POROVISCOELASTIC MODELINGOF PROTEIN HYDROGELS

1

Ruben Mercadé-PrietoJoaquim Lopez, Xiao Dong Chen

Soochow University, ChinaIQS Engineering School, Spain

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Water in Food - Viscoelasticity

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• Many kinds of foods have large water content• Water defines many characteristics of foods (from texture

to preservation – hence the importance of drying)• Food mechanical properties important for quality, hence

extensively characterized in food engineering.• Foods are multiphasic materials, complex…• … so many simplifications (empirical models)

Del Nobile et al. J. Food Eng. (2007) 78:978

AgarMozzarella

BreadCom

pres

sive

stre

ssTime dependencyof mechanicaldeformation:viscoelasticity(e.g. general

Maxwell models)

Water in Food - Poroelasticity

• Viscoelasticity assumes some degree of homogeneity• Phase separation? Porous solid matrix with solvent• Poroelasticity considers the flow of fluid inside a porous

media – Force/stress relaxation• Scarcely used in food engineering, why?

Complex analysis, geometry dependent, etc.• Recent work on biomaterials by Prof. Suo (Harvard) and

Prof. Oyen (Cambridge): simple experimental design,coupled with simple analysis from master curvesobtained with FEM.

Small perturbations to gels in swelling equilibrium• Obtain meaningful physical parameters (permeability,

solvent diffusivity, Flory-Huggins solvent interaction , etc.)

Experimental procedure – Protein hydrogels

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1. Heat set whey protein gels: 15 wt% protein 80°C 20 min

2. Cut gelsinto 2-8

mm disks

3. Swellingin 0.1-0.3M NaCl

4. Followswelling

untilequilibrium

±0.001 N

5. Indentation ofgels in solution

using acylindrical punch

WPI gel

Loading andtime relaxation

Loading

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The compression load with a cylinder of radius R for asmall indentation h is obtained from Hertz

=4

=GE Shear modulusE Elastic modulus

Poisson ratioOnly valid for semi-infinite samples. For small samples, acorrection factor is needed to take into account the substrate

= ,

where , is obtained from FEM

Cao et al Acta. Biomater. (2009) 5:240

Loading very quick, no time dependent effects

Viscoelasticity

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Typically consider empirical Prony series

GVE(t )= GE(1j= 1

N VE

gVEj(1 e t VEj)))

where gVEj and VEj are the relative shear modulus and therelaxation times of the viscoelastic deformation

Normalized relaxation force with time

FVE(t)FE

= 1j = 1

NVE

gVEj(1 e t VEj))

The parameters gVEj and VEj are obtained by non-linearregression of the experimental relaxation

Poroelasticity

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For a solvent saturated isotropic elastic gel, the relationshipbetween the strains and the stress is

= 2 +2

2 +

d Poisson ratio of the drained gelij Kroneker delta

P Pressure of the pore fluid (local osmotic pressure)

Initially, the gel is in mechanical equilibrium

= 0

Lin and Hu J. non-Cryst. Solids (2006) 352:4034

Poroelasticity

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The pore pressure gradient is related to the flux of the gelsolvent J by Darcy’s law:

= k Gel permeabilitySolvent viscosity

Considering the mass balance of the pore fluid

= =2

(1 2 )Solvent diffusivity in the swollen gel

Poroelastic equations must be solved numerically(complex boundary conditions)

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Poroelasticity

Most FEM packages can simulate poroelasticity (Abaqus,Ansys, Comsol, etc.) .

