Defining temperatures of granular powders analogously with thermodynamics to understand jamming phenomena

Tian Hao; Tian Hao

doi:10.3934/matersci.2018.1.1

AIMS Materials Science

2018, Volume 5, Issue 1: 1-33. doi: 10.3934/matersci.2018.1.1

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Defining temperatures of granular powders analogously with thermodynamics to understand jamming phenomena

Tian Hao ^,

Nutrilite Health Institute, 5600 Beach Boulevard, Buena Park, CA 90621, USA

Received: 10 November 2017 Accepted: 15 December 2017 Published: 21 December 2017

For the purpose of applying laws or principles originated from thermal systems to granular athermal systems, we may need to properly define the critical “temperature” concept in granular powders. The conventional environmental temperature in thermal systems is too weak to drive movements of particles in granular powders and cannot function as a thermal energy indicator. For maintaining the same functionality as in thermal systems, the temperature in granular powders is defined analogously and uniformly in this article using kinetic energy connections. The newly defined granular temperature is utilized to describe and explain one of the most important phenomena observed in granular powders, the jamming transition, by introducing jamming temperature and jamming volume fraction concepts. The predictions from the equations of the jamming volume fractions for several cases like granular powders under shear or vibration are in line with experimental observations and empirical solutions in powder handlings. The equations are mainly for hard sphere systems without frictional forces among particles, but can be easily extended to frictional granular systems with frictional energy term included in. The goal of this article is to lay a foundation for establishing similar concepts in granular powders, allowing granular powders to be described with common laws or principles we are familiar with in thermal systems. Our intention is to build a bridge between thermal systems and granular powders to account for many similarities already found between these two systems.
- jamming,
- granular temperatures,
- thermodynamics,
- granular powders
Citation: Tian Hao. Defining temperatures of granular powders analogously with thermodynamics to understand jamming phenomena[J]. AIMS Materials Science, 2018, 5(1): 1-33. doi: 10.3934/matersci.2018.1.1

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Abstract

For the purpose of applying laws or principles originated from thermal systems to granular athermal systems, we may need to properly define the critical “temperature” concept in granular powders. The conventional environmental temperature in thermal systems is too weak to drive movements of particles in granular powders and cannot function as a thermal energy indicator. For maintaining the same functionality as in thermal systems, the temperature in granular powders is defined analogously and uniformly in this article using kinetic energy connections. The newly defined granular temperature is utilized to describe and explain one of the most important phenomena observed in granular powders, the jamming transition, by introducing jamming temperature and jamming volume fraction concepts. The predictions from the equations of the jamming volume fractions for several cases like granular powders under shear or vibration are in line with experimental observations and empirical solutions in powder handlings. The equations are mainly for hard sphere systems without frictional forces among particles, but can be easily extended to frictional granular systems with frictional energy term included in. The goal of this article is to lay a foundation for establishing similar concepts in granular powders, allowing granular powders to be described with common laws or principles we are familiar with in thermal systems. Our intention is to build a bridge between thermal systems and granular powders to account for many similarities already found between these two systems.

