Consider a network of polymers in contact with a solvent, subject to a mechanical load and geometric constraint, under a constant temperature. The schematic representation is as shown in Figure 1. If we take the stress-free dry network as the reference state, the deformation gradient of the network is defined as:

(1)

where, are the network coordinates of a gel system at reference state and is the network coordinate of a gel system at a current or deformed state [24,25,26].

In the current state, denotes the number of solvent molecules in an elemental volume with a solvent concentration in the gel, . The combination of the two fields and describes the state of the gel system in which the field describes the deformation of the network, while describes the distribution of the solvent molecules in the gel system [24,25]. Let be the external mechanical body force applied on the elemental volume, and the external mechanical force applied on an elemental area. When the network deforms by a small amount, , (a small virtual displacement), the field of mechanical load does work . The integrals extend over the volume and the surface of the network in the reference state. If we assume to be the Helmholtz free energy density, the gel free energy in the elemental volume will be When the field of concentration in the gel changes by the external solution does work , where is the chemical potential of the solvent molecules. Thermodynamics dictates that the change in the free energy of the gel should equal the sum of the work done by the external mechanical force and the external solvent:

(2)

Thus, the gel is in a state of equilibrium characterized by two fields, and . The free energy density of the gel, W, is a function of the deformation gradient of the network, F, and the concentration of the solvent in the gel, C. When the gel equilibrates with the solvent and the mechanical load, the chemical potential, µ, of the solvent molecules is homogeneous in the external solvent and in the gel [24]. Hence the chemical potential will be:

(3)

Introduce a new free-energy function by using a Legendre transformation:

(4)

is a function of the deformation gradient of the network and the chemical potential of the solvent molecules, that is (F, µ). Combining Equations 2 and 4 gives us:

(5)

Equation (5) has the same form as that in solid mechanics. Once the function (µ) is prescribed, it can be solved via finite element method [24].

The material behavior of hydrogel is analogous to a compressible hyperelastic solid and a material model characterized by a free-energy function developed by Flory and Rehner [27,28], Ricka and Tanaka [29], and Brannon-Peppas and Peppas [30]. In the neutral state with the assumption of incompressibility of individual polymeric and water molecules, the free energy form can be expressed as [24]

(6)

is due to the network stretch, and results from the solvent-network blending. The constraint between the deformation field and concentration field can be derived from the assumption of the incompressibility of individual polymeric and water molecules. This constraint is written as , where is the volume per solvent molecule [24]. The free energies of stretching the network and mixing the solvent with the network are:

(7)

(8)

where is the number of polymeric chains per reference volumn and is a dimensionless measure of the enthalpy of mixing. When , the solvent molecules’ diffusion is less energetically favorable. For pH-sensitive gel, the detailed energy form is derived by Marcombe et al. [21]. Following previous work [21,22,23,24], an idealized model is adopted, assuming that the free-energy density of the gel is a sum of several contributions:

(9)

is the energy due to solvent-ion mixing, and is derive from the acidic group dissociation.

The state of the inhomogeneously swollen gel is specified by the following independent fields: and , where is the nominal concentration of the hydrogen ions, is the nominal concentration of the counter-ion that carries charge of sign opposite to the fixed charge, is the nominal concentration of the co-ion that bears a charge of the same sign as the fixed charge. We stipulate that the nominal density of free energy is . According to Marcombe et al. [21], the concentrations of the mobile ions are taken to be low, so that their contribution to the free energy is due to the entropy of mixing, namely,

(10)

where is a reference value of the concentration of species.

The contribution due to the dissociation of the acidic groups is as follows:

(11)

The expression comprises the entropy and the enthalpy of dissociation, where γ is the increase in the enthalpy when an acidic group dissociates. Note that and are the nominal concentrations of the fixed charge and of the associated acidic group. They are constrained by , where is the ratio between acidic groups on a polymer chain and the total number of monomers on the chain and is the volume per monomer. and can be expressed using independent fields as (electroneutrality) and .

The number of particles of species in the gel at the current state divided by the volume of the dry network defines the nominal concentration of the species, C_a. The same number divided by the volume of the gel in the current state defines the true concentration of the species, c_a. The two definitions are related, as . Recall that when the number of particles is counted in units of the Avogadro number, N_A = 6.023 × 10²³, the molar concentration of the species α is designated by [α]; for example,

Recall a relation in continuum mechanics connecting the true stress σ_ij and the nominal stress :, for pH-sensitive hydrogel it can be derived as [21]

(12)

Using the function as specified above, equation (12) becomes

(13)

(14)

(15)

is the osmotic pressure due to the imbalance of the number of ions in the gel and in the external solution, is the osmotic pressure due to the mixing of the network and the solvent. Using the above governing equations, the free-energy function can be coded into subroutine UHYPER for ABAQUS simulation. More complex boundary value problems can also be solved with the same subroutine.

