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Soft Matter
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Dept. of Chemical and Biological Engineer
Boulder, CO 80309, USA. E-mail: arthi.jaya
† Electronic supplementary informa10.1039/c3sm00144j
Cite this: Soft Matter, 2013, 9, 6876
Received 13th January 2013Accepted 1st March 2013
DOI: 10.1039/c3sm00144j
www.rsc.org/softmatter
6876 | Soft Matter, 2013, 9, 6876–68
Polydisperse homopolymer grafts stabilize dispersionsof nanoparticles in a chemically identical homopolymermatrix: an integrated theory and simulation study†
Tyler B. Martin and Arthi Jayaraman*
This paper presents a computational study of the effect of polydispersity in grafted polymers on the
effective interactions between polymer grafted nanoparticles in a polymer matrix, when graft and
matrix polymers are chemically identical. The potential of mean force (PMF) between grafted particles,
calculated using a self-consistent PRISM theory-Monte Carlo simulation approach, shows that graft
polydispersity weakens the attractive well at intermediate inter-particle distances, eliminating the well
completely at high polydispersity index (PDI). The elimination of the mid-range attractive well is due to
the longer grafts in the polydisperse distribution that introduce steric repulsion at large distances, and
the increased wetting of the grafted layer by matrix chains arising from reduced monomer crowding
within the polydisperse grafted layer. Trends in how the PMF changes as a function of grafting density,
ratio of matrix to graft length, and packing fraction of polymer matrix seen for monodisperse grafts are
preserved for polydisperse grafts. Comparison of a log-normal distribution to a bidisperse distribution of
chain lengths (with equal number of short and long chains) with the same PDI and average length,
shows that the polydisperse distribution can better stabilize dispersions than the bidisperse distributions
because of the longer chains in the polydisperse distribution. Additionally, in a bidisperse distribution,
with all chains shorter than the matrix chain length, there is a reduction in the mid-range attraction,
thus confirming the role of reduced monomer crowding in the bidisperse grafted layer in increasing the
grafted layer wetting by the matrix chains, and, as a result, improving miscibility of grafted particles and
matrix.
1 Introduction
Controlling the morphology of nanoscale additives in a polymermatrix is critical for tuning the macroscopic properties of theresulting polymer nanocomposite. For example, superiormechanical properties of polymer nanocomposites can be ach-ieved via good dispersion of nanoscale llers in the polymermatrix. One approach to control the morphology of the nano-composite is via functionalization of the nanoparticle surfacewith ligands, such as polymers and surfactants, that tune theeffective interactions between the particles in the matrix leadingto the target morphology. In particular, to achieve good disper-sion of nanoparticles in a polymer matrix, the nanoparticlesurface is graed with polymers that are chemically identical tothe matrix polymer. The chemical similarity between the graand matrix improves the effective miscibility of the graedparticles in the matrix over that seen with bare (organic or
ing, UCB 596 University of Colorado at
tion (ESI) available. See DOI:
89
inorganic) nanoparticles in the matrix. Extensive theoretical andexperimental studies, both at high and low graing density, forthese chemically identical gra and matrix systems, have shownthat themolecular weights of the graed andmatrix polymer playa critical role in dictating whether the graed nanoparticlesaggregate or disperse.1–10 At high graing density, where thegraed chains are stretched in the “brush” regime, nanoparticlesdisperse (aggregate) if the gramolecular weight is higher (lower)than matrix molecular weight. This is explained by the wettingand dewetting of the graed layer by the matrix chains. In thecase where the graed polymer is the same chemistry as thematrix polymer, the wetting/dewetting of the graed chains bythe matrix chains is driven purely by the entropy of the system.When the gra molecular weight is larger than the matrixmolecular weight, the conformational entropy of the graedchains dominate, and the matrix chains penetrate (or wet) thegraed layer. When the gra molecular weight is smaller thanthe matrix molecular weight, the conformational entropy of thematrix chain dominates, and the matrix chains deplete thegraed layer. The exact value of the matrix to gra chainmolecular weight ratio where this transition from wetting todewetting occurs is a function of the graing density and the
This journal is ª The Royal Society of Chemistry 2013
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curvature of the particle.1 In contrast to the high graing densitycase, at low graing density, dispersion/aggregation is dependenton the amount of exposed nanoparticle surface. Larger gramolecular weight chains can better shield nanoparticle surfacesfrom direct particle–particle contacts and lead to dispersion ofgraed particles in the polymer matrix.11 Despite the importanceof gra molecular weight for controlling the morphology, thereare very few experimental and theoretical studies on polymergraed nanoparticles that have investigated the role of poly-dispersity in the graed chains and its effect on miscibility ofgraed chains with matrix chains.
While there have been less than a handful of studies onpolydisperse polymers graed on nanoparticle surfaces (withnite curvature), there have been many studies focused at the twoextremes in the range of curvature – polymers graed on atsurfaces (zero curvature) and star polymers (cores with innitecurvature). Polydispersity in chain lengths graed on atsurfaces12,13 have been shown to alter chain conformations andthe overall height of the graed layer on these surfaces (with nocurvature). In case of star polymers with polydisperse arms, theeffective force, F, between polydisperse star polymers in a goodsolvent has been shown to have a drastically different expressionas compared to monodisperse star polymers.14,15 To understandbehavior of polydisperse graed polymers on a surfaces of nitecurvature, Dodd and Jayaraman,16 using Monte Carlo simula-tions, studied a single spherical polymer graed nanoparticlewith polydisperse graed chains, in an implicit solvent, at apurely athermal limit, for varying polydispersity indices (PDI ¼1–2.5), particle diameter, and graing density. Dodd and Jayara-man showed that the conformations of the polydisperse chainsgraed chains on spherical nanoparticles (5–8 nm in diameter)deviate from themonodisperse counterpart, and approach that ofa single graed chain on the same particle size because of poly-dispersity-induced relief in monomer crowding. Specically, theradius of gyration of the short chains was lower at PDI > 1 than atPDI ¼ 1 (monodisperse), and the long chains were less stretchedat distances away from particle surface at PDI > 1 than at PDI¼ 1.
These past studies demonstrate that graed chain confor-mations are signicantly affected by polydispersity in the graedchain lengths. This leads to the question: In the presence of apolymer matrix, is the effect of polydispersity on graed chainconformations large enough to alter how matrix chains wet/dewet/deplete the graed layer? If yes, is this change in matrixwettability of the graed layer predictable so that one coulddeliberately introduce polydispersity as a design knob to tailorinter-particle interactions? Past studies on at surfaces haveshown improved mixing of the graed layer with the free chainsin the presence of polydispersity.17–19 It was observed that whengra–matrix interaction is repulsive, polydispersity does not affectthe width of the interface between graed brush and matrix.18
However, for attractive or athermal brush–matrix interaction,there is increased stretching of the gra chains into the matrix asthe polydispersity (measured by PDI ¼ DPw/DPn) increases from1.0 to 3.0, indicating enhanced mixing between the matrix andhighly polydisperse graed chains with increasing poly-dispersity18 when brush monomer–matrix monomer interactionsare attractive or athermal. These results for the at brush case
This journal is ª The Royal Society of Chemistry 2013
cannot be extrapolated directly to the case of chains graed on ahigh (convex) curvature nanoparticle surface. This is because theavailable volume per graed chain on a at surface ismuch lowerthan that on a convex surface, which changes the amount ofcrowding among the graed chains, and results in signicantlydifferent conformations for the graed chains.
