Article pubs.acs.org/Macromolecules
Concentrated Polymer Solutions are Different from Melts: Role of Entanglement Molecular Weight Qian Huang,† Olga Mednova,‡ Henrik K. Rasmussen,§ Nicolas J. Alvarez,† Anne L. Skov,† Kristoffer Almdal,‡ and Ole Hassager*,† †
Department of Chemical and Biochemical Engineering, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark Department of Micro- and Nanotechnology, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark § Department of Mechanical Engineering, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark ‡
ABSTRACT: We compare viscoelastic properties of several polystyrene solutions and melts with the same number of entanglements. It is demonstrated that the modulus and time can be shifted such that the linear viscoelastic properties are identical provided the number of entanglements are identical. However the nonlinear properties in strong extensional flow are different with polymer solutions showing markedly stronger extensional hardening than the corresponding melts. While increased chain extensibility for solutions may provide part of the explanation, it is demonstrated that other mechanisms are needed for a full explanation for the differences between solutions and melts.
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INTRODUCTION The reason for the observed qualitative difference in extensional steady-state viscosity between polymer solutions and melts is still an open question. Experiments on entangled polystyrene solutions1 show that when the strain rate is larger than the inverse Rouse time, the steady-state extensional viscosity increases with increasing strain rate. In contrast, experiments on entangled polystyrene melts2,3 show that the steady-state extensional viscosity decreases monotonically even when the strain rate is larger than the inverse Rouse time. This contradiction is confounded by the lack of consistent model predictions. For example, the tube model1 that includes the mechanisms of chain stretch and convective constraint release is capable of describing the non-monotone behavior of the polystyrene solutions, but cannot capture the monotonic thinning behavior observed in polystyrene melts. By contrast, the interchain pressure model4,5 captures the monotonic thinning of polystyrene melts, but fails to capture the observed polystyrene solution behavior. According to the tube model, several parameters are affected when a highly entangled polymer melt is diluted, namely, the entanglement molecular weight, Me , the number of entanglements per chain, Z, and the finite extensibility, λmax. This paper examines these parameters individually in order to determine how each parameter influences the linear and nonlinear rheological response of a material and to investigate the origin of the observed discrepancies between polymer melts and solutions. We restrict the investigation to small amounts of diluent to compare melts with highly concentrated solutions. When an entangled polymer melt is diluted, the number of entanglements per chain Z decreases and the entanglement molecular weight Me increases correspondingly. The number of © XXXX American Chemical Society
entanglements per chain in a polymer solution is defined as Z = Mw/Me , where Mw is the molecular weight of the polymer. The entanglement molecular weight for a polymer solution is given by Me(ϕ) = Me(1)ϕ−α
(1)
where ϕ is the volume fraction of the polymer in the solution, α is the dilution exponent, and Me(1) is the entanglement molecular weight of the polymer melt with ϕ = 1. The value of the dilution exponent is an open research topic.6−8 α is stated to vary between 1 and 4/3, depending on the model used to analyze the data and the architecture of the sample (cf. α = 11,9,10 and α = 4/3).11,12 There is also a suggestion that the value of α depends on the relative range of frequency, α = 1 for high frequencies and α = 4/3 for low frequencies.8 What can be agreed upon is the difficulty in discerning the value of α from experimental data and model predictions. In addition to increasing Me , dilution of a polymer chain in a solvent increases the number of Kuhn segments between entanglements Ne = Me/M0 , where M0 is the molecular weight of a Kuhn segment. Since the maximum stretch ratio is given as λmax = (Ne)1/2, solutions with larger values of Me can be stretched to larger ratios than their corresponding melts. One hypothesis is that differences in λmax determine the nonlinear response of a material, such that materials with small values of λmax reach a steady state faster and are less prone to strain hardening than molecules with larger values. Received: April 24, 2013 Revised: May 24, 2013
A
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Me and Z are typically determined from rheological experiments in the linear viscoelastic (LVE) regime. For example, a decrease in Me has the direct effect that G′ and G″ are shifted to lower values due to the reduction in the plateau modulus, G0N. Generally, the value of Me is directly determined from G0N by the relationship, GN0 ≡
4 ρϕRT 5 Me
is because polystyrene and styrene oligomer are completely miscible. The two melts and three solutions are measured in both shear and extensional flows. Two additional polystyrene solutions diluted in dibutyl phthalate (DBP), which is a commonly used good solvent for polystyrenes, are also prepared and measured in shear flows for comparison. We show that melts and concentrated solutions with same values of Z behave identically in linear rheology, but remarkably different in nonlinear rheology.
