- Email: [email protected]
On evaluation of a limit separating amorphous and crystalline states based on thermal expansion measurements
Polymer Testing 21 (2002) 135–138 www.elsevier.com/locate/polytest
Data Interpretation
On evaluation of a limit separating amorphous and crystalline states based on thermal expansion measurements B. Jurkowski
a,*
, B. Jurkowska b, K. Andrzejczak
b, c
a b
Plastic and Rubber Processing Division, Poznan University of Technology, Piotrowo 3, 61-138 Poznan, Poland Research and Development Centre for the Tire Industry (OBRPO)“Stomil,” Starolecka 18, 61-361 Poznan, Poland c Institute of Mathematics, Poznan University of Technology, Piotrowo 3A, 60-965 Poznan, Poland Received 23 February 2001; accepted 25 May 2001
Abstract A ratio of the thermal expansion coefficient in a liquid (or high-elastic) state a2 to that in a glassy state a1 could be accepted as a measure of whether the substance is in an amorphous or crystalline state. Amorphous materials are characterised by a ratio a2/a1⬍6.24 or ⬍5.12, respectively, depending on the method used for evaluation, if the thermal expansion coefficients are randomly distributed in the interval given by Ferry. Consequently, for crystalline and semicrystalline materials being more ordered than amorphous ones and, as a result, with more compact structure, during expansion up to melting the ratio a2/a1⬎6.24 or ⬎5.12. However, the lower magnitude of this limit (5.12) is better founded. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Thermal expansion; Free volume; Dilatometry; TMA; Crystalline state; Amorphous state
1. Introduction X-ray diffraction, DSC, NMR, DMTA, TMA, dilatometry and some other techniques are commonly used to evaluate mobility of different structure elements of polymeric materials [1–3]. Some of them give information also on the degree of crystallinity and the state of order of structures intermediate between crystalline and amorphous. It is also known that interactions between neighbouring segments of the polymer chains determine the free volume of the material and related thermal expansion. This suggests using thermal expansion characteristics obtained over a wide temperature interval to evaluate whether a material is in crystalline or amorphous state or could be characterised by regions
* Corresponding author. Tel.: +48 61 6652 771; fax: +48 61 6652 217. E-mail addresses: [email protected] (B. Jurkowski), [email protected] (B. Jurkowska), [email protected] (K. Andrzejczak).
differing in state of order. These regions or portions (complex structures differing in thermal expansion characteristics) in our previous articles were called topological blocks. The presence of such regions was observed in block copolymers, homopolymers differing in polarity along the chain and composites. In the last two cases the thermal expansion curves have a shape similar to that for block copolymers. This allows using the same nomenclature to describe a structure of all the polymeric materials mentioned above. In Ferry’s book [4] it is written: “For the present, the glass transition temperature Tg of any amorphous substance, whether polymeric or not, may be defined as the point where the thermal expansion coefficient undergoes a discontinuity. Above this temperature, it has the magnitude generally associated with liquids—6 to 10×10⫺4 deg⫺1. Decrease in temperature is accompanied by collapse of the free volume, which is made possible by configurational adjustments. Eventually, the free volume becomes so small that further adjustments are extremely slow or even impossible; then it no longer decreases and further contraction in total volume with decreasing tem-
0142-9418/02/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 9 4 1 8 ( 0 1 ) 0 0 0 5 9 - 9
136
B. Jurkowski et al. / Polymer Testing 21 (2002) 135–138
perature is much less, so it drops suddenly to between 1 and 3×10⫺4 deg⫺1.” This phenomenon could allow determining, based on thermal expansion measurements, whether the tested material is amorphous or not. However, to date we have not found any information on how to determine the most probably upper limit of the ratio of a2/a1⫽q
(1)
for an amorphous substance, where a1 is the coefficient of linear thermal expansion in a glassy state and a2 that in a high elastic state. For materials with ordered structure we should accept that q is larger than that for amorphous ones because now a2 has higher magnitude. In our previous articles [3,5–14] it was accepted as a preliminary that for amorphous substances q⬍6. However, there is no statistical evidence for this claim. It is very important to find evidence as to whether it is true or not because during TMA tests [3] one should know this magnitude in order to determine whether particular regions (topological blocks) of any substance could be accepted as amorphous or crystalline. Because of this, the aim of this article is to evaluate the most probable limit of such a ratio separating crystalline and amorphous materials or such regions in a complex structure.
