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Fracture of ultrafine fibers in the flow of mixtures of non-newtonian polymer melts
Journal of Non-Newtonian
Fluid Mechanics, 31 (1989) l-26
1
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
FRACTURE OF ULTRAFINE FIBERS IN THE FLOW OF MIXTURES OF NON-NEWTONIAN POLYMER MELTS
M.V. TSEBRENKO
and G.P. DANILOVA
Kiev Technological Institute of Light Industry, 2.52011 Kiev (U.S.S.R.)
A.YA. MALKIN Research Institute of Plastics, Perovskii pr. 35, 111 I I2 Moscow (U.S.S.R.)
(Received April 20, 1988)
Experimental and theoretical investigations into the fracture of liquid streams resulting from the coextrusion of mixtures of polymer melts are generalized. It is shown that the fracture of a liquid cylinder (fiber) for viscoelastic polymer systems differs from the known theoretical predictions concerning Newtonian fluids. This is expressed by the following facts:
(4 the wavelength of the destructive disturbance and the size of the resulting drops for polymer systems are greater than for Newtonian fluids; tb) the growth of amplitude of the destructive wave slows down at the final stages of fracture; (4 theoretical and experimental values of the lifetime of a polymer stream differ by 2-3 orders. These discrepancies between experiment and theory are due to non-Newtonian effects caused by the elasticity of polymer melts. Ultrafine fibers produced during the flow of mixtures of polymer melts through dies are maximally unstable if the viscosity ratio of the initial polymer melts is close to unity. The relation between the lifetime of a stream and the ratio of viscosities of both phases is determined by the matrix viscosity. The relative radius of the drops which results from microfiber fracture is independent of the chemical nature of mixed polymers and equals 2.2.
0377-0257/89/$03.50
0 1989 Elsevier Science Publishers B.V.
2 1. Introduction Utilization of polymer mixtures, in addition to many interesting applications, opens a new way to obtain fibers of small diameter (from several microns to fractions of a micron), i.e. ultrafine synthetic fibers or microfibers [l-4]. The essence of this phenomenon is that when mixtures of melts of incompatible polymers flow through dies, a disperse phase polymer (fiber-forming) produces in the mass of another (matrix) polymer a great number of fibers strictly oriented in the direction of extrusion. These fibers can be extracted from the matrix component by a solvent which is inert to the fiber-forming polymer. Here, unlike conventional methods of producing chemical fibers, a complex fiber from tens and hundreds of thousands of microfibers results from the extrusion of a mixture melt through one hole [3,5]. In this case we may speak about a fundamentally different process of fiber formation of one of the polymers of the mixture under the influence of rheological forces, when the number of filaments in a fiber is not determined by the number of holes in a die. The phenomenon just described has been called ‘specific fiber formation’ [6]. A typical example illustrating it is shown in Figs. 1, 2, which show an electron microphotograph of an extrudate transversal fracture and microphotographs
Fig. 1. Scanning electron microscopic photograph of an extrudate transversal fracture of the POM/CPA mixture (30 : 70).
Fig. 2. SEM pictures of (a) transverse and (b) longitudinal POM/CPA mixture.
sections of the extrudate of the
of a transversal and a longitudinal section of extrudate of poly(oxymethylene)-copolyamide mixture. A great number of microfibers of one polymer in the matrix of another polymer is clearly seen. The transverse section (Fig. 2a) exhibits a typical disperse structure. This very remarkable phenomenon, which is closely connected with the characteristics of the rheological properties of the coextruded fluids, has attracted the attention of many researchers. Thus, a mechanism of microfiber formation has been suggested [4], the influence of the conditions for deforming mixture melts on fiber formation has been studied [3,7-91, and features of properties and structure of ultrafine fibers have been estimated [5,10,11]. However, the fracture mechanism of the fibers formed from the mixture coextrusion is only poorly understood. Evidently, the problem here amounts to estimating the conditions for fracture of viscoelastic fluids. In spite of numerous works describing the fracture of liquid streams of Newtonian fluids, we have few ideas about viscoelastic systems and they are in the course of development [12]. In this paper the authors have tried to compile a critique of the experimental data, both their own and from the literature, concerning the fracture of ultrafine polymer streams produced in the matrix of another polymer, and to compare the data with well-known theoretical predictions. To this
4 end, the authors first present a brief review of the known theoretical concepts concerning Newtonian fluids and then develop the specific behaviour of non-Newtonian media. 2. Deformation of drops and fracture of liquid streams in the mixtures of two Newtonian fluids 2.1. Fracture of a drop Under conditions of dispersion flow of liquid drops, the stresses arising in the dispersion medium tend to deform and orient a drop. According to Taylor’s classical theory [13], the force responsible for the deformation is the difference between the pressures inside and outside the drop API’. For Couette flow, the pressure difference is expressed as sin 2+,
APp = 4917
(1)
where y is the rate of shear; n2 is the viscosity of the dispersion medium; K = ql/q2 is the ratio of the viscosities of the melt of the drop phase vi and the medium q2; and 9 is the angle determining the orientation of the drop in the direction of the flow. For low rates of shear, the drop assumes the shape of an ellipsoid; with high p the drop turns into a liquid cylinder (stream). In addition to the force APT, which tends to deform the drop, the force created by the interphase tension AP,, also acts on the drop. This force is determined by the Laplace equation:
where u is the interphase tension and r is the radius of an undeformed drop. Under equilibrium conditions, the deformation D of the drop is given by D=
19K+16. 16K+16
we ’
where We is the Weber number. Thus, according to Taylor’s theory, the deformation of the drop in another fluid medium, depends on two parameters: the viscosity ratio K = q1/q2 of the drop and the medium and the Weber Number We, i.e. the ratio of the product of the local shear stress and the drop radius to the interphase tension: We = q2-+r/a
(4)
Taylor [13] supposed that the drop fractures when the surface tension,
5 which seeks to keep the drop in spherical form, can no longer be balanced by viscous forces, which attempt to fragment the drop. Therefore, the parameters K and We that control the deformation must also control the critical conditions for fracture. According to Taylor, the deformation E, of the drop during fracture (critical value of deformation) equals 0.5 and is calculated by the formula E,=(L-B)/(L+B),
(5)
where L and B are the major and minor axes of the deformed drop, respectively. A great number of theoretical and experimental examinations have been carried out on the fracture of a liquid cylinder in steady uniaxial or shear flow, in transient conditions, and in a resting fluid after the cessation of flow [14-271. Practically all these works consider drop emulsions of a Newtonian fluid in another Newtonian fluid. Han [28] has generalized the experimental data of some authors on the fracture of Newtonian drops located in another Newtonian fluid which was exposed to an homogeneous shear flow. According to his conclusions, in all cases, the dependence of critical deformation E, on the ratio between the phase viscosities K passes through a broad minimum lying in the range of values of K 2: 0.1-1.0. Different authors give different data on the critical value of E, in the minimum region, though there are results close to the theoretical limit EC = 0.5 [18,23,28]. Beyond the limits of the given range of values of K, the value of the viscous forces necessary for the drop’s fracture grows very quickly and almost unboundedly. Thus, for very low or very high values of K, the deformed drop is stable in an homogeneous shear flow. Similar results have been obtained in a number of works [16,18,23]. For example, Rumscheidt and Mason [16] point out that the drops of a Newtonian fluid were deformed and fractured in Couette flow in the range K = 0.01-2. For larger values of K the drops were not fractured. In this case, as i, increases, the limiting deformation of ellipsoidal drops is achieved and their major axes are oriented along the direction of the flow. Torza, Cox, and Mason [18] have shown that the drops of a Newtonian fluid were not fractured for K> 3. Karam and Bellinger [23] observed the fracture of the deformed drops in the range of the viscosity ratio 0.005 G K G 3.0. The authors of [16] describe four types of drop fracture of Newtonian fluids according to the value of K. If the disperse phase is low-viscosity (K-c 0.2), then at a critical value of 9 the drop becomes S-shaped and tiny drops break away from its ends. As the viscosity of the drop increases (K > 0.2), the deformed drop breaks through the centre with the formation of two large drops and three satellite drops. For K < 2 the drop extends into a long cylinder and eventually falls into many small drops. For K > 2 the
6 drop becomes deformed and is oriented along the flow without fracture even at 9 up to 40 s-t. 2.2. Fracture of a cylinder A liquid cylinder (stream) is thermodynamically unstable due to an unfavourable ratio of the surface to the volume. The authors of [29] note that when the ratio of the length I of a liquid cylinder to its diameter d, reaches 4.5, the lateral surface of the cylinder becomes equal to the surface of two spheres with the total volume being equal to the volume of the initial cylinder. Further increase of the cylinder length is unfavourable from an energetic point of view and it splits into drops under the action of surface tension. Thus, a liquid stream with Z/d, > 4.5 is a nonequilibrium system and its fracture is determined by the local conditions of the flow, and the rheological and interfacial properties of the mixed fluids. The authors of [18,30] have discovered that the fracture of the deformed drops is determined not only by the ratio of the phase viscosities K, but also by the rate of growth of i, with time t, i.e. by the quantity dy/dt. Moreover, in many papers it has been noted that a liquid cylinder becomes varicose before fracture (Fig. 3): periodics swellings and &innings appear on it. Both of these factors (dependence of fracture conditions on dy/dt and stream varicosity) give grounds to believe that a drop fracture is determined by hydrodynamic disturbances. Varicosity (capillary waves) may appear for various reasons such as fluctuations of density and viscosity and vibration of the equipment [29]. Therefore, the fracture of a liquid stream is, by its nature, a transient phenomenon. Any attempt to develop the theory of this phenomenon requires the solution of balance equations for an usteady flow. On the basis of the classical works of Rayleigh and Weber, Tomotika [19,20] has analyzed the hydrodynamic stability of a liquid stream and explained Taylor’s experimental observations. Beginning with Rayleigh’s works [26], the fracture of a liquid cylinder is analyzed for a fluid at rest. According to the existing concepts, the appearance of the wave disturbances on the
u-
-
-t
t
-u__
Fig. 3. Liquid cylinder exposed to the action of a capillary wave.
7 surface of a liquid cylinder is responsible for its fracture (Fig. 3). The fracture is caused by the disturbance whose amplitude grows with the maximum rate. The stream is destroyed when the disturbance amplitude becomes equal to the stream radius. This is attained in time t,, called the lifetime of a liquid cylinder, or the fracture time. A simple empirical expression [31,32] is suggested for an approximate estimate of the lifetime of a stream: tb =
qlR/a,
(6)
where R is the radius of the initial liquid cylinder. Varicosity enlarges the surface of the cylinder and, consequently, increases its surface energy. Therefore, the fracture is caused by the attempt of the system to reduce its free surface energy. The radius rd of the resulting drops is related to the wavelength h, of the destructive disturbance by the ratio: r, = (3RZX,/4)1’3. According to Tomotika’s theory [19,20], the relationship between the wavenumber of the disturbance leading to fracture and log K is represented by a curve having a maximum in the region K = 0.28 (see below). Such a relationship points to the fact that a definite wavelength is typical for each value of K = const, and when this wave propagates, a liquid cylinder becomes unstable to the maximum extent. It is also evident that the cylinder’s stability sharply increases both for very low and very high values of K. Since the size of the resulting drops is determined by the destructive wavelength, the dependence of r, on log K is represented by a curve with a minimum in the region K = 0.28. Fracture of a liquid cylinder in a flow is characterized by a number of features and this process is undoubtedly more complicated than that in a liquid at rest. These features are as follows. The stability of a liquid cylinder in a flow sharply increases [19,20]. This is associated with an irregular increase in the amplitude of the destructive wave and, as a result, the cylinder in the flow breaks down nonsimultaneously along the entire length, and the resulting drops have different sizes [15]. Therefore, concerning the fracture of liquid cylinders in a flow, Tomotika’s theory is confirmed only qualitatively. For example, the dependence of the size of the drops resulting from the fracture of liquid cylinders on log K is represented by a curve having a minimum, but the coordinates of the minimum do not correspond to those given by the theory [15]. In the process of polymer flow, there is often an elongational deformation. Han [28] has compared the critical value of the deformation for which the deformed drop fractures in shear and uniaxial flows. He has shown that, in a uniaxial field, the drops fracture more effectively and with a lower
8
consumption of energy. This is expressed in the fact that for shear, the EC versus log K curve passes through a minimum at K = 0.1-1.0, for elongation at K z: 1-5; the minimum value of EC for shear is = 0.6 and for elongation it is only 0.2. If in a shear flow the ratio of the phase viscosities is larger than 3.5, the drops can no longer be fractured, while a uniaxial flow causes the fracture of liquid cylinders even for larger values of K. 3. Fracture of viscoelastic liquid streams 3.1. Stresses in Poiseuille jlow for a viscoelastic fluid The foregoing theoretical works discuss Newtonian drops or cylinders dispersed in a Newtonian fluid. If one or both fluids possess viscoelastic properties, the fracture mechanism for liquid streams may be different. It seems, from inspection of the few works devoted to this problem [24,30-401, that at present the influence of polymer elasticity in a mixture on stream fracture is only poorly understood. The shape of drops in the flow of a mixture of two viscoelastic fluids is determined by shear stresses and pressure distribution around a drop. The stress tensor T in Poiseuille flow for a viscoelastic fluid is written as [41,42] T,, = -P, - ( u2 - ul) T,,=
J0
2,
TOO= -P,-a,--
,
(8)
r
d
‘a, J0
J,
T, = +fr, P, = P, +
r
d
ral
-P,-
*ul dr /0
r
(10) (11)
ru, d /0
--f(o,-2u,), r
02)
where z is the direction of flow; r is the direction of the velocity gradient; 8 is the direction perpendicular to the (r, z)-plane; f is the partial derivative in the direction of flow; p, is the isotropic pressure, which, as is seen from (12) depends on radial position in a capillary tube; and ui, a, are the normal stresses, which are given as L-
&=a,,
T,,-
T,,=u,.
