Hydrogels in charged solvents

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Hydrogels in charged solvents

Peter Koˇsovan, Christian Holm, Tobias Richter

Institute for Computational Physics

Pfaffenwaldring 27 D-70569 Stuttgart Germany

27. May 2013

Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 1 / 21

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Outline

1 Introduction

2 Motivation

3 Simulation Setup

4 Results

5 Conclusions

6 References

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Introduction

Hydrogels are charged polymer-networks

Hydrogels can swell tremendously in aqueous solutions Swelling behavior can be influenced by external parameters

Pressure pH

Concentration of electrolytes

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Motivation

Experiment [1]

Johannes H¨ opfner at KIT was

investigating the desalination capacity of Hydrogels

Poly-acrylic-acid Idea:

Swelling Hydrogel in salty water solution

extract Hydrogel from salt solution Deswell/Compress Hydrogel and squeeze out the incorporated water

Figure 1: Experimental setup,

taken from [1]

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Motivation

Experimental Results The process is reversible Desalination capacity per cycle is up to 35% [1]

Open Questions

How does the desalination work on the microscopic level

What influences the desalination capacity

Crosslinking density Charge density

Figure 2: Salt concentration of the extracted water, taken from [1].

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Motivation

Experimental Results The process is reversible Desalination capacity per cycle is up to 35% [1]

Open Questions

How does the desalination work on the microscopic level

What influences the desalination capacity

Crosslinking density Charge density

Figure 2: Salt concentration of the

extracted water, taken from [1].

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Motivation

Experimental Results The process is reversible Desalination capacity per cycle is up to 35% [1]

Open Questions

How does the desalination work on the microscopic level

What influences the desalination capacity

Crosslinking density Charge density

Figure 2: Salt concentration of the extracted water, taken from [1].

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Simulation Setup

Simulation setup

Tetrafunctional polymer-network with 16 chains and 8 nodes and equally spaced charged on the chains

Periodic boundary conditions Explicit counterions

Langevin thermostat → Brownian dynamics

Simulation in (NVT) and (µVT) ensemble as well as testing those together with the “npt-isotropic” barostat in ESPResSo

Translation of experiment to computer simulation

1 Determine chemical potential and pressure in salt solution

2 Determine equilibrium swelling and internal salt concentration at a certain external salt concentration c _s → fixed external chemical potential

equilibrium conditions p = p

salt

(c

s

) and

µ

i

= µ

^?_i

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Simulation Setup

Simulation setup

Tetrafunctional polymer-network with 16 chains and 8 nodes and equally spaced charged on the chains

Periodic boundary conditions Explicit counterions

Langevin thermostat → Brownian dynamics

Simulation in (NVT) and (µVT) ensemble as well as testing those together with the “npt-isotropic” barostat in ESPResSo

Translation of experiment to computer simulation

1 Determine chemical potential and pressure in salt solution

2 Determine equilibrium swelling and internal salt concentration at a certain external salt concentration c _s → fixed external chemical potential

equilibrium conditions p = p

_salt

(c

_s

) and µ

i

= µ

^?_i

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Simulation Setup

Figure 3: Snapshot of the simulated system

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Simulation Setup

Simulation in the grand canonical ensemble Using the (NVT) ensemble simulation ESPResSo already provides Adding particle exchange with external reservoir → Monte Carlo steps Acceptance probability according to [2]

acc(N → N + 1) = min

1, V

Λ ³ (N + 1) exp (−β (−µ _out + ∆U ))

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acc(N → N − 1) = min

1, Λ ³ N

V exp (−β (µ _out + ∆U ))

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Rewritten acceptance rates

acc(N → N + 1) = min

1, 1

% _in exp (−β (−µ ^ex _out − 1/β ln % _out + ∆U ))

(3) acc(N → N − 1) = min (1, % in exp (−β (µ ^ex _out + 1/β ln % out + ∆U ))) (4)

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Simulation Setup

Simulation in the grand canonical ensemble

acc(N → N + 1) = min

1, V

Λ ³ (N + 1) exp (−β (−µ _out + ∆U ))

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acc(N → N − 1) = min

1, Λ ³ N

V exp (−β (µ out + ∆U ))

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Elimination of Λ

with µ _out = µ ^ex _out + k _B T ln(% _out Λ ³ )

Rewritten acceptance rates

acc(N → N + 1) = min

1, 1

% in

exp (−β (−µ ^ex _out − 1/β ln % _out + ∆U ))

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acc(N → N − 1) = min (1, % in exp (−β (µ ^ex _out + 1/β ln % out + ∆U ))) (4)