(2)
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where ρ is the density of the polymer melt, R is the gas constant, and T is the temperature. We follow the ‘G definition’ of the entanglement spacing,13,14 according to the Doi− Edwards (DE) model.15 Once G0N is determined, then the value of Me follows from eq 2 and Z = Mw/Me. There are several methods to determine G0N from LVE data and they are summarized in Liu et al.14 Substitution of eq 1 into eq 2 gives the relationship between the solution plateau modulus, G0N(ϕ), and the melt plateau modulus, G0N(1), GN0 (ϕ) = GN0 (1)ϕ1 + α
EXPERIMENTAL DETAILS
Synthesis and Chromatography. The two polystyrenes PS-285k and PS-545k with narrow molecular weight distributions have been synthesized by living anionic polymerization according to the standard procedure.16 Sec-butyllithium was employed as the initiator, and the reaction was carried out for 3 h at 30 °C in cyclohexane. Size exclusion chromatography (SEC) was employed for sample characterization. The SEC system consists of a SIL-10AD injector (Shimadzu), a triple Viscotek detector and two PLgel Mixed C and Mixed D columns. Stabilized tetrahydrofuran (THF) was used as the eluent and the flow rates have been adjusted according to Irganox signals. The styrene oligomer OS-2k was bought from Sigma-Aldrich. Table 1 summarizes the weight-average molecular weight Mw, the polydispersity index PDI and the glass transition temperature Tg of the synthesized polystyrenes and the oligomer.
(3)
The value of α is often determined from a plot of as a function of ϕ, where the slope is equivalent to (1 + α). However, measuring or determining G0N is nontrivial and its value will certainly vary depending on which method is used.6,14 Moreover for the limited range of concentrations investigated here, G0N is not particularly sensitive to the value of α and it is therefore difficult to discern a value of α with confidence. Given these complications, we present an alternative method for determining α that is particularly suited for monodisperse melts with two clearly separated crossover frequencies for G′ and G″. In terms of the tube model,15 the inverse of the lower cross frequency, ωx,l , is proportional to disengagement time, τd. Conversely the high frequency crossover, ωx,h , is proportional to the single segment equilibration time, τe. Therefore, the ratio of τd/τe ∝ ωx,h/ωx,l ∝ Z3.4. Since Z(ϕ) = Z(1)ϕα, where Z(ϕ) and Z(1) are the number of entanglements per chain for a solution and a melt, respectively, substitution gives ωx , h τ ∝ d ∝ Z(1)3.4 ϕ3.4α ωx , l τe (4) G0N
Table 1. Molecular Weight and Glass Transition Temperature of the Polystyrenes and the Styrene Oligomer sample name
Mw [g/mol]
PDI
Tg [°C]
PS-545k PS-285k OS-2k
545 000 285 000 1920
1.12 1.09 1.08
106.5 107.5 60.5
Preparation of Solutions. Three polystyrene solutions were made using either PS-285k or PS-545k diluted in OS-2k. The solutions were prepared by dissolving both the polystyrene and the oligomer in THF and stirring at room temperature overnight. When the components were well dissolved and mixed, the THF solution was cautiously put into methanol drop by drop and the blends were recovered by precipitation. Finally the blends were dried under vacuum at 50 °C for a week. Considering that the methanol may partly dissolve OS-2k during precipitation at room temperature, the concentrations of all the three polystyrene solutions were determined by the peak areas of the bimodal curve in SEC. For each polystyrene solution, two randomly picked parts were checked in SEC in order to ensure the concentration is homogeneous throughout the sample. In addition to the three solutions diluted in OS-2k, another two solutions were prepared using PS-545k diluted in DBP, following the procedure described in Bhattacharjee et al.1 Table 2 summarizes the components, the weight fraction ϕ and the glass transition temperature Tg of the solutions. Mechanical Spectroscopy. The linear viscoelastic properties of the polystyrene melts and solutions were obtained from small amplitude oscillatory shear flow measurements. An 8 mm plate− plate geometry was used on an ARES-G2 rheometer from TA Instruments. The measurements for the two melts and the three
From eq 4, the value of α can be determined by a plot of either ωx,h/(ωx,lZ(1)3.4) or of τd/(τeZ(1)3.4) versus ϕ. Both methods will be used in the following. The value of Z(1) is determined from G0N(1), which was measured from experimental melt data. Once α is determined, the values of Me(ϕ), and Z(ϕ) follow from eq 1 and Z(ϕ) = Z(1)ϕα, respectively. Note that the proportionality in eq 4 suggests that polymer melts and solutions with the same value of Z have the same ratio of ωx,h/ ωx,l. To the best of our knowledge this observation has not been confirmed experimentally. The purpose of the present work is to make well-defined experiments to directly evaluate the influence of Z, Me , and λmax on the rheological properties of entangled polymer melts and solutions in the specific concentration range, 0.4 ≤ ϕ ≤ 1, where hydrodynamic interactions are avoided. We carefully synthesized two nearly monodisperse polystyrenes with molecular weight of 285 and 545 kg/mol. Three binary blends were prepared from one of these two polystyrenes and a 2 kg/ mol styrene oligomer. The oligomer is far below the entanglement molecular weight (Me = 13.3 kg/mol) and therefore the three blends are considered polymer solutions. The main reason for choosing a styrene oligomer as the solvent
