Three-Dimensional Model of Whipping Motion in the Processing of

Ind. Eng. Chem. Res. 2011, 50, 1099–1109

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Three-Dimensional Model of Whipping Motion in the Processing of Microfibers Yafeng Sun, Yongchun Zeng, and Xinhou Wang* College of Textiles, Donghua UniVersity, Songjiang, Shanghai 201620, People’s Republic of China

Electrospinning and melt blowing are the most commonly used processes to produce microfibers from extruded polymer solution or melt. The present work deals with the dynamic modeling of the whipping instability and related processes during production of microfibers. A bead-viscoelastic element fiber model is employed in modeling three-dimensional paths of the fiber motion in the processes of electrospinning and melt blowing. The simulation results provide a reasonable comparison between these two processes and point out that the Coulomb force in electrospinning always has the function to sustain and increase the bending instability; while in the melt blowing process, whether the aerodynamic force increases the bending instability or not depends on several factors. 1. Introduction One of the most significant developments in recent years has been the technology to extrude extremely fine fibers while maintaining all of the characteristics expected by textile manufacturers and consumers. Electrospinning, melt blowing, and melt spinning are the most commonly used processes to produce microfibers. Electrospinning can make fibers with diameters in the range of 100 nm and even less; while the fibers produced via conventional melt spinning are usually not smaller than 10 µm. Melt blowing is a rapid, single-step process used to produce fibers with diameters in the range of 1-2 µm. Multiorifice commercial melt blowing lines have the advantage of much lower cost than the multiorifice commercial electrospinning production lines. If melt blowing technology could be extended to nanoscale fiber sizes, then it would penetrate new markets and enhance current product offerings. Therefore, a major focus on melt blowing research should be to extend the technology to nanofibers. The process of electrospinning uses a high-voltage source to create an electrostatically driven jet of polymer solution that thins and elongates as it is driven toward an electrically grounded target. The physical mechanism of the electrospinning process is explained and described following several researchers.1-7 And it is shown that the bending instability, which is called whipping, is the key physical element of the electrospinning process responsible for enormously strong stretching and the formation of nanofibers. In the melt blowing process, fibers are produced by extruding a polymer melt through the spinneret and drawing down with a jet of high-velocity hot air. Figure 1 shows the schematic of diagram of melt blowing. During melt blowing, the fiber undergoes bending instability (or fiber vibration), which is attributed to the high velocity of the air. In this work, bending instability in the melt blowing process is also called whipping, which was previously adopted by Ellison et al.8 in their research on melt-blown nanofibers. Compared to the electrospinning research, much less has been done on the whipping dynamics in the melt blowing process. The features of fast, chaotic, and complicated make it difficult to study the melt blowing process theoretically as well as experimentally. Shambaugh and co-workers did a series work on understanding the melt blowing process. Uyttendaele and Shambaugh,9 Rao and Shambaugh,10 and Marla and Shambaugh,11 respectively, developed 1-D, 2-D, and 3-D models to predict the fiber motion * Corresponding author e-mail:[email protected].

Figure 1. Schematic diagram of melt blowing process.

in the melt blowing process. The 2-D and 3-D models took into account the fiber vibrations. In relation to their experimental work, Shambaugh and co-workers10,12-16 reported experimental measurements of fiber motion and fiber diameter using a singleorifice melt-blowing die. For example, Wu and Shambaugh12 used laser Doppler velocimetry (LDV) to measure the cone diameters of fiber vibrations. Rao and Shambaugh10 and Chhabra and Shambaugh13 measured the fiber vibration with multiple-image flash photography. Recently, Beard et al.17 used a high-speed camera (150 000 frame/s) to record the motion of a fiber below both a melt-blowing slot die and a melt-blowing swirl die. The photograph taken under the operation of a swirl die recorded a spiral motion of the fiber, which is the typical shape of the whipping motion in electrospinning.1,2,18 Electrospinning and melt blowing are analogous in the processes of drawing fibers. The polymer jets are drawn in the external fields. For electrospinning, an electric field is applied

10.1021/ie101744q  2011 American Chemical Society Published on Web 12/01/2010

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Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

