Dynamical and Rheological Properties of Ultrasoft Colloids under

Article pubs.acs.org/Macromolecules

Dynamical and Rheological Properties of Ultrasoft Colloids under Shear Flow Sunil P. Singh,† Apratim Chatterji,‡ Gerhard Gompper,† and Roland G. Winkler†,* †

Theoretical Soft Matter and Biophysics, Institute for Advanced Simulation and Institute of Complex Systems, Forschungszentrum Jülich, 52425 Jülich, Germany ‡ Physics Department, Indian Institute of Science Education and Research (IISER) , Pune 411021, India ABSTRACT: The dynamical and rheological properties of ultrasoft colloids and star polymers are investigated in dilute and semidilute solutions under linear shear flow. We apply a hybrid mesoscale hydrodynamics simulation approach, which combines molecular dynamics simulations for the solute with the multiparticle collision dynamics approach for the solvent. We investigate the effect of concentration on relaxation, diffusion, and the rheological properties of the star polymers. We find that the relaxation time of a star-polymer arm is a universal function of a concentration-dependent Weissenberg number. The center-of-mass mean square displacements of the star polymers are anisotropic under shear flow. At high shear rate, we find shear-induced enhanced center-of-mass displacements along the vorticity and gradient directions. Moreover, we determine the shear viscosity and normal stress coefficients as a function of concentration. The shear viscosity exhibits shear thinning with a weak functionality dependence.

I. INTRODUCTION The rheological behavior of soft deformable materials has attracted considerable attention in the past few decades, both, from fundamental research as well as industrial and biomedical applications.1,2 Soft matter systems are typically composed of nano- to micrometer size objects ranging from simple hardsphere colloidal particles to complex structures such as (bio)polymers, membranes, vesicles, and cells. These systems are often far from equilibrium and driven by spatially and temporally varying forces, which renders their understanding even more difficult. Star polymers are particular interesting, because their interactions can be tuned from polymer- to colloidal-like, by varying the functionality f, i.e., the number of polymer arms.3,4 Consequently, their non-equilibrium behavior has been studied intensively,4−7 driven by their technological importance in oil industry8 and potential applicability in drug delivery.9−11 Theoretically, various attempts have been undertaken to derive an effective pair potential between interacting star polymers.12−18 These studies imply a complex potential due to the radially inhomogeneous structure. At small distances, the effective potential exhibits a logarithmic distance dependence.13−16,18 Different functions have been predicted for large distances. On the one hand, a rather soft Yukawa-type potential was proposed12,14−16 and, on the other hand, a more rigid Gaussian potential was suggested.15,18 In any case, the structural, dynamical, and rheological properties of suspensions of star polymers depend on the functionality, the polymer length, and the concentration.19−21 The variation of the functionality reveals an interesting equilibrium phase diagram. © XXXX American Chemical Society

On the basis of the Yukawa-type potential, for stars with f < 34, only a liquid phase is found, whereas for higher functionalities fcc and bcc crystalline phases are stable,16,22,23 and reentrant melting appears in dense suspensions.16,24 The influence of the large-distance potential on the phase behavior has not been investigated. The non-equilibrium behavior of stars is governed by the dynamics of individual polymer arm as well as collective effects due to their linkage and excluded-volume interactions. As far as individual linear flexible polymers are concerned, experiments and simulations reveal large conformational changes under flow. In steady state, they continuously undergo stretching and compression cycles, which is denoted as tumbling dynamics.25−28 The individual arms of a star exhibit a similar dynamics. The dynamics of the star, however, depends on its functionality.29,30 For f < 5, a star exhibits a tumbling type motion, whereas for f > 5 a tank treading-like motion has been observed, which implies a rotation of the colloid with a concentration- and functionality-dependent frequency.29,30 In dilute suspensions, ultrasoft colloids undergo Brownian motion. However, in dense suspension caging emerges, as is well-known for colloidal glasses, with slow relaxation of the complex environment.31−34 This leads to a non-Markovian and non-Gaussian dynamics of a tracer particle at short times.31,32 An applied external field, e.g., a shear flow, perturbs the environment of a tracer particle and leads to enhanced Received: July 26, 2013 Revised: September 2, 2013

A

dx.doi.org/10.1021/ma401571k | Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

Article

B. Multiparticle Collision Dynamics (MPC). The explicit solvent is modeled by Ns point particles with continuous positions ri and velocities υi (i = 1, ..., Ns). Their dynamics proceeds in discrete time increments h, denoted as collision time, by alternating streaming and collision steps.36−38 In the streaming step, the solvent particles of mass m move ballistically with their respective velocities and their positions are updated according to

diffusion. This generic behavior has been studied for glassy systems31−34 but not so far for ultrasoft particle systems. In this article, we analyze the dynamical behavior of ultrasoft colloids at equilibrium and under shear flow. We address the universal nature of the arm relaxation time and overall rotational motion in terms of a concentration and functionality dependent Weissenberg number ϕf Wic, where ϕf accounts for the functionality dependence and Wic = γ̇τ(c) is the concentration dependent Weissenberg number; γ̇ is the shear rate and τ(c) the concentration dependent relaxation time. Moreover, we shed light on the equilibrium dynamical behavior of colloids in dense suspensions whose viscosity increases and diffusion coefficient decreases dramatically with concentration. We demonstrate that the random displacements of a colloid in a crowed environment are no longer described by a Gaussian distribution. Moreover, we study the non-equilibrium polymer relaxation and colloid diffusion behavior in dense suspensions under shear flow. The rheological behavior of linear polymers is a universal function of Wi c . 35 Here, we address the corresponding behavior of ultrasoft colloids and characterize their rheological behavior in terms of the shear viscosity and normal stress coefficients. In addition, we address the orientational behavior of the stars in terms of a concentration and functionality dependent Weissenberg number ϕf Wic. We establish a relation between the structural properties and rheological behavior of a suspension. The article is organized as follows. In section II, we outline the model of the star polymer and the multiparticle collision dynamics method, which we apply to describe the solvent. In section III, we discuss results for the relaxation behavior of polymer arm at equilibrium and in flow. Section IV, contains results on the shear-enhanced diffusion of colloids and its conformational properties. Section V presents shear stress, shear viscosity and normal stress coefficients. The results are summarized in section VI.

ri(t + h) = ri(t ) + hυi (t )

