Hydrogels—Wet, Elastic Materials

Ronald A. SiegelPharmaceutics/Biomedical Engineering

University of Minnesota

CollapsedImpermeable

SwollenPermeable

wet

dry

http://www.spectroscopynow.com/ftp_images/2NMR3‐swellgel.jpg

Useful Properties•Soft but Firm, Slippery

•Low Interfacial Tension with Water—Often Compatible with Biological Tissues

•Elastic—Restore Shape after Deformation

•Exert Force against Confining or Attached Structures

•Retain or Reject Solutes, Affect Transport Properties of Molecules

•Many Biological Tissues are Hydrogels—Cartilage, Cornea, Intercellular Matrix etc. 

Hydrogels

Applications of Hydrogels (Partial List)

Present

• Soft Contact Lenses• Gel Permeation Chromatography• Ion Exchangers• Gel Electrophoresis• Drug Delivery (e.g. Capsules, 

Matrices)• Cosmetics• Food Additives• Fragrances• Dessicants• Soft Tissue Replacements• Cosmetic Surgery• Scar block• Toys

Future

• (Bio)sensors• Stimuli‐Sensitive Drug Delivery• Switchable Chromatography• “Active” Separations• Tissue Culture, Tissue Engineering, 

and Bioartifical Organ Supports• Burn, Wound Dressings• Catheter, Suture Coatings• Artificial Muscle• Microparticulate Drug and Gene 

Carriers• Fire Retardants

Reversible Swelling/Collapse Transitions

Temperature

pHGlucose

UreaAntigen/Antibody

DNA HybridElectricityMagnetism

Light

CollapsedImpermeable

SwollenPermeable

Osmotic Pressure

Polymer Elasticity

Polymer Solvent Interaction

Polyelectrolyte Hydrogel Tug of War

Swelling Pressure = ∆ πmix + ∆ πelast + ∆ πion = 0

Theory of Hydrogel Swelling: Flory-Rehner-Donnan theory

COO-Na+

Na+

Na+

Na+

Polymer-solventmixing

Network (Rubber)Elasticity

Ion Osmotic (Donnan)Pressure

at equilibrium

Flory‐Huggins Flory‐Rehner Donnan

Mixing Free Energy: Flory‐Huggins Lattice Model∆Fmix = ∆Fconf + ∆Fadj

0

0

confconf

conf

STF

S

0)1ln()1(ln

PTNkF

B

conf

+Water + Small Solute

+0

0

confconf

conf

STFS

Water + Polymer

Configurational Part (∆Fconf)

For polymers of length P (P=8 in above example), it can be shown that

For hydrogels P → ∞ , so )1ln()1(

TNkF

B

conf

N = total # of lattice sites (N=25 above); φ = volume fraction of polymer (φ=16/25 above)

Adjacency (Contact) Part (∆Fadj)

gww gpp gwp

2ppww

wpB

ggg

Tkz

z = # neighbors on lattice (z=4 for square lattice)

Repulsive: > 0, Gadj > 0

Attractive: < 0, Gadj < 0

“Athermal”: = 0, Gadj = 0

These may depend on temperature, 

gxy = hxy ‐ Tsxy

Mixing Free Energy: Flory‐Huggins Lattice Model∆Fmix = ∆Fconf + ∆Fadj

)1(

TNkF

B

adj

Contact Free Energy

)1()1ln()1(

TNkF

B

mix

Assuming random mixing:

Combining:

Neighbor “type” +

Simplified Polymer Elasticity Theory

2/30

2

2/3

2)(

022

ReRp

RR

02

20

2

22/3

23ln

23lnlnln 0

22

RRTkConstTkHF

RRkConstkekConstkS

BBchainchain

BBRR

BBchain

Single polymer chains execute random walks

End‐to‐end distance, R, is a random variable, which isspecified by a normal, or Gaussian (3D) distribution,

where <R2>0 is the mean‐square value of R for an unstretchedchain.  Entropy and Gibbs Free Energy of chain are

R

Constant

It has been shown that enthalpy change plays virtually no role in polymer elasticity.  Stretching chain from R0 to R, then, 

1

23

23

20

2

02

202

02

02 R

RRTRkRR

RTkF BB

chain

Simplified Polymer Elasticity TheoryNow assume single chain is embedded in a hydrogel containing numerous chains that behave in the same way, but are independently configured.  Also assume that swellingis affine (all end‐to‐end vectors are all scaled by the same constant).

