Shear rheology and structural properties of chemically identical dendrimer-linear polymer blends through molecular dynamics simulations Elnaz Hajizadeh, B. D. Todd, and P. J. Daivis Citation: The Journal of Chemical Physics 141, 194905 (2014); doi: 10.1063/1.4901721 View online: http://dx.doi.org/10.1063/1.4901721 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/19?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A molecular dynamics investigation of the planar elongational rheology of chemically identical dendrimer-linear polymer blends J. Chem. Phys. 142, 174911 (2015); 10.1063/1.4919654 Nonequilibrium molecular dynamics simulation of dendrimers and hyperbranched polymer melts undergoing planar elongational flow J. Rheol. 58, 281 (2014); 10.1122/1.4860355 Nonequilibrium molecular dynamics of the rheological and structural properties of linear and branched molecules. Simple shear and poiseuille flows; instabilities and slip J. Chem. Phys. 123, 054907 (2005); 10.1063/1.1955524 Coarse-grained molecular dynamics simulations of polymer melts in transient and steady shear flow J. Chem. Phys. 118, 10276 (2003); 10.1063/1.1572459 Rheological Properties of Liquid Crystalline Copolyester Melts. II. Comparison of Capillary and Rotary Rheometer Results J. Rheol. 29, 539 (1985); 10.1122/1.549830

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THE JOURNAL OF CHEMICAL PHYSICS 141, 194905 (2014)

Shear rheology and structural properties of chemically identical dendrimer-linear polymer blends through molecular dynamics simulations Elnaz Hajizadeh,1,a) B. D. Todd,1,b) and P. J. Daivis2,c) 1

Department of Mathematics, Faculty of Science, Engineering and Technology, and Centre for Molecular Simulation, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia 2 School of Applied Sciences, RMIT University, Melbourne, Victoria 3001, Australia

(Received 27 August 2014; accepted 3 November 2014; published online 18 November 2014) We present nonequilibrium molecular dynamics (NEMD) simulation results for the miscibility, structural properties, and melt rheological behavior of polymeric blends under shear flow. The polymeric blends consist of chemically identical linear polymer chains (187 monomers per chain) and dendrimer polymers of generations g = 1–4. The number fraction x of the dendrimer species is varied (4%, 8%, and 12%) in the blend melt. The miscibility of blend species is measured, using the pair distribution functions gDL , gLL , and gDD . All the studied systems form miscible blend melts under the conditions investigated. We also study the effect of shear rate γ˙ and dendrimer generation on inter-penetration between blend species for different blend systems. The results reveal that shear flow increases the interpenetration of linear chains toward the core of the dendrimers. We also calculate the shear-rate dependent radius of gyration and ratios of the eigenvalues of the gyration tensor to study the shear-induced deformation of the molecules in the blend. Melt rheological properties including the shear viscosity and first and second normal stress coefficients obtained from NEMD simulations at constant pressure are found to fall into the range between those of pure dendrimer and pure linear polymer melts. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4901721] I. INTRODUCTION

Dendrimers are unique nanoscopic tree-like macromolecules, possessing high number of end functional groups, that show very interesting properties, such as low shear1 and elongational viscosity2 compared to linear chain polymers of the same molecular weight. Therefore, these tree-like polymers have attracted significant interest as potential polymer processing aids (PPA) for high molecular weight linear polymers.3, 4 The flow properties of dendrimers1 and linear polymer melts2, 5 have been studied previously using nonequilibrium molecular dynamics (NEMD) simulation techniques. Under shear, similar to the conventional linear chain polymers, dendrimers in the melt undergo a transition from the Newtonian regime characterized by the shear independent viscosity to the non-Newtonian shear-thinning regime.1 The onset of shear thinning for dendrimers occurs at higher strain rate, and the rate of shear thinning is smaller1, 6 compared to the linear chain polymers of the same molecular weight. Bosko et al.7 calculated the shear rate dependent radius of gyration for dendrimers and showed that shear induced stretching occurs and this effect is particularly pronounced for large dendrimers. They suggested that the deformation of molecular shape is mainly responsible for the macroscopically observed shear thinning of the dendrimer melt. The onset of both shape deformation and shear thinning occurred at a specific value of strain rate, which depended on the generation of the dendrimer.7 a) [email protected] b) [email protected] c) [email protected]

0021-9606/2014/141(19)/194905/15/$30.00

In 1992, Kim and Webster8 reported that the melt shear viscosity of polystyrene melts decreased in the presence of a small amount of bromo-functional hyperbranched polymer. However, they did not discuss the phase behavior of the blend system. A similar reduction in viscosity was observed for solutions of hyperbranched polymers and their blends with linear polymers.9 Nunez et al.9 suggested that the reduction in viscosity, after replacing linear chains with hyperbranched polymers of lower viscosity, is also a consequence of a reduction in the number of entanglements between linear chains in the presence of hyperbranched polymers. A few years after Kim and Webster’s8 observation, it was reported that the addition of small perfect dendritic polyesters10 and modified hyperbranched polyesters11 to poly (ethylene terephthalate) and poly (carbonate), respectively, can yield miscible blends. Massa et al.12 studied the miscibility of blends of both hydroxyl-terminated and acetate-terminated hyperbranched polyesters and linear polymers. They showed that besides the chemical structure and interaction, the architecture of the species in the blend plays a role in polymer miscibility. Fredrickson and Liu13 incorporated entropic corrections to the Flory-Huggins theory of polymer blends in order to take into account the architectural and conformational effects of blend species on the free energy of mixing. Based on their modified theory for conventional mixtures of linear and star polymers with modest numbers of truly polymeric arms, they predicted that phase separation due to architectural differences alone is unlikely. The effect of hyperbranched polymer content on the blend viscosity and miscibility of a polyamide 6 and

141, 194905-1

© 2014 AIP Publishing LLC

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hyperbranched aramid blend at 5–30 wt. % of hyperbranched polymer was investigated by Monticelli et al.14 Viscosity and glass transition Tg data of the blends proved the miscibility of the components in the composition range studied. They attributed the observed miscibility to the possible hydrogen bonding between amide groups and hyperbranched functional end groups. In recent work, Li et al.15 investigated the melt rheology and mechanical properties of blends of linear poly (ether ether ketone) and hyperbranched poly (ether ether ketone). They reported that the blend shows only one glass transition temperature, indicating single phase (miscible) blend formation. They suggested that this is due to the similar chemical structure of the two blend species. In addition to experimental observations, there have been a few computational studies using different simulation techniques to study the miscibility and rheological properties of blends of dendritic and linear polymers. For example, Lee and McHugh16 applied Brownian dynamics simulations to study the effect of the mole fraction of linear segments in the hyperbranched polymers on the rheological properties of linear-hyperbranched polymers. They found that the first normal stress difference is more sensitive to the addition of linear segments in the perfectly branched architecture than the zero-shear viscosity. In addition, they observed that there is a crossover in rheological behavior from hyperbranched-like behavior (low viscosity) to linear-chain like behavior (high viscosity) at a transitional mole fraction of linear segments of 0.8. Theodorakis et al.17 studied effects of the chain architecture on the miscibility of symmetric linear/linear and star/star polymer blends using Monte Carlo simulations. They found that star/star blends are more miscible than linear/linear ones, at least for short chains. With an increase in the number of arms, the miscibility was enhanced significantly. They suggested that this comes mainly from the shielding effect of the star cores, which reduces the number of heterocontacts. Consequently, there was a smaller reduction in star polymer dimensions relative to their unperturbed state than there was in the respective dimensions in linear/linear blends. In order to take advantage of potential applications of dendrimers and hyperbranched polymers upon blending, one should be able to predict the phase behavior of these blends. Kosmas and Vlahos18 included long range interactions into the Flory-Huggins model where the excluded volume interactions between units belonging to different chains are not cancelled and might lead to a nonrandom mixing or even demixing. They applied their new model to analyse the stability of blends of chemically identical and different homopolymers in the bulk and in a film. They successfully explained some of the experimentally observed phase diagrams in 2D and 3D systems. Vlahos and Kosmas19 also applied this approach to study the effective interaction parameters for chemically identical star/star, ring/ring, and ring/linear blends. They found no phase separation in 3D for blends composed of ring/ring and ring/linear chains at any disparity or volume fraction. The same was found for star/star or star/linear polymer blends with a few to moderate number of arms, which was in agreement with experimental and theoretical results.