(13)
From the given eqns. (8)-(12) it is clear that the general solutions that determine the conditions for fracture are scarcely possible due to the wide
9 range of opportunities of varying rheological equations of state for both fluids in a mixture. Therefore, at present we can only point out the peculiarities of the stressed state of viscoelastic drops and reveal these peculiarities for typical viscoelastic media in an experiment. 3.2. Experimental
data
The authors of [7,9,43-471 were the first to carry out extensive systematic quantitative and qualitative investigations into the phenomenon of ‘a specific fiber formation’ according to the nature and rheological properties of mixed polymers, the mixture composition, and the degree of dispersion of a polymer of the disperse phase. According to the results obtained, the first condition for achieving fiber formation during the flow of a mixture melt is the dispersion morphology type of the initial mixture. To meet this condition, polymers must be thermodynamically incompatible. The disperse phase is usually formed by the polymer with the higher density of cohesion energy or greater surface tension. The first attempt to substantiate and predict the morphology types of the two-phase polymer system was Van Oene’s theory [41], according to which a more elastic phase produces drops in a less elastic phase. For each pair of polymers, there exists an optimum relationship between the components and an optimum degree of dispersion such that fiber formation is the most clearly expressed [43-451. The second condition for achieving fiber formation is the deformation of drops into liquid streams during the flow of mixture melts. From this point of view the degree of component interaction in the interphase layer (thermodynamic affinity) must be sufficient for the transfer of stresses arising in the dispersion medium, through the interface to the polymer drops of a disperse phase. Otherwise, with all the other conditions being favourable, fiber formation will not be achieved. We can indirectly estimate the supposed level of interaction of the mixed polymers according to the values of their specific surface energy, solubility parameters and Flory-Huggins interaction parameters I/Q. To ensure phase separation and, at the same time, sufficient adhesion at the interface, the difference between the solubility parameters for the polymers must be near 0.3 [48]. Since the shear stress is continuous at the interface [13], it follows that for 9, = r/z then th e polymers of a disperse phase and dispersion medium deform to the same degree, which promotes the fiber formation of one polymer in the mass of the other [7,45]. The deviation of the ratio n1/n2 from unity causes a different degree of deformation of the mixture components and a worsening of fiber formation. A sharp difference in elasticity of the melt components acts in the same way. Viscoelastic drops are more resistant to stretching and shear than Newtonian drops.
10
The authors of [7,9,43-471 have established the relationship between the swelling ratio B of extrudates of polymer mixtures and ‘specific fiber formation’: the more fibers are formed and the thinner they are, the higher B is. Thus, swelling, being an indirect characteristic of the rubbery elasticity of the mixture melt, also correlates with ‘a specific fiber formation’. Taking into account the fact that the variables which control the degree of drop deformation (K and We, for Newtonian systems) also determine the mechanism of its fracture, we analyzed the fracture of liquid viscoelastic polymer streams according to the viscosity ratio, K, of the melts of mixed components, the chemical nature of the polymers, and the temperature [38,40]. 3.2.1. Experimental The investigation was concerned with the following mixtures: poly(oxymethylene) (POM) with copolymer of ethylene and vinyl acetate (CEVA); POM with spirit-soluble copolyamide (CPA); POM with polystyrene (PS); low-pressure polyethylene (LPE) with CPA; CPA with copolymer of ethylene and butylene (CEB); POM/CEB; CEVA/CPA; POM with polycapramide (PCA); POM with poly(viny1 alcohol) (PVA). Polymers and methods of mixture preparation are described in [7, 9,49-511 and Tables 1-3. The components of the mixture are in the ratio 20 : 80. With the help of a constant-pressure capillary viscometer we have obtained extrudates of mixtures under conditions such that the fiber formation is most clearly defined. For a quantitative estimate of fiber formation a procedure was developed [7] which made it possible to measure and evaluate all the types of structure found in the extrudate of a polymer mixture and to determine their number and mass fraction. To study the fracture of microfibers, longitudinal sections of the extrudates of polymer mixtures (thickness of the section is lo-20 pm) were placed in an immersion medium on the heating stage of a microscope at a definite temperature, depending on the nature of the polymers, or at room temperature, followed by an increase in temperature. Different stages of fiber fracture of polymer of the disperse phase were photographed. As described in [38] (Table l), we treated the microscope data in terms of the variables of Tomotika’s theory [18,19]. In this case, we determined the wavelength X m and the wave number 27rR/X, of the destructive disturbance, the radius R of the initial liquid cylinder (fiber) in terms of eqn. (7) the radius r, of the drops resulting from the fracture and its reduced value rd/R, the lifetime t, of a liquid cylinder and its reduced value tb/R. The initial experimental data were treated by the methods of mathematical statistics. The error in determining the values of r, and h, was + 3.5% with the confidence level 0.95. The elasticity of polymer mixture B and of a
TABLE 1 Parameters No.
i. 2. 3. 4. 5.
of the process
Viscosity of melt, Pass
of fracture
of ultrafine
Swelling of extrudate
poly(oxymethylene) K= n,/r~t
CEVA
POM
CEVA
POM
790 790 790 790 790
280 720 830 1320 3270
1.4 1.4 1.4 1.4 1.4
1.3 1.3 1.2 1.2 1.3
0.35 0.91 1.05 1.67 4.14
fibers in the extrudates
of the POM/CEVA
*& pm
A,,,, pm
R, pm
*d/R
2.5 3.0 3.3 3.4 2.8
16.2 17.7 18.6 21.6 22.1
1.16 1.46 1.60 1.54 1.13
2.19 2.09 2.06 2.20 2.40
211R AIn 0.45 0.52 0.54 0.44 0.32
mixtures
at 165 o C
tb,
tb/R>
S
S/w
15 75 120 300 450
13.2 52.8 75.0 195.0 400.2
12 disperse phase B, and a matrix B, was estimated from the size of swelling of the extrudates after annealing by the method given in [52]. 3.2.2. The role of rheological properties of mixed polymers Analysis of the role of rheological factors in the fracture of polymer streams is quite complicated because, when a matrix and disperse phase vary, different factors such as viscosity, elasticity, and polymer compatibility usually change simultaneously. Therefore, while for Newtonian fluids the deciding factors are determined quantitatively (parameters K and We), for polymer systems, unfortunately, it is necessary to use semi quantitative or qualitative estimates. Therefore a correct selection of the parameter being varied plays a decisive role in the uniqueness of the conclusion. Taking POM/CEVA mixtures as an example, we have studied the influence of the viscosity ratio K = nl/qZ of mixed polymers on the fracture mechanism of POM microfibers in a CEVA matrix. The viscosity ratio K was varied in the range 0.20-10.78. Here the samples employed of POM and CEVA differed very little in melt elasticity (with a range of B from 1.2-1.5) [7]. In one series of experiments we mixed POM of different molecular weights with one sample of CEVA (variant 1); in the second series we mixed one sample of POM with the CEVA samples with different melt viscosity (variant 2). In the first variant ncEvA = const, in the second nr,oM = const. From the results obtained it follows that the external pattern of the initial stages of destabilization and fracture of a liquid cylinder for viscoelastic polymer systems does not differ qualitatively from the mechanism described for low molecular weight systems, i.e. the fracture mechanism of a viscoelastic cylinder also consists in the formation of wave disturbances on its surface. Therefore it is of fundamental importance to consider critical conditions for fracture in the same terms as for Newtonian fluids. This will enable us to reveal immediately whether the fracture really occurs according to the same mechanism. Our experiments indicate that, at the POM melting point, microfibers at first become varicose and then (after the lifetime tb) fracture into a chain of drops (Fig. 4). The curves 2xR/X, vs. log K (Fig. 5, curves 2) obtained from the experimental data are situated below Tomotika’s theoretical curve. This means that the disturbance wavelength (and the size of the resulting drops) is greater for viscoelastic fluids than for Newtonian fluids. The same conclusions are drawn in [34-361, in which the authors analyzed the fracture of a viscoelastic liquid stream in another viscoelastic fluid, taking as an example mixtures of polystyrene and poly(dimethylvinyl)siloxane elastomer in a common solvent [34], poly(methyl)siloxane oligomers of different MW in a matrix of polybutadiene oligomer [35], a melt of the mixture of poly(methylmethacrylate) and polystyrene.
13
Fig. 4. SEM pictures of longitudinal at (a) 150°C and (b) 165“C.
sections
of the extrudates
of the POM/CEVA
mixtures
From Fig. 5 it follows that the experimental curves have a maximum in the range of values of K = 1. We think that this result is of fundamentally importance, since it implies that a mixture of two polymers, one of which is in the form of fibers, is maximally unstable when fiber formation of a polymer in the mixture with another polymer is most clearly defined, and this, according to our data, occurs for K = 1 [7]. The established mechanism is in agreement with the conclusions of Chin and Han [33] that the variables which control a drop’s deformation also determine the conditions for fracture of a liquid cylinder. A maximum instability of fibers in the extrudates of mixtures with K = 1 is supported by the fact that, for these mixtures, swelling of the extrudate is a maximum [7].
14
-4
m Al
-2
0
2
-4
-2
0
qK
on the viscosity ratio between the Fig. 5. Dependence of the wavenumber x = 2qR/X, phases of a drop and a dispersion medium: 1 - Tomotika’s theoretical curve, 2 - experimental data for the melt of the POM/CEVA mixture for (a) qIIcEvA = const and (b) vpoM = const.
