Osmotic pressure of permeable ionic microgels: Poisson-Boltzmann theory and exact statistical mechanical relations in the cell model

For over a century, Poisson-Boltzmann (PB) theory has been widely applied to model interfaces between charged surfaces and ionic solutions.^1–3 By accounting for thermal motion of ions, the theory explains the origin of a diffuse electrical double layer comprising fixed surface charges and mobile microions (counterions and coions) in solution. The densities of microions are described by the PB equation, which results from combining the Poisson equation for the electrostatic potential with the Boltzmann distributions for the microion charge densities. Despite neglecting interparticle correlations through its underlying mean-field approximation, PB theory often reasonably describes the distributions of weakly correlated (usually monovalent) microions in solution around macroions of various types, including charged colloids, polyelectrolytes, and charged lipid bilayers.

Combining the PB theory of electrostatic interactions between charged surfaces with the Hamaker theory of van der Waals interactions yields the famous Derjaguin-Landau-Verwey-Overbeek (DLVO) theory,^4,5 which explains the stability of charge-stabilized colloidal suspensions.^1–3 PB theory also provides a basis for modern theories of effective electrostatic interactions in soft matter.^6–10 Following important elaborations of the original Gouy-Chapman formulation^11,12 of PB theory by Stern¹³ and Grahame¹⁴ to account for the nonzero size, hydration, and surface adsorption of ions, recent refinements of the theory have addressed ion-specific effects by incorporating nonuniform dielectric permittivity and non-Coulombic interactions.^15–22 

In practical applications, PB theory is often implemented in the cell model,^23–25 which abstracts from a bulk suspension a single macroion and its associated microions. The shape of the macroion dictates the appropriate geometry of the cell (planar, cylindrical, and spherical), the symmetry of which eases the computational solution of the nonlinear PB equation. Predictions for microion densities and osmotic pressure prove to be most accurate for systems with a relatively low salt concentration.^9,26

Aside from determining the electrical properties of charged surfaces in ionic solutions, such as differential capacitance and electrostatic interactions between surfaces, microion densities have relevance for the bulk thermodynamic properties of charge-stabilized colloidal suspensions, such as the pressure and phase behavior. In fact, within the cell model, the pressure is proportional to the total microion density at the edge of the cell. This important relation was first established in the framework of PB theory²³ but was subsequently proven to be exact in the cell model,²⁴ independent of PB theory.

In recent years, PB theory has been widely applied to penetrable macroions, which are distinguished by their permeability to solvent and microions. The most intensively studied are microgels—soft colloidal particles consisting of crosslinked polymer networks—whose ability to swell and deswell in solution leads to unusual materials properties with a multitude of practical applications, including drug delivery, water filtration, and biosensing.^27–30 Predictions of PB theory for the distribution of monovalent counterions in the cell model of ionic microgels often prove accurate when compared with molecular simulations.^31–35 Relatively, little attention has been devoted, however, to modeling thermodynamic properties, such as osmotic pressure.^36–39 The electrostatic contribution to the interior osmotic pressure of a permeable macroion, defined as the difference in pressure between the inside and outside, is determined by the density profiles of the fixed charge and mobile microions. The interior osmotic pressure, in turn, determines the equilibrium swollen size of an ionic microgel through a balance between electrostatic, elastic, and mixing contributions to the total osmotic pressure.^40–42 Accurately predicting the structure and phase behavior of concentrated microgel suspensions requires consistently incorporating deswelling of the compressible particles.^43,44

Swelling of ionic microgels has been probed in recent years by a variety of experimental methods, including light scattering,^45–52 neutron scattering,^52,53 electrophoresis,⁵⁴ and osmometry.⁵⁵ Measurements of the swollen radii of pNIPAM microgels in deionized solutions^50–52 indicate that deswelling sets in at concentrations well below random close packing, where particles start to overlap, qualitatively consistent with theoretical predictions for coarse-grained models.^34,44 Such observations point to the vital role of electrostatic interactions in driving deswelling of ionic microgels.

While theories of macroscopic ionic gels are well established,^56–61 a correspondingly comprehensive description of ionic microgels is still lacking. In particular, the important distinction between the osmotic pressure of a bulk suspension and the osmotic pressure of a single microgel is neither widely appreciated nor fully understood. As a result, simplified approximations are often adopted for the electrostatic component of the interior osmotic pressure in lieu of a more rigorous theory capable of quantitative predictions. The purpose of this paper is to fill this lacuna by deriving general expressions for the electrostatic component of the osmotic pressure of permeable macroions and to discuss implications for swelling and thermodynamic phase behavior of microgels.