J. Chem. Phys. 141, 194905 (2014)

Freed and Dudowicz20 extended the Flory-Huggins theory for linear chains to account for the chain architecture effect on the phase behavior of the system, with the so-called lattice cluster theory (LCT). Enders et al.21 applied LCT to predict the phase behavior of blends of linear and hyperbranched polymers. The theory predicts large demixing regions and a small mixing gap, which depends on different structural parameters. However, the LCT needs to be validated by comparing with experimental results because there have been several cases where dendritic and linear polymers form miscible blends.10, 11, 15 Akten and Mattice22 proposed a method to assess miscibility of linear binary systems by calculating the intermolecular radial distribution functions (RDFs). They stated that if the intermolecular RDFs of unlike pairs are higher than those of like pairs, the systems are miscible; otherwise, the systems are immiscible. This concept has now been used by several authors.23, 24 Mackay et al. conducted several experiments in order to study the non-Einstein-like decrease in viscosity,25 miscibility,26 and structural changes of linear polystyrene molecules27 in the presence of spherical cross-linked polystyrene (PS) and dendritic polyethylene (PE) nanoparticles. They observed a very surprising result that although linear PS-linear PE blends are a classic phase-separating system, branched PE nanoparticles disperse uniformly in PS. This means architecture and size both make a clear difference in the miscibility of this system. They showed that thermodynamically stable dispersion of nanoparticles into the polymeric liquid was enhanced when the radius of gyration of the linear polymer is greater than the radius of the nanoparticle.26 Addition of nanoparticles to polymer reduced the blend viscosity, which they found scales with the change in the free volume introduced by the nanoparticles and not with the decrease in entanglement. They reported that the entanglement was not affected at all, suggesting an unusual polymer dynamics.25 Additionally, they found a 10%–20% increase in the radius of gyration of PS when the nanoparticles are homogeneously dispersed in the polymer, an effect that occurs only when the radius of gyration of the polymer is larger than the nanoparticle radius.27 Their studied system composed of PS nanoparticles dispersed in the linear PS, resembles our blend systems composed of dendrimer molecules and linear chains of the same chemical nature. We will discuss this further in Sec. III. The main objective of this current paper is to investigate the performance of dendrimers as rheology modifiers for high molecular weight linear polymers. Rheology modifiers need to be miscible with the bulk polymer in order to reduce the bulk viscosity effectively. Therefore, we need to confirm that our blend systems at conditions studied are miscible. To the best of our knowledge this current paper is the first NEMD study to investigate the miscibility, rheological and structural properties of blends of chemically identical dendrimers and linear chains under shear flow. More details about the structural and rheological properties of these polymers can be found in our previous work.2, 6 The remainder of this paper is organized as follows. Section II A describes the molecular model we used to design

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J. Chem. Phys. 141, 194905 (2014)

TABLE I. Number, reduced molecular mass of blend species, dendrimer number x, and mass percentage c, size of both blend species Rg,D and Rg,L , size ratio Rg,L /Rg,D , and box length a for all the simulated systems. The leading number in the system column is the number fraction of dendrimers as a percentage of the blend system. System

ND

NL

MD

ML

x

c

Rg,D

Rg,L

Rg,L /Rg,D

a

4D19B 8D19B 12D19B 4D43B 8D43B 12D43B 4D91B 8D91B 12D91B 4D187B 8D187B 12D187B

5 10 15 5 10 15 5 10 15 5 10 15

120 115 110 120 115 110 120 115 110 120 115 110

19 19 19 43 43 43 91 91 91 187 187 187

187 187 187 187 187 187 187 187 187 187 187 187

4 8 12 4 8 12 4 8 12 4 8 12

0.42 0.88 1.4 0.95 2 3 2 4.1 6.2 4 8 12

1.55 1.6 1.67 2.15 2.25 2.35 2.72 2.82 2.92 4.25 4.35 4.45

8.05 8.03 8.02 8.00 7.95 7.87 7.98 7.90 7.83 7.95 7.86 7.77

5.20 5.01 4.81 3.72 3.53 3.35 2.93 2.80 2.68 1.87 1.81 1.75

29.93 29.55 29.17 29.82 29.66 29.34 30.10 29.88 29.67 30.29 30.23 30.30

our polymers. Section II B presents the NEMD algorithm . In Sec. III A, structural properties of the blends are discussed, and in Sec. III B we present miscibility studies. In Sec. III C, we present the rheological behavior of blend systems undergoing shear flow and in Sec. IV we present the conclusions of this work. II. METHODOLOGY A. Dendrimer and hyperbranched models

Dendrimer polymers were simulated using a coarsegrained uniform bead-spring model. This model has been frequently used to model polymers, especially rheological studies of polymer melts,6, 29, 42 where the basic units (beads) correspond to the linear units or branching points of the molecule and are interconnected to create treelike structures. Beads along the chain can rotate and vibrate freely. All beads (monomers) are identical and indistinguishable except for their position in the molecule. The total number of monomers in a dendrimer can be defined as N = fb((f−1)g + 1 − 1)/(f − 2) where f is the functionality of the end groups, b is the number of monomers in the chain units, and g is the generation number. With the choice of f = 3 and b = 2, dendrimers of generation 1, 2, 3, and 4 will have 19, 43, 91, and 187 monomers, respectively. Linear polymers of 187 beads are built by successively attaching the same monomers. More details can be found in Bosko et al.1 Details of blended systems can be found in Table I. It is important to note that dendrimers and linear molecules constituting our blend systems have identical chemical nature. The main reason to study such blend systems is to solely study the effect of molecular architecture disparity on the blend miscibility without chemical structure imposing extra complexities. This is because there are situations that even blending species of identical chemical nature can cause blend immiscibility. The classic example of this is blends of the PE family.30 In this model, only bond and pairwise interactions are taken into account. Monomers comprising the melt interact via the Weeks-Chandler-Anderson (WCA) potential, which is a shifted and truncated Lennard-Jones (LJ) potential, and bonds are presented by a finitely extensible nonlinear elastic