Table 2. Components of the Polystyrene Solutions sample name PS-285k/2k-72 PS-285k/2k-44 PS-545k/2k-58 PS-545k/DBP-63 PS-545k/DBP-52 B
components 285k 285k 545k 545k 545k
+ + + + +
2k 2k 2k DBP DBP
ϕ [wt %] 72% 44% 58% 63% 52%
(±1%) 285k (±1%) 285k (±1%) 545k 545k 545k
Tg[°C] 94.0 85.0 91.0 − −
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solutions diluted in OS-2k were performed at temperatures between 110 and 170 °C under nitrogen. For each polystyrene sample, the data was shifted to a single master curve at 130 °C using the time− temperature superposition procedure. The measurements for the two solutions diluted in DBP were performed at temperatures between 20 and 70 °C under nitrogen. For each sample, the data were shifted to a single master curve at 70 °C. The shift factor aT is reported in Table 3 for different temperatures.
⟨σzz − σrr ⟩ =
πR(t )2 1 × 1 + (R(t )/R 0)10/3 exp(−Λ 30)/(3Λ 20)
110 to 130 °C
120 to 130 °C
150 to 130 °C
170 to 130 °C
PS-545k PS-285k PS-285k/2k-72 PS-285k/2k-44 PS-545k/2k-58 sample name
− − 180.06 56.03 −
− − 9.56 − 7.57 20 to 70 °C
0.017 0.017 0.039 − 0.051 30 to 70 °C
0.0011 0.0011 − − 0.0058 50 to 70 °C
− 415.94
391.79 81.78
10.37 6.49
PS-545k/DBP-63 PS-545k/DBP-52
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RESULTS Linear Viscoelasticity. We analyze the LVE data by fitting to the five parameter continuous Baumgaertel-SchausbergerWinter (BSW) relaxation spectrum20 as illustrated in Figure 1.
Extensional Stress Measurements. The extensional stress was measured by a filament stretching rheometer (FSR) equipped with an oven to allow measurements from room temperature to 200 °C.17 Measurements were performed for the two melts and the three solutions diluted in OS-2k, but not for the two solutions diluted in DBP due to experimental difficulties. Prior to making a measurement, all polystyrene samples were molded into cylindrical test specimens with a fixed radius R0 = 2.7 mm. To ensure no air bubbles were trapped in the samples, the mold was connected to a vacuum pump. The initial length L0 of the cylindrical test specimens was controlled by the addition of a given mass of the sample into the mold. The L0 of each cylindrical test specimen was varied between 1.3 mm and 1.6 mm, giving an aspect ratio Λ0 = L0/R0 between 0.48 and 0.59. The polystyrene melts were pressed at approximately 150 °C and annealed at this temperature for 15 min under vacuum. The duration of 15 min was chosen to ensure that the polymer chains were completely relaxed. The polystyrene solutions were pressed at approximately 130 °C and annealed for 15 min under vacuum. The samples were checked by SEC again after the extensional stress measurements to ensure that there was no degradation or concentration change. All the polystyrene samples were pre-stretched to a radius Rp ranging from 1.5 mm to 2 mm at either 160 °C (for the melts) or 140 °C (for the solutions) prior to the elongational experiments to increase the maximum obtainable stress in the middle of the filament.2 After pre-stretching, the temperature was decreased to 130 °C for the extensional stress measurements of the melts, and between 110 and 120 °C for the solutions depending on the Tg. Nitrogen was used in the whole procedure. During the extensional measurements, the force F(t) is measured by a load cell and the diameter 2R(t) at the midfilament plane is measured by a laser micrometer. At small deformation in the startup of the elongational flow, part of the stress difference comes from the radial variation due to the shear components in the deformation field. This effect may be compensated by a correction factor as described in Rasmussen et al.18 The Hencky strain and the mean value of the stress difference over the mid-filament plane are then calculated as
ε(t ) = − 2 ln(R(t )/R 0)
(6)
where mf is the weight of the filament and g is the gravitational acceleration. The strain rate is defined as ε̇ = dε/dt. To ensure a constant strain rate, the diameter at the mid-filament plane is required to decrease exponentially during stretching. A recently updated control scheme19 is employed in the FSR to ensure accurate constant strain rate. The extensional stress growth coefficient is defined as η̅+ = ⟨σzz − σrr⟩/ε̇.
