Sphere-to-Cylinder Transitions in Thin Films of Diblock Copolymers

May 7, 2012 - methods using bead−spring models are better suited to describe ...... (9) Harrison, C.; Park, M.; Chaikin, P. M.; Register, R. A.; Ada...
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Sphere-to-Cylinder Transitions in Thin Films of Diblock Copolymers under Shear: The Role of Wetting Layers Alexandros Chremos,† Paul M. Chaikin,‡ Richard A. Register,† and Athanassios Z. Panagiotopoulos*,† †

Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, United States Center for Soft Condensed Matter Research and Department of Physics, New York University, New York, New York 10003, United States

‡

ABSTRACT: Shear-induced sphere-to-cylinder transitions in diblock copolymer thin films have been studied using coarse-grained Langevin dynamics simulations. Parameters of the coarse-grained model were chosen to represent a polystyrene−polyisoprene copolymer with molecular weights of the blocks equal to 68 and 12 kg/mol, respectively, matching the system studied experimentally by Hong et al. [Sof t Matter 2009, 5, 1687]. At zeroshear conditions and below the order−disorder transition temperature, thin films form a monolayer or bilayer of spheres. The minority block has higher affinity for the confining surfaces, thus forming wetting layers whose chains interpenetrate those forming the microdomain layer(s). Once a shear field is applied and above a critical shear rate, the spheres elongate and merge with their neighbors to form cylinders. We find that shear-induced cylinder formation is closely related to stretching of individual diblock chains. Our simulations suggest that a higher stress is required to achieve the sphere-to-cylinder transition in monolayer versus bilayer thin films because the wetting layers transfer momentum into the film by stretching the chains, which in turn causes higher shear stress for a given surface velocity. This observation is in agreement with experimental findings. In addition to the effects of shear, the impact of temperature was investigated with respect to chain stretching and the formation of cylinders under shear.

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INTRODUCTION Block copolymers comprise two or more chemically distinct blocks covalently attached to each other. Below the order− disorder transition temperature and above their glass transition temperature, organized structures are generated by the microphase separation driven by the chemical differences between the components. Block copolymers are useful because by choosing the polymer segment chemical character (impacting the Flory−Huggins interaction parameter χ), the chain length (the degree of polymerization N), and composition (monomer volume fraction f) they self-assemble into a rich variety of microstructures. The topologies of the microstructures strongly influence their mechanical, rheological, and optical properties, making block copolymers excellent materials for many applications, including but not limited to microelectronics,1,2 magnetic storage,3 drug delivery,4 and lithographic chemical patterning.5 Transitions between different ordered phases in a single diblock copolymer are of particular interest for enhancing our understanding and for many potential applications. Such order−order transitions are typically achieved by changing a state variable, such as temperature. However, regular annealing processes tend to be slow and long-range orientational order is rarely achieved in practice.6−8 Other approaches include using confinement: morphology changes are often obtained when the confining walls strongly interact with the film. Thin films of cylinder-forming diblock copolymers show a complex phase behavior including lamellae, perforated lamellae, cylinders, and many other morphologies.9−15 Accelerating the alignment and © 2012 American Chemical Society

promoting long-range order can be accomplished via the use of external fields, for example electrostatic16 or shear.17,18 Previous studies have shown that using shear improves the long-range order of the microdomains in the bulk (spheres, cylinders, and lamellae). Recent experiments by Hong et al.19 showed that steady shear can also trigger a sphere-to-cylinder order−order transition in thin films, producing a high degree of long-range order. Several methods have been developed for simulations of polymeric systems, such as the time-dependent Ginzburg− Landau method,20−22 dynamic density functional theory,23 cell dynamics method,24 and dissipative particle dynamics.25,26 These methods have successfully described the structural and dynamical properties of simple and complex polymers at equilibrium. Away from equilibrium, molecular dynamics methods using bead−spring models are better suited to describe the behavior of polymers under shear at length scales comparable to the intermolecular distances. Even though there is extensive literature on thin films of block copolymers at equilibrium and thin films of homopolymers under shear, the study of thin films of diblock copolymers under shear has become an active field only in the past decade. To perform a credible simulation of diblock copolymers under shear, the simulated system size must not only be larger than the individual chains but also bigger than the microstructures that Received: February 23, 2012 Revised: April 25, 2012 Published: May 7, 2012 4406

