Colloidal Plastic Crystals in a Shear Field - Langmuir (ACS Publications)

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Colloidal Plastic Crystals in a Shear Field Fangfang Chu,†,‡ Nils Heptner,†,‡ Yan Lu,‡ Miriam Siebenbürger,‡ Peter Lindner,§ Joachim Dzubiella,*,†,‡ and Matthias Ballauff*,†,‡ †

Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany Soft Matter and Functional Materials, Helmholtz-Zentrum Berlin, Hahn-Meitner Platz 1, D-14109 Berlin, Germany § Institut Laue-Langevin, 71 avenue des Martyrs - CS 20156 - 38042 Grenoble Cedex 9, France ‡

S Supporting Information *

ABSTRACT: We study the structure and viscoelastic behavior of 3D plastic crystals of colloidal dumbbells in an oscillatory shear field based on a combination of small-angle neutron scattering experiments under shear (rheo-SANS) and Brownian dynamics computer simulations. Sterically stabilized dumbbell-shaped microgels are used as hard dumbbell model systems which consist of dumbbell-shaped polystyrene (PS) cores and thermosensitive poly(N-isopropylacrylamide) (PNIPAM) shells. Under increasing shear strain, a discontinuous transition is found from a twinnedfcc-like crystal to a partially oriented sliding-layer phase with a shear-molten state in between. In the novel partially oriented sliding-layer phase, the hard dumbbells exhibit a small but finite orientational order in the shear direction. We find that this weak correlation is sufficient to perturb the nature of the nonequilibrium phase transition as known for hard sphere systems. The discontinuous transition for hard dumbbells is observed to be accompanied by a novel yielding process with two yielding events in its viscoelastic shear response, while only a single yielding event is observed for sheared hard spheres. Our findings will be useful in interpreting the shear response of anisotropic colloidal systems and in generating novel colloidal crystals from anisotropic systems with applications in colloidal photonics.

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INTRODUCTION Suspensions of spherical colloidal particles provide excellent model systems for studying the structure and phase behavior of condensed matter because they exhibit the fluid-to-crystal transition1 as well as the fluid-to-glass transition.2,3 For particles with a judiciously chosen size, Brownian motion ensures the attainment of local equilibrium within a typical experimental time scale. Compared to atomic and molecular systems, the shear modulus of such a suspension is of the order of kBT/a3, where a is the radius of the particles and kBT is the thermal energy, allowing easy manipulation and control of its material properties by external fields. For particles with a radius in the colloidal domain, that is, of the order of several hundred nanometers, the application of an oscillating shear field, for instance, may lead to the crystallization of a dense fluid while steady shear can lead to the melting of a colloidal crystal if the applied shearing force is strong enough.4−6 Thus, nonequilibrium phases with novel properties may be formed by colloidal suspensions of spherical particles that have been the subject of intense research in recent years. The combination of experimental work with simulations has led to detailed out-ofequilibrium phase diagrams.6 A shear-induced transition has been reported for hard spheres in an oscillatory shear field.4,6,7 A so-called twinned-fcc phase is formed at low strain, where two fcc crystal twins with different stacking are alternating in one cycle. A sliding-layer phase is induced by high strain in © XXXX American Chemical Society

which the crystal planes reorient and the sliding hexagonal layers are oriented with their most densely packed directions along the shear direction. However, much less is known about these nonequilibrium phases of particles with small anisotropy. Densely packed particles with an axial ratio slightly above unity are predicted to form a plastic crystal in equilibrium in which the centers of gravity are fully ordered whereas the long axes of the particles can rotate freely.8−14 These plastic crystals present highly interesting intermediates between solids and liquids. They feature a variety of fascinating material properties on molecular scales12,15 as well as on colloidal scales11 with possible applications in photonics due to their complete band gaps.16−20 Mildly anisotropic hard-core model colloids form such a plastic crystal with a known theoretical equilibrium phase diagram.14 Until now, the number of experimental studies on both equilibrium and nonequilibrium behavior of plastic crystals is still very small. We are aware of only the early pioneering studies of Mock and Zukoski who used dumbbellshaped particles made by emulsion polymerizations.21,22 The application of a shear field led to the appearance of Bragg spots that could be indexed by sliding hexagonally packed layers. Liu Received: December 19, 2014 Revised: January 29, 2015

A

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Langmuir et al.23 studied plastic crystals formed by charged colloidal rods and their alignment in an electrical field. Recently, we provided clear evidence that thermosensitive core−shell dumbbell-shaped particles can serve as hard dumbbell model systems for forming plastic crystals.24 Figure 1a gives a schematic representation of this model system. The

