Effect of reactive compatibilization on droplet coalescence in shear flow

J. Non-Newtonian Fluid Mech. 145 (2007) 139–149

Effect of reactive compatibilization on droplet coalescence in shear flow Jacques Huitric a,∗ , Michel Moan a , Pierre J. Carreau b , Nicolas Dufaure b a

Laboratoire de Rh´eologie, Universit´e de Brest, 29285 Brest Cedex, France b CREPEC, Ecole Polytechnique of Montreal, Que. H3C 3A7, Canada

Received 15 February 2007; received in revised form 11 June 2007; accepted 11 June 2007

Abstract The evolution of the morphology of blends of polyethylene and polyamide with a droplet–matrix morphology, generated by transient shear flows, has been determined using a method based on quenching following deformation of the samples kept between the parallel plates of a rheometer. Droplet size evolution and transient viscosity of blends compatibilized with different amounts of polyethylene-graft-maleic anhydride were measured as a function of the total strain. The effects of applied shear rates and concentration of the reactive compatibilizer were investigated. The results indicate that initially the droplet size was governed by coalescence, whereas, when the droplet size became comparable to the break-up limit, no further significant increase was observed at large strain values. A significant coalescence inhibition was only observed when a relatively high amount of compatibilizer was used. This corresponded to an important interfacial coverage of the interface by the grafted copolymer chains. In that situation, the intensity of the coalescence became independent of the applied shear rate. On the contrary, the coalescence was favoured by the absence of compatibilizer or the use of a low compatibilizer concentration and increased with increasing shear rate. The rather “complex” effect of the compatibilizer addition on coalescence phenomena has been analyzed in terms of blend microstructure changes induced by the compatibilizer addition in relation to droplet collision frequency. Predictions of the droplet size evolution and viscosity under transient shear flows using a modified version of Lee and Park model is shown to describe the coalescence and break-up phenomena under large deformation shear flow. The predictions of the droplet size evolution are greatly improved when an additional parameter for non-affine deformation (slip parameter), is introduced in the model. On the other hand, the predictions are much less satisfactory for the transient viscosity data. © 2007 Elsevier B.V. All rights reserved. Keywords: Polymer blend; Reactive compatibilization; Coalescence; Transient shear flow

1. Introduction The mixing of two immiscible polymers in the molten state often consists of a minor phase dispersed in the form of droplets in a continuous matrix of the major component. The final droplet size distribution, under mixing, results from the balance between the effects of the break-up and coalescence of the droplets induced by flow. However, the production of a fine dispersion may require the addition of a block copolymer, called compatibilizer. This latter is either simply added prior to mixing or created during mixing by a grafting reaction. This last process, also known as reactive compatibilization, consists in forming in situ a block or grafting copolymer by blending suitably functionalized polymers. Whatever the mode of compatibilization, the copolymer, which is active at the surface of the droplets, lowers the interfacial tension [1,2] and, hence, facilitates break-

∗

Corresponding author. Tel.: +33 2 98 01 66 85; fax: +33 2 98 01 79 30. E-mail address: [email protected] (J. Huitric).

0377-0257/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2007.06.001

up, but, above all, inhibits significantly the coalescence [3–7]. In flow-driven coalescence, the droplets are brought together by the shear flow and then the drainage of the matrix film between colliding droplets decreases the film thickness until the interface ruptures and coalescence occurs. Various mechanisms can be likely relevant for the coalescence inhibition [8]. Among them, two mechanisms, which can occur between two approaching droplets have been received the most attention: repulsive steric interactions resulting from the compression between the block copolymer layers present at the interface and Marangoni stresses induced by a gradient of block copolymer concentration at interfaces, which is caused by the squeezing flow between two droplets (film drainage). In this last mechanism, the block copolymers are partially removed from the gap towards the backside of the droplets, whereas in the case of steric repulsion it is assumed that the block copolymers do not move on the interface, as expected for static coalescence. Moreover, these two mechanisms might simultaneously contribute to the suppression of the coalescence. In these conditions, the degree of interface coverage, in relation to the mobility of block copolymers,