2. First method of evaluation As we mentioned above, according to Ferry [4], for amorphous substances a2 has the magnitude between 6 and 10×10⫺4 deg⫺1. However, there is no information about statistical distribution of experimental results within this interval. If we accept that these results have a normal statistical distribution, for 99% of them it could be written that a2=a¯ 2±3Sa1=8±2 (average a¯ 2=8 and empirical standard deviation Sa2=2/3). Below Tg the a drops suddenly to a magnitude between 1 and 3×10⫺4 deg⫺1, which fact could be written as a1=a¯ 1±3Sa1=2±1 (average a¯ 1=2 and empirical standard deviation Sa1=1/3). However, a question arises of how to determine a standard deviation of a ratio a2/a1? From literature surveys [15,16], it is known that the standard deviation of such a quotient Sq could be calculated from: Sq ⫽
S 2a2 a¯ 22
冪 a¯ +a¯ S 2 1
4 1
2 a1
.
(2)
This is a special case of a law of error propagation widely used by metrology engineers in indirect measurements. We accepted that both a1 and a2 values are independent variables (however, in the general case they may be statistically dependent) with normal distribution. It allows estimating their expected values as averages a1 and a2. It is commonly accepted that for results with normal statistical distribution 99% of them are in the limits a¯ ±3 standard deviations, which is known as the three-
sigma law. From this we can find their standard deviations S1 and S2. Then, using these data and Eq. (2), we can indirectly evaluate both an expected value of a ratio (1) and its standard deviation. This reasoning is correct because it is concluded from two facts given below, which are more general. Fact 1. If Z=f(X1, X2, …, Xn) is a function of independent random variables X1, …, Xn and mi measurements were performed of such variables Xi, for i=1, 2, …, n, then the arithmetic mean (average) z¯ of indirect measurements of variable Z could be written as z¯⫽f(x1, …, xn)
(3)
where xi, i=1, 2, …, n is an arithmetic mean value of measurements of variable Xi. Evidence of this could be obtained from expansion of a function of indirect measurements in a Taylor series and taking into account terms of the lowest row only. This is well founded for the fast convergence of series for very low errors in comparison with arithmetic mean values. Based on Fact 1, it is possible to obtain the law of error propagation for a single dimensional data transformation. Fact 2 called Law of error propagation. If Z=f(X1, X2, …, Xn) is a function of independent random variables X1, …, Xn and for i=1, 2, …, n mi measurements of variable Xi were performed, then the empirical standard deviation of dependent variable Z is described by:
冪冘f ⬘(x ,…,x )S n
Sz ⫽
x1
1
n
2 xi
(4)
i⫽1
where fx1⬘ is the first partial derivative of f with respect to xi. Using formula (4) for a ratio q (1) we obtain Eq. (2). Finally, the indirectly evaluated arithmetic mean and empirical standard deviation of a ratio q could then be used to determine the upper limit of q using the threesigma law. Based on the fact that dependence (1) was proved mathematically, we can use it for calculation of a standard deviation of a ratio of the thermal expansion coefficients in high-elastic and glassy states, with magnitudes in the mentioned interval as determined by Ferry, as follows:
冪冉 冊
Sq ⫽
2/3 2 82 + 4(1/3)2⫽ 2 2
冪1/9+16(1/9)⫽3冑5⫽0.7454. 64
1
From this we obtain that q=a2/a1=(a2/a1)±3Sq=4±2.236. This means that a measure in an amorphous state is its maximal magnitude, which could be expressed as a2/a1ⱕ6.24. For ordered structures, the free volume is smaller than that for amorphous ones. Because of this, for crystalline
B. Jurkowski et al. / Polymer Testing 21 (2002) 135–138
and semi-crystalline materials during their expansion up to melting the ratio a2/a1 has to be higher, namely more than 6.24.