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A coarse-grained bead−spring model is employed to represent the constituent polymers. In the model, each polymer is composed of N beads connected by springs to form a linear chain. Each chain has two blocks; one block with f N = 13 beads of type A representing the PI block and another block with (1 − f)N = 65 beads of type B representing the PS block. The composition f is chosen to match the volume fraction of the polymer used in the experiments, f ≈ 0.17. The nonbonded interactions between the polymer beads are described by the cut-and-shifted Lennard-Jones (LJ) potential with a cutoff distance, rc = 2.5σ.

form. This can be accomplished by significantly coarse-graining the system. For example, a recent cell dynamics simulation for thin films of diblock copolymers under shear displayed a rich structural behavior.27 Nevertheless, such approaches neglect the influence of shear on the chain conformations. With recent advances in computer hardware, simulations can model physical systems at large enough length and time scales to gain useful insights. Thin films of sphere-forming diblock copolymers under shear have been recently investigated computationally by some of us.28 It was found that the spherical domains elongate in the direction of shear and merge with their neighbors to form cylinders. The results were rationalized by constructing maps of different morphologies observed in the parameter space of composition, segregation strength, and shear rate. However, the model of ref 28 was not tailored to match specific experimental systems and did not take into account the role of the film thickness, which was kept fixed, and the formation of wetting layers via the adsorption of the minority block on the surface. The main objective of the present work is to investigate how shear affects thin films of different thickness, in the presence of wetting layers. We use Langevin dynamics of a coarse-grained “bead−spring” model with suitable interactions to promote microphase separation. The model has been tailored to match a specific experimental system19 and is able to reproduce many of the experimental observations. The simulations provide valuable insights into the microstructures and polymer conformations and their relation to macroscopic properties of the film.

⎧ ⎡ 12 12 6 ⎪ 4ε ⎢⎜⎛ σ ⎟⎞ − ⎜⎛ σ ⎟⎞ − ⎛⎜ σ ⎞⎟ r ≤ r c ⎪ ij⎢⎝ r ⎠ ⎝r⎠ ⎝ rc ⎠ ⎣ ⎪ ⎪ ⎛ σ ⎞6 ⎤ Vij(r ) = ⎨ ⎪ +⎜ ⎟⎥ ⎝ rc ⎠ ⎥⎦ ⎪ ⎪ ⎪0 r > rc ⎩

(1)

where εij and σ are energy and range parameters, with i, j = A or B. The nonlinear finitely extensible (FENE) potential of the springs is given by ⎛ 1 r2 ⎞ ⎟ VFENE(r ) = − kR 0 2 ln⎜1 − 2 R 02 ⎠ ⎝

(2)

