0
Research Papers

Mechanics of Supercooled Liquids

[+] Author and Article Information
Jianguo Li

School of Engineering and Applied Sciences,
Kavli Institute for Bionano Science
and Technology,
Harvard University,
Cambridge, MA 02138;
International Center for Applied Mechanics,
State Key Lab for Strength and Vibration
of Mechanical Structures,
School of Aerospace Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China

Qihan Liu

School of Engineering and Applied Sciences,
Kavli Institute for Bionano Science
and Technology,
Harvard University,
Cambridge, MA 02138

Laurence Brassart

Institute of Mechanics,
Materials and Civil Engineering,
Université Catholique de Louvain,
Louvain-la-Neuve 1348, Belgium

Zhigang Suo

Fellow ASME
School of Engineering and Applied Sciences,
Kavli Institute for Bionano Science
and Technology,
Harvard University,
Cambridge, MA 02138
e-mail: suo@seas.harvard.edu

1These authors contributed equally to this work.

2Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 27, 2014; final manuscript received September 12, 2014; accepted manuscript posted September 18, 2014; published online September 24, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(11), 111007 (Sep 24, 2014) (8 pages) Paper No: JAM-14-1401; doi: 10.1115/1.4028587 History: Received August 27, 2014; Revised September 12, 2014; Accepted September 18, 2014

