Dynamics of Phase Separation in Polymeric Systems - American

Chapter 16

Dynamics of Phase Separation in Polymeric Systems Downloaded via TUFTS UNIV on July 13, 2018 at 05:15:52 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

G. J. A. Sevink, Α. V. Zvelindovsky, and J. G. Ε. M. Fraaije Leiden University, Soft Condensed Matter Group, P . O . Box 9502, 2300 RA Leiden, The Netherlands

We review the dynamic density functional (DDFT) method for 3D pattern formation in complex amphiphilic systems. The fo cus is on the supra-molecular or mesoscopic level. The building blocks consist of sequences of dissimilar monomers, connected in copolymer chain molecules. Internal factors such as com position and architecture of the polymers, but also external factors such as applied electricfieldsor confinement control the self-organisation phenomena. The specific examples presented here focus on the validation of our method by experimental data and its predictive value. The ambitious goal is the inven tion of methods for the rational design of truly complex biomimicing materials, in which we combine principles of chemical engineering, physics, chemistry and biology. The keyword is self-organisation. However, autonomous self-organisation leads to trouble, modulated self-organisation leads to beauty.

258

© 2003 American Chemical Society

Case and Alexandridis; Mesoscale Phenomena in Fluid Systems ACS Symposium Series; American Chemical Society: Washington, DC, 2003.

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Introduction For a long time, chemical engineers have analyzed macroscale properties using a variety of continuum mechanics models. In the last decade molecular modeling has grown to an essential tool for research and development in the chemical and pharmaceutical industry. Despite the considerable success of both molecular and macroscale modeling, in the past few years it has become increasingly apparent that in many materials the structures on the mesoscale level determine the material properties to a very large extend. Mesoscale structures are typically of the size of 10 to 1000 nm. Even more, recent developments in nanotechnology deal with tailored structures exactly in this range of scales. The industrial relevance of mesoscale modeling is therefore obvious. Nevertheless the development of the necessary general purpose computational tools is still at its infancy. We are developing general purpose methods that are aimed at combining molecular detail and the computational efficiency of continuum approaches. The resulting molecular field method is now widely used for mesoscale soft condensed matter computer simulations. It is based on a functional Langevin approach for mesoscopic phase separation dynamics of complex polymeric liquids. The project aims to develop tools and facilitate answers for a wide range of factors of utmost importance in ûano-technologieal applications, such as chemical reactions, convection and flow effects, surfaces and boundaries, applied electric field, special sample preparations etc. The morphology formation in complex liquids has been studied by many authors using time-dependent Landau- * ° general diblock copolymer, A f B A f Ν = NA+ Νβ for a symmetric triblock copolymer). In case of solvent, it presence can be easily taken into account (1?). The volume of the system is V . In this volume the following fields are used: concentration fields p/(r), external potentials Ui(r) and intrinsic chemical potentials /*/(r), where I denotes the bead type (say, A or B). Imagine that on a course grained time scale, there is a certain collective concen tration field pi(r). Given this concentration field a free energy functional F{p] can be defined as follows (10-13) ra

A

NA 2

NB

B

NA 2y

(i) Here Φ is the partition function for the ideal Gaussian chains in the external field Uj, and F \p] is the mean-field contribution due to the non-ideal interactions. The external potentials and the concentration fields are bijectively related via a density functional for ideal Gaussian chains: nid

Ν

(2) where 5f , is a Knonecker delta with value 1 if bead $' along the Gaussian chain is of type J and 0 otherwise. The trace Tr is limited to an integration over the a

c


261 coordinates of one chain Tr ()=C

(3)

J {-)\{dR. yΝ

c

β==1

with C a normalization constant. The single chain distribution function

«*

,

(4)

Ν G

with H

2

=

Σ (Re-Rs-i)

the Gaussian chain Hamiltonian of an unperturbed

s=2

chain, and α the Gaussian bond length parameter, is such that the distribution φ is optimal at each coarse-grained stage in time. There is no known closed analytical expression for the inverse density functional, but for our purpose it is sufficient that the inverse functional can be iteratively determined by an efficient numerical procedure. The mean field part F **^] of the total free energy contains several terms, including a part F [p] for the cohesive interactions and excluded volume interactions by^F [p], taking care of the incompressibility of the system. As external factors, we can have a contribution due to the energetic interactions with (filler) particles, giving rise to F [p], or an applied electric field denoted by F [p]. We can then write the non-ideal part as 11

c

e

CM

E

nid

c

F \p]