Table 3. Temperature Shift Factor aT for the Polystyrene Melts and Solutions sample name
F(t ) − mf g /2
Figure 1. Parameters in BSW spectrum. For ne = 0.23 and ng = 0.70, c1 ≈ 1.8 and c2 ≈ 1.4.
The BSW relaxation modulus G(t) is found from the continuous spectrum H(τ), given by G (t ) =
∫0
∞
H (τ ) exp( −t /τ ) dτ τ
H(τ ) = He(τ ) + Hg(τ )
(7) (8)
where He(τ ) =
⎛ τ ⎞ne ⎟ h(1 − τ /τm) ⎝ τm ⎠
neGN0 ⎜
⎛ τ ⎞−ng Hg(τ ) = neGN0 ⎜ ⎟ h(1 − τ /τm) ⎝ τc ⎠
(9)
(10)
where h(x) is the Heaviside step function. For the nearly monodisperse polystyrene melts and solutions, the values of ne and ng are fixed to 0.23 and 0.70, respectively.21 The meaning of the remaining BSW parameters G0N, τc , and τm and their relation to tube model parameters and the crossover frequencies is explained in the following. Originally the BSW model was proposed with two individual cut-off times.20 However in later studies,22−24 it was shown that a single cut-off time, τm , is sufficient. First of all, the BSW spectrum separates the relaxation time spectrum into He that represents the viscoelastic behavior and
(5)
and C
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Table 4. Material Properties Obtained from the BSW Spectrum sample name
T [°C]
ne
ng
G0N [Pa]
τc [s]
τm [s]
PS-545k PS-285k PS-285k/2k-72 PS-285k/2k-44 PS-545k/2k-58 PS-545k/DBP-63 PS-545k/DBP-52
130 130 130 130 130 70 70
0.23 0.23 0.23 0.23 0.23 0.23 0.23
0.7 0.7 0.7 0.7 0.7 0.7 0.7
256 700 252 040 133 370 45 650 84 140 94 070 62 200
0.419 0.444 0.0736 0.0268 0.0513 0.000 790 0.000 141
58 750 6846 381.5 24.29 1112 21.13 2.425
ne τmω + ··· 1 + ne
× × × × × × ×
109 108 106 105 107 105 104
The adjustable parameters G0N, τc , and τm are found by leastsquares fitting to the LVE data with the results given in Table 4. The zero shear rate viscosity η0 is calculated using eq 8 in Nielsen et al.26 and the value is also included in Table 4. Figures 2−4 present the corresponding BSW predictions.
Hg that represents the glassy behavior. If we similarly let Ge′ and G″e represent the parts of the storage and loss modulus derived from He the following asymptotic are obtained at low frequency, ne Ge′(ω) = GN0 (τmω)2 + ··· 2 + ne (11) Ge″(ω) = GN0
η0 [Pa·s] 2.823 3.242 9.613 2.145 1.756 3.728 2.833
(12)
Hence, the intersection of the two asymptotes occurs at ωa,l = c1/τm where c1 (= (2 + ne)/(1 + ne)) is a constant of order unity that depends only on ne. As illustrated in Figure 1, ωa,l is approximately equal to the low crossover frequency ωx,l. It follows that the BSW parameter τ m is equal to the disengagement time τd within a factor of order unity. Conversely, if G′g and G″g represent the storage and loss modulus derived from Hg, the high frequency asymptotes are given by neπ G′g (ω) ∼ GN0 (τcω)ng , for ω → ∞ 2 − ng 2 sin 2 π
(
)
neπ
G″g (ω) ∼ GN0
(
2 sin
1 − ng 2
π
)
(13)
(τcω)ng ,
for ω → ∞
Figure 2. LVE data fitted with the BSW spectrum for PS-545k and PS285k at 130 °C.
