Spinnability of Dilute Polymer Solutions - Macromolecules (ACS

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Spinnability of Dilute Polymer Solutions A. Ya. Malkin,*,† A. V. Semakov,† I. Yu. Skvortsov,† P. Zatonskikh,† V. G. Kulichikhin,† A. V. Subbotin,†,‡ and A. N. Semenov§ †

A.V. Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences, Leninskii Prospect 29, Moscow 119991 Russia A.N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Leninskii Prospect 31, Moscow 119071, Russia § Institut Charles Sadron, CNRS-UPR 22, Universite de Strasbourg, 23 rue du Loess, BP 84047, 67034 Cedex 2 Strasbourg, France ‡

S Supporting Information *

ABSTRACT: Experiments carried out on a series of seven different polymers with molecular weights varying over a wide range have allowed us to confirm that stable jets can be obtained at concentrations much below the crossover point. A jet was considered as stable if its lifetime exceeds the Plateau−Rayleigh time by several orders of magnitude. The systematic study carried out for poly(ethylene oxide) solutions in a wide range of high molecular weight showed that the lowest concentration at which a stable fiber can still be formed is scaled by [η]−2.14±0.3 or M−1.63±0.29. However, for the domain of not so high M, the spinnability concentration corresponds to the onset of entanglements and scales as M−0.70±0.14, which is the same as the dependence of the crossover concentration on molecular weight. The difference in the scaling exponents reflects two possible regimes of stable fiber formation in fiber spinning. These exponents are close to those obtained by Palangetic et al. [Polymer 2014, 55, 4920] for other polymer solutions in the electrospinning experiments. Several examples of spinnability at very low concentrations for other polymer solutions are demonstrated. A possibility of the formation of stable jets from dilute solutions is explained by an increase of the intermolecular interactions of extended macromolecular chains, resulting in the phase separation and leading to the formation of fibers created by oriented macromolecules. The theoretical considerations show that there are two sources of jet stabilization at low concentrations (high M), namely, the coil−stretch transition and demixing of the polymer solution.

1. INTRODUCTION

a four-roll mill for creating alignment of dilute solutions and the birefringence method for detecting the macromolecule conformations.4−6 De Gennes considered an uncoiling of macromolecular chain under the longitudinal velocity gradient, regarding it as the coil−stretch transition which can be abrupt (the first-order transition) or continuous (the second-order transition).7 Effects of such kind have been really observed; they reveal, in particular, an abrupt alignment of the chains at the critical value of the Weissenberg number5,7,8 Wi = ε̇τ∼ 1 (determined as the product of the elongation rate, ε̇, and the maximal relaxation time τ which can be considered using different molecular models).9

The possibility of fiber spinning is a unique property of polymers well-known in natural phenomena (e.g., spiders’ webs and silk formation) as well as numerous technological processes such as the artificial fiber production and electrospinning. The physics standing behind this effect is directly related to the development of large rubbery (elastic) deformation as a consequence of the coil to extended chain transition of macromolecules.1 The basic effect of extension in macromolecular systems is orientation of a polymeric chain resulting in their significant anisotropy. Perhaps, Peterlin was the first who revealed on anisotropy of segmental interaction in uniaxial extension and connected the birefringence with the degree of anisotropic polarizability.2,3 The conformation behavior of macromolecules in extension has been extensively studied by the Bristol research group using © XXXX American Chemical Society

Received: April 6, 2017 Revised: September 15, 2017

A

DOI: 10.1021/acs.macromol.7b00687 Macromolecules XXXX, XXX, XXX−XXX

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also used. (2) Aqueous solutions of acrylamide copolymers. Four samples with molecular weights in the range of 6−15 MDa were produced by SNF Floerger. This series included one anionic (charge level 30%) and three cationic (charge levels of 3, 14, and 35%) species. One sample was produced in RF. (3) Solutions of polyacrylonitrile (PAN) in dimethyl sulfoxide. A series of 17 samples of homo- and copolymers of PAN (all produced in RF) with molecular weights from 54 to 836 kDa and polydispersity from 1.7 to 8.1 were used. Nine samples were synthesized by polymerization in supercritical CO2. Five samples were prepared by controlled free-radical copolymerization. Three samples were commercial products.16 (4) Aqueous solutions of hydroxypropyl cellulose (HPC). This series included six samples with molecular weights of 80−1150 kDa. All samples were produced by Klucel. (5) Aqueous solutions of poly(vinyl alcohol) (PVA). The series included three samples produced by Cevol and one sample supplied by Spectrum Chemical. (6) Aqueous solutions of polyvinylpyrrolidone (PVP). Two samples with molecular weights of 30 and 90 kDa were produced by ISP Technologies. (7) Solutions of cellulose in cadoxen. Three samples were studied. The first was produced by the Lyocell Technology on the laboratory stand using Baikal Lake cellulose (RF). The degree of polymerization was 500. The second sample was provided by the Kotlass pulp and a mill for viscose process. Its degree of polymerization was 650. The third sample was an industrial grade of cellophane with a degree of polymerization of 200. All samples under study were characterized by their intrinsic viscosity, [η]. Viscometric measurements were carried out in the Ubbelohde viscometer at 30 ± 0.1 °C. The characteristic time of the viscometer was not less than 60 s. Calculations of the intrinsic viscosity were made with standard protocol taking into account an average of five experiments. The Mark−Houwink−Kuhn equation was used for the correlation between the intrinsic viscosity values, [η], and molecular weight, M:

The alignment of macromolecules leads to their selforganization which can be so high as to result in extensioninduced crystallization.6 In extension of solutions of amorphous polymers, the alignment happening at Wi ∼ 1 leads to formation of supramolecular structures as has been observed by Kalashnikov and Tsilauri for planar elongational flows of dilute polymer solutions; this effect has been documented for polymers with the intrinsic viscosity [η] exceeding some critical limit.10 Kinetic aspects of this transition in dilute solutions were considered by Cifre and Torre,11 who showed that the kinetics of the transition are related to the relaxation time distribution and time dependence of the fraction of remaining coils. Some more complicated situations taking into account the hysteresis of the transition and superposition of shearing and longitudinal flows were discussed.12,13 The examination of the mechanism of the electrospinning of polymer solutions which was identified in analogy to uniaxial extension of a fluid jet14 emphasized the role of elasticity in the formation of electrospun fibers where elastic (rubbery) deformations are accompanied by the growth of the apparent elongational viscosity.15,16 Meanwhile, we note that the coil−stretch condition is necessary but is insufficient for fiber spinning because the fiber integrity provided by intermolecular interactions is also required. Usually, the uniaxial extension of polymeric fibers is considered for polymer melts or concentrated solutions. However, a reasonable and important question is what is the minimum concentration of a solution at which fiber formation is still possible. The obvious answer might be that fiber formation is possible above the viscosity crossover concentration, C > C** ≈ 5/[η], corresponding to the onset of macromolecular entanglements. Experimental data obtained in our previous study carried out on a single polymer−solvent pair seem to confirm this view.14 Meanwhile, there are some evident exceptions, since stable jets (fibers) for very high molecular weight samples could exist at concentrations well below the overlap concentration of polymer coils C < C* ≈ 1/[η], i.e., in the dilute solution regime.17 A possibility to obtain fibers from polymer solutions at concentrations below the crossover point C* has been also demonstrated.14,17 This study aims to provide an assessment on the possibility of fiber spinning for different polymer−solution pairs in a wide molecular weight and concentration ranges above and below the crossover point. We did not intend to measure the rheological properties of dilute solutions as was done already in many interesting and important publications (see a comprehensive review18 where the authors also succeeded in measuring relaxation times of weakly viscoelastic polymer solutions under extensional flows). Here we establish and analyze the critical conditions for fiber formation in the studied systems and discuss the mechanism standing behind this phenomenon in the domain of dilute and very dilute polymer solutions.

