Polymeric suspensions in shear flow: Relaxation and normal stress differences

Journal of Non-Newtonian Fluid Mechanics 239 (2017) 28–34

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Polymeric suspensions in shear flow: Relaxation and normal stress differences Yuan Lin a, Nhan Phan-Thien b,∗, Boo Cheong Khoo b a b

Institute of Ocean Engineering and Technology, Ocean College, Zhejiang University, Zhoushan 316021, China Department of Mechanical Engineering, NUS-Keppel Corporate Laboratory, National University of Singapore, 117576, Singapore

a r t i c l e

i n f o

Article history: Received 5 July 2016 Revised 12 December 2016 Accepted 17 December 2016 Available online 19 December 2016 Keywords: Polymeric suspension Relaxation Normal stress difference

a b s t r a c t Viscometric properties of non-colloidal polymeric suspensions are investigated, showing a longer relaxation time with increasing volume fraction of the particulate phase, due to the influence of particles that form a pairwise close-contact microstructure. This results in a shift of the onset of the shear thinning to a lower shear rate, with a corresponding shift in the first normal stress difference. The effective relaxation time derived from the first normal stress difference and the shear stress is investigated as a function of the particle volume fraction. The relative relaxation time (normalized by the relaxation time of pure polymeric fluid) is independent of the polymeric suspending mediums used in our experiment (various silicone oils of different molecular weights).

1. Introduction Viscoelastic properties of polymeric suspensions can be greatly modified with the addition of particles, and its understanding is therefore very important in industrial processes in crude oil transport, injection molding of filled polymeric materials, etc. The interaction between polymer molecules and nano-particles in polymeric nanocomposites has been studied both experimentally and numerically [1–10], at low Peclet number, Pe ≈ 0 (Pe = γ˙ a2 /D, where γ˙ is shear rate, D = kT /6π η0 a is the self-diffusivity of a particle of radius a and thermal energy kT in a fluid of viscosity of η0 ). The conformations of a polymer chain can be significantly altered, compared to the polymers in bulk, if a typical length scale of the confined space is less than a diameter of gyration, 2Rg , of the polymer molecules. This behavior is subject to many factors, such as the end effect of polymers at the interface, interactions between walls and polymers, decrease of entanglement and segregation of polymer chains [4]. In nanocomposites, at medium and high volume fractions of the particulate phase, since the gap between particles is smaller than a diameter of gyration of the molecular chains (Rg ∼ 10 nm), the motion of the macromolecules is significantly affected by the particles. The entangled molecular chains are found to be significantly disentangled, which results in a transition from the confinement between molecular chains to the confinement by particles [9]. With increasing particles volume fraction, φ ,

∗

Corresponding author. Fax: +65-68723069. E-mail address: [email protected] (N. Phan-Thien).

http://dx.doi.org/10.1016/j.jnnfm.2016.12.005 0377-0257/© 2016 Published by Elsevier B.V.

© 2016 Published by Elsevier B.V.

the relaxation behavior of polymeric nanocomposites is considerably slowed down due to the confinement effect of the particles, and results in the considerable change of the rheological behavior [8]. For polymeric suspensions with non-colloidal (micron-sized) particles, by contrast, the behavior of polymer phase is unlikely to be affected by particles at static state (Pe ≈ 0), due to the large gap between particles at equilibrium [11] (Fig. 1(a)). Nevertheless, in shear flow (Pe  1), the pairwise particle configuration becomes completely different. Some works have been done on the motion of particles in the viscoelastic matrix and the interaction between the particle phase and the polymeric phase [12–18], Hwang et al. [12] simulate single-, two- and many-particle motions in simple shear flow in an Oldroyd-B fluid, and find that two neighboring particles exhibit a tumbling behavior. Yoon et al. [18] study the two-particle interactions, and find three types of interactions: pass (particles approach and then rotate around each another and continue on moving in their original direction after interaction), return (particles approach one another and return), and tumble (particles approach and continually rotate around one another). They also find that particles are pushed closer with increasing elasticity of the fluid. Experimentally, Snijkers et al. [15] find that the fore– aft symmetry (an equal probability of a second particle lying on a trajectory where it is approaching (fore) or receding (aft) relative to a reference particle) of the open trajectory of an isolated pair of particles in a Newtonian fluid is distorted in a shear-thinning viscoelastic suspending medium. The particles is less likely to tumble compared to simulation. The particle microstructure (configura-