(FENE) potential (details can be found in papers by Kröger and Hess28 and also Kremer and Grest29 ). The WCA potential is given as ⎧  12  6  r ⎨ 4 σ +  for σij < 21/6 , − rσ rij ij UijW CA = r ⎩ 0 for σij ≥ 21/6 , (1) where rij is the separation between the sites represented by monomers i and j,  is the potential well depth, and σ is the effective diameter of the monomers. This potential results in a purely repulsive force that includes the effect of excluded volume. The FENE potential is expressed as 

for rij ≤ R0 , −0.5kR02 ln 1 − (rij )/R02 F ENE = Uij ∞ for rij ≥ R0 , (2) where R0 is a finite extensibility and k is a spring constant. In this work, R0 and k were set to 1.5 and 30, respectively, as is typical (Kröger and Hess,28 Kremer and Grest29 ). For this choice of parameters, the maximal extent of bonds is short enough to prevent crossing of chains, whereas the magnitude of the bonding force is small enough to enable simulations with relatively large time steps. Nonbonded monomers only have WCA potential interactions whereas bonded monomers have both FENE and WCA interactions which creates a potential well for the flexible bonds that maintains the architecture of the molecules. In the remainder of this paper, all quantities are expressed in terms of site reduced units in which the reduction parameters are the Lennard-Jones interaction parameters  and σ and the mass, miα of bead α in molecule i. The reduced temperature is given by T ∗ = kB T/, the density by ρ ∗ = ρσ 3 , the pressure tensor by P∗ = Pσ 3 /, and strain rate by γ˙ ∗ = γ˙ (mσ 2 /)1/2 . For simplicity of notation, the asterisk will be omitted hereafter. In all simulations, we set σ =  = miα = kB = 1. B. NEMD simulation

Our NEMD simulations are based on the molecular version of the homogeneous isothermal-isobaric SLLOD equations of motion (details can be found in Todd and

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Daivis31 and Bosko et al.1 ), p r˙ iα = iα + iγ˙ yi + ζ˙ ri , miα

J. Chem. Phys. 141, 194905 (2014)

(a)

(3)

p˙ iα = Fiα − i(miα /Mi )γ˙ pyi − α(miα /Mi )pi − ζ˙ (miα /Mi )pi , (4) where riα and piα represent the position and thermal momentum of monomer α on molecule i, γ˙ is the shear rate; ri Nα = α=1 miα riα /Mi is the position of the molecular centre of Nα mass of molecule i, Mi = α=1 miα is the mass of molecule Nα i, and pi = α=1 piα is the momentum of the molecular centre of mass of molecule i. Fiα is the inter-monomeric force on monomer α on molecule i imposed by all other monomers in the system. The simulations were performed at constant temperature using a molecular version of the Gaussian thermostat with a constraint multiplier α given by N N i=1 (Fi ·pi ) − γ˙ i=1 (pix piy ) − ζ˙ , (5) α= N 2 i=1 (pi ) where N is the number of molecules in the system. This expression for α is derived from Gauss’s principle of least constraint and keeps the molecular centre of mass kinetic temperature constant. All simulations were performed at a molecular temperature TM = 1.25. Constant pressure simulations were accomplished by coupling the system to an extended degree of freedom ζ˙ , which can be obtained via solving an additional differential equation given as (p − p0 )V , ζ¨ = Qp N kB T

(6)

where Qp is a damping factor for which the optimal value will depend on the type of system studied, p is the instantaneous isotropic pressure p = 13 Tr(PM ), and p0 is the target pressure (for more details see our previous paper2 ). The equations of motion were integrated using a fifthorder Gear predictor-corrector differential equation solver with reduced time step t = 0.001. After blend systems of 125 molecules were generated at low density, they were compressed to the monomer number density of ρ = 0.84. They were then equilibrated for typically several million time steps and the pressure was plotted against time to ensure the system had reached equilibrium. Finally, shear flow was applied, and after reaching the nonequilibrium steady state, ensemble averages of all properties of interest were calculated by averaging over typically 30 independent NEMD steady-state trajectories. Fig. 1 shows a snapshot in time of the blend system 12D187B (notation as given in Table I) at equilibrium and under low shear rate, where one can see how the box becomes deformed in the flow direction, especially at very high shear rate. III. RESULTS AND DISCUSSION A. Structural analysis

The extension of a molecule in space can be characterized by its radius of gyration. The average tensor of gyration is

(b)

FIG. 1. Snapshot configuration of blend system 12D187B at (a) equilibrium and (b) γ˙ = 0.0001. Red balls represent beads of dendrimer molecules while the blue ones show beads of linear chains.

given by the expression 

n α=1 mα (rα − rCM )(rα − rCM ) , Rg Rg ≡ n α=1 mα

(7)

where rα is the position of monomer α, rCM is the position of the molecular centre of mass, and the angle brackets denote an ensemble average. The value of the radius of gyration, which is defined as  the square root of the trace of the tensor of gyration (Rg = T r(Rg Rg )), characterizes the size of the molecule. In Fig. 2, the radius of gyration of both linear and dendrimer molecules in the blend melt are normalized with their corresponding value Rg,0 in the pure melt of the linear and dendrimer molecules, respectively, and are plotted against the number fraction of the dendrimer in the blend. The equilibrium molecular sizes Rg,0 for dendrimers of generations 1–4 and linear polymers with 187 beads per chain in their pure melt, are shown in Table II. Fig. 2 shows that the radius of gyration of linear molecules increases in the presence of dendrimers. This increase is less when the content of dendrimer increases in the blend. The amount of size increase for linear molecules is higher when blended with dendrimers of lower generations. On the other hand, dendrimer molecules are slightly contracted compared to their size in their pure melt at low x and increases when their content increases in the blend. Chai et al.38 performed NMR experiments and concluded that in good solvents conformations with extended outward facing branches are in the majority, while in poor solvents conformations with folded branches are dominant.

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1.15

J. Chem. Phys. 141, 194905 (2014)

1.15

(a)

1.05 1 0.95 0.9

2

4

6

8 x

10

12

1

0.9

14

2

1.15

(c)

4

6

8 x

10

12

14

4

6

8 x

10

12

14

(d)

1.1 Rg /Rg,0

1.1 Rg /Rg,0

1.05

0.95

1.15

1.05 1 0.95 0.9

(b)

1.1 Rg /Rg,0

Rg /Rg,0

1.1

1.05 1 0.95

2

4

6

8 x

10

12

14

0.9

2

FIG. 2. Radius of gyration for L187 and dendrimer species in the blend normalized with their corresponding values in their pure melt against dendrimer number fraction in the blend, at equilibrium. L187 , D . (a) xD19B, (b) xD43B, (c) xD91B, (d) xD187B.

