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Measurement of the thermal glass transition of polystyrene in a cooling rate range of more than six decades
G Model TCA 76892 No. of Pages 7
Thermochimica Acta xxx (2014) xxx–xxx
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Measurement of the thermal glass transition of polystyrene in a cooling rate range of more than six decades Jürgen E.K. Schawe * Mettler Toledo AG, Sonnenbergstrasse 74, CH-8603 Schwerzenbach, Switzerland
A R T I C L E I N F O
A B S T R A C T
Article history: Received 2 April 2014 Received in revised form 15 May 2014 Accepted 20 May 2014 Available online xxx
The thermal glass transition of polystyrene (PS) is measured in a range between 0.003 K/s and 4000 K/s using a conventional DSC 1 and a Flash DSC 1 from Mettler-Toledo. The dependency between the cooling rate and the glass transition temperature is in accordance with the Vogel–Fulcher–Tammann–Hesse (VFTH) equation, as it is usually used for description of the frequency–temperature relationship of the dynamic glass transition (or main relaxation process). Furthermore we discuss the influence of the thermal lag on the measurement and develop correction methods. ã 2014 Elsevier B.V. All rights reserved.
Keywords: Vitrification kinetics Glass transition Flash DSC Thermal lag correction
1. Introduction The glass transition occurs in an amorphous or semicrystalline materials in a supercooled melt. Usually, the glass transition is described as a relaxation process related to the cooperative molecular rearrangements. This process is accompanied by a heat capacity change, Dcp [1,2]. The transformation from the structural equilibrated liquid state to a non-equilibrated glassy state (and vice versa) is the thermal glass transition. During cooling, the related molecular fluctuations freeze-in in the region of the thermal glass transition; this means that the fluctuations become too slow to be measured in the experimental time [1]. The material vitrifies. The temperature range of the glass transition depends on the experimental conditions. With increasing cooling rate, bc, the glass transition shifts to higher temperatures [8]. In the past, the cooling rate dependence of the glass transition temperature was usually measured in a cooling rate range of about two or three decades. In this range the cooling rate dependency of the glass transition temperature can be described by empirical functions. The simplest proposed relationship is the linear approximation [3,4]. logjbc j ¼ a1 þ a2 T g
(1)
This equation has no theoretical background, but it fits well to experimental results in a limited cooling rate range of about three
decades and fits to the rule of thumb, that the thermal glass transition of polymers shifts about 2–5 K per decade cooling rate [4,5]. Another approximation of the cooling rate dependence of Tg is the Barteniev–Ritland (BR) equation [6–9] logjbc j ¼ b1 þ b2 =T g
This equation can be derived from the Arrhenius equation of a thermally activated process. This equation should describe the behavior of strong glass formers [10]. In the case of fragile glass formers like most polymers, Eq. (2) should be valid only in a limited cooling rate range. For fragile glass formers measured in a limited cooling rate range, the Eqs. (1) and (2) are practically equivalent at glass transition temperatures above 200 K. The dynamic glass transition is the thermal relaxation process in the structural equilibrated melt measured by a small temperature perturbation [11,12]. This relaxation process is directly measured by the frequency dependent complex heat capacity c* (v,T) (v is the angular frequency) [13,14]. Measurements of the dynamic glass transition were performed in a frequency range between 103 Hz and 106 Hz [15,16]. In a large range this data accord with the Vogel–Fulcher–Tammann–Hesse (VFTH) equation [17–19]: logvðTÞ ¼ A
* Tel.: +41 44 806 7438. E-mail address: [email protected] (J.E.K. Schawe).
(2)
B T TV
(3)
where A and B are a constants and T V is the so called Vogeltemperature (the extrapolated temperature for v ! 0).
http://dx.doi.org/10.1016/j.tca.2014.05.025 0040-6031/ ã 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: J.E.K. Schawe, Measurement of the thermal glass transition of polystyrene in a cooling rate range of more than six decades, Thermochim. Acta (2014), http://dx.doi.org/10.1016/j.tca.2014.05.025
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The correspondence between the thermal and the dynamic glass transition can be expressed be the Frenkel–Kobeko–Reiner (FKR) equation [2]:
bc ¼C v
(4)
The shift factor C is usually assumed to be a constant [1,2]. This approach is frequently used to combine conventional DSC data with data of the dynamic glass transition measured by thermal but also dielectric and mechanical techniques [1,20–22]. In this paper we use the FKR equation as an empirical model function. A discussion of the theoretical background is given in Ref. [6]. From Eqs. (3) and (4) follow a VFTH equation for that relation between cooling rate and glass transition temperature: logjbc ðTÞj ¼ Ab
B Tg TV
(5)
with Ab ¼ A þ logC:
(6)
Eq. (5) was taken from Hensel at al. [22] to analyze the correspondence between the cooling rate dependence of the thermal glass transition and the frequency dependence of the dynamic glass transition. These measurements cover almost three orders in magnitude of cooling rate. Because of the limited dynamic range this measurements could not confirm the validity of Eqs. (4) and (5) without reasonable doubt. The necessary wide dynamic range can be covered since the development of the nonadiabatic chip calorimeter (or high rate calorimeter (HRC)) by Schick [23–26]. Minakov et al. combined the first time such high cooling rate measurements with conventional DSC measurements to measure Tg as a function of the cooling rate for some polymers [27]. However, there is a significant scanning rate range gap between the highest rates in the conventional DSC and the lowest rates of the chip calorimeter. This gap can be closed using the Flash DSC 1 by Mettler-Toledo [28,29]. Measurements of thin polystyrene films in a cooling rate range between 101 K/s and 103 K/s were recently published by the group of S. Simon [30]. We present in this paper measurements of the thermal glass transition temperature in a cooling rate range of more than six decades (3 103–4 103 K/s) without any data gaps. The measurements were performed on polystyrene (PS) by using a conventional DSC and a fast scanning DSC (Flash DSC 1). We discuss the correction of the thermal lag for measurements of the glass transition. The resulting data are used to verify the validity of the Eqs. (1), (2) and (5) in the scanning rate range of investigation. 2. Experimental 2.1. Sample and instruments The polystyrene was PS168N from BASF (r = 1.047 g/cm3, Mn = 95,000 g/mol, Mw = 270,000 g/mol). The DSC measurements in the scanning rate range between 0.2 K/min and 60 K/min were performed with a Mettler-Toledo DSC 1 with IntraCooler and nitrogen as purge gas. The temperature was calibrated using water, indium and tin up to heating rates of 100 K/ min. The heat flow was adjusted by specific melting enthalpies measurements at 20 K/min. The measurements were performed using a flat 5 mg sample. The sample was cut from a granulate grain and placed in a 20 ml aluminum crucible. During the measurements the sample was first cooled from 180 C to 50 C at bc and immediately afterwards heated to 180 C at bh with |bc| = |bh|.
