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Bulk and Shear Rheology of a Symmetric Three-Arm Star Polystyrene Jiaxi Guo,1 Luigi Grassia,2 Sindee L. Simon1 1

Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409

2

Department of Aerospace and Mechanical Engineering, The Second University of Naples, Via Roma 19, 81031 Aversa (CE), Italy

Correspondence to: S. L. Simon (E-mail: [email protected]) Received 23 April 2012; revised 20 May 2012; accepted 4 June 2012; published online 16 July 2012 DOI: 10.1002/polb.23113

ABSTRACT: The bulk and shear rheological properties of a symmetric three-arm star polystyrene were measured using a selfbuilt pressurizable dilatometer and a commercial rheometer, respectively. The bulk properties investigated include the pressure–volume–temperature behavior, the pressure-dependent glass transition temperature (Tg), and the viscoelastic bulk modulus and Poisson’s ratio. Comparison with data for a linear polystyrene indicates that the star behaves similarly but with slightly higher Tgs at elevated pressures and slightly higher limiting bulk moduli in glass and rubbery states. The Poisson’s ratio shows a minimum at short times similar to what is observed for the linear chain. The horizontal shift factors above Tg obtained from reducing the bulk and shear viscoelastic

responses are found to have similar temperature dependence when plotted using T Tg scaling; in addition, the shift factors also exhibit a similar temperature dependence to linear polystyrene. The retardation spectra for the bulk and shear responses are compared and show that the long time molecular mechanisms available to the shear response are unavailable to the bulk. At short times, the two spectra have similar slopes, but the short-time retardation spectrum for the shear response is significantly higher than that for the bulk, a finding that is, C 2012 Wiley Periodicals, Inc. J Polym Sci as yet, unexplained. V Part B: Polym Phys 50: 1233–1244, 2012

INTRODUCTION The underlying molecular mechanisms for the viscoelastic bulk and shear deformations have been suggested to be different by Leaderman,1 with the bulk modulus originating from intramolecular relaxations and the shear modulus originating from the intermolecular or intersegmental relaxations. This hypothesis has been backed up by several studies.2–8 For example, the temperature-dependence of the shift factors used to construct reduced curves differ for the bulk and shear viscoelastic responses, and the dispersion of the bulk compliance is narrower than that for the shear with respect to spectral width.2–6 Similar differences were also observed in calculations by Tschoegl,7 as well as by Yee and Takemori.8

(i.e., the bulk response is narrower), they hypothesized that the long-time chain mechanisms which are available to the shear response are unavailable to the bulk. Work from our laboratory12,13 also demonstrated that the shear response is wider than that for the bulk and that the bulk response ends when chain modes start to dominate, backing up Bero and Plazek’s hypothesis. It is noted that this interpretation is also consistent with some work2,3,5,7 which came to a different conclusion. Although we found that the slopes of the retardation spectra for the bulk and shear responses are similar at short times, we were able to calculate the absolute magnitude of the retardation spectra and found a difference in the magnitudes. The difference in the magnitude was initially ascribed to differences in pressure since the bulk viscoelastic response was measured at elevated pressures and the shear response was performed at ambient pressure.12,13 However, when the bulk retardation spectra are calculated from the time-dependent bulk modulus, which is more correct than determining them from the pressure relaxation response as was done in our earliest work,12 the differences in the magnitudes of the short-time retardation spectra appears to be unrelated to differences in pressure. As part of this work, we calculate the retardation spectra from the viscoelastic bulk modulus of linear polystyrene14 rather than from the pressure relaxation data12 to further examine this issue.

On the other hand, evidence that the bulk and shear responses have the same underlying molecular mechanisms has also been reported.9–13 For example, the shift factors for the bulk and shear responses have been observed to be the same within the errors of the measurements for a polyisobutylene sample,9 for epoxy samples,10,11 and for a linear polystyrene,12 as well as for two polycyanurate networks.13 In addition, based on a comparison between the slopes of the retardation spectra for the bulk and shear deformations, Bero and Plazek10 suggested that the bulk and shear responses have the same molecular mechanisms at short times, but since the bulk response cuts out before the shear

KEYWORDS: dilatometry; polystyrene; rheology; star polymers

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JOURNAL OF POLYMER SCIENCE PART B: POLYMER PHYSICS 2012, 50, 1233–1244

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styrene sample.12,34 We finally calculate the viscoelastic Poisson’s ratio and the retardation spectra for the bulk and shear relaxations and compare the results to those in the literature for the linear polystyrene.12,14,34,35 We end with discussion and conclusions. EXPERIMENTAL

FIGURE 1 Chemical polystyrene.

