales
characteristic of the mixture and obtaining a complete
description of the polymer blend structure at a defined
temperature and composition.
The key to this procedure lies in extracting the
monomer-monomer correlation function, hmm(k) from
the coarse-grained correlation function, hcc(k), by inverse
mapping, using the reciprocal relation:
?
ωcm
hmm
αβ(k) =
ωmm
αα(k)ωmm
αα(k)ωcm
ββ(k)
ββ(k)
?
hcc
αβ(k) .(11)
Here, hcc(k) is obtained from the mesoscale simulation,
while the intramolecular form factors, ω(k), entering
Equation 11 are calculated directly from short UA MD
simulations. Since UA-MD simulations are needed only
to obtain the intramolecular distributions, these simu-
lations need only be for system box sizes on the order
of the radius of gyration, which is a significant advan-
tage over obtaining the global structure from UA-MD
alone which requires much larger box sizes to accurately
calculate the intermolecular distributions.
lated hmm(k) is valid only for small k values, where the
coarse-grained description applies, and begins to diverge
as ωcm(k) approaches zero at large k. This is shown in
Figure 1(b), where we compare directly hmm(k) calcu-
lated using Equation 11 with UA-MD data of the same
system.
To include local scale information, the total correla-
tion function hmm(k) at small k, obtained following the
described procedure, is combined with hmm(k) from UA
simulation for large values of k. The key point in the
procedure is to define a cut-off length, kcut∝ 1/rcut, at
which the two descriptions are to be combined.
Because the local scale information is averaged out in
the coarse-grained description, it is important to ensure
that the whole intramolecular description is accounted
for by the UA simulation. To estimate the length scale
at which local intramolecular contributions become neg-
ligible, we evaluate the fraction of intra to total site/site
contacts. For a given component of type α in the mix-
ture, this function is defined as
The calcu-
fsαα(r) =Nsαα(r)
Ntotal(r),(12)
where Nsαα(r) is the number of α type intramolecular
contact sites
?r
Nsαα(r) = 4πρ
0
(r?)2ωmm
αα(r?)dr?,(13)
Page 7
7
while the total number of site/site contacts is given by
Ntotal(r) =4
3πρr3.(14)
In our procedure, the integral in Equation 13 is computed
numerically using UA data for ωmm
αα(r).
Selecting the correct cut-off distance is an important
step of the procedure. Choosing a larger cutoff radius
rcut, smaller kcut, includes more information from UA
simulations, increasing the precision of the calculation,
but it increases the computational time, partially de-
feating the purpose of adopting a multiscale procedure.
In the case of polymer melts we observed that a value
of fs(r) = 0.025 gives good precision in the calculated
hmm(r) when compared with full united atom simula-
tion data, while retaining a reasonable speeding up of the
calculations.[18] For polymer blends we adopt the same
criterium and we evaluate a posteriori the agreement ob-
tained for the blend samples analyzed in this paper. In
Table III we report, for both AA and BB type inter-
actions, the effective radius, rcut, for combining simula-
tions, which lies in the intermediate length scale on the
order of a few Rgunits and is unique for each blend. Also
reported is the total number of site/site contacts within
the radius determined by the value fs(r) = 0.025, evalu-
ated using Equation 14 at r = rcut. Dividing Ntotalby
the number of united atoms per chain (96 in our case)
gives the number of chains, n, needed in a UA simulation
to produce the statistical information necessary to obtain
the local structure. This is important since it determines
how “short” the UA-MD simulations must be without
losing pertinent information about local correlations.
Large-scale and local-scale information were combined
at the chosen kcut= 2π/rcut, and the procedure was re-
peated for each of the mixtures shown in Table I. The
correlation coefficient between hmm(r) determined from
our multiscaling procedure and from full UA simulations
is calculated for the case when the multiscale simulations
are combined at fs= 0.025. These values are also pre-
sented in Table III, showing that once rcutis defined, the
multiscale procedure provides an accurate way of obtain-
ing the total correlation function for binary blends while
circumventing the need for prohibitively large atomistic
simulations. For the case of AB type interactions, an av-
erage radius between AA and BB data is used to deter-
mine where the simulations should be combined. While
this approach allows one to explore a large range in the
degree of polymerization, the method becomes impracti-
cal for liquids of long chains, for which intramolecular ef-
fects will remain long-ranged, since the cutoff length scale
occurs on the order of Rg. Moreover, for long chains, en-
tanglement effects enter the dynamics and have to be ac-
counted for. Therefore, for large N systems it is advisable
to include a third, intermediate, level of coarse-graining,
with a cut-off of the order of (and larger than) the per-
sistence length and the entanglement length scale.