The remainder of the paper is organized as follows: In Sec. II, we define the primitive and cell models of permeable macroions. In Sec. III, we derive from Poisson-Boltzmann theory, the pressure tensor general expressions for the bulk pressure and the electrostatic component of the osmotic pressure of ionic microgels in the planar, cylindrical, and spherical cell models. In Sec. IV, we apply PB theory to derive explicit expressions for the electrostatic component of the osmotic pressure of surface- and volume-charged microgels. In Sec. V, we derive exact statistical mechanical relations in the cell model for the electrostatic osmotic pressure of ionic microgels. In Sec. VI, we validate our results by numerically solving the PB equation to compute pressures and comparing predictions of PB theory with the corresponding exact relations. We also discuss the relevance of our results for modeling swelling of ionic microgels and interpreting experiments. In Sec. VII, we summarize and conclude with an outlook for future work.

We consider an aqueous suspension of N_m ionic microgels in a volume V at temperature T. Each particle is modeled as a swollen slab, cylinder, or sphere, composed of a crosslinked network of polymer chains. Dissociation of Z counterions from ionizable sites on the polymers leaves each microgel with a charge −Ze, where e is the electron charge. The spatial distribution of fixed charge on the polymers depends on the distribution of monomers. The presence of salt in solution contributes coions and additional counterions. In chemical (Donnan) equilibrium with an electrolyte reservoir of salt ion pair density n₀, the suspension contains N_s dissociated salt ion pairs. For simplicity, we assume the microions are monovalent and pointlike. From the condition of electroneutrality, the total number of counterions is N₊ = ZN_m + N_s and coions N₋ = N_s.

The microgels are permeable to both microions and solvent (water). In the primitive model, the solvent is a dielectric continuum of uniform dielectric constant ϵ, which reduces the strength of the bare Coulomb interaction between a pair of ions at separation r to $v$(r) = e²/(ϵr) (Gaussian units). The neglect of solvent structure limits the primitive model to phenomena that do not depend significantly on ion hydration effects.

The cell model^23–25 represents a bulk suspension of macroions, by a single macroion, along with mobile microions, in a cell of commensurate symmetry (Fig. 1). In the planar cell model, a slab of ionic gel of thickness a (infinite in the other two dimensions) with fixed charge distribution −en_f(x) is placed at one side of a cell of width L, representing a suspension of planar microgels of thickness 2a and volume fraction ϕ = a/L. In the cylindrical cell model, an infinitely long cylinder of ionic gel of cross-sectional radius a with axially symmetric fixed charge distribution −en_f(r) is centered in a cylindrical cell of radius R, representing a suspension of cylindrical microgels⁶² of volume fraction ϕ = (a/R)². In the spherical cell model, a sphere of ionic microgel of radius a with spherically symmetric fixed charge distribution −en_f(r) is centered in a spherical cell of radius R, representing a suspension of spherical microgels of volume fraction ϕ = (a/R)³.

The counterions and coions have equilibrium number density profiles n₊(r) and n₋(r), respectively, and the corresponding charge density profiles ±en_±(r), which are determined by the fixed charge distribution. The electroneutrality of the system dictates that the electric field vanishes at the cell boundaries: x = 0 and L for the planar cell and r = R for the cylindrical and spherical cells. Combining the cell model with the primitive model, the solvent is represented by a dielectric continuum. While a significant abstraction from a bulk suspension of macroions, the cell model has proven accurate, compared with multi-macroion simulations and experiments, in predicting osmotic pressures of deionized suspensions of charged colloids^8,9 and ionic microgels.³³ Suspensions of charged clay platelets have been successfully modeled using the Wigner-Seitz cell model,⁶³ with circular or square platelets placed at the center of a cylindrical or parallelepipedic cell. In the latter geometry, the fixed charge distribution, electric field, and microion distributions are not axially symmetric.