The fast scanning DSC measurements were performed using the Mettler-Toledo Flash DSC 1 [29,31]. The Flash DSC 1 is equipped with the UFS 1 MultiSTAR sensor. This sensor is a microchip embedded in a ceramic support. The sensor consists of two separate calorimeters (for sample and reference) which are operated as a complete power-compensated DSC. The actual sensor consists of two identical quadratic silicon nitride membranes with a length of 1.6 mm. The thickness of the membrane is 2 mm. The sample and reference area with a diameter of 0.5 mm each is coated with aluminum so that a homogeneous temperature profile is achieved. The Flash DSC 1 is equipped with an IntraCooler to control the sensor support temperature, Tss, and the temperature of the surrounding gas. The minimum temperature is about 100 C. The surrounding gas was nitrogen. After the sample measurements a small indium sample was placed on the reference side. The onset of the melting peak was measured at 100 K/s. This temperature was used for the final temperature correction. Because of the 1 ms time constant no significant thermal lag is expected at this heating rate [29]. Samples of about 35 ng, 90 ng and 220 ng were used for the scanning rate range between 2 K/s and 4000 K/s. For preparation a 10 mm thick slice was cut from a part of a granulate grain using a razor blade microtome. The slice was divided in small samples with a knife. The samples are placed on the sensor using a hair. Afterwards the samples were heated to 180 C to form a good thermal contact to the sensor. The 220 ng sample has formed a thick meniscus after the first heating. This sample was smeared to a thin film on the sensor at 200 C using a thin soft copper wire. After this procedure the sample thickness was below 10 mm. Preheating generates a good thermal and mechanical contact between sample and membrane. The sample thickness was estimated from the density and the contact area. The sensitivity of DSC measurements decreases with increasing scanning rate. Therefore the small samples do not show a nice glass transition curve at rates below 10 K/s. To get an overlap between the DSC and the flash DSC data we used larger samples with a size of about 700 ng and 5 mg. With large samples the mechanical coupling after preheating can produce stresses to the membrane if the sample becomes hard in the glassy state. This can be avoided by using a contact medium. We used highly viscous silicon oil (Mettler-Toledo 30069922) for this purpose [32]. By multiple heating to 350 C a stable thin oil film was obtained on the sample and reference side. Afterwards the sample was placed on the sensor and preheated. These samples are measured at rates between 0.05 K/s and 1000 K/s. For the measurements the sensor support temperature, Tss, was selected to be 95 C (ready temperature) to accelerate the equilibration of Tss. The stability of Tss is important for the temperature accuracy of the measurement, because Tss is the cold junction temperature of the thermocouples on the sensor. This is also the reason why Tss should be constant and equal to the ready temperature during the correction procedure (the calibration of the thermocouple in respect to Tss). 2.2. Mass estimation for the Flash DSC samples The sample mass needed for Flash DSC measurements sample cannot be weighed anymore with a conventional balance. Thus the sample mass has to be estimated using thermal properties like melting enthalpy or heat capacity in the melt [33]. In our case the step height of the glass transition (intensity) is used. The evaluation of the conventional DSC curves delivers the change of the specific heat capacity at the glass transition temperature Dcp. Before the actual measurements the sample mass was set to 1 ng in the software. From the measurement at 100 K/s the apparent intensity, Dcp,a was determined. The sample mass is
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m¼
Dcp;a 1ng Dc p
(7)
3. Results and correction procedures 3.1. The glass transition temperature As mentioned before the thermal glass transition is the transformation of the structurally equilibrated supercooled liquid to the non-equilibrated glassy state. According to Tool [34] this transformation can be described by the fictive temperature, Tf, which characterizes the structure of the glass. Above the glass transition Tf is identical to the sample temperature T and in the glassy state (at suppression further structural relaxation) the fictive temperature is constant. There are many different procedures to determine the glass transition temperature, Tg. We use the procedure proposed by Richardson and Savill [35] and Moynihan et al. [36]. The glass transition temperature is determined in such a way, that the indicated area below Tg equals to the area difference above Tg (see Fig. 1). The procedure is implemented in the instrument software. This Tg is also called thermodynamic glass transition temperature, it is equal to the fictive temperature of the glass after cooling or before heating, the so-called limiting fictive temperature. Advantages of this procedure are: The glass transition temperature is directly linked to the glassy
structure, The temperatures measured in cooling and subsequent heating
are identical (see the proof in Ref. [37]). For a reproducible and accurate determination of these glass transition temperatures, it is necessary to select a sufficient low temperature for the tangent determination, because the low temperature broadening of the glass transition is due to vitrification (see Fig. 1). 3.2. Thermal lag Temperature gradients occurring in sample and instrument during the measurement influence the experimental results [38]. The gradients shift the apparent glass transition temperature and broaden the transition. This effect is also called smearing. Principally dynamic and static temperature gradients may occur.
Fig. 1. Procedure to determination of the glass transition temperature as fictive temperature showed on a Flash DSC curve measured at a heating rate of 3000 K/s.
3
The dynamic temperature gradients are caused by the temperature scan. The resulting thermal lag depends on the scanning rate. It can be neglected at sufficient low rates and increases with increasing scanning rate [39,40]. The thermal lag contribution of the instrument (and crucible) is usually measured by the heating rate dependence of the melting temperatures (onset temperatures) of pure calibration materials [33]. For cooling measurements a test on a symmetric behavior is required [41]. In the case of glass transition measurements dynamic smearing causes a rate increasing differences between glass transition temperatures measured in cooling and subsequent heating at the same rates (bh = bc) [42]. Static temperature gradients do not disappear at low scanning rates. They may occur additionally to the dynamic gradients. These gradients may induce a permanent temperature shift. The influences of the temperature gradients on the measured curves are different in the DSC and the Flash DSC. In the DSC the furnace is relatively large and the sample is imbedded in a crucible. Thus the sample is heated from all sides. The temperature gradient is generated during heating and the lowest temperature is inside the sample and can be significantly large at relative high scanning rates [43]. The sample is basically thermal insolated to the surrounding. Static temperature gradients are usually irrelevant. In the Flash DSC the heat capacity of the furnace is relatively low. It is good coupled to the cooled surrounding gas to provide the high cooling rates. Consequently, static temperature gradients along the sample could occur. To reduce this, the sample should be as thin as possible. As a result dynamic smearing effects appear at relatively high scanning rates. The normal thermal lag correction is significant only above 1000 K/s [29]. 3.2.1. Correction of dynamic heat transfer effects The dynamic (rate dependent) temperature gradient causes smearing of the DSC curve, which increasing by increasing scanning rate [39,40]. Different methods are proposed to correct the dynamic thermal lag glass transition evaluation [4,22,24,40,44]. Our approach based on the cooling and heating symmetry of the dynamic thermal lag. As a consequence of the energy conservation low the thermodynamic glass transition temperature is identical in cooling and subsequent heating [37]. For the case |bc| = |bh| smearing symmetrically effects the measured glass transition temperature for heating and cooling because dynamic smearing depends on time (and not temperature) [38]. Therefore the true cooling dependent glass transition temperature at the rate bc, Tg(bc), is the average of the measured glass transition temperatures at the cooling rate bc and subsequent heating at bc [42]. This simple procedure compensates all dynamic smearing effects in instrument and sample regarding Tg(bc). Fig. 2 shows the measured apparent glass transition temperatures of the 5 mg DSC sample and the 90 ng Flash DSC sample as a function of the scanning rate. For these samples significant deviations between the cooling and heating runs occur at rates above 0.1 K/s (DSC) and 1000 K/s (Flash DSC), respectively. Fig. 2 shows further the good agreement of averaged data from both measurements. 3.2.2. Correction of static effects in the Flash DSC Using larger Flash DSC-samples achieves a better overlap between the DSC and the Flash DSC data. To show the upper sample size limits of the Flash DSC large sample with a mass of about 700 ng and 5 mg are selected. As it is mentioned before, for larger samples the use of silicon oil as a contact medium is recommended [32]. After the first measurements the samples form relative high meniscuses. Because of the oil film it was not possible to flatten the samples.