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will help elucidate which mechanism plays the most important role. Lyu et al. [9] examined the effect of adding a block copolymer of polystyrene and polyethylene (PS–PE) on the coalescence in semi-concentrated blends of polystyrene (PS) and high-density polyethylene (HDPE). They monitored the droplet size change with the strain at different shear rates and the results indicated that a minimum concentration of copolymer was required to suppress coalescence. This minimum concentration was found to decrease with the applied shear rate and with increasing molecular weight of the copolymer. For example, at a shear rate of 0.1 s−1 , the minimum concentration was 0.28 vol.% corresponding to a surface coverage of the interface Σ = 0.2 chain/nm2 for a copolymer of molecular weight 20 kg/mol per block. So, Lyu et al. [9] concluded that these results were inconsistent with Marangoni stresses but supported the notion of steric repulsion. Van Hemelrijck et al. [6] studied the shear-induced coalescence in semi-concentrated blends of polydimethylsiloxane (PDMS) and polyisoprene (PI) with a droplet morphology in the presence of a block copolymer PI–PDMS. They found that the form relaxation time of the droplets remained constant during shearing (4.8 and 1.2 s−1 ) for a copolymer concentration around 1% (Σ = 0.2 chain/nm2 ). This means that, in those conditions, the droplet size became independent of the shearing time and, hence, the coalescence was completely suppressed. However, Van Hemelrijck et al. [6] did not conclude about the mechanism that was responsible for the coalescence suppression. In an another work on similar systems, Van Hemelrijck et al. [7] have shown that the coalescence suppression was more effective when the overall molecular weight of the block copolymer was increased and, also, when the longest block of the copolymer was located in the matrix. Some theoretical studies based on the Marangoni effect [10,11] led also to a minimum interfacial coverage, which was proportional to the shear rate. However, the effect was independent of the molecular structure of the block copolymer, such as the molecular weight and the composition. On the other hand, Macosko et al. [3] considered that the block copolymers acted at the interface like steric stabilizers do for colloidal particles. By equating the van der Waals force with the steric force, they have estimated that the minimum coverage of the block copolymer was inversely proportional to the molecular weight of the copolymer. However, the eventual effect of the shear rate was not taken into account. As mentioned above most of the experimental studies have examined the effect of the simple addition of a block copolymer on the coalescence in semi-concentrated blends (φ ∼ 10%). Except for the study of Sundararaj and Macosko [12] on drop break-up and coalescence during melt blending using an internal mixer (high shear rates), the effect of grafted copolymers made by reactive blending has not been investigated, at least for low shear rates. So, the objective of this paper is to elucidate the effect of a reactive compatibilization on the coalescence of droplets in a concentrated blend (φ = 30%) under shear flows. For that purpose two very different compatibilizer concentrations, corresponding respectively to a low coverage of the interface and to a quasi-saturation by grafted compatibilizer chains were investigated. The values of these concentrations have been partly

chosen on the basis of the previous investigations mentioned above. The evolution of the morphology of the blends, generated by transient shear flows was determined using a method based on quenching following deformation of molten samples. In addition, the effect of the shear rate on the droplet size evolution strain was examined. Finally, these experimental results are compared to the predictions calculated using a modified version of the Lee and Park model, which is able to describe the coalescence and break-up phenomena under relatively large deformation shear flow. 2. Materials and experimental methods The blends consisted of a linear low-density polyethylene (PE) and a polyamide 12 (PA) fully immiscible, supplied by Enichem (Flexirene FG 20F) and Arkema (Aechvo), respectively. All the blends, used in this work, contained a weight percentage φ = 30% of PA as the dispersed phase and presented a droplet morphology. The viscosity ratio of these blends, i.e., the ratio of the viscosity of the dispersed phase to that of the matrix was 0.21. The compatibilizer used was a polyethylene-graftmaleic anhydride (PE-g-ma) obtained from Arkema (orevac 18302). The well-known reaction between a maleic anhydride group and an amine end group of polyamide occurred during the mixing and the in situ formed graft copolymer acts as a compatibilizer between PE and PA. The characteristics of the pure components of these blends are given in Table 1. All the blends were prepared using an internal mixer (Rheocord, Haake). The blending conditions were the same for all blends: temperature of 200 ◦ C; speed of the blades of 32 rpm, which corresponded to a shear rate of about 100 s−1 . The compatibilizer was mixed together with the two other components. Mixing was maintained until the torque reached a steady-state regime, where the final morphology of the blend was supposed to be obtained. The time required to reach steady state was about 5 min for all the blends. The concentration, φc , of the compatibilizer (expressed as weight percentage of the blend) was varied from 0 to 2%, but the weight percentage of the dispersed phase was maintained constant at φ = 30%. After mixing, the blends were quenched in cold water and then compressed molded into 2 mm thick sheets, using a press under 20 MPa at 200 ◦ C. Because of the hygroscopic nature of PA, the blends were dried for 4 h at 85 ◦ C in a vacuum oven before experiments. The morphology of the blends was observed using a scanning electron microscope (SEM) on cryofractured surfaces. The rheological characterization was performed on a Rheometrics Dynamic Analyzer (RDA II) with parallel plate Table 1 Characteristics of the pure components Material

Mw (g/mol)

Mn (g/mol)

η0 (Pa s)a

PE PA PE-g-ma

140,000 37,000 92,500

37,000 20,000 20,100

10,750 2,270 23,000b

a b

Measured at 200 ◦ C. Determined by fitting the viscosity vs. shear rate data with the Carreau model.