R

Felast

Volume V0 , Polymer volume fraction φ0 Volume V ,,Polymer vol fract φ < φ0

Conservation of polymer: φV = φ0V0

Also, V/R3=V0/R03 ,  so (R/R0)2= (φ/φ0)‐2/3

Let  be number of chains in hydrogel.   By definition,  /200

2

chainsRR

Combining,

12

33/2

0

TkFF B

chainchainelast

R0

Combination of Mixing and ElasticityWe did not use the lattice model to derive elastic free energy, so we need to set up proper correspondence.  To do so, simply define the lattice site volume as that of solvent (water), vw.  Then V=Nvw, and we may write

0

3/1

00

3/2

000

3/2

0

v231

2v31

23

VVVNV

TNkF ww

B

elast

0 elastmix

0

TNkF

TNkF

BB

Earlier, it was stated that equilibrium should be expressed in a net osmotic pressureequal to zero (no ionic contribution here):

Set  F=Fmix+Felast.  Then an equivalent condition is

Substituting,

0v

)1ln()()(3/1

00

2

V

vf welastmixw

Combination of Mixing and Elasticity

02

v)1ln()(

0

3/1

00

2

V

f w

An expression that makes better predictions when the polymer shrinks (φ > φ0) is

We take vw=0.018 L/mol.  Evidently, equilibrium swelling depends on , /V0, and φ0.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

20

Free Swelling curve

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

f(φ)

φ

=1.0

=0.9

=0.8

=0.7

=0.3

Φ0=0.1, /V0=0.1 M

1/φA

C

B

A: Φ0=0.1, /V0=0.1 M

B: Φ0=0.1, /V0=0.5 M

C: Φ0=0.05, /V0=0.1 M

Swelling Ratio

Charged (e.g. pH‐sensitive) Hydrogels

)( 'i

iiion ccRT

0)1( ii

icz

0 ionelastmix

‐

‐+

+‐

‐‐

‐

‐

‐+

++

+

+

+

Excess mobile ion concentration inside charged gel, neutralizing fixed charges, leads to ion osmotic swelling pressure.

σ = fixed charge concentration inside hydrogel zi = ion valence  (1 for simple salt, e.g. NaCl)ci = ion concentration in hydrogelci’ = ion concentration in external bath

Electroneutrality:

Hydrogel External Bath

Van’t Hoff’s law:

Now require

Ideal Donnan assumption

'i

zi cc i

Donnan ratio

Polyacid Hydrogels

)(

00

101/

)1( pKapHAc

Density (mol/gel volume) of ionizablgroups at synthesis

Langmuir Isotherm

KaAH A‐ +   H+

Flory‐Rehner‐Donnan‐Langmuir Model(NaCl external, neglect H+ and OH‐)

021v2

v)1ln( '

0

3/1

00

2

NaClww c

V

0=

101/ 1)1(

0)(

'0'

apKpH

NaClNaCl

cc

Osmotic pressure balance

Electroneutrality inside hydrogel

AH

AHa c

ccK

aa KpK 10log

Solve for , λ given hydrogel parameters (, ν/V0, σ0) and external pH, c’NaCl

pH

Swelling

pKa pH

Swelling

pKa

Depends on  , 0 ,  c’NaCl

Monotonic or Bistable (Hysteretic) Behaviors

Swelling Ratio

pHΔFpH

Swelling Ratio

pHΔFpH

Towards Closed Loop Insulin Delivery MinimedParadigm® (Medtronic)