CM

e

(5)

E

= F [p] +F [p] + F \p] + F [p)

with F C

e

W = \ Σ / / "(\r - A)f>i{r)pj{r')drdr' (6) Λ·/ ν V which is a two-body mean field potential, where eu(\r — r'|) are the reciprocal cohesive interactions between beads of type I at position r and J at r', often taken as a Gaussian kernel 2

etjQr - r'|) = o , ( - ^ ) l - * ( - - ' - ' ) . e

(7)

e

For a diblock system AN BN , the values of the e° parameters are easily related to the more familiar Flory-Huggins χ parameter by the relation χ = 1/2βι/~~ [ε° + %Α ~ ΑΛ - BB) = fi ~ AB ( ^ c l volume), since ε° = = 0 in most cases. In some case, rigid (filler) particles that are stationary in time can be part of the system, like electrodes, substrates and long rods. We have then an extra contribution to the free energy for the energetic interaction between beads of A

B

ι

ΑΒ

Ε

Ε

£

U

LE

w i t

Ν

t n e

D e a

ΑΑ


262 different types and the rigid particle material. The extra term in the free energy due to this contribution can be written as (18) CM

F \P)

=\ Σ//

ε

^{\τ

- r^prtfpMWdrdr'.

(8)

ν ν Inside the filler particles, occupying a total volume V° C V, the concentration fields pj(r) of the different bead types is equal zero. The constant density field PM (where M represents 'auxiliary' beads of thefillerparticle type, and we allow different interaction values EIM for different filler particles) that appears in (8), is defined as p (r) = 1 for r € V° and p (r) = 0 elsewhere (r € V/V°). The interaction kernels are chosen equal to the ones used in (7). Although more rigorous approaches have been considered (19), the excluded volume interaction is included by a phenomenological Helfand penalty function M

M

(9) with KH the Helfand compressibility parameter and pj is the reference concentration field of bead type I in complete mixing. This term provides a mathematically simple and numerically stable way to account for compressibility effects in the system (19). In the case that an electric field is acting on the volume V, there is an extra contribution to the free energy. Under the assumption of a fully incompressible diblock copolymer (A — B) system and expanding the dielectric constants up to second order in the order parameterfieldip(r) (and neglecting higher orders) (20, 21), this term can be written in Fourier space as function of this order parameter field as E

Fm

X

= jgpr *B

J #(W(-fe)(|l - eifdk

(10)

V

where is the unit vector in the direction of the electricfield,and Β is a dimensionless parameter

where \E \ is the strength of the electricfieldand €/ is the dielectric constant of the pure component I (A or B). The intrinsic chemical potentials μι are defined by the functional derivatives of thefreeenergy Q

f

ΡΊ(Γ)

ν

C

CM

e

B

OF , 6F = δρι j— = - ^ +6pi(r) 7-^ +SF 7-^ +SF7-7-7 +6F Sp/(r) Spr(r) 5pi(r) r

=

r

λ

^ / ( r ) + M Ï ( r ) + / i ^ ( r ) + MÎ(r)+ f(r) M


(12)

263 with Mr)

^f*"(\r~-r'\)pj(r')dr'

(r)

vi*>H I Y]vj{pj(r)

-Pj)dr

pf(k)

(13)

Finding the minimum offreeenergy Several methods can be employed to find the minimum of free energy (1) and equilibrium concentration fields pj (r). They can roughly be divided into static and dynamic methods, although a number of hybrids exist which are generally referred to as quasi-dynamic methods (for instance {22))· A rather complete and recent review is given in Ref (23). In this overview article, we focus on a number of different dynamic schemes that have been developed within our group for the minimization of the free energy (1). An advantage of the dynamic schemes is that that they consider a dynamic pathways towards a free energy minimum, including visits to long-living metastable states. In this sense, the model can be seen to mimic the experimental reality when compared to static schemes, which are optimizations, based upon mathematical arguments. In equilibrium pi(r) is constant; this yields the familiar self-consistent field equations for Gaussian chains, given a proper choice of F [p]. When the sys tem is not in equilibrium, the gradient of the intrinsic chemical potential — Vpj acts as a thermodynamic force which drives collective relaxation processes. In the standard method of so-called Mixed Dynamics employed in our group, we write stochastic partial differential equations that describe the collective diffusion of the concentration fields (24) 9

nid

dt

= M / V · prVpj + ηι

(14)

where we have used Onsager or local transport coefficients that are linear in concen tration, Mi is a constant mobility for bead I and m(r) is a noise field, distributed according to the fluctuation-dissipation theorem. In some cases, it is sufficient to


