For laminar, incompressible flow inside the microchannel, conservation of mass and momentum are given by the Navier–Stokes equation: (1) ∇ ⋅ v ( f ) = 0 , (2) ρ 0 ( f ) ∂ v ( f ) ∂ t + ρ 0 ( f ) ( v ( f ) ⋅ ∇ ) v ( f ) = − ∇ p + μ ( f ) ∇ 2 v ( f )where ρ 0 ( f ) is the true density of fluid, μ^(f) is the dynamic viscosity of the fluid, p is the fluid pressure and v^(f) is the fluid velocity.

We account for ion transfer inside the channel and the hydrogel with the Nernst–Planck equation and electroneutrality condition [32–34], which can be expressed as (3) ∂ c k ∂ t + ∇ ⋅ ( c k v ( f ) ) = D k ∇ 2 c k + z k FD k ℜ T ∇ ( c k ∇ ψ ) (4) z f c f + ∑ k N z k c k = 0where F is Faraday's constant, ℜ is the universal gas constant, ψ is the electric potential, D_k is the diffusive coefficient, z_k and c_k are valence and concentration of the ion species k (= H⁺, Na⁺, OH⁻and Cl⁻) respectively; ϵ₀ is the permittivity for vacuum, ϵ is the dielectric constant of medium relative to vacuum, z_f is the valence of fixed charge and c_f is the fixed charge concentration inside the hydrogel.

For the hydrogel, conservation of mass is solved for a biphasic mixture comprising the solid and fluid phase, whereas conservation of momentum for the hydrogel is considered in terms of Navier's equation with infinitesimal deformations and generalized Darcy's law for a moving porous medium [35–37]; that is (5) ∇ ⋅ v ( p ) = − 1 ρ 0 ( f ) ∇ ⋅ q (6) ∇ p = − μ ( f ) κ ( p ) ρ 0 ( f ) q (7) ∇ ⋅ σ = 0where q =ρ^(f)v^(r) is the Eulerian relative flow vector of the fluid phase with respect to the polymer phase, v^(r) = v^(f) − v^(p) is the velocity of fluid relative to the solid phase velocity, v⁽ⁱ⁾ is the intrinsic velocity of phase i (solid and fluid), κ^(p)is the permeability of polymer phase, σ is the mixture stress tensor, and ρ ( i ) = ϕ ( i ) ρ 0 ( i ) is the density of phase i, where ρ 0 ( i ) is the true density of each phase. The velocity of the solid phase can be defined as the rate of deformation; that is, v^(p) = ∂u^(p) /∂t, where u is the deformation of the hydrogel.

The mixture stress tensor for the hydrogel is given by [35] (8) σ = − p I + σ eff ( p )where σ eff ( p ) is the elastic stress tensor of the polymer phase in the hydrogel. We treat the polymer phase as an isotropic elastic material, whence the elastic stress tensor [32,38] of the polymer phase can be expressed as (9) σ eff ( p ) = λ s ( E : I ) I + 2 μ s Ehere, the Lamé coefficients, λ_s and μ_s and the elastic strain tensor of the solid phase, E are defined as [39] respectively; E₀ is Young's modulus and ν is the Poisson ratio.

(10) λ s = ν E 0 ( 1 + ν ) ( 1 − 2 ν ) (11) μ s = E 0 2 ( 1 + ν ) (12) E = 1 2 [ ∇ u + ( ∇ u ) T ]

The osmotic pressure inside the hydrogel comprises the mixing and ionic contributions, which can be expressed as (13) p ( osm ) = − k B T V m ( ϕ ( p ) + χ ϕ ( p ) 2 + ln ( 1 − ϕ ( p ) ) ) ︸ mixing + ℜ T ∑ k N ( c k − c k * ) ︸ ionicwhere k_B is Boltzmann's constant, V_m is the equivalent volume occupied by one monomer, N_x is the degree of polymerization, χ is the polymer-solvent interaction parameter, and c k * is the concentration in the fluid channel (outside hydrogels) of the ion species k. The polymer-solvent interaction parameter, χ(T, ϕ^(p)), is generally expressed as a function of temperature and polymer volume fraction [40–42]; i.e., (14) χ = − Δ s k B + Δ h k B T + χ 2 ϕ ( p )where Δh and Δs denote the changes in enthalpy and entropy, and χ₂ is a parameter to express the polymer volume fraction dependence of the interaction parameter.