structure

of

three-arm

star

shaped

The effects of star-shaped structures on the shear rheological properties of the polymers have been widely investigated due to their unique architectures, particularly in the terminal and softening regimes.15–29 The general finding is that the relaxation spectrum broadens for star polymers relative to their linear counterparts in the terminal regime15–22,25–29 due to the fact that arm retraction is a hierarchical process;23,24 similar results are observed for long-chain branching,30,31 comb polymers,32 and dendritic stars.33 Time-temperature superposition (TTS) has been observed to fail in the terminal regime for star hydrogenated polyisoprene and polyisoprene17,20 and for polyethylene with long chain branching,31 although it is valid for their linear samples. On the other hand, TTS was successfully applied to both linear and star-branched polyisobutylene17 and poly(ethylene-altpropylene)22 and to polyethylene with short chain branching.30 Very few articles have examined the entire relaxation spectrum, but Hatzikiriakos et al.27 found that the viscoelastic shear response in the short-time regime is similar for polybutadiene stars and linear chains. In contrast to the shear response, the bulk viscoelastic response of star-shaped polymers has never been studied to the best of our knowledge, and this is the aim of the current work. The article is organized as follows. First we present the experimental methodology, followed by the results of the pressure–volume–temperature (PVT) data. Then we show the pressure relaxation curves and their corresponding viscoelastic bulk moduli, as well as the dynamic shear measurement results. We will also compare the bulk and shear shift factors together with the counterparts for a linear poly-

Materials The material studied is a symmetric three-arm star polystyrene with weight average molar mass (Mw) of 117.5 kg/mol for each arm (Polymer Source). The chemical structure of the polymer is shown in Figure 1. The polydispersity index (PDI) of the sample is 1.07. The density was measured to be 1.047 g/cm3 at 21.2 C and 1 atm. The physical parameters for the star polystyrene are shown in Table 1. Also shown are those for a linear sample, Dylene 8,12,34,36 that will be used to compare to the star sample in this work; we note that the star arm molar mass was chosen to give a longest span similar to the weight-average molecular weight of the linear chain since we had already had shear and bulk viscoelastic data for that material. The glass transition temperature (Tg) of star polystyrene is 98.2 6 1.7 C measured by differential scanning calorimetry (DSC) at 10 K/min heating rate after cooling at 10 K/min. For pressurizable dilatometry experiments, the sample was molded at 170 C and annealed at 115 C under vacuum conditions and was machined to a rod shape with diameter 4.0 mm; for the shear rheology experiments, the sample was molded in the rheometer at 170 C under nitrogen atmosphere. Dilatometric Measurements The custom-built pressurizable dilatometer used in the study has been described elsewhere;14 here, we briefly describe the instrument for the sake of completeness. The star polystyrene sample, having mass of 0.82 g, was loaded in a sample cell surrounded by about 5 g of fluorinated synthetic oil (Krytox GPL107 from DuPontTM). The oil is chemically inert and stable between 30 and 288 C. The sample cell is stainless steel 316 with an inner diameter of 4.8 mm, a length of 10 cm, and a volume of 2.0 cm3. The sample cell is immersed in a Hart Scientific oil bath which maintains temperature with a standard deviation of 60.2 K over the course of several days; the accuracy of the temperature measurement is better than 0.1 K.14 The system is pressurized by a piston controlled by a stepper motor and the position of the piston is measured by a linear variable differential transducer (LVDT). A Laboratory Virtual Instrumentation Engineering Workbench (LabVIEW) program controls the

TABLE 1 Physical Properties for the Star and Linear Polystyrenes Polymer Star PS a

Linear PS

Arms

q (g/cm3)

Mw,a (kg/mol)b

Mw,bb (kg/mol)b

PDI

Tgc ( C)

3

1.047

117.5

235

1.07

98.2 6 1.7

2.38

99.1 6 0.2

1.045

a

The physical parameters for linear PS, Dylene 8, are obtained from ref. 12. Mw,a and Mw,bb are arm and backbone weight average molecular weights for the polystyrenes, respectively. b

1234


221 c

36

Tg for star PS and linear PS (Dylene 8) were measured by DSC at 10 K/min heating rate after cooling at 10 K/min. Both data were obtained at atmospheric pressure.


stepper motor through the stepper motor controller and collects the pressure, temperature, and LVDT voltage values. Isobaric temperature scans were performed from 20 to 30 K above Tg(P) to 30 K below Tg(P) at a cooling rate of 0.1 K/ min at pressures ranging from 28 to 83 MPa. A slow cooling rate of 0.1 K/min was used to minimize temperature gradients in the sample. The pressure-dependent Tg [Tg(P)] is obtained from the intersection of the extrapolated glass and liquid lines based on the LVDT voltage versus temperature plot. In addition to temperature scans, isothermal pressure scans were performed, and these were made on increasing pressure at temperatures of 120 and 130 C. Isothermal pressure relaxation was measured as a function of time after volume jumps at two starting pressures, Po ¼ 24 6 0.2 MPa and 41 6 0.2 MPa, for various temperatures in the vicinity of Tg(P). The volume jump was 200,000 steps of the stepper motor, which resulted in volumetric strains of less than 0.6% in the region of the glassy response and less than 0.7% in the region of the rubbery response; the results are expected to be in the linear viscoelastic range based on previous work14 on linear polystyrene. The sample was heated to Tg(P) þ 15 C or to Tg(P) þ 20 C for the measurements performed at Po of 24 and 41 MPa, respectively, held for at least 4 h to reach equilibrium density, and then cooled down to the test temperature at 0.1 K/min. For runs made at temperatures below Tg(P) þ 7K, Struik’s protocol37 was applied, that is, the testing time was limited to 1/10 of the aging time, to ensure that there is no effect of physical aging on the pressure relaxation experiments. Aging times were 3 days for the pressure relaxation measurements made from 21 MPa and 7 days for the measurements made from 41 MPa. Pressure relaxation data are reported only after 102.5 seconds due to the electronic transient response and the transition period resulting from adiabatic heating caused by the volume jump. In previous work,12 we have shown that the linear polystyrene sample was not adversely affected by the surrounding fluorinated synthetic oil by swell tests and by Tg measurements before and after the experiment. In this study for the star polystyrene, Tg at 41.4 MPa was 110.4 6 0.9 C based on three runs at the very first beginning of the experiment, and was 110.7 C 8 months later, after all of the measurements were complete. In addition, repeated pressure relaxation runs were also performed several months apart and showed good reproducibility (shown later in the results), further indicating that the sample has not been adversely affected by the confining fluid. Dynamic Shear Stress Measurements The shear dynamic modulus for the star polystyrene was measured using a commercial rheometer (Anton Paar, MCR 501) with 8-mm parallel platens with a sample thickness of 1.8 mm. The dynamic studies were performed at frequencies ranging from 0.0006 to 100 rad/s. The testing temperatures for the shear measurements ranged from 95 to 170 C, with the lowest frequencies used only at 140 and 160 C to obtain the terminal relaxation behavior in the absence degradation. For the measurements at temperatures above 110 C, the sample was held


