phase behavior and structure formation in linear...
TRANSCRIPT
Phase behavior and structure formation in linear multiblock copolymersolutions by Monte Carlo simulation
Marian E. Gindy,1,a� Robert K. Prud’homme,1 and Athanassios Z. Panagiotopoulos1,2,b�
1Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544 USA2Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton,New Jersey 08544, USA
�Received 18 December 2007; accepted 12 March 2008; published online 24 April 2008�
The solution phase behavior of short, strictly alternating multiblock copolymers of type �AnBn�m wasstudied using lattice Monte Carlo simulations. The polymer molecules were modeled as flexiblechains in a monomeric solvent selective for block type A. The degree of block polymerization n andthe number of diblock units per chain m were treated as variables. We show that within the regimeof parameters accessible to our study, the thermodynamic phase transition type is dependent on theratio of m /n. The simulations show microscopic phase separation into roughly spherical aggregatesfor m /n ratios less than a critical value and first-order macroscopic precipitation otherwise. Ingeneral, increasing m at fixed n, or n at fixed m, promotes the tendency toward macroscopic phaseprecipitation. The enthalpic driving force of phase change is found to universally scale with chainlength for all multiblock systems considered and is independent of the existence of a true phasetransition. For aggregate forming systems at low amphiphile concentrations, multiblock chains areshown to self-assemble into intramolecular, multichain clusters. Predictions for microstructuraldimensions, including critical micelle concentration, equilibrium size, shape, aggregationparameters, and density distributions, are provided. At increasing amphiphile density, interaggregatebridging is shown to result in the formation of networked structures, leading to an eventualsolution-gel transition. The gel is swollen and consists of highly interconnected aggregates ofapproximately spherical morphology. Qualitative agreement is found between experimentallyobserved physical property changes and phase transitions predicted by simulations. Thus, a potentialapplication of the simulations is the design of multiblock copolymer systems which can beoptimized with regard to solution phase behavior and ultimately physical and mechanicalproperties. © 2008 American Institute of Physics. �DOI: 10.1063/1.2905231�
I. INTRODUCTION
Linear multiblock copolymers, like their diblock ana-logs, undergo microphase separation when the blocks aresufficiently incompatible. The behavior of these molecules inconcentrated solutions and melts has been extensively ex-plored experimentally,1–8 where the main applications ofmultiblock copolymers have been in the areas of thermoplas-tic elastomers, blend compatibilizers, and barrier materials.Theoretically, a number of formalisms have been applied to-ward the description of microphase separation or the order-disorder transition �ODT� in multiblock copolymer melts.Models based on confined chain statistics as developed byKrause,9 Meier,10,11 and Henderson and Williams12,13 andself-consistent mean-field and random phase approximationtheories, as proposed by Helfand,14 Zielinski and Spontak,15
and Ohta and Kawasaki16 have proven to be particularly use-ful in elucidating relationships between molecular character-istics and microstructural dimensions in the strong-segregation limit, where only fully developed microdomainsare considered. In the weak-segregation regime, where mi-
crostructure domains are wide, microstructure formation nearthe ODT was first modeled by Leibler17 in the case ofdiblock copolymers by using the random phase approxima-tion of de Gennes.18 Benoit and Hadziioannou19 extended theformalism of Leibler17 to calculate the spinodal line for mi-crophase separation. Their work additionally addressed themicrostructural characteristics of the microphase separatedstate to obtain estimates of microdomain periodicity as afunction of the number of block pairs per molecule. Kavas-salis and Whitmore20 later employed the functional integralmethods of Hong and Noolandi21 to examine the lamellarphase in the weak-segregation limit. Most recently, themean-field theory of Leibler17 has been adapted by Mayesand Olvera de la Cruz22 to triblock copolymers and laterextended by Wu et al.1 to establish an asymptotic relation-ship for the ODT in symmetric multiblock copolymer melts.Since then, modifications to the formalisms have been pro-posed to further account for block polydispersity,20,23–25 non-uniform block sequence distribution,23,26,27 and blocklooping.19,28–31 On the computational side, investigations ofmultiblock copolymer melts by Houdayer and Muller,32 Do-brynin and Erukhimovich,33 Semler and Genzer,34 and Bal-azs et al.35 have provided complementary insight into theODT and resulting microdomain architectures. In general,
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predictions from both theoretical and simulation models haveyielded good agreement with experimental results.
Investigations of multiblock copolymer phase behaviorin dilute, selective solvents have primarily focused on co-polymers of low block sequences. To date, the majority ofresearch efforts have been directed toward di- and triblockcopolymers, where upon self-assembly, the formation ofnearly spherical core-shell type microstructures has beenpredicted theoretically36,37 and demonstrated viaexperiments38,39 and computer simulations.40–43 Microstruc-tures based on these architectures have been used in a num-ber of advanced technologies ranging from drug delivery44 tothe patterning of nanosurfaces.45 More recently, progress inthe field of synthetic polymer chemistry has made the prepa-ration of low molecular weight, linear multiblock copoly-mers with high degree of structural regularity possible. Cur-rent experimental research efforts aim to exploit the solutionphase self-assembly of low molecular weight multiblock co-polymers in novel applications, particularly in the field ofprotein-based drug delivery.46–54 Amphiphilic multiblock co-polymers are thought to offer more flexibility in tailoringproperties of a delivery system than comparable di- and tri-block copolymers. The use of multiple short amphiphilicsegments in one polymer has been shown to prevent therapid and preferential loss of one polymer type, allowing formodulated drug release profiles and leading to an overallenhanced stability of the microstructure.46,51,53,54 In general,the multiblock copolymer chains employed in these applica-tions are relatively short, ranging from tetrablocks to de-cablocks, with individual block sizes below 2000 Da. Forexample, Bikram et al.52 have described the formation ofmultiblock microstructures based on copolymers of poly�eth-ylene glycol� conjugated to cationic poly�L-lysine� as carri-ers of therapeutic genes. The multiblock complexes wereshown to efficiently encapsulate plasmid DNA into structureswith an average particle size of approximately 200 nm. Morerecently, Quaglia et al.48 reported the preparation of micronsized protein-loaded particles based on the self-assembly ofamphiphilic poly�ethylene glycol�-poly��-caprolactone�multiblock copolymers. The authors observe significant vari-ability in protein distribution within microparticle interiorsand protein release profiles depending on the distribution ofpoly�ethylene glycol� blocks along the chain. These resultssuggest that fine control over the structural parameters ofamphiphilic linear multiblocks can be employed for the ra-tional design of microstructures for the optimized delivery ofdiverse therapeutic proteins.
