How to calculate Deborah number De viscoelasticity polymers

Calculate Deborah number De = lambda/t_obs ratio relaxation time material vs observation time + viscoelasticity polymers. De<<1 viscous liquid; De~1 viscoelastic transition; De>>1 elastic Hookean solid. Reiner 1964 Bible Judges. NIST + MIT + ExxonMobil + Dow + Goodyear + 3M.

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How to calculate Deborah number (De) dimensionless rheology ratio measuring material relaxation time (lambda) to observation time (t_obs): De = lambda / t_obs.

Regime classification: De1000 pure solid Hookean.

Markus Reiner (Israel 1886-1976) Technion Haifa 1964 Physics Today 17:62 + biblical reference Judges 5:5 'mountains flowed before the Lord'.

USA universities + national labs: NIST Boulder CO + Gaithersburg MD 1901 3000 thermophysical properties + REFPROP database + Polymer Division; MIT Cambridge MA 1861 11000 + Mechanical Engineering + Robert Cohen + Patrick Doyle microfluidics + Bradley Olsen polymer; Stanford 1885 17000 + Chemical Engineering + Curtis Frank polymer; Caltech Pasadena 1891 2300 + GALCIT + John Brady transport + Mark Davis catalysis; Harvard SEAS 1847 + David Weitz soft matter + Howard Stone fluid dynamics + microfluidic; UC Berkeley LBNL 1868 45000 + Polymers + Materials Science; Princeton 1746 + Howard Stone soft matter + Athanassios Panagiotopoulos; University Wisconsin Madison 1848 + Bird Stewart Lightfoot 1960 textbook polymers + Macosko rheology textbook; Cornell 1865 + Polymer Outreach; UPenn 1740 + Howard Stone microfluidics; UMN 1851 + Polymer; Texas A&M; Michigan; Georgia Tech; Akron OH 1870 rubber + Goodyear + tire research; Massachusetts Amherst + polymer; Lehigh University polymer; RIT Rochester.

USA national labs: NREL Golden CO 3000 + bioplastics; ORNL Oak Ridge TN 5000 + Spallation Neutron Source SNS + polymer characterization; ANL Argonne 5000 + Advanced Photon Source APS + polymer X-ray scattering; LLNL Lawrence Livermore CA 1952; SNL Sandia 14000 + Combustion Research Facility CRF; LANL Los Alamos NM 12000 + materials; PNNL Pacific Northwest 5000; INL Idaho 5000; BNL Brookhaven NY + RHIC.

USA chemical industry polymers + viscoelasticity: ExxonMobil Spring TX 62000 + Chemical 22000 + Baytown TX 590k bpd + Beaumont + Baton Rouge + Joliet refineries + LDPE LLDPE HDPE PE PP polymer + olefins; Chevron 45000 + Phillips Chemical 8000 + olefins polymers; Phillips 66 14000; Marathon 17000; Valero 10000; Halliburton 40000; Schlumberger 99000; Baker Hughes 55000; Dow Chemical Midland MI 1897 36000 + plastics + polyethylene + polypropylene + ethylene oxide + EVA + Saran + viscoelastic; DuPont Wilmington DE 1802 23000 + Kevlar Teflon Mylar Tyvek + Nylon + viscoelastic polymer; LyondellBasell Houston TX 19000 + polypropylene polyethylene + viscoelastic; Chemours Wilmington DE 6000 + TiO2 + Krytox; Sherwin-Williams 60000 paints + atomization + viscoelastic; PPG Industries Pittsburgh PA 50000 paints + glass + viscoelastic; Eastman Chemical Kingsport TN 14000 specialty chemicals + cellulosics + acetate + viscoelastic; Honeywell 13000 + UOP catalysts; Trinity Industries; Air Products 19000 + cryogenic LNG + Engineering Hydrogen.

USA tires + rubber industry: Goodyear Tire Rubber Akron OH 1898 71000 + tires + rubber + Goodyear Engineered Rubber Products + Goodyear Aviation; Cooper Tire Findlay OH 1914 10000 + tires + rubber; Bridgestone USA Nashville TN subsidiary Japanese parent 50000 + tires; Michelin North America Greenville SC subsidiary French parent + tires; Continental USA Fort Mill SC subsidiary German parent + tires; Pirelli USA Atlanta GA subsidiary Italian parent + tires; Sumitomo USA + Yokohama USA + Hankook USA + Kumho USA + Nokian Tyres USA.

