pubs.acs.org/MacromoleculesPublished on Web 10/22/2010r 2010 American Chemical Society

Macromolecules 2010, 43, 9567–9577 9567

DOI: 10.1021/ma102046q

Computer Simulation of Controlled Radical Polymerization:Effect of Chain Confinement Due to Initiator Grafting Densityand Solvent Quality in “Grafting From” Method

Salomon Turgman-Cohen and Jan Genzer*

Department of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh,North Carolina 27695-7905, United States

Received September 3, 2010; Revised Manuscript Received October 11, 2010

ABSTRACT: We use stochasticMonte Carlo simulation following the bond fluctuation model to study theeffects of grafting density of surface-anchored initiators and solvent quality on controlled radical polymer-ization (CRP) from flat impenetrable substrates under good and poor solvent conditions. Our CRP modelincludes a mechanism for activation/deactivation of the chains and neglects termination and chain transferreactions. The system is, thus, “truly living”. We find that under these conditions, surface-initiatedpolymerizations at low grafting densities resemble those in the bulk. In contrast, at high initiator graftingdensities, these surface-initiated polymerizations result in gradients of the free monomer and chain-endconcentrations, which lead to an uneven growth of the chains and ultimately yield polymers with broadmolecular weight distributions. Poor solvent conditions exacerbate this problem by collapsing the chains andin some cases forming chain aggregates, which further restrict the access of free monomers by the activepolymer chain ends and contribute to their uneven growth and ultimately broader length distributionsrelative to good solvent conditions. While at low grafting densities the molecular weight distributions can bedescribed by the conventional Schulz-Zimm distribution function, at high grafting densities this approachfails to describe accurately the dispersity in chain lengths.

Introduction

During the past decade, numerous research reports havedemonstrated that specialty end-tethered polymer layers on sur-faces can be generated by “grafting from” polymerization usingcontrolled radical polymerization (CRP) schemes.1-10 Equilibriumbetween active polymerizing chains and inactive dormant chains inCRPs is shifted heavily towards the dormant chains, which leads tocontrolled molecular weights and molecular weight distributions(MWDs).11,12 The “grafting from” scheme consists of growingpolymers directly from initiator-covered surfaces. In contrast tothe “grafting onto” approach, in which polymers with reactiveend groups attach to the surface, the “grafting from” strategy isbelieved to yield dense brusheswith controlled thickness, graftingdensities, and relatively narrow MWDs characterized by a smallpolydispersity index (PDI). Despite the common use of CRP forthe synthesis of end-tethered layers, there is a dearth of researchcharacterizing the PDI of polymers produced from initiating sur-faces. The current approach for the estimation of PDI in tetheredlayers involves the addition of free bulk initiators to the poly-merization mixture and the subsequent analysis of the solution-synthesized polymers by size-exclusion chromatography (SEC)or an equivalent technique. Themajor drawbackof this approachis that it neglects possible effects of surface confinement anddiffusion of the reactive components to and from the propagatingradical ends, which, in turn, affect the ability of CRP to yieldgrafted polymers with low PDI.

Surface-initiated atom transfer radical polymerization (SI-ATRP)and its various derivatives (i.e., ARGETATRP) are probably themost frequently used CRP schemes employed in the formation

of polymeric grafts on surfaces using the “grafting from”method.13,14 The popularity of thesemethodologies has stemmedfrom their robustness, insensitivity to the presence of smallamounts of impurities (most notable oxygen) that tend toterminate the radical processes, and their ability to createcopolymers and polymers with complex topologies. In spite of asizable amount of work on this topic, our understanding of the“grafting from” process and complete characterization of theproperties of grownmacromolecular grafts has been hindered bymany technical limitations. As a result, contradicting reportshave not been reconciled conclusively by means of our currentknowledge.15-19 For example, experimental studies performedby cleaving the surface-initiated polymers15-19 report bothincreases15,16,18 and similar17 PDIs of the surface-grafted poly-mers relative to polymers grown in the solution. In addition,while most reports noted that “grafting from” polymerizationresulted in polymers whose molecular weights were lower thanpolymers grown under identical experimental conditions in thebulk,Koylu et al. reported recently on the opposite trend.18 Theseand other findings are not easy to understandwithout performinga comprehensive set of additional experiments involving SECmeasurements with very sensitive concentration detectors. How-ever, even with the available SEC measurements of surface-grown polymers, more information about the structure of thegrafted layer is needed in order to comprehend the effect of thevarious process parameters that influence the “grafting from”mechanism. In spite of tremendous progress in various instru-mentation tools, no currently available analytical technique iscapable of characterizing completely the structure of the graftedlayer. Computer simulations and various molecular theoriesmay prove useful because they provide monomer-level insightinto the polymerization process that may not be accessible to

*To whom correspondence should be addressed. Tel.: þ1-919-515-2069. E-mail: jan_genzer@ncsu.edu.

9568 Macromolecules, Vol. 43, No. 22, 2010 Turgman-Cohen and Genzer

experiments. These approaches further offer the opportunity tovary systematically the parameters of the polymerization systemand to characterize completely the properties of the moleculargrafts without the limitations involved with experiments.