264

consider constant transport coefficients and the diffusion equation becomes ^

= Μ Δμ +» ; /

/

(15)

? /

A numerically efficient scheme including non-local kinetic coupling can be arrived at by considering External Potential Dynamics instead of Mixed Dynamics (25). In this case, the governing equations for the dynamics read ^Εΐ^-Μίάμι

+ ηι

(16)

A full discussion about the validity of the approximations used to arrive at equation (16) can be found in Ref (25). Recently, an alternative scheme was developed that has been derived from the External Potential Dynamics (16). In the notation of (25), the External Potential Dynamics algorithm is obtained by left-commutation of mobility and gradient operators V - P / J V ^ J -¥ F / J V / ^ J . A n alternative algorithm is obtained by right-commutation V - P / J V / ^ J -> V P / j £ t j , which is valid provided μj « 1. This gives rise to the dynamic equations 2

2

2

^=M V (p -p yi ]), I

I

I

nid

(17)

where pi[pnid](r) is the concentration functional of the polymer (or solvent) and can be calculated from equation (2) by replacing the external potential fields Uj by the intrinsic chemical potentials due to the non-ideal interactions. As the algorithm is fully explicit in the concentration variables, and thereby avoids the calculation of the external potentials, it is a computationally efficient scheme compared to the other ones. This algorithm is well suited for avoiding the compressibility problems that normally arise when concentration fields are zero or close to zero in a certain part of the simulation volume.

External fields Recently, polymeric systems received increased interest because of their use in many applications in nanotechnology. One important aspect in nanotechnology is the ability of tailoring micro-phases for industrial needs by various external and internal conditions. The application of external fields, such as applied flow fields, applied electric fields or confinement in thin films, is known to have a strong effect on the structuring of micro-phases on a macro-scale level; it gives rise to a global orientation. In some case, experiments as well as simulations have shown that even the micro-phase type can be altered by the external conditions. For this reason, the behavior of systems under an external field is an extensively studied topic both experimentally and theoretically. On the other hand, also the preparation pathway can lead to altered phases. A n example is polymeric vesicles, which can only be obtained following a specific preparation technique [26-28).


265 Earlier we considered the extra terms in the free energy due to the electric fields and confinement. In the dynamic scheme, application of an electric field results in ^

= Μ / Δ μ / + Mrf^vBVfa

+ ηι.

(18)

where an electric field applied in the (i ~x,y or z) direction. In case of geometry constraints (the inclusion of (filler) particles), the dynamic equations are not altered by extra terms. However, for the particles we consider boundary conditions, in accordance with the conservation laws, that allow no flux through the solid particle surface (18), i.e. ν μ / · η = 0, (19) where η is the normal pointing towards the particle. The same boundary conditions apply for the calculation of the noise, that should satisfy the fluctuation-dissipation theorem. At all boundaries where no filler particles are present, periodic boundary conditions apply. In the presence of a steady shear flow, with v = jy and v = v = 0, the microphase separation is described by (31,32) x

^

2

= MrV pi

- jyV pi x

y

z

+m

(20)

which is equal to equation (15) apart from an extra term that accounts for the applied shear (where 7 is the shear rate (the time derivative of the shear strain 7) and sheared boundary conditions apply (29-32)).

Examples In this section we consider three systems, two of which are directly related to experiments (33,34) carried out in the group of Georg Kraus and Robert Magerle (Bayreuth University, Germany). For the third system of polymeric particles in highly selective solvents, extensive experimental knowledge is present in the group of Adi Eisenberg (McGill University, Canada) (#5). A comparison to their experiments is the focus of current collaboration. It is worth mentioning that even in the calculation of the phase diagram of sections B , the dynamics plays a role. In our method, there are no means to distin guish between equilibrium states and states that are stable within the time range considered in our simulations. The same holds for the experimental data, as the S T M data is all acquired at some fixed time.