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0¼
@2 T Q þ @x2 l
(9)
If we neglect the contact resistance between sample and sensor (T(x0) = Tsens) flows no heat at x0 = 0: lð@T=@xÞjx0 ¼ 0: This leads to
@T Q ¼ x @x l
(10)
and TðxÞ ¼ T sens
Q
2l
x2
(11)
At x = x1 and T(x1) = Tsens DTs is
Q¼
2lDT s x21
(12)
From Eqs. (11) and (12) follows the temperature profile along the sample: 2 x (13) TðxÞ ¼ T sens DT s x1 Fig. 2. Glass transition temperature measured in cooling and subsequent heating as a function of the scanning rate. The measured and average data of the DSC (5 mg sample) and the Flash DSC (90 ng sample) measurements are shown. Furthermore the averaged data of the 700 ng and the 5 mg sample are plotted.
The 5 mg sample is principally too large for normal Flash DSC measurements. This sample covers almost the total sensor area. The thickness is in the order of 0.1 mm. The large samples can be measured from 0.1 K/s to 1000 K/s and from 0.05 K/s to 200 K/s, respectively. The average Tg data of the cooling and heating measurements are plotted in Fig. 2. It is obvious, that the resulting glass transition temperatures are significantly higher than those from the other samples. The reason is a static temperature gradient in the relative thick sample. To estimate the influence of this gradient on the experimental results we discuss simple model. The sample is flat and thick. The contact area between sensor and sample is large enough to neglect boundary effects (Fig. 3). At x0 = 0 the sample is in contact with the sensor. The sensor temperature Tsens is controlled by the heater. The sample thickness is x1. At x1 (upper side of the sample) the temperature is Tsens DTs. The temperature of the surrounding gas equals Tss (the sample support temperature). The heat transfer coefficient between the sample and the gas is k. The thermal conductivity and diffusivity of the sample are l and a, respectively. The one dimensional Fourier-equation of the heat transfer is
@T @2 T a ¼a 2þ Q (8) l @t @x where Q is the volumetric heat flux. For the static case follows
Fig. 3. Calculation (left) and measurement (right) of the static temperature gradient of thick samples in the Flash DSC.
The heat flux out of the sample into the cooled gas (with the constant temperature Tss) is @T l jx1 ¼ k T sens DT s T ss @x
(14)
Introducing Eqs. (12) and (14) in (10) delivers the temperature difference between the bottom and the upper side of the sample
DT s ¼
T sens T ss 1 þ ð2l=kx1 Þ
(15)
The quotient 2l/k is defined by the sample material and the gas. DTs can be reduced by decreasing the sample thickness and the transfer coefficient (i.e. using another gas) as well as by increasing the sensor support temperature Tss. In practice, this means that the sample should be as thin as possible and Tss should be set as high as possible, whereby the maximum cooling rate has still to be maintained down to the lowest temperature needed. To measure thick samples a gas with poor heat conduction is recommended. The measured glass transition temperature is the sensor temperature Tsens. The true glass transition temperature, Tg, corresponds to the average sample temperature T: T¼
1 x1
Zx1 TðxÞdx
(16)
0
Using Eq. (13) this leads to T g ¼ T ¼ T sens
DT s 3
(17)
Thus the correct Tg is the measured glass transition temperature reduced by one third of the temperature gradient along the sample. We cannot use Eq. (15) for estimation of DTs, because the denominator is unknown. However, DTs can be estimated experimentally. For this purpose a small indium sample was placed on upper side of the PS sample and another behind the sample (or on the reference furnace). Fig. 4 shows a measured curve at 100 K/s with a small amount of indium on the upper side of the sample and some other indium placed on the reference side. The difference of the onset temperatures of the indium melting peaks (20.3 K) is the temperature gradient in the sample at this heating rate. Such measurements were performed at different heating rates. For the 700 ng sample the onset temperatures of the indium melting peaks are plotted versus the heating rate in Fig. 5. The static temperature gradient DTs is the difference of the onset temperatures extrapolated to heating rate zero. For this sample is
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Fig. 4. Measured curve of the 700 ng sample at 100 K/s with an indium sample on the upper side of the sample and another indium sample on the reference side.
DTs = 18.5 K. This means that the measured Tg has to be corrected by 6.1 K. For the 5 mg sample we determined from similar measurements DTs = 21.9 K and thus a Tg correction of 7.3 K. The sample thicknesswasroughlyestimated fromthesamplearea:60 mm(5 mg) and 40 mm (700 ng), respectively, using the density of 1 g/cm3. 3.3. Cooling rate dependence of the thermal glass transition temperature The influence of the dynamic thermal lag on the measured curve is compensated by determining the thermodynamic glass transition temperature (Fig. 1) upon cooling and subsequent heating and by averaging the two respective measured glass transition temperatures. For information about the Tg reproducibility 35 ng and 220 ng samples were measured additionally. As described before, the measurement of thick samples by Flash DSC have to be corrected additionally for the static temperature gradient. In Fig. 6 the properly corrected data are displayed. The following conclusions can be drawn: The Flash DSC data of thin samples (below 10 mm thickness)
match together. The largest difference between the Tg-value at the same cooling rate is 2.8 K. These measurements are performed on different sensors.
Fig. 6. Glass transition temperature as a function of the cooling rate. The solid symbols represent average data of heating and cooling measurements. The open symbols are Flash DSC data of the thick samples (700 ng and 5 mg) after correction of the static temperature gradient. The curves are fitting results using the VFTH equation (see text).
The DSC and Flash DSC measured data fit well together. The
overlap of the cooling rate range of both techniques is more than one decade. The correction of the static temperature gradient for thick Flash DSC samples is appropriate. The corrected data of the 700 ng samples agree well with the averaged data of the thin samples.
4. Discussion The cooling rate dependence of the glass transition temperature is usually described by three different model functions: As a linear function (Eq. (1)). The BR equation (Eq. (2)). Using the VFTH equation (Eq. (5)).
To determine which model is in best agreement with the experimental results the data from Fig. 6 were fitted with the models. In the case of the VFTH model we use T V = 333 K [15]. This value was not varied during the fitting procedure. The resulting fit parameters are summarized in Table 1. To qualify the agreement of the model functions with the experimental dataset, the differences between the measured glass transition temperatures, Tg, and the calculated temperature, Tfit, are plotted as a function of the logarithmic cooling rate in Fig. 7. The data scattering is due to experimental uncertainties. As better the model function corresponds to the dataset as more the average of (Tg Tfit) in a certain abscissa interval tends to zero. This means for a proper model function the (Tg Tfit) scatter around zero, for wrong models the local average differs from zero. The local average of the data in Fig. 7 can be simply estimated by fitting a quadratic Table 1 Fit parameters derived from the data in Fig. 6. The values of the fixed parameters are taken from Weyer et al. [15].
Fig. 5. Onset temperatures of indium on the reference side and on the upper side of the 700 ng sample for various heating rates.