The variations in PDI investigated herein are relevant giventhat properties such as the brush height and the polymerconcentration profile vary significantly when comparing graftedpolymers with equal number averagemolecular weights (ÆNæ) butdiffering MWDs.20-29 For instance, using numerical self-consis-tent mean field theory de Vos and Leermakers29 demonstratedrecently that broadening ofMWDs leads to changes of the shapesof the brush concentration profiles from parabolic to linearto concave and to corresponding increases in the brush layerthickness under good solvent conditions. The authors also notedthat, interestingly, the scaling exponents for the brush height withrespect to the molecular weight and the grafting density areidentical for both monodisperse and polydisperse polymers. In alater study, the authors reported on the effect of PDI on theantifouling properties of tethered polymers, which is an impor-tant application of these layers.30 In this contribution, we buildupon previous reports and investigate the effect of the initiatorgrafting density and the solvent quality on the properties ofpolymers initiated from flat substrates; and how they compare tobulk initiated reactions in athermal solvents. The results provideinformation about themolecular weight, PDI, and concentrationprofiles for grafted polymers grown by CRP reactions.

Reports simulating and modeling surface-initiated polymeri-zation have appeared previously. Wittmer et al.31 studied thediffusive growth of a polymer layer by in situ polymerizationusing a mean-field treatment and scaling theories of polymers ingood solvents. Computer simulations utilizing the bond-fluctua-tion model (BFM) were used to verify the developed theory.Wittmer and co-workers considered a system in which initiators,which could commence irreversible growth of linear unbranchedchains, covered densely an impenetrable wall. The authorsfurther assumed that the chain growth was slow and, therefore,the chains retained their local equilibrium conformation duringthe polymerization process. An infinitesimal flux of incomingmonomers then provided the reactants from which the polymerscould grow. Wittmer et al. predicted that for diffusion-limitedreactive growth the formed chains would be polydisperse butnonetheless strongly stretched away from the surface.

Numerous computer simulation studies have been performedon bulk- and surface-initiated equilibrium, “living” polymeriza-tions, in which polymerization and depolymerization reactionswere accounted for and the polymers were in chemical equilib-rium with a population of free monomers. Using off-lattice andBFMMonte Carlo (MC) simulations, Milchev and co-workers32

reported briefly on the effect of growing polymers from low andhigh initiator grafting densities that resulted in differences in theirMWDs. Matyjaszewski et al.13 also implemented computersimulations to investigate surface-initiated polymerizations. Theyconstructed concentration profiles and computed the MWD ofthe simulation-grown polymers.

Recently, Liu and co-workers33 reported on the effects of thepolymerization rate and the density of initiating centers on theproperties of macromolecules grown by “grafting from” poly-merization from flat substrates bymeans of coarse-grainedmolec-ular dynamics. Liu et al. modeled a “true” living/controlledpolymerization process by not considering any termination (orchain transfer) in their simulations. They demonstrated thatdecreasing the rate of polymerization resulted in polymeric graftswith lower PDI, relative to polymerization performed at higherrates. Liu et al. also reported on the interplay between the densityof the initiators on the surface, their effectiveness of initiation,and polymerization rates. Specifically, the authors pointed outthat in systems with low initiator density nearly all initiators

started polymerization. At slow polymerization rates, all chainspropagated at approximately the same rate. In contrast, at highdensities of the initiator centers, initiation efficiency decreasedwith increasing rate of polymerization, which, in turn, resulted ina higher fraction of inactive initiators and higher PDIs.

A few years ago, one of us developed andpresented a computersimulation model of controlled radical polymerization resem-bling processes occurring in ATRP by employing a stochasticMC scheme.34We described how ÆNæ and theMWDof polymersgrown in bulk and on flat impenetrable surfaces depended on anumber of parameters, such as, the concentrations of differentspecies and the simulation equivalents to kinetic rates and equi-librium constants. We demonstrated that increasing the termina-tion probability and decreasing the initial probability of additionof a newmonomer to a growing chain broadened theMWD.Ourresults also indicated that confinement of surface-initiated poly-mers resulted in increased numbers of early terminations, which,in turn, led to higher PDIs. Upon increasing the grafting densityof the initiators on the surface, we noted further increases in thePDIs of the grafted polymers.

The current work builds upon our previous MC study34 andthe work of Liu and co-workers33 to investigate the effect thatdecreasing the solvent quality and/or grafting the initiators to thesurface has on the broadness of the MWDs. By turning off thetermination reaction and decreasing the rate of polymerization(in contrast to our previous work), we aim to model the “trueliving”/controlled radical process. In contrast to previousstudies, we include a mechanism for activation/deactivation ofthe propagating species that resembles most CRP schemes inour simulations. We will demonstrate that even in “truly living”polymerizations, confining the growing chains (either by graftingto the surface or by collapsing the chains in poor solvents) mayresult in appreciable increases in the PDI of the surface-tetheredpolymers. We also explore the effect of confinement due to chaincollapse on the MWDs. Our motivation for investigating poly-merization in poor solvents comes from experimental studies inwhichmethanol/water/monomermixtures were used for SI-ATRPof methyl methacrylate (MMA)19,35 and n-isopropylacrylamide(NIPAAm)36-40 from planar substrates. If large monomer con-centration gradients develop in these reactions, they can exposethe grafted polymers to a water/methanol mixture, which is apoor solvent at the experimental conditions.

Simulation Model

Computer simulations of CRP reactions are performed byimplementing a stochastic MC algorithm34 in conjunction withthe BFM.41 The polymers and monomers reside in a three-dimensional cubic lattice in which the monomers can be bondedaccording to the following vector families: P(2,0,0), P(2,1,0),P(2,1,1), P(2,2,1), P(3,0,0), and P(3,1,0).42 These vector setsprevent any bonds from crossing and monomers (and polymers)from overlapping as long as the only allowed moves in thesimulation are nearest neighbor ones. The fluctuating characterof the bond-lengths in this model allows for close approximationof continuum behavior while at the same time preserving theadvantages of lattice models such as integer arithmetic.