[η] = K ηM α

(1)

The values of its parameters (Kη and α) are collected in Table 1.

19

Table 1. Values of the Parameters in Eq 1 (Solvent = Water; T = 30 °C) polymer

α

Kη × 104, dL/g

PVP PEO, PEG PAA PVA PAN

0.59 0.78 0.8 0.64 0.768

3.93 1.25 0.631 5.43 2.685

The results of the [η] measurements and a comparison of our polymer characterization with the producers’ data are given in the Supporting Information. Molecular-weight distribution of most samples is not wide. Of course, some fraction of high-molecular-weight chains is always present. However, the effect of the high-MW tails is rather weak, and despite the fact that the C* value for those fractions is lower than for the average MW, their concentration is too low for the entanglement formation. So, we suppose that it is reasonable to neglect the effect of the high-MW fractions. 2.2. Methods. Optics. The main experimental method was highspeed photography. This was realized with a Nikon v2 camera, providing the possibility of 1200 shots/s. Therefore, the time gap between two consecutive shots was ∼10−3 s. The shooting of a jet (fiber) was carried out under contour lighting: a sample was positioned against the light source. A LED matrix with power 10 W and size 10 × 10 mm was used, which created a bright uniform white backlight of a screen. Macro pictures were taken with the zoom lens Tokina 100f 2.8D and the maximum possible magnification. Estimations and real practice showed that the macro-optics and the LED lighting used allowed us to obtain clear pictures of the fibers with a size of 320 × 120 pixels and

2. EXPERIMENTAL SECTION 2.1. Samples. We used several series of different polymers with a wide range of molecular weights (see Table in the Supporting Information): (1) Aqueous solutions of poly(ethylene oxide), PEO, and poly(ethylene glycol), PEG. The range of molecular weights of PEO covered 40−8000 kDa and included 12 samples. Six of them were supplied by Sigma-Aldrich, three were produced by Dow, and the remaining three samples were produced in Russia (designation RF). One sample of PEG was produced by Arcos, and two samples were made in Russia. Two low-molecular-weight fluid samples of PEG were B


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the order 103 s−1 (acceleration = 400 m/s2, time of stretching is of the order of 5 ms, length of a fiber at rupture is of the order of 1−3 mm). This method is treated as “fast stretching”. According to the second protocol (method B), the needle was lifted up with the minimum possible acceleration, bearing in mind to have a rate of stretching much lower than the rate of fiber thinning obliged by the capillary forces. This method allowed for a low rate of deformation (i.e., “slow stretching”). In this case, the length of a fiber at rupture is of the same order though rupture is a statistical phenomenon, and in some experiments the length at rupture in slow stretching was higher than in fast stretching, but the difference was not significant. The time of elongation is of the order of 50 ms, and the velocity is approximately equal to 0.02 m/s; i.e., roughly, the velocity is 10 times lower and the time of stretching before rupture is 10 times higher than in the method A. The slow stretching deformation rates are of the order of 102 s−1. The values of the deformation rates can slightly vary for different samples under study, but the order of values is the same. The experimental procedure was as follows. Initially, a polymer solution with the given concentration was prepared. A part of this sample was taken for the stretching experiments. If the specimen with this concentration were able to form a fiber, the remaining solution was gradually diluted, and the stretching experiment was repeated. This procedure continued until reaching the concentration at which fiber spinning became impossible. The experiments were repeated at least 10 times for every concentration. All experiments were performed at 29 ± 1 °C.

the physical resolution of the picture of 16 μm per pixel (the real physical size of a photograph is 5.1 × 1.9 mm). Interpolation of the pictures allowed us to observe objects with diameters larger than 5 μm. The time resolution in the process of stretching was 0.83 μs. Devices and Experimental Protocol. The central goal of the work was to establish the minimum (limiting) concentration of a solution at which stable (on the time scale of the Rayleigh instability) fiber spinning is still possible. The general scheme of the experiment consisted in stretching a fiber from a droplet. This scheme is close to the one proposed in the original publication20 and afterward used in different modifications in numerous publications.21 The picture of an experimental device is shown in Figure 1, and the scheme illustrating the experimental procedure of fiber formation drawn from a droplet is shown in Figure 2.

3. RESULTS AND DISCUSSION 3.1. Measuring Intrinsic Viscosity. The controlling estimation of molecular weight of the polymer samples under study has been carried out via the standard protocol for measuring the intrinsic viscosity. In all the cases, the dependencies of the reduced viscosity on concentration were linear, which allowed us to find the intrinsic viscosity to the necessary accuracy. The overall results of measuring the intrinsic viscosity, the comparison with the producers’ data, and designations of samples used in this study are collected in the Table given in the Supporting Information. In the further discussion, the results of our measurements of the intrinsic viscosities of the samples will be used. Generally speaking, PEO is a surface-active agent for water. However, direct measurements of the surface tension of the PEO/water solutions have shown that the variation of the surface tension for the concentration range used here does not exceed 10% which is not significant for the results of this study discussed below.22 It is worth mentioning that the viscosity vs concentration dependencies for PEO have been the subject of numerous investigations. It is therefore interesting to compare our results to the most systematic data.23 This comparison is presented in Figure 3 with specific viscosity ηsp as the function and nondimensional concentration C[η] as the argument shown on the x-axis. It is seen that there is a very good correlation of our data with the results of a systematic study of PEO solutions.23 Moreover; we widened the range of concentration to the lowconcentration domain. Thus, Figure 3 presents the results of measuring the viscosity of PEO solutions over a very wide concentration range. Therefore, the comparison of our data with previously published data allowed us to precisely identify the viscosity-based crossover concentration: it is equal to C** ≈ 5/[η], in good agreement with the classical estimates.24 3.2. Choosing Conditions of Fiber Spinning. A qualitative picture of the behavior of a droplet in stretching

Figure 1. Main elements of the experimental device: 1 = rigid frame, 2 = lower cylinder, 3 = sample, 4 = cannula, 5 = trigger, 6 = elastic element, 7 = upper frame, and 8 = LED backlighting.

Figure 2. Scheme of stretching (fiber formation) with droplet with d = 8 mm. The 1 mL sample of a preliminary prepared polymer solution was dripped on the surface of the lower cylinder. In the initial state a droplet had the shape of a hemisphere with diameter of 8 mm and height of 2 mm. A cannula with diameter of 0.6 mm was moving down along the vertical guides to approach and touch the droplet. When touching, the needle is wetted with a solvent and is fixed at the starting position. The needle is then connected to the elastic spring, which creates the calibrated vertical force while the needle moves upward. Two experimental procedures realizing different rates of stretching were used. According to the first protocol (method A), stretching is realized at a given acceleration in the range of 500−2000 m/s2. This method allowed for the creation of high rates of deformation. The rough estimation of the deformation rate at rupture gives the values of C


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higher for slow stretching. Therefore, it is useful to highlight the spread in the observed results. This effect is illustrated by the examples collected in Figure 5. The reason for these results is rather interesting. Indeed, higher rates normally lead to the formation of more stable fibers. Just because of this, high rate stretching results in formation of longer filaments. It is well-known that rupture is a probabilistic phenomenon, and an increase in the fiber length increases the likelihood of its breakage. Most probably, it is this effect that is reflected in Figure 5. Let us now examine the PEG−PEO samples because this is the most complete series with a very wide range of molecular weights. The low molecular weight oligomer PEG 400 (sample P0.4k) with a rather low viscosity can be easily stretched. Snapshots of the consecutive stages of stretching realized by the two methods are shown in Figure 6. An oligomer with this rather low molecular weight can still demonstrate spinnability. Meanwhile, a sample with a lower molecular weight (P0.2k) cannot form filaments; i.e., it does not have enough rubbery elasticity. Therefore, the lowest molecular weight providing the formation of stable jets for undiluted PEG is close to Mmin ∼ 400. An important task is to construct the general correlation diagram defining the limit of spinnability. To this end it is necessary to choose the appropriate coordinates. The reciprocal intrinsic viscosity 1/[η] can be used for the x-axis since [η] is related to the hydrodynamic macromolecular radius RH27