Fit Fit Fit Fit
1 2 3 4
Eq. Eq. Eq. Eq.
(1) (2) (5), T V is constant (5), T V and B are constant
a1 = 95.5 b1 = 98.2 Ab = 13.2 Ab = 13.5
a2 = 0.25 K1 b2 = 36,884 K B = 556 K B = 570 K
T V = 333 K T V = 333 K
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the shift factor will be presented elsewhere. A comparison of the measurements presented with the frequency dependence of the dynamic glass transition, indicate that dynamic glass transition follows a similar relaxation behavior like the thermal glass transition. The curvature factor, B, and the Vogel temperature, T V, seem to be the same for the dynamic and the thermal glass transition. It should be mentioned that this behavior is found within a temperature range of about 25 K. Especially at higher temperatures, a deviation of the described behavior could occur due to the change of the relaxation behavior close to the splitting point [45]. Also at very low temperatures there are indications for changes of the relaxation kinetics [46]. 5. Conclusions
Fig. 7. Deviations between the experimental data and the best fit using the linear equation (Eq. (1)), the Barteniev–Ritland equation (Eq. (2)) and the VFTH equation (Eq. (3)) to describe the cooling rate dependence of the glass transition temperature. The dashed lines are quadratic fit functions.
function. The dashed curves in Fig. 7 are these average functions. For the VFTH equation (fit 3) the average function is practically zero in the range of investigation. The maximum deviation from zero is less than 0.2 K. For the other models these deviations are more than ten times larger. As expected, the BF equation describes the experimental results not significantly better than the empirical linear function for the present conditions (Tg at about 650 K and a variation of Tg of about 25 K). This leads to the conclusion that the VFTH equation describes for PS the cooling rate dependence of the glass transition in the cooling rate range between 103 and 104 K/s. The dashed curve in Fig. 6 represents the result of fit 3 in Table 1. Now we discuss the shift factor C from Eq. (4) by comparison the cooling rate dependency of the thermal glass transition with the frequency dependent behavior of the dynamic glass transition. Weyer et al. [15] published data of the (frequency dependent) dynamic glass transition in a similar large dynamic range. The VFTH parameters were determined to be A = 12, B = 570 K and T V = 333 K. According to the FKR equation C is constant. This means that only the parameter A is different for the dynamic and the thermal glass transition. The related fit (fit 4 in Table 1) results the solid curve in Fig. 6. The maximum deviation between fit 3 and fit 4 is about 0.5 K at 104 K/s. This is less than the experimental accuracy. The shift factor is according to Eq. (6) logC = 1.5 or C = 31.6 K. This value agrees with model calculations in Ref. [15] and measurements of Minakov et al. [27]. Slightly smaller values (logC = 1.3) was founded from TMDSC and conventional DSC measurements in a small dynamic range by Hensel et al. [22]. A detailed discussion of
Using conventional DSC and the fast scanning Flash DSC the cooling rate dependence of the glass transition of polystyrene is measured a scanning rate range of more than six decades between 3 103 K/s and 4 103 K/s. We showed that the cooling rate dependency of the thermal glass transition temperature follows a similar VFTH-behavior as it was founded for the dynamic glass transition. The relation between cooling rate and frequency for PS is estimated to be bc = 31.6 K v. For the evaluation of the glass transition temperature dynamic and static heat effects have to be considered. Dynamic effects are due to the formation of a temperature profile during scanning. This effect becomes significant at rates above 0.1 K/s (DSC) and 1000 K/s (Flash DSC). This limit also depends on the sample size. A simple correction procedure is to average the glass transition temperatures measured during cooling and subsequent heating at bc = bh. For this procedure it is recommend to determine Tg as the thermodynamic glass transition temperature according to Richardson and Savill, and Moynihan et al. [35,36]. High cooling rates in the Flash DSC require large temperature differences between the sample and the surrounding gas. This is the reason for potential static temperature gradients in the sample. These gradients can be suppressed by using thin samples with a typical thickness in the order of 10 mm or lower. For thicker samples a simple correction method is introduced. This method bases on the determination of the temperature gradient along the sample. For this purpose a reference material (such as indium) is placed on the upper side of the sample as well as on the sensor. The difference of the melting temperature of the sample characterizes the temperature gradient. Acknowledgement The author thanks Prof. Christoph Schick, University of Rostock, for providing the PS material. References [1] E. Donth, The Glass Transition, Springer-Verlag, Berlin, 2001. [2] I. Gutzow, J. Schmelzer, The Vitreous State, Springer, Berlin, 1995. [3] F.R. Schwarzl, F. Zahradnik, The time temperature position of the glass-rubber transition of amorphous polymers and the free volume, Rheol. Acta 19 (1980) 137–152. [4] J.R. Saffell, The effect of heating and cooling rate on the characteristics of the calorimetric glass transition for glassy polymers, Thermochim. Acta 36 (1980) 251–264. [5] H.-J. Bittrich, H.-J. Schad, H. Tanneberger, Untersuchung des volumentemperatur-zeit-verhaltens amorpher polymere im glasübergangsbereich, Acta Polym. 33 (1982) 736–740. [6] J.W. Schmelzer, Kinetic criteria of glass transition and the pressure dependence of the glass transition temperature, J. Chem. Phys. 136 (2012) 074512. [7] G.M. Bartenev, O , Dokl. Akad. Nauk SSSR 69 (1951) 227–230.