Each bead is considered to represent the center of the mass of an element. The mass of bead i, mi, is contributed by its adjacent fiber elements (i-1, i) and (i, i+1), thus, mi is described as follows: 1 mi ) Ff (Ai-1,ili-1,i + Ai,i+1li,i+1) 2

(2)

where Ff is the fiber (polymer) density, Ai-1,i and Ai,i+1 are the fiber cross-sectional areas of fiber elements (i-1, i) and (i, i+1), respectively, and are described as follows: 1 2 Ai-1,i ) πdi-1,i 4

(3a)

1 2 Ai,i+1 ) πdi,i+1 4

(3b)

where di-1,i and di,i+1 are the diameters of fiber elements (i-1, i) and (i, i+1), respectively. For the first bead, bead 1, m1 is contributed by the element (1, 2). 2.2. Lagrange Equations for Fiber Motion. When spinning, the beads are exerted by the following forces: (1) external force Fex such as gravity, aerodynamic force, or electric force; (2) internal force Fin such as viscoelastic force or Coulomb force; and (3) bending restoring force Fb to restore the rectilinear shape of the bending part of the jet. Therefore, we obtain the equation governing the motion of bead i in the following form: mi

Figure 2. Schematic of a bead-viscoelastic element fiber model.

to create the drawing force; while for the melt blowing process, an airflow field is used. During drawing processes, both electrospun jet and meltblown jet undergo whipping instability. Whipping in the electrospinning process is an electrically driven bending instability; while in the melt blowing process, it is an aerodynamic-driven bending instability. The theory of aerodynamically driven jet bending was described by Entov and Yarin.19 According to the above experimental observation and modeling work of Shambaugh’s group,9-16 it could be concluded that whipping in the melt blowing process is not as significant as in electrospinning. It is assumed that the suppression of whipping is the key physical element for impeding the formation of melt blown nanofibers. In this work, we will compare the characteristics of whipping behavior during electrospinning and melt blowing by modeling the two spinning processes. A mathematical model is formulated to describe the fiber motion in the processing of microfibers.

d2ri dt2

) Fexi + Fini + Fbi

where ri ) ixi + jyi + kzi is the position of the ith bead; Fexi, Fini, and Fbi are external force, internal force, bending restoring force acting on bead i, respectively; i, j, and k are unit vectors in the Cartesian coordinate system. 2.2.1. Aerodynamic Force. As with the case for the mass of bead i, the air drag for bead i is contributed by fiber elements (i-1, i) and (i, i+1). The air drag force on bead i may be calculated by the following: 1 i i Fdi ) (Fdi-1,i + Fdi,i+1 ) 2

2.1. Mathematical Description of a Fiber. The polymeric jet in the spinning process behaves as a viscoelastic non-Newton fluid. We model the jet as a series of beads connected by viscoelastic elements, as shown in Figure 2a. The enlarged view of the part in the red window is shown in Figure 2b. A pair of adjacent beads, i -1 and i, form the fiber element (i-1, i), and the length of the element li-1,i is given by the following: li-1,i ) [(xi - xi-1)2 + (yi - yi-1)2 + (zi - zi-1)2]1/2

(1) The total number of beads, N, increases over time as new beads are inserted at the top of Figure 2 to represent the flow of melt polymer into the fiber.

(5)

where Fdi is the air drag on bead i, Fidi-1,i is the air drag on bead i contributed by fiber element (i-1, i), and Fidi,i+1 is the air drag on bead i contributed by (i, i+1). For the element (i-1, i), the air drag can be written as follows: i i i Fdi-1,i ) Ffi-1,i + FPi-1,i

2. Mathematical Model

(4)

(6)

where Fifi-1,i is the skin friction drag on element (i-1, i) for bead i, and Fipi-1,i is the pressure drag on element (i-1, i) for bead i. The direction of the friction drag is parallel to the fiber direction; while the pressure drag is normal to the fiber direction. To calculate the air drag on the fiber element, the local relative velocity between the air and the fiber is resolved into two components: axial and normal to the direction of the fiber element. Here, we introduce a fiber vector, which was previously adopted by Marla and Shambaugh.11 Figure 3 shows a fiber element (i, i-1). A unit vector ft that is parallel to the fiber axis is as follows: ft ) sin(φ)cos(R)i + sin(φ)sin(R)j + cos(φ)k

