Temperature-sensitive grafted polymers have been exploited to achieve flow regulation in synthetic membranes. A majority of previous studies involving thermoresponsive permeability control has exploited the LCST behavior of PNIPAAM [7,8,9,10,11]. For example, Akerman et al. [9] grafted PNIPAAM chains into poly(vinylidene fluoride) (PVDF) membranes and demonstrated that transport of model compounds across the grafted membranes can be controlled as a function of temperature. The flow regulation was shown to be a function of grafting density, degree of polymerization, and ionic concentration. In another study, Lokuge et al. [11] have achieved actively controlled thermoresponsive, size-selective transport control by grafting PNIPAAM brushes onto a Au-coated nanocapillary membrane through the use of atom transfer radical polymerization (ATRP). While the open-close transition of PNIPAAM functionalized membranes occurs at 32 °C which corresponds to its LCST, Xie et al. [32] have demonstrated that by adding varying amounts of hydrophilic or hydrophobic monomers into PNIPAAm it is possible to increase or decrease the transition temperature. The LCST was found to increase linearly with increasing molar percentage of the hydrophilic monomer and decrease linearly with increasing molar percentage of the hydrophobic monomer. Also, Rao et al. [33] fabricated functional membranes containing elastin-like polypeptides in the place of PNIPAAM. Elastin-like polypeptides may exhibit a wide range of lower critical solution temperature values, depending on the primary sequence and the length of these materials. Thus it is possible to design and fabricate a temperature-responsive gating membrane with the desired transition temperature specific to a given application.

pH-sensitive materials have also been utilized to achieve flow regulation in synthetic membranes. Ito et al. [12] have investigated pH-triggered water permeation control through membranes grafted with three different polyacids. They demonstrated the change in permeation occurred at pH values of 3.0, 4.0 and 6.8 for poly(acrylic acid), poly(methacrylic acid) and poly(ethacrylic acid) grafted membranes, respectively. Subsequently, they demonstrated a pH-responsive nanometer-scale gate, in which a polypeptide brush was grafted into a gold plated nanoporous membrane [15]. Water permeation through the material was regulated by helix-coil transformation of grafted poly-(L-glutamic acid) chains in response to pH. At low pH values, grafted poly-(L-glutamic acid) has a helical conformation. As the pH is increased, the helical conformation expands to a random coil conformation. This transition reduces the effective pore diameter and hence the system permeability. Mika et al. [14] synthesized smart membranes with inverse pH behavior. They reported that polypropylene microfiltration membranes containing poly(4-vinylpyridine) anchored within the pores exhibited very large chemical valve effects, specifically, an increase pressure-driven permeability by more than three orders of magnitude when the pH was increased from two to five [14]. Qu et al. [34] demonstrated a novel composite membrane system that utilized both grafted polymers and gels to produce a large pH-responsive release. Their membrane consisted of a porous polyvinylidene fluoride (PVDF) membrane grafted with positively pH-responsive poly(methacrylic acid) (PMAA) linear chains. These smart pores in the membrane acted as flow control valves to a reservoir inside which a crosslinked, negatively pH-responsive poly(N,N-dimethylaminoethyl methacrylate) (PDM) hydrogel functioned as a functional pumping element. The cooperative action of the pH-responsive gating and pumping systems produced responsive release rates much higher than either of the mechanisms used alone. Next generation nanoporous materials are envisioned with multiple biological functionalities, including size screening, responsive flow regulation, and dynamic pore sizing.

Our model considers flow through a cylindrical nanopore grafted with polymer chains (Figure 6a). Following earlier continuum calculations [40], this problem is treated as flow through a porous medium and the grafted polymer chains are approximated as a polymer brush layer with permeability related to the monomer volume fraction. The resulting Brinkman equation [41] for the system can be written as

