Share This Article:

Dynamic and Configurational Approach to the Glass Transition by Nanoscale Cooperativity

Abstract Full-Text HTML Download Download as PDF (Size:269KB) PP. 88-100
DOI: 10.4236/ojbiphy.2012.23012    6,642 Downloads   10,817 Views   Citations
Author(s)    Leave a comment

ABSTRACT

Here we examine the findings obtained for disaccharide/water mixtures near glass transition that involves cooperative relaxation features on kinetic by viscosity and on thermodynamic behaviour by neutron scattering. Then to address cooperative phenomena that mitigate the Debye-Waller behaviour we invoke Adam-Gibbs’ idea of a cooperative rearranging region. Neutron results suggest that the excess mean square displacement behaves as free volume and is closely connected to an elementary step of the structural relaxation. Then viscosity data evidence a breakdown of the Einstein-Debye relation, decoupling attributed to the intermolecular cooperativity.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

G. Romeo, "Dynamic and Configurational Approach to the Glass Transition by Nanoscale Cooperativity," Open Journal of Biophysics, Vol. 2 No. 3, 2012, pp. 88-100. doi: 10.4236/ojbiphy.2012.23012.

References

[1] F. J. Chavarri, M. De Paz and M. Nueez, “Cryoprotective Agents for Frozen Concentrated Starters from Non-Bitter Streptococcus Lactis Strains,” Biotechnology Letters, Vol. 10, No. 1, 1988, pp. 11-16. doi:10.1007/BF01030016
[2] J. K. Kaushik and R. Bhat, “Why Is Trehalose an Exceptional Protein Stabilizer? An Analysis of the Thermal Stability of Proteins in the Presence of the Compatible Osmolyte Trehalose,” Journal of Biological Chemistry, Vol. 278, No. 29, 2003, pp. 26458-26465.doi:10.1074/jbc.M300815200
[3] A. Morana, P. Stiuso, G. Colonna, M. Lamberti, M. Carten and M. De Rosa, “Stabilization of S-Adenosyl-L-Methionine Promoted by Trehalose,” Biochimica et Bio-physica Acta, Vol. 1573, No. 2, 2002, pp. 105-108.
[4] L. S. Limaye and V. P. Kale, “Cryopreservation of Human Hematopoietic Cells with Membrane Stabilizers and Bioantioxidants as Additives in the Conventional Freezing Medium,” Journal of Hematotherapy and Stem Cell Research, Vol. 10, No. 5, 2001, pp. 709-718.doi:10.1089/152581601753193931
[5] A. B. Richards, S. Krakowka, L. B. Dexter, H. Schmid, A. P. M. Wolterbeek, D. H. Waalkens Berendsen, et al., “Trehalose: A Review of Properties, History of Use and Human Tolerance, and Results of Multiple Safety Studies,” Food and Chemical Toxicology, Vol. 40, No. 7, 2002, pp. 871-898. doi:10.1016/S0278-6915(02)00011-X
[6] A. D. Elbein, Y. T. Pan, I. Pastuszak and D. Carroll, “New Insights on Trehalose: A Multifunctional Molecule,” Glycobiology, Vol. 13, No. 4, 2003, pp. 17R-27R.doi:10.1093/glycob/cwg047
[7] P. H. Yancey, M. E. Clark, S. Hand, R. Bowlus and G. N. Somero, “Living with Water Stress: Evolution of Osmolyte Systems,” Science, Vol. 217, No. 4566, 1982, pp. 1214-1222. doi:10.1126/science.7112124
[8] J. F. Carpenter and J. H. Crowe, “Modes of Stabilization of a Protein by Organic Solutes during Desiccation,” Cryobiology, Vol. 25, No. 5, 1988, pp. 244-255.doi:10.1016/0011-2240(88)90032-6
[9] J. H. Crowe, L. M. Crowe and S. A. Jackson, “Preservation of Structural and Functional Activity in Lyophilized Sarcoplasmic Reticulum,” Archives of Biochemistry and Biophysics, Vol. 220, No. 2, 1983, pp. 477-484.doi:10.1016/0003-9861(83)90438-1
[10] L. M. Crowe, D. S. Reid and J. H. Crowe, “Is Trehalose Special for Preserving Dry Biomaterials?” Biophysical Journal, Vol. 71, No. 4, 1996, pp. 2087-2093.doi:10.1016/S0006-3495(96)79407-9
[11] S. Magazù, G. Maisano, H. D. Middendorf, P. Migliardo and V. Villari, “Hydration and Transport Properties of Aqueous Solutions of α-α-Trehalose,” Journal of Chemical Physics, Vol. 109, No. 3, 1998, pp. 1170-1174.doi:10.1063/1.476662
[12] C. Branca, S. Magazù, G. Malsano, F. Migliardo, P. Migliardo, G. Romeo and E. Vorobieva, “Hydration Properties of Disaccharide Aqueous Solutions,” Molecular Crystals and Liquid Crystals, Vol. 372, No. 1, 2002, pp. 25-35.doi:10.1080/10587250127596
[13] C. Branca, S. Magazu and F. Migliardo, “New Perspectives on Bioprotectant Complex Molecules: Spectroscopic Findings,” Recent Research Development in Physical Chemistry, Vol. 6, 2002, pp. 35-73.