(7)


1101

Vrti ) vri · ft

(11a)

Vrni ) vri · fn

(11b)

Therefore, the friction drag and the pressure drag at bead i are given as follows: 1 i 2 Ffi-1,i ) CfFaVrti πdi-1,ili-1,i · ft 2

(12)

1 i 2 FPi-1,i ) CPFaVrni di-1,ili-1,i · fn 2

(13)

and

where Fa is the air density. In eqs 12 and 13, Cf and CP are the friction drag coefficient and the pressure drag coefficient. Cf is recommended to be a function of Renolds number Rel. Matsui20 developed a form of a turbulent boundary layer a theoretical relation as follows: Cf ) β(Redt)-n

(14)

in this relation for Cf there is no dependence on Rel. Matsui attributed this independence to random vibrations of the filament in melt spinning which cause there to be a constant average boundary layer thickness along the filament. Majumdar and Shambaugh21 determined that β ) 0.78 and n ) 0.61 for Melt blowing conditions. We adopt this relation and the values for our system. And Redt is defined as

Figure 3. Orientation of a fiber element in the 3D coordinate system.

Redt )

FaVrtidi-1,i µa

(15)

in which µa is the air dynamic viscosity. CP is given from a plot of the pressure drag coefficient versus the Reynolds number Redn for a cylinder placed with its axis perpendicular to the direction of motion.22 Redn is defined as Redn ) Figure 4. Relationship of ft, fn, and vr.

where the angle φ is the angle between the fiber axis and the z axis. The projection of the fiber element upon the x-y plane makes an angle R with respect to x axis. The relative velocity at bead i is as follows: vri ) vai - vfi

(8)

where vai and vfi are the air and fiber velocities at bead i, respectively. To determine fn, the unit vector normal to the fiber axis, we need first to define the plane in which fn lies. The vectors vr and ft determine the plane (u) that contains both the parallel and normal components of the air drag on the fiber, as shown in Figure 4. Let us define u ) f t × vr

dσi-1,i G 1 dli-1,i - σi-1,i )G dt li-1,i dt µ

ft × u ft × u

(10)

The axial and normal components of the relative velocity at bead i with respect to the fiber element axis are as follows:

(17)

where t is time, G and µ are the elastic modulus and viscosity of the fiber, respectively. Therefore, the net viscoelastic force acting on bead i is as follows: FVei

fn )

(16)

2.2.2. Viscoelastic Force. Each fiber element is modeled as a viscoelastic Maxwellian liquid jet, consisting of an ideal elastic spring attached to an ideal dashpot (shown in Figure 2a). Maxwell model is widely used to represent the rheological behavior of a polymer.23 The tensile stress acting on the fiber element (i-1, i), σi-1,i, is given by the following:

(9)

Then

FaVrnidi-1,i µa

2 2 πdi,i+1 πdi-1,i σi,i+1ft σi-1,ift ) 4 4

(18)

2.2.3. Electric Force. The electric force imposed on the ith bead by the electric field created by the potential difference between the spinneret and the collector is as follows: Fei ) eiEi

(19)

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where ei is the charge possessed by bead i, and Ei ) Exii + Eyij + Ezik is the electric strength at bead i. 2.2.4. Coulomb Force. Reneker and his co-worker ever researched Coulomb force in their electrospinning model.2 Here we introduce their method to analyze the Coulomb force on each bead. The net Coulomb force acting on the ith bead from all the other beads is given by the following:

∑

Fci )

j)1,N

[

yi - yj zi - zj e2 xi - xj i +j +k 2 Rij Rij Rij Rij

]

(20)