References

[1]	Edwards SF, Mehta A (1989) Statistical mechanics of powder mixtures. Physica A 157: 1091–1100. doi: 10.1016/0378-4371(89)90035-6
[2]	Edwards SF, Oakeshott RBS (1989) Theory of powders. Physica A 157: 1080–1090. doi: 10.1016/0378-4371(89)90034-4
[3]	Edwards SF (2001) Can one learn glasses from advances in granular materials? J Non-Cryst Solids 293–295: 279–282.
[4]	Reis PM, Ingale RA, Shattuck MD (2007) Caging Dynamics in a Granular Fluid. Phys Rev Lett 98: 188301. doi: 10.1103/PhysRevLett.98.188301
[5]	Reis PM, Ingale RA, Shattuck MD (2007) Forcing independent velocity distributions in an experimental granular fluid. Phys Rev E 75: 051311. doi: 10.1103/PhysRevE.75.051311
[6]	Pacheco-Vázquez F, Caballero-Robledo GA, Ruiz-Suárez JC (2009) Superheating in Granular Matter. Phys Rev Lett 102: 170601. doi: 10.1103/PhysRevLett.102.170601
[7]	Reis PM, Ingale RA, Shattuck MD (2006) Crystallization of a Quasi-Two-Dimensional Granular Fluid. Phys Rev Lett 96: 258001. doi: 10.1103/PhysRevLett.96.258001
[8]	Coniglio A, De Candia A, Fierro A, et al. (2004) On Edwards' theory of powders. Physica A 339: 1–6. doi: 10.1016/j.physa.2004.03.038
[9]	Song C, Wang P, Makse HA (2008) A phase diagram for jammed matter. Nature 453: 629–632. doi: 10.1038/nature06981
[10]	Onoda GY, Liniger EG (1990) Random loose packings of uniform spheres and the dilatancy onset. Phys Rev Lett 64: 2727–2730. doi: 10.1103/PhysRevLett.64.2727
[11]	Liu C, Nagel SR, Schecter DA, et al. (1995) Force Fluctuations in Bead Packs. Science 269: 513–515. doi: 10.1126/science.269.5223.513
[12]	Edwards SF, Grinev DV (1999) Statistical Mechanics of Stress Transmission in Disordered Granular Arrays. Phys Rev Lett 82: 5397. doi: 10.1103/PhysRevLett.82.5397
[13]	Edwards SF (2005) The full canonical ensemble of a granular system. Physica A 353: 114–118. doi: 10.1016/j.physa.2005.01.045
[14]	Henkes S, O'Hern CS, Chakraborty B (2007) Entropy and Temperature of a Static Granular Assembly: An Ab Initio Approach. Phys Rev Lett 99: 038002. doi: 10.1103/PhysRevLett.99.038002
[15]	Henkes S, Chakraborty B (2009) Statistical mechanics framework for static granular matter. Phys Rev E 79: 061301. doi: 10.1103/PhysRevE.79.061301
[16]	Majmudar TS, Sperl M, Luding S, et al. (2007) Jamming Transition in Granular Systems. Phys Rev Lett 98: 058001. doi: 10.1103/PhysRevLett.98.058001
[17]	Tighe BP, Vlugt TJH (2011) Stress fluctuations in granular force networks. J Stat Mech-Theory E 2011: P04002.
[18]	Wu Y, Teitel S (2015) Maximum Entropy and the Stress Distribution in Soft Disk Packings Above Jamming. Phys Rev E 92: 022207. doi: 10.1103/PhysRevE.92.022207
[19]	Xia C, Cao Y, Kou B, et al. (2014) Angularly anisotropic correlation in granular packings. Phys Rev E 90: 062201. doi: 10.1103/PhysRevE.90.062201
[20]	Chandler D (1978) Introduction to Modern Statistical Mechanics, New York: Oxford University.
[21]	Bertin E, Dauchot O, Droz M (2006) Deﬁnition and relevance of nonequilibrium intensive thermodynamic parameters. Phys Rev Lett 96: 120601. doi: 10.1103/PhysRevLett.96.120601
[22]	Makse HA, Kurchan J (2002) Testing the thermodynamic approach to granular matter with a numerical model of a decisive experiment. Nature 415: 614–617. doi: 10.1038/415614a
[23]	Song C, Wang P, Makse HA (2005) Experimental measurement of an effective temperature for jammed granular materials. P Natl Acad Sci USA 102: 2299–2304. doi: 10.1073/pnas.0409911102
