Electric-field induced alignment of nanoparticle-coated channels in thin-film polymer membranes Paul C. Millett Citation: The Journal of Chemical Physics 140, 144903 (2014); doi: 10.1063/1.4870471 View online: http://dx.doi.org/10.1063/1.4870471 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in How do evaporating thin films evolve? Unravelling phase-separation mechanisms during solvent-based fabrication of polymer blends Appl. Phys. Lett. 105, 153104 (2014); 10.1063/1.4898136 Mean field theory for a reversibly crosslinked polymer network J. Chem. Phys. 137, 024906 (2012); 10.1063/1.4732149 Colloidal gold nanosphere dispersions in smectic liquid crystals and thin nanoparticle-decorated smectic films J. Appl. Phys. 107, 063511 (2010); 10.1063/1.3330678 Electric field versus surface alignment in confined films of a diblock copolymer melt J. Chem. Phys. 125, 164716 (2006); 10.1063/1.2360947 Asymmetric block copolymers confined in a thin film J. Chem. Phys. 112, 2452 (2000); 10.1063/1.480811
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THE JOURNAL OF CHEMICAL PHYSICS 140, 144903 (2014)
Electric-field induced alignment of nanoparticle-coated channels in thin-film polymer membranes Paul C. Milletta) Department of Mechanical Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA
(Received 3 January 2014; accepted 19 March 2014; published online 11 April 2014) Microscopic phase separation in immiscible polymer melts can be significantly altered by the presence of dispersed nanoparticles and externally applied electric fields. Inducing order or directionality to the resulting microstructure can lead to novel materials with efficient synthesis. Here, the coupled morphology of an immiscible binary polymer blend with dispersed nanoparticles in a thin-film geometry is investigated under the influence of an applied electric field using a unique mesoscale computational approach. For asymmetric binary blends (e.g., 70–30), the resulting microstructure consists of columnar channels of the B-phase perpendicular to the major plane of the film (aligned with the electric field), with the particles segregated along the channel interfaces. The simulations reveal the variability of the average channel diameter and the interfacial arrangement of the particles. The high density of exposed particles makes these structures viable candidates for catalytically active porous membranes or macromolecular manipulation devices. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4870471] I. INTRODUCTION
The development of new material systems with novel nanoscale or mesoscale architectures, which directly result in enhanced functionality, is a highly pursued topic in the broader field of materials science and engineering. Polymer nanocomposites, consisting of nanoparticles within a polymer matrix, are receiving growing attention due to their unique microstructural and property characteristics.1, 2 Nanoparticles dispersed in immiscible polymer blends or block copolymers that undergo phase separation hold even greater promise for the controlled design of a rich diversity of mesoscale structures.3–5 In such systems, the dispersed particles can be tailored to reside either in one of the bulk phases or along the phase interfaces.6, 7 For the latter, the interfacial arrangement of particles can dramatically lower the interfacial area and curvature thereby eliminating the driving force for coarsening, thus stabilizing a variety of interesting structures such as colloidally jammed bijels8, 9 and Pickering emulsions.10 These morphologies are particularly interesting in thin-film geometries, in which the film thickness can dictate if the phase separation is continuous or discrete, as shown by the work of Composto and colleagues11, 12 using 20 nm particles, grafted with polymer ligands, that segregate to the interfaces in immiscible homopolymer blends. However, our understanding of the mutual interactions between particles and multi-phase polymer melts during phase separation, and how to impart order or directionality to their morphology, is still quite limited. Over the last two decades, computational mesoscale methods have developed into particularly useful approaches to study the co-evolution of particles in phase separating fluids (including polymers). For example, dissipative particle dynamics have been used by Hore and Laradji13–16 and a) Electronic mail: [email protected]
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others17 to study various particle assemblies in binary phaseseparating mixtures. Hybrid models, on the other hand, represent a different approach, in which the fluid or polymer phase is represented by concentration or density fields on a discrete mesh, while the dispersed particles evolve in continuous space by discrete moves informed by their instantaneous force or energy, the earliest example from Kawakatsu et al.18 With regards to particles in phase-separating mixtures, hybrid models have utilized the Lattice Boltzmann method8 to represent low-viscous fluids, as well as Cahn-Hilliard (CH)18–22 and Self-Consistent Field23 theories to represent viscous polymer melts. A good review of these approaches, and their application to polymer nanocomposites, can be found in the recent literature.2, 24 All of these approaches have predicted the segregation of neutral particles to the phase interfaces leading to retarded coarsening and eventual arrest, given the particles are sufficiently larger than the interfacial thickness. Recent demonstrations of nanoparticle-induced reactions, including the conversion of CO2 gas to methane with Cu–Au nanoparticles,25 and H2 production from water splitting using Si nanoparticles,26 illustrate the promise for nanoparticle materials in energy and environmental applications. An important challenge to overcome involves the design and synthesis of suitable scaffolds, including membranes, which can support a large density of nanoparticles and simultaneously provide a large exposure area for interaction with a flowing gas or liquid. Self-assembled polymernanoparticle composites may potentially be well-suited for such applications. This article presents the results of a computational study that characterizes the evolution of a two-phase immiscible homopolymer melt containing dispersed spherical particles in a thin-film geometry with an applied electric field. The computational approach used is a mesoscale Brownian Dynamics (BD)/CH model.21, 22 The simulations evolve the system from
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an initial state consisting of a random distribution of particles within a homogeneous solution of two homopolymers (represented by concentration field variables), throughout the decomposition process to a state of stabilization. The phase separation, and therefore the final structure, is altered by the presence of an electric field,27 which effectively aligns the microstructural domains in the direction of the field.4, 28 The simulations presented here emphasize the relationships between the average channel sizes and the relative polymer and particle compositions, as well as the particle configurations that ultimately form on the polymer-polymer interfaces.