http://www.minimed.com/images/realtime/realtime_system.jpg

http://www.medtronic‐diabetes‐me.com/tl_files/UK/prt/x22_features_callout.jpg

Glucose + O2 + H2O Glucolactone + H2O2Glucose Oxidase

Enzyme‐Based Glucose Sensing

Glucose + O2 + H2O

Glucolactone + H2O2

½O2 + H2O

Glucose Oxidase

Measure O2 Depletion with O2 ElectrodeGluconic Acid

Gluconate‐ H+

Gluconolactonase

pH‐Sensitive Hydrogel

•Under free swelling conditions, change in hydrogel diameter of volume is measured

•When hydrogel is confined, it develops a swelling pressure or stress against confining structure

Problems: •Long term instability of enzymes in situ•Inefficiency due to physiologic buffering

Reduce H2O2 to e‐ at Electrode, Measure Current

)/][1log(0 sapp KSpKpK

Enzyme Free Glucose Sensitive Hydrogel

A. Matsumoto et al. Chem. Commun. 2010, 46, 2203‐2205. 

pKa=8.86 KsB OH

NHO ONH2

OH

OH-

Na+

]n[

Glucose sensitive moiety

MethacrylamidophenylboronicAcid ( MPBA): 20 mol%

Acrylamide (AAm): 80 mol%

Crosslinker: 0.125 mol%Methylene Bisacrylamide

]m[

+ 2H2O

sugar

Normal Range (3-7 mM)

Diabetic Range (2-20mM)

0

50

100

150

200

250

0 20 40 60 80 100Glucose concentration (mM)

Volume chan

ge (%

)

PBS (pH 7.4)

R.A. Siegel et al., J. Controlled Release (2010).

Glucose Sensitive, Hydrogel Driven Cantilever Beams

Hydrogel

Cantilever Beam

20

21

22

23

24

25

26

27

28

0 20 40 60 80 100 120

Glucose Concentration (mM)

Cantilever D

eflections (

m)

Sensitivity = 0.1 m/mM gluscose concentration(1nm/0.01mM)

M. Lei et al, J. Nanosci Nanotech. 7, 780 (2007).

Glucose Diffusion Hydrogel 

Swelling/Deswelling

41.2

41.4

41.6

41.8

42

42.2

42.4

42.6

42.8

0 20 40 60 80 100 120

Glucose concentration (mM)

Freq

uenc

y (M

Hz)

Glucose Concentration (mM)

Resonant Frequ

ency (M

Hz)

41.9

42

42.1

42.2

42.3

42.4

42.5

42.6

42.7

0 30 60 90 120 150 180 210 240 270

Time (min)

Freq

uenc

y (M

Hz)

0mM 20mM 0mM

Resonant Frequ

ency (M

Hz)

Microcoil

SiliconGlass

M. Lei et al, Diabetes Technol. Therap. 8, 112 (2006).

Skin

Subcutaneous Tissue

Wireless Transmitter Device

Wireless Transmitter

1/ 2 LC

Resonant Frequency

Wireless Glucose Sensor

Implanted Sensor

pH x Sugar Effects on Free SwellingMPBA‐co‐AAm (20/80) Hydrogels in PBS 

Fructose                                       pKa=8.86

Glucose

pH

Matsumoto et al. Alexeev, V. L. et al., Anal. Chem. 2003, 75, 2316‐2323. 

ionelastmix ΔπΔπΔπΔπ

Modeling Response to pH and FructoseFlory‐Rehner‐Donnan‐Langmuir (FRDL) Model

Crosslink Density: polymer‐solvent interaction parameter

Polymer‐solvent mixing Network elasticity

PBA‐Na+

Na+

Na+

Na+

Ion Osmotic Pressure

Fixed charge density, salt concentration

Fructose, OH‐ Binding

pH, fructose conc.