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[28] V. Mathot, M. Pyda, T. Pijpers, G. Van den Poel, E. van de Kerkhof, S. van Herwaarden, F. van Herwaarden, A. Leenaers, The Flash DSC 1, a power compensation twin-type, chip-based fast scanning calorimeter (FSC): first findings on polymers, Thermochim. Acta 522 (2011) 36–45. [29] J.E.K. Schawe, Influence of processing conditions on polymer crystallization measured by fast scanning DSC, J. Therm. Anal. Calorim. 116 (2014) 1165–1173. [30] S. Gao, Y.P. Koh, S.L. Simon, Calorimetric glass transition of single polystyrene ultrathin films, Macromolecules 46 (2013) 562–570. [31] J.E.K. Schawe, The revolutionary new Flash DSC 1: optimum performance for metastable materials, UserCom 32 (2010) 12–16www.mt.com/ta-usercoms. [32] J.E.K. Schawe, Practical aspects of the Flash DSC 1: sample preparation for measurements of polymers, Mettler Toledo Therm. Anal. UserCom 36 (2012) 17–24www.mt.com/ta-usercoms. [33] S. Sarge, W. Hemminger, E. Gmelin, G.W.H. Höhne, H. Cammenga, W. Eysel, Metrologically based procedures for the temperature, heat and heat flow rate calibration of DSC, J. Therm. Anal. 49 (1997) 1125–1134. [34] A.Q. Tool, Relation between inelastic deformability and thermal expansion of glass in its annealing range, J. Am. Ceram. Soc. 29 (1946) 240–253. [35] M.J. Richardson, N.G. Savill, Derivation of accurate glass transition temperatures by differential scanning calorimetry, Polymer 16 (1975) 753–757. [36] C.T. Moynihan, A.J. Easteal, M.A. DeBolt, J. Tucker, Dependence of the fictive temperature of glasses on cooling rate, J. Am. Ceram. Soc. 59 (1976) 12–16. [37] J.E.K. Schawe, An analysis of the meta stable structure of poly(ethylene terephthalate) by conventional DSC, Thermochim. Acta 461 (2007) 145– 152. [38] G.W.H. Höhne, J.E.K. Schawe, Dynamic behavior of power compensated differential scanning calorimeters. Part 1. DSC as a linear system, Thermochim. Acta 229 (1993) 27–36. [39] J.E.K. Schawe, C. Schick, Influence of the heat conductivity of the sample on DSC curves and its correction, Thermochim. Acta 187 (1991) 335–349. [40] J.E.K. Schawe, C. Schick, G.W.H. Höhne, Dynamic behavior of powercompensated differential scanning calorimeters. Part 4. The influence of change in material properties, Thermochim. Acta 244 (1994) 49–59. [41] S. Neuenfeld, C. Schick, Verifying the symmetry of differential scanning calorimeters concerning heating and cooling using liquid crystal secondary temperature standards, Thermochim. Acta 446 (2006) 55–65. [42] J.E.K. Schawe, Description of thermal relaxation of polystyrene close to the glass transition, J. Polym. Sci.: Part B: Polym. Phys 36 (1998) 2165–2175. [43] B. Wunderlich, Thermal Analysis of Polymeric Materials, Springer, Berlin, 2005 Chapter 4.3. [44] J. Morikawa, T. Hashimoto, Estimation of frequency from a temperature scanning rate in differential scanning calorimetry at the glass transition of polystyrene, Polymer 52 (2011) 4129–4135. [45] H. Huth, M. Beiner, E. Donth, Temperature dependence of glass-transition cooperativity from heat capacity spectroscopy: two post-Adam-Gibbs variants, Phys. Rev. B 61 (2000) 15092–15101. [46] D. Cangialosi, V.M. Boucher, A. Alegria, J. Colmenero, Direct evidence of two equilibrium mechanisms in glassy polymers, Phys. Rev. Lett. 111 (2013) 095701.
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Thermochimica Acta xxx (2014) xxx–xxx
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Measurement of the thermal glass transition of polystyrene in a cooling rate range of more than six decades Jürgen E.K. Schawe * Mettler Toledo AG, Sonnenbergstrasse 74, CH-8603 Schwerzenbach, Switzerland
A R T I C L E I N F O
A B S T R A C T
Article history: Received 2 April 2014 Received in revised form 15 May 2014 Accepted 20 May 2014 Available online xxx
The thermal glass transition of polystyrene (PS) is measured in a range between 0.003 K/s and 4000 K/s using a conventional DSC 1 and a Flash DSC 1 from Mettler-Toledo. The dependency between the cooling rate and the glass transition temperature is in accordance with the Vogel–Fulcher–Tammann–Hesse (VFTH) equation, as it is usually used for description of the frequency–temperature relationship of the dynamic glass transition (or main relaxation process). Furthermore we discuss the influence of the thermal lag on the measurement and develop correction methods. ã 2014 Elsevier B.V. All rights reserved.
Keywords: Vitrification kinetics Glass transition Flash DSC Thermal lag correction
1. Introduction The glass transition occurs in an amorphous or semicrystalline materials in a supercooled melt. Usually, the glass transition is described as a relaxation process related to the cooperative molecular rearrangements. This process is accompanied by a heat capacity change, Dcp [1,2]. The transformation from the structural equilibrated liquid state to a non-equilibrated glassy state (and vice versa) is the thermal glass transition. During cooling, the related molecular fluctuations freeze-in in the region of the thermal glass transition; this means that the fluctuations become too slow to be measured in the experimental time [1]. The material vitrifies. The temperature range of the glass transition depends on the experimental conditions. With increasing cooling rate, bc, the glass transition shifts to higher temperatures [8]. In the past, the cooling rate dependence of the glass transition temperature was usually measured in a cooling rate range of about two or three decades. In this range the cooling rate dependency of the glass transition temperature can be described by empirical functions. The simplest proposed relationship is the linear approximation [3,4]. logjbc j ¼ a1 þ a2 T g
(1)
This equation has no theoretical background, but it fits well to experimental results in a limited cooling rate range of about three
decades and fits to the rule of thumb, that the thermal glass transition of polymers shifts about 2–5 K per decade cooling rate [4,5]. Another approximation of the cooling rate dependence of Tg is the Barteniev–Ritland (BR) equation [6–9] logjbc j ¼ b1 þ b2 =T g
This equation can be derived from the Arrhenius equation of a thermally activated process. This equation should describe the behavior of strong glass formers [10]. In the case of fragile glass formers like most polymers, Eq. (2) should be valid only in a limited cooling rate range. For fragile glass formers measured in a limited cooling rate range, the Eqs. (1) and (2) are practically equivalent at glass transition temperatures above 200 K. The dynamic glass transition is the thermal relaxation process in the structural equilibrated melt measured by a small temperature perturbation [11,12]. This relaxation process is directly measured by the frequency dependent complex heat capacity c* (v,T) (v is the angular frequency) [13,14]. Measurements of the dynamic glass transition were performed in a frequency range between 103 Hz and 106 Hz [15,16]. In a large range this data accord with the Vogel–Fulcher–Tammann–Hesse (VFTH) equation [17–19]: logvðTÞ ¼ A
* Tel.: +41 44 806 7438. E-mail address: [email protected] (J.E.K. Schawe).
(2)
B T TV
(3)
where A and B are a constants and T V is the so called Vogeltemperature (the extrapolated temperature for v ! 0).
http://dx.doi.org/10.1016/j.tca.2014.05.025 0040-6031/ ã 2014 Elsevier B.V. All rights reserved.
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The correspondence between the thermal and the dynamic glass transition can be expressed be the Frenkel–Kobeko–Reiner (FKR) equation [2]:
bc ¼C v
(4)
The shift factor C is usually assumed to be a constant [1,2]. This approach is frequently used to combine conventional DSC data with data of the dynamic glass transition measured by thermal but also dielectric and mechanical techniques [1,20–22]. In this paper we use the FKR equation as an empirical model function. A discussion of the theoretical background is given in Ref. [6]. From Eqs. (3) and (4) follow a VFTH equation for that relation between cooling rate and glass transition temperature: logjbc ðTÞj ¼ Ab
B Tg TV
(5)
with Ab ¼ A þ logC:
(6)
Eq. (5) was taken from Hensel at al. [22] to analyze the correspondence between the cooling rate dependence of the thermal glass transition and the frequency dependence of the dynamic glass transition. These measurements cover almost three orders in magnitude of cooling rate. Because of the limited dynamic range this measurements could not confirm the validity of Eqs. (4) and (5) without reasonable doubt. The necessary wide dynamic range can be covered since the development of the nonadiabatic chip calorimeter (or high rate calorimeter (HRC)) by Schick [23–26]. Minakov et al. combined the first time such high cooling rate measurements with conventional DSC measurements to measure Tg as a function of the cooling rate for some polymers [27]. However, there is a significant scanning rate range gap between the highest rates in the conventional DSC and the lowest rates of the chip calorimeter. This gap can be closed using the Flash DSC 1 by Mettler-Toledo [28,29]. Measurements of thin polystyrene films in a cooling rate range between 101 K/s and 103 K/s were recently published by the group of S. Simon [30]. We present in this paper measurements of the thermal glass transition temperature in a cooling rate range of more than six decades (3 103–4 103 K/s) without any data gaps. The measurements were performed on polystyrene (PS) by using a conventional DSC and a fast scanning DSC (Flash DSC 1). We discuss the correction of the thermal lag for measurements of the glass transition. The resulting data are used to verify the validity of the Eqs. (1), (2) and (5) in the scanning rate range of investigation. 2. Experimental 2.1. Sample and instruments The polystyrene was PS168N from BASF (r = 1.047 g/cm3, Mn = 95,000 g/mol, Mw = 270,000 g/mol). The DSC measurements in the scanning rate range between 0.2 K/min and 60 K/min were performed with a Mettler-Toledo DSC 1 with IntraCooler and nitrogen as purge gas. The temperature was calibrated using water, indium and tin up to heating rates of 100 K/ min. The heat flow was adjusted by specific melting enthalpies measurements at 20 K/min. The measurements were performed using a flat 5 mg sample. The sample was cut from a granulate grain and placed in a 20 ml aluminum crucible. During the measurements the sample was first cooled from 180 C to 50 C at bc and immediately afterwards heated to 180 C at bh with |bc| = |bh|.
