Hydrogels in charged solvents

Hydrogels in charged solvents Peter Košovan, Christian Holm, Tobias Richter 27. May 2013 Institute for Computational Physics Pfaffenwaldring 27 D-70569 Stuttgart Germany Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 1 / 21

Outline Outline 1 Introduction 2 Motivation 3 Simulation Setup 4 Results 5 Conclusions 6 References Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 2 / 21

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 Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 3 / 21

Motivation Experiment [1] Johannes Höpfner 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] Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 4 / 21

Motivation Experimental Results The process is reversible Desalination capacity per cycle is up to 35% [1] Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 5 / 21

Motivation Experimental Results The process is reversible Desalination capacity per cycle is up to 35% [1] Figure 2: Salt concentration of the extracted water, taken from [1]. Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 5 / 21

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]. Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 5 / 21

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 Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 6 / 21

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 Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 6 / 21

Simulation Setup Figure 3: Snapshot of the simulated system Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 7 / 21

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] ( ) V acc(n N + 1) = min 1, Λ 3 (N + 1) exp ( β ( µ out + U)) (1) ( ) acc(n N 1) = min 1, Λ3 N V exp ( β (µ out + U)) (2) Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 8 / 21

Simulation Setup Simulation in the grand canonical ensemble ( ) V acc(n N + 1) = min 1, Λ 3 (N + 1) exp ( β ( µ out + U)) ( ) acc(n N 1) = min 1, Λ3 N V exp ( β (µ out + U)) (1) (2) Elimination of Λ with µ out = µ ex out + k B T ln(ϱ out Λ 3 ) Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 8 / 21

Simulation Setup Simulation in the grand canonical ensemble ( ) V acc(n N + 1) = min 1, Λ 3 (N + 1) exp ( β ( µ out + U)) ( ) acc(n N 1) = min 1, Λ3 N V exp ( β (µ out + U)) (1) (2) Rewritten acceptance rates ( ) 1 acc(n N + 1) = min 1, exp ( β ( µ ex out 1/β ln ϱ out + U)) ϱ in acc(n N 1) = min (1, ϱ in exp ( β (µ ex out + 1/β ln ϱ out + U))) (4) (3) Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 8 / 21

Simulation Setup µ ex [k B T] 1 0.5 0 0.5 1 1.5 Widom method extended Debye Hückel Experiment 2 10 4 10 3 10 2 10 1 10 0 10 1 c s [mol/l] Figure 4: Chemical potential obtained with for Bjerrum-length equal 2 together with a fit of the extended Debye-Hückel model and experimental activity coefficients measurements done by Truesdell [3]. Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 9 / 21

Simulation Setup µ ex [k B T] 1 0.5 0 0.5 1 1.5 Widom method extended Debye Hückel Experiment 2 10 4 10 3 10 2 10 1 10 0 10 1 c s [mol/l] Figure 4: Chemical potential obtained with for Bjerrum-length equal 2 together with a fit of the extended Debye-Hückel model and experimental activity coefficients measurements done by Truesdell [3]. 3 2πλ 2 b ϱ µ DB = 1 + 8πa ϱ Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 9 / 21 (5)

Simulation Setup Check of the Grand canonical Algorithm: Running the simulation with an empty box, no hydrogel and measure the salt density outside density: 10 3 µ ex obtained from Widom particle insertion Density 0.0012 a 0.0011 0.001 0.0009 0.0008 0.0007 0.0006 0.0005 0.0004 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Simulation time Figure 5: Empty box test. Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 10 / 21

Simulation Setup Check of the Grand canonical Algorithm: Running the simulation with an empty box, no hydrogel and measure the salt density outside density: 2.7e 3 µ ex obtained from Widom particle insertion c in 3.2e-03 3.0e-03 2.8e-03 2.6e-03 2.4e-03 2.2e-03 2.0e-03 1.8e-03 1.6e-03 1.4e-03 2.5e-03 2.0e-03 1.5e-03 0 500 1000 1500 0 5000 10000 15000 20000 t 79 1/4 Figure 5: Development of the salt concentration inside a hydrogel (N=79 f=1/4 c s = 0.1mol/L ˆ=2.7e 3) Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 10 / 21