Figure 3. Comparison of our results for specific viscosity ηsp values for PEO samples as a function of the reduced concentration C[η] with data from Ebagninin et al.23

looks similar in both explored methods of observation. A thin bridge between the two parts is usually formed before rupture. The length of the formed filament only slightly depends on the method of stretching. On the average, the filament lengths obtained from dilute solutions are the same. Meanwhile, the stability of the jet obviously depends on experimental conditions. However, it is important to stress that the lifetime of a jet in all cases is much longer than the corresponding times of Rayleigh instability. Indeed, the lifetime τPR associated with the Plateau−Rayleigh capillary instability of a filament of radius r is approximately τPR ≈ max(τin,τvis), where25,26 τin = 2.9 ρr 3/γ ,

τvis = 6ηr /γ

⎛ 3M[η] ⎞1/3 RH = ⎜ ⎟ ⎝ 10πNA ⎠

(3)

where M is the molecular weight (MW) of the polymer and NA is the Avogadro number. Thus, [η] reflects both the molecular weight and the macromolecular coil size. It is important to compare the lowest concentration providing spinnability with the concentration corresponding to the intermacromolecule contacts. According to the existing concepts,28,29 the dimensionless product C[η] is considered as a measure of the volume filled by macromolecular coils in a solution. At C*[η] ≈ 1 the contacts between coils become possible, and this concentration C* (usually called the crossover concentration) corresponds to transition from dilute to semidilute solution where such contacts should be present. The second important concentration point is C**[η] ≈ 5...6, marking the transition from semidilute to more concentrated regime characterized by formation of a special three-dimensional entanglement network. This point roughly coincides with the change in the slope of the viscosity vs concentration

(2)

are the lifetimes defined by inertial and viscous forces, respectively, ρ is fluid density, and γ is its surface tension. In this work we consider a jet as stable if its lifetime is at least 1 order of magnitude longer than τPR. For dilute solutions, fast stretching leads first to a deformation of the fluid drop. The limiting stage is either the appearance of a thin neck or rupture. For slow stretching, deformation immediately results in the formation of a thin neck or rupture rather than uniform stretching. This difference is clearly visible in the examples presented in Figure 4. The concentration corresponding to the onset of the principal ability to form stable fiber does not depend on the method of stretching. However, by decreasing the concentration below some limit where the fiber spinning becomes probable, the possibility to observe thin threads is remarkably

Figure 4. Polymer fiber snapshots illustrating the difference in the lifetime of a thin thread for fast (a) and slow (b) stretching of the dilute (2 × 10−3%) solution of poly(acrylamide) AA2 (M ∼ 6 × 106). The time from the start of stretching is shown below the samples (the coil size Rcoil ∼ 0.3 μm and Zimm time τZimm ∼ 3 ms). D


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Figure 5. Probability of fiber formation in stretching realized by two methods. The red and green columns correspond to the results obtained by the fast and slow stretching methods, respectively.

Figure 6. Snapshots of the consecutive stages of stretching of pure (without any solvent) PEG 400 (sample P0.4k) for fast (a) and slow (b) stretching.

dependence. As seen from Figure 3, it happens at C**[η] ≈ 5...6 for PEO solutions. Both conditions C*[η] ≈ 1 and C**[η] ≈ 5...6 are shown in Figure 7 (built for the PEO solutions) by solid straight lines, and the corridor between them corresponds to the transient situation of semidilute solutions where entanglements are not yet important. The condition of macromolecular contacts can be derived from eq 3. It is seen that RH is proportional to (M[η])1/3. According to eq 1, for PEO solutions, [η] ∝ M0.78 and RH is

scaled as M0.59. This coincides with the exponent value obtained by Devanand and Selser.27 Therefore, the upper line in Figure 7 is scaled as C ** ∝ M −0.78

(4)

Experimental points in this figure correspond to the lowest concentration, Cstab, of PEO solutions which allows for the formation of stable polymer jets on the time scale much longer than the Rayleigh instability time. This diagram can be also presented as the dependence of Cstab vs molecular weight (Figure 8). This diagram cleary demonstrates that there are two ranges of molecular weights with different types of conditions for spinnability. Below, these two ranges will be discussed separately.

Figure 7. Correlation diagram for the concentration limit of spinnability providing the possibility of fiber formation vs intrinsic viscosity for PEO solutions in the whole range of MWs. Lower and upper straight lines correspond to conditions C*[η] = 1 and C**[η] = 5, respectively.

Figure 8. Dependence of the minimal concentration necessary for spinnability on molecular weight of PEO. E


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However, in our opinion, the approach developed by Palangetic et al.17 is not totally consistent: They used the dumbbell model (cf. eq 5 of ref 17), which is not appropriate to describe the viscosity change upon chain stretching since this simplified model does not properly reproduce the effect of hydrodynamic interactions (HDI). The HDI effect strongly changes as polymer chain is stretched, giving rise to a stronger dependence of the effective viscosity on the chain size Rz in the flow direction: η ∝ CRz3/M. In the nearly fully stretched chain regime emerging above the coil−stretch transition the chain size is Rz = sLN ∝ M since the stretching degree s ≥ 0.5; therefore, the viscosity η ∝ CM2 (cf. Appendix). The latter result differs from the incorrect law η ∝ [η]CM2(1−ν) ∝ CM1+ν proposed in ref 17. Once this point is corrected, the M−2 scaling for Cstab is obtained (i.e., our result; see the next section). The inconsistency of the M−(1+ν) law predicted in ref 17 is also evident from the mere fact that it involves the Flory exponent ν which is not relevant for stretched chains: ν originates from a balance between chain elasticity and intramolecular interactions which are strongly suppressed for fully extended chains. It is noteworthy that our result for the extensional viscosity of dilute solutions of extended chains (η ∝ CM2) is in full agreement with the theory of ref 33. In fact, their result for the drag force is Fdrag ∼ ε̇ηsRz2, where ηs is the solvent viscosity (see eq 4 of ref 33 where we omit numerical and log factors and used our notation replacing their L with Rz = L). Obviously, the polymer-related normal stress difference is N1 ∼ cchainFdragRz, where cchain is the number concentration of polymer chains; hence, the extensional viscosity η ∼ N1/ε̇ ∼ cchainηsRz3. We thus arrive at the result quoted above (η ∝ CM2) since cchain ∝ C/M and Rz ∝ M. Now, let us consider the domain of low MW (high concentrations), i.e., the left part of Figure 8. The experimental data are presented in Figure 11. One can see that the

Let us now consider only the domain of dilute solutions lying below the line of C[η] = 1. These ten experimental points are presented in Figure 9.

Figure 9. Correlation diagram for the limit of spinnability providing the possibility of fiber formation for dilute PEO solutions. [η] is intrinsic viscosity.

The linear approximation gives the slope of −2.14 ± 0.3, i.e. Cstab ∝ [η]−2.14 ± 0.3

(5)

Figure 10. Correlation diagram for the limit of spinnability providing a possibility of fiber formation from dilute solutions of high MWs. The dashed line corresponds to the M−2 dependence (see the Theoretical section).

This correlation can be presented as a function of MW as shown in Figure 10. The linear approximation gives the following scaling relation: Cstab ∝ M −1.63 ± 0.3

Figure 11. Correlation diagram for the limit of spinnability providing a possibility of fiber formation from solutions of relatively low MWs.