bivariate normal distribution and E(a1)ⱖs1 (where E is the operator of the expected value and s1 is the standard deviation of a1), then to a good degree of approximation the following dependence is valid:
2.1. Conclusions Applying the law of error propagation allowed evaluating indirectly a magnitude of the quotient q by use of the three-sigma law. Theory suggests and practice confirms that the law can only be applied for essentially normal statistical distributions. Computer simulations performed by us of distribution of a quotient of variables characterised by a normal statistical distribution with parameters evaluated for a1 and a2, showed its strong asymmetry (large positive values of empirical skewness). This asymmetry is a reason for excessively high magnitude of standard deviation of a quotient q. In this case, the most reasonable approach is to evaluate the distribution of a quotient q accepting that distributions of random variables a1 and a2 are given. For more detailed description of this method of determination the distributions need to use special non-linear transformations of random variables; in this article they are omitted. It is a reasonable presumption that variables a1 and a2 have normal statistical distribution. Any assumption that they have other distributions than normal, for instance uniform distribution, has to be well founded (has to be empirically determined). Distributions other than normal could be caused by: (i) non-systematic deregulation of measuring devices, (ii) magnitudes of the thermal expansion coefficients in glassy and high-elastic states depending on each other, (iii) these coefficients depending on other disturbing factors.
3. Second method of evaluation Analytical determination of the statistical distribution of a quotient for random variables with given distributions is difficult or even impossible. However, for many cases, statistical distribution of a quotient is known. For instance, a quotient of two variables with independent standardised normal distributions has the Cauchy distribution [17]. As far as we know, for this case there does not exist the mean value of a quotient, which is needed to calculate a limit separating amorphous and crystalline states according to the previous method. However, from this fact a question could be formulated: Whether for a quotient of arbitrary normal distributions an expected value exists? Because of these reasons, we cannot calculate the accurate distribution of a quotient for different distributions of a1 and a2, but as a second route for evaluation of an upper limit of such a quotient q we use the next fact described below [18]. Fact 3. If a1 and a2 are variables, which have a joint
137
P
冉 冊
a2 ⬍k ⬇⌽(g) a1
(5)
where k is any real number, ⌽ is a cumulative standardised normal distribution and g is a real number, which could be obtained from equation: g⫽
kE(a1)−E(a2)
冑s −2rs s +s 2 1
1
2
2 2
(6)
where si is standard deviation of variable ai (i=1, 2) and r is the correlation coefficient of variables a1 and a2. This fact helps to solve the problem of evaluation of the upper limit of q by applying the following estimation: the probability k that q=a2/a1⬍k is very high (close to 1), i.e., ⌽(3)=0.99865 According to Fact 3, ⌽ is a cumulative distribution function of a standardised normal distribution. For this distribution the standard deviation s=1. Because of this, we use dependencies (5) and (6) for g=3s=3. 3.1. Solution From assumption that a1 and a2 are independent variables we have r=0. Using Fact 3 we obtain kE(a1)−E(a2)
冑s +s 2 1
2 2
⬇3.
(7)
Substituting into this equation estimators of the distribution parameters (a1, a2, Sa1, Sa2) and transforming it, we obtain k⬇
3冑S 2a1+S 2a2+a2 . a1
(8)
After introduction of the estimated magnitudes of such parameters into (8), we obtain: k⬇
3·0.7454+8 ⫽5.1181. 2
This means that the upper limit of a ratio a2/a1 equals 5.12. 3.2. Conclusions The result obtained means that P(a2/a1⬍5.12)ⱖ0.99, which is equivalent to the three-sigma law applied in the first method. The reasons for error of estimation in the second method in comparison with the first one are that the assumption that variables a1 and a2 are independent, or
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B. Jurkowski et al. / Polymer Testing 21 (2002) 135–138
that they have normal statistical distributions, could not be fulfilled. Because of this, both of these assumptions should be verified empirically in each particular case. To do so, it is necessary to have experimental data and information about conditions of the thermal expansion experiments.
4. Conclusions Accepting that the thermal expansion coefficients are randomly distributed, statistical evidence was found that for the amorphous state a2/a1ⱕ6.24. However, if we accept another route of evaluation, based on Fact 3, we obtain that a2/a1ⱕ5.12. Consequently, for crystalline and semi-crystalline materials during expansion up to their melting temperature with probability of 0.99, two different critical values for the ratio a2/a1⬎6.24 or ⬎5.12 were obtained. The source of the large difference in evaluated magnitudes is usage of different evaluation routes. In the two methods being compared we accepted in advance that variables a1 and a2 have normal statistical distribution and that they are independent. In our opinion, strong asymmetry of the statistical distribution of the quotient q is the main reason for the difference obtained. Such asymmetry substantially influences the result obtained by the first method. The second method is less dependent on or even independent of asymmetry of the distribution and, due to this, the result obtained is more reliable than that from the first route. If the combined distribution of random variables (a1, a2) is known, for this particular case it is possible to calculate the accurate distribution of the quotient q and to evaluate its characteristics. In the case of difficulties in analytical calculation of a quotient distribution, we always have the possibility of using a method of computer simulation to generate its empirical distribution and to test its properties. However, this discussion is very theoretical. Because of this, to solve this problem properly such evaluations should be compared with experimental data in each case.