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Nomenclature a D D e Fpp gbl (r ) g0 (r ) g∞ (r ) G k L Mn Mw N1 N2 Pe R Rg SH bl H Sbl (xy )

T uT We γ˙ γ˙ c γ˙ R

ε

η η∞ λ λ0 μ μ0 μp ξ τ τ xy φ

radius of particle self-diffusivity deformation-rate tensor gap between two nearest particles normal force measured by parallel-plate geometry pair-distribution function at boundary layer of particles equilibrium pair distribution function for hard spheres pair distribution in flow with high Peclet number rigidity modulus Boltzmann’s constant effective velocity gradient number average molecular weight weight average molecular weight first normal stress difference second normal stress difference Peclet number radius of plate in parallel-plate and cone–plate geometries radius of gyration hydrodynamic stress tensor at boundary layer of particles hydrodynamic shear stress at boundary layer of particles absolute temperature velocity gradient in shear flow Weissenberg number shear rate critical shear rate accounting for shear thinning rim shear rate parameter related to elongation behavior of polymeric fluid constant viscosity coefficient dynamic viscosity at high frequency characteristic relaxation time of suspension characteristic relaxation time of pure fluid from experiment viscosity of polymeric suspension viscosity of pure fluid viscosity of polymer matrix from PTT model parameter accounts for slip between molecular network and continuum medium stress tensor shear stress volume fraction

tion of a pair of interacting particles) in a polymeric suspension in shear is found to be similar to that in a Newtonian suspension of rough particle, as illustrated in Fig. 1(b) [19,20]. There is an excess of particles along the compressional axes (approaching the reference particle) and relatively a smaller number of particles in the extensional quadrants (leaving the reference particle). Therefore, with approaching of particles in shear flow (Fig. 1(b)), a lubrication force arises tending to separate particles by a gap that could be smaller than 2Rg . This could then affects the dynamics of the polymer matrix by confining a fraction of polymer chains in the microstructure formed by the approaching particles. As a result, the relaxation behavior of a polymeric suspension is expected to be different from that at static state [11]; the approach of particles can also induce local shearing in the microstructure [21], both of which result in the complicated

Fig. 1. A sketch of the particle microstructure in suspension (a) in static state and (b) in shear flow with Pe  1.

rheological behavior. Zarraga et al. [22] find that the magnitude of the first normal stress difference, N1 , of the non-colloidal polymeric suspension can be attributed to the viscoelasticity of the suspending fluid, while the second normal stress difference, N2 , at high concentration approaches that measured for a similar Newtonian suspension. Tanner and Qi [23] point out that the stress tensor of the polymeric suspension composes of the stress from a Newtonian part and a viscoelastic part, and the relaxation time of the viscoelastic part increases with increasing volume fraction. It is found in a previous study [11] that, the shift of N1 with increasing φ in a non-colloidal polymeric suspension, which has been reported and much studied [22,24–27], is caused by a change of the relaxation behavior of polymer molecules in the confined space between particles. Furthermore, the local shear-rate enhancement in the confined space may also contribute to the shift of the onset of shear thinning behavior. At high volume fractions, the second normal stress difference, N2 , mainly arises from the hydrodynamic interactions of the particle phase, and can be described by the theory of Brady and Morris [19] for the Newtonian suspensions. In this paper, we study the rheological behavior of non-colloidal polymeric suspensions using three types of silicone oils with different molecule weights as suspending mediums. Based on the particle microstructure that is similar to that in Newtonian suspensions of rough particles, the relation between the effective relaxation time and the particle volume fraction is identified. Normal stress differences are also investigated. 2. Theoretical background There are many representative constitutive equations for polymer solutions. We choose the Phan–Thien–Tanner (PTT) model as a convenient means to discuss our experimental data. This choice is dictated by convenient for discussing our experimental data, and by no means, a necessary choice. In the PTT model, the extra stress is written as ∇

f (tr(τ ))τ + λ τ = 2ηD,

(1)

where τ and D are the extra-stress and deformation-rate tensors, tr is the trace operation, λ is a relaxation time, η = Gλ is the con∇

stant viscosity coefficient, where G the rigidity modulus. Here, τ is upper convected derivative defined as ∇