The relationships between the wavenumber x = 2?rR/h, and the viscosity ratio of the components have a different nature depending on the method by which the change in K is achieved, i.e. as a result of a variation of the viscosity of the melt of the phase or the dispersion medium (see Fig. 5). For variant 1 (Fig. 5a), the relationship differs significant from Tomotika’s curve. With increasing MW of POM not only does the viscosity grow, but the melt elasticity of POM also changes. The fact that curve 2 passes through a maximum indicates that the elasticity of a polymer of a disperse phase has a different effect according to the ratio of the viscosities of the phases. The lifetime of a liquid cylinder found experimentally (Fig. 6) for variant 1 increases with the melt viscosity of POM (which causes the growth of K), as required by eqn. (6). This means that if a drop phase is very viscous, the drop can be elongated to a low-radius cylinder, which will remain stable for a long time. For variant 2, the experimental curve x vs. log K (Fig. 5b) is asymmetric, its left brach almost coincides with the left brach of curve 2 for variant 1. However, when the viscosity of the matrix component (CEVA) becomes lower than the viscosity of POM (K > l), the experimental points are located very close to Tomotika’s curve, but are somewhat lower. Comparing two curves 2 in Fig. 5a and b, enables us to stress the importance of elasticity of the dispersion medium for the fracture of a liquid stream. While the elasticity of the medium is large, the fracture mechanism differs greatly from the theoretical data, irrespective of the way in which K changes. As the elasticity of CEVA decreases (low-viscosity samples), the fracture of fibers in a CEVA matrix actually occurs according to Tomotika’s mechanism. The curve of the lifetime of a liquid cylinder for variant 2 (Fig.
15
-0,4
0
0.4
0.8
12
Fig. 6. Dependence of the reduced lifetime of the liquid cylinder on the viscosity ratio of the phases for (1) qCEVA= const and (2) qpoM = const.
6, curve 2) is the inverse of the curve for variant 1 (Fig. 6, curve I): with increasing K, t, decreases. This indicates that the way in which either value of K is achieved is important. The absolute value of the viscosity of the medium is capable of qualitatively affecting the stability of the stream, all other factors being equal. The authors of [35,36] have also obtained a different relationship between the lifetime of a liquid cylinder and K, depending on the method by which the change in the viscosity ratio of the polymers was attained. Thus, viscoelastic properties of the medium affect the stability of liquid streams more than viscoelastic properties of the disperse phase (drops or cylinders being deformed). 3.2.3. Features of fracture kinetics of viscoelastic liquid streams According to the classical Rayleigl-Weber-Tomotika theory [19,20,26,27], the disturbance amplitude a (Fig. 3) during stream fracture grows exponentially with time t
a = a0 exp (4th
(14)
where a, is the initial disturbance amplitude; which is a complex function of the wavenumber ratio of the phases K. q =
&F(-K
K),
q is the instability factor, x = 27rR/X, and viscosity
(15)
-1 _
-2
-3 _
-41
r 40
80
120
I60
200 t.S
Fig. 7. Variation of the disturbance amplitude microfibers in the CEVA matrix at 168 o C.
In logarithmic straight line
coordinates
In a = In a, + qt,
of the destructive
wave with time for the POM
the a vs. t relationship
is represented
by a
(16)
the slope of this line to the X-axis is the instability factor q. The analysis of the fracture kinetics of the POM microfibers in the CEVA matrix at 168” C has shown that the experimental time dependence of the disturbance amplitude a deviates from a straight line at the final stages of fracture (Fig. 7). The degree of deviation grows with an increase in the elasticity of the polymer of the disperse phase. The reduction of growth of the destructive wave amplitude for the polymer of the disperse phase which possesses viscoelasticity in a melt was observed by the authors of [35]. The features of the fracture kinetics of viscoelastic liquid streams described there are caused by the effect of the elasticity of polymers, which will be discussed in detail in Section 3.2.4. Equation (15) establishes the relationship between the instabilitiy factor of the stream and interfacial tension u - the most important macroscopic characteristic for a mixture of polymers. The surface energy, structure, and length of interphase layer can be judged from the value of u. On the basis of this, Chappelear [53] has suggested a method for measuring the interfacial tension in a mixture of two polymers, one of which forms liquid streams (fibers) in another polymer. Experimental measurement of CTin polymer mixtures (especially in melts) is extremely complicated. Therefore, application of the theory of destabilization of a liquid cylinder for determining u is of particular interest. On the basis of data on the investigation of the fracture kinetics of
17 TABLE 2 Results of determining interfacial the POM/CEVA mixture No. of system ’
1. 2. 3. 4.
Viscosity of initial components, Pa. s POM,
CEVA,
rll
v2
280 720 1320 3270
790 790 790 790
r 1, 2, 4 - annealing
at 180°C,
tension
according
K=
2R,
L,,
7h/7~2
pm
v
0.35 0.91 1.67 4.14
6.51 4.57 6.17 7.93
35.36 30.58 23.48 46.82
3 - annealing
to fracture
4
kinetics
F(x,K)
of liquid streams
2aR xl?l
0.0140 0.0243 0.0196 0.0159
0.125 0.067 0.053 0.035
0.58 0.47 0.56 0.53
in
D x103, N/m
0.58 1.30 1.90 2.84
at 168OC.
microfibers in the polymer mixtures analyzed, we have calculated the values of interfacial tension (Table 2). The values of u obtained in Table 2 are low, which is typical for polymer mixtures [53-561 and is explained by the presence of a transition layer at the interface of the components. However, in determining u by the method of destabilization of a liquid cylinder, it is necessary to remember that the theory used takes no account of the effect of elasticity, which manifests itself in the fact that the experimental time dependence of the disturbance amplitude deviates from a straight line in the final stages. 3.2.4. The role of the chemical nature of polymers in mixtures In addition to the viscoelastic properties of polymers in mixtures, the degree of polymer interaction at the interface, has quite an important effect on the fracture mechanism of liquid streams. Let us consider the results of investigation into the fracture mechanism of ultrafine fibers in extrudates of mixtures of polymers which are different in chemical nature (Table 3). Pairs of polymers were chosen in such a way that K was close to unity, this being a necessary condition for the preferred formation of microfibers by a polymer of the disperse phase. The fracture of microfibers in sections of extrudates was analyzed at 165OC, except for mixture No. 6, for which the experimental (annealing) temperature was 175°C. The analysis of the experimental data (Table 3) enables us to draw the following conclusions. The stability of fibers of each fiber-forming polymer depends on the nature of the matrix polymer. Thus, the lifetime of the POM fibers in the CEVA matrix is 2-4 times longer than in CPA (Table 3, mixtures no l-5). Evidently, this is due to the different character of the interphase interaction of POM with CPA and CEVA. Indeed, the formation of hydrogen bonds between POM and CPA in a mixture is confirmed by the method of reverse phase gas chromatography [57]. Therefore, the stresses appearing in the
TABLE 3 The effect of the nature of mixed polymers on the parameters of fracture of ultrafine fibers in the extrudates of mixtures No.
Mixture
Fiber-forming polymer
Matrix polymer
7)1*
927
4
Pa.s 1.
2. 3. 4. 5. 6. 7. 8. 9. 10.
POM/CPA POM/CPA POM/CEVA POM/CPA POM/CEVA LPE/CPA’ LPE/CPA POM/PS CPA/CEB CEB/CPA
’ annealing at 175OC.
830 1090 830 1320 1320 1400 1160 650 730 1010
K= vt/nz
rd, pm
A,,,, pm
R, pm
‘d
R
2nR X
m
tb,
tdR1
S
S/pm
eqn.
J32
S
Pa.s 1.2 1.3 1.2 1.2 1.2 2.4 2.4 1.2 1.4 1.8
960 960 790 960 790 960 975 670 1010 730
1.0 1.0 1.4 1.0 1.4 1.0 1.4 2.1 1.8 1.4
tb according to
0.86 1.13 1.05 1.37 1.67 1.46 1.19 0.97 0.72 1.38
2.2 2.5 3.3 2.2 3.4 5.8 6.1 1.6 3.7 5.0
12.8 15.4 18.6 14.7 21.4 38.0 42.0 8.9 23.2 32.1
1.0 1.2 1.6 1.0 1.5 2.6 2.7 0.8 1.7 2.3
2.2 2.1 2.1 2.2 2.3 2.2 2.3 2.0 2.2 2.2
0.49 0.49 0.54 0.43 0.44 0.43 0.41 0.54 0.46 0.45
20 45 120 90 300 672 960 240 270 90
20 38 75 90 200 259 356 300 159 39
0.17 0.26 0.27 0.26 0.40 0.73 0.63 0.10 0.25 0.46
(6),
19 matrix during the flow of the POM/CPA mixture are transmitted to a drop, and, as a result, fine but stressed fibers are formed. Besides this, it is known [58] that application of CEVA as a matrix polymer (or as a third component) increases the homogeneity and kinetic stability of a dispersion, and improves the compatibility of polymers. It is precisely such properties of CEVA that ensure the greater stability of the POM microfibers in it. The sample of PS employed judging from the size of swelling of the extrudate, possesses high melt elasticity. The data from Table 3 show that the POM microfibers in the PS matrix are characterized by long lifetimes ( tb/R = 300 s/pm), exceeding the given lifetime for the POM microfibers in the CPA medium by more than an order of magnitude. A detailed quantitative analysis of the microstructure of extrudates of the POM/PS mixtures indicates that short fibers and particles are not formed during the flow of this mixture. This means that the elasticity typical of the PS melt stabilizes the resulting POM fibers. The established regularity agrees with the general conclusions of reference [30], which shows that, for 77i/n2 < 1, the elasticity of the medium stabilizes the deformed drop, whereas, for high n/q,, it destabilizes the drop. In our case ~i/r/~= 0.97 < 1, which leads to stabilization of the stream. As regards the degree of compatibility of the polymers given in Table 3, LPE and CPA have the lowest affinity for one another [9], which determines a small thickness of the interphase layer and a weak interaction of the components in it and, as a result, a smaller degree of deformation of the fiber-forming polymer as compared to the matrix polymer and lower tension in the resulting microfibers. In view of the foregoing, the microfibers from the melt of the LPE/CPA mixture are characterized by the largest lifetimes. is 356 s/ym for the PE Thus, for close values of vi, qZ, and K, t,/R microfibers and it is 200 and 90 s/pm for the POM microfibers in the CEVA and CPA matrices, respectively (Table 3, mixture Nos. 4, 5, 7). The samples of LPE and CEB employed, in comparison with POM, are characterized by high elasticity (Table 3). The results obtained point to the fact that the maximum values of R, r,, and X, are precisely typical of these polymers (Table 3). Thus, the increased elasticity of the polymer of the disperse phase does not assist in fiber formation. From eqn. (6) it follows that, for a given pair of polymers, the lifetime increases with the viscosity of the polymer melt of the disperse phase (Table 3, mixtures Nos 1, 2, 4, and Nos 3, 5). Analyzing the results of Table 3, we see that if we vary the chemical nature of the polymers in a mixture and the viscoelastic properties of their melts, the values of rd/R are a constant quantity and amount to 2.2 on average. This points to a common fracture mechanism of liquid polymer streams in the matrix of another polymer.
20 Theoretical values of the lifetime of a liquid cylinder calculated from eqn. (6) for dilute solutions of polymers, characterized by low viscosity, yield a quantity equal to tenths of a microsecond. For such values of t, the existence of a liquid cylinder could not be observed experimentally, and since the formation of liquid polymer streams is registered experimentally without any difficulty, we can draw a conclusion on the decisive influence of non-Newtonian effects. Obviously, the initial stages of the generation of disturbances are similar in all cases and they scarcely differ for polymer streams and for streams of Newtonian fluids. But polymer fluids are well known for their ability to accumulate large rubbery deformations upon stretching [12]. These large rubbery deformations, accumulated during the formation of a liquid cylinder, cause the macromolecules to become oriented. As a result, the junctions generated between the drops do not tear apart but, on the contrary, they stabilize the cylinder being stretched, and the stream acquires a typical bead structure. The structure of the stream in the form of a string of beads at the final stages of fracture of viscoelastic streams has been described in [24,37], where it was also noted that in the stream of a Newtonian fluid the links between the drops do not arise, regardless of the type of fracture. By virtue of such features of the fracture of viscoelastic liquid streams at the final stages, there is a deviation of the disturbance amplitude versus time curve from a straight line (Fig. 7), as Tomotika’s classical theory requires. For the mixtures of the polymers analyzed (Table 3), the values of t,, calculated from eqn. (6), range from 0.1 to 0.73 s. In such a case, at the outlet of the forming hole, the microfibers would fracture earlier than the extrudate was cooled to the required degree which makes it possible to fix the resulting fibers by the growth of viscosity. However, the experiment shows (see the comparison between the experimental and theoretical values of tb in Table 3) that microfibers are conserved even during free flow of the stream, when axial deformation, which prevents fracture, is very small. The values of t, found experimentally for ultrafine fibers of POM, PE, CPA, CEB are 2-3 orders greater than the t, values found theoretically. Such a sharp disagreement between theory and experiment can be explained only by the effect of the rubbery elasticity of polymer melts. Van Oene [41] believes that one of the reasons for the stability of fibers (streams) of a polymer of the disperse phase flowing along a capillary is the fact that when fibers fracture into n drops of the same radius as the fiber, the total surface area increases by a factor of 3n (1 + 212). We have shown experimentally (Tables 1 and 3) that the radius of drops resulting from the fracture of a stream exceeds the radius of the initial fiber by more than twice, i.e. the total surface does not increase. Thus, the explanation of fiber stability suggested by Van Oene is not supported by the experimental results.
