Impact of Entanglement Density on Solution Electrospinning: A

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Impact of Entanglement Density on Solution Electrospinning: A Phenomenological Model for Fiber Diameter Chi Wang,*,† Yu Wang,† and Takeji Hashimoto*,‡,§ †

Department of Chemical Engineering, National Cheng Kung University, Tainan 701, Taiwan ROC Kyoto University, Kyoto 606-8501, Japan § Quantum Beam Science Center, Japan Atomic Energy Agency, Ibaraki 319-1195, Japan ‡

S Supporting Information *

ABSTRACT: The rheological properties of poly(N-isopropylacrylamide, PNIPAM) in dimethylformamide solvent were investigated and correlated with solution electrospinnability. The jet diameter was measured by using the light scattering technique during the electrospinning in the straight jet region prior to the jet whipping. The diameter of the straight jet end is independent of the solution concentration (or viscosity within the range of 15−2000 mPa·s). Thus, the final fiber diameter df observed on the grounded collector is dominantly controlled by the jet-whipping process. According to the present PNIPAM solution and other different polymer solutions, df is correlated with the solution concentration ϕ. A master curve is constructed by using the following equation: df/df,e = (ϕ/ϕe)2.5, where df,e is the diameter of the fibers electrospun from the solutions with an entanglement concentration of ϕe, above which the specific viscosity starts to increase with ϕ according to ϕ3.7 or ϕ4.7, depending upon the given polymer/solvent pair. The derived exponent of 2.5 is in good agreement with the theoretical exponent value of 2.3 provided that df is proportional to the entanglement density υ(ϕ) ∼ ϕ2.3 (entangled strands per unit volume of the solution). Our results imply that the plateau modulus (elasticity) of the entangled polymer solution rather than its viscosity plays a major role in determining the final fiber diameter. The entangled polymer solutions behave like elastic swollen gels during electrospinning because of the high deformation rates. We propose that the deformation-induced structure formation in the jet eventually results in the fiber with the concentration-dependent diameter.

1. INTRODUCTION Electrospinning is a sophisticated process for preparing polymeric fibers with a submicrometer diameter. Many nanofibers of different polymers have been readily obtained by using this technique.1−3 During electrospinning, the polymer solution is delivered at a given flow rate into a capillary (needle) connected to a high voltage source. At a critical potential that is sufficiently high to overcome surface tension the liquid meniscus at the capillary end forms a conical shape known as the Taylor cone. Moreover, an electrified liquid jet is issued from the cone apex to develop a tapered straight jet. The straight jet with ca. several millimeters long is followed by the jet-whipping process at its end because of the “bending instability” phenomenon.4 In this region, the majority of the solvent evaporates, thus leaving the charged solid fibers to be collected by a grounded collector. Several electrospinning © XXXX American Chemical Society

modes are developed depending on the applied voltage at a given flow rate. The cone−jet mode, which is a well-controlled process, is the most desirable electrospinning mode that provides good reproducibility of the fiber properties. The governing parameters for determining the morphology and mechanical strength of electrospun fibers can be generally divided into two groups:1−3 solution properties (e.g., viscosity, conductivity, and surface tension) and processing variables (e.g., applied voltage, tip-to-collector distance, and solution flow rate). Previous studies have focused on the direct relationship between the governing parameters and fiber diameter, whereas the jet diameter along the spinning line Received: March 12, 2016 Revised: September 27, 2016

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DOI: 10.1021/acs.macromol.6b00519 Macromolecules XXXX, XXX, XXX−XXX

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terminal relaxation time τd was estimated by τd = η0J0s . It is reminded that τd determined in this manner (also termed weight-average relaxation time) is strongly dependent upon the polydispersity of polymers when commercial polymers are used. The complex viscosity η* was calculated by using [G′(ω)2 + G″(ω)2]0.5/ω.16 All the solution properties were measured and determined at 25 °C. 2.2. Processing and Measurements. The homogeneous polymer solution was delivered by a syringe pump (Cole−Parmer) at a controlled flow rate of 0.5 mL/h through the PTFE tubing into the stainless needles (Hamilton, o.d. = 1.47 mm). High electrical voltage (Bertan, 205B) was applied in the needles. To construct a point-plate electrode configuration, we used a steel net (30 × 30 cm2) to collect electrospun fibers at a tip-to-collector distance of 14 cm below the needle tip. During electrospinning, the shape variation of the Taylor cone at the needle tip was monitored by a CCD. The dimensions of the Taylor cone were measured on the basis of the recorded images. To monitor the diameter profile dj(z) along the liquid jet beneath the Taylor cone, we applied the light scattering technique. A 1 mW He−Ne laser with a pinhole of 1 mm diameter was used as a light source, and the scattering patterns on the screen were collected by using another CCD for further analysis. The intensity profile along the equator was plotted as a function of magnitude of the scattering vector q (q = 4π sin(θ/2)/λ, where θ and λ are the scattering angle and the wavelength of the incident beam in the solution, respectively, the He−Ne laser light of the wavelength of 632.8 nm in vacuum being used). The liquid jet diameter (dj) was determined at the position of the first scattering maximum (qm,1) by using dj = 10.22/qm,1, provided that the Rayleigh−Gans−Born (RGB) theory is valid.17 The morphology and diameter of as-spun fibers (i.e., dried whipping jets spread on the collector) were observed and measured by using a scanning electron microscope (SEM, Hitachi S4100).

remains largely unexplored. However, manipulating the free surface of an electrified jet is the first and determinant step to control the diameter of electrospun fibers. Thus, understanding the jet diameter profile, with respect to the distance z from the needle end along the jet, in the “straight jet region” is important to determine the final fiber diameter. Jet diameter has been measured by using imaging techniques;5−8 however, difficulties have been reported in terms of image resolution because the electrospinning jet can be thinner than 10 μm. Analyses of jet images have shown that the jet diameter profile along the straight jet dj(z) follows a power-law relationship dj(z) ∼ z−n, n = 0.2−0.5, which depends on the solution properties. Predicting the fiber diameter is the ultimate goal in the development of the electrospinning process. Electrospun fibers are formed from the whipping jets in air which are randomly spread on the grounded collector after solvent evaporation. The measured fiber diameter has been correlated with the governing parameters based on some theoretical calculations.5,6,9−11 Although successful correlations have been occasionally reported, a precise analysis has not been completed because several sophisticated interactions, such as electrofluid dynamics, solution rheology, and drying kinetics, are involved. Among all governing parameters, solution viscosity is one of the most important parameters in determining fiber diameter. A minimum polymer concentration (solution viscosity) is generally required to obtain uniform bead-free fibers.12 Moreover, increasing the solution viscosity will produce fibers with larger diameters. Fiber formation is attributed to the existence of entanglement in polymer solutions. Chain entanglements are important, but its effect on fiber diameter has been rarely reported.12−14 Despite all the efforts, a clear and integrated picture of the jet diameter profile from the cone to the straight jet end is still nonexistent. In this article, the effects of entanglement density on jet and fiber diameters were reported. The jet diameters were measured by using a light scattering technique to resolve the resolution issue of optical imaging. For a given polymer solution, the straight jet end diameter is independent of the polymer concentration (entanglement density). Moreover, the fiber diameter is approximately proportional to the entanglement density according to the results of 10 different polymer solutions. Some general scaling laws are presented to provide deep insights into the manipulation of the charged jet, thus obtaining fibers with a small diameter. In the final section, a molecular view of nonlinear rheology for the electrospinning process is provided.