(6)

Surely, there is a direct conformity between the exponents in both scaling relations determined by eq 1 via the exponent in the Mark−Houwink−Kuhn equation for PEO which is equal to 0.78. These sets of experimental data and the observed correlation could be compared with the theoretically predicted by the Palangetic et al.17 scaling law Cspin ∝ M−(1+ν), where Cspin is the low concentration limit providing spinnability. Evidently, this value is equivalent by its physical sense to Cstab considered in this study (but they are not identical by their definition, as will be shown below). The factor ν is the excluded volume exponent which is related to the exponent α in the Mark− Houwink−Kuhn equation by the relation α = 3ν − 1.30−32 In the case of PEO, α = 0.78 and ν ≈ 0.59, and consequently, the following result is expected: Cstab ∝ Mη−1.59. Indeed, this correlation is very close to our results expressed by eq 6.

dependence of Cstab on MW is scaled by M−0.70±0.14; i.e., the conditions of spinnability are quite different than in the case of weak solutions. It is worth reminding that the crossover concentration is scaled by M−0.78 (eq 4). At the same time, the dependence of the spinnability concentration on MW for the range of low MW is scaled by M−0.765 and M−0.665 for PMMA of two types.17 These exponents are almost the same as obtained in this work and presented in Figure 11. So, following Palangetic et al.,17 we identify two essentially different regimes of fiber spinning corresponding to entangled macromolecules above the crossover point and individual macromolecules in dilute solutions. The mechanism of fiber spinning in dilute polymer solutions is uncovered in the next section. The main general conclusion of the theory outlined in section 4 is that F


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the fiber, inhomogeneous light scattering spots appear, and a part of the sample becomes opaque. These results of such an experiment can be considered as a direct proof of phase separation in a dilute soluion at strong elongations (or normal stresses). It has therefore been shown that fiber spinning can occur through demixing of the solution. An example of the application of this method is demonstrated in Figure 13 for the fiber formation from PEO

there are two mechanisms of jet stabilization, namely, due to the coil−stretch transition and due to demixing. The concentration Cstab is connected with the coil−stretch transition and scales as M−2 with MW in the dilute regime. Note that in the general case we can define Cspin as the minimal concentration at which stable fibers (with practically infinite lifetime) can be spinned. Obviously, this requirement is stronger than the condition for Cstab: Cspin > Cstab. As discussed in the next section, the ultimate fiber stability can be achieved as a result of solvent evaporation or by extension induced polymer/solvent demixing. The latter process can occur if the current concentration C exceeds Cdem ∝ 1/M (see eq 18). Thus, Cstab < Cspin < Cdem. It is therefore not surprising that the exponent in the MW dependence for Cspin obtained in ref 17 (around −1.6) is higher than that (−2) for Cstab. Apparently, Cspin depends on the solvent evaporation rate and approaches Cstab when the rate is high. The concentration Cspin is determined in electrospinning experiments17 where the solvent evaporation rate is considerable and solid fibers are formed. In our experiments the evaporation rate is low, and we determine Cstab rather than Cspin. The aforementioned protocol was applied for all examined objects. However, the limited set of molecular weights did not allow to construct reliable correlation curves for solutions of some polymers. So, the data discussed below should be considered only as the proof that very different polymer solutions can form stable fibers (be spinnable) at concentrations much below the crossover point. Turning to cellulose-based polymers, we found that some HPC solutions demonstrate a qualitatively different behavior compared to the other polymer solutions. The HPC solutions in the range of the used concentrations have much higher viscosities than the other examined solutions and consequently much longer times of surface stability. A typical series of snapshots obtained during stretching of the HPC solutions are shown in Figure 12. One can see that the stretching of the HPC

Figure 13. Consecutive stages of stretching of 0.5% aqueous solution of PEO (sample P50). Slow mode of extension. The time step was 1/ 24 c. The sequence of the snapshots goes from left to right and from the upper row to the lower row.

solution. When a fiber begins to form, the sample loses transparency and becomes opaque, while a bright glow appears. Figure 13 shows the consecutive stages of this process. Phase separation was described many years ago for shear flows35 and subsequently observed by many authors. It can be very significant, and the shift of the cloud point can reach 30 K.36 The emergence of liquid droplets on the surface of a fiber under extension was clearly demonstrated by Frenkel’ et al.37 Phase separation as a result of stretching of a polymer solution (jet) has been well documented.38 This effect is quite similar to “blistering” described by Dunlap and Leal,39 where formation of isolated droplets has been clearly demonstrated. Figure 14 is reproduced here as an illustration of the phase separation phenomenon for a fiber of larger diameter. One can see the different stages of the phase separation: solvent release

Figure 12. Snapshots obtained during stretching of 5% aqueous solution of the HPC Klucel LF (sample HPC1.4). Fast stretching mode.

samples leads to the continuous thinning of the sample; it appeared impossible to detect the critical (lowest) concentration with this method. It is worth saying some words about the physics of the processes taking place during the stretching of a fiber produced from polymer solutions. The experimental technique and some principal results were published in our earlier paper.34 The idea of our experimental method consists in using an elongating fiber as an optical line. One of the most evident consequences of the phase separation is the emergence of an apparent opacity of the solution. A homogeneous medium (polymer solution) does not scatter light, but as the solvent begins to release out of

Figure 14. Solvent layer released on the surface of a strongly elongated fiber and the resulting drop of pure solvent are shown for a semidilute solution (C = 0.075) of polyacrylonitrile (MW = 516 kDa) in dimethyl sulfoxide (like in ref 34). G


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Macromolecules during fiber extension and the resulting formation of separate solvent droplets. These fluid drops can be simply removed by cotton wool. So, the phenomenon of phase separation induced by jet stretching has been definitely demonstrated for many systems. Thus, experiments confirm that there is a direct relation between an elastic deformation that is necessary for fiber formation and polymer/solvent phase separation during solution jet stretching. Shortly speaking, the following experimental facts confirm that we meet with the demixing during jet stretching rather than the unified droplets-on-spring structure. Applying a new technique of using a fiber as a light guide allowed us to fix the point where light scattering appears, and a bright glow reflects the formation of inhomogeneous domains due to the appearance of a new phase (demixing). So, demixing does take place in the process of stretching but not after it. Direct observations confirm formation of separate droplets on a fiber surface which can be taken off by cotton or blotting paper (as was documented in our former publication34). It is worth mentioning that the droplet formation in ref 38 was observed on nanofibers, but in our case the fiber diameter was much larger and this allowed us to observe droplets directly. All the above features evidence that we meet with the real demixing during the process of stretching. Figure 15 presents the final results of this study for most samples together. The points in this figure correspond to the

Figure 16. Correlation diagram for the limit of spinnability providing a possibility of fiber formation from dilute solutions of cellulose.

The experimental data presented in Figures 7 and 15 lead to the rather evident conclusion. There are two types of jet behavior in fiber formation. At high values of 1/[η] (the right part of the correlation diagram), the stable jet formation (spinnability) is determined by the existence of an entanglement network that is reflected by points lying in the corridor between two parallel lines corresponding to the criterion of the coil overlap crossover and entanglements formation. However, the situation can be quite different, and the requirement for entanglements is not absolutely necessary for spinnability. This is presented in the left part of the diagram in Figure 15 where the experimental points lie below the crossover concentration, and the difference in concentration can reach several orders of magnitude. A physical understanding of the latter regime is hinged on the possibility of extension thickening and another possibility that the continuum rubbery deformation of jets can be accompanied by phase separation due to extension and orientation of macromolecular chains (see the next section). The assumption of an enhanced interaction for highly extended polymeric chains was first proposed by Dunlap and Leal39 and recently found a more detailed physical explanation.41−43 This concept implies that extension-induced orientation of macromolecules can drastically reduce the energy of their repulsion, so the attraction between the chains gains more importance. This effect can lead to a tendency for formation of a concentrated polymer phase and a concomitant polymer−solvent segregation. The demixing becomes possible above a certain deformation rate. As shown experimentally, a stable jet formation can be realized at very low concentrations, much below the crossover C*. In this sense, the strongly stretched dilute solutions cannot always be considered as genuinely dilute (viz. a solution without intermolecular interactions between polymers). This statement was clearly demonstrated for nearly monodisperse polystyrene solutions.44 It was shown that the lowest concentrations necessary for the experimental manifestation of the effects of polymer viscoelasticity are much lower that the crossover concentration, C*, and that the relevant relaxation times are substantially longer than those expected from the standard small-amplitude oscillatory shear experiment or the Zimm molecular model. This conclusion is completely in accordance with the results reported above for a large variety of polymers; it is also supported by the theoretical studies.41−43,45 The experimental results reported here suggest that extended macromolecular chains can start to interact in stretching, forming a continuous phase that does not exist in dilute solutions at equilibrium or in shearing. It is important to stress that fiber spinning from a polymer solution can happen via a