21
3.2.5. The effect of temperature on microfiber stability We have used polymers that are characterized by different melting points ranging from 50 to 230 o C (Table 4). Therefore, the analysis of the action of temperature on fiber stability in extrudates of polymer mixtures of different chemical nature is of some interest. The analysis of the results shows (Table 4) that fracture becomes possible only for T 2 T,“, where TL is the melting
TABLE 4 The action of temperature on microfiber stability in the extrudates of polymer mixtures Extrudate of mixture
Fiberforming polymer
Melting point of fiberforming “C
Melting point of matrix polymer, “C
Temperature of experiment, OC
PresenceJ + ) or absence( - ) of fracture of the main part of microfibers
POM/CPA 548
POM
165
164
150
-
165 170 150 160
+ + _ -
170 140
+ -
170 170 180 200 210
+ _ -
220 170 180 190 215
+ + + -
50
POM/CEVA
POM
165
POM/PS
POM
165
T,=loo°C
POM/PCA
POM
165
215
POM/PVA
POM
165
230
PE
PE/CPA 548
CEVA/CPA
CEVA/CPA
54
CEVA
CPA 54
132
164
225 230 130 150 160
50
181
170 165 170
+ _
180 185 165 170
+ + -
180 185
+ +
181
50
22 point of a fiber-forming polymer. However, this condition is necessary but not sufficient. Thus, for the mixtures of POM/PVA, PE/CPA, POM/PCA, and CEVA/CPA, when the melting point of a polymer of the disperse phase is achieved (POM, PE, CEVA), fracture of the fibers does not occur, it is retained by the unmelted matrix. The formation of drops (from fibers) is observed for T 2 T/ of the matrix. Therefore, for example, the POM microfibers in the mixtures with CPA, CEVA, and PS are stable at temperatures below 165 o C and in the mixture with PVA up to 220 o C. Thus, the fracture of microfibers can take place only if the melting point of fiber-forming and matrix polymers is attained. 3.2.6. Some conclusions from the experimental investigations In a similar manner to Newtonian fluids, the fracture mechanism for a viscoelastic liquid cylinder also consists of the formation of wave disturbances on its surface. Parameters controlling a drop deformation also determine the critical conditions for fracture of the resulting liquid cylinder. The viscosity ratio of the melts of mixed polymers and the Weber number are such parameters. However, we observe only qualitative agreement with the theory developed for Newtonian fluids. This is expressed by the following facts: the disturbance wavelength and the size of the resulting drops are greater for polymer systems than for Newtonian liquids; the growth of amplitude of the destructive wave slows down at the final stages of fracture; theoretically calculated and experimentally found values of the lifetime of a polymer stream differ by 2-3 orders of magnitude; for one and the same value of K the viscosity of the medium determines the lifetime of a polymer stream, other things being equal. The established disagreement between experiment and theory is due to the influence of non-Newtonian effects caused by the elasticity of polymer melts. A mixture of two polymers, one of which is in the form of fibers in the matrix of another polymer, is maximally unstable if the viscosity ratio of the polymer melts is close to unity. During the flow of such a mixture the fiber formation of one polymer in the mass of another polymer is most clearly expressed. The chemical nature of the mixed polymers determines the stability of microfibers (the lifetime of a liquid cylinder) through the interaction of components at the interface. When the viscosity ratio between the fiber-forming and the matrix polymers is less than unity, the enhanced elasticity of the melt of the matrix polymer stabilizes liquid streams of a polymer of the disperse phase (i.e. the time of fracture grows). The reduced radius of the drops resulting from the fracture of microfibers is independent of the chemical nature of the analyzed polymers and amounts
23 to 2.2. This points to a general mechanism of fracture of liquid streams of one polymer in the matrix of another polymer. During the fracture of a polymer stream (fiber) the radius of the resultant drops exceeds the radius of the initial stream by more than twice; i.e. the total surface does not increase. The microfiber fracture can occur at temperatures equal to or exceeding melting points of fiber-forming and matrix polymers. The conclusions that we have obtained agree with the data from the literature [l&30,33,34]. Thus, the authors of [34] have obtained the relationship between the reduced lifetime of a liquid cylinder and the viscosity ratio K of the phases, which is represented by a curve with the minimum at K = 1. Torza and coauthors [lg] have established for a mixture of Newtonian fluids that the dependence of jl~~/u (the measure of the ratio of viscous and interphase surface forces) on K has a minimum in the range 0.3 G K G 0.9 upon the fracture of a liquid cylinder in a flow. They note that this range of values of K, corresponding to the easiest fracture of the drops, is in good agreement with the theoretical values of K = 0.632, at which value the energy scattering passes through a maximum inside the deformed drop [59]. The authors of [60] have investigated the fracture mechanism of drops in concentrated emulsions of a solution of one polymer in a solution of another. The resultant relationship between the degree of deformation of the drop and K is represented by a curve having a maximum at K = 1. The curve rd vs K has a minimum in the region K = 1. These results do not agree with Tomotika’s theory, according to which the smallest particle size must be observed at K = 0.28. The foregoing confirms the general character of the conclusion (which we have established in Section 3.2.2.) that ultrafine fibers in a melt of the matrix polymer are unstable to a maximum extent at K= 1. We should dwell on the fundamental investigation into the fracture of a deformed drop carried out by Han and Chin [30,33] for flow along a cylindrical tube, when two fluids possess viscoelastic properties. They determined the critical conditions for the fracture of drops in terms of the rheological properties of both phases, the initial drop size, and the rate of shear at the wall of the tube. The main statements of their work amount to the following. The fracture of a deformed drop always occurs after a liquid thread-like cylinder passes through the entrance zone of the cylindrical tube, i.e. in the region where the elongated drops begin to relax. Factors that determine the degree of deformation also determine the critical conditions for fracture. These parameters are the viscosity ratio K of the phases, Weber number We, and the elasticity of the phases which were estimated by the elasticity parameter 6 = /IjlE/q12 ( qE is the elongational velocity gradient; j3, is a
24 material constant). An increased viscosity of the medium stablizes the liquid cylinder, while an increased interphase tension destabilizes it. The rate of growth of disturbances increases with any of the three parameters: elasticity of the medium, elasticity of a drop, or interphase tension. In other words, if the disturbance is introduced into the system and there is instability, then two factors, namely the elasticity of both fluids and interphase tension, cause an increase in the rate of growth of the disturbance. However, the authors of [33] are correct to note that the results of the analysis of fracture depend on the selection of a rheological model. Han and Chin have employed a linear Maxwellian model. Employing other models can result in different conclusions. 4. Conclusion Fracture of liquid streams in binary polymer systems is one of the most interesting microrheological processes, which is realized in melts and solutions of polymer mixtures both at the stage of preparing polymer compositions and at the stage of their processing into finished goods. The lifetime of a liquid stream determines the possibility of realizing the specific fiber formation during the flow of polymer mixtures and the temperature of microfiber fracture, i.e. the conditions for their use. The known theories of the fracture of streams refer to Newtonian fluids, while fracture in viscoelastic media is different in many respects. When developing the corresponding theoretical models, it is necessary to take into account the basic peculiarities of the fracture of viscoelastic streams, as formulated in Section 3.2.6. References 1 A.L. Breen and W. Chester, U.S. Patent No. 3,382,305 (1968). 2 B.T. Hayes, Chem. Eng. Progr., 65 (1969) 50. 3 M.V. Tsebrenko, A.V. Yudin, M.Yu. Kuchinka, G.V. Vinogradov and K.A. Zubovich, Vysokomolek. Soedin., B 15 (1973) 566 (in Russian). 4 M.V. Tsebrenko, A.V. Yudin, T.I. Ablazova and G.V. Vinogradov, Polymer, 17 (1976) 831. 5 M.V. Tsebrenko, A.I. Benzar and A.V. Yudin, Khimicheskie volokna, 4 (1978) 48 (in Russian). 6 A.V. Yudin, M.V. Tsebrenko and M. Yakob, in: Preprints of International Symposium on Chemical Fibers, Vol. 3, Kahnin, 1974 (in Russian). 7 M.V. Tsebrenko, N.M. Rezanova and G.V. Vinogradov, Polymer Eng. Sci., 20 (1980) 1023. 8 K. Dietrich and H. Versaumer, Faserforsch. u. Textiltechn., 26 (1975) 347. 9 M.V. Tsebrenko, Khimicheskie volokna, 3 (1983) 28 (in Russian). 10 U.S. Patent No. 3,549,734 (1967).
25 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
M.V. Tsebrenko, Khimicheskie volokna, 5 (1980) 33 (in Russian). A.Ya. Malkin and G.V. Vinogradov, Vysokomolek. Soedin., 27A (1985) 227 (in Russian). G.I. Taylor, Proc. R. Sot. London, A 146 (1934) 501. H.L. Goldsmith and S.G. Mason, in: F.R. Eirich (Ed.), Rheology. Theory and Applications, Vol. 4, Acad. Press, New York, 1967. T. Mikami, R.G. Cox and S.G. Mason, Int. J. Multiphase Flow, 2 (1975) 113. F.D. Rumscheidt and S.G. Mason, J. Colloid Sci., 16 (1961) 210. F.D. Rumscheidt and S.G. Mason, J. Colloid Sci., 17 (1962) 260. S. Torza, R.G. Cox and S.G. Mason, J. Colloid Interface Sci., 38 (1972) 395. S. Tomotika, Proc. R. Sot. London, A150 (1935) 322. S. Tomotika, Proc. R. Sot. London, Al53 (1936) 302. A. Acrivos and T.S. Lo, J. Fluid. Mech., 86 (1978) 641. J.M. Rallison and A. Acrivos, J. Fluid Mech., 89 (1978) 91. H. Karam and J.C. Bellinger, Ind. Eng. Chem. Fundam., 7 (1968) 576. M. Gordon, J. Yerushalmi and R. Shinner, Trans. Sot. Rheol., 17 (1973) 303. G. Hestroni and S. Haber, Rheol. Acta, 9 (1970) 488. L. Rayleigh, Philos. Mag., 34 (1892) 145. C. Weber, Z. Angew. Math., 11 (1931) 136. C.D. Han, Multiphase Flow in Polymer Processing, Academic Press, New York, 1981. A.T. Serkov and P.V. Egorova, in: A.T. Serkov (Ed.), Theory of Formation of Chemical Fibers, Khiniya, Moscow, 1975. (in Russian). C.D. Han and K. Funatsu, J. Rheol., 22 (1978) 113. N. Hirai, J. Chem. Sot. Jpn., 75 (1954) 1019. S.P. Papkov, Khimicheskie volokna, 3 (1964) 34 (in Russian). H.B. Chin and C.D. Han, J. Rheology, 24 (1980) 1. Yu.P. Miroshnikov, M.K. Petrosyan and V.N. Kuleznev, Kolloidnyi Zh., 38 (1976) 279 (in Russian). Yu.P. Miroshnikov, M.L. Kaminskii and V.N. Kuleznev, Kolloid. Zh., 41 (1979) 1112 (in Russian). Yu.P. Miroshnikov, A.M. Gol’man and V.N. Kuleznev, Kolloid. Zh., 41 (1979) 1120 (in Russian). M. Goldin, J. Yerushalmi, R. Preffer and R. Shinnar, J. Fluid Mech., 38 (1969) 689. M.V. Tsebrenko and G.P. Danilova, Vysokomolek. Soedin., A26 (1984) 2562 (in Russian). J.J. Elmendorp, Polymer Eng. Sci., 26 (1986) 418. M.V. Tsebrenko and G.P. Danilova, Vysokomolek. Soedin., B29 (1987) 268 (in Russian). H. van Oene, J. Colloid Interface Sci., 40 (1972) 448. G.V. Vinogradov and A.Ya. Malkin, Rheology of Polymers, Mir Publishers, Moscow Springer Verlag, Berlin-Heidelberg-N.-Y., 1980. M.V. Tsebrenko, M.Yu. Kuchinka M. Yakob et al., Int. J. Polym. Mater., 3 (1974) 99. M.V. Tsebrenko, Vysokomolek. Soedin., B20 (1978) 600 (in Russian). M.V. Tsebrenko, Int. J. Polym. Mater., 10 (1983) 83. M.V. Tsebrenko, Khimicheskaya Tekhnologiya, 6 (1983) 3 (in Russian). T.I. Sizevich and M.V. Tsebrenko, Vysokomolek. Soedin., 28 (1986) 31 (in Russian). F.C. Stehling, T. Huff, C.S. Speed and G. Wissler, J. Appl. Polym. Sci., 26 (1981) 2693. T.I. Ablazova, M.V. Tsebrenko, A.V. Yudin and G.V. Vinogradov, J. Appl. Polym. Sci., 19 (1975) 1781. M.V. Tsebrenko, AI. Benzar, A.V. Yudin and G.V. Vinogradov, Vysokomolek. Soedin., A21 (1979) 830 (in Russian). M.V. Tsebrenko and N.N. Osadchaya, Izvestiya vuzov. Khimiy i khimicheskaya tekhnologiya, 28 (1985) 92 (in Russian).