Here r is the bead−bead separation, R0 is the maximum possible bonded bead−bead separation, and k is the spring constant. We use spring parameters from earlier work,29 namely R0 = 1.5σ and k = 30εij/σ2. A useful length scale in polymer systems is the end-to-end distance, ⟨R2⟩. The apparent size of a diblock chain in solution is understood as the sum of two contributions: (a) the sum of each block’s size; (b) the distance between the center of mass of the two blocks.30 The apparent sizes of symmetric diblock copolymers have been experimentally studied in the past.31−33 We assume that each block has a size equal to that of a homopolymer chain of the same length in a melt. Moreover, due to the fact that PI dislikes PS, we assume that the overall size is scaled by a factor of 1.3, which is consistent with what has been observed for PS−PI diblock copolymer chains in dilute chlorobenzene solution.34 In the experimental system, the PS block has molar mass equal to 68 kg/mol (648 monomers), and we want to map the PS block onto 65 beads. The end-to-end distance of a PS chain of 648 monomers in a melt can be estimated from the tabulation of Fetters et al.35 as ⟨R2PS⟩1/2 = 17.2 nm. Similarly, we get for the PI block (176 monomers) ⟨R2PI⟩1/2 = 8.5 nm. Thus, a chain of 78 beads has an end-to-end distance of 11.7σ, which gives us a length scale σ = 2.2 nm. Despite the large difference in relaxation times between polystyrene and polyisoprene, we expect that the minority block (PI), which has a small volume fraction ( f ≈ 0.17), will not significantly affect the kinetics of the system. We can get an estimation of the unit of time, τ, by mapping the diffusion coefficient versus chain length from our simulation data for homopolymers onto experimental results for polystyrene (from ref 30). We find τ = 5.5 × 10−6 s. With this approach, for the length of the chains that we are interested in, any entanglement effects on the diffusion coefficients are incorporated into τ. The acceleration in time scales is larger than that previously observed in ref 29 because we use a higher degree of coarse-

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SIMULATION METHODS Model. The model system boundaries consist of two parallel walls separated by distance H. The direction normal to the walls coincides with the z-axis; hence, the walls are located at z = ±H/2. The diblock copolymer film occupies the space between the walls (see Figure 1).

Figure 1. Zero-shear monolayer film at (χN)eff = 193. (a) Side view of the film. (b) Top view of the film. B-type beads in (a) and (b), and the top wall along with its wetting layer in (b) are rendered invisible for clarity. 4407

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× (H + 3σ), where the periodic boundary conditions and the minimum-image convention was applied only in the directions parallel to the walls. The box size was L = 80σ = 176 nm, H = 30σ = 65 nm for the monolayer and H = 46.4σ for the bilayer films. The total number of interaction centers was 206 838 and 311 826 for monolayer and bilayer films, respectively. In order to perform long simulations within a reasonable amount of wall clock time for these large systems, we have employed graphics processing units (GPUs). This was done in conjunction with a general-purpose, open-source molecular dynamics code written specifically for GPUs, called “highly optimized object-oriented many-particle dynamics” (HOOMD).43 Flory−Huggins Parameter. The Flory−Huggins χ parameter describes the degree of segregation in binary polymer mixtures and is a function of temperature. For PS−PI diblocks it was empirically determined44 as

graining in our model, mapping 10 polystyrene monomers into a single bead. The group contribution method of Constantinou and Gani37 was used to estimate the critical temperature of chains with a given number of monomers; a bead of type B corresponds to 10 PS monomers, and an A-type bead corresponds to 13.2 PI monomers. Thus, the energy interaction parameters between like beads (A−A and B−B) are assigned the following values: εBB = TCB/TCB = 1.0, εAA = TCA/TCB = 0.951. The cross-energy interaction parameter was set to εAB = 0.7. Typically in simulations with asymmetric energy interactions the crossinteraction is determined by the Lorentz−Berthelot law. In our case, values of εAB larger than 0.7 would require lower temperature for phase separation to occur, making the simulations significantly slower. The value of 0.7 was chosen to produce a temperature range where the polymers phase separate in the liquid phase and where the simulations are also computationally feasible. In experiments the film is in contact with the SiOx and one air surface, and previous work has shown that the PI block tends to wet both interfaces.38 In the simulations the interfaces are represented as two parallel and planar walls composed of a single layer of beads arranged in a square pattern with lattice spacing δ = 1.56. The surface bead interaction parameters chosen are σAS = σBS = σ, and εAS = 0.98, εBS = 0.67, which produce coverage of the surface by type A (PI) beads of 66%. The surface coverage in the simulations remained at the same level for different values of temperature. Integration. The Langevin dynamics method, which includes random and frictional forces in addition to conservative forces, was used to simulate the bead−spring polymer chains. A thermostat is required because without one the system temperature would increase without limit once shear is applied (work is done on the system). Langevin dynamics is an appropriate isothermal simulation procedure for thermodynamic properties near or away from equilibrium.39 Nevertheless, artifacts may be created for systems under an external field due to the nonconservation of momentum in Langevin dynamics. Methods such as dissipative dynamics40 and SLLOD algorithm41 successfully reproduce the Navier−Stokes type hydrodynamic equation. For the conditions that we examine the Langevin dynamics was sufficient, as demonstrated by simulations with alternate thermostats, which produced effectively the same shear rates at the conditions of interest. The equations of motion are given by m

