rheology of nano-cellulose fiber suspension

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Article

Nihon Reoroji Gakkaishi Vol.45, No.1, 3~11(Journal of the Society of Rheology, Japan)©2017 The Society of Rheology, Japan

1. INTRODUCTION

Cellulose, a representative biomass, is non-soluble in most of liquids and has been used as a constructional material exhibiting a very small thermal expansibility and a high modulus ( per unit mass). Recent discovery of solvents for cellulose1-5) enables detailed molecular characterization, which has led to deeper understanding of the properties of pristine cellulose. Nevertheless, industrial application of cellulose is largely relying on processing of cellulose fiber suspensions in water, because these fibers are not molecularly dissolved in water and preserve their outstanding, pristine properties. Thus, it is important to study the properties of those cellulose suspensions in relation to the primary and higher-order structures of the cellulose fibers, the latter referring to a “mesh” structure formed by those fibers. For example, for the

storage and loss moduli of pulp fiber suspensions, Tatsumi and Matsumoto6,7) revealed a characteristic power-law type dependence on the fiber concentration and related the power-law exponent to the scarceness (contact density) in the mesh.

Recently, Abe, Yano, and coworkers8,9) developed a mechanical method for preparation of cellulose nanofibers: They utilized a high-speed blender to crack macroscopic cellulose fibers (having an optically visible diameter) such as the pulp fibers into nanofibers having a meso-scale length (> 1 mm) but a nano-scale diameter (= 10-100 nm).8,9) It is of interest to examine properties of such nanofibers in suspensions.

Thus, we have examined rheological properties of suspensions of nano-cellulose (NC) fibers made from pristine cellulose and calboxymethyl nano-cellulose (CM-NC). It turned out that those NC suspensions containing a mesh of NC fibers behaved as elastic solids under small strains. The NC fibers were curvy and had a multiple-branch structure, and the elasticity of the suspensions was attributed to strain-

Rheology of Nano-Cellulose Fiber Suspension

Yumi MatsuMiya*, Hiroshi Watanabe*,†, Kentaro abe**, Yasuki MatsuMura***, Fumito tani***, Yasuo Kase****, Shojiro KiKKaWa****, Yasushi suzuKi*****, and Nanase ishii*****

* Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan** Research Institute for Sustainable Humanosphere, Kyoto University, Uji, Kyoto 611-0011, Japan

*** Graduate School of Agriculture, Kyoto University, Uji, Kyoto 611-0011, Japan**** Nissei Co. Ltd., 510-7 Aza-Suwa, Oaza-Shida, Taga-cho, Inukami-gun, Shiga 522-0314, Japan

***** Saraya Co. Ltd., 24-12 Tamate-cho, Kashiwara, Osaka 582-0028, Japan(Received : August 23, 2016)

Rheological behavior was examined for aqueous suspensions of nano-cellulose (NC) fibers. The plain-NC and carboxymethyl NC (CM-NC) fibers of an average diameter of ≅ 20 nm and a length of > 1 mm were used. These fibers had a curvy, multiple-branch (branch-on-branch) structure, and some of those branches were bundled into thicker trunks in particular for CM-NC, as revealed from TEM. Both plain-NC and CM-NC suspensions, having a low NC concentration of 0.62 wt%, exhibited essentially elastic behavior under small strain in the linear regime. This elastic behavior was attributable to elastic bending of the NC fibers that formed a mesh in the suspensions. Comparison of the elastic modulus data with the modified Doi-Kuzuu model prediction suggested that the curvy, multiple-branch structure of the NC fibers significantly increases effective contacts between the fibers thereby enhancing the elasticity, in particular for the plain-NC fiber having scarcer trunk portions compared to the CM-NC fiber. Corresponding to this origin of the enhanced elasticity, the suspensions exhibited significant nonlinearity under large amplitude oscillatory shear (LAOS) and steady shear flow. Namely, the modulus and steady state viscosity of the suspensions decreased roughly in proportion to reciprocal of the strain and shear rate, respectively, although strain-hardening was also noted for the CM-NC suspension under moderately small strains. This nonlinear decrease of the modulus and viscosity, phenomenologically classified as the yielding, would have resulted from strain/flow-induced slippage (decrease of the effective contacts) between the fibers. Key Words: Nano-cellulose suspension / Fibber bending elasticity / Fiber-fiber contact slippage

† to whom correspondence should be addressed [email protected]

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Nihon Reoroji Gakkaishi Vol.45 2017

induced bending of those fibers between their mutual contacts being enhanced by this curvy, multiple-branch structure. Under large strain and/or flow, the NC suspensions exhibited significant nonlinearity (mostly softening) attributable to strain/flow-induced slippage of effective fiber-fiber contacts. Details of these results are presented in this article.