26 52 M.V. Tsebrenlco, T.I. Ablazova, G.V. Vinogradov and A.V. Yudin, Vysokomolek. Soedin., Al8 (1976) 420 (in Russian). 53 D.C. Chappelear, Amer. Chem. Sot. Polym. Prepr., 5 (1964) 363. 54 S. Wu, J. Polym. Sci., C34 (1971) 19. 55 S. Wu, J. Macromol. Sci., Cl0 (1974) 1. 56 H.W. Kammer, Z. Phys. Chem., 261 (1980) 519. 57 Yu.S. Lipatov, A.E. Fainerman and O.V. Anokhin, Doklady AN SSSR, 231 (1976) 381 (in Russian). 58 I.M. Fedotkin, V.V. Bakhnova, F.G. GiIimyanov and V.I. Sivetskii, Khimicheskaya Tekhnologiya, 4 (1981) 40 (in Russian). 59 C.E. Chaffey and S.G. Mason, J. Colloid Interface Sci., 21 (1966) 254. 60 A. Silberberg and W. Kuhn, J. Polym. Sci., 13 (1954) 21.
Fluid Mechanics, 31 (1989) l-26
1
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
FRACTURE OF ULTRAFINE FIBERS IN THE FLOW OF MIXTURES OF NON-NEWTONIAN POLYMER MELTS
M.V. TSEBRENKO
and G.P. DANILOVA
Kiev Technological Institute of Light Industry, 2.52011 Kiev (U.S.S.R.)
A.YA. MALKIN Research Institute of Plastics, Perovskii pr. 35, 111 I I2 Moscow (U.S.S.R.)
(Received April 20, 1988)
Experimental and theoretical investigations into the fracture of liquid streams resulting from the coextrusion of mixtures of polymer melts are generalized. It is shown that the fracture of a liquid cylinder (fiber) for viscoelastic polymer systems differs from the known theoretical predictions concerning Newtonian fluids. This is expressed by the following facts:
(4 the wavelength of the destructive disturbance and the size of the resulting drops for polymer systems are greater than for Newtonian fluids; tb) the growth of amplitude of the destructive wave slows down at the final stages of fracture; (4 theoretical and experimental values of the lifetime of a polymer stream differ by 2-3 orders. These discrepancies between experiment and theory are due to non-Newtonian effects caused by the elasticity of polymer melts. Ultrafine fibers produced during the flow of mixtures of polymer melts through dies are maximally unstable if the viscosity ratio of the initial polymer melts is close to unity. The relation between the lifetime of a stream and the ratio of viscosities of both phases is determined by the matrix viscosity. The relative radius of the drops which results from microfiber fracture is independent of the chemical nature of mixed polymers and equals 2.2.
0377-0257/89/$03.50
0 1989 Elsevier Science Publishers B.V.
2 1. Introduction Utilization of polymer mixtures, in addition to many interesting applications, opens a new way to obtain fibers of small diameter (from several microns to fractions of a micron), i.e. ultrafine synthetic fibers or microfibers [l-4]. The essence of this phenomenon is that when mixtures of melts of incompatible polymers flow through dies, a disperse phase polymer (fiber-forming) produces in the mass of another (matrix) polymer a great number of fibers strictly oriented in the direction of extrusion. These fibers can be extracted from the matrix component by a solvent which is inert to the fiber-forming polymer. Here, unlike conventional methods of producing chemical fibers, a complex fiber from tens and hundreds of thousands of microfibers results from the extrusion of a mixture melt through one hole [3,5]. In this case we may speak about a fundamentally different process of fiber formation of one of the polymers of the mixture under the influence of rheological forces, when the number of filaments in a fiber is not determined by the number of holes in a die. The phenomenon just described has been called ‘specific fiber formation’ [6]. A typical example illustrating it is shown in Figs. 1, 2, which show an electron microphotograph of an extrudate transversal fracture and microphotographs
Fig. 1. Scanning electron microscopic photograph of an extrudate transversal fracture of the POM/CPA mixture (30 : 70).
Fig. 2. SEM pictures of (a) transverse and (b) longitudinal POM/CPA mixture.
sections of the extrudate of the
of a transversal and a longitudinal section of extrudate of poly(oxymethylene)-copolyamide mixture. A great number of microfibers of one polymer in the matrix of another polymer is clearly seen. The transverse section (Fig. 2a) exhibits a typical disperse structure. This very remarkable phenomenon, which is closely connected with the characteristics of the rheological properties of the coextruded fluids, has attracted the attention of many researchers. Thus, a mechanism of microfiber formation has been suggested [4], the influence of the conditions for deforming mixture melts on fiber formation has been studied [3,7-91, and features of properties and structure of ultrafine fibers have been estimated [5,10,11]. However, the fracture mechanism of the fibers formed from the mixture coextrusion is only poorly understood. Evidently, the problem here amounts to estimating the conditions for fracture of viscoelastic fluids. In spite of numerous works describing the fracture of liquid streams of Newtonian fluids, we have few ideas about viscoelastic systems and they are in the course of development [12]. In this paper the authors have tried to compile a critique of the experimental data, both their own and from the literature, concerning the fracture of ultrafine polymer streams produced in the matrix of another polymer, and to compare the data with well-known theoretical predictions. To this
4 end, the authors first present a brief review of the known theoretical concepts concerning Newtonian fluids and then develop the specific behaviour of non-Newtonian media. 2. Deformation of drops and fracture of liquid streams in the mixtures of two Newtonian fluids 2.1. Fracture of a drop Under conditions of dispersion flow of liquid drops, the stresses arising in the dispersion medium tend to deform and orient a drop. According to Taylor’s classical theory [13], the force responsible for the deformation is the difference between the pressures inside and outside the drop API’. For Couette flow, the pressure difference is expressed as sin 2+,
APp = 4917
(1)
where y is the rate of shear; n2 is the viscosity of the dispersion medium; K = ql/q2 is the ratio of the viscosities of the melt of the drop phase vi and the medium q2; and 9 is the angle determining the orientation of the drop in the direction of the flow. For low rates of shear, the drop assumes the shape of an ellipsoid; with high p the drop turns into a liquid cylinder (stream). In addition to the force APT, which tends to deform the drop, the force created by the interphase tension AP,, also acts on the drop. This force is determined by the Laplace equation:
where u is the interphase tension and r is the radius of an undeformed drop. Under equilibrium conditions, the deformation D of the drop is given by D=
19K+16. 16K+16
we ’
where We is the Weber number. Thus, according to Taylor’s theory, the deformation of the drop in another fluid medium, depends on two parameters: the viscosity ratio K = q1/q2 of the drop and the medium and the Weber Number We, i.e. the ratio of the product of the local shear stress and the drop radius to the interphase tension: We = q2-+r/a
(4)
Taylor [13] supposed that the drop fractures when the surface tension,
5 which seeks to keep the drop in spherical form, can no longer be balanced by viscous forces, which attempt to fragment the drop. Therefore, the parameters K and We that control the deformation must also control the critical conditions for fracture. According to Taylor, the deformation E, of the drop during fracture (critical value of deformation) equals 0.5 and is calculated by the formula E,=(L-B)/(L+B),
(5)
where L and B are the major and minor axes of the deformed drop, respectively. A great number of theoretical and experimental examinations have been carried out on the fracture of a liquid cylinder in steady uniaxial or shear flow, in transient conditions, and in a resting fluid after the cessation of flow [14-271. Practically all these works consider drop emulsions of a Newtonian fluid in another Newtonian fluid. Han [28] has generalized the experimental data of some authors on the fracture of Newtonian drops located in another Newtonian fluid which was exposed to an homogeneous shear flow. According to his conclusions, in all cases, the dependence of critical deformation E, on the ratio between the phase viscosities K passes through a broad minimum lying in the range of values of K 2: 0.1-1.0. Different authors give different data on the critical value of E, in the minimum region, though there are results close to the theoretical limit EC = 0.5 [18,23,28]. Beyond the limits of the given range of values of K, the value of the viscous forces necessary for the drop’s fracture grows very quickly and almost unboundedly. Thus, for very low or very high values of K, the deformed drop is stable in an homogeneous shear flow. Similar results have been obtained in a number of works [16,18,23]. For example, Rumscheidt and Mason [16] point out that the drops of a Newtonian fluid were deformed and fractured in Couette flow in the range K = 0.01-2. For larger values of K the drops were not fractured. In this case, as i, increases, the limiting deformation of ellipsoidal drops is achieved and their major axes are oriented along the direction of the flow. Torza, Cox, and Mason [18] have shown that the drops of a Newtonian fluid were not fractured for K> 3. Karam and Bellinger [23] observed the fracture of the deformed drops in the range of the viscosity ratio 0.005 G K G 3.0. The authors of [16] describe four types of drop fracture of Newtonian fluids according to the value of K. If the disperse phase is low-viscosity (K-c 0.2), then at a critical value of 9 the drop becomes S-shaped and tiny drops break away from its ends. As the viscosity of the drop increases (K > 0.2), the deformed drop breaks through the centre with the formation of two large drops and three satellite drops. For K < 2 the drop extends into a long cylinder and eventually falls into many small drops. For K > 2 the
6 drop becomes deformed and is oriented along the flow without fracture even at 9 up to 40 s-t. 2.2. Fracture of a cylinder A liquid cylinder (stream) is thermodynamically unstable due to an unfavourable ratio of the surface to the volume. The authors of [29] note that when the ratio of the length I of a liquid cylinder to its diameter d, reaches 4.5, the lateral surface of the cylinder becomes equal to the surface of two spheres with the total volume being equal to the volume of the initial cylinder. Further increase of the cylinder length is unfavourable from an energetic point of view and it splits into drops under the action of surface tension. Thus, a liquid stream with Z/d, > 4.5 is a nonequilibrium system and its fracture is determined by the local conditions of the flow, and the rheological and interfacial properties of the mixed fluids. The authors of [18,30] have discovered that the fracture of the deformed drops is determined not only by the ratio of the phase viscosities K, but also by the rate of growth of i, with time t, i.e. by the quantity dy/dt. Moreover, in many papers it has been noted that a liquid cylinder becomes varicose before fracture (Fig. 3): periodics swellings and &innings appear on it. Both of these factors (dependence of fracture conditions on dy/dt and stream varicosity) give grounds to believe that a drop fracture is determined by hydrodynamic disturbances. Varicosity (capillary waves) may appear for various reasons such as fluctuations of density and viscosity and vibration of the equipment [29]. Therefore, the fracture of a liquid stream is, by its nature, a transient phenomenon. Any attempt to develop the theory of this phenomenon requires the solution of balance equations for an usteady flow. On the basis of the classical works of Rayleigh and Weber, Tomotika [19,20] has analyzed the hydrodynamic stability of a liquid stream and explained Taylor’s experimental observations. Beginning with Rayleigh’s works [26], the fracture of a liquid cylinder is analyzed for a fluid at rest. According to the existing concepts, the appearance of the wave disturbances on the
u-
-
-t
t
-u__
Fig. 3. Liquid cylinder exposed to the action of a capillary wave.