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geometry (25 mm diameter and 2 mm spacing). All experiments were carried out at a temperature of 200 ◦ C, under a continuous purge of dry nitrogen, in order to avoid sample degradation and absorption of moisture. The oscillary shear measurements were performed at a strain value equal to 0.04, which is smaller than the limit of the linear viscoelastic regime of all the blends. A controlled stress rheometer (Bohlin CSM), with a similar parallel plate geometry, was also used for coalescence study. The morphological and rheological characterization of very similar PE/PA blends (compatibilized or not) have been described in detail elsewhere [13,14]. 3. Reactive compatibilization of polymer blends The reduction in the dispersed phase size with compatibilizer addition is illustrated by the micrographs of Fig. 1. As expected the size of the dispersed droplets decreased significantly as the compatibilizer concentration was increased (compare Fig. 1a to b and c). As shown in Fig. 2, the number average diameter dn of the droplets decreased very rapidly with the addition of a very small quantity of compatibilizer and it stabilized at concentrations close to 2%. For the rest of this work, as mentioned in the introduction, only two compatibilizer concentrations, φc = 0.2 and 2%, were used. The effect of the compatibilizer addition on the complex viscosity as a function of frequency is shown in Fig. 3. The effect is not significant for the lowest compatibilizer concentration (φc = 0.2%) and the curve is very close to that of the uncompatibilized blend on the whole range of investigated frequencies. On the other hand, a much larger viscosity is observed at the lowest frequencies for the largest compatibilizer concentration (φc = 2%). The simplified version of the Palierne model [15] can be used to determine the interfacial tension, α, from the frequency corresponding to the form relaxation mechanism of the droplets. However, the compatibilized blends clearly show, at least for a sufficient compatibilizer concentration, another slower relaxation mechanism. This can be related to the dynamics of a droplet–matrix interphase, created by the presence of a sufficient amount of copolymer at the interface. The viscosity enhancement observed for φc = 2% is probably due to the presence of this interphase [14]. This second relaxation mechanism has also been related, on the basis of the general Palierne model [16], to an interfacial shear modulus [6,17,18]. Table 2 shows the effect of the compatibilizer addition on some characteristics of the blends, such as the number average diameter dn of the droplets, the polydispersity dv /dn (dv , being the volume average diameter) and the interfacial tension α. The interfacial coverage Σ, corresponding to the number of compatibilizer chains per surface area [3], is an important parameter for Fig. 1. SEM of cryofractured PA: (a) φc = 0%, (b) φc = 0.2%, and (c) φc = 2%. Table 2 Characteristics of the blends φc (%)

dn (␮m)

dv /dn

α (mN m)

Σ (chains/nm2 )

h (␮m)

0 0.2 2

4.4 2.0 1.2

1.50 1.31 1.26

10 3.2 2

0 0.06 0.30

0.90 0.40 0.25

analyzing coalescence inhibition. The values calculated for the two compatibilizer concentrations φc = 0.2 and 2% are also given in Table 2. Σ = 0.30 chain/nm2 obtained for the largest compatibilizer concentration could correspond to a quasi-saturation of the interface, as suggested by the limiting value of the droplet size observed for 2%. The average interparticle distance is a

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Fig. 2. Number average diameter vs. volume fraction of compatibilizer.

function of the volume fraction of the dispersed phase but also of the droplet size. It is certainly another interesting parameter for discussing static and dynamic coalescence phenomena. We have estimated the average distance at rest, h, and the values are reported in Table 2. These results show that even for these concentrated blends, h is relatively important, and, in any case for the blend containing 2% of compatibilizer, it is much larger than the interphase thickness which is of the order of 10 nm [14]. 4. Evolution of the blend morphology during shear flow To follow indirectly the morphology evolution of the blend in situ under shear we froze the morphology by a rapid cooling of the sample kept within the parallel plates of the rheometer. In fact, a rapid and efficient cooling (1.5 ◦ C/s) was obtained by blowing cold dry nitrogen on the fixture after a predetermined shearing time. Martin et al. [19] have shown that, for the cooling rate used in this work, the relaxation of the blend morphology was negligible during cooling and, hence, the “real” shearinduced blend morphology was accessible. This technique has been successfully used to examine the dynamic coalescence of