TheBFMis, by its nature, self-avoiding and as such it describespolymer behavior under good solvent conditions. To simulatepoor solvent conditions we have incorporated a set of inter-molecular, intramolecular and bond-length potentials.43 The firstpotential is of the Lennard-Jones (LJ) form and is truncated toonly act between beads that find themselves within the maximumdistance allowed by the above-mentioned vector families (bondlength = 101/2). A bond-length potential is also included todescribe the increase in chain stiffness with decreasing tempera-ture. In our system, the LJ and bond-length potentials only act

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between bonded beads in the simulation box and not between thebeads representing free monomers. We neglect the polymer-monomer and monomer-monomer interactions to prevent ag-gregation of free monomers that would describe an unlikelyphysical condition in which the monomers are not soluble inthe polymerization medium. It was demonstrated previously thatincreasing the interaction parameter (ε) in these potentials re-lative to temperature led to collapsed chain conformations.43,44

We confirm independently chain collapse in the bulk- and sur-face-initiated cases (see Supporting Information) and performreactive simulations in both good (ε = 4 � 10-7 kBT) and poor(ε = 4 � 10-1 kBT) solvents.

The MC algorithm used to simulate the polymerization reac-tions has been described previously.34 The total number ofinitiators and monomers is fixed for all simulations performed.We vary the density of initiators on the surface while maintainingnearly constant the overall volume of the lattice by adjustingthe dimensions of the cubic lattice (Lx=Ly, Lz). In this manner,we can investigate the effect of the grafting density of initiatorson the polymerization while keeping other relevant parameters(i.e., monomer concentration) unchanged. Table 1 tabulates theLx = Ly and the Lz dimensions of the lattices used in this studyand the resulting surface density of the initiators (σ, where σ=1represents a fully covered surface in the BFM). In bulk-initiatedpolymerizations, we implement periodic boundary conditions inthe three spatial directions. For surface-initiated polymeriza-tions, two impenetrable walls are located at z= 0 and z= Lz

representing the substrate and the limit of the simulation box,respectively; the periodic boundary conditions are employed onlyin the xy plane of the lattice.

The initial configuration for bulk-initiated polymerizationscorresponds to the monomers and initiators distributed uni-formly throughout the volume of the simulation box. Eachinitiator corresponds to two bonded polymer beads, of whichone is a propagating center (i.e., chain end). In the case of surface-initiated polymerizations, the inactive bead of the initiator islocated at equally spaced intervals on the substrate and at z=1,while the propagating bead is placed directly above of theanchoring bead at z = 3. In the surface-initiated case, the beadadjacent directly to the surface is not movable. The numbers ofmonomers and initiators have been fixed at Mo = 12500 andIo = 400, respectively; this corresponds to a lattice occupancy of10.64%. We perform a preliminary long equilibration run con-sisting of ≈109 attempted Monte Carlo steps (MCS, defined asMC steps per simulation bead) to obtain a randomly distributedconfiguration. The equilibrated configuration serves as input tothe MC simulations.

We initialize the simulations by a short equilibration period of106 MCS. This is equivalent to every bead in the simulation boxundergoing, on average, 75 attempted moves. Once the secondpre-equilibration finishes the reactiveMCalgorithm commences.The reactiveMCschemecontinuesuntil either 80%of themonomeris exhausted (i.e., polymerized) or a predeterminedmaximumnumber of MCS elapses. The latter premature end of the simula-tions before reaching high conversionswas necessary because, in thehigh surface density and poor solvent regimes probed, the poly-merization rate is very slow and the simulation would not reach80% conversion in a reasonable time.

TheMCalgorithm implemented here is similar to our previouswork34 except for a few changes that we describe next. Previously,the probability of reaction versus motion was set a priori toPr =0.5. In the current work, we investigate the effect of this param-eter on the results of the simulation and determine that, for thestudy of different levels of confinement and poor solvent condi-tions, a more appropriate value of Pr is 0.01 (see SupportingInformation). Reducing the reaction rate (propagation onlyin the present work since we prohibit termination and chaintransfer) further assures controlled polymerization growth,which is consistent with the approach of Liu and co-workers.Reducing Pr also shifts the system behavior from a diffusion-limited reaction to a kinetically limited one, in turn,mitigating theeffects of gradients in the free monomer volume fraction andallowing the polymer chains to approach their equilibriumconformations. Although it would be desirable to slow thereaction rate enough to achieve full kinetic control, balancingthis need with feasible simulation times was required. The seconddifference between the current simulations andourpreviousworkis that the probability to add a monomer (Pa) is no longerproportional (Pa=Pa,o[M/Mo], where Pa,o is a constant andM is the instantaneous number of free monomers) to the numberof free monomers left in the simulation box. This proportionalitywas redundant because the probability that an active chain and afree monomer approach to within the reactive distance is alreadyproportional to the instantaneous monomer concentration. Wetherefore modify our approach in the current work and set Pa =Pa,o throughout the simulation run. Both of these changes affectthe observed results in an absolute fashion, but all the observedtrends and conclusions of our previouswork34 remain unchangedin this revision of the algorithm.