[14] C. Branca, S. Magazu, G. Maisano, F. Migliardo and G. Romeo, “Vibrational Versus Relaxational Contribution for Disaccharide-Water Glass Formers: Neutron Scattering Evidence,” Philosophical Magazine Part B, Vol. 82, No. 3, 2002, pp. 347-355.doi:10.1080/13642810208221314
[15] C. Branca, S. Magazu, G. Maisano, F. Migliardo and G. Romeo, “α,α-Trehalose/Water Solutions. 5. Hydration and Viscosity in Dilute and Semidilute Disaccharide Solutions,” Journal of Physical Chemistry B, Vol. 105, No. 41, 2001, pp. 10140-10145. doi:10.1021/jp010179f
[16] C. Branca, S. Magazu, G. Maisano and F. Migliardo, “Vibrational and Relaxational Contributions in Disaccharide/ H2O Glass Formers,” Physical Review B, Vol. 64, No. 22, 2001, Article ID: 2242041. doi:10.1103/PhysRevB.64.224204
[17] S. Magazu, P. Migliardo, A. M. Musolino and M. T. Sciortino, “α,α-Trehalose-Water Solutions. 1. Hydration Phenomena and Anomalies in the Acoustic Properties,” Journal of Physical Chemistry B, Vol. 101, No. 13, 1997, pp. 2348-2351. doi:10.1021/jp961139s
[18] S. Magazu, V. Villari, P. Migliardo, G. Maisano and M. T. F. Telling, “Diffusive Dynamics of Water in the Presence of Homologous Disaccharides: A Comparative Study by Quasi Elastic Neutron Scattering. IV,” Journal of Physical Chemistry B, Vol. 105, No. 9, 2001, pp. 1851-1855.doi:10.1021/jp002155z
[19] C. Branca, A. Faraone, S. Magazu, G. Maisano, F. Migliardo, P. Migliardo and V. Villari, “Structural and Dynamical Properties of Trehalose-Water Solutions: Anomalous Behaviour and Molecular Models,” Recent Research Developments in Physical Chemistry, Vol. 3, 1999, pp. 361-403.
[20] C. Branca, S. Magazu, G. Malsano and P. Migliardo, “Anomalous Cryoprotective Effectiveness of Trehalose: Raman Scattering Evidences,” Journal of Chemical Physics, Vol. 111, No. 1, 1999, pp. 281-287.doi:10.1063/1.479288
[21] M. C. Donnamaria, E. I. Howard and J. R. Grigera, “Interaction of Water with α,α-Trehalose in Solution: Molecular Dynamics Simulation Approach,” Journal of Chemical Society, Faraday Transactions, Vol. 90, No. 18, 1994, pp. 2731-2735. doi:10.1039/ft9949002731
[22] C. A. Angell, P. H. Poole and J. Shao, “Glass-Forming Liquids, Anomalous Liquids, and Polyamorphism in Liquids and Biopolymers,” Il Nuovo Cimento D, Vol. 16, No. 8, 1999, pp. 993-1025. doi:10.1007/BF02458784
[23] C. A. Angell, “Glassforming Liquids with Microscopic to Macroscopic Two-State Complexity,” Progress of Theoretical Physics, Suppl. 126, pp. 1-8.doi:10.1143/PTPS.126.1
[24] M. Bée, “Quasielastic Neutron Scattering: Principles and Applications in Solid State Chemistry, Biology and Material Science,” Adam Hilger, Bristol, 1988.