semi-concentrated and concentrated blends sheared at imposed constant stress [19] or constant shear rate [9]. The advantage of a creep experiment is that the shear stress applied to the blend can be maintained constant during the cooling process. In this work, we have performed both creep experiments (Bohlin CSM controlled stress rheometer) at a shear stress of 2000 Pa and stress growth experiments (Rheometrics RDA II) for three different shear rates of 0.7, 0.1 and 0.05 s−1 measured at the rim of the fixture (parallel plate geometry, 25 mm diameter and 2 mm gap). In this range of deformation rates, the blends show a shear thinning regime, as displayed by Fig. 3. The experiments have been done for accumulated strain values going up to 250 strain units (measured at the rim of the fixture), corresponding to relatively long shear times of 1 h or more. After cooling within the rheometer, the sample was removed and fractured in liquid nitrogen in a plane perpendicular to the radial direction. In order to avoid edge effects, the sample was fractured at half of the disk radius. Consequently, the values of the shear rate and of the strain, used later on, have been determined at the radial position of the fracture. Scanning electron microscopy analysis has been performed in the central region of the fractured surface. In order to verify the morphological stability of the blends in absence of flow, annealing tests were performed by holding various samples at 200 ◦ C between the plates of the rheometer up to 2 h, in the absence of applied shear. No significant changes in dn and dv with time were observed, indicating that the morphology of the blends was stable with time. This result ensured that the observed changes in the blend microstructure would be induced only by shear. The capillary number, Ca, is an important parameter for examining the possibility of break-up of the droplets in a shear field, although it is defined for an isolated droplet (dilute blend) in absence of compatibilizer. The capillary number is the ratio between shear forces and interfacial forces: Ca =

˙ ηM γd 2α

(1)

where ηM is the matrix viscosity, γ˙ the shear rate and d is the droplet diameter. A critical capillary number Cac can be introduced in order to define, for example, the upper limit of the droplet size for break-up at a fixed shear rate. In fact, for Ca < Cac the droplets take an equilibrium shape, whereas for Ca  Cac , the droplets deform in filaments, which break-up into smaller domains. Cac is a function of the viscosity ratio and, in our situation, a value Cac = 0.51 can be estimated from the work of Grace [20]. Table 3 shows the values of the capillary number ratio, Ca/Cac , calculated using the initial droplet size, for different conditions of compatibilization and shear flow. We see that Table 3 Initial values of the capillary number ratio Ca/Cac

Fig. 3. Complex viscosity vs. frequency: () PE/PA 70/30, () PE/PA/PE-g-ma 69.8/30/0.2, and (䊉) PE/PA/PE-g-ma 68/30/2.

φc (%)

0.025 s−1

0.05 s−1

2000 Pa

0.35 s−1

0 0.2 2

0.12 0.18 0.14

0.22 0.35 0.26

0.92 (0.2 s−1 a ) 1.28 (0.18 s−1 a ) 0.40 (0.075 s−1 a )

1.63 2.45 1.88

a Values of the shear rate determined at the radial position R/2, using the steady-state viscosity measured in creep at 2000 Pa (see Fig. 7).

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except for the largest shear rate, the initial values of Ca were smaller than Cac . 5. Coalescence in uncompatibilized blends Fig. 4 shows an example of the transient viscosity of the PE/PA 70/30 uncompatibilized blend determined from startup and creep tests. Whatever the type of tests, an overshoot is observed at small strain (very short time). Taking into account the fact that in these experiments the shear rates were small, this overshoot is simply the result of the deformation of the droplets dispersed in the matrix [21]. At large strains and if the shear rate was not too small, the viscosity reached an equilibrium value (steady-state viscosity). For the lowest shear rates (0.05 and 0.025 s−1 ), there were not real equilibrium and we observe a progressive decrease probably due to coalescence with a large increase of the droplet size and a decrease of number of droplets by volume unit. It can be noted that the stress, σ = 2000 Pa, used in the creep test corresponds to a shear rate of 0.20 s−1 , which is slightly smaller than the largest shear rate used in the startup tests. As shown in Fig. 4, the respective position of the two viscosity curves is coherent with this observation. So, we can consider that the results of the creep and start-up tests are therefore in a good agreement and, for the following only the shear rate results will be given. Fig. 5 shows the variation of the ratio dn /dn0 as a function of the total strain for the PE/PA 70/30 uncompatibilized blend; dn and dn0 are the droplet sizes for a given strain and the initial value, respectively. The use of this ratio allows comparing the results of experiments performed for different values of dn0 , in relation to the addition of compatibilizer. It must be noted that the morphology evolution was not monitored continuously, but observed at some different strain values from separate experiments. So, the lines in Fig. 5 and the following are to show the tendencies. For the smallest shear rate of 0.05 s−1 , the experiment has been only performed for the maximum strain of 250. It is also worth noting that elongated particles were observed at relatively large strain, γ ≥ 50, for the lowest shear rates used. However, the majority of particles remained approximately spherical, and,

Fig. 5. Variation of dn /dn0 vs. strain for the PE/PA 70/30 uncompatibilized blend: () 0.05 s−1 , () 0.35 s−1 , and () 2000 Pa.