d2ri dt 2

= −∇ri V − m Γ

dri + Wi (t ) dt

χ = 71.4/T − 0.0857

where T is temperature in kelvin. Establishing a relation between χ and the reduced system temperature, T*, for our simulations would allow us to set the energy scale of the model. To establish such as a relationship, we follow the work by Groot et al.45 and later by Glotzer et al.46 We start from the general Flory−Huggins free energy expression for a twocomponent polymer mixture ϕ 1 − ϕA F = A ln ϕA + ln(1 − ϕA ) + χϕA (1 − ϕA ) kT NA NB (6)

where φA is the volume fraction and NA and NB are the number of beads in each A and B block. Minimizing eq 6 for monomers, NA = NB = 1, we obtain that at two-phase coexistence χ=

ln(1/ϕA − 1) 1 − 2ϕA

(7)

On the basis of this relation of χ and φA, we can perform simulations to calculate φA as a function of the system’s temperature. We set up a simulation box of size 47.6σ × 12.7σ × 12.7σ and 7200 beads, of which half the beads are of type A. The interaction parameters are the same as with our polymer system except there are no bonds. We perform a Monte Carlo simulation which consists of two types of moves: random displacements and particle identity swaps. For a given set of values of the interaction parameters there is a temperature range over which the beads segregate into two phases. Once equilibrium is reached, we measure the concentration profile along the long dimension of the simulation cell; see Figure 2a for a representative concentration profile. Repeating the process for different temperatures, we get φA as a function of T* and from eq 7 a relationship between reduced temperature and χ. Figure 2b shows this approximately linear relationship

(3)

where Γ is the friction coefficient, Wi(t) describes the Brownian forces acting on the bead, and V = ∑i i

(12)

We looked at ⟨R2g⟩ for two groups of diblock chains; the first group contains those chains whose minority block is adsorbed on the surface, and the second contains those chains forming the layer(s) of spherical domains. In both groups variation of temperature over the range 1.2 ≤ T* ≤ 1.8 (or 103 ≤ (χN)eff ≤ 193) produced no significant changes to ⟨R2g⟩. The overall behavior suggests that at zero shear temperature and (χN)eff have little effect on the density profile of the thin film and do not significantly affect the conformation of the diblock chains. This can be understood from the fact that the chains are in a melt and not stretched. Thin Films with Shear. The effects of shear are studied by moving each wall along the x-axis at certain velocity, ±vs, as seen in Figure 1a. As the walls move, collisions with the polymer beads occur, which create a Couette-like shear flow. Initial structures were the equilibrium structures obtained at zero-shear at the desired temperature, as described above. After a steady state was reached for a given surface velocity, vs, a production run of 1.5 × 107 time steps was performed. The surface velocity is an input parameter in our simulations, but since there is a linear velocity profile within the film, we will use the shear rate in the middle of the film, γ̇, to analyze the results. Figure 6 shows screenshots of simulations of a bilayer film at (χN)eff = 193 (T = 223 K) for different shear rates. At zero-shear spherical domains (along with some globular structures) are formed (see Figure 6a). Small shear rates do not cause any significant changes (see Figure 6b). However, above a critical shear rate, the spherical domains elongate, adopting an ellipsoidal shape, and nearby domains start to merge. The newly formed objects continue to elongate in the direction of shear, and finally cylinder-like structures begin to appear (see Figure 6c). For higher values of shear rate, longer and better shaped cylinders are formed (see Figure 6d). The above mechanism of cylinder formation is consistent with our previous simulation work,28 but the current model allows us to examine the system in more detail. Figure 7 shows the shear rate measured in the middle of the film versus the surface velocity. Shear rate increases with surface velocity, but the trends are not the same between films of different thicknesses. This is better seen in the inset to Figure 7, where the ratio of the shear rate to the ideal shear rate, γ̇0 = 2vs/ H, is plotted as a function of surface velocity. Bilayer films have

