Dynamic shear rheology of high molecular weight polydimethylsiloxanes: comparison of rheometry and ultrasound1

J. Non-Newtonian Fluid Mech., 76 (1998) 213 – 232

Dynamic shear rheology of high molecular weight polydimethylsiloxanes: comparison of rheometry and ultrasound1 P.Y. Longin, C. Verdier, M. Piau * Laboratoire de Rhe´ologie 2, BP 53, Domaine Uni6ersitaire, 38041 Grenoble, Cedex 9, France Received 16 June 1997; received in revised form 20 July 1997

Abstract The viscoelastic properties of three linear polydimethylsiloxanes (PDMS) of high molecular weights are investigated using rheometrical as well as ultrasonic tests over a large range of temperatures. Classical shear rheometrical measurements are carried out in the low frequency range from 10 − 1 to 102 rad s − 1 between −50 and +20°C. The frequency range is enlarged using the time–temperature superposition principle, allowing coverage of about 6 – 7 decades of pulsation. Ultrasonic tests use an inclined incidence wave reflection technique to measure the complex shear mechanical impedance from 1.5 to 25 MHz, between − 10 and +50°C. Rheometrical and ultrasonic experiments are then combined for the three PDMSs at the same reference temperature. They give the reduced shear elastic and loss moduli for reduced frequencies covering 10 decades. A discrete relaxation time spectrum is first deduced from the master curve in each case. More accurate predictions may be obtained using a molecular weight distribution and BSW (Baumgaertel–Schausberger– Winter) model for polydisperse systems. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Polydimethylsiloxane; Relaxation; Mechanical impedance; Rheometry; Shear waves; Ultrasound

1. Introduction It is well known that conventional rheometers do not allow the characterization of materials at high frequencies. Nevertheless, by use of the time–temperature superposition principle, it is possible to enlarge the frequency range covered. For some polymers, the entire viscoelastic behaviour from the terminal region to the glassy state has been investigated. Recently, Palade et * Corresponding author. Fax.: +33 04 76825164. 1 Dedicated to the memory of Professor Gianni Astarita. 2 Universite´ Joseph Fourier, Grenoble I, Institut National Polytechnique de Grenoble, CNRS (UMR 5520). 0377-0257/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 0 2 5 7 ( 9 7 ) 0 0 1 1 9 - 5

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al. [1] obtained the loss and storage shear moduli for polybutadienes over 15 logarithmic decades of frequency and over a wide range of temperatures (200°C). However, for some polymers crystallization reduces the application of the time–temperature superposition principle to a smaller range of temperatures. This is the case for polydimethylsiloxanes (PDMSs), for which a better understanding of the moderate to high frequency properties is still required, due the large variety of their industrial applications. At a given temperature, the development and growth of crystalline domains is observed [2,9] (below − 50°C) and this modifies the mechanical characteristics of these products. The time–temperature superposition principle is no longer valid. By dynamic and creep methods, Plazek et al. [2] investigated a series of very high molecular weight polydimethylsiloxanes (MW =0.41×106 – 4.9 × 106 g mol − 1) between − 49 and +75°C, from a maximum frequency of 600 Hz to a maximum time of 36 days. Their measurements encompassed only the terminal and plateau zones of the viscoelastic time scale because crystallization occurred at −49°C. Several efforts have been made recently to investigate the missing range of frequencies. For example, a rheometer was developed by Thomas et al. [3] to measure the viscoelastic properties of polymer melts at ultrasonic frequencies, but this instrument operates only at pulsations ranging from 0.25× 106 –1.28× 106 rad s − 1. Eggers and Richmann [4] described a rheometer which allows shearing of a liquid sample in the frequency range 1 Hz–10 kHz. They realized that this instrument must be improved (sensitivity and reduction of parasitic noise at high frequencies) to reach a standard of equipment comparable to ultrasonic techniques. The use of non intrusive techniques such as ultrasound can be a very useful means of extending master curves of the storage and loss moduli (G%, G¦) for molten polymers [5,6]. By use of ultrasonic shear waves, the velocity and attenuation of sound are directly connected with the complex shear modulus G* of the medium studied. Herzfeld and Litovitz [7] showed that the measurement of these acoustic parameters is difficult except at very low frequencies, due to the high attenuation. In fact, the complex shear modulus at high frequencies may be obtained by determining the mechanical shear impedance of the material. Investigations on polymer melts using ultrasound have been developed on different occasions over the past 40 years, in the case of polydimethylsiloxanes. Zeqiri [8] has even carried out measurements on Dow Corning 710 silicone to establish its suitability as a reference material for ultrasonic attenuation. Barlow et al. [9] studied six polydimethylsiloxanes of viscosity grades ranging from 0.1 to 100 Pa s in the frequency range 10 kHz–78 MHz, at temperatures between −50 and +50°C. They observed that crystallization occurred in the polymer when the temperature fell below −50°C and investigated the transition zone from a rubber to a glass-like consistency, showing that in this transition the viscoelastic behaviour becomes clearly independent of molecular weight for liquids with viscosities above 0.35 Pa s. Barlow et al. [9] also showed that the theory of Rouse seems to describe the viscoelastic behaviour adequately for low molecular weight polydimethylsiloxanes; when entanglements become important, this theory is inadequate. Therefore, they developed a method to describe the properties of polydisperse entangled polymers, using the molecular weight distribution and the variations of viscosity with molecular weight. In the case of PDMSs, this method agrees well with experimental data. The authors described the relaxation curve over 4 – 5 decades of pulsation, although some intervals are missing. Lamb and Lindon [10] completed these experiments for the same polydimethylsiloxanes from the terminal to the