hence, the elongated particles were not taken into consideration to determine the average diameter. Except for the largest shear rate (0.35 s−1 ), Fig. 5 shows that the droplet size changed relatively rapidly at first and then stabilized practically at a limiting value, all the more rapidly than the shear rate was large. It is obvious that coalescence occurred, as the droplet size increased with strain (or time). As shown in Table 4, due to the increase in the droplet size, Ca increased with strain and became comparable to Cac . This result suggests that initially the droplet size evolution was governed by coalescence, and when the droplet size became comparable to the break-up limit, no further increase was observed. So, the limiting size observed at large strains seems to be the result of a break-up/coalescence equilibrium. Fig. 5 also shows that the increase in droplet size is very small for the shear rate of 0.35 s−1 (see Table 5). In fact, the evolution of the size with strain is quite complex: an initial decrease followed by a slight increase was observed. Ca was initially larger than Cac at this shear rate, hence break-up and coalescence governed the droplet size evolution on the whole range of strain. Note that whatever the shear rate a relatively important decrease of the polydispersity was observed in these experiments. For example, at a shear rate of 0.05 s−1 , dv /dn decreased from 1.5 to 1.23. Table 4 Variation of the capillary number ratio, Ca/Cac , with strain at different shear rates γ˙ (s−1 )

0.025

0.05

0.2

0.35

0 25 50 100 125

0.12 – – – 0.86

0.22 0.45 0.57 0.69 0.73

0.92 1.55 1.47 1.63 1.61

1.63 1.37 1.45 1.92 1.69

Table 5 Variation of (dn /dn0 )lim. with shear rate for different compatibilizer concentrations

Fig. 4. Transient shear viscosity vs. strain for the PE/PA 70/30 uncompatibilized blend: (♦) 0.025 s−1 , () 0.05 s−1 , () 0.35 s−1 , and () 2000 Pa, approximately corresponding to 0.20 s−1 under steady-state conditions.

γ˙ (s−1 )

0.025

0.05

0.075

0.18

0.2

0.35

φc = 0% φc = 0.2% φc = 2%

7.6 5 1.5

3.2 5 1.75

– – 1.75

– 3 –

1.75 – –

1.1 2.5 1.6

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rates used. Note that, in the creep experiment (Fig. 7), the shear rate corresponding to the steady-state viscosity decreased as the compatibilizer concentration was increased. Due to the simultaneous decrease of the droplet size and of the interfacial tension, the initial values of the capillary number remained close to that of the uncompatibilized blend (see Table 3). Fig. 8 reports the effect of compatibilizer on the variations of droplet size with strain for different shear rates or stress. This evolution, displayed by the variation of dn /dn0 versus strain,

Fig. 6. Variation of (dn /dn0 )lim. vs. shear rate for different compatibilizer concentrations: () φc = 0%, () φc = 0.2%, and (䊉) φc = 2%.

All these results suggest that the limiting value of dn /dn0 at large strain values, noted (dn /dn0 )lim. , can be used to quantify the intensity of the coalescence, despite the fact that both deformation and break-up were effectively present. Table 5 and Fig. 6 show that, in the case of the uncompatibilized blend, the intensity of coalescence increases as the shear rate is decreased. Similar results have been reported by Vinckier et al. [22,23] for semiconcentrated blends of polyisobutene and polydimethylsiloxane (PIB/PDMS) and by Martin et al. for concentrated blends of polystyrene and high-density polyethylene (PS/HDPE) [19]. 6. Effect of compatibilization on coalescence Fig. 7 shows an example of the transient viscosity as a function of the strain for different compatibilizer concentrations. The viscosity increased with the compatibilizer concentration. And it is particularly important for the largest compatibilizer concentration. This can be related, as in the case of the complex viscosity (see Fig. 3), to the existence of a thick droplet–matrix interphase. On the other hand, the strain-dependence of the viscosity was very similar to that observed for the uncompatibilized blend and, in particular, no steady state was reached for the smallest shear

Fig. 7. Transient shear viscosity vs. strain for different compatibilizer concentrations: () φc = 0%, () φc = 0.2%, and (䊉) φc = 2%. σ = 2000 Pa.

Fig. 8. Variation of dn /dn0 vs. strain for different compatibilizer concentrations: () φc = 0%, () φc = 0.2%, and (䊉) φc = 2%; (a) γ˙ = 0.35 s−1 , (b) σ = 2000 Pa, and (c) γ˙ = 0.05 s−1 .

J. Huitric et al. / J. Non-Newtonian Fluid Mech. 145 (2007) 139–149

is qualitatively similar to that observed for the uncompatibilized blend. For the blend containing 0.2% of compatibilizer coalescence was important and strangely more important than in absence of compatibilizer. However, this effect was less pronounced when the shear rate was increased (compare Fig. 8a–c). On the other hand, for the blend containing 2% of compatibilizer, coalescence was almost suppressed compared to the blend containing 0.2% and compared to the uncompatibilized blend at small shear rates (see Fig. 8c). The data of Fig. 8c are reported in Fig. 9 in terms of dn , instead of dn /dn0 , versus strain. It is interesting to note that the final droplet size for the compatibilized blends always remained smaller than that of the uncompatibilized blend, even for the blend containing 0.2%, which was more affected by coalescence under shear flow than the uncompatibilized blend as shown in Fig. 8c Note that a significant polydispersity decrease was only observed for the blend containing 2% of compatibilizer: dv /dn decreased from 1.26 to 1.12 at a shear rate of 0.05 s−1 . The efficiency of the compatibilizer in coalescence inhibition, as displayed by the data of Figs. 8 and 9, should be analyzed more precisely by considering the important changes of the blend microstructure induced by the compatibilization. Indeed, due to the droplet size decreases with compatibilizer, the number of droplets per volume unit, n, increased and the separation distance between droplets, h, decreased at a constant volume fraction of the dispersed phase. Under flow-driven coalescence, this results in an increase of the frequency of the collisions responsible for coalescence. Indeed, at a given shear rate, the collison frequency C is given by Smoluchowski [24]: C ∼ d 3 n2

