# Rheological Considerations in Processing Self-Reinforced Thermoplastic Polymer Nanocomposites: A Review

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. From Polymer Blends to Nano-Fibrillar Polymer–Polymer Composites: Rheological Fundamentals

## 3. Miscibility of Polymers

## 4. Summary of Droplet Deformation and Breakup Theories

#### 4.1. Newtonian Fluids in Well-Defined Flow Fields

_{d}/η

_{m}), and the Weber (or capillary) number, Ca, which is a dimensionless parameter representing the ratio of the viscous stresses exerted on the droplet by the external flow field to the interfacial tension forces restoring the particle to a spherical shape. The capillary number is defined as

_{m}is the viscosity of the continuous phase, $\dot{\mathsf{\gamma}}$ is the shear rate, R is the radius of the droplet prior to deformation, and Γ is the interfacial tension between the particle phase and its continuous counterpart. At a specific shear rate ($\dot{\mathsf{\gamma}}$), the viscous forces are greater than the interfacial one, causing the droplet to break up. The (Ca) corresponding to the critical shear rate (${\dot{\mathsf{\gamma}}}_{\mathrm{c}}$) is called the critical capillary number and is denoted by Ca(crit) with

_{burst}), under steady-state shearing flows (i.e., flows with a gradually increased shear rate and negligible inertia), is equal to the critical capillary number Ca(crit) within a range of viscosity ratios 0.1 ≤ k ≤ 1 with Def

_{burst}≈ Ca(crit) ≈ 0.5. It should be noted that the term f(k) ranged from 1 to 1.187 as k is increased from 0 to ∞, and that Def = 0 for a sphere. For Def ≥ 0.5 or L ≥ 3B, the droplet broke up [16,25].

- Class (a): for k < 0.2; the particle takes on a sigmoidal shape and tiny drops break away from its ends (tip streaming phenomenon).
- Class (b): 0.2 ≤ k < 1; the particle’s central portion suddenly extends into a cylindrical shape, creating a neck in the middle (necking mechanism). The neck becomes progressively thinner until two identical daughter droplets and three satellite droplets are formed.
- Class (c): 1 ≤ k < 4; the droplet extends into a long thread that gets progressively longer until breaking up into a large number of fine particles.
- Class (d): k > 4; no burst occurs regardless of the applied shear rates. The particle deforms into an ellipsoid and orients along the flow without showing any signs of disintegration even at the upper limit of Ca of the apparatus ($\dot{\mathsf{\gamma}}$ up to 40 s
^{−1}). This is predicted by $\mathrm{Def}=\frac{5}{2\mathrm{k}+3}$, yielding Def < 0.3 for k > 4, which can be regarded as insufficient for breakup.

- Ca < 0.1 Ca(crit): no droplet deformation occurs. The interfacial energy dominates.
- Ca(crit) ≤ Ca < Ca(crit): there is slight droplet deformation without break-up, and a stable form is reached.
- Ca(crit) ≤ Ca ≤ 2 Ca(crit): the interfacial stress is dominated by the viscous stress causing the droplet to become unstable and breakup to occur as a splitting of the particle into two equal parts before elongation into a filament can be achieved. The radius of the drops can be calculated as:$${\mathrm{R}}_{\mathrm{drops}}=\frac{\mathrm{C}\mathrm{a}\left(\mathrm{crit}\right)}{{\mathsf{\eta}}_{\mathrm{m}}\dot{\mathsf{\gamma}}/\mathsf{\Gamma}}{2}^{-1/3}$$
- Ca > 4 Ca(crit): the shear stress is much stonger than the interfacial stress, causing the droplets to be deformed into long fibrils that do not break, but rather rotate in the flow field. In this case, the formation of a stable fibrillar structure can be obtained under specific conditions [17,34,35,36] (see later).