In numerical calculations, we have assumed that the volume per monomer equals the volume per solvent molecule , and the following parameters and values are used in the simulation. The number f is the number of acidic groups on a polymer chain divided by the total number of monomers on the chain, f = 0.05; is the number of polymer chains per unit volume of the dry network, v is the volume per monomer, 1/Nv is the number of monomers per polymer chain, and Nv = 0.001; χ is a dimensionless measure of the enthalpy of mixing, an indicator of the ease of polymer-solvent interaction, χ = 0.1; dissociation constant pK_a = −log₁₀K_a = 4.3, initial stretch = 3.39; is the concentration in external solution, = 6.02 × 10⁻⁵. Besides these basic parameters, the following parameters are involved: kT is the temperature in the unit of energy, kT = 4 × 10⁻²¹ J and kT/v = 4 × 10⁷ Pa at room temperature. A representative value of the volume per molecule is v = 10⁻²⁸ m³. The elastic modulus of the dry network is NkT. For Nv = 10⁻³, the elastic modulus is NkT = 4 × 10⁴ Pa. These values are adopted to study the behavior of hydrogel as a microfluidic valve elaborated in section 3.

Fluid-structure interactions (FSI), the interactions of deformable structures with a surrounding fluid flow, have significant effects on modeling and computational issues. It is also a challenging multi-physics problem. In earlier micro-fluidics study, the gel material was regarded as a solid structure for micro-fluidic valves. Since gel material is very soft, it can easily deform under pressure, and FSI should be considered for the design of gel valve. In the present simulation, the evaluation of fluid flow coupled with the function of the gel valve structure when it swells are addressed. In the coupling system, the nonlinear gel deformation has been carried out by using a finite elements method through commercial software ABAQUS and gel subroutine. The governing equations for incompressible viscous fluid flow under an Arbitrary Lagrangian-Eulerian (ALE) coordinate system can be formulated as follows:

(16)

(17)

In the above-mentioned continuity and momentum equations, , and denote the fluid velocity, grid velocity and gravitational acceleration respectively. The stress tensor is defined as , and e denotes the velocity strain tensor, which is in the form of . In these formulae, p and denote pressure and dynamic viscosity respectively.

The kinematic and dynamic conditions applied to the fluid-structure interface are as follows:

(18)

(19)

where denotes the fluid stresses, and and denotes the solid gel displacement and stress respectively at the FSI interface.

It should be noted that the moving boundary conditions are updated with the latest structural displacements at the interface. The movement of boundary nodes may reduce the mesh quality, resulting in lower computational efficiency and accuracy. Thus, it becomes necessary to adjust the interior nodes to keep good mesh quality. To achieve this, we solve the following Laplace Equation:

(20)

where denotes the increment of the displacement. The latest displacement is then updated by adding the incremental solutions.

A finite element method has been employed to solve the Laplace equation, based on either the latest nodal positions or the initial nodal positions. In the simulation, either two-way coupling method or a one-way coupling method can be used to solve FSI problem. To simplify the process, we use a one-way FSI coupling approach in this study. FLUENT and ABAQUS are used to solve gel FSI problems. At each stage, the deformed configuration of the gel valve under a particular pH value is used to initiate the configuration for computational fluid dynamic (CFD) simulation to obtain the corresponding flow pattern inside the pipe, together with the pressure distribution on the gel valve. In this study, FLUENT was used to perform CFD simulation. The pressure field obtained from CFD is then applied as boundary conditions on to the hydrogel in ABAQUS to calculate the deformation of the gel valve. The deformed geometry will be used as input for CFD simulation to calculate the pressure on the deformed gel valve, which will be used in the mechanical simulation to get further deformation of the gel. We iterate between the fluid flow (CFD) and gel deformation simulations until differences at each step are negligible.

To investigate the swelling behavior of pH-sensitive hydrogel as a micro-fluidic valve, the free swelling ratio as a function of pH at a fixed salt concentration is studied via inhomogeneous gel theory [21]. Figure 2 shows the free swelling ratio of gel volume as a function of pH at a fixed salt concentration. The hydrogel remains steady at low pH values and swells at pH values higher than the pK_a value. This pH-induced swelling behavior can be attributed to the presence of acidic groups bounded to the polymer network, and two limits in Figure 2 should be emphasized: (i) fully associated limit and (ii) fully dissociated limit with respect to the pK_a value.

The swelling of the hydrogel is marginal as pH value varies below pK_a. However, there is a slight increase as the pH value approaches pK_a. The hydrogel expands at a much greater rate when the pH value is greater than the pK_a value, as the osmotic force induces the hydrogel to swell for pH ranging between 4 and 7. When the ionization process reaches its saturation point, a further increase in pH value does not affect the swelling behavior of the hydrogel. This can be observed in Figure 2 where the hydrogel ceases to swell when pH > 7.