3. RESULTS AND DISCUSSION 3.1. Determination of Overlap Concentration ϕ* and Entanglement Concentration ϕe. In the electrospinning process, the importance of chain entanglement on fiber formation has been first demonstrated in the pioneering studies of Long and Wilkes.12,13 A fiber-like structure begins to be detected in the electrospinning of polymer solutions with increasing concentrations higher than ϕ*.13 However, a minimum concentration of 2.0−2.5 ϕe is required to obtain bead-free fibers.12 Other research showed that this minimum concentration for bead-free fibers could be as low as ϕe,18 depending on the polymer/solvent pair. Thus, for a given polymer solution, such as the PNIPAM/DMF used in this study, determining ϕ* and ϕe is essential for understanding the solution spinnability; that is, the rheological properties of the electrospinning solution are better studied prior to electrospinning. A capillary viscometer was used to obtain the specific viscosity ηsp of dilute polymer solutions ( ϕe and at 25 °C studied in our laboratory.19−26 The two groups of the classified solution characteristics are as follows: the solutions of PS/THF,19 PAN/ DMF,20 PDLLA/DMF,21 and PVA/H2O25 exhibit θ-solventlike characteristics, whereas the solutions of PET/TFA,23 PBT/ TFA,26 Nylon6/FA,22 Nylon46/FA,24 and PNIPAM/DMF present good solvent characteristics. Table 1 also shows ϕe and the corresponding η0 for these polymer solutions determined from the log−log plot of ηsp vs ϕ (Figure 2). The electrospinnability of these solutions have been investigated19−26 and compared with the present PNIPAM/DMF solution in the following sections. 3.2. Determination of the Jet Diameter by Light Scattering. Understanding jet stretching is important in predicting the final diameter of the solid fibers. After the polymer solution emerges in the needle end to form a Taylor cone, two main stages of liquid jet stretching occur; the first stage is in the cone−jet region, whereas the other stage is in the jet-whipping region (jet bending instability). Considering the instability nature, controlling the whipping jet is more difficult than manipulating the cone−jet region. A thinner straight jet prior to bending instability produces thinner electrospun fibers on the grounded collector.23 Theoretically, the jet diameter profile dj(z) (z is the distance from the needle end) in the straight jet region has been extensively studied by solving sophisticated fluid dynamics, involving inertial term, rheological stresses, and electrical stresses. Some useful scaling laws of dj(z) have been derived.6,27−29 However, measuring dj experimentally is difficult because the jet diameter can be as small as 10 μm, which is lower than the resolution of the optical imaging techniques.5−8 Therefore, we applied the light scattering technique to analyze the scattering pattern obtained by shinning a laser light on the electrospinning jet. The laser light position on the straight jet was carefully controlled by using the two step motors connected orthogonally (Figure 3c). In this procedure, the scattering pattern of the liquid jet at different z values can be obtained. Figure 4a shows the typical scattering pattern for the laser light shining on the jet end (prior to jet whipping) of the 4 wt % electrospinning solution. This pattern could be used to determine the straight jet end diameter dj,e. On the equator, a scattering maximum is detected from the intensity profile (Figure 4b). On the basis of the RGB theory, valid under the assumption of (2π/λ0)dj|nj − 1| ≪ 1, the scattering intensity function of a perfectly oriented cylindrical jet with a radius of R (= dj/2) is derived to be I(qR) ∼ (nj − 1)2[J1(qR)/qR]2, where J1 is the Bessel function of the first order and nj is the refractive index of jet. The validity of the RGB theory could be readily checked by the presence of successive scattering maxima asymptotically decreasing with the relation of Im ∼ qm−3, where Im and qm are the intensity and the magnitude of the scattering vector for the mth-order scattering maximum. It should be noted that the scattering function obtained by RGB scattering is not strict compared to the rigorous one derived from Mie

Figure 1. (a) Dynamic moduli measured in small-amplitude oscillatory shear experiments for PNIPAM/DMF solutions with two concentrations at 25 °C. At low ω, G′ (open symbols) is found to be proportional to ω2, whereas G″ (closed symbols) is proportional to ω, indicating that terminal flow region is reached. Based on the dynamic modulus data in the terminal region, solution characteristics (such as η0, J0s , and τd) are derived. (b) Complex viscosity η* versus frequencies. The zero-shear viscosity η0 can be also derived from the Newtonian region at low frequencies.

solutions does not scale with the concentration ϕ as τd ∼ ϕ1.6 for a good solvent, as predicted for the monodisperse polymer solution. The complex viscosities of the PNIPAM solutions at different applied frequencies are shown in Figure 1b. Apparently, η0 increases with increasing polymer concentration. Figure 2 shows the log−log plot of the specific viscosity ηsp versus the polymer volume fraction ϕ with the low viscosity data obtained from the capillary viscometer. All data were measured at 25 °C. A constant slope of 1.02 is detected in the dilute solution regime. Above ϕ*, the slope rapidly increases and finally reaches a constant value of 3.67, thus suggesting that the entrance of the semidilute entangled solution regime. The

Figure 2. Specific viscosity ηsp versus the volume fraction ϕ for the PNIPAM/DMF solutions. The arrow indicates the overlap concentrations ϕ* and entanglement concentration ϕe of 0.81 and 7 wt %, respectively. The final slope that reached the entangled solution regime is 3.67. C

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Macromolecules Table 1. Solution properties at 25 °C and Electrospinning Variables Used for the Polymer Solutions Studieda polymer/solvent

