Dynamic mechanical properties of linear and cross-linked polyurethane

Dynamic mechanical properties of linear and cross-linked polyurethane Frederic Prochazka, Dominique Durand, and Taco Nicolaia) Chimie et Physique des Mate´riaux Polyme`res, UMR CNRS, Universite´ du Maine, 72085 Le Mans Cedex 9, France (Received 6 April 1999; final revision received 19 July 1999)

Synopsis Linear polyurethane melts were prepared by a polycondensation reaction of poly共oxypropylene兲 共POP兲 diol with a diisocyanate. Covalently cross-linked gels were obtained using three-armed star POP triol. The glass transition temperature and the viscoelastic properties were investigated as a function of the molar mass of the POP precursors. The variation of T g is dominated by the density of urethane links. The loss peak of the shear modulus at high frequencies or low temperatures broadens with increasing density of urethane links. The gel modulus of end-linked POP triol decreases linearly with increasing molar mass of the precursors. The loss shear modulus of end-linked POP triol has a power law frequency dependence at low frequencies. The exponent of the power law dependence decreases with increasing molar mass of the precursors. Gels formed with POP triol with molar mass larger than 6 kg/mol show the effect of entanglements at intermediate frequencies. © 1999 The Society of Rheology. 关S0148-6055共99兲00906-2兴

I. INTRODUCTION Dynamic shear measurements on poly共oxypropylene兲 共POP兲 melts show a relaxation at high frequencies or equivalently low temperatures characterizing the transition from solid to liquid behavior 关Randrianantoandro and Nicolai 共1997兲兴. This so-called ␣ relaxation is due to motion on the scale of a few segments. At lower frequencies the segmental relaxation crosses over in the chain backbone conformational relaxation. Recently, it was shown 关Nicolai and Floudas 共1998兲兴 that the conformational relaxation of linear POP diol and three-armed star POP triol can be well described by the so-called Rouse model up to a spanning molar mass M s ⬇ 4000 g/mol. M s is the molar mass of two arms for star polymers and is equal to the total molar mass of POP diol and two thirds of POP triol. The Rouse model describes the chain backbone motion in terms of a superposition of normal modes and assumes that hydrodynamic interactions are screened 关Doi and Edwards 共1990兲兴. The frequency dependence of the shear modulus of POP melts fits well to the sum of a stretched exponential for the ␣ relaxation plus the spectrum of normal modes. For M s ⬎ 4000 g/mol the polymers entangle and it is necessary to include the relaxation process of disentanglement. POP diol and triol can be end linked with a diisocyanate to form polyurethane. In the first case the result is an entangled melt of long linear chains while in the second case a three-dimensional polymer network is formed. It is obvious that end linking has important consequences for the mechanical properties. Earlier we made an extensive study of a兲

Author to whom all correspondence should be addressed; electronic mail: [email protected]

© 1999 by The Society of Rheology, Inc. J. Rheol. 43共6兲, November/December 1999

0148-6055/99/43共6兲/1511/14/$20.00

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PROCHAZKA, DURAND, AND NICOLAI TABLE I. Characteristics of POP triols and diols. M n (g/mol)

M w /M n

T g (K)

426 1010 2040 4060 7400 11000 260 720 2590 6200 7600

1.06 1.04 1.08 1.2 1.2 1.1 1.1 1.08 1.08 1.2 1.2

201 203 205 207 207 207 219 212 207 204 204

D425 D1000a D2200b D4200b D8200b D12200b T260a T720b T2500b T6300b T8000c a

Purchased from Aldrich. Gift from Arcol. c Gift from ICI. b

one small POP triol sample as a function of cross-link density, both in situ during the reaction 关Prochazka et al. 共1996兲兴 and for different amounts of added cross-linking agent after complete reaction 关Tabellout et al. 共1995兲 and Nicolai et al. 共1997a,b兴. Here we look at fully end-linked linear and star POP over a range of molar masses.