_{d}= 12 Pa·s) was used in a polydimethylsiloxane matrix (η

_{m}= 9 Pa·s), and the radius of the droplet was 0.48 mm, whereas the interfacial tension was equal to 4 mN/m. In both experiments, the capillary number was first increased from zero to 1.3 Ca(crit) and then decreased back to zero. In the first experiment, the maximum capillary number reached 42 s after the start of the experiment: the droplet deformed as the capillary number increased and then retracted back to a sphere again. In the second experiment, the capillary number was increased at a slower rate and the maximum capillary number was reached in 46 s. In this case, the drop did not retract back to a sphere but continued to elongate until breakup occurred. It should be noted that the viscosity ratio k = 1.3 with (Ca/Ca(crit) < 2) in both experiments.

_{0}, see Figure 7), very small sinusoidal disturbances appear on the surface of the fibril see Figure 8. Distortions with a wavelength, λ, larger than the original circumference of the fibril, 2πR

_{0}, give rise to a reduction in interfacial surface area and as a result only these distortions are able to grow (see Figure 7).

_{0}/λ

_{0}is the amplitude of the distortion at time t = 0. A lower limit of ε

_{0}can be obtained by thermal fluctuations and was estimated by [40]:

_{B}is the Boltzmann constant, T is the absolute temperature and Γ is the interfacial tension. According to [40], ε

_{0}≈ 10

^{−9}m for Γ = 10 mN.m

^{−1}. Ref. [41] gaves a higher estimate of 10

^{−8}to 10

^{−7}m.

_{m}is the matrix viscosity, R

_{0}is the initial radius of the thread (R

_{0}= B/2; see Figure 7) and Ω(k, X) is a complex function of the characteristic wave number, X, of the perturbation and the viscosity ratio, k, of the system in question. When the function Ω(k, X) is at its maximum, breakup of the thread occurs. Values of Ω(k, X) can be calculated from Tomotika’s original equation [39] and Figure 9 shows the function Ω(k, X) for two immiscible liquids with a viscosity ratio k = 0.91. Tomotika observed that the breakup of the fiber took place at an unique value for the dominant wave number, X

_{m}= 0.568 with a value of Ω(k, X

_{m}) = 0.074.

_{m}) and the dominant wave number X

_{m}vs. the viscosity ratio k [17,20,42]. For k→0, the dominant growth rate function Ω(k, X

_{m}) approached unity and for k ≈ 100, Ω(k, X

_{m}) it was equal to zero. For k ≈ 0.3, the dominant wavelength λ

_{m}was at its minimum (the wave number X

_{m}was maximal) indicating a maximum amount of capillary instabilities in this region of the viscosity ratio.

_{m}):

_{0}= −2.588, b

_{1}= −1.154, b

_{2}= 0.03987, b

_{3}= 0.0889, and b

_{4}= 0.01154.

_{m}= 0.568, so that the diameter of the droplets formed after breakup of the fiber would be approximately twofold that of the original fiber.

_{b}, can be determined using the equation

_{Rayleigh}can be determined as the diameter of the fiber divided by the breakup time:

_{0}), it was evaluated by experimental results [43,44]. A value of 20 was found to give the best agreement between theory and experiment.

_{b}

^{*}), have been the subjects of several studies [17,38,42]. Elemans et al. [17] compared the results concerning the dimensionless time for breakup (t

_{b}

^{*}) with data from similar research by Grace [25] and Figure 11 summarizes the findings. Grace demonstrated that (t

_{b}

^{*}) became lower as Ca/Ca(crit) increased and the viscosity ratio k decreased. The following empirical formula was obtained from Figure 11:

_{b}

^{*}).

^{−6}and 950) for the rupture of an initially spherical drop in a quasi-steady homogeneous flow. From Grace’s data one can conclude that the critical capillary number Ca(crit) depends both on the viscosity ratio k and on the flow type. Ca(crit) is, regardless of k, lower for a hyperbolic flow (2D elongation) than for a simple shear flow and it is thus easier to deform and rupture a particle under extensional flow than under shear flow.

_{i}values for shear and extensional flows are given in Table 1.