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Simulation Setup

Simulation in the grand canonical ensemble

acc(N → N + 1) = min

1, V

Λ ³ (N + 1) exp (−β (−µ _out + ∆U ))

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acc(N → N − 1) = min

1, Λ ³ N

V exp (−β (µ out + ∆U ))

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Rewritten acceptance rates

acc(N → N + 1) = min

1, 1

% in

exp (−β (−µ ^ex _out − 1/β ln % out + ∆U ))

(3) acc(N → N − 1) = min (1, % _in exp (−β (µ ^ex _out + 1/β ln % _out + ∆U ))) (4)

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Simulation Setup

−2

−1.5

−1

−0.5 0 0.5 1

10 ⁻⁴ 10 ⁻³ 10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ µ ex [k B T]

c _s [mol/L]

Widom method extended Debye Hückel Experiment

Figure 4: Chemical potential obtained with for Bjerrum-length equal 2 together with a

fit of the extended Debye-H¨ uckel model and experimental activity coefficients

measurements done by Truesdell [3].

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Simulation Setup

−2

−1.5

−1

−0.5 0 0.5 1

10 ⁻⁴ 10 ⁻³ 10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ µ ex [k B T]

c _s [mol/L]

Widom method extended Debye Hückel Experiment

Figure 4: Chemical potential obtained with for Bjerrum-length equal 2 together with a fit of the extended Debye-H¨ uckel model and experimental activity coefficients measurements done by Truesdell [3].

µ _DB = −

√ 2πλ

3 2

b

√ % 1 + 8πa ^? √

% (5)

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Simulation Setup

Check of the Grand canonical Algorithm:

Running the simula- tion with an empty box, no hydrogel and measure the salt density

outside density:

10 ⁻³

µ _ex obtained from Widom particle insertion

0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 0.0011 0.0012

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Density

Simulation time a

Figure 5: Empty box test.

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Simulation Setup

Check of the Grand canonical Algorithm:

Running the simula- tion with an empty box, no hydrogel and measure the salt density

outside density:

2.7e − 3 µ ex obtained from Widom particle insertion

1.4e-03 1.6e-03 1.8e-03 2.0e-03 2.2e-03 2.4e-03 2.6e-03 2.8e-03 3.0e-03 3.2e-03

0 5000 10000 15000 20000

cin

t

79 1/4

1.5e-03 2.0e-03 2.5e-03

0 500 1000 1500

Figure 5: Development of the salt concentration inside a hydrogel (N=79 f=1/4 c

s

= 0.1mol/L ˆ =2.7e − 3)

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Simulation Setup

Determining the pressure dependence on Volume for a given external salt solution

0.009 0.0095 0.01 0.0105 0.011 0.0115 0.012

30 32 34 36 38 40 42 44

Internal pressure

Box length [ σ ] N=59 f=1/4 external pressure Fit

Figure 6: Total pressure dependence on volume for a salt solution with concentration

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Simulation Setup

Donnan Theory

Philipse et. al derived an equation for colloid system in contact with an infinite reservoir of salt [4].

% −

% ext

= − % c

2% ext

+ s

1 + % c

2% ext

2 , (6)

% − is the salt concentration in the hydrogel

% _c = ^{mN f} _V

hg

, m is the number of polymer chains and N the number of monomers per chain, f is the charge fraction of the hydrogel

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Simulation Setup

Osmotic Donnan Model

Combining Donnan theory for the salt partitioning with an equation for the pressure balance

Equation of state

Minimal model: the chains in the hydrogel behave as ideal gaussian chains

(i) : mN f

V _hg + % − = % + (7)

(ii) : % ² _ext = % − % + (8)

(iii) :

% ₊ + % − − 1 N b ² R e

k _B T + p _ext = 2% _ext k _B T (9)

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Simulation Setup

Osmotic Donnan Model

Combining Donnan theory for the salt partitioning with an equation for the pressure balance

Equation of state

Minimal model: the chains in the hydrogel behave as ideal gaussian chains

p _in = p _out p ^osm _in + p _elastic + p extern = p ^osm _salt

(i) : mN f

V hg

+ % − = % ₊ (7)

(ii) : % ² _ext = % − % ₊ (8)

(iii) :

% + + % − − 1 N b ² R _e

k B T + p ext = 2% ext k B T (9)

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Simulation Setup

Osmotic Donnan Model

Combining Donnan theory for the salt partitioning with an equation for the pressure balance

Equation of state

Minimal model: the chains in the hydrogel behave as ideal gaussian chains

(i) : mN f

V _hg + % − = % ₊ (7)

(ii) : % ² _ext = % − % + (8)

(iii) :

% ₊ + % − − 1 N b ² R e

k _B T + p _ext = 2% _ext k _B T (9)