In the collision step, the simulation box is partitioned in cubic cells of side length a to define the multiparticle collision environment. The solvent particles are sorted into these cells and their relative velocities, with respect to the center-of-mass velocity of the cell, are rotated around a randomly oriented axis by an angle α, i.e. υi (t + h) = υi (t ) + (9(α) − 0)(υi (t ) − υcm (t ))

ks (|R i , i + 1| − l)2 2

N

c 1 υcm (t ) = ( mυi (t ) + ∑ mNc + MNcm i = 1

Ncm

∑ Mυj (t )) j=1

(5)

Here, Nmc is the number of beads in the considered cell. Thereby, momentum is redistributed between solvent and monomers in the same cell. Lees−Edwards boundary conditions are applied to impose shear flow.42 This yields a linear fluid velocity profile υx = γ̇y in the flow direction (x-axis) as a function of the particle positions along the gradient direction (y-axis). A local cell-based Maxwellian thermostat (MBS) is applied by which velocities are scaled to maintain the desired temperature of the system.43 C. Simulation Parameter. For the MPC fluid, we use the parameters h/(ma2/(kBT))1/2 = 0.1, α = 130°, and the average number of fluid particles in a collision cell ⟨Nc⟩ = 10, which yields the solvent viscosity ηs = 8.7(mkBT/a4)1/2 and the Schmidt number Sc ≈ 17. For the polymer, we set ε = kBT and l = a. We use the spring constant ksl2/(kBT) = 103, the diameter of a bead σ/l = 0.8 and its mass M = 10m. The velocity Verlet algorithm is used to integrate Newton’s equations of motion of the star polymers with the time step hm = 5 × 10−3(ml2/(kBT))1/2. A cubic simulation box of length L = 100l is used and periodic periodic conditions are applied. The number of stars Nsp depends on concentration and is given in Table 1 together with other parameters. We will present results for the polymer length Nm = 30 and the functionalities f = 10, 20, 30, and 50. The polymer concentration is measured relative to the overlap concentration,

(1)

where l is the equilibrium bond length and Ri,i+1 = Ri+1 − Ri is the bond vector. The spring constant ks is chosen such that even under strong shear flow the change in the equilibrium bond length remains less than a few percent. Excluded-volume interactions between non-bonded beads are taken into account by the repulsive, truncated, and shifted Lennard-Jones (LJ) potential ⎡⎛ σ ⎞12 ⎛ σ ⎞6 1⎤ VLJ(r ) = 4 ∈ ⎢⎜ ⎟ − ⎜ ⎟ + ⎥Θ(21/6σ − r ) ⎝r⎠ 4⎦ ⎣⎝ r ⎠

(4)

where 9 is the rotation matrix, 0 is the unit matrix, and υcm = ΣNj=1c υj/Nc is the center-of-mass velocity of the cell with Nc particles. In this stochastic process mass, momentum, and energy are conserved, which ensures that hydrodynamic behavior emerges on larger length scales.36,39 The transport properties of the solvent depend on the collision time h, the rotation angle α, and the average number of particles ⟨Nc⟩ per cell. Tuning these variables allows us to attain solvents with a high Schmidt number, where momentum transport dominates over mass transport. The coupling of the star polymers and the fluid occurs in the collision step, where the velocities of the polymer beads are rotated according to eq 4,40,41 but with the center-of-mass velocity of the cell

II. METHOD AND MODEL A. Star Polymer. Each of the Nsp considered star polymers is composed of f linear polymers, which are linked to a common center by one of their ends. A polymer comprises Nm beads of mass M, wich are consecutively connected by the bond potential Vb , i =

(3)

(2)

where Θ(r) is the Heaviside step function (Θ(r) = 0 for r < 0 and Θ(r) = 1 for r ≥ 0). The equilibrium bond length of each center-arm connection lc and LJ diameter of the central bead σc are taken to be twice as large as those of other monomers, while the mass of the center bead is the same as for any other monomer. B


Macromolecules

Article

Table 1. List of Simulation Parameters for Nsp Star Polymers of Functionality f a f

Nsp

Rg0/l

Rh/l

c/c*

10 20 30 50

100−750 50−375 34−250 20−150

7.1 7.9 8.5 9.2

7.1 9.1 10.7 13.2

0.15−1.14 0.16−2.37 0.17−2.57 0.19−2.92

a Rg0 and Rh are the equilibrium radii of gyration and hydrodynamic radii in dilute solution, and c indicates the range of considered concentrations with c* denoting the overlap concentration.

⎤−1 ⎡4 c * = ⎢ πR h 3 ⎥ ⎦ ⎣3

(6)

where Rh is the hydrodynamic radius, which is obtained from the diffusion coefficient of a star in the dilute solution extrapolated to infinitely large systems44 (see Table 1). The strength of the shear flow is either characterized by the Weissenberg number Wi = γ̇τZ, or by the concentrationdependent Weissenberg number Wic = βγ̇τZ, where τZ = 3 ηsN3ν m l /kBT is the Zimm relaxation time of a polymer arm and β = β(c/c*) is a concentration dependent scale factor. Under good solvent conditions, the scaling exponent is ν ≈ 0.6.45 In our system, the scaling exponent is somewhat larger, ν≈0.63, due to the relatively short arm length.6 We introduce the factor β to account for the concentration dependence of the characteristic relaxation time of a star polymer, i.e., βτZ is up to a constantequal to the concentration dependent relaxation time. To determine β, we consider the shear rateand concentration-dependent radius of gyration tensor component along the flow direction and calculate the ratio between the shear-rate-dependent data at a given concentration and the comparable data for the lowest concentration given in Table 1.6 The fit by a quadratic function yields the relation β ≈ 1 + 0.97c/c* + 0.39(c/c*)2. The same idea has also been applied in refs 5 and 35 to study the concentration-dependent properties of linear and star-polymer solutions for stars with f = 10. In order to achieve reliable and better data quality, we perform several independent simulation runs for fixed solution conditions.