transmission electron microscopy (Figure 1b) demonstrates that these particles are nearly monodisperse and may be looked upon as short spherocylinders in an aqueous dispersion. As shown by Chu et al.,24 suspensions of these particles in water form a crystalline phase which was evidenced by the Bragg reflections in the visible range and by rheological measurements. From a comparison with the theoretical phase diagram of hard dumbbells,13,14 it was conjectured that these particles form plastic crystals. However, there is no detailed study on the equilibrium and nonequilibrium phase behavior of colloidal plastic crystal from hard dumbbells with a fixed aspect ratio. In this study, the core−shell dumbbell-shaped microgels as shown in Figure 1a,b are used as hard dumbbell model systems, and the core−shell spherical microgels (Figure 1c) are used as a reference system of hard spheres. We present the equilibrium phase diagram of hard dumbbells with L* = 0.24 that is in accord with the theoretical prediction.11,13,14 On the basis of a combination of rheo-SANS measurements and nonequilibrium Brownian dynamics (BD) simulations, we make the first report on the structural evolution and the corresponding nonequilibrium phase behavior of colloidal plastic crystals from hard dumbbells with L* = 0.24 under oscillatory shear. We observe that hard dumbbells exhibit a shear-induced phase transition from the twinned-fcc phase to the sliding layer phase, which is analogous to sheared hard spheres.6,7,27,28 However, hard dumbbells in an oscillatory shear field show a discontinuous phase transition and form a novel partially oriented sliding-layer phase at high strain. We further demonstrate that this novel nonequilibrium transition is accompanied by a viscoelastic yielding behavior with two yielding events which have not been observed in sheared colloidal crystals before.

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Figure 1. Experimental systems. (a) Scheme of a dumbbell-shaped core−shell microgel model system. Rc and RH denote the hydrodynamic radius of one sphere in the dumbbell-shaped core and microgels, respectively. LH is the thickness of the PNIPAM shell. L* is defined as the ratio of the center-to-center distance (L) to the diameter of one sphere in the dumbbell-shaped microgels. Cryo-TEM images of dumbbell-shaped microgels (b) and spherical microgels (c) that are measured at 15 °C (scale bar 250 nm).

MATERIALS AND METHODS

Materials. Sodium dodecyl sulfate (SDS), potassium persulfate (KPS), N-isopropylacrylamide (NIPAM), N,N-methylenebis(acrylamide) (BIS), tetrahydrofuran (THF), 3-(trimethoxysilyl) propyl methacrylate (MPS), 2,2′-azobis (2-methylpropionitrile) (AIBN), styrene, and inhibitor remover were purchased from SigmaAldrich. Styrene was purified by inhibitor remover and stored in the refrigerator before use. Water used in this work was purified through reverse osmosis and ion exchange (MilliRo, Millipore). The other reagents were used as received without further purification. Synthesis. Dumbbell-Shaped Microgels. On the basis of the phase-separation technique,24,25 monodisperse dumbbell-shaped polystyrene (PS) core particles were first fabricated. Using dumbbellshaped PS particles as seeds, a thermosensitive PNIPAM shell crosslinked by BIS was attached to the surface of core particles.24 In a typical run, a 2 L three-necked glass flask served as the reactor, where 150.6 g of a 7.13 wt % PS core latex was mixed with 145 g of water containing a mixture of NIPA and BIS monomers. After complete mixing (via stirring for ca. 20 min), the reactor was charged with 0.3 g of KPS dissolved in 5 g of H2O under a nitrogen atmosphere. The reaction started by increasing the temperature to 80 °C and ran for 4.50 h. Afterward, the dumbbell-shaped microgels were cleaned by ultrafiltration for around 1 month. The purified dumbbell-shaped microgels were redispersed in a 50 mM KCl solution, which is sufficient to screen the residual charges from synthesis. Spherical Microgels. Spherical microgels have been synthesized in two main steps. PS spheres were first prepared through emulsion polymerizations using KPS as the initiator.29 Next, a cross-linked PNIPAM shell was attached to the surface of spherical seeds through seeded polymerizations. In a 2 L glass flask as the reactor, 7.9 g of PS seeds was mixed with 8.4 g of NIPAM and 0.9 g of BIS in 250 mL of water. After stirring for ca. 20 min, 0.3 g of KPS dissolved in 10 mL of H2O was added to the reactor under the protection of nitrogen. The

dumbbell-shaped core consists of two partially fused polystyrene spheres, which has been prepared on the basis of the phase-separation technique.25 The distance between the centers of these spheres is denoted as L. Onto the surface of this core particle, a thermosensitive network of poly(N-isopropylacrylamide) (PNIPAM) cross-linked by N,N′-methylenebis(acrylamide) (BIS) has been grafted. Suspended in cold water, this network will swell and thus increase the effective volume fraction of the particles. Heating the suspension to temperatures below the volume transition of the PNIPAM network will lead to a shrinking of the network and a concomitant decrease in the effective volume fraction which can be adjusted precisely in this way. Spherical core−shell particles with this architecture have been the subject of intense research recently,26 and it can be shown that their interaction can be well approximated by a hard sphere potential (cf. Siebenbürger et al.3 and further references therein). The thickness LH of the network and the diameter D can be obtained precisely for a given temperature from light-scattering experiments and corresponding calculations.24 Hence, the aspect ratio, defined here as L* = L/D, can be determined, and the effective volume fraction ϕeff can be calculated precisely. A micrograph of the particles taken by cryogenic B