The fast scanning DSC measurements were performed using the Mettler-Toledo Flash DSC 1 [29,31]. The Flash DSC 1 is equipped with the UFS 1 MultiSTAR sensor. This sensor is a microchip embedded in a ceramic support. The sensor consists of two separate calorimeters (for sample and reference) which are operated as a complete power-compensated DSC. The actual sensor consists of two identical quadratic silicon nitride membranes with a length of 1.6 mm. The thickness of the membrane is 2 mm. The sample and reference area with a diameter of 0.5 mm each is coated with aluminum so that a homogeneous temperature profile is achieved. The Flash DSC 1 is equipped with an IntraCooler to control the sensor support temperature, Tss, and the temperature of the surrounding gas. The minimum temperature is about 100 C. The surrounding gas was nitrogen. After the sample measurements a small indium sample was placed on the reference side. The onset of the melting peak was measured at 100 K/s. This temperature was used for the final temperature correction. Because of the 1 ms time constant no significant thermal lag is expected at this heating rate [29]. Samples of about 35 ng, 90 ng and 220 ng were used for the scanning rate range between 2 K/s and 4000 K/s. For preparation a 10 mm thick slice was cut from a part of a granulate grain using a razor blade microtome. The slice was divided in small samples with a knife. The samples are placed on the sensor using a hair. Afterwards the samples were heated to 180 C to form a good thermal contact to the sensor. The 220 ng sample has formed a thick meniscus after the first heating. This sample was smeared to a thin film on the sensor at 200 C using a thin soft copper wire. After this procedure the sample thickness was below 10 mm. Preheating generates a good thermal and mechanical contact between sample and membrane. The sample thickness was estimated from the density and the contact area. The sensitivity of DSC measurements decreases with increasing scanning rate. Therefore the small samples do not show a nice glass transition curve at rates below 10 K/s. To get an overlap between the DSC and the flash DSC data we used larger samples with a size of about 700 ng and 5 mg. With large samples the mechanical coupling after preheating can produce stresses to the membrane if the sample becomes hard in the glassy state. This can be avoided by using a contact medium. We used highly viscous silicon oil (Mettler-Toledo 30069922) for this purpose [32]. By multiple heating to 350 C a stable thin oil film was obtained on the sample and reference side. Afterwards the sample was placed on the sensor and preheated. These samples are measured at rates between 0.05 K/s and 1000 K/s. For the measurements the sensor support temperature, Tss, was selected to be 95 C (ready temperature) to accelerate the equilibration of Tss. The stability of Tss is important for the temperature accuracy of the measurement, because Tss is the cold junction temperature of the thermocouples on the sensor. This is also the reason why Tss should be constant and equal to the ready temperature during the correction procedure (the calibration of the thermocouple in respect to Tss). 2.2. Mass estimation for the Flash DSC samples The sample mass needed for Flash DSC measurements sample cannot be weighed anymore with a conventional balance. Thus the sample mass has to be estimated using thermal properties like melting enthalpy or heat capacity in the melt [33]. In our case the step height of the glass transition (intensity) is used. The evaluation of the conventional DSC curves delivers the change of the specific heat capacity at the glass transition temperature Dcp. Before the actual measurements the sample mass was set to 1 ng in the software. From the measurement at 100 K/s the apparent intensity, Dcp,a was determined. The sample mass is
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m¼
Dcp;a 1ng Dc p
(7)
3. Results and correction procedures 3.1. The glass transition temperature As mentioned before the thermal glass transition is the transformation of the structurally equilibrated supercooled liquid to the non-equilibrated glassy state. According to Tool [34] this transformation can be described by the fictive temperature, Tf, which characterizes the structure of the glass. Above the glass transition Tf is identical to the sample temperature T and in the glassy state (at suppression further structural relaxation) the fictive temperature is constant. There are many different procedures to determine the glass transition temperature, Tg. We use the procedure proposed by Richardson and Savill [35] and Moynihan et al. [36]. The glass transition temperature is determined in such a way, that the indicated area below Tg equals to the area difference above Tg (see Fig. 1). The procedure is implemented in the instrument software. This Tg is also called thermodynamic glass transition temperature, it is equal to the fictive temperature of the glass after cooling or before heating, the so-called limiting fictive temperature. Advantages of this procedure are: The glass transition temperature is directly linked to the glassy
structure, The temperatures measured in cooling and subsequent heating
are identical (see the proof in Ref. [37]). For a reproducible and accurate determination of these glass transition temperatures, it is necessary to select a sufficient low temperature for the tangent determination, because the low temperature broadening of the glass transition is due to vitrification (see Fig. 1). 3.2. Thermal lag Temperature gradients occurring in sample and instrument during the measurement influence the experimental results [38]. The gradients shift the apparent glass transition temperature and broaden the transition. This effect is also called smearing. Principally dynamic and static temperature gradients may occur.
Fig. 1. Procedure to determination of the glass transition temperature as fictive temperature showed on a Flash DSC curve measured at a heating rate of 3000 K/s.