266

Dynamics of electric field alignment In recent experiments Bôker et al. (34) identified two distinct microscopic mechanisms of electric field induced alignment of diblock copolymer microdomains. The system they considered was a dissolved (in toluene) polystyrene- Wodb-polyisoprene (SI) diblock copolymer of nearly symmetric compositions, forming lamellae under normal conditions. Initially (before switching on the electric field) the lamellae align parallel to the electrodes (with a small distribution in orientation) due to the preferential attraction of the PS block to the gold surfaces. Bôker et al. observed that after the electric field is switched on, the lamellae start to orient in the direction of the electric field. Interestingly, two different underlying mechanisms were identified. Close to order-disorder transition (ODT) the scattering peak from the parallel to the electrodes lamellae quickly disappears and a new,peak from the perpendicular lamellae emerges. Further from ODT the peak merely rotates from the initial position towards the final one. We used two-dimensional D D F T simulations (with free energy (1), dynamic equations (15) and the appropriate term for the^electric field contribution) to see whether we are able to reproduce the observed scattering behavior. It should be mentioned that our dynamics is purely diffusive, and does not include any hydrodynamic contributions. In the simulations, the initial orientation parallel to the electrodes can be obtained by preferentially attracting surfaces (18). A n alternative route is the application of shear for a certain amount of time. This seems reasonable, since, in the experiments, the scattering data was obtained from the center part of the sample, far away from the confining electrodes. This route facilitates an easy way to obtain the roughly aligned lamellae of the experimental study. One structure after shear is shown in plate 1 (at t = τ/AT = 5). After cessation of shear, the electric field is applied in the direction orthogonal to shear direction. In their article Bôker et al argue that the two mechanisms (grain boundary migration and grain rotation) are distinguished due to kinetic factors (different initial grain size, different mobility and viscosity). Our simulation shows that the behaviors are discriminated by energetics, namely by the balance of the ponderomotive force (20) and the surface tension (and higher terms like bending) of a single lamella, which is related to the block interaction €AB, the only varied parameter. The initial grain size, plate 1 (at t = 5) was nearly the same. Close to ODT the surface tension cannot withstand the electric field, and undulations appear in lamellae, plate 1 (waves at t = 5.2). In addition to existing grains new perpendicular grains form in the body of the parallel lamellae. They appear via a transition to an intermediate micellar or/and bicontinuous phase. As this phase is a structured one (not "selective melting" into disorder) there is no considerable broadening of the scattering peak, as was already observed in (34). The migration of grain boundaries follows the same transition (row of micelles in the square, plate 1). Further from ODT the undulation instability is suppressed and the lamellae collectively rotate.


267

Plate 1: Top: the scattering intensity as a function of the azimuthal angle and di mensionlesstime(t = τ/ATxl(P)fora AiBi-mett in 2D. Below: simulation snap shots. Left: closer to ODT (χΝ = 19.2J. Right: further from ODT (χΝ = 25.6;. The axis in which the electric field is applied is vertical. The electricfieldstrength parameter Β = 0.2 (Reprinted with permission from Zvelindovsky et al. Phys. Rev. Lett 2003, 90, 049601. Copyright 2003, American Physical Society). (See page 9 of color insert.)


268

The proeess is assisted by single defect migration and binary annihilation. The ki netic factors (34) will only modify the time scale of the processes. As to the initial misalignment (here created on purpose by shearing), it is indeed responsible for the choice of rotation direction (34).

Confined polymer melts Most of the studies performed in confined polymer systems so far have focussed on the orientational phase transitions of the system of certain bulk symmetries (lamel lae, cylinders and spheres) under some external field. It should be realized that the presence of external surfaces and the confinement of the material to certain film thicknesses considerably alters the microphase behavior and can stabilize novel, non-bulk, structures. It is important to study the phase behavior as the alteration increases the number of possible thin-film structures for a given polymeric system. This results in complex phase behavior that may be exploited in potential applica tions. We have used our model (the free energy (1) and dynamic equations (14)) previously (37,38) to calculate the phase diagram for a confined cylinder-forming diblock copolymer (which was modeled as a A$BQ chain and SAB - 7.1). We found that non-cylindrical structures are stabilized in thin films by the preferential at traction of one type of block to the surface (the surface field). In particular, we found a sequence of phase transitions, with increasing surface field, from cylinders oriented parallel to the surface via hexagonally perforated lamellae to lamellae. In a recent study, Knoll et al (33) have performed similar calculations for the tri block analog of the polymer used in Huinink et al (37,38), a cylinder forming triblock copolymer modeled as a A3B12A3 chain, and revealed phase behavior quite similar to the diblock copolymer case (37,38). These simulation results were sup ported by experiments on thin films of concentrated solutions of a polystyrene-Wacibpolybutadiene- Woefc-polystyrene (SBS) triblock copolymer in chloroform. Films of different thickness were exposed to a well-controlled partial pressure of chloroform vapor. As chloroform is a relatively neutral solvent for both chemical blocks, the effective strength of the surface interactions was varied via the solvent concentration in the polymer film, giving rise to an experimental thin-film phase diagram. The effective interaction was then determined using the experimental data as SAB = 6-5 (all other parameters are as in (87,38)) and the phase behavior simulated as a func tion of slit width H and polymer-particle interaction difference EM = ε AM — £BMA stunning agreement between the calculated and the measured phase diagram was found (33). In particular, non-cylindrical structures were indeed observed in certain, well-defined regions of the experimental phase diagram. As an example, figure 1 shows a comparison of the experimental results (top and middle pictures) and the simulations (bottom picture) for a gpven strength of the surface field (CM = 6). The simulation box contains two filler particles: a horizontal planar particle (one gridcell thick) at the bottom and a tilted planar particle (also