To understand how polydispersity in graed polymer affectsmiscibility of polymer graed nanoparticles with nite convexcurvature in a polymer matrix of same chemistry, in a recentletter, Jayaraman and coworkers20 explored how polydispersity inpolymer chains graed on nanoparticles affects the potential ofmean force (PMF) between the polymer graed nanoparticles atvarying graing densities (e.g. low, intermediate, and high), in adense solution of matrix polymers and melt-like polymer matrixat varying matrix lengths (e.g. less than and greater than averagegra length) using Polymer Reference Interaction Site Model(PRISM) –Monte Carlo (MC) approach. One of the key results wasthat, at high graing density, polydispersity in the graed poly-mers removed the attractive well in the potential of mean force,suggesting that polydispersity could stabilize dispersions in amonodisperse polymer matrix at conditions where correspond-ing monodisperse polymer graed particles would exhibitaggregation. In this paper, we present detailed results from theabove PRISM-MC study of innitely dilute concentration ofpolydisperse polymer graed nanoparticles in a monodispersehomopolymer matrix at varying graing density, matrix packingfraction and particle diameter, along with some results fromBrownian dynamics simulations of a single polydisperse polymergraed nanoparticle in an explicit monodisperse homopolymermatrix. We nd that, in the case of a polydisperse (log-normal)distribution of gra lengths and matrix lengths longer than gralengths, the presence of the long chains and the improvedwetting of the graed layer by the matrix chains brought about bythe reduced monomer crowding in the graed layer, togetherremove the mid-range attraction seen at the correspondingmonodisperse limit. Direct comparison of the effects of a statis-tically polydisperse distribution of gra lengths to a bidispersedistribution of gra lengths on the potential of mean force helpsus understand how the chains lengths that are longer and shorterthan the average chain length, and the reduced monomercrowding in the graed layer contribute to the overall shape andfeatures of the potential of mean force.
This paper is organized as follows. In the method section weprovide details of the model, theory and simulation, and anal-ysis methods used in this work. In the results section we presentresults from PRISM-MC approach that show the effect of poly-dispersity of gra lengths on potential of mean force (PMF)between two polymer graed nanoparticles in a homopolymermatrix at varying graing density, matrix chain length, matrixpacking fraction, and particle diameter. We also present incertain cases the corresponding BD simulation results forconcentration proles and chain conformations for a singlepolymer graed nanoparticles in a homopolymer matrix. Weconclude the result section with a direct comparison of poly-disperse graed system with a bidisperse graed system. In theconclusions section we summarize this work, and present theimplications of the key results and future directions.
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2 Method2.1 Model
Wemodel a homopolymer graed spherical nanoparticle as a hardspherical nanoparticle of diameter D with freely jointed chainspermanently attached randomly on the particle surface. Throughvisual analysis we ensure that the chains are approximately equallyspaced apart, and not clustered regionally on the particle. Wemodel each matrix homopolymer chain as a freely jointed chain.Both the graed and matrix homopolymer chains consist ofmonomer beads of diameter d chosen tomimic a Kuhn segment ofa linear synthetic polymer. While chains in the Monte Carlo (MC)simulations have a constant bond length of 1.4d, monomers in theBrownian Dynamics (BD) simulations are connected together withharmonic bonds with force constant k ¼ 30 and bond rest lengthr0¼ 1.4d. The graed homopolymer chains aremonodisperse withchains of equal number of monomers (Ngra¼ 20), or polydispersewith log-normal distribution of chain lengths or bidisperse with abimodal distribution consisting of (equal number of) short andlong chains. We quantify the polydispersity via the polydispersityindex (PDI). For systems with a polydisperse gra length distri-bution with PDI greater than 1 we t the distribution of chainlengths to a log-normal distribution with minimum and numberaverage gra lengthsNgra,min¼ 8 andNgra,avg¼ 20. The details ofthe PDI calculation and the allocation of chain length distributionare given in the ESI.† The matrix is maintained monodispersethroughout this study and its chain length is denoted by Nmatrix.The total packing fraction h is the volume fraction of the systemoccupied by the matrix chains and the polymer-graed nano-particles (llers). The volume fraction of h occupied by the poly-mer-graed nanoparticles is denoted by the ller fraction f. In thispaper we choose f ¼ 0.001 to calculate potential of mean forcebetween the graed particles at the innitely dilute (or 2-particle)ller limit. We maintain athermal interactions Uij(r) between allpairs of monomers (on graed and matrix chains), particle andparticle, and particle and monomers (on graed and matrixchains). In the PRISM-MC calculations, we model these athermalinteractions using hard sphere potentials, while in the BD simu-lations, we use shied-truncated LJ potentials (i.e. Weeks–Chan-dler–Andersen or WCA potential), dened as:
VLJðrÞ ¼43
��sr
�12
��sr
�6�þ 43
"�s
rcut
�12
��
s
rcut
�6#
0
r\rcutr $ rcut
8>><>>:
where 3 is the interaction well depth, s is the bead diameter, andrcut (¼ 21/6s) is the cutoff distance for the potential. The choice ofathermal interactions is appropriate to mimic experimentalsystems where the gra and matrix monomers have similarchemistry, and particle–monomer interactions are negligible.Additionally, the choice of athermal interactions ensures that wecan capture the effective interactions resulting from entropiceffects purely.
2.2 PRISM-MC method
We use a self-consistent Polymer Reference Interaction SiteModel theory and Monte Carlo simulation (PRISM-MC)
6878 | Soft Matter, 2013, 9, 6876–6889
approach recently developed by Nair and Jayaraman forstudying polymer graed particles,21 to calculate the potential ofmean force between monodisperse and bidisperse polymergraed particles in a homopolymer matrix. We use this inte-grated theory-simulation approach because it is computation-ally much faster than either pure Monte Carlo simulations ormolecular dynamics simulations of polymer graed particles inan explicit polymer matrix, thus allowing us to scan a largeparameter space in a reasonable time.