α

PS/THF19 PAN/DMF20 PDLLA/DMF21 PVA/H2O25

4.60 4.78 4.68 4.76

14 5 10 4.6

PET2/TFA23 PET1/TFA23 nylon-6/FA22 nylon-46/FA24 PBT/TFA26 PNIPAM/DMF

3.82 3.56 3.77 3.79 3.87 3.67

10.5 12 6 10 7 7

surface tension (dyn/cm)

electrical conductivity (μS/cm)

V (kV)

H (cm)

Q (mL/h)

ϕmin (wt %)

60 47 151 100

24.2 36.3 36.1 50.0

1.2 36 6.4 1017

10 6 12.5 11

14 7 28 21

3 0.3 1 0.2

20 6 19 7

175 125 81 151 131 46

21.1 21.1 38.6 38.6 21.0 36.2

1.4 1.4 4700 4200 2.3 2.0

9.3 9.3 20 20 9.3 6

14 14 7 7 14 14

1 1 0.3 0.3 1 0.5

8 12 8 10 9 14

ϕe (wt %) η0 at ϕe (mPa·s)

a For a given polymer/solvent pair, the processing variables of applied voltage (V), tip-to-collector distance (H), and solution flow rate (Q) are fixed for electrospinning. ϕmin is the minimum polymer concentration to obtain the bead-free fibers. The i.d. and o.d. of the spinneret are 1.07 and 1.47 mm, respectively, for all the solutions except PET2 and PET1 solutions with 0.508 and 0.813 mm, respectively.

Figure 3. Setup of the light scattering apparatus to measure the jet diameter during electrospinning. Two step motors are used to precisely position the laser beam on the straight jet at different z positions. The diameter of the pinhole is 1 mm. The diameter of the straight jet end is denoted as dj,e. The scattering pattern on the screen is captured by a CCD for further analysis. Images (a) and (b) are obtained from a conventional CCD and a high-speed camera, respectively.

Figure 4. (a) Light scattering pattern of the straight jet end obtained from the electrospinning of 4 wt % solution and (b) the equatorial intensity profile versus the magnitude of the scattering vector q of the straight jet end for solutions with different weight concentrations indicated by ϕw (%). The straight jet end diameter dj,e is determined by 10.22/qm,1 where qm,1 is the q at the first-order scattering maximum. Each scattering profile was vertically shifted to avoid overlapping of them and uncorrected for the background scattering due to instruments themselves.

scattering;17 however, it provides an approximated but convenient guide in evaluating the relative change of the diameter of the electrospinning jet with distance z and concentration ϕ. Since a distinct first-order scattering maximum is observed (Figure 4b), we calculated dj by using 10.22/qm,1 according to the RGB theory, where qm,1 is the position of the first-order scattering maximum. qm,1 is relatively independent of ϕ, thus suggesting similar dj,e values (ca. 6.4 ± 0.2 μm) for the electrospinning solutions within a wide range of PNIPAM concentrations. All solutions are electrospun under the same processing variables of 6 kV, 0.5 mL/h, and a tip-to-collector distance of 14 cm. Moreover, the surface tension and solution conductivity of these solutions are relatively constant at 36.2 dyn/cm and ∼2 μS/cm, respectively. The major difference is the solution viscosity, which is 5 mPa·s for the 2 wt % solution and 615 mPa·s for the 14 wt % solution. To validate the dj measurement, we measured the jet diameter profiles dj(z) during the electrospinning of the

PNIPAM solutions with different concentrations (4−21 wt %). Figure 5a shows the results for the 4, 15, 17, and 19 wt % solutions at a given flow rate of 0.5 mL/h, while Figure S2a shows those for the 9, 17, and 21 wt % solutions at another flow rate of 1.0 mL/h. A general trend can be obtained from these plots; dj gradually decays with increasing distance z from the needle end and then reaches an asymptotic value prior to the jet-whipping process. The asymptotic jet diameter dj,e is kept only over a small z range of approximately 1−2 mm, which is well expected in consideration of the presence of the downstream charged whipping jet (Figure 3b). dj,e is independent of the solution concentration but increases with increasing flow rate. However, as the solution concentration increases, a bigger Taylor cone along with a thicker jet develops along the spin line. The decaying rate of dj with z follows a scaling law of dj(z) to z−n. The exponent n, which is independent of the flow rate but depends on the solution viscosities, was evaluated to be 0.18, 0.30, 0.38, 0.51, 0.64, and D

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Figure 5. (a) Jet diameter profile along the jet length for the electrospinning of solutions with different concentrations. The dashed line indicates the straight jet end diameter dj,e. (b) Calculated jet velocity profile along the jet length. Symbols of # and ∗ indicate the end of straight jet and the demarcation of region I and region II, respectively. The processing variables are 0.5 mL/h, 6 kV, and a tip-to-collector distance of 14 cm.

Figure 6. SEM images of the PNIPAM fibers electrospun from the PNIPAM/DMF solutions with different concentrations. A magnification of 800× is used to reveal the morphology of the majority of the fibers, and a higher magnification of 10000× is shown by the inset. The uniform fibers fibers begin to be obtained with increasing the concentration larger than 14 wt %, which is 2 times higher than the entanglement concentration ϕe.

inertial term and electrical stresses become dominant for less viscous solutions. Thus, the electrical force is counterbalanced with the inertial force in the present 4 wt % PNIPAM solution with a viscosity of 16 mPa·s. As the concentration increases up to 17 wt % with a viscosity of 1115 mPa·s, the rheological

0.53 for the 4, 9, 15, 17, 19, and 21 wt % solutions, respectively. Our derived exponent n is consistent with those obtained from theoretical derivations, which is either 0.56,29 in the case when the rheological and electrical stresses are the dominant factors for more viscous solutions or 0.2527,29 in the case when the E