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Langmuir polymerization was started by increasing the temperature to 80 °C, which lasted 4.5 h. In the end, the spherical microgels were purified by ultrafiltration for around 1 month. The purified spherical microgels dispersed in a 50 mM KCl solution were used as a reference system. Dynamic light scattering experiments have been carried out on PS spherical cores at room temperature (T ≃ 25 °C) and on core−shell spherical microgels at various temperatures. The radius of the PS spherical core was determined to be 75 nm, and the thickness of the PNIPAM shell was linearly dependent on temperature (LH = −1.37T nm/°C + 174 nm, with T in units of °C). Volume fractions of the spherical microgels at various temperatures have been calculated from the corresponding dimensional information and solid concentration.30 Characterization. Cryogenic Transmission Electron Microscopy (Cryo-TEM). Cryo-TEM specimens were prepared by the vitrification of thin liquid films supported on a TEM copper grid with a lacey carbon film (200 mesh, Science Services, Munich, Germany) in liquid ethane at its freezing point using a Vitrobot Mark IV (FEI, Eindhoven, The Netherlands). The specimens were inserted into a cryo-transfer holder (Gatan 914, Gatan, Munich, Germany) and transferred into a JEOL JEM-2100 (JEOL GmbH, Eching, Germany). The TEM measurement was carried out at an acceleration voltage of 200 kV. All images were recorded digitally by a bottom-mounted 4k CMOS camera system (TemCam-F416, TVIPS, Gauting, Germany) and processed with a digital imaging processing system (EM-Menu 4.0, TVIPS, Gauting, Germany). The cryo-TEM measurements were performed to investigate the morphology of microgels dispersed in water. Dynamic Light Scattering Measurements. Dynamic light scattering (DLS) and depolarized dynamic light scattering (DDLS) experiments were carried out on an ALV/DLS/SLS-5000 compact goniometer system equipped with a He−Ne laser (with a wavelength of 632.8 nm). The temperature of the samples was controlled by a thermostat (Rotilabo, ±0.1 °C). For every temperature, at least 1 h was set for the sample to reach equilibrium. All DLS measurements were performed at scattering angles of between 20 and 150° in steps of 5°. For DDLS measurements, an angular step of 1° was set for scattering angles of between 20 and 30°, and a step of 2.5° was set for scattering angles of between 30 and 50°. For both DLS and DDLS measurements, three runs per angle were measured and averaged. The intensity autocorrelation functions were evaluated using the CONTIN software provided by the manufacturer (ALV Correlator Software 3.0). Rheology and SANS. Rheology. Rheological measurements have been performed with a stress-controlled rheometer (Physica MCR 301, Anton Paar) equipped with cone−plate geometry (CP50). Yielding behavior has been investigated in an oscillatory shear field with a fixed frequency (f = 1 Hz) but various shear strains. The shear strains were increased with a logarithmic ramp from 0.1 to 1000%, where 60 points have been measured with a measurement time of 100 s per point. For all measurements, samples were presheared at a shear rate of 100 s−1 for 200 s, and a waiting time of 60 s was followed to guarantee that all measurements started with the same nonequilibrium initial state. Rheo-SANS. The rheo-SANS experiments were carried out at the D11 instrument of the Institut Laue-Langevin (ILL) in Grenoble in order to investigate the structural evolution of hard dumbbell suspensions under oscillatory strain. The rheo-SANS setup was built on the basis of the combination of SANS and a stress-controlled rheometer (Physica MCR 501, Anton Paar) with a Searle cell. A polychromatic beam from the cold source was monochromated by a helical slot velocity selector (ASTRIUM), which selected neutrons of ±9% about the mean wavelength as determined by the rotational speed of the drum. The wavelength used in this study was 13 Å. The instrument was equipped with a 96 × 96 cm2 3He gas detector (CERCA) with a 7.5 × 7.5 mm2 resolution. The distance between the detector and the sample can be varied from 1.2 to 39 m. In this study, all experiments were done with a detector distance of 39 m except for several transmission measurements and static experiments, which were done at an 8 m detector distance. The Anton Paar rheometer was mounted to perform rheological measurements on samples with the CTD 200/GL measuring system.

The used Searle geometry was composed of a fixed quartz cup with a diameter of 50 mm and a rotating cylinder with a diameter of 49 mm. The gap between both cylinders was 0.5 mm. The Ti cylinder (ME49/ Ti/SANS) of the same size was used to provide good solvent protection. The dumbbell-shaped microgels were first loaded into the shear cell, and half an hour was set for the temperature equilibrium at a fixed temperature. Prior to the SANS measurements on the sample in the equilibrium state, at least 1 h was set to allow the crystallization to set in. For experiments under shear, SANS measurements were made at various shear strains along the dependence of the storage modulus (G′) and loss modulus (G″). For each measurement, 600 oscillations (10 min) were first applied to induce the corresponding structure, and the following 900 oscillations (15 min) were averaged to yield the scattering patterns. In this work, SANS experiments were carried out with the incident radiation propagating in the radial beam configuration as schematically shown in Figure 2a. The coordinate