2. EXPERIMENTAL

2.1 Materials

Nanofibers of pristine cellulose were prepared from a 0.75 wt% aqueous suspension of Japanese cypress pulp by strongly agitating this starting material with a high-speed blender equipped at Research Institute for Sustainable Humanosphere, Kyoto University. Details of this blender (such as a high-power ABS-BU motor), and the operation condition were described elsewhere.8,9) For comparison, a 2.4 wt% aqueous suspension of carboxymethyl (CM) cellulose fiber was prepared at Nissei Co. Ltd with a high-pressure homogenizer. The CM group was introduced (through a reaction with the hydroxyl group of cellulose) mostly on the fiber surface, and the reaction efficiency was below 0.3.

These mother suspensions were dried in vacuum, coated with platinum by an ion sputter coater, and then subjected to Transmission Electron Micrographic (TEM) observation with JSM-7800F Prime (JEOL Ltd., Tokyo, Japan). Fig. 1 shows TEM images of the plain (pristine)-NC and CM-NC fibers thus obtained. These images suggest that both plain-NC and CM-NC fibers had an average diameter of ≅ 20 nm and a length > 1 mm (not accurately determined because the fibers were longer than the width of TEM view field). More importantly, the plain-NC and CM-NC fibers are curvy and have a branch-on-branch type multiple-branch structure. Some of those branches are bundled into thicker trunks, and those trunk portions are scarcer in the plain NC fiber than in the CM-NC fiber. This curvy, multiple-branch structure becomes a key in our later discussion of rheological behavior.

The mother suspensions thus obtained above were diluted with water to prepare samples subjected to rheological measurements, the plain-NC and CM-NC suspensions having the same NC concentration of 0.64 wt%. The mother suspension of CM-NC (2.4 wt%) was also subjected to the measurements.

2.2 Measurements

For the NC and CM-NC aqueous suspensions prepared as above, dynamic oscillatory and star-up flow measurements were conducted with a laboratory rheometer, ARES-G2 (TA

Instruments), at 25 ˚C (room temperature). A cone-plate fixture with the diameter of 25 mm and the gap angle of 0.1 radian was used. In the dynamic measurements made at the angular frequencies w = 0.1 - 100 (s-1), the oscillatory strain amplitude, g 0, was varied to examine the nonlinearity of the modulus. (Linear behavior was observed for sufficiently small g0.) In the start-up flow measurements, the shear rate g4 was varied from 0.1 to 30 (s-1) to examine the nonlinearity under flow.

3. RESULTS AND DISCUSSION

3.1 Dynamic Behavior

Figure 2 shows the dynamic oscillatory moduli data measured for the 0.62 wt % aqueous suspension of plain-NC at 25 ˚C. For sufficiently small strain amplitudes, g0 ≤ 0.05, the suspension exhibited the moduli being insensitive to g 0. The moduli for such small g0 reduce to the storage and loss moduli, G' and G", defined in the linear viscoelastic regime. As noted in Fig. 2 for g0 ≤ 0.05, G' is considerably larger than G" and insensitive to the angular frequency w . Namely, essentially elastic behavior is observed in the linear regime.

Fig.1 TEM images of plain-NC and CM-NC fibers. Fig. 1. TEM images of plain-NC and CM-NC fibers.

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MATSUMIYA • WATANABE • ABE • MATSUMURA • TANI • KASE • KIKKAWA • SUZUKI • ISHII : Rheology of Nano-Cellulose Fiber Suspension

In contrast, a nonlinearity prevails for larger g 0 > 0.05, as noted from changes of the moduli with g 0. Although the moduli for such large g0 are no longer identical to the storage and loss moduli defined in the linear regime, we hereafter denote, for simplicity, those nonlinear moduli also as G' and G". This g0 dependence is much more significant for G' than for G", in particular at low w , as noted in Fig. 2. Nevertheless, G' at low w still remains insensitive to w , suggesting that some elastic feature remains even under large strains. For a test of this nonlinear elasticity, we may utilize the G' data at the lowest w examined (w = 0.1 s-1) as the nonlinear static modulus Ge under large strain. As shown in Fig. 3, Ge of the plain-NC suspension (squares) remains independent of g0 for small g0 ≤ 0.05 (which characterizes the equilibrium elasticity in the linear regime) but decreases almost in proportion to g 0