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Results

Equilibrium swelling

0 50 100 150 200 250 300 350 400 450

0.05 0.1 0.15 0.2 0.25 0.3

Swelling degree

c_s [mol/L]

N=39 f=1/8 theory N=39 f=1/8 N=39 f=1/4 theory N=39 f=1/4 N=39 f=1/2 theory N=39 f=1/2

0 100 200 300 400 500 600 700 800

0.05 0.1 0.15 0.2 0.25 0.3

Swelling degree

c_s [mol/L]

N=59 f=1/4 theory N=59 f=1/4 N=59 f=1/2 theory N=59 f=1/2

Figure 7: Dependence of the equilibrium swelling degree on the salt concentration of the external reservoir. The dashed lines are the predictions of the osmotic Donnan model.

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Results

Donnan theory

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 cint [mol/L]

f cp [mol/L]

Donnan 0.01 mol/L N=39 N=59 N=79 f=1/8 f=1/4 f=1/2

0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 cint [mol/L]

f c_p [mol/L]

Figure 8: Testing the Donnan prediction for the salt partitioning against the simulation for two intermediately high salt concentrations: left: 0.01 mol/L, right: 0.1 mol/L.

% −

= − % _c +

s 1 +

% _c 2

, (10)

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Results

0.29 0.295 0.3 0.305 0.31 0.315 0.32 0.325

0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 cint [mol/L]

f c_p [mol/L]

Donnan 0.37 mol/L N=59 f=1/4

0.13 0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 cint [mol/L]

f c_p [mol/L]

Donnan 0.1 mol/L N=59 f=1/4

Figure 9: Testing the Donnan prediction for salt partitioning for c

s

= 0.37mol/L (left) and 0.2 mol/L with randomly placed charges on the hydrogel with N=59 f=1/4.

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Results

Donnan theory in the limit of no electrostatics (λ

B

= 0)

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17

0.05 0.1 0.15 0.2 0.25 0.3

cint [mol/L]

f c_p [mol/L]

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Results

Simulation in µpT

Figure 11: Pressure versus volume for the N=59 f=1/4 hydrogel. The circles indicate the start configurations for testing simulations in (µpT ).

0.0094 0.0095 0.0096 0.0097 0.0098 0.0099 0.01 0.0101 0.0102 0.0103

28 30 32 34 36 38 40 42 44

pressure

Box length [σ] external pressure

0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 0.0055

0 5000 10000 15000 20000 25000 30000 cin

t

Above Below

Figure 12: Left: The final volumes that were obtained using the (µpT ) ensemble simulation for the two different start configurations. Right: The salt density inside the hydrogels for the simulations in (µpT ).

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Results

Simulation in µpT

0.0094 0.0095 0.0096 0.0097 0.0098 0.0099 0.01 0.0101 0.0102 0.0103

28 30 32 34 36 38 40 42 44

pressure

Box length [σ] external pressure

0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 0.0055

0 5000 10000 15000 20000 25000 30000 cin

t

Above Below

Figure 11: Left: The final volumes that were obtained using the (µpT ) ensemble

simulation for the two different start configurations. Right: The salt density inside

the hydrogels for the simulations in (µpT ).

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Conclusions

The equilibrium swelling for hydrogels in various external salt concentrations could be determined

The Osmotic Donnan model and the simulations qualitatively agree The Donnan theory for the salt partitioning show increasing

deviations for increasing charge fractions of the hydrogel

The linear charge density on the hydrogel chains breaks the assumption of homogeneity

This seems to remain true even for very high salt concentrations (strong screening of the electrostatics) and randomly placed charges on the hydrogel backbone

In the case of no electrostatics (the salt behaves almost as ideal gas) the donnan theory and the simulations fully agree

The barostat in combination with the grand canonical ensemble does not work to predict the equilibrium swelling volume

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Conclusions

Outlook

Improvement of the “Donnan”–model

Poisson-Boltzmann Cell model to treat the elastic properties in an electrolyte solution

Track down the origin of the deviations

Having a closer look at the distribution of ions around the chains in

the hydrogel

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References

[1] Johannes H¨ opfner, Christopher Klein, and Manfred Wilhelm.

A novel approach for the desalination of seawater by means of reusable poly(acrylic acid) hydrogels and mechanical force.

Macromol. Rapid Commun., 31:1337, 2010.

[2] Daan Frenkel and Berend Smit.

Understanding Molecular Simulation.

Academic Press, San Diego, second edition, 2002.

[3] Alfred H. Truesdell.

Activity coefficients of aqueous sodium chloride from 15

^◦

to 50

^◦

Hydrogels in charged solvents