[24]	Wang P, Song C, Briscoe C, et al. (2008) Particle dynamics and effective temperature of jammed granular matter in a slowly sheared 3D Couette cell. Phys Rev E 77: 061309. doi: 10.1103/PhysRevE.77.061309
[25]	Saksenaa RS, Woodcock LV (2004) Quasi-thermodynamics of powders and granular dynamics. Phys Chem Chem Phys 6: 5195–5202. doi: 10.1039/b407699k
[26]	Ciamarra MP, Coniglio A, Nicodemi M (2006) Thermodynamics and Statistical Mechanics of Dense Granular Media. Phys Rev Lett 97: 158001. doi: 10.1103/PhysRevLett.97.158001
[27]	Casas-Vazquez J, Jou D (2003) Temperature in non-equilibrium states: a review of open problems and current proposals. Rep Prog Phys 66: 1937–2023. doi: 10.1088/0034-4885/66/11/R03
[28]	Lu K, Brodsky EE, Kavehpour HP (2008) A thermodynamic uniﬁcation of jamming. Nat Phys 4: 404–407. doi: 10.1038/nphys934
[29]	Chen Q, Hou M (2014) Effective temperature and fluctuation-dissipation theorem in athermal granular systems: A review. Chinese Phys B 23: 074501. doi: 10.1088/1674-1056/23/7/074501
[30]	Hao T (2015) Understanding empirical powder flowability criteria scaled by Hausner ratio or Carr index with the analogous viscosity concept. RSC Adv 5: 57212–57215. doi: 10.1039/C5RA07197F
[31]	Hao T (2015) Analogous Viscosity Equations of Granular Powders Based on Eyring's Rate Process Theory and Free Volume Concept. RSC Adv 5: 95318–95333. doi: 10.1039/C5RA16706J
[32]	Fermi F (1956) Thermodynamics, New York: Dover.
[33]	Atkins P (2007) The four laws that drive the universe, New York: Oxford University Press.
[34]	Janssen HA (1895) Versuche uber Getreidedruck in Silozellen. Z Ver Deut Ing 39: 1045–1049.
[35]	Schulze D (2008) Powders and Bulk Solids: Behavior, Characterization, Storage and Flow, Berlin, Heidelberg: Springer-Verlag.
[36]	Savage S, Jeffery D (1981) The stress tensor in a granular flow at high shear rates. J Fluid Mech 110: 255–272. doi: 10.1017/S0022112081000736
[37]	Jenkins JT, Richman MW (1985) Grad's 13-moment system for a dense gas of inelastic spheres. Arch Ration Mech An 87: 355–377.
[38]	Lun C, Savage SB, Jeffrey DJ, et al. (1984) Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J Fluid Mech 140: 223–256. doi: 10.1017/S0022112084000586
[39]	Lun C (1991) Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. J Fluid Mech 233: 539–559. doi: 10.1017/S0022112091000599
[40]	Goldshtein A, Shapiro M (1995) Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations. J Fluid Mech 282: 75–114.
[41]	Sela N, Goldhirsch I, Noskowicz SH (1996) Kinetic theoretical study of a simply sheared two-dimensional granular gas to Burnett order. Phys Fluids 8: 2337–2353. doi: 10.1063/1.869012
[42]	Brey JJ, Moreno F, Dufty JW (1996) Model kinetic equation for low-density granular flow. Phys Rev E 54: 445–456. doi: 10.1103/PhysRevE.54.445
[43]	Brey JJ, Dufty JW (2003) Hydrodynamic modes for granular gases. Phys Rev E 68: 030302. doi: 10.1103/PhysRevE.68.030302
[44]	Brey JJ, Ruiz-Montero MJ (2004) Simulation study of the Green-Kubo relations for dilute granular gases. Phys Rev E 70: 051301. doi: 10.1103/PhysRevE.70.051301
[45]	Brey JJ, Dufty JW (2005) Hydrodynamic modes for a granular gas from kinetic theory. Phys Rev E 72: 011303. doi: 10.1103/PhysRevE.72.011303
[46]	Lutsko JF (2006) Chapman-Enskog expansion about nonequilibrium states with application to the sheared granular fluids. Phys Rev E 73: 021302. doi: 10.1103/PhysRevE.73.021302
[47]	Loeb LB (2004) The Kinetic Theory of Gases, New York: Dover.
[48]	Shabana AA (1995) Theory of Vibration: An Introduction, Springer.