τi ) εEi · fn

(26)

where is dielectric constant of the ambient air. Melt blowing applied a jet of high-velocity hot air for drawing the polymer jet (fiber). During the process, the fiber temperature varies greatly. Assuming heat transfer within the fiber is neglected, the energy equation requires: miCi

di-1,i + di,i+1 li-1,i + li,i+1 dTi ) -hπ (Ti - Tai) dt 2 2

(27)

j*i

where η is the surface tension coefficient, ki is the fiber part curvature. 2.2.6. Gravity. Although ref 2 shows that when modeling the electrospinning process, the effect of gravity is negligibly small in comparison with electrical forces, it cannot be ignored in the melt blowing process. We express it as follows:

where Ci is the polymer heat capacity, Ti is the temperature of bead i, Tai is the air temperature at bead i, and h is the convective heat transfer coefficient. 2.4. Airflow Field and Electric Field. To solve the model, the velocity and temperature of the airflow filed, as well as the strength of the electric field need to be known. For our study, we obtained these data by numerical simulation. Many researchers27-32 have used computational fluid dynamics software to simulate the airflow field and the simulation results were accordant favorably with experimental results. In this work, the melt blowing airflow field was simulated by using the commercial software Fluent 6.2. The boundary conditions can be found in ref 32. Figure 5 shows the velocity contour of a melt blowing slot die. To obtain the electric field, Maxwell SV (Ansoft Corporation) was applied. Figure 6 shows the vectors of the strength of the electric field. The air velocity and temperature fields, as well as the strength of the electric field, were imported to calculate the fiber motion in the spinning processes. 2.5. Perturbation of the Rectilinear Fiber Segment. Perturbations of the lateral position and lateral velocity of the fiber lead to the development of the bending instability, and become the observed whipping motion. Yarin and his co-workers2,19 studied the bending dynamics of thin liquid jets in air for the first time, and they considered the helical disturbance by virtue of the linearity of the problem for small bending disturbances. In our model, the initial perturbation is added by inserting small displacements to the x and y coordinates of bead i,

Fgi ) migk

(23)

xi ) asin(ωt)

(28a)

2.3. Conservation Equations. The fiber is governed by conservation of mass. For each bead, the mass conservation requires:

yi ) acos(ωt)

(28b)

and Rij ) [(xi - xj)2 + (yi - yj)2 + (zi - zj)2]1/2

(21)

2.2.5. Restoring Force. Consider a fiber segment (i-1, i, i+1), the equilibrium shape of the fiber segment is rectilinear. When the fiber segment is bent, there is a surface tension force acting on bead i, tending to restore the rectilinear shape of the bending part of the fiber. In fact, the surface tension always counteracts the bending because bending always leads to an increase of the area of the fiber surface.19 Reneker2 introduced surface tension to analyze the electrically charged liquid jets. In this work, the theory is adopted in our model. The surface tension force on bead i is calculated by the fiber elements (i-1, i) and (i, i+1), and given by the following:

(

ηπ Fbi )

)

di-1,i + di,i+1 2 ki 2 [i|xi |sign(xi) + j|yi |sign(yi)] 4(x2i + y2i )1/2 (22)

FfAi,i+1li,i+1 ) m0

(24)

where Ff is the fiber density; m0 is the initial fiber mass and is determined by the polymer flow rate. For the electrospinning process, we assume that each bead possesses a charge e and it will not change with time. Hohman et al.4,5 and Feng24,25 used a slender-body theory to analyze the charged jets. They believed that the charge on the jets is composed of two parts: surface charge and inner current of the jets. Therefore, we obtain the charge possessed by each bead as follows: e ) πd20KE0z + πd0l0τ0

(25)

where K is the conductivity of the liquid, d0, l0, E0, and τ0 are the initial diameter, fiber element length, electric strength, and surface charge density at t ) 0. On the basis of the research of Feng24 and Gannan-Calvo,26 the surface charge density of bead i can be expressed:

here a is the initial perturbation amplitude, and ω is the initial perturbation frequency. 3. Results and Discussion 3.1. Trajectory of Whipping Motion. According to the mathematical model described above, we begin with the calculation of the trajectory of the fiber motion. The developments of perturbations into a whipping motion in the processes of electrospinning and melt blowing were simulated. Figure 7 shows the fiber whipping trajectory of electrospinning. The polymer parameters used were for 4% PEO solution, which were given by Doshi33 and Fong et al.34 Figures 8 and 9 show the fiber whipping development in the melt blowing process. The polymer parameters used were for Molten PP, which is a commonly used material in melt blowing. Table 1 lists the specification of the polymer properties and processing parameters used in the simulation. The whipping motion, which is triggered by lateral perturbations, is the key physical element of the spinning process responsible for strong stretching and formation of microfibers.


1103

Figure 5. Velocity contour of a slot die.

