Mapping and Reverse-Mapping of the Morphologies for a Molecular

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Mapping and Reverse-Mapping of the Morphologies for a Molecular Understanding of the Self-Assembly of Fluorinated Block Copolymers ¨ zen* Gokhan Kacar, Canan Atilgan, and Alimet Sema O Materials Science and Engineering Program, Faculty of Engineering and Natural Sciences, Sabanci UniVersity, Orhanli 34956, Tuzla, I˙stanbul, Turkey ReceiVed: August 28, 2009; ReVised Manuscript ReceiVed: October 20, 2009

A multiscale computational approach employing quantum mechanical, atomistic, and coarse-grained simulations has been adapted to reveal the self-assembly patterns of a styrene-co-fluorinated acrylate oligomer. Mesoscopic morphologies were determined using the dissipative particle dynamics method and were found to change from spherical micelles to hexagonal cylinders and lamella with increasing oligomer concentration. Quantum mechanics calculations were performed at the MP2/6-31(d) level of theory on the subsegments constituting the co-oligomer as well as on the shortest oligomer chain to determine the intermolecular interactions leading to the observed morphologies and self-assembly behavior. Atoms-in-molecules theory was employed to understand the nature of the noncovalent interactions between the styrene rings and between the fluorinated segments. A proportionality relation between the solubility parameters determined by the atomistic simulations and density functional theory-based reactivity descriptors such as global and local hardness is revealed. 1. Introduction A major challenge in current materials science is how to make structures of the size, precision, and complexity of biological construction without using biological catalysts or the coded information in the genes. Therefore, a thorough understanding of the self-assembly process, which is the fundamental mechanism of organization in biological systems, is of great importance.1-5 This mechanism involves organization of different precursor molecules in such a way to form heterogeneous and hierarchical structures in scales varying from nano- to micrometer levels. Generally, the components or the subassemblies constituting the system are held together by a web of noncovalent intermolecular interactions. Competitive and reversible nature of these interactions might provide the system a stimuli-responsive functionality such that its dynamical and mechanical properties can be strongly modified by relatively weak external fields.2,3,5 The same interactions are responsible for differentiating the ordered self-assembly systems from irreversible aggregation of molecules such as disordered glasses.3 This nature-inspired strategy for new materials requires not only the design and synthesis of proper molecular components, but also an absolute control over the molecular interactions among them. Block copolymers consist of two or more polymeric repeat units with different chemical composition joined covalently.4,6-8 If there is high degree of chemical incompatibility between the blocks, the subassemblies tend to phase separate due to the longrange repulsive forces. However, the covalent bond between them frustrates their tendency to phase separate macroscopically. Thereof, counterplay of repulsive and attractive forces in block copolymers leads to ordered structures of complex mesoscopic morphologies by self-assembly. Chemical constitution, length, and sequence of the blocks may be selected to achieve the desired functionality.8-12 Partially fluorinated polymers constitute an important group of polymer blocks.13,14 They are highly hydrophobic. When used * Corresponding author. E-mail: [email protected].

in membranes, they provide a high selectivity in gas-separation processes. Because of their unique properties such as selfassembly, forming low-energy surfaces, bioinertness, low absorbance to UV light, and solubility in supercritical carbon dioxide, they are very useful in advanced technology applications. In the future, they may replace toxic antifouling coatings in marine applications, they may be employed in biomedical coatings, or they may enable the replacement of hazardous solvents in microlithography with supercritical carbon dioxide.15,16 The structural hierarchy of the self-assembly systems, and, among them, complex fluids such as block copolymers, in different length and times scales demands inevitably a multiscale computational approach.17-26 The dynamic processes and structural changes occurring on those different length and time scales are strongly correlated, and, in principle, multiscale modeling covers a wide methodological spectrum from quantum mechanical (QM) and electronic structure calculations to the finite element modeling of the macroscopic world. In practice, the intermediate ranges such as the fine-grain atomistic molecular dynamics (MD) simulations on the nano scale17,24,27 and the coarse-grained (CG) methods reducing the groups of atoms into interacting beads on the meso scale17,23,28-32 are utilized more frequently. Most of these approaches require the application of certain assumptions related to the physical world and certain degrees of parametrization based on fitting to empirical data. On the other hand, many system properties may be determined by employing quantum mechanics calculations without any need for such experimental input. The limitations are the properties that depend on complexities at larger length scales.33 Therefore, it is self-evident that the degree of efficiency in bridging the QM calculations, atomistic simulations, coarse-grained, and finite element methods will shape the future of computational materials science.21 One of the primary goals of this study is to extend the time scale for the investigation of the polymer dynamics by mapping and reverse-mapping of the oligomer morphologies. It is also aimed to introduce QM calculations for a molecular understanding of the self-assembly process and to investigate the relation

10.1021/jp908324d  2010 American Chemical Society Published on Web 11/25/2009

Self-Assembly of Fluorinated Block Copolymers

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Figure 1. Organization of the multiscale study employed in this study. Starting with MD simulations to determine solubility parameters (upper left corner), one can feed the obtained data into coarse-grained simulations to determine the microphase separated morphologies. By reverse-mapping the atomistic detail back onto these morphologies followed by MD simulations, one can focus on the detailed intra- and intermolecular interactions. These interactions may further be interrogated via calculations on the QM scale. Such a multiscale approach not only provides a deeper understanding of self-organization on different scale, but also may lead to improved parameterization of the systems for another cycle of this process.