(14)
Hence, the intersection of the high frequency asymptote of G″g with G0N occurs at ωa,h = c2/τc where c2 (=(2 sin ((1 − ng)π/2)/ neπ)1/ng) is a constant of order unity. With the values for ne and ng commonly used for monodisperse polymers, τc becomes practically equal to the Rouse time for a single segment τe.25 Moreover, as illustrated in Figure 1, 1/τc is also approximately equal to the high crossover frequency ωx,h, albeit somewhat smaller. The BSW parameter G0N is exactly the plateau modulus, which may be seen from the following two relations: lim Ge′(ω) = GN0
ω→∞
2 π
∫0
∞
Ge″(ω) dω = GN0 ω
The fitted values of G0N for the two melts in Table 4 are in agreement with the value reported by Bach et al. (G0N(1) = 250 kPa at 130 °C).2 However the time constants do not agree with those reported by Bach et al. since those samples were slightly plasticized due to residual solvent. As a result the time constants reported by Bach et al. are smaller by a factor of about 1.4 (for PS200K) and 1.7 (for PS390K) than the time constants would be for the corresponding pure polystyrene melts. We take Me(1) = 13.3 kg/mol for polystyrene melts as reported by Bach et al.2 Consequently the number of entanglements per chain is Z(1) = 41.0 for PS-545k and Z(1) = 21.4 for PS-285k. Figure 5(a) plots the plateau modulus G0N from Table 4 as a function of the polymer volume fraction ϕ. The slope of the solid line in the figure is 2 (with α = 1), and the slope of the dashed line is 2.33 (with α = 4/3). It can be seen that the plateau modulus approximately follows the power law scaling as shown in eq 3. However, whether α = 1 or 4/3 cannot be clearly determined from Figure 5(a), since the slope has insufficient sensitivity to the value of α in the limited range of concentrations investigated. The open symbols in Figure 5b represent the ratio of the two frequencies ωx,h and ωx,l as a function of ϕ. The values of ωx,h and ωx,l are directly taken from the experimental data at the crossover points in Figures 2−4. The closed symbols in Figure 5b represent the ratio of τm and τc from Table 4 as a function of ϕ. Both the open symbols and the
(15)
(16)
Equation 15 is illustrated in Figure 1. Equation 16 is similar to the relation commonly used to determine the plateau modulus from experimental data.14 However there is one key difference. In the application directly to experimental data, one is faced with the problem of terminating the integral at some upper frequency to avoid contributions from the glassy modes. Conversely, in the fitting of the BSW spectrum to experimental data eq 16 is not used explicitly but the viscoelastic and glassy modes are automatically separated from each other due to the form of eq 8. D
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Figure 5. Determination of the dilution exponent α. (a) Plateau modulus G0N as a function of ϕ. The slope of the solid line in the figure is 2 (α = 1), and the slope of the dashed line is 2.33 (α = 4/3). Note that the temperature for the two DBP solutions (70 °C) is different from the other five samples (130 °C). (b) Ratio of the time scales as a function of ϕ. The slope of the solid lines in the figure is 3.4 (α = 1) and the slope of the dashed line is 4.53 (α = 4/3).
Figure 3. LVE data fitted with the BSW spectrum at 130 °C for (a) PS-285k/2k-72 and PS-285k/2k-44 and (b) PS-545k/2k-58. PS-285k data and PS-545k data are taken from Figure 2.
the open symbols, because 1/τc and 1/τm are not equivalent to the crossover frequencies as shown in Figure 1. The slope of the solid lines in Figure 5(b) is 3.4 (α = 1) and the slope of the dashed line is 4.53 (α = 4/3). Based on Figure 5b, we conclude that α = 1 gives the best description of our data in the range ϕ ∈ [0.44;1.00]. We therefore use α = 1 in the following. Table 5 lists the values of Me calculated from eq 1 with α = 1, as well as the values of Z for all the melts and solutions. In addition to the parameters in Table 4, we need a time scale for stretch relaxation of the chains. Since the single Table 5. Entanglement Molecular Weight Me and the Number of Entanglements per Chain Z for the Polystyrene Melts and Solutions
Figure 4. LVE data fitted with the BSW spectrum for PS-545k/DBP63 and PS-545k/DBP-52 at 70 °C.