(2)

where n can be related to the droplet size, d, and volume fraction, φ, i.e., φ ∼ nd3 . Hence, for φ = const., relation (2) becomes: C ∼ d −3

(3)

From this relation we can estimate the collision frequency ratio K = C(φc )/C(φc = 0), C(φc ) and C(φc = 0) being the collision frequency for the compatibilized blends and for the uncompatibilized blend, respectively. From the values of K cal-

Fig. 9. Variation of dn vs. strain for different compatibilizer concentrations: () φc = 0%, () φc = 0.2%, and (䊉) φc = 2%. γ˙ = 0.05 s−1 .

145

Fig. 10. Variation of (dn /dn0 )/K vs. strain for different compatibilizer concentrations: () φc = 0%, () φc = 0.2%, and (䊉) φc = 2%. γ˙ = 0.05 s−1 .

culated using the initial droplet sizes, C for the blend containing 0.2% of compatibilizer is 11 times that of the uncompatibilized blend and 50 times for the blend containing 2% compatibilizer. Hence, the collision frequency strongly increases with the compatibilizer addition. So, in order to display an “intrinsic” efficiency of the compatibilization, we plotted in Fig. 10 dn /dn0 divided by K as function of strain using the data of Fig. 8c. The ratio (dn /dn0 )/K is shown to decrease significantly with increasing compatibilizer concentration. This means that the “intrinsic” efficiency increases with compatibilization. Even, if the collision frequency should be multiplied by the coalescence probability (fraction of collisions, which actually leads to coalescence), these findings remain significant, especially since coalescence probability strongly increases as the droplet size decreases [25], which is the case when the compatibilizer concentration is increased. This analysis suggests that changes of the blend microstructure should be taken into consideration in the discussion of the coalescence data in terms of efficiency of compatibilization to prevent coalescence. Fig. 6 and Table 5 show that for a low compatibilizer concentration, as already observed for the uncompatibilized blend, coalescence increased as the shear rate was decreased and became relatively important at the smallest shear rates used. On the other hand, for the high compatibilizer concentration, coalescence was significantly inhibited and was almost independent of the shear rate. In order to verify the observed effect of the shear rate on coalescence, we have compared the data obtained on a blend sample fractured at half and three-fourth of the disk radius. These experiments have been performed at a given shear rate of 0.1 s−1 at the rim of the fixture corresponding to different values of the shear rate, γ˙ eff. , experienced by the blend. The data obtained for a blend without compatibilizer and for a blend containing 2% of compatibilizer are summarized in Table 6. We see that in absence of compatibilizer the intensity of the coalescence increased as the radial position of the fracture decreased, whereas no position effect is observed for the compatibilized blend. These results agree very well with the above observations on the coalescence-shear rate dependence. These results also show that when coalescence is important a radial

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Table 6 Variation of (dn /dn0 )lim. with the radial position of the fracture, at γ˙ R = 0.1 s−1 Radial position

R/2

3R/4

(s−1 )

0.05 7.6 1.75

0.075 3.2 1.65

γ˙ eff. (dn /dn0 )lim. (φc = 0%) (dn /dn0 )lim. (φc = 2%)