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which is approximately 2ϕe. The beaded fibers are frequently observed within the concentration range of ϕ* and 2ϕe. These findings are consistent with previous reports,12,13 and other electrospinning solutions such as those shown in Table 1, studied in this laboratory. However, the maximum concentration available for continuous electrospinning is 21 wt %. At concentrations higher than 21 wt %, the entangled solution is too viscous to be stretched by the electric field established for the fixed voltage of 6 kV and a tip-to-collector distance of 14 cm. Thus, only limited solution concentrations with η0 ranging from 615 to 2000 mPa·s are available for determining the viscosity effect. This result is the general trend for electrospinning. Figure 7 shows the measured fiber diameter plotted as a function of η0. The measured dj,e by light scattering is also

stresses associated with the high solution viscosity should be considered. Based on the measured dj, jet velocity vj is calculated by 4Q/ πdj2 under the assumption that solvent evaporation is negligible. Figure 5b and Figure S2b show the jet velocity profiles, from which the strain rate is estimated by dvj/dz. Along the straight jet line, the strain rate is time (position) dependent, and three distinct regions can be identified; the strain rate is the highest near the Taylor cone apex (region I), decreases quickly to reach a constant value, and then remains unchanged (region II) until approaching the jet end (region III), at which the strain rate significantly drops to be nearly zero, as presented in Figure S3d. Thus, the chain deformation involved in region I and II is likely to relax in region III where the jet attained the zero strain rate. For the 4 wt % solution, a significant drop of strain rate is seen from 1970 s−1 in region I to 510 s−1 in region II. With increasing polymer concentration, not only the strain rate is reduced but also a mild strain-rate transition is detected; the strain rate varies from 565 to 430 s−1 for the 19 wt % solution. The ultimate strain rate near the cone apex is due to the largest electrical stresses induced by the highest electrical field which significantly pulls the fluid element out of the cone. When the solution viscosity is too high, the electrical stress cannot significantly pull the fluid element so that the demarcation of region I and region II becomes indistinct; this is seen for the 21 wt % solution (Figure S2b). Near the straight jet end (region III), the rapid strain-rate drop is associated with the Coulombic electrical repulsion between the straight jet segment and the whipping jet that follows. Previous studies6,30 used particle tracing velocimetry to show that the fluid velocity in the straight jet region significantly increases and reaches a high value of ca. 2−5 m/s within a short distance of several millimeters. Thus, the calculated strain rate of the fluid element dramatically increases from the cone apex, reaches a maximum of about 600 s−1,30 and then rapidly decreases near the jet end. In the whipping jet region, the whipping loop can expand rapidly from a diameter of 1 to 8 mm in 7 ms, giving rise to an estimated strain rate of 103 s−1.30 These experimental findings reveal that electrospinning is an unsteady transient process. After entering the apex of the Taylor cone, the fluid element experiences different strain rates (mainly extension strain) along the spin line until the solvent evaporates to form solid fibers collected on the grounded collector. Our experimental evidence that the straight jet in region III is stable against the bending instability (the whipping) of the jet, suggesting that the chain stretching in regions I and II are strong so that the chains in region III, where the strain rate goes down to zero, are still remained to be stretched sufficiently high to avoid the bending instability. This experimental discovery on the chain stretching in region III of the straight jet motivated us to compare “qualitatively” the experimentally assessed strain rates and the longest Rouse relaxation rates as a function of polymer concentration (discussed later in section 3.5). 3.3. Fiber Diameter as a Function of Solution Viscosity. It is well-known that the diameter of electrospun fibers increases with solution viscosity.12,13,18 However, only bead-free fibers should be considered for comparison. Figure 6 shows the fiber morphology obtained from the electrospinning of PNIPAM solutions with various concentrations. A fiber-like structure (beaded fibers) starts to be detected for solutions with concentrations higher than ϕ*. The minimum PNIPAM concentration (ϕmin) for obtaining uniform fibers is 14 wt %,

Figure 7. Effects of solution viscosity on the “straight jet end” diameter dj,e (open symbols) and fiber diameter df (closed symbols) for PNIPAM/DMF. The number besides the symbols indicates the corresponding solution concentration in weight percent.

included in the figure. Considering that dj,e is independent of η0 (and the concentration), we conclude that the magnitude of df is independent of dj,e and hence of the stretching of the fluid element in the straight jet region but is strongly associated with the fiber formation process in the jet-whipping process. Moreover, a scaling law between df and η0 is derived to be df ∼ η00.8, which is in agreement with previous finding for the PET/TFA solution.23 An intriguing question is immediately raised: is the η0 the suitable “variable” to scale with the diameter of electrospun fibers? If that is the case, the exponent derived (∼0.8) should be consistent with all other solution systems. For a detailed comparison, the log−log plots of df that depend on η0 for the electrospinning of different polymer solutions19−26 at 25 °C is shown in Figure 8. The exponent for each solution is also determined and presented in the figure. Two distinct groups of exponent are obtained (Table 2). The first group is ca. 0.41− 0.53 for the polymers in the θ solvent and ca. 0.61−0.80 for the polymers in good solvent. Other research groups have also studied the η0 dependence of df. The derived exponent is approximately 0.5 for the PAN/DMF solution,31 0.80 for the poly(ethylene terephthalate-co-ethylene isophthalate) (PET-coPEI) solution in the mixture of chloroform and DMF (CF/ DMF),12 and 0.71 for the poly(methyl methacrylate)/DMF solution.13 These results indicate that η0 may not be an appropriate variable to scale with df because of the scattered exponents (0.41−0.80). Figure 8 also shows that the polymer solutions with viscosities of 102−104 mPa·s may be electrospun to yield fibers with diameters of 50−7000 nm depending on the polymer−solvent system. 3.4. Fiber Diameter as a Function of Entanglement Density (Reduced Polymer Concentration). To explore F

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Figure 8. Effects of solution viscosity on the fiber diameter df. Ten polymer solutions are included for comparison. The slope for each solution is presented. Two groups of the slope are obtained: one group is within the range of 0.41−0.53 for the PS, PAN, PDLLA, and PVA solutions, whereas the other group is within the range of 0.61−0.80 for the PET, PBT, nylon-6, nylon-46, and PNIPAM solutions.

Figure 9. Reduced concentration (ϕ/ϕe) effect on the fiber diameter df. The slope for each polymer solution is displayed. The fiber diameter obtained from the electrospinning of polymer solutions with the entanglement concentration ϕe is denoted as df,e. Depending on the entangled chain network, df,e is either experimentally obtained or derived from the extrapolation of the linear line.