closed symbols approximately follow the power law scaling as shown in eq 4. The level of the closed symbols is different from E
sample name
Me[g/mol]
Z
PS-545k PS-285k PS-285k/2k-72 PS-285k/2k-44 PS-545k/2k-58 PS-545k/DBP-63 PS-545k/DBP-52
13 300 13 300 18 472 30 227 22 931 21 111 25 577
41.0 21.4 15.4 9.4 23.8 25.8 21.3
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segment Rouse time is approximately given by the BSW parameter τc , a simple estimate for the chain Rouse time is Z2τc. Table 6 gives the resulting values. To demonstrate consistency Table 6. Evaluated Time Scales for the Polystyrene Melts and Solutions at 130 °C sample name
2τ1 [s], eq 17
2τ1 [s], eq 18
τLM R [s]
τR [s]
Z2τc [s]
PS-545k PS-285k PS-285k/2k-72 PS-285k/2k-44 PS-545k/2k-58
850.2 232.5 20.6 2.28 33.2
776.8 216.3 19.6 2.33 30.2
779 216 19.0 2.40 31.2
802 222 19.7 2.34 31.5
704.7 203.2 17.5 2.38 29.1
with other methods, however we present three other methods for determination of the longest stretch relaxation time τ1 of the Rouse model. First, τ1 can be evaluated from the coefficient a with27,28 ⎛ aM w ⎞2 τ1 = ⎜ ⎟ ⎝ 1.111ρϕRT ⎠
(17)
where Mw is the molecular weight of the polymer, and a is obtained from G′ in the power law range G′(ω) = aω1/2. Second, τ1 can be evaluated from the zero shear rate viscosity η0 with29 2.4 6M w η0 ⎛ Mc ⎞ τ1 = 2 ⎜ ⎟ π ρϕRT ⎝ M w ⎠
(18)
when Mw > Mc , where Mc is the critical molecular weight with the value 35 kg/mol for polystyrene melts.3,30 As for the polystyrene solutions, the value of Mc can be assumed to be proportional to the entanglement molecular weight Me.28,31 The evaluated Rouse time τ1 for polystyrene melts and solutions from both eq 17 and eq 18 are summarized in Table 6. It should be noted that here the Rouse time τ1 is a factor of 2 smaller than the value of the Rouse rotational relaxation time τR described in Larson et al.13 Finally, the Rouse rotational relaxation time τLM R evaluated from the Likhtman−McLeish theory32 is also listed in Table 6. These three methods are in good agreement with each other. From this point onward, the average Rouse time, τR , of these three methods is selected. Note that the τR is in close agreement with the simple estimate from Z2τc. Startup and Steady-State Elongational Flow. Figure 6a shows the measured extensional stress growth coefficient η̅+ as a function of time at 130 °C for the two polystyrene melts. The solid lines in the figure are predictions from the LVE parameters listed in Table 4. The deviation of the experimental data from the LVE at small Hencky strains (typically less than 0.03) is due to limitations in the dynamic control of the FSR at startup. The lowest strain rate for PS-285k was measured at 170 °C and shifted to 130 °C with the shift factor in Table 3. Figure 6b plots the same η̅+ from Figure 6a as a function of Hencky strain. It is clearly seen that η̅+ reaches a steady-state value ηs̅ teady for each strain rate at large Hencky strain (typically bigger than 4). The extensional stress growth coefficient η̅+ for the polystyrene solutions at 130 °C are shown in Figure 7. PS285k/2k-72 and PS-545k/2k-58 were measured at 120 °C, while PS-285k/2k-44 was measured at 110 °C. In Figure 7 the data are shifted to 130 °C with the respective shift factors listed
Figure 6. The measured extensional stress growth coefficient for PS285k and PS-545k at 130 °C. (a) The measured extensional stress growth coefficient as a function of time. Strain rate for PS-285k (from left to right): 0.03, 0.01, 0.003, 0.001, 0.0003, 0.00003 s−1. Strain rate for PS-545k (from left to right): 0.01, 0.003, 0.001 s−1. (b) The measured extensional stress growth coefficient as a function of Hencky strain. Strain rate for PS-285k (from top to bottom): 0.03, 0.01, 0.003, 0.001, 0.0003, 0.00003 s−1. Strain rate for PS-545k (from top to bottom): 0.01, 0.003, 0.001 s−1.
in Table 3. From these figures it is evident that the strain hardening effect is more pronounced especially at low concentrations in polystyrene solutions compared with melts. This is better seen in Figure 8, where the extensional steadystate viscosity η̅steady is plotted as a function of the strain rate ε̇ for all the polystyrene samples at 130 °C. The η̅steady shows a monotonic thinning for the two melts PS-285k and PS-545k, as previously reported in Bach et al.2 However, the three solutions show very different behavior. The η̅steady of PS-285k/2k-72 seems to have three regions. η̅steady initially decreases with increasing ε̇ in the region of low strain rate. At moderate strain rate, η̅steady does not change with increasing ε̇. Finally at large strain rate, η̅steady decreases with increasing ε̇. PS-285k/2k-44 and PS-545k/2k-58 only show a plateau region in the measured range of strain rates. At large values of ε̇, ηs̅ teady of PS-285k/2k44 shows a slight increase with increasing strain rate. F
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Figure 8. Extensional steady-state viscosity as a function of the strain rate for all the polystyrene samples at 130 °C.
Figure 9. Comparison of G′ and G″ normalized by the plateau modulus as a function of the normalized frequency for PS-285k melt and the two DBP solutions.