from values of one to zero (or vice-versa) within a finite width inside the interface. The free energy density function consists of bulk and interfacial terms (3) F = [f (φ1 , φ2 ) + κ(∇φ1 )2 + κ(∇φ2 )2 ] dV ,
II. MODELING APPROACH
where ψ is a variable stored on the grid that equals one inside of any particle and zero outside of all particles. Equation (4) has been used previously to model particles within two-phase fluids.29, 30 Outside of the particles, the function is a double well with two symmetric minima at (φ 1 ,φ 2 ) = (1,0),(0,1), see Fig. 1. The exact shape of the double well functional has been shown to not meaningfully alter the scaling function and exponents in late-stage phase separation,33, 34 thus Eq. (4) can be assumed to be an appropriate substitution for the FloryHuggins energy31
The BD/CH computational approach used here simultaneously evolves the suspended nanoparticles and the mesoscale phase separation occurring within the two-phase polymer melt. Here, we assume the particles are chemically neutral, i.e., they do not prefer one phase over the other and therefore will preferentially segregate to the polymerpolymer interfaces, similar to the experimental conditions prescribed in Refs. 11 and 12. The approach utilizes algorithms typically used in both particle-based and grid-based methods, with additional steps required to couple the particle and liquid interactions. The method is quite similar to previous studies,21, 22, 27 however some distinctions exist, and a detailed description is given below. The polymer melt phase separation is evolved in space and time by solving the CH equation on a discrete cubic grid using a standard finite differencing scheme.31 The CH equation, with an additional term to represent electric field-driven diffusion,28 is given by δF ∂φi = Mi ∇ 2 + k∇z2 φi , ∂t δφi
(1)
where φ i and Mi are the volume fraction and mobility of homopolymer species i, respectively, F is the free energy density of the multi-phase mixture, and k indicates the strength of the electric field applied in the z-direction defined as27, 28 k=
o υo 2Eo2 (A − B )2 , kB T (A + B )
(2)
where o is the vacuum permittivity, υ o is the volume of one monomer, Eo is the real strength of the electric field, and A and B are the dielectric constants of domains A and B. In the typical experimental system of PS-PMMA, a value of k = 0.1 corresponds to an electric field of 8 V/μm.32 A principal distinction between the current model and previous BD/CH simulations21, 22, 27 is the number of concentration fields used to represent the binary polymer blend. In the previous studies, a single phase variable is generally used that assumes a value of either 1 or −1 in the bulk phases. In the simulations here, there are two concentration variables evolved, φ 1 and φ 2 , which is motivated by the fact that using two variables simplifies the computation of the interfacial capillary forces on the particles, as will be shown later. Within any particular phase, the concentrations are constant (after the initial phase separation). The polymer-polymer and polymerparticle interfaces are diffuse, hence the variables transition
where f(φ 1 , φ 2 ) is the bulk free energy density, and the gradient terms contribute to the interfacial energy. The bulk free energy density is given by f (φ1 , φ2 ) = w 3φ14 − 4φ13 + 3φ24 − 4φ23 +6φ12 φ22 + 6ψ φ12 + φ22 , (4)
f φ2 φ1 ln φ1 + ln φ2 + χ φ1 φ2 , = kB T N1 N2
(5)
where N1 and N2 are the degrees of polymerization of polymers 1 and 2, respectively, and χ is the interaction parameter. Figure 1 illustrates how Eq. (4) compares with the FloryHuggins model for N1 = N2 = 20 and χ = 0.59. On the other hand, inside a particle, the function assumes a single well shape with a minima at φ 1 = φ 2 = 0, thereby eliminating polymer concentration within the solid particles. Central finite differencing is used to solve the spatial gradients of Eqs. (1) and (3), and a forward difference is used to solve the time gradient in Eq. (1).35 The CH approach to modeling polymer separation assumes that the molecular transport is purely governed by diffusion, and that hydrodynamic transport is negligible (although, successful efforts have been made to couple the CH equation with the Navier-Stokes equations, see, e.g., Ref. 21). This assumption is generally valid for high viscosity (low mobility) polymer separation, particularly when the domain size is on the order of tens to hundreds of nanometers and the separation is not in the very late stages.36 The trajectories of the suspended nanoparticles are updated throughout time with discrete time steps using standard particle-based algorithms used in, for example, Molecular Dynamics (MD). The polymer melt medium is assumed to have a high viscosity; therefore, the particle motions are dissipative and consequently the particle forces are related to velocities rather than accelerations, consistent with Brownian dynamics.37 Each particle is considered a point mass with an associated radius and mobility. The overall particle forces and velocities are obtained from pp
cap
FTi = Fi + Fi
+ Fstoc , i
vi = ηi FTi ,
(6) (7)
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FIG. 1. (a) Free energy density plot of the immiscible binary polymer melt as a function of the two volume fractions, φ 1 and φ 2 , illustrating the double-well functional with minima at φ 1 = 1, φ 2 = 0 and φ 1 = 0, φ 2 = 1. (b) Comparison of Eq. (4) (assuming φ 1 + φ 2 = 1) with the Flory-Huggins model with the stated assumed parameters. The red dashed line in (b) corresponds with that in (a). (c) The free energy density within a particle changes to a single well function with a minimum at φ 1 = φ 2 = 0.
where vi and ηi are the velocity and mobility of particle i. Neighbor lists are constructed for each particle, which are used to calculate short-range particle-particle steric repulsive forces defined by the following function: pp
Fi = A(rij − 2(rp + rhalo ))n ,
(8)
where rij is the distance between particles i and j, rp is the particle radius, and rhalo is an additional halo distance around the particles that is used to improve the numerical computations of particle interactions with the polymer interfaces.38 The pre-factor A and the exponent n dictate the magnitude of the short-range repulsion force. The stochastic forces on the particles induced by the polymer molecules leading to Brownian motion are defined as stoc stoc stoc , (9) Fstoc = B R1 Fi,x + R2 Fi,y + R3 Fi,z i where R1 , R2 , and R3 are random numbers between −0.5 and 0.5. The constants A and B in Eqs. (8) and (9) are force constants. For chemically neutral particles, interfacial segregation occurs to lower the overall energy of the system by reducing the polymer-polymer interfacial area, which has been observed even for polymer-grafted nanoparticles with diameters of 20 nm in homopolymer blends.11, 12 Capillary forces act to maintain interfacial particle attachment. In the present model, the particle-polymer interactions include the capillary forces that the polymer-polymer interfaces exert on the particles, and the dependency of the free energy density on the particles, described by the last term in Eq. (4) which effectively induces a pinning force on polymer-polymer interfaces where particles exist. To incorporate both of these elements in the model, the set of particle positions must be mapped onto the finite difference grid used to solve the CH equation. In order to do this appropriately, the particle radii rp must be larger than the grid spacing to adequately define the particle shape, size, and interaction with the polymers. As mentioned above, the variable ψ is used to represent the particles’ positions on the grid ⎧ rij < rp ⎨1, (10) ψ = exp(−(rij − rp )), rp < rij < rp + rint . ⎩ 0, rij > rp + rint
On the grid, the particle surfaces are therefore also diffuse with a width of rint . The capillary forces are calculated using the two polymer concentration variables and the ψ variable
cap dV , (11) Fi = D ψi φ1 φ2 rij k − rcm i where ψ i is the ψ field for particle i, rij k is the vector position is the vector position of the center-ofof grid point ijk, rcm i mass of particle i, and D is a force constant. Here, the brackets . . . signify that the vector within is a unit vector. The capillary force is therefore non-zero only at the triple junction lines where φ 1 > 0, φ 2 > 0, and ψ > 0. Equation (11) represents a more simplified expression for the capillary force computation for the specific case of a 90◦ contact angle, compared with the author’s previous work,30 which is valid for varying contact angles. We assume no explicit interaction between the electric field and the particles, i.e., electrostatic forces on particles are neglected. The values of all of the numerical parameters are given in Table I. The motivation for some of these parameter choices is not completely rigorous. For example, A and n govern the short-range particle-particle repulsions, and the chosen values represent a rather steep repulsive energy curve that spans TABLE I. BD/CH simulation parameters used in this study. Parameter
Value
x t Mi κ k w ηi A B D n rp rhalo rint
1.0 0.01 1.0 1.0 0.1 0.15 0.4 50.0 20.0 2.0 − 7.0 4.0 5.0 3.0
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roughly 1/8 the particle diameter. Also, the relative values of B and D, which scale the Brownian motion and capillary forces, respectively, are chosen such that the particles’ thermal energy is generally insufficient to detach a particle that is absorbed on a polymer-polymer interface. Small variations of these parameters is not expected to qualitatively change the simulation results, however, a complete parametric analysis has not been performed. The simulations are performed in reduced length and time units. Each time step is executed by the following sequential steps: (1) mapping the particle positions on the grid by calculating the ψ variable, (2) updating the polymer domains by solving the CH equation on the grid, (3) calculating the particle forces including those from the polymer-polymer interfaces, and (4) updating the particle positions. Steps (1) through (4) are repeated, with the occasional update to the particle neighbor lists. To ensure proper particle interactions with the polymer interfaces, the particle diameter must be larger than the diffuse interface width,39 which is ∼ 5 x. (Here, the effective particle diameter is ∼ 10 x, with rp = 4 x and rint = 3 x.) Following micro-phase separation, the polymer domain widths are typically 30–70 x, hence evolving a system of 2000 particles requires grid sizes of approximately 400 × 400 × 100. The simulations are therefore executed in parallel, in which worker processors update the grid computations (using domain decomposition), and a master processor updates the particle neighbor lists, interactions, and positions. The capillary forces exerted on a particle attached to a polymer-polymer interface depend on several factors including the interfacial energies, the particle contact angle, and the particle’s shape, size, and position. For a spherically shaped particle interacting with an otherwise flat horizontal interface, the capillary force perpendicular to the interface has the following closed-form, analytical solution:40 F cap = 2π rp σ12 sin φc sin (φc + θ ),
(12)
where the angles φ c and θ are defined visually in Fig. 2, and σ 12 is the interfacial energy of a fluid-fluid interface. To validate that the BD/CH model described above reproduces Eq. (12), several simulations were performed involving a spherical particle attached to an interface at different positions (i.e., different φ c angles), with a fixed contact angle of θ = 90◦ . For this analysis, the spherical particle had a chosen radius of 18 x. Figure 2 shows the calculated capillary forces versus φ c in the simulations compared with Eq. (11), illustrating a good quantitative agreement. The amplitude of the curve in Fig. 2 is dependent on rp and σ 12 . In the simulations, the capillary force is scaled by the force constant, D. A value of D = 2.0 is used, which considering that the maximum amplitude in Fig. 2 is 50.07, correlates to an interfacial energy of σ 12 = 0.88. The interfacial energy derived from Eq. (3) is 0.78 using the parameters given in Table I. III. RESULTS
Confined within a thin-film geometry, the binary polymer-nanoparticle composite evolution, under an electric field, depends strongly on the relative blend ratio of the A
FIG. 2. The capillary force exerted on a single spherical particle by a polymer-polymer interface intersecting the particle at different locations relative to the particle center-of-mass. The particle radius is 18 x and the assumed contact angle is θ = 90◦ . The calculated capillary forces, represented by the blue data points, correlate well with the analytical solution represented by the solid curve. The inset provides a visual schematic displaying the angles θ and φ c , the latter correlating to the x-axis in the plot.