II. EXPERIMENT Polyurethanes were formed by polycondensation of polyoxypropylene triols and diols with hexamethylene diisocyanate 共HMDI兲 共Aldrich兲. The polyols used in this study have been characterized by size exclusion chromatography 共SEC兲 using both refractive index and UV absorption detection 关Foucault 共1998兲兴. The combination of these two detection techniques allows determination of the size distribution and the number of hydroxyl groups per polymer. With high resolution columns the individual component of the smallest triol 共T260兲 and diol 共D425兲 can be resolved. These samples consist of a series of fully functionalized oligomers with a number of oxypropylene segments between 2 and 6 for T260 and between 5 and 10 for D425. The samples T720, D1000, and D2200 are also fully functionalized, but larger diols and triols contain a fraction of monofunctional material f m with lower molar mass. f m was estimated to be less than 5% for T2500 and between 5% and 10% for the larger samples. The number averaged molar mass M n and the polydispersity index M w /M n are summarized in Table I. We note that the polydispersity indices are overestimated due to intrinsic dispersion effects in the SEC columns. To obtain fully end-linked systems the ratio of isocyanate groups to hydroxyl groups, r ⫽ 关 NCO兴 / 关 OH兴 , needs to be unity. The amount of HMDI needed to obtain r ⫽ 1 was calculated on the basis of the hydroxyl content of the samples. 2⫻10⫺3 g dibutyl tin dilaurate catalyst was added for every gram of HMDI. After mixing and complete homogenization of the reaction components, the samples were cured at 40 °C until complete consumption of the isocyanate groups. We have determined the fraction of soluble material using SEC after adding a large excess of THF to the gels. For all the samples studied here the fraction of soluble material was negligible.

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TABLE II. Characteristics of end-linked POP diols.

D425 D1000 D2200 D4200 D8200 D12200

DPw

DPw /DPn

T g (K)

T g v (K)

167 51 67 10 16 13

3.5 2.2 2.3 2.1 3.1 2.4

245 222 215 210 210 208

244 219.3 211.5 207.8 205.3

Glass transition temperatures T g were measured using differential scanning calorimetry 共DSC兲. The samples were cooled rapidly to 170 K and then heated at a rate of 10°/min. Values of T g were taken as the midpoint of the transition. The reproducibility is ⫾2°. Dynamic shear measurements were done on a Rheometric Scientific RDA II dynamic spectrometer using parallel-plate geometry at temperatures between 200 and 340 K. In the so-called hold mode the gap is corrected for temperature variations of the sample volume. The plate size 共diameter between 50 and 4 mm兲 and the imposed deformation 共0.2%–20%兲 were adjusted to obtain an accurate torque response while remaining in the linear regime. The shear modulus could be measured in the range 10– 109 Pa. We were able to measure very large moduli by using the maximum stable sample thickness 共2–2.5 mm兲 in combination with a small plate size. III. RESULTS A. Segmental relaxation We have studied fully end-linked POP diol with molar masses between 425 and 11 000 g/mol and POP triol between 260 and 8000 g/mol 共see Table I兲. The glass transition temperatures T g were measured by DSC. The molar mass dependence of T g of precursor POP diol and triol is much smaller than might be expected from the effect of free ends 共see Table I兲. This is probably due to the effect of hydrogen bonding which reduces the mobility of the end groups. In fact for POP triol the hardening effect of the core is more important than the softening effect of the free ends. For the fully end-linked samples T g increases with decreasing molar mass of the precursor 共see Tables II and III兲. Figure 1 suggests that the increase is mainly due to the isocyanate fraction. Since the mobility of free ends is reduced anyway by hydrogen bonding, the loss of free ends after end linking is not important. The effect of the core plays a role in the small POP triols. Figure 2 shows the radial frequency dependence of the normalized loss modulus G ⬙ of end-linked POP diol and triol with different molar masses. The master curves were

TABLE III. Characteristics of end-linked POP triols. G 0 (⫻105 Pa) a

a

z

T g (K)

T g v (K)

26 5.5 3.4 2.5

0.62 0.39 0.54 0.47

0.88 0.48 0.32 0.28

301 261 221 215 210

303.3 257 214.3 207.7 206

T260 T720 T2500 T6300 T8000 a

At T ⫽ 323 K.