References [1] A.L. Kovarski, Molecular Dynamics of Additives in Polymers, VSP, Utrecht, 1997. [2] G.M. Bartenev, A.G. Barteneva, Relaxationnyje svojstva polimerov, Khimia, Moscow, 1992. [3] B. Jurkowska, Y.A. Olkhov, B. Jurkowski, J. Appl. Polym. Sci. 74 (14) (1999) 490. [4] J.D. Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 1970. [5] Y.A. Olkhov, B. Jurkowski, J. Appl. Polym. Sci. 65 (1997) 499. [6] B. Jurkowski, Y.A. Olkhov, J. Appl. Polym. Sci. 65 (1997) 1807. [7] B. Jurkowska, Y.A. Olkhov, B. Jurkowska, J. Appl. Polym. Sci. 68 (1998) 2159. [8] B. Jurkowska, Y.A. Olkhov, B. Jurkowski, O.M. Olkhov, J. Appl. Polym. Sci. 71 (5) (1999) 729. [9] B. Jurkowski, S.S. Pesetskii, Y.A. Olkhov, Y.M. Krivoguz, K. Kelar, J. Appl. Polym. Sci. 71 (1999) 1771. [10] B. Jurkowska, Y.A. Olkhov, B. Jurkowski, O.M. Olkhov, J. Appl. Polym. Sci. 74 (14) (1999) 3305. [11] B. Jurkowska, Y.A. Olkhov, B. Jurkowski, O.M. Olkhov, J. Appl. Polym. Sci. 75 (5) (2000) 660. [12] B. Jurkowska, Y.A. Olkhov, B. Jurkowska, J. Appl. Polym. Sci. 78 (4) (2000) 732. [13] B. Jurkowski, Y.A. Olkhov, K. Kelar, O.M. Olkhova, J. Appl. Polym. Sci., in press. [14] S.S. Pesetskii, B. Jurkowski, Y.M. Krivoguz, Y.A. Olkhov, J. Appl. Polym. Sci., in press. [15] B.M. Jaworski, A.A. Dietaf, Fizyka Poradnik encyklopedyczny (Lexicon in Physics), PWN, Warsaw, 1995, p. 642, Translation from Russian edition, Nauka, Moscow, 1990. [16] S. Brandt, Statistical and Computational Methods in Data Analysis, 3rd ed., Springer, New York, 1997, Polish translation: Analiza danych, Metody statystyczne i obliczeniowe, PWN, Warsaw, 1998, p. 73. [17] B.G. Volodin, M.P. Ganin, I.J. Diner, L.B. Komarov, A.A. Swieshnikov, K.B. Starobin, Rukovodstvo dla inzhenerov po resheniu zadach teorii vierojatnostiej, Gosudarstviennoje Vsesojuznoje Izdatielstvo Sudostroitielnoj Promyshliennosti, Leningrad, 1962, Polish translation: Problemy rachunku prawdopodobien˜ stwa (Problems of Probability), PWN, Warsaw, 1966, p. 147. [18] N.L. Johnson, F.C. Leone, Statistics and Experimental Design, vol. I, Wiley, New York, 1977, Russian translation, Mir, Moscow, 1980, p. 187.