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plateau region. They used an electromagnetic system over the frequency range 20–1500 Hz. In 1974, Barlow and Erginsav [11] determined the dynamic behaviour of five linear short-chain PDMSs. Measurements were made at very high frequency (459 MHz) and at temperatures as low as − 140°C. They found that the elastic modulus G% tends to a limit G (109 Pa). Later, Rahalkar et al. [12,13] studied a series of linear PDMSs with molecular weights greater than the critical value of entanglement Mc ( 21 000 g mol − 1) at a temperature of +23.2°C. They extended the measurements of Barlow et al. [9] to the terminal regime using classical rheometry (10 − 1 – 102 Hz) and to the end of the transition zone using a normal incidence wave reflection technique at 450 MHz. The relaxation curve was studied over 10 decades of pulsation, but some intervals are still missing. They also showed that the total viscoelastic spectrum cannot be modelled with a single model. The response must be divided into three parts associated with the terminal region (Dot–Edwards theory), the intermediate region (Rouse theory) and the high frequency contribution. The aim of the present study was to combine such experiments in order to describe the entire relaxation curve for high molecular weight PDMS from terminal to the transition regime. Three linear PDMSs are studied using rheometrical as well as ultrasonic tests. The main characteristics of the fluids are given in the first section. In the next section, we describe measurements of the mechanical shear impedance, determined from the complex reflection coefficient of an incident oblique shear wave at a solid–polymer interface. We also present rheometrical measurements, in the low frequency range (10 − 1 –102 rad s − 1). The use of the time – temperature superposition principle (temperatures from −50 to +20°C) allows coverage of about 6–7 decades of pulsation. In the last section, the ‘IRIS’ software developed by Baumgaertel and Winter [14] is used to deduce a discrete relaxation time spectrum from the master curve; the BSW model [15] is then considered. This model uses a continuous relaxation function H(l) and has been extended to polydisperse melts [16] by use of a molecular weight distribution to predict the mechanical behaviour of the materials studied. The agreement with experimental curves was quite good. 2. Materials The three polydimethylsiloxanes studied were commercial samples supplied by Rhoˆne-Poulenc. These products are linear polymers having different molecular weights (their main characteristics are given in Table 1). The letters A, B and C denote the three PDMSs, ranging by order of increasing viscosity. Table 1 Physical characteristics of the three PDMSs Sample

Viscosity (Pa s) at 298°K

r (g cm−3) at 298°K

Mn (g mol−1)

Mw (g mol−1)

I = Mw/Mn

A B C

60 100 500

0.973 0.973 0.973

62 400 73 300 105 800

116 500 133 000 199 000

1.86 1.81 1.88

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Fig. 1. Variation of the steady state viscosity for different PDMSs as a function of molecular weight (Mw) at 298°K.