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FIG. 1. Glass transition temperatures of fully end-linked POP diol 共open兲 and triol 共filled兲 as a function of the weight fraction of HMDI.

obtained using time–temperature superposition of data at different temperatures. In a semilogarithmic representation one only observes the ␣ relaxation. No significant variation of the shape with temperature was observed except for the smallest triol 共T260兲. It is difficult to obtain reproducible results for fully end-linked T260 and we will not discuss this sample in the following. It is clear that the width of the peak broadens with decreasing molar mass of the precursors. We have fitted the loss and storage moduli simultaneously to the generalized exponential relaxation time distribution 共GEX兲 G⬘共␻兲 ⫽ G⬁

冕

G⬙共␻兲 ⫽ G⬁

冕

␻2␶2

⬁

0

Ag共␶兲

1⫹␻2␶2

␻␶

⬁

0

Ag共␶兲

1⫹␻2␶2

d␶, 共1兲 d␶,

with A共␶兲 ⫽

p s 兩s兩•␶⫺p g ␶ exp关⫺共␶/␶g兲 兴

⌫

冉冊 p

,

共2兲

s

where G ⬁ is the high frequency modulus, and k is a normalization constant such that 兰 A g ( ␶ )d ln ␶ ⫽ 1. The GEX function is equivalent to a stretched exponential relaxation in the time domain: G(t) ⫽ G ⬁ exp关⫺(t/␶w)␤兴 for ␤ ⬍ 0.6 if p ⫽ ␤ and s ⫽ ␤ /(1 ⫺ ␤ ) 关Nicolai et al. 共1996兲兴. Stretched exponential relaxation gives a good description of the data for ␻ ⬍ 3 ␻ max , but deviates at higher frequencies. A similar deviation is observed for the precursors. ␤ decreases with decreasing molar mass and is mainly related to the concentration of urethane links 共see Fig. 3兲. The lower value of ␤ of T720 compared to D425 shows the influence of the core. For the precursors we found

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FIG. 2. 共a兲 Frequency dependence of the normalized loss shear modulus of fully end-linked POP diol with different molar masses as indicated in the figure. The lines represent a stretched exponential decays with ␤ ⫽ 0.49 共solid兲 and 0.39 共dashed兲. 共b兲 Frequency dependence of the normalized loss shear modulus of fully end-linked POP triol with different molar masses. The lines represent a stretched exponential decays with ␤ ⫽ 0.49 共solid兲 and 0.33 共dashed兲.

␤ ⫽ 0.49 independent of the molar mass 关Randrianantoandro and Nicolai 共1997兲兴. This value is the same as for large end-linked POP for which the concentration of urethane links can be neglected. The temperature dependence of ␻ max is plotted in Fig. 4 as a function of T ⬘ ⫺T g v . T g v is defined as the temperature where ␻ max ⫽ 1 rad/s. T g v can be seen as an alternative value of the glass transition temperature and is roughly 4° above T g as determined by DSC, 共see Tables II and III兲. Due to the limited dynamical range of rheometers, ␻ max can only be determined over a range of about 4 decades and thus only over a very limited temperature range. In this range the variation of the temperature dependence with the molar mass of the precursor is small. The temperature dependence of the POP with M s ⬎ 1000 is the same before and after end linking.

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FIG. 3. Stretched exponent ␤ of fully end-linked POP diol 共open兲 and triol 共filled兲 as a function of the weight fraction of HMDI.

B. Chain backbone relaxation Ideally, end linking linear POP diol with diisocyanate leads to the formation of a single polymer if the stoichiometric ratio of the functional groups is exactly unity. In practice a polydisperse solution of chains is obtained with varying degrees of polymerization 共DP兲 共see Table II兲. Impurity of the starting products, the presence of material with lower functionality, and competing side reactions can all lead to a decrease of DP.