Flory‐HugginsFlory‐Rehner

Flory‐Rehner‐DonnanFlory‐Rehner‐Donnan‐Langmuir

Total Swelling Pressure

Flory‐Rehner‐Donnan‐Langmuir (FRDL) Model

)2/1(/)1ln( 3/100

2 swww

cvvvRT

0)/()/1()1( 00 fcs

)]/1/(101/[1 )( 0FF

pKpH Kcf

Swelling Pressure

Electroneutrality Inside Hydrogel

Binding Isotherm

L/mol) (0.018 water of memolar volu

)5.8(sugar of absencein groupPBA ofacidity energy malmolar ther

mol/L) (0.15ion concentratsalt externalsynthesisat

groupsMPBA of hydrogel) of (mol/Lion concentratsynthesisat polymer offraction volume

0

0

0

w

s

v

pKRT

c

ionconcentrat fructosepH external

FcpH

synthesisat hydrogel) (mol/Ldensity chain activeparametern interactiolvent polymer/so

0

FcpH ,given at polymer offraction volume

hydrogelin Cl ,Na ,)//( ratioDonnan ionizedfraction

-ClNa

ccccf

ss

Fixed Parameters

Fitted Parameters

Model Inputs

Model Outputs, Compare with Data

Model Intermediates, at given pH, cF

Model Fit

Parameter Estimates (±95% CI)

= 0.606 ± 0.012

ρ0  = 0.0066 ± 0.0052 mol/L

pK0 = 8.18 ± 0.14

Kf = 0.09 mM

135.085.035.0168.085.068.01

Correlation Matrix

Strongly Charged Hydrogels in Water

)/(RTσRTc 00 ion

‐

‐‐

‐

‐

+

++

+

+

+

σ = σ0(φ/φ0) = fixed charge concentration inside hydrogel

Electroneutrality: σ0(φ/φ0) – (1‐φ)c+ = 0

Hydrogel External Bath

0)/(v2

v)1ln()( 000

3/1

00

2

ww

Vf

[omitting (1 ‐ φ) term]

Models for  parameter based on data from poly(NIPAAm‐co‐SA) gels

2ppww

wpB

ggg

Tkz gxy = hxy ‐ Tsxy

BB ks

Tkh

Simplest model (Hirotsu, Hirokawa, Tanaka):

Best fit gives 11311 108.1;104.5 Kergsergh

Independent measurements: ergh 14107.8

2

BB ks

Tkh

Next order model (Hirotsu):

Best fit: 6.0;105.4;103.1 211613 Kergsergh

Nature of Transition—First Order? (Hirotsu)

Effect of dimension (cylinder) (Hirotsu)

Inhomogeneities during transition

Hirotsu Sato‐Matsuo and Tanaka

Okajima, Harada, Nishio, and Hirotsu, J. Chem. Phys. 116, 9068 (2002)

Kinetics of Volume Phase Transition in Poly(N‐isopropylacrylamide ) Gels

↑ T

Simple Bending

p‐NIPAAm

p‐Acrylamide

Zhibing Hu et al.

Klein, Efrati and Sharon, Science 315,1116 (2007)

Shaping of Elastic Sheets by Prescription of Non‐Euclidean Metrics

Rel. length at 5

0 C

NIPA concentration

Designed Responsive Buckled Surfaces by Halftone Gel Lithography

Kim et al., Science, 335, 1201 (2012)

Hydrogel CoatingsSubstrate

Hydrogel Coating

COMMON TECHNIQUES

Spin coating followed by photo(UV)polymerizationKnife castingPlasma polymerizationMonomer vapor deposition with (photo)initiators, crosslinkersLayer by layer polymer depositionSpray coating of polymer

APPLICATIONS

Lubricity enhancement/BioadhesionTemperature controlCoating of tablets to prevent acid or enzyme attach in GI systemMatch mechanical, surface properties of host tissueAntifouling, antibacterial (e.g. catheters), suppression of foreign body responseSensing/actuating in response to environmental stimuli (physical, chemical)Tissue culture/Tissue engineering—Soft tissue modelingImmunoprotection of transplanted cells (e.g. pancreatic beta cells)