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for at least 15 min to remove the thermal history; for those at temperatures 110 C, Struik’s protocol37 was also used to ensure no effects of physical aging. Aging times were 120 min. In the glassy state, the initial strain was set to be 0.05%; whereas in the rubber state, the value was 5%. The dynamic shear data are corrected for instrument compliance following the method of Schroeter et al.:38 1 1 1 ¼ þ ; Kmeas Ksam Kinstr

(1)

and K ¼G

pR4 ; 2h

(2)

where K*meas is the measured complex torsional stiffness, K*sam is the actual complex torsional stiffness for the sample. 1/Kinstr is the instrument torsional compliance and is 7.27 103 rad/Nm determined by Zheng et al.39 using the same instrument to measure the dynamic shear behavior for polymer/ oligomer blends, G is the shear modulus, R is the radius of the platen, and h is the gap of the upper and lower platens. Calculation of Poisson’s Ratio The Poisson’s ratio, which we determine from the bulk and shear viscoelastic responses, is defined as vðtÞ ¼

e2 ðtÞ e0

(3)

where e2 is the time-dependent transverse strain when an uniaxial step strain deformation, e1 ¼ e0 h(t), is applied to the sample, where h(t) is the unit step function. In the framework of the linear viscoelasticity, the viscoelastic Poisson’s ratio can be calculated from the shear and bulk responses:35 ^vðsÞ ¼

^ 1 3K^ ðsÞ 2GðsÞ ^ ^ þ 2GðsÞ s 6KðsÞ

(4)

^ ^ where ^vðsÞ, KðsÞ, and GðsÞ are the Laplace transforms of the viscoelastic Poisson’s ratio, m(t), the bulk relaxation modulus, K(t), and the shear relaxation modulus, G(t), respectively. The bulk response is obtained from the experiments directly in the form of relaxation modulus, K(t), whereas the dynamic shear response is measured here as storage, G0 (x), and loss, G00 (x), moduli. The time-dependent shear relaxation modulus, G(t), can be equivalently calculated from G0 and G00 with the following relations: 2 GðtÞ ¼ p 2 GðtÞ ¼ p

Z1 0

Z1 0

G0 ðxÞ sin xt dx x

(5)

G00 ðxÞ cos xt dx x

(6)


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Equations 5 and 6 contain a highly oscillatory integrand. They are computed using the sequence convergence acceleration for the sum of the sequence that consists of each of the integrals with regions between two consecutive zeros of the integrand. For example, the first one is calculated as follows N 2X N!1 p i¼1

Zzi

GðtÞ ¼ lim

zi1

G0 ðxÞ sin xt dx x

(7)

where zi ¼ ti p; i 2 N are the zeros of the oscillating function sinxt. The limit appearing in eq 7 is computed using Wynn–Rho’s extrapolation method,40 which gives a good estimate of the limit of the sequence (7) by calculating only few elements of it. Calculations of G(t) are performed using both eqs 5 and 6, with the results for the natural logarithm of G(t) differing by less than 0.4%. The Laplace transforms of the shear relaxation modulus and of the bulk relaxation modulus are numerically evaluated only on the real line, as ^ ¼ GðsÞ

Z1 GðtÞ Exp½st dt

(8)

KðtÞ Exp½st dt

(9)

0

^ KðsÞ ¼

Z1 0

where s is a real and positive number. The Laplace transform of the viscoelastic Poisson’s ratio, ^vðsÞ, is, then calculated on the positive real line using eq 4. The Gaver–Wynn–Rho (GWR) algorithm41 in the framework of multiprecision computational environments is used to transform back m(s) into the time domain, which requires the evaluation of the Laplace transform ^vðsÞ only for real values of the complex variables. The algorithm is based on the sequence of the Gaver functionals42 with convergence accelerated with the Wynn–Rho algorithm.41 Practically speaking, the GWR algorithm gives an approximation of m(t) when numerical values of m(s) are known. The algorithm works as follows:41