Despite these advances, relatively little is known aboutthe specific effect of block copolymer architecture on phaseseparation, micellization, and the resulting micellar proper-ties. Experimentally, there is a large parameter space to ex-plore, and it becomes difficult to independently consider theeffects of factors such as monomer type, block size, andchain molecular weight on the observed solution phase be-havior. In addition, the dilute concentrations at which phaseseparation has been observed in these systems further com-plicate experimental investigations, whereby it becomes ex-ceedingly difficult to purely separate equilibrium effectsfrom kinetic influences. For multiblock copolymers, the issue
of equilibration becomes more pronounced relative to di- andtriblock copolymers. In the regions where microstructure for-mation takes place, the state of a multiblock chain can be-come kinetically trapped in one of multiple local minima inthe free energy landscape.39,55 Similarly, the predictive capa-bilities of theoretical models in the dilute regime are rela-tively limited. The complexity of macromolecule associationin selective solvents precludes reliable prediction of thestructures formed at thermodynamic equilibrium. Conse-quently, the complexity of the phase diagram, which mayinclude regions of homogeneity, aggregation, gel formation,or precipitation, cannot be accounted fully by theoretical rep-resentations. Toward this end, researchers have employedcomputational methods to explore the rich phase behavior ofpolymers with complex architectures.40,56,57 Atomistic mod-els in which the solvent is explicitly considered are often toocomputationally costly and are, thus, only appropriate for theexploration of single microstructure properties. In contrast,coarse-grain models which implicitly account for the solventare more suitable for phase coexistence calculations.
In this work, we present a Monte Carlo study of theself-assembly of strictly alternating, low molecular weightmultiblock amphiphiles of type �AnBn�m in dilute solutions,where the solvent is implicitly considered. Because they canbe conveniently synthesized experimentally through poly-merization methods that employ coupling of diblock chains,compositionally symmetric linear multiblock copolymers of-fer a simple model upon which relationships between co-polymer architecture and solution phase behavior determinedvia simulations can be validated. In particular, we systemati-cally explore the influence of block size and number ofblocks per chain on the tendency toward microstructure for-mation and macroscopic phase separation.
The manuscript is arranged as follows. In Sec. II, wedescribe the model and methodologies employed to distin-guish between macro- and microscopic phase separationsand obtain estimates of critical parameters and aggregationproperties. In Sec. III, we present our data and compare ourresults to available lattice data and to qualitative trends ob-served experimentally in dilute, nonionic polymer solutions.Finally, we draw some conclusions and suggest physical in-sights into the general characteristics of phase separation inmultiblock solutions, which may be applied to the study oflow molecular weight copolymers of more complex architec-ture.
II. MODEL AND SIMULATION METHOD
The simulation model is designed to represent a solutionof a mutiblock copolymer of type �AnBn�m in a selective sol-vent. Copolymer architecture is defined by two parameters;the size of the amphiphilic A and B blocks, designated by thesubscript n, and the number of repeating �AnBn� diblock unitsdesignated by the subscript m. In all systems, the volumefraction of A and B blocks in the chain is held constant andequal to 0.5.
As in previous works,41,42,58 polymers are represented aslinear chains of connected sites on a cubic lattice in accor-dance with the model of Larson et al.59 Each monomer of the
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polymer occupies a single lattice site and all unoccupied sitesare considered as solvent. Links between successive seg-ments along the chain are confined along the vectors �0,0,1�,�0,1,1�, and �1,1,1� and their reflections on the cubic lattice,resulting in a coordination number of 26. Attractions existbetween sites along these 26 possible directions. Interactionsof strength �AA occur between two solvophilic groups, a sol-vophilic group and a solvent molecule, and two solvent mol-ecules. Interactions of strength �AB occur between solvo-philic or solvent groups and solvophobic groups, whileinteractions of strength �BB occur between solvophobicgroups. The interaction between B-B groups is set to −2, andall other interactions are set to 0. The total interaction be-tween two lattice sites is calculated from pairwise summa-tion of site-site interactions, where all sites have the samemass and interaction range. The simulation model, thus, rep-resents a solution of �AnBn�m chains in a solvent poor for Bunits and athermal for A units.
We use the grand canonical Monte Carlo method withmultihistogram reweighting as described in previousworks.41,42,58 Simulations are performed on a simple cubiclattice of dimensions L�L�L under periodic boundary con-ditions. The inverse temperature �=1 /kT, chemical potential�, and volume V=L3 are defined per simulation run as inputparameters. The temperature, energy, and chemical potentialare nondimensionalized by the nearest neighbor interactionstrength, while length is nondimensionalized with the latticespacing of unit size.
Data from multiple runs are combined to obtain a globalfree energy function according to the methodology proposedof Ferrenberg and Swendsen.60,61 This approach is based onperforming a number of simulations for a given polymer atselected temperatures and specified chemical potentials tocollect particle density and energy probability histograms.Minimization of differences between the predicted and ob-served histograms is then used for data integration. Onceconvergence has been achieved, all thermodynamic proper-ties for the system over the range of densities and energiescovered by the histograms can be obtained.