Figure 2. Schematic illustration of the rheo-SANS setup (a) equipped with a Searle cell. The coordinate frame (b) with velocity (x), velocity gradient (y), and vorticity (z) directions is defined to describe the scattering planes for the available geometries. frame (Figure 2b) with the velocity (x), velocity gradient (y), and vorticity (z) was defined to describe possible scattering planes. Because the incident beam runs along the gradient direction, the structure factor was thus projected into the vorticity−velocity scattering plane. Brownian Dynamics Simulations. To analyze the SANS results in more detail, we have performed particle-resolved BD simulations of plastic crystals under oscillatory shear. Here, N = 864 dumbbell particles have been modeled as Yukawa hard-sphere-like segment models31 subject to Lees−Edwards32 periodic boundary conditions. The BD simulations have been carried out using Ermak’s method33 for interacting particles in solution with an additional term to account for the oscillatory shear force. Hydrodynamic interactions are not included, which has been empirically justified for related dense systems in the low-frequency range.6 The particles interact via two spherical beads with a steep Yukawa potential (decay length κ−1 = 0.05) constraint at the constant distance L which is defined by

V (r ) = ϵ

⎧ ⎛r ⎞⎫ σ exp⎨ − κ ⎜ − 1⎟⎬ ⎝ ⎠⎭ ⎩ r σ

(1)

where ϵ = kBT and σ set the energy and length scales, respectively. The parameter κ allows us to tune the softness of the interaction and has been chosen to maintain computational performance and resemble the hard particle behavior. We have employed a forward Euler scheme in order to integrate the equations of motion as follows: The time scale is set to the Brownian time τ = σ2/DS0 of a single bead of diameter σ and diffusivity DS0 = kBT(3πηsσ)−1 with a solvent viscosity ηs. The parallel and perpendicular center-of-mass (COM) coordinates are updated according to n+1

R⃗ i , C

n

= R⃗ i , + Δt

D kBT

n

Fi⃗ , + δri , ui⃗ n

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n

R⃗ i , ⊥ = R⃗ i , ⊥ + Δt

R⃗ i

n+1

=

n+1 R⃗ i ,

+

D⊥ n Fi⃗ , ⊥ + δri ,1ei⃗ n,1 + δri ,2ei⃗ n,2 kBT

n+1 R⃗ i , ⊥

+

Δtγ(̇ t )R ynex⃗

⟨P2(uα , i)⟩cycle (t ) = ⟨P2(cosθα , i)⟩cycle (t ) (3)

where θα,i represents the angle between the long axis of particle i and direction α and the instantaneous values are averaged over all steadystate cycles. The full average P2(uα) has been taken over all time steps in the steady state. Furthermore, the translational order parameter Q4 evaluates the quality and type of the average local order of fcc crystal structures in the three-dimensional neighborhoods of all particles36 and is defined as

(4)

The shear flow affects only the COM transport in the x direction through the last term in eq 4. The directors are updated via ui⃗ n + 1 = ui⃗ n + Δt

Dr n Ti⃗ × ui⃗ n + δx1ei⃗ n,1 + δx 2ei⃗ n,2 kBT

(5)

Ql =

where T⃗ ni is the total torque exerted on particle i at time t = nΔt and random variates δri,α, α ∈{||, 1, 2}, δx1 and δx2 have zero mean and the Gaussian variances are ⟨δri,||2⟩ = 2D||Δt, ⟨δri,{1,2}2⟩ = 2D⊥Δt, and ⟨δxj2⟩ = 2DrΔt. The torque is composed of the interparticle and backgroundflow contributions.34 The time-dependent strain is given by

γ(t ) = γmax sin(2πft )

ql̅ (i) =

qlm(i) =

The imposed frequency equals f = 15.3 Hz on the experimental time scale. Because crystal nucleation is a few times slower in dumbbell systems than for hard spheres, we needed to simulate at higher frequencies in order to average over a sufficient number of cycles. The systems have been simulated in periodic boxes in the canonical NVT ensemble with box sizes Lx = 7.7259D, Ly = 8.1946D, and Lz = 13.3817D initialized in a crystalline (equilibrium) fcc state. Each simulation has been sheared immediately for 100τ at a frequency of f = 5τ−1 resulting in a total of 500 strain cycles. The averages have been calculated in the steady state over the last 250 strain cycles. To map between computational and experimental scales, we have set the interaction length scale to σ = D = 2RH. The Brownian time of a free suspended sphere (RH = 235 nm) in water at T = 10 °C = 283.15 K and viscosity η = 1.3059 × 10−3 Pa·s is τ = D2/D0 = 0.3269 s, where D0 = 6.78 × 10−13 m/s2. Thus, the average translational = diffusion coefficient of a free dumbbell with L* = 0.24 is DDB 0 0.90133D0, and the experimental strain frequency equals f = 1 Hz ≈ 1/ (3τ). The parallel (D|| = 0.92586DS0), perpendicular (D⊥ = 0.88907DS0), and rotational (Dr = 0.68927DSr = 0.68927(3DS0)/D2) diffusivities have been obtained by the shell bead model method and matched to the experimental data described in our previous work.24 The rotational Brownian time scale is thus τr = 0.68927D2/(3DS0). Simulation Analysis. From our simulations the corresponding full-2D intensity patterns I(qx, qz), as measured in SANS, have been calculated in the steady state. The center-of-mass positions R⃗ i and orientations ui⃗ yield the full scattering function I(qx, qy = 0, qz) including the orientation-dependent scattering amplitudes in the reciprocal velocity−vorticity plane as

l

∑

|qlm(i)|2

m =−l

(12)