III. RESULTS
A. Total Pair Correlation Functions of Athermal
Reference Blends
The method just discussed gives a systematic way of
merging simulations to optimize the tradeoff between the
gain in accuracy due to inclusion of UA simulation data
and the gain in efficiency due to the coarse grained meso-
scopic picture. This procedure works well as it yields to-
tal correlation functions in excellent agreement with UA
simulations at a reduced computational cost.
TABLE III: Effective Radius for Combining Simulations and
the Number of Total Site/Site Contacts Evaluated at fs(r) =
0.025
Type rcut[˚ A]fs=0.025 Ntotal n Corr. Coeff
hhPP/sPPAA29.1
hhPP/sPPBB28.7
hhPP/PE96 AA 28.7
hhPP/PE96 BB27.6
PIB/PE96 AA28.8
PIB/PE96BB27.2
PIB/sPPAA29.1
PIB/sPPBB29.6
iPP/PE96AA29.3
iPP/PE96BB27.6
System
3427 36
3288 34
3288 34
2924 31
3432 36
2891 30
3536 37
3721 39
3458 36
2891 30
0.9999
0.9999
0.9998
0.9999
0.9999
0.9999
0.9995
0.9999
0.9999
0.9998
Results from the multiscale procedure are reported in
Figures 2 - 6. In Figure 2, the left panels show the to-
tal correlation function, hmm(k), obtained by combining
UA simulations for the short length scales (large k) with
mesoscale simulations for the long length scales (small k),
for the first mixture in Table I, hhPP/sPP. The vertical
dashed line in the left panels represents the cut-off length
scale at which the two simulations are combined, which
corresponds to fs= 0.025. The insert in the left panels
depicts the peak representing the local structure, which
is accurately determined by UA simulation. This peak
depends on the geometry of the monomeric structure and
changes as a function of the type of polymers involved in
the mixture.
The right panels in Figure 2 show hmm(r), i.e. the
Fourier transform of the corresponding total correlation
function, hmm(k). As in the case for melts, we adopted a
sampling step of ∆k = 0.01 in reciprocal space, as this is
of the same order as the discontinuity in hmm(k), which
results from joining the two simulations. Because of the
amplitude of the chosen sampling step, there is no ef-
fect on the Fourier transformed function hmm(r). The
pair distribution function calculated from the multiscale
modeling procedure is identical to the coarse-grained an-
alytical expression in the large-scale regime, but also in-
cludes the local scale solvation shells, which come from
UA-MD and are not captured by the PRISM thread ex-
Page 8
8
pressions. Both pieces of information are needed to pro-
vide the complete structural and thermodynamic picture
of the system.
Figure 2 illustrates the spirit of our multi-scale model-
ing approach in which independent simulations represent-
ing the same system at different levels of coarse-graining
can be combined to provide a complete description of
the polymer. Analogous plots for the hhPP/PE and the
PIB/PE mixture are reported in Figure 3 and Figure
4, respectively. Finally, results for the two mixtures of
PIB/sPP and iPP/PE are presented in Figures 5 and 6.
All systems in real space show quantitative agreement
with UA-MD data but are obtained at a much more
efficient computational time than running the full UA
MD simulation. Furthermore, the procedure to obtain
the pair correlation functions is entirely analytical, hence
we do not utilize any optimization procedure or numeri-
cal fitting scheme to obtain consistency between the two
descriptions, and thus the method is fully transferable.
Figures 2-6 demonstrate the versatility of the approach
for studying mixtures of polymers with subtly different
chemical architecture, since the same multiscale proce-
dure is applied in all cases.