Fig. 1 shows the change in the shape of the viscosity versus molecular weight curve for different linear PDMSs. This figure includes the results of earlier investigations [2,9,11,12,17,18], as well as a curve provided by Rhoˆne-Poulenc and the position of samples A, B and C. This confirms the value of the critical molecular weight Mc (28 000 g mol − 1). At viscosities greater than 0.4 Pa s, the effect of polymer chain entanglement becomes important. It is clearly shown that PDMSs A, B and C are characterized by molecular weights greater than the critical value mentioned above. For silicones A, B and C, the molecular weight distribution (Fig. 2) was measured by GPC (Rhoˆne-Poulenc): Samples all showed similar molecular weight distributions and the presence of small low molecular weight tails. In order to detect the appearance of crystallization, a differential scanning calorimetry (DSC) analysis was performed using a Perkin–Elmer apparatus at a heating rate of 5°C min − 1. This scanning rate was a compromise between good peak resolution and the sensitivity of the apparatus. The proper range of temperature ( −150 to −20°C) was explored in order to cover the phase transitions (Fig. 3, for sample B). The glass transition temperature was found in the vicinity of −130°C (sudden small rise in specific heat). At − 95°C, a crystallization peak was observed. Two consecutive melting peaks appear at a temperature of about − 50°C, indicating the transition of a semicrystalline state to a liquid state. The onset of the first melting peak indicates the temperature at which crystals start to disappear throughout the sample; this temperature is near −53°C. Consequently, for an isothermal experiment at temperatures lower than −53°C, PDMSs crystallized, while for temperatures as low as − 53°C, no crystals expanded within the samples. This was in good agreement with previous studies [2,9]. The DSC analysis for samples A and C presents an identical shape and the same locations for the different peaks.

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Fig. 2. Molecular weight distribution for samples A, B and C.

3. Experimental procedure

3.1. Acoustic measurements: mechanical shear impedance 3.1.1. Working formulae The propagation of shear waves in a medium gives direct access to the velocity 6s and

Fig. 3. DSC analysis of sample B. Heating rate: 5°C min − 1.

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attenuation as of these waves. The complex shear moduli G* may be deduced from these measurements. For polymer melts, direct measurements of shear wave speeds and attenuation coefficients are difficult, due to the extremely large values of the shear wave attenuation as involved at high frequencies. In particular, at ultrasonic frequencies ( f in the MHz range) shear wave speeds in PDMSs are close to 50 m s − 1 and values obtained for shear wave attenuation may be 105 times larger than those of longitudinal waves. Therefore, an indirect method must be used. The complex mechanical shear impedance Z *s is related to the acoustic parameters (velocity and attenuation) through the following relations: Z*s = Rs + iXs = (rG*(v))1/2 or Z*s = r6*

where

1 1 as = +i , 6* 6s v

(1)

where v is the angular frequency (v = 2pf), r is the density and Rs and Xs are respectively, the resistive and the reactance components of the shear mechanical impedance Z *. s The expressions for the storage and the loss moduli G% and G¦ are easily deduced: G%=

R 2s − X 2s , r

(2)

2RsXs . r

(3)

G¦ =

The method employed to measure both components is based upon the measurement of the complex reflection coefficient of an incident shear wave at a solid–polymer interface, the wave being polarized so that the shear displacement is parallel to the plane of the interface. Thus, reflection does not create extra compressional waves in addition to the reflected shear waves. This technique comes from previous works of Mason et al. [19] and McSkimin [20]. Briefly, the principles of the measuring technique are as follows. If a shear wave is reflected at a quartz – polymer interface at normal incidence, the complex reflection coefficient is given by R* = re − iu =