Figure 15. Cumulative correlation diagram for the limit of spinnability, providing the possibility of fiber formation. The PAN data were partly taken from Semakov et al.16

samples which allowed us to obtain fibers with probability higher than 50%. Just such concentration is conventionally defined as the “concentration of stable spinning” or spinnability. The lower limits of the bars correspond to the concentrations below which the formation of fibers appears completely impossible. Some spread of experimental points can be explained by the difference in polydispersity of the samples.17 Indeed, polymers having high molecular weght fractions in their composition can create much higher rubbery deformations,40 and therefore “highly polydisperse polymer solutions are desirable for electrospinning”.17 Note that the number of points for other polymers than PEO is too low, and the spread of points is too wide to use other polymers for building correlation dependences. Only three points obtained for cellulose samples can be considered. In this case (see Figure 16) the slope can be estimated as −2.06 ± 0.64, which roughly agrees with the data presented in Figure 10. H


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Macromolecules separation of the solvent.34 A nonthermodynamic approach to this phenomenon was proposed by Cromer et al.46 where the initial solution state was treated as a mixture of two fluids, and the concentration fluctuations grow under extension due to the stress−concentration coupling (SCC) effect.47 However, the SCC theory does not predict any phase separation in unentangled solutions. Moreover, it was argued that SCC effect in unentangled polymer systems is always subdominant with respect to the extension-induced attraction of polymer segments.41,42 Finally, it is worth emphasizing the very close exponents in the scaling laws obtained in this work for Cstab and by Palangetic et al.17 for Cspin in spite of quite different scheme of stretching. A possible cause is that in the electrospinning experiments the solvent evaporation rate is high enough so that the concentration Cspin is close to Cstab.

apply the slender body approximation leading to the momentum and incompressibility equations48 ∂ ∂ ⎛⎜ 2 2 ∂v ⎞ ∂r ρr v − 3ηr 2 ⎟ = γ ; (ρr 2v 2) + ⎝ ⎠ ∂t ∂z ∂z ∂z ∂r 2 ∂ 2 + (r v ) = 0 ∂t ∂z

(9)

where r = r(z), ν = ν(z) is flow velocity along the jet (z-axis), and η is solution viscosity. The boundary conditions reflect the continuity of the momentum flux along the jet through the boundaries z1 and z2 between the zones. In the zone 1 (ν = 0, H = 0, p = p0) the momentum flux is J1 = −2πγa (we set p0 = 0 for simplicity). The momentum flux in the zone 2 is J = πr 2ρv 2 − 2πγr − πr 2σzz

where σzz is the longitudinal mechanical stress. The first normal stress difference N1 = σzz − σrr has viscous nature in dilute solutions, N1 = 3ηε̇, where ε̇ = ∂v/∂z is the local elongation rate. Noting also that σrr = −2Hγ = −γ/r, we get

4. THEORETICAL ANALYSIS OF FIBER FORMATION IN DILUTE SOLUTIONS The aim of this section is to analyze the formation and stability of a thin and long polymer solution jet. We anticipate that the jet can be formed by an extension of a neck developed as a result of a capillary instability. The neck structure is schematically shown in Figure 17. It consists of three zones:

⎛ γ ∂v ⎞ J ∼ πr 2⎜ρv 2 − − 3η ⎟ ⎝ r ∂z ⎠

The condition J = J1 at z = z1 then becomes ρv 2 +

γ ∂v − 3η =0 r ∂z

at z = z1

(10)

and the same condition applies at another end, z = z2. A general analysis of above equations is difficult. For simplicity we first neglect the inertial terms, assuming that viscoelastic forces dominate. The first eq 9 then simplifies as Figure 17. Jet structure; r = r(z), ν = ν(z).

∂v ∂ ⎛⎜ 2 ∂v ⎞⎟ 3ηr = −γ ∂z ⎝ ∂z ⎠ ∂z

two menisci (1: z < z1; 2: z > z2) and the cylindrical jet (3: z1 < z < z2). Meniscus 1 is at rest, and meniscus 2 is moving as a whole with velocity νm. Zone 1 can be considered approximately neglecting the fluid motion there. The force balance then says that the pressure p1 in this zone must be equal to the pressure in the mother fluid droplet which in turn is nearly equal to the atmospheric pressure p0 (we neglect the low curvature of the droplet surface): p1 ≈ p0. Therefore, the capillary pressure 2γH at the meniscus surface must be p1 − p0 = 0, hence its mean curvature H = 0 (γ is the surface tension). The shape of the droplet is then defined by the equation ⎛ rrz″ ⎞ ⎜1 − ⎟ = 2H 1 + rz′2 ⎠ r 1 + rz′2 ⎝

It can be integrated yielding (on using also the boundary condition (10))

r 2(3ηε ̇ + γ /r ) = 2γa

The latter equation is compatible with another simplifying assumption of the uniform jet radius in the zone 2: r(z, t) = a(t). In this case it yields N1 = 3ηε ̇ = γ /r

(12)

The fluid incompressibility implies ε̇ = −

1

which after integration can be presented in the form r = a + H (r 2 − a 2 ) 2 1 + rz′

(11)

Hence,

(7)

2 dr r dt

dr = dt 49

(13) γ

− 6η , and we arrive at a linear law for capillary

thinning:

r(t ) = r0 − (8)

γt 6η

(14)

The strain rate defined in eq 13, ε̇ = γ/3ηr(t), strongly increases as the jet becomes thinner. The time of this initial thinning stage is limited by t = tmax ≈ 6ηr0/γ. This time is nearly the viscosity-defined growth time of the Plateau−Rayleigh instability (cf. eq 2). Therefore, the instability cannot develop significantly during this time tmax. The simple result, eq 12, is valid if the viscosity η does not change, which is true for ε̇ ≤ ε̇c ∼ 1/τ, where τ is the polymer coil relaxation time. At some moment (tc) the elongation rate

leading to the well-known result ⎛ z − z1 ⎞ ⎟ r(z) = a cosh⎜ ⎝ a ⎠

where a = r(z1) and r(z) is the jet radius in cylindrical coordinates (r, z). Zone 3 is characterized by the similar shape with z2 instead of z1 (for simplicity, we also assume the symmetry: r(z2) = r(z1)). In the cylindrical region (zone 2) we I


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Macromolecules exceeds ε̇c, and the coil−stretch transition takes place;7,41−43,45 it happens once r < rc = γ/(3ηε̇c). The chains become nearly fully stretched as a result of this transition, and the effective viscosity of the solution increases to a much higher value η* (the factor η*/η depends on polymer concentration and MW; it scales as CM2 in the dilute solution regime). The jet thinning γ dr slows down accordingly: dt = − 6η * for t > tc. However, even

10r, the characteristic thinning time τin ∼ 3 ρr 3/γ (compare with eq 2) and extension rate ε̇ ∼

Thus, the jet radius decreases, while ε̇ ∝ r increases indefinitely, so the coil−stretch transition and fiber formation must be expected at some point in complete analogy with the case of no inertia considered above (during the transition the viscous force increases dramatically, so eventually the inertial effects turn negligible). To be more precise, let us estimate the critical jet radius rc corresponding to the onset of coil stretching. It is defined by the condition ε̇ ∼ 1/τ, where τ is the coil relaxation time, and ε̇ is defined in eq 16. In the dilute solution regime τ is the Zimm time τ = τZimm = 0.325