FIG. 4. Dependence of the peak position of the loss shear modulus of fully end-linked POP diol 共open兲 and triol 共filled兲 as a function of T⫺T g v . T g v is defined as the temperature where ␻ max ⫽ 1 rad/s. Symbols are as in Fig. 2. The solid line shows the behavior of the precursors taken from Ref. 2.

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Using mean field theory 关Stockmayer 共1943兲; Gordon 共1962兲; Macosko and Miller 共1976兲, and Durand and Bruneau 共1979兲; 共1982兲兴 we can calculate the influence of r, the reaction extent p, and the functionality on DP. An uncertainty in r and p of only a few percent explains the observed values of DP of the samples D425, D1000, and D2200. The low values of DP for the larger precursors are due to the presence of a fraction of monofunctional POP. We have measured the frequency dependence of the loss and storage shear moduli of each sample at different temperatures. It was shown in detail 关Randrianantoandro and Nicolai 共1997兲兴 that the ␣ relaxation and the slower conformation relaxation have a slightly different temperature dependence for T⫺T g v ⬍ 30 K. As the temperature approaches T g the conformational relaxation merges with the ␣ relaxation. This means that we cannot apply time–temperature superposition to all the data. Nevertheless, it is possible to show the whole frequency dependence at given reference temperature T ref by superimposing data at high frequencies for T ⬍ T ref and data at low frequencies for T ⬎ T ref . T ref was chosen so that both relaxation modes are visible in the frequency window. All master curves shown in this article were obtained in this way. Master curves of different systems can be compared directly because we have chosen T ref at the same distance from T gv : T ref⫺T g v ⫽ 5 K. Figure 5 shows the effect of end linking for D12000. For this molar mass the relaxation at high frequencies is slightly modified by end linking because the fraction of urethane links is small. D12000 is already weakly entangled before end linking, but doesn’t show a plateau in G ⬘ or a maximum in G ⬙ . After end linking the terminal relaxation time and thus the viscosity ␩ is increased by more than 5 orders of magnitude. There is a very broad crossover from the elastic response (G ⬘ ⬀ ␻ 0 ) at high frequencies to the purely viscous response at low frequencies (G ⬘ ⬀ ␻ 2 ) characteristic for liquids. The broad crossover is due to polydispersity of the end-linked systems. For entangled melts of monodisperse linear polymers the time needed for disentanglement t r increases strongly with the molar mass t r ⬀ M 3.4. Therefore even a small polydispersity gives a relatively broad relaxation time distribution. Figure 6 compares the frequency dependence of G ⬘ of end-linked POP diols with different molar masses. The plateau modulus G 0 is independent of the molar mass of the precursor except for the smallest precursor. This means that the molar mass between entanglements M e is not strongly modified by the incorporation of urethane bonds. The crossover to the liquidlike behavior at low frequencies depends on the molar mass distribution. It varies from one sample preparation to another because it is sensitive to small differences in r. The temperature dependence of the low frequency relaxation is shown in Fig. 7. It is almost independent of the molar mass of the precursor and can be fitted to the so-called Vogel Fulcher Tamman 共VFT兲 equation log共aT兲 ⫽ ⫺6.40⫹

595 共 T⫺T g v 兲 ⫹43.4

T ref ⫽ T g v ⫹50.

共3兲

As shown in Fig. 7 the temperature dependence before and after end linking is similar. Ideally, end-linking POP triol leads to the formation of a regular polymer network, but, in practice a small fraction of chain ends will not be linked together, for the same reasons that end-linking POP diol does not lead to a single long polymer. Figure 8 shows the effect of end linking for T8000. As in the case of large POP diol, end linking large POP triol does not modify the segmental relaxation. However, after end linking, G ⬘ relaxes to a frequency independent gel modulus G 0 . In Fig. 9 we compare end-linked

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FIG. 5. Comparison of the storage 共top兲 and loss 共bottom兲 moduli of D12000 before 共triangles兲 and after 共circles兲 end linking.