• set the precision to 2.1M where M is an even number • compute the Gaver functionals according to the equation: fn ðtÞ ¼

lnð2Þn t

2n n

X n n ^v½ði þ nÞlnð2Þ=t ð1Þi i i¼0 1nM

ð10Þ

• apply the following recursive algorithm that translates the Wynn–Rho acceleration scheme ðnÞ

ðnÞ

q1 ¼ 0; q0 ¼ fn ðtÞ; n 0 ðnÞ

k

ðnþ1Þ

qk ¼ qk2 þ

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FIGURE 2 Specific volume as a function of temperature at four pressures for star polystyrene. The data point at 0.1 MPa and 25.3 C was obtained by density measurement. Three measurements were performed at 41.4 MPa and are shown in different symbols with the same color. The solid lines are the best fit of Tait equations to the data in both liquid and glassy states; the dashed line is the pressure-dependent glass transition temperature [Tg(P)]. Only a fraction of data is shown.

ðnþ1Þ qk1

ðnÞ qk1

; k 1

(11)


• obtain the approximation to m(t) as ð0Þ

vðtÞ vðt; MÞ ¼ qM

(12)

Multiprecision computing is required because eqs 10 and 11 are characterized by round-off error propagation when a fixed precision is used. We comment that the GWR algorithm contains only one free parameter (namely M) and that the accuracy in determination of m(t) increases as M increases. Here, we use M of 32 as suggested by Valk o and Abate.41 Calculation of the Retardation Spectra The relaxation spectra of the bulk response were calculated from the second-order approximation by Schwarzl and Staverman43 based on the bulk relaxation modulus K(t), which is fit by the Kohlrausch–Williams–Watts (KWW)44,45 function to more accurately calculate the first and second derivatives and reduce the effect of the noise on the results. The relaxation spectra for the shear response is calculated using Tschoegl’s second-order approximation46 from the dynamic shear modulus G0 (x). The retardation spectra of both bulk and shear responses are calculated from the interrelations between the relaxation and retardation spectra.47 RESULTS

PVT Behavior The specific volume for the star polystyrene measured on cooling at 0.1 K/min at various pressures is shown in Figure 2. Also shown is the volumetric data measured at atmospheric pressure and 21.2 C, and the pressure-dependent


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TABLE 2 Tait Parameters for Star and Linear Polystyrenes in the Liquid and Glassy States Linear PS14

Star PS

ao (g cm3)

Liquid State

Glassy State

Liquid State

Glassy State

1.079

1.053

1.077

1.052

a1 (104 g cm3 C1)

5.02

n/a

4.99

n/a

a1 (104 C1)

n/a

2.30

n/a

2.37

a2 (107 g cm3 C2)

2.30

n/a

2.08

n/a

Bo (MPa)

246

270

217

209

B1 ( C1)

0.00352

0.00135

0.0035

0.001

C

0.0894

0.0894

0.0894

0.0894

glass transition temperatures [Tg(P)]. Three measurements were made at 41.4 MPa and are shown in the same color with different symbols; the repeat runs indicate good reproducibility of the data. The PVT data in Figure 1 are fitted to the Tait equation48 given by, V ðP; TÞ ¼ V ð0; TÞ 1 C ln 1 þ

P Bo expðB1 TÞ

;

(13)

where C, Bo, and B1 are material-dependent constants, and C is generally taken to be 0.0894.49 In the liquid state, the specific volume at zero pressure [V(0,T)] can be described by a polynomial function of temperature,

whereas in the glassy state, V(0,T) is described by an exponential function of temperature,50 V ð0; TÞ ¼

1 ao expða1 TÞ

(15)

The parameters of the Tait equation for the star polystyrene are listed in Table 2. Also shown are the Tait parameters for the linear polystyrene.14 As indicated by the similarity of the Tait parameters, the star and linear polystyrene show similar PVT behavior. The most significant difference is the value of Bo, which is 10% higher for star polystyrene than that for the linear polystyrene. An increase in Bo of 10% reflects a similar increase in the liquid bulk modulus at low to moderate pressures, since all other terms being equal, the liquid bulk modulus is directly proportional to the term P þ BoeB1T. The higher value of Bo for our star sample is consistent with findings51 for branched and linear polyethylene where Bo was found to be 4% higher for the branched sample. ln V , are plotThe coefficients of thermal expansion, a ¼ @ @T P ted in Figure 3 as a function of pressure in both liquid and glassy states and compared with those for the linear polystyrene14 also shown in Figure 3. Within the error of the measurements, the star and linear polystyrenes have the same coefficients of thermal expansion in both liquid and glassy states and as a function of pressure. Moreover, the thermal expansion coefficients predicted from the Tait equation fits to the PVT data reasonably describe the data, as shown by the solid and dashed lines in Figure 3.