Pure substances can often be cooled below their melting points and still remain in the liquid state. For some supercooled liquids, a further cooling slows down viscous flow greatly, but does not slow down self-diffusion as much. We formulate a continuum theory that regards viscous flow and self-diffusion as concurrent, but distinct, processes. We generalize Newton's law of viscosity to relate stress, rate of deformation, and chemical potential. The self-diffusion flux is taken to be proportional to the gradient of chemical potential. The relative rate of viscous flow and self-diffusion defines a length, which, for some supercooled liquids, is much larger than the molecular dimension. A thermodynamic consideration leads to boundary conditions for a surface of liquid under the influence of applied traction and surface energy. We apply the theory to a cavity in a supercooled liquid and identify a transition. A large cavity shrinks by viscous flow, and a small cavity shrinks by self-diffusion.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Dienes, G. J., and Klemm, H. F., 1946, “Theory and Application of the Parallel Plate Plastometer,” J. Appl. Phys., 17(6), pp. 458–471. [CrossRef]
Cotts, R. M., Hoch, M. J. R., Sun, T., and Markert, J. T., 1989, “Pulsed Field Gradient Stimulated Echo Methods for Improved NMR Diffusion Measurements in Heterogeneous Systems,” J. Magn. Reson., 83(2), pp. 252–266. [CrossRef]
Smith, R. S., Dohnálek, Z., Kimmel, G. A., Stevenson, K. P., and Kay, B. D., 2000, “The Self-Diffusivity of Amorphous Solid Water Near 150 K,” Chem. Phys., 258(2–3), pp. 291–305. [CrossRef]
Mills, P. J., Green, P. F., Palmstrøm, C. J., Mayer, J. W., and Kramer, E. J., 1984, “Analysis of Diffusion in Polymers by Forward Recoil Spectrometry,” Appl. Phys. Lett., 45(9), pp. 957–959. [CrossRef]
Einstein, A., 1905, “Uber die von der Molekularkinetischen Theorie der Warme Geforderte Bewegung von in Ruhenden Flussigkeiten Suspendierten Teilchen (English translation: On the Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular-Kinetic Theory of Heat),” Investigations on the Theory of the Brownian Movement, Dover, New York.
Edward, J. T., 1970, “Molecular Volumes and the Stokes–Einstein Equation,” J. Chem. Educ., 47(4), pp.261–270. [CrossRef]
Borucka, A. Z., Bockris, J. O'M., and Kitchener, J. A., 1957, “Self-Diffusion in Molten Sodium Chloride: A Test of the Applicability of the Nernst–Einstein Equation,” Proc. R. Soc. A, 241(1227), pp. 554–567. [CrossRef]
Ediger, M. D., Angell, C. A., and Nagel, S. R., 1996, “Supercooled Liquids and Glasses,” J. Phys. Chem., 100(31), pp. 13200–13212. [CrossRef]
Rosenfeld, D., and Woodley, W. L., 2000, “Deep Convective Clouds With Sustained Supercooled Liquid Water Down to −37.5 °C,” Nature, 405(6785), pp. 440–442. [CrossRef] [PubMed]
Shi, Z., Debenedetti, P. G., and Stillinger, F. H., 2013, “Relaxation Processes in Liquids: Variations on a Theme by Stokes and Einstein,” J. Chem. Phys., 138(12), p. 12A526. [CrossRef] [PubMed]
Angell, C. A., Ngai, K. L., McKenna, G. B., McMillan, P. F., and Martin, S. W., 2000, “Relaxation in Glassforming Liquids and Amorphous Solids,” Appl. Phys. Rev., 88(6), pp. 3113–3157. [CrossRef]
Debenedetti, P. G., and Stillinger, F. H., 2001, “Supercooled Liquids and the Glass Transition,” Nature, 410(6825), pp. 259–267. [CrossRef] [PubMed]
Mapes, M. K., Swallen, S. F., and Ediger, M. D., 2006, “Self-Diffusion of Supercooled o-Terphenyl Near the Glass Transition Temperature,” J. Phys. Chem. B, 110(1), pp. 507–511. [CrossRef] [PubMed]
Swallen, S. F., Mapes, M. K., Kim, Y. S., McMahon, R. J., Ediger, M. D., and Satija, S., 2006, “Neutron Reflectivity Measurements of the Translational Motion of Tris(Naphthylbenzene) at the Glass Transition Temperature,” J. Chem. Phys., 124(18), p. 184501. [CrossRef] [PubMed]
Swallen, S. F., Traynor, K., McMahon, R. J., Ediger, M. D., and Mates, T. E., 2009, “Self-Diffusion of Supercooled Tris-Naphthylbenzene,” J. Phys. Chem. B, 113(14), pp. 4600–4608. [CrossRef] [PubMed]
Fujara, F., Geil, B., Sillescu, H., and Fleischer, G., 1992, “Translational and Rotational Diffusion in Supercooled Orthoterphenyl Close to the Glass Transition,” Z. Phys. B Condens. Matter, 88(2), pp. 195–204. [CrossRef]
Cavagna, A., 2009, “Supercooled Liquids for Pedestrians,” Phys. Rep., 476(4–6), pp. 51–124. [CrossRef]
Ediger, M. D., and Harrowell, P., 2012, “Perspective: Supercooled Liquids and Glasses,” J. Chem. Phys., 137(8), p. 080901. [CrossRef] [PubMed]
Ediger, M. D., 2000, “Spatially Heterogeneous Dynamics in Supercooled Liquids,” Annu. Rev. Phys. Chem., 51(1), pp. 99–128. [CrossRef] [PubMed]
Berthier, L., 2011, “Dynamic Heterogeneity in Amorphous Materials,” Physics, 4, p. 42. [CrossRef]
Karmakar, S., Dasgupta, C., and Sastry, S., 2014, “Growing Length Scales and Their Relation to Timescales in Glass-Forming Liquids,” Annu. Rev. Condens. Matter Phys., 5(1), pp. 255–284. [CrossRef]
Sillescu, H., Böhmer, R., Diezemann, G., and Hinze, G., 2002, “Heterogeneity at the Glass Transition: What Do We Know?” J. Non-Cryst. Solids, 307–310, pp. 16–23. [CrossRef]