269 one gridcell thick) at the top. This simulation volume was chosen to mimic the lateral increase in film thickness of the experimental observation (schematic middle picture in fig 1). In figure 1, a rich variety of structures is observed as the film thickness increases from left to right. With increasing film thickness, both experiment and simulation show the same sequence of thin-film phases: a featureless film surface for the smallest thickness, isolated domains of the A phase, parallel-oriented A cylinders, perforated A lamella, parallel-oriented A cylinders of ellipsoidal cross section, perpendicularly oriented A cylinders, and finally two layers of paralleloriented cylinders. The phase transitions occur at well-defined film thickness, as can be seen from the white iso-thickness lines that have been calculated from the scanning force microscopy (SFM) height images. From a fundamental point of view, these study nicely demonstrates that block copolymer thin films are governed by the interplay of two parameters: i) the surface field can either orient the bulk structure or stabilizes non-bulk structures, so-called surface reconstructions (36) ii) the film thickness modulates the stability regions of the different phases via interference and confinement effects. The range of the surface reconstructions seems to be limited to a near-surface region comparable to a characteristic microdomain spacing (33,37,38). However, even for thicker films the surface regions can be of crucial importance for the function of the film, as they establish the contact between the film and its surrounding media, such as electrodes, electrolytes or etching media. The mere existence of surface reconstructions leads to the important general remark that it is not straightforward to draw conclusions about the volume structure of a thin film from the surface structure. The above simulation study considers 'hard' films where both plates or particles are inpenetrable and completely incompressible. Here, we did not consider 'halfsoft' films where one of the confining sides is highly compressible, for instance in the presence of air. The results for the 'hard' films can be extrapolated (37,38) to films that are able to adjust their shape, as in the experiments, to the local tensions in the film. A n alternative route, that is currently under investigation, is to introduce neutral solvent. Moreover, in experiments of supported films, dissimilar substrates often exist, i.e. at one particle surface the favored microdomain orientation or surface reconstruction is different than at the other particle surface (for a simulation example, see figure 2). In such a situation a hybrid structure may form and a grain boundary is stabilized in the middle of the film (39). Two experimental examples of such hybrid structures reported for cylinder-forming block copolymer are 'a perforated lamellae with spheres' (40) and 'cylinders with necks' (^1). Both of them exhibit the same surface structure. However, due to the different underlying structures, the mechanical response to external forces will be quite different. The phase diagram for this type of system with dissimilar substrates has been calculated and a detailed article will be submitted soon. The method and concepts presented here, together with recent developments in experimental three-dimensional imaging


270

Figure î: a,b) Tapping-mode SFM phase images of thin SBSflmson silicon substrates after annealing m chloroform vapor. The surface is covered with a hom geneous « 10 nm thick polybutadiene (PB) layer. Bright [dark] corresponds to polystyrene (PS) [PB] microdomains below this top PB layer. Contour lines calc lated from the corresponding height images aire superimposed, c) Schematic he profile of the phase images shown in (a,b). d) Simulation of on A^BtiAz block copolymerfihnin one large simulation box withfilmthickness increasing from le toright.The iso-concentration surface ρA = 0.5 is shown. (Reprinted with permis sion from Bôker et oL (34) Copyright 2002, American Physical Society).