Following the convention used in a previous work,42 thesize dependence of the calculated osmotic pressures andphase envelopes for a given amphiphile are used to charac-terize the nature of the phase transition. In binary systemsconsisting of a mixture of amphiphile and solvent, where thelattice is fully occupied, the pressure calculated from simu-lations is equivalent to the osmotic pressure of the am-phiphile in the system. For amphiphiles undergoing macro-scopic phase separation, the slope of the osmotic pressurewithin the two-phase region is finite in a finite-size system,and the phase transition is rounded. As the system size isincreased, the transition approaches a horizontal line. In ad-dition, the apparent phase diagram is essentially independentof simulation system size away from the critical point. Theseattributes indicate the presence of a true first-order phasetransition. By contrast, for systems undergoing micellization,the osmotic pressure behavior is essentially identical for allsystem sizes. The slope of the osmotic pressure curve is re-lated to the number of independent kinetic aggregates and is,thus, expected to be finite and independent of system size for
systems that form micelles. Correspondingly, there is astrong finite-size effect on the phase envelope.42
For systems undergoing microscopic phase separation,or micellization, the critical micelle concentration �cmc� in asimulation can be defined as the volume fraction at which thebreak in osmotic pressure curve is observed.41,42 This volumefraction is indicative of the point at which the first micelleappears in the thermodynamic limit. In practice, the cmc isdefined as the volume fraction at which the second derivativeof the osmotic pressure curve is a maximum.41,42
In macroscopic phase-separating systems, critical param-eter estimates for vapor-liquid phase transitions are madeusing the mixed-field finite scaling method.58,62 According tothe finite-size scaling theory, an un-normalized ordering op-erator combining the number of particles and energy is de-fined such that a nonuniversal field mixing parameter con-trols the strength of coupling between energy and densityfluctuations near the critical point. The probability distribu-tion of this operator is obtained by combining simulationresults of a series of runs performed near the expected criti-cal point using the multihistogram reweighting approach ofFerrenberg and Swedsen.60,61 The ordering parameter is thennormalized to zero mean and unit variance. Comparison ofthe resulting distribution to the Ising-class limiting distribu-tion and subsequent optimization to minimize the absolutedeviations between the observed and Ising-class distributionyield an estimate of the nondimensionalized critical tempera-ture �T
c*� and chemical potential ��
c*�.
In all systems, statistical uncertainties for selected con-ditions were obtained as the standard deviation �2�� of re-sults obtained by performing independent runs at identicalthermodynamic conditions but varying random number se-quences.
III. RESULTS AND DISCUSSION
A. Macro-versus microphase separation
The phase transition behavior of alternating, low mo-lecular weight multiblock copolymers of type �AnBn�m in asolvent selective for block A is systematically investigated asa function of block degree of polymerization �n� and numberof blocks per chain �m�. The fraction of A and B monomersper molecule is held constant and equal to 0.5 in all systemsstudied. The interaction energy parameter is fixed such thatthe monomer-solvent incompatibility is invariant and is rep-resentative of a regime in which B monomers are stronglyattractive. The phase transition is, thus, driven by a variationin the temperature, or equivalently, the strength of the attrac-tive potential.
The computed phase and aggregation behaviors for allsystems examined in this work are summarized in Table I.Aggregate forming systems are indicated by cells in boldtype, while systems undergoing first-order, macroscopicphase separation are designated by cells in nonbold type. Theapproach employed to distinguish between aggregation andphase separations was briefly explained in the previous sec-tion and is primarily based on the finite-size dependence ofcalculated osmotic pressure versus density as detailed in ear-lier work by Panagiotopoulos et al.42 In general, for multi-
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blocks of equal molecular weight and overall composition,an increase in the degree of block polymerization reduces thetendency for macroscopic phase separation and favors theformation of aggregates. When the degree of block polymer-ization is held constant and the number of blocks in a mol-ecule is increased, the driving force toward microphase sepa-ration is diminished. A roughly diagonal demarcation lineexists between aggregate forming systems and those under-going macroscopic precipitation. We define a correlation pa-rameter �, such that �=m /n characterizes the architecture ofa molecule of constant composition. The simulations thenpredict microphase separation for � less than a critical valueand macroscopic separation otherwise. For �AnBn�m multi-block chains, the critical value of � is seen to increase withthe size of constituent blocks. Systems located near the de-marcation lines drawn in Table I will display evidence ofapproaching crossover to the opposite behavior, as will besubsequently shown.
Experiments that investigate the association behavior ofmultiblock copolymers in dilute, apolar organic solutions arelimited.63–65 An example is the work by Nie and et al.65 thatstudied the structure and dynamics of a polystyrene �PS�-polybutadiene �PB� pentablock copolymer �S-B-S-B-S� inheptane, a strongly selective solvent for PB and a poor sol-vent for PS. In dilute solutions, no micellization of the pent-ablock was detected. Instead, the pentablock copolymer wasfound to undergo macroscopic phase separation into two liq-uid phases. By contrast, a structurally similar triblock co-polymer �S-B-S� of comparable molecular weight and PScomposition experiences microscopic phase separation toyield spherical micelles.64 This effect can be explained in theframework of the present work. As a first assessment, theenthalpic contribution to the interaction energies in the ex-perimental and simulation systems can be compared. Experi-mentally, the Flory–Huggins interaction parameter � is de-termined by using the values of solubility parameters of PS,PB, and n-heptane ��PS=9.1, �PB=8.4, �heptane
=7.4 �cal /cm3�1/2� and the molar volume of n-heptane �243�10−24 cm−3�, yielding �heptane,PS�217 /T and �heptane,PB
�75 /T. Thus, at a temperature of 300 K, the � parametersrepresent solvent systems poor for PS and good for PB.Comparatively, the interaction set ��=�AB− �1 /2��AA
−1 /2�BB� employed in the present simulation model is rep-resentative of a solvent system poor for B units ��BB=−2�and athermal for A units ��AA=0, �AB=0�. This interactionset was selected to match the previous work by Panagioto-poulos et al.42 to allow comparisons with diblock and tri-block copolymers studied earlier. Based on the calculatedFlory–Huggins interaction parameters for PS and PB inn-heptane, interaction energy values �BB�−1.8, �AA�−0.6,and �AB=0 would instead be required to more accuratelyreflect critical temperature estimates of PS homopolymer in
n-heptane as well as the quality of the solvent for the PB.Experimentally, the crossover in phase transition from mi-celle formation to macroscopic phase separation was ob-served going from S-B-S triblock copolymers to S-B-S-B-Sheptablock copolymers. In our simulations, this transition isbest represented by �A2B2�m model systems, where a similarcrossover is expected to occur for approximate values of mbetween 2 and 3. Thus, the qualitative agreement betweenexperimental and simulation results supports the potentialapplication of this simple model to the study of solutionphase behavior for multiblocks of more complex architec-ture.