PRISM theory consists of a matrix of Ornstein–Zernike-likeintegral equations that relate the total site–site inter-molecular pair correlation function, hij(r), to the inter-moleculardirect correlation function, cij(r), and intra-molecular paircorrelation function, uij(r). The PRISM equations in Fourierspace are
H(q) ¼ U(q) C(q) [ U(q) + H(q) ] (1a)
Hij(q) ¼ rirjhij(q) (1b)
UijðqÞ ¼ rXNi
a¼1
XNj
b¼1
Uaibj ðqÞ (1c)
where H(q), C(q) and U(q) in this study are matrices of size 3 � 3for the following 3 types of sites: gra monomers, particle, andmatrix, with the matrix elements dened in eqn (1b) and (c). Wenote that, despite the chemistry of the gra and matrix beingthe same, these sites are physically identied separately as graor matrix. In the above equations, Ni and ri are respectively thenumber and number density of site i, r is the molecular numberdensity, and Uij(q) the intra-molecular pair correlation functionbetween sites i and j within a certain molecule in Fourier space.To solve eqn (1), we use closure relations connecting the realspace cij(r), hij(r) (¼ gij(r) � 1) and interaction potentials Uij(r).The choice of closures depends on the system being studied.Previous work on a mixture of nanoparticles and polymers22–31
shows that the Percus–Yevick (PY) closure for polymer–polymerand polymer–particle, and the hypernetted chain (HNC) closurefor particle–particle work well. We have used the same combi-nation of atomic closures, since this work also consists ofpolymers and nanoparticles. Given that sij is the distance ofclosest approach between sites i and j, i.e. sij ¼ d for monomer–
monomer pairs and sij ¼ Dþ d2
(as stated earlier, d and D are
the monomer and particle diameters, respectively) and particle–monomer pairs, the impenetrability condition applies insidethe hard core:
gij(r) ¼ 0, r < sij (2a)
Outside the hard core, the PY approximation describes thedirect correlation function between all pairs of sites (exceptparticle–particle):
cij(r) ¼ (1 � ebUij(r)) gij(r), r > sij (2b)
and the HNC closure handles the particle–particle directcorrelation function:
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cPP(r) ¼ hPP(r) � ln gPP(r) � bUPP(r), r > D (2c)
To efficiently solve this system of coupled nonlinear integralequations we employ the KINSOL algorithm32 with the linesearch optimization strategy. Attaining convergence forcomplex nonlinear integral equations is much easier withKINSOL as compared to the Picard technique, the method usedin prior PRISM theory work. The solution of the PRISM equa-tions yields pair correlation functions, gij(r), and the partialcollective structure factors, Sij(q). We note that some sets ofparameters, especially those involving larger particle sizes orlonger polymer chains (gra or matrix) do not yield any solu-tions due to numerical issues.
We use a self-consistent approach linking PRISM theory andMonte Carlo simulations,21 where the intra-molecular paircorrelation function Uij that is input to PRISM are provided byMC simulations of a single polymer-graed nanoparticle or asingle matrix chain in an external medium-induced potentialobtained from PRISM theory. The interdependence of the chainconformations (Uij) and the medium-induced potential gives riseto the self-consistency in the approach. The advantage of thisapproach is that it can tackle non-ideal conformations along thegraed andmatrix polymers that have been neglected in previousstudies of homopolymer-graed nanoparticles.28–31 Furthermore,in contrast to previous self-consistent PRISM-MC studies onhomopolymer melts alone,33–38 or on bare particles in a homo-polymer melt,39,40 where the self-consistent loop involved MCsimulations of only a single matrix polymer chain, we use alter-nate self-consistent loops for a single polymer-graed particleand a single matrix chain. This ensures that we account for non-idealities in both the graed and matrix chain conformations.
Since the steps involved in this self-consistent approach aredetailed in our previous papers,21,41 where we describe thisapproach in detail and its application to study polymer graedparticles in polymer matrix we present here only a brief overviewof this method. First, the pairwise-decomposed medium-induced solvation potential, Djij(r), is obtained from the PRISMequations; this describes the interaction between any two sites iand j as mediated by all the remaining sites in the system, i.e.,including the matrix, gras and particles themselves. The formof the solvation potential depends on the approximation usedin its derivation33–35,42–44 and we use the PY-form
DjPYij (r) ¼ �kTln[1 + cik(r)*skk0 0(r)*ck0j(r)] (3a)
S(q) ¼ U(q) + H(q) (3b)
where ‘*’ in eqn (3a) denotes a convolution integral in spatialcoordinates, k is the Boltzmann constant, and T is the tempera-ture. S(q) in eqn (3b) is the structure factor in terms of intra- andinter-molecular pair correlation functions in Fourier space. Thesolvation potential Djij(r) is then fed to the MC simulation of asingle polymer-graed particle or a single matrix chain. In theMC simulation, the model of the polymer-graed particle or thematrix chain is the same as that used in PRISM theory. Thegraed chain length distribution (e.g. monodisperse, bidisperseor polydisperse) is assigned in the initial conguration of the
This journal is ª The Royal Society of Chemistry 2013
polymer graed nanoparticle in the MC simulation, and theeffect of the polydisperse or bidisperse distribution of gralengths is present in the PRISM section in the terms in the intra-molecular pair correlation function associated with the polymergraed particle calculated in the MC simulation. In the simula-tion the total interaction between sites i and j separated by adistance r, Utot
ij (r) (i, j¼ gramonomer, particle, or matrix), is thesum of Uij(r) and the solvation potential, Djij(r), obtained fromthe preceding PRISM step. We alternately simulate (a) a singlepolymer-graed nanoparticle or (b) an isolatedmatrix chain withthe set of solvation potentials from the most recent iteration ofPRISM calculations. During the production stage of the MCsimulation, the intra-molecular structure factors between sitepairs are sampled every 5 � 105 steps and the ensemble averageof the intra-molecular structure factors is calculated, whichserves as the new input for the following iteration of PRISMcalculations. The self-consistent PRISM-MC iterations arecontinued until convergence of Djij(r) between iterations.
2.3 Brownian dynamics method
Even though the system we study with PRISM-MC is polymergraed nanoparticles in an explicit polymer matrix, in the MCsimulation portion of PRISM-MC we alternately simulate asingle polymer graed nanoparticle and a single matrix chain inan effective solvation potential. This does not allow for simul-taneous visualization of the polymer graed nanoparticle in anexplicit polymer matrix. To visualize and characterize certainphysical aspects (e.g. matrix monomer concentration from thesurface of the graed particle) we simulate, for a select fewsystems, a single polymer graed nanoparticle placed in anexplicit monodisperse polymer matrix using NVT Browniandynamics simulations on the HOOMD-blue platform.45 Usingthe HOOMD-blue code we are able to access faster simulationtime on Graphical Processing Units (GPUs) than possible withtraditional CPU-based codes. To initialize our simulations, werst build a graed nanoparticle, with chains extending radiallyfrom the particle surface, in the absence of any matrix chains. Ashort simulation with strong Lennard-Jones monomer–mono-mer attraction is then run to compress the graed chains. Thiscompressed graed nanoparticle, along with 1500 chains oflengthNmatrix, are then placed in a large cubic box, which is thencompressed to reach the target system packing fraction. Thereare 4 stages of initialization, compression and cooling beforethe nal equilibration and production stage. The rst stage isrun for four million steps at a reduced temperature of T* ¼ 5,and box length of L ¼ 300 in order to help eliminate any biasintroduced in the system during the initialization. The system isthen compressed over 1 million steps to a volume fraction ofh ¼ 0.1 at T* ¼ 5. The third stage is another randomizationstage at T* ¼ 5 for 5 million steps to allow the system toequilibrate at h ¼ 0.1. During stage 4, the system is cooledlinearly, from T* ¼ 5 to T* ¼ 1, over a period of ve millionsteps. During the nal stage, the system is sampled for 5 millionsteps at h ¼ 0.1 and T* ¼ 1. The data collected in the nal stageis veried to be equilibrated by ensuring the total energy of thesystem is constant. To be sure that we only use statistically
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independent samples in our averages, we calculate the corre-lation time of various data during the sampling period (nalstage). We nd that correlation times for our systems for all datarange from 5000 to 8000 timesteps, so we sample every 10 000timesteps.