For a suspension of fixed volume and temperature in Donnan equilibrium with a salt reservoir, imposing a salt ion chemical potential of μ₀ = k_BT ln n₀, the appropriate thermodynamic potential is the semigrand potential Ω, a Legendre transform of the Helmholtz free energy F,

In a mean-field approximation, the electrostatic part of the semigrand potential functional can be expressed as

where the exact ideal-gas functional (in k_BT units) is

and the electrostatic potential φ is related to all of the ion densities via

with the integrals extending over the system volume V. To simplify notation, we henceforth express all energies in thermal (k_BT) units. For given distributions of fixed and mobile charges, φ obeys the Poisson equation,

or, upon introducing the reduced electrostatic potential, ψ ≡ eφ/(k_BT), and Bjerrum length, λ_B = e²/(ϵk_BT),

Minimizing Ω with respect to the microion densities, i.e., requiring δΩ/δn_± = 0, generates the Boltzmann approximations, n_± = n₀ exp(∓ψ), which when substituted into Eq. (6) yield the PB equation,^64,65

where $κ=8πn0λB$ is the Debye screening constant. Combining Eqs. (2), (3), and (7), we have

As the electric field (−∇ψ) vanishes on the boundary S,

In applications to hard charged colloids, the fixed charge is usually confined to a surface, while for ionic gels, n_f may be a volume-distributed charge. Legendre transforming to the Gibbs free energy, G = Ω + PV, shifts the constraint from fixed volume V to fixed thermodynamic pressure P (in k_BT units), which may be determined by variational minimization of G with respect to ψ.^15–17 In cell models with planar, cylindrical, or spherical symmetry, P is the normal force per unit area acting on the outer boundary of the cell—a surface whose normal is parallel to the electric field⁶⁶—corresponding to the bulk pressure of the suspension.

It is important to distinguish the bulk pressure from the pressure tensor in the cell, the elements of which represent forces per unit area exerted on surfaces whose normal vectors are oriented parallel to different coordinate axes. In a dielectric medium with electric field E and total number density of mobile ions n(r), the electrostatic pressure tensor (in k_BT units) is given by^{35,63,67–70}

where 1 is the unit tensor. The first term on the right side is the Maxwell pressure tensor (negative of the Maxwell stress tensor), associated with the electric field generated by both the fixed charge and the mobile microions. The second term is the kinetic pressure tensor, associated with the momentum transported by the mobile microions. The normal electrostatic pressure, acting on a surface whose normal is parallel to E, is

while the electrostatic pressure in transverse directions is

In a gel medium, the total pressure tensor is the sum of the electrostatic and gel pressure tensors, P = P_e + P_g. The bulk pressure equals the normal component of the total pressure at the cell edge, P = P_∥(R) = P_rr(R).

In matrix notation, the total pressure tensor can be expressed as

where we choose the first coordinate parallel to E and off-diagonal elements are zero in the absence of shear forces. The condition of mechanical equilibrium is that the net force density at any point must vanish, which is equivalent to vanishing of the divergence of the total pressure tensor, ∇ · P = 0. As shown below, this condition implies that the elements of the total pressure tensor, P_∥ and P_⊥, are spatially constant in the planar cell model, but spatially varying in cylindrical and spherical cells.

It is also essential here to distinguish between the terms “pressure” and “osmotic pressure,” which are often used interchangeably in soft matter physics. Let us avoid confusion by defining the osmotic pressure of a suspension as the difference in bulk pressure between the suspension and an electrolyte reservoir (the pressure itself being measured relative to vacuum). This bulk property is distinct from the osmotic pressure of a single microgel, defined as the change in the normal component of the total pressure tensor across the surface of the gel. In equilibrium, the polymer network swells until the single-microgel osmotic pressure vanishes, implying continuity of the normal component of the total pressure at the microgel surface. Next, we derive a number of fundamental expressions for the osmotic pressure in cell models of various geometries.

In a planar cell of width L, we can express the Gibbs free energy functional per unit area in the form

where

depending explicitly on x through the fixed charge n_f(x). To emphasize the distinction between the bulk pressure and the total pressure inside the cell, we denote the former by P(L) and the latter by P(x), although, as it turns out in planar geometry, P(x) = P(L). Minimizing the functional with respect to ψ, i.e., replacing ψ(x) with ψ(x) + αη(x), with arbitrary perturbation function η(x), and requiring dG/dα = 0, leads to⁷¹ 

where Δx represents a variation of the endpoint (x = L), which allows the cell to adjust its volume to reach equilibrium at the externally applied normal pressure. Equation (16) immediately yields the Euler-Lagrange equation

the boundary conditions

and the constraint

Substituting g from Eq. (15) into Eqs. (17)–(19) recovers the PB equation [Eq. (7)] and leads to ψ′(0) = ψ′(L) = 0 (implying electroneutrality) and also the thermodynamic pressure acting at the edge of the cell,

Assuming n_f(L) = 0 (no fixed charge at the cell edge), we finally obtain

which is the well-known planar cell theorem relating the bulk pressure of the suspension to the ion density at the cell edge.^23,24 The osmotic pressure of the suspension is obtained by subtracting the pressure of the reservoir.