Figure 1. Star polymer conformations (top) and monomer density distributions at equilibrium (left) and under shear flow (Wic = 78, right). The functionality is f = 50 and the concentration c/c* = 0.97.

superscript indicates the equilibrium value at the particular concentration. In the dilute regime ⟨Gγγ′⟩ = Rg02/3, and the radius of gyration Rg02 obeys the scaling relation Rg02 ∼ l2Nm2νf1−ν in terms of arm length and functionality,12,46−48 with the scaling exponent ν ≈ 0.6. As mentioned before, our simulations yield the somewhat larger exponent ν ≈ 0.63. The star polymer alignment is measured by the angle χG between the largest eigenvalue of the average radius of gyration tensor and the flow direction. It can be expressed as tan(2χG ) =

1 N

N

∑ Δri ,γ Δriγ′

(8)

IV. NON-EQUILIBRIUM DYNAMICS A. Polymer Relaxation. To characterize the relaxation behavior of an individual polymer of a star, we determine the correlation function

(7)

i=1

⟨Gxx⟩ − ⟨Gyy⟩

by the components of the radius of gyration tensor. As stressed in ref 6, we obtain a universal behavior for the radius of gyration tensor components and the alignment angle as a function of a shear-rate dependent Weissenberg number Wic and a functionality dependent factor ϕf, where ϕf is presented in the inset of Figure 2. Figure 2 shows tan (2χG) as a function of shear rate. In the limit Wic → 0, χG is close to the equilibrium value π/4. Since ⟨Gxy⟩ ∼ ϕf Wic and ⟨Gxx⟩ − ⟨Gyy⟩ ∼ (ϕf Wic)2 in the linear response regime, tan (2χG) decreases initially as tan(2χG) ∼ Wic−1.5,6,28,49 The increasing deformation for Wic > 1 and finite polymer extensibility28,50 implies a crossover to a slower decay of the alignment, where tan(2χG) ∼ (ϕf Wic)−δ with the exponent δ ≈ 0.43. In section V, we will discuss the universal properties of tan (2χG) in terms of rheological quantities.

III. NON-EQUILIBRIUM STRUCTURAL PROPERTIES The conformational and structural properties of star polymers have been discussed in refs 5 and 6. Here, we briefly summarize the basic findings and discuss aspects, which are relevant for the analysis of the rheological properties. As has been shown, star polymers undergo substantial conformational changes in shear flow. The arms are stretched along the flow direction and a star assumes on average an ellipsoidal shape with a preferred alignment of its major axis with respect to the flow direction. At the same time, the star substantially shrinks along the gradient direction. This is illustrated in Figure 1. The deformation of a star can be characterized by the radius of gyration tensor, which is defined as Gγγ ′ =

2⟨Gxy⟩

where Δri is the position of the i monomer relative to the star center of mass, γ, γ′∈{x,y,z}, and N = f Nm + 1 is the total number of monomers in a star.5,6,35 At equilibrium, all diagonal components of Gγγ′ are equal, i.e., ⟨Gγγ⟩ = ⟨G0γγ⟩, where the th

Cceγ (t ) = C

⟨Rceγ (t )Rceγ (0)⟩ ⟨Rceγ (0)Rceγ (0)⟩

(9)


Macromolecules

Article

Figure 4. Equilibrium relaxation times τeq of polymer arms for star polymers in dilute solution. The line indicates the power-law fit τeq ∼ f 0.6.

Figure 2. Alignment angles tan (2χG) of star polymers as a function of ϕf Wic for various concentrations and the functionalities f = 10 (bullets) (c/c* = 0.15, 0.45, 0.76, 1.14), f = 20 (squares) (c/c* = 0.16, 0.47, 0.79, 1.18, 2.37), f = 30 (diamonds) (c/c* = 0.17, 0.51, 0.86, 1.28, 2.57), and f = 50 (triangles) (c/c* = 0.2, 0.29, 0.49, 0.68, 0.97, 1.46, 2.92). Lines indicate power laws with the exponents −1 (dashed) and −0.43 (solid). The inset displays the functionality dependence of the scale factor ϕf.

cyclic stretching and collapse motion similar to the tumbling motion of linear polymers.25−28,52 Since, star polymers assume an anisotropic shape under shear flow, we expect that their relaxation behavior is distinctly different along the various spatial directions.53 However, due to coupling of the flow in the shear-gradient plane, the relaxation behavior is rather similar in these spatial directions. At low Weissenberg numbers Wic ≪ 1 the shape of a star polymer is isotropic and close to equilibrium, thus the dynamics is essentially isotropic. As shown in Figure 5,

of the vector Rce = Re − Rc connecting the star polymer center Rc and the end bead Re of the considered polymer arm; γ ∈ {x,y,z} denotes the Cartesian components. At equilibrium, the correlation function decays exponentially in the long-time limit, as shown in Figure 3 for the various

Figure 5. Center-to-end vector correlation functions of polymer arms under shear flow along the flow (solid black), gradient (dashed red), and vorticity (solid green, top) direction. The functionality is f = 50, the concentration c/c* = 0.29, and the Weissenberg number Wic = 78.

Figure 3. Center-to-end vector correlation functions of polymer arms for stars of functionality f = 50 in dilute solution (bottom curve, black) and the concentrations c/c* = 0.29, 0.68, 0.97, and 1.46 (bottom to top).

in the limit Wic ≫ 1, the correlation functions show a damped oscillatory behavior along the flow and gradient directions. However, in the vorticity direction, the correlation Cz(t) decays always exponentially. The shear-rate dependence of the relaxation time τ(1) z along the vorticity direction is presented in Figure 6. For Wic > 1, τ(1) z decreases rapidly with increasing shear rate, in a similar manner as for linear polymers.28,53 We observe a weak concentration dependence. However, over a wide range of concentrations the correlation functions decay in a universal manner. The straight −2/3 line indicates the power law τ(1) . The same exponent z ∼ Wic is found for linear polymers in experiments,25,26 theory,27,28 and simulations.52−54 Evidently, the relaxation times along the flow and gradient directions are shorter than that along the vorticity direction. To extract a decay rate and frequency of the correlation functions along the flow and gradient directions, we fit the simulation results by the function

concentrations. We denote the corresponding concentrationdependent relaxation time as τeq(c). The functionality dependence of the relaxation times τeq of dilute systems is presented in Figure 4. A power-law fit yields the dependence τeq ∼ f 0.6, which is in close agreement with the scaling prediction τeq ∼ Nm3νf 3(1 − ν)/2 = Nm3νf 0.6