Data Interpretation
On evaluation of a limit separating amorphous and crystalline states based on thermal expansion measurements B. Jurkowski
a,*
, B. Jurkowska b, K. Andrzejczak
b, c
a b
Plastic and Rubber Processing Division, Poznan University of Technology, Piotrowo 3, 61-138 Poznan, Poland Research and Development Centre for the Tire Industry (OBRPO)“Stomil,” Starolecka 18, 61-361 Poznan, Poland c Institute of Mathematics, Poznan University of Technology, Piotrowo 3A, 60-965 Poznan, Poland Received 23 February 2001; accepted 25 May 2001
Abstract A ratio of the thermal expansion coefficient in a liquid (or high-elastic) state a2 to that in a glassy state a1 could be accepted as a measure of whether the substance is in an amorphous or crystalline state. Amorphous materials are characterised by a ratio a2/a1⬍6.24 or ⬍5.12, respectively, depending on the method used for evaluation, if the thermal expansion coefficients are randomly distributed in the interval given by Ferry. Consequently, for crystalline and semicrystalline materials being more ordered than amorphous ones and, as a result, with more compact structure, during expansion up to melting the ratio a2/a1⬎6.24 or ⬎5.12. However, the lower magnitude of this limit (5.12) is better founded. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Thermal expansion; Free volume; Dilatometry; TMA; Crystalline state; Amorphous state
1. Introduction X-ray diffraction, DSC, NMR, DMTA, TMA, dilatometry and some other techniques are commonly used to evaluate mobility of different structure elements of polymeric materials [1–3]. Some of them give information also on the degree of crystallinity and the state of order of structures intermediate between crystalline and amorphous. It is also known that interactions between neighbouring segments of the polymer chains determine the free volume of the material and related thermal expansion. This suggests using thermal expansion characteristics obtained over a wide temperature interval to evaluate whether a material is in crystalline or amorphous state or could be characterised by regions
* Corresponding author. Tel.: +48 61 6652 771; fax: +48 61 6652 217. E-mail addresses: [email protected] (B. Jurkowski), [email protected] (B. Jurkowska), [email protected] (K. Andrzejczak).
differing in state of order. These regions or portions (complex structures differing in thermal expansion characteristics) in our previous articles were called topological blocks. The presence of such regions was observed in block copolymers, homopolymers differing in polarity along the chain and composites. In the last two cases the thermal expansion curves have a shape similar to that for block copolymers. This allows using the same nomenclature to describe a structure of all the polymeric materials mentioned above. In Ferry’s book [4] it is written: “For the present, the glass transition temperature Tg of any amorphous substance, whether polymeric or not, may be defined as the point where the thermal expansion coefficient undergoes a discontinuity. Above this temperature, it has the magnitude generally associated with liquids—6 to 10×10⫺4 deg⫺1. Decrease in temperature is accompanied by collapse of the free volume, which is made possible by configurational adjustments. Eventually, the free volume becomes so small that further adjustments are extremely slow or even impossible; then it no longer decreases and further contraction in total volume with decreasing tem-
0142-9418/02/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 9 4 1 8 ( 0 1 ) 0 0 0 5 9 - 9
136
B. Jurkowski et al. / Polymer Testing 21 (2002) 135–138
perature is much less, so it drops suddenly to between 1 and 3×10⫺4 deg⫺1.” This phenomenon could allow determining, based on thermal expansion measurements, whether the tested material is amorphous or not. However, to date we have not found any information on how to determine the most probably upper limit of the ratio of a2/a1⫽q
(1)
for an amorphous substance, where a1 is the coefficient of linear thermal expansion in a glassy state and a2 that in a high elastic state. For materials with ordered structure we should accept that q is larger than that for amorphous ones because now a2 has higher magnitude. In our previous articles [3,5–14] it was accepted as a preliminary that for amorphous substances q⬍6. However, there is no statistical evidence for this claim. It is very important to find evidence as to whether it is true or not because during TMA tests [3] one should know this magnitude in order to determine whether particular regions (topological blocks) of any substance could be accepted as amorphous or crystalline. Because of this, the aim of this article is to evaluate the most probable limit of such a ratio separating crystalline and amorphous materials or such regions in a complex structure.
2. First method of evaluation As we mentioned above, according to Ferry [4], for amorphous substances a2 has the magnitude between 6 and 10×10⫺4 deg⫺1. However, there is no information about statistical distribution of experimental results within this interval. If we accept that these results have a normal statistical distribution, for 99% of them it could be written that a2=a¯ 2±3Sa1=8±2 (average a¯ 2=8 and empirical standard deviation Sa2=2/3). Below Tg the a drops suddenly to a magnitude between 1 and 3×10⫺4 deg⫺1, which fact could be written as a1=a¯ 1±3Sa1=2±1 (average a¯ 1=2 and empirical standard deviation Sa1=1/3). However, a question arises of how to determine a standard deviation of a ratio a2/a1? From literature surveys [15,16], it is known that the standard deviation of such a quotient Sq could be calculated from: Sq ⫽
S 2a2 a¯ 22
冪 a¯ +a¯ S 2 1
4 1
2 a1
.