The fixed charge concentration is given by [31] (15) c f = { 1 ℌ c f 0 K a ( K a + c H + ) , inside the hydrogel , 0 , outside the hydrogelwhere c f 0 and c_H+ are the initial fixed charge and hydrogen ion concentrations, respectively, K_a is the dissociation constant of the fixed charge group and ℌ is the hydration state of the hydrogel, which is defined as the ratio of the volume of the fluid phase to the volume of the polymer phase inside the hydrogel, ℌ = V_f/V₀. For axially restrained cylindrical hydrogels, hydration can be related to the strain of hydrogel as (16) ℌ = E r r + E θ θwhere E_rr and E_θθ are the radial and tangential strains, given by [43] (17) E r r = ∂ u r ∂ r (18) E θ θ = u r rrespectively Note that—compared to De et al [31]—an additional term for hydration has been included: viz., tangential strain, since the deformation in the radial direction will trigger strain in the tangential direction [44]; hence,hydration should take into account total strain in radial and tangential direction.

The dynamic viscosity of the fluid phase can be expressed as [35] (19) μ ( f ) = a 1 ( T + a 2 ) a 3and the permeability of the polymer network is given by [45] (20) κ ( p ) = κ 0 ( p ) ( ϕ ( f ) ϕ ( p ) ) nwhere _i, κ 0 ( p ) and are constants summarized in Table 1.

The effective diffusivity of ion inside the hydrogel is taken into account by Bruggeman equation [46]: (21) D k ( eff ) = ( ϕ ( f ) ) 3 / 2 D kwhere D_k is the diffusive coefficient of ion species in water.

Here, pH and pKa are the negative logarithm of hydrogen ion concentration and dissociation constant given by (22) p H = − log 10 ( c 1 × c H + ) (23) p K a = − log 10 ( c 1 × K a )respectively; ₁ is a constant presented in Table 1.

The boundary conditions can be summarized as follows:

At the inlet of the channel, we prescribe (24) c k = c k , in , ψ = ψ in , p = p in

Initial conditions invoked are (29) c k = c k , 0 , ψ = ψ 0 , p = p out , u = c 2 x

Here, n is a unit vector normal to the given surface, |₋ and |₊ denote condition inside and outside the hydrogel, and ₂ is a constant presented in Table 1. The boundary condition in the centre of hydrogel is necessary in order to prevent translational movement of the hydrogel and corresponds to the way the hydrogel is attached to the flow channel; see e.g., [14,15].

Before we study the behavior of a hydrogel and its effect on the overall fluid flow in a microfluidic T-junction, we calibrate Δh, Δs, and χ₂ with the steady-state swelling curve for a diameter of 300 μm (training set) from the experiments by De et al. [31], as shown in Figure 2, and validate the deformation of the pH-sensitive hydrogel HEMA with experimental hydrogels with a diameter of 500 and 700 μm (test set). Overall, good agreement is achieved between the model prediction and the experiments. Clearly, the pHEMA hydrogel collapses at low pH and swells at high pH: This pH-induced swelling behavior can be attributed to the presence of acidic groups bound to the polymer chains, which become highly ionized at certain pH value [16,50]. The acidic group inside the hydrogel is only slightly ionized when the pH drops below the pKa of the hydrogel—in this case the pKa is 5. As a result, swelling of the hydrogel at pH changes below pKa is marginal, which is mirrored by the slight increase in the hydrogel radius for pH ≲ 4 in Figure 2. As the pH increases and approaches the pKa value, the acidic functional group becomes near to fully ionized by deprotonation, which results in an increase in the fixed charge density, c_f The increase in the fixed charge density, in turn, appears in the osmotic pressure (driving force) and causes a swelling, as depicted in Figure 2 for the pH range 4–6. When the ionization process reaches its saturation point, an increase in pH does not affect swelling behavior of the pHEMA hydrogel [51]. This can be observed in Figure 2 where the hydrogel stops swelling at leading order when pH > 7.

Turning our attention towards the deformation kinetics of the hydrogel, we first calibrate the permeability constant, k₀, for the shrinking of a 300 μm hydrogel when subjected to pH changes from 6 to 3 (training set), as illustrated in Figure 3a. We then validate the deformation kinetics with the corresponding swelling (see Figure 3b) with reasonably good agreement. Overall, we note that the shrinking is approximately ten times faster than the swelling: shrinking and swelling require around 1500 s and 18,000 s respectively in order to reach the new steady state.

Aside from calibration and validation purposes, it is of interest to study equilibrium swelling behaviorand deformation kinetics of a hydrogel since these are two important factors in designing hydrogels for microfluidic flow control: From the equilibrium swelling behaviour, we can identify and modify the properties of hydrogels that affect the swelling ratio and synthesize a hydrogel with the desired swelling ratio for flow control purposes; and from the deformation kinetic behaviour, we cam estimate the response time of the hydrogel when it is employed as microfluidic flow controller, after which we can design or synthesize a hydrogel with a sufficiently fast response for any given microfluidic flow control system.