and B polymer species, as shown in Fig. 3. Here, the simulations start from a uniform disordered state with φ 1 and φ 2 having small initial fluctuations in the range of ± 0.05 about the global average. The initial particle positions are random, with the condition that their hard spheres do not overlap. The start of the simulation corresponds to an instant quench below the critical temperature, thereby initiating phase separation. For symmetric blends (i.e., 50–50), the resulting geometry resembles a two-dimensional jammed bijel8 – a spinodally decomposed arrangement of polymer domains with interfacially segregated particles that assume a close-packed hexagonal arrangement (see Ref. 8). For the system shown in Fig. 3(a), the coarsening has terminated and the interconnected ligaments, which can have rather high curvature, are stable. For this simulation, the grid size is 400 × 400 × 100 with 2000 uniformly sized particles (particle volume fraction = 0.065), essentially all of which are attached to the interfaces. When the blend ratio is changed to 36–64, the system morphology changes from a spinodal pattern to a columnar arrangement aligned in the z-direction with the electric field. From this structure, a continuous membrane film with a high directional porosity can be obtained with the selective removal of the minority polymer phase.41, 42 Once again, the particles are largely segregated to the interfaces and assume a nearly close-packed configuration. In general, the interfacial particle density for the columnar phase was observed to be not quite as uniform as the spinodal phase, i.e., many columns achieved nearly close-packed densities while some columns were somewhat less than close-packed. This is likely attributable to the fact that the interfaces in the spinodal regime are continuous and therefore allow particle redistribution to accommodate high particle density and uniformity. The columns, on the other hand, are not connected with each other and therefore cannot readily exchange particles from columns with high particle density to those with low density. Hence, particles that do not locate on an interface
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FIG. 3. BD/CH simulations of membrane structures. (a)–(c) A 50-50 polymer blend, with particle volume fraction = 0.065, under an applied electric field after phase separation and particle stabilization, resulting in a columnar bijel structure with a nearly close-packed monolayer of particles along the interfaces ((c) is an xy-plane view). (d)–(f) A 36-64 blend is shown (same particle volume fraction), resulting in columnar channels again with interfacial particle segregation. (g) On the channel interfaces, the particles assume a hexagonal arrangement with a tilted helical angle (this channel is indicated by the arrow in (f)).
early in the phase separation may have to travel a relatively long distance to find a channel interface that can accommodate them. As seen in Fig. 3(f), there are a noticeable number of particles located in the bulk in the majority phase. All in all, however, the interfacial particle coverage is quite high in the simulations described here (usually up to 80% of the ideal close packing) and most columns have nearly close-packed particle configurations. A perpendicular view of one of the columnar channels in Fig. 3(f) (indicated by the arrow) is shown in Fig. 3(g). Interestingly, the particles assume a close-packed structure, however the hexagonal arrangement is tilted with a certain helical angle. This helical tilt was observed for basically all of the channels in the film. A more detailed analysis is given in Fig. 6 below. For comparison purposes, Fig. 4 shows the morphology of the same composition albeit without the influence of an electric field. Particle segregation to the interfaces still occurs as before, however interfaces are ori-
FIG. 4. Thin film morphologies for the same simulation conditions as in Fig. 3, however with no electric field. (a) and (b) A 50-50 blend with and without particles illustrated, and (c) and (d) a 36-64 blend with and without particles illustrated. Without an electric field, the domains are not aligned in the z-direction.
ented in all directions and there is no discernible alignment of the domains. All things considered, the degree to which vertical columnar channels form in the system will be a function of the electric field strength, the film thickness, and the particle volume fraction. A systematic study of these interrelationships will be left for future work. Here, the particle volume fraction and film thickness are large enough such that vertical channels are not ubiquitous without the applied field. Raman et al.43 recently modeled superparamagnetic nanoparticles in block copolymers and found that an applied magnetic field can orient the domains due to particle alignment. The interfacial particle segregation effectively stabilizes the channels from coarsening due to Ostwald ripening,44 which naturally occurs in phase-separating systems without particles (Fig. 5). In Fig. 5, and for the rest of the results, the electric field is turned back on. Analogous to the bijel structure,8 the particles eliminate a large portion of the interfacial area, i.e., the driving force for coarsening. Coarsening can only occur if particles detach from the interfaces – a phenomenon that is rarely observed in the later stages of the phase separation. Figure 5(h) shows how the fraction of total particles located on an interface, and the interfacial particle coverage (given as a fraction of the ideal close-packed configuration), evolve during time as the phase separation progresses. Throughout time, the fraction of total particles touching an interface decreases initially but gradually steadies to a value of 0.85. Conversely, the particle interfacial packing fraction increases from the beginning of the simulation to a rather steady-state value of 0.72. A potentially desirable attribute of such films may be a controllable channel diameter and channel areal density (the number of channels per area of film). A parametric analysis was performed for two key variables: the particle volume fraction and the blend ratio. Simulation cell sizes of 200 × 200 × 70 were utilized, in which these two parameters were systematically varied. For each composition, three independent simulations were executed (starting from different initial distributions), and the resulting structural
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FIG. 5. Channel diameter stabilization due to particles. Top-down view of the time evolution of a 36-64 polymer blend with an electric field applied in the out-of-plane direction (a)–(c) without particles, and (d)–(f) with particles (volume fraction = 0.074), illustrating the stabilizing effect of the nanoparticle suspensions against coarsening. The simulation times are 1000, 2000, and 3000 for (a) and (d), (b) and (e), and (c) and (f), respectively. (g) The average channel diameter throughout time both with and without particles. (h) The time evolution of the fraction of all particles segregated to the interfaces and the interfacial particle coverage, given as a fraction of the perfect close-packed density.
characteristics were averaged to provide somewhat better statistical insight (admittedly, a sample size of three is small, however the computational burden of these simulations is limiting). Figure 6 illustrates the dependency of the average channel diameter and density as these two variables are independently changed. Each data point is an average of three runs with different initial states and the error bars show the maximum and minimum values from that set of three. The channel diameter increases with decreasing particle volume fraction and increasing B-phase ratio, while the channel density varies in the opposite manner. A higher density of particles stabilizes the separation process earlier when the domain sizes are smaller, and also permits a higher interfacial coverage. Upon varying the B-phase ratio, increasing diameters were observed especially above 30%, which corresponds approximately with the nucleation/spinodal transition concentration. The arrangement of particles on the channel interfaces was observed to be close-packed hexagonal, however with a helical tilt angle, as mentioned above. Nearly all of the channels that possessed a high density of segregated particles exhibited this helical tilt, which was also observed in the spinodal structure shown in Fig. 3(a). On a flat surface, no particular orientation of an infinite close-packed ensemble of spheres is favored over another. Hence, the curvature of the channels must influence the minimum energy configuration. To verify this, a set of simulations was performed in which particles were randomly positioned on the interface of a single cylindrical channel and allowed to evolve in time. Different channel radii were chosen for each simulation ranging from
J. Chem. Phys. 140, 144903 (2014)
FIG. 6. Tuning the channel diameter by particle and polymer composition. (a) The average channel diameter and (b) the areal channel density for a 3664 blend as a function of particle volume fraction (i.e., varying particle number density) after stabilization. (c) and (d) Same except the B-phase ratio is varied and the particle volume fraction is fixed at 5.6%. In general, the average channel diameter increases with decreasing particle volume fraction and increasing B-phase ratio. Each data point is an average of three runs with different initial states and the error bars show the maximum and minimum values from the set.