(10)

for ν ≈ 0.6, based on the blob model. As discussed in ref 51, the center-to-end relaxation behavior is determined by the rotational diffusion of the whole star molecule due to the strong interactions with the surrounding polymers. This time is much longer than the characteristic time of shape fluctuations, for which scaling considerations predict the functionality dependence f(2−3ν)/2 ≈ f 0.1 in the presence of hydrodynamic interactions.51 In linear shear flow, star polymers exhibit a tank-treading-like rotation.29 Thereby, an individual polymer arm undergoes a 51

D


Macromolecules

Article

indicates a memory of the center-to-end vector correlation function over more than a cycle. Interestingly, the frequency exhibits the same shear-rate dependence as the inverse relaxation time τ(1) z for Weissenberg numbers Wic > 1. The product ωτ(c) is a universal function of γ̇τ(c) for all concentrations. Only at large Weissenberg numbers and high concentrations, we observe a deviation from the universal behavior. Most importantly, ω or ωτ(c) depends on the functionality and ω increases sublinearly with increasing functionality. B. Diffusion. 1. Equilibrium. The diffusive dynamics of the star polymers depends strongly on their concentration. We expect that dense suspensions exhibit gel-like or glassy behavior. To unravel the dynamical properties, we consider the center-of-mass mean square displacement (MSD) of individual colloids

Figure 6. Nonequilibrium relaxation times τ(1) z /τeq along the vorticity direction of polymer arms under shear flow as a function of the concentration dependent Weissenberg number Wic. The functionalities and concentrations are f = 10 (black), c/c* = 0.15 (bullets), 0.45 (squares), 0.76 (diamonds), 1.14 (triangles); f = 30 (red), c/c* = 0.17 (bullets), 0.51 (squares), 0.85 (diamonds), 1.28 (triangles); and f = 50 (green), c/c* = 0.29 (bullets), 0.68 (squares), 0.97 (diamonds), 1.46 (triangles). The solid line indicates the power-law dependence τ(1) z ∼ Wic−2/3. (1)

t →∞

2 ΔR cm (t ) = ⟨(R cm(t ) − R cm(0))2 ⟩ ⎯⎯⎯⎯→ 6Dt

(12)

where D is the self-diffusion coefficient of a finite-size periodic system. As is well-known, the actual diffusion coefficient is obtained in the limit of an infinitely large simulation box due to long-range hydrodynamic interactions. However, we focus here mainly on dense suspension, where we expect that long wavelength hydrodynamic modes are screened.39 Examples for equilibrium mean square displacements are presented in Figure 8 for various concentrations. At short times,

(2)

Cceγ (t ) = aγ e−t / τγ [cos ωγ t + bγ sin ωγ t ] + (1 − aγ )e−t / τγ

(11)

γ ∈ {x,y}, as suggested in refs 53, 55, and 56. aγ and bγ are amplitudes of the correlation functions, and ωγ and τ(1) are γ frequencies and relaxation times, respectively. Fitting yields the following general relations for the various shear rates and concentrations. The amplitudes aγ depend weakly on the functionality f and ax ≈ ay ≈ 1 for f ≳ 20. Only for f ≲ 10, we find ax ≈ 0.9, i.e., a value smaller than unity. Interestingly, the frequencies ωx = ωy are independent of the functionality. The amplitudes bγ are negative and nearly independent of the shear rate. The relaxation time τ(2) γ is an order of magnitude smaller than τ(1) γ , hence the second term decays much faster than first term. In studies of phantom linear polymers53 this term was absent. Therefore, we believe that this is due to excludedvolume interactions between the beads. Figure 7 shows the shear-rate dependence of the frequency ω = ωx for all functionalities and concentrations. In the limit Wic → 0, the frequency vanishes, because of the pure exponential decay of the correlation function at equilibrium. A non-zero ω

Figure 8. Mean square displacements of the center of mass of star polymers of functionality f = 50 in dilute solution (black) and for the concentrations c/c* = 0.68 (red), c/c* = 0.97 (green), and c/c* = 1.46 (blue) (top to bottom). The dashed line indicates the asymptotic linear dependence.

the colloid dynamics is govern by inertia. Here, the MSD is independent of concentration. In the vicinity of t/(ma2/ (kBT))1/2 ≈ 103, a crossover occurs to a power-law behavior. At low concentrations, the MSD increases linearly in time, i.e., the stars exhibit Brownian diffusion. With increasing concentration, we observe a decrease in the slope and hence a sub-diffusive dynamics, which is well know for supercooled liquids. In the long-time limit, the Brownian diffusive regime will be reached. However, due to limitations in the computational time, we are not able to reach this regime for higher concentrations. At equilibrium, the probability distribution function of the displacements is Gaussian in the diffusive regime. In the subdiffusive regime, the displacements are no longer Gaussian. The latter reflects the caging of the colloidal particles in a glassy environment. We characterize the non-Gaussianity by the ratio

Figure 7. Frequencies ω obtained from the fit of the simulation data Cγce by eq 11 for the functionalities f = 10 (black, bottom), 30 (red, middle), and f = 50 (green, top). The symbols correspond to the concentrations of Figure 6. The solid line (blue) is a guide to the eye with the dependence γ̇2/3. E


Macromolecules

Article

α2 of the forth moment and the square of the second moment of the displacement distribution function,57,58 i.e., α2(t ) =

3⟨ΔR cm(t )4 ⟩ 5⟨ΔR cm(t )2 ⟩2

−1 (13)

For free diffusion, α2 is zero. Figure 9 displays α2(t) as a function of time for various concentrations. As expected, the

Figure 9. Non-Gaussian parameter α2 as a function of time for the concentrations c/c* = 0.68 (red), c/c* = 0.97 (green), and c/c* = 1.46 (blue) (bottom to top). The functionality is f = 50.

parameter is zero at short times independent of concentration. In an intermediate time regime significantly larger than the Zimm time, α2 grows slowly, reaches a maximum and decreases again. We expect α2(t) → 0 for long times. The maximum value increases with increasing concentration and the time at the maximum shifts toward smaller values. In ref 59, non-Gaussian parameters are presented for a supercooled Lennard-Jones liquid. The maximum values presented in this article are of the same order as those of Figure 9. However, the maxima of ref 59 shift to larger times with increasing non-Gaussianity. This indicates an interesting dynamical behavior of star polymers already at equilibrium. Restrictions by neighboring colloids imply a concentration-dependent temporal confinement. 2. Shear Enhanced Diffusion. The interesting question is now how shear flow modifies the colloid dynamics. To resolve this issue, we compute the mean square displacements of the star polymers center-of-mass under flow in the vorticity ΔRy2 and gradient ΔRz2 directions. Figure 10 displays results for stars with f = 50 arms and several shear rates at the concentration c/ c* = 0.97. The corresponding local slopes ζγ =