Table 2. Scaling Parameters Derived to Correlate the Diameter of Electrospun Fibers exponent for df ∼ η0a

exponent for df ∼ (ϕ/ϕe)b

df,e (nm)

PS/THF PAN/DMF PDLLA/DMF PVA/H2O

0.41 0.48 0.45 0.53

2.73 2.07 2.49 2.59

553 142 75 67

PET2/TFA PET1/TFA nylon-6/FA nylon-46/FA PBT/TFA PNIPAM/DMF PET-co-PEI/(CF/DMF)12

0.78 0.68 0.61 0.62 0.73 0.80 0.80

2.61 2.05 2.36 2.25 2.83 2.54 2.60

375 226 44 41 145 104 180

polymer/solvent

range of 2.05−2.83 (Table 2). In this study, commercial polymers with a relatively broad molecular weight distribution are used. Thus, shorter chains may not be involved in the entangled network, and longer chains may possess a higher number of entanglements per chain. Moreover, the loss of entanglement during electrospinning is prone to taking place for the lightly entangled solutions due to the mechanism of the convective constrain release. These factors may influence the exponent of 2.3 in eq 1 derived from the monodisperse polymer. To further explore the genuine exponent between df and ϕ, we plotted the reduced fiber diameter df/df,e as a function of ϕ/ϕe and shown in Figure 10. df,e is the diameter of the fibers electrospun from the solutions with a concentration of ϕe. Table 2 presents the df,e values, which are experimentally measured or extrapolated for the various polymer solutions (Figure 9). The solutions with higher conductivity (such as nylon-6, nylon-46, and PVA solutions) possess lower df,e. All data are superimposed together to form a master curve with a

the appropriate scaling variable for the fiber diameter, we considered the entanglement density υ(ϕ), which is the number of the entangled strands per unit volume. For a polymer solution with a volume concentration of ϕ (≥ϕe), the entanglement density is estimated as follows:16 υ(ϕ) = G N0(ϕ)/kT = (ρp NA /Me(1))ϕ2.3

(1)

G0N(ϕ)

where is the plateau modulus of the entangled polymer solution, k is the Boltzmann constant, T is the absolute temperature, ρp is the polymer density, Me(1) is the entanglement molecular weight at melt (undilute) state, and NA is Avogadro’s number. Theoretically, the exponent of 2.3 in eq 1 is valid to polymers in the θ and good solvents. Experimentally, the exponent is between 2.0 and 2.5 for the monodisperse polymers.32 For a fixed polymer/solvent pair with different volume concentrations, the entanglement density ratio of the solution with a concentration of ϕ with respect to that of ϕe is expressed by the following: υ(ϕ)/υ(ϕe) = G N0(ϕ)/G N0(ϕe) = (ϕ/ϕe)2.3

(2)

Figure 9 shows the log−log plots of df versus the reduced concentration ϕ/ϕe based on the conjecture of df ∼ ν(ϕ). In spite of the significant differences in the solution properties (i.e., surface tension and electrical conductivity) and processing variables (Table 1), the derived exponent is within the narrow

Figure 10. Plot of the reduced fiber diameter (df/df,e) versus the reduced solution concentration (ϕ/ϕe). The diameter of the fibers electrospun from the given polymer solution with a concentration of ϕe is denoted as df,e. G

DOI: 10.1021/acs.macromol.6b00519 Macromolecules XXXX, XXX, XXX−XXX

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to high strain rates) for the entangled solution (ϕ > ϕe) with a higher concentration. Moreover, melts in the absence of appreciable amount of high-MW component show little, or even no, strain hardening.33 For lightly entangled solution commonly used for electrospinning, the average number of entanglements per chain is low (ca. 2−8) for the present polydisperse PNIPAM sample. Longer chains may have only a slightly larger Z than the shorter chains; however, the difference in Z is low. Thus, the role of molecular weight distribution may not be important during uniaxial extension of lightly entangled polymer solutions. Based on the above arguments, it is of importance to obtain the concentration dependence of τd and τR in order to compare with the strain rates to which the fluid elements are subjected during electrospinning. As shown in Figure 5b or more explicitly in Figure S3d, the fluid element experiences timedependent strain rates in the cone/jet section prior to jet whipping; thus, the nonequilibrium state of the chain conformations is expected during electrospinning. The strain rate is extremely small inside the Taylor cone. However, near the cone apex, a significant jet stretching is initiated by the electrical stresses after the stresses outweighing the surface tension. For the selected electrospinning solutions (4, 9, 15, 17, 19, and 21 wt %), the derived strain rates near the cone apex (region I) as well as that in the main jet (region II) are shown in Figure 11. For entangled solutions, the strain rate is found to decrease with increasing the polymer concentration. However, the unentangled 4 wt % solution exhibits a different trend. Also included in Figure 11 are the τd of polymer solutions obtained from the terminal region at low frequencies since there was no crossover between the G′ and G″ curves for our commercial polymers used (Figure 1a). As expected, τd−1 is decreased with

scaling relationship of df/df,e = (ϕ/ϕe)2.5 (Figure 10). The exponent of 2.5 is consistent with the theoretical value of 2.3, provided that df is proportional to the entanglement density υ(ϕ). On the basis of eq 2, our results infer that for a given polymer/solvent pair the key rheological property in determining the fiber diameter is the plateau modulus of the entangled polymer solution rather than its viscosity. The selected polymer solutions with different concentrations are subjected to a similar electrical stress to stretch the fluid element provided that the applied voltage (V) and tip-tocollector distance (H) are fixed. Considering that the straight jet end diameter is independent of the solution concentration (Figure 7), we further conclude that the stretching of the elastic chain network in the jet-whipping region caused by the induced electrical stress is the dominant process in determining the fiber diameter. When subjected to a similar electrical stress induced by the electric field of V/H, the fluid element composed of a swollen entangled network with a high plateau modulus is difficult to be stretched; the dissipation or relaxation processes of the stored energy in the deformed fluid element may strongly influence the fiber formation process and fiber diameter, clarification of which deserves future works. 3.5. Molecular Aspect Underlying the Proposed Universality. Figure 10 shows that the fiber diameter is approximately scaled with the plateau module G0N(ϕ), which is a linear viscoelastic parameter. At first glance, it is quite surprising since the electrospinning generally exhibits the strong flow and the fast processing time so that the nonlinear rheological behavior is expected. To understand this unique fact, a molecular view is provided as follows. For solutions with a concentration higher than ϕe, two intrinsic time scales (τd and τR) should be considered in order to characterize polymer chain conformation during the electrospinning. τR is the longest Rouse relaxation time of the backbone chain, and its inverse (τR−1) represents the polymer stretch relaxation rate. τd is the time required for a polymer chain to reptate the length of a fictitious tube. It represents the hindrance to the chain motion due to the presence of entanglements. Hence, τd−1 stands for the polymer orientation relaxation rate. For monodisperse polymers, the separation of these two time scales is related to the number of entanglements per chain Z = Mw/Me(ϕ) (where Me(ϕ) = Me(1)/ϕ1.3 in our case) in the given solution, and a simple relation of τd ∼ τRZ1.4 is theoretically derived.33 Hence, the reptation is slower than the relaxation of polymer stretch, if the polymer chain contains many tube segments (entanglements). Depending upon the strain rate ε̇ setup by the uniaxial extensional flow in the electrospinning process, three situations could be encountered.34 When ε̇ < τd−1, linear viscoelasticity is valid and the steady extensional viscosity assumes a value of 3η0. In this regime, the stress (σ) scales linearly with the strain rate, i.e., σ/G0N(ϕ) = 3ε̇τd. When τd−1 < ε̇ < τR−1, the polymer chains become increasingly aligned in the principal stress direction, but the chain stretching is not significant. In this regime, the steady extensional viscosity decreases with increasing strain rate (i.e., extensional thinning), and the stress is expressed by σ/G0N(ϕ) ∝ (ε̇τd)0.5.34 At still higher strain rates when ε̇ > τR−1, the chain stretching begins and the steady extensional viscosity increases, giving rise to the extensional thickening behavior. Significant strain-hardening is often observed for the dilute solution (ϕ < ϕ*) with long polymer chains;35 however, this effect is gradually reduced (or delayed