15 to 80 x, corresponding to 1.5–8 particle diameters. Under the influence of the capillary forces and short-range particleparticle repulsions, the particles evolved towards the configurations shown in Fig. 7. The helical tilt angle of the hexagonal lattices was found to increase with decreasing channel diameter, up to a value of 20◦ . At a channel radius of 80 x (10 particle diameters), the favorability of a tilt angle was no
FIG. 7. Interfacial particle configurations for different channel diameters. These simulations were initialized with a single circular channel and a random distribution of particles near the interface, which evolved to the structures shown. The channel radii are (a) 15 x, (b) 25 x, (c) 40 x, (d) 60 x, and (e) 80 x. (f) The helix angle increases with decreasing channel radius to a limit of 20◦ –25◦ .
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longer observed to exist. This occurrence of a tilted lattice of spherical particles assembled on a cylinder was recently analyzed in depth by Wood et al.,45 who demonstrate that a two-dimensional lattice on a cylinder must have a periodic structure that is commensurate with the cylinder circumference. This constraint leads to a frustration that favors a helical tilt angle. Wood et al.45 assumed particles were exactly centered on the cylinder surface, which is not required for the particles attached to the polymer-polymer interfaces considered here, however the capillary forces here are strong enough to result in a similar arrangement. IV. CONCLUSIONS
The mutual morphology of an immiscible binary homopolymer mixture containing suspended particles can be guided by an external electric field to form a unique selfassembled membrane structure. By selectively dissolving the minority polymer domains, the membrane will contain aligned pore channels with partially exposed particles that can react with an external fluid media that passes through the channels. Furthermore, the particles were observed to arrange themselves in a tilted hexagonal lattice around the channel interfaces, a configuration that may enable controlled rotational manipulation of macromolecules, or the synthesis of selfpropelled rods if the majority polymer phase is dissolved. The thin film morphologies analyzed in this paper essentially represent metastable states that can be repeatedly formed when the initial state is a homogeneous mixture. However, it is not clear, and perhaps unlikely, that the nanoparticle-coated channels would assembly from an initial configuration with phase separation already developed and particles uniformly dispersed, or one in which nanoparticles are clustered and the polymer concentration is uniform. The processing route described here is not material specific, and can lead to novel functional membranes in many applications including ultrafiltration systems, chemical separation or sensing, biomedical devices, and nanofluidic energy conversion devices. ACKNOWLEDGMENTS
P.C.M. would like to acknowledge fruitful discussions with Yu Wang (Michigan Tech University) and the computational resources at the Idaho National Laboratory and the University of Arkansas. 1 A. C. Balazs, T. Emrick, and T. P. Russell, Science 314, 1107–1110 (2006). 2 L.-T.
Yan and X.-M. Xie, Prog. Polym. Sci. 38, 369–405 (2013). R. Bockstaller, R. A. Mickiewicz, and E. L. Thomas, Adv. Mater. 17, 1331–1349 (2005).
3 M.
J. Chem. Phys. 140, 144903 (2014) 4 H. Hori, O. Urakawa, O. Yano, and Q. Tran-Cong-Miyata, Macromolecules
40, 389–394 (2007). He, V. V. Ginzburg, and A. C. Balazs, J. Polym. Sci. B: Polym. Phys. 44, 2389–2403 (2006). 6 J. J. Chiu, B. J. Kim, E. J. Kramer, and D. J. Pine, J. Am. Chem. Soc. 127, 5036–5037 (2005). 7 H.-J. Chung, J. Kim, K. Ohno, and R. J. Composto, ACS Macro Lett. 1, 252 (2012). 8 K. Stratford, R. Adhikari, I. Pagonabarraga, J. C. Desplat, and M. E. Cates, Science 309, 2198–2201 (2005). 9 E. M. Herzig, K. A. White, A. B. Schofield, W. C. K. Poon, and P. S. Poon, Nat. Mater. 6, 966–971 (2007). 10 T. Saigal, H. Dong, K. Matyjaszewski, and R. D. Tilton, Langmuir 26(19), 15200–15209 (2010). 11 H.-J. Chung, K. Ohno, T. Fukuda, and R. J. Composto, Nano Lett. 5, 1878 (2005). 12 S. Gam et al., Soft Matter 7, 7262 (2011). 13 M. J. A. Hore and M. Laradji, J. Chem. Phys. 126, 244903 (2007). 14 M. J. A. Hore and M. Laradji, J. Chem. Phys. 128, 054901 (2008). 15 M. Laradji and M. J. A. Hore, J. Chem. Phys. 121, 10641 (2004). 16 M. Laradji, J. Chem. Phys. 120, 9330 (2004). 17 M. Huang, Z. Li, and H. Guo, Soft Matter 8, 6834 (2012). 18 T. Kawakatsu, K. Kawasaki, M. Furusaka, H. Okabayashi, and T. Kanaya, J. Chem. Phys. 99, 8200 (1993). 19 V. V. Ginzburg, G. Peng, F. Qiu, D. Jasnow, and A. C. Balazs, Phys. Rev. E 60, 4352 (1999). 20 B. P. Lee, J. F. Douglas, and S. C. Glotzer, Phys. Rev. E 60, 5812 (1999). 21 G. Peng, F. Qiu, V. V. Ginzburg, D. Jasnow, and A. C. Balazs, Science 288, 1802–1804 (2000). 22 L. T. Yan, E. Maresov, G. A. Buxton, and A. C. Balazs, Soft Matter 7, 595–607 (2011). 23 S. W. Sides, B. J. Kim, E. J. Kramer, and G. H. Fredrickson, Phys. Rev. Lett. 96, 250601 (2006). 24 V. Ganesan and A. Jayaraman, Soft Matter 10, 13 (2014). 25 Z. Xu, E. Lai, Y. Shao-Horn, and K. Hamad-Schifferli, Chem. Commun. 48, 5626–5628 (2012). 26 F. Erogbogbo et al., Nano Lett. 13, 451–456 (2013). 27 L. T. Yan, H. G. Schoberth, and A. Böker, Macromol. Chem. Phys. 210, 1003–1010 (2009). 28 M. Pinna and A. V. Zvelindovsky, Soft Matter 4, 316–327 (2008). 29 T.-L. Cheng and Y. U. Wang, Langmuir 28, 2696 (2012). 30 P. C. Millett and Yu. U. Wang, J. Colloid Interface Sci. 353, 46 (2011). 31 T. Matsuoka, in Polymer Blends and Alloys, edited by G. O. Shonaike and G. P. Simon (Marcel Dekker, New York, 1999), Chap. 7. 32 K. S. Lyakhova, A. V. Zvelindovsky, and G. J. A. Sevink, Macromolecules 39, 3024–3037 (2006). 33 Q. H. Zeng, A. B. Yu, and G. Q. Lu, Prog. Polym. Sci. 33, 191–269 (2008). 34 M. Bahiana and Y. Oono, Phys. Rev. A 41, 6763–6771 (1990). 35 J. Kim, Commun. Comput. Phys. 12, 613–661 (2012). 36 S. C. Glotzer, in Annual Review of Computational Physics, edited by D. Stauffer (World Scientific, Singapore, 1995), Vol. 2. 37 J. C. Chen and A. S. Kim, Adv. Colloid Interface Sci. 112, 159–173 (2004). 38 T.-L. Cheng and Y. U. Wang, J. Colloid Interface Sci. 402, 267–278 (2013). 39 F. Jansen and J. Harting, Phys. Rev. E 83, 046707 (2011). 40 G. Kaptay, J. Mater. Sci. 40, 2125–2131 (2005). 41 B. A. Miller-Chou and J. L. Koenig, Prog. Polym. Sci. 28, 1223–1270 (2003). 42 I. Vukovic, G. Brinke, and K. Loos, Polymer 54, 2591–2605 (2013). 43 V. Raman, A. Bose, B. D. Olsen, and T. A. Hatton, Macromolecules 45, 9373–9382 (2012). 44 C. K. Haas and J. Torkelson, Phys. Rev. E 55, 3191–3201 (1997). 45 D. A. Wood, C. D. Santangelo, and A. D. Dinsmore, Soft Matter 9, 10016– 10024 (2013). 5 G.