7 surface of a liquid cylinder is responsible for its fracture (Fig. 3). The fracture is caused by the disturbance whose amplitude grows with the maximum rate. The stream is destroyed when the disturbance amplitude becomes equal to the stream radius. This is attained in time t,, called the lifetime of a liquid cylinder, or the fracture time. A simple empirical expression [31,32] is suggested for an approximate estimate of the lifetime of a stream: tb =
qlR/a,
(6)
where R is the radius of the initial liquid cylinder. Varicosity enlarges the surface of the cylinder and, consequently, increases its surface energy. Therefore, the fracture is caused by the attempt of the system to reduce its free surface energy. The radius rd of the resulting drops is related to the wavelength h, of the destructive disturbance by the ratio: r, = (3RZX,/4)1’3. According to Tomotika’s theory [19,20], the relationship between the wavenumber of the disturbance leading to fracture and log K is represented by a curve having a maximum in the region K = 0.28 (see below). Such a relationship points to the fact that a definite wavelength is typical for each value of K = const, and when this wave propagates, a liquid cylinder becomes unstable to the maximum extent. It is also evident that the cylinder’s stability sharply increases both for very low and very high values of K. Since the size of the resulting drops is determined by the destructive wavelength, the dependence of r, on log K is represented by a curve with a minimum in the region K = 0.28. Fracture of a liquid cylinder in a flow is characterized by a number of features and this process is undoubtedly more complicated than that in a liquid at rest. These features are as follows. The stability of a liquid cylinder in a flow sharply increases [19,20]. This is associated with an irregular increase in the amplitude of the destructive wave and, as a result, the cylinder in the flow breaks down nonsimultaneously along the entire length, and the resulting drops have different sizes [15]. Therefore, concerning the fracture of liquid cylinders in a flow, Tomotika’s theory is confirmed only qualitatively. For example, the dependence of the size of the drops resulting from the fracture of liquid cylinders on log K is represented by a curve having a minimum, but the coordinates of the minimum do not correspond to those given by the theory [15]. In the process of polymer flow, there is often an elongational deformation. Han [28] has compared the critical value of the deformation for which the deformed drop fractures in shear and uniaxial flows. He has shown that, in a uniaxial field, the drops fracture more effectively and with a lower
8
consumption of energy. This is expressed in the fact that for shear, the EC versus log K curve passes through a minimum at K = 0.1-1.0, for elongation at K z: 1-5; the minimum value of EC for shear is = 0.6 and for elongation it is only 0.2. If in a shear flow the ratio of the phase viscosities is larger than 3.5, the drops can no longer be fractured, while a uniaxial flow causes the fracture of liquid cylinders even for larger values of K. 3. Fracture of viscoelastic liquid streams 3.1. Stresses in Poiseuille jlow for a viscoelastic fluid The foregoing theoretical works discuss Newtonian drops or cylinders dispersed in a Newtonian fluid. If one or both fluids possess viscoelastic properties, the fracture mechanism for liquid streams may be different. It seems, from inspection of the few works devoted to this problem [24,30-401, that at present the influence of polymer elasticity in a mixture on stream fracture is only poorly understood. The shape of drops in the flow of a mixture of two viscoelastic fluids is determined by shear stresses and pressure distribution around a drop. The stress tensor T in Poiseuille flow for a viscoelastic fluid is written as [41,42] T,, = -P, - ( u2 - ul) T,,=
J0
2,
TOO= -P,-a,--
,
(8)
r
d
‘a, J0
J,
T, = +fr, P, = P, +
r
d
ral
-P,-
*ul dr /0
r
(10) (11)
ru, d /0
--f(o,-2u,), r
02)
where z is the direction of flow; r is the direction of the velocity gradient; 8 is the direction perpendicular to the (r, z)-plane; f is the partial derivative in the direction of flow; p, is the isotropic pressure, which, as is seen from (12) depends on radial position in a capillary tube; and ui, a, are the normal stresses, which are given as L-
&=a,,
T,,-
T,,=u,.
(13)
From the given eqns. (8)-(12) it is clear that the general solutions that determine the conditions for fracture are scarcely possible due to the wide
9 range of opportunities of varying rheological equations of state for both fluids in a mixture. Therefore, at present we can only point out the peculiarities of the stressed state of viscoelastic drops and reveal these peculiarities for typical viscoelastic media in an experiment. 3.2. Experimental
data
The authors of [7,9,43-471 were the first to carry out extensive systematic quantitative and qualitative investigations into the phenomenon of ‘a specific fiber formation’ according to the nature and rheological properties of mixed polymers, the mixture composition, and the degree of dispersion of a polymer of the disperse phase. According to the results obtained, the first condition for achieving fiber formation during the flow of a mixture melt is the dispersion morphology type of the initial mixture. To meet this condition, polymers must be thermodynamically incompatible. The disperse phase is usually formed by the polymer with the higher density of cohesion energy or greater surface tension. The first attempt to substantiate and predict the morphology types of the two-phase polymer system was Van Oene’s theory [41], according to which a more elastic phase produces drops in a less elastic phase. For each pair of polymers, there exists an optimum relationship between the components and an optimum degree of dispersion such that fiber formation is the most clearly expressed [43-451. The second condition for achieving fiber formation is the deformation of drops into liquid streams during the flow of mixture melts. From this point of view the degree of component interaction in the interphase layer (thermodynamic affinity) must be sufficient for the transfer of stresses arising in the dispersion medium, through the interface to the polymer drops of a disperse phase. Otherwise, with all the other conditions being favourable, fiber formation will not be achieved. We can indirectly estimate the supposed level of interaction of the mixed polymers according to the values of their specific surface energy, solubility parameters and Flory-Huggins interaction parameters I/Q. To ensure phase separation and, at the same time, sufficient adhesion at the interface, the difference between the solubility parameters for the polymers must be near 0.3 [48]. Since the shear stress is continuous at the interface [13], it follows that for 9, = r/z then th e polymers of a disperse phase and dispersion medium deform to the same degree, which promotes the fiber formation of one polymer in the mass of the other [7,45]. The deviation of the ratio n1/n2 from unity causes a different degree of deformation of the mixture components and a worsening of fiber formation. A sharp difference in elasticity of the melt components acts in the same way. Viscoelastic drops are more resistant to stretching and shear than Newtonian drops.
10
The authors of [7,9,43-471 have established the relationship between the swelling ratio B of extrudates of polymer mixtures and ‘specific fiber formation’: the more fibers are formed and the thinner they are, the higher B is. Thus, swelling, being an indirect characteristic of the rubbery elasticity of the mixture melt, also correlates with ‘a specific fiber formation’. Taking into account the fact that the variables which control the degree of drop deformation (K and We, for Newtonian systems) also determine the mechanism of its fracture, we analyzed the fracture of liquid viscoelastic polymer streams according to the viscosity ratio, K, of the melts of mixed components, the chemical nature of the polymers, and the temperature [38,40]. 3.2.1. Experimental The investigation was concerned with the following mixtures: poly(oxymethylene) (POM) with copolymer of ethylene and vinyl acetate (CEVA); POM with spirit-soluble copolyamide (CPA); POM with polystyrene (PS); low-pressure polyethylene (LPE) with CPA; CPA with copolymer of ethylene and butylene (CEB); POM/CEB; CEVA/CPA; POM with polycapramide (PCA); POM with poly(viny1 alcohol) (PVA). Polymers and methods of mixture preparation are described in [7, 9,49-511 and Tables 1-3. The components of the mixture are in the ratio 20 : 80. With the help of a constant-pressure capillary viscometer we have obtained extrudates of mixtures under conditions such that the fiber formation is most clearly defined. For a quantitative estimate of fiber formation a procedure was developed [7] which made it possible to measure and evaluate all the types of structure found in the extrudate of a polymer mixture and to determine their number and mass fraction. To study the fracture of microfibers, longitudinal sections of the extrudates of polymer mixtures (thickness of the section is lo-20 pm) were placed in an immersion medium on the heating stage of a microscope at a definite temperature, depending on the nature of the polymers, or at room temperature, followed by an increase in temperature. Different stages of fiber fracture of polymer of the disperse phase were photographed. As described in [38] (Table l), we treated the microscope data in terms of the variables of Tomotika’s theory [18,19]. In this case, we determined the wavelength X m and the wave number 27rR/X, of the destructive disturbance, the radius R of the initial liquid cylinder (fiber) in terms of eqn. (7) the radius r, of the drops resulting from the fracture and its reduced value rd/R, the lifetime t, of a liquid cylinder and its reduced value tb/R. The initial experimental data were treated by the methods of mathematical statistics. The error in determining the values of r, and h, was + 3.5% with the confidence level 0.95. The elasticity of polymer mixture B and of a
TABLE 1 Parameters No.
i. 2. 3. 4. 5.
of the process
Viscosity of melt, Pass
of fracture
of ultrafine
Swelling of extrudate
poly(oxymethylene) K= n,/r~t
CEVA
POM
CEVA
POM
790 790 790 790 790
280 720 830 1320 3270
1.4 1.4 1.4 1.4 1.4
1.3 1.3 1.2 1.2 1.3
0.35 0.91 1.05 1.67 4.14
fibers in the extrudates
of the POM/CEVA
*& pm
A,,,, pm
R, pm
*d/R
2.5 3.0 3.3 3.4 2.8
16.2 17.7 18.6 21.6 22.1
1.16 1.46 1.60 1.54 1.13
2.19 2.09 2.06 2.20 2.40
211R AIn 0.45 0.52 0.54 0.44 0.32
mixtures
at 165 o C
tb,
tb/R>
S
S/w
15 75 120 300 450
13.2 52.8 75.0 195.0 400.2
12 disperse phase B, and a matrix B, was estimated from the size of swelling of the extrudates after annealing by the method given in [52]. 3.2.2. The role of rheological properties of mixed polymers Analysis of the role of rheological factors in the fracture of polymer streams is quite complicated because, when a matrix and disperse phase vary, different factors such as viscosity, elasticity, and polymer compatibility usually change simultaneously. Therefore, while for Newtonian fluids the deciding factors are determined quantitatively (parameters K and We), for polymer systems, unfortunately, it is necessary to use semi quantitative or qualitative estimates. Therefore a correct selection of the parameter being varied plays a decisive role in the uniqueness of the conclusion. Taking POM/CEVA mixtures as an example, we have studied the influence of the viscosity ratio K = nl/qZ of mixed polymers on the fracture mechanism of POM microfibers in a CEVA matrix. The viscosity ratio K was varied in the range 0.20-10.78. Here the samples employed of POM and CEVA differed very little in melt elasticity (with a range of B from 1.2-1.5) [7]. In one series of experiments we mixed POM of different molecular weights with one sample of CEVA (variant 1); in the second series we mixed one sample of POM with the CEVA samples with different melt viscosity (variant 2). In the first variant ncEvA = const, in the second nr,oM = const. From the results obtained it follows that the external pattern of the initial stages of destabilization and fracture of a liquid cylinder for viscoelastic polymer systems does not differ qualitatively from the mechanism described for low molecular weight systems, i.e. the fracture mechanism of a viscoelastic cylinder also consists in the formation of wave disturbances on its surface. Therefore it is of fundamental importance to consider critical conditions for fracture in the same terms as for Newtonian fluids. This will enable us to reveal immediately whether the fracture really occurs according to the same mechanism. Our experiments indicate that, at the POM melting point, microfibers at first become varicose and then (after the lifetime tb) fracture into a chain of drops (Fig. 4). The curves 2xR/X, vs. log K (Fig. 5, curves 2) obtained from the experimental data are situated below Tomotika’s theoretical curve. This means that the disturbance wavelength (and the size of the resulting drops) is greater for viscoelastic fluids than for Newtonian fluids. The same conclusions are drawn in [34-361, in which the authors analyzed the fracture of a viscoelastic liquid stream in another viscoelastic fluid, taking as an example mixtures of polystyrene and poly(dimethylvinyl)siloxane elastomer in a common solvent [34], poly(methyl)siloxane oligomers of different MW in a matrix of polybutadiene oligomer [35], a melt of the mixture of poly(methylmethacrylate) and polystyrene.