ηsR coil 3 30 , kBT

where Rcoil is the coil size (rms

end-to-end distance). The size Rcoil can be obtained based on the intrinsic viscosity using the relation30,43 [η] ≈ 0.425

NAR coil 3 M

leading to

τZimm ≈ 0.765

ηsM[η] R GT

where RG = NAkB is the gas constant. The condition ε = 1/τZimm then gives ⎛ γ ⎞1/3 rc ∼ ⎜ ⎟ τZimm 2/3 ⎝ ρ⎠

(17)

For a dilute solution of PEO in water (M ∼ 400 kDa, polymerization index N ∼ 104, [η] = 4 dL/g; cf. Supporting Information Table 2), we then get the critical radius rc ∼ 50 μm and the jet length Lc ∼ 10rc ∼ 500 μm, in reasonable agreement with the observed range of jet thickness/length. Let us discuss the critical concentration Cstab required for polymer solution jet stability (which is also the necessary condition for spinnability). As explained above, a polymer fiber can be formed due to a flow-induced phase separation (solvent demixing from the polymer phase caused by capillary thinning of the jet). As shown in ref 43, the driving force for this process is proportional to the nondimensional parameter X = ϕ|k|LN/d (kBT·X is the energy gain per chain during the first stage of the phase separation43), where LN = Nl1 is the polymer chain contour length, d is the chain effective diameter, ϕ is polymer volume concentration in the initial solution, and k is the thermodynamic parameter defined in eq A1 (cf. Appendix). The phase separation occurs if the interchain attraction dominates, that is, k < 0 (which can be true for stretched chains), and if X = ϕ|k|LN/d > const ∼ 1 (cf. Appendix).41−43 The latter condition defines the minimum polymer concentration required for polymer separation (demixing)

defined in eq 2 as τin = 2.9 ρr 3/γ and τvis = 6ηr/γ,25,26 where r is jet radius and η is solution viscosity. In dilute solutions, η ≈ ηs (ηs ≈ 1 mPa·s, γ ≈ 72 mN/m, and ρ = 103 kg/m3 for water). Thus, τin ∼ 10−5 s and τvis ∼ 10−6 s for r = 10 μm, so τin > τvis meaning that inertial effects are indeed important for r ≥ 10 μm. Neglecting now the viscous term in the first eq 9, we get (15)

The meaning of this equation is simple: during the jet thinning the surface tension energy is converted into the kinetic energy of the fluid motion along the tube. The fluid velocity ν can be estimated based on the balance of the second term on the lefthand side and the capillary term on the right-hand side : ρr2ν2 ∼ γr, so v ∼ (γ/ρr)1/2. The extension rate is ε̇ = ν/L, where L is the jet length; hence ε̇ ∼

(16) −3/2

this slower process can become virtually arrested due to the demixing phenomenon.41−43 This effect can be briefly described as follows: the segments of stretched chains become significantly oriented due to high-rate extension flow (in particular, if the jet thins well below rc); the excluded volume repulsion of segments then gets suppressed due to their alignment, so the unbalanced van der Waals attraction drives the chains to aggregate, i.e., to demix from the solvent (see Appendix for more details). As a result, the system becomes strongly inhomogeneous, and a dense, strong oriented fiber can be formed eventually. Such solidified fiber can prevent the jet from further stretching and stabilize it for a long time.34,42 The fiber may turn virtually permanent or may undergo a rupture somewhat later if its strength is not high enough. So far we assumed for simplicity that the central jet part is cylindrical. This is not exactly the case in practice (see Figures 4, 6, and 14): normally, the jet is thinner somewhere in the middle (or sometimes closer to one of its ends). Therefore, the demixing and polymer fiber formation process do not occur simultaneously along the whole jet: they obviously must start in the thinnest jet part (where r drops below rc first) and then spread over the rest of the jet in the course of capillary thinning. This phenomenon is clearly visible in the images of Figure 14: the jet is initially transparent, then a part of it becomes white (due to turbidity related to the demixing process), and then the white region spreads along the jet (note also the droplets of pure solvent formed on the jet surface). Let us now briefly address the inertial effects (which are important in some dilute aqueous polymer solutions considered in section 2). The characteristic capillary instability times are

∂ ∂ ∂r (ρr 2v) + (ρr 2v 2) = γ ∂t ∂z ∂z

⎛ γ ⎞1/2 2 ∼ ⎜ ⎟ r −3/2 τin ⎝ ρ⎠

1/2 1⎛ γ ⎞ ⎜ ⎟ L ⎝ ρr ⎠

ϕdem ∼

d 1 ∝ L N |k | N

(18)

Note that for long chains (large N) the concentration ϕdem can be much lower than the coil overlap crossover concentration

Obviously, the extension rate is higher (thinning time shorter) for shorter L. This means that the dominant (fastest) thinning mode corresponds to the shortest L ∼ const·r; that is, we arrive at the classical inertial Plateau−Rayleigh instability with L ∼

ϕ* ∝ N1 − 3ν J

(19) DOI: 10.1021/acs.macromol.7b00687 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules where we consider the general case (arbitrary solvent quality): coil size Rcoil ∝ Nν with ν = 0.58 in good solvent, and ν = 0.5 is the marginal or theta solvent regimes. Thus, ϕdem ≪ ϕ* for sufficiently long chains (N ≫ 1); hence, the jet-extensioninduced demixing is possible also in dilute solutions of highMW polymers.43 Noteworthily, ϕdem depends on the solvent Θ quality (and temperature): |k| = χ (T ) T − I(s) increases as the solvent quality decreases (from good to marginal or thetasolvent regimes); hence, ϕdem decreases in parallel. Therefore, better fiber stability (due to a stronger tendency for demixing) may be expected close to the theta-solvent conditions (as compared to the good solvent). Interestingly, a lower bound for ϕdem can be obtained by simply demanding that the stretched chains are nearly overlapping. (Otherwise, the chain aggregation would not be possible as the chains would not have enough time to diffuse and interact with the neighboring chains during the jet thinning process.) The volume occupied by the stretched coil is Vch ∼ LNR02, and the intrinsic chain volume is Vin ∼ LNd2; hence, ϕdem = Vin/Vch ∼ d2/R02 ∼ 1/N. Turning to the critical concentration Cstab (and the corresponding volume concentration ϕstab), the question we should address is, does ϕdem correspond to ϕstab? According to the definition adopted in this paper, the jet is stable if its lifetime exceeds the Plateau−Rayleigh time by a factor 10 (or more). This sort of stability can be naturally obtained as a result of the coil−stretch transition. Indeed, the polymer intrinsic viscosity in the regime of stretched chains increases as Rz3; hence, the apparent extensional viscosity η in this nonlinear regime rises in parallel as ⎛ ϕR 3 ⎞ η ≈ ηs⎜1 + const z ⎟ Nv1 ⎠ ⎝

Figure 18. Schematic diagram illustrating the main regimes of stable and unstable jet behavior. The boundary concentration correspond to three stabilization mechanisms: at low molecular weight M the critical concentration C stab corresponds to C = C** = const1·M −0.7 (entanglement stabilization in a semidilute solution), at intermediate M it follows the demixing concentration C = Cdem = const2·M−1 (stabilization by flow-induced polymer/solvent phase separation), and at high M it is defined as C = Cstab = const3·M−2(dilute solution regime of stabilization by extension hardening). Here const1, const2 and const3 are prefactors depending on physical parameters: In particular, note that const2 depends on temperature T (more precisely, on the distance between T and theta temperature), while const3 depends on solvent parameters like viscosity and surface tension.

related to entanglements, demixing, and viscosity increase as the chains become stretched by the flow. Entanglements are required for polymers of relatively low molecular weight (this may sound a bit paradoxically but just mean that concentrations beyond the entanglement onset are demanded for such polymers), whereas extension hardening and demixing are important for dilute solutions of very high molecular weight polymers where the effect of chain stretching can be very strong. We expect that the condition ϕ > ϕstab (C > Cstab) may be sufficient for spinnability in the case of thin enough jets and/or rapid enough solvent evaporation. However, if the solvent volatility is low, the flow-induced demixing may be necessary. In this case the stronger criterion for spinnability should be applied: ϕspin > ϕdem (Cspin > Cdem). In the general case (of high molecular weight polymers) we expect the minimal polymer concentration required for spinnability, Cspin, to be in the range Cstab ≤ Cspin ≤ Cdem.