-1 for larger g 0 > 0.05. Phenomenologically, this decrease of Ge is indicative of softening due to the yielding, and the suspension can be classified as an elasto-plastic material. The linear viscoelastic G' and G" data were reproduced when the small amplitude dynamic test (with g0 = 0.02) was conducted immediately after the large amplitude test (with g0 = 0.5 and 1) and/or the start-up flow test explained later for Fig. 8. Thus, the elasto-plastic feature of the plain-NC suspension was accompanied by no detectable thixotropy and rheopexy. This

was the case also for the CM-NC suspension explained below.For comparison with the plain-NC suspension, Fig. 4 shows

the dynamic oscillatory moduli data of the 0.62 wt % CM-NC suspension having the same NC concentration. The behavior of this CM-NC suspension, decreases of G' with increasing g0 (> 0.1) that are much more pronounced compared to changes of G" with g 0, is qualitatively similar to the behavior of the plain-NC suspension (Fig. 2).

Fig.2 Dynamic oscillatory moduli data measured for 0.62 wt% plain-NC suspension in water at 25˚C. The strain amplitude γ0 was varied from 0.01 to 1.

Fig.3 Nonlinear static modulus of plain-NC and CM-NC suspensions with w = 0.62 wt%.

Fig. 2. Dynamic oscillatory moduli data measured for 0.62 wt% plain-NC suspension in water at 25 ˚C. The strain amplitude g 0 was varied from 0.01 to 1.

Fig.4 Dynamic oscillatory moduli data measured for 0.62 wt% CM-NC suspension in water at 25˚C. The strain amplitude γ0 was varied from 0.02 to 1.

Fig. 4. Dynamic oscillatory moduli data measured for 0.62 wt% CM-NC suspension in water at 25 ˚C. The strain amplitude g 0 was varied from 0.02 to 1.

Fig.2 Dynamic oscillatory moduli data measured for 0.62 wt% plain-NC suspension in water at 25˚C. The strain amplitude γ0 was varied from 0.01 to 1.

Fig.3 Nonlinear static modulus of plain-NC and CM-NC suspensions with w = 0.62 wt%.

Fig. 3. Nonlinear static modulus of plain-NC and CM-NC suspensions with w = 0.62 wt%.

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Despite this qualitative similarity, we also note quantitative differences between the plain-NC and CM-NC suspensions: As shown in Fig. 3, Ge of the CM-NC suspension (circles), evaluated as G' at w = 0.1 s-1, firstly increases with increasing g 0 up to 0.1 and then decreases on a further increase of g 0. Namely, this suspension exhibits the strain-hardening followed by softening, the former not being observed for the plain-NC suspension. This strain-hardening behavior may be intimately related to the mechanism of elasticity of the NC suspensions discussed in the next section. At the same time, we also note that Ge for small g 0 is considerably smaller for the CM-NC suspension than for the plain-NC suspension. Thus, before discussing the mechanism of elasticity, it is informative to examine the moduli data of a more concentrated CM-NC suspension having a larger Ge. For this purpose, Fig. 5 shows the G' data (moduli data of our interest) of a concentrated CM-NC suspension with w = 2.4 wt%, and Fig. 6, the g 0 dependence of Ge (G' at w = 0.1 s-1) of this suspension. The equilibrium modulus of this suspension, Ge

o ≅ 1000 Pa (obtained as g0-independent Ge value for small g0; cf. Fig. 6), is significantly larger than Ge

o (≅ 25 Pa; cf. Fig. 2) of the 0.64 wt% plain-NC suspension. Nevertheless, the concentrated CM-NC suspension exhibits elasticity under large strains (Fig. 5) that is characterized by the strain-hardening followed by softening (yielding), as noted in Fig. 6. This result, combined with the result seen in Fig. 3, suggests that the hardening feature emerges more easily for the CM-NC suspension than for the plain-NC suspension. This point is discussed below in relation the mechanism of elasticity.

3.2 Mechanism of Elasticity of NC Suspension

3.2.1 OverviewFor molecular solutions of cellulose, the stress in long time

scales can be related to entropy elasticity of cellulose chains exhibiting active thermal motion. The models10) for ordinary polymeric liquids should be applicable to those cellulose solutions. In fact, the molecularly dissolved cellulose solutions prepared from cotton behave very similarly to ordinary polymer solutions1,7), and this behavior can be explained by standard entanglement models (such as the reptation model10)) for polymeric liquids.