B. Microscopic phase separation
The behavior of aggregate forming multiblocks is moreclosely examined here as a function of block size and overallchain length. In Fig. 1, the polymer volume fractions at thecmc �cmc� for �A3B3�m and �A5B5�m aggregate forming sys-tems are plotted as a function of inverse temperature, 1 /T*.Points from simulations are fitted by using an equation of theform
ln�cmc� = A + B/T*, �1�
where A and B are parameters. In a first-order phase change,parameter B is proportional to the enthalpy change per chainupon transition from a dilute homogeneous phase to a densephase, assuming that solution nonidealities can beneglected.41 While micellization is not strictly a first-ordertransition, it can be seen that all data in the temperatureranges investigated are well fitted by Eq. �1� �R20.999�.Thus, by analogy, parameter B in aggregate forming systems
TABLE I. Micellization vs phase separation for systems studied. Cells in bold type correspond to systems thatfrom aggregates, while cells in nonbold type designate systems undergoing macroscopic phase separation.
(A1B1)1 �A1B1�2 �A1B1�3 �A1B1�4 �A1B1�6 �A1B1�15 �A1B1�30
(A3B3)1 (A3B3)2 (A3B3)3 �A3B3�4 �A3B3�5 �A3B3�6 �A3B3�10
(A5B5)1 (A5B5)2 (A5B5)3 (A5B5)4 (A5B5)5 (A5B5)6
FIG. 1. Critical micelle concentration �cmc� as a function of reciprocaltemperature �1 /T*� for selected aggregate forming multiblocks as indicatedin legend. Points are from simulations, with estimated 2� statistical uncer-tainties in measured data represented by error bars. Lines are fitted to pointsaccording to Eq. �1�.
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can be taken to represent the enthalpy change per chain upontransition from a dilute homogeneous solution to a denseaggregate core. Calculated values of parameter B for �A3B3�m
and �A5B5�m multiblocks as a function of the total number ofsolvophobic monomers per chain s, s=2 nm, are shown inFig. 2. Values for additional diblock systems, as previouslydetailed by Panagiotopoulos et al.42 are shown for reference.In all cases, the standard enthalpies of transition are negativeindicating that, as modeled, chain association is an enthalpy-driven process. The enthalpy of transition for both diblockand multiblock copolymer chains is found to scale linearlywith s, such that the phase separation is facilitated by in-creasing the molecular weight of the copolymer. However,the quantitative scaling of parameter B with s differs amongthe heteropolymer chain types examined. As plotted, theslopes of data in Fig. 2 represent the enthalpy of phase tran-sition per unit solvophobic monomer. It is, thus, particularlyinteresting that simulation data for all �A3B3�m and �A5B5�m
systems investigated are well matched by a single linear fit,the slope of which is approximately half of its diblock coun-terpart. Therefore, for copolymers of equal composition andmolecular weight, the simulations reveal a reduced tendencytoward phase separation with the addition of diblock unitsper copolymer chain. Furthermore, within the limited rangeof block sizes examined, the enthalpic scaling of multiblockchains is seen to be essentially independent of the size ofconstituent blocks. Finally, B parameter values of macro-scopic phase separating �A3B3�4 and �A3B3�5 multiblock co-polymers are also plotted in Fig. 2. As shown, these valuesfall on the same line as aggregate forming �A3B3�1,2,and 3 sys-tems. Hence, the observed instability of the microscopicphase separated state with the addition of diblock units, asshown in Table I, is unlikely to result from inherent dis-similarity in the enthalpic driving force toward phase sepa-ration. Instead, the instability can be indirectly attributed toentropic contributions, which as will be subsequently dem-onstrated, stem from excluded volume effects of loop chains
in aggregate formation as the number of blocks per chain isincreased.
Despite the short chain lengths employed in the presentsimulations, our results are in qualitative agreement with pre-dictions from theoretical treatments examining microphaseseparation of symmetric multiblock copolymers in the melt.Specifically, both the thermodynamic model of Krause9 andthe functional integral methods of Kavassalis andWhitmore20 indicate that microphase separation �into alamellar morphology� becomes less favored with an increasein the number of AB block pairs in copolymers of equalmolecular weight. The application of analogous theoreticalformalisms to the association of multiblock chains in dilutesolutions is more complex. In the latter case, domains ofvarious sizes and shapes are possible, making a completeequilibrium theoretical model with exhaustive statistical ac-counting of all possible microstructures impractical. Presenttheoretical studies66,67 that examine multiblock chain asso-ciation in dilute solutions employ phenomenological modelsto construct mean-field treatments of copolymer chain col-lapse in simplified systems for which restrictions on blocksize, number of blocks, and chain conformations have beenmade. Our simulations impose no a priori restrictions on thestates a multiblock chain can adopt at thermodynamic equi-librium. We are, thus, able to explicitly evaluate chain-structure dependent effects on multiblock copolymer phasestability.
C. Aggregation properties
Amphiphilic copolymer chains in solution adopt con-figurations in which interactions of insoluble segments withthe solvent are minimized. In amphiphilic multiblock co-polymers, competition between intra- and intermolecular as-sociations allows for a large number of possible chain to-pologies, in which insoluble segments can adopt loop,bridged, and dangling configurations.68,69 The fraction ofchains which take on a particular configuration depends on adelicate balance of enthalpic and entropic contributions tothe free energy.
In this section, equilibrium properties of aggregatesformed by microphase separating multiblock chains are in-vestigated. The aggregate size from simulations is dependenton both temperature and overall amphiphile concentration�r�. To enable comparisons across copolymers of differentarchitectures, T* is selected to be near the upper temperaturelimit for aggregate formation as defined in a previous work.41
The overall polymer volume fraction at T* is specified, suchthat it is approximately twice the value of cmc for the systemunder consideration. This ratio is chosen, such that the poly-mer concentration is as near the cmc as is possible for thesimulation system sizes studied. In analyses of the simulationdata, we define an aggregate, or equivalently a cluster, asfollows. We consider that two solvophobic B monomers be-long to the same cluster if the distance between them is lessthan or equal to the interaction range of �3 times the latticespacing. Once a cluster has been defined, the number ofchains comprising the cluster is calculated. Thus, the distri-
FIG. 2. Parameter B values as determined from Eq. �1� as a function of totalsolvophobic monomers per chain �s� for multiblocks ��, �� �this work� anddiblocks ��� �Ref. 42�. Points are from simulations, with estimated 2�statistical uncertainties in measured data represented by error bars. Linearregression fit of data represented by solid line for �AnBn�1 and dashed linefor �A3B3�m and �A5B5�m systems �for m�1�.