1 Nb(i)

Nb(i)

∑ Ylm(R⃗ ij) j=1

(13)

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RESULTS AND DISCUSSION Equilibrium Phase Behavior. On the basis of our previous characterization of the dumbbell-shaped microgels by DLS and DDLS measurements, L is determined to be 105.12 nm, Rc is 98.80 nm, and LH is linearly dependent on temperature (LH = −1.57T nm/°C + 137.39 nm, with T in units of °C).24 Within the experimental temperature range between 10 and 20 °C, the aspect ratio is determined to be 0.24 ± 0.02, which is assumed to be constant at 0.24.24 The formation of an ordered phase at high packing fractions can be seen directly from the occurrence of Bragg reflections in the visible range (cf. the inset in Figure 3a for a photograph of a sample in the biphasic region). Crystallization takes place after several hours and is thus much slower than the crystallization of spherical core−shell particles of comparable size.23 Indeed, umbrella sampling computer simulations by Ni and Dijkstra37 showed much slower nucleation rates of dumbbells in the plastic crystal phase at low supersaturation when compared to hard spheres.37 The crystalline phase is slightly more dense than the fluid phase, and clear separation into a crystalline phase coexisting with a fluid phase can be observed (cf. the inset in Figure 3a). Figure 3 displays the resulting volume fraction of the crystalline phase as the function of the overall dumbbell volume fraction. The black filled circles mark the theoretical values for the phase boundary which is shown in Figure 3b for a wider range of aspect ratios.11,13,14 The experimental biphasic region in Figure 3b, indicated by the open red squares at L* = 0.24, is slightly larger than the one predicted by simulations, but the deviations are hardly beyond experimental uncertainty. So we conclude that our system forms plastic crystals at certain volume fractions. To the best of our knowledge, this is the first quantitative measurement of the phase behavior of a colloidal plastic crystal for a given aspect ratio. The shear response of this thermodynamic plastic crystal will be analyzed in the following text. Nonequilibrium Phase Behavior. Rheo-SANS. We first discuss the viscoelastic response of the colloidal crystals in an oscillatory shear field for both hard sphere (L* = 0) and hard

(8)

where 1

∫−L* dt cos(q ⃗·ui⃗ RH[t + L*])

4π 2l + 1

where Ylm(R⃗ ij) represents the spherical harmonics for normalized separation vectors Rij and Nb(i) is the number of the ith particle’s next neighbors.

∑ A(q ⃗; ul⃗ ) A(q ⃗; uj⃗ )e−iq ⃗·(R⃗l − R⃗ j)

A(q ⃗ ; ui⃗ ) = 4πRH 3

(11)

i=1

is defined by the components (7)

j,l

N

∑ ql̅ (i)

where index i runs over all particles, l = 4, and

thus the linear solvent velocity profile is

I(q ⃗) =

1 N

(6)

vx(y , t ) = γ(̇ t )y

(10)

(9)

is the scattering amplitude of a dumbbell with orientation u⃗i. The scattering vectors are restricted to qα = n2π/Lα with integer values of n = 1, 2, 3,... and a box length of Lα (α = x, y, z). The brackets ⟨...⟩ denote an ensemble average over all particles and steady-state trajectories. We have also determined appropriate translational and orientational order parameters to characterize the crystalline structures in detail. We use order parameter P2(uα,i(t)) as a measure of the orientational order in direction α = x, y, z defined as the second Legendre polynomial of the usual scalar product between (normalized) dumbbell orientation u⃗i and direction vector e⃗α via 35

D

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Figure 4. Dependence of storage modulus G′ (filled circles) and loss modulus G″ (open circles) of hard spheres at ϕeff = 0.57 (a) and hard dumbbells at ϕeff = 0.60 (b) on increasing shear strains at a fixed frequency f = 1 Hz. Strain positions I−III along the G curves indicate the strains used in the rheo-SANS experiments, which are chosen based on the positions corresponding to the two yielding events. See the text.

Figure 3. (a, b) Experimental fluid−crystal coexistence boundaries (open red squares) of hard dumbbells at L* ≈ 0.24 compared to the theoretical prediction (filled black circles and gray coexistence regions) for hard dumbbells.13,14 In (a), a photograph of the experimental sample at coexistence is shown as the inset. In (b), our investigated state points by rheology for spherical colloids (open blue circle) and dumbbells (filled red square) are included.