Zq − Z*s Zq + Z*s

(4)

where Zq (close to a real number, due to very small attenuation in quartz) and Z *s are the shear mechanical impedances of the quartz and polymer, respectively. From Eq. (4) we deduce Z*s = Zq

1 − r 2 + 2ir sin u . 1 + 2r cos u + r 2

(5)

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O’Neil [21] showed that enhanced effect (better precision) can be obtained by inclining the incident wave at an angle F with respect to the normal to the plane. The expression in Eq. (5) becomes

 

cos F 1− r 2 + 2ir sin u , cos b 1+ 2r cos u + r 2

Z*s = Zq

(6)

where b is the refraction angle of the refracted wave transmitted into the polymer and is given by sin b=

6s sin F. 6q

(7)

6s and 6q are the shear velocities in the polymer and in the quartz. This refracted wave is highly attenuated and does not modify the reflected wave flight. When the shear wave is reflected at a quartz – air interface, r=1 and u =0. When applying the fluid to the quartz, r decreases and u increases. The real and imaginary parts of Z *s and hence the components G% and G¦ of the shear modulus, are determined by measuring the changes in amplitude r and phase u of the reflected wave arising from the contact between the polymer and the quartz.

3.1.2. Principle Successive echoes may be obtained with two different configurations: with air and with polymer (Fig. 4). r and u can be obtained by comparison of echoes An and Bn and by measurement of Dt through the following relations: r=

 

Dt=

Bn An

1/2n

2nu . v

,

(8) (9)

The changes in amplitude, r 2n and the delay time Dt are larger when working with a distant echo (n large) and thus the precision of the method is enhanced.

3.1.3. Experimental setup The experimental system is shown in Fig. 5. The electronic system is similar to that described by Verdier and Piau [22]: “ Waves were generated using a pulse/function generator (Hewlett Packard 8116A, DC-50 MHz). “ Waves were amplified through a wide band amplifier (Kalmus Engineering, LP300H, 0.1–100 MHz). “ The transducer performed the role of both emitter and receiver (Matec contact shear wave transducers/radius: 8 mm/model: MCM 0204 SH, MCM 0504 SH and MPD 1504 SH). “ A DIP-3 (diplexing transformer Matec) was used for separating the echoes, in order to prevent them mixing with the incident wave. “ Signals were amplified once more (Preamplifier Matec, model 253).

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Fig. 4. Evolution of the echoes with air or PDMS ( f = 15 MHz, sample A, T= + 20°C).

Signals were visualized on an oscilloscope (Tektronics TDS 540, 1 GS s − 1, bandwidth 500 MHz) and the smallest vertical scale was used for each echo in order to improve accuracy. “ Acquisition and data processing were carried out using a Macintosh Quadra 800. A program (under Labview™) has been developed which allows continuous visualization of the values of r, u, G% and G¦. To study the range of frequencies 1.5–25 MHz, three broadband transducers having resonant frequencies 2.25, 5 and 15 MHz respectively, were used with the inclined incidence method (F= 80°) for the first two transducers in order to increase accuracy and with the normal incidence method for the third. The 15 MHz transducer has a small buffer rod (5 mm thick) at the front which induces numerous echoes. Therefore, normal incidence works better. Errors in G% and G¦ arise mostly from the error measurements in r and u and also on the echo number n. The error in r depends on the scale used and that in u on the time sampling chosen, as well as on the temperature variations. Errors in the storage values and the loss modulus are estimated to be less than 9 10 and 95%, respectively. “

3.1.4. Results Measurements were carried out between −10 and +50°C. Results are shown in Figs. 6–8 for samples A, B and C at a reference temperature of +20°C. The time–temperature superposition principle was used to enlarge the frequency range of shear measurements using

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both aT and bT [23]. Both shift factors translate the isothermal data along the axes in order to obtain master curves at a reference temperature. The vertical shift factor bT is calculated from the variations of the temperature and the density versus the temperature: bT =

r(Tref)Tref r(T)T

(10)