τ=

d τ − L · τ − τ · LT , dt

where the effective velocity gradient, L, is defined by

L = ∇ u T − ξ D. In these equations, ∇ uT is the velocity gradient and the parameter ξ accounts for the slip between the molecular network and the continuum medium. The linearized form of f(tr(τ )) is used in this study

f (tr(τ )) = 1 +

ελ tr(τ ), η

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Y. Lin et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 28–34

Table 1 Properties of silicone oils used in the experiment. Silicone oil

μ0 (Pa s )

Mn (g/mol )

Mw (g/mol )

Density (g/cm3 )

KF-96H-10 0 0 0 KF-96H-60 0 0 0 KF-96H-50 0 0 0 0

10.2 61.8 528

22,338 41,039 33,694

71,835 113,564 202,404

0.975 0.976 0.978

in which ε is a parameter related to the elongational behavior of the model. ξ and ε are parameters to be determined experimentally. For a shear flow along the x-direction and with the velocity gradient along the y-direction, the shear viscosity for the polymeric fluid, μ p (γ˙ ), as well as the first and second normal stress differences at steady state can be derived analytically

μ p (γ˙ ) = N1 =

τxy η − λξ τxx = , γ˙ 1 + [2ελ(1 − ξ )/η (2 − ξ )]τxx

2 ξ τxx , N2 = − τxx , 2−ξ 2−ξ

(2) (3)

where τ xx can be found explicitly in terms of γ˙ [28]. For a pair of particles with some certain roughness, or undergoing Brownian motion in a Newtonian suspension, when Pe  1, as illustrated in Fig. 1(b), a fore–aft asymmetry in its microstructure is formed [19]. There is a thin boundary layer of the thickness of O(aPe−1 ) (a is the radius of particles) around a generic particle along the compressional axis, in which an excess of particles exists statistically. The microstructure is described by the pair distribution function of particles, gbl (r ). The stress tensor from this microstructure, which originates from a two-body lubrication force, assuming a Newtonian solvent, is dominant in the particle contributed stress. The stress tensor scales as [19,20]  2 ∞ SH bl ∼ γ˙ η ∞ (φ )φ g (2; φ ),

(4)

where η∞ is the dynamic viscosity at high frequency, the contribu-

tion to the viscosity resulting from the particles’ hydrodynamic interactions that are present in the equilibrium configuration of particles, and it is also a measure of the enhancement of the shear force due to the added particles [20]. g∞ (2; φ ) is the value of the pair-distribution function just outside the boundary layer at Pe → ∞. The steady viscosity of this Newtonian suspension, μ(φ ), can be estimated as

μ(φ ) − η∞ (φ ) ∼ η ∞ (φ )φ 2 g∞ (2; φ ).

(5)

Since η∞ , which is used for Newtonian suspensions, cannot be measured directly in small amplitude oscillatory shear (SAOS) experiment for our polymeric suspensions because of the nonNewtonian character of the polymer matrix, η∞ is thus deduced via [29]

(η∞ /η0 − 1 )/(η∞ /η0 + 3/2 ) = φ [1 + S(φ )], φ2

− 2.3φ 3 .

(6)

in which S(φ ) = φ + For φ < 50%, φ ) can be approximately represented by the equilibrium pair distribution function for hard spheres [20], g0 (2; φ ) = (1 − 0.5φ )/(1 − φ )3 . g∞ (2;

3. Experiment In this study, borosilicate glass spheres (Dantec Dynamics) with an average diameter of 3.35 μm are suspended in three types of silicone oils (PDMS, Shin-Etsu). The suspended particles are polydispersed in size as measured in the previous work [30], with diameters distributed from approximately 1–7 μm. The zero shear viscosity, μ0 , molecular weight and density of the silicone oils used are listed in Table 1. Experimental data for silicone oil KF96H-60 0 0 0 from our previous study [11], are also included here.