However, the NC fibers including those examined in this study have a diameter much larger than the molecular diameter of cellulose chains and are much longer than individual chains; cf. Fig. 1. Thus, in the usual experimental window of rheological measurements, for example in the range of w /s-1 = 0.1-100 examined in this study, the thermal motion should not be important for those NC fibers. Then, the elasticity of NC suspensions should originate from the strain-induced deformation (bending) of the NC fibers. In fact, considering a layered stack structure of fibers and focusing on the momentum transport velocity, Tatsumi7) deduced an expression of the equilibrium modulus Ge

o of those layers, Geo

= Ef c5p2 with Ef, c, and p being the Young’s modulus, mass

concentration, and the aspect ratio of the fiber. The absolute temperature representing the energy of the thermal motion does not appear in this expression, indicating the non-thermal nature of the elasticity of the fiber stacks.

Our NC suspensions do not have such a stacked layer structure: Even the fully dried suspensions included no stacked structure as judged from the TEM images (Fig. 1), and this should have been the case also for the rather dilute suspensions examined in this study. Instead, Fig. 1 suggests that the suspensions should have included a mesh of randomly


Fig.6 Nonlinear static modulus of 2.4 wt% CM-NC suspension. Fig. 5. Dynamic oscillatory moduli data measured for 2.4 wt% CM-NC

suspension in water at 25 ˚C. The strain amplitude g 0 was varied from 0.001 to 1.


Fig.6 Nonlinear static modulus of 2.4 wt% CM-NC suspension.

Fig. 6. Nonlinear static modulus of 2.4 wt% CM-NC suspension.

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oriented, curvy NC fibers each having multiple branches (and some of the fibers being bundled in thicker trunks in particular for the CM-NC fibers), and the strain-induced bending of the fibers in the mesh should have resulted in the elasticity of the suspensions. This mesh structure is somewhat similar, though not identical, to the structures modeled by Veenstra et al.11) and by Doi and Kuzuu.12) Thus, in the followings, we firstly revisit these models as a starting point of our discussion and then discuss the effect(s) of the multiple-branch structure of the NC fibers on the suspension rheology.

3.2.2 Available modelsAs a model for bicontinuous polymer composites, Veenstra

et al.11) considered rod-like (cylindrical) domains of a viscoelastic material embedded in a viscoelastic medium. These domains are fused (connected) with each other at their ends to form an infinitely large cubic mesh similar to a jungle gym, and the complex modulus of the system as a whole is expressed in terms of the moduli of the cylindrical domain and the surrounding medium.11) For our NC suspensions, the medium (water) is extremely softer than the NC fibers, and the result obtained by Veenstra et al.11) reduces to a very simple expression of the equilibrium modulus in the linear viscoelastic regime:

Ge

o 2E f /3 with 3 2 2 3 f (1)

Ge[DK]

E f

9rf

4f

Nc ( )Lf

4

(2)

Nc () 4

{1 F1()}5 F2()

F1() ( 4 /8 for ≤ 1) (3)

F1() 12 2 4

1/ 2

(4a)

F2() ( 2 2) 2 4 (4b)

Geo

Q4

9

E ff

5 with Q 1/2 (5)

Geo

Q4

9

E ff5

f 4 with Q 1/2 (6)

(1)

Here, Ef is the Young’s modulus of the NC fiber (Ef ≅ 100 GPa for cellulose),13) and a is a reduced length (corresponding to the fiber radius) determined by the fiber volume fraction f f. The 0.62 wt% plain-NC suspension had f f = 0.0041 (calculated with the assumption of volume additivity of NC and water), which gives an estimated by Eq. (1), Ge

o ≅ 50 kPa. This estimate is more than 1000 times larger than the data of this suspension, Ge

o = 25 Pa (cf. Fig. 3). This significant overestimation, noted also for the CM-NC suspensions, should reflect the difference between the mesh structures assumed in the model and included in the actual NC suspension. The NC fibers have multiple branches but are not mutually fused, which is quite different from the fused network assumed in the model. The strain applied to such an unfused mesh is not efficiently transmitted to individual fibers, which should results in much weaker bending of the fibers than considered in the model.

An opposite situation is found for the Doi-Kuzuu (DK) model12) formulated for an unfused mesh of rods having no rod-rod contact in the quiescent state. The DK model, mimicking biological tissues, gives no equilibrium elasticity (Ge

o = 0) in the linear viscoelastic regime because the model assumes that an infinitesimal strain does not bring the rods

in contact and thus activates no rod-bending. Obviously, this model prediction significantly underestimates the Ge

o data (> 0) of the NC suspensions, because the actual NC fibers should have mutual contacts even in the quiescent state: The strain-induced bending of the NC fibers between their contact points results in the equilibrium elasticity reflected in the Ge

o data, as discussed later in more detail.