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butions of aggregates are independently computed as a func-tion of solvophobic B monomers per aggregate �NB� andnumber of chains per aggregate �Nchain�.
Aggregation parameters for selected microphase separat-ing �A3B3�m and �A5B5�m multiblocks in the dilute regime�r /cmc�2� are shown in Table II. Corresponding averageaggregate number distributions are illustrated in Fig. 3. Asshown, the aggregate number distributions of all multiblock
chain types examined exhibit a single, well-distinguishedpeak, signifying the formation of well-defined aggregates.The most probable aggregation numbers, denoted as N
B* and
Nchain* , are defined by the peak in the Gaussian portion of the
respective distribution. To ascertain the effect of diblock re-peat units on microstructure properties, the simulation dataare grouped into two series, such that �A3B3�1,2, or 3 systemsare collectively examined, separately from �A5B5�1,2, or 3 sys-
TABLE II. Properties of aggregating multiblock systems. T* is the temperature and cmc is the critical micelleconcentration. Micellar aggregate size distributions were measured at T* and volume fraction of copolymer r.N
B* is the number of solvophobic sites corresponding to the maximum of the monomer aggregate distribution,
while Nchain* is the number of chains corresponding to the maximum in the chain aggregate distribution. Esti-
mated 2� statistical uncertainties in units of the last digit reported are given in parentheses.
System T* cmc r /cmc Nchain* �T* ,r� N
B* �T* ,r�
�A3B3�1 6.6 0.029 �2� 1.9�2� 61�1� 174 �8��A3B3�2 6.6 0.011 �1� 2.1�3� 30�1� 168 �8��A3B3�3 6.8 0.0073�3� 1.8�3� 17�3� 147�10��A5B5�1 9.1 0.0204�1� 1.9�2� 48�3� 240�20��A5B5�2 9.1 0.0145�1� 1.8�1� 21�2� 195�20��A5B5�3 9.1 0.0086�2� 2.5�6� 13�3� 185�45�
FIG. 3. �Color online� Representative aggregate size distributions for �A� �A3B3�m and �B� �A5B5�m systems at the conditions of Table II, calculated as afunction of solvophobic monomer number NB and number of chains Nchain. All points are from simulations.
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tems. As can be seen in Table II, for chains of defined blocksize, self-assembly results in the formation of aggregates forwhich N
chain* decreases uniformly as the number of diblock
units per chain is systematically varied from 1 to 3. Thistrend is confirmed for both the �A3B3�1,2, or 3 and�A5B5�1,2, or 3 series, where each successive addition of adiblock unit results in an approximately 50% reduction in theN
chain* value. By contrast, calculated values of N
B* are essen-
tially unaffected by m. For instance, the value of NB* for
�A3B3�m chains ranges from 147�10 to 174�8, while for�A5B5�m chains, the value of N
B* ranges among 185�45 and
240�20, where the uncertainty represents 2� in measuredquantities for multiple runs generated at each condition.Thus, for each series, N
B* values are statistically equivalent
and independent of the number of diblock units per chain.The minimum number of chains required to form an ag-
gregate of a given multiblock chain type can be externallycalculated based on the value of N
B* obtained from simula-
tions, assuming that only intrachain association occurs. Forexample, in the case of �A3B3�2 aggregates, the simulationresults indicate that the most probable clusters are composedof approximately 168�8 solvophobic B units, as shown inTable II, column 6. Because each �A3B3�2 chain consists ofsix B units, the minimum possible number of chains requiredto construct an aggregate of this chain type is approximatelyequal to 28�2��168�8� /6�. This value is essentiallyequivalent to the value of N
chain* independently determined
through simulations, as reported in Table II, column 5. Con-sequently, we can infer that at low r, the majority of multi-block chains comprising clusters adopt a fully looped con-figuration in which essentially all solvophobic units of asingle molecule participate in the aggregate core. The highprobability of loop formation is intuitively attributed to theentropic penalty of chain stretching that would be requiredfor interchain association of insoluble units in solutions oflow amphiphile density. In addition, the enthalpic penalty ofsolvophobe-solvent interactions associated with danglingchains will limit the frequency of their occurrence.
The average number of chains per diblock micelle islimited according to geometric constraints on the radius, de-fined by the extended length of the insoluble segment withinthe micellar core.70 In the case of multiblock chains, theaverage aggregate size will be determined by contributionsfrom both insoluble looped segments and extended chainends within the aggregate core. For chains with a large num-ber of diblock units, contributions from loop segments areexpected to dominate. However, when the number of repeatunits is low, such as for systems examined in this work,packing of stretched insoluble chain ends within the core islikely to dictate the aggregate size. Because in the core onlya few extended chain ends are required to define the spheri-cal aggregate size, the radius of a multiblock aggregate canreach that of the extended solvophobic chain end without amajor free energy penalty.70 Therefore, for chains of speci-fied block size, the mean radius of low molecular weightmultiblock copolymer aggregates will, in essence, be deter-mined by the radius of the equivalent diblock micelle. Simu-lation data in support of this statement are provided in thefollowing section.
Equivalence in mean radii of aggregates across systemsof a particular series additionally implies equality in aggre-gate surface areas and, by extension, aggregation numbers.However, in multiblock copolymers where chains adopt loopconfigurations to form aggregates, the area occupied by A-Bjunction sites at the core-corona interface is expected to re-duce the surface area available for incorporation of addi-tional chains. The number of A-B junction sites at the inter-face is related to the number of loops that chains adopt. If, assuggested earlier, chains adopt loop configurations in the di-lute regime such that nearly all insoluble segments of amultiblock chain participate in cluster formation, then thenumber of A-B junction sites at the interface of a clustershould be proportional to the number of A-B junction pointsper chain. This is clearly evidenced in Table II, where forchains of defined block size, the value of N
chain* is shown to
decrease linearly in proportion to the number of diblock unitsper chain, thus providing additional support of the predomi-nance of the loop topology of multiblock chains in the diluteregime.