Figure 10. Mean square displacements of the center of mass of star polymers in (a) gradient and (b) vorticity direction for the functionality f = 50, c/c* = 0.97, and Wic = 1.9 (black), 8.9 (red), 17.8 (green), 89 (blue), 178 (purple), and 444 (magenta) (bottom to top). The bottom curve (dark maroon) is the equilibrium MSD. The dashed lines indicate the asymptotic linear dependence.

diffusive dynamics. For longer times, the MSDs approach the diffusive regime. Accompanied with the appearing maximum is the formation of two minima M1 and M2 in the ζγ, one at shorter times (M1) and another one at longer times (M2). Both minima shift to even shorter times with increasing Wic, similar as the maximum of the ζγ. Above Wic ≳ 1, the minimum M1 disappears, whereas the maximum and the minimum M2 shift to shorter and shorter times. Finally, we find slopes on the order of 2, as for ballistic motion at high Wic and short times. For times above the time of the minimum M2, the MSDs slowly approaches the diffusive behavior with the slope unity. Thereby, the minimum M2 for ζy along the vorticity direction is significantly more pronounced than that along the gradient direction. Correspondingly, the Brownian regime is assumed at longer times only for the MSDs along the gradient direction. A similar behavior has been reported for the diffusive dynamics in glasses.31−34 The dynamical behavior can be rationalized as follows. At higher concentration, close spatial proximity of others stars lead to caging of an individual star. The respective star rattles in its cage until a certain rearrangement of the neighborhood opens a route to escape. Above the “escape time”, the stars exhibit Brownian motion. Shear promotes fast and considerable rearrangements of stars, particularly since stars a dragged along the flow direction by shear. Thus, the stars can escape easily from the local neighborhood, which is reflected in the shear enhanced dynamics. At higher shear rates, we can extract diffusion coefficients of the star polymers along the various spatial directions. Figure 12 shows the effective diffusion coefficients obtained by fitting a

d[log ΔR γ 2] d[log t ]

(14)

with γ ∈ {x,y}, are presented in Figure 11. Shear-enhanced diffusion is clearly present for all Weissenberg numbers. Without shear, the local slopes approach the value ζy = ζz ≈ 0.6 at t/τz ≈ 10, the maximum time of the simulation. For Weissenberg numbers Wic ≲ 5 and short times, the MSDs are close to the equilibrium curve. Shear leads to an increase in the MSDs and super-diffusive behavior, which is reflected in the increasing local slopes. At very low Weissenberg numbers, the slopes increase in the vicinity of t/τz ≈ 1 first, because there the diffusion dynamics at zero shear is significantly slowed down. The slopes ζy and ζz exhibit a maximum at t/τz ≈ 1 above a certain (small) Weissenberg number. In the vicinity of the maxima, the ζγ are larger than unity, corresponding to a superF


Macromolecules

Article

linear function at larger times. At low shear rates, the Dγ are nearly independent of shear rate. The corresponding diffusion coefficients are equal to the self-diffusion coefficients at the particular concentrations, independent of the spatial direction. Above Wi ≳ 10, the coefficients increase rapidly with the shear rate in both directions. Thereby, we obtain somewhat larger diffusion coefficients along the vorticity direction than along the gradient direction. Interestingly, we observe only a weak dependence of the shear-rate-dependent diffusion coefficients on the concentration. The same trend is observed for other functionalities. Only for f = 10, Dy seems to be less sensitive to shear for Wi > 102.

V. RHEOLOGICAL PROPERTIES A. Stress Tensor. To determine the rheological properties of the star-polymer suspension, we compute various components of the stress tensor, in particular σxy, σxx, σyy, and σzz, using the Irving−Kirkwood formula for the virial.60−63 Our focus is here on the polymer contributions to the virial, i.e., the bond (spring potential) and excluded-volume (LJ potential) contributions. The shear viscosity η and normal-stress coefficients Ψ1 and Ψ2 are calculated according to σxy η(γ )̇ = γ̇ (15) Ψ1(γ )̇ =

Figure 11. Local slopes (eq 14) of the MSDs of Figure 10 along (a) the gradient and (b) the vorticity direction for the functionality f = 50, c/c* = 0.97, and Wic = 0 (dark maroon), 1.9 (black), 8.9 (red), 17.8 (green), 89 (blue), 178 (purple), and 444 (magenta).

Ψ2(γ )̇ =

N1 2

γ̇

N2 2

γ̇

= =

σxx − σyy γ 2̇

(16)

σyy − σzz γ 2̇

(17)

where N1 and N2 are the normal stress differences. The shear-rate dependence of σxy is displayed in Figure 13 for stars with f = 50 and various concentrations. The data sets

Figure 13. Normalized shear stress σxy as a function of Wic for the concentrations c/c* = 0.2, 0.29, 0.49, 0.68, 0.97, 1.46, 2.92 (top to bottom) and the functionality f = 50. The dashed and solid lines indicate the power-laws σxy ∼ Wic and σxy ∼ Wic0.6, respectively. The inset displays the dependence of the scale factor σ0xy on concentration.

are scaled by the factors σ0xy(c/c*) such that they superimpose with each other as much as possible. The factor σ0xy(c/c*) itself is shown in the insetit increases with increasing concentration in a non-linear way. At low concentrations and for Wic < 1, the shear stress increases linearly with Wic. With increasing shear rates, we find concentration-dependent deviations and the scaled data seem to approach a limiting curve over the considered range of shear

Figure 12. Effective diffusion coefficients of star polymers under shear in (a) the gradient and (b) the vorticity direction for the concentrations c/c* = 0.29 (bullets), 0.68 (squares), 0.97 (diamonds), 1.46 (triangles) as a function of the Weissenberg number Wi = γ̇τZ. The functionality is f = 50.

