System: (Polymer gel network + solvent inside) inside the control volume (c.v.)., i.e., the control volume is changing as the get network is expanding due to absorbing the surrounding solvent fluid.
Additional Modeling Assumptions:
The Donnan membrane equilibrium phenomena is due to the balance between the diffusion tendencies and the electrical attraction/repulsion among ions/fixed charge groups. Since the equilibrium condition require the ions in both regions to have equal potential while also requiring to satisfy the electroneutrality condition, the electric potential develop across the gel/membrane interface. It should be noted that the macroscopic electroneutrality is valid for dimensions much greater than the Debye length.
Consider the ionic gel immersed in ionic solvent bath as shown above. Assuming the concentration of the mobile ions in both the gel and the solvent bath are much smaller that the concentration of polymer or solvent, the mobile ion concentration can be considered as dilute, and the electrochemical potential of the ions can be described by
where
where
or, since is the same for the g and b phases, we have
or
where is , the electrical potential difference across the gel-bath interface. Since this electrical potential difference is common to all ions, the activity coefficient ratio across the interface of different ions, say i and j, can be related as
For the 1:1 electrolyte where z_{i} = 1 and z_{j} = -1, the above expression reduces to
For the case where the activity coefficients in the g- and b-phases are equal or for the ideal dilute solution case, we can also write
or
The electroneutrality conditions of both the bath and the gel are
for external bath
for inside the gel/membrane
The Donnan equilibrium and electroneutrality conditions, along with charge conservation relations, can be used to compute individual ion concentrations. Stictly speaking, the above result applies to true equilibrium case. However, in general, it can also be applied to non-equilibrium case as long as the net transport is much slower than the diffusion or the drift components of the transport process.
The transport of ions due to diffusion and migration in the electrolytic medium is described by widely used relation (neglecting the connective contribution)
Assuming both the concentration and electrical potential gradient inside the gel near its interface is linear and considering only one dimensional case for simplification, then the average ion transport into the gel at the interface can be written as
where , is the ion concentration and electrical potential differences between inside the gel near its interface and outside bath, i.e. , and is the distance in between. The first term describes the diffusion component due to the concentration gradient, and the second term describes the migration component due to the electric potential gradient. Since the electric potential gradient is commonly experienced by all ions, it is the source of coupling effect among ions. Studying the expression for the ion transport above, one can easily represent this using the bond graph structure
The ions with greater diffusivity will tend to move across the medium at faster rate than the ions with lower diffusivity. However, such tendency for charge separation induces local electric field among ions, and it in turn acts to retard the faster ions and to accelerate the slower ions. For instance, if the faster ions are positively charged whereas the slower ions are negatively charged, then positive local electric potential gradient will build up to retard the positive ions and to accelerate the negative ions. Such "local" electric field, which acts in the scale of a few Debye length, is quite large and it is rather difficult to induce charge separation in macroscopic scale. Therefore, in macroscopic scale, the electroneutrality prevails.
The electrical potential coupling among ions can be modeled as a simple ideal parallel plate capacitor (Oster)
where
The bond graph representation is simply
Note that the induced electric potential here only models the aporoximate effect of bulk electroneutrality in uniform medium, and therefore it does not model the electric potential difference between two regions of different medium.
In addition to the macroscopic electroneutrality coupling among ions in each region of uniform electrolytic medium (bulk electroneutrality), there also exists electric potential gradient between two regions of unequal ion concentrations. For example, the Donnan equilibrium membrane potential is the electric potential difference across the membrane induced by the balance between the individual ion diffusion tendencies and the bulk electroneutrality in each region. In order to describe the electric potential profile exactly, one needs to solve the Gauss' law which relates the electric field gradient to the net charge at a point
where is the net charge (in fact, the electroneutrality effect in previous section may also be solved using Gauss' law directly). However, not only the analysis is non-trivial and computation intensive, the exact electric potential profile is not of concern for our purpose. Therefore, we will model the electric potential difference between two uniform media region as the average of the individual ion specie's contribution at near equilibrium (quasistatic) condition. Defining the reference electrical potential to be at the surrounding bath when it is electrically neutral, the individual ion contribution and the average electric potential at a region are modelled as
The total electric potential at a region then may be expressed as
Applying this expression to the surrounding bath, we get
and the electrical potential of surrounding bath becomes zero, i.e. reference potential, when it is electrically neutral as defined previously. The bond graph representation of the ion transport across two region based on the above results is
Applying the above bond graph structures to the case of 1:1 electrolyte transport at the gel/membrane interface, we then have
In most cases, the external solvent bath may be modelled as the ideal ions/solvent source as well as the electric potential reference ground where the bulk electroneutrality is always satisfied, in which case the bond graph representation then simpliflies to
Chemical Capacitances:
With the assumption of dilute solution, the chemical potential of ion in the gel may be described by
Treating the polymer molecule as solvent/ion size molecule with equivalent scaling factor, x, the effective mole fraction of ion, x_{i} is then
and
Safely assuming the solvent concentration changes at much slower rate than the individual ion concentration, the rate of change of ion's chemical potential may be approximated by
Electrical Capacitances:
As mentioned previously, the capacitance field describing the Donnan potential and the electroneutrality with respect to surrounding bath as the reference potential is expressed as
or
The rate of change of electric potential is then
Resistance:
The ion transport kinetics of the resistive element is simply
Comparing this expression to the generally used ion transport equation
we get the expression for the ion's transport coefficient Ki
For our model, we will use the average value of the above Ki's.