A lower limit on Nchain* is not easily defined. For the
multiblock chains investigated in this work, for which theblock size is relatively low, the formation of stable clusterswith progressively lower N
chain* values is preempted by mac-
roscopic phase separation. When the block size is increased,the crossover to macroscopic phase separation is suppressed�see Table I�. Consequently, in the limit of sufficiently largeblock size, the formation of single chain, unimolecular ag-gregates is predicted by the simulations, in agreement withtheoretical scaling predictions by Halperin.66
Experimentally, the formation of unimolecular micellesby amphiphilic copolymers has traditionally been observedfor dendrimer and graft copolymers in selective solvents.71,72
To the best of our knowledge, the formation of unimolecularmicelles by linear, strictly alternating multiblock copolymerswas only very recently realized. Zhou et al.73 have reportedon the formation of unimolecular micelles by multiblock co-polymers based on N ,N-dimethylacrylamide �PDMA� andN-isopropylacrylamide �PNIPAM� in aqueous solutions.More interestingly, the authors demonstrate that when thenumber of diblock pairs per multiblock chain is fixed �avalue of �4 is reported73�, the copolymers can self-assembleinto either unimolecular micelles or multichain aggregates,depending on the lengths of the PDMA and PNIPAM blocks.These preliminary experimental data are in agreement withthe simulation results of our present model and, thus, confirmthe influence of multiblock architecture, in particular, blocklengths, on copolymer chain self-assembly.
D. Aggregate size, shape, and density distributions
Multiblock chain molecular architecture can impose con-straints that force microphase separated aggregates to adopt anumber of morphologies, which cannot be predicted a priori.In this section, the geometric characterization of stable multi-block copolymer aggregates is considered. The size and mor-phology of aggregates are quantified by calculating the ra-dius of gyration, where the radius of gyration tensor isdefined by74
164906-7 Multiblock copolymer solutions J. Chem. Phys. 128, 164906 �2008�
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Rxixj
2 =1
N�k−1
N
�xi,k − xi,cm��xj,k − xj,cm� , �2�
where xi,k and xj,k are the ith and jth components of thepositions of the monomer k and N is the total number ofmonomers in the aggregate. The coordinates of the center ofmass of an aggregate are given by75
xi,cm =1
N�k−1
Nn
xi,k. �3�
The eigenvalues of the tensor defined by Eq. �2� yield thethree principal radii of gyration, ordered such that R1
2 R22
R32. The average values of the ordered eigenvalues over all
aggregates are computed to produce the average principalradii of gyration, denoted as �R1
21/2, �R221/2, and �R3
21/2. Inthe simulations, the micelles do not have distinct boundaries.As such, their sizes are characterized by the root meansquared radius of gyration, defined as
�R2mean1/2 = ��R1
21/2 + �R221/2 + �R3
21/2�1/2. �4�
We describe the morphology of aggregates through cal-culation of a shape, or asphericity factor �S, as previouslyemployed,75 where
�s =��R1
2 − �R22�2 + ��R1
2 − �R32�2 + ��R2
2 − �R32�2
12 ��R1
2 + �R22 + �R3
2�2. �5�
Values of �S can range from 0 for a perfect sphere to amaximum of 1 in the limit of a long, thin cylinder. By usingthe convention of Kenward and Whitmore,75 we designateaggregates with asphericity values �0.1 as spherical.
Calculated mean cluster radii �R2mean1/2 for �A3B3�m and
�A5B5�m systems at the conditions of Table II are reported inFig. 4. Only the most probable aggregates, those in the vi-cinity of N
chain* , are considered in this analysis. In general, for
chains of specified block size, the addition of diblock unitsyields only slight increases in radius. Alternatively, when thenumber of blocks per chain is fixed, �R2mean
1/2 scales in pro-
portion to the ratio of block sizes in approximate accordanceto n1/3, consistent with previously reported76,77 effect ofblock size on the dimension of diblock copolymer micelles.Thus, as suggested in the previous section, for strictly alter-nating multiblock chains of equal composition and in thestrong-segregation limit, the radius of microphase separatedaggregates is primarily governed by the size of the constitu-ent blocks and is relatively independent of the number ofblocks per chain. Finally, asphericity values for �A3B3�m and�A5B5�m aggregates in the vicinity of N
chain* are calculated
according to Eq. �5�. For all systems investigated, the valuesof �s do not exceed 0.1 and the most prevalent clusters re-sulting from multiblock chain association are effectivelyspherical.
The behavior of aggregate forming multiblocks at thelimit of micro- to macroscopic phase transition is moreclosely examined. Multiblocks close to this boundary are ex-pected to display evidence of approaching crossover to phaseseparation. As an illustration, the aggregate size distributionsof �A3B3�3 chains are studied. Despite the low amphiphiledensity considered in Fig. 3�a�, the emergence of a secondpeak at higher aggregation numbers can be detected. We de-note this aggregation peak by N
chain** , and note that it lies at a
value of approximately 60. Principal and mean radii of gy-ration of �A3B3�3 aggregates in the vicinity of N
chain** are re-
ported in Fig. 5�a�, along with values for aggregates nearN
chain* . Ratios of the characteristic lengths for clusters near
Nchain** indicate elongation of aggregates in two of the three
principal directions, and an �s value of approximately 0.22 iscalculated for these clusters. The transition to larger aggre-gates with elongated morphology is also evident in the cor-responding snapshot of Fig. 5�b�. Accordingly, the transitionof stable microstructures to macroscopic phase precipitationis preceded by the broadening of the aggregate size distribu-tions and the formation of clusters of higher aggregationnumbers.