heterogeneity of the blend microstructure occurs due to the nonhomogeneous strain field of the parallel plate geometry. So, it is likely that the rheological data obtained for droplet morphology blends are affected by this phenomenon. This study shows that a significant inhibition is only observed when the interfacial coverage is important (Σ = 0.30 chain/nm2 ) and probably close to saturation. This does not support a dominant contribution of a Marangoni effect to coalescence inhibition. Indeed, this high concentration of compatibilizer at the interface does not facilitate the movement of the chains from the film gap to the backside of the droplets, especially since the mobility of these chains is reduced by their grafting. In fact, for reasons of mobility, the Marangoni effect should preferably appear for the lowest compatibilizer concentration used, especially as the interfacial coverage, related to the droplet size, decreases strongly as the strain is increased. This is not obviously observed. So, we think that steric repulsion plays a predominant role in coalescence inhibition observed for the largest compatibilizer concentration in relation to the droplet–matrix interphase. The fact that the degree of coalescence inhibition is independent of the shear rate is also in favour of this steric barrier concept. In this context, the “apparent” inefficiency observed for the smallest compatibilizer concentration could be explain by the flattened conformation of the compatibilizer chains at the interfaces due to a low concentration of chains, which are individually grafted to several chains of PA. 7. Modeling The Doi and Ohta model [26] for mixture of equiviscous Newtonian fluids have been used to predict the transient rheological response of concentrated blends in relation to their morphology evolution [27–29]. Generally these studies focussed on the modeling of the transient rheological properties, with little attention to the droplet size evolution with time. The Doi and Ohta model consists of a constitutive equation relating the stresses to the interfacial structure and kinetic equations for the evolution of the interfacial area per unit volume Q (= 6/dv ) and of an interface tensor, which characterizes the anisotropy of the interface. Lee and Park [30] have modified the stress equation to take into account the mismatch of the viscosities of the two polymers. They have also introduced an additional term in the kinetic equations. This term is related to different mechanisms of relaxation corresponding to three fitting parameters, d1 , d2 and d3 (λ, μ and η, respectively in Lee and Park), which are characteristics of the degree of total relaxation, degree of size relaxation and degree of break-up and shape relaxation, respectively. Lacroix et al. [21,31] have modified this model by introducing a simple linear mixing rule based on the volume fraction of each component.

In the Doi and Ohta model as well as in the Lee and Park model, the development of the interface governing equations assumes affine deformation. Lacroix et al. [32] have proposed to take into consideration the non-affine deformation of the interface by introducing a slip parameter ξ. The modified Lee and Park model is then expressed by the following governing equations [32]:     ξ ξ ξ ∂qij = −qki κkj 1 − + qki κjk − qkj κki 1 − ∂t 2 2 2 ξ 2 Q + qkj κik + (1 − ξ)δij κlm qlm − (1 − ξ)γ˙ ij 2 3 3   α qlm κlm +(1 − ξ) Qqij qij − d1 Q ηM   α qlm qlm −d1 d3 qij ηM Q ∂Q α 2 α = −κij qij (1 − ξ) − d1 d2 Q − d 1 d3 qij qij ∂t ηM ηM σij = ηM

1 + 3/2H (1 − ξ)γ˙ ij − αqij 1−H

(4)

(5) (6)

where H is given by H =φ

2(ηI − ηM ) 2ηI + 3ηM

(7)

α is the interfacial tension; ηM and ηI the viscosities of the matrix and of the inclusions, respectively; φ the volume fraction of the dispersed phase; γ˙ ij = κij + κji the components of the rate of deformation tensor; κij the components of the velocity gradient tensor; δ the Kronecker symbol; σ ij the components of the stress tensor and P is the pressure. Setting ξ = 0, we retrieve in Eq. (4) the affine deformation governed by the lower convected derivative, whereas for ξ = 2, the deformation is governed by upper convected derivative. For ξ = 1 (corresponding to total slip) the flow is purely rotational and there is no contribution of the interface to the stress tensor. The parameters determination is explained in detail in [21,30,31]. The parameter d1 , which is related to the kinetics of the relaxation of the interfacial area and its anisotropy under the effect of the interfacial tension, is determined from the linear viscoelastic data at low frequencies. Indeed, considering that no coalescence takes place in small amplitude oscillatory flow, d2 is set to 0 and d3 is taken equal to 1 − φ as proposed by Lee and Park [30]. So, in the case of affine deformation, d1 is the only fitting parameter. We can notice that the linear viscoelastic behavior of the uncompatibilized blend and of the blend containing 0.2% of compatibilizer are well described by the Lee and Park model with a linear mixing rule. However, for the blend containing 2% of compatibilizer, it was not possible to correctly fit the data at the lowest frequencies used and, hence, a d1 value has been determined without considering the low frequency data. This discrepancy is due, as mentioned in Section 3, to the second relaxation mechanism, which occurs at sufficiently high compatibilizer concentration. Finally, the parameters d2 and ξ were

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147

Fig. 11. Comparison between morphological data () and model predictions for ξ = 0 (- - -) and ξ = 0.57 (—). φc = 0% and γ˙ = 0.35 s−1 .