Reinsberg, S. A., Qiu, X. H., Wilhelm, M., Spiess, H. W., and Ediger, M. D., 2001, “Length Scale of Heterogeneous Dynamic Heterogeneity in Supercooled Glycerol Near Tg,” J. Chem. Phys., 114(17), pp. 7299–7302. [CrossRef]
Qiu, X. H., and Ediger, M. D., 2003, “Length Scale of Dynamic Heterogeneity in Supercooled D-Sorbitol: Comparison to Model Predictions,” J. Phys. Chem. B, 107(2), pp. 459–464. [CrossRef]
Berthier, L., Biroli, G., Bouchaud, J.-P., Cipelletti, L., Masri, D., L'Hôte, D., Ladieu, F., and Pierno, M., 2005, “Direct Experimental Evidence of a Growing Length Scale Accompanying the Glass Transition,” Science, 310(5755), pp. 1797–1800. [CrossRef] [PubMed]
Furukawa, A., and Tanaka, H., 2011, “Direct Evidence of Heterogeneous Mechanical Relaxation in Supercooled Liquids,” Phys. Rev. E, 84(6), p. 061503. [CrossRef]
Furukawa, A., 2013, “Simple Picture of Supercooled Liquid Dynamics: Dynamic Scaling and Phenomenology Based on Clusters,” Phys. Rev. E, 87(6), p. 062321. [CrossRef]
Yamamoto, R., and Onuki, A., 1998, “Heterogeneous Diffusion in Highly Supercooled Liquids,” Phys. Rev. Lett., 81(22), pp. 4915–4918. [CrossRef]
Berthier, L., Chandler, D., and Garrahan, J. P., 2005, “Length Scale for the Onset of Fickian Diffusion in Supercooled Liquids,” Europhys. Lett., 69(3), pp. 320–326. [CrossRef]
Kumar, S. K., Szamel, G., and Douglas, J. F., 2006, “Nature of the Breakdown in the Stokes–Einstein Relationship in a Hard Sphere Fluid,” J. Chem. Phys., 124(21), p. 214501. [CrossRef] [PubMed]
Chang, I., and Sillescu, H., 1997, “Heterogeneity at the Glass Transition Translational and Rotational Self-Diffusion,” J. Phys. Chem. B, 101(43), pp. 8794–8801. [CrossRef]
Swallen, S. F., and Ediger, M. D., 2011, “Self-Diffusion of the Amorphous Pharmaceutical Indomethacin Near Tg,” Soft Matter, 7(21), pp. 10339–10344. [CrossRef]
Royal Society of Chemistry, 2014, “ ChemSpider Online Database,” Royal Society of Chemistry, Cambridge, UK, http://www.chemspider.com/
Batchelor, G. K., 1967, An Introduction to Fluid Dynamics, Cambridge University, Cambridge, MA.
Darken, L. S., 1948, “Diffusion, Mobility and Their Interrelation Through Free Energy in Binary Metallic Systems,” Trans. AIME, 175(1), pp. 184–201.
Stephenson, G. B., 1988, “Deformation During Interdiffusion,” Acta Metall., 36(10), pp. 2663–2683. [CrossRef]
Suo, Z., 2004, “A Continuum Theory That Couples Creep and Self-Diffusion,” ASME J. Appl. Mech., 71(5), pp. 646–651. [CrossRef]
Mullins, W. W., 1959, “Flattening of a Nearly Plane Solid Surface Due to Capillarity,” J. Appl. Phys., 30(1), pp. 77–83. [CrossRef]
Herring, C., 1950, “Diffusional Viscosity of a Polycrystalline Solid,” J. Appl. Phys., 21(5), pp. 437–445. [CrossRef]
Kingery, W. D., 1959, “Surface Tension of Some Liquid Oxides and Their Temperature Coefficients,” J. Am. Ceram. Soc., 42(1), pp. 6–10. [CrossRef]
Tracht, U., Wilhelm, M., Heuer, A., Feng, H., Schmidt-Rohr, K., and Spiess, H. W., 1998, “Length Scale of Dynamic Heterogeneities at the Glass Transition Determined by Multidimensional Nuclear Magnetic Resonance,” Phys. Rev. Lett., 81(13), pp. 2727–2730. [CrossRef]
Graessley, W. W., 2009, “On Dynamic Heterogeneity in Supercooled Liquids,” J. Chem. Phys., 130(16), p. 164502. [CrossRef] [PubMed]
Russel, E. V., and Israeloff, N. E., 2000, “Direct Observation of Molecular Cooperativity Near the Glass Transition,” Nature, 408(6813), pp. 695–698. [CrossRef] [PubMed]
Pharr, M., Zhao, K., Suo, Z., Ouyang, F., and Liu, P., 2011, “Concurrent Electromigration and Creep in Lead-Free Solder,” J. Appl. Phys., 110(8), p. 083716. [CrossRef]
Biot, M. A., 1941, “General Theory of Three-Dimensional Consolidation,” J. Appl. Phys., 12(2), pp. 155–164. [CrossRef]
Hui, C. Y., Lin, Y. Y., Chuang, F. C., Shull, K. R., and Lin, W. C., 2006, “A Contact Mechanics Method for Characterizing the Elastic Properties and Permeability of Gels,” J. Polym. Sci. Part B Polym. Phys., 44(2), pp. 359–370. [CrossRef]
Galli, M., Comley, S. C., Shean, T. A. V., and Oyen, M. L., 2009, “Viscoelastic and Poroelastic Mechanical Characterization of Hydrated Gels,” J. Mater. Res., 24(3), pp. 973–979. [CrossRef]
Yoon, J., Cai, S., Suo, Z., and Hayward, R. C., 2010, “Poroelastic Swelling Kinetics of Thin Hydrogel Layers: Comparison of Theory and Experiment,” Soft Matter, 6(23), pp. 6004–6012. [CrossRef]
Hu, Y., and Suo, Z., 2012, “Viscoelasticity and Poroelasticity in Elastomeric Gels,” Acta Mech. Solida Sin., 25(5), pp. 441–458. [CrossRef]
Brassart, L., and Suo, Z., 2013, “Reactive Flow in Solids,” J. Mech. Phys. Solids, 61(1), pp. 61–77. [CrossRef]
Szamel, G., and Flenner, E., 2006, “Time Scale for the Onset of Fickian Diffusion in Supercooled Liquids,” Phys. Rev. E, 73(1), p. 011504. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Arrhenius plots of (a) viscosity η, (b) self-diffusivity D, and (c) the quantity ηD/kT. Here, T is the temperature, Tm is the melting point, and Tg is the glass transition temperature defined as the temperature at which the viscosity of a given substance reaches η=1012 Pa·s. Included are data for several substances [31-33]: silica, 1,3-bis-(1-naphthyl)-5-(2-naphthyl)benzene (TNB), o-terphenyl (OTP), and 1-(p-chlorobenzoyl)-5-methoxy-2-methyl-indole-3-acetic acid (IMC).