3
The dynamic temperature gradients are caused by the temperature scan. The resulting thermal lag depends on the scanning rate. It can be neglected at sufficient low rates and increases with increasing scanning rate [39,40]. The thermal lag contribution of the instrument (and crucible) is usually measured by the heating rate dependence of the melting temperatures (onset temperatures) of pure calibration materials [33]. For cooling measurements a test on a symmetric behavior is required [41]. In the case of glass transition measurements dynamic smearing causes a rate increasing differences between glass transition temperatures measured in cooling and subsequent heating at the same rates (bh = bc) [42]. Static temperature gradients do not disappear at low scanning rates. They may occur additionally to the dynamic gradients. These gradients may induce a permanent temperature shift. The influences of the temperature gradients on the measured curves are different in the DSC and the Flash DSC. In the DSC the furnace is relatively large and the sample is imbedded in a crucible. Thus the sample is heated from all sides. The temperature gradient is generated during heating and the lowest temperature is inside the sample and can be significantly large at relative high scanning rates [43]. The sample is basically thermal insolated to the surrounding. Static temperature gradients are usually irrelevant. In the Flash DSC the heat capacity of the furnace is relatively low. It is good coupled to the cooled surrounding gas to provide the high cooling rates. Consequently, static temperature gradients along the sample could occur. To reduce this, the sample should be as thin as possible. As a result dynamic smearing effects appear at relatively high scanning rates. The normal thermal lag correction is significant only above 1000 K/s [29]. 3.2.1. Correction of dynamic heat transfer effects The dynamic (rate dependent) temperature gradient causes smearing of the DSC curve, which increasing by increasing scanning rate [39,40]. Different methods are proposed to correct the dynamic thermal lag glass transition evaluation [4,22,24,40,44]. Our approach based on the cooling and heating symmetry of the dynamic thermal lag. As a consequence of the energy conservation low the thermodynamic glass transition temperature is identical in cooling and subsequent heating [37]. For the case |bc| = |bh| smearing symmetrically effects the measured glass transition temperature for heating and cooling because dynamic smearing depends on time (and not temperature) [38]. Therefore the true cooling dependent glass transition temperature at the rate bc, Tg(bc), is the average of the measured glass transition temperatures at the cooling rate bc and subsequent heating at bc [42]. This simple procedure compensates all dynamic smearing effects in instrument and sample regarding Tg(bc). Fig. 2 shows the measured apparent glass transition temperatures of the 5 mg DSC sample and the 90 ng Flash DSC sample as a function of the scanning rate. For these samples significant deviations between the cooling and heating runs occur at rates above 0.1 K/s (DSC) and 1000 K/s (Flash DSC), respectively. Fig. 2 shows further the good agreement of averaged data from both measurements. 3.2.2. Correction of static effects in the Flash DSC Using larger Flash DSC-samples achieves a better overlap between the DSC and the Flash DSC data. To show the upper sample size limits of the Flash DSC large sample with a mass of about 700 ng and 5 mg are selected. As it is mentioned before, for larger samples the use of silicon oil as a contact medium is recommended [32]. After the first measurements the samples form relative high meniscuses. Because of the oil film it was not possible to flatten the samples.
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0¼
@2 T Q þ @x2 l
(9)
If we neglect the contact resistance between sample and sensor (T(x0) = Tsens) flows no heat at x0 = 0: lð@T=@xÞjx0 ¼ 0: This leads to
@T Q ¼ x @x l
(10)
and TðxÞ ¼ T sens
Q
2l
x2
(11)
At x = x1 and T(x1) = Tsens DTs is
Q¼
2lDT s x21
(12)
From Eqs. (11) and (12) follows the temperature profile along the sample: 2 x (13) TðxÞ ¼ T sens DT s x1 Fig. 2. Glass transition temperature measured in cooling and subsequent heating as a function of the scanning rate. The measured and average data of the DSC (5 mg sample) and the Flash DSC (90 ng sample) measurements are shown. Furthermore the averaged data of the 700 ng and the 5 mg sample are plotted.
The 5 mg sample is principally too large for normal Flash DSC measurements. This sample covers almost the total sensor area. The thickness is in the order of 0.1 mm. The large samples can be measured from 0.1 K/s to 1000 K/s and from 0.05 K/s to 200 K/s, respectively. The average Tg data of the cooling and heating measurements are plotted in Fig. 2. It is obvious, that the resulting glass transition temperatures are significantly higher than those from the other samples. The reason is a static temperature gradient in the relative thick sample. To estimate the influence of this gradient on the experimental results we discuss simple model. The sample is flat and thick. The contact area between sensor and sample is large enough to neglect boundary effects (Fig. 3). At x0 = 0 the sample is in contact with the sensor. The sensor temperature Tsens is controlled by the heater. The sample thickness is x1. At x1 (upper side of the sample) the temperature is Tsens DTs. The temperature of the surrounding gas equals Tss (the sample support temperature). The heat transfer coefficient between the sample and the gas is k. The thermal conductivity and diffusivity of the sample are l and a, respectively. The one dimensional Fourier-equation of the heat transfer is
@T @2 T a ¼a 2þ Q (8) l @t @x where Q is the volumetric heat flux. For the static case follows
Fig. 3. Calculation (left) and measurement (right) of the static temperature gradient of thick samples in the Flash DSC.
The heat flux out of the sample into the cooled gas (with the constant temperature Tss) is @T l jx1 ¼ k T sens DT s T ss @x
(14)
Introducing Eqs. (12) and (14) in (10) delivers the temperature difference between the bottom and the upper side of the sample
DT s ¼
T sens T ss 1 þ ð2l=kx1 Þ
(15)
The quotient 2l/k is defined by the sample material and the gas. DTs can be reduced by decreasing the sample thickness and the transfer coefficient (i.e. using another gas) as well as by increasing the sensor support temperature Tss. In practice, this means that the sample should be as thin as possible and Tss should be set as high as possible, whereby the maximum cooling rate has still to be maintained down to the lowest temperature needed. To measure thick samples a gas with poor heat conduction is recommended. The measured glass transition temperature is the sensor temperature Tsens. The true glass transition temperature, Tg, corresponds to the average sample temperature T: T¼
1 x1
Zx1 TðxÞdx
(16)
0
Using Eq. (13) this leads to T g ¼ T ¼ T sens
DT s 3
(17)
Thus the correct Tg is the measured glass transition temperature reduced by one third of the temperature gradient along the sample. We cannot use Eq. (15) for estimation of DTs, because the denominator is unknown. However, DTs can be estimated experimentally. For this purpose a small indium sample was placed on upper side of the PS sample and another behind the sample (or on the reference furnace). Fig. 4 shows a measured curve at 100 K/s with a small amount of indium on the upper side of the sample and some other indium placed on the reference side. The difference of the onset temperatures of the indium melting peaks (20.3 K) is the temperature gradient in the sample at this heating rate. Such measurements were performed at different heating rates. For the 700 ng sample the onset temperatures of the indium melting peaks are plotted versus the heating rate in Fig. 5. The static temperature gradient DTs is the difference of the onset temperatures extrapolated to heating rate zero. For this sample is
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Fig. 4. Measured curve of the 700 ng sample at 100 K/s with an indium sample on the upper side of the sample and another indium sample on the reference side.
DTs = 18.5 K. This means that the measured Tg has to be corrected by 6.1 K. For the 5 mg sample we determined from similar measurements DTs = 21.9 K and thus a Tg correction of 7.3 K. The sample thicknesswasroughlyestimated fromthesamplearea:60 mm(5 mg) and 40 mm (700 ng), respectively, using the density of 1 g/cm3. 3.3. Cooling rate dependence of the thermal glass transition temperature The influence of the dynamic thermal lag on the measured curve is compensated by determining the thermodynamic glass transition temperature (Fig. 1) upon cooling and subsequent heating and by averaging the two respective measured glass transition temperatures. For information about the Tg reproducibility 35 ng and 220 ng samples were measured additionally. As described before, the measurement of thick samples by Flash DSC have to be corrected additionally for the static temperature gradient. In Fig. 6 the properly corrected data are displayed. The following conclusions can be drawn: The Flash DSC data of thin samples (below 10 mm thickness)
match together. The largest difference between the Tg-value at the same cooling rate is 2.8 K. These measurements are performed on different sensors.