(20)

where const is a numerical constant, Rz is the chain size in the flow direction, ν1 is the intrinsic volume per monomer unit, and ϕ/Nv1 is the number of chains per unit volume. A derivation of eq 20 was outlined in the paragraph below Figure 10 (only the most important polymer contribution corresponding to the second term in eq 20 was considered there). Equation 20 can be also easily obtained from the general equation for the ϕ macroscopic stress of polymer solution σzz ∼ ηsε ̇ + Nv FzR z 30

5. CONCLUSIONS Visualization of the process of stable jet formation (fiber spinning) from dilute polymer solutions carried out for a large number of polymer−solvent pairs over a wide range of molecular weights has allowed us to confirm that stable fibers (with respect to the time scale of the Plateau−Rayleigh instability) can be formed from weak polymer solutions in the concentration range lying well below the crossover point determined by the condition C[η] = 1. So, two different regimes of spinnability can exist, first, at concentrations C ≥ C** providing the formation of an entanglement network and, second, at much lower concentrations C = min[Cstab, Cdem] ≪ C**, where the concentration Cstab is related to extension hardening due to the coil−stretch transition and scales as M−2, whereas Cdem is determined by the extension-induced polymer/ solvent demixing and scales as Cdem ∝ 1/M (see Figure 18). The physics standing behind the spinnability of dilute solutions is related to the enhanced intermolecular interactions of elongated macromolecules in solutions that are treated as “dilute” in shearing. The latter conception has been proposed by Clasen et al.44 We found that fiber formation in dilute solutions of high-MW polymers can be accompanied by the

1

using slender body approximation for the hydrodynamic drag force Fz, Fz ∼

ηsε̇R z 2 33,41,50 ln(R z / d)

(logarithmic term in (20) is

omitted). The dilute solution viscosity is η0 = ηs at low strain rates. However, the apparent viscosity can strongly increase after the coil−stretch transition (for Rz ∼ LN) up to η ∼ ηs

ϕLN 3 Nv1

∼ ηsϕN 2 , so that η/ηs ∼ ϕN2. As a result, the jet

lifetime (cf. the second eq 3) increases significantly if ϕN2 ≫ 1. The minimum concentration for jet stability is then ϕstab ∼ const/N 2

(21)

where const is a large numerical constant. The latter scaling dependence of ϕstab on N (implying the same dependence of Cstab on M) is in rather good agreement with the experimental results shown in Figures 10 and 16. Obviously, ϕstab is much lower than ϕdem for long chains. The schematic diagram of Figure 18 shows the regions of stable (relative to the Plateau−Rayleigh time) and unstable jets. We identified three stabilization mechanisms, namely those K


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Macromolecules

chain stretching s) in the regime of really strongly stretched chains (s → 1) due to the finite extensibility effect. As a result, both Δ2 and the chain relaxation time τchain decrease with s in this regime. We take this effect quantitatively into account for a realistic wormlike chain model. 4 1 The result is k TL Fel ≈ B 2l N F (̃ R z /LN ), where F̃ is implicitly defined by the following equations involving auxiliary parameter A: F̃ = A coth A − 1 and Rz/LN = coth A − 1/A. Using the above equations, we get

phase separation. The possibility of spinnability from weak polymer solutions was observed for several different polymers. A quantitative evaluation of the lowest concentration Cstab at which the spinnability is still possible was obtained for a series of PEO solutions in a wide molecular weight range. This concentration is scaled with intrinsic viscosity with the exponent of −2.14 ± 0.3 or with molecular weight with the exponent of −1.63 ± 0.29. For the domain of entangled macromolecules, the spinnability concentration is scaled as M−0.70±0.14, which is nearly the same as the dependence of the crossover concentration C* on molecular weight. Our results correlate well with the theoretical prediction and experimental data for the other polymers.17 It is important to note that generally Cspin ≥ Cstab as a longtime jet stability (which may still be followed the jet rupture) is not always enough to ensure spinnability (formation of robust polymer fibers): the latter phenomenon requires the solvent removal from the fiber which can be achieved, in particular, by solvent evaporation (in the case of highly volatile solvent and/or thin fiber) or by flow-induced polymer/solvent demixing (the condition C ≥ Cdem). Therefore, we generally expect that

τchain ∼ 4(1 − s)2 τR where

τR =

2 π ηsLN l 18 kBTkH

(A1b)

is the characteristic relaxation time of a nearly fully stretched chain with Rz ∼ LN/2 (with s ∼ 0.5), ηs is the solvent viscosity, and the factor kH is defined in the next paragraph. Note that τR scales with the chain length as LN2 like the standard Rouse time. Equations A1a and A1b show that τchain = τR in the regime of interest, for nearly fully stretched chains with s ∼ 0.5−0.7 (in this range ζchain can decrease by a factor of just 3). In eq A1b it was taken into account that the chain friction factor is ζchain ∼ ηsRz/kH as explained below. In fact, the total chain friction factor is ζchain = ζRz where ζ is the effective hydrodynamic friction constant per unit length along the z-axis: ζ defines the friction (drag) force on a chain element of length Δz, ΔFdrag ≈ ζνzΔz, where νz is the relative polymer segment/ solvent velocity. The factor ζ is proportional to the solvent viscosity: ζ ∼ ηs/kH, where kH is the hydrodynamic factor taking into account the effect of hydrodynamic interactions on the chain relaxation. We are interested in the regime of strongly stretched chains (s ≥ 0.5). In this regime not only the chain as a whole is aligned (along the z-axis defined by the flow) but also the chain segments are strongly extended so the hydrodynamic factor kH is only logarithmic like in the case of rigid rods:30 kH ≈ ln(ξH/d) where ξH is the relevant range of hydrodynamic interaction and d is the chain thickness. In a very dilute solution, ϕ ≪ (d/LN)2, the length ξH is simply the chain length, ξH = LN. At higher concentrations ξH is the hydrodynamic screening length which is comparable with the lateral distance between the neighboring chains, ξH ∼ d/(ϕs)1/2. The above expression for ΔFdrag, which is valid in the logarithmic approximation, was obtained41,43 using the Oseen tensor formalism in agreement with the classical theoretical results on hydrodynamic interactions in polymer solutions.30 In the regime of interest, s ≥ 0.5 and ϕ > (d/LN)2, the hydrodynamic factor therefore is

Cstab ≤ Cspin ≤ Cdem

It is also reasonable to anticipate that in the case of thick fibers and nonvolatile solvents the minimum concentration for spinnability Cspin ∼ Cdem.