B. Applications to Thermal Blends with Realistic
χ Parameters
In the multiscale modeling procedure presented above,
we set χ = 0 in our soft-colloidal representation, and
the UA simulations to which we combine our mesoscale
simulations are assumed to be well-mixed. This provides
an efficient means of obtaining pair correlation functions
for polymer mixtures close to the athermal limit. In this
section we examine the assumption that the blends are
well-mixed and discuss the application of our multiscale
modeling procedure to modeling thermal mixtures where
subtle local features such as monomer shape, branching,
and site energetic asymmetries gives rise to non-trivial
divergent fluctuations.
Blends of polypropylenes of different tacticities have
been studied by Woo et al., and the χ parameter of blends
of iPP/sPP and aPP/sPP was found to be nearly zero
over the temperature range T=423-453 K.[34] Thus, for
the hhPP/sPP blend of Figure 2 it is reasonable to as-
sume the effective χ parameter will be small, which is
supported by the overall quantitative agreement in Fig-
ure 1 and 3. For the other systems investigated in this
publication, many of the pertinent χ parameters may
be determined from the available literature on polyolefin
blends which is presented in Table IV.
To demonstrate the extension of our approach to ther-
mal mixtures, we use the literature values for the χ pa-
rameter evaluated at the simulation temperature, T =
453K and perform MS-MD simulations for the mixtures:
hhPP/PE, PIB/PE, and hhPP/PIB. The hhPP/PIB
blend is particularly interesting because it exhibits a
lower critical solution temperature (LCST) phase dia-
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hAA(k)
hhPP/sPP " = 0.5
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hAA(r)
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hBB(k)
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hBB(r)
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hAB(k)
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hAB(r)
FIG. 2: (Left) Multiscale modeling: The left panels show the total correlation function, hmm(k),
FIG. 2: (Left) Multiscale modeling: The left panels show the
total correlation function, hmm(k), for AA (top), BB (mid-
dle), and AB (bottom) interactions, for a mixture of 50:50
hhPP/sPP. The data over the range of small k values was ob-
tained by mesoscale simulation, whereas over the large k range
it was obtained by UA MD simulation. The inset depicts the
local structure. The dashed line indicates the value at which
the two simulations were combined. (Right) The correlation
function, hmm(r), after Fourier transform (solid red line) is
compared with results from the full UA MD simulation (open
symbols).
for AA (top), BB (middle), and AB (bottom) interactions, for a mixture of 50:50 hhPP/sPP. The
data over the range of small k values was obtained by mesoscale simulation, whereas over the large
k range it was obtained by UA MD simulation. The inset depicts the local structure. The dashed
line indicates the value at which the two simulations were combined. (Right) The correlation
function, hmm(r), after Fourier transform (solid line) is compared with results from the full UA
22
MD simulation (filled symbols).
gram, which can in principle be modeled using our mul-
tiscale approach as well.[11] Since the parameters used in
our mutiscale modeling procedure are determined using
a UA description where each CHxgroup is represented
as a effective site, care must be taken when calculating
the effective parameter from Table IV where χ is de-
fined on a monomer basis, for which several CHxgroups
are grouped as one monomer unit. Since χ is propor-
tional to the free energy of mixing per site, the χ pa-
TABLE IV: Temperature Dependence of Polyolefin Blends
Blend [A/B]
hhPP/PE
PIB/PE
iPP/PE
hhPP/PIB
χ(T) χ/χs (453K)
0.077
0.163
0.360
0.018
−0.0294 + 17.58/Ta
0.00257 + 4.99/Tb
0.01c
0.027 − 11.4/Td
aReference 26bReference 27cReference 25dReference 23
Page 9
9
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hAA(k)
hhPP/PE "=0.5
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hAA(r)
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hBB(k)
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hBB(r)
0.5
1
1.5
2
k (Å-1)
-80
-60
-40
-20
0
!hAB(k)
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hAB(r)
FIG. 3: Same as in Figure 2 for a blend of 50:50 hhPP/PE.
FIG. 3: Same as in Figure 2 for a blend of 50:50 hhPP/PE.
23
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hAA(k)
PIB/PE " = 0.5
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hAA(r)
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hBB(k)
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hBB(r)
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hAB(k)
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hAB(r)
FIG. 4: Same as in Figure 2 for a blend of 50:50 PIB/PE.
FIG. 4: Same as in Figure 2 for a blend of 50:50 PIB/PE.