Qiao et al.14) pointed out that the expression of the static modulus deduced from the DK model11) reads

Ge

o 2E f /3 with 3 2 2 3 f (1)

Ge[DK]

E f

9rf

4f

Nc ( )Lf

4

(2)

Nc () 4

{1 F1()}5 F2()

F1() ( 4 /8 for ≤ 1) (3)

F1() 12 2 4

1/ 2

(4a)

F2() ( 2 2) 2 4 (4b)

Geo

Q4

9

E ff

5 with Q 1/2 (5)

Geo

Q4

9

E ff5

f 4 with Q 1/2 (6)

(2)

where Ef, rf, Lf, and f f denote the Young’s modulus, radius, length, and volume fraction of the rods, respectively, and Nc( g) is the average number of contacts between a focused (representative) rod and the other surrounding rods under a given shear strain g . In the DK model, Nc(0) = 0 and thus Ge

o = 0 as explained above, but Nc( g) increases with g from this limiting value of Nc(0) = 0. Namely, the DK model exhibits the strain-hardening, a characteristic feature of biological tissues. This hardening behavior of the DK model under shear is described by12)

Ge

o 2E f /3 with 3 2 2 3 f (1)

Ge[DK]

E f

9rf

4f

Nc ( )Lf

4

(2)

Nc () 4

{1 F1()}5 F2()F1()

( 4 /8 for ≤ 1) (3)

F1() 12 2 4

1/ 2

(4a)

F2() ( 2 2) 2 4 (4b)

Geo

Q4

9

E ff

5 with Q 1/2 (5)

Geo

Q4

9

E ff5

f 4 with Q 1/2 (6)

(3)

with

(4a)

Ge

o 2E f /3 with 3 2 2 3 f (1)

Ge[DK]

E f

9rf

4f

Nc ( )Lf

4

(2)

Nc () 4

{1 F1()}5 F2()

F1() ( 4 /8 for ≤ 1) (3)

F1() 12 2 4

1/ 2

(4a)

F2() ( 2 2) 2 4 (4b)

Geo

Q4

9

E ff

5 with Q 1/2 (5)

Geo

Q4

9

E ff5

f 4 with Q 1/2 (6)

(4b)

Doi and Kuzuu12) also pointed out that the rods with a finite diameter could be in mutual contact to have Nc(0) > 0 in the quiescent state, given that the rod volume fraction f f is larger than the overlapping threshold, f f*. Qiao et al.14) estimated this Nc(0) by calculating a probability of rod-rod contact in that state. The corresponding expression of the equilibrium modulus in the linear regime (obtained with the aid of a relationship, f f = vf prf

2Lf with vf being the number density of the rods) is summarized as14)

Ge

o 2E f /3 with 3 2 2 3 f (1)

Ge[DK]

E f

9rf

4f

Nc ( )Lf

4

(2)

Nc () 4

{1 F1()}5 F2()

F1() ( 4 /8 for ≤ 1) (3)

F1() 12 2 4

1/ 2

(4a)

F2() ( 2 2) 2 4 (4b)

Geo

Q4

9

E ff

5 with Q 1/2 (5)

Geo

Q4

9

E ff5

f 4 with Q 1/2 (6)

(5)

where Q is a numerical factor obtained from the rod-rod contact probability mentioned above. Equation (5) well describes the equilibrium modulus of concentrated silk fiber composites having f f ≥ 0.04.14) Nevertheless, for our 0.62 wt%

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NC suspensions, Eq. (5) gives Geo ≅ 3 × 10-4 Pa, which is

order-of-magnitudes smaller than the Geo data (cf. Fig. 3). This

difference in the validity of Eq.(5) can be attributed to the fiber structure: The silk fibers had smooth rod-like (or wire-like) shape14) similar to that considered in the model, whereas the NC fibers are curvy and have multiple branches of branch-on-branch type, as explained for Fig. 1. An effect(s) of this NC fiber structure on the equilibrium elasticity is briefly discussed below together with the effect on the nonlinear elasticity under large strains.