We consider 2 boundary conditions: BC1 and BC2

Solvent

Rgel

(~6-7 mm)

R (2 mm)

t0

(~6 mm)Gel

Cylindrical punch(~500 µm)

Free solventflow

No flow( p

z= 0)

BC1,BC2, p= 0

pz = 0

(p= 0)

Poroelasticity FEM - Loading

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Misses Stress

Max S/E = 0.2

Max P/E = 0.2

Example to determine ,

Pore pressure

= 0.1R/t0 = 0.33

= 2.53

Loading very quick, no solventdiffusion

Poroelastic Relaxation

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The normalized poroelastic relaxation in semi-infinitesamples has been found only to depend on the type ofindenter

)) = ,

Where =

• It does not depend on the indentation depth h

• But in real (small) samples, it depends on(

)

Master curves

Hu et al. Appl. Phys. Lett. (2010) 121904

Poroelasticity FEM - Relaxation

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Misses Stress

Pore pressure

Max S/E = 0.2

Max P/E = 0.2

d = 0.1R/t0 = 0.33

Poroelasticity FEM - Relaxation

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Pore pressureBC1 boundary condition

No solvent flow from the bottom

Each frame limitsd = 0.1

R/t0 = 0.33

(0)) = 1.8

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Poroelasticity FEM - Relaxation

d = 0R/t0 = 0.33

BC2 boundary conditionSolvent flow from the bottom

(0)) = 2

Each frame limits

Poroelasticity FEM - Relaxation

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( ) =

()

0(

)

Poroelastic Relaxation

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(0)) = 2

The extent of force relaxation in poroelasticity is given by thedrained Poisson ratio

Poroelastic Relaxation

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Poroelastic analysis of compression data:Need to determine 3 parameters: G (or ), and (or k)

• G (or E ) is calculated from the loading• is calculated from F(0)/F( )• D is obtained by non-linear regression of the relaxation

force with time, ( )

Only one adjustable parameter

0.1 M NaClGE = 15.3 kPa

Experimental results - Loading

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Whey protein gels at swelling equilibrium in 3 different [NaCl][NaCl] / M 0.1 0.2 0.3Swelling ratio SR

(mean ± SD)0.123a ± 0.017

swelling-0.091b ± 0.02

shrinkage-0.134c ± 0.02

shrinkageNumber of tests 34 19 28GE / kPa 15a ± 3 32b ± 5 36c ± 7

Experimental results – Viscoelastic Relaxation

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10-1 100 101 102 1031.5

2

2.5

3

3.5

4x 104

She

arM

odul

us(P

a)

1 Exp fit2 Exp fit3 Exp fit4 Exp fitExperimental

10-1 100 101 102 103-1000

01000

time (s)

Res

idua

l(P

a)

GVE(t )= GE(1j= 1

N VE

gVEj(1 e t VEj)))

0.3 M NaCl

Good fit NVE 3# parameters 6!

Experimental results – Viscoelastic Relaxation

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[NaCl] /M

gVE1 VE1/ s

gVE2 VE3/ s

gVE3 VE3/ s

Av.MSE

0.1 0.12a 10a 0.13a 107a 0.17a 860a 5x10-5

0.2 0.15b 10a 0.15b 130ab 0.20a 1100ab 2x10-5

0.3 0.17b 10a 0.17c 125b 0.25b 1000b 2x10-5

Consider 3 exponentials (NVE = 3)

Very long relaxationviscoelasticity?

GVE(t)= GE(1j= 1

N VE

gVEj(1 e t VEj)))

Experimental results – Poroelastic Relaxation

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[NaCl] / M 0.1 0.2 0.3F( )/FE

(PE)0.59a ± 0.060.14a ± 0.09

0.50b ± 0.07-0.01b ± 0.2

0.44c ± 0.08-0.2c ± 0.3

Poroelastic relaxation profiles

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 1 10 100 1000

(F(t

)-F(

))/(

F E-F

())

Relaxation time (s)

0.1 M NaClD = 3.6x10-9 m2 s-1

Some (<9% tests) are good…

00.10.20.30.40.50.60.70.80.9

1

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01

(F(t)

-F(

))/(

F E-F

())

PE

0.3 M NaClD = 9.3x10-9 m2 s-1

… majority are not.

Viscousrelaxation

Poroviscoelasticity

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Combining poro- and viscoelasticity in FEM is verystraightforward.