Figure 11. Comparison of the strain rate of the electrospinning jet (presented by the symbols ×), polymer orientation relaxation rate τd−1 (△), and polymer stretch relaxation rate τR−1 (= τd−1Z1.4, ▼) for the PNIPAM/DMF solutions. τd−1 is experimentally determined from the small-amplitude oscillatory shear measurement. Dashed line is the concentration dependence of τR−1 derived from eq 3 by using the measured solution shear viscosity. For a given solution jet, two strain rates are displayed; the upper strain rate is that in the vicinity of Taylor cone apex (region I), and the lower one is that in the main jet (region II). Beaded fibers are obtained from the 4 and 9 wt % solutions, whereas uniform fibers are obtained from the 15, 17, 19, and 21 wt % solutions. H

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is of importance to note that the strain history of jet segments is not homogeneous along the spin line because the fluid element undergoes time-dependent strain rates with intermittent processes of the chain stretching and the chain/tube relaxation. Our results infer that the position- or time-dependent ε̇ in the straight jet region (regions I and II) gives rise to ε̇τd value much larger than 1.0 (ca. 11−20) as revealed in Figure 11, except in the jet segment near the jet end. Hence, the chain orientation/ stretch occurs, and the nonlinear behavior of polymer solution is likely. It is also noted that not only the magnitude of ε̇τd but also the Hencky (total) strain of the fluid element determines the flow behavior of the electrospinning solution. The strain of the jet is evaluated by ∫ ε̇/νj dz. The cumulative strain in the region I is subjected to uncertainties due to the presence of limited data available in the small z gap (Figure S3), whereas those in the region II can be correctly determined to be 0.29, 0.77, 0.88, 1.41, 1.63, and 2.12 for the 4, 9, 15, 17, 19, and 21 wt % solution, respectively. It is intriguing to note that despite the high strain rate, the flowing time in the main jet (region II) is very short, ca. 1−4 ms. Under the assumption that the deformation mode in strain is additive, the Hencky strains for the 4, 9, 15, 17, 19, and 21 wt % solution are estimated to be ca. 0.48, 1.87, 1.30, 2.36, 2.42 and 2.87, respectively. Upon reaching the region III of zero strain rate with a short duration (ca. ∼1 ms), a relaxation of stretched chains may proceed, to some extent, to reduce the cumulative strain. Provided that the Hencky strain is larger than 1.0−1.4,38 the nonlinear viscoelastic flow behavior is frequently seen for the moderately or highly entangled solution. For the lightly entangled solution (Z = 13.1 and 15), however, Sridhar et al.34 have shown that G0N(ϕ) is a suitable parameter to describe the steady extensional stress at constant strain rates, the rate of which were varied over an appropriate range; the simple relation of σ/G0N(ϕ) = 3ε̇τd is applicable from linear viscoelastic region (ε̇ < τd−1) even up to the chain stretch region (i.e., ε̇ ≈ τR−1 as shown in Figure 4d of ref 34). Judging from the high strain rate (τd−1 < ε̇ ≈ τR−1) induced by the electrical force in the straight jet region, we believe the fiber formation processes are related to the nonlinear extensional viscosity growth function which controls the flow and deformation behavior of the solution. Even under the nonlinear elongation flow process in the straight jet and in the whipping jet, the entanglement density of the lightly entangled solutions (Z = 2−8) may serve as a key parameter which controls the fiber formation process leading to the observed fiber diameter on the collector as a function of ϕ/ϕe (Figure 10). The nonlinear stretching of the entangled strands involves by and large the stretching-induced disentanglements. However, the number of the residual entanglements existing in various stages of the fiber formation process is intimately correlated with the number of the entangled strands originally existing in the solution. Thus, the plateau modulus G0N(ϕ) characterized by the linear viscoelasticity influences the whole fiber formation process. 3.6. Proposed Hypotheses for Structural Evolution in Electrospinning Jet. Based on our findings in Figures 7 and 10, two important facts deserve more attention. The first is that the jet diameter in region III does not scale with the plateau modules (i.e., independent of the modulus). The second is that the fiber diameter observed after the jet whipping process scales with the plateau modules. These two facts may infer the following strikingly important hypotheses.

increasing polymer concentration. The strong concentration dependence of τd−1 may reflect the change of segmental friction as well as the nature of polydisperse polymers used. It is known that the segmental friction would have increased on an increase of ϕ above 0.1 or 0.2, depending upon the polymer/solvent interaction. It should be also reminded that τd is strongly dependent upon the molecular weight distribution of polymer used; τd is pronouncedly increased with the presence of high molecular weight components. Regarding the Rouse time, Osaki et al.36 have proposed a precise expression to deduce the τR of an entangled polymer solution; τR = (aMw/1.11cRT)2, where a is the prefactor of the Rouse modulus expression G′(ω) = aω1/2, obtained from fitting of the storage modulus at high frequencies ω just beyond the rubbery plateau regime. However, the rubbery plateau regime is rather difficult to reach for common electrospinning solutions which possess merely few numbers of entanglements per chain. In spite that the concentration−temperature superposition principle may be applied to construct the master curve in order to deduce the prefactor a, precaution should be taken for the likeness of solution phase separation, which would occur when the oscillatory shear measurement is performed at temperatures close to the upper (or lower) critical solution temperature. In the absence of data for the full G′ spectrum other than the zeroshear viscosity η0, an estimate of τR can be found from the relationship between η0 and polymer molecular weight Mw:36,37 τR = (6η0M w /π 2ρp ϕRT )[Me(ϕ)/M w ]2.4 = 0.61M w −1.4Me(1)2.4 ϕ−4.12η01.0 /ρp RT