[25] G. Adam and J. H. Gibbs, “On the Temperature Dependence of Cooperative Relaxation Properties in GlassForming Liquids,” Journal of Chemical Physics, Vol. 43, No. 1, 1965, pp. 139-146. doi:10.1063/1.1696442
[26] E. Donth, “Can Dynamic Neutron Scattering Help to Understand a Thermodynamic Variant of an Internal Quantum-Mechanical Experiment in the Angstrom Range?” European Physical Journal E, Vol. 12, No. 1, 2003, pp. 11-18. doi:10.1140/epje/i2003-10051-5
[27] H. Solunov, “Significance of the Intramolecular Degrees of Freedom on the Glass-Forming Process,” Journal of Optoelectronics and Advanced Materials, Vol. 7, No. 1, 2005, p. 365.
[28] E. Donth, “The Glass Transition. Relaxation Dynamics in Liquids and Disordered Materials,” Springer, Berlin, 2001.
[29] H. Sillescu, “Heterogeneity at the Glass Transition: A Review,” Journal of Non-Crystalline Solids, Vol. 243, No. 2-3, 1999, pp. 81-108. doi:10.1016/S0022-3093(98)00831-X
[30] T. Kanaya, T. Kawaguchi and K. Kaji, “Local Dynamics of Cis-1,4-Polybutadiene near the Glass Transition Temperature Tg,” Physica B: Condensed Matter, Vol. 182, No. 4, 1992, pp. 403-408. doi:10.1016/0921-4526(92)90043-R
[31] T. Kanaya, T. Kawaguchi and K. Kaji, “Low-Energy Excitation and Fast Motion near Tg in Amorphous Cis-1, 4-Polybutadiene,” Journal of Chemical Physics, Vol. 98, 1993, pp. 8262-8270. doi:10.1063/1.464531
[32] J. Ferry, “Viscoelastic Properties of Polymers,” Wiley, New York, 1980.
[33] S. Magazù, G. Romeo and M. T. F. Telling, “Temperature Dependence of Protein Dynamics as Affected by Sugars: A Neutron Scattering Study,” European Biophysics Journal, Vol. 36, No. 7, 2007, pp. 685-691.doi:10.1007/s00249-007-0190-y
[34] M. H. Cohen and D. Turnbull, “Molecular Transport in Liquids and Glasses,” Journal of Chemical Physics, Vol. 31, No. 5, 1959, pp. 1164-1169. doi:10.1063/1.1730566
[35] U. Buchenau and R. Zorn, “A Relation between Fast and Slow Motions in Glassy and Liquid Selenium,” Europhysics Letters, Vol. 18, No. 6, 1992, p. 523.doi:10.1209/0295-5075/18/6/009
[36] A. P. Sokolov, et al., “Dynamics of Strong and Fragile glass Formers: Differences and Correlation with LowTemperature Properties,” Physics Review Letters, Vol. 71, No. 13, 1993, pp. 2062-2065.doi:10.1103/PhysRevLett.71.2062
[37] T. Kanaya, et al., “Microscopic Basis of Free-Volume Concept as Studied by Quasielastic Neutron Scattering and Positron Annihilation Lifetime Spectroscopy,” Physical Review E, Vol. 60, No. 2, 1999, pp. 1906-1912.doi:10.1103/PhysRevE.60.1906
[38] S. Magazu, et al., “Mean-Square Displacement Relationship in Bioprotectant Systems by Elastic Neutron Scattering,” Biophysical Journal, Vol. 86, No. 5, 2004, pp. 3241-3249. doi:10.1016/S0006-3495(04)74372-6
[39] J. Hansen, T. Kanaya, K. Nishida, K. Kaji, K. Tanaka and A. Yamaguchi, “Role of Vibrational Softening in Fast Dynamics of an Amorphous Polyimide Far Below Tg,” Journal of Chemical Physics, Vol. 108, No. 15, 1998, pp. 6492-6497. doi:10.1063/1.476055
[40] G. P. Johari and E. Whalley, “Dielectric Properties of Glycerol in the Range 0.1 105 Hz, 218 357 K, 0 53 kb,” Faraday Symposia Chemical Society, Vol. 6, 1972, pp. 23-41.