between the noncovalent interactions that govern disorder-order phase transitions and the mesoscopic morphologies of the block copolymers. Hence, it is intended to extend the methodology in our previous study34 to a new level of theory by bridging QM calculations, atomistic simulations, and coarse-grain methods. It is important to derive general chemical rules that allow predicting the properties of the self-assembly system using information from noninteracting subassemblies, as in a previous study on the hydrogen bonds.35 Chemical reactivity descriptors such as electronegativity,36 electronic chemical potential,37 hardness,38,39 and softness can be the technique of choice to achieve this task. Principles derived within the framework of conceptual density functional theory (DFT),40,41 Sanderson’s electronegativity equalization principle,42 Pearson’s hard and soft acids and bases principle,38,39 and maximum hardness principle,38,39 have served for a better understanding of the nature of chemical reactions as well as to predict the intermolecular and intramolecular reactivity trends. Conceptual DFT principles and descriptors might be useful as a tool for a fine-tuning of the MD and CG parameters in systems where noncovalent interactions are not described properly. This Article adapts the methodological order of mapping

reverse-

molecular

mapping

understanding

MD 98 CG 98 MD 98 QM and is organized as follows: Theoretical background related to the coarse-gained method dissipative particle dynamics (DPD) and DFT-based reactivity descriptors are summarized in the second section. Details of the atomistic (MD), coarse-grained (DPD), and quantum mechanics (QM) methodologies are described in the third section. The fourth section discusses the parametrization of the interactions for the coarse-grained DPD methodology. It mentions possible use of the DFT-based reactivity descriptors hand-in-hand with the MD derived solubility parameters for a better description of the intermolecular interactions. Mesoscopic morphologies of the fluorinated cooligomers obtained by the DPD method are presented in the fifth section. Next, a reverse-mapping scheme to implement the atomistic details back into the coarse-grained system is de-

scribed, and results of the MD simulations on the reversemapped systems are discussed. The seventh section aims to explain the self-assembly behavior of the fluorinated cooligomers from the previous sections at the molecular level by the help of the atoms-in-molecules (AIM) theory within the realm of QM. The organization of the present multiscale study is shown in Figure 1. 2. Theoretical Background DPD17,28,29,43-45 simulations are performed on a collection of particles, interacting with Newton’s equations of motion:

db ri dV bi )b V i, ) bf i dt dt

(1)

where b r i, b Vi, and bfi are the position vector, velocity, and force acting on the particle i, respectively. All particle masses are assumed to be equal and set to unity. With a mass of 1, the force acting on a particle in reduced units is equal to its acceleration. The force bfi is a sum of three pairwise additive components:44

bf i )

∑ (FbijC + bFijD + bFijR)

(2)

j*i

where the summation is over all other particles j that are within a certain cutoff radius b rc of particle i. This cutoff radius is used as the unit of length in the subsequent treatment, rc ) 1. The first of these component forces is the conservative force, b FCij . It is a soft repulsion acting along the line connecting the centers of the particles i and j, represented by the equation

b F ijC )

{

aij(1 - rij)rˆij (rij < 1) (rij g 1) 0

(3)

where aij is a maximum repulsion between particles i and j, b rij )b ri - b rj, rij ) |r bij|, and rˆij ) b rij/|r bij|. The second is the dissipative force, b FDij , which is proportional to the relative velocities of the particles i and j with respect to each other. It acts so as to reduce their relative momentum. The third is a random force, b FRij , which maintains the system temperature. The dissipative and random forces also act along the line of centers and conserve linear and angular momentum.

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In the DPD method, the particles do not correspond to individual atoms or molecules, but to beads, which represent coarse-grained groups of atoms. Atomic-level interactions are replaced by bead-bead interactions. This system is extended to polymers by the introduction of springs between consecutive beads along the chain. The dissipative and random forces act in unison, and their combined effect is a thermostat. Therefore, it is the conservative soft repulsive force that embodies the essential chemistry of the system. The parameters aij are referred to as bead-bead repulsion parameters or simply as DPD interaction parameters, and they depend on the underlying atomistic interactions. The DPD method can be used for the simulation of liquid-liquid and liquid-solid interfaces. In this way, it is similar to the Flory-Huggins mean field theory of polymers6,7,46 and can be viewed as a continuous version of the lattice model introduced therein. The mean field theory for the polymer solution explains the miscibility of the polymer with a given solvent by comparing the free energy of the polymer-solvent system before and after mixing. A similar theory describes thermodynamics of polymer blends and, with some modifications, diblock copolymers and their blends with homopolymers. The energy of mixing is related to the dimensionless FloryHuggins interaction parameter, χ, which is the change in energy in units of kBT, when a segment of A (a polymer) is taken from an environment of pure A and swapped with a segment B in an environment of pure B (a solvent or another polymer):

1 z εAB - (εAA + εBB) 2 χ) kBT

[

]

(4)

In eq 4, z is the lattice coordinate, and εAA, εBB, and εAB represent the interaction energies between neighboring polymer-polymer, solvent-solvent, and polymer-solvent molecules, respectively. The soft sphere interactions of DPD can be mapped onto Flory-Huggins theory through the χ parameter.28,29 If the system has A and B components or beads interacting with each other FCii , and if it is chosen that aAA ) aBB for the conservative force b then, according to Groot and Waren,28 the mapping relation is

χ)

2R(aAB - aAA)(FA + FB) kBT

(5)

where R is a parameter related to the pair-correlation function gj(r), which is expressed as a function of the reduced coordinate jr ) b/r r C, aAB is the repulsion parameter between A and B beads in DPD simulations, and FA + FB ) F is the number density (number of beads per unit volume) of the system. The enthalpic contribution to the χ parameter can be calculated from the solubility parameters29 by the equation

χH )