[49]	Dong RG, Schopper AW, McDowell TW, et al. (2004) Vibration energy absorption (VEA) in human fingers-hand-arm system. Med Eng Phys 26: 483–492. doi: 10.1016/j.medengphy.2004.02.003
[50]	Santos A, Montanero JM, Dufty JW, et al. (1998) Kinetic model for the hard-sphere fluid and solid. Phys Rev E 57: 1644–1660.
[51]	Garzó V, Dufty JW (1999) Homogeneous cooling state for a granular mixture. Phys Rev E 60: 5706–5713. doi: 10.1103/PhysRevE.60.5706
[52]	Dufty JW, Baskaran A, Zogaib L (2004) Gaussian kinetic model for granular gases. Phys Rev E 69: 051301. doi: 10.1103/PhysRevE.69.051301
[53]	Kumaran V (2005) Kinetic Model for Sheared Granular Flows in the High Knudsen Number Limit. Phys Rev Lett 95: 108001. doi: 10.1103/PhysRevLett.95.108001
[54]	Jenkins J, Richman M (1985) Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys Fluids 28: 3485–3494. doi: 10.1063/1.865302
[55]	Coussot P (2014) Rheophysics: Matter in all its States, Springer.
[56]	Müller CR, Holland DJ, Sederman AJ, et al. (2008) Granular temperature: Comparison of Magnetic Resonance measurements with Discrete Element Model simulations. Powder Technol 184: 241–253. doi: 10.1016/j.powtec.2007.11.046
[57]	Jaeger HM (2015) Celebrating Soft Matter's 10th Anniversary: Toward jamming by design. Soft Matter 11: 12–27. doi: 10.1039/C4SM01923G
[58]	Bi D, Zhang J, Chakraborty B, et al. (2011) Jamming by shear. Nature 480: 355–358. doi: 10.1038/nature10667
[59]	Trappe V, Prasad V, Cipelletti L, et al. (2001) Jamming phase diagram for attractive particles. Nature 411: 772–775. doi: 10.1038/35081021
[60]	Zhang Z, Xu N, Chen DTN, et al. (2009) Thermal vestige of the zero-temperature jamming transition. Nature 459: 230–233. doi: 10.1038/nature07998
[61]	Silbert LE, Ertas D, Grest GS, et al. (2002) Analogies between granular jamming and the liquid-glass transition. Phys Rev E 65: 051307. doi: 10.1103/PhysRevE.65.051307
[62]	Hao T (2005) Electrorheological Fluids: The Non-aqueous Suspensions, Amsterdam: Elsevier Science.
[63]	Hao T (2015) Tap density equations of granular powders based on the rate process theory and the free volume concept. Soft Matter 11: 1554–1561. doi: 10.1039/C4SM02472A
[64]	Kuwabara S (1959) The Forces experienced by Randomly Distributed Parallel Circular Cylinders or Spheres in a Viscous Flow at Small Reynolds Numbers. J Phys Soc Jpn 14: 527–532. doi: 10.1143/JPSJ.14.527
[65]	Torquato S, Stillinger FH (2010) Jammed hard-particle packings: From Kepler to Bernal and beyond. Rev Mod Phys 82: 2633–2672. doi: 10.1103/RevModPhys.82.2633
[66]	Hao T (2015) Derivation of stretched exponential tap density equations of granular powders. Soft Matter 11: 3056–3061. doi: 10.1039/C4SM02892A
[67]	Vivanco F, Rica S, Melo F (2012) Dynamical arching in a two dimensional granular flow. Granul Matter 4: 563–576.
[68]	Duran J (2000) Sands, Powders, and Grains, An Introduction to the Physics of Granular Materials, Springer.
[69]	Behringer RP, Dijksman J, Ren J, et al. (2013) Jamming and shear for granular materials. AIP Conf Proc 1542: 12–19.
[70]	Peters IR, Majumdar S, Jaeger HM (2016) Direct observation of dynamic shear jamming in dense suspensions. Nature 532: 214–217. doi: 10.1038/nature17167
[71]	Lu K, Brodsky EE, Kavehpour HP (2007) Shear-weakening of the transitional regime for granular flow. J Fluid Mech 587: 347–372.
[72]	Glasstone S, Laidler K, Eyring H (1941) The theory of rate process, New York: McGraw-Hill.
[73]	Kou B, Cao Y, Li J, et al. (2017) Granular materials flow like complex fluids. Nature 551: 360–363.

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