2.4 Analysis
In the self-consistent PRISM-MC approach, at the end of theself-consistent loop we obtain the equilibrium inter-molecularpair correlation function, gij(r) ¼ hij(r) + 1, that characterizes thelocal structure of the graed nanoparticles and the matrixpolymer. The potential of mean force (PMF) between twonanoparticles, WPP(r), is calculated from the particle–particlepair correlation function, gPP(r) as follows:
WPP(r) ¼ �kTlngPP(r). (4)
Using the g(r), we can also calculate the second virial coef-cient, B2, a measure of tendency for particle aggregation ordispersion.
B2 ¼ 2pÐr2(1 � g(r))dr (5)
where r is a radial coordinate and g(r) is the value of the radialdistribution function at radial coordinate r.
To calculate the matrix penetration depth we begin by rsttruncating and shiing the partial pair distribution functionbetween the nanoparticle and the matrix beads, gPM, obtainedfrom PRISM-MC so that the domain of the function variesbetween the particle surface and the height of the graed layer,hg. We then calculate the square root of the normalized secondmoment of gPM in this domain to obtain the matrix penetrationdepth, l.
l ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðhg0
r2gPMðrÞdr
ðhg0
gPMðrÞdr
vuuuuuuuuuut(6)
We also calculate concentration proles of the gra (in theMC simulations within the PRISM-MC approach, and in the BDsimulations) and matrix monomers (only in BD simulations)from the nanoparticle center or surface. The number ofmonomer beads of type Y – graed or matrix – in concentricshells of width Dr radiating outwards from the particle center isrecorded. Themonomer concentration CY(r) at a distance r fromthe nanoparticle surface is calculated by dividing the averagepopulation nY(r) in a spherical shell by its volume:
CYðrÞ ¼ nYðrÞ4pr2Dr
(7)
2.5 Parameters
The system presented in this paper consists of homopolymergraed spherical nanoparticles, in an explicit homopolymer
6880 | Soft Matter, 2013, 9, 6876–6889
matrix, where the graed polymers are either monodisperse,polydisperse, or bidisperse, and the matrix polymers aremonodisperse. The polydispersity of the graed polymers isquantied by polydispersity index (PDI) is varied from 1(monodisperse) to 2.5. In the case of the polydisperse graedpolymers, the graed polymer lengths are chosen from a lognormal distribution of chain lengths representing a specicPDI, while maintaining the average graed polymer length,Ng,avg, to be 20 Kuhn segments and minimum length to be 8Kuhn segments. To study matrix polymer lengths lower andhigher than the average gra polymer lengths, the matrix lengthis varied from 10 Kuhn segments to as high as 300 Kuhnsegments, however much of the discussion in this paper focuseson 10 and 40 Kuhn segments. The spherical nanoparticlediameter is either 5d (where d is the diameter of a Kuhnsegment or “monomer”) or 8d. The polymer graing density, s,on these particles is varied from high (0.65 chains per d2) brush-like, to intermediate (0.25 chains per d2) and low (0.1 chain perd2) values. The total system packing fraction, h, is either 0.1 tomimic dense solution of matrix polymers or 0.3 to mimic melt-like polymer matrix. The volume fraction of the graed parti-cles, f, is maintained low (f ¼ 0.001) to model an innitelydilute concentration of the graed particles.
3 Results
In Fig. 1, we present the PMF between polymer graed nano-particles with monodisperse (PDI ¼ 1) and polydisperse (PDI > 1)homopolymer chains graed on a spherical particle of size D ¼5d at graing densities of 0.65, 0.25 and 0.1 chains per d2 andplaced in a dense solution (h ¼ 0.1) of monodisperse homopol-ymer matrix at athermal interactions between all species. At thehigher graing density of 0.65 chains per d2 (Fig. 1a), withmonodisperse (PDI¼ 1.0) gras ofNg,avg¼ 20 andmonodispersematrix chains of Nmatrix ¼ 10 (red solid symbols), the PMFexhibits a repulsion at contact and weak attraction at interme-diate distances. The corresponding PMF inmatrix of Nmatrix ¼ 40(red open symbols) also exhibits a repulsion at contact and anattractive well at intermediate distances (or mid-range attractivewell) that is stronger than that seen at Nmatrix¼ 10. The repulsionat contact is attributed to the graed monomers on the particlesterically repelling the other polymer graed particle as theparticle surfaces approach each other, and is observed at bothmatrix lengths. The attractive well at intermediate distances isattributed to the overlap in graed layers brought about by thedewetting of themonodisperse graed layer by themonodispersematrix chains, which is considerable when the matrix chainlengths are greater than the gra chain length. This behavior ofincreasing mid-range attraction strength with increasing ratio ofmatrix lengths to gra lengths seen in Fig. 1a has been seen inprior theoretical and experimental work for polymer graednanoparticles at brush-like graing densities at the mono-disperse limits.1 Here we see that, as polydispersity in gras (PDI)increases from 1 to 2.5, the steric repulsion at contact weakens by1–2 kT and the attractive well at contact is eliminated, for bothmatrix chain lengths. Additionally, as gra PDI increases, therepulsive tail in the PMF increases in strength and extends to
This journal is ª The Royal Society of Chemistry 2013
Fig. 1 Potential of mean force (in units of kT) versus inter-particle distance, r� D(in units of d), between grafted nanoparticles (D ¼ 5d) at s ¼ 0.65 (a), 0.25 (b),and 0.10 chains per d2 and PDI ¼ 1.0 (circles), 1.5 (squares), 2.0 (upward facingtriangles), and 2.5 (downward facing triangles) with Ng,avg ¼ 20, in a densesolution (h ¼ 0.1) of monodisperse homopolymer matrix chains with Nmatrix ¼ 10(solid symbols) and Nmatrix ¼ 40 (open symbols). The insets have the same axeslabels as the main plots.
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larger inter-particle distances, for both matrix lengths. At theintermediate graing density of 0.25 chains per d2 (Fig. 1b), therepulsion at contact and the attractive well at the monodisperselimit are weaker in strength than the corresponding mono-disperse values at higher graing density (Fig. 1a). This reductionin contact repulsion and mid-range attraction with decreasinggraing density is also in agreement with prior theoretical andexperimental work for polymer graed nanoparticles withmonodisperse gras.1,3,6,7 We see that the effect of increasingpolydispersity at 0.25 chains per d2 is qualitatively similar, but
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quantitatively weaker than that seen at 0.65 chains per d2. At thelow graing density of 0.1 chains per d2 (Fig. 1c), the PMFexhibits repulsion at contact and no attractive well at interme-diate inter-particle distances, which is also in agreement withpast theoretical studies for these lightly graed systems (seereview articles1,46) where wetting/dewetting of the graed layer bymatrix chains does not occur due to absence of a graed brush.At the lowest graing density, as the gra PDI increases, the PMFbecomes slightly less repulsive at contact and slightly morerepulsive at larger distances (Fig. 1c inset).