Fig. 6. Glass transition temperature as a function of the cooling rate. The solid symbols represent average data of heating and cooling measurements. The open symbols are Flash DSC data of the thick samples (700 ng and 5 mg) after correction of the static temperature gradient. The curves are fitting results using the VFTH equation (see text).
The DSC and Flash DSC measured data fit well together. The
overlap of the cooling rate range of both techniques is more than one decade. The correction of the static temperature gradient for thick Flash DSC samples is appropriate. The corrected data of the 700 ng samples agree well with the averaged data of the thin samples.
4. Discussion The cooling rate dependence of the glass transition temperature is usually described by three different model functions: As a linear function (Eq. (1)). The BR equation (Eq. (2)). Using the VFTH equation (Eq. (5)).
To determine which model is in best agreement with the experimental results the data from Fig. 6 were fitted with the models. In the case of the VFTH model we use T V = 333 K [15]. This value was not varied during the fitting procedure. The resulting fit parameters are summarized in Table 1. To qualify the agreement of the model functions with the experimental dataset, the differences between the measured glass transition temperatures, Tg, and the calculated temperature, Tfit, are plotted as a function of the logarithmic cooling rate in Fig. 7. The data scattering is due to experimental uncertainties. As better the model function corresponds to the dataset as more the average of (Tg Tfit) in a certain abscissa interval tends to zero. This means for a proper model function the (Tg Tfit) scatter around zero, for wrong models the local average differs from zero. The local average of the data in Fig. 7 can be simply estimated by fitting a quadratic Table 1 Fit parameters derived from the data in Fig. 6. The values of the fixed parameters are taken from Weyer et al. [15].
Fig. 5. Onset temperatures of indium on the reference side and on the upper side of the 700 ng sample for various heating rates.
Fit Fit Fit Fit
1 2 3 4
Eq. Eq. Eq. Eq.
(1) (2) (5), T V is constant (5), T V and B are constant
a1 = 95.5 b1 = 98.2 Ab = 13.2 Ab = 13.5
a2 = 0.25 K1 b2 = 36,884 K B = 556 K B = 570 K
T V = 333 K T V = 333 K
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the shift factor will be presented elsewhere. A comparison of the measurements presented with the frequency dependence of the dynamic glass transition, indicate that dynamic glass transition follows a similar relaxation behavior like the thermal glass transition. The curvature factor, B, and the Vogel temperature, T V, seem to be the same for the dynamic and the thermal glass transition. It should be mentioned that this behavior is found within a temperature range of about 25 K. Especially at higher temperatures, a deviation of the described behavior could occur due to the change of the relaxation behavior close to the splitting point [45]. Also at very low temperatures there are indications for changes of the relaxation kinetics [46]. 5. Conclusions
Fig. 7. Deviations between the experimental data and the best fit using the linear equation (Eq. (1)), the Barteniev–Ritland equation (Eq. (2)) and the VFTH equation (Eq. (3)) to describe the cooling rate dependence of the glass transition temperature. The dashed lines are quadratic fit functions.
function. The dashed curves in Fig. 7 are these average functions. For the VFTH equation (fit 3) the average function is practically zero in the range of investigation. The maximum deviation from zero is less than 0.2 K. For the other models these deviations are more than ten times larger. As expected, the BF equation describes the experimental results not significantly better than the empirical linear function for the present conditions (Tg at about 650 K and a variation of Tg of about 25 K). This leads to the conclusion that the VFTH equation describes for PS the cooling rate dependence of the glass transition in the cooling rate range between 103 and 104 K/s. The dashed curve in Fig. 6 represents the result of fit 3 in Table 1. Now we discuss the shift factor C from Eq. (4) by comparison the cooling rate dependency of the thermal glass transition with the frequency dependent behavior of the dynamic glass transition. Weyer et al. [15] published data of the (frequency dependent) dynamic glass transition in a similar large dynamic range. The VFTH parameters were determined to be A = 12, B = 570 K and T V = 333 K. According to the FKR equation C is constant. This means that only the parameter A is different for the dynamic and the thermal glass transition. The related fit (fit 4 in Table 1) results the solid curve in Fig. 6. The maximum deviation between fit 3 and fit 4 is about 0.5 K at 104 K/s. This is less than the experimental accuracy. The shift factor is according to Eq. (6) logC = 1.5 or C = 31.6 K. This value agrees with model calculations in Ref. [15] and measurements of Minakov et al. [27]. Slightly smaller values (logC = 1.3) was founded from TMDSC and conventional DSC measurements in a small dynamic range by Hensel et al. [22]. A detailed discussion of
Using conventional DSC and the fast scanning Flash DSC the cooling rate dependence of the glass transition of polystyrene is measured a scanning rate range of more than six decades between 3 103 K/s and 4 103 K/s. We showed that the cooling rate dependency of the thermal glass transition temperature follows a similar VFTH-behavior as it was founded for the dynamic glass transition. The relation between cooling rate and frequency for PS is estimated to be bc = 31.6 K v. For the evaluation of the glass transition temperature dynamic and static heat effects have to be considered. Dynamic effects are due to the formation of a temperature profile during scanning. This effect becomes significant at rates above 0.1 K/s (DSC) and 1000 K/s (Flash DSC). This limit also depends on the sample size. A simple correction procedure is to average the glass transition temperatures measured during cooling and subsequent heating at bc = bh. For this procedure it is recommend to determine Tg as the thermodynamic glass transition temperature according to Richardson and Savill, and Moynihan et al. [35,36]. High cooling rates in the Flash DSC require large temperature differences between the sample and the surrounding gas. This is the reason for potential static temperature gradients in the sample. These gradients can be suppressed by using thin samples with a typical thickness in the order of 10 mm or lower. For thicker samples a simple correction method is introduced. This method bases on the determination of the temperature gradient along the sample. For this purpose a reference material (such as indium) is placed on the upper side of the sample as well as on the sensor. The difference of the melting temperature of the sample characterizes the temperature gradient. Acknowledgement The author thanks Prof. Christoph Schick, University of Rostock, for providing the PS material. References [1] E. Donth, The Glass Transition, Springer-Verlag, Berlin, 2001. [2] I. Gutzow, J. Schmelzer, The Vitreous State, Springer, Berlin, 1995. [3] F.R. Schwarzl, F. Zahradnik, The time temperature position of the glass-rubber transition of amorphous polymers and the free volume, Rheol. Acta 19 (1980) 137–152. [4] J.R. Saffell, The effect of heating and cooling rate on the characteristics of the calorimetric glass transition for glassy polymers, Thermochim. Acta 36 (1980) 251–264. [5] H.-J. Bittrich, H.-J. Schad, H. Tanneberger, Untersuchung des volumentemperatur-zeit-verhaltens amorpher polymere im glasübergangsbereich, Acta Polym. 33 (1982) 736–740. [6] J.W. Schmelzer, Kinetic criteria of glass transition and the pressure dependence of the glass transition temperature, J. Chem. Phys. 136 (2012) 074512. [7] G.M. Bartenev, O , Dokl. Akad. Nauk SSSR 69 (1951) 227–230.
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Please cite this article in press as: J.E.K. Schawe, Measurement of the thermal glass transition of polystyrene in a cooling rate range of more than six decades, Thermochim. Acta (2014), http://dx.doi.org/10.1016/j.tca.2014.05.025