We additionally consider the radial density distributions�� of A and B monomers within aggregates as a function ofdistance �r� from the center of mass. As already reported, themean radius of the most probable aggregates slightly in-creases as a function of the number of diblock units perchain, while at the same time, no variation in the aggregationnumber N
B* is detected. When jointly considered, this implies
variability in the spatial distribution of monomers within ag-gregates of different chain types. To better understand thiseffect, monomer distributions for �A3B3�m clusters are exam-ined at the conditions of Table II, considering only aggre-gates in the vicinity of the reported N
chain* values. Typical
representations of total A and total B monomers as a functionof r are shown in Fig. 6�a�. In all cases, the density profilesin the aggregate cores �r approximately 2.8� are essentiallyuniform, and the cores consist almost entirely of B unitssurrounded by solvated A monomers. However, as the num-ber of diblock units per chain increases, the aggregate coresbecome less defined. The core regions of �A3B3�2 and�A3B3�3 aggregates exhibit a reduction in B monomer densityas well as a corresponding, nearly equivalent increase in thedensity of A monomers. In all cases, the solvent remains theminority component.
FIG. 4. Root mean squared radius of gyration �R2mean1/2 �see Eq. �4�� of
�A3B3�m ��� and �A5B5�m ��� systems of aggregate forming multiblocks asa function of m. Values are calculated for aggregates in the vicinity of N
chain*
and at the conditions of Table II. Estimated 2� uncertainties in values arerepresented by error bars.
164906-8 Gindy, Prud’homme, and Panagiotopoulos J. Chem. Phys. 128, 164906 �2008�
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The distribution of monomers is alternatively examinedby considering the probability that a monomer of type A ortype B is located at a distance r from the center of an aggre-gate. This probability is proportional to the quantity r2�r�.Density distributions, plotted as a function of r2�r�, areshown in Fig. 6�b�. When plotted in this manner, the overlapin the distributions of total A and B monomers in the inter-facial region is more clearly discerned. The area of coinci-dence among the distributions is shown to increase with m.For �A3B3�1 chains, the fraction of total A and B monomerswithin the area of overlap is approximately 22%, while for�A3B3�2 and �A3B3�3 systems, the fraction increases to ap-proximately 31% and 36%, respectively. The greater degreeof A and B monomer intermingling additionally indicates theformation of aggregates with less defined core-corona struc-tures, larger interfacial regions, and a reduced ability to seg-regate incompatible monomer types. Accordingly, the dem-onstrated tendency toward macroscopic phase separationwith block addition is attributed to the formation of micro-structures, in which steric stabilization of solvent incompat-ible monomers is significantly diminished. The simulations,thus, suggest that for multiblock chains of low molecular
weight blocks, the loss of entropy associated with looping ofsolvophilic segments becomes prohibitive as the number ofblocks per chain is increased and, consequently, precludesthe formation of stable aggregates. Finally, the ability to gen-erate stable polymeric aggregates in which the aggregatecore polarity �degree of A and B monomer overlap� can betailored through copolymer architecture offers a highlyadaptable platform for drug delivery, particularly for the de-livery of peptide and protein therapeutics. The altered chemi-cal and physical nature �i.e., porosity� of microstructure in-teriors has experimentally been shown to facilitatehydrophilic drug incorporation and modulate the kinetics ofdrug release.78,79 Nonetheless, a deeper understanding of thepotential of such copolymer systems requires a carefulknowledge of how molecular structure �composition, blocklength, and architecture� affects the overall microstructurefeatures, influences which can be elucidated through coarse-grained simulations such as those discussed in this work.
E. Network formation
The ability of multiblock chains to bridge multiple in-soluble segments allows for the formation of connectedstructures at sufficiently high concentrations.24,67,80 Thischaracteristic feature has been extensively explored for
FIG. 5. �Color� �A� Principal radii of gyration �Ri21/2 of �A3B3�3 multiblocks
in the vicinity of Nchain* �circles� and N
chain** �triangles�, calculated at the
conditions of Table II. Corresponding root mean squared radii of gyrationfor each aggregate type are reported in legend, with estimated 2� statisticaluncertainties in the last digit reported in parentheses. �B� Representativeconfiguration of aggregates in the vicinity of N
chain** at the conditions of
Table II and a simulation box size of L*=35.
FIG. 6. �A� Volume fraction distributions �� of total solvophobic B mono-mers �open symbols� and total solvophilic A monomers �filled symbols� as afunction of the distance to the center of mass r for aggregate forming�A3B3�m systems at the conditions of Table II �B� Radially averaged prob-ability distributions r2 as a function of r for �A3B3�m systems at the con-ditions of �A�. Only aggregates in the vicinity of N
chain* are considered in
these analyses.
164906-9 Multiblock copolymer solutions J. Chem. Phys. 128, 164906 �2008�
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multiblock chains in the melt15,30,69,81–83 and in solutions ofselective solvents.65,75,84 In this work, the qualitative natureof multiblock network formation is examined with respect tothe type of thermodynamic phase equilibrium attained in di-lute solutions.
As an illustration, representative snapshots of the mac-roscopic phase separating �A3B3�10 system are shown in Fig.7. The simulation temperature is selected such that it is sig-nificantly below T
c*, and equilibrium exists between low and
high density polymer phases. Interchain bridging of col-lapsed solvophobic segments can clearly be seen in the lowr snapshot of Fig. 7�a�. However, under this condition, themajority of chains are essentially independent and cross-linking is limited to a low fraction of chains, as evidenced bythe corresponding chain distribution plot in Fig. 7�a�. Alter-natively, at higher r, the probability of segment bridgingincreases and a connected network results, as shown in Fig.7�b�. Computationally, the network is characterized as a gelwhen nearly all the chains in a system have bridging seg-ments, yielding a single cluster spanning the box boundariesin the x, y, and z directions. With periodic boundary condi-tions in all three directions, these bridging segments producean infinitely large three dimensional network or gel. Thechain distribution of Fig. 7�b� confirms the presence of asingle, relatively monodisperse collection of chains, whosepeak coincides with the average number of molecules in thesimulation system. Thus, at semidilute concentrations, mac-roscopic phase separating multiblocks are shown to formswollen gels. Because no aggregates are present in dilutesolutions of this system, gel formation results as a conse-
quence of random association of solvophobic segmentsacross collapsed microdomains and not as a consequence ofbridging across multichain aggregate cores.