Vbead (δ - δB)2 kBT A

(6)

where Vbead is the average molar volume of the beads, and δA and δB are the solubility parameters of beads A and B, respectively. We take χ as a purely enthalpic term. In reality, it may also include entropic contributions. For example, according to the equation of state theory, there is also a free-volume term in addition to this enthalpic interaction term.47 Because the former is strongly molecular weight dependent, its contribution is larger for polymer-solvent systems than for polymer-polymer blends. However, the systems of concern in the present work are low molecular weight oligomers, and it has been assumed

that the contribution of the free-volume term is not as large as that of the polymer-solvent systems. The values of solubility parameters depend on the chemical nature of the species in question and can be obtained by MD simulations within the framework of Hildebrand’s definition:48

δ)

( ) ∆Eν Vm

1/2

) (CED)1/2

(7)

where ∆Eν and CED are the molar energy of vaporization and cohesive energy density, respectively. To understand the self-assembly behavior of the fragments and for a better parametrization of DPD simulations, DFT-based reactivity descriptors suitable for hard-hard interactions have been evaluated. Among these, hardness is a global property describing the resistance to changes in electronic charge:38-41

η)

∂µ ∂N

( )

V(b) r

)

(

∂2E[N, V(b)] r ∂N2

)

(8) V(b) r

In eq 8, µ is the electronic chemical potential identified as b)] is the the negative of the electronegativity,36,37 and E[N,V(r energy of the system as a function of N, number of electrons, and V(r b), the external potential (i.e., due to the nuclei). In the finite difference approximation, this is equivalent to

η≈I-A

(9)

where I and A are the vertical ionization energy and electron affinity, respectively. For closed-shell molecules, it can be further approximated as the HOMO-LUMO energy gap. Local hardness, a local counterpart of the global hardness, can be defined as

η(b) r )

( δF(δµb)r )

V(b) r

(10)

or in other words as the response of chemical potential µ with respect to a change in the electron density, F(r b). Within the Thomas-Fermi-Dirac (TFD) approach to the DFT and taking into account the exponential falloff of the density in the outer regions of the system, local hardness can be approximated as

ηDTFD(b) r ≈-

1 r V (b) 2N el

(11)

where Vel(r b) is the electronic part of the molecular electrostatic potential (MEP). The use of the electronic part of the MEP as an approximation to the local hardness has been documented in the literature.49,50 3. Computational Methodology A. Atomistic Simulations for DPD Parametrization. Atomistic simulations were performed to calculate the solubility parameters and cohesive energy densities (eq 7). To optimize the geometries of the beads, successive applications of steepest descents, conjugate gradients, and Newton minimization methods for a total of 20 000 steps was utilized as implemented in Materials Studio Program.51 MD simulations were performed using the Amorphous Cell module of the same program. Simulation boxes containing 10 beads of the same type with a density of 1.0 were constructed. Following the 1 ps equilibration step at 298 K, 10 ps MD simulations were performed on these boxes. We note that such short simulations in small systems are sufficient to determine the interaction parameters between bead types. Periodic boundary conditions with a cutoff radius of 8.5 Å for all nonbonded interactions were employed in the canonical ensemble (NVT). Initial velocities have been assigned from a Maxwell-Boltzmann distribution in such a way that


Figure 2. Molecular structure of the styrene-co-fluorinated oligomer system and partitioning of the beads for coarse-grained simulations.

the total momentum in all directions sums up to zero. COMPASS52 force field was used in geometry optimizations and MD simulations. Molar volumes were calculated using the ACDLabs/ChemSketch 5.0 software package.53 B. DPD Simulations. During the DPD simulations, oligomer chain architecture was constructed as A10B1D7 (Figure 2). DPD cubic boxes of size 10 × 10 × 10 rc3 are constructed with a density of F ) 3 DPD units where rc is the cutoff radius. The spring constant was chosen as 4.0 for the oligomer chain beads.28 Simulation temperature and bead masses were set to unity for simplicity. The total number of beads, including oligomer chains and THF molecules, was set to 3000. Different concentrations of the oligomer were equilibrated for 20 000 DPD steps followed by 150 000 steps of data collection. At F ) 3, aii and aij parameters were calculated according to the relationship established by Groot and Madden29 where aii ) 25kBT and aij ≈ aii + 3.27χij. Simulations are carried out at a series of concentrations spanning 10-100% of oligomer in the system.

J. Phys. Chem. C, Vol. 114, No. 1, 2010 373 C. QM Calculations. The Gaussian 0354 program was employed for the calculation of geometries and energies. Optimizations were performed at the MP2/6-31G(d) level of theory. Ground-state structures were confirmed by frequency analyses. Solvent optimizations were performed using polarizable continuum model (PCM)55-57 within the self-consistent reaction field theory. Wave function files were generated with the Gaussian 03 program at the level MP2/6-31G(d) prior to the analysis for the electron density contours and topological critical points using the AIM 2000 implementation of Bader’s AIMPAC suite of programs.58,59 WebLab ViewerPro60 program was used for the graphical presentation of the geometries and critical points. D. Atomistic Simulations on Reverse-Mapped DPD Systems. Although mesoscopic simulation techniques are very fundamental in obtaining morphologies, they have a significant limitation that they do not contain atomistic detail. Hence, to study the related dynamics, it is of great importance to have atomistic detail implemented on beads that have been coarsegrained. There are several reverse-mapping strategies to the coarse-grained systems.30,61,62 In our algorithm (Figure 3), energy-minimized template structures with atomistic detail of the beads are used for the fitting procedure.63 The centers of mass of these templates were translated on the coordinates of the DPD beads (Figure 3a). Next, the templates are rotated around their centers of mass to satisfy the condition of minimizing the distance between the first atom of the following template and the last atom of the previous template (Figure 3b). Finally, the structure is minimized with the COMPASS force field to satisfy features of the molecules (Figure 3c). The minimization step targets optimizing the hardest degrees of freedom (i.e., the bond lengths followed by bond angles), which

Figure 3. Steps of the reverse-mapping of atomistic detailed single styrene-co-PFA oligomer chain structures whose templates are shown on the lower left. (a) Translation of the centers of mass of the templates on those of the corresponding beads. (b) Rotation of the templates so as to minimize the bond length deviation of the connections with the previous and the following repeat units. (c) Energy-minimized final structure.