The decrease in repulsion at contact with increasing PDI isdriven by the polydisperse graed polymers relieving somemonomer crowding in the graed layer. The relief in crowding iscaused by a change in graed chain conformations to maximizethe overall conformational entropy upon introduction of poly-dispersity, as seen in recent Monte Carlo simulation (MC) studyin implicit matrix.16 This MC study also showed that the effect ofpolydispersity on chain conformations is relatively minor at lowgraing densities and more drastic at higher graing densitieswhere chain crowding in the graed layer is strong at themonodisperse limit. In an explicit homopolymer matrix and highgraing density, (0.65 chains per d2), Brownian dynamics simu-lations results (ESI†) show that the graed chain end-monomerconcentration proles show larger values near the particlesurface with increasing polydispersity, implying higher accessi-bility of the particle surface. The higher accessibility of theparticle surface by end monomers is due to the presence ofshorter chains in the distribution, as well as the small changes inchain conformations due to reduced monomer crowding result-ing from a wider graed chain length distribution. Theincreasing relief in monomer crowding with increasing PDI,especially at higher graing densities, manifests in the PMF asmore signicant reduction in repulsion at contact at 0.65 chainsper d2 for Nmatrix ¼ 40 (Fig. 1a) than at smaller graing densities.
Most interestingly, at 0.65 chains per d2 and Nmatrix > Ng,avg
(open symbols in Fig. 1a) the attractive well of�0.1 kT in the PMFat intermediate distances seen in monodisperse systems iscompletely eliminated at PDI of 1.5 and above. Additionalcalculations at smaller PDI (1.05–1.4) (ESI†) found that attractivewell is not eliminated at all PDI > 1, and that there is a minimum,or a critical, PDI needed to eliminate the attractive well. Since thestrength of mid-range attraction is dependent on the graingdensity, particle size, and average gra and matrix length, onecan expect the exact value of the minimum or critical PDI neededto eliminate this attractive well to also be a function of theseparameters. The attractive well is eliminated at higher PDIbecause the longer chains in the polydisperse chain lengthdistribution (a) sterically repel the longer chains on the othergraed particle, and (b) shi the entropic contributions moreheavily towards the graed chains than matrix chains, thusdriving matrix chains to wet the graed layer. This increasedwetting is also captured via concentration proles of the matrixmonomers from the particle surface in the Brownian dynamicssimulations (ESI†). ESI† shows that, as PDI increases, the matrixmonomer concentration prole goes further into the graedlayer (characterized by the graed monomer concentration) ascompared to monodisperse gras.
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Notably, many of the trends in how the effective interactionschange with graing density and ratio of matrix chain length togra chain length for monodisperse gras1 remain the sameeven in the presence of polydispersity in gras.
The most exciting aspect of the above results is that, sincethe mid-range attractive well has been attributed to drive thenanoscale additives in polymer nanocomposites towardsaggregation, eliminating the attractive well should stabilizedispersions of particles in systems where monodisperse graswould drive aggregation. The second virial coefficient B2 (Fig. 2),which characterizes the propensity of particles to assemble(negative B2) or disperse (positive B2), also shows that, for allgraing densities, as PDI increases, the B2 value increases, withthe net increase being largest at the highest graing density.Our emphasis here is on the qualitative trend of increasing B2
value with increasing PDI, and not the value of B2 itself, as thatwould depend on the specic matrix and gra lengths, particlesize, and the direct particle–particle attractive interactions aswell (which are maintained as athermal here). Here the B2 valueis positive even at the monodisperse limit, because the matrixchains are only 40 segments long, which is only twice that of theaverage gra length and the nanoparticle–nanoparticle inter-actions are athermal. Past studies have shown that, for curvedsurfaces, when the ratio of matrix length to gra length is ve tosix, the particles aggregate at the monodisperse gra limit.1,47
While we only show results for Nmatrix¼ 10 and 40 in this article,we have conrmed at the monodisperse gra limit that asNmatrix increases the attractive well at intermediate distancesdeepens (ESI†), thereby increasing tendency for particle aggre-gation, in agreement with past studies. Since the attractive wellis stronger at intermediate distances, one might also needlarger PDI to eliminate that attraction. To test this hypothesisfor a few cases, we conducted PRISM-MC calculations of poly-mer graed particles with graed chain polydispersity of PDI ¼2 at the highest graing density at large matrix molecularweights (ESI†). We observe that, at PDI ¼ 2, as the matrix chainlength increases, the mid-range attractive well depth is noteliminated completely. This observation, along with stronger
Fig. 2 Second virial coefficient for polymer grafted nanoparticles in a polymermatrix as a function of graft polydispersity (PDI) at grafting densities of s ¼ 0.10chains per d2(solid line), s ¼ 0.25 chains per d2(dashed line), and s ¼ 0.65 chainsper d2(dotted line), and total system volume fraction of h ¼ 0.1. All data is forparticle diameter D ¼ 5d, average graft length Ng,avg ¼ 20, and monodispersematrix length Nmatrix ¼ 40.
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mid-range attraction at the monodisperse limit at high matrixchain lengths (ESI†), conrm our expectation that, at largermatrix chain lengths, larger PDI is needed to eliminate theattractive well. We could not conduct an extensive systematiccalculation of PMF with increasing PDI at larger Nmatrix becauseof lack of numerical convergence in PRISM-MC runs at longmatrix lengths arising from numerical issues at these parame-ters. In this regard, we also note that, for a few systems, at thehighest graing density (0.65 chains per d2) only, where matrixchains are expected to deplete/dewet large regions in the graedlayer near the particle surface, the choice of Percus–Yevickclosure leads to negative values in gmatrix–particle(r) at low r (wheregmatrix–particle(r) should be 0), due to numerical issues. We alsoadd that, for these specic systems, all other pair correlationfunctions are devoid of this issue and do not exhibit anynegative values. Despite this issue, the PRISM-MC results weshow here correctly predicts all known (qualitative and somequantitative) trends in monodisperse systems – (a) withincreasing graing density the mid-range attractive welldeepens and shis to higher inter-particle distances (Fig. 1); (b)with increasing matrix chain length the attractive well depthdeepens (ESI†); (c) the value of the well depth seen at 0.65chains per d2 is of the same order of magnitude (�0.3 to 0.5 kTin ESI†) as that seen for similar systems in recent simulationstudies3,48 on systems with gra length of 10 monomers andmatrix lengths of 10–70 monomers, and particle sizes approxi-mately 10 times monomer size at high graing density (�0.76chains per nm2). The ability of PRISM-MC to predict the samequalitative trends as prior monodisperse studies, and, in certaincases, show quantitative agreement with prior simulations formonodisperse gras, suggests that this approach is capable ofpredicting correct qualitative trends for the polydisperse poly-mer graed nanoparticles as well.
Continuing with our discussion of reduction in attractivewell depth with increasing polydispersity in gras, while thegra and matrix concentration proles in the ESI† show theincreased region of overlap between the gra monomerconcentration and matrix monomer concentration prole, todemonstrate the increased wetting of the polydisperse graedlayer by the matrix chains we present the penetration depth ofthe matrix chains into graed layer, l, (Fig. 3). Fig. 3 showsincreasing l with increasing PDI, conrming increased wettingof the graed layer by the matrix chains with increasing PDI forboth matrix lengths. Also, there is a larger effect of poly-dispersity on improving wetting at higher graing densitiesthan lower graing densities. The reason behind this is thesame as mentioned earlier. Since the change in chain confor-mations due to increasing polydispersity is more drastic at thehigher graing densities than lower graing densities, theresulting improved wetting of the graed layer by the matrix isalso more drastic at higher graing density.