The shift factor aT is adjusted graphically to obtain agreement between the curve at a given temperature and the master curve. From −10 to +20°C, the same values for aT were found for ultrasonic and rheometrical tests. It must be noted that due to the different mechanisms involved (at low and high frequencies), the shift factors probably do not obey one single WLF law [1]. Consequently, no law was considered to superpose the relaxation curves. The three polymers appeared to show similar behaviour in the transition zone: at reduced frequencies larger than 3.107 rad s − 1, the loss and storage moduli G% and G¦ behaved in a similar manner, showing nearly parallel variations and having a common slope of the order of 0.85.

3.1.5. Temperature effects A major factor enhancing precision for such ultrasonic measurements is temperature stability. Considerable effort was devoted to achieving thermal and mechanical isolation: a system

Fig. 5. Experimental ultrasonic system.

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Fig. 6. Sample A. Reduced shear moduli vs aTv. Tref = + 20°C.

ensuring changes in temperature of less than 0.1°C between the reference measurement (quartz– air interface) and the measurement with the sample (quartz–PDMS interface) were used. A pump pushed the fluid through a thin tube (also located in the thermal chamber), causing the fluid to flow onto the quartz surface. The measurement was carried out at the same temperature, after adequate contact was made (Fig. 5).

Fig. 7. Sample B. Reduced shear moduli vs aTv. Tref = + 20°C.

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Fig. 8. Sample C. Reduced shear moduli vs aTv. Tref = + 20°C.

Quartz was chosen because of its reasonably low attenuation and its thermal stability. Calibration of the quartz velocity of propagation and density was carried out at different temperatures and these values were corrected each time when operating between − 10 and + 50°C.

3.2. Dynamic shear rheometry A first set of experiments was described by Verdier et al. [24]. Viscoelastic measurements were carried out in the dynamic regime using a stress-controlled rheometer, the Carri-med CSL 100. A plate – plate configuration (19 mm diameter) was used during all tests and for all PDMSs. The range of the oscillatory angular frequency varied between 0.1 and 100 rad s − 1 for each temperature. The temperature range was −50 to +20°C and the temperature was regulated using Nitrogen gas such that stabilization was obtained within 0.1°C. As usual, the linear viscoelastic domain region was determined. To avoid slip phenomena, a rough adhesive was glued to the upper and lower plates without modifying the properties of the polymer. The system could also extend or shrink, leading to changes in the gap value when working over a wide temperature range. These effects were corrected by use of an automatic gap compensation. A typical value for the system at hand was 2 mm °C − 1. Rheometrical tests were performed at temperatures below −50°C but proved to be unsuccessful, owing to crystallization of the polymers. Therefore, measurements were limited to cases where no crystallization occurred (T\ − 50°C). Fig. 9 clearly shows the stability of the shear moduli at temperatures above − 50°C and the slight increase of the shear moduli versus time when the temperature fell below −50°C: crystallization occurred within the sample and created hardening of the material. This point is confirmed by DSC as described previously.

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Fig. 9. Sample A. Variation of the shear moduli vs time at −50°C (lower two curves) or − 60°C (v = 1 rad s − 1).

For a monodisperse polymer, a decrease in G¦ was obtained at the beginning of the rubbery region. In this study, a slight decrease in the G¦ of the positive slopes was observed, corresponding to polydisperse polymers. Nevertheless, at the end of the rubbery region, the storage moduli G% had the same magnitude of about 500 000 Pa in all three samples. Rheometrical and ultrasonic experiments were then combined for the three PDMSs at the same reference temperature of 20°C. The continuity between rheometrical and ultrasonic tests

Fig. 10. Comparison of the experimental results with the predictions of IRIS for sample C. Tref = + 20°C.