Therefore, based on their observations, increased molecular size of dendrimers in the presence of linear chains in our blend systems could be a sign of miscibility of blend species. Miscibility will be discussed in more detail in Sec. III B. Qualitatively, the dendrimer molecules behave as a solvent to swell the polymer chains, which is correlated with the molecular mass and the number fraction of the dendrimers in the blend. This means that the low dendrimer mass fractions and lower generations promote linear molecules swelling, presumably due to being a better solvent in this limit. These trends resemble the trends observed for the dendritic PS nanoparticles dispersed in the linear PS reported by MacKay et al.27 They attributed this behavior to the excluded volume effects and nanoparticle mobility and mixing entropy in the system. A similar mechanism would be attributed to our athermal blend systems. Nakatani et al. in their study of polysilicate nanoparticles blended with PDMS also noted that the condition Rg,L > Rg,D is necessary for chain swelling and they also observed that low nanoparticle concentrations promote chain swelling, probably due to better dispersion of nanoparticles at lower concentrations.39 Fig. 3 shows the radius of gyration as a function of shear rate for L187 species and dendrimers of generation 1–4 (a)– (d) in the blend systems; 12D19B, 12D43B, 12D91B, and 12D187B. At lower shear rates, the size of both dendrimer and linear molecules tend to fluctuate around a constant value TABLE II. Equilibrium molecular size Rg,0 of dendrimers of generations 1–4 and linear molecules with 187 beads per chain in their pure melt used to normalize molecular size of blend species in the blend. System

D19

D43

D91

D187

L187

Rg,0

1.67

2.35

2.92

4.46

7.4

and after a certain shear rate, which corresponds to the shear rate at which viscosity shear thins (see Sec. III C) their size increases. Constant size at lower shear rates can be explained in terms of competing mechanisms, i.e., rotation due to vorticity and stretching. Apparently, at lower shear rates these mechanisms compensate each other’s effects on molecular size, and only at higher shear rates does the molecular size increase. However, the increase in size for both of the blend species is not significant due to the low Weissenberg number range in this study, which captures just the start of the thinning region and molecules are not fully stretched. The shear rate at which the radius of gyration increases, decreases by increasing the generation number. The shear rate at which stretching occurs for both linear and dendrimer species are compared with their corresponding values in their pure melt.6 Results (note that x = 12%) reveal that stretching starts approximately at the same shear rate, dendrimer molecules are more stretched in the blend than they are in their pure melt, but linear molecules have approximately the same size (for example, at γ˙ = 0.01 the radius of gyration of D19, D43, D91, D187, and L187 molecules equal 1.75, 2.45, 3.32, 4, and 10, respectively6 ), which we discussed earlier in terms of the solvent effect. Fig. 4 shows the shear-rate dependent size ratio between linear and dendrimer molecules for the blend systems at x = 12%; 12D19B, 12D43B, 12D91B, and 12D187B. It is clear that the size ratio increases with increasing shear rate. However, this is only true above a critical strain rate for each system. The shear-induced increase in size ratio moves our blend systems further toward the miscibility region in the Mackay et al.26 phase diagram. Therefore, this suggests that shear flow improves the miscibility. The radius of gyration is an averaged quantity over the three directions. We can further investigate the flow-induced

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J. Chem. Phys. 141, 194905 (2014)

10 10

(b)

(a)

8 6

6

Rg

Rg

8

4

4

2

2

0 10

−4

10

0

−3

−4

−3

10

γ˙

10

γ˙ 10

10

(d)

(c) 8

6

6

Rg

Rg

8

4

4

2

2

0 10

−4

0

−3

−5

−4

10

10

−3

10

10

γ˙

γ˙

FIG. 3. Root mean-squared radius of gyration for L187 and different generations of dendrimer against the strain rate in the blend systems at equilibrium. L187 , D . (a) 12D19B, (b) 12D43B, (c) 12D91B, (d) 12D187B.

molecule, of each type, regardless of its orientation. Changes in these values with γ˙ quantitatively describe the flow induced stretching of the molecules. In addition, the asymmetry of molecules is characterized by the ratio of the eigenvalues of the average gyration tensor, which we define here as L12 = L1 /L2 , L13 = L1 /L3 , and L23 = L2 /L3 . If the ratios of the eigenvalues are closer to 1, the molecules have greater spherical symmetry.

5

5

4

4

Rg,L /Rg,D

Rg,L /Rg,D

deformation of molecules under shear flow and upon blending by calculating ratios of the eigenvalues of the tensor of gyration. For each system mentioned above, the eigenvalues of the tensor of gyration for each dendrimer and linear molecule in the blend (in descending order, L1 , L2 , and L3 ) were computed and averaged over all molecules of each type. These eigenvalues can be interpreted as the linear dimensions of the instantaneous ellipsoid occupied by the average

3

3

(a)

(b) 2

2 10

−4

γ˙

10

−3

−3

γ˙

10

−2

10

5 4.5

(c)

4

Rg,L /Rg,D

Rg,L /Rg,D

10

10

5 4.5

−4

−2

3.5 3 2.5

(d)

4 3.5 3 2.5

2

2 10

−4

γ˙

10

−3

−2

10

−4

10

−3

γ˙

10

−2

10

FIG. 4. Size ratio between blend species against the shear-rate. (a) 12D19B, (b) 12D43B, (c) 12D91B, (d) 12D187B.

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Hajizadeh, Todd, and Daivis

J. Chem. Phys. 141, 194905 (2014)

2

2

1.9

1.9

1.8

1.8

L13,L

L13,L

194905-7

1.7 1.6

(a)

1.7 1.6

1.5

−4

−3

10

10

(b)

1.5

−2

10

10

−4

10

−3

−2

10

γ˙

γ˙ 2 2 1.9

L13,L

L13,L

1.9

1.8

1.7

1.6

1.8

1.7

(c) −4

−3

10

10

10

−2

γ˙

1.6 −5 10

(d) 10

−4

−3

10

γ˙

FIG. 5. The L13 values of the tensor of gyration for linear molecules against the shear rate for blend systems at constant dendrimer number fraction x = 12%: (a) 12D19B, (b) 12D43B, (c) 12D91B, and (d) 12D187B.

As an example, changes in L13 as a function of γ˙ for linear molecules in the blend systems; 12D19B, 12D43B, 12D91B, and 12D187B are shown in Fig. 5. The values of L13 , which can be interpreted as the aspect ratio of the molecules, fluctuate around a constant value at very low shear-rates and decrease up to intermediate γ˙ values and then increase at higher γ˙ values. Rotation of the molecules at lower shear rates is responsible for the initial decrease in L13 , and the following increase suggests that after a certain shear rate (corresponding to the shear rate at which thinning occurs), stretching is the dominant phenomenon under shear flow. Both the L12 and the L23 show similar patterns as L13 does, which for the sake of space are not shown here.

where riα,jβ =| rjβ − riα |, riα is the position of bead α in dendrimer molecule i and rjβ is the position of bead β in linear molecule j. Ns,D and Ns,L are the number of beads for the dendrimer and linear species, respectively. ND and NL are the number of dendrimer and linear molecules in the blend, respectively. V is the averaged volume of the system. In this function, the distribution of the beads of linear chains from the beads of the dendrimers is calculated. In this way, the mixing of beads belonging to the dendrimer molecules and beads belonging to linear molecules can be analysed. In addition, we define the gLL distribution function as   Nα NL N V i=1 α=1 j =Li α=β δ(r − rjβ,iα ) , (9) gLL (r) = 4π r 2 NL Ns,L