USA adhesives + sealants + concrete admixtures: 3M St Paul MN 1902 95000 + adhesives + sealants + Scotch tape + Post-it Notes + Velcro + viscoelastic; Henkel USA Stamford CT subsidiary German parent + Loctite + Bordon adhesives; Sika USA Lyndhurst NJ subsidiary Swiss + concrete admixtures; Mapei USA Deerfield Beach FL subsidiary Italian parent + adhesives; Dow Chemical Adhesives 36000; HB Fuller St Paul MN 1887 + adhesives.

USA bitumen + asphalt + roads viscoelasticity: ExxonMobil Bitumen + Chevron Asphalt + Marathon Asphalt + Valero Asphalt; Sherwin-Williams + PPG Roadmark + Ennis Paint road striping; Caterpillar Peoria IL 1925 105000 + asphalt paving equipment; Vulcan Materials Birmingham AL 9000 + aggregates + Pavers; Martin Marietta Aggregates Raleigh NC; HeidelbergCement USA + Lehigh Hanson Pennsylvania + cement; Holcim USA Lafarge subsidiary; CRH Americas + Oldcastle USA.

USA pharmaceutical drug delivery viscoelastic: Pfizer NYC 80000 + Lipitor + drug delivery viscoelastic; Merck 70000 + drug delivery; Johnson Johnson 130000 + Janssen + viscoelastic; AbbVie 50000 + Humira; Bristol Myers 30000 + Eliquis; Eli Lilly 35000 + Trulicity Mounjaro; Gilead 16000 + Veklury; Regeneron 12000 + EYLEA; Moderna 5000 mRNA Lipid Nanoparticles LNP viscoelastic; Vertex 5000.

USA food rheology + dough + chocolate viscoelasticity: General Mills Minneapolis MN 1856 31000 + cereals + flour + dough; Kraft Heinz Chicago IL 1869 36000 + dairy + cheese viscoelastic; PepsiCo Purchase NY 1898 309000 + chips + crackers + Frito-Lay + Doritos; Coca-Cola Atlanta GA 1886 86000; Hershey Hershey PA 1894 16000 + chocolate temper viscoelastic; Mars McLean VA 1911 130000 family-owned + chocolate + Snickers M&M's + viscoelastic; Mondelez Chicago IL 90000 + Oreo Cadbury chocolate; Nestle USA Vevey Switzerland 30000; Tyson Foods Springdale AR 142000 meat protein viscoelastic; ConAgra Brands Chicago IL 18000 + frozen dough; Mondelez International + Kellogg's Battle Creek MI 30000 + cereals; Cargill Wayzata MN 160000 + food + bioplastics + lecithin viscoelastic.

USA glass industry viscoelasticity Tg: Corning Painted Post NY 1851 60000 + glass + Pyrex + Gorilla Glass + LCD + viscoelastic Tg; PPG Industries 50000 + glass + Pittsburgh; Owens-Illinois Toledo OH 1929 22000 + container glass; Saint-Gobain Norton MA 170000 worldwide + glass + ceramics; Schott North America Elmsford NY subsidiary German parent + Schott Zerodur + Schott Ceran; AGC Glass Industries USA + Asahi Glass + AGC Glass; Vitro Mexico USA market + glass; ACE Glass + Wheaton Industries.

Deborah formula

De = lambda/t_obs. Maxwell lambda = mu/G. Reiner 1964 Physics Today 17:62 + biblical Judges 5:5.

History and context

Markus Reiner Israel 1886-1976 Technion Haifa fondatore + 1928 Reiner-Rivlin viscoelastic equation + 1949 textbook 'Deformation Strain Flow' + 1962 'Lectures on Theoretical Rheology' + 1964 Physics Today 17:62 'The Deborah number' classic eponym + biblical Judges 5:5 'mountains flowed before the Lord'. Bird-Armstrong-Hassager 1987 Wiley 'Dynamics Polymeric Liquids' classic + Macosko 1994 Wiley 'Rheology' + Larson 1999 Oxford + Tanner 2000 Oxford. USA universities: NIST Boulder + Gaithersburg 1901 3000 + REFPROP + Polymer Division; MIT 1861 11000 + Cohen + Doyle + Olsen polymer; Stanford 1885 17000 + Frank polymer; Caltech 1891 2300 + GALCIT Brady + Davis; Harvard SEAS 1847 + Weitz + Stone; UC Berkeley LBNL 1868 45000 + Polymers + Materials; Princeton 1746 + Stone + Panagiotopoulos; Wisconsin Madison 1848 + Bird Stewart Lightfoot textbook + Macosko rheology textbook; Cornell 1865 + Polymer Outreach; UPenn 1740 + Stone; UMN 1851; Texas A&M; Michigan; Georgia Tech; Akron OH 1870 rubber tire + Goodyear; UMass Amherst polymer; Lehigh; RIT. USA national labs: NREL Golden 3000 + bioplastics; ORNL Oak Ridge 5000 + SNS polymer characterization; ANL Argonne + APS X-ray polymer; LLNL Livermore + SNL CRF + LANL + PNNL + INL + BNL RHIC. USA chemical: ExxonMobil 62000 + Chemical 22000 + Baytown 590k + LDPE LLDPE HDPE PE PP olefins; Chevron 45000 + Phillips 8000; Phillips 66 14000 + Marathon 17000 + Valero 10000 + Halliburton 40000 + Schlumberger 99000 + Baker Hughes 55000;