G


Macromolecules

Article

rates. For concentrations c/c* ≳ 1, we find a nonlinear shearrate dependence for Wic → 0, which points toward the presence of yield stress in such systems. B. Viscosity. Results for the corresponding shear viscosities are presented in Figure 14. In dilute solution, the shear

for star polymers with f = 10 and various arm lengths. However, there is a certain dependence on the functionality. The data for f < 50 are rather close, but for f = 50 the η0 values are consistently smaller. There could be a functionality dependence of the zero shear viscosity, which is not absorbed in the variable ϕf Wic. C. Alignment. The shear-rate dependence of the alignment angle tan (2χG) [eq 8] can be expressed by rheological quantities. Instead of the Weissenberg number Wic, the ratio between the first normal stress difference N1 eq 16 and the shear stress σxy eq 15 can be exploited to characterize the strength of the applied flow. In Figure 15, the alignment angle is plotted as a function of the ratio N1/σxy. Evidently, we obtain a universal behavior for

Figure 15. Alignment angles tan (2χG) of star polymers as a function of N1/σxy for the functionalities f = 10 (bullets), f = 20 (squares), f = 30 (diamonds), and f = 50 (triangles). The concentrations are the same as in Figure 2. Figure 14. (a) Shear viscosities ηp/η0 as a function of ϕf Wic for dilute systems and the functionalities f = 10 (black bullets, c/c* = 0.15), f = 20 (red squares, c/c* = 0.16), f = 30 (green diamonds, c/c* = 0.17), and f = 50 (blue triangles, c/c* = 0.2). Inset: Concentration dependence of the zero-shear viscosity η 0 for the various functionalities. (b) Viscosities for f = 50 and the concentrations c/c* = 0.2, 0.29, 0.49, 0.68, 0.97, 1.46, 2.92 (bottom to top at ϕf Wic = 1). Inset: Viscosities for the highest concentrations and the functionalities f = 10 (black bullets, c/c* = 1.14), f = 20 (red squares, c/c* = 1.18), f = 30 (green diamonds, c/c* = 1.28), and f = 50 (blue triangles, c/c* = 1.46). The dashed line corresponds to the power-law Wic−0.4.

the various concentrations and functionalities. tan (2χG) itself decreases as (N1/σxy)−δ, with the exponent δ ≈ 1.0. The nonlinear properties of the alignment are absorbed in the ratio N1/σxy. Its shear rate dependence is presented in Figure 16. The

viscosities exhibit a Newtonian plateau for ϕf Wic < 1 and shear thinning for larger Weissenberg numbers. Correspondingly, the viscosities are scaled by the zero-shear viscosities η0. Shear thinning is also observed for higher concentrations, but no Newtonian regime is reached anymore within the considered range of shear rates. We rather find a power-law decay of the viscosity at large concentrations for all functionalities. This is clearly visible for high concentration data (cf. inset of Figure 14b). Since there is no zero-shear plateau anymore, we scale the viscosity data such that the various curves match with the lower shear rate data in the shear-thinning regime. The apparent power-law decrease of the viscosity depends on the concentration. At low concentrations, the exponent is equal to approximately −0.3 (Figure 14a), while at higher star concentrations c/c* > 1 it is about −0.4 (inset Figure 14b). The scale factor η0, which corresponds at low concentrations to the zero shear viscosity, increases with increasing concentration in a nonlinear manner (cf. Figure 14a). It exhibits a similar dependence as the factors presented in ref 5

Figure 16. Ratios of first normal stress differences and shear stresses N1/σxy as a function of ϕf Wic for f = 10 (bullets), f = 20 (squares), f = 30 (diamonds), and f = 50 (triangles). The concentrations are the same as in Figure 2.

curves for the various functionalities exhibit a rather similar dependence on the shear rate. At small Weissenberg numbers, they grow as Wic1/2 with the shear rate and turn into the weaker shear rate dependence Wic1/4 at larger Wic. With respect to functionality, we observe an offset between the curves for the H


Macromolecules

Article

various f, where the ratios for increasing f become smaller at the same ϕf Wic. D. Normal Stress Coefficients. The shear-rate dependence of the first and second normal-stress coefficients is displayed in Figure 17. The curves of the Ψi for the various

Figure 19. Concentration dependence of the scale factors for the first and second normal stress coefficients for f = 10 (bullets), f = 20 (squares), f = 30 (diamonds), and f = 50 (triangles). Filled and open symbols correspond to the first and second normal stress coefficient, respectively.

comparable magnitude points to a similar dependence of the zero-shear-rate plateau values of Ψ1 and Ψ2 on concentration. There are two major contributions to the second normal stress coefficient, excluded-volume interactions between the monomers and hydrodynamic interactions.35,65,68,69 For linear polymers in dilute solution, hydrodynamic interactions dominate and Ψ2 is small. Since Ψ2 is comparable with Ψ1 for our star polymer solutions, even at low concentrations, we conclude that both, inter- and intra-star, excluded-volume interactions contribute significantly to the second normal stress coefficient.

Figure 17. Scaled first normal stress coefficient Ψ1 for various functionalities and concentrations. The same notation as in Figure 2 is used. The dashed line indicates the power law Ψ1 ∼ Wic−1.

concentrations are scaled by factors Ψ0i , which are shown in Figure 19. As for the viscosity, we find a plateau for Wic < 1 and low concentrations. For the higher concentrations, Ψ1 exhibits the power-law decay Wic−1 over a broad range of ϕf Wic values. Thereby, the slope changes for ϕf Wic ≳ 102 toward the value −4/3 predicted by theory and observed for linear flexible polymers26,28,35,64−67 and star polymers of shorter arm length.5 The corresponding normalized second normal-stress coefficients Ψ2 are presented in Figure 18. The power-law region at high Wic with the exponent −1.4 for all functionalities is close to the theoretically expected value for Ψ1.