Figure 4. Density profile of zero-shear bilayer at (χN)eff = 193 as a function of height: p(z) is the normalized probability of finding a bead (A and B) at certain point on the z-axis; pA and pB are the corresponding probabilities of finding A- and B-type beads, respectively.

profiles are normalized so that ∫ ∞ 0 p(z) dz = 1. The sum p(z) = pA(z) + pB(z) is nearly constant, except near the walls. A-type (PI) beads wet the confining walls more than B-type (PS) beads but do not completely cover the surface, leaving the surface to be partially wet by B-type beads as well. The resulting structure comes from the balance of two opposite effects. On one hand, type A beads have a higher attraction with the surface beads and so for enthalpic reasons are more likely to be adsorbed on the surface, but on the other hand, the minority block will have the tendency to avoid the surface for entropic reasons, as reported in previous studies.12,14,28 Near the walls, the density is peaked corresponding to well-defined fluid layers. Beyond a distance of order 8σ the oscillations become negligible and the total density approaches that of the bulk liquid. The role of temperature in the structure of thin films can be seen in Figure 5 where we plot the density profile of the adsorbed chains for a bilayer film at different temperatures as a function of the distance from the confining wall. A decrease in (χN)eff results in the density profile displaying reduced structural oscillations near the surface. Moreover, the tail of the distribution also shows a change. At lower temperatures the brush is less extended and closer to the confining wall. In addition to the density profile, we are also interested in the conformations of the diblock copolymers. A common way to quantify the conformation of a polymer chain is to measure the radius of gyration 4410

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momentum to the rest of the film. In contrast, in bilayer films the fraction of adsorbed chains drops to about one-third and the adsorbed chains do not cover the whole film, suggesting that transferring momentum to the middle of the film becomes more difficult. Less momentum transferred means there is less flow of chains within the film and bilayer films thus tend to slip more than monolayer films. Indeed, the same is observed in homopolymer films. Homopolymer chains are less stretched than the diblock films for the same temperature and surface velocity, and thus there is less momentum transfer. We calculate the ratio of the average force acting on the walls over the walls’ area, ⟨Fx⟩/A, versus the shear rate. Results are shown in Figure 8 for diblock copolymer thin films at (χN)eff =

Figure 6. Bilayer thin film of diblock copolymers at (χN)eff = 193 at different shear rates: (a) zero shear; (b) vs = 0.003στ−1 (γ̇ = 0.00026τ−1); (c) vs = 0.005στ−1 (γ̇ = 0.00051τ−1); (d) vs = 0.008στ−1 (γ̇ = 0.0001126τ−1). The top half of the film and all B-type beads are made invisible for clarity.

Figure 8. Average force acting on the walls, divided by the walls’ area, versus the shear rate, γ̇, for diblock copolymer thin films at (χN)eff = 193 (solid symbols) and (χN)eff = 155 (open symbols). Circles and squares correspond to monolayer and bilayer films, respectively.