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TABLE 1: Flory-Huggins Interaction Parameters, χij, Calculated through Eq 7 by Using the Solubility Parameters δ ((cal/cm3)1/2) and Molar Volume Information of the Beads Vm (cm3/mol), Presented Here Together with DPD Interaction Parameters, aij, Calculated through Flory-Huggins Interaction Parameters, χij A δ ) 7.99 Vm ) 115.3 A

0.00 (χij) 25.00 (aij)

B

B δ ) 9.33 Vm ) 109.6 0.29

D δ ) 3.92 Vm ) 166.4

bead type

global hardness, η (au)

local hardness, η(r) (au)

A

0.405

B

0.495

0.052 (edge) 0.056 (ring) 0.044 (CdO) 0.055 (C-O) 0.033 (mid-chain) 0.028 (end-chain) 0.054 (C-O)

4.10

0.21

D

0.690

7.24

0.00

F

0.639

25.95

0.00 (χij) 25.00 (aij)

38.42

48.69

0.00 (χij) 25.00 (aij)

25.67

25.00

36.27

D

F δ ) 9.31 Vm ) 79.7

TABLE 2: MP2 Calculated Chemical Reactivity Descriptors of the Beads As Compared to the Solubility Parameters

F

3.45 0.00 (χij) 25.00 (aij)

automatically acts on large changes in the soft degrees of freedom (i.e., torsions), thus enabling efficient search on many possible conformations of these “interesting” degrees of freedom. Thus, the last step of the procedure is similar in spirit to the local torsional deformations method used in the conformational search of molecules under restraint.64,65 In addition, because this algorithm is based on minimizing distances between molecules, it is possible to study different types of molecules having different number of atoms with little additional computational effort. MD simulations have been performed on the reverse-mapped systems at 330 K for 1.5 ns after they were soaked in a box of THF molecules. Periodic boundary conditions with a cutoff radius of 9.5 Å for all nonbonded interactions have been employed in the canonical ensemble (NVT). Initial velocities have been assigned from a Maxwell-Boltzmann distribution in such a way that the total momentum in all directions sums up to zero. COMPASS52 force field has been used. Reversemapped structures are visualized by the program WebLab Viewer Pro60 and Vega ZZ.66

Mulliken charges

solubility parameters (cal/cm3)1/2 7.99

-0.599 (O) -0.657 (O) -0.388 (F) -0.348 (F) -0.625 (O)

9.33 3.92 9.31

4. Parameterization of the Interactions for the Coarse-Grained DPD Methodology Figure 2 shows the molecular structure and DPD bead partitioning of the fluorinated cooligomer systems of interest in this study. In addition to the beads A, B, and D presented therein, the solvent THF molecule constitutes the bead F without any segmentation. The extent of the mutual miscibility or compatibility within different segments (beads) of the copolymer, as well as within the solvent molecules and any segment of the copolymer, depends on the solubility parameters δ of the Hildebrand type. Therefore, atomistic simulations were performed to calculate the solubility parameters from eq 7. These solubility parameters were fed into eq 6 to find the FloryHuggins interaction parameters χ between the segments or subassemblies A, B, D, and F (Table 1). A positive χ denotes that contacts between different bead types are less favored as compared to those between beads of the same type. A negative χ value means that contacts between the different segments of the polymer or segment-solvent contacts are favored, promoting miscibility of the two blocks or solvation of the polymer. For the system under study, the results presented in Table 1 directly lead to the conclusion that fluorine-containing segments show a tendency for microphase separation in the presence of acrylate-

Figure 4. Mesoscopic morphologies of the fluorinated co-oligomers at different concentrations obtained by the DPD method.



Figure 5. Reversed-mapped systems composed of 27 chains for (a) a cylindrical morphology with 80% oligomer concentration in THF, (b) a spherical micelle morphology with 10% oligomer concentration in THF, and (c) intrinsic lamellar morphology without solvent. Fluorine atoms are colored red.

and styrene-containing segments as well as the THF molecules, giving the co-oligomer an “amphiphilic” character, which is further related to self-assembly. Optimized geometries of the beads A, B, D, and F were employed for finding their global and local reactivity descriptors in search for a possible relation to solubility parameters (Table 2). In that manner, it is intended to bridge the conceptual DFT principles such as “hard prefers hard” (Pearson’s hard and soft acids and bases principle, HSAB,38,39 and Sanderson’s electronegativity equalization principle)42 and the solution theory principles such as “like dissolves in like”. According to Table 2, beads A (styrene) and B (methacrylate), whose solubility

parameters are close to each other, are softer than beads D (fluorine-containing segment) and F (THF). Bead D is not only the hardest species, as expected due to the presence of fluorine atoms, but also the one with the lowest solubility parameter. On the other hand, while bead F’s solubility parameter is in the range of beads A and B, it is almost as hard as bead D. One possible explanation might be that beads D and F attain these hardness values for different reasons. Bead F is a stable fivemembered ring and the smallest molecule as compared to the other beads. These observations hint at the possibility of an inverse relationship between the hardness values and solubility parameters with additional contributions from local interactions.