VI. SUMMARY AND CONCLUSIONS We have studied the dynamical and rheological properties of star polymers under shear flow for various concentrations and functionalities. The simulations reveal a strong influence of shear flow on the star-polymer dynamics, both, the intramolecular dynamics and the diffusive motion. At equilibrium, the polymer center-to-end vector correlation function shows the power-law dependence τeq ∼ f 0.6, in agreement with theoretical predictions.12,46−48,51 Shear flow leads to a decrease of the relaxation times with increasing shear rate, where we obtain a shorter relaxation time along the strongly coupled shear and gradient directions than along the vorticity direction. The coupling, which is more pronounced than in linear polymers,53 is a consequence of the tank-treading-like rotation of the whole star for f ≳ 5.6,29 Experimentally, nonequilibrium properties of systems under shear are often characterized by the first normal stress difference N1 and the shear stress σxy. We indeed find a universal linear dependence of the alignment angle on the ratio N1/σxy for all functionalities and concentrations. As far as a linear flexible polymer is concerned, this behavior is easily explained. Assuming that σxy and N1 are determined by the longest polymer relaxation time τ(γ̇), their ratio becomes N1/ σxy ≈ γ̇τ(γ̇).28 The Weissenberg number is defined by the relaxation time τ(0) at equilibrium, i.e., Wi = γ̇τ(0). Hence, with the ratio μ = τ(0)/τ(γ̇), we obtain N1/σxy ≈ Wi/μ(γ̇), which is the theoretical expression for the shear-rate dependence of the alignment angle.28 Hence, tan (2χG) is a linear function of N1/σxy. The nonequilibrium conformational, dynamical, and rheological properties of linear polymers can be characterized by a concentration dependent Weissenberg number Wic = γ̇τ(c),

Figure 18. Scaled second normal stress coefficient Ψ2 for various concentrations and functionalities. The same notation as in Figure 2 is used. The dashed line indicates the power law Ψ1 ∼ Wic−4/3.

The scale factors Ψ01 and Ψ02 are shown in Figure 19. These factors are obtained by scaling the normal-stress coefficients of the various concentrations to the lowest considered concentration. Under the assumption of a universal scaling of Ψ1 and Ψ2 for various concentrations, the factors Ψ01 and Ψ02 are proportional to the corresponding zero-shear-rate plateau values of the normal-stress coefficients. The factors Ψ01 and Ψ02 are rather similar for a given functionality. They show a functionality dependence and decrease with increasing f. Their I


Macromolecules

Article

where the relaxation time τ(c) follows, e.g., from the end-to-end vector correlation function and corresponds to the longest relaxation time of the polymer.28,35,52 For star polymers, we in addition expect and predict a dependence of the Weissenberg number and the relaxation time on the functionality. Interestingly, our simulations suggest a universal dependence on the scaling variable ϕfγ̇τ(c), where ϕf decreases with increasing f. Thus, based on the relation Wic = γ̇τ(c,f), we propose a relaxation time τ(c,f), which decreases with increasing f, since ϕf decreases. This is in contrast to the functionality dependence of the center-to-end vector relaxation time τeq, which increases with increasing f. Hence, τeq is not proportional to τ(c,f) and cannot be used to characterize the nonlinear response. There are a number of other longer relaxation times of a star polymer such as the rotational diffusion time. The latter is predicted to scale as f 3(1−ν)/2 in the presence of hydrodynamic interactions,51 which also yields f 0.6 for ν = 0.6. Hence, accepting the idea of a universal behavior in terms of the scaling variable γ̇τ(c,f), we lack a theoretical understanding of the relevant relaxation time τ(c,f). Here, further theoretical studies are encouraged.

■

(10) Qiu, L. Y.; Bae, Y. H. Pharm. Res. 2006, 23, 1. (11) Wiltshire, J. T.; Qiao, G. G. Aust. J. Chem. 2007, 60, 699. (12) Daoud, M.; Cotton, J. P. J. Phys. (Paris) 1982, 43, 531. (13) Witten, T. A.; Pincus, P. A. Macromolecules 1986, 19, 2509. (14) Likos, C. N.; Löwen, H.; Watzlawek, M.; Abbas, B.; Jucknischke, O.; Allgaier, J.; Richter, D. Phys. Rev. Lett. 1998, 80, 4450. (15) Jusufi, A.; Watzlawek, M.; Löwen, H. Macromolecules 1999, 32, 4470. (16) Likos, C. N. Phys. Rep. 2001, 348, 267. (17) Rubio, A. M.; Freire, J. J. Comp. Theory Polym. Sci. 2000, 10, 89. (18) Hsu, H.-P.; Grassberger, P. Europhys. Lett. 2004, 66, 874. (19) Fetters, L. J.; Kiss, A. D.; Pearson, D. S.; Quack, G. F.; Vitus, F. J. Macromolecules 1993, 26, 647. (20) Stellbrink, J.; Lonetti, B.; Rother, G.; Willner, L.; Richter, D. J. Phys.: Condens. Matter 2008, 20, 404206. (21) Vlassopoulos, D.; Fytas, G. Adv. Polym. Sci. 2010, 236, 1. (22) Ishizu, K. Prog. Polym. Sci. 1998, 23, 1383. (23) Laurati, M.; Stellbrink, J.; Lund, R.; Willner, L.; Richter, D.; Zaccarelli, E. Phys. Rev. Lett. 2005, 94, 195504. (24) McConnell, G. A.; Gast, A. P. Macromolecules 1997, 30, 435. (25) Schroeder, C. M.; Teixeira, R. E.; Shaqfeh, E. S. G.; Chu, S. Phys. Rev. Lett. 2005, 95, 018301. (26) Schroeder, C. M.; Teixeira, R. E.; Shaqfeh, E. S. G.; Chu, S. Macromolecules 2005, 38, 1967. (27) Winkler, R. G. Phys. Rev. Lett. 2006, 97, 128301. (28) Winkler, R. G. J. Chem. Phys. 2010, 133, 164905. (29) Ripoll, M.; Winkler, R. G.; Gompper, G. Phys. Rev. Lett. 2006, 96, 188302. (30) Ripoll, M.; Winkler, R. G.; Gompper, G. Eur. Phys. J. E 2007, 23, 349. (31) Foss, D. R.; Brady, J. F. J. Fluid Mech. 1999, 401, 243. (32) Foss, D. R.; Brady, J. F. J. Rheol. 2000, 44, 629. (33) Krüger, M.; Weysser, F.; Fuchs, M. Eur. Phys. J. E 2011, 34, 88. (34) Harrer, C. J.; Winter, D.; Horbach, J.; Fuchs, M.; Voigtmann, T. J. Phys.: Condens. Matter 2012, 24, 464105. (35) Huang, C.-C.; Winkler, R. G.; Sutmann, G.; Gompper, G. Macromolecules 2010, 43, 10107. (36) Malevanets, A.; Kapral, R. J. Chem. Phys. 1999, 110, 8605. (37) Kapral, R. Adv. Chem. Phys. 2008, 140, 89. (38) Gompper, G.; Ihle, T.; Kroll, D. M.; Winkler, R. G. Adv. Polym. Sci. 2009, 221, 1. (39) Huang, C.-C.; Gompper, G.; Winkler, R. G. Phys. Rev. E 2012, 86, 056711. (40) Malevanets, A.; Yeomans, J. M. Europhys. Lett. 2000, 52, 231. (41) Ripoll, M.; Mussawisade, K.; Winkler, R. G.; Gompper, G. Europhys. Lett. 2004, 68, 106. (42) Lees, A. W.; Edwards, S. F. J. Phys. C 1972, 5, 1921. (43) Huang, C.-C.; Chatterji, A.; Sutmann, G.; Gompper, G.; Winkler, R. G. J. Comput. Phys. 2010, 229, 168. (44) Westpahl, E.; Singh, S. P.; C.-C. Huang, Gompper, G.; , and Winkler, R. G.; , preprint. (45) Doi, M.; and Edwards, S. F., The Theory of Polymer Dynamics; Clarendon Press: Oxford, U.K, 1986. (46) Birshtein, T. M.; Zhulina, E. B. Polymer 1984, 25, 1453. (47) Birshtein, T. M.; Zhulina, E. B.; Borisov, O. V. Polymer 1986, 27, 1078. (48) Grest, G. S.; Kremer, K.; Witten, T. A. Macromolecules 1987, 20, 1376. (49) Aust, C.; Kröger, M.; Hess, S. Macromolecules 1999, 32, 5660. (50) Kowalik, B.; Winkler, R. G. J. Chem. Phys. 2013, 138, 104903. (51) Grest, G. S.; Kremer, K.; Milner, S. T.; Witten, T. A. Macromolecules 1989, 22, 1904. (52) Huang, C.-C.; Sutmann, G.; Gompper, G.; Winkler, R. G. Europhys. Lett. 2011, 93, 54004. (53) Huang, C.-C.; Gompper, G.; Winkler, R. G. J. Phys.: Condens. Matter 2012, 24, 284131. (54) Huang, C.-C.; Gompper, G.; Winkler, R. G. J. Phys.: Conf. Ser. 2012, 392, 012003. (55) Jose, P. P.; Szamel, G. J. Chem. Phys. 2008, 128, 224910.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank D. A. Fedosov, J. K. G. Dhont, D. Richter, M. Ripoll, J. Stellbrink (Jülich), J. Vermant (Leuven), and D. Vlassopoulos (FORTH Crete) for stimulating discussions. Financial support by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center “Physics of Colloidal Dispersions in External Fields” (SFB TR6), by the EU through the Collaborative Research Project “NanoDirect” (NMP4-SL2008-213948), and the EU through FP7-Infrastructure ESMI (Grant 262348) is gratefully acknowledged. A.C. acknowledges the use of the compute-cluster available for the Nano-Science unit in IISER, which is funded by the DST, India (Project No. SR/NM/NS-42/2009).