[41] G. P. Johary, “Glass Transition and Secondary Relaxations in Molecular Liquids and Crystals,” Annals of the New York Academy of Sciences, Vol. 279, 1976, pp. 117140. doi:10.1111/j.1749-6632.1976.tb39701.x
[42] S. Kahle, et al., “Confirmation of a Calorimetric Peculiarity in the Crossover Region of Glass Transition in Poly(n-hexyl Methacrylate) by Differential Scanning Calorimetry,” Journal of Molecular Structure, Vol. 479, No. 2-3, 1999, pp. 149-162.doi:10.1016/S0022-2860(98)00866-7
[43] S. Magazu, C. Mondelli and G. Romeo, “Landscape Excitation Profiles and Excess Thermodynamic Properties of Disaccharide Aqueous Solutions,” Journal of Biological Physics, Vol. 32, No. 2, 2006, pp. 145-151.doi:10.1007/s10867-006-9009-9
[44] W. Doster, S. Cusack and W. Petry, “Dynamical Transition of Myoglobin Revealed by Inelastic Neutron Scattering,” Nature, Vol. 337, 1989, pp. 754-756.doi:10.1038/337754a0
[45] W. Kauzmann, “The Nature of the Glassy State and the Behavior of Liquids at Low Temperatures,” Chemical Reviews, Vol. 43, No. 2, 1948, pp. 219-256.doi:10.1021/cr60135a002
[46] U. Mohanty, et al., “Supercooled Liquids,” Advances in Chemical Physics, Vol. 89, 1995, p. 89.doi:10.1002/9780470141489.ch2
[47] U. Mohanty, “On the Nature of Supercooled and Glassy States of Matter,” Physica A: Statistical Mechanics and Its Applications, Vol. 177, No. 1-3, 1991, pp. 345-355.doi:10.1016/0378-4371(91)90172-9
[48] M. Goldstein, “Viscous Liquids and the Glass Transition: A Potential Energy Barrier Picture,” Journal of Chemical Physics, Vol. 51, No. 9, 1969, pp. 3728-3739.doi:10.1063/1.1672587
[49] C. Leon and K. L. Ngai, “Rapidity of the Change of the Kohlrausch Exponent of the α-Relaxation of Glass-Forming Liquids at TB or Tβ and Consequences,” Journal of Physical Chemistry B, Vol. 103, No. 20, 1999, pp. 40454051. doi:10.1021/jp983756h
[50] C.-Y. Wang and M. D. Ediger, “Anomalous Translational Diffusion: A New Constraint for Models of Molecular Motion,” Journal of Physical Chemistry B, Vol. 104, 2000, p. 1724.
[51] M. Beiner, H. Huth and K. Schroter, “Crossover Region of Dynamic Glass Transition: General Trends and Individual Aspects,” Journal of Non-Crystalline Solids, Vol. 279, No. 2-3, 2001, pp. 126-135.doi:10.1016/S0022-3093(00)00409-9
[52] S. Kahle, J. Korus, E. Hempel, et al., “Glass-Transition Cooperativity Onset in a Series of Random Copolymers Poly(n-butyl methacrylate-stat-styrene),” Macromolecules, Vol. 30, No. 23, 1997, pp. 7214-7223.doi:10.1021/ma961933b
[53] C. A. Solunov, “Cooperative Molecular Dynamics and Strong/Fragile Behavior of Polymers,” European Polymer Journal, Vol. 35, No. 8, 1999, pp. 1543-1556.doi:10.1016/S0014-3057(98)00226-2
[54] L. Abate, I. Blanco, C. Branca, S. Magazù, G. Malsano, F. Migliardo and G. Romeo, “Homologous Disaccharide Properties at Low Temperatures,” Journal of Molecular Liquids, Vol. 103-104, 2003, pp. 177-180.doi:10.1016/S0167-7322(02)00137-X
[55] C. A. Solunov, “Cooperative Molecular Dynamics and Strong/Fragile Behavior of Polymers,” European Polymer Journal, Vol. 35, No. 8, 1999, pp. 1543-1556.doi:10.1016/S0014-3057(98)00226-2
[56] C. A. Solunov, “The Apparent Activation Energy and Relaxation Volume from the Point of View of Adam-Gibbs Theory,” Journal of Physics: Condensed Matter, Vol. 14, No. 31, 2000, p. 7297. doi:10.1088/0953-8984/14/31/302
[57] C. A. Solunov, In: S. J. Rzoska, V. P. Zhelezny, Eds., Nonlinear Dielectric Phenomena in Complex Liquids, Kluwer Academic Publications, Netherlands, 2004, p. 275.