13
Fig. 4. SEM pictures of longitudinal at (a) 150°C and (b) 165“C.
sections
of the extrudates
of the POM/CEVA
mixtures
From Fig. 5 it follows that the experimental curves have a maximum in the range of values of K = 1. We think that this result is of fundamentally importance, since it implies that a mixture of two polymers, one of which is in the form of fibers, is maximally unstable when fiber formation of a polymer in the mixture with another polymer is most clearly defined, and this, according to our data, occurs for K = 1 [7]. The established mechanism is in agreement with the conclusions of Chin and Han [33] that the variables which control a drop’s deformation also determine the conditions for fracture of a liquid cylinder. A maximum instability of fibers in the extrudates of mixtures with K = 1 is supported by the fact that, for these mixtures, swelling of the extrudate is a maximum [7].
14
-4
m Al
-2
0
2
-4
-2
0
qK
on the viscosity ratio between the Fig. 5. Dependence of the wavenumber x = 2qR/X, phases of a drop and a dispersion medium: 1 - Tomotika’s theoretical curve, 2 - experimental data for the melt of the POM/CEVA mixture for (a) qIIcEvA = const and (b) vpoM = const.
The relationships between the wavenumber x = 2?rR/h, and the viscosity ratio of the components have a different nature depending on the method by which the change in K is achieved, i.e. as a result of a variation of the viscosity of the melt of the phase or the dispersion medium (see Fig. 5). For variant 1 (Fig. 5a), the relationship differs significant from Tomotika’s curve. With increasing MW of POM not only does the viscosity grow, but the melt elasticity of POM also changes. The fact that curve 2 passes through a maximum indicates that the elasticity of a polymer of a disperse phase has a different effect according to the ratio of the viscosities of the phases. The lifetime of a liquid cylinder found experimentally (Fig. 6) for variant 1 increases with the melt viscosity of POM (which causes the growth of K), as required by eqn. (6). This means that if a drop phase is very viscous, the drop can be elongated to a low-radius cylinder, which will remain stable for a long time. For variant 2, the experimental curve x vs. log K (Fig. 5b) is asymmetric, its left brach almost coincides with the left brach of curve 2 for variant 1. However, when the viscosity of the matrix component (CEVA) becomes lower than the viscosity of POM (K > l), the experimental points are located very close to Tomotika’s curve, but are somewhat lower. Comparing two curves 2 in Fig. 5a and b, enables us to stress the importance of elasticity of the dispersion medium for the fracture of a liquid stream. While the elasticity of the medium is large, the fracture mechanism differs greatly from the theoretical data, irrespective of the way in which K changes. As the elasticity of CEVA decreases (low-viscosity samples), the fracture of fibers in a CEVA matrix actually occurs according to Tomotika’s mechanism. The curve of the lifetime of a liquid cylinder for variant 2 (Fig.
15
-0,4
0
0.4
0.8
12
Fig. 6. Dependence of the reduced lifetime of the liquid cylinder on the viscosity ratio of the phases for (1) qCEVA= const and (2) qpoM = const.
6, curve 2) is the inverse of the curve for variant 1 (Fig. 6, curve I): with increasing K, t, decreases. This indicates that the way in which either value of K is achieved is important. The absolute value of the viscosity of the medium is capable of qualitatively affecting the stability of the stream, all other factors being equal. The authors of [35,36] have also obtained a different relationship between the lifetime of a liquid cylinder and K, depending on the method by which the change in the viscosity ratio of the polymers was attained. Thus, viscoelastic properties of the medium affect the stability of liquid streams more than viscoelastic properties of the disperse phase (drops or cylinders being deformed). 3.2.3. Features of fracture kinetics of viscoelastic liquid streams According to the classical Rayleigl-Weber-Tomotika theory [19,20,26,27], the disturbance amplitude a (Fig. 3) during stream fracture grows exponentially with time t
a = a0 exp (4th
(14)
where a, is the initial disturbance amplitude; which is a complex function of the wavenumber ratio of the phases K. q =
&F(-K
K),
q is the instability factor, x = 27rR/X, and viscosity
(15)
-1 _
-2
-3 _
-41
r 40
80
120
I60
200 t.S
Fig. 7. Variation of the disturbance amplitude microfibers in the CEVA matrix at 168 o C.
In logarithmic straight line
coordinates
In a = In a, + qt,
of the destructive
wave with time for the POM
the a vs. t relationship
is represented
by a
(16)
the slope of this line to the X-axis is the instability factor q. The analysis of the fracture kinetics of the POM microfibers in the CEVA matrix at 168” C has shown that the experimental time dependence of the disturbance amplitude a deviates from a straight line at the final stages of fracture (Fig. 7). The degree of deviation grows with an increase in the elasticity of the polymer of the disperse phase. The reduction of growth of the destructive wave amplitude for the polymer of the disperse phase which possesses viscoelasticity in a melt was observed by the authors of [35]. The features of the fracture kinetics of viscoelastic liquid streams described there are caused by the effect of the elasticity of polymers, which will be discussed in detail in Section 3.2.4. Equation (15) establishes the relationship between the instabilitiy factor of the stream and interfacial tension u - the most important macroscopic characteristic for a mixture of polymers. The surface energy, structure, and length of interphase layer can be judged from the value of u. On the basis of this, Chappelear [53] has suggested a method for measuring the interfacial tension in a mixture of two polymers, one of which forms liquid streams (fibers) in another polymer. Experimental measurement of CTin polymer mixtures (especially in melts) is extremely complicated. Therefore, application of the theory of destabilization of a liquid cylinder for determining u is of particular interest. On the basis of data on the investigation of the fracture kinetics of
17 TABLE 2 Results of determining interfacial the POM/CEVA mixture No. of system ’
1. 2. 3. 4.
Viscosity of initial components, Pa. s POM,
CEVA,
rll
v2
280 720 1320 3270
790 790 790 790
r 1, 2, 4 - annealing
at 180°C,
tension
according
K=
2R,
L,,
7h/7~2
pm
v
0.35 0.91 1.67 4.14
6.51 4.57 6.17 7.93
35.36 30.58 23.48 46.82
3 - annealing
to fracture
4
kinetics
F(x,K)
of liquid streams
2aR xl?l
0.0140 0.0243 0.0196 0.0159
0.125 0.067 0.053 0.035
0.58 0.47 0.56 0.53
in
D x103, N/m
0.58 1.30 1.90 2.84
at 168OC.
microfibers in the polymer mixtures analyzed, we have calculated the values of interfacial tension (Table 2). The values of u obtained in Table 2 are low, which is typical for polymer mixtures [53-561 and is explained by the presence of a transition layer at the interface of the components. However, in determining u by the method of destabilization of a liquid cylinder, it is necessary to remember that the theory used takes no account of the effect of elasticity, which manifests itself in the fact that the experimental time dependence of the disturbance amplitude deviates from a straight line in the final stages. 3.2.4. The role of the chemical nature of polymers in mixtures In addition to the viscoelastic properties of polymers in mixtures, the degree of polymer interaction at the interface, has quite an important effect on the fracture mechanism of liquid streams. Let us consider the results of investigation into the fracture mechanism of ultrafine fibers in extrudates of mixtures of polymers which are different in chemical nature (Table 3). Pairs of polymers were chosen in such a way that K was close to unity, this being a necessary condition for the preferred formation of microfibers by a polymer of the disperse phase. The fracture of microfibers in sections of extrudates was analyzed at 165OC, except for mixture No. 6, for which the experimental (annealing) temperature was 175°C. The analysis of the experimental data (Table 3) enables us to draw the following conclusions. The stability of fibers of each fiber-forming polymer depends on the nature of the matrix polymer. Thus, the lifetime of the POM fibers in the CEVA matrix is 2-4 times longer than in CPA (Table 3, mixtures no l-5). Evidently, this is due to the different character of the interphase interaction of POM with CPA and CEVA. Indeed, the formation of hydrogen bonds between POM and CPA in a mixture is confirmed by the method of reverse phase gas chromatography [57]. Therefore, the stresses appearing in the
TABLE 3 The effect of the nature of mixed polymers on the parameters of fracture of ultrafine fibers in the extrudates of mixtures No.
Mixture
Fiber-forming polymer
Matrix polymer
7)1*
927
4
Pa.s 1.
2. 3. 4. 5. 6. 7. 8. 9. 10.
POM/CPA POM/CPA POM/CEVA POM/CPA POM/CEVA LPE/CPA’ LPE/CPA POM/PS CPA/CEB CEB/CPA
’ annealing at 175OC.
830 1090 830 1320 1320 1400 1160 650 730 1010
K= vt/nz
rd, pm
A,,,, pm
R, pm
‘d
R
2nR X
m
tb,
tdR1
S
S/pm
eqn.
J32
S
Pa.s 1.2 1.3 1.2 1.2 1.2 2.4 2.4 1.2 1.4 1.8
960 960 790 960 790 960 975 670 1010 730
1.0 1.0 1.4 1.0 1.4 1.0 1.4 2.1 1.8 1.4
tb according to
0.86 1.13 1.05 1.37 1.67 1.46 1.19 0.97 0.72 1.38
2.2 2.5 3.3 2.2 3.4 5.8 6.1 1.6 3.7 5.0
12.8 15.4 18.6 14.7 21.4 38.0 42.0 8.9 23.2 32.1
1.0 1.2 1.6 1.0 1.5 2.6 2.7 0.8 1.7 2.3
2.2 2.1 2.1 2.2 2.3 2.2 2.3 2.0 2.2 2.2
0.49 0.49 0.54 0.43 0.44 0.43 0.41 0.54 0.46 0.45
20 45 120 90 300 672 960 240 270 90
20 38 75 90 200 259 356 300 159 39
0.17 0.26 0.27 0.26 0.40 0.73 0.63 0.10 0.25 0.46
(6),
19 matrix during the flow of the POM/CPA mixture are transmitted to a drop, and, as a result, fine but stressed fibers are formed. Besides this, it is known [58] that application of CEVA as a matrix polymer (or as a third component) increases the homogeneity and kinetic stability of a dispersion, and improves the compatibility of polymers. It is precisely such properties of CEVA that ensure the greater stability of the POM microfibers in it. The sample of PS employed judging from the size of swelling of the extrudate, possesses high melt elasticity. The data from Table 3 show that the POM microfibers in the PS matrix are characterized by long lifetimes ( tb/R = 300 s/pm), exceeding the given lifetime for the POM microfibers in the CPA medium by more than an order of magnitude. A detailed quantitative analysis of the microstructure of extrudates of the POM/PS mixtures indicates that short fibers and particles are not formed during the flow of this mixture. This means that the elasticity typical of the PS melt stabilizes the resulting POM fibers. The established regularity agrees with the general conclusions of reference [30], which shows that, for 77i/n2 < 1, the elasticity of the medium stabilizes the deformed drop, whereas, for high n/q,, it destabilizes the drop. In our case ~i/r/~= 0.97 < 1, which leads to stabilization of the stream. As regards the degree of compatibility of the polymers given in Table 3, LPE and CPA have the lowest affinity for one another [9], which determines a small thickness of the interphase layer and a weak interaction of the components in it and, as a result, a smaller degree of deformation of the fiber-forming polymer as compared to the matrix polymer and lower tension in the resulting microfibers. In view of the foregoing, the microfibers from the melt of the LPE/CPA mixture are characterized by the largest lifetimes. is 356 s/ym for the PE Thus, for close values of vi, qZ, and K, t,/R microfibers and it is 200 and 90 s/pm for the POM microfibers in the CEVA and CPA matrices, respectively (Table 3, mixture Nos. 4, 5, 7). The samples of LPE and CEB employed, in comparison with POM, are characterized by high elasticity (Table 3). The results obtained point to the fact that the maximum values of R, r,, and X, are precisely typical of these polymers (Table 3). Thus, the increased elasticity of the polymer of the disperse phase does not assist in fiber formation. From eqn. (6) it follows that, for a given pair of polymers, the lifetime increases with the viscosity of the polymer melt of the disperse phase (Table 3, mixtures Nos 1, 2, 4, and Nos 3, 5). Analyzing the results of Table 3, we see that if we vary the chemical nature of the polymers in a mixture and the viscoelastic properties of their melts, the values of rd/R are a constant quantity and amount to 2.2 on average. This points to a common fracture mechanism of liquid polymer streams in the matrix of another polymer.