In Fig. 8, representative snapshots of the �A3B3�3 micro-scopic phase separating multiblock are presented as a func-tion of r. At an amphiphile density of approximately 3%�Fig. 8�a��, interconnections between solvophobic aggregatecores are clearly visible. However, most clusters remain in-dependent and possess a relatively well-defined aggregationnumber, as evidenced by the peak in the corresponding chaindistribution. At an intermediate density �Fig. 8�b��, highlypolydisperse clusters of large aggregation numbers dominate,yet the presence of small aggregates can still be detected�indicated by arrow�. Above a certain density, which is notquantitatively determined in this work, there are enoughbridged segments connecting a sufficient number of aggre-gates to span the entire simulation system, and gelation oc-curs. This is signified by the presence of a single peak in thechain distribution, as shown in Fig. 8�c�. Qualitative exami-nation of aggregate structure as a function of r indicatesthat the elongated morphology of individual clusters at lowr �Figure 8�a�� remains relatively unperturbed for the highlyinterconnected clusters of the gel phase �Fig. 8�c��. Thus, thegel can be viewed as a swollen array of elongated aggregatesconnected by bridged chains, where the insoluble domainsize is primarily determined by the characteristic scale of themicrophase separated aggregate core. These results are quali-tatively similar to previous studies in triblock copolymers.85
However, gels formed by bridging of multiblock copolymeraggregates may be better suited as viscoelastic modifiers
FIG. 7. �Color� Representative snapshots and chain distributions of �A3B3�10 multiblock for simulations at the following conditions: �A� T*=8.4, �=−65.3,L*=40, and =0.016�1� and �B� T*=8.4, �=−65.2, L*=40, and =0.348�3�.
164906-10 Gindy, Prud’homme, and Panagiotopoulos J. Chem. Phys. 128, 164906 �2008�
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than comparable triblock micellar gels. The increased num-ber of aggregate junctions possible in a multiblock chainmay potentially yield a gel of higher elastic modulus andenhanced mechanical properties in contrast to materials withlow number of blocks.3,69,83 Finally, the observed variabilityin solvophobic domain size and distribution in gels of multi-block copolymers depending on the phase transition type inthe dilute regime can be exploited in the design of novel drugdelivery vehicles, where release kinetics of entrapped drugscan potentially be tailored through specification of gel micro-structure.
IV. CONCLUSIONS
In the present study, we have examined the dilute phasebehavior of perfectly alternating, amphiphilic �AnBn�m typemultiblock copolymers. We show that a simple lattice modelof the multiblock chains can qualitatively account for experi-mentally observed chain-structure dependence on copolymersolution phase behavior.64,65,73 In particular, when the archi-tecture of chains is characterized through a correlation pa-rameter �=m /n, we find that the phase transition behavior ofdilute solutions of �AnBn�m depends on the value of �. Forchains of a specified value of n, the simulations predict mi-croscopic phase separation below a critical � value and mac-roscopic phase precipitation otherwise. The location of thecritical correlation parameter was shown to increase with the
molecular weight of constituent blocks. Alternatively, forchains in which m is specified, our simulations predict areduction in the enthalpic contribution toward phase separa-tion when the chains are subdivided into greater number ofblocks. Thus, diblocks are found to phase separate muchmore readily than tetrablocks, which, in turn, do so morereadily than hexablocks and so on. This tendency is evidentin all multiblock chains examined, irrespective of the type ofphase transition experienced.
Equilibrium microstructural properties were studied as afunction of multiblock chain architecture and amphiphileconcentration. In the dilute regime, the formation of well-defined, near spherical, multichain, core-shell type aggre-gates was predicted for all microphase separating multi-blocks studied. Under these conditions, multiblock chainswere shown to predominantly adopt loop configurations,such that nearly all insoluble segments of a single chain par-ticipate in cluster formation. When the block size is speci-fied, the mean radius of multiblock copolymer aggregateswas primarily determined by the characteristic scale of cor-responding diblock micelles. As a consequence, characteris-tic aggregation parameters of the diblock copolymer micellealso defined the mean cluster aggregation numbers of multi-block copolymer aggregates. Calculated values of N
B* were
nearly constant among clusters of multiblock chains withspecified block size. Alternatively, N
chain* was shown to lin-
FIG. 8. �Color� Representative snapshots and aggregate size distributions of �A3B3�3 multiblocks for simulations at �A� T*=6.8, �=−52.7, L*=35, and
=0.03�1�, �B� T*=6.8, �=−52.6, L*=35, and =0.21�3�, and �C� T*=6.8, �=−52.0, L*=35, and =0.384�3�. Points are from simulations, with lines drawnfor visual clarity.
164906-11 Multiblock copolymer solutions J. Chem. Phys. 128, 164906 �2008�
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early decrease in proportion to the number of added diblockunits per chain. Thus, for microphase separating multiblockcopolymers of sufficient block size, the formation of unimo-lecular aggregates is predicted in the limit of high blocknumber, in agreement with the theoretical predictions ofHalperin.66
As the polymer concentration is increased, network for-mation as a consequence of bridging across the insolublesegments of multichain clusters was observed. A progressivetransition from solutions with few, generally noninteractingelongated aggregates to systems of increasing aggregate den-sity and bridging was found. At a sufficiently high concen-tration, a transient gel was formed, whereby all insolubleaggregate cores were bridged. In comparison, gels of macro-scopic phase separating multiblock chains were formed as aconsequence of association between collapsed insoluble seg-ments of the primarily unassociated chains. These distinc-tions are expected to result in significant differences in gelelastic and mechanical behavior.1,3
The unique behavior of multiblock copolymers makesthem useful in various fields as thermoplastic elastomers,86
viscoelastic modifiers,1 drug delivery nanostructures,46,87 andblend compatibilizers.88 However, experimental limitationsoften limit systematic analyses of copolymer structural ef-fects on the rheological, morphological, and mechanicalproperties in dilute solutions. Thus, a practical implication ofthe simulations presented in this work is the design of rela-tively low molecular weight multiblock copolymer systemswhich can be tailored with regard to solution phase behavior,microstructure formation, and gelation properties.
ACKNOWLEDGMENTS
Financial support of this work was provided by the Na-tional Science Foundation through a Graduate Research Fel-lowship to M.G. and by the NSF NIRT Center on Nanopar-ticle Formation. Additional support was provided by theDepartment of Energy, Office of Basic Energy Sciences�Grant No. DE-FG02-01ER15121�. M.G. also gratefully ac-knowledges financial support from Merck and Companythrough a Doctoral Research Fellowship.
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