[58] C. A. Angell, In: J. C. Dore and J. Teixeira Kluwer, Eds., Hydrogen-Bonded Liquids, Academic Publishers, Dordrecht, 1991, pp. 59-79.
[59] C. A. Angell, “Two-State Thermodynamics and Transport Properties for Water from ‘Bond Lattice’ Model,” Journal of Physical Chemistry, Vol. 75, No. 24, 1971, pp. 3698-3705. doi:10.1021/j100693a010
[60] C. A. Angell, In: K. L. Ngai and G. B.Write, Eds., Relaxation in Complex Systems, National Technology Information Service, Washingtong DC, 1984, pp. 3-11.
[61] V. Velikov, S. Borick and C. A. Angell, “The Glass Transition of Water, Based on Hyperquenching Experiments,” Science, Vol. 294, No. 5550, 2001, pp. 2335-2338.doi:10.1126/science.1061757
[62] E. Donth, “The size of Cooperatively Rearranging Regions at the Glass Transition,” Journal of Non-Crystalline Solids, Vol. 53, No. 3, 1982, pp. 325-330.doi:10.1016/0022-3093(82)90089-8
[63] F. Fujara, B. Geil, H. Sillescu and G. Fleisher, “Translational and Rotational Diffusion in Supercooled Orthoterphenyl Close to the Glass Transition,” Zeitschrift für Physik B Condensed Matter, Vol. 8, No. 2, 1992, pp. 195-204.
[64] M. T. Cicerone, F. R. Blackburn and M. D. Ediger, “How Do Molecules Move near Tg? Molecular Rotation of Six Probes in o-Terphenyl across 14 Decades in Time,” Journal of Chemical Physics, Vol. 102, No. 1, 1995, pp. 471479. doi:10.1063/1.469425
[65] E. Donth, “Phenomenological Treatment of Dynamic Glass Transition Heterogeneity,” Acta Polymerica, Vol. 50, No. 7, 1999, pp. 240-251.doi:10.1002/(SICI)1521-4044(19990701)50:7<240::AID-APOL240>3.0.CO;2-H
[66] M. H. Cohen and D. Turnbull, “Molecular Transport in Liquids and Glasses,” Journal of Chemical Physics, Vol. 31, No. 5, 1959, pp. 1164-1169. doi:10.1063/1.1730566
[67] E. Donth, “The Size of Cooperatively Rearranging Regions at the Glass Transition,” Journal of Non-Crystalline Solids, Vol. 53, No. 3, 1982, pp. 325-330.doi:10.1016/0022-3093(82)90089-8
[68] F. H. Stillinger and J. A. Hodgdon, “Translation-Rotation Paradox for Diffusion in Fragile Glass-Forming Liquids,” Physical Review E, Vol. 50, No. 3, 1994, pp. 2064-2068doi:10.1103/PhysRevE.50.2064
[69] A. Einstein, “über Die von der Molekularkinetischen Theorie der W?rme Geforderte Bewegung von in Ruhenden Flüssigkeiten Suspendierten Teilchen,” Annalen der Physik, Vol. 322, No. 8, 1905, pp. 549-560.doi:10.1002/andp.19053220806
[70] P. Debye, “Polar Molecules,” Dover, New York, 1929.