20 Theoretical values of the lifetime of a liquid cylinder calculated from eqn. (6) for dilute solutions of polymers, characterized by low viscosity, yield a quantity equal to tenths of a microsecond. For such values of t, the existence of a liquid cylinder could not be observed experimentally, and since the formation of liquid polymer streams is registered experimentally without any difficulty, we can draw a conclusion on the decisive influence of non-Newtonian effects. Obviously, the initial stages of the generation of disturbances are similar in all cases and they scarcely differ for polymer streams and for streams of Newtonian fluids. But polymer fluids are well known for their ability to accumulate large rubbery deformations upon stretching [12]. These large rubbery deformations, accumulated during the formation of a liquid cylinder, cause the macromolecules to become oriented. As a result, the junctions generated between the drops do not tear apart but, on the contrary, they stabilize the cylinder being stretched, and the stream acquires a typical bead structure. The structure of the stream in the form of a string of beads at the final stages of fracture of viscoelastic streams has been described in [24,37], where it was also noted that in the stream of a Newtonian fluid the links between the drops do not arise, regardless of the type of fracture. By virtue of such features of the fracture of viscoelastic liquid streams at the final stages, there is a deviation of the disturbance amplitude versus time curve from a straight line (Fig. 7), as Tomotika’s classical theory requires. For the mixtures of the polymers analyzed (Table 3), the values of t,, calculated from eqn. (6), range from 0.1 to 0.73 s. In such a case, at the outlet of the forming hole, the microfibers would fracture earlier than the extrudate was cooled to the required degree which makes it possible to fix the resulting fibers by the growth of viscosity. However, the experiment shows (see the comparison between the experimental and theoretical values of tb in Table 3) that microfibers are conserved even during free flow of the stream, when axial deformation, which prevents fracture, is very small. The values of t, found experimentally for ultrafine fibers of POM, PE, CPA, CEB are 2-3 orders greater than the t, values found theoretically. Such a sharp disagreement between theory and experiment can be explained only by the effect of the rubbery elasticity of polymer melts. Van Oene [41] believes that one of the reasons for the stability of fibers (streams) of a polymer of the disperse phase flowing along a capillary is the fact that when fibers fracture into n drops of the same radius as the fiber, the total surface area increases by a factor of 3n (1 + 212). We have shown experimentally (Tables 1 and 3) that the radius of drops resulting from the fracture of a stream exceeds the radius of the initial fiber by more than twice, i.e. the total surface does not increase. Thus, the explanation of fiber stability suggested by Van Oene is not supported by the experimental results.
21
3.2.5. The effect of temperature on microfiber stability We have used polymers that are characterized by different melting points ranging from 50 to 230 o C (Table 4). Therefore, the analysis of the action of temperature on fiber stability in extrudates of polymer mixtures of different chemical nature is of some interest. The analysis of the results shows (Table 4) that fracture becomes possible only for T 2 T,“, where TL is the melting
TABLE 4 The action of temperature on microfiber stability in the extrudates of polymer mixtures Extrudate of mixture
Fiberforming polymer
Melting point of fiberforming “C
Melting point of matrix polymer, “C
Temperature of experiment, OC
PresenceJ + ) or absence( - ) of fracture of the main part of microfibers
POM/CPA 548
POM
165
164
150
-
165 170 150 160
+ + _ -
170 140
+ -
170 170 180 200 210
+ _ -
220 170 180 190 215
+ + + -
50
POM/CEVA
POM
165
POM/PS
POM
165
T,=loo°C
POM/PCA
POM
165
215
POM/PVA
POM
165
230
PE
PE/CPA 548
CEVA/CPA
CEVA/CPA
54
CEVA
CPA 54
132
164
225 230 130 150 160
50
181
170 165 170
+ _
180 185 165 170
+ + -
180 185
+ +
181
50
22 point of a fiber-forming polymer. However, this condition is necessary but not sufficient. Thus, for the mixtures of POM/PVA, PE/CPA, POM/PCA, and CEVA/CPA, when the melting point of a polymer of the disperse phase is achieved (POM, PE, CEVA), fracture of the fibers does not occur, it is retained by the unmelted matrix. The formation of drops (from fibers) is observed for T 2 T/ of the matrix. Therefore, for example, the POM microfibers in the mixtures with CPA, CEVA, and PS are stable at temperatures below 165 o C and in the mixture with PVA up to 220 o C. Thus, the fracture of microfibers can take place only if the melting point of fiber-forming and matrix polymers is attained. 3.2.6. Some conclusions from the experimental investigations In a similar manner to Newtonian fluids, the fracture mechanism for a viscoelastic liquid cylinder also consists of the formation of wave disturbances on its surface. Parameters controlling a drop deformation also determine the critical conditions for fracture of the resulting liquid cylinder. The viscosity ratio of the melts of mixed polymers and the Weber number are such parameters. However, we observe only qualitative agreement with the theory developed for Newtonian fluids. This is expressed by the following facts: the disturbance wavelength and the size of the resulting drops are greater for polymer systems than for Newtonian liquids; the growth of amplitude of the destructive wave slows down at the final stages of fracture; theoretically calculated and experimentally found values of the lifetime of a polymer stream differ by 2-3 orders of magnitude; for one and the same value of K the viscosity of the medium determines the lifetime of a polymer stream, other things being equal. The established disagreement between experiment and theory is due to the influence of non-Newtonian effects caused by the elasticity of polymer melts. A mixture of two polymers, one of which is in the form of fibers in the matrix of another polymer, is maximally unstable if the viscosity ratio of the polymer melts is close to unity. During the flow of such a mixture the fiber formation of one polymer in the mass of another polymer is most clearly expressed. The chemical nature of the mixed polymers determines the stability of microfibers (the lifetime of a liquid cylinder) through the interaction of components at the interface. When the viscosity ratio between the fiber-forming and the matrix polymers is less than unity, the enhanced elasticity of the melt of the matrix polymer stabilizes liquid streams of a polymer of the disperse phase (i.e. the time of fracture grows). The reduced radius of the drops resulting from the fracture of microfibers is independent of the chemical nature of the analyzed polymers and amounts
23 to 2.2. This points to a general mechanism of fracture of liquid streams of one polymer in the matrix of another polymer. During the fracture of a polymer stream (fiber) the radius of the resultant drops exceeds the radius of the initial stream by more than twice; i.e. the total surface does not increase. The microfiber fracture can occur at temperatures equal to or exceeding melting points of fiber-forming and matrix polymers. The conclusions that we have obtained agree with the data from the literature [l&30,33,34]. Thus, the authors of [34] have obtained the relationship between the reduced lifetime of a liquid cylinder and the viscosity ratio K of the phases, which is represented by a curve with the minimum at K = 1. Torza and coauthors [lg] have established for a mixture of Newtonian fluids that the dependence of jl~~/u (the measure of the ratio of viscous and interphase surface forces) on K has a minimum in the range 0.3 G K G 0.9 upon the fracture of a liquid cylinder in a flow. They note that this range of values of K, corresponding to the easiest fracture of the drops, is in good agreement with the theoretical values of K = 0.632, at which value the energy scattering passes through a maximum inside the deformed drop [59]. The authors of [60] have investigated the fracture mechanism of drops in concentrated emulsions of a solution of one polymer in a solution of another. The resultant relationship between the degree of deformation of the drop and K is represented by a curve having a maximum at K = 1. The curve rd vs K has a minimum in the region K = 1. These results do not agree with Tomotika’s theory, according to which the smallest particle size must be observed at K = 0.28. The foregoing confirms the general character of the conclusion (which we have established in Section 3.2.2.) that ultrafine fibers in a melt of the matrix polymer are unstable to a maximum extent at K= 1. We should dwell on the fundamental investigation into the fracture of a deformed drop carried out by Han and Chin [30,33] for flow along a cylindrical tube, when two fluids possess viscoelastic properties. They determined the critical conditions for the fracture of drops in terms of the rheological properties of both phases, the initial drop size, and the rate of shear at the wall of the tube. The main statements of their work amount to the following. The fracture of a deformed drop always occurs after a liquid thread-like cylinder passes through the entrance zone of the cylindrical tube, i.e. in the region where the elongated drops begin to relax. Factors that determine the degree of deformation also determine the critical conditions for fracture. These parameters are the viscosity ratio K of the phases, Weber number We, and the elasticity of the phases which were estimated by the elasticity parameter 6 = /IjlE/q12 ( qE is the elongational velocity gradient; j3, is a
24 material constant). An increased viscosity of the medium stablizes the liquid cylinder, while an increased interphase tension destabilizes it. The rate of growth of disturbances increases with any of the three parameters: elasticity of the medium, elasticity of a drop, or interphase tension. In other words, if the disturbance is introduced into the system and there is instability, then two factors, namely the elasticity of both fluids and interphase tension, cause an increase in the rate of growth of the disturbance. However, the authors of [33] are correct to note that the results of the analysis of fracture depend on the selection of a rheological model. Han and Chin have employed a linear Maxwellian model. Employing other models can result in different conclusions. 4. Conclusion Fracture of liquid streams in binary polymer systems is one of the most interesting microrheological processes, which is realized in melts and solutions of polymer mixtures both at the stage of preparing polymer compositions and at the stage of their processing into finished goods. The lifetime of a liquid stream determines the possibility of realizing the specific fiber formation during the flow of polymer mixtures and the temperature of microfiber fracture, i.e. the conditions for their use. The known theories of the fracture of streams refer to Newtonian fluids, while fracture in viscoelastic media is different in many respects. When developing the corresponding theoretical models, it is necessary to take into account the basic peculiarities of the fracture of viscoelastic streams, as formulated in Section 3.2.6. References 1 A.L. Breen and W. Chester, U.S. Patent No. 3,382,305 (1968). 2 B.T. Hayes, Chem. Eng. Progr., 65 (1969) 50. 3 M.V. Tsebrenko, A.V. Yudin, M.Yu. Kuchinka, G.V. Vinogradov and K.A. Zubovich, Vysokomolek. Soedin., B 15 (1973) 566 (in Russian). 4 M.V. Tsebrenko, A.V. Yudin, T.I. Ablazova and G.V. Vinogradov, Polymer, 17 (1976) 831. 5 M.V. Tsebrenko, A.I. Benzar and A.V. Yudin, Khimicheskie volokna, 4 (1978) 48 (in Russian). 6 A.V. Yudin, M.V. Tsebrenko and M. Yakob, in: Preprints of International Symposium on Chemical Fibers, Vol. 3, Kahnin, 1974 (in Russian). 7 M.V. Tsebrenko, N.M. Rezanova and G.V. Vinogradov, Polymer Eng. Sci., 20 (1980) 1023. 8 K. Dietrich and H. Versaumer, Faserforsch. u. Textiltechn., 26 (1975) 347. 9 M.V. Tsebrenko, Khimicheskie volokna, 3 (1983) 28 (in Russian). 10 U.S. Patent No. 3,549,734 (1967).
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