24
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hAA(k)
PIB/sPP " = 0.5
1
1.5
2
-1
0
1
5
10
15
r (Å)
-1
-0.5
0
hAA(r)
0
0.5
1
1.5
k (Å-1)
-60
-40
-20
0
!hBB(k)
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hBB(r)
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hAB(k)
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hAB(r)
FIG. 5: Same as in Figure 2 for a blend of 50:50 PIB/sPP.
FIG. 5: Same as in Figure 2 for a blend of 50:50 PIB/sPP.
25
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hAA(k)
iPP/PE " = 0.75
1
1.5
2
-0.5
0
0.5
0
5
10
15
20
r (Å)
-1
-0.5
0
hAA(r)
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hBB(k)
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hBB(r)
0.5
1
1.5
k (Å-1)
-80
-60
-40
-20
0
!hAB(k)
1
1.5
2
-1
0
1
0
5
10
15
20
r (Å)
-1
-0.5
0
hAB(r)
FIG. 6: Same as in Figure 2 for a blend of 75:25 iPP/PE.
FIG. 6: Same as in Figure 2 for a blend of 75:25 iPP/PE.
26
Page 10
10
rameter must be normalized by the average number of
united atom sites per monomer[4] as defined in the lit-
erature (6 for hhPP/PE[35], 4 for PIB/PE[36], and 4.8
for hhPP/PIB[4]). For hhPP/PE this corresponds to a
value of χ = 0.0016; for PIB/PE a value of χ = 0.0034;
and hhPP/PIB a value of 3.8 × 10−4. For the temper-
ature range at which the united atom simulations were
performed, the magnitude of χ in all cases is small, sup-
porting our initial modeling of χ = 0 in the previous
section. In Table IV we present the thermodynamically
relevant parameter, χ/χs at 453K, which provides an
indication of how far the mixture is from the spinodal.
We calculated the total correlation function for
the thermal mixtures of hhPP/PE, PIB/PE, and
hhPP/PIB, using the multiscale modeling approach de-
scribed above. These results are presented in the sup-
plemental material[37] and look nearly identical to the
athermal results shown in Figures 3 and 4. Although
the correlation functions are nearly identical with the
athermal limit because the blends are well mixed, sub-
tle differences in the structure as a result of increased
concentration fluctuations should be manifest in the dis-
tributions. To assess the changes in structure that re-
sult from increased fluctuation, the monomer level par-
tial static structure factors can be calculated from the
total correlation function, hmm(k),
Smm
Smm
Smm
AA(k) = φωmm
BB(k) = (1 − φ)ωmm
AB(k) = ρφ(1 − φ)hmm
where the monomer form factors were determined from
UA simulations as in Equation 11 above. The structure
factor measuring correlations in the relative concentra-
tion, Sφφ(k), which diverges as the mixture approaches
the spinodal, is expressed as a linear combination of these
partial structure factors,
AA(k) + ρφ2hmm
BB(k) + ρ(1 − φ)2hmm
AB(k) ,
AA(k) ,
BB(k) ,(15)
Sφφ(k) = (1−φ)2Smm
AA(k)+φ2Smm
BB(k)−2φ(1−φ)Smm
AB(k).
(16)
In small angle neutron scattering (SANS) experiments,
the χ parameter is determined from fitting the par-
tial structure factor to the random phase approximation
(RPA) equation of de Gennes[38]
1
S(k)=
1
φωmm
AA(k)+
1
(1 − φ)ωmm
BB(k)− 2χ,
evaluated at χ = 0.0016 (solid line) and χ = 0.00 (dashed line) is shown. The inset shows the
cles) for hhPP/PE (φ = 0.5) at T = 453K. For compari-
son the RPA equation evaluated at χ = 0.0016 (solid line)
and χ = 0.00 (dashed line) is shown. Filled circles represent
the structure factor from the full UA simulation. (B) The
same as part (A), except for PIB/PE (φ = 0.5), for which
the RPA equation, evaluated at χ = 0.0034 (solid line) and
χ = 0.00 (dashed line), is shown. (C) the same except for
the hhPP/PIB blend, and the RPA equation is evaluated at
χ = 0.00038 (solid line) and χ = 0.00 (dashed line).
(17)
where for convenience the monomer site volumes were set
equal to one.