Fig. 2. Viscosity and the first normal stress difference for silicone oils used as suspending liquids. The solid lines are the results of the PTT model. The experimental data of KF-96H-60 0 0 0 is from the previous study [11].

The experimental temperature is kept at 25 ◦ C. Curves of viscosity and N1 for those silicone oils are shown in Fig. 2. The glass spheres are hollow, and have an effective density of 1.1 g/cm3 . Due to the high viscosity of silicone oils, the suspensions are stable; no sedimentation is observed during the experiments. The suspensions are mixed manually for a long period (at least 30 min). Then samples taken from different part of the suspension are tested in the rheometer, and the viscosity data are compared to each other in order to make sure that the suspension is homogeneous. Measurements are carried out using a HAAKE MARS III rotational rheometer (Thermo Scientific) in the shear-rate control mode. The cone– plate and parallel-plate geometries with 35 mm diameter are employed. The cone angle is 1.984° and it is truncated at 104 μm from the vertex. The gap used in the parallel-plate geometry is 500 μm. The suspension viscosity and N1 are measured directly using the cone–plate geometry. The second normal stress difference, N2 , is deduced from the measured N1 − N2 using parallel-plate geometry according to the equation [31],

N1 (γ˙ R ) − N2 (γ˙ R ) =

Fpp π R2



2+



d ln Fpp . d ln γ˙ R

(7)

Here, Fpp is the normal force measured by the parallel-plate geometry, R is the radius of the plate and γ˙ R is the rim shear rate. Fpp versus γ˙ R is first fitted by a power law equation and d ln Fpp /d ln γ˙ R is then deduced accordingly. N2 can be accurately determined by this method if its magnitude is comparable to N1 . At high shear rates, edge fracture may occur in our experiments, resulting in a decreasing in the viscosity with shearing time in an otherwise steady shear-rate flow; it may also produce shear thinning behavior in experiments with step increments in shear rates [32,33], especially at high particles volume fraction. In order to mitigate the influence of edge fracture at higher shear rates, we reduce the shearing time to make sure that the viscosity is stable or just slightly reduced in the shearing period between shear rate increments, during which no obvious sample loss is found at the edge of the geometry. The normal stress differences are averaged over a steady shear period that lasted for at least 5 s (with at least 40 data points recorded). Also, the normal stresses are averaged over three samples at each concentration, with the fluctuations between different samples less than 10% of the average value. Detailed experimental procedure can be found in our previous work [11].

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Fig. 3. (a) Dependence of the magnitude of complex viscosity on frequency in the SAOS experiments; (b) steady shear viscosity of suspensions with volume fraction of 0%, 25%, 40%, 45% and 49%, respectively. (c) Enlargement of the viscosity curves for suspension with volume fraction of 0% and 25%, respectively; the solid lines are used to help determine the onset of the second shear thinning region caused by polymer suspending phase. The polymeric matrix is silicone oil KF-96H-10 0 0 0.

