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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
Okajima, Harada, Nishio, and Hirotsu, J. Chem. Phys. 116, 9068 (2002)
Kinetics of Volume Phase Transition in Poly(N‐isopropylacrylamide ) Gels
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
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