During a typical polymerization run, wemonitor themonomerconversion (xm), polydispersity index (PDI), radius of gyration(|Rg

2|1/2), volume fraction profiles (φi(z)), and the number ofnearest neighbor site per active chain-end occupied bymonomers(θm), polymers (θp), and chain ends (θe).We evaluate thePDI,Rg,and φi(z) at steps of 1% monomer conversion. We average thesequantities for at least three independent runs to gain statisticalcertainty. The number of the nearest neighbors to the active chainends are computed every 10000 MCSs and plotted as a functionof monomer conversion. These data are fitted with a third degreepolynomial to discern the trends between the different runs.

Results and Discussion

Solvent Quality. Computer simulations of polymerizationreactions were performed in good and poor solvents forbulk-initiated systems. The working hypothesis was thatwith decreasing the solvent quality, the coil becomes morecompact thus making the interior of the coil inaccessible tomonomers and hiding the growing chain inside the collapsedcoil. As a result, relative to polymerization under goodsolvent conditions, the rate of polymerization under poorsolvent conditions will be slower and the PDI will increase.Figure 1 depicts the evolution of the |Rg

2|1/2 (a) and PDI (b) asa function of monomer conversion for polymerizationsperformed in good (blue squares) and poor (red circles)solvent conditions. Compared to polymers grown in goodsolvents, under poor solvent conditions the resulting macro-molecules possess smaller dimensions for identical averagemolecular weights (or monomer conversions). Moreover, thePDI of polymers grownunder poor solvent conditions increasesat higher conversions relative to the good solvent case.

While these observations confirm that chain collapseunder poor solvent conditions and the resulting confinementof the propagating centers lead to broader MWDs, to gaininsight into the reasons for this broadening, we have to

Table 1. Lattice Dimensions and the Corresponding Values of σ

Lx = Ly Lz lattice Volume σ

160 39 998400 0.06140 51 999600 0.08120 69 993600 0.11100 100 1000000 0.1680 156 998400 0.2560 278 1000800 0.44

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monitor the environment around the propagating chainends. Specifically, for every 10000 MCSs we determine thenumber of nearest neighbor sites per active chain endoccupied by monomers (θm), polymer beads belonging tothe observed chain end (θp,s) or to other chains (θp,o), and theremaining empty sites (θe). We least-squares fit these data tofacilitate the identification of differences between good andpoor solvent conditions (Figure 2). In contrast to polymer-ization under good solvent conditions, in poor solvents thereis a lower likelihood that an active chain-endwill encounter amonomer and undergo reaction as reflected by the data for θm.Concurrently, θp,s and θp,o increase in going from goodto poor solvent conditions. The increase in θp,s indicates thatthe chains are collapsing and that chain ends are more likelyto neighbor other beads in the same chain. The increase in θp,osuggests that the chains tend to aggregate in poor solventconditions, thus, further shielding the active chain ends andhindering the polymerization reaction. Both of these effectscan disrupt monomer delivery to the chain ends, which canexplain the decreased rate of polymerization (not shown) andthe increase in PDI. One explanation for the increase in PDIthat we cannot currently verify is that each chain mightcollapse with its chain end located farther or closer to theinterphase between the collapsed globule and the solution.The dynamics of a collapsed chain are slow enough that thechain will not change its configuration significantly through-out the polymerization. This would result in uneven growthof chains depending on the specific collapse configuration.

While the aforementioned observations are specific toour model reaction, they provide insight into the factors

influencing a real system. Take, for example, ATRP reac-tions, in which a terminal halogen atom can transfer revers-ibly to a neighboring metal/organic ligand complex protec-ting and deprotecting the chains.45 In this case, not only willthe collapse of the chain obstruct the monomer from reach-ing the reaction centers but it will also hinder the metal/organic ligand complex and the catalyst crucial for thereaction. One can therefore envision situations in whichcollapsed polymers do not alternate efficiently between theactive (living) and inactive (dormant) forms resulting indeterioration of reaction control. Many other factors maycome into play under poor solvent conditions (i.e., alteredhalogen transfer between the metal/ligand complex and thegrowing chain) and, in this sense, our simulations represent asimplified scenario providing insight into the factors affect-ing polymerization control in poor solvents.

Surface Density. Computer simulations of polymeriza-tions were performed with bulk and surface-bound initiatorsunder good and poor solvent conditions. Our benchmarksystem corresponds to a lattice defined byLx=Ly=Lz=100with the initial number of monomers equal to 12500 and

Figure 1. (a) Radius of gyration and (b) polydispersity index of bulkpolymers in good (blue squares, ε=4� 10-7 kBT) andpoor (red circles,ε = 4 � 10-1 kBT) solvent conditions as a function of monomerconversion.

Figure 2. Average number of nearest neighbor sites per active endoccupied by free monomers (θm), polymers (θp), polymers belonging toa different chain (θp,o), polymer belonging to the observed chain (θp,s),and the remaining empty sites (θe) for bulk polymerization under good(solid, ε=4� 10-7 kBT) and poor (dashed, ε=4� 10-1 kBT) solventconditions.