Figure 7 presents Sφφ(k) for the three different ther-
mal blends obtained from our multiscale modeling pro-
cedure.The static structure factor calculated in this
manner exhibits good agreement when compared to the
RPA equation, which was evaluated at χ = 0.0016 for
hhPP/PE, χ = 0.0034 for PIB/PE, and χ = 3.8 × 10−4
for hhPP/PIB, using the intramolecular form factors
from UA MD simulations. The relevance of this com-
parison to the RPA formula, which is known to fit low
00.20.4
0.6
k (Å-1)
0
5
10
15
20
25
30
S!!(k)
" = 0.0016
" = 0.00
A)
00.20.4
0.6
k (Å-1)
0
5
10
15
20
25
30
S!!(k)
" = 0.0034
" = 0.00
B)
00.20.4
0.6
k (Å-1)
0
5
10
15
20
25
30
S!!(k)
" = 0.00038
" = 0.00
C)
FIG. 7: (A) The concentration fluctuation structure factor obtained from the multiscale modeling
FIG. 7: (A) The concentration fluctuation structure factor
obtained from the multiscale modeling procedure (open cir-
procedure (open circles) for hhPP/PE (φ = 0.5) at T = 453K. For comparison the RPA equation
structure factor from the full UA simulation. (B) The same as part (A), except for PIB/PE
28
(φ = 0.5), for which the RPA equation, evaluated at χ = 0.0034 (solid line) and χ = 0.00 (dashed
line), is shown
wavevector scattering curves well, is that there is clearly
a better fit using the experimental χ parameter than with
Page 11
11
the χ = 0 case, which demonstrates the consistency of the
description and that our approach is able to capture the
fluctuations in concentration that arise in thermal poly-
mer mixtures even at the relatively high temperatures of
these simulations. Also, the advantage of a multiscale ap-
proach is exhibited by Figure 7 since the low wave vector
regime is determined by mesoscale simulations so that
UA simulations only need to be performed on systems at
length scales up to the cut-off length (dashed line). This
is important since only the initial stages of the divergent
behavior need to be captured by united atom simulations,
thus the need for prohibitively large simulation boxes is
circumvented. For comparison, S(k), calculated for the
full UA MD simulation is also shown (offset for clarity)
and agrees with our multiscale results, demonstrating the
consistency between the two approaches.
When experimental values of the χ parameter are un-
available, a commonly used alternative is to estimate the
interaction parameter from the solubility parameters cal-
culated from a pure melt. These may be obtained from
the cohesive energy determined from MD simulations of
the individual melts. For the systems investigated here,
we used united atom simulations of pure melts provided
by Jaramillo et al.[4] The cohesive energy density can be
calculated from the radial distribution functions for each
UA site type, gij, where i,j designate a particular C, CH,
CH2, or CH3group, according to
Ucoh= 2π
?
ij
ρiρj
?
vij(r)gij(r)r2dr,(18)
where vij is the potential between nonbonded sites, for
which we employ a Lennard-Jones potential with the
corresponding TraPPE-UA parameters of Martin and
Siepmann[39]. In Equation 18 the integration is carried
out over the attractive branch of the potential. The sol-
ubility parameter (δ =√−Ucoh), calculated for each of
the polyolefins in this study is presented in Table V. For
comparison we also present results obtained by P¨ utz, et
al.[40] from UA simulations of chains with 24 backbone
carbons.
TABLE V: Solubility Parameter for UA simulations of poly-
olefin melts (N=96) at T = 453K. Values for δMD−2 are
computed from Ref. [40] for chains with 24 backbone car-
bons.