The resulting system equations for the case of 1:1 binary electrolyte are then
For the purpose of testing the ion transport model at the gel/membrane interface, the change in solvent concentration of the control volume is not included. In addition, the diffusivities and mobilities of H+ and Cl- ions in the gel control volume is assumed to be in the same order of magnitude as in the aqueous solution. Note that both the diffusivity and the mobility of H+ ion is much greater than of Cl- ion. In general, this implies much faster transport of H+ than of Cl- in neutral medium. However, in a highly positively charged medium, the H+ will be electrically repelled from the medium, and the Cl- concentration in the gel is thus expected to be much higher than H+'s. Therefore, one can expect the K+ to be much smaller than the K- since the average concentration of the H+ will be much smaller than the Cl- at the interface.
The computer simulation results based on these assumptions and above parameters are shown in the above figures with two different time scales. The dashed lines are the theoretically expected equilibrium values whereas the solid lines are the simulation result values. As expected, the average transport coefficient of Cl- obtained from the simulation was much greater than of H+ with values 7.05x10-3 and 3.23x10-4 respectively, resulting in overshoot of Cl- concentration before settling to equilibrium value. The situation is reversed for the case of negatively charged gel/membrane system, i.e. zm = -1, as shown in the computer simulation results below.
The simulation values converged to the expected values very well, and the overall ion transport process took roughly 1x10-11 seconds which is many magnitudes smaller than the typical response time of the gel/membrane. Therefore, this simulation result justifies the assumption of quasi-equilibrium between the surrounding bath and the region near the interface (assuming the typical gel/membrane are thicker than the Debye length by several orders of magnitude). For such quasi-equilibrium case with fixed charge density, the above system equations then reduces to the Donnan equilibrium condition and the bulk electroneutrality at the interface region.
It should be noted that the transport rate of establishing Donnan equilibrium condition is extremely fast due to large local electric field coupling among ions/fixed charged groups within the Debye length of the solvent bath/gel interface. Therefore, in most case, the Donnan equilibrium condition and the bulk electroneutrality is valid even in the non-equilibrium case with net transport, and they may be employed as the boundary conditions for the states of gel/membrane's bulk units. In such cases, the combined effect of Donnan equilibrium condition and bulk electroneutrality can be treated as effort (electric potential) source, and the bond graph representation further reduces to
The electric potential source term on the right hand side exerts the necessary electric potential on the ions based on the concentrations of the both positive and negative ions and membrane fixed charges to achieve both the Donnan equilibrium condition and bulk electroneutrality, i.e., combining the bulk electroneutrality and Donnan equilibrium condition yields the mole fractions and the electric potential of the ions
The bulk of the gel/membrane model is identical to the gel/membrane interface model discussed in previous section. It is assumed to be quasistatic with uniform state properties such as ion/solvent concentrations, fixed charge density, and temperature, and as with the interface, the strong electric field coupling among ions and fixed charged prevails inducing macroscopic bulk electroneutrality at equilibrium
and ion transport is described by
The bond graph structure of the gel/membrane interface and two gel/membrane bulk units connected in series is
According to above bond graph model, the electric potential differences can exist even among the bulk units during the transport process since the internal pH and subsequently the fixed charge density of each unit may well differ from others. This electric potential differences would vanish at equilibrium though when the uniform fixed charge density is achieved among interface/bulk units. The governing equations of each bulk unit are identical to the ones for the interface unit. The governing dynamic equations for the above system of the interface unit connected to two bulk units in series are
where
where the solid and dotted lines represent the responses of the bulk units 1 and 2 respectively. The responses of the interface unit, which are not shown in the graphs above, settled to near the equilibrium values in the order of 1x10-9 seconds as with the Donnan interface model simulation case discussed previously, and this results provide justification for treating the interface unit as a quasistatic one of which the states are simply determined by Donnan interface condition and the electroneutrality law. This simplified system model with quasistatic interface model incorporated is computer simulated, and the resulting responses of the bulk units 1 and 2 are plotted below.
As expected, the interface unit's much faster kinetic has negligible influence on the bulk units' behaviors, and the results are almost identical to the results of the initial model which incorporates the interface unit dynamics. The simplified model is further simulated to t = 1x10-4 seconds where the states converge to the equilibrium values well as shown below.