4. Result and discussion The viscosity and N1 for suspensions in silicone oil KF-96H10 0 0 0 are shown in Figs. 3 and4, respectively. From Fig. 3(a), we find that in small amplitude oscillatory shear (SAOS) flow where the strain amplitude is in the linear viscoelastic range, the complex viscosity of the suspension has the same form as that of the pure polymer fluid, except for a vertical shift, similar to the previous studies [11,34]. Since linear viscoelastic properties reflect the suspension microstructures at the equilibrium state; thus, in static state, the relaxation behavior seems independent of the volume fraction of particles in contrast to nano-particle suspensions [8]. As has been pointed out in our previous study [11], the gap between 3 particles at static state is estimated as e ≈ a (1 − φ ) /6φ (2 − φ ) [35]. For micro-particles, the gap between neighboring particles for a suspension with volume fraction of 50% is approximately 50 nm, which is still much larger than 2Rg . Therefore, the relaxation behavior of the polymer phase is expected not to be affected by micron-sized particles within the concentration range examined. In shear flow, we observe from Fig. 3(b) and (c) that for the pure polymeric fluid, shear thinning occurs at a critical shear rate, γ˙ c = 79.6 s−1 , with a characteristic relaxation time λ0 = 1/γ˙ c = 0.013 s. The shear thinning behavior accounts for the orientation of polymer chains due to flow [36]. For polymeric suspensions, with increasing particle volume fraction, two shear thinning regions are clearly observed. At low shear rates (γ˙ < 0.5 s−1 ), the first shear thinning appears and becomes more pronounced with increasing φ , similar to that observed in a Newtonian suspension [30,37]. Lin et al. [38] point out that the shear thinning behavior at low shear rates is caused by the forming of aggregative structures of particles in steady shear, the size of which decreases with increasing shear rate. At higher shear rates, a second shear-thinning behavior is observed, the onset of which is shifted to a lower shear rate with increasing φ . For the first normal stress differences, as shown in Fig. 4(a), a positive N1 is observed. When φ ≤ 25%, N1 decreases with increasing φ if compared at constant values of the shear stress, τ ; N1 (τ ) can be represented by a power-law function: N1 (τ ) ∼ τ n1 , and n1 ≈ 2 is independent of φ . This behavior agrees with the finding of Mall-Gleissle et al. [27] using similar Boger fluids as matrix. For φ > 25%, we find N1 becomes insensitive to φ , and n1 decreases compared to the one at low φ , while Ohl and Geissle [25] find that N1 still decreases with increasing φ . Meanwhile, if we plot the data as a function of shear rate, as shown in Fig. 4(b), a positive N1 is observed, which increases with increasing φ . This is consistent with the finding of Haleem and Nott [24]. N1 arises with the second shear thinning, similar to that of the pure polymeric fluid (Fig. 2). Since in a Newtonian suspension, N1 from the particle hydrodynamic interaction is negative and small in magnitude (see for

Fig. 4. The first normal stress difference of suspensions as a function of (a) shear stress, and (b) shear rate, with various φ . The inset shows the rescaling of the data by We = γ˙ /γ˙ c . The polymeric matrix is silicone oil KF-96H-10 0 0 0.

example, Zarraga et al. [39] and Dai et al. [40]), it is considered that the positive N1 is attributed to the orientation and stretching of the polymeric phase in the suspension [11]. The dependence of the characteristic relaxation time, λ(φ ) (or γ˙ c−1 ), of the polymeric suspension on the particle volume frac-

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Y. Lin et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 28–34 Table 2 The critical shear rate, γ˙ c , of suspensions with different silicone oils and at various φ . The data for KF-96H-60 0 0 0 are taken from our previous study [11]. Silicone oil

0%

KF-96H-10 0 0 0 KF-96H-60 0 0 0 KF-96H-50 0 0 0 0

79.6 s−1 18.6 s−1 1.05 s−1

15%

25%

15.1 s−1

51.2 s−1 12.8 s−1 0.77 s−1

33%

40%

45%

49%

9.7 s−1 0.65 s−1

16.7 s−1 6.2 s−1 0.38 s−1

11.5 s−1 3.6 s−1

8.7 s−1 2.0 s−1

of μp from polymeric phase due to increasing λ, should be of the same order as the shear stress originating from the two-body luH brication force, Sbl (xy ) . If this physical picture is correct, we expect that the two stresses are of the same order leading to, H Sbl (xy ) /γ˙ ∼ μ p − μ0 = μ0 (λ (φ )/λ0 − 1 ).

(8)

Noting Eq. (4), we deduce that

λ(φ )/λ0 − 1 ∼ φ 2 η∞ (φ )g∞ (2; φ )/μ0 .