(14)

The instantaneous bulk modulus was also determined in the liquid state at 120 and 130 C from isothermal pressure

FIGURE 3 Thermal expansion coefficients for the star and linear polystyrenes in liquid and glassy states as a function of pressure. The data are determined from the experimental PVT data shown in Figure 2; the lines are calculated from the Tait equations with best fits to the data. The linear polystyrene data and fits are obtained from Meng et al.14

FIGURE 4 The instantaneous liquid bulk modulus for the star polystyrene as a function of pressure measured from isothermal pressure scans at 120 and 130 C. The solid and dashed lines are calculated from the Tait equation for 120 and 130 C, respectively. The dashed–dotted (blue) line is calculated from the Tait equation for the linear polystyrene14 at 130 C.

V ð0; TÞ ¼

1 ; ao þ a1 T þ a2 T 2



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at temperatures ranging from 70 to 123.5 C, whereas in Figure 6(b), Po ¼ 41 MPa and measurements were carried out at temperatures ranging from 81 to 133.5 C. The pressure relaxation curves shown as P(t) Po in the figures are directly related to the viscoelastic bulk modulus: at low temperatures, the glassy response is obtained; in the vicinity of Tg(P), pressure relaxation is observed; and at high temperatures or long times, the rubbery response is obtained. Repeat runs made after 8 months and designated by filled and unfilled symbols of the same color show good reproducibility. The noise at long times is attributed to temperature fluctuations.

FIGURE 5 Glass transition temperatures (Tg) of star polystyrene as a function of pressure obtained at the cooling rate of 0.1 K/min. Also shown are pressure-dependent Tgs for the linear polystyrene measured by Meng and Simon.12

scans. The data are shown in Figure 4. The error bars of approximately 0.2 GPa are based on the standard deviation found using several different methods to fit the volume versus pressure data to obtain the derivative (and hence, K); the data shown are fit to a power law, V ¼ (a þ bP)c, but we also obtained K from a three-point derivative and by fitting ln V versus P to a second order polynomial. The Tait equation prediction for the liquid elastic bulk modulus, shown as solid and dashed curves, are slightly lower than the data at 120 and 130 C. The Tait equation prediction for the bulk modulus for the linear chains at 130 C, shown as the dashed–dotted line is 10% lower than that for the star polystyrene, as might well be expected given the 10% difference in Bo between the two materials. The glass transition temperatures for star polystyrene are plotted as a function of pressure in Figure 5. Tg at each pressure is taken by the intersection of the liquid and glassy lines for PVT data obtained on cooling at 0.1 K/min, as shown in Figure 2. As indicated, Tg increases with increasing pressure with a slope dTg/dP ¼ 0.38 6 0.02 K/MPa for the star polystyrene. Also shown are comparisons of the star polystyrene data to the Tg data for the linear polystyrene Dylene 8.12 Although the Tgs of the two polystyrenes are similar at low pressures, the value of the slope dTg/dP for the linear sample is 0.30 6 0.01 K/MPa, slightly lower than that for the star polystyrene. The fact that the star has higher Tgs at elevated pressures may be related to its higher bulk modulus and local crowding at the star center. Viscoelastic Bulk Modulus Isothermal pressure relaxation curves [P(t) Po] are shown as a function of logarithmic time in Figure 6(a,b) for the star polystyrene. The measurements were carried out in the vicinity of Tg(P) after volume jumps for two pressure ranges. In Figure 6(a), the initial pressure from which the jump was made is Po ¼ 24 MPa and relaxation curves were measured 1238


FIGURE 6 Pressure relaxation curves [P(t) Po] as a function of logarithmic time for star polystyrene carried out at approximately (a) 36 and (b) 55 MPa, for Po ¼ 24 and 41 MPa, respectively. The relaxation curves were measured at temperatures in the vicinity of 108.5 C at 36 MPa and 113.5 C at 55 MPa as indicated in the figures. Two pressure relaxation runs are shown at 90, 110, and 117 C at 36 MPa, and at 106.5, 120, 123.5, and 133.5 C at 55 MPa, one as filled circles and the other as open circles. Only a fraction of data is shown for clarity.


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K ðtÞ ¼ V

DPðtÞ j D Vo T

(16)

where DP(t) ¼ P(t) Po, is the pressure relaxation response as a function of time and DVo is the step change in volume applied. The viscoelastic bulk moduli at 36 and 55 MPa for the star polystyrene are plotted as a function of logarithmic time and compared with that for the linear polystyrene14 measured at 42 MPa. The three reduced curves are shown in Figure 8, where the reference temperature for each curve is Tg(P). We note that we also calculated the time-dependent bulk modulus from each pressure relaxation curve as a function of measurement temperature, and shifted the K(t) curves to obtain a reduced K(t) curve rather than calculating the curve from the reduced P(t) curve; identical results were obtained. Although the limiting moduli for the star polystyrene are higher, the strength and width of the relaxation of the bulk response appear to be quite similar for star and linear polystyrenes. To further quantitatively compare the viscoelastic bulk moduli, the data are fit to the KWW44,45 function: b t KðtÞ ¼ Kr þ ðKg Kr Þexp s

FIGURE 7 Pressure relaxation curves as a function of time after application of vertical shifts for star polystyrene at temperatures ranging from (a) 70 to 123 C at 36 MPa and (b) 81 to 133.5 C at 55 MPa. The reduced curve on the right of each figure is obtained by TTS.