3.2.3 Effect of curvy, multiple-branch structure on elasticity

Figure 7 schematically illustrates the curvy, multiple-branch structure of the plain-NC fiber. (The bundled trunk portion explained for Fig. 1 is scarce for the plain-NC fibers and is not considered in Fig. 7.) The elasticity of NC suspensions reflects the strain-induced fiber bending that occurs between the fiber-fiber contact points. The branching points depicted in Fig. 7 work as the fiber-fiber contact points. In addition, physical contact points between neighboring fibers not connected through branching, being somewhat similar to trapped entanglements in rubbers, also serve as the fiber-fiber contact points. (Those physical contact points are not shown in Fig. 7.) In principle, description of the NC suspension elasticity requires us to calculate the probability of these two types of fiber-fiber contact and evaluate the contact number Nc(0) in the quiescent state from this probability, as in the previous study.14) However, for our NC fibers, this calculation is enormously difficult because of the curvy and irregularly branched shape of the fibers.

Considering this difficulty, we here examine a very crude but simple approximation of regarding the curvy and branched fiber as an effective, smooth fiber as depicted with the dotted cylinder in Fig. 7. This effective fiber has the Young’s modulus Ef,eff being smaller than the intrinsic modulus of cellulose, Ef ≅ 100 GPa,13) because the effective fiber contains vacancies filled with water in the suspension. At the same time, the volume fraction f f,eff of the effective fiber is larger than the

nominal f f because of those vacancies. Utilizing a volume fraction f of the real NC fiber in the effective fiber, we may express E f,eff and f f,eff as E f,eff ≅ f Ef and f f,eff ≅ f f /f. Then, we may use these E f,eff and f f,eff instead of Ef and f f in Eq.(5) to find

Ge

o 2E f /3 with 3 2 2 3 f (1)

Ge[DK]

E f

9rf

4f

Nc ( )Lf

4

(2)

Nc () 4

{1 F1()}5 F2()

F1() ( 4 /8 for ≤ 1) (3)

F1() 12 2 4

1/ 2

(4a)

F2() ( 2 2) 2 4 (4b)

Geo

Q4

9

E ff

5 with Q 1/2 (5)

Geo

Q4

9

E ff5

f 4 with Q 1/2 (6)

(6)

As noted in Fig. 1, the thick trunk portion bundling the curvy branches is scarcer in the plain-NC fiber than in the CM-NC fiber. Then, ƒ would be smaller for the plain-NC suspension than for the CM-NC suspension having the same f f (= 0.0041 that corresponds to w = 0.62 wt%), as can be easily noted from Fig. 7 (where some of the branches are to be replaced by thicker trunks for the CM-NC suspension). Correspondingly, Ge

o evaluated from Eq.(6) should be larger for the plain-NC suspension, which is in accord to the data shown in Fig. 3. This result in turn suggests that the curvy, multiple-branch structure of NC fibers enhances the elasticity of the NC suspension by increasing the fiber-fiber contact in the quiescent state.

Despite this qualitatively reasonable discussion, it is not easy to accurately evaluate ƒ from the TEM images in Fig. 1. Nevertheless, it is still informative to examine a very rough estimate of ƒ = 0.05 (95% vacancy) for the plain-NC: This ƒ value roughly corresponds to the structure illustrated in Fig. 7. For the 0.62 wt% plain-NC suspension, Eq. (6) with ƒ = 0.05 and Ef ≅100 GPa13) gives an estimate of Ge

o ≅ 40 Pa. This estimate is in the same order of magnitude as the Ge

o data of this suspension (≅ 25 Pa; cf. Fig. 3), lending qualitative support to the use of Eq. (6).

Now, we turn our attention to the nonlinearity observed in Figs. 3 and 6. The strain-hardening observed for the CM-NC suspensions at small g0 would be attributed to the strain-induced increase of the fiber-fiber contact number Nc(g 0), as explained for the DK model (cf. Eq. (2)). Nevertheless, the plain-NC suspension does not exhibit detectable strain-hardening. This difference could be partly related to the stiffness of the effective fiber explained above. Rigid fiber-fiber contacts induced by the strain, being required for the strain-hardening behavior, can occur more easily for a stiffer effective fiber, that is, for the CM-NC fiber having a larger Ef,eff = f Ef compared to the plain-NC fiber, as observed.

However, the CM-NC suspensions begin to exhibit the strain-hardening at very small g0 (~ 0.02 and ~ 0.005 for the 0.62 and 2.4 wt% suspensions) compared to the DK model: Equation (3) suggests that the hardening of detectable magnitude (~ 5% increase of Ge) emerges at large g ~ 0.4, which reflects a very strong g dependence of the factor

Fig.7 Schematic illustration of plain-NC fiber structure.

Fig. 7. Schematic illustration of plain-NC fiber structure.