H Fig. 5. Changes of λ/λ0 − 1 with φ and in comparison with Sbl (xy ) /γ˙ scaled as φ 2 η∞ (φ )g∞ (2; φ )/μ0 . The experimental data of KF-96H-60 0 0 0 are from the previous study [11].

tion can be deduced from the onset of the second shear thinning or from the shift of N1 . We have shown in our previous study [11] that it is more straightforward to determine γ˙ c from the horizontal shift of N1 . With increasing volume fraction, N1 shifts towards increasingly lower shear rates as shown in Fig. 4(b). The critical shear rates, γ˙ c (φ ), can be deduced by overlapping the N1 curve using the Weissenberg number, W e = γ˙ /γ˙ c (φ ) as shown in the inset of Fig. 4(b). For pure silicone oil (φ = 0), γ˙ c = 79.6 s−1 is determined as the shear rate at the onset of shear thinning, as shown in Fig. 3(c). Table 2 shows γ˙ c (φ ) for suspensions with various silicone oils and φ . The characteristic relaxation time, λ(φ ) = γ˙ c−1 (φ ) can be obtained. The relative relaxation time, λ(φ )/λ0 − 1, is plotted as a function of φ , shown in Fig. 5, in which λ0 is the relaxation time of the pure polymeric fluid. We can see in Fig. 5 that λ(φ )/λ0 appears independent of the suspending polymeric fluids used in our experiment. In the PTT model, with increasing relaxation time, λ, the coefficient η = Gλ increases, leading to an increase in the viscosity, μp (see Eq. (2)). According to the PTT model, μ p = μ0 λ(φ )/λ0 at low shear rates (before the shear thinning of the polymeric phase). The increase of μp with λ(φ ) may be explained by the fact that an increase in λ is accompanied by an increase in the number of entanglements of polymer chains, thereby giving higher resistance against the chain motion. Thus, for polymer melts, η is a measure of the resistance felt by a polymer chain moving through the tube formed by its neighbors. While in a polymeric suspension, the polymer chains, confined by the close-contact particles, are significantly disentangled and the motion of polymer chains is now constrained by “particle entanglements” instead of the polymer-chain entanglements [9,10]. Therefore, the increase of η (and thus of μp ) due to increasing λ is regarded as a measure of the resistance to the chain motion provided by pairs of close-contact particles, which may be described by the theory of Brady and Morris [19]. In other words, the additional shear stress arises from the increase

(9)

This indicates that λ(φ ) increases with particles volume fraction, because of the increase in the lubrication force between particles, estimated as φ 2 η∞ (φ )g∞ (2; φ ). Since φ 2 η∞ (φ )g∞ (2; φ )/μ0 is a function of φ , independent of the viscosity of the pure polymeric fluid μ0 , one can infer from Eq. (9) that λ(φ )/λ0 should form a master curve for suspensions regardless of the various polymer matrix used. Eq. (9) is used to estimate the relaxation time and compared with our experimental data. The agreement is good as shown in Fig. 5, providing the consistency in the physics. Therefore, Eq. (9) is deemed a good estimate of the relaxation of polymer suspension in shear flow. The additional shear stress (μ p − μ0 )γ˙ from H confined polymers approaches the shear stress, Sbl (xy ) , from the lubrication force of particles in shear flow. We show that the relaxation of the polymeric suspension becomes slower with increasing volume fraction of particles. In a stress relaxation experiment, Aral and Kalyon [26] found the relaxation of the polymeric suspension become slower with increasing φ , consistent with our results. The slowing down of the relaxation of polymeric suspensions is considered to originate from the confinement of polymer chains in the close-contact regions of particles in shear flow, and thus has direct connection with the lubrication force, as is indicated by Eq. (8). On the other hand, the local shearrate enhancement could also contribute to this as well. In the suspension model from Foss and Brady [20] (Eq. (4)), the shear rate in the suspension is considered homogeneous and equals to the average or bulk shear rate. However, the local shear rate between particles is not homogeneous [21]. Indeed the increase of the characteristic relaxation time from Eq. (9) implies the local shear rate enhancement by a factor of λ(φ )/λ0 . Here, we may propose that the shift in N1 and shear thinning viscosity are combined effects to slow down the relaxation behavior of the polymeric phase and enhance the local shear rate between approaching particles. In previous studies, the stress tensor for a polymeric suspension is considered to be a sum of the stress arising from hydrodynamic interactions from the particle phase and the stress from polymeric phase with an effective relaxation time modified by particle volume fraction. The dominant contribution to N1 comes from the polymeric matrix, while the second normal stress difference, N2 , is dominated by the stress from particle phase at high φ [11,22,23]. N2 for our polymeric suspensions is shown in Fig. 6 at different volume fractions. However, we cannot measure N2 reliably for the pure polymeric matrix. The magnitude of N2 for the pure matrix is not more than O(10) Pa, based on instrument/sensor consideration. It can be seen that N2 for the polymeric suspension is negative, the magnitude of which increases with increasing φ ; N2 ∼ γ˙ is similar to that of Newtonian suspension, which is consistent with the finding of Zarraga et al. [22]. To attempt to quantify the contribution of the particle phase on N2 for the suspension, N2 curves are