■

APPENDIX Let us consider a dilute polymer solution undergoing uniaxial elongational flow with extension rate ε̇. Depending on the value of ε̇, polymer chains can be either in the coiled state or in the stretched state, and the coil−stretch transition is sharp.7 The coil−stretch transition in unentangled solutions of semiflexible polymer chains of diameter d, Kuhn segment length l, and contour length LN = Nl1 has been recently considered by Subbotin and Semenov.41 Orientation of chain segments in the stretched state is defined by the balance between the chain elastic energy Fel and the work of the hydrodynamic friction forces elongating the chain. It is characterized by the order parameter s = ⟨cos θ⟩, where θ is the angle between a chain segment and the flow axis (z-axis) and angular brackets imply averaging with respect to the chain conformations. The order parameter s is related to the chain size Rz in the flow direction: s = Rz/LN. Thus, the degree of stretching is given by the parameter s: s = 1 corresponds to fully stretched chains, while we are mostly interested in the regime of rather strongly stretched chains with s ≥ 0.5. The chain relaxation time (in particular, for the lateral relaxation mode) in this regime is defined by the competition between the polymer/solvent friction (ζchain) and elastic forces: τchain

(A1a)

kH ≈ 0.5 ln(1/ϕ)

ζ ∼ chain Δ2 kBT

(A2)

Alerted by the comments of the reviewers of this paper, we would like to emphasize that the finite extensibility effect was taken into account in our estimate of the solution extensional viscosity in the regime s ≥ 0.5 (see the text below Figure 10). Indeed, in this case the elastic force fel must be balanced by the drag force Fdrag ∼ ζ∥νzRz with νz ∼ ε̇Rz; hence fel ∼ Fdrag ∼ (ηsRz/kH)ε̇Rz. The contribution of one chain to the normal stress difference therefore is η ̇ z3 ΔN1 ∼ fel R z ∼ s εR kH (A3)

where Δ2 is the variance of the lateral chain size Rx and T is the temperature. For s ≥ 0.5 the fluctuation Δ is defined by the chain tension fel Δ2 ∼ kBTLN /fel

where fel = ∂Fel/∂Rz. Here Fel = Fel(R) is the elastic free energy of a chain with the end-to-end distance R ≈ Rz. It is noteworthy that fel strongly increases with R (that is, with the degree of L


Article

Macromolecules As a result we get ΔN1 ∼

ηs kH

tions is delayed. The condition ε̇τZimm > 1 is therefore necessary to guarantee that the jet does not break before the coil−stretch transition completes. The kinetics of phase separation has been analyzed by Semenov and Subbotin43 and includes three main stages: (i) amplification of concentration fluctuations and formation of oriented domains (protofibrils) with characteristic size

εL̇ N 3 since Rz ∼ LN. This result

leads to the following polymer contribution to the extensional η ϕ viscosity: η ∼ k s LN 3 L d2 ∝ ϕM2 , which is in agreement with H

N

the main text (where the log-factor kH is omitted for simplicity). Equation A3 highlights the relevance of Rz for the polymer-induced extensional stress: in the full stretch regime the polymer-specific viscosity is defined geometrically by polymer configuration, in particular, by the chain size along the flow.51,52 This conclusion is in accord with the well-known rheological behavior of solutions of hard spheres or rigid rods whose specific viscosity is defined by geometrical parameters of the solute molecules, while the molecular rigidity, however large, does not play a major role.30 It is useful to define the relevant Weissenberg number as Wi > ε̇τR. Then, the order parameter s can be considered as a function of the Weissenberg number Wi: s = s(Wi). It was shown that when Wi < Wic ≈ 3.3, the coiled state is stable, whereas at Wi > Wic, the stretched conformation is stabilized: the coiled state is metastable in the last case being separated from the stretched state by a potential barrier.41−43 The order parameter at the transition point is sc = s(Wic) ≈ 0.64; it increases with increasing the Weissenberg number. The steric repulsion between the extended chains decreases with increasing their orientation.53 Therefore, the balance between repulsive and attractive interactions in the polymer solution is shifted toward attractions as Wi is increasing. As a result, the polymer/solvent phase separation occurs eventually 1 ∂Π once the osmotic modulus κ = k T ∂c becomes negative. Here

ξ∼

πl 1 + 1 ϕk N 2d

■

2

) is much shorter than τ ; therefore, R

ASSOCIATED CONTENT

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b00687. Table 2 (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (A.Y.M.). ORCID

A. Ya. Malkin: 0000-0002-7065-7898 Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS Partial financial support from the Russian Foundation for Basic Research through Grant No. 16-03-00259 is acknowledged.

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REFERENCES

(1) Malkin, A. Ya.; Arinstein, A.; Kulichikhin, V. G. Polymer extension flows and instabilities. Prog. Polym. Sci. 2014, 39, 959−978. (2) Peterlin, A.; Č opič, M. Gradient Dependence of the Intrinsic Viscosity of Linear Macromolecules. J. Appl. Phys. 1956, 27, 434−438. (3) Peterlin, A. Hydrodynamics of linear macromolecules. Pure Appl. Chem. 1966, 12, 563−586. (4) Pope, D. P.; Keller, A. Alignment of macromolecules in solution by elongational flow; a study of the effect of pure shear in a four roll mill. Colloid Polym. Sci. 1977, 255, 633−643. (5) Keller, A.; Odell, J. A. The extensibility of macromolecules in extension; a new focus for macromolecular science. Colloid Polym. Sci. 1985, 263, 181−201. (6) Keller, A.; Machin, M. J. Oriented crystallization in polymers. J. Macromol. Sci., Part B: Phys. 1967, 1, 41−91. (7) de Gennes, P. G. Coil-stretch transition of dilute flexible polymers under ultrahigh velocity gradients. J. Chem. Phys. 1974, 60, 5030−5042. (8) Brestkin, Yu. V.; Dyakonova, N. E.; Frenkel’, S. YA.; Petropavlovslii, G. A.; Kholmuminov, A. A.; Bochek, A. M.; Agranova, S. A. Behavior of semi-rigid macromolecules in longitudinal hydrodynamic field. Vysokomol. Soed. A 1992, 34 (5), 15−23. (9) Ö ttinger, H. C. Stochastic Processes in Polymeric Liquids; Springer Verlag: Berlin, 1996. (10) Kalashnikov, V. N.; Tsiklauri, M. G. Supermolecular structures and flow birefringence. Colloid Polym. Sci. 1996, 274, 1119−1128.

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where ϕ = (π/4)d l1c is volume fraction of polymer in the solution. When the modulus becomes negative (κ < 0), the polymer solution starts to separate in polymer and solvent phases. The minimal concentration at which the phase separation occurs is given by d 1 ∝ Nl1|k| N

d ϕLN |k|

S Supporting Information *

2

ϕdem ∼

(

for polymer concentrations ϕ > ϕdem the phase separation could fully develop during the stretching regime.

The parameter χ generally depends on temperature and characterizes nonsteric interactions between different segments and between the segments and the solvent, Θ is Θ-temperature (χ(Θ) = 1), and the function I(s) = (4/π)⟨sin β⟩ accounts for anisotropy of the steric repulsion between polymer segments (β is the angle between two randomly chosen segments).41 Thus, the osmotic modulus of the polymer solution can be written as κ=

perpendicular to the z-axis; (ii) lateral collapse

two stages tc ∼ τR

Π is the osmotic pressure, and c is concentration of the monomer units. The osmotic pressure involves the ideal gas term and contribution due to interactions Θ k = I (s ) − χ (T ) T

ld ϕ |k |

of the protofibrils and formation of a network of highly oriented and stiff fibrils having diameter df ∼ (ld)1/2/|k|, df ≪ ξ; (iii) collapse of the network accompanied by solvent release and maturation of the fiber. The characteristic time of the first

B

⎛c ⎞ π Π = kBT ⎜ + dl12c 2k ⎟ , ⎝N ⎠ 4

1/2

( )

(A6)

It is inversely proportional to the MW of the chains. Thereby the polymer solutions having volume concentration ϕ < ϕdem do not demix above the coil−stretch transition, but they can demix if ϕ > ϕdem. We stress that at 3.3/τR < ε̇ < 1/τZimm the coiled state is metastable and is separated from the extended state by the potential barrier which decreases with increasing the extension rate. Therefore, at 3.3/τR < ε̇ < 1/τZimm, the transition between the coiled and extended chain conformaM


Article

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Spinnability of Dilute Polymer Solutions - Macromolecules (ACS

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