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THE JOURNAL OF CHEMICAL PHYSICS 140, 144903 (2014)
Electric-field induced alignment of nanoparticle-coated channels in thin-film polymer membranes Paul C. Milletta) Department of Mechanical Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA
(Received 3 January 2014; accepted 19 March 2014; published online 11 April 2014) Microscopic phase separation in immiscible polymer melts can be significantly altered by the presence of dispersed nanoparticles and externally applied electric fields. Inducing order or directionality to the resulting microstructure can lead to novel materials with efficient synthesis. Here, the coupled morphology of an immiscible binary polymer blend with dispersed nanoparticles in a thin-film geometry is investigated under the influence of an applied electric field using a unique mesoscale computational approach. For asymmetric binary blends (e.g., 70–30), the resulting microstructure consists of columnar channels of the B-phase perpendicular to the major plane of the film (aligned with the electric field), with the particles segregated along the channel interfaces. The simulations reveal the variability of the average channel diameter and the interfacial arrangement of the particles. The high density of exposed particles makes these structures viable candidates for catalytically active porous membranes or macromolecular manipulation devices. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4870471] I. INTRODUCTION
The development of new material systems with novel nanoscale or mesoscale architectures, which directly result in enhanced functionality, is a highly pursued topic in the broader field of materials science and engineering. Polymer nanocomposites, consisting of nanoparticles within a polymer matrix, are receiving growing attention due to their unique microstructural and property characteristics.1, 2 Nanoparticles dispersed in immiscible polymer blends or block copolymers that undergo phase separation hold even greater promise for the controlled design of a rich diversity of mesoscale structures.3–5 In such systems, the dispersed particles can be tailored to reside either in one of the bulk phases or along the phase interfaces.6, 7 For the latter, the interfacial arrangement of particles can dramatically lower the interfacial area and curvature thereby eliminating the driving force for coarsening, thus stabilizing a variety of interesting structures such as colloidally jammed bijels8, 9 and Pickering emulsions.10 These morphologies are particularly interesting in thin-film geometries, in which the film thickness can dictate if the phase separation is continuous or discrete, as shown by the work of Composto and colleagues11, 12 using 20 nm particles, grafted with polymer ligands, that segregate to the interfaces in immiscible homopolymer blends. However, our understanding of the mutual interactions between particles and multi-phase polymer melts during phase separation, and how to impart order or directionality to their morphology, is still quite limited. Over the last two decades, computational mesoscale methods have developed into particularly useful approaches to study the co-evolution of particles in phase separating fluids (including polymers). For example, dissipative particle dynamics have been used by Hore and Laradji13–16 and a) Electronic mail: [email protected]
0021-9606/2014/140(14)/144903/7/$30.00
others17 to study various particle assemblies in binary phaseseparating mixtures. Hybrid models, on the other hand, represent a different approach, in which the fluid or polymer phase is represented by concentration or density fields on a discrete mesh, while the dispersed particles evolve in continuous space by discrete moves informed by their instantaneous force or energy, the earliest example from Kawakatsu et al.18 With regards to particles in phase-separating mixtures, hybrid models have utilized the Lattice Boltzmann method8 to represent low-viscous fluids, as well as Cahn-Hilliard (CH)18–22 and Self-Consistent Field23 theories to represent viscous polymer melts. A good review of these approaches, and their application to polymer nanocomposites, can be found in the recent literature.2, 24 All of these approaches have predicted the segregation of neutral particles to the phase interfaces leading to retarded coarsening and eventual arrest, given the particles are sufficiently larger than the interfacial thickness. Recent demonstrations of nanoparticle-induced reactions, including the conversion of CO2 gas to methane with Cu–Au nanoparticles,25 and H2 production from water splitting using Si nanoparticles,26 illustrate the promise for nanoparticle materials in energy and environmental applications. An important challenge to overcome involves the design and synthesis of suitable scaffolds, including membranes, which can support a large density of nanoparticles and simultaneously provide a large exposure area for interaction with a flowing gas or liquid. Self-assembled polymernanoparticle composites may potentially be well-suited for such applications. This article presents the results of a computational study that characterizes the evolution of a two-phase immiscible homopolymer melt containing dispersed spherical particles in a thin-film geometry with an applied electric field. The computational approach used is a mesoscale Brownian Dynamics (BD)/CH model.21, 22 The simulations evolve the system from
140, 144903-1
© 2014 AIP Publishing LLC
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J. Chem. Phys. 140, 144903 (2014)
an initial state consisting of a random distribution of particles within a homogeneous solution of two homopolymers (represented by concentration field variables), throughout the decomposition process to a state of stabilization. The phase separation, and therefore the final structure, is altered by the presence of an electric field,27 which effectively aligns the microstructural domains in the direction of the field.4, 28 The simulations presented here emphasize the relationships between the average channel sizes and the relative polymer and particle compositions, as well as the particle configurations that ultimately form on the polymer-polymer interfaces.
from values of one to zero (or vice-versa) within a finite width inside the interface. The free energy density function consists of bulk and interfacial terms (3) F = [f (φ1 , φ2 ) + κ(∇φ1 )2 + κ(∇φ2 )2 ] dV ,
II. MODELING APPROACH
where ψ is a variable stored on the grid that equals one inside of any particle and zero outside of all particles. Equation (4) has been used previously to model particles within two-phase fluids.29, 30 Outside of the particles, the function is a double well with two symmetric minima at (φ 1 ,φ 2 ) = (1,0),(0,1), see Fig. 1. The exact shape of the double well functional has been shown to not meaningfully alter the scaling function and exponents in late-stage phase separation,33, 34 thus Eq. (4) can be assumed to be an appropriate substitution for the FloryHuggins energy31
The BD/CH computational approach used here simultaneously evolves the suspended nanoparticles and the mesoscale phase separation occurring within the two-phase polymer melt. Here, we assume the particles are chemically neutral, i.e., they do not prefer one phase over the other and therefore will preferentially segregate to the polymerpolymer interfaces, similar to the experimental conditions prescribed in Refs. 11 and 12. The approach utilizes algorithms typically used in both particle-based and grid-based methods, with additional steps required to couple the particle and liquid interactions. The method is quite similar to previous studies,21, 22, 27 however some distinctions exist, and a detailed description is given below. The polymer melt phase separation is evolved in space and time by solving the CH equation on a discrete cubic grid using a standard finite differencing scheme.31 The CH equation, with an additional term to represent electric field-driven diffusion,28 is given by δF ∂φi = Mi ∇ 2 + k∇z2 φi , ∂t δφi
(1)
where φ i and Mi are the volume fraction and mobility of homopolymer species i, respectively, F is the free energy density of the multi-phase mixture, and k indicates the strength of the electric field applied in the z-direction defined as27, 28 k=
o υo 2Eo2 (A − B )2 , kB T (A + B )
(2)
where o is the vacuum permittivity, υ o is the volume of one monomer, Eo is the real strength of the electric field, and A and B are the dielectric constants of domains A and B. In the typical experimental system of PS-PMMA, a value of k = 0.1 corresponds to an electric field of 8 V/μm.32 A principal distinction between the current model and previous BD/CH simulations21, 22, 27 is the number of concentration fields used to represent the binary polymer blend. In the previous studies, a single phase variable is generally used that assumes a value of either 1 or −1 in the bulk phases. In the simulations here, there are two concentration variables evolved, φ 1 and φ 2 , which is motivated by the fact that using two variables simplifies the computation of the interfacial capillary forces on the particles, as will be shown later. Within any particular phase, the concentrations are constant (after the initial phase separation). The polymer-polymer and polymerparticle interfaces are diffuse, hence the variables transition
where f(φ 1 , φ 2 ) is the bulk free energy density, and the gradient terms contribute to the interfacial energy. The bulk free energy density is given by f (φ1 , φ2 ) = w 3φ14 − 4φ13 + 3φ24 − 4φ23 +6φ12 φ22 + 6ψ φ12 + φ22 , (4)
f φ2 φ1 ln φ1 + ln φ2 + χ φ1 φ2 , = kB T N1 N2
(5)
where N1 and N2 are the degrees of polymerization of polymers 1 and 2, respectively, and χ is the interaction parameter. Figure 1 illustrates how Eq. (4) compares with the FloryHuggins model for N1 = N2 = 20 and χ = 0.59. On the other hand, inside a particle, the function assumes a single well shape with a minima at φ 1 = φ 2 = 0, thereby eliminating polymer concentration within the solid particles. Central finite differencing is used to solve the spatial gradients of Eqs. (1) and (3), and a forward difference is used to solve the time gradient in Eq. (1).35 The CH approach to modeling polymer separation assumes that the molecular transport is purely governed by diffusion, and that hydrodynamic transport is negligible (although, successful efforts have been made to couple the CH equation with the Navier-Stokes equations, see, e.g., Ref. 21). This assumption is generally valid for high viscosity (low mobility) polymer separation, particularly when the domain size is on the order of tens to hundreds of nanometers and the separation is not in the very late stages.36 The trajectories of the suspended nanoparticles are updated throughout time with discrete time steps using standard particle-based algorithms used in, for example, Molecular Dynamics (MD). The polymer melt medium is assumed to have a high viscosity; therefore, the particle motions are dissipative and consequently the particle forces are related to velocities rather than accelerations, consistent with Brownian dynamics.37 Each particle is considered a point mass with an associated radius and mobility. The overall particle forces and velocities are obtained from pp
cap
FTi = Fi + Fi
+ Fstoc , i
vi = ηi FTi ,
(6) (7)
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J. Chem. Phys. 140, 144903 (2014)
FIG. 1. (a) Free energy density plot of the immiscible binary polymer melt as a function of the two volume fractions, φ 1 and φ 2 , illustrating the double-well functional with minima at φ 1 = 1, φ 2 = 0 and φ 1 = 0, φ 2 = 1. (b) Comparison of Eq. (4) (assuming φ 1 + φ 2 = 1) with the Flory-Huggins model with the stated assumed parameters. The red dashed line in (b) corresponds with that in (a). (c) The free energy density within a particle changes to a single well function with a minimum at φ 1 = φ 2 = 0.