Simulation Setup Determining the pressure dependence on Volume for a given external salt solution 0.012 0.0115 N=59 f=1/4 external pressure Fit Internal pressure 0.011 0.0105 0.01 0.0095 0.009 30 32 34 36 38 40 42 44 Box length [σ] Figure 6: Total pressure dependence on volume for a salt solution with concentration 0.2mol/L. Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 11 / 21

Simulation Setup Donnan Theory Philipse et. al derived an equation for colloid system in contact with an infinite reservoir of salt [4]. ϱ = ϱ ( ) 2 c ϱc + 1 +, (6) ϱ ext 2ϱ ext 2ϱ ext ϱ is the salt concentration in the hydrogel ϱ c = mnf 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 Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 12 / 21

Simulation Setup Osmotic Donnan Model Combining Donnan theory for the salt partitioning with an equation for the pressure balance Equation of state Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 13 / 21

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 osm in p in = p out + p elastic + p extern = p osm salt Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 13 / 21

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) : mnf V hg + ϱ = ϱ + (7) (ii) : ϱ 2 ext = ϱ ϱ + (8) ( (iii) : ϱ + + ϱ 1 ) Nb 2 k B T + p ext = 2ϱ ext k B T (9) R e Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 13 / 21

Results Equilibrium swelling Swelling degree 450 400 350 300 250 200 150 100 50 0 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 0.05 0.1 0.15 0.2 0.25 0.3 c s [mol/l] Swelling degree 800 700 600 500 400 300 200 100 0 N=59 f=1/4 theory N=59 f=1/4 N=59 f=1/2 theory N=59 f=1/2 0.05 0.1 0.15 0.2 0.25 0.3 c s [mol/l] 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. Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 14 / 21

Results Donnan theory c int [mol/l] 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 Donnan 0.01 mol/l N=39 N=59 N=79 f=1/8 f=1/4 f=1/2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 f c p [mol/l] c int [mol/l] 0.09 0.085 0.08 0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04 Donnan 0.1 mol/l N=39 N=59 N=79 f=1/8 f=1/4 f=1/2 0.035 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 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. ϱ ϱ ext = ϱ c 2ϱ ext + 1 + ( ϱc 2ϱ ext ) 2, (10) Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 15 / 21

Results 0.325 0.32 Donnan 0.37 mol/l N=59 f=1/4 0.17 0.165 Donnan 0.1 mol/l N=59 f=1/4 0.315 0.16 c int [mol/l] 0.31 0.305 c int [mol/l] 0.155 0.15 0.145 0.3 0.14 0.295 0.135 0.29 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 f c p [mol/l] 0.13 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 f c p [mol/l] 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. Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 16 / 21

Results Donnan theory in the limit of no electrostatics (λ B = 0) c int [mol/l] 0.17 0.16 0.15 0.14 0.13 Donnan 0.2 mol/l N=39 N=59 N=79 f=1/8 f=1/4 f=1/2 0.12 0.11 0.1 0.05 0.1 0.15 0.2 0.25 0.3 f c p [mol/l] Figure 10: Comparison of the partitioning prediction of Donnan and simulation data for c = 0.2mol/L. Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 17 / 21

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 ). Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 18 / 21

Results Simulation in µpt 0.0103 0.0102 external pressure 0.0055 0.005 Above Below 0.0101 0.0045 pressure 0.01 0.0099 0.0098 0.0097 c in 0.004 0.0035 0.003 0.0096 0.0025 0.0095 0.002 0.0094 28 30 32 34 36 38 40 42 44 Box length [σ] 0.0015 0 5000 10000 15000 20000 25000 30000 t 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 ). Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 18 / 21

Conclusions 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 Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 19 / 21

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 Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 20 / 21

References [1] Johannes Höpfner, 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 c measured with a glass electrode. Science, 161(3844):884 886, 1968. [4] A Philipse and A Vrij. The donnan equilibrium: I. on the thermodynamic foundation of the donnan equation of state. Journal of Physics: Condensed Matter, 23(19):194106, 2011. Tobias Richter (ICP, Stuttgart) Hydrogels 27. May 2013 21 / 21