Effect of increasing matrix packing fraction
All the results presented so far were at a total packing fraction ofh ¼ 0.1, which we characterize as a dense polymer solution,rather than a melt, through calculations of the compressibility
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Fig. 3 Penetration depth (in units of d) of thematrix chains into the grafted layerof nanoparticles (D¼ 5d) with polydisperse chains (Ng,avg ¼ 20) at s¼ 0.10 chainsper d2 (circles) and s ¼ 0.25 chains per d2 (triangles) in a dense solution (h ¼ 0.1)with (a) Nmatrix ¼ 10 and (b) Nmatrix ¼ 40.
Fig. 4 Potential of mean force (in units of kT) versus inter-particle distance, r� D(in units of d), between grafted nanoparticles (D ¼ 5d) at s ¼ 0.65 (a), 0.25 (b),and 0.10 chains per d2 and PDI ¼ 1.0 (circles), 1.5 (squares), 2.0 (upward facingtriangles), and 2.5 (downward facing triangles) with Ng,avg ¼ 20, in a melt-like (h¼ 0.3) matrix of monodisperse homopolymer chains with Nmatrix ¼ 10 (solidsymbols) and Nmatrix ¼ 40 (open symbols). The insets have the same axes labels asthe main plots.
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from the structure factor S(k) as k / 0.28 At a melt-like packingfraction of h ¼ 0.3, the matrix polymers have been shown toinduce depletion like attractions between both bare andmonodisperse polymer graed nanoparticles at innitely diluteconcentrations.30,49,50 At high graing densities, this matrix-induced depletion-like attraction signicantly reduces the stericrepulsion at contact and deepens the mid-range attractive well(inset of Fig. 4a versus Fig. 1a). At low graing densities, thematrix-induced depletion-like attraction manifests itself in thePMF as attraction at contact (Fig. 4c versus Fig. 1c). Comparingh ¼ 0.3 and h ¼ 0.1 at high graing density (Fig. 4a versusFig. 1a), the repulsion at contact is less sensitive to PDI ath ¼ 0.3, and a larger PDI is needed to eliminate the strongerattractive well at intermediate distances at h ¼ 0.3 (inset ofFig. 4a). At low graing density (Fig. 4c), the effects of poly-dispersity are reduced at h ¼ 0.3 as compared to h ¼ 0.1, as thevalues of attraction at contact (�3 kT) dominate at all PDI. Thisimplies that, in melt-like polymer matrices, one can stabilizedispersions using polydispersity only at high graing densities,and the extent of polydispersity needed to stabilize dispersionsis higher as compared to the gra polydispersity needed tostabilize dispersions in dense polymer solutions. In a melt-likepolymer matrix at low graing densities, any effect of poly-dispersity in gras will be overcome by the dominant
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matrix-induced depletion attraction that will enhance thetendency for aggregation of particles. This is further conrmedby comparing the second virial coefficient at h ¼ 0.3 (Fig. 5) ascompared to h ¼ 0.1 (Fig. 2). At high graing densities (dottedlines in Fig. 5 and 2) at each PDI, the B2 is much smaller for h ¼0.3 as compared to h ¼ 0.1. Similar behavior is seen at inter-mediate graing density (dashed line in Fig. 5 and 2). At lowgraing densities the value of B2 is weakly positive, and does notchange with increasing PDI as seen at h ¼ 0.1. Comparison ofpenetration depth at h ¼ 0.3 (ESI†) and h ¼ 0.1 (Fig. 3) shows
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Fig. 5 Second virial coefficient for polymer grafted nanoparticles in a polymermatrix as a function of graft polydispersity (PDI) at grafting densities of s ¼ 0.10chains per d2(solid line), s ¼ 0.25 chains per d2(dashed line), and s ¼ 0.65 chainsper d2(dotted line), and total system volume fraction of h ¼ 0.3. All data is forparticle diameter D ¼ 5d, average graft length Ng,avg ¼ 20, and monodispersematrix length Nmatrix ¼ 40.
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that the matrix penetrates a lot less at high matrix packingfraction than in dense solutions. This lower penetration depthcan be explained partly by the reduction in the thickness of thegraed layer at higher matrix packing fraction due tocompression of the graed monomers by the melt-like matrixchains.51 Overall, the net effect of polydispersity is lowered athigh melt-like matrix packing fraction due to matrix-induceddepletion interactions dominating the physics in these systems.
Fig. 6 Potentials of mean force (in units of kT) versus inter-particle distance,r � D (in units of d) between nanoparticles of size D ¼ 8d grafted with poly-disperse chains at PDI¼ 1.0 (circles), PDI¼ 1.5 (squares), PDI¼ 2.0 (upward facingtriangles), and PDI ¼ 2.5 (downward facing triangles) with Ng,avg ¼ 20 at agrafting density of s ¼ 0.1 chains per d2 (parts a and b) and 0.25 chains per d2
(part c) in a dense solution (h ¼ 0.1, part a and c) and melt (h ¼ 0.3, part b) ofmonodisperse homopolymers with Nmatrix ¼ 40. The insets have the same axeslabels as the main plots.
Effect of particle diameter or curvature
So far we have observed that the effect of polydispersity isenhanced at conditions where there is large monomer crowdingat the monodisperse limit (e.g. higher graing density) andtherefore the relief to that crowding brought about by poly-dispersity is more signicant. Based on that observation, onecould expect that, at constant graing density, since themonomer crowding is larger on surfaces with lower curvature,the effect of polydispersity induced relief in crowding wouldalso be larger on surfaces with lower curvature. To test this, wecalculated the PMFs for graed particles with diameter D ¼ 8dwith polydisperse gras at 0.1 and 0.25 chains per d2 in a densehomopolymer matrix (Fig. 6a and c) and at 0.1 chains per d2 in amelt-like homopolymer matrix (Fig. 6b) with Nmatrix ¼ 40. In adense solution, upon comparing the PMF for D ¼ 8d (Fig. 6aand c) to the corresponding PMF for D ¼ 5d (open symbols inFig. 1c and b) we observe the following: (i) at the monodispersegra limit the repulsion at contact andmid-range attractive wellare stronger for D¼ 8d as compared to D¼ 5d when Nmatrix¼ 40for both graing densities. Fig. 7a presents a direct comparisonof the monodisperse gras on D ¼ 5d to D ¼ 8d at 0.25 chainsper d2 graing density in a dense solution of matrix of lengthNmatrix ¼ 40 and conrms the above trend of higher repulsion atcontact and stronger mid-range attraction in D ¼ 8d ascompared D ¼ 5d. Fig. 7a also shows the slight shi of the mid-range attraction to larger inter-particle distances. These trendsare in accord with previous studies of monodisperse gras6,11
that showed that decreasing curvature increases the monomer
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crowding near the particle surface, increases the graed layerthicknesses, and decreases the propensity of matrix to wet thegraed layer. (ii) As gra polydispersity increases, the repulsionat contact is reduced and the attractive well at intermediatedistances is replaced by a long repulsive tail for D ¼ 8d, similarto D ¼ 5d. Table 1 present a direct comparison of the attractivewell depth for D ¼ 8d and D ¼ 5d at PDI ¼ 1.5 and 2, graingdensity of 0.25 chains per d2 and Nmatrix ¼ 40 which demon-strates that, at PDI ¼ 1.5 in case of D ¼ 8d, we can reduce a
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Fig. 7 Potentials of mean force, PMF (in units of kT) versus inter-particle distance,r � 5 (in units of d) between nanoparticles of size D ¼ 8d (open symbol) and D ¼5d (solid symbols) grafted with polydisperse chains at (a) PDI ¼ 1.0, (b) PDI ¼ 1.5,and (c) PDI¼ 2.0 with Ng,avg¼ 20 at a grafting density of s¼ 0.25 chains per d2, ina dense solution (h ¼ 0.1) of monodisperse homopolymers with Nmatrix ¼ 40. Theinsets have the same axes labels as the main plots.