376


Figure 6. Common noncovalent interaction motifs observed in the reversemapped systems after minimization and MD simulations of 1.5 ns.

Comparison of local hardness values as generally related to the electrostatic interactions and solubility parameters might give an idea about the possible interaction sites on different beads. For miscibility trends, the idea of local hardness matching works together with the idea of minimum solubility difference between the beads (Table 2). Favorable interactions between beads dominate the smaller is the difference between solubility parameters/ local hardness values. Accordingly, bead D, having much different local hardness and solubility parameter values, is immiscible with beads A, B, and F. This is also confirmed by the microphase separation within the morphologies. These preliminary results on

Kacar et al. the use of DFT descriptors in describing solubility trends seem to be very promising. Global and local hardness values might be considered as a second-order parameter on determination of the miscibility between different polymer segments where the firstorder parameter is the solubility parameters. One unexpected and interesting result from the local hardness calculations is that the fluorine atom in bead D appears to be relatively soft (Table 2). On the other hand, local softness and hardness do not necessarily indicate the softest and hardest regions of a molecule. Instead, they represent pointwise measures of the “local abundance” of the corresponding global quantity.67 According to this, fluorine atom is not a soft atom. It only contributes less to the global hardness of bead D. The next step is the supply of the information obtained through atomistic simulations to coarse-grained simulations of DPD according to the mapping relation given in eq 5. DPD interaction parameters a are tabulated in Table 1. To have the compressibility of water at room temperature, it was found that the diagonal repulsion terms, aii, have to be chosen according to the equation aiiF ) 75kBT.28 For F ) 3, this relation gives aii ) 25kBT. The off-diagonal DPD interaction terms have been calculated using χ parameters through the relation aij ≈ aii + 3.27χij at a density F ) 328 and are presented in Table 1. 5. Mesoscopic Morphologies of the Fluorinated Co-oligomers Using the interaction parameters data in Table 1, DPD simulations were performed to find the mesoscopic morphologies of the fluorinated co-oligomers in THF at different concentrations as shown in Figure 4. The formation of spherical

Figure 7. Sandwich and parallel types of π-stacking (interaction 1 of Figure 6) encountered in the reverse-mapped systems of (a) micellar, (b) cylinder, and (c) lamellar morphologies, during MD simulations. (Molecules displayed in wireframe correspond to the bigger systems where the information is extracted.)


J. Phys. Chem. C, Vol. 114, No. 1, 2010 377 In the literature, there are many experimental studies on the block copolymers of styrene with perfluorinated acrylates of different chain lengths.71 The spherical and cylindrical micellar morphologies of these polymers in THF and in other solvents support the microphase separated morphologies obtained in the present study and confirm the selectivity of THF for one block. 6. Reverse-Mapping of the Atomistic Details

Figure 8. Intrachain F · · · F short-contacts (interaction 3 of Figure 6) observed in reverse-mapped systems during simulation.

micelle-like structures is observed in dilute solutions such as 10-20%. As the concentration of the oligomers increases, other morphologies appear. Spherical micelles formed in the dilute solution with a cubic symmetry become egg-shaped micelles with increasing concentration and later turn into cylinders at a concentration of 50%. At 90% concentration, the intrinsic copolymer morphology, which is lamella, appears. The information about these morphologies might be essential for understanding the origins and extent of “nanoscale roughness” for the design of the superhydrophobic surfaces of higher quality.68-70

The reverse-mapping algorithm used in this study is based on fitting atomistic detail into structures obtained from mesoscale DPD simulations (see Figure 3 and section 3D).63 The algorithm was applied to micellar, cylindrical, and lamellae morphologies of styrene-co-PFA, which correspond to 10%, 80%, and 100% of oligomer concentrations in THF, respectively. Figure 5 shows some reverse-mapped systems, each composed of 27 chains. Solvent molecules are excluded for the sake of simplicity in the representation. The amphiphilic character of the block cooligomer reveals itself in these structures. For example, in Figure 5b, the fluorinated part of the chains, which are insoluble in the solvent THF, form the core of the micelle where styrene groups are directed toward the solvent, forming the corona. In Figure 5c, a layer of fluorinated chains is sandwiched between layers of styrene-containing chains, resulting in a lamellar morphology. For a better understanding of the dynamics of these systems, MD simulations of 1.5 ns length were performed on cooligomers obtained from the reverse-mapped structures. Each system was composed of subsets of 10 oligomer chains and soaked into a solvent box made of THF molecules, except for the lamellar morphology, which does not contain any solvent molecules. To identify the interactions that are predominant in each morphology while keeping the system size manageable, the 10 chains are selected to maintain the relevant curvature of the sphere/cylinder/lamella along the surface. Each box is a cube of dimension 42 Å containing 2500 oligomer and 1300 solvent atoms. Noncovalent interactions that might be responsible for the self-assembly of these systems were examined in detail from the locally relaxed structures obtained during the simulations. Figure 6 represents the main interaction motifs on the structure of an oligomer chain stripped from a reverse-mapped system after the MD simulations. Among the motifs of interactions in Figure 6, π-π stacking between the styrene rings

Figure 9. Atoms-in-molecules theory bond (pink balls), ring (yellow balls), and cage (blue balls) critical points for distyrenes (a) A2 (stacking rings on the same chain), (b) A · · · A (stacking rings on neighbor chains), and (c) tristyrene A3, where both parallel and edge-shaped stacking interactions are observed.