271

Figure 2: Representative simulation result in case of dissimilar substrates located at the top and the bottom of the simulation box: one substrate favors a perpendicular microstructrure, the other a parallel (perforated lamellae) one. Left: crop of the field in the vicinity of the hard particles for iso-concentration surface ρA = 0.5, right: totalfield(same iso-value).


272 techniques {40,42) provide the means to control and understand block copolymer thin-film structures of various symmetry.

Smart nanoparticles Through simulation we discovered remarkable bicontinuous structures in dispersed droplets of polymer surfactant (figures 3-4). The nanogels structures are soft and fragile: the molecular bonding energy is weak, derivedfromself-assembly of polymer molecules. Nanogels may have many applications in soft nanotechnology, in drug delivery, templates for heterogeneous catalysts, aerosols and personal care prod ucts. Patterned nanostructures are the key in all these applications, and a better understanding of their formation is of paramount interest. In these calculations, we generate the nanogels by quenching a homogenous droplet of polymer surfactant in an aqueous bath. The free energy model is given by (1) and the dynamic equations by (17). The simulation parameters are for diblock polymer surfactant AN-Μ Β M with Ν = 20 in weakly selective solvent and mild segregation, XABN = 40,- XAS ~ XBS = 0.3, so that A is solvo-phobic and Β is solvo-phylie. These are essentially the parameters we verified before by comparison with experimental microphase diagrams of concentrated Poly Propylene Oxide - Poly Ethylene Oxide aqueous solutions in ambient conditions (17), in which case each bead or statistical unit corresponds to 3-4 monomers. Of course one should realize any polymer surfactant solution with the same properly scaled interaction parameters will behave in exactly the same way, according to the mean-field model. The free energy model is that for an nVT ensemble, and accordingly we do not calculate the global equilibrium in an open system, but rather a local equilibrium morphology of an isolated droplet. The situation is analogous to that of the classic study of the shape of an isolated lipid vesicle, when interactions between vesicles are less relevant. With the selected values of the Flory-Huggins parameters, and Ν and M , the polymers are all insoluble, hence the deformations of the droplets are at constant mass of polymer. The simulations proceed by a sudden quench of a homogenous droplet in a sol vent bath. Following the quench, the droplet takes up or releases additional solvent locally and globally, depending on the particular morphology being formed. The polymer concentration outside the droplet is zero, and this remains so during the morphology adaptation. The simulations are stopped when the order parameters do not change any more: the free energy is then steady in a minimum, and the pattern is a solution to the self-consistent field condition pi = ρι\μψ% We have found it ad vantageous to add a small uncorrelated white noise field to the mean-field chemical potentials. The uncorrelated noise does not obey the fluctuation-dissipation theo rem, but it helps small barrier crossings in the free energy landscape. The droplets (initial radius R° cell units, each cell has the size of the statistical unit) are placed in the centre of the box with sufficient space, N/2 cells, between the droplet surface


273

Figure 3: Morphologies of AH-MBM polymer surfactant nanodroplets (isosurfaces partly removed for visualisation). Solvophylic A concentrationfieldfor different block ratios f = M/N. 0.35(a), 0.30(b), 0.25(c), 0.20(d), 0.15(e), 0.10(f). (Reprin with permission from Fraaije et al (accepted) Copyright 2003, American Chemical Society).

Figure 4: Morphologies of AN^MBM polymer surfactant nanogels (f = M/N = 0.25) gels for different initial radii RP. From left to right: R? = 20,23,26,30,33 (in units of polymer statistical bead size). Notice that in this viewing mode the outer fuzzy shell is not visible. (Reprinted with permission from Fraaije et al (accepted) Copyright 2003, American Chemical Society).