Figure 6. Vectors of an electric field.

Figures 7(a)-(d), 8(a)-(d), and 9(a)-(d) illustrate that the small perturbation, eqs 28, develop into a complex path of a whipping motion. The main difference between these three trajectories is

the amplitude of whipping motion with time evolution. The whipping amplitude is defined as the maximum lateral displacement at position z and is calculated as (x2max + y2max)1/2. Figure

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Figure 7. The time evolution of a whipping development for the simulation of electrospinning. (a) t ) 0.0014 s, (b) 0.12 s, (c) 0.26 s, and (d) 0.4 s.

10 shows the comparison of whipping amplitudes among the electrospinning process and the melt blowing process. Electrospinning shows an increase in amplitude of fiber whipping with distance from the spinneret. However, our simulation results show that the whipping instability in the melt blowing process is complicated, strongly depending on the air flow field. The whipping amplitude of the melt blowing process with high air velocity (280 m/s) increases with distance from the spinneret; while the melt blowing process with low air velocity (110 m/s) shows a rapid increase of the amplitude close to the spinneret and decays rapidly afterward. The predicted whipping amplitude in electrospinning is larger than that in melt blowing with low air velocity (110 m/s) except in the area close to the spinneret. As mentioned above, electrospinning and melt blowing are analogous in the processes of drawing fibers. This section would focus on the comparison between electrospinning and melt blowing with low air velocity (110 m/s). For electrospinning, the drawing force is the electric force created by the applied high voltage. As for the melt blowing process, the aerodynamic force creates the drawing force to attenuate the polymer jet. Figure 11 shows the drawing force on the fiber axis as a function of the distance from the spinneret. The drawing forces, including the aerodynamic force in melt blowing and the electric force in

electrospinning, decrease with the increasing distance from the spinneret. The small fluctuation of the aerodynamic force near the spinneret may be contributed to the recirculation zone of the airflow field. In the melt blowing process, the aerodynamic force depends on the local relative velocity between the air and the fiber. The decreased relative velocity leads to the decreased drawing force with the increasing distance from the spinneret. In electrospinning, each bead is considered to possess a same charge e, so the electric force on every bead is dependent on the electric strength. Figure 6 shows the decrease of the electric strength from the spinneret to the collector, which leads to the decreased electric force with the increasing distance. Figure 11 also shows the ratio of the air force to the electric force. In our model, the air force exerts on the meltblown fiber is by 1 to 2 orders of magnitude higher than the electric force on the electrospun fiber. The ratio increases from 40 to 170 near the spinneret, then decrease with further increasing distance. Figure 12 illustrates that the viscoelastic force for electrospinng is of the same order of magnitude as that for the melt blowing process. As is the case for viscoelastic force, the surface tension forces for the two processes are also of the same order of magnitude. The drawing force created by the airflow field is much larger than the drawing force created by the electric field.


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Figure 8. The time evolution of a whipping development for the simulation of melt blowing process. (a) t ) 0.255 s, (b) 0.265 s, (c) 0.275 s, and (d) 0.295 s.

However, it is known that electropsun fibers are much finer than melt blown fibers. We calculated the final fiber diameter of the electrospun and melt blown fibers. For an initial diameter of 949 µm, the predicted final diameter of a melt blown fiber is 88.9 µm, while the predicted diameter of an electrospun fiber is 10.4 µm. Several studies on electrospinning35 have demonstrated that whipping motion is the key physical element responsible for the strong stretching and therefore the formation of nanofibers. The experimental observations show that the whipping motion in the melt blowing process is not as significant as in electrospinning. The image of the electrically driven bending instability in a single-jet electrospinning process taken by Theron et al.18 shows that the whipping amplitudes of electrospinning approach centimeters. Yang et al.36 recorded the whipping process in electrospinning and measured the bending parameters. Their experimental results also illustrate that the whipping amplitudes approach centimeters. Beard et al.15,17 measured fiber vibration during melt blowing, and revealed that the amplitude of vibration increases from around 1 to 2.5 mm. We consider it as the main reason for the formation of larger fibers via the melt blowing process. In the following section, we compare the mechanisms of the bending perturbation development in the two spinning processes, and therefore explain