(2)

where r is the radial distance from the center of the pore, μ is the viscosity of the solvent, ∆P/∆z is the pressure drop driving the flow, v_z is the z-component of solvent velocity, and k is the permeability of the brush layer. This continuum equation completely describes flow through a cylindrical nanopore grafted with a polymer brush by assigning an infinite k value for the central polymer-free region of the pore (which reduces equation (2) to the Navier-Stokes equation), and a finite value for permeability through the brush layer that is related to the monomer volume fraction Φ by some function k(Φ) to be described. It is important to point out that Φ is not a constant across the grafted layer thickness (step function profile), and in fact has a parabolic profile under good solvent conditions as predicted by self-consistent mean-field theories [24,42] and supported by neutron reflectivity measurements [43]. This implies that the permeability k through the grafted layer is a function of r. The monomer volume fraction profile Φ(r) in the brush layer is determined by extending a mean field analytic brush model [42] to polymer chains grafted to a concave surface. This model takes as input the pore size, polymer grafting density, number of monomers per chain, monomer repeat length, and Kuhn length. To incorporate the effect of solvent quality on Φ(r), the mean-field theory of polymer brushes is used. Equation (2) is then solved numerically to determine the velocity profile and hence the fluid flow rate through the pore as a function of pH.

All self-consistent mean field theories of polymer brushes involve expressing the free energy as a functional of the monomer volume fraction profile in the grafted layer and minimizing this functional under appropriate constraints to obtain the equilibrium volume fraction profile [24]. Here, we calculate Φ(r) by extending the mean field analytic theory of Zhulina and co-workers [42] developed for planar brushes to the case of a concave cylindrical brush. It is worthwhile to point out that a computationally more demanding numerical SCF approach based on Scheutjens and Fleer [44] can be employed to determine the structure of grafted polymers inside cylindrical pores by Egorov et al. [45]. We will determine the dependence of the equilibrium structure of the concave polymer brush on the grafting density, length of grafted chains, curvature of the cylinder, and solvent quality. In the present work, we will ignore the effect of flow field on the brush structure. While at relatively high Weissenberg numbers the flow field is known to cause shear induced deformation of the brush [36], it is reasonable to consider the equilibrium brush profile at low shear values.

We consider polymer chains of length N monomers grafted to an impermeable concave cylindrical surface of radius R (Figure 6a,b), with a grafting density of one chain per an area of s. The monomer size and the number of residues in the Kuhn length are taken as the effective monomer size a and thenumber of residues p in the effective Kuhn length respectively. The quantity r is the radial distance measured from the axis of the cylinder and h = R − r is the distance from the grafting surface. To determine the monomer density profile, ϕ(h), as a function of distance from the grafting surface we first define the total conformational free energy of the concave brush,

(3)

which contains the volume interaction energy,

(4)

and elastic stretching energy

(5)

where the function g(h^’) is the average fraction of chains ending at a radial distance h^’ and E(h, h^’) gives the local stretching dh/dn at h for a chain ending at h^’. The parameter υ = πa³/6 represents the excluded volume and H is the distance from the grafting surface beyond which the density of monomers is zero (Figure 4b). The volume fraction of monomers ϕ(h) is expressed by the functions g(h^’) and E(h, h^’) as

(6)

Minimizing the free energy ΔF as a functional of unknown functions g(h^’) and E(h, h^’) under the constraints

(7)

and

(8)

gives the expression for the function of local chain stretching

(9)

and for volume fraction of monomers in a layer

(10)

where

(11)

and λ is a undetermined Lagrangian multiplier that can be determined from the normalization Equation (7). The function g(h^’) is determined by solving the integral Equation (6), with g(h^’) as an unknown function. Equations (9), (4) and (5) were then used to determine the free energy ΔF as a function of height H. The equilibrium value of brush height, H₀ was determined by minimizing the free energy with respect to H. Once H₀ is known, the monomer volume fraction profile is easily computed.

As an illustration, we have calculated the monomer density profile for a polymer brush grafted inside a cylindrical nanopore of radius R = 100 a for solvent quality parameter values of ν = 1.0 and 0.4. The values N = 400 and a grafting density = 0.01 a⁻² were used. The monomer density c(r) = ϕ(r)/πa³/6 plotted in Figure 7 for different values of ν shows spreading of the grafted layer as the solvent quality is decreased. This indicates that for the brush considered in the calculation increasing ν decreases the pore opening. It is important to note that unlike the monomer density profile obtained from MD simulations (Figure 4a), which increases from zero at the grafting surfaces to reach a maximum value before decreasing, the density profile from mean field calculations is maximum at the grafting surface and then decreases parabolically. This discrepancy arises due to the fact that the mean-field theory calculation does not take into account the interaction between the grafting surface (pore interior) and the monomers. It is possible to include such an interaction in numerical self-consistent field calculations [45].