where vi and ηi are the velocity and mobility of particle i. Neighbor lists are constructed for each particle, which are used to calculate short-range particle-particle steric repulsive forces defined by the following function: pp
Fi = A(rij − 2(rp + rhalo ))n ,
(8)
where rij is the distance between particles i and j, rp is the particle radius, and rhalo is an additional halo distance around the particles that is used to improve the numerical computations of particle interactions with the polymer interfaces.38 The pre-factor A and the exponent n dictate the magnitude of the short-range repulsion force. The stochastic forces on the particles induced by the polymer molecules leading to Brownian motion are defined as stoc stoc stoc , (9) Fstoc = B R1 Fi,x + R2 Fi,y + R3 Fi,z i where R1 , R2 , and R3 are random numbers between −0.5 and 0.5. The constants A and B in Eqs. (8) and (9) are force constants. For chemically neutral particles, interfacial segregation occurs to lower the overall energy of the system by reducing the polymer-polymer interfacial area, which has been observed even for polymer-grafted nanoparticles with diameters of 20 nm in homopolymer blends.11, 12 Capillary forces act to maintain interfacial particle attachment. In the present model, the particle-polymer interactions include the capillary forces that the polymer-polymer interfaces exert on the particles, and the dependency of the free energy density on the particles, described by the last term in Eq. (4) which effectively induces a pinning force on polymer-polymer interfaces where particles exist. To incorporate both of these elements in the model, the set of particle positions must be mapped onto the finite difference grid used to solve the CH equation. In order to do this appropriately, the particle radii rp must be larger than the grid spacing to adequately define the particle shape, size, and interaction with the polymers. As mentioned above, the variable ψ is used to represent the particles’ positions on the grid ⎧ rij < rp ⎨1, (10) ψ = exp(−(rij − rp )), rp < rij < rp + rint . ⎩ 0, rij > rp + rint
On the grid, the particle surfaces are therefore also diffuse with a width of rint . The capillary forces are calculated using the two polymer concentration variables and the ψ variable
cap dV , (11) Fi = D ψi φ1 φ2 rij k − rcm i where ψ i is the ψ field for particle i, rij k is the vector position is the vector position of the center-ofof grid point ijk, rcm i mass of particle i, and D is a force constant. Here, the brackets . . . signify that the vector within is a unit vector. The capillary force is therefore non-zero only at the triple junction lines where φ 1 > 0, φ 2 > 0, and ψ > 0. Equation (11) represents a more simplified expression for the capillary force computation for the specific case of a 90◦ contact angle, compared with the author’s previous work,30 which is valid for varying contact angles. We assume no explicit interaction between the electric field and the particles, i.e., electrostatic forces on particles are neglected. The values of all of the numerical parameters are given in Table I. The motivation for some of these parameter choices is not completely rigorous. For example, A and n govern the short-range particle-particle repulsions, and the chosen values represent a rather steep repulsive energy curve that spans TABLE I. BD/CH simulation parameters used in this study. Parameter
Value
x t Mi κ k w ηi A B D n rp rhalo rint
1.0 0.01 1.0 1.0 0.1 0.15 0.4 50.0 20.0 2.0 − 7.0 4.0 5.0 3.0
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J. Chem. Phys. 140, 144903 (2014)
roughly 1/8 the particle diameter. Also, the relative values of B and D, which scale the Brownian motion and capillary forces, respectively, are chosen such that the particles’ thermal energy is generally insufficient to detach a particle that is absorbed on a polymer-polymer interface. Small variations of these parameters is not expected to qualitatively change the simulation results, however, a complete parametric analysis has not been performed. The simulations are performed in reduced length and time units. Each time step is executed by the following sequential steps: (1) mapping the particle positions on the grid by calculating the ψ variable, (2) updating the polymer domains by solving the CH equation on the grid, (3) calculating the particle forces including those from the polymer-polymer interfaces, and (4) updating the particle positions. Steps (1) through (4) are repeated, with the occasional update to the particle neighbor lists. To ensure proper particle interactions with the polymer interfaces, the particle diameter must be larger than the diffuse interface width,39 which is ∼ 5 x. (Here, the effective particle diameter is ∼ 10 x, with rp = 4 x and rint = 3 x.) Following micro-phase separation, the polymer domain widths are typically 30–70 x, hence evolving a system of 2000 particles requires grid sizes of approximately 400 × 400 × 100. The simulations are therefore executed in parallel, in which worker processors update the grid computations (using domain decomposition), and a master processor updates the particle neighbor lists, interactions, and positions. The capillary forces exerted on a particle attached to a polymer-polymer interface depend on several factors including the interfacial energies, the particle contact angle, and the particle’s shape, size, and position. For a spherically shaped particle interacting with an otherwise flat horizontal interface, the capillary force perpendicular to the interface has the following closed-form, analytical solution:40 F cap = 2π rp σ12 sin φc sin (φc + θ ),
(12)
where the angles φ c and θ are defined visually in Fig. 2, and σ 12 is the interfacial energy of a fluid-fluid interface. To validate that the BD/CH model described above reproduces Eq. (12), several simulations were performed involving a spherical particle attached to an interface at different positions (i.e., different φ c angles), with a fixed contact angle of θ = 90◦ . For this analysis, the spherical particle had a chosen radius of 18 x. Figure 2 shows the calculated capillary forces versus φ c in the simulations compared with Eq. (11), illustrating a good quantitative agreement. The amplitude of the curve in Fig. 2 is dependent on rp and σ 12 . In the simulations, the capillary force is scaled by the force constant, D. A value of D = 2.0 is used, which considering that the maximum amplitude in Fig. 2 is 50.07, correlates to an interfacial energy of σ 12 = 0.88. The interfacial energy derived from Eq. (3) is 0.78 using the parameters given in Table I. III. RESULTS
Confined within a thin-film geometry, the binary polymer-nanoparticle composite evolution, under an electric field, depends strongly on the relative blend ratio of the A
FIG. 2. The capillary force exerted on a single spherical particle by a polymer-polymer interface intersecting the particle at different locations relative to the particle center-of-mass. The particle radius is 18 x and the assumed contact angle is θ = 90◦ . The calculated capillary forces, represented by the blue data points, correlate well with the analytical solution represented by the solid curve. The inset provides a visual schematic displaying the angles θ and φ c , the latter correlating to the x-axis in the plot.