B. Miscibility and interpenetration

where rjβ,iα =| rjβ − riα |, riα is the position of bead α on a linear molecule i and rjβ is the position of bead β in other linear molecules j. Similarly, the gDD distribution function is defined as   ND Nα N V i=1 α=1 j =Di α=β δ(r − rjβ,iα ) gDD (r) = , (10) 4π r 2 ND Ns,D

In order to investigate the miscibility and interpenetration of blend species, a series of different pair distribution functions g(r) are commonly employed.22–24 These functions represent the probability of finding a pair of atoms at a separation r relative to the bulk phase in a completely random distribution. This can provide insights into the specific atomic and/or molecular arrangements in the blend melt. Three different pair distribution functions were used to analyze the miscibility of dendrimer/linear (D/L) blend systems. The gDL (r) function is defined as   Nβ ND Nα NL V i=1 α=1 j =1 β=1 δ(r − riα,jβ ) gDL (r) = , (8) 4π r 2 ND Ns,D NL Ns,L

where rjβ,iα =| rjβ − riα |, riα is the position of bead α on a dendrimer molecule i and rjβ is the position of bead β in other dendrimer molecules j. Fig. 6 represents the gDL , gLL , and gDD distribution functions at equilibrium for blend systems 12D19B, 12D43B, 12D91B, and 12D187B, where the notation is as defined in Table I. Figs. 6(a) and 6(b) show that for the blend systems

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194905-8

Hajizadeh, Todd, and Daivis

J. Chem. Phys. 141, 194905 (2014)

1.5

1.5 (a)

(b)

1

g(r)

g(r)

1 gDL

0.5

gDL 0.5

gLL

gLL gDD

gDD 0

0

5

r

10

0

15

0

2

4

r

6

1.5

1.5

(d)

(c) 1

g(r)

g(r)

1 gDL 0.5

gDL 0.5

gLL

gLL

gDD 0

8

0

2

4

r

6

gDD 8

0

0

5

r

10

15

FIG. 6. The gDL , gLL , and gDD distribution functions for blend system; (a) 12D19B, (b) 12D43B, (c) 12D91B, and (d) 12D187B, at equilibrium.

containing dendrimers of lower generations (D19 and D43), gDL is always greater than gLL and at smaller separations is greater than gDD . According to Akten and Mattice,22 this implies the formation of a miscible blend. The Akten and Mattice criterion in the case of linear-linear blend systems shows that gDL is greater than both gLL and gDD at all the separation distances. However, this is not the case for blends containing dendrimers with an architectural barrier, which makes some distances not easily accessible for penetration. Therefore, we limit this criterion for the proximity of the first neighbours, i.e., r = 0.97. On the other hand, for the blend containing dendrimer of generation 4 (D187) (Fig. 6(d)), only after a specific distance (approximately, equal to the radius of gyration of the dendrimer, see Table I), does gDL become greater than both gLL and gDD , which reveals a very interesting aspect of the tree-like structure of dendrimers and architecturally different D/L polymer blend species. The blend system containing D91 shows unique distribution functions, where it has similarities to both blend categories containing lower and higher generations. Fig. 6(c) shows that gDL is lower than gLL at all the separations and at r = 0.97 all g(r) functions are less than unity similar to blend system containing a higher generation of the dendrimer, while gDL is greater than gDD at lower to intermediate separations similar to blends containing lower generations of the dendrimers. This system 12D91B manifests itself as a crossover from a random to a non-random mixing, in which the generation of dendrimer species plays a determining role. More importantly, at r = 0.97 as the generation of dendrimer species is increased gDL becomes smaller than gLL and smaller than unity. This suggests reduced miscibil-

ity for blends containing dendrimers of higher generations, which agrees with findings of Vlahos and Kosmas19 for chemically identical blend species with different architecture. In addition, for the blend systems and at higher distances, gDD is greater than gDL , where the height of the peaks decreases when the generation (or molecular weight) of the dendrimer is increased and peaks tend to move toward higher values of the separation and become weaker for higher generations. The location of the second peak correlates approximately with twice the radius of gyration of the dendrimer molecules (see Table I) in the blend (note that the crossover blend system 12D91B is an exception in these observed trends possibly because it has similarities to both the blend categories, i.e., random and non-random mixing systems). This observation, in combination with the fact that dendrimers have a crowded surface due to the high number of end groups which doubles with each generation, suggests that dendrimer beads (more specifically, higher generations) are not inter-penetrating deeply toward the lower shells of the other dendrimer molecules and are correlated only at higher separations. This also makes it difficult for linear chains to penetrate into the interior of dendrimer molecules of higher generations. Consequently, it leads to the higher gLL than the gDL at low distances for 12D91B and 12D187B blend systems, as linear chains do not have such an architectural barrier for inter-penetration. Therefore, a higher gLL or gDD than gDL at some specific distances does not necessarily imply that D/L blends are not miscible and higher gDD at some specific distances indicates that dendrimer species are naturally segregated due to their dendritic architecture (impenetrable interior due to the layered architecture and large number of end groups) and are correlated with each

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Hajizadeh, Todd, and Daivis

where ri1,j α =| rj α − ri1 |, ri1 is the position of the core of dendrimer molecule i and rj α is the position of bead α in linear molecule j. Ns,D is the number of beads for the dendrimer species. ND and NL are the number of dendrimer and linear molecules in the blend, respectively. In this function, the distribution of the beads of linear chains from the core bead of the dendrimers is calculated. In this way, the penetration of the beads belonging to linear molecules toward the interior of dendrimer molecules, is analysed. Fig. 7 compares the gc,DL function for blends with dendrimers of different generations; 12D19B, 12D43B, 12D91B, and 12D187B, at equilibrium. It shows that the interpenetration between D-L pairs strongly depends on the molecular weight of the dendrimer species in the blend and increases when the generation of dendrimer species in the blend is reduced. As already mentioned, the surface of dendrimer molecules become more compact with increasing the generation number.1 With the increase in the density of the outermost shell of a dendrimer with generation number due to the high number of end groups, the interior becomes less accessible. This explains why the peaks become weaker in the gDL distribution upon increasing the generation (molecular mass) of the dendrimer species in the DL blend systems. In addition, as first predicted by Goddard et al.,32 dendrimers are subject to a congestion-induced molecular shape transform from flat, loose conformations to robust spheroids as the generation number increases. The shape crossover was succeedingly