Dow 36000 + plastics + EVA Saran viscoelastic; DuPont 23000 + Kevlar Teflon Mylar Nylon viscoelastic; LyondellBasell 19000 + polypropylene polyethylene; Chemours 6000; Sherwin-Williams 60000 + atomization viscoelastic; PPG 50000 + glass viscoelastic; Eastman 14000 + cellulosics; Honeywell 13000 UOP + Air Products 19000. USA tires rubber: Goodyear Akron 1898 71000 + tires + Engineered Rubber + Aviation; Cooper Findlay OH 10000; Bridgestone USA 50000 Nashville; Michelin Greenville + Continental Fort Mill + Pirelli Atlanta + Sumitomo + Yokohama + Hankook + Kumho + Nokian. USA adhesives sealants: 3M St Paul 1902 95000 + Scotch + Post-it + viscoelastic; Henkel USA Stamford + Loctite + Bordon; Sika USA + concrete admixtures; Mapei USA Deerfield FL + adhesives; Dow Adhesives; HB Fuller St Paul 1887. USA bitumen asphalt: ExxonMobil Bitumen + Chevron Asphalt + Marathon + Valero + Sherwin-Williams Roadmark + Ennis Paint + Caterpillar 105000 + Vulcan 9000 aggregates + Martin Marietta + HeidelbergCement + Holcim + CRH. USA pharma viscoelastic: Pfizer 80000 + Merck 70000 + JJ 130000 + AbbVie 50000 + Bristol Myers 30000 + Eli Lilly 35000 + Gilead 16000 + Regeneron 12000 + Moderna 5000 mRNA LNP + Vertex 5000. USA food: General Mills 31000 + Kraft Heinz 36000 + PepsiCo 309000 + Coca-Cola 86000 + Hershey 16000 chocolate temper + Mars 130000 + Mondelez 90000 Oreo Cadbury + Tyson Foods 142000 protein + ConAgra 18000 + Kellogg's 30000 + Cargill 160000 + Nestle USA 30000.

USA glass: Corning 1851 60000 + Pyrex Gorilla Glass + LCD viscoelastic; PPG 50000 + Pittsburgh + Owens-Illinois Toledo 22000; Saint-Gobain Norton 170000; Schott Elmsford + Zerodur + Ceran; AGC Glass USA + Asahi Glass + Vitro Mexico USA market; ACE Glass + Wheaton.

Deborah table

  • De = lambda/t_obs
  • Maxwell lambda = mu/G
  • Reiner 1964 Physics Today 17:62
  • Biblical Judges 5:5 Deborah
  • De<0.001 pure liquid Newtonian
  • 0.001-0.1 liquid-like viscous
  • 0.1-1 viscoelastic transition
  • 1-10 elastic-like
  • 10-1000 solid-like
  • De>1000 pure solid Hookean
  • Water lambda ~1e-12 s
  • Engine oil ~1e-9 s
  • Polymer melts ~0.01-1 s
  • Bitumen ~1-100 s
  • Silly putty ~100 s
  • Glass room T lambda ~1e15 s
  • Rocks geological ~1e22 s
  • Reiner Israel 1886-1976
  • Reiner-Rivlin 1928 viscoelastic
  • Bird Armstrong Hassager 1987 Wiley
  • Macosko 1994 Wiley Rheology
  • USA NIST 1901: 3000 + Polymer Division
  • USA MIT 1861: 11000 + Cohen Doyle Olsen
  • USA Wisconsin BSL textbook + Macosko
  • USA Akron OH 1870: rubber tire + Goodyear
  • USA ExxonMobil 62000: Baytown 590k LDPE
  • USA Dow 36000: plastics PE PP EVA Saran
  • USA DuPont 23000: Kevlar Teflon Mylar Nylon
  • USA Goodyear Akron 1898: 71000 tires
  • USA 3M St Paul 1902: 95000 Scotch Post-it
  • USA Hershey 16000: chocolate temper
  • USA Corning 1851: 60000 Pyrex Gorilla Glass

Numerical examples

Example 1 - Water flow. lambda=1e-12, t_obs=1. De=1e-12. Pure liquid. USA: ExxonMobil Baytown + processes industriales.