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Figure 10. Atoms-in-molecules theory bond (pink balls) and ring (yellow balls) critical points for the fluorinated segments perfluorinated ethane dimer and bead D showing (a) intrachain F · · · F and (b) interchain interactions, respectively. (c) Laplacian plot of bead D showing C-F and F · · · F interactions. (d) HOMO of the fluorinated segment bead D optimized at the MP2/6-31(d) level showing charge localization.

(interactions 1 and 2), especially at the consecutive positions on the same chain, were observed for all of the morphologies (Figure 7). Inter-ring distances from atomistic simulations are within the range of 2.9-3.5 Å. The rings mostly adopt sandwich or parallel-displaced stacking configurations composed of two or three phenyl units,72 probably due to their entrapment on the backbone (interaction 1). On the other hand, there are some occasional T-shaped and edge-faced T-shaped configurations72 observed (interaction 2), mostly within the same chain and at nonconsecutive positions due to steric effects forbidding C-H · · · π interactions at the consecutive positions. The nature and strength of these interactions were examined at the QM level and are discussed in the next section. Fluorine-fluorine short-contacts, where the distance between the interacting fluorine atoms is shorter than the combination of their van der Waals distance of 1.47 Å,73 were observed at the fluorinated part of the acrylate group (interaction 3 in Figure 6). Kowalik et al. discussed the same type of interactions in the crystal structure of 6-perfluorohexylsulfonyl-2-naphthol.74 These interaction motifs can be held responsible for the bending and self-assembly of the long perfluoroalkyl chains. They have both intra- (Figure 8) and intermolecular variants. QM calculations were performed to reveal the nature of F · · · F nonbonding interactions and are discussed in the next section. In addition, there are also some rare C-H · · · F (interaction 4) and C-O · · · F (not shown in Figure 6) interactions observed again within the same oligomer chain in the reverse-mapped structures and modeled at the QM level. An unusual type of interaction was observed between fluorine atoms and carbons of the styrene ring. Those C-F/π interactions, described by short van der Waals contacts, are examined at the QM level in the next section.

One interesting result of this section is that there is no preference of one interaction type over the other among different morphologies. In other words, the type of noncovalent interactions is unbiased with respect to the amount of the solvent present. The interplay between the entropic effects coming from the change in the polymer concentration and the energetics with contributions of the noncovalent interactions might result in different morphologies for these systems.7,75,76 7. Molecular Understanding of the Self-Assembly of Fluorinated Block Copolymers To understand the molecular basis of the self-assembly in fluorinated co-oligomers, QM calculations were performed at the MP2/6-31(d) level. Atoms-in-molecules (AIM) theory was employed to reveal the nature of the noncovalent interactions such as π-π stacking and F · · · F short contacts that might be essential for the self-assembly of the subassemblies. According to this theory,77 an extremum in the electron density function, F(r), whether it is a maximum, a minimum, or a saddle point, is called a critical point. Any critical point, (r, s), can be defined by the number of nonzero eigenvalues or local intrinsic curvatures (rank, r) and the sum of the signs of these eigenvalues (signature, s). For example, a bond critical point (bcp) is a saddle point with three nonzero eigenvalues (rank ) 3), which correspond to maxima in two directions and a minimum in one direction (signature ) -1), and it is represented as (3, -1). Figure 9a and b shows the critical points of the electron density topologies for the dimer formed by the consecutive styrene side groups on the same oligomer chain and for the dimer formed by the styrene side groups on neighboring oligomer chains, respectively. Figure 10a and

0.2837 0.1726 0.2047

0.9222 0.0623 0.0275

-0.74021 -0.00860 -0.00313

-0.37946 0.00136 0.00108


Values in parentheses correspond to the optimizations in THF. a

0.36075 0.00996 0.00421 -0.07485 0.04531 0.00528 1.37 0.25694 2.81-2.85 0.00801 3.23-3.28 0.00487

1.4040 1.2434 0.8645

0.00147

H(r) ) G(r) + V(r) V(r)

-0.01257 0.0775 1.7929 1.1888 0.01404 0.05610 2.67-2.70 0.01181

0.0377(0.0367) 0.0216(0.0213) 0.0023 0.0229 0.9121(0.9185) 0.0759(0.0767) 0.8707

λ3 ε

0.4796(0.4812) 0.6395(0.6682) 0.9045 0.3337 0.2135(0.2126) 1.7986(1.4434) 0.2207 0.6516(0.6512) 0.5836(0.5837) 0.6347 0.6281 1.3693(1.3721) 1.1670(1.1900) 1.3522

G(r)/F(r) G(r)

0.01016(0.00992) 0.03140(0.03063) 0.00662(0.00646) 0.00622(0.00615) 0.01715(0.01697) 0.00363(0.00359) 0.00553 0.01748 0.00351 0.00613 0.01839 0.00385 0.26318 (0.26364) -0.12928(-0.12738) 0.360383(0.36174) 0.01335(0.01158) 0.06609(0.06084) 0.01558 (0.01378) 0.25895 -0.13596 0.35014 3.62(3.66) 3.72(3.66) 2.95 3.04 1.34-1.36 2.65-2.75 1.36

A2 parallel stacking A · · · A parallel stacking A3 T-shaped stacking A3 parallel stacking D C-F (mid-chain) D F · · · F (mid-chain) ABD C-F in oligomer (mid-chain) ABD F · · · F in oligomer (mid-chain) C-F on different chains F · · · F on different chains A · · · CF4 C · · · F

∇2F(r) F(r) R (Å)