_{elongation}and Ca(crit)

_{shear}. Experimental data have been confirmed by numerical studies on the critical capillary number under various types of flow [46].

_{DC}, which represents the ratio between the crystallization (or quenching) time and the breakup time of the filaments. If λ

_{DC}>> 1, a nodular morphology is expected.

- If λ
_{DC}= 1, thin fibers form nodules and thick fibers adopt a more or less pronounced wavy shape. - If λ
_{DC}<< 1, a fibrillar morphology is expected.

#### 4.2. Newtonian Polymer Blends: Effect of Elasticity

_{1}) remained the most resistant to breakup, while the more elastic matrices gave rise to increasingly unstable drops. Milliken and Leal [21] conducted an experimental deformation/breakup study of isolated viscoelastic droplets made up of an aqueous polymer solution in a Newtonian fluid matrix subjected to a planar extensional flow generated by a four-roll-mill apparatus. They found that the viscoelastic particles deformed to a lesser extent than their Newtonian counterparts at a given capillary number and that the critical capillary number increased as compared with a Newtonian system at an equivalent viscosity ratio.

_{1}divided by the shear stress σ at a given deformation rate ($\dot{\mathsf{\gamma}}$):

_{d}and one for the matrix denoted Wi

_{m}. Since the elastic stresses in the drop are dependent on the strength of its internal flow, which in turn depends on the viscosity ratio (particles with higher viscosity have weaker internal flows), it is clear that there generally exists a connection between the viscosity ratio and the strength of the elastic forces in the particle.

_{d}) of the particle phase (also known as the characteristic elastic time),

_{m}) with

_{d}/λ

_{m}) ≤ 4, the critical capillary number Ca(crit) for droplet breakup under conditions of steady shearing became greater with increasing k’, reaching a plateau of Ca(crit) ≈ 1.75 at the high elasticity ratio (k’ ≈ 4). This can be compared with Ca(crit) ≈ 0.5 for Newtonian drops (Figure 13b). Thus, the particle’s resistance to deformation and breakup was greater with a higher elasticity ratio between the particle and the matrix phase.

_{d}≤ 0.02) in a Newtonian medium, and established that the droplet elasticity gave rise to a small increase in Ca(crit) up to 20%. The droplet’s elasticity gave rise to a reduced degree of deformation at any given shear rate and a higher critical deformation at breakup, which resulted in an increased Ca(crit). However, when the Weissenberg number was at its highest (Wi

_{d}≥ 1), this effect appeared to saturate, leading to only a modest increase in Ca(crit).

^{dynamic}is the dynamic interfacial tension of a droplet of fluid d in a matrix m, Γ

^{steady}is the interfacial tension of a quiescent polymer blend (in the absence of flow), D is the droplet diameter, N

_{2,d}is the second normal stress difference of the dispersed phase, and N

_{2,m}is the second normal stress difference of the matrix phase, which is dependent on the molecular weight, the molecular weight distribution and the shear stress. Vanoene’s results suggest that the interfacial tension under dynamic flow differed from what it would be under static flow. He demonstrated that, under dynamic flow conditions, the differences in elasticity between a blend’s components may cause the interfacial tension (known as the dynamic interfacial tension) to vary, and the obtained value could be quite different from its counterpart in the absence of flow.

^{dynamic}should decrease as the shear increases, and inversely should become greater when the melt elasticity of the blend matrix is smaller than that of the dispersed phase.

^{−1}), N

_{1}> 2G′, but the two quantities are proportional to each other. As a first approximation to obtain the qualitative behavior, the two sides of Equation (37) are equated to obtain an expression for the drop diameter,

_{1}was approximated by 2G′, and by equating the two sides, the drop diameter equation becomes:

^{E}, (the elastic capillary number) that they expressed as:

^{E}> 1.