1

0.8

gc,DL

other only beyond a specific distance comparable to their size. Additionally, our dendrimer and linear molecules have the same chemical nature and there are no energy differences between these two species (athermal limit) to cause immiscibility. Based on the above arguments, we conclude that our systems form miscible blends, ranging from random to nonrandom mixing as the generation of dendrimer species is increased. This is supported by the fact that at long distances all of the distribution functions approach 1, as there are no long-range correlations, implying the absence of phaseseparated regions. Additionally, Mackay et al.26 derived a polymer radius of gyration-branched polymeric nanoparticle radius phase diagram, in which they highlighted the importance of the size ratio between the linear and branched polymeric nanoparticle in phase-separation, that Rg,L > Rg,D is necessary to obtain a miscible blend. It is important to note that dendrimers, due to their globular constrained shape and unique inter-connected structure, show particle like behavior along with their macromolecular characteristics. The blend species size ratio Rg,L /Rg,D (presented in Table I) for our blend systems locates in the miscible region of the phase diagram reported in a work by Mackay et al.26 and further supports our argument. To describe interpenetration between molecules in the melt, we define a modified version of the radial distribution function previously proposed for pure dendrimer systems.6 It is defined as   Nα ND NL V i=1 j =i α=1 δ(r − ri1,j α ) gc,DL (r) = , (11) 4π r 2 ND Ns,D NL

J. Chem. Phys. 141, 194905 (2014)

0.6

0.4

12D19B 12D43B 12D91B

0.2

12D187B 0 0

2

4

6

8

r

10

12

FIG. 7. The gc,DL distribution functions for blend systems; 12D19B, 12D43B, 12D91B, and 12D187B, at equilibrium.

proven by different measurements (Turro et al.33 and Hawker and Fréchet34 ), which further supports a reduced chance of inter-penetration due to the increased generation (molecular mass). The current consensus on dendrimer structure predicts a dense core,6, 35–37 highly back-folded branches, and functional end groups distributed throughout the dendrimer interior. Therefore, the reduced inter-penetration between linear chains and higher generations of dendrimers was expected based on the model of Boris and Rubinstein.35 Fig. 8 shows the effect of shear rate (γ˙ ) on gc,DL for the 12D187B blend system. It reveals that the gc,DL distribution is much greater for blend systems under the influence of shear compared to its value at equilibrium, and inter-penetration increases upon increasing γ˙ , which is evident through the larger first peak. Under shear, the dendrimer molecules become stretched, therefore more open, and parts of other molecules can come closer to their centres. In addition, it shows that when the shear rate is increased, the secondary and tertiary peaks at higher distances tend to disappear while the first peak increases, and this is because lower distances become more accessible for linear chains. One of the characteristic features of dendrimers is the large number of their terminal groups.

1

0.8

gc,DL

194905-9

0.6

0.4

γ˙ γ˙ γ˙ γ˙

0.2

= 0.0 = 0.00004 = 0.0004 = 0.002

0 0

1

2

3

4

5

r

6

7

8

9

10

FIG. 8. The gc,DL distribution functions for blend system of 12D187B at equilibrium and different shear rates.

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194905-10

Hajizadeh, Todd, and Daivis

J. Chem. Phys. 141, 194905 (2014)

0.18

12D187B

0.16

12D91B 12D43B

0.14

12D19B

0.12

0.1

D43

0.1

gend (r)

gend (r)

0.12

0.08

0.08 0.06 0.04

0.06

0.02 0.04

0

0

1

2

3

4

5

r 0.02

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

r

FIG. 9. Distribution of terminal groups gend of dendrimer molecules in the blend systems; 12D19B, 12D43B, 12D91B, and 12D187B, at γ˙ = 1 × 10−5 . Inset shows the gend of dendrimer molecules of generation 2 in their pure melt.

of generation 2 in their pure melt, at the same shear rate. The most significant feature of these distribution functions is that by increasing the generation number of dendrimer in the blend, the intensity of both first and secondary peak increases. This means that back-folding of end groups occurs significantly for dendrimers of higher generations. Therefore, dendrimers of lower generations have more empty interior space, which makes inter-penetration easier compared to blend systems containing dendrimers of higher generations. On the other hand, comparing the end group distribution for dendrimers of generation 2 in the pure melt (inset plot) to its distribution in the blend melt reveals that back-folding is suppressed significantly upon blending dendrimers with linear chains. This further confirms the opened up configuration of dendrimers in the presence of linear chains and that interpenetration is higher when blending linear chains with dendrimers of lower generations. C. Rheological behavior

In order to get a deeper understanding of configurational rearrangement at the molecular level upon blending, such as back-folding, it is important to understand the spatial distribution of these groups. The location of these groups through the molecular interior affects the penetration of linear molecules into their interior. The distribution of terminal groups from the central bead is defined by  N  D i=1 α δ(r − ri1,iα ) , (12) gend (r) = 4π r 2 ND where ri1,iα =| riα − ri1 |, and α runs over outermost beads only. Fig. 9 shows the gend (r) distribution functions for dendrimer molecules in the blend systems; 12D19B, 12D43B, 12D91B, and 12D187B, at γ˙ = 1 × 10−5 . The inset plot shows the gend distribution function for dendrimer molecules

1. Shear viscosity

The shear viscosity is defined as

 Pxy + Pyx η= − , 2γ˙

whereas the first and second normal stress coefficients are 

Pyy − Pxx (14) 1 = γ˙ 2 and

Pzz − Pyy

 .

γ˙ 2

(15)

Here, Pα,β is the αβ component of the molecular pressure tensor.2 In Fig. 10, the shear viscosity is plotted against

10

(b)

2

η

η

2 =

(a)

2

10

(13)

1

10

10

−5

10

−4

10

10

−3

−2

10

1

−1

−5

10

10

−4

10

−3

−2

10

γ˙

10

−1

10

γ˙ (c)

2

10

(d)

2

η

η

10

1

10

10

−5

10

−4

10

10

−3

−2

γ˙

10

−1

10

1

−5

10

−4

10

−3

−2

10

10

−1

10

γ˙

FIG. 10. Strain-rate dependence of shear viscosity for pure linear (L187), pure dendrimer (D), and for L/D blend systems for 4%, 8%, and 12% of dendrimer molecules with 19 (a), 43 (b), 91 (c), and 187 (d) monomers per molecule. Symbols in each subplot represents: L +, 4DgB , 8DgB , 12DgB , pure D .

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Hajizadeh, Todd, and Daivis

J. Chem. Phys. 141, 194905 (2014)

shear-rate for dendrimer-linear blend melts. Blends are composed of linear chains with 187 beads per chain and dendrimers with (a) 19, (b) 43, (c) 91, and (d) 187 beads per molecule at three different number fractions (x) of 4%, 8%, and 12%. For comparison, the shear viscosity data for the melt of pure linear (L187) and pure dendrimers (D19, D43, D91, and D187) are also presented. All of the systems studied exhibit a transition from the Newtonian regime for small shear rates to the non-Newtonian regime for high shear rates. Melt rheological properties including the shear viscosity (Fig. 10) and first and second normal stress coefficients obtained from constant pressure simulations were found to fall into the range between those of pure dendrimer and pure linear polymer melts. Replacing linear polymers with dendrimers in blends results in a drastic reduction in blend viscosity. We suggest that two factors can be responsible for this reduction. The first and most important factor is related to the replacement of high molecular weight linear chains with dendrimers of lower viscosity. The second factor involves the increased free volume in the system introduced by the dendrimers, along with their effect on the change in the conformation of linear chains (discussed earlier). Mackay et al.25 observed this non-Einsteinlike decrease in viscosity in the presence of branched polymer nanoparticles. They reported that entanglement did not change at all, suggestive of unusual polymer dynamics induced by nanoparticles.25 In addition, Merkel et al.40 also reported the critical role of the free volume on determining the rheological behavior of their nanocomposite system.40 They showed that the fractional free volume is increased by approximately 3φ/Rg,D ,25 where  is the thickness of an assumed spherical excluded volume around each dendrimer, Rg,D is the radius of gyration of the dendrimer, and φ is the volume fraction of dendrimers in the system. It is clear that free volume fraction increases in the system when size of the dendrimer decreases. This further supports our viscosity data that lower generations of the dendrimers are more effective in reducing the blend viscosity and reduction in viscosity is higher for blends containing a higher volume fraction of the dendrimers. The flow curves can be characterized by the zero shear rate viscosity η0 = lim η(γ˙ ), the value of the exponent in the γ˙ →0