The pressure relaxation curves obtained at different temperatures are superposed to form a reduced curve after applying horizontal shifts and small vertical shifts accounting for the temperature dependence of the relaxation times and the limiting bulk moduli, respectively. The results are shown in Figure 7(a,b) for the relaxation curves after jumping from Po ¼ 24 and 41 MPa, respectively. The data after applying a vertical shift are shown on the left-hand side of the two figures, and the reduced curves after the application of TTS are shown on the right-hand sides. Both reduced curves are shifted to Tref ¼ Tg(P), where P is taken midway through the dispersion and Tg(P) is that measured at 0.1 K/min; for the lower pressure range, P 36 MPa and Tg ¼ 108.5 C, whereas for the higher pressure range, P 55 MPa and Tg ¼ 113.5 C. The viscoelastic bulk modulus [K(t)] is then calculated from the pressure relaxation response using the definition given by,


(17)

where Kr is the rubbery bulk modulus, Kg is the glassy bulk modulus, s is the average relaxation time, and b is the Kohlrausch stretching exponent. The KWW parameters for the bulk modulus are shown in Table 3. In addition to the higher values of Kg and Kr for the star polystyrene, the b values are also higher for the star sample (0.33–0.34 vs. 0.24), suggesting a narrower relaxation time distribution. In Table 3, the KWW parameters are also provided for the fitting of the viscoelastic bulk modulus of the linear sample at 76 MPa; these data are not shown in Figure 8 for purposes of clarity.

FIGURE 8 Comparison of viscoelastic bulk moduli for the star and linear14 polystyrenes measured at the pressures indicated. The reference temperature for each reduced curve is Tg(P). Lines are fits to the KWW equation.


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TABLE 3 KWW Fitting Parameters for Pressure Relaxation Response and Viscoelastic Bulk Modulus of Star and Linear Polystyrenes Measured at Different Pressures Linear PS14

Star PS P (MPa)

36

55

42

76 3.48 6 0.002

Kg (GPa)

3.67 6 0.006

3.80 6 0.006

3.34 6 0.003

Kr (GPa)

2.24 6 0.004

2.40 6 0.002

1.97 6 0.003

2.22 6 0.002

b

0.331 6 0.004

0.345 6 0.004

0.241 6 0.001

0.266 6 0.001

916 6 20

1125 6 22

1056 6 16

1432 6 15

108.5

113.5

107.5

117.0

s(s)

Tref ( C)

Dynamic Shear Relaxation Measurements The storage (G0 ) and loss (G00 ) moduli as a function of frequency for the star polystyrene are shown in Figure 9(a,b); the reduced curves obtained by TTS of the data with small vertical shifts are shown in Figure 9(c). Both of the reduced curves are obtained at reference temperature Tref ¼ 93.8 C. For the storage modulus, at low temperatures or high frequencies, the glassy response is observed; as temperature increases or frequency decreases, the a-relaxation or softening dispersion occurs in the vicinity of Tg(x); at high temperatures, the rubbery response is observed, followed by terminal flow. The loss modulus goes through two transitions as temperature increases (or frequency decreases) through the a-relaxation and terminal regions, respectively. The reduced curves were obtained by first reducing the tan d data using only horizontal shifts since no vertical shifts are needed because the temperature dependences in G0 and G00 cancel out; then we use the horizontal shifts from tan d along with small vertical shifts to reduce G0 and G00 . We do note that although the superposition of G0 and G00 are satisfactory, the reduction of tan d in the region of the a-relaxation shows a small degree of thermo-rheological complexity with the magnitude of the tan d peak depending on temperature. A similar lack of thermo-rheological simplicity was also observed by Plazek and coworkers from reduced tan d for linear polystyrene and other chains in the softening dispersion.52

for that sample; for the star polystyrene, we assume Gg ¼ 0.906 GPa, as measured here. The short time limit of m(t) depends on the ratio of Kg/Gg, increasing as the ratio increases—this explains why the star sample appears to

The glassy modulus (Gg) for the star polystyrene is obtained assuming Andrade creep at short times, that is, that G0 (x) ! x1/3 and that Gg is the limiting value of G0 (x) at x ¼ 0. From the Andrade analysis, we obtain that Gg ¼ 906 MPa, which is the similar to that obtained for the linear polystyrene (Dylene 8) studied by Agarwal34 where Gg was found to be 952 6 126 MPa based on the short time recoverable compliance measurements made at 96.6, 100.2, and 103.3 C. The time dependence of the viscoelastic Poisson’s ratio for the star polystyrene and linear polystyrene Dylene 8 are reported in Figure 10 for P ¼ 0.1 MPa and T ¼ 93.8 C. The Poisson’s ratio of the linear polystyrene was calculated in our previous work35 where Gg was assumed to be 1.0 GPa, which is consistent with the Agarwal’s measurements of Gg 1240


FIGURE 9 Dynamic shear storage modulus (a) and loss modulus (b) as a function of frequency for the star polystyrene at temperatures ranging from 95 to 170 C. The reduced curves at Tref ¼ 93.8 C are shown on the right-hand side of the figures.