_{r}= G

_{d}/G

_{m}, where G

_{d}and G

_{m}are respectively the elastic modulus of the drop and the matrix. Based on the simple assumptions that stretching in the hoop direction is greater than that in the thickness direction and that the second normal stress difference is proportional to the first normal stress difference, Levitt and Macosko were able to derive an approximate simplified equation for the drop thickness after deformation according to:

_{n}

^{max}is half of the maximum thickness.

^{compat}, corresponding to the critical capillary number in the case where a droplet is modified by an interfacial active agent:

_{d}/η

_{m}) values greater than unity and the (−) sign in the exponent applies to values of k below unity. He also put forward an empirical correlation relating the particle size of the suspended phase D

_{n}to the viscosity ratio k for several extruded immiscible polymer blends.

_{n}corresponds to the minimum of Ca(crit) and thus to a viscosity ratio k = 1.

_{d}) and (φ

_{m}) are the volume fraction of respectively the suspended phase and the matrix. Serpe was able to confirm Wu’s equation for PE/PA6 blends by using this modified viscosity ratio. He demonstrated that Ca(crit) increased along with the concentration of the suspended phase (Figure 15).

_{crit}corresponds to the critical droplet radius as calculated from Ca(crit); α represents the probability of coalescence of the drops after collision; f

_{1}is the slope of a function describing the frequency of droplet breakup at Ca(crit); and Φ is the volume fraction of the suspended phase. This relationship still contains several parameters that are not readily quantifiable for the blending of viscoelastic polymers.

_{1d}/N

_{1m}, between the suspended phase and the medium, on the critical capillary number in uncompatibilized immiscible polymer blends under a simple shear flow. The authors found that in 80/20 w/w polyethylene/polystyrene blends sheared in a rheometer, the critical capillary numbers ranging from 2 to 30 depended on the relative magnitudes of the normal stress differences in the droplet and matrix phases as well as on the viscosity ratio (0.5, 1 and 2). These capillary numbers were between 4- and 80-fold their counterpart for breakup of a Newtonian drop in a Newtonian matrix. According to the authors, this large increase in critical capillary number (and hence droplet size) was assigned to the role of viscoelasticity. Breakup of viscoelastic particles in a viscoelastic medium is harder than for Newtonian drops in a Newtonian matrix. This is due to the contribution of both the particle elasticity and the shear-thinning of the polymer matrix. For all investigated blends, the critical capillary numbers were seen to increase with N

_{1d}/N

_{1m}, and were correlated by a power law in N

_{1d}/N

_{1m}(Figure 16), with

_{1}are problematic and most investigations of fibrillar morphology have therefore focused on the effect of the viscosity ratio, neglecting the importance of the elasticity ratio between the dispersed and continuous phase. This has led to contradictions and ambiguity when it comes to drawing a clear parallel between the viscosity ratio and the optimal fiber formation conditions as well as temporal stability in the case of viscoelastic immiscible polymer blends. According to Min and White [72], fibrils were obtained at a viscosity ratio 0.3 < k = η

_{d}/η

_{m}≤ 1 for blends of undrawn melt-mixed polyethylene/polystyrene. However, Berger et al. [52] studied poly (ethylene terephthalate)/polyamide blends and found that pure shearing did not create a droplet-fiber transition when the viscosity ratio k ≤ 1. The undrawn fibrous material appeared as a dispersion of spherical particles in the polymer matrix. Fibril-in-matrix structures were only for k = 3.7 in the undrawn fibrous material. The authors confirmed that it was therefore possible to create fibril-in-matrix structures by pure shearing only when the viscosity of the dispersed phase exceeded that of the continuous one. Drawing of the fibrous material always induces fibril-in-matrix structures.

_{1d}/N

_{1m}< 1). At the same time, the viscosity ratio should be larger than two (k ≥ 2). These results are consistent with the experimental findings of Mighri et al. [60] and Berger et al. [52].