fitted to obtain the best agreement between the model predictions and the experimental evolution of the droplet sizes during shear flow. The knowledge of the interfacial area Q allowed us to determine an average diameter if the dispersed phase was present as spherical particles. The break-up and coalescence phenomena occurring under finite strain flow are quantitatively predicted by the model if we assume non-affine deformation. This is illustrated in Fig. 11 where the predictions for the uncompatibilized blend using ξ = 0 and 0.57 are compared. Obviously, the predicted results for nonaffine deformation are much closer to the values determined by SEM, indicating that the non-affine deformation is a more realistic situation. Fig. 12 compares the morphological changes and the model predictions for different compatibilizer concentrations under two different shear rates, γ˙ = 0.05 s−1 (Fig. 12a) and γ˙ = 0.35 s−1 (Fig. 12b). The parameters are reported in Table 7. We see that whatever the compatibilizer concentration the agreement between the model predictions and experimental data is quite good both for the largest and the smallest shear rates used. Table 7 shows that the parameter d2 , which is related to coalescence, increases when a small quantity of compatibilizer is added (0.2%) then decreases when a higher concentration is used (2%). This is in a qualitative agreement with the effect of the compatibilizer concentration on the coalescence intensity discussed above (see Section 6). Table 7 also shows that the slip parameter ξ significantly decreases when the compatibilizer concentration is increased. This result can be explained by the presence for the compatibilized blends of a droplet–matrix interphase created by the formation in situ of a copolymer at the interface. This interphase improves the interfacial adheTable 7 Parameters used for predicting the morphology evolution, at two different shear rates with d3 = 0.7 φc (%)

0 0.2 2

γ˙ = 0.35 s−1

γ˙ = 0.05 s−1

d1

d2

ξ

d1

d2

ξ

0.4 0.4 0.5

0.05 0.43 0.2

0.44 0.075 0.075

0.4 0.4 0.5

0.017 0.15 0.026

0.57 0.2 0.1

Fig. 12. Comparison between morphological changes and model predictions (—) for different compatibilizer concentrations: () φc = 0%, () φc = 0.2%, and (䊉) φc = 2%; (a) γ˙ = 0.05 s−1 and (b) γ˙ = 0.35 s−1 .

sion between the droplets and matrix and, hence decreases the slip. The modified Lee and Park model has also been used to describe the transient viscosity data, keeping constant the values of the parameters d2 and ξ previously used to fit the morphological data. The model appears to be inadequate for predicting the transient viscosity, except in the case of the uncompatibilized blend as shown in Fig. 13, especially at the largest shear rate of 0.35 s−1 (Fig. 13a). Even if the peak of the overshoot is underestimated, this good description is probably due to the very small droplet size changes occurring during shearing of this blend. In all the other cases, as shown in Fig. 13b for the uncompatibilized blend at the low shear rate (0.05 s−1 ), the model underestimates the viscosity even if the position for the overshoot is well predicted. Moreover, the model predicts a steady-state viscosity which is not experimentally observed. The disagreement becomes more and more important as the compatibilizer concentration is increased. The discrepancies between the model predictions and the transient viscosity data are partly due to the radial heterogeneity of the blend microstructure, which occurs during shearing in the parallel plate geometry and, which is not included in the model. The discrepancies may be also due to the fact that the interactions (hydrodynamic and steric) between droplets, which govern the viscosity for these concen-

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effect is proposed. It is based on the variation of droplet collision frequency associated to the blend microstructure changes induced by the compatibilizer addition. The observed effects of both the shear rate and the interfacial coverage suggest that steric repulsions between grafted copolymer chains at the interface contribute more largely than Marangoni forces to the coalescence inhibition. The droplet size evolution is well described by using a modified version of the Lee and Park model, which takes into account non-affine deformation of the interface. On the other hand, the predictions for the transient viscosity are not totally satisfactory. The discrepancies are believed to be largely due to the radial heterogeneity of the dispersed droplets in the blend and to droplet interactions, which were not taken into account in the modified model. The modified Lee–Park has been shown to be quite useful in predicting the morphology changes of immiscible blends under simple shear flow. It is obvious that more work is required to assess fully the model using other blends and more complex flows. Also this model needs to be improved to account for elastic properties of both components and orientation as well as interactions between domains. References

Fig. 13. Comparison between stress growth viscosity data (- - -) and model predictions (—). φc = 0%; (a) γ˙ = 0.35 s−1 and (b) γ˙ = 0.05 s−1 .

trated blends, are not taken into consideration in the model. Moreover, the magnitude of these interactions can be greatly affected by the compatibilizer addition in relation to the changes in the average distance between droplets due to droplet size changes. The modification of these interactions during shearing was not considered. 8. Concluding remarks The effect of adding small amounts of a reactive interfacial agent as a compatilizer to PE/PA concentrated immiscible blends on the droplet coalescence induced by shear was investigated. A quenching method, based on a rapid cooling of a sample placed between parallel plates of a rheometer was applied to follow morphological changes during start-up and creep experiments. Morphological changes and coalescence intensity have been analyzed as functions of the applied shear rate (or shear stress) and of the compatibilizer concentration. A significant coalescence inhibition was only observed when a relatively high amount of compatibilizer (2%) was added, corresponding to an important interfacial coverage. In this situation the coalescence intensity became independent of the applied shear rate. On the other hand, the coalescence intensity was larger in absence of compatibilizer or when a low compatibilizer concentration (0.2%) was added and increased as the shear rate was decreased. A possible semi-quantitative explanation of this rather complex

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