Example 2 - Polymer melt extrusion. lambda=0.5, t_obs=1. De=0.5. Viscoelastic transition. USA: ExxonMobil Chemical 22000 + Dow Chemical 36000 + DuPont 23000 + LyondellBasell 19000.

Example 3 - Silly putty bounce. lambda=100, t_obs=0.1. De=1000. Solid Hookean elastic. USA: laboratorio MIT + Stanford + Caltech demonstration.

Example 4 - Silly putty slow flow. lambda=100, t_obs=3600. De=0.028. Liquid-like. USA: research universities Wisconsin BSL Macosko + Akron rubber.

Example 5 - Glass room T. lambda=1e15, t_obs=1. De=1e15. Pure solid. USA: Corning 1851 60000 Pyrex Gorilla Glass + PPG 50000 + Owens-Illinois 22000 + Saint-Gobain Norton.

Limits

  1. Single relaxation time spectrum Maxwell-Generalized.
  2. Linear vs nonlinear viscoelasticity.
  3. Temperature Arrhenius + WLF.
  4. Glass transition Tg.
  5. Yield stress Bingham.
  6. Weissenberg Wi distinct.
  7. Aging materials lambda varies.
  8. Multiphase composite emulsion.
  9. Frequency-dependent G' G''.
  10. Numerical CFD POLYFLOW.

Common errors

De vs Wi: time-scale vs shear unsteadiness.

Single lambda: real spectrum.

Temperature Arrhenius WLF.

Glass solid vs liquid Tg.

Asphalt bitumen geological.

Practical applications

  1. Polymer extrusion processing: USA ExxonMobil Chemical 22000 + Dow 36000 + DuPont 23000 + LyondellBasell 19000 + Chemours 6000 + Eastman 14000.
  2. Tires rubber industry: Goodyear Akron 1898 71000 + Cooper Findlay 10000 + Bridgestone USA Nashville 50000 + Michelin Greenville + Continental Fort Mill + Pirelli Atlanta + Sumitomo + Yokohama + Hankook.
  3. Bitumen asphalt roads viscoelastic: ExxonMobil Bitumen + Chevron Asphalt + Marathon + Valero + Caterpillar 105000 paving + Vulcan 9000 + Martin Marietta + HeidelbergCement + Holcim.
  4. Adhesives sealants concrete: 3M St Paul 1902 95000 Scotch + Henkel USA Loctite + Sika USA + Mapei USA + Dow Adhesives + HB Fuller 1887 St Paul.
  5. Drug delivery viscoelastic: Pfizer 80000 + Merck 70000 + JJ 130000 + AbbVie 50000 + Bristol Myers 30000 + Eli Lilly 35000 + Moderna 5000 mRNA LNP.
  6. Food rheology dough chocolate: General Mills 31000 + Kraft Heinz 36000 + PepsiCo 309000 + Hershey 16000 chocolate temper + Mars 130000 + Mondelez 90000 + Tyson 142000 + Cargill 160000 + Kellogg's 30000.
  7. Cardiovascular research wall stiffness: NIH NHLBI Bethesda + Cleveland Clinic 70000 + Mayo 73000 + MGH Boston 26000 + Hopkins 30000 + Stanford Health 14000 + UCLA 22000 + UCSF 31000.
  8. Tissue engineering biomaterials: NIH NIBIB Biomedical Imaging Bioengineering + 10x Genomics + MIT + Caltech + Stanford + Harvard SEAS biomaterials.
  9. Glass ceramics manufacturing: Corning 1851 60000 Pyrex Gorilla Glass + PPG 50000 + Owens-Illinois Toledo 22000 + Saint-Gobain Norton 170000 + Schott Zerodur Ceran + AGC + Vitro.
  10. University research polymers viscoelasticity: NIST Polymer Division + MIT 11000 + Wisconsin BSL Macosko textbook + Akron OH rubber tire + UMass Amherst + Lehigh + Princeton Stone + UPenn + Cornell + UMN.