## 5. Manufacturing of In Situ Nano-Fibrillar Composites (NFC)

- −
**Melt extrusion (mixing step)**: Melt blending followed by extrusion of the two immiscible polymers having sufficiently different melting temperatures. The matrix and/or reinforcing polymers are dried (to avoid the hydrolytic degradation) and mixed before being compounded.- −
**Drawing**: The blend extrudate undergoes melt or cold drawing through roller pairs with the drawing ratio defined as the relation between the linear speeds (S_{2}/S_{1}) of the two sets of rollers used to draw the filament. It gives an indication of the amount of alignment imparted to the blend. The filament is then either collected on a spool or pelletized.- −
**Matrix consolidation through thermal processing (isotropization step)**: The drawn filaments or pellets are injection- or compression-molded at a temperature T_{proc}above Tm of the component with the lower melting temperature and below the Tm of the higher-melting one. This converts the major phase into an isotropic matrix, while still retaining the oriented nanofibrillar structure of the component with the higher melting temperature. If T_{Proc}is too high, the nanofibrils melt and may revert to their original spherical shape, in which case the reinforcing effect might be lost.

#### 5.1. Melt Extrusion and Hot-Stretching Process

_{m}of the higher melting and already fibrillated component. In doing so, the oriented crystalline structure of the latter can be preserved, whereby the reinforcing elements of the NFC are formed.

#### 5.2. Melt Extrusion and Solid-State Cold Stretching

_{1}) is maintained identical to the speed of the extrudate. Beyond the nip rolls are a pair of stretch rolls of the same diameter as the nip rolls, but whose speed (V

_{2}) can be varied to attain different draw ratios and, thereby reduce the cross sectional dimensions of the strands. The ratio between the stretch roll to nip roll velocities (V

_{2}/V

_{1}) corresponds to the draw (or stretching) ratio. Finally, the strands are pelletized and subsequently processed by injection or compression molding at a temperature below the T

_{m}of the reinforcing component.

## 6. Morphology Development of Nanofibrillar Nanocomposites during Processing

#### 6.1. Effect of Coalescence on the Morphology of NFCs Prepared by Hot-Stretching

_{cr}(in nanometers). Following the guidelines presented by Tjahjadi and Ottino [83], Perilla and Jana calculated a

_{cr}at the various share rates used. It was seen that the a

_{cr}values were much smaller than the radii of the nanofibrils, regardless of the share rate studied, indicating that the nanofibrils remained stable under the imposed flow conditions.

#### 6.2. Morphology Development of NFCs Prepared by Solid-State Cold Stretching

## 7. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Scheme of the counter-rotating cylindrical Couette system. Adapted from reference [24].

**Figure 2.**Schematic view of a four-roller mill apparatus. Adapted from reference [27].

**Figure 3.**Definition of length (L), width (B), and orientation angle (α) of a deformed spherical droplet with an initial radius ‘a’ in simple shear flow. Adapted from reference [27].

**Figure 4.**Various deformation modes of a particle subjected to a simple shear flow for (Ca ≥ Ca(crit)). (

**a**) k = 2 × 10

^{−4}; (

**b**) k = 1; (

**c**) k = 0.7 and Ca = Ca(crit)); (

**d**) k = 6. Adapted from [28].

**Figure 5.**Droplet response to a triangular flow rate–time profile where the maximum capillary number was reached in 42 (

**top**) and 46 (

**bottom**) seconds. Adapted from [37].

**Figure 6.**Scalar measures of deformation and orientation. Adapted from [38].

**Figure 7.**Left: schematic representation of capillary instability (sinusoidal distortion) taking place as a cylindrical thread with an initial radius R

_{0}is extended. Here, $\overline{\mathrm{R}}=\sqrt{{\mathrm{R}}_{0}{}^{2}-{\mathsf{\epsilon}}^{2}/2}$ (Equation (19)) is the average radius, ε is the amplitude of the distortion, λ is the wavelength and z is the Cartesian coordinate along the cylinder’s main axis. Right: Length (L), width (B), and orientation angle (α) of a deformed spherical droplet with an initial radius ‘a’ in simple shear flow. Adapted from [39].