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for the polymeric phase is small in magnitude. In a polymeric suspension at low φ , N2 from the particle phase is still comparable to that from the polymer phase. Therefore, the power law behavior, N2 ∼ γ˙ n2 with n2 close to 2 in the polymeric phase, shifts to n2 close to 1 at high volume fraction, as shown in previous studies [11,22]. At high φ , N2 from the particle phase dominates, which is also confirmed in this study. Acknowledgments The support from the Agency for Science, Technology and Research (A∗ STAR) through Grant #102 164 0147 is gratefully acknowledged. References

Fig. 6. The second normal stress differences, N2 , of suspensions with various φ . The inset shows N2 scaled by the Weissenberg number. The polymeric matrix is silicone oil KF-96H-10 0 0 0.

also scaled using the Weissenberg number, We, as shown in the inset of Fig. 6. Since N1 , which is mainly from the polymer phase, can be scaled into a master curve by We, shown in Fig. 4(b), the polymeric contribution to N2 is also expected to be independent with φ when scaled with We. From the inset of Fig. 6, we find that N2 increases with φ and approaches a “master curve” at high φ when scaled with We. This can be identified as the contribution to N2 from the particles. If we equate N2 to the particles-contributed stress according to Eq. (8), we have

N2 ∼ μ0 (1/λ0 − 1/λ(φ ))W e.

(10)

As shown in Table 2, for the polymeric matrix of KF-96H-10 0 0 0, the term 1/λ0 dominate 1/λ(φ ) at high concentration, results in Eq. (10) becoming nearly independent of φ , at high φ , in agreement with the result shown in Fig. 6. We thus conclude that the behavior of N2 is dominated by the particle phase at high volume fractions, consistent with the previous works [11,22]. It should be noted that the conclusion proposed here is expected to be relevant for polymeric suspensions in which the motion of a generic pair of particles is similar to that in Newtonian suspension. Due to the edge-fracture phenomenon at higher shear rates in the rheological test, we cannot extend our study into the intensively shear-thinning range, where the particle microstructure may be completely changed. Particles could form strings which align with the flow [14,17,41,42]. Choi and Hulsen [13] performed 2D simulations for a Giesekus fluid, and found that particles can form strings which grows longer with increasing We. String formation is observed to occur at approximately N1 /τ xy > 1, in dilute to moderately concentrations. The proposed framework is not applicative if strings form. Also, this model may not be suitable for very dilute polymeric suspensions (φ  1%) where the lubrication force is not dominant. The change of normal stress differences may be caused by the tension of streamlines around the particle in flow in this case [43]. 5. Conclusion In polymeric suspensions, the stress tensor is the sum of the contributions from both the polymeric phase and the particle phase. Since N1 from the particle phase is negative and small in magnitude, an observed positive N1 for polymeric suspension thus comes from a dominant contribution from the polymeric phase. The relaxation behavior of the polymeric phase is slower by the particle phase, and this behavior can be well predicted, for example, by the PTT model with the relaxation time (from Eq. (9)). N2

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