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number of initiators equal to 400. In the surface-initiatedpolymerization, this corresponds to σ = 0.16. We alsoexplore how varying σ affects the polymerization reaction.While a logical way to change σ would seem to be to addmore initiators to the surface, this approach would alter theratio between numbers of monomers and initiators, whichwould consequently influence the kinetics and PDI of thereaction. To achieve different values of σ without changingthe numbers of monomers and initiators, one should thusmodify the dimensions of the lattice to vary the area perinitiator molecule while keeping the total volume of thelattice approximately constant. Given that Lz varies forsimulations at different σ, all runs are inspected to ensurethat the perpendicular component of the radius of gyration(|Rg,^

2 |1/2) does not approach Lz.Figure 3 shows the parallel (|Rg, )

2 |1/2) and perpendicular(|Rg,^

2 |1/2) components of the radius of gyration as a functionof the monomer conversion for bulk- (dashed line) andsurface-initiated (symbols) systems under good (left column)and poor (right column) solvent conditions at three differentσ equal to 0.08 (top), 0.16 (middle), and 0.44 (bottom). Wemust note that the averages computed here are not ensemble

averages for configurations of a single chain, but the aver-ages of the instantaneous configurations of a population ofchains with varying degrees of polymerization (N). Bulkpolymers exhibit equal dimensions in every direction becauseof orientational averaging over many conformations and aretherefore represented by a single line. The average size of thebulk-initiated polymers increases with increasing conver-sion; the increase is slower in poor solvents relative to goodsolvents, as noted previously. Surface-initiated polymers ingood solvents exhibit larger dimensions in the directionperpendicular to the substrate than parallel to it; this effectbecomes more pronounced with increases in σ. This shapeasymmetry is due to repulsive excluded volume interactionsbetween neighboring chains, which force the polymers toextend away from the tethering surface. Most of the shapeasymmetry manifests itself as an increase in |Rg,^

2 |1/2, while the|Rg, )

2 |1/2 undergoes only a small decrease from its bulk value.Under poor solvent conditions and at low σ |Rg,^

2 |1/2 becomessmaller than |Rg, )

2 |1/2.With increasingσ, the difference between|Rg,^

2 |1/2 and |Rg, )

2 |1/2 vanishes and eventually |Rg,^2 |1/2> |Rg, )

2 |1/2.These observations reveal that under poor solvent conditionsthe chains try to minimize their contact with the solvent. Atlow grafting densities, they achieve this by collapsing ontothe substrate and stretching toward their neighbors. Athigher grafting densities, the horizontal confinement in-creases and the only direction the chains can extend to isthat perpendicular to the substrate.

In an ideal system, one would want all initiator moleculesto activate simultaneously so that the propagation reaction

Figure 3. Parallel (Rg, ), closed symbols) and perpendicular (Rg,^, opensymbols) components of the average radius of gyrationof grafted chainsas a function of monomer conversion for good (left column, ε = 4 �10-7 kBT) and poor (right column, ε = 4 � 10-1 kBT) solventconditions at three different grafting densities (σ): 0.08 (top row,magenta), 0.16 (middle row, green), and 0.44 (bottom row, black).The dashed lines represent corresponding data for chains polymerizedin bulk.

Figure 4. Initiator efficiency as a function of monomer conversion fora) good (ε = 4 � 10-7 kBT) and b) poor (ε = 4 � 10-1 kBT) solventconditions at various grafting densities (σ): 0.06 (dark green left-pointing triangle), 0.08 (pink diamond), 0.11 (blue downward triangle),0.16 (light green triangle), 0.25 (red circle), and 0.44 (black square).

9572 Macromolecules, Vol. 43, No. 22, 2010 Turgman-Cohen and Genzer

commences instantaneously for all growing polymers. Inreality, not all initiators are activated at the same time (oractivated at all).19 There are multiple reasons for this behav-ior. Some of them are associated with the chemical nature ofthe initiator and the environmental trigger that activates it(typical examples include azo-based initiators for free radicalpolymerizations46-48) or with steric hindrance of the initiatormolecules in confined systems. Because the properties, partic-ularly the PDI, of the polymeric tethers depend crucially onthe initiator efficacy, it is important tomonitor the fraction ofactive initiators that participate in the reaction.Although sucha task is challenging to carry out experimentally, computersimulations can handle monitoring this parameter very easily.In Figure 4 we plot the fraction of initiators that undergopolymerization as a function of monomer conversion (xm).For nearly all values of σ, complete (i.e., 100%) initiationefficiency is achieved by xm = 0.20. The sole exceptions aresystems with very high grafting density where the initiatorefficiency plateaus earlier, σ g 0.44 in good solvents and σ g0.25 in poor solvents. In poor solvent conditions and at thehighestσ, the initiator efficiency reaches a plateau value of 0.8,which might have implications for polymerization of realsystems if large gradients in monomer concentration develop(i.e., methacrylates in water/alcohol mixtures).19,35

One of the main benefits of computer simulations appliedto polymerization is that one can determine theMWDof thegrown polymers in situ. Although SEC experiments are inprinciple capable of providing similar information, theamount of material cleaved off from the substrate is oftenbelow the sensitivity limit of most SEC detectors. Further-more, even if SEC traces show a polymer peak, they mightmiss (or skew) low andhighmolecularweight tails, leading tounderestimation of the broadness of the distribution. Com-puter simulations overcome this limitation by calculating theMWD from the size (N) of each chain. The data in Figure 5represent MWDs as a function of xm and σ for good (leftcolumn) and poor (right column) solvent conditions. Thevalues in the ordinate represent conversions and the blacklines represent slices of the MWDs at the labeled conver-sions. While relatively narrow peaks represent theMWDs atlow σ, at high σ, a highmolecular weight tail develops. Goingfrom σ=0.08 (top) to σ= 0.44 (bottom) the MWD broad-ens significantly. Although at low σ the peak moves steadilyto largermolecular weight areas, at high σ, most of the chainsremain short. Polymerizations under poor solvents exacerbatethese effects, possibly due to chain collapse and obstruction ofthe access of monomers to chain ends.