**Figure 8.**Sinusoidal distortions for a thread (diameter 55 µm) of polyamide 6 embedded in a PS matrix at 230 °C, adapted from [17]. The photographs were taken at: t = 0, 15, 30, 45, 60 s.

**Figure 9.**Curve of Ω(k, X) for a viscosity ratio k = 0.91. Adapted from [39].

**Figure 11.**Effect of higher Ca/Ca(crit) values on the dimensionless burst time t

_{b}

^{*}(from Grace [1982] and Elemans [1988]). ■: k= 0.0002; +: k = 0.0018; *: k = 0.0169; □: k = 0.107; x: k = 0.135 (Elemans); ▲: k = 0.933 (Elemans). The dashed line was obtained by plotting t

_{b}

^{*}at Ca/Ca(crit) = 1 on the vertical axis and assuming an identical slope as for the solid lines. Reproduced from [17] with permission from Elsevier.

**Figure 12.**The critical capillary number Ca(crit) as a function of the viscosity ratio k (also known as the Grace curve) for the rupture of an initially spherical drop in quasi-steady homogeneous shear flow (under simple shear and a plane hyperbolic flow). Adapted from reference [25].

**Figure 13.**(

**Left**). Drop deformation as a function of the capillary number, Ca: effect of the drop elasticity. (

**Right**). Variation of the critical capillary number, Ca(crit), with the elasticity ratio k’. Adapted from reference [60].

**Figure 14.**The critical capillary number as a function of the viscosity ratio. Comparison between viscoelastic systems blended in a co-rotating twin screw extruder and Newtonian liquids under steady uniform shear and in an elongational (hyperbo1ic) flow. Adapted from reference [68].

**Figure 15.**Critical capillary number vs. viscosity ratio and concentration of the dispersed phase (Φ

_{d}) for PE-PA6,6 blends for a wide range of mixing conditions (G: Shear rate, T: Temperature). Adapted from [69].

**Figure 16.**The dependence of the critical capillary number on the first normal stress difference ratio for blend systems with varying viscosity ratios. Adapted from [71].

**Figure 17.**NFC manufacturing process by a melt extrusion hot-stretching-quenching process and the structural evolution of the fibrillated component through sequent steps: slit die extrusion, hot stretching, quenching, and pelletizing. Adapted from [78].

**Figure 18.**Schematic diagram of preparation process of TP/TP in situ nanofibrillar nanocomposites by a melt extrusion–solid state cold drawing process.

**Figure 21.**Representation of the nanofibrillation process in the entrance zone and in the die. Adapted from [84].

**Figure 22.**Schematic illustration of the nanofibril formation mechanism in immiscible polymer blends during cold drawing (transformation of the spherical particles into nanofibrils via coalescence under transverse contraction). Adapted from [85].

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | |
---|---|---|---|---|---|

Shear flow | −0.5060 | −0.0994 | 0.1240 | −0.1150 | −0.6110 |

Elongational flow | −0.64853 | −0.02442 | 0.02221 | −0.00056 | −0.00645 |

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**MDPI and ACS Style**

Yousfi, M.; Samuel, C.; Soulestin, J.; Lacrampe, M.-F.
Rheological Considerations in Processing Self-Reinforced Thermoplastic Polymer Nanocomposites: A Review. *Polymers* **2022**, *14*, 637.
https://doi.org/10.3390/polym14030637

**AMA Style**

Yousfi M, Samuel C, Soulestin J, Lacrampe M-F.
Rheological Considerations in Processing Self-Reinforced Thermoplastic Polymer Nanocomposites: A Review. *Polymers*. 2022; 14(3):637.
https://doi.org/10.3390/polym14030637

**Chicago/Turabian Style**

Yousfi, Mohamed, Cédric Samuel, Jérémie Soulestin, and Marie-France Lacrampe.
2022. "Rheological Considerations in Processing Self-Reinforced Thermoplastic Polymer Nanocomposites: A Review" *Polymers* 14, no. 3: 637.
https://doi.org/10.3390/polym14030637