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appearing in Eq.(3), F( g) ≡ {1 - F1( g)}5F2( g)/{gF1( g)}, that is numerically similar to high-exponent power-law dependence, F( g) ~ g4. This difference of g for the onset of strain-hardening could be qualitatively related to the radius of the effective fiber. A local strain g [local]

eff for the effective fiber (cf. Fig. 7) is larger than the local strain g [local]

rod for the rod (unbranched fiber) considered in the DK model, and the strain at the onset of hardening considered in the model decreases by a factor of g [local]

rod /g [local] eff for the effective fiber. It is difficult to evaluate this

g [local] rod /g [local]

eff ratio quantitatively, but a rough, order-of-magnitude estimate would be still informative: Utilizing the radii of the real and effective fibers, r and rf -1/2 with f being the volume fraction of the real fiber in the effective fiber explained above, we may express the ratio as g [local]

rod /g [local] eff =

(x-rf -1/2)/(x-r) where x is an average center-to-center distance between the effective fiber. If r ≅ 0.3x and f ≅ 0.1, this ratio is of the order of 0.1 and the strain at the onset of hardening deduced from the DK model, g ~ 0.4, can be reduced to g ~ 0.04. Thus, the hardening of the CM-NC suspensions occurring at small strains of the order of 0.01 (Fig. 6) can be expected from the crude but simple idea of effective fiber.

Finally, we focus on the strain-softening (or yielding) commonly observed for the plain-NC and CM-MC suspensions (cf. Figs. 3 an 6). This softening is attributable to strain-induced slippage of the fiber-fiber contacts (not considered in the DK model) that can naturally occur when the stress reaches the static frictional force acting on the fiber-fiber contact points. This slippage could be tuned by some additives thereby controlling the processability of NC suspensions. A further study is desired for this effect of additives.

3.3 Flow Behavior

Figures 8 and 9, respectively, show the stress growth behavior of the 0.62 wt% plain-NC and CM-NC suspensions at 25 ˚C on start-up of shear flow at various shear rates g4 as indicated. In panels (a), the viscosity growth function, h+(t) = s+(t)/g4 with s+(t) being the growing stress at time t, is plotted against t, and in panels (b), s+(t) is plotted against the strain, g = g4t. The dotted lines indicate the elastic behavior expected from the equilibrium modulus Ge

o in the linear regime, h + el =

Geog /g4 = Ge

ot and s + el = Ge

og . The data of both plain-NC and CM-NC suspensions follow these lines at short t and/or small g , reflecting their elastic behavior before the yielding. At long t and/or large g , the suspensions exhibit yielding (downward deviation from the elastic lines) and then flow plastically. For both suspensions, the steady state viscosity h characterizing this plastic flow behavior strongly decreases with increasing g4, as shown in Fig. 10.

Fig.8 Stress growth of 0.62 wt% plain-NC suspension in water on start-up of shear flow at 25˚C. The shear rate was varied from 0.1 to 30 s-1.

Fig.9 Stress growth of 0.62 wt% CM-NC suspension in water on start-up of shear flow at 25˚C. The shear rate was varied from 0.1 to 30 s-1.

Fig. 8. Stress growth of 0.62 wt% plain-NC suspension in water on start-up of shear flow at 25 ˚C. The shear rate was varied from 0.1 to 30 s-1.

Fig.8 Stress growth of 0.62 wt% plain-NC suspension in water on start-up of shear flow at 25˚C. The shear rate was varied from 0.1 to 30 s-1.

Fig.9 Stress growth of 0.62 wt% CM-NC suspension in water on start-up of shear flow at 25˚C. The shear rate was varied from 0.1 to 30 s-1.

Fig. 9. Stress growth of 0.62 wt% CM-NC suspension in water on start-up of shear flow at 25 ˚C. The shear rate was varied from 0.1 to 30 s-1.

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Thus, the flow behavior is qualitatively similar for the plain-NC and CM-NC suspensions. The yielding and strong decrease of h with g4, observed for both suspensions, can be attributed to the flow-induced decrease of the effective fiber-fiber contacts due to slippage at the contact points under flow, which well corresponds to the decrease of Ge with strain amplitude seen for the dynamic behavior (Fig. 3). The slippage under flow appeared to be followed by instantaneous recovery of intact contact on cessation of flow, as judged from an observation that the linear viscoelastic G' and G" data were recovered when the small amplitude dynamic test was conducted immediately after the start-up flow test.