The problem becomes how to easily determine modelparameters from experimental data (reverse engineering)

Key assumption: (Strange et al. Appl. Phys. Lett (2013) 102:031913)

Poro- and viscoelasticity can be decoupled.

FPVE(t )=FVE( t )FPE( t )

FE

FPVE )F E

=(1j= 1

NVE

gVEj)( 12(1 d))

Where the extent of relaxation is expected to be

Poroviscoelasticity FEM - Relaxation

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gVE1 0.1

= 0.003

Misses Stress

Max S/E = 0.2

Each frame limits

Pore pressure

d = 0.1R/t0 = 0.33

Poroviscoelasticity FEM – Relaxation Master curves

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01

(FPV

E(t)-

F PVE

())

/(F E

-FPV

E())

PE (=Dt/R2)

PVE, = 0.4PVE, = 0.3PVE, = 0.2PVE, = 0.1PVE, = 0.0PVE, = -0.2PE BC1

gVE1 0.1

Decoupling assumption provides good results in mostconditions – Easy non-linear regression

FPVE(t)=FVE( t)FPE( t)

FE

Experimental results – Poroviscoelasticity

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-0.10

0.10.20.30.40.50.60.70.80.9

1

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01

(F(t)

-F(

))/(

F E-F

())

PE

Exp. 3 PEPE BC1Exp. 3 PVEOptim. PVE BC2

Residuals

PVE BC2D = 4.9x10-9 m2 s-1

gVE1 = 0.10VE1 = 6 s

PE BC 1D = 9.3x10-9 m2 s-1

Experimental results – Poroviscoelasticity

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[NaCl] / M 0.1 0.2 0.3D / 10-9 m2 s-1 4.0a ± 2 3.2ab ± 1.7 3.0b ± 1.1

gVE1 0.082a ± 0.04 0.12b ± 0.06 0.10ab ± 0.04VE1 / s 13a ± 6 12a ± 5 11a ± 4

d 0.21a ± 0.1 0.10b ± 0.2 -0.08c ± 0.3Av. MSE 1.1x10-4 a 6.5x10-5 b 7.8x10-5 ab

Key parameter affectedby swelling was thedrained Poisson ratio d

D ~ self diffusivity of freewater (~2x10-9 m2 s-1)

Similar viscoelasticparameters

Experimental results – Negative Poisson’s ratio

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• Negative d have been reported inother PVE biomaterials (agar)

• Experimental confirmation (could beanother viscoelastic relaxation withsimilar VE)

• Need to perform micro- or nano-indentations to separate clearly poro-and visco- elasticity.

AgarGelatine

Strange and Oyen, J. Mech. Behav.Biomed. Mat. (2012) 11:16

Evans and Alderson.Adv. Mater. (2000) 12:617

Experimental results – Negative Poisson’s ratio

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Poly(vinyl acetate) (PVA) Hydrogels

Closed pores

Interconnected pores

Ma et al. J. Mech. Behav.Biomed. Mat. (2013) 23:22

Whey protein hydrogelsare homogeneous at the

micron levelAuxetic behavior due to

nanostructure?CLSM SEM

20µm 2µm

100 nm TEM

Experimental results – Poroviscoelasticity

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For swollen polymers (e.g. PDMS), Flory-Huggins theory

Hu et al. J. Mater. Res. (2011) 26:785

Protein gelssystem

0.5

If the low and even negative d are confirmed, weneed new theoretical framework…

Conclusions

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• FEM can be used to obtain master curves for PE andPVE as well, easy subsequent analysis.

• Simple food model (protein hydrogel) can be wellcharacterized well using only 1 adjustable parameterin PE, 3 in PVE, compared to 6 for viscoelasticity.

• D estimated are very reasonable.• Poisson ratio – highly swelling or salt dependent.• Negative drained d values need better experimental

confirmation – Auxetic materials?• … could this methodology be extended to initial

conditions not in equilibrium?