Acid-induced protein gels: from gelation to stress-induced failure Mathieu Leocmach Laboratoire de physique, Ecole Normale Supérieure de Lyon  June  

Texture as important as taste  

Mouth feel A very complex, multidimensional sensation Behaviour of the food at large deformation? How does food break into pieces? Ductile fracture Irreversible deformation rubber lead butter Brittle fracture concrete asphalt glass  

The simplest Yoghurt Acid-set protein gel Water ( ◦C) Sodium caseinate (milk protein) % stable solution Glucono-δ-lactone (GDL) % ⇒ slow homogeneous acidification     isoelectric point pH ≈ . pH                 gelation time (h) G ,G (Pa) Kaláb (), Roefs & van Vliet (), Lucey & Singh ()  

Linear rheology: Power law (visco)elastic solid stress → σ = Gγ ← strain storage →G + ıG ← loss −      G ∼ G ∼ f. Frequency (Hz) Storage and loss moduli (Pa)  

The yoghurt creep experiment σ A Taylor- Couette cell where the yogurt is made in situ ultrasonic velocimetry optical imaging A rheometer imposes a constant stress σ and records the strain γ(t)  

The yoghurt creep experiment power-law creep nucleation explosive regime σ = Pa −     − − − − t (s) strain rate ˙ γ (s−)  

Different (constant) stresses: universality −       − − − − − −   time (s) Strain rate ˙ γ (s−)  Pa     Normalisation τf failure time ˙ γmin minimum strain rate ⇒ Physical origin? − −       t/τf ˙ γ/ ˙ γmin power-law creep  . . . .      t/τf fracture nucleation − −       (τf − t)/τf explosive regime  

Power-law creep and plasticity “Andrade” creep () Power-law creep known in ductile solids for over a century Classically explained by disinclination dynamics in crystals Recently explained by plastic events in glasses Miguel et al., PRL  Any plastic event in yoghurt? Ultrasound velocimetry ⇒ local velocity maps T. Gallot et al., Rev. Sci. Instrum. ,  () homogeneous strain field (no wall slip, no shear band) if present, plastic events are below our resolution (a few µm)  

Is power-law creep damaging the structure? Stop creep much before (γ < .) fracture nucleation (γ ≈ ) Let relax ⇒ No complete recovery Reset deformation and start again ⇒ superimposable          . . . σ ← Pa σ ←  γ ←  time (h) strain γ Undamaged structure ⇒ Plasticity cannot explain power law Viscous dissipation  

Power-law creep as linear response −       G ∼ G ∼ f. Frequency (Hz) G , G (Pa) σ ∝ γfα Fourier − − − − − − → σ ∝ γt−α Constant σ γ(t) ∝ σtα d dt − − → ˙ γ(t) ∝ σtα− − − − − −       ˙ γ(t) ˙ γmin ∼ t τf −. t/τf ˙ γ/ ˙ γmin Power-law creep is the linear regime of the material up to % strain! explained without plasticity  

Explaining the explosive regime Fracture length − − −   . . . . . H ∼ ln τf −t τf (τf − t)/τf /H γ ∝ ∝ ln τf − t τf ˙ γ ∝ τf − t τf − − − − − −       ˙ γ(t) ˙ γmin ∼ τf τf −t (τf − t)/τf ˙ γ/ ˙ γmin ⇒ Explosive regime dominated by fracture growth  

Failure time & master curve           τf ∼ σ−. No yield stress! σ (Pa) τf (s) Basquin law τf ∼ σ−β fatigue (oscillatory stress) heterogeneous solids  . . . .  .  .  .  .  τmin t/τf ˙ γ/ ˙ γmin Master curve ˙ γ(t) ˙ γmin = λ t τf −. linear response + µ  − t/τf fractures ⇒ no room for plasticity  

Fibre bundle models σ model elastic fibres local yield strain local coupling Jagla et al., PRE  ˙ γ ˙ γ But yield stress built in power-law creep no master curve − −     t/τf ˙ γ/ ˙ γmin − −     (τf − t)/τf ˙ γ/ ˙ γmin  

Fibre bundle models Kun et al. PRL  Halász et al., PRE  elastic fibres local yield strain local coupling + damage accumulation model But no power-law creep too slow divergence D fractures           σ (Pa) τf (s)  

Creep and yielding summary failure involves a single time scale τf ∼ σ−. Basquin law reversible, homogeneous creep → irreversible fracture growth a single expression captures all the global rheological response − −       t/τf ˙ γ/ ˙ γmin  . . . .      t/τf − −       (τf − t)/τf a model soft solid well captured by fibre-bundle models  

Can we generalise to biogels? (physical gels Ebond kB T) Cell mechanics Electrophoresis Cosmetics Bacterial culture Food Sources Wikimedia Commons www.madaboutscience.com www.keautystore.com  

Merciless yoghurt breakers Leocmach et al. PRL ,  (), ArXiv . Christophe Perge Thibaut Divoux Sebastien Manneville PhD student CNRS researcher Professor E.N.S. Lyon CRPP Bordeaux E.N.S. Lyon Special thanks to Alan Parker (Firmenich) for prompting this study and providing casein & GDL  

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Creep rheology: Three regimes primary secondary tertiary −       . . . . . time (s) strain γ  Pa     Failure at γ ≈  for a well defined time τf 