Ph.D. thesis (pdf) - dirac

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134 Mean squared displacement I P is of the order of magnitude 4 for fragile liquids at T g (τ = 100s) while T g α P ∼ 0.1, meaning that the last term can be neglected. Using the general relation between the conventional steepness index and the Olsen index (equation 2.2.6) it subsequently follows that the model predicts a proportionality between fragility and the relative change of 〈u 2 〉 with relative change in temperature: m P ( ) ( ) τg τg ∂ log〈u 2 〉 = log 10 (1 + I P ) = log τ 10 0 τ 0 ∂ log T ∣ (7.5.5) P = 16 ∂ log〈u2 〉 ∂ log T ∣ . (7.5.6) P The last equality follows if all values are evaluated at T g defined by τ g = 100 s and if it is assumed that τ 0 = 10 −14 s. Hence the elastic model predicts a correspondence between the slope seen in figure 7.13 and the fragility found from the temperature dependence of the alpha relaxation time. Figure 7.14 tests this relation, using fragilities and T g ’s taken from literature (see appendix for values and references). We also include some ∂ log〈u2 〉 ∂ log T ∣ (T = T g ) P calculated on the basis of mean square displacements reported in literature. The value of ∂ log〈u2 〉 ∂ log T ∣ (T = T g ) is in all cases calculated in a narrow range temperature P range from T g to ∼ 1.1T g , because this corresponds to the range where fragility is determined. It has to be stressed when considering this figure that the elastic model not only predicts a proportionality between m P and ∂ log〈u2 〉 ∂ log T ∣ , the elastic model P predicts the proportionality constant as well, hence the line is not a fit nor a guide to the eye. The line appearing in the figure is a parameter free prediction of the elastic model. It is therefore quite striking and not at all trivial, that the order of magnitude is correct. Secondly it also appears that the variations in m P follow the variations in ∂ log〈u2 〉 ∂ log T ∣ except for the very fragile liquids. P A further test of the predicted correlation, could in principle be to consider the pressure dependence of fragility and ∂ log〈u2 〉 ∂ log T ∣ . This correlation is of the type discussed P in section 3.4, that is a correlation between fragility and the temperature dependence of another quantity. This means that both quantities are path dependent, and the ∣ (T = T g ) is hence expected to follow the pressure dependence of P isobaric ∂ log〈u2 〉 ∂ log T the isobaric fragility. The isobaric fragility of DBP is not pressure dependent in the relevant range. It is therefore consistent that we find that the whole 〈u 2 〉 temperature dependence collapses after scaling with ρ (−2/3) g and T g (P) (figure 7.10). The isobaric fragility of cumene increases with pressure. This means that the correlation implies that the slope of 〈u 2 〉 should be lower above T g . Figure 7.11 does not support this prediction, it could even be argued that it contradicts it. However, the
7.6. Relaxational contributions 135 dlog / dlog T 10 8 6 4 prediction glycerol tpp m−toluidiene otp pib B 2 O 3 DHIQ cumene DBP 2 0 0 50 100 150 m P Figure 7.14: ∂ log〈u 2 〉 ∂ log T ∣ as a function of isobaric fragility. The line shows the predic- P tion of the elastic model (equation 7.5.6). . scatter in the data points is too large for us to draw any conclusions on this basis. In glycerol it has been observed that the fragility increases with pressure [Cook et al., 1994; Paluch et al., 2002]. This should mean a larger temperature dependence of 〈u 2 〉 above T g . This is not seen in figure 7.12. But the expected change is again very small compared to the precision of the data. / (T g ) 4 3 2 1 DHIQ Glycerol 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 T/T g Figure 7.15: 〈u 2 〉 scaled to 〈u 2 〉 Tg for the most fragile sample, DHIQ, and the least fragile sample, glycerol. The temperature is scaled to T g . 7.6 Relaxational contributions The elastic model is based on the idea that the barrier height is related to curvature of the harmonic potential, and the decreasing barrier height above T g is rationalized
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THÈSE présentée par Kristine NIS
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Acknowledgement First of all I woul
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Contents Abstract iii Acknowledgeme
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CONTENTS ix 9 Summarizing discussio
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2 Introduction fragility with other
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Résumé du chapitre 2 De manière
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8 Slow and fast dynamics glass. The
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10 Slow and fast dynamics energy in
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12 Slow and fast dynamics increasin
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14 Slow and fast dynamics (linear)
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16 Slow and fast dynamics The dynam
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18 Slow and fast dynamics T > T 2
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20 Slow and fast dynamics as theore