Figure 7. Measured extensional stress growth coefficient as a function of the time for the polystyrene solutions at 130 °C. The plots for PS545k and PS-285k are taken from Figure 6a. (a) Strain rate for PS285k/2k-72 (from left to right): 0.57, 0.29, 0.096, 0.029, 0.0096 s−1. Strain rate for PS-285k/2k-44 (from left to right): 5.6, 3.36, 1.68, 0.56, 0.168, 0.056 s−1. (b) Strain rate for PS-545k/2k-58 (from left to right): 0.45, 0.23, 0.076, 0.023, 0.0076 s−1.
Figure 10a, PS-545k/2k-58, Z = 23.8, is compared with the melt PS-285k, Z = 21.4, in the same way as in Figure 9. The slight mismatch between the curves for PS-285k and PS-545k/2k-58 at low frequency is due to the imperfect matching of Z. And finally Figure 10b compares PS-285k/2k-72, Z = 15.4, with the melt PS-200k, Z = 15.0. The shear data and the BSW parameters for PS-200k are taken from Bach et al.2 It appears from the figures, that provided the systems have the same number of entanglements, the modulus and time can be scaled such that the linear viscoelastic properties are identical. This then is a good confirmation, that the basic tube theory for melts carries over to solutions provided the tube dilation is adjusted for. We now turn to the behavior of PS-545k/2k-58 and PS-285k in extensional flow as compared in Figure 11a, also under nondimensional parameters scaled the same way as in Figure 10a. A similar comparison for PS-285k/2k-72 and PS-200k is presented in Figure 11b. The extensional data of PS-200k is again taken from Bach et al.2 The thin lines in parts a and b of Figure 11 are the LVE predictions. Departure from the LVE signifies non-linear strain hardening. It is apparent that solutions and melts show similar linear behavior in extension,
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DISCUSSION We are now in a position to compare the linear and nonlinear properties of polystyrene solutions and melts with the same number of entanglements but different Me. In Figure 9, PS545k/DBP-52, Z = 21.3, is compared with the melt PS-285k, Z = 21.4, in small amplitude oscillatory shear flow. Since the glass transition temperatures Tg of the solution and the melt are quite different, they are compared using dimensionless parameters with ω̃ = ωτc, G̃ ′ = G′/G0N, and G̃ ″ = G″/G0N, where τc and G0N can be found in Table 4. In this way, the crossover point at high frequency for PS-545k/DBP-52 in Figure 4, and the similar point for PS-285k in Figure 2, are shifted to overlap each other in Figure 9. It can be seen that with the same values of Z, G′ and G″ data of melts and solutions can be superimposed creating a master curve for a given Z. In contrast, in Figure 9 the G′ and G″ data of PS545k/DBP-63 which has a higher value of Z (Z = 25.8) cannot overlap the data of PS-285k at low frequencies. Likewise in G
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Figure 11. Comparison of normalized η̅+ as a function of normalized time in the startup of extensional flow for (a) PS-545k/2k-58 and PS285k and (b) PS-285k/2k-72 and PS-200k (τc = 0.31s).
Figure 10. Comparison of G′ and G″ normalized by the plateau modulus as a function of the normalized frequency in small amplitude oscillatory shear flow for (a) PS-545k/2k-58 and PS-285k, and (b) PS285k/2k-72 and PS-200k (τc = 0.31s).
The higher steady-state viscosity observed in polystyrene solutions at WiR > 1 may be due to their higher maximum stretch ratios, λmax. The values of Ne for some typical polymer melts, such as polyethylene (PE), isotactic polypropylene (iPP) and polyisoprene (PI), can be found in Fetters et al.33 Their maximum stretch ratios λmax are listed in Table 7. The value of Ne for atactic polystyrene (a-PS) can not be directly found in Fetters et al.,33 but can be calculated from
but very different non-linear strain hardening behavior, as already mentioned in the section Startup and Steady-State Elongational Flow. In the LVE regime, a change in Me only changes the magnitude of G0N, cf. Equation 2. Thus, normalizing G′ and G″ by G0N eliminates the dependence on Me, collapsing the vertical axis, cf. Figure 10. Contrary in the nonlinear regime, normalization of η̅+ with G0Nτc does not collapse the curves for solution and melt, cf. Figure 11, signifying that Me plays a more complicated role in the nonlinear regime. In Figure 12, we compare the dimensionless steady-state viscosity of the melts and solutions determined from Figure 11, normalized by the time scale τR. Note that this is equivalent to normalizing the data by τc, since the values of Z are similar and τR ≈ Z2τc as shown in Table 6. The Weissenberg number is defined as WiR = ε̇τR. It can be seen from both parts a and b of Figure 12 that at WiR > 1, the non-dimensional steady-state viscosity of the solutions are higher than the melts. It can also be directly seen from Figure 11(a) that the non-dimensional stress growth coefficient in the startup of the flow is almost the same for PS-545k/2k-58 and PS-285k, due to their similar Weissenberg numbers as shown in Figure 12a. However, PS285k reaches a steady state much earlier than PS-545k/2k-58.