Polymer δ (MPa)
PE
hhPP
iPP
PIB
sPP
1
2
δMD−2
14.4
13.4
12.5
13.9
13.0
13.7 (T = 448K)
13.3
12.4
14.1
12.2
From these solubility parameters, the Flory-Huggins
parameter may be estimated according to
χH=
(δA− δB)2
?ρ◦
Aρ◦
BkBT
(19)
where the densities, ρ◦, are the pure melt densities of
species A or B. Table VI shows a comparison between the
Flory-Huggins χ parameter determined from this conven-
tional solubility parameters approach and the observed
experimental values. A major limitation of this solubil-
ity approach is that the value of χ is always positive
and UCST phase behavior is always predicted. Further-
more, because non-combinatorial entropy is ignored, the
concentration dependence of χ cannot be predicted. De-
spite these assumptions, the solubility approach has been
demonstrated to hold up reasonably well for a large vari-
ety of polyolefin blends[41]. For comparison, in Table VI
we also present the χ parameter computed directly from
the full UA simulation of a polymer blend, by an extrap-
olation of the three lowest wave vector points to k = 0 on
an Ornstein-Zernike plot (1/S(k) vs. k2), analogous to
that obtained in typical SANS experiments. The lowest k
vector is determined by the size of the simulation box and
in this case is k = 0.4˚ A. Due to this limitation, the effec-
tive χ parameter can be shifted by about ±0.002 while
still maintaining a reasonable fit to the blend structure
factor.[5]
TABLE VI: Comparison between experiment, solubility pa-
rameter, and RPA approaches to determine the χ parameter
(T = 453K) for the polyolefin blends in Figure 7
Blend [A/B]
hhPP/PE
PIB/PE
hhPP/PIB 0.00038 0.0012 0.0017
χexp
0.0016 0.0048 0.0022
0.0034 0.0012 0.0027
χH
χOZ plot
Finally we note that much work has been done using
the PRISM theory of Schweizer and Curro to determine
both the qualitative and quantitative miscibility behav-
ior of polymer blends.[42–44] For polyolefin blends of the
type investigated here, realistic models utilizing PRISM
expressions for the pair distribution function along with
UA potential parameters have been shown to give rea-
sonable estimates of χ for various polypropylene and
polyethylene blends.[5, 45, 46]; however this approach is
not pursued further here since the purpose of the current
publication is to demonstrate the implementation of the
multiscale modeling procedure once the χ parameter is
obtained.
IV.CONCLUSION AND OUTLOOK
In this paper we present a multiscale modeling pro-
cedure to simulate mixtures of polymer chains at large
scales. The method is completely general and applies to
Page 12
12
mixtures of polymers with different molecular structures,
at different thermodynamic conditions of temperature,
mixture composition, and chain length. The procedure
combines large-scale information from a mesoscale sim-
ulation with a more detailed model which captures the
local structure of the blend. For the detailed model, we
use a united atom representation which captures the lo-
cal polymer structure. Our mesoscale model represents
the blend as a mixture of soft colloidal particles and
accurately describes the large-scale structure as exhib-
ited in the center of mass radial distribution function.
Once the mesoscale simulations are performed, the rele-
vant chemical details are reinserted by implementation of
the Ornstein-Zernike formalism. We then combine this
information with short united atom simulations to ob-
tain a complete description over all length scales of in-
terest. In this article we test the approach by applying
our method to several different polyolefin mixtures. We
check the validity of our procedure by direct compari-
son of the pair correlation function against full united
atom molecular dynamic simulations. Because the UA-
MD simulations available for comparison are limited to
high temperatures where the samples can be more eas-
ily equilibrated, the test of our procedure includes only
high temperature samples. However our mesoscale simu-
lations are easily performed at temperatures approaching
the demixing temperature. Furthermore, the multiscale
modeling procedure provides a straightforward method
to circumvent large simulations when modeling thermal
mixtures, which has the potential to extend the range
currently available to computer simulations of polymers.
At low temperatures, where equilibration of even the lo-
cal UA-MD simulation may be difficult, our two step pro-
cedure could be implemented “on the fly” where informa-
tion is transferred from the local UA-MD (which provides
the packing information) to the global scale MS-MD and
vice versa until consistency between the two is reached.
One advantage of the proposed method is that no op-
timized parameterization scheme is required to maintain
consistency between the different levels of description.
The approach is thus transferable to different systems
and different conditions, while remaining quantitative
enough to distinguish even subtle differences in chemical
structure. Considering the advantages of the proposed
multiscale approach, it should provide an important tool
for investigating the dynamics of large-scale phenom-
ena via MD simulation. Future work should focus on
performing simulations of large systems approaching the
spinodal where full UA MD simulations are prohibitively
difficult.
V.ACKNOWLEDGEMENTS
We acknowledge support from the National Science
Foundation through grant DMR-0804145.
tional resources were provided by LONI through the Ter-
aGrid project supported by NSF.
Computa-
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