TABLE III. Carreau-Yasuda model parameters for DL blend systems. System 4D19B 8D19B 12D19B 4D43B 8D43B 12D43B 4D91B 8D91B 12D91B 4D187B 8D187B 12D187B

η0 η= , [1 + (λγ˙ )2 ]p

(16)

λ

p

90.71 61.96 54.05 91.24 64.43 47.90 130.21 110.32 54.72 130.36 98.35 76.22

1250 1150 1000 1850 1650 1400 2300 2100 1800 2500 2300 2100

0.30 0.25 0.26 0.32 0.30 0.29 0.56 0.58 0.31 0.59 0.6 0.61

presented in Table III. For all the blend systems, the exponents in the power law region increase with the molecular mass of the dendrimer (MD ) in the blend at all the x values, which is in agreement with the results obtained by Bosko et al.1 for pure dendrimer melts. We also observed that the higher values of x for the dendrimers in the blend result in the lower power law exponent values for the blend. In other words, this trend confirms findings of other authors that the absolute values of the power law exponents are larger for linear polymers compared to dendrimers.1, 42–44 Therefore, based on these findings we can conclude that by increasing the amount of the dendrimers in the blend shear thinning behavior becomes less pronounced compared to a pure linear melt. An interesting feature of the shear viscosity versus shear rate for blend systems is that the blend melts have smaller relaxation time (presented in Table III) compared to pure linear (λ 1000) and longer compared to corresponding pure dendrimer melts (67, 332, 514, and 1625 for dendrimers of generation 1–4, respectively). Dendrimer-induced free volume manifests itself through reduction in longest relaxation time of the system compared to the pure linear melt and its subsequent relation to the monomeric friction factor. Fig. 11

2800

−n

power law region η ∝ γ˙ , and the cross-over shear rate γ˙c above which the viscosity becomes shear rate dependent. In order to obtain the zero shear rate viscosity of melt fluids through NEMD simulations, we computed the shear viscosity as a function of the strain rate and then extrapolated to zero shear rate to extract η0 . The Carreau-Yasuda model41 of the form

η0

xD187B

xD91B

xD43B

xD19B

2600

2400

2200

2000

λ

194905-11

1800

1600

1400

where η0 is the zero shear viscosity, λ is defined as the relaxation time of the system, which is calculated through a numerical fitting to the simulation data (see Table III). p is the power law exponent and the exponents in the power law region were obtained from the linear region in the log-log plot of the viscosity versus strain rate curve. Their values as well as the zero-shear rate viscosities for all studied systems are

1200

1000 3

4

5

6

7

8

x

9

10

11

12

13

FIG. 11. Relaxation time λ versus number fraction x for different blend systems. Data for relaxation times obtained from fitting viscosity data to the Carreau-Yasuda model.

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194905-12

Hajizadeh, Todd, and Daivis

J. Chem. Phys. 141, 194905 (2014)

0

0

10

η/η0

η/η0

10

−1

10

(a)

−1

10

(b)

−2

10

−2

10

−2

10

−1

10

0

10

1

2

10

−2

10

10

−1

0

10

λγ˙

10

1

10

2

10

λγ˙

0

10

0

10

η/η0

η/η0

−1

10

−1

10

−2

10

(d)

(c) −2

−2

10

−1

10

0

10

1

10

2

10

λγ˙

10

−2

10

−1

10

0

10

1

10

2

10

λγ˙

FIG. 12. Normalized shear viscosity against Weissenberg number for pure linear (L187), pure dendrimer (D), and for L/D blend systems for 4%, 8%, and 12% of dendrimer molecules with 19 (a), 43 (b), 91 (c), and 187 (d) monomers per molecule. Symbols in each subplot represents: L +, 4DgB , 8DgB , 12DgB , pure D .

presents the time constants obtained from the shear viscosity data fitted to the Carreau-Yasuda model41 versus number fraction of the dendrimers in the blend. The time constant is proportional to the longest relaxation time of molecules composing fluids.1 Fig. 11 shows that by increasing the number fraction of dendrimers in the blend the relaxation time decreases. This further implies the importance of increase in the free volume of the system which is directly related to the number and molecular weight of dendrimer molecules in the blend. Although the absolute values of the viscosity depend on the actual thermodynamic state point, the flow curves can be superimposed on one master curve41, 45 as also shown in Fig. 12. Therefore, though the simulations were performed only at one single thermodynamic point (single pressure, temperature), the results obtained are characteristic for particular molecules comprising the fluid. Fig. 12 shows the shear viscosity normalized by its corresponding zero-shear rate viscosity against Weissenberg number (W i = λγ˙ ) and also fitted to the Carreau-Yasuda model for different dendrimer-linear blends. For better clarity, the master curve has been divided into 4 sub-figures (each sub-figure corresponds to blends composing different number fractions of a specific generation of dendrimer). The most important feature of these plots is that the nonlinear shear thinning regions for the blend systems are not collapsing on top of each other (compared to the pure melts of dendrimers of generations 1–42 ), revealing that the mass fraction of dendrimer in the blend has a great impact on the rheological behavior of these blend systems. One should note that the strain-rate has been normalized with

respect to the longest relaxation time of the blend to define a single Weissenberg number, and this Weissenberg number is not correlated with the relaxation of any of the single components in the blend. Therefore, different shear-thinning levels for blends containing different mass fraction of dendrimers is due to this non-unique relaxation time. In fact, it is not possible to obtain a master curve for these systems, because two different molecules with different relaxation times are involved in the blend dynamics. 2. Normal stress coefficients

In Fig. 13, the log-log plot of the dependence of the first normal stress coefficient on strain rate for dendrimerlinear blends and pure systems of linear polymer L187 and dendrimers of generations 1–4 is shown. Fig. 14 illustrates the second normal stress coefficient versus strain rate for the same systems. The absolute values of first and second normal stress coefficients for the blend systems lie between those of pure linear and pure dendrimer data. However, for the blend systems at higher shear rates, the second normal stress coefficients tend to show lower values compared to their corresponding pure dendrimer melt. This crossover behavior has been previously observed for dendrimer-linear polymer blends.46 In addition, Fig. 13 shows a crossover behavior at very high shear rates when pure dendrimer and pure linear melts are compared. This could be because the pure dendrimer melt has only just stared its thinning behavior while blends are well into their thinning region. In this region, linear chains of the blend melt compared to dendrimer molecules