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Fig. 11. Experimental results and predictions of BSW for sample A. Tref = + 20°C.

was good (Figs. 10–13). These master curves confirmed previous conclusions (Barlow et al. [9], Rahalkar et al. [12,13]) that the viscoelastic behaviour of high molecular weight PDMSs is independent of chain length for frequencies greater than about 3×107 rad s − 1. These curves give the reduced shear storage and loss moduli for reduced frequencies covering 10 decades quasi continuously and enhance the validity of the rheometrical and ultrasonic measurements.

Fig. 12. Experimental results and predictions of BSW for sample B. Tref = + 20°C.

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Fig. 13. Experimental results and predictions of BSW for sample C. Tref = + 20°C.

4. Model In the following section, two mathematical models have been introduced for analysis of the mechanical data: the first gives access to a discrete relaxation time spectrum [14] and the second to a continuous relaxation spectrum [15].

4.1. The IRIS software 4.1.1. The discrete relaxation time spectrum In this model, the relaxation modulus, G(t), can be expressed as a discrete sum of exponentials: N

G(t) = % gi exp(− t/li ).

(11)

i=1

The N relaxation modes are defined by their relaxation moduli gi and their associated relaxation times li. The dynamic moduli, G% and G¦, can be expressed by N

G%(v) = % gi i=1 N

G¦(v) = % gi i=1

(vli )2 , 1+ (vli )2

(12)

vli . 1+ (vli )2

(13)

From these equations, the discrete relaxation time spectrum may be determined from measured values of the dynamic moduli.

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The IRIS software [14] allows calculation of the coefficients gi and relaxation times li by fitting the above-mentioned equations to the G%, G¦ data. These coefficients are determined so that the average square deviation between the predicted G% and G¦ values and measured values is minimum. In this numerical program, the initial number of relaxation modes N is chosen empirically between 1 and 2 per decade and the spacing between the relaxation times is not necessarily equidistant. The values of gi and li are given in Table 2, while in Fig. 10, the continuous line shows the result of this fit for sample C and gives an excellent agreement with the data for the samples. The curves obtained were in good agreement, but an integral relaxation time fit will certainly improve the comparison and has the advantage of relating the parameters of the model to the characteristics of the polymer more closely. This is attempted in the next subsection.

4.2. The Baumgaertel–Schausberger–Winter (BSW) model 4.2.1. The continuous relaxation time spectrum To obtain more accurate data predictions, the BSW model was chosen because it has been used with success by different authors for predictions of polymer behaviour from the melt to the transition regime. It must be mentioned that the BSW model is limited to the transition region and cannot model the glassy zone because of the asymptotic behaviour of the dynamic moduli [25]: G%(v) and G¦(v) vary in the same manner as v ng as v“ . The relaxation modulus, G(t), can be expressed as a continuous function of time t: G(t)=

&



H(l) exp(− t/l)

0

dl , l

(14)

where H(l) is the continuous relaxation spectrum function. The dynamic moduli, G% and G¦, become Table 2 Relaxation times li and relaxation moduli gi Sample A

Sample B

Sample C

li (s)

gi (Pa)

li (s)

gi (Pa)

li (s)

gi (Pa)

3.6×10−2 4.3×10−3 4.9×10−4 4.8×10−4 6.9×10−5 1.1×10−5 2.5×10−6 6.4×10−7 1.2×10−8 8.1×10−10

1.3×102 5.5×103 4.5×104 1.7×104 9.6×104 1.3×105 1.7×105 2.8×105 2.3×106 2.4×107

4.1×10−2 4.2×10−3 6.3×10−4 1.6×10−4 2.8×10−5 1.2×10−5 7.3×10−6 1.8×10−7 4.4×10−8 1.5×10−9

4.8×102 1.4×104 2.3×104 3.2×104 6.0×104 2.6×104 1.1×105 1.2×105 6.6×105 1.8×107

2.3×10−1 4.1×10−2 5.7×10−3 7.2×10−4 7.2×10−5 1.6×10−5 1.9×10−6 1.7×10−8 1.3×10−9 1.3×10−9

1.8×102 6.0×103 3.1×104 5.8×104 6.2×104 5.6×104 1.1×105 1.8×106 4.7×105 1.4×107

228

G%(v) =

& &

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H(l)

v 2l 2 dl , 1+ v 2l 2 l

(15)

H(l)

dl vl . 2 2 1+ v l l

(16)



0

G¦(v) =

0

Assuming a linear superposition of the entanglement and the transition behaviours, Baumgaertel et al. [15] proposed the following spectrum: H(l) =

!