Table 1 The maximum value of the magnitude of the mid-range attraction wellin the potentials of mean force (in units of kT) as a function of graft PDI andparticle size D ¼ 8d and D ¼ 5d. The polydisperse chains have Ng,avg ¼ 20 and aregrafted on the particles with a grafting density of s ¼ 0.25 chains per d2. Thegrafted particles are place in a dense solution (h ¼ 0.1) of monodisperse homo-polymers with Nmatrix ¼ 40
PDI D ¼ 5d D ¼ 8d
1 �0.0436 kT �0.1363 kT1.5 �0.0002 kT �0.0032 kT2 �0.0001 kT �0.0000 kT
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much larger attractive well seen at PDI ¼ 1 as compared to therelatively smaller attractive well of D ¼ 5d. At 0.1 chains per d2
and h ¼ 0.3 (Fig. 6b for D ¼ 8d, Fig. 4c for D ¼ 5d) we do not seeany signicant qualitative differences in polydispersity effectsfor D ¼ 8d and D ¼ 5d, as expected since the crowding isminimal at that low graing density and the matrix induceddepletion-like attraction starts to dominate. The trends seen sofar suggest that with increasing diameter or decreasing curva-ture, we can expect polydispersity in gras to play a larger role inreducing the tendency of graed particle aggregation. For low
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curvatures, as the well-depth deepens signicantly in themonodisperse limit with increasing matrix length beyond thosepresented here, we might also need larger polydispersity toeliminate the relatively stronger mid-range attraction.
Effect of distribution of chain lengths
The key result in this paper is that when polydisperse homo-polymer chains with a broad chain length distribution aregraed on nanoparticle surfaces they eliminate or reduce themid-range attractive well in the PMF that has been shown tocause aggregation in monodisperse limits. To ensure that ourchoice of a specic discretized chain length distribution, thatmimics a continuous log-normal distribution, does not bias theabove results, we calculated the PMF for ve different chainlength distributions (all log-normal) for a select few systems. InFig. 8, we present the PMF between nanoparticles of size D ¼ 5dwith polydisperse chains of PDI ¼ 1.0 (circles) and ve differentlog-normal distributions with PDI ¼ 2.0 (other symbols) andNg,avg¼ 20 at a graing density of s¼ 0.65 chains per d2 and in adense solution or h ¼ 0.1 (Fig. 8a) and melt-like matrix h ¼ 0.3(Fig. 8b) of monodisperse homopolymers with Nmatrix ¼ 40.Clearly, for both h ¼ 0.1 and 0.3, the PMF between the particleswith polydisperse chain length distribution are quantitativelysimilar for the ve distributions, and quantitatively distinctfrom the corresponding PMF between particles with mono-disperse gras. We also expect that the choice of another formof distribution (different from log normal) would not changethe effect of polydispersity removing the mid-range attraction,because the distribution we have chosen is a discretized versionof a continuous log-normal distribution and another discretizedbroad distribution of chain lengths should bring about thesame effect. This then begs the question – how will the PMFobtained for particles with a bimodal distribution of gralengths or bidisperse gra length distribution compare withthat of the PMFs seen so far with a statistically polydispersedistribution.
Polydisperse versus bidisperse
In Fig. 9, we compare PMFs from particles graed with a log-normal chain length distribution (solid circles) to those graedwith a bidisperse chain length distribution (open circles) at thesame PDI ¼ 1.5, and the corresponding monodisperse graswith same average gra length (no markers). These bidispersechain length distributions have equal number of monodisperse
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Fig. 8 Potentials of mean force between nanoparticles of size D ¼ 5d with poly-disperse chains of PDI ¼ 1.0 (circles) and PDI ¼ 2.0 (other symbols) with Ng,avg ¼ 20and Ng,min ¼ 8 at a grafting density of s ¼ 0.65 chain per d2, for five different log-normal distributions, in (a) a dense solution (h ¼ 0.1) and (b) melt (h ¼ 0.3) ofmonodisperse homopolymers with Nmatrix ¼ 40. Each symbol (besides circle)corresponds to a different discretization of the log-normal chain distribution.
Fig. 9 Potentials of mean force (in units of kT) versus inter-particle distance, r �D (in units of d), between grafted nanoparticles (D ¼ 5d) with monodispersehomopolymer grafts with PDI ¼ 1.0 (solid line), polydisperse homopolymer graftsPDI¼ 1.5 (solid circle), and bidisperse homopolymer grafts PDI¼ 1.5 (open circle),all with Ng,avg ¼ 20 and grafting density s¼ 0.65 (left), and 0.25 (right) chains perd2 in a dense solution (h ¼ 0.1) (top row) or melt-like (h ¼ 0.3) (bottom row)monodisperse homopolymer matrix with Nmatrix ¼ 40. The insets have the sameaxes labels as the main plots.
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short (¼ 6 segments) and monodisperse long (¼ 36 segments)chains with Ngra,avg ¼ 20 and PDI ¼ 1.5. At both 0.65 and 0.25chains per d2 and both h ¼ 0.1 and 0.3, we observe distinctbehavior between the monodisperse, polydisperse (log-normal)and bidisperse PMFs both at contact and at intermediatedistances, when Nmatrix is greater than Ngra,avg. At high graingdensity (0.65 chains per d2) and dense solution matrix (Fig. 9a),the repulsion at contact is most reduced by a bidispersedistribution while the attractive well at intermediate distancesis most reduced by the polydisperse distribution. The former isbecause half the chains in the bidisperse distribution are short(6 monomers), while there are only six chains in the poly-disperse distribution (PDI ¼ 1.5) that are less than 10 mono-mers long. The latter is because, in the polydispersedistributions, the long chains are signicantly longer than thelong chains in the bidisperse distribution (Ngra,long ¼ 36monomers). We have attributed the elimination of mid-rangeattraction in the PMF to (i) the presence of these signicantlylong chains sterically hindering other long chains and (ii)increased gra chains mixing with the matrix chains due toreduced graed layer crowding in the polydisperse system. Thefact that the long chains in the bidisperse distribution areshorter than the Nmatrix (systems where aggregation is observed)and yet the bidisperse distribution is able to reduce the
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mid-range attraction emphasizes the role of lowered monomercrowding (brought about having short and long chains) inreducing the mid-range attraction.