To quantify the broadness of theMWDs at different σ anddifferent solvent conditions, we monitor the PDI of thesurface-initiated chains as a function of xm and plot theresults in Figure 6.We compute the PDI, defined as the ratioof the mass-average molar mass (second moment) to the

Figure 5. Chain length distributions for surface-initiated polymeriza-tions as a function of monomer conversion (xm, ordinate) for good(left column, (a, ε = 4 � 10-7 kBT) and poor (right column, ε = 4 �10-1 kBT) solvent conditions at three grafting densities (σ): 0.08 (top,magenta), 0.16 (middle, green) and 0.44 (bottom, black). The dark areasrepresent high frequency of a specific chain length and light colors lowfrequency. The lines represent slices of these distributions and theirordinate value represent the frequency at a given chain length.

Figure 6. Polydispersity index (PDI) of grafted chains as a function ofmonomer conversion for good (a, ε= 4� 10-7 kBT) and poor (b, ε=4 � 10-1 kBT) solvent conditions at various grafting densities (σ): 0.06(dark green left-pointing triangle), 0.08 (pink diamond), 0.11 (bluedownward triangle), 0.16 (light green triangle), 0.25 (red circle), and0.44 (black square). The dotted black lines represent the PDI values forchains polymerized in bulk.

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number-average molar mass (first moment) of the distribu-tion, by taking into account the length of all chains includingthose of initiators that have not yet polymerized (N=2). Ingood solvents (Figure 6a) and at low σ, the reaction remainscontrolled; very low PDI values are attainable at all mono-mer conversions. Furthermore, the PDIs for low σ areidentical to those obtained for bulk polymerizations (dashedline). This indicates polymerizations from surfaces popu-lated sparsely with initiators may achieve similar control asthe bulk-initiated CRP (we caution that some differencesbetween the two polymerization geometries may still exist inreal systems even at low σ because effects not considered inthis model, such as mobility of the metal/organic ligandcomplex, the activation reaction, etc). Increasing σ leads tohigher PDI values indicating that polymers grown fromdense initiator assemblies have broader MWDs. Given thatchain termination is neglected, the monomer concentrationis unchanged and all the relevant kinetic parameters of themodel are constant, we attribute this gradual increase in PDIto the increase in initiator density on the surface and there-fore to chain crowding. This observation is consistent withprevious experimental studies15,16,18,19 and thework in Liu etal.,32 which report higher PDIs for surface grown polymersrelative to polymers grown in solution.

Computer simulations performed under poor solventconditions yield similar trends as those observed in goodsolvents. However, we observe pronounced increases in thePDI with increasing monomer conversion, especially at highσ (Figure 6b). At σ= 0.08, we detect an increase of ≈5% inthe PDI at the maximum conversion achieved (xm = 0.8) ingoing from good to poor solvent condition; this valueincreases to 26% (xm = 0.27) at σ = 0.44. Overall, thereare synergistic effects when both high σ and poor solventconditions are present that result in very poor control ofpolymerization reactions. In addition, for the bulk-initiatedsystems, the increase in PDI when going from good to poorsolvent conditions is≈7.8%,which is larger than the value atσ = 0.08. Polymers grown at σ = 0.08 and under poor sol-vent conditions represent the only case, in which we observea lower PDI for grafted chains relative to those in the bulk.A possible explanation for this behavior is that althoughbulk chains can readily form aggregates through intermolec-ular interactions, tethered chains cannot diffuse to oneanother because of their attachment to the surface. The latterprevents chain aggregation and therefore reduces the chain-end confinement on the surface when compared to the bulk.

To explain the increases in PDIs as a function of σ, werecur to the average occupation of the nearest neighbor sitesof the active chain-ends. In Figure 7, we plot θm, θp, and thefraction of surface sites near the chain ends (θw) as a functionof xm for polymerizations performed under good (leftcolumn) and poor (right column) solvent conditions at threeσ values: 0.08 (top), 0.16 (middle), and 0.44 (bottom). Weobserve that with increasing monomer conversion, θmdecreases due to monomer consumption, θp increases dueto the higher molar mass of polymers, and θw decreases dueto the chain ends moving farther away from the surface. Atthe lowest σ, the initial value of θm (xm=0) is about half thatof the bulk-initiated case and it decreases further at highervalues of σ. We attribute this behavior to the presence of thesubstrate and to the increased crowding of the initiators athigher σ. As the polymerization progresses and the chainends migrate away from the substrate (at about xm=0.31),θm for the low σ case and the bulk-initiated case becomeequal, indicating similar environments of the active chainends in these systems. Further, θp increases with increasing σ,indicating that the polymers are occupying the space near the

active chain ends and blocking their access to free monomer.Under poor solvent conditions (right column), this effect ismore pronounced suggesting that chain collapse leads toadditional masking of the active chain ends by the polymerand additional disruption of the chain-ends access to mono-mer. Finally, θw decreases as a function of xm except for thelargest σ in which its value is zero for all xm.When σ=0.44,the initiators almost fully cover the surface and within theBFM it is impossible for macromolecules growing undersuch conditions to access the sites adjacent to the wall.