and B polymer species, as shown in Fig. 3. Here, the simulations start from a uniform disordered state with φ 1 and φ 2 having small initial fluctuations in the range of ± 0.05 about the global average. The initial particle positions are random, with the condition that their hard spheres do not overlap. The start of the simulation corresponds to an instant quench below the critical temperature, thereby initiating phase separation. For symmetric blends (i.e., 50–50), the resulting geometry resembles a two-dimensional jammed bijel8 – a spinodally decomposed arrangement of polymer domains with interfacially segregated particles that assume a close-packed hexagonal arrangement (see Ref. 8). For the system shown in Fig. 3(a), the coarsening has terminated and the interconnected ligaments, which can have rather high curvature, are stable. For this simulation, the grid size is 400 × 400 × 100 with 2000 uniformly sized particles (particle volume fraction = 0.065), essentially all of which are attached to the interfaces. When the blend ratio is changed to 36–64, the system morphology changes from a spinodal pattern to a columnar arrangement aligned in the z-direction with the electric field. From this structure, a continuous membrane film with a high directional porosity can be obtained with the selective removal of the minority polymer phase.41, 42 Once again, the particles are largely segregated to the interfaces and assume a nearly close-packed configuration. In general, the interfacial particle density for the columnar phase was observed to be not quite as uniform as the spinodal phase, i.e., many columns achieved nearly close-packed densities while some columns were somewhat less than close-packed. This is likely attributable to the fact that the interfaces in the spinodal regime are continuous and therefore allow particle redistribution to accommodate high particle density and uniformity. The columns, on the other hand, are not connected with each other and therefore cannot readily exchange particles from columns with high particle density to those with low density. Hence, particles that do not locate on an interface
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Paul C. Millett
J. Chem. Phys. 140, 144903 (2014)
FIG. 3. BD/CH simulations of membrane structures. (a)–(c) A 50-50 polymer blend, with particle volume fraction = 0.065, under an applied electric field after phase separation and particle stabilization, resulting in a columnar bijel structure with a nearly close-packed monolayer of particles along the interfaces ((c) is an xy-plane view). (d)–(f) A 36-64 blend is shown (same particle volume fraction), resulting in columnar channels again with interfacial particle segregation. (g) On the channel interfaces, the particles assume a hexagonal arrangement with a tilted helical angle (this channel is indicated by the arrow in (f)).
early in the phase separation may have to travel a relatively long distance to find a channel interface that can accommodate them. As seen in Fig. 3(f), there are a noticeable number of particles located in the bulk in the majority phase. All in all, however, the interfacial particle coverage is quite high in the simulations described here (usually up to 80% of the ideal close packing) and most columns have nearly close-packed particle configurations. A perpendicular view of one of the columnar channels in Fig. 3(f) (indicated by the arrow) is shown in Fig. 3(g). Interestingly, the particles assume a close-packed structure, however the hexagonal arrangement is tilted with a certain helical angle. This helical tilt was observed for basically all of the channels in the film. A more detailed analysis is given in Fig. 6 below. For comparison purposes, Fig. 4 shows the morphology of the same composition albeit without the influence of an electric field. Particle segregation to the interfaces still occurs as before, however interfaces are ori-
FIG. 4. Thin film morphologies for the same simulation conditions as in Fig. 3, however with no electric field. (a) and (b) A 50-50 blend with and without particles illustrated, and (c) and (d) a 36-64 blend with and without particles illustrated. Without an electric field, the domains are not aligned in the z-direction.
ented in all directions and there is no discernible alignment of the domains. All things considered, the degree to which vertical columnar channels form in the system will be a function of the electric field strength, the film thickness, and the particle volume fraction. A systematic study of these interrelationships will be left for future work. Here, the particle volume fraction and film thickness are large enough such that vertical channels are not ubiquitous without the applied field. Raman et al.43 recently modeled superparamagnetic nanoparticles in block copolymers and found that an applied magnetic field can orient the domains due to particle alignment. The interfacial particle segregation effectively stabilizes the channels from coarsening due to Ostwald ripening,44 which naturally occurs in phase-separating systems without particles (Fig. 5). In Fig. 5, and for the rest of the results, the electric field is turned back on. Analogous to the bijel structure,8 the particles eliminate a large portion of the interfacial area, i.e., the driving force for coarsening. Coarsening can only occur if particles detach from the interfaces – a phenomenon that is rarely observed in the later stages of the phase separation. Figure 5(h) shows how the fraction of total particles located on an interface, and the interfacial particle coverage (given as a fraction of the ideal close-packed configuration), evolve during time as the phase separation progresses. Throughout time, the fraction of total particles touching an interface decreases initially but gradually steadies to a value of 0.85. Conversely, the particle interfacial packing fraction increases from the beginning of the simulation to a rather steady-state value of 0.72. A potentially desirable attribute of such films may be a controllable channel diameter and channel areal density (the number of channels per area of film). A parametric analysis was performed for two key variables: the particle volume fraction and the blend ratio. Simulation cell sizes of 200 × 200 × 70 were utilized, in which these two parameters were systematically varied. For each composition, three independent simulations were executed (starting from different initial distributions), and the resulting structural
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Paul C. Millett
FIG. 5. Channel diameter stabilization due to particles. Top-down view of the time evolution of a 36-64 polymer blend with an electric field applied in the out-of-plane direction (a)–(c) without particles, and (d)–(f) with particles (volume fraction = 0.074), illustrating the stabilizing effect of the nanoparticle suspensions against coarsening. The simulation times are 1000, 2000, and 3000 for (a) and (d), (b) and (e), and (c) and (f), respectively. (g) The average channel diameter throughout time both with and without particles. (h) The time evolution of the fraction of all particles segregated to the interfaces and the interfacial particle coverage, given as a fraction of the perfect close-packed density.
characteristics were averaged to provide somewhat better statistical insight (admittedly, a sample size of three is small, however the computational burden of these simulations is limiting). Figure 6 illustrates the dependency of the average channel diameter and density as these two variables are independently changed. Each data point is an average of three runs with different initial states and the error bars show the maximum and minimum values from that set of three. The channel diameter increases with decreasing particle volume fraction and increasing B-phase ratio, while the channel density varies in the opposite manner. A higher density of particles stabilizes the separation process earlier when the domain sizes are smaller, and also permits a higher interfacial coverage. Upon varying the B-phase ratio, increasing diameters were observed especially above 30%, which corresponds approximately with the nucleation/spinodal transition concentration. The arrangement of particles on the channel interfaces was observed to be close-packed hexagonal, however with a helical tilt angle, as mentioned above. Nearly all of the channels that possessed a high density of segregated particles exhibited this helical tilt, which was also observed in the spinodal structure shown in Fig. 3(a). On a flat surface, no particular orientation of an infinite close-packed ensemble of spheres is favored over another. Hence, the curvature of the channels must influence the minimum energy configuration. To verify this, a set of simulations was performed in which particles were randomly positioned on the interface of a single cylindrical channel and allowed to evolve in time. Different channel radii were chosen for each simulation ranging from
J. Chem. Phys. 140, 144903 (2014)
FIG. 6. Tuning the channel diameter by particle and polymer composition. (a) The average channel diameter and (b) the areal channel density for a 3664 blend as a function of particle volume fraction (i.e., varying particle number density) after stabilization. (c) and (d) Same except the B-phase ratio is varied and the particle volume fraction is fixed at 5.6%. In general, the average channel diameter increases with decreasing particle volume fraction and increasing B-phase ratio. Each data point is an average of three runs with different initial states and the error bars show the maximum and minimum values from the set.