D12000 with end-linked T6000. The comparison shows that the plateau modulus of linear POP melts is close to the gel modulus of POP networks with molar mass between cross links M c ⬇ 4000 g/mol. This means that M e ⬇ 4000 g/mol in agreement with measurements on the precursors. As discussed in Mark and Erman 共1988兲 the dynamics on length scales smaller than M e or M c are the same for entangled melts and gels. Two extreme assumptions have been made to explain the low frequency modulus G 0 of cross-linked polymer networks 关Mark and Erman 共1988兲兴. In the so-called phantom model it is assumed that the cross links are fully mobile, while in the so-called affine model the cross links are assumed immobile. G 0 is in both cases inversely proportional to the molar mass between cross links: G 0 ⫽ a ␳ RT/M c , where R is the gas constant, ␳ is the density, and T is the absolute temperature. The affine model gives a ⫽ 1 while for the phantom model a depends on the functionality of the cross links and gives a ⫽ 1/3 for f ⫽ 3. Experimental values of a are generally situated between these two extremes. If M s is larger than M e trapped entanglements can act as cross links and increase G 0 , but the effect of entanglements on G 0 is negligible even for the largest precursor used here.

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FIG. 6. Comparison of the storage moduli of end-linked POP diol with different molar masses. Symbols are as in Fig. 2共a兲.

Values of G 0 at T ⫽ 323 K, and a of end-linked POP triol with different molar masses are given in Table III. In the calculation we have used ␳ ⫽ 1.0 g/ml and M c ⫽ M s 共triol兲⫹M 共HMDI兲. a is about 0.5 independent of M s . The value of a is closer to that of a phantom network, but any nonideality reduces the number of elastically active chains and thus a. It is therefore difficult to conclude which model is best suited to describe the present system.

FIG. 7. Temperature dependence of the shift factors used to superimpose the low frequency data of the shear modulus of fully end-linked POP diol 共open兲 and triol 共filled兲. Symbols are as in Fig. 2. The solid line represents a nonlinear least squares fit to the VFT equation.

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FIG. 8. Comparison of the storage 共top兲 and loss 共bottom兲 moduli of T8000 before 共triangles兲 and after 共circles兲 end linking.

That the gels are not ideal becomes evident when we look at the frequency dependence of G ⬙ 共see Fig. 8 and 10兲. Ideal polymer networks are fully elastic which means that G ⬙ ⬀ ␻ at low frequencies. In fact we find at low frequencies G ⬙ ⬀ ␻ z with z depending on the molar mass of the precursors, but always smaller than 1. A power law dependence of G ⬙ with z ⬍ 1 has been reported earlier for other polymer networks 关Adolf and Martin 共1991兲兴. If G ⬙ has a power law dependence on ␻ then G ⬘ should show the same frequency dependence after subtraction of G 0 . Figure 10 shows that this is indeed the case. The exponent z decreases with increasing molar mass of the precursor 共see Table III兲. The value of z for the sample based on T720 is slightly larger than reported earlier for a different gel based on the same precursor 关Nicolai et al. 共1996兲兴. The difference probably reflects the dependence of the nonideality of the net-

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FIG. 9. Comparison of the storage 共top兲 and loss 共bottom兲 moduli of end-linked D12000 共triangles兲 and T6000 共circles兲.

work on the exact value of r and p. This is analogous to the variation of the degree of polymerization of POP diol for different preparations at r ⫽ 1. IV. DISCUSSION It appears that the nonideality of gels leads to a power law relaxation time distribution at low frequencies at least over the range accessible in the experiment. From Eq. 共1兲 it follows that G ⬘ ⬀ G ⬙ ⬀ ␻ z implies that the relaxation time distribution also has a power law dependence: A( ␶ )d ln(␶) ⬀ ␶⫺zd ln(␶). Any nonideality with the following two properties gives rise to power law dependence of A( ␶ ): 共1兲 the number of defects decreases exponentially with size m:

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FIG. 10. Frequency dependence of the loss modulus 共circles兲 and the storage modulus before 共squares兲 and after 共triangles兲 subtraction of G 0 of fully end-linked POP triol with different molar masses indicated in the figure.

N共m兲dm ⬀ exp共⫺a1•m兲dm,

共4兲

␶ ⬀ exp共a2•m兲.