Calculation of the velocity profiles used as an input Ф(r) obtained from the mean-field method was explained in the preceding section. Due to the variation of polymer volume fraction as a function of r, Equation (2) must be solved numerically. The calculation was performed by dividing the brush region into n annular shells of thickness dr. A value of dr = a was used. The Brinkman equation was written for i = 1, n shells separately with permeability value k_i related to the monomer density of the corresponding shell as k_i = a²/Ф_i² [39]. For shells free of polymers, equation (2) reduces to the Navier-Stokes equation. The solution for the velocity profile involved making an initial guess for the velocity at the center of the pore. Equation (1) was then applied to shells i = 1 (r = 0) to n (r = R) sequentially, subject to boundary conditions of constant velocity and shear stress at the interface between shell i-1 and i to obtain the velocity profile inside the pore. The composite velocity profile across the nanopore was obtained by successively improving the guess until the boundary condition that velocity at r = R vanishes was met. As an example, the velocity profiles at ν values 1 and 0.4 for a pore of R = 100 a, grafting density = 0.01 a⁻² and degree of polymerization N = 400 are plotted in Figure 8. The velocity profile is normalized by v_z0, the velocity at the center of an unmodified pore under the same flowing conditions. When the grafted layer is expanded it reduces the flow region and hence, also, reduces the fluid velocity at the center as compared to the collapsed brush layer. The overall effect is that as the solvent quality improves the flow rate through the nanopore decreases.

This is illustrated in Figure 9 where the normalized flow rate as a function of υ is plotted for R = 100 a, N = 400 and grafting density values of 0.01 a⁻², 0.005 a⁻² and 0.0033 a⁻². The analysis also illustrates that a higher grafting density leads to a lower open pore permeability as well as a higher on/off ratio. A similar analysis of the permeability change with solvent quality can be performed for varying chain lengths. In principle, the smart nanopore can be designed to have any desired ratio of flow rate in open and closed configuration by choosing an appropriate pore radius and polymer content P_c (the amount of grafted polymer per unit pore volume = 2N/sR).

The model described above can be applied to real flow control systems. The key is to incorporate the effect of signal responsive behavior of the structure of the brush layer. In the case of pH responsive systems, it is instructive to incorporate protonation/deprotonation as a function of pH and self consistently determine the monomer density profile in the polyelectrolyte layer. For example, recently Tagliazucchi et al. analyzed the pH-sensitive ionic conductivity through poly(4-vinyl pyridine) grafted nanopores. In particular, they determined the pH-sensitive conformation of the grafted brush by minimizing a free energy functional that incorporates interactions among the polymer, solvent, and the ions. Previously, we extended the model described in section 3.2 and 3.3 to analyze water permeation control through a nanoporous membrane grafted with poly(L-glutamic acid) chains [39]. We analyzed the helix-coil transition according to the Zimm-Bragg model to determine the monomer density profile of the grafted polypeptide layer inside a cylindrical nanopore as a function of pH. The flow rate through the nanopore was then calculated as a function of pH and the total permeability of the nanopore is plotted in Figure 10. These calculated pH-induced permeability change had a sigmoidal behavior and was in very good agreement with experimental results reported by Ito and coworkers [15]. The findings establish that polymer statistical mechanical models combined with a continuum porous layer treatment of flow through the polypeptide grafted nanopore can be used successfully to model smart flow control systems. The effect of grafting parameters on permeability change was analyzed. The optimum grafting parameters correspond to those for which the span of the monomer density profile is approximately equal to the pore radius in the closed state. The method described here can be easily extended to other polymer-solvent systems.

To design a smart flow control system, it is necessary to analyze the gating effect as a function of pore size, degree of polymerization, and grafting density. The performance of the nanoporous system at a given grafting condition is determined by the maximum change in permeability between the closed and open states. With reference to equation 1, we characterized the change in solvent permeability by the difference ΔK between permeability values at pH = 3 and 7. In Figure 11, ΔK = K_pH=3 − K_pH=7 is plotted as a function of the degree of polymerization N for two different grafting densities in pores of R = 800 and 1,600 Å. At low degrees of polymerization the value of ΔK increases with N. When the grafted layer thickness in the closed state is comparable to the pore radius, an increase in N continues to decrease K in the open state without causing comparable reduction in K in the closed state. Consequently, ΔK reaches a maximum value and then begins to decrease with increasing N as indicated by the results. The N value at which ΔK reaches a maximum increases with decreasing grafting density and increasing R.