15 to 80 x, corresponding to 1.5–8 particle diameters. Under the influence of the capillary forces and short-range particleparticle repulsions, the particles evolved towards the configurations shown in Fig. 7. The helical tilt angle of the hexagonal lattices was found to increase with decreasing channel diameter, up to a value of 20◦ . At a channel radius of 80 x (10 particle diameters), the favorability of a tilt angle was no
FIG. 7. Interfacial particle configurations for different channel diameters. These simulations were initialized with a single circular channel and a random distribution of particles near the interface, which evolved to the structures shown. The channel radii are (a) 15 x, (b) 25 x, (c) 40 x, (d) 60 x, and (e) 80 x. (f) The helix angle increases with decreasing channel radius to a limit of 20◦ –25◦ .
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Paul C. Millett
longer observed to exist. This occurrence of a tilted lattice of spherical particles assembled on a cylinder was recently analyzed in depth by Wood et al.,45 who demonstrate that a two-dimensional lattice on a cylinder must have a periodic structure that is commensurate with the cylinder circumference. This constraint leads to a frustration that favors a helical tilt angle. Wood et al.45 assumed particles were exactly centered on the cylinder surface, which is not required for the particles attached to the polymer-polymer interfaces considered here, however the capillary forces here are strong enough to result in a similar arrangement. IV. CONCLUSIONS
The mutual morphology of an immiscible binary homopolymer mixture containing suspended particles can be guided by an external electric field to form a unique selfassembled membrane structure. By selectively dissolving the minority polymer domains, the membrane will contain aligned pore channels with partially exposed particles that can react with an external fluid media that passes through the channels. Furthermore, the particles were observed to arrange themselves in a tilted hexagonal lattice around the channel interfaces, a configuration that may enable controlled rotational manipulation of macromolecules, or the synthesis of selfpropelled rods if the majority polymer phase is dissolved. The thin film morphologies analyzed in this paper essentially represent metastable states that can be repeatedly formed when the initial state is a homogeneous mixture. However, it is not clear, and perhaps unlikely, that the nanoparticle-coated channels would assembly from an initial configuration with phase separation already developed and particles uniformly dispersed, or one in which nanoparticles are clustered and the polymer concentration is uniform. The processing route described here is not material specific, and can lead to novel functional membranes in many applications including ultrafiltration systems, chemical separation or sensing, biomedical devices, and nanofluidic energy conversion devices. ACKNOWLEDGMENTS
P.C.M. would like to acknowledge fruitful discussions with Yu Wang (Michigan Tech University) and the computational resources at the Idaho National Laboratory and the University of Arkansas. 1 A. C. Balazs, T. Emrick, and T. P. Russell, Science 314, 1107–1110 (2006). 2 L.-T.
Yan and X.-M. Xie, Prog. Polym. Sci. 38, 369–405 (2013). R. Bockstaller, R. A. Mickiewicz, and E. L. Thomas, Adv. Mater. 17, 1331–1349 (2005).
3 M.
J. Chem. Phys. 140, 144903 (2014) 4 H. Hori, O. Urakawa, O. Yano, and Q. Tran-Cong-Miyata, Macromolecules
40, 389–394 (2007). He, V. V. Ginzburg, and A. C. Balazs, J. Polym. Sci. B: Polym. Phys. 44, 2389–2403 (2006). 6 J. J. Chiu, B. J. Kim, E. J. Kramer, and D. J. Pine, J. Am. Chem. Soc. 127, 5036–5037 (2005). 7 H.-J. Chung, J. Kim, K. Ohno, and R. J. Composto, ACS Macro Lett. 1, 252 (2012). 8 K. Stratford, R. Adhikari, I. Pagonabarraga, J. C. Desplat, and M. E. Cates, Science 309, 2198–2201 (2005). 9 E. M. Herzig, K. A. White, A. B. Schofield, W. C. K. Poon, and P. S. Poon, Nat. Mater. 6, 966–971 (2007). 10 T. Saigal, H. Dong, K. Matyjaszewski, and R. D. Tilton, Langmuir 26(19), 15200–15209 (2010). 11 H.-J. Chung, K. Ohno, T. Fukuda, and R. J. Composto, Nano Lett. 5, 1878 (2005). 12 S. Gam et al., Soft Matter 7, 7262 (2011). 13 M. J. A. Hore and M. Laradji, J. Chem. Phys. 126, 244903 (2007). 14 M. J. A. Hore and M. Laradji, J. Chem. Phys. 128, 054901 (2008). 15 M. Laradji and M. J. A. Hore, J. Chem. Phys. 121, 10641 (2004). 16 M. Laradji, J. Chem. Phys. 120, 9330 (2004). 17 M. Huang, Z. Li, and H. Guo, Soft Matter 8, 6834 (2012). 18 T. Kawakatsu, K. Kawasaki, M. Furusaka, H. Okabayashi, and T. Kanaya, J. Chem. Phys. 99, 8200 (1993). 19 V. V. Ginzburg, G. Peng, F. Qiu, D. Jasnow, and A. C. Balazs, Phys. Rev. E 60, 4352 (1999). 20 B. P. Lee, J. F. Douglas, and S. C. Glotzer, Phys. Rev. E 60, 5812 (1999). 21 G. Peng, F. Qiu, V. V. Ginzburg, D. Jasnow, and A. C. Balazs, Science 288, 1802–1804 (2000). 22 L. T. Yan, E. Maresov, G. A. Buxton, and A. C. Balazs, Soft Matter 7, 595–607 (2011). 23 S. W. Sides, B. J. Kim, E. J. Kramer, and G. H. Fredrickson, Phys. Rev. Lett. 96, 250601 (2006). 24 V. Ganesan and A. Jayaraman, Soft Matter 10, 13 (2014). 25 Z. Xu, E. Lai, Y. Shao-Horn, and K. Hamad-Schifferli, Chem. Commun. 48, 5626–5628 (2012). 26 F. Erogbogbo et al., Nano Lett. 13, 451–456 (2013). 27 L. T. Yan, H. G. Schoberth, and A. Böker, Macromol. Chem. Phys. 210, 1003–1010 (2009). 28 M. Pinna and A. V. Zvelindovsky, Soft Matter 4, 316–327 (2008). 29 T.-L. Cheng and Y. U. Wang, Langmuir 28, 2696 (2012). 30 P. C. Millett and Yu. U. Wang, J. Colloid Interface Sci. 353, 46 (2011). 31 T. Matsuoka, in Polymer Blends and Alloys, edited by G. O. Shonaike and G. P. Simon (Marcel Dekker, New York, 1999), Chap. 7. 32 K. S. Lyakhova, A. V. Zvelindovsky, and G. J. A. Sevink, Macromolecules 39, 3024–3037 (2006). 33 Q. H. Zeng, A. B. Yu, and G. Q. Lu, Prog. Polym. Sci. 33, 191–269 (2008). 34 M. Bahiana and Y. Oono, Phys. Rev. A 41, 6763–6771 (1990). 35 J. Kim, Commun. Comput. Phys. 12, 613–661 (2012). 36 S. C. Glotzer, in Annual Review of Computational Physics, edited by D. Stauffer (World Scientific, Singapore, 1995), Vol. 2. 37 J. C. Chen and A. S. Kim, Adv. Colloid Interface Sci. 112, 159–173 (2004). 38 T.-L. Cheng and Y. U. Wang, J. Colloid Interface Sci. 402, 267–278 (2013). 39 F. Jansen and J. Harting, Phys. Rev. E 83, 046707 (2011). 40 G. Kaptay, J. Mater. Sci. 40, 2125–2131 (2005). 41 B. A. Miller-Chou and J. L. Koenig, Prog. Polym. Sci. 28, 1223–1270 (2003). 42 I. Vukovic, G. Brinke, and K. Loos, Polymer 54, 2591–2605 (2013). 43 V. Raman, A. Bose, B. D. Olsen, and T. A. Hatton, Macromolecules 45, 9373–9382 (2012). 44 C. K. Haas and J. Torkelson, Phys. Rev. E 55, 3191–3201 (1997). 45 D. A. Wood, C. D. Santangelo, and A. D. Dinsmore, Soft Matter 9, 10016– 10024 (2013). 5 G.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 205.170.15.173 On: Tue, 09 Dec 2014 14:26:18