To study the mechanical behavior of hydrogel used as a microfluidic valve, the valve structure is constructed as shown in Figure 3. In this assembly, a spherical hydrogel inside the cylindrical pipe is considered. The hydrogel is coated on a solid particle of radius 0.25 mm. The latter is used to hold the hydrogel in place, creating the microfluidic valve from pH-sensitive hydrogel to regulate the flow of an aqueous solution in the pipe. The spherical hydrogel (excluding solid core) has an initial thickness of 0.75 mm coated on the core. The pipe is made of hard material such as steel with an inner radius of 1.7 mm. The fluid flows through the pipe that can also be used to constrain the swelling of the hydrogel. The thickness of the steel pipe is less relevant as we can regard the steel pipe as a rigid body when compared with the soft hydrogel sphere. According to free swelling properties of pH-sensitive hydrogel, the hydrogel sphere will expand at high pH value of the external solution to block the flow and shrink at low pH value to ease the flow. The hydrogel sphere will swell until it touches the inner surface of the pipe to block the flow of fluid. The schematic representation of a hydrogel valve assembly is shown in Figure 3.

Marcombe et al. [21] simulated a similar case of the hydrogel coated on a rigid pillar in a microfluidic channel. They observed the same swelling behavior to regulate the flow but their study did not consider the flow influence on the soft hydrogel deformation (FSI effects). There is an array of the same type of hydrogel posts in a channel; the hydrogel coated on the perimeter will expand at high pH to block the channel. At a lower pH value, the hydrogel shrinks to allow the fluid to pass the microfluidic valve [2]. So far, several studies on hydrogel microfluidic valves have been carried out, but the studies did not consider the FSI effects on hydrogel deformation. Under the FSI condition, the spherical hydrogel will not deform homogeneously upon exposure to a pH value in the environment. FSI occurs when a fluid interacts with the soft hydrogel and exerts pressure on it. This normally causes further deformation in the hydrogel valve altering the fluid flow. As the hydrogel is very soft, it is crucial to consider FSI effects in the design of the hydrogel microfluidic valve, and hence we have performed the study of swelling behavior of the hydrogel with respect to pH values and FSI effects. The solid gel deformation, fluid pressure distribution and fluid velocity distribution are investigated for microfluidic valve with and without FSI effects for comparison. We assume the inlet fluid pressure to be 2000 Pa.

The deformed radii of the spherical gel considered the increase in pH value without FSI effects at various pH values are tabulated in Table 1. We select six particular outer radii of the gel, based on its original outer radius of 1mm and the constraint pipe with a radius of 1.7 mm. The imposed inlet pressure induces pressure on the hydrogel sphere surface which alters further the free swelling shape of the spherical gel. To study the effects of FSI on the gel valve, the deformed configurations of the gel at the above six representative stages are simulated.

At each stage, we provide the pH value with the corresponding deformed configuration of the gel valve, the pressure and the velocity caused by the FSI. Owing to the cylindrical symmetry, the 3-D model can be simplified to a 2-D axis symmetrical model, and we can obtain the pressure on the surface of the hydrogel by simulating fluid flow in the pipe using FLUENT. Figure 4 presents the pressure on the surface of the gel as a function of x at the original state with an inlet pressure of 2,000 Pa. The pressure values obtained on the gel surface are imposed as the boundary conditions to the hydrogel in ABAQUS causing additional gel deformation. We iterate between the fluid flow and gel deformation simulations until differences at each step are negligible. The pressure patterns at various stages are illustrated in Figure 5.

Figure 6 illustrates the deformation shapes and contours of the hydrogel at various stages with the 2,000 Pa inlet pressure.

The deformed configurations of the hydrogel at six selected stages are depicted in Figure 7. The original hydrogel sphere expanded homogeneously along the radius direction without FSI, whereas the shape of the hydrogel sphere is distorted once FSI is considered in the analysis. The shape variations at various pH values can be observed in Figure 7, which compares the deformed configurations of the hydrogel at these six stages.

The deformation of the hydrogel due to FSI increases the effective blocking area. The effective radii of the blocking area corresponds to the maximum vertical distance from the gel centroid, as shown in Figure 7. The effective radii obtained from the analyses with and without FSI are compared in Figure 8. The flow area reduction (FAR) is defined as the ratio between the reduced cross-sectional area due to FSI and the non-pressurized flow area. The variation of FAR at various pH values is presented in Figure 9.

As shown in Figure 9, the flow area reduction follows an exponential relationship with the solvent pH value, which implies that FSI effects on FAR increase more rapidly as the pH value approaches the fully dissociated value. If we let the flow area reduction be 1, the pH value is approximately 6.18. At this particular pH value, solvent can still flow through the pipe, without considering FSI effects. However, this is not actually possible when we predict the deformed configurations of the hydrogel taking FSI into consideration.

The velocity distribution is also affected by FSI and consequently the flow rate. We can simulate the velocity distribution at various stages and calculate the overall flow rate via a surface integration over the cross-sectional area. The velocity distribution at pH = 4.8 is shown in Figure 10 and the maximum velocities of flow at various stages with and without FSI are shown in Figure 11.

Table 1 also indicates detailed results collected at various pH values with and without FSI effects. Significant increases in FAR values are observed at higher pH values, which imply more significant effects of FSI at higher pH values. The valve blocks the flow nearly completely at pH value of 5.60. The findings demonstrate that FSI effects have to be considered in the design of hydrogel for microfluidic valves.