This definition has structural origins, as will become clear further in the following. Note that viscoelastic processes with two yielding events have been reported widely for colloidal glasses of particles with a short-range interparticle attraction.38,39 Because the dumbbell-shaped microgels used in this work are purely sterically stabilized, these two yielding events should not be related to the effect of attractions between particles as discussed by Kramb and Zukoski40,41 as well as by Koumakis and Petekidis.38 The work of Zukoski40,41 and of Schweizer42 has revealed that the shape anisotropy also affects the yielding process of anisotropic systems in the glassy state due to coupled translational and rotational correlations. Translation−rotation coupling is known to have substantial effects on equilibrium phase transitions and elasticity in molecular systems.12 The degree of structural translational and orientational order will now be illuminated by SANS measurements and computer simulations. Figure 5 gives a summary of the SANS measurements made in the plastic crystalline phase of the dumbbells. As shown in Figure 5a, the plastic crystal of hard dumbbells shows a purely isotropic scattering pattern. Its corresponding SANS intensity I(q) versus q (the magnitude of the scattering vectors) in Figure S1 indicates the existence of primary, secondary, and tertiary scattering rings at 0.0013, 0.0021, and 0.0026 Å−1, respectively. Combined with the observed iridescence, we conclude that colloidal plastic crystals in their equilibrium state have a polycrystalline structure. After the application of increasing oscillatory shear, new nonequilibrium patterns become visible as displayed by the representative image patterns labeled b−f in the same figure. The two scattering spots of the primary scattering ring as shown in Figure 5b indicate the onset of crystallization at γmax = 23.6%. With the

dumbbell systems (L* = 0.24) under increasing shear amplitudes but a fixed frequency. The effective volume fraction, ϕeff, for hard dumbbells is 0.60 (12.05 wt %, 10 °C), while ϕeff is 0.57 for the hard sphere reference system (12.76 wt %, 13 °C) Figure 4a,b displays the dependence of storage and loss moduli, G′ and G″, on increasing strain amplitudes at a frequency of 1 Hz. Figures 4a refers to a suspension of equivalent but strictly spherical core−shell particles whereas Figure 4b displays the same experiment for a suspension of hard dumbbells. In both cases, the respective volume fractions were chosen so that the systems are fully crystallized (filled sphere and square symbols in Figure 3b). As we easily see, the viscoelastic response of the two systems is very different. In particular, the dumbbells display a double yielding process at strain magnitudes in the range of roughly 20 to 100%. By defining yielding as the onset of the nonlinear shear response, we estimated the yielding stress (G′γ) as 22.79 Pa at γmax ≃ 13.14% and 22.94 Pa at γmax ≃ 51.51%. It is interesting that these two yielding stresses correspond to different shear strains but show comparable values. In the case of spherical particles, there is only one yielding peak with a distinct maximum of G″ related to the dissipation of energy when melting the colloidal crystals of hard spheres.4 Figure 4b suggests that the second yielding for the dumbbell system proceeds through a yet unobserved intermediate state that is characterized by a local peak in G″, resulting in two yielding events under increasing shear strain. In the following text, we will define this intermediate structure as state II, flanked by states I and III. E

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Figure 5. Structural evolution of the plastic crystal (a) formed from the hard dumbbells with L* ≈ 0.24 and ϕeff = 0.60 under oscillatory shear at fixed frequency ( f = 1 Hz) but increasing strain amplitudes, γmax. The applied strain amplitudes are chosen on the basis of the dependence of G′ and G″ on shear strains at γmax = 23.6% (b), γmax = 50% (c), γmax = 60% (d), γmax = 116% (e), and γmax = 1000% (f), respectively. The scattering patterns are recorded in the velocity− vorticity scattering plane.

Figure 6. State I for colloidal plastic crystals at ϕeff = 0.60 under oscillatory shear at a fixed frequency f = 1 Hz and strain γmax = 50%. (a) Experimental scattering pattern. (b) Scattering results from BD simulations in the fully crystalline phase with ϕ = 0.55 and L* = 0.24 with f = 15.3 Hz and γmax = 15% (I). (c) Snapshots taken at the half time of a cycle from the BD simulations. (Particles are scaled to half size for better visibility, and additional snapshots are available in Figure S2 in the SI.) (d) Orientational order parameter Pi2 in i = velocity (filled triangles), vorticity (open squares), and gradient (filled circles) directions.

increase in the applied oscillatory shear, two pronounced hexagonal patterns with sharp Bragg peaks are seen at γmax = 50% (Figure 5c) and γmax = 116% (Figure 5e), respectively. There is an intermediate state in between at γmax = 60% as shown in Figure 5d. The interpretation of the scattering data will be discussed in detail in the following sections. Structures c−e in Figure 5 correspond to distinct states I−III, as defined in the rheology of Figure 4b. Obviously, intermediate state II constitutes a disordered structure between two very ordered structures as indicated by the clear Bragg peaks in the scattering patterns of the latter two. Hence, this finding points to a strong structural (nonequilibrium) transition as the reason for the novel double yielding event. To analyze and understand the structural information from SANS results in more detail, we have performed particleresolved BD computer simulations of plastic crystals under oscillatory shear, as described in the Methods section. In the following text, we discuss experiments and simulations referring to three different strains marked by red arrows in Figure 4b. The respective experimental strains are (I) 50, (II) 60, and (III) 116%. In the simulations, we find transition state II at a lower shear strain of 20%. The reason is probably the higher frequency we have to use in the simulation leading to a transition already at smaller strains. We therefore define states I and III in the simulations as in the experiments relative to the transition state, namely, 15 and 30%. Selected results of the experimental rheo-SANS experiments are displayed in Figures 6 (state I), 7 (state III), and 8 (state II) together with the respective simulation runs. Figure 6a displays the experimental SANS pattern for state I that refers to the vorticity−velocity plane (cf. Figure 2a and b), and Figure 6b shows the corresponding scattering patterns calculated by BD simulations. At the beginning of the experiment the SANS pattern of the sample does not exhibit any orientation (Figure 5a). Only after applying an oscillating shear with a strain of γmax = 50% is the sample induced to form an ordered structure as is obvious from the SANS pattern shown for state I in Figure 6a. We see that the experimental