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comes directly from the Williams–Landel–Ferry (WLF) equation53 given by log

FIGURE 10 Calculated time-dependent Poisson’s ratio for star polystyrene (red) and linear polystyrene (blue)35 at 0.1 MPa and 93.8 C. The small figures on the right show the short-time minimum in the Poisson’s ratio (solid line), along with calculations from a sensitivity analysis (dashed lines) in which the reduced bulk modulus K(t) is shifted 0.5 decades to shorter or longer times without changing the position of the reduced shear modulus G(t) for the calculation; shifting K(t) shifts the position of, but does not eliminate, the minimum in m.

have a higher short time limit of m(t). However, we estimate that the error in m(t ¼ 0) is 10 %, and hence, differences in the short time limits of the two samples are within the error of the measurements. For both polymers, the long time limit of m(t) 0.5 because in the rubbery state the bulk modulus is much larger than the shear one. As can be observed in Figure 10, the time-dependent Poisson’s ratio seems to undergo a weak minimum at short times before it increases to the liquid-like value of 0.5. The presence of this minimum is independent of the limiting value of m(t) and is unaffected by the errors in Kg and Gg; rather, it is related to differences in the relaxation time distributions of K(t) and G(t). To assess whether the nonmonotonic behavior of m(t) at short times is a result of measuring the shear and bulk properties using different samples such that the relative positions of the relaxations at the temperature and pressure of the calculation (0.1 MPa and 93.8 C) have some error, the calculations have been performed again shifting the bulk modulus by half a decade on the time axis, both backward and forward. The results, displayed in the small figures on the right of Figure 10, show that the Poisson’s ratio continues to show a weak minimum at short time. This sensitivity analysis confirms the presence of the short-time minimum in m(t) because the errors in constructing the shear and bulk modulus master curves are considerably less than the result from shifting the curves a half decade in time. Similar nonmonotonic behavior in m(t) is calculated for a polycarbonate although not for two crosslinked polycyanurates;35 the origin of the nonmonotonic behavior is addressed further in the discussion. Comparisons between Bulk and Shear Responses The logarithmic horizontal shift factors for the bulk and shear responses of star and linear polystyrenes12,34 are plotted as a function of T Tg(P) in Figure 11. This scaling WWW.MATERIALSVIEWS.COM

s sTg

¼

C1 ½T Tg ðPÞ C2 þ ½T Tg ðPÞ

(18)

where C1 and C2 are material-dependent constants. In the event that the constants C1 and C2 do not depend on pressure and are similar for star and linear polystyrene, one expects that plotting the shift factors versus T Tg(P) will result in superposition of data obtained at different pressures. This is, in fact, observed: in the liquid state, all of the bulk and shear shift factors for the two samples measured at different pressures superpose as a function of T Tg(P), consistent to our previous findings in two polycyanurate networks having different crosslink densities.13 A fit of the data in the liquid state to the WLF equation yield C1 ¼ 18.2 6 0.5 and C2 ¼ 42.8 6 2.1 K. On the other hand, approaching the glassy state, the shift factors for the star polystyrene deviate from the equilibrium WLF relationship at higher temperatures than do those for the linear polystyrene from Meng and Simon12 in spite of similar aging histories: the star polymers were aged for 3 or 7 days and the linear samples were aged for 7–10 days. The origin of this behavior is not understood but it may be related to the narrower distribution of relaxation times for the star sample. The retardation time spectra of the viscoelastic bulk and shear responses for the star polystyrene and the linear sample34 are plotted in Figure 12. The shear response is observed to have long-time mechanisms, whereas for the bulk, the retardation time spectra are cut off at short times, consistent with our previous results12,13 and those of Bero and Plazek10—these results indicate, perhaps not surprisingly, that chain modes are not active in the bulk volumetric response. In the short-time region of the retardation spectra,

FIGURE 11 Logarithm of horizontal shift factors obtained from the pressure relaxation studies and shear dynamic measurements for the star polystyrene with their counterparts for the linear polystyrenes from creep compliance measurements of Agarwal,34 and pressure relaxation measurements of Meng and Simon.12 The data above Tg are fitted to the WLF equation.


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shear data, Tg is similar for the linear and star polymers and hence, the two scaling methods are identical; for the bulk data, Tg increases by less than 5% and given the small range of the data, no differences in superposition can be discerned.

FIGURE 12 Comparison of the bulk and shear retardation time spectra for the star polystyrene. Spectra for the linear polystyrene are calculated from data in the literature.14,34 The bulk retardation time spectra are calculated from the KWW function fit to the viscoelastic bulk modulus.

both the star and linear materials show a regime where log L(k) varies with log k with a slope of 1/3; this is known as Andrade creep.54 The bulk retardation time spectra in the Andrade regime have similar slopes to the shear spectra, but they are approximately one order of magnitude lower in absolute value. Assuming that the retardation spectra are physically meaningful and that the retardation times correspond to specific mechanisms, the results indicate that only a small fraction of the Andrade mechanisms available for the shear response are effective in the bulk response or that the contribution of the Andrade modes to the bulk response is much weaker than for the shear. DISCUSSION