共5兲

共2兲 the relaxation time of a defect increases exponentially with m:

Inserting Eq. 共5兲 into Eq. 共4兲 we obtain for the relaxation time distribution, A共␶兲d ln ␶ ⬀ ␶⫺a1 /a2d ln ␶,

a1 ⬍ a2,

共6兲

which means that z ⫽ a 1 /a 2 . One source of nonideality is dangling chains larger than the distance between cross links. Such chains relax by disentanglement of dangling chains from the network. As one end of the dangling chain is fixed to the network disentanglement can only occur through chain retraction. The time needed to relax through a chain retraction process increases exponentially with the degree of polymerization of the dangling chain 关Rubinstein et al. 共1990兲 and Curro et al. 共1985兲兴. In addition, using mean field theory, we find that the number of dangling chains decreases exponentially with their length for the present system. This means that dangling chains could give rise to a power law relaxation time distribution. In order to estimate the relative contribution to the shear modulus we need to calculate the fraction of 共branched兲 dangling chains with backbone larger than M c . For the fully end-linked samples studied here, dangling chains occur because r is not exactly 1 and because the samples contain a small fraction of monofunctional material f m . Using mean field theory we have calculated the weight fraction of dangling chains with backbone larger than M c for the present system in a way analogous to the calculation of Rubinstein et al. 关Rubinstein et al. 共1990兲兴. In Fig. 11 the result is compared to the fraction of segments in the elastic network as a function of r. Clearly, the fraction of dangling chains larger that M c is too small to give rise to the slow relaxation shown in

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FIG. 11. Comparison of the weight fraction of segments in the elastic network 共circles兲 and in the dangling chains with backbone molar mass larger than M c 共squares兲 as a function of the stoichiometric ratio r.

the experiment unless r is close to r c and f m is large. Since for r ⬎ 0.6 and small f m the sol fraction is even smaller than that of the dangling chains it cannot be invoked either to explain the slow relaxation. Note that the gels we are investigating are very different from gels formed by the lightly cross linking large highly entangled polymers (M c Ⰷ M e ) which also show a power law relaxation 关Curro et al. 共1985兲兴. A second, more likely, source of the slow relaxation of the shear modulus is restructuring of the gel under the influence of stress involving a cooperative motion of a domain containing m segments. In real gels the spatial distribution of cross links is inhomogeneous 关Bastide et al. 共1988兲兴. Redistribution of the relative positions of the cross links could possibly release stress. Structural relaxation gives rise to a power law relaxation time distribution if it becomes both exponentially slower and exponentially less likely with increasing m. Whatever the origin of the slow relaxation the present results show that it is more prominent if the molar mass between cross links is larger. In an earlier study on gels based on T720 关Nicolai et al. 共1996兲兴, we showed that the slow relaxation is also more prominent if we reduce the connectivity extent. In this context it is important to clearly distinguish the power law frequency dependence of G ⬙ close to the gel point from that of more densely connected systems showing a clear gel modulus. At the gel point the relaxation is due to conformational relaxation of the polydisperse sol fraction. The percolation model in conjunction with Rouse dynamics predicts z ⫽ 0.67, not far from the values observed for a number of systems at the gel point. With increasing connectivity extent the gel fraction becomes more and more important. The relaxation of the gel fraction on length scales smaller than the distance between cross links also gives rise to a power law frequency dependence, but with a smaller value of z because polydispersity does not influence the value. This explains why the observed values of z initially decrease with increasing r. Only for r ⭓ 0.65 could we begin to observe the power law relaxation

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on length scales larger than the distance between cross links. The power law exponent of this relaxation increases with increasing r for r ⭓ 0.65 when the fraction of the sol and dangling chains larger than the average distance between cross links is negligible. The increase of z could mean that restructuring is less likely for a denser gel. Gels made with the two largest precursors show an intermediate weaker frequency dependence between the normal modes relaxation between cross links and the final power law dependence. Possibly this intermediate regime is related to the effect of entanglements as M e becomes larger than M c .

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