274

and the boundaries of the computational box, thereby avoiding artifacts resulting from the periodic boundary conditions. In all cases the droplets develop an outer fuzzy layer of the solvo-phylic Β block, but since the confinement of the polymers is not hard but soft, the droplet surface is not necessarily spherical. The surface topography is that of small valleys, ridges and bumps, reflecting the underlying gel morphology, very much like earths topography is an image of deeper events. The nanogel internal structures, shown in figure 3, depend on the size ratio / = M/N similar to bulk block copolymer systems. More symmetric polymers / = 0.35 form into an onion structure of alternating A and Β layers (figure 3a); slightly less sym metric polymers / = 0.30 - 0.25 form a bicontinuous phase (figure Zb-c), then at / = 0.20 a cylindrical phase (figure Zd) and an inverted micellar phase (figure 3e) at / = 0.15. Too asymmetric polymers / = 0.1 do not form any internal structure (figure 3f). In equilibrium the gels contain an appreciable amount of solvent (on the average ca. 15%), distributed inhomogeneously over the solvo-phobic A and solvo-phylic 2?-rich layers. In the case of / = 0.25, figure 4, the layers are perforated with pores, and the entire structure is bicontinuous. The gel strikingly resembling a buckyball, with in each inner Β layer a mixed pore pattern of pentagons, hexagons and septagons. In bulk solution or melt systems, perforated lamellae consist ideally of a perfectly hexagonal array of pores. In the curved nanogels, by rule of geometry, a perfect array of hexagons is impossible to form, and the perforation is mixed. In bulk, a mixture of 85% of this particular polymer surfactant and 15% solvent forms a gyroid bicontinuous structure (data not shown). An interesting finding for the morphologies of figure 4 is that the equilibrium structures are more or less radial invariant. From figure 5, where the radial positions of internal maxima of Β concentration are plotted as a function of initial droplet size 0, it can be observed that, with increasing B?, the position of the Β rich shells shifts linearly. The system adds shells at the center if the confinement allows it. When an additional layer can be formed, the inner layers move outward. Since the domain size is more or less constant, the pore density in each layer is constant too, and the number of pores in each shell is determined from its radius. There is no intrinsic magic W number, as in Ceo* associated with additional stabil ity of the nanogels. The resemblance with a carbon buckyball is purely coincidental, based on a geometrical rule for packing pores in a spherical shell. The experimental investigation of these systems is at its infancy. Perforated lamellar or similar bicontinuous structures have been observed experimentally in thin polymer films (33), and in simulations of confined systems (33,37,3S). In the area of polymersomes (26), closely related to the systems of interest here, results so far indicate hollow bilayer vesicles. At first glance the system of bicontinuous cubic phases or cubosomes (see also the article by Pat Spicer in this volume) resembles the buckyball structures in this article. Although this looks very promising, the cubosomes are cubic liquid crystalline phases that autonomously self-assemble from


275

Figure 5: Shell position versus initial radius for different layers, (Repnnted with permission from Fraaije et al. (accepted) Copyright 2003, American Chemical Society).


276

solution, whereas our structures depend on the processing conditions (starting from dispersed nano droplets) to a high degree and are made of rather floppy polymers (due to the Gaussian chain model involved). Therefore, more work should be carried out to determine the leading factors. Very recently it was shown that polymersomes may fold in giant high-genus vesicles (27), which also superficially resemble the bicontinuous structures we have found. But in the perforated polymersome case the holes are very large, much larger than single cou domain-sizes we observe in our gels. In solutions of crew-cut amphiphilic polymers Eisenberg and co-workers have determined a entire wonderland of structures (3$). Some exhibit onion phases, others are bicontinuous, and the raspberry droplets with the inverted micelles (figure 3e) resemble so-called Large Compound Vesicles. A l l agreements at this point are qualitative. A more detailed comparison, using realistic experimental parameters, is in progress.

Conclusions In a recent review, it was remarked that field-theoretic computational methods are a powerful alternative to particle simulations in soft matter (23). The results shown in our examples section emphasize this remark and offer a glimpse of the wide range of research areas where this type of methods is valuable. We showed that our molecular field method is: 1) capable of capturing the diffusive dynam ics of phase separation in experimental systems and provide a detailed microscopic picture of the underlying phenomena, 2) able to reproduce the static experimental phase behavior of thin films and allow to distinguish determining factors, 3) suit able for the prediction of new phenomena, based upon the proven reliability of the method. Future developments include an extension to hydrodynamic contributions: this will open a new era in soft condensed matter research.

Acknowledgment The authors acknowledge NWO-DFG project D N 72-216. Part Β of the exam ples was carried out in close collaboration with Armin Knoll (experiments), A n driana Horvat and Robert Magerle (simulations) at Bayreuth University: they are gratefully acknowledged. We acknowledge Katya Lyakhova (Leiden University) for providing figure 2 of this article. The supercomputer resources were provided by N C F (Nationale Computer Faciliteiten) at the national computer facilities at SARA, Amsterdam for the electric field alignment and the High-Performance Computing Centre, University of Groningen, for the smart particles. We thank Adi Eisenberg (McGill University), Alexander Bôker and Georg Krausch (Bayreuth University) for stimulating discussions.


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