Grahic Jump Location
Fig. 2

A supercooled liquid may form a dynamic structure, consisting of regions in which molecules rearrange at very different rates. Here, we represent slow regions by dark circles and fast regions by gray circles. The size of these regions is large compared to the size of individual molecules. Viscous flow proceeds by disrupting the dynamic structure, but self-diffusion proceeds as individual molecules migrate through the liquid. As the temperature drops, the dynamic structure increasingly jams viscous flow, but does not retard self-diffusion as much.

Grahic Jump Location
Fig. 3

A composite thermodynamic system consists of a piece of liquid, a set of external forces, and a reservoir of molecules. The piece evolves through a sequence of homogeneous states, represented by a parallelepiped that changes its shape and volume at a rate of deformation dij, and absorbs molecules from the reservoir at rate R per unit volume. The external forces apply to the piece of liquid a state of stress σij. The reservoir and the piece of liquid exchange the species of molecules that constitute the liquid, and the chemical potential of the species of molecules in the reservoir is μ.

Grahic Jump Location
Fig. 4

The definition of self-diffusion flux using the net flux of molecules and the velocity of markers. Imagine a plane fixed in space, with n being the unit vector normal to the plane. The net flux of molecules N is a vector, such that N·n is the number of molecules crossing the plane per unit area and per unit time. Dispersed in the liquid are markers moving at the velocity v. The net flux of molecules and the velocity of markers can be independently measured. Define the self-diffusion flux by J=N-v/Ω, where Ω is the volume per molecule.

Grahic Jump Location
Fig. 5

A composite thermodynamic system consists of a body of a pure liquid, the surface of the body, a set of external forces, and a set of reservoirs. The reservoirs can inject molecules of the same species as those constitute the liquid. Molecules in the interior of the body can diffuse out and plate onto the surface, but cannot leave the liquid. As the surface of the body moves, the area of the surface changes, and the total surface energy changes.

Grahic Jump Location
Fig. 6

Arrhenius plots of (a) the characteristic length Λ, and (b) the characteristic time Γ. Here, T is the temperature, and Tg is the glass transition temperature. Data are from Refs. [31-33,40-33,40].

Grahic Jump Location
Fig. 7

In a supercooled liquid, a spherical cavity shrinks under the influence of surface energy. The rate of shrinking is a function of the radius of the cavity. A small cavity shrinks by self-diffusion, a large cavity shrinks by viscous flow, and a cavity of an intermediate size shrinks by coupled self-diffusion and viscous flow.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In