[71] C. F. Behrens, T. G. Christiansen, T. Christensen, J. C. Dyre and N. B. Olsen, “Comment on ‘Dynamic Viscosity of a Simple Glass-Forming Liquid’,” Physical Review Letters, Vol. 76, No. 9, 1996, p.1553.doi:10.1103/PhysRevLett.76.1553
[72] J. Dufour, L. Jorat, A. Bondeau, A. Sibilini and G. Noyel, “Shear Viscosity and Dielectric Relaxanon Time of Dibutyl Phthalate Down to Glass Transition Temperature,” Journal of Molecular Liquids, Vol. 62, No. 1-3, 1994, pp. 75-82. doi:10.1016/0167-7322(94)00764-0
[73] E. Cornicchi, G. Onori and A. Paciaroni, “PicosecondTime-Scale Fluctuations of Proteins in Glassy Matrices: The Role of Viscosity,” Physical Review Letters, Vol. 95, No. 15, 2005, Article ID: 158104.doi:10.1103/PhysRevLett.95.158104
[74] I. Chang and H. Sillescu, “Heterogeneity at the Glass Transition: Translational and Rotational Self-Diffusion,” Journal of Physical Chemistry B, Vol. 101, No. 43, 1997, pp. 8794-8801. doi:10.1021/jp9640989
[75] E. W. Fisher, E. Donth and W. Steffen, “Temperature Dependence of Characteristic Length for Glass Transition,” Physical Review Letters, Vol. 68, No. 15, 1992, pp. 23442346. doi:10.1103/PhysRevLett.68.2344
[76] E. R?ssler, “Indications for a Change of Diffusion Mechanism in Supercooled Liquids,” Physical Review Letters, Vol. 65, No. 13, 1990, pp. 1595-1598.doi:10.1103/PhysRevLett.65.1595
[77] R. Bohmer, “Nanoscale Heterogeneity if Glass-Forming Liquids: Experimental Advances,” Current Opinion in Solid State and Materials Science, Vol. 3, No. 4, 1998, pp. 378-385. doi:10.1016/S1359-0286(98)80048-X
[78] R. Bohmer, et. al., “Nature of the Non-Exponential Primary Relaxation in Structural Glass-Formers Probed by Dynamically Selective Experiments,” Journal of NonCrystalline Solids, Vol. 235-237, 1998, pp. 1-9doi:10.1016/S0022-3093(98)00581-X
[79] H. Sillescu, “Heterogeneity at the Glass Transition: A Review,” Journal of Non-Crystalline Solids, Vol. 243, No. 2-3, 1999, pp. 81-108.doi:10.1016/S0022-3093(98)00831-X
[80] S. C. Glotzer, “Spatially Heterogeneous Dynamics in Liquids: Insights from Simulation,” Journal of Non-Crystalline Solids, Vol. 274, No. 1-3, 2000, pp. 342-355.doi:10.1016/S0022-3093(00)00225-8
[81] J. P. Garrahan and D. Chandler, “Geometrical Explanation and Scaling of Dynamical Heterogeneities in Glass Forming Systems,” Vol. 89, No. 3, 2002, Article ID: 035704. doi:10.1103/PhysRevLett.89.035704
[82] J. P. Garrahan and D. Chandler, “Coarse-Grained Microscopic Models of Glass Formers,” Proceedings of the National Academy of Science USA 100, 3 August 2003, pp. 9710-9714.
[83] E. W. Fisher, E. Donth and W. Steffen, “Temperature Dependence of Characteristic Length for Glass Transition,” Physical Review Letters, Vol. 68, No. 15, 1992, pp. 23442346. doi:10.1103/PhysRevLett.68.2344
[84] I. Chang and H. Sillescu, “Heterogeneity at the Glass Transition: Translational and Rotational Self-Diffusion,” Journal of Physical Chemistry B, Vol. 101, No. 43, 1997, pp. 8794-8801. doi:10.1021/jp9640989
[85] J. D. Ferry, L. D. Grandine and E. R. Fitzgerald, “The Relaxation Distribution Function of Polyisobutylene in the Transition from Rubber-Like to Glass-Like Behavior,” Journal of Applied Physics, Vol. 24, No. 7, 1953, p. 911. doi:10.1063/1.1721401
[86] S. Sastry, “The Relationship between Fragility, Configurational Entropy and the Potential Energy Landscape of Glass-Forming Liquids,” Nature, Vol. 409, 2001, pp. 164167. doi:10.1038/35051524
[87] A. Q. Tool, “Relation between Inelastic Deformability and Thermal Expansion of Glass in Its Annealing Range,” Journal of American Ceramic Society, Vol. 29, No. 9, 1946, pp. 240-253.doi:10.1111/j.1151-2916.1946.tb11592.x
[88] A. Q. Tool and C. G. Eichlin, “Variations Caused in the Heating Curves of Glass by Heat Treatment,” Journal of American Ceramic Society, Vol. 14, No. 4, 1931, pp. 276308. doi:10.1111/j.1151-2916.1931.tb16602.x

  
comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.