Despite the above qualitative similarity between the plain-NC and CM-NC suspensions, we also note a quantitative difference. For the plain-NC suspension, the stress s+ in the steadily flowing state does not increase with g4 significantly (Fig. 8(b)) and thus h decreases almost in proportion to g4-1 (Fig. 10). In contrast, the increase of the steady state stress at high g4 (≥ 3 s-1) is considerably stronger for the CM-NC suspension than for the plain-NC suspension, as noted from Figs 8(b) and 9(b). Correspondingly, the decrease of h at such high g4 is weaker for the CM-NC suspension (Fig. 10). This difference between the two suspensions could reflect a difference of the stiffness of the “effective” fiber discussed in the previous section. The CM-NC fibers are bundled into the thick trunks more significantly compared to the plain-NC fibers, thereby having a larger modulus Ef,eff = f Ef. Thus, the fiber-fiber contact points under fast flow would slip less easily for the stiffer “effective” CM-NC fiber so that h at high g4 would have decreased less significantly for the CM-NC suspension. In addition, the surface friction under fast flow might be larger for CM-NC fibers than for the plain-NC fibers

(because of interaction between CM groups) thereby reducing the slippage efficiency for the former fibers.

Finally, we note in Fig. 3 that the decrease of Ge at g0 = 0.2-1 is similar for the plain-NC and CM-NC suspensions. This similarity might look inconsistent with the above argument for the difference of these two suspensions. However, this is not the case. The decrease of Ge at g0 = 0.2-1 characterizes the nonlinearity at low w (= 0.1 or 0.2 s-1), and the corresponding shear rate, g4 = g0w , is in the range of g4 = 0.02-0.2 s-1. At such small g4, the h data of the two suspensions under steady flow (Fig. 10) exhibit a very similar decrease with g4 (h ~ g4-1), which suggests a consistency between the h and Ge data.

4. CONCLUDING REMARKS

We have examined rheological behavior of plain-NC and CM-NC suspensions in water. The suspensions behaved as elastic solids under small strains either in the dynamic oscillatory or start-up flow measurements, whereas the yielding/plastic flow behavior was noted under large strains. Thus, the NC suspensions were phenomenologically classified as the elasto-plastic material.

The elasticity under small strain was attributed to the bending of NC fibers between their mutual contact points existing in the quiescent state. The equilibrium modulus (Ge

o) data of the suspensions in the linear viscoelastic regime were orders of magnitude larger than the estimate from the Doi-Kuzuu (DK) model (modified by Qiao et al) formulated for a mesh of rods. This result was related to the structure of NC fibers: The NC fibers were curvy and had multiple branches of branch-on-branch type (as revealed from TEM), and the fiber-fiber contact should have been enhanced by this structure as compared to the contacts between the rods considered in the modified DK model. This curvy, multiple-branch structure of the actual NC fiber was crudely but simply approximated as an “effective” fiber containing vacancies filled with water, and Ge

o of a mesh composed of such effective fibers was estimated from the modified DE model. The order of magnitude difference explained above was largely removed for this estimate. Furthermore, the analysis with the modified DK model suggested that the difference of the Ge

o data of the plain-NC and CM-NC suspensions (smaller for the latter) reflected a difference of the structure, the trunk portion bundling the fibers being richer in the CM-MC fibers (that effectively reduced the contact between effective fibers at equilibrium).

As a consequence of the analysis of the linear Geo data

explained above, the nonlinear yielding/plastic flow behavior

Fig.10 Steady state viscosity of 0.62 wt% plain-NC and CM-NC suspensions at 25˚C.

Fig. 10. Steady state viscosity of 0.62 wt% plain-NC and CM-NC suspensions at 25 ˚C.

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under large strain/flow, observed for both plain-NC and CM-NC suspensions, was attributed to the strain/flow-induced slippage of fiber-fiber contacts (reduction of effective contact number). In addition, quantitative differences between the plain-NC and CM-NC suspensions, the strain-hardening observed only for the latter and shear-thinning at high g4 weaker for the latter, were attributable to the difference of the stiffness of the effective fibers. The CM-NC fiber appeared to be stiffer compared to the plain-NC fiber (because of the bundled trunks richer in the former), which possibly resulted in more intact fiber-fiber contacts under large strain/fast flow to give the strain-hardening and weak thinning explained above.

All above results suggest that the rheological behavior of NC suspensions is strongly affected by the curvy, multiple-branch structure of the NC fiber partly bundled with each other. Control of this structure, expected to be essential for application of NC suspensions, deserves further attention.

ACKNOWLEDGMENT

This work was supported by the joint research project of NARO Bio-oriented Technology Research Advancement Institution.

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