(2)
This is a special case of a law of error propagation widely used by metrology engineers in indirect measurements. We accepted that both a1 and a2 values are independent variables (however, in the general case they may be statistically dependent) with normal distribution. It allows estimating their expected values as averages a1 and a2. It is commonly accepted that for results with normal statistical distribution 99% of them are in the limits a¯ ±3 standard deviations, which is known as the three-
sigma law. From this we can find their standard deviations S1 and S2. Then, using these data and Eq. (2), we can indirectly evaluate both an expected value of a ratio (1) and its standard deviation. This reasoning is correct because it is concluded from two facts given below, which are more general. Fact 1. If Z=f(X1, X2, …, Xn) is a function of independent random variables X1, …, Xn and mi measurements were performed of such variables Xi, for i=1, 2, …, n, then the arithmetic mean (average) z¯ of indirect measurements of variable Z could be written as z¯⫽f(x1, …, xn)
(3)
where xi, i=1, 2, …, n is an arithmetic mean value of measurements of variable Xi. Evidence of this could be obtained from expansion of a function of indirect measurements in a Taylor series and taking into account terms of the lowest row only. This is well founded for the fast convergence of series for very low errors in comparison with arithmetic mean values. Based on Fact 1, it is possible to obtain the law of error propagation for a single dimensional data transformation. Fact 2 called Law of error propagation. If Z=f(X1, X2, …, Xn) is a function of independent random variables X1, …, Xn and for i=1, 2, …, n mi measurements of variable Xi were performed, then the empirical standard deviation of dependent variable Z is described by:
冪冘f ⬘(x ,…,x )S n
Sz ⫽
x1
1
n
2 xi
(4)
i⫽1
where fx1⬘ is the first partial derivative of f with respect to xi. Using formula (4) for a ratio q (1) we obtain Eq. (2). Finally, the indirectly evaluated arithmetic mean and empirical standard deviation of a ratio q could then be used to determine the upper limit of q using the threesigma law. Based on the fact that dependence (1) was proved mathematically, we can use it for calculation of a standard deviation of a ratio of the thermal expansion coefficients in high-elastic and glassy states, with magnitudes in the mentioned interval as determined by Ferry, as follows:
冪冉 冊
Sq ⫽
2/3 2 82 + 4(1/3)2⫽ 2 2
冪1/9+16(1/9)⫽3冑5⫽0.7454. 64
1
From this we obtain that q=a2/a1=(a2/a1)±3Sq=4±2.236. This means that a measure in an amorphous state is its maximal magnitude, which could be expressed as a2/a1ⱕ6.24. For ordered structures, the free volume is smaller than that for amorphous ones. Because of this, for crystalline
B. Jurkowski et al. / Polymer Testing 21 (2002) 135–138
and semi-crystalline materials during their expansion up to melting the ratio a2/a1 has to be higher, namely more than 6.24.
bivariate normal distribution and E(a1)ⱖs1 (where E is the operator of the expected value and s1 is the standard deviation of a1), then to a good degree of approximation the following dependence is valid:
2.1. Conclusions Applying the law of error propagation allowed evaluating indirectly a magnitude of the quotient q by use of the three-sigma law. Theory suggests and practice confirms that the law can only be applied for essentially normal statistical distributions. Computer simulations performed by us of distribution of a quotient of variables characterised by a normal statistical distribution with parameters evaluated for a1 and a2, showed its strong asymmetry (large positive values of empirical skewness). This asymmetry is a reason for excessively high magnitude of standard deviation of a quotient q. In this case, the most reasonable approach is to evaluate the distribution of a quotient q accepting that distributions of random variables a1 and a2 are given. For more detailed description of this method of determination the distributions need to use special non-linear transformations of random variables; in this article they are omitted. It is a reasonable presumption that variables a1 and a2 have normal statistical distribution. Any assumption that they have other distributions than normal, for instance uniform distribution, has to be well founded (has to be empirically determined). Distributions other than normal could be caused by: (i) non-systematic deregulation of measuring devices, (ii) magnitudes of the thermal expansion coefficients in glassy and high-elastic states depending on each other, (iii) these coefficients depending on other disturbing factors.