Similarly, the computer simulation results for the case of negatively charged gel/membrane system are shown below.
As the ion moves into the gel/membrane network, it may also react with the fixed charged side groups. Namely diffusing H+ or OH- in solvent can react with corresponding conjugate basic or acidic structures on the network, and it was shown that such diffusion-reaction can significantly slow down the overall ion transport process (Grodzinsky). As the ions are moving into the gel/membrane network while reacting with the fixed charges of network simultaneously, not only the overall kinetic of the ion's transport is affected, also the concentrations of both the mobile ions and the fixed charge groups are changed as well, subsequently changing the overall equilibrium behavior. Therefore, proper integration of the dynamic model of ion diffusion/reaction process is necessary.
Assuming the predominant chemical reaction is the acid/base reaction between the H^{+}/OH^{-} and the fixed charge groups, we will limit the discussion to acid/base reaction only. However, the results can easily be translated to other types of reactions as well.
Let's consider the case of gel/membrane with negatively charged fixed side groups. For the side groups with the pKa below 7, the side group is positively charged when the internal pH is above the pKa, whereas the side group is neutral when the internal pH is below the pKa. When the pH is equal to the pKa, this implies exactly half of the total side group is dissociated. In acidic environment, there are two reactions occuring simultaneously. First, the neutral side group of gel/membrane network can dissociate into the anion and the hydronium ion
where
The chemical reaction is represented by a two-port resistance field, and the thermal (entropy generation) port is not included. In the case where the thermal port is of importance, it can be modified as
Second, the hydrogen ion diffusing in from outside can associate with the network-bound anion, A- to form the neutral side group.
where
Combining the above two reactions, we have
Since Hd+ is identical to Hb+, the overall bond graph representation of diffusion/reaction simplies to
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According to the simple mass-action law for elementary chemical reaction such as
where
where the equilibrium constant Keq in this case would then be
For simplicity, let's model both the fixed sidegroups and the hydrogen ions as dilute solutions, i.e.,
where xAH, xA- and xH+ are mole fractions of -AH, -A- and H+ respectively. Since concentrations of -AH, -A- and H+ are much smaller than the concentration of solvent, the mole fractions can be related to corresponding concentrations as
where vs is the molar volume of the solvent. In addition, the rate of concentration changes of -AH is related to the rate of mole of -AH change by
The reaction kinetic expression can then be expressed as
where
In many cases, simplifications can be made with assumption that the reaction occurs at much faster rate than the diffusion, and thus can be considered as "instantaneous" and "always" at equilibrium (quasi-equilibrium). This does not imply that the concentrations of -A-, H+ or -AH do not vary with time, but they change at a rate much slower than the reaction rate, limited by the diffusion of the hydrogen ion into the gel/membrane unit. The quasi-equilibrium reaction model says at every instant of time
Since the total possible number of the binding site -A is fixed
the equilibrium constant Keq can also be written as
Differentiating this with respect to time, we get
For such quasi-equilibrium reaction, the bond graph representation of the diffusion-reaction then reduces to
Note that the bond graph model shows only two out of total three capacitances have integral causality implying only two capacitances are independent energy storage elements whereas the other one is dependent. This result is in agreement with the above analysis since the -AH and -A- concentrations are dependent of each other. As with the Donnan quasi-equilibrium model considered previously, the dissipated energy (heat of reaction) and the entropy generation are assumed to be unimportant in the overall dynamics of the ion's transport for the quasi-equlibrium chemical reaction model.
Combining the models of the gel/membrane's Donnan interface and the bulk with the diffusion limited chemical reaction described above for the case of binary solution (H+, Cl-), we arrive at the following model. The chloride ion is a completely dissociated ion of the strong acid, HCl, and is assumed not to react or bind with other ions or fixed charges. The dissociation of gel bound groups -AH produces fixed charges -A- and hydronium ion H+ which in turn influences the ion transport in the gel through changing both the H+ concentration and the electric potential. The electric field coupling, however, does not affect the dissociation chemistry of -AH since the valences of produced charges are exactly opposite (i.e. +1 for H+ and -1 for -A-) and therefore their electrical effects cancel each other out exactly. For the complete model of polyelectrolyte gel, following model needs to be incorporated into the model describing the swelling behavior of the gel.
Similarly to the nonelectrolyte gel model, the quasistatic differential energy change of the control volume due to the mixing/diffusion within the gel itself can be expressed as
Since the ion concentration inside the gel is much less than the concentrations of either the polymer or the solvent, the contribution of ions to the change in free energy is assumed to be additive (Flory), i.e.,
Similary, the entropy change due to mixing is
or
where
(van't Hoff equation)
The bond graph representation of the polyelectrolyte gel's mixing process derived above can be expressed as
Combining the above results for the binary mobile ion specie case, we have