To comprehend the increases in PDI as σ increases, weconstruct volume fraction profiles of the polymers (φp(z), leftcolumn), free monomers (φm(z), center column), and chainends (φe(z), right column) as a function of distance from thesubstrate (z, abscissa) and xm (ordinate) for good (Figure 8)and poor (Figure 9) solvents. As stated previously, the datain the top, middle, and bottom rows correspond to σ equal to0.08, 0.16, and 0.44, respectively. The colored areas corre-spond to high concentrations, while the white color denoteszero concentration. The black lines represent sample slices ofthe φi(z) at a specific xm. In general, φp(z) and φe(z) extendaway from the surface with increasing xm, as expected. Inaddition, φm(z) develops a depletion layer near the sub-strate as xm increases, indicating that within the polymerbrush the concentration of monomer decreases appreciably.

Figure 7. Average number of nearest neighbor sites per active endoccupied by monomers (θm, top row), polymers (θp, middle row), andwall (θw, bottom row) as a function of monomer conversion forgood (left column, ε=4� 10-7 kBT) and poor (right column, ε=4�10-1 kBT) solvent conditions. The solid lines represent values corre-sponding to various grafting densities: 0.08 (magenta), 0.16 (green), and0.44 (black). The dark gray dashed lines represent the correspondingvalues for chains polymerized in bulk.

9574 Macromolecules, Vol. 43, No. 22, 2010 Turgman-Cohen and Genzer

As σ increases, the monomer depletion continues to developand reaches a point where the populations of chain ends andmonomers no longer overlap. In more detail, Figure 10shows φp(z), φm(z), and φe(z) on the same scale at threedifferent σ, xm = 0.2 and at good (left column) and poor(right column) solvent conditions. At σ = 0.08 (top), φm(z)remains almost constant even very close to the surface; thereis appreciable overlap between the free monomer and chain-end populations. At σ = 0.16 (middle), φm(z) undergoes asteeper decrease near the substrate and in poor solvents it isvery close to zero. Finally, at σ = 0.44 (bottom), φm(z)reaches zero farther away from the substrate under both

good and poor solvent conditions. The populations of chainends and free monomers have minimal overlap leading touneven growth in the high σ case.We therefore conclude thatat higher σ any differences in the chain lengths of thepolymers lead to preferential growth of the longer chainsversus the shorter ones, a situation that is detrimental to thecontrolled growth of macromolecules from surfaces.

As alluded to above, properties such as the brush heightand the volume fraction profile of the polymers and chainends depend on the broadness of the MWDs. A briefexploration of these differences can be seen in Figure 11,which shows φp(z) for grafted layers created by our reactive

Figure 8. Volume fraction of grafted chains (left column, black), monomers (middle column, red), and chain ends (right column, green) expressed as afunction of the distance from the wall (abscissa, z) vs monomer conversion (ordinate, xm) under good (ε= 4� 10-7 kBT) solvent conditions at threegrafting densities: 0.08 (top row), 0.16 (middle row), and 0.44 (bottom row).

Figure 9. Volume fraction of grafted chains (left column, black), monomers (middle column, red) and chain ends (right column, green) expressed as afunction of themonomer conversion (xm) vs the distance from thewall (z) under poor (ε=4� 10-1 kBT) solvent conditions at three grafting densities:0.08 (top row), 0.16 (middle row), and 0.44 (bottom row).

Article Macromolecules, Vol. 43, No. 22, 2010 9575

simulation (up-triangles) and those with PDI = 1 (squares).Samples with N=10.0 (10.1 for polydisperse system) cor-respond to xm=0.26 and forN=20.0 (20.1 for polydispersesystem) correspond to xm = 0.58. As σ increases from 0.11(top) to 0.25 (bottom), the PDI of the synthesized layersgrows from1.10 to 1.22, respectively, atN=10.Although atσ = 0.11 the monodisperse and polydisperse concentrationprofiles are nearly identical, there are substantial differencesat σ = 0.25. Relative to the monodisperse layers, the poly-disperse ones exhibit lower φp(z), close to the substrate, butextend farther away from it, indicating larger thicknesses. Ingeneral, this effect is more notable as the PDI increases and itcan be observed for the chains in poor solvents as well.

de Vos and Leermakers recently published a detailedaccount of the effect of chain length polydispersity on char-acteristics of end-tethered polymers.29 They used a numericalself-consistent fieldmodel to calculate concentration profilesof brushes that had bimodal and continuous distributions(the latter was assumed to follow the Schulz-Zimm dis-tribution). The authors reported that, for both distributionsstudied, increases in PDI of the brushes in good solventsresult in changes in the shape of the concentration profilesfrom parabolic to linear to concave. They also noted that theaverage height of the brush increased with increasing PDI;for instance, an increase in PDI from 1.0 to 1.1 resulted in a

12% increase in brush height. de Vos and Leermakers notedthat, in polydisperse brushes, chain ends segregate, depend-ing on the molecular weight of the chain as opposed to themonodisperse case, in which the locations of the chain endsfluctuate strongly through the entire thickness of the graftedlayer. Our simulations allow us to generate populations ofchains grown directly from the surface and determine if theSchulz-Zimm distribution describes correctly these popula-tions. Table 2 shows the least-squared fitted parameters ofthe Schulz-Zimm distribution to data for several of ourgrafted polymer populations. Although the Schulz-Zimmdistribution describes the populations at σ = 0.08 and σ =0.16 well, at σ = 0.44 the quality of the fit deteriorates.Moreover, the values obtained for the PDI andN in the highσ fits deviate from the true values for these populations. Weposit that the Schulz-Zimm distribution fails to describe thechain-length distribution at high σ because the chains are nolonger growing with equal probability, that is, small chainshave a lower probability of growth than larger ones. We canalso verify independently the conclusions reached by de Vosand Leermakers without assuming any a priori form of thedistribution of chain lengths. First, regardless of solventquality, increasing σ leads to higher PDI and higher brushheights. Second, polydisperse brushes in good solvents aremore extended than those in poor solvents, and the latterexhibit larger increases in PDI relative to systems under goodsolvent conditions. Third, for a given combination ofsolvent quality, σ, and N, a crossover point exists whereall volume fraction profiles intersect regardless of the PDI.