Ne = a 2 /b2 , ⟨R2⟩0 ⟨R2⟩0 M ⟨R2⟩0 nmb ⟨R2⟩0 = = = R max M R max M nl cos(θ /2) M mb , l cos(θ /2)
b=
(19)
where a is the tube diameter, b is the Kuhn length, ⟨R ⟩0 is the mean square end-to-end distance of the polymer chain, Rmax is the fully extended size of the chain, M is the molecular weight of the chain, mb is the average molecular weight per backbone bond, l is the backbone bond length, and θ is the backbone bond angle. According to Fetters et al.,33 for a-PS at 413 K, a = 85.2 Å, ⟨R2⟩0/M = 0.437 Å2, l = 1.5 Å, cos (θ/2) = 0.83, and mb 2
H
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steady-state viscosity also decreases monotonically as a function of strain rate. This then is not surprising, since the value of λmax of PE melts is even lower than that of a-PS melts. For fully stretched molecules, the steady stress level should scale as σsteady ∼ λmax2 if we assume λmax. Hence if finite extensibility is to explain the non-linear data alone, then we expect that the difference in the ratio of steady-state viscosity between solutions and melts should be proportional to the ratio of λmax2. However, as seen in Figure 12, the ratio of ηs̅ teady between PS-285k/2k-72 and PS-200k is not constant. From Figure 12 it is clear that the ratio of solution to melt viscosities is much larger than the corresponding λmax2 ratios. This observation shows that the finite extensibility of the molecule is in itself insufficient to explain the trend in ηs̅ teady, and suggests that other parameters such as monomeric friction as reported by Yaoita et al.,36 may also influence the non-linear behavior.
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CONCLUSIONS From the results presented, here four conclusions can be identified. (1) It is possible to scale the time constant and the plateau modulus such that polystyrene melts and concentrated polystyrene solutions with the same Z have identical LVE properties both in shear and extension. (2) The dilution exponent, α, is more readily determined from a plot of τm/τc (ωx,h/ωx,l) versus ϕ due to the sensitivity of τm/τc to α. And thus the plateau modulus (G0N) scales with the square of the polymer volume fraction in the concentrated domain investigated. (3) Polystyrene melts and concentrated solutions with the same Z differ in the non-linear extensional region. Strain hardening increases with the number of Kuhn steps per entanglement (Ne). In addition, the power law dependence of steady-state extensional viscosity on strain rate originally observed for monodisperse polystyrene melts is not observed for concentrated polystyrene solutions. (4) The finite extensibility of the molecule is insufficient to capture the reported trend in steady-state extensional viscosity.
Figure 12. Comparison of normalized extensional steady-state viscosity as a function of Weissenberg number for (a) PS-545k/2k58 and PS-285k and (b) PS-285k/2k-72 and PS-200k.
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Corresponding Author
Table 7. Maximum Stretch Ratio λmax for Polymer Melts and Solutions sample name
T [K]
Ne
λmax
PE i-PP PI a-PS PS-285k/2k-72 PS-285k/2k-44 PS-545k/2k-58
413 463 298 413 413 413 413
6.89 36.5 46.5 21.8 30.3 49.5 37.6
2.6 6.0 6.8 4.7 5.5 7.0 6.1
AUTHOR INFORMATION
*E-mail: (O.H.)
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The FSR control scheme was developed by Dr. J. M. R. Marı ́n. The research leading to these results has received funding from the European Union Seventh Framework Programme [FP7/ 2007−2013] under Grant Agreement No. 214627-DYNACOP and the Danish Council for Independent Research − Technology and Production Sciences Grant no. 10-082409. O.M. and K.A. were supported by the VKR centre of Excellence NAMEC.
= 52. Inserting the values into eq 19, we get Ne = 21.8, which is in agreement with Fang et al.34 If the molecular weight of a Kuhn segment is M0, then Me = NeM0. The values of M0 for polystyrene melts and solutions are assumed to be the same. Therefore, according to eq 1, the values of Ne for polystyrene solutions are calculated using Ne,s = Ne,mϕ−1 = 21.8ϕ−1. These values are listed in Table 7. If the hypothesis is that λmax is the relevant parameter that determines non-linear behavior, PS285k/2k-44 should behave similarly to PI melts and PS-545k/ 2k-58 should behave similarly to i-PP melts. However such comparison have to our knowledge not been made. The previous experiments on PE melts35 have shown that the
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