A more general expression for the bulk pressure, applicable to any fixed charge distribution, follows by first using Eq. (17) to simplify the total derivative of g,

Integrating from position x to the cell edge (x = L) yields

Substituting g from Eq. (15) into Eq. (23), noting that $∂g/∂u=nf′(u)ψ(u)$⁠, integrating by parts, and making use of Eqs. (20) and (21), we obtain

where the first two terms on the right side are the Maxwell pressure and the kinetic pressure of the mobile microions and the third term,

is the counteracting gel pressure required to stabilize the fixed charge against the electric field (−ψ′), thus ensuring the mechanical stability of the gel. For charged surfaces, the third term in Eq. (24) is often implicit, subsumed into the boundary conditions,¹⁷ while for ionic gels with volume-distributed fixed charge it must be explicitly included. Note that Eq. (25) is consistent with the force-balance condition for mechanical equilibrium of the fixed charge in the presence of an electric field,

In microgels, the gel pressure would be provided by the swollen polymer network, while for a membrane, P_g could be supplied by the restraining force of an external support or a lipid bilayer. The normal component of the electrostatic pressure is then identified as

consistent with Eq. (11) in which n(x) = 2n₀ cosh ψ(x) is the total microion number density. From Eqs. (21) and (25), we can also write

Note that in the planar cell, the condition for mechanical equilibrium, ∇ · P = 0, implies dP(x)/dx = 0, and thus the total pressure inside the cell is spatially constant, i.e., P(x) = P(L). In contrast, in cylindrical and spherical cell models, the total pressure inside the cell is spatially varying, as shown in Sec. III C.

A similar development yields the bulk pressure of a suspension modeled by an infinitely long, axially symmetric, cylindrical cell of cross-sectional radius R. The Gibbs free energy functional per unit length along the cylinder axis now takes the form

where r is the radial distance from the axis and

To again emphasize the distinction between the bulk pressure and the total pressure inside the cell, we denote the former by P(R) and the latter by P(r). The additional factor of r in Eq. (30), from axial symmetry in cylindrical geometry, modifies the results derived in planar geometry. Minimizing the functional with a variable endpoint (r = R)⁷¹ yields the Euler-Lagrange equation

the boundary conditions

and the constraint

Substituting g from Eq. (30) into Eqs. (31)–(33) recovers the PB equation [Eq. (7)] in cylindrical polar coordinates, the electroneutrality condition, ψ′(R) = 0, and further yields the bulk pressure (acting on the cell boundary),

In passing, we note that this result is consistent with the condition for mechanical equilibrium of the cell,⁷² 

when the fixed charge of the ionic gel is included in the total charge density. Assuming n_f(R) = 0 (no fixed charge at the cell edge) leads to the cell theorem for the bulk pressure,^23,24

As in planar geometry, we can derive a more general expression for the bulk pressure by first using Eq. (31) to simplify the total derivative of g,

Integrating Eq. (37) from radius r to the cell edge (r = R) and using Eq. (30) to evaluate ∂g/∂r, we have

Substituting g from Eq. (30) on both sides of Eq. (38), using Eq. (34), integrating by parts, and solving for P(R), we obtain the following completely general expression for the bulk pressure in the cylindrical cell model,

where n(r) ≡ n₊(r) + n₋(r) = 2n₀ cosh ψ(r) is the total microion number density. Setting r = R in Eq. (39) recovers the cell theorem [Eq. (36)], while evaluating at r = 0 yields

Interestingly, in cylindrical geometry, the bulk pressure can be expressed as an integral over the radial coordinate but is not obtained by simply substituting r for x in the corresponding expression in planar geometry [Eq. (24)]. Numerical evaluation confirms that Eqs. (39) and (40) give exactly the same bulk pressure as Eq. (36) [Sec. VI]. For