the different dynamic behaviors of whipping motion in electrospinning and melt blowing. 3.2. Comparison of Whipping Developments. Whipping in the electrospinning process is an electrically driven bending instability; while in the melt blowing process, it is an aerodynamic-driven bending instability. Reneker and his co-workers2 have analyzed the instability mechanism that is relevant in the electrospinning context. They chose a fiber element (i-1, i, i+1) in rectilinear part of the polymer jet and treated it as three pointlike charges, each with a value e, shown in Figure 13a. If a small perturbation causes point i to move off the line by a distance δ to i′, then a net Coulomb force Fci acts on i in a direction perpendicular to the rectilinear jet, and tends to cause i to move further away from the line in the direction of the perturbation. The Coulomb force sustains the increase of the perturbation. The viscoelastic force Fvei and the surface tension force tend to counteract the bending perturbation. If the Coulomb force is larger than the viscoelastic force and the surface tension force, then the bending perturbation continues to grow. Here we use this method to analyze the instability mechanism in the melt blowing process. A small perturbation is added to bead i of the straight fiber element (i-1, i, i+1), and makes i to move off the line by a distance δ to i′. The fiber elements (i-1,

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Figure 9. The time evolution of a whipping development for the simulation of melt blowing process at high air velocity slot die. (a) t ) 0.04 s, (b) 0.05 s, (c) 0.06 s, and (d) 0.07 s Table 1. Specification of the Polymer Properties and Processing Parameters materials

4% PEO solution

molten PP

density (kg/m3) viscosity (Pa · s) elastic modulus (Pa) surface tension (kg/m) conductivity (Ω/m) heat capacity temperature (°C) initial diameter (µm) initial length (µm) volume flow rate (mL/min) voltage (V) air velocity (m/s) air jet temperature at spinneret air density (kg/m3) air dynamic viscosity (Pa · s) air convective heat transfer coefficient (W/m2 · K) distance to collector (m) perturbation amplitude (µm) perturbation frequency (Hz)

1.2 × 103 12.5 1 × 105 78.132 × 10-4 4.902 × 10-3 N/A 23 949 94.9 4.24 10 000 0 23 1.293 0.297 × 10-4 0.026

598.28 38.38 2.8 × 104 0.7 N/A 1.78 × 103 310 949 94.9 4.24 N/A 110/280 310 1.293 0.297 × 10-4 0.026

0.2 94.9/1000 10 000

0.2 94.9/1000 10 000

i′) and (i′, i+1) are subjected to the aerodynamic force. The aerodynamic force includes two parts: the friction drag, which is along the fiber axis; and the pressure drag, which is normal to the fiber axis. Due to the fiber curvature, a distributed aerodynamic force, which is vertical to the line, may enhance perturbation and make the perturbation grow. If the viscoelastic

Figure 10. Amplitude of the two processes.

force and surface tension force fail to oppose the dynamic action of the air, then the bending perturbation will grow. Our simulation results show that the bending perturbation in electrospinng tends to grow and subsequently become whipping motion; while the development of the bending perturbation in the melt blowing process is more complicated. In this work, we applied our model to two melt blowing airflow fields: the first case is an annular die with 110 m/s air velocity, and the second case is a slot die with the air velocity used in commercial


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Figure 11. Drawing force on jet axis and the ratio of air force to electric force.

Figure 12. Viscoelastic forces and Surface tension of these two processes.

production, say, 280 m/s. The predicted whipping amplitudes of theses two cases are shown in Figures 8 and 9, respectively. Unlike the profile shown in the case of low air velocity (110 m/s), in the case of high air velocity (280 m/s), we predict a gradually growing whipping as time develops. In electrospinning, the Coulomb force is responsible for the growth of the bending perturbation; while in the process of melt blowing, the aerodynamic force sustains the perturbation. In Figure 13a, when the fiber segment is perturbed, the directions of the two Coulomb forces are downward and outward, and upward and outward. Thus, the resultant of these forces Fci is in a radial direction with respect to the straight jet and grew exponentially in time as the radial displacement of the segment increased. From Figure 13c, we can see that the direction of the resultant force of aerodynamic forces from the above and the below elements of bead i, is relevant to the position of bead i. When its direction is downward and outward, it may sustain the bending perturbation. On the contrary, when the force is directed downward and inward, it may suppress the perturbation. We believe that this mechanism is responsible for the difference of the observed whipping developments in electrospinning and melt blowing.