In the melt-like matrix (Fig. 9b) we also see a shi in the mid-range attraction to the lower distances in the case of poly-disperse distribution as compared to bidisperse distribution. Inthe melt-like matrices the matrix induced depletion attractionpushes the graed particles together. The inter-particledistance where the two graed layers start to overlap due to thisdepletion-like attraction corresponds to the position of the mid-range attraction in the PMF. This overlap between the graedlayers occurs at the outer-most region of the graed layer in themonodisperse case, and therefore corresponds to the distancewhere the gra concentration prole drops close to zero. Thegra concentration prole is slightly different in the poly-disperse case when compared to the bidisperse case (ESI†).Close to the particle surface, the polydisperse and bidispersegra monomer concentrations proles have similar values. Atintermediate distances from the particle surface, the bidispersecase has the lowest concentration; the region where the longchains of the bidisperse distribution have a “stem-like”conformation51 and polydisperse chains have a distribution ofchain conformations. At larger distances from the particlesurface (outer region of the graed layer), the bidisperseconcentration is higher than polydisperse and monodispersebecause of the “crown” region of the long chains in the bidis-perse region.51 Due to these features in the graed layermonomer distribution, the overlap of two graed layers and thecorresponding mid-range attractive well occurs at slightlydifferent inter-particle distances for polydisperse, bidisperse,
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and monodisperse gra distributions. The above trendsbetween bidisperse and polydisperse distributions on the mid-range attraction are also observed at 0.25 chains per d2 (Fig. 9band d).
These results in Fig. 9 suggest that, for this specic type ofbidisperse distribution (equal number of short and long chains),with the same average gra length and PDI as polydispersedistribution, the differences in monomer arrangement withinthe graed layer between polydisperse and bidisperse grasleads to polydisperse gras being able to stabilize particledispersion better than bidisperse gras, in dense polymermatrix. However, we note that, if we relaxed some of theconditions chosen in Fig. 9 (e.g. equal number of short and longchains) or change the average gra length in the bidispersedistribution, there could be cases where the bidisperse orbimodal distributions provide better dispersion of particles ascompared to a polydisperse distribution. In a melt-like polymermatrix, the effect of bidispersity or polydispersity is lower thanthe corresponding dense polymer matrix; the attractive well inthe PMF is not removed in melt-like matrix at the PDI where thecorresponding dense solution matrix has been eliminated bypolydispersity/bidispersity. The different positions for the mid-range attraction in the PMF between graed particles withbidisperse and polydisperse graed chains suggests that thechoice of gra length distribution can be used as a way to tunethe structure (e.g. inter-particle spacing) within the aggregatesof these graed particles.
4 Conclusions and future directions
In summary, this article presents a theory and simulation studythat demonstrates how polydispersity in polymers graed onspherical nanoparticles affects the effective inter-particleinteractions between homopolymer graed nanoparticles in achemically identical homopolymer matrix. Gra polydispersityreduces the strength of repulsion at contact and weakens theattractive well at intermediate inter-particle distances in thepotential of mean force (PMF) between graed particles. Thiseffect is attributed to polydispersity in graed chain lengthsreducing the graed layer monomer crowding seen at mono-disperse limits, which in turn increases wettability of the graf-ted layer by the matrix chains. The reduction/elimination of theattractive well suggests that gra polydispersity can reduce thetendency for particle aggregation, even stabilizing dispersion insome cases where the monodisperse gras would cause aggre-gation. As the graing density decreases from a brush-likegraing density, the effect of polydisperse gras on the poten-tial of mean force reduces because the relief in monomercrowding brought about by polydisperse gras is insignicantat low graing density. As the matrix packing fraction increases,the matrix-induced depletion-like attraction between the graf-ted particles becomes dominant, reducing the above grapolydispersity effects. At high graing density, this matrixinduced depletion-like attractions results in a need for higherpolydispersity to eliminate the mid-range attraction completely,while at low graing density the effect of polydispersity iscompletely removed. At the monodisperse gra limit, the larger
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particles (lower curvature) have increased monomer crowding,especially at high graing densities, compared to smaller(highly curved) particles, and, as a result, the effect of poly-dispersity is enhanced at lower curvature surfaces.
Comparison of bidisperse and polydisperse distributions ofgra lengths elucidate the role of long chains in removing theattractive well by essentially adding a repulsive tail to the PMFthrough long gras sterically repelling the gras on the otherparticle. At the same PDI and average gra length, the presenceof longer “long” chains in the polydisperse distribution elimi-nates the attractive well in the PMF better than the shorter“long” chains in the bidisperse distribution. In a bidispersedistribution with both short and long chains that are shorterthan the matrix chains, we see reduction in mid-range attrac-tion attributed to the lowered monomer crowding in thebidisperse graed layer, that in turn increases the graed layerwetting by the matrix chains. In a melt-like matrix, where thematrix-induced depletion attraction starts to dominate over thepolydispersity effects, the different conformations of the graedchains in the bidisperse and polydisperse distribution,captured through the gra monomer concentration proles,change the location of the attractive well in the potential ofmean force.
The implications of these results, that polydispersity canstabilize particle dispersions, are exciting since much of thepolymer synthesis community has been striving to achieve lowpolydispersity. This study motivates synthetic efforts to bedirected towards obtaining controlled gra molecular weightdistributions to tailor inter-particle interactions of polymergraed nanoparticles in a polymer matrix. In this regard, recentexperiments by Rungta et al.52 and Li et al.53 that involvedsynthesis of bimodal polymer graed nanoparticles, conrmthe ability of bidispersity in polymers graed on nanoparticlesto improve dispersion of the nanoparticles in the matrix. Inaddition to gra polydispersity induced morphological effects,the next interesting direction would be to study the segmentaldynamics in the presence of gra polydispersity and the glasstransition temperature of the material, as well as the viscousresponse of the nanocomposite, as done recently in experi-ments and simulations of monodisperse gra and matrixsystems.54,55,56 The increased wetting of the graed layer by thematrix chains in the presence of gra polydispersity couldenhance the entanglement of the long gra chains with thematrix chains, when gra and matrix molecular weights aregreater than the entanglement molecular weight. Therefore, itwould be worth investigating systematically how polydispersityin gras impacts the mechanical reinforcement of the nano-composite. Lastly, while we focused on a monodisperse matrixin this study, it would be important to understand how poly-dispersity in the matrix chains affect these effective interactionsof monodisperse polymer graed particle and polydispersepolymer graed nanoparticles, and the above thermal andmechanical properties of the nanocomposite.
In conclusion, one of the key ndings in this paper is thatpolydispersity can stabilize dispersions even when the averagegra polymer molecular weight is lower than matrix polymermolecular weight, conditions that would cause particle
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aggregations for monodisperse gras. This study motivatessynthetic efforts to be directed towards obtaining controlledpolydispersity in chain lengths as a design tool to programinter-particle interactions in a polymer matrix, and in turn themorphology of the polymer nanocomposite.
Acknowledgements
The authors acknowledge nancial support by Department ofEnergy Early Career Award Program under Grant no. DE-SC0003912.
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