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Hajizadeh, Todd, and Daivis

10 10

Ψ1

10 10 10 10

J. Chem. Phys. 141, 194905 (2014)

10

10

10

8

8

10

6

6

10

Ψ1

194905-13

4

4

10

2

2

10

(a) 0

(b)

0

−5

−4

10

10

10

−3

−2

10

−1

10

−5

10

−4

10

10

−3

10

γ˙

10

Ψ1

10 10 10 10

−1

10

γ˙

10

10

10

8

8

10

6

6

10

Ψ1

10

−2

10

4

4

10

2

2

10

(c) 0

(d)

0

−5

10

−4

10

10

−3

−2

10

−1

10

10

10

−5

−4

10

−3

10

γ˙

−2

−1

10

10

γ˙

FIG. 13. First normal stress coefficient against shear rate for L/D blend systems for 4%, 8%, and 12% of dendrimer molecules with 19 (a), 43 (b), 91 (c), and 187 (d) monomers per molecule. Symbols in each subplot represents L +, 4DgB , 8DgB , 12DgB , pure D .

can easily align with the flow field while dendrimers start to deform and consequently the difference between pressure components are larger for dendrimers compared to the blend system, which leads to a lower  2 for blend melt compared to dendrimer melt. For all systems, the normal stress coefficients have large power law regions. Linear functions were fitted to the data in the power law region, and from the slopes, the exponents of

−Ψ2

10 10 10 10 10

1 ∝ γ˙ −α

(17)

| 2 | ∝ γ˙ −β

(18)

and

were derived. The values obtained are presented in Table IV. The absolute values of the first and second normal stress

10

8

10 (a)

8

(b) 6

10 6

−Ψ2

10

the asymptotic dependences

4

4

10

2

10

2

0

0

−5

10

−4

10

−3

10

−2

10

10

10

−1

−5

10

10

−4

γ˙

10

−3

−2

10

γ˙ 8

8

10

10

(d)

(c) 6

6

10 −Ψ2

10 −Ψ2

−1

10

4

10

4

10

2

2

10

10

0

0

10

−5

10

−4

10

10

−3

γ˙

−2

10

−1

10

10

−5

10

−4

10

−3

10 γ˙

10

−2

−1

10

FIG. 14. Second normal stress coefficient against shear rate for L/D blend systems for 4%, 8%, and 12% of dendrimer molecules with 19 (a), 43 (b), 91 (c), and 187 (d) monomers per molecule. Symbols in each subplot represents: L +, 4DgB , 8DgB , 12DgB , pure D .

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194905-14

Hajizadeh, Todd, and Daivis

J. Chem. Phys. 141, 194905 (2014)

TABLE IV. Estimated values of the exponents in the power law regions of the first and second normal stress coefficients for DL blends and pure systems (ρ = 0.84, T = 1.25). System D19 D43 D91 D187 L187 4D19BL 8D19BL 12D19BL 4D43BL 8D43BL 12D43BL 4D91BL 8D91BL 12D91BL 4D187BL 8D187BL 12D187BL

α

β

1.089 1.099 1.099 1.065 1.448 1.355 1.353 1.351 1.377 1.372 1.369 1.387 1.384 1.383 1.398 1.395 1.392

0.99 1.03 1.003 1.04 1.507 1.378 1.376 1.372 1.390 1.388 1.385 1.475 1.473 1.470 1.490 1.487 1.483

coefficients are usually smaller for dendrimers than for linear molecules.1 A similar decrease in the normal stresses of branched polymers, compared to linear polymers, was also observed by Jabbarzadeh et al.47 This can be attributed to the globular shape of the dendrimers and their internal bond constraints, which prevent large stretching of the molecules and in turn lead to smaller differences between the diagonal elements of the stress tensor. Table IV shows that these exponents for the blend systems compared to the pure linear system are slightly affected by the addition of the dendrimers and decrease with increasing x of the dendrimers in the blend and also increase by the mass (or generation number) of dendrimers. The values of these two exponents lay in the range reported for linear chains by Kröger et al.42 and Bosko et al.6 IV. CONCLUSION

We used nonequilibrium molecular dynamics simulations to investigate the miscibility, structural properties, and rheological behavior of polymeric blends based on linear polymer chains (187 monomers per chain) and dendrimer polymers of generations (g) 1–4 at different number fractions of dendrimers (x) (4%, 8%, and 12%) undergoing shear flow in the NPT ensemble. We studied the miscibility of the blends, using the radial distribution function gDL , gLL , and gDD obtained from configurational snapshots after reaching a steady state. For all the systems studied, gDL was higher than both gLL , and gDD at some specific distances, which demonstrates that linear chains and dendrimers under the conditions investigated are miscible and their properties are independent of time. The size ratio between linear and dendrimers is a determining parameter on blend miscibility, which has been reported earlier by several authors.25, 39 The Rg,L > Rg,D condition is required to get a miscible blend, which was the case for all our blend systems. From a thermodynamic point of

view, since our blend species had the same chemical nature, the entropy of mixing was a dominant term in the free energy of mixing, which has been promoted in the presence of lower molecular weight dendrimers. The gc,DL pair distribution functions were analysed to study the effect of dendrimer molecular mass and shear rate on interpenetration of linear chains toward the dendrimer interior. We observed higher interpenetration between linear and dendrimer molecules for lower generations of dendrimer, due to the surface congestion effect and a higher amount of back-folding for higher generations. In addition, shear flow increased the inter-penetration between linear and dendrimer species, as stretched dendrimer molecules have more open spaces compared to their undisturbed architecture. We further studied flow-induced molecular deformation of both linear and dendrimer molecules by calculating the radius of gyration and eigenvalues of the tensor of gyration under shear flow. Shear-induced molecular deformation is a consequence of competition between two effects: rotation and stretching. We observed that at lower shear rates, molecular rotation dominates and the molecular aspect ratio decreases, but after a certain shear rate, stretching is dominant and the molecular size increases at a greater rate for linear chains compared to dendrimers. This was expected, as we showed in our previous paper2 that linear molecules can be stretched more easily compared to dendrimers of more constrained geometry. Melt rheological properties including the shear viscosity and first and second normal stress coefficients obtained from constant pressure simulations were found to fall into the range between those of pure dendrimer and pure linear polymer melts. A small amount of dendrimer added to the melt of the linear chains significantly reduced the shear viscosity of the blend system compared to the pure high molecular weight linear melt. This drop in the viscosity was correlated with the number fraction of dendrimers in the blends and also with the geometry and generation of the dendrimer molecules. Blends containing a higher amount of dendrimer molecules of lower generations show much lower melt viscosities. The free volume introduced by the globular shape dendrimers in the system was found to be an important factor in reducing the blend viscosity and was reported earlier by Mackay et al.25 and Merkel et al.40 for similar systems. The amount of the dendrimer-induced free volume in the system is higher for dendrimers of lower generations, which further explains trends observed in the rheological behavior of dendrimer/linear blend systems. 1 J.

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