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22 Slow and fast dynamics and sugge
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24 Slow and fast dynamics The boson
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26 Slow and fast dynamics modulus 3
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Chapter 3 What we learn from pressu
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3.2. Empirical scaling law and some
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3.3. Correlations with fragility 37
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3.4. Temperature dependences 39 mea
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3.5. Summary 41 assumption that the
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Chapter 4 Experimental techniques a
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4.2. Dielectric spectroscopy 47 4.1
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4.2. Dielectric spectroscopy 49 fie
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4.3. Inelastic Scattering Experimen
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Résumé du chapitre 5 La relaxatio
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72 Alpha Relaxation the dielectric
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74 Alpha Relaxation Measuring the c
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76 Alpha Relaxation 4 2 0 This work
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78 Alpha Relaxation for the 1 s iso
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80 Alpha Relaxation log10(τ α ) 2
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82 Alpha Relaxation 2 1 0 216.4K th
Page 94 and 95: 84 Alpha Relaxation function has a
Page 96 and 97: 86 Alpha Relaxation 2.5 2.5 2 2 log
Page 98 and 99: 88 Alpha Relaxation 0.4 log 10 Im(
Page 100 and 101: 90 Alpha Relaxation keeping the rel
Page 102 and 103: 92 Alpha Relaxation correlation bet
Page 105 and 106: Chapter 6 High Q collective modes I
Page 107 and 108: 6.1. Inelastic X-ray scattering 97
Page 109 and 110: 6.2. Sound speed and attenuation 99
Page 115 and 116: 6.3. Nonergodicity factor 105 where
Page 117 and 118: 6.3. Nonergodicity factor 107 1 0.9
Page 119 and 120: 6.3. Nonergodicity factor 109 high
Page 121 and 122: 6.4. Nonergodicity factor and fragi
Page 127 and 128: 6.5. Summary 117 d log (e(ρ))/ d l
Page 129: Résumé du chapitre 7 Dans ce chap
Page 132 and 133: 122 Mean squared displacement colle
Page 134 and 135: 124 Mean squared displacement 1.5 I
Page 136 and 137: 126 Mean squared displacement 1.4 1
Page 138 and 139: 128 Mean squared displacement a cen
Page 140 and 141: 130 Mean squared displacement cumen
Page 142 and 143: 132 Mean squared displacement / a
Page 146 and 147: 136 Mean squared displacement as be
Page 149: Résumé du chapitre 8 Le pic de bo
Page 152 and 153: 142 Boson Peak The FEC-model sugges
Page 154 and 155: 144 Boson Peak the Qout Q in factor
Page 156 and 157: 146 Boson Peak We are mainly intere
Page 158 and 159: 148 Boson Peak 1.5 x 10−3 ω [meV
Page 160 and 161: 150 Boson Peak comparison of the ev
Page 162 and 163: 152 Boson Peak In figure 8.6 we sho
Page 164 and 165: 154 Boson Peak Predictions/explanat
Page 166 and 167: 156 Boson Peak meaning that the Deb
Page 168 and 169: 158 Boson Peak g(ω)/ω 2 [arb. uni
Page 170 and 171: 160 Boson Peak are equivalent becau
Page 172 and 173: 162 Boson Peak m P /m ρ 2 1.8 1.6
Page 174 and 175: 164 Boson Peak S(ω) [arb.unit] S(
Page 176 and 177: 166 Boson Peak 3 2.5 S(ω) [arb. un
Page 178 and 179: 168 Boson Peak perature effect on t
Page 181 and 182: Chapter 9 Summarizing discussion Wh
Page 183 and 184: 173 temperature in the harmonic app
Page 185: Chapter 10 Perspectives In this wor
Page 188 and 189: 178 BIBLIOGRAPHY Angell, C. A. [199
Page 190 and 191: 180 BIBLIOGRAPHY Casalini, R. and R
Page 192 and 193: 182 BIBLIOGRAPHY Farago, B., Arbe,
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184 BIBLIOGRAPHY Inamura, Y., Arai,
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186 BIBLIOGRAPHY Monaco, A., Chumak
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188 BIBLIOGRAPHY Patkowski, A., Gap
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190 BIBLIOGRAPHY Sekula, M., Pawlus
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Appendix A Details on the samples T
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A.1. Cumene 195 the value found fro
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A.2. PIB 197 The temperature depend
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A.4. DBP 199 peak differently at di
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A.6. Other samples 201 NH 2 tempera
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Appendix B Data compilations B.1 Da
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B.1. Data compilation 205 DHIQ = De
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B.1. Data compilation 207 Triphenyl
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Appendix C Dielectric setup Figure
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Abstract The degree of departure fr
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