Figure 7. High-strain state III of hard dumbbells with L* ≈ 0.24 at γmax = 116%. (a) SANS intensity pattern. (b) Scattering results from BD simulations in the fully crystalline phase with ϕ = 0.55 and L* = 0.24 with f = 15.3 Hz and γmax = 30% (III). (c) Snapshots taken at the half time of a cycle from the BD simulations. (Particles scale as above.) (d) Orientational order parameter Pi2 in i = velocity (filled triangles), vorticity (open squares), and gradient (filled circles) directions.

F

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However, it is a not-yet-described nonequilibrium quasi-plastic version of the twinned-fcc phase with weakly correlated degrees of rotation and translation. Let us now turn to state III. In the experiments, a second hexagonal scattering pattern at high strain as shown in Figure 7a indicates that hard dumbbells are induced to form ordered hexagonal layers under larger oscillatory shear with a strain magnitude of 116%. This hexagonal scattering pattern is very similar to that reported for the shear-induced “sliding layer phase” of colloidal hard spheres, where the crystalline planes are rotated by 30° versus the twinned fcc and the most densely packed particle direction is oriented in the velocity direction.7,27,28 This experimental structure factor with sixfold symmetry agrees with the results of the BD simulation that has been performed on hard dumbbells in the plastic crystalline phase under oscillatory shear of f = 15.3 Hz and γmax = 30%. The ratio of the intensities of the stronger scattering spots to the relative weaker ones on the first ring is 1.75 ± 0.02 in the experiment (Figure 7a) and 2.34 in the calculated pattern (Figure 7b). Compared to model calculations43 of structure factors for a variety of possible stacking sequences, we conclude that the present layers follow a strongly registered “zigzag” trajectory, as the peaks in the vorticity region would be expected to vanish for straight sliding layers. The correlation between positions in subsequent layers as in the twinned fcc is lost, as can be seen in the simulation snapshot in Figure 7c. However, orientational order is observed, as indicated by order parameter P2 plotted in Figure 7d, namely, a dumbbell orientation in the velocity direction, where P2 ≈ 0.05 is constant and overall higher than in the twinned fcc phase. Hence, we call this new nonequilibrium phase formed in colloidal plastic crystals under shear the “partially oriented sliding layer phase”. The experimental intensity pattern of intermediate stage II is shown in Figure 8a. We see that state II exhibits a significant breakdown in order as is clear from the very smeared out pattern. In Figure 8b, we investigate the idea of a coexistence of nonequilibrium states I and III and a disordered molten state in the sample volume in the intermediate state. Here, we assumed that the experimental scattering pattern in Figure 8a is a result of a linear superposition of the different coexisting states. The calculation shows that the superposition result of 0.3 times pattern I, 0.25 times pattern II, and 0.45 times the isotropic pattern (Figure 8b) is most comparable to the experimental pattern. Hence, the comparison of Figure 8a,b confirms that for state II we observe a coexistence of states I and III together with a very prominent molten phase. Our simulations confirm the loss of structure as well because the twinned-fcc crystal fully melts away upon strain increase from state I to a totally isotropic pattern at strain stage II. The corresponding isotropic structure factor and a snapshot are shown in Figure 8c,d. The snapshot as shown in Figure 8d proves that hard dumbbells in this stage are free of translational and orientational order. For a more comprehensive picture, orientational and translational crystal order parameters P2 and Q4 are plotted versus strain in Figure 9a. The order parameters jump to vanishing values around state II, and no order was detectable. Thus, both experiments and simulations strongly point to a significantly disordered, partially molten state II. This behavior is in stark contrast to our computer simulations of the spherical reference system. Here we find that intermediate state II (located at a strain of 50%) does not

Figure 8. Intermediate stage II of hard dumbbells with L* ≈ 0.24 in the plastic crystalline phase under shear strain. (a) Experimental scattering pattern of colloidal dumbbells at ϕeff = 0.60 under oscillatory shear of γmax = 60% and (b) corresponding superposition by the LAMP-SANS software. The calculation result is a superposition result of 0.3 times the pattern at 50%, 0.25 times the pattern at 116%, and 0.45 times the pattern of the shear-melt stage at a strain of 1000%. (c) Intensity patterns from Brownian dynamics (BD) simulations, which have been carried out on the hard dumbbells in the fully crystalline phase with ϕ = 0.55 and L* = 0.24 at γmax = 20% and (d) the corresponding snapshot taken at the half time of a cycle from BD simulations (particle scale; see above).