193. At higher values of γ̇ shear thinning is observed. Our results are in reasonable agreement with experimental data. Specifically, viscosities obtained from the simulations for a monolayer with a shear stress of ⟨Fx⟩/A = 0.04ε/σ3 ≈ 9.4 kPa are ∼1.2 × 103 Pa·s, while experimental values for bulk diblocks of similar composition and molecular weight at 125 °C are found to be of order 105 Pa·s.51 An interesting observation is that the minimum shear stress required to trigger a sphere-tocylinder transition is different between films of different thicknesses. For (χN)eff = 193 monolayer films require 1.4 times more shear stress than bilayer films, which is in good agreement with the experiments of Hong et al.19 In the experiments, the required stress to observe the transition was between 10 and 15 kPa for monolayers, while for bilayer films the transition occurred between 5 and 10 kPa. The formation of cylinders from spherical domains requires some further examination. We first identified the objects formed in the middle of the film based on the criterion that two chains are considered to belong to the same object if the distance between any of their A-type beads is smaller than 1.2σ. Once the objects in the middle of the film have been identified, and the number of chains, Nc, that they are composed of measured, then we measure the asphericity of each object, As(Nc). Asphericity allows us to further understand the influence of shear on the shape of the objects and is computed as follows:

Figure 7. Shear rate versus surface velocity for diblock copolymer films (solid symbols) at (χN)eff = 193 (equivalent to T* = 1.2) and homopolymer films (open symbols) at T* = 1.2. Circles and squares correspond to monolayer and bilayer films, respectively. The dashed lines indicate the approximate boundaries between the formation of spheres and cylinders in diblock copolymer films. We consider that cylinders are formed and become dominant structural feature of the film when ⟨As⟩ > 0.4. The inset shows the ratio of the shear rate to the ideal shear rate as a function of the surface velocity.

significantly smaller ratio values than monolayers, which means there is more slip. To understand this behavior better, we examined homopolymers (composed of B-type beads) of the same length in films of the same thickness and at the same surface velocity. Homopolymer films display more slip at high surface velocities in comparison to diblock films (see Figure 7). This suggests that the origin of the slip difference between the different film thicknesses is closely related to the structure and conformation of the chains within the film. Indeed, in diblock monolayer films half the chains are adsorbed on the confining surfaces with a highly directional conformation perpendicular to the surface (details on the conformation are discussed below). When shear is introduced into the film, the adsorbed chains would prefer to move along with the surface, but since they have a conformation spanning half the film, they transfer

d

As = 4411

∑ j > i ⟨(Li2 − Lj2)2 ⟩ d

(d − 1)⟨(∑i = 1 Li2)2 ⟩

(13)

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where Li are the principal components of the squared radius of gyration of the object (considering only the beads of type A) and d is the number of principal components (equal to the number of spatial dimensions). The quantity As is bounded within the range 0 ≤ As ≤ 1. When As takes the value 1, then the object extends in one dimension (rod-like), while when it takes the value 0, the object has a spherical shape. The objects were broken and re-formed throughout the course of the simulations. This results in numerous small clusters and a small number of large clusters; we are interested in the behavior of the larger clusters. Therefore, asphericity was averaged in each time instance as follows: ⟨A s⟩ =

1 Nm

Nm

∑ j=1

energy, affecting the chain dynamics. In Figure 10, the meansquare radius of gyration is plotted as a function of shear rate.

N

∑i =01 NcA s(Nc) N

∑i =01 Nc

(14)

where N0 is the number of objects at a given time instance and Nm is the number of measurements. ⟨As⟩ takes into account all the clusters formed in a film but with an emphasis on larger clusters. Figure 9 shows the behavior of ⟨As⟩ versus the shear

Figure 10. Radius of gyration of adsorbed and nonadsorbed chains as a function of shear rate for (a) monolayer and (b) bilayer films. The dashed lines approximately indicate the sphere-to-cylinder order− order transition at ⟨R2g⟩ = 28σ2.