■

REFERENCES

(1) Larson, R. G., The structure and rheology of complex fluids; Oxford University Press: Oxford, NY, 1999. (2) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of polymeric liquids:Fluid mechanics, 2nd ed.; Wiley: New York, 1987. (3) Stellbrink, J.; Allgaier, J.; Monkenbusch, M.; Richter, D.; Lang, A.; Likos, C.; Watzlawek, M.; Löwen, H.; Ehlers, G.; Schleger, P. In Trends in Colloid and Interface Science XIV; Buckin, V., Ed.; Progress in Colloid and Polymer Science 115; Springer: Berlin and Heidelberg, Germany, 2000; pp 88−92. (4) Vlassopoulos, D.; Fytas, G.; Pakula, T.; Roovers, J. J. Phys.: Condens. Matter 2001, 13, R855. (5) Fedosov, D. A.; Singh, S. P.; Chatterji, A.; Winkler, R. G.; Gompper, G. Soft Matter 2012, 8, 4109. (6) Singh, S. P.; Chatterji, A.; Winkler, R. G.; Gompper, G. J. Phys.: Condens. Matter 2012, 24, 464103. (7) Singh, S. P.; Winkler, R. G.; Gompper, G. Phys. Rev. Lett. 2011, 107, 158301. (8) Grest, G. S.; Fetters, L. J.; Huang, J. S.; Richter, D. Adv. Chem. Phys. 1996, 94, 67. (9) Peppas, N. A.; Keys, K. B.; Torres−Lugo, M.; Lowman, A. M. J. Controlled Release 1999, 62, 81. J


Macromolecules

Article

(56) Aust, C.; Hess, S.; Kröger, M. Macromolecules 2002, 35, 8621. (57) Rahman, A. Phys. Rev. 1964, 136, A405. (58) Kanaya, T.; Tsukushi, I.; Kaji, K. Prog. Theor. Phys. Suppl. 1997, 126, 133. (59) Kob, W.; Donati, C.; Plimpton, S. J.; Poole, P. H.; Glotzer, S. C. Phys. Rev. Lett. 1997, 79, 2827. (60) Irving, J. H.; Kirkwood, J. G. J. Chem. Phys. 1950, 18, 817. (61) Winkler, R. G.; Morawitz, H.; Yoon, D. Y. Mol. Phys. 1992, 75, 669. (62) Winkler, R. G.; Hentschke, R. J. Chem. Phys. 1993, 99, 5405. (63) Winkler, R. G.; Huang, C.-C. J. Chem. Phys. 2009, 130, 074907. (64) Lyulin, A. V.; Adolf, D. B.; Davies, G. R. J. Chem. Phys. 1999, 111, 758. (65) Petera, D.; Muthukumar, M. J. Chem. Phys. 1999, 111, 7614. (66) Jendrejack, R. M.; de Pablo, J. J.; Graham, M. D. J. Chem. Phys. 2002, 116, 7752. (67) Hsieh, C.-C.; Larson, R. G. J. Rheol. 2004, 48, 995. (68) Bird, R. B.; Hassager, O.; Armstrong, R. C.; Curtiss, C. F. Dynamics of Polymeric Liquids; John Wiley Sons: New York, 1987; Vol. 2. (69) Ö ttinger, H. C. Stochastic Processes in Polymeric Fluids; Springer: Berlin, 1996.

K


Dynamical and Rheological Properties of Ultrasoft Colloids under

Recommend Documents