Figure 10. Volume fraction profiles of grafted chains (black square),monomers (red circle), and chain ends (green triangle) for good (leftcolumn, ε=4� 10-7 kBT) and poor (right column, ε=4� 10-1 kBT)solvent conditions at three grafting densities: 0.08 (top row), 0.16(middle row), and 0.44 (bottom row).

Figure 11. Volume fraction profiles for monodisperse (squares) andpolydisperse (triangles, PDI listed in the legend) for good (left column,ε=4� 10-7 kBT) and poor (right column, ε=4� 10-1 kBT) solventsat three grafting densities: 0.11 (top row), 0.16 (middle row), and 0.25(bottom row). Profiles were obtained at xm values of 0.26 and 0.58 forN equal to 10.1 and 20.1, respectively.

9576 Macromolecules, Vol. 43, No. 22, 2010 Turgman-Cohen and Genzer

This observation was also made by de Vos and Leermakers,but not discussed in their paper. We will provide a detaileddiscussion of this effect in a subsequent publication.49

Conclusions

WeemployedMonteCarlo simulations to investigate the effectthat grafting density and solvent quality on surface-initiatedliving polymerizations. Our computer simulations neglectedterminations and chain transfer while incorporating an idealizedmechanism for activation anddeactivation of the growing chains.We have shown that even under such “true living” conditions“grafting from” polymerization from planar impenetrable sub-strates results in the broadening ofmolecular weight distributionsrelative to bulk-initiated polymerizations and that this broad-ening increases as the grafting density of initiators increases. Weattribute this broadening at high grafting densities to the unevenaccess of free monomers for short and long chains due to thedepletion of free monomer close to substrate. In addition, thePDI of both bulk- and surface-initiated polymers produced byCRP increasedwith decreasing solvent quality. This behavior hasbeen attributed to the collapsed conformation of the coil andpossible chain aggregates that confine the growing chain end andlimit the access of free monomers to the growing end. Thecombination of poor solvent conditions and high grafting den-sities further exacerbates the broadening of the molecular weightdistribution. For instance, relative to good solvent conditions,low grafting density systems (σ=0.08) underwent an increase inPDI of 5%under poor solvent conditions. This number increasedto 26% at the highest grating density (σ=0.44) studied.We alsoshowed that, although the Schulz-Zimm distribution describescorrectly the chain-length probability density of surface-initiatedpolymers at small and moderate grafting densities, at highgrafting densities (σ = 0.44), the description is no longer valid.We attribute the latter behavior to the difference in the prob-ability of short (low) and long (high) chains to grow, which is adirect result of the separation of the monomer and chain-endpopulations under these conditions.

Although the effect of initiator density and solvent quality onsurface-initiated CRP is clear, additional work is needed tounderstand the process of living polymerization from surfacesin detail and perhaps a in amore realisticmanner. One importantaspect we have ignored in this work is that solvent quality willdepend on the monomer concentration. The interaction energiesused in this study could in principle be a function of the distanceaway from the substrate and depend on the local monomerconcentration. This adjustment would resemble experimentalconditions more closely. Experimentally, one works with largeexcess of monomer and stops the reaction in its early stages tokeep the PDI relatively low. Computer simulations describingsuch situations should be developed; this would effectively meanswitching from the NVT simulation system to one that considersa constant chemical potential of free monomers in the bulk.A more thorough study is necessary to explore the validity of theSchulz-Zimm distribution for layers polymerized to higher

average degree of polymerization and to check for a crossoverdegree of polymerization atwhich the Schulz-Zimmdistributionceases to be valid. The effect of the substrate curvature on themolecularweight (MW) andPDI should also be extendedbeyondflat surfaces and compared to similar systems polymerized inbulk. In particular, it would be interesting to explore whetherstrong confinement of chains polymerized from surfaces withnegative curvature (i.e., concave surfaces) leads to increases inPDI and decreases in MW. Living polymerization from surfaceswith positive curvature (i.e., convex) should also be considered.Here the effect expected is the opposite to polymerization inconcave spaces. Specifically, with increasing the positive curva-ture, the polymers grown from the surface should start toresemble their counterparts polymerized in solution. While theeffect of confinement is, in general, relatively trivial to imagine, itis difficult to provide numerical data about the combined effect ofcurvature, initiator density, and solvent quality. This is wherecomputer simulations will come in very handy.

Acknowledgment. Acknowledgment is made to the Donorsof the American Chemical Society Petroleum Research Fund forsupport of this research under Grant No. 45688-AC7. Partialsupport from theNational Science Foundation though theGrantNo. DMR-0906572 is also acknowledged.

Supporting Information Available: Themapping of the coil-to-globule transition and the investigation of the probability ofreaction versus motion is presented. This material is availablefree of charge via the Internet at http://pubs.acs.org.

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