3. Second method of evaluation Analytical determination of the statistical distribution of a quotient for random variables with given distributions is difficult or even impossible. However, for many cases, statistical distribution of a quotient is known. For instance, a quotient of two variables with independent standardised normal distributions has the Cauchy distribution [17]. As far as we know, for this case there does not exist the mean value of a quotient, which is needed to calculate a limit separating amorphous and crystalline states according to the previous method. However, from this fact a question could be formulated: Whether for a quotient of arbitrary normal distributions an expected value exists? Because of these reasons, we cannot calculate the accurate distribution of a quotient for different distributions of a1 and a2, but as a second route for evaluation of an upper limit of such a quotient q we use the next fact described below [18]. Fact 3. If a1 and a2 are variables, which have a joint
137
P
冉 冊
a2 ⬍k ⬇⌽(g) a1
(5)
where k is any real number, ⌽ is a cumulative standardised normal distribution and g is a real number, which could be obtained from equation: g⫽
kE(a1)−E(a2)
冑s −2rs s +s 2 1
1
2
2 2
(6)
where si is standard deviation of variable ai (i=1, 2) and r is the correlation coefficient of variables a1 and a2. This fact helps to solve the problem of evaluation of the upper limit of q by applying the following estimation: the probability k that q=a2/a1⬍k is very high (close to 1), i.e., ⌽(3)=0.99865 According to Fact 3, ⌽ is a cumulative distribution function of a standardised normal distribution. For this distribution the standard deviation s=1. Because of this, we use dependencies (5) and (6) for g=3s=3. 3.1. Solution From assumption that a1 and a2 are independent variables we have r=0. Using Fact 3 we obtain kE(a1)−E(a2)
冑s +s 2 1
2 2
⬇3.
(7)
Substituting into this equation estimators of the distribution parameters (a1, a2, Sa1, Sa2) and transforming it, we obtain k⬇
3冑S 2a1+S 2a2+a2 . a1
(8)
After introduction of the estimated magnitudes of such parameters into (8), we obtain: k⬇
3·0.7454+8 ⫽5.1181. 2
This means that the upper limit of a ratio a2/a1 equals 5.12. 3.2. Conclusions The result obtained means that P(a2/a1⬍5.12)ⱖ0.99, which is equivalent to the three-sigma law applied in the first method. The reasons for error of estimation in the second method in comparison with the first one are that the assumption that variables a1 and a2 are independent, or
138
B. Jurkowski et al. / Polymer Testing 21 (2002) 135–138
that they have normal statistical distributions, could not be fulfilled. Because of this, both of these assumptions should be verified empirically in each particular case. To do so, it is necessary to have experimental data and information about conditions of the thermal expansion experiments.
4. Conclusions Accepting that the thermal expansion coefficients are randomly distributed, statistical evidence was found that for the amorphous state a2/a1ⱕ6.24. However, if we accept another route of evaluation, based on Fact 3, we obtain that a2/a1ⱕ5.12. Consequently, for crystalline and semi-crystalline materials during expansion up to their melting temperature with probability of 0.99, two different critical values for the ratio a2/a1⬎6.24 or ⬎5.12 were obtained. The source of the large difference in evaluated magnitudes is usage of different evaluation routes. In the two methods being compared we accepted in advance that variables a1 and a2 have normal statistical distribution and that they are independent. In our opinion, strong asymmetry of the statistical distribution of the quotient q is the main reason for the difference obtained. Such asymmetry substantially influences the result obtained by the first method. The second method is less dependent on or even independent of asymmetry of the distribution and, due to this, the result obtained is more reliable than that from the first route. If the combined distribution of random variables (a1, a2) is known, for this particular case it is possible to calculate the accurate distribution of the quotient q and to evaluate its characteristics. In the case of difficulties in analytical calculation of a quotient distribution, we always have the possibility of using a method of computer simulation to generate its empirical distribution and to test its properties. However, this discussion is very theoretical. Because of this, to solve this problem properly such evaluations should be compared with experimental data in each case.
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