We proceed further by examining the sensing and actuating behavior of a 300 μm hydrogel in a T-junction when the solution pH is changed between 3 and 7, as illustrated in Figure 4. This configuration represents a simple microfluidic flow controller based on a stimuli-responsive hydrogel—also commonly referred to as resistance-based flow control [15]. Initially, at low pH, the hydrogel is in its shrunken state and thus allows fluid flow (Figure 4a) between itself and the walls of the microchannel. A step change in pH from 3 to 7 is then applied to the system, for which the hydrogel starts to swell towards the new equilibrium and block the channel, as depicted in Figure 4b–d. The mass flow rate of the fluid at the inlet and outlets of the channel as the hydrogel deforms is presented in Figure 5. When the hydrogels reach new equilibrium at t ∼ 200 min, a step change in pH from 7 to 3 is applied; thus, the hydrogel starts to shrink towards the initial condition. Here, several features are apparent: First, the mass flow rate at the inlet of the channel decrease as the hydrogel swells, similar to that at the left outlet; second, the response of the hydrogel is rather slow—it takes around 120 min for the hydrogel to fully close the channel—which can defeat the purpose of flow control. The reason for the first observation is simple: as the hydrogel swells, it obstructs the flow and creates high resistance for the fluid to flow because of its low permeability. The second observation suggests that we should modify the hydrogel microfluidic system to obtain higher response rates.

In light of the second observation, we demonstrate two strategies in improving a hydrogel's response rate: first, by replacing a single larger hydrogel with multiple smaller hydrogels, and second, by employing a hydrogel with higher permeability (macroporous hydrogel).

For the first strategy, we implement two hydrogels with the size of 150 μm and three hydrogels with the size of 100 μm to replace the 300 μm, as shown in Figure 6. In doing so, we find that the response time is approximately 3 times faster (for 150 μm hydrogels) and 6 times faster (for 100 μm hydrogels) compared to the corresponding case with a single 300 μm hydrogel, as depicted in Figure 5. The reason for this response time enhancement is the fact that by reducing the size of hydrogel, we shorten the diffusion path of the penetrating fluid, which, in turn, leads to a faster response by the hydrogel. It should be noted, however, that by reducing the size of the hydrogels, we may reduce the mechanical strength and stability, which are necessary for a microfluidic flow controller [16,27]; therefore, careful consideration has to be taken to ensure an optimum design.

The second strategy is achieved by implementing a 300 μm hydrogel with 10 and 100 times higher permeability than the base-case hydrogel, which can be realized by utilizing macroporous hydrogels; see, e.g., [53,54]. With this approach, we find that the response times are around 10 and 95 times faster compared to the flow control system with a lower permeability, as presented in Figure 7. The faster response can intuitively be explained by the fact that swelling and shrinking kinetics mainly depend on the permeability of the hydrogel: on one hand, a low permeability induces a high resistance to the penetrating fluid flowing into the hydrogels, which in turn result in slow deformation response; on the other hand, a hydrogel with a high permeability allows for easier fluid penetration [35]. The utilized macroporous hydrogels, however, should possess a sufficiently low permeability, because hydrogels with too high permeability might allow a non-negligible amount of fluid to flow through them, which would defeat the purpose of flow control.

In this configuration, two hydrogels with 10 times higher permeability than the base-case hydrogel with opposite behavior are introduced in the microchannel: a positive pH-responsive hydrogel, which swells as the pH increases, and a negative pH-responsive hydrogel, which shrinks as the pH increases. The fluid flow is thus either directed to the left or the right channel depending on the pH, as illustrated in Figure 8. Recalling that the shrinking is approximately ten times faster than the swelling for the conditions and HEMA hydrogel considered in this study, we expect that the switching between the positive and negative response hydrogels will not be symmetric. This is indeed the case, as during the first few minutes, the left channel starts to open before the right channel is fully closed. This, in turn, leads to fluid flow through both branches and causes an increase in the mass flow rate at the inlet, as shown in Figure 9. Clearly, one has to be careful when designing a flow sorter with hydrogels since the latter may still allow fluid flow through an undesired channel during the first few minutes (depending on response rate).

As the pH-positive hydrogel swells further, the right branch starts to be blocked; therefore, we see a decrease in mass flow rate at the inlet. When the positive pH-sensitive hydrogel reaches a new equilibrium (t ∼25 min), the entire right branch has been blocked, forcing all the fluid through the left branch (Figure 8d), for which we observe that the mass flow rate at the inlet is equal to that of the left branch. A step change in pH from 7 to 3 is then applied for which the positive pH-sensitive hydrogel shrinks whereas the negative pH-sensitive hydrogels swells towards the new equilibrium conditions.

In this configuration, each hydrogel plays the role of a sensor, regulator and an actuator for the fluid flow commonly handled by three different components. This particular configuration could be implemented in chemical or biochemical applications, where precise pH control is required; for example, in sequence determination of protein and DNA analysis [21].