H el ne + Hgl − ng 0

for l Bl max, for l\l max.

(17)

The first term in the above relation, represents the entanglement and terminal regime with ne 0 ng He = neG 0Nl − max and the second term is related to the transition regime with Hg =ngG Nl 0 . The parameters ne and ng are related to the slope of G¦ in the entanglement regime and to the slope of G¦ (or G%) in the transition regime. G 0N is the plateau modulus, l0 is the relaxation time for the onset of the glass transition and lmax is the longest relaxation time. For monodisperse polymers, the longest relaxation time is given by [26] lmax = l0





Mw Me(T)

A

.

(18)

A is the scaling exponent of zero shear viscosity with molecular weight and has values between 3.3 – 3.7 for most polymers. Mw is the average molecular weight and Me the molecular weight between two entanglement points at a given temperature. Me is related to the plateau modulus G 0N (usually deduced from the experimental data of the storage modulus) by [26]: Me = gN

rRTref G 0N

at Tref = + 20°C,

(19)

where R is the ideal gas constant and gN a numerical factor close to unity. Note that the zero-shear viscosity h0 is given by h0 = lmaxG 0N

ne . 1+ ne

(20)

The samples are polydisperse polymers as indicated in Table 1 (Mw/Mn B2 for the three samples) and as shown in the previous section. A molecular weight distribution must be introduced in order to predict the behaviour of such systems.

4.2.2. Slightly poly-disperse melts Jackson and Winter [16] proposed to simplify a generalized linear blending rule giving the relaxation time spectrum. For highly entangled systems, time shifts may be neglected due to the slight polydispersity; then, the resulting expression H(l) is N



N



H(l) = % wi wi + 2 % wj Hi (l) i=1

j\i

assuming Mi + 1 \Mi \Mc,

(21)

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where wi is the weight fraction corresponding to the molecular weight Mi. It is also assumed that this blending rule will only be valid when all the blend components are above the critical value of appearance of entanglements Mc. Considering the low dispersity index and despite the presence of small amounts of low molecular weights in the samples, the same blending rule was applied and parameters of the model were determined for each sample. The complete relaxation time spectrum was calculated by summing the contributions of each molecular weight fraction. It is important to note that only the entanglement component He(l) is affected by the longest relaxation time lmax In formula (18), the longest relaxation time lmax must be replaced by the relaxation time li associated with the molecular weight Mi. In Eq. (22), Me can be considered as an ‘apparent’ molecular weight between two entanglement points:



li = l0

Mi Me(T)



A

.

(22)

4.2.3. Application of the model Six parameters must be determined in this model: A, ne, ng, l0, G 0N and Me. The scaling exponent A is obtained by plotting the zero-shear viscosity versus Mw on a bilogarithmic graph and fitting a line to the data (Fig. 1). For Mw \Mc, the value of 3.5 is found, which is usual for high molecular weight silicones. The glass transition exponent ng =0.85 is fixed for the samples corresponding to the common slope of the loss moduli. The crossover relaxation time l0 is adjusted such that the onset of the transition regime for G% and G¦ agrees with the data. From the limiting value of the storage modulus G%(v) at the end of the entanglement regime, the same value G 0N =500 000 Pa is deduced for all samples. The following assumption was made: Me is supposed to be related to the G 0N value and deduced from Eq. (19) at Tref = + 20°C with gN =1; the fixed value of 4760 g mol − 1 was used. The coefficient ne must be adjusted to achieve good agreement between the model and the data in the terminal zone. Values of ne were obtained for samples A (0.13), B (0.11) and C (0.12). ne appeared to be independent of chain length, whereas l0 appeared to slightly decrease with molecular weight. The theoretical curves drawn in this case compare well with the experimental results, as shown in Figs. 11–13. The approach adopted by Jackson and Winter [16] in the case of polystyrene, polycarbonate and poly(vinyl methyl ether) is slightly different. They proposed using a value of 0.23 for the entanglement exponent, for all linear flexible polymers. Then A, ne and ng are fixed (these parameters were not further adjusted in the iteration procedure). G 0N and l0 were evaluated to obtain the proper height of G%(v) in the entanglement plateau region and agreement with the data in the glass transition region. Finally Me was adjusted to agree with the terminal zone and entanglement regime. In this case, no relationship between Me and the plateau modulus was assumed, with Me being taken as an independent parameter. If this procedure is applied to samples A, B and C with a constant ne of 0.23, good agreement is also found. Moreover, it is important to note that a value of 5800 g mol − 1 was obtained for Me with the three samples. This value can then be related to the plateau modulus through Eq. (19) with gN coefficients slightly higher than 1.