The chains are not significantly affected at small shear rates. However, above a critical shear rate the chains are significantly stretched. Adsorbed chains were more stretched on average than the nonadsorbed diblock chains. This difference was more pronounced in bilayer relative to monolayer films. The stretching of the diblock chains is not isotropic, so it is necessary to consider the different directions of the chains with respect to the confining walls. We resolve the vector ri − rj in eq 12 into components along x, y, and z. The x and y components describe the stretching of the diblock chains parallel to the confining walls, with x parallel and y perpendicular to the shear direction. The z component describes the stretching perpendicular to the confining walls. Results for a bilayer film at (χN)eff = 193 are shown in Figure 11. As mentioned above, below a shear rate threshold the diblock chains do not stretch significantly. Above that threshold the chains start being significantly stretched and in particular the stretching occurs in the x-direction. In the y-direction there was no significant stretching. In the z-direction there is a decrease, which implies

Figure 9. Average value of asphericity versus the average force acting on the walls, divided by the walls’ area, in films at (χN)eff = 193. Circles and squares correspond to monolayer and bilayer films, respectively. Two regions have been identified: in (I), spheres are unaffected by stress, while in (II) stress causes the spheres to elongate and form cylinders. The dashed line outline the approximate boundaries between the two regions.

stress for films with different thickness at (χN)eff = 193. Two regions are identified; in the first region (I), the stress applied to the system does not affect the shape of the objects, i.e., ⟨As⟩ has small values close to zero. This signifies that most objects have nearly spherical shape. In the second region (II), the stress significantly affects the shape of the objects, which become elongated and form cylinders. There is, however, a difference between monolayer and bilayer films. Although the spheres elongate at the same stress, the formation of long cylinder-like objects in bilayer films requires less stress than in monolayer films. We consider that the formation of cylinders starts when ⟨As⟩ > 0.4. To better understand the conditions under which the sphereto-cylinder transition occurs, we focus on the effects of shear on the conformation of diblock chains. Previous work has demonstrated that chain stretching due to shear flow significantly alters the dynamics and morphology for phaseseparated polymer blends52 and diblock copolymers.53 The main reason behind this is that stretched chains store elastic

Figure 11. Components of the radius of gyration for nonadsorbed diblock chains in the bilayer film at (χN)eff = 193. The dashed line marks the approximate location of the sphere-to-cylinder transition, which is based on the behavior of asphericity (⟨As⟩ > 0.4 corresponds to cylinders). 4412

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that although a diblock chain gets stretched in the direction of shear, it adopts configurations to minimize the chain’s exposure to different velocity layers. If a chain spans different velocity layers, the segments of the chain will have on average different velocities, causing the chain to stretch. In other words, to minimize the stretching effects, the diblock chains adopt configurations that are more parallel to the shear direction and to the confining walls. Our results indicate that the sphere-to-cylinder transition is triggered when the radius of gyration exceeds a critical value, ⟨R2g⟩ = 28σ2. This value was independent of film thickness and temperature, indicating that the shear-induced sphere-tocylinder transition is related to chain conformational properties: there is a point at which polymer chain stretching favors cylinders over spheres. This resonates with Zhulina’s work54 on the shape of micelles in solution (at equilibrium) based on scaling arguments and the composition of diblock chains, though that work is not directly applicable to our case because it assumes that both blocks have a persistence length equal to the monomer size, while in our case, the chains are stretched significantly in one direction, implying persistence lengths much larger than a single monomer. Figure 12 shows the effects of temperature on the stretching of diblock chains under shear. For monolayer and bilayer films,

Figure 13. (a) Screenshot of a bilayer film at (χN)eff = 155. (b) Screenshot of a bilayer film with well-formed cylinders at (χN)eff = 193. The B-type beads and a fraction of the A-type beads are rendered invisible for clarity.

neighboring spheres or cylinders to reform cylinders with the diagonal orientation. The effects of temperature on the system can also be observed in the density profile of the adsorbed diblock chains. We have already shown that at zero shear temperature makes the wetting layer increase in thickness (see Figure 5). In Figure 14, a comparison is shown for a bilayer film at (χN)eff = 193