Above Tg, the shift factors obtained by bulk and shear measurements for the star and linear polystyrenes can be reduced by plotting as a function of T Tg(P). The result is consistent with the literature where shift factors for polystyrenes with different molecular weights,55 polybutadienes with different architectures,56 and two polycyanurate networks having different crosslink densities13,57 can be reduced plotting as a function of 1/(T To), or T Tg (or To). Although this behavior is understandable since the local chemical structure either for the polystyrenes or for the polybutadienes or between the crosslinks in the polycyanurates are similar, it is in contrast to Roland and Casalini58 where the relaxation times for the polystyrenes with different molecular weights studied by dielectric measurements could not be reduced as a function of T Tg, where Tg is taken at s ¼ 100 s. The underlying meaning of plotting shift factors as a function of T Tg or 1/(T To) for reducing the data obtained at different pressure and/or for materials having similar microstructures have been discussed somewhere else.59 Here, we also note that plotting versus T/Tg satisfactorily reduces the data; this is perhaps not surprising because T/Tg scaling is equivalent to T Tg scaling when Tg is constant. For the 1242


The retardation time spectra for the bulk and shear responses of star and linear polystyrenes are compared and the results indicate that the bulk and shear retardation time spectra have similar slopes at short times, but the mechanisms that are available for the shear spectra at long times are unavailable for the bulk, consistent with earlier observations.10,12,13 In addition, the magnitudes of the bulk retardation time spectra at short times are found to be lower than those for the shear. In previous publications,12,13 the differences at short times were ascribed to pressure effects, because the shear measurements were carried out at atmospheric pressure; whereas the bulk responses were performed at higher pressure. However, in the linear polystyrene study,12 the retardation time spectra for the bulk were calculated from the pressure relaxation responses which are induced not only by the sample, but also by the confined fluid. In this study, the magnitudes for the bulk retardation spectra calculated from the bulk modulus depend only slightly on pressure (with the slope of the log L versus log k decreasing slightly with decreasing pressure) indicating that the differences between the shear and bulk retardation spectra cannot be attributed to pressure effects. Assuming that the retardation spectra has physical meaning, the higher magnitude of the shear retardation time spectra indicates that the shear relaxation response arises from a greater number of Andrade short-time relaxation modes or that the contribution of each mode is greater for shear deformation. Finally, the presence of a minimum at short time in m(t) can be explained by means of the relative position on the time axis of the bulk and shear response. We showed in ref. 35, using a simple analytical model, that the viscoelastic Poisson’s ratio can be a monotonic or increasing function of time when sG/sK is on the order of unity, or a nonmonotonic function of time showing a minimum at short time when sG/sK 1 (sG and sK are the principal shear and bulk relaxation times, respectively). As shown in Figure 12 for both the star and linear polystyrenes, even if the bulk and shear response share similar timescales at short times, the shear response is broader than the bulk one and contains long-time mechanisms that are unavailable to the bulk response such that sG/sK 1. Hence, the minimum observed in m(t) in Figure 10. CONCLUSIONS

The bulk and shear rheological properties for a three-arm star polystyrene have been investigated using a self-built pressurizable dilatometer and a commercial rheometer. The results are compared with a counterpart linear polystyrene sample having a similar molecular weight as the star span molar mass. The pressure-dependent glass transition temperature, Tg(P), obtained from isobaric cooling runs at 0.1 K/min cooling rate, is found to increase with increasing pressure at a rate of dTg(P)/dP ¼ 0.38 6 0.02 K/MPa for



the star polystyrene, slightly higher than that for the linear chain. Isothermal pressure relaxation measurements were made at temperatures in the vicinity of Tg(P) at 36 and 55 MPa, respectively, and reduced curves were constructed by TTS. The reduced viscoelastic bulk moduli are compared with those for the linear sample; the results show that the absolute values of the limiting moduli are higher for the star sample and the relaxation time distribution is narrower giving a higher KWW b. The dynamic shear storage and loss moduli of the star polystyrene were measured at temperatures ranging from 95 to 170 C, and shifted to form reduced curves. The viscoelastic Poisson’s ratio was calculated, and it is showed to be a nonmonotonic function of time with a weak minimum at short times, consistent with the observations related to the breadth of the shear and bulk retardation spectra. Above Tg, the horizontal shift factors (aT) of the bulk and shear responses for the star and linear polystyrenes are the same within experimental error when plotted as a function of T Tg(P). In addition, the retardation time spectra for the bulk and shear responses of the star and linear polystyrenes indicate that the bulk and shear spectra have similar slopes at short times, but the retardation spectra for the bulk response show a lower magnitude, indicating that the shear response may arise from more Andrade creep modes at short times or that the contribution of these modes to the shear response may be greater. The bulk response cuts out when chain modes start to dominate the shear response, indicating that the long-time mechanisms available for the shear response are not available for the bulk relaxation. ACKNOWLEDGMENTS

The authors thank the funding from the National Science Foundation Division of Materials Research grants, DMR-0606500 and 1006972. In addition, helpful suggestions from Gregory B. McKenna, as well as help from Miao Hu with the shear rheological measurements and help from Paul O’Connell with repairs of the pressurizable dilatometer are acknowledged.

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