TABLE 3: Bond Critical Point Data in Atomic Unitsa

-0.00539(-0.00526) 0.00123(0.00120) -0.00296(-0.00293) 0.00067(0.00066) -0.00264 0.00087 -0.00311 0.00074 -0.75309(-0.75532) -0.39271 (-0.39358) -0.01464(-0.01235) 0.00097(0.00143) -0.73428 -0.38414


c shows the topology and Laplacian plot, respectively, of the fluorine-containing bead D. Interatomic interactions can be classified as shared or closedshell interactions using AIM theory parameters. Accordingly, a shared (covalent) interaction is one where the Laplacian of electron density,∇2F(r), at the (3, -1) bond critical point is negative (electron concentration) with a F(r) value of the order 10-1 au (0.675 e Å-3).77 A closed-shell (noncovalent) interaction is one where ∇2F(r) is positive (electron depletion) with a F(r) value of the order 10-2 au (0.068 e Å-3). There are also intermediate interactions where there is a positive ∇2F(r) value with a reasonably high F(r). According to these definitions, ionic bonds, hydrogen bonds, bonds in van der Waals molecules, and noble gas clusters are closed-shell interactions, whereas covalent or polar bonds are shared interactions. Therefore, in search for a proper classification, geometric parameters, such as relevant interring or interatomic distances, and AIM parameters corresponding to the bcp observed between two atoms or rings, such as F(r), ∇2F(r), kinetic energy density G(r), the ratio G(r)/F(r), potential energy density V(r), and total energy density H(r) are tabulated in Table 3 for the styrene dimers A2 and A · · · A, styrene trimer A3, bead D, oligomer ABD, perfluorinated ethylene dimers, and A · · · CF4 dimer. In the case of the styrene dimers shown in Figure 9, the small electron density values, the positive Laplacian of electron density at the (3,-1) BCPs, and the dominance of the kinetic energy over the potential energy (hence a positive value for the total energy) are typical of closed-shell interactions.78 π-π stacking is observed between styrene rings both on the same chain and on neighboring chains, the interaction energy for the former being larger. The interaction energy for rings on the same chain is 0.6 kcal/mol as calculated by the energy difference between parallel type stacked and nonstacked dimers. Atomistic simulations on the reverse-mapped systems showed only rare cases of interchain stacking, while there are multiple intrachain stacking interactions in almost every chain (interaction I). These stacking interactions might be responsible for the packing of the bead A. In QM calculations, the existence of a reactive field due to the solvent THF did not alter the AIM parameters much. However, energetically, the A2 system is stabilized in the presence of solvent by 4.0 kcal/mol, as calculated by the difference of energies in vacuum and in solvent, in accord with the solubility parameters. Styrene trimer A3 molecule in Figure 9c represents the “distorted” edge-shaped stacking interaction motif of the reverse-mapped systems between rings 1 and 3 (interaction 2) together with a parallel-displaced stacking on the same chain between rings 1 and 2. Even though AIM parameters point to a weak interaction, the energy difference between stacked and nonstacked trimers is 4.9 kcal/mol, which is much stronger than a parallel type-only stacking of the dimer. On the other hand, edge-shaped and T-shaped stacking motif is not as common as the parallel or sandwich type of stacking in the reverse-mapped systems studied. This observation might be explained by the introduction of the steric effects. Because benzene rings are captured on the backbone, they have more chance to interact with their neighbors than being exposed to the other chains. On the other hand, interacting with the neighbors results only in parallel type of stacking interactions because of the steric hindrance. The electron density topology of bead D shows some interesting features (Figure 10a). First, the (3,-1) BCPs for C-F bonds have a high value of F(r) and negative∇2F(r). The BCPs are located very close to the carbon atoms, only 0.41 Å away, and therefore the associated bonded charge concentration is

380


Kacar et al.

Figure 11. (a) Optimized geometry of the ABD chain at the MP2/6-31(d) level, and (b) atoms-in-molecules theory bond (pink balls) and ring (yellow balls) critical points for the ABD chain.

located within the boundary of the fluorine atom due to the charge transfer from carbon to fluorine.79 This corresponds to shared or covalent interactions but also a polarized bond. The strong polar character is revealed by G(r)/F(r) >1 and a negative H(r).78 The Laplacian plot in Figure 10c also confirms the highly polar character of the bond between carbon and fluorine atoms.81 The HOMO sketch of bead D shown in Figure 10d can be considered as another evidence for the localization of electron density on the C-C bonds and fluorine atoms. Interchain F · · · F contacts observed in the reverse-mapped systems were modeled by a smaller system of two perfluorinated ethane molecules (Figure 10b). Here, the interaction energy is found to be 3.4 kcal/mol. On the other hand, this value was calculated as the energy difference between the dimer and the individual perfluorinated ethane molecules. So, it contains some contribution coming from the hydrogen-bonding interactions. F · · · H hydrogen-bond energy was calculated as 1.6 kcal/mol by AIM analysis from G(r) and ∇2F(r).80 The interaction energy for F · · · F contacts, therefore, is expected to be less than 2 kcal/ mol. The AIM analysis at the bond critical point gives low F(r) and positive ∇2F(r) values, indicating a closed-shell interaction within the same category of hydrogen bonds and van der Waals interactions. Kinetic energy density dominates over the potential energy density. Hence, H(r) > 0 and G(r)/F(r) > 1. There are also intrachain closed-shell F · · · F interactions between the fluorine atoms located on the alternating carbon atoms in bead D again with low F(r), positive ∇2F(r), positive H(r), and G(r)/ F(r) > 1 (Figure 10a). The Laplacian plot in Figure 10c represents the depletion of electron density between fluorine atoms. On the other hand, by comparison of the AIM parameters such as electron density and total energy density, intrachain interactions are stronger than interchain F · · · F close contacts. These nonbonded interactions were also observed in the literature for different fluorine compounds and held responsible fordirectingtheself-assemblyof,forexample,metalcomplexes.74,81-85 Matta et al., using AIM theory, characterized closed-shell F · · · F interactions in aromatic compounds, and they found results similar to those of the current study.81