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230

Table 3 BSW parameters of samples A, B and C calculated from slightly polydisperse data Procedure

ne

ng

l0 (s)

G 0N (MPa)

Me (g mol−1)

gN

G 0N =f(Me) A B C

0.13 0.11 0.12

0.85 0.85 0.85

2.1×10−8 1.9×10−8 1.6×10−8

0.5 0.5 0.5

4760 4760 4760

1 1 1

Jackson and Winter A 0.23 B 0.23 C 0.23

0.85 0.85 0.85

2.3×10−8 2.1×10−8 1.8×10−8

0.48 0.46 0.45

5800 5800 5800

1.17 1.12 1.10

Parameters corresponding to both methods are listed in Table 3. The curves obtained were very close to each other. With the method used in the present paper, the slope of G¦ in the entanglement regime is slightly higher and G%-G¦ curves cross at a lower frequency at the beginning of the rubbery plateau. The correlation of the plateau modulus G 0N to an apparent value Me used in this paper had the advantage of being related to the physics of the polymer. Therefore, the BSW model has proved satisfactory for predicting the dynamic behaviour of high molecular weight PDMSs.

5. Conclusion Using a combination of rheometrical and ultrasonic techniques, continuous dynamic master curves covering ten decades for three high molecular weight silicones were obtained. These results were correlated to the predictions of the BSW model applied to polydisperse systems and agreement was quite good. This method is very useful for the investigation of high frequency behaviour of polymers undergoing crystallization at low temperatures. It may be used further to analyse the glassy transition zone and the validity of the time–temperature superposition principle for other non crystalline polymeric systems.

Acknowledgements We would like to thank the Rhoˆne-Poulenc company for determination of the molecular weight distribution of samples A, B and C.

Appendix A. Nomenclature A aT, bT

scaling exponent horizontal and vertical shift factors

P.Y. Longin et al. / J. Non-Newtonian Fluid Mech. 76 (1998) 213–232

f gN gi G* G% G¦ G 0N H(l) He(l) Hg(l) i, j Mw Mn Mc Me Mi n ne ng r Rs R R* T 6* 6s wi Xs Z *s Zq as b Dt F li lmax l0 h0 r u v

frequency numerical coefficient relaxation moduli complex shear modulus storage modulus loss modulus plateau modulus relaxation spectrum function relaxation function for the entanglement and terminal regime relaxation function for the transition regime component indexes weight average molecular weight number average molecular weight critical molecular weight molecular weight between two entanglement points molecular weight of component i echo number slope of G¦ in the entanglement and terminal regime slope of G¦ in the transition regime amplitude resistive component of the mechanical impedance ideal gas constant complex reflection coefficient temperature complex velocity shear velocity weight fraction of component i reactance component of the mechanical impedance shear mechanical impedance of the polymer shear mechanical impedance of the quartz attenuation coefficient refraction angle delay time reflection angle relaxation times maximum relaxation time relaxation time for the onset of the glass transition zero-shear viscosity density phase angular frequency

231

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