Mechanical Instabilities of GelsJulien Dervaux and Martine Ben Amar, Annu. Rev. Condens. Matter Phys. 2012. 3:311–32

Gel layer

Rigid substrate

Swelling

Swollen hydrogel on soft substrate

Kinetic study of swelling‐induced surface pattern formation and ordering in hydrogel films with depth‐wise crosslinking gradient Murat Guvendiren , Jason A. Burdick and Shu Yang  Soft Matter, 2010, 6, 2044‐2049

The control of stem cell morphology and differentiation by hydrogel surface wrinklesMurat Guvendiren and Jason A. Burdick,  Biomaterials, 31, 6511‐6518 (2010).

36

out

Glucose in

T= 22 – 25 0CNIPA

Magnetic Stirrer

Glucose Glucose oxidase (GOX)

T=24 - 43.90 CDonor Cell (A) Receptor Cell (B)

Temperature Modulated Oscillations

50 mM Saline 50 mM Saline

NaOH addition to maintain pH constant

NIPA GEL

37

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35 40

Time (Hrs)

Vol N

aOH

(0.0

5N) (

ml

42.443.840.838.5

20 mM Glucose/ NIPA membrane

T -> Cell B

24

Cell B Temp Modulation

T

24

T

24

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0 200 400 600 800 1000 1200 1400 1600

Time (Minutes)

d(VN

aOH )/

dt

38

27.5

28

28.5

29

29.5

30

30.5

0 2 4 6 8 10 12 14 16 18 20

NIPA Temperature Modulation/ 20 mM Glucose

43.4 C

26 C

43.4 C

Time (Hours)

Vol.

NaO

H (m

L

Gel at 11 hr Gel at

90 min

39

T = 24T = 24 T = 24 T = 43.4 T = 24 T = 43.4

Glucose

SwollenMembrane

Collapsed Skin

Glucose

Porous Skin

Glucose

Glucose

TIME

A B

Membrane in swollen permeable state

Membrane with collapsed, impermeable skin with development of stress in skindue to clamping

Collapsed skin in with “pores”due to phase separation in responseto stress, leading to intermediateglucose permeability

40

Surgical Endoscope

41

Temperature Oscillator + Endoscope

42

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 50 100 150 200 250 300 350 400

Time (Min)

Volu

me

0.05

N N

aOH

(mL

A

B C DD

Cooling

E F G

H

I

Heating

JK

43

43.4 C

A B

43.4 C

C

43.4 C

D

43.4 C

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 50 100 150 200 250 300 350 400

Time (Min)

Volu

me

0.05

N N

aOH

(mL

A

B C DD

Cooling

E F G

H

I

Heating

JK

440.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 50 100 150 200 250 300 350 400

Time (Min)

Volu

me

0.05

N N

aOH

(mL

A

B C DD

Cooling

E F G

H

I

Heating

JK

43.4 C

D

D

D D

43.4 C 43.4 C

43.4 C

45

41.6 C

A

37.6 C

D

E F

G

34 C 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 50 100 150 200 250 300 350 400

Time (Min)

Volu

me

0.05

N N

aOH

(mL

A

B C DD

Cooling

E F G

H

I

Heating

JK

46

25 38 C

H I J

42.6 C K

43.4 C

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 50 100 150 200 250 300 350 400

Time (Min)

Volu

me

0.05

N N

aOH

(mL

A

B C DD

Cooling

E F G

H

I

Heating

JK

Conclusions• Hydrogels are crosslinked polymer networks that swell substantially in water

• Swelling degree can be controlled by external variables such as temperature, pH, and sugar concentration

• The Flory‐Rehner‐Donnan‐Langmuir (FRDL) model  can account for, at least approximately, the degree of swelling achieved at equilibrium as a function of the external variables.

• Some applications presented

• 3‐d folding can be programmed into 2‐D hydrogels

• Gradients in temperature across hydrogel lead to surface patterns