Figure 13. Illustrations of bending instability in electrospinning and melt blowing process.

as well as momentum balances. All kinds of forces imparted on the fiber element are summarized into three kinds of forces, namely external force, internal force and bending restoring force. This model can predict the fiber diameter, fiber vibration amplitude, and fiber trajectory. This model also gives a method to compare the whipping dynamic of these two spinning processes. The results show that although the aerodynamic force is one or two orders larger than the electric force on each fiber element, the drawing ratio in the melt blowing process is less than that in electrospinning, and fiber whipping in the melt blowing process is not as significant as in electrospinning. The main reason is that the Coulomb force in electrospinning always has the function to sustain and increase the bending instability; while in the melt blowing process, whether the aerodynamic force increases the bending instability or not depends on its value and direction at relevant fiber element. Now that the bending instability is the main reason for such a large drawing ratio for electrospinning, we believe that one way to increase the drawing ratio of the melt blowing process is to add extra bending during the fiber attenuation, or introduce another force which can function just like the Coulomb force in electrospinning.

4. Conclusions A comprehensive model has been developed for the melt blowing process and electrospinning process. This model involves the simulation of conservation law of mass and charge

Nomenclature di-1,i ) the diameter of the fiber elements (i-1, i) e ) the charge of bead

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ft ) the unit vector that is parallel to the fiber axis fn ) the unit vector that is vertical to the fiber axis li-1,i ) the length of the fiber element (i-1, i) mi ) the mass of bead i h ) the convective heat transfer coefficient ki ) the fiber part curvature u ) a vector perpendicular to the plane which contains parallel and normal components of the air drag force x ) Cartesian coordinate; see Figures 2 and 3 y ) Cartesian coordinate; see Figures 2 and 3 z ) Cartesian coordinate; see Figures 2 and 3 µ ) the viscosity of the fiber σi-1,i ) the stress acting along the fiber element (i-1, i) τi ) the surface charge density of bead i ε ) dielectric constant of the ambient air Fa ) the air density Ff ) the fiber (polymer) jets density a ) initial perturbation amplitude ω ) initial perturbation frequency η ) the surface tension coefficient Ai-1,i ) the fiber cross-sectional areas of fiber elements (i-1, i) Ci ) the polymer heat capacity Cf ) the friction drag coefficient CP ) the pressure drag coefficient Ei ) the electric field at ith bead Fex ) external force Fin ) internal force Fb ) bending restoring force Fdi ) the air drag force on bead i Fifi-1,i ) the skin friction drag on element (i-1, i) for bead i FiPi-1,i ) the pressure drag on element (i-1, i) for bead i FVe,i ) the net viscoelastic force acting on bead i Fe,i ) the electric force imposed on the ith bead Fc,i ) the net Coulomb force acting on the ith bead from all the other beads Fb,i ) the surface tension force on the ith bead Fg,i ) the gravity imposed on the ith bead G ) the elastic modulus of the fiber K ) the conductivity of the liquid N ) the total number of bead Rel ) Reynolds number Redt ) the Reynolds number based on the fiber diameter and the axial component of relative velocity of air with respect to the fiber Redn ) the Reynolds number based on the fiber diameter and the normal component of relative velocity of air with respect to the fiber Rij ) the distance between ith and jth bead Ti ) the temperature of fiber bead i Tai ) the air temperature at bead i vr) relative velocity of air with respect to the fiber vt ) the axial component of any vector (V) with respect to the fiber element direction vn ) the normal component of any vector (V) with respect to the fiber element direction vai ) air and fiber velocities at bead i vfi ) fiber velocities at bead i Acknowledgment This work is financially supported by National Natural Science Foundation of China; Contract Grant No. 50976091. Foundation for the Author of National Excellent Doctoral Dissertation of the People’s Republic of China; Contract Grant

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ReceiVed for reView August 18, 2010 ReVised manuscript receiVed November 16, 2010 Accepted November 17, 2010 IE101744Q

Three-Dimensional Model of Whipping Motion in the Processing of

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