Figure 14. Density profile of adsorbed diblock chains for films at (χN)eff = 193 under different shear rates.

between zero-shear and high-shear conditions. For higher shear rates, the wetting layer becomes thinner and denser. The overall behavior is similar to that seen in the conformations of the diblock chains, where increased temperature opposes the effect of shear. Another way to measure the effect of shear on the wetting layers is by calculating the bond orientation of the adsorbed chains. This is quantified by calculating the angle between sequential bonded beads starting from one end of the chain (we picked the end where the minority block starts) to the other end. Then the results are averaged and plotted along the z-axis (see Figure 15). The results show that at zero shear the adsorbed chains have a directionality perpendicular to the surface. This effect remains the same throughout the wetting layer, up to the point where a layer of spheres (or cylinders) is reached at about z = 10−15σ; the results can be compared with Figure 4. When shear is introduced, the bond z-orientation decreases because the adsorbed minority block moves with the surface and drags along the rest of the chain.

Figure 12. Radius of gyration of diblock chains as a function of the shear rate. The dashed line marks the approximate location of the sphere-to-cylinder transition at ⟨R2g⟩ = 28σ2.

an increase in temperature (decrease in χ) results not only in smaller shear rates (as discussed above) but also in less stretching at high shear. In general, entropic effects are more important at higher temperatures. When polymer chains are stretched, they lose a significant number of potential configurations, making stretching entropically unfavorable.55 Thus, temperature has an effect opposite to that of shear rate. Temperature also influences the shear rate needed for a certain surface velocity: the shear rate decreases as (χN)eff decreases, and the shape of the cylinders deteriorates by not being parallel to the surface. Figure 13a,b shows screenshots of a bilayer film at different values of (χN)eff. At (χN)eff = 193 the cylinders formed are parallel to the surface while for lower values of (χN)eff, the cylinders adopt an orientation diagonal to the surfaces and can span both layers. Such a diagonal orientation should be unstable with a linear velocity profile: a cylinder would break since different parts are moving in opposite directions. The broken parts must merge with

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CONCLUSIONS In this paper, we have used Langevin dynamics simulations of a bead−spring model to investigate the shear-induced sphere-tocylinder transition of thin films of diblock copolymers. We focus on films forming one or two layers of spheres. The minority block wets the film’s interfaces forming a brush-like 4413

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Figure 15. Bond orientation of the adsorbed chains perpendicular to the walls (z-direction) for a bilayer film at (χN)eff = 193 for different shear rates.

structure. Above a critical shear rate, the spheres transform into cylinders. Our model successfully captures many features observed in recent experiments and provides insight into the mechanism of the sphere-to-cylinder transition. Once shear is introduced, the spheres elongate and merge with each other to form cylinders parallel to the shear direction. There are two factors that contribute to the transition: The first is that the diblock chains are significantly stretched in the direction of shear and are pulled along the direction of shear. This leads to the melting of the spherical domains and the formation of cylinders. The second factor is the role of the wetting layers. They shrink under shear, leaving more space for the spherical layers in the middle of the film, but less space near the surface. The wetting layers in monolayer films span a significant part of the film thickness and penetrate into the middle of the film. In bilayer films the wetting layers are similar to those in monolayer films but span a smaller fraction of the film thickness, which means less momentum transfer to the middle of the film. The latter provides an understanding as to why the sphere-to-cylinder transition was observed to require higher stresses in monolayer rather than in bilayer films. Even though temperature (within the range explored) does not significantly affect the structure at zero shear, such as the density profile and the diblock chain conformation; it becomes a significant factor for films under shear. Higher temperatures enhance the entropic effects of chain stretching and oppose the stretching caused by shear.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support for this work was provided by the Princeton Center for Complex Materials (PCCM), a U.S. National Science Foundation Materials Research Science and Engineering Center (Grant DMR-0819860).

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