(3)

for the entangled solution with the relation of Me(ϕ) = Me(1)/ ϕ1.3. When density of PNIPAM of 1.1 g/cm3, Mw of 650 000 g/ mol, and Me(1) = 54 000 g/mol (Figure S4) are substituted, τR for the entangled solution at 25 °C can be estimated. Equation 3 is subjected to certain uncertainties but enable a priori estimation of τR using only the zero-shear viscosity of the electrospinning solution. Based on eq 3, the concentration dependence of τR−1 is shown in Figure 11 by the dashed line. Presumably for monodisperse polymers, it reveals that τR−1 is ca. 800−103 s−1 for the entangled solution regime (ϕ > ϕe). It is noted that eq 3 is empirical and suitable for highly entangled solutions with a large Z so that the separation between τd−1 and τR−1 is sufficient. For commercial polymers with an average molecular weight of Mw, the τR−1 derived from eq 3 is taken as the upper bound magnitude since the high molecular weight species may cause the Rouse relaxation rate smaller. Because lightly entangled solution (Z = 2−8) is used for electrospinning, τR−1 may be better estimated by τd−1Z1.4 using the experimentally determined τd; the obtained data are also shown in Figure 11 by the closed symbols. It is found that τd−1 < ε̇ ≈ τR−1 (at least for the high molecular weight species) is prevailing in the major straight jet region (regions I and II) for the entangled solutions, which produce bead-free fibers. This finding is supported by the fact that no bending instability is seen in region III. In other words, the chains in region III, after experiencing an effectively high stretching rate in regions I and II, are not yet fully relaxed, even though the strain rate is zero in this region. The residual elongation must be large enough so that the viscoelastic longitudinal retractive force along the jet is larger than the Coulomb force induced by the charged jet segments to cause the bending instability,4 thereby suppressing the jet whipping and promoting the straight jet in region III. It I

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Macromolecules i. Hypothesis I. The jet in region III may not have a homogeneous structure composed of the stretched entangled strands swollen uniformly with the solvent but rather have the modulus-dependent internal string-like structures which are composed of bundles of stretched chains having their internal polymer concentration higher than the polymer concentration of their surrounding medium within the jet. They are aligned parallel to and dispersed in the jet solution having the concentration-independent jet diameter. The larger is the plateau modulus, the larger is the total cross-section of the strings which is anticipated to eventually control the fiber diameter. The string-like structures may be formed via flowinduced phase separation,39−41 the driving force of which depends on the stress level (but not the total strain) imposed on the entangled polymer solution. This stress level is controlled by the plateau modulus G0N(ϕ), which may provide a key factor to account for our experimental finding on df/df,e = (ϕ/ϕe)2.5 ∼ G0N(ϕ)/G0N(ϕe). There will be a sufficient time for the flow-induced phase separation to occur in the system via the solvent-squeezing mechanism inherent to the dynamic asymmetry system.41 In the case of the wet fiber spinning process reported in ref 41, an onset of the phase separation occurred in the time scale of order of 20 ms. ii. Hypothesis II. The jet in region III may still have a homogeneous structure composed of the stretched entangled strands swollen with the solvent. The higher is the polymer concentration, the larger the concentration of the stretched entangled strands is within the homogeneous jet with the concentration-independent diameter. The homogeneous jet undergoes fiber formation through the jet-whipping process which involves further nonlinear stretching of the network strands, solvent evaporation, and the syneresis of the solvent from the jets. The syneresis occurs as a consequence of the stress-induced macroscopic phase separation between polymers and solvent and thereby depends on the plateau modulus. The phase separation in the jet whipping process in hypothesis II may occur through either a one-step or twostep process. In the one-step process, the phase separation may occur from the stretched strands of the entangled polymer chains uniformly swollen by the solvent. On the other hand, the two-step process may involve first formation of the string-like structures dispersed in the medium within the jet as in the case of hypothesis I, which is followed by a second-step process involving the syneresis of the solvent from the jet via the phase separation between the string-like structures and their medium composed mostly of the solvent. The phase separation in either hypothesis I or II is indispensable for the fiber formation process. This is because the phase separation compensates the free energy cost due to the reduction of a very large conformational entropy loss encountered by the stretched chains swollen and isolated by solvent molecules by the gain of the free energy due to a reduction of the enthalpy via intermolecular associations of the chains into the bundles of the stretched chains. It is crucial to investigate the proposed hypotheses and the roles of G0N(ϕ) on them. It is also important to clarify where the flow-induced phase separation causing the nonlinear behavior of the polymer solutions takes place along the jet, i.e., in region I, II, or III in the straight jet or in the whipping jet. However, the investigation and the clarification are far beyond the scope of the present work but definitely deserve future work. Nevertheless, we believe from general viewpoints of structural evolution processes that the linear viscoelastic parameters (ϕe

and G0N) play important roles in transient fiber formation processes, thereby on the final fiber structure and fiber diameter. We like to stress the point that entangled polymer solutions behave like elastic gels because of the high deformation rate during electrospinning; therefore, the final fiber diameter is phenomenologically considered to be mainly scaled with the plateau modulus rather than the solution viscosity. In addition, we would like to remark that the deformation-induced liquid−liquid phase separation, the driving force of which depends on ϕ and thereby ν(ϕ), is likely to occur in the electrospinning jet and evolves a series of various dissipative structures, which eventually results in the formation of the fibers observed on the collector. A more advanced work of in-situ light scattering experiments is required for a better understanding of the evolution of the dissipative structures in the straight jet. In this paper we would like to leave the clarification of the physical processes giving rise to the relationship between the fiber formation and the observed scaling law, df/df,e = (ϕ/ϕe)2.5, as an important future work in this field.