Updated July 2026

Frequently Asked Questions

What is the Deborah number?

De = lambda / t_obs dimensionless rheology ratio between material relaxation time lambda and observation time t_obs. De<<1 Newtonian viscous liquid (water lambda~1e-12 s); De~1 viscoelastic transition (polymers + bitumen + tissues); De>>1 Hookean elastic solid (silly putty bounce + glass + steel). Markus Reiner Israel 1964 Physics Today 17:62 + biblical Judges 5:5 'mountains flowed before the Lord'. USA: NIST Polymer Division + MIT Cohen Doyle Olsen + Wisconsin BSL Macosko textbook + ExxonMobil 62000 polymers + Goodyear Akron 71000 tires.

How to calculate Deborah?

Procedure: (1) De = lambda / t_obs; (2) lambda relaxation time s (Maxwell lambda = mu/G); (3) t_obs observation time s. Examples: water lambda=1e-12 + 1s: De=1e-12 pure liquid; polymer melts lambda=0.5 + 1s extrusion: De=0.5 viscoelastic transition; silly putty lambda=100 + 0.1s bounce: De=1000 solid-like. USA: ExxonMobil Chemical 22000 + Dow 36000 + DuPont 23000 + Goodyear 71000 polymer + tire rubber rheology.

What flow regimes does De classify?

De<0.001 pure liquid: Newtonian (water + air); 0.001-0.1 liquid-like: viscous dominate (oils + blood); 0.1-1 viscoelastic transition: mixed (polymer melts + bitumen + tissues); 1-10 elastic-like: elastic dominate (silly putty short obs); 10-1000 solid-like: elastic solid (silly putty long obs + asphalt); De>1000 pure solid: Hookean (steel + glass + diamond + rocks geological). USA: industry ExxonMobil + Dow + DuPont + Goodyear + 3M + Hershey chocolate temper.

Who was Markus Reiner?

Markus Reiner (Czernowitz Bukovina Austria-Hungary 1886 - Haifa Israel 1976) University Vienna 1908 PhD chemistry + emigrated Palestine 1922 + Technion Israel Institute Technology Haifa 1923-1976 founder Israeli chemical engineer pioneer rheology + viscoelasticity + plasticity; 1928 Reiner-Rivlin equation viscoelastic non-linear + 1949 textbook 'Deformation Strain and Flow' classic + 1964 Physics Today 17:62 'The Deborah number' classic 1 page eponym + biblical Judges 5:5 Cantico Deborah 'mountains flowed before the Lord'; co-founder International Committee Rheology + 1958 Bingham Medal Society of Rheology.

What is the biblical Deborah origin?

Judges 5:5 Cantico of Deborah Old Testament Bible + 'mountains flowed before the Lord' + Markus Reiner 1964 Physics Today 'The Deborah number' interpretation 'on a sufficiently long time-scale even mountains flow' + Deborah biblical prophetess judge Old Testament; concept time-scale dependent solid-fluid behavior depending observation duration; sufficiently long observation time even rocks flow geological time mountains flow. USA: research NIST + MIT + Wisconsin BSL Macosko textbook + ExxonMobil polymer rheology.

Difference De vs Weissenberg Wi?

Deborah De = lambda/t_obs ratio relaxation time/observation measures time-scale memory + transient. Weissenberg Wi = lambda * shear_rate dimensionless shear unsteadiness scaling normal stresses + non-Newtonian response. Distinct: De time-scale; Wi flow magnitude. For steady high shear flows Wi>1 nonlinear; transient flow De>1 elastic memory. USA: research MIT + Wisconsin Madison BSL Macosko + Akron rubber tire + Goodyear research.

Applications of De?

Applications: polymer extrusion processing USA ExxonMobil Chemical 22000 + Dow 36000 + DuPont 23000 + LyondellBasell 19000 + Eastman 14000; tires rubber Goodyear Akron 1898 71000 + Cooper 10000 + Bridgestone USA 50000 + Michelin Greenville + Continental + Pirelli Atlanta; bitumen asphalt ExxonMobil + Chevron + Caterpillar 105000 paving + Vulcan 9000 aggregates; adhesives sealants 3M St Paul 95000 Scotch Post-it + Henkel Loctite + Mapei USA; food dough chocolate General Mills 31000 + Hershey 16000 chocolate temper + Mars 130000 + Mondelez 90000 + Cargill 160000; glass Corning 60000 Pyrex Gorilla Glass + PPG 50000 + Saint-Gobain Norton 170000 + Schott Zerodur.

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