Another rare but interesting type of interaction that is observed in MD simulations and modeled at the QM level is C-F/π contact. These weak interactions are described in the AIM analysis by a bond path linking fluorine atoms with carbon atoms of the styrene ring accompanied by a bond critical point. F(r) and ∇2F(r) values presented in Table 3 actually correspond to a halogen bond.86 For a better understanding of the details of the self-assembled fluorinated polymers, beads A, B, and D are linked to form the smallest oligomer of this study. Figure 11a represents the optimized geometry of the molecule ABD at the MP2/6-31(d) level. AIM values for the oligomer are tabulated in Table 3, and bond critical points are shown in Figure 11b. Because of extensive hydrogen-bond interactions, each of which is weak (1-2 kcal/mol) between the styrene, methacrylate, and fluorinated segments such as C-H · · · π and C-F · · · H,84 the molecular structure of ABD is not linear. C-F · · · H interactions also result in the weakening of the F · · · F interactions by comparison of the total energy density H(r). To see the compatibility of these interactions on a different scale, MD simulations were performed on the ABD single-chain with and without the solvent THF. The same bending pattern was observed during the simulations. Thus, the hydrogen-bonding interactions at the QM level are also reproducible at the atomistic MD level. On the other hand, the same type of bending was occasionally observed in the reverse-mapped systems. 8. Conclusion The aim of this study is to investigate the relation between the noncovalent interactions that govern disorder-order phase transitions and the mesoscopic morphologies of the block copolymers. A multiscale approach collecting structural and energetic information from the quantum mechanics, molecular dynamics, and coarse-grained DPD methods has been adapted. The co-oligomer, composed of styrene blocks and a fluorinated tail in THF, showed morphologies ranging from spherical micelles to cylinders with increasing concentration. The intrinsic

Self-Assembly of Fluorinated Block Copolymers copolymer morphology was lamellar. The information about these morphologies could be essential for understanding the “nanoscale roughness” for the design of the superhydrophobic surfaces of higher quality.70 A reverse-mapping algorithm was developed to introduce the atomistic details back into the coarsegrained system to form the molecular basis of the nanoscale roughness. Noncovalent interactions that might govern the oligomer morphologies were investigated using the atoms-in-molecules theory. The π-stacking interactions were found between the styrene rings both on the same chain and on the neighbor chains. Furthermore, F · · · F interactions observed on the fluorinated segment might result in the clustering of the fluorinated parts and further contribute to stabilize the incompatibility of the two segments. We find that these interactions are morphology independent. The same type of interactions was observed in all three reverse-mapped systems investigated. The entropic effects due to the presence of the solvent molecules are effective in determination of the polymer morphology.7,75,76 It is also intended to bridge the conceptual DFT principles such as “hard prefers hard” and the solution theory principles such as “like dissolves in like”. An inverse relationship between the hardness values and solubility parameters was found in that context with local factors contributing to this observation. In fact, local hardness matching and minimum solubility difference between the beads gave parallel results for predicting the solubility trends. Reactivity descriptors were found to be promising for the modification of the solubility parameters so as to better describe the noncovalent interactions better. A large set of structures must be systematically studied with both methods to determine the general features that contribute to the relationship between hardness and solubility. Furthermore, for the systems with hydrogen bonds in which solubility parameters of Hildebrand type are not adequate to describe the whole miscibility picture, global and local hardness calculations can be introduced as a correction or modification to the DPD interaction parameters. This will increase the number of systems that can be studied by DPD method without any need for the experimental data for parametrization. How this modification can be performed requires further work with different co-oligomer systems and is beyond the scope of this study, yet within the realm of our group’s current interests. Acknowledgment. This work was supported by the Turkish Scientific and Technical Research Council project no. TBAG106T522. We gratefully acknowledge fruitful discussions with A. R. Atilgan, Y. Z. Menceloglu, and Eren Simsek. References and Notes (1) Whitesides, G.; Mathias, J.; Seto, C. Science 1991, 254, 1312. (2) Muthukumar, M.; Ober, C. K.; Thomas, E. L. Science 1997, 277, 1225. (3) Whitesides, G. M.; Boncheva, M. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 4769. (4) Fo¨rster, S.; Plantenberg, T. Angew. Chem., Int. Ed. 2002, 41, 688. (5) Bucknall, D. G.; Anderson, H. L. Science 2003, 302, 1904. (6) Jones, R. A. L.; Richards, R. W. Polymers at Surfaces and Interfaces; Cambridge University Press: Cambridge, 1999. (7) Jones, R. A. L. Soft Condensed Matter; Oxford University Press: Oxford, 2006. (8) Klok, H. A.; Lecommandoux, S. AdV. Mater. 2001, 13, 1217. (9) Discher, D. E.; Eisenberg, A. Science 2002, 297, 967. (10) Hamley, I. W. Soft Matter 2005, 1, 36. (11) Mecke, A.; Dittrich, C.; Meier, W. Soft Matter 2006, 2, 751. (12) Acatay, K.; Simsek, E.; Ow-Yang, C.; Menceloglu, Y. Z. Angew. Chem., Int. Ed. 2004, 43, 5210. (13) Fluoropolymers 1: Synthesis; Antonietti, M., Oestreich, S., Eds.; Plenum Press: New York, 1999.

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JP908324D

Mapping and Reverse-Mapping of the Morphologies for a Molecular

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