4. CONCLUSIONS To determine the importance of the entanglement density on fiber formation, we electrospun different polymer solutions with various properties to investigate the concentration effect on the fiber diameter. A simple scaling law between the “reduced fiber diameter” and “reduced concentration” is developed on the basis of the 10 solutions studied. The fiber diameter is scaled with the polymer concentration with an exponent of 2.5, which is consistent with the theoretical value of 2.3 for the scaling relation between the plateau modulus of the entangled solution and polymer concentration. The disparity of the two exponents may result from the polydispersity of polymers used; at the given entanglement concentration (ϕe) shorter chains may be inactive in the formation of entangled network. Nevertheless, our results imply that the plateau modulus of the polymer solution rather than the solution viscosity plays a key role in determining the fiber diameter. We conclude that the rapid stretching of the elastic gel consisting of the network chain developed by the entanglements is the main mechanism of formation of the fibers during electrospinning. The level of the stretching force is relevant with the electric field and solution conductivity. For an effective stretching, the stretching rate is higher than the Rouse relaxation rate of the entangled network. The detail process and mechanism of the likely stretching-induced structure formation into the fibers deserves future work. It should be reminded that the fibers are dried, becoming free from the solvent, during the jet whipping process and after forming the sheet-like fabric. The solvent removal during the jet whipping process may be due to not only the solvent evaporation but also stress-induced syneresis of the solvent from the stretched swollen gel (stress-induced phase separation between solvent and polymer).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b00519. Figures S1−S4 (PDF) J

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Polymer Solutions: Good Solvent, Non-specific Polymer-polymer Interaction Limit. Polymer 2005, 46, 3372−3384. (15) Zeng, F.; Tong, Z.; Sato, T. Molecular Chain Properties of Poly(N-isopropyl acrylamide). Sci. China, Ser. B: Chem. 1999, 42, 290−297 . In this paper, the Mark−Houwink equation for monodisperse PNIPAM with a molecular weight range of 22−1300 kg/mol in THF solvent at 25 °C is derived to be [η] = 6.90 × 10−5Mw0.73. According to the polymer supplier, the intrinsic viscosity of PNIPAM in THF solvent measured at 25 °C is 1.202 dL/g. Hence, the corresponding viscosity average molecular weight is about 650 000 g/ mol.. (16) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: London, 2003. (17) Uemura, Y.; Fujimura, M.; Hashimoto, T.; Kawai, H. Application of Light-Scattering from Dielectric Cylinder Based upon Mie and Rayleigh-Gans-Born Theories to Polymer Systems. 1. Scattering from a Glass-Fiber. Polym. J. 1978, 10, 341−351. (18) Wang, C.; Cheng, Y. W.; Hsu, C. H.; Chien, H. S.; Tsou, S. Y. How to Manipulate the Electrospinning Jet with Controlled Properties to Obtain Uniform Fibers with the Smallest Diameter?-a Brief Discussion of Solution Electrospinning Process. J. Polym. Res. 2011, 18, 111−123. (19) Wang, C.; Hsu, C. H.; Lin, J. H. Scaling Laws in Electrospinning of Polystyrene Solutions. Macromolecules 2006, 39, 7662−7672. In this paper, rheological properties and electrospinning of atactic polystyrene (PS) in tetrahydrofuran (THF) solvent were investigated. For the PS polymer used, the weight average molecular weight and polydispersity index were 300 000 g/mol, and 2.3, respectively. (20) Wang, C.; Chien, H. S.; Hsu, C. H.; Wang, Y. C.; Wang, C. T.; Lu, H. A. Electrospinning of Polyacrylonitrile Solutions at Elevated Temperatures. Macromolecules 2007, 40, 7973−7983. In this paper, rheological properties and electrospinning of poly(acrylonitrile) (PAN) in DMF solvent were investigated. The viscosity average molecular weight of PAN was 141 400 g/mol. (21) Wang, C.; Chien, H. S.; Yan, K. W.; Hung, C. L.; Hung, K. L.; Tsai, S. J.; Jhang, H. J. Correlation Between Processing Parameters and Microstructure of Electrospun Poly(D,L-lactic acid) Nanofibers. Polymer 2009, 50, 6100−6110. In this paper, rheological properties and electrospinning of poly(D,L-lactic acid) (PDLLA) in DMF solvent were investigated. For the PDLLA polymer used, the weight-average molecular weight and polydispersity index were 178 000 g/mol and 2.1, respectively. (22) Tsou, S. Y.; Lin, H. S.; Wang, C. Studies on the Electrospun Nylon 6 Nanofibers from Polyelectrolyte Solutions: 1. Effects of Solution Concentration and Temperature. Polymer 2011, 52, 3127− 3136. In this paper, rheological properties and electrospinning of polyamide 6 (nylon-6) in formic acid solvent (FA, 99 vol % purity) were investigated. The weight-average molecular weight of nylon-6 was 35 000 g/mol. The nylon-6/FA system is a polyelectrolyte solution. At low concentrations, due to the electrostatic charge repulsion along the backbone chains, the typical characteristic for the polyelectrolyte is obtained ηsp ∼ ϕ0.5. When the nylon-6 concentration is high, the electrostatic interactions between backbone chains and solvents are highly screened and neutral solution dynamics are recovered, i.e., ηsp ∼ ϕ3.75. For a simple comparison with other solutions, we apply a consistent approach (as shown in Figure 2, the initial concentration at which the final linear domain is reached) to determine the ϕe for our nylon-6 solution. It should be noted that our determined value corresponds to the onset of the concentrated regime defined traditionally for the polyelectrolyte solution. (23) Wang, C.; Lee, M. F.; Wu, Y. J. Solution-Electrospun Poly(ethylene terephthalate) Fibers: Processing and Characterization. Macromolecules 2012, 45, 7939−7947. In this paper, rheological properties and electrospinning of poly(ethylene terephthalate) (PET) with two average molecular weights (PET1 and PET2) in trifluoroacetic acid (TFA) solvent were investigated. The viscosity-average molecular weights for the PET1 and PET2 polymers were 14 260 and 25 960 g/mol, respectively.

AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (C.W.). *E-mail [email protected] (T.H.). Notes

The authors declare no competing financial interest. T.H.: Professor Emeritus of Kyoto University.

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ACKNOWLEDGMENTS The authors are grateful to the Ministry of Science and Technology, Taiwan, R.O.C., for the research grant (MOST103-2221-E-006-262-MY3, NSC101-2221-E-006-093) that supported this work. This research was, in part, supported by the Ministry of Education, Taiwan, R.O.C. The Aim for the Top University Project to the National Cheng Kung University (NCKU). C.W. thanks Prof. B. Chu for a critical reading of the first-draft manuscript and Prof. G. C. Rutledge for the hospitality during his stay in MIT in 2012. The authors are also thankful to the anonymous reviewers for the helpful comments on the nonlinearity and the Rouse relaxation rate.

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