rheo-SANS pattern is in good agreement with the one calculated by the simulations in Figure 6b. Figure 6c shows a simulation snapshot along the velocity−gradient plane. All of the data demonstrate directly that an oscillating shear field can order a colloidal plastic crystal fully. From the well-developed Bragg spots, the scattering in state I can be indexed by the socalled twinned face-centered cubic (fcc) crystal. This phase, where two fcc “twin” states with different stacking are alternating, is known already for the spherical reference case.7,27,28 Because the first peak appears at q ≃ 0.0013 A−1, the lattice parameter d that refers to the spacing between adjacent lattice planes has been calculated to be 472.5 ± 5 nm from d = 2π/q. Figure 6d displays orientational order parameter P2 from the simulations that probe the average alignment of the dumbbells with respect to the various directions as a function of the time (within one cycle). Positive values of P2 indicate parallel order, while negative values indicate antiparallel order. The small values for P2 in the range of 0.04 to 0.02 for the twinned-fcc demonstrate that the long axes of the dumbbells with the centers of gravity being located on an fcc lattice cannot rotate completely freely. The slight dependence of P2 on instant time (strain) as shown in Figure 6d indicates that the weak alignment in the shear velocity direction is indeed coupled to the translational motion due to the collisions within the oscillating layers. Thus, we essentially recover the nonequilibrium twinned fcc phase of the hard sphere system. G

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possess weak but clearly detectable orientation correlations. A novel, partially oriented sliding-layer phase has been observed to be formed from colloidal plastic crystals under oscillatory shear. Those ordered degrees of freedom, missing in spherical systems, are evidently enough to induce a strong, discontinuous transition in the system, as clearly shown by the combination of rheo-SANS experiments and BD simulations. The shearinduced structural evolution and discontinuous phase transitions are the underlying structural mechanism for the novel double yielding behavior of colloidal crystals from hard dumbbells. Hence, even weak anisotropies of colloids have a significant impact on the phase transitions of colloidal crystals in external fields. This study is important in understanding the viscoelastic shear response of anisotropic colloidal systems and opens up new opportunities to create novel colloidal crystals with controlled orientations.16−19,23 To clarify the effect of anisotropy, investigations on the equilibrium and nonequilibrium phase behavior of slightly longer dumbbells with L* ≈ 0.30 are in progress.

Figure 9. BD simulation results for orientational order P2 (filled triangles) and translational crystal order Q4 (filled squares) of sheared dumbbells (HD) along the velocity direction versus shear strain for a packing fraction of ϕ = 0.55 and L* = 0.24. I−III correspond to the three experimental states along the G″ curve in Figure 4b. Also shown is the BD result for Q4 for the hard sphere (HS) system (open circles) which has its transition state at about γmax = 50%. The solid lines are guides to the eyes.

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ASSOCIATED CONTENT

* Supporting Information S

Experimental scattering pattern of the plastic crystal in equilibrium formed from hard dumbbells and its corresponding SANS intensity plotted against the magnitude of the scattering vectors. Typical BD snapshots at 15% strain amplitude from different planes. Computer simulation snapshot of a spherical reference system at intermediate strain amplitude. This material is available free of charge via the Internet at http://pubs.acs.org.

exhibit any loss of translational order (cf. the Q4 curve in Figure 9a which continuously decreases with strain). In fact, we find that for spheres, state II is a fully ordered, crystalline hybrid of states I and III, which easily forms itself in the simulations. (For a snapshot, see Figure S3 in the SI.) Hence, both experiments and simulations demonstrate that intermediate state II is much more disordered for the anisotropic dumbbells than it is for the isotropic, spherical system for which the phase transition proceeds in a continuous fashion. The discontinuous shearinduced phase transition in the dumbbell case must thus be made responsible for the novel yielding behavior with a dip in G″ as noted by state II in Figure 4b.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: matthias.ballauff@helmholtz-berlin.de.

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Notes

The authors declare no competing financial interest.

CONCLUSIONS We presented the first detailed study of the equilibrium and nonequilibrium phase behavior of a colloidal plastic crystal with a fixed aspect ratio (L* = 0.24). A summary of the occurring phases is given in Figure 10. Under oscillatory shear, the experimentally observed phases are shown by simulations to

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ACKNOWLEDGMENTS We acknowledge financial support by the Deutsche Forschungsgemeinschaft within the Schwerpunktprogramm “Hydrogele” (SPP1259).

Figure 10. Illustration of the nonequilibrium states encountered for increasing shear in this study. (I) Snapshots of distinct positions in the strain cycle, bridge stacking at zero strain γ(t) = 0 (left), and ABC stacking at maximum strain (right). (II) Disordered system which is found to be coexisting with I and III in the experiment. (III) Strongly registered sliding layers with a dense direction parallel to the velocity (x). For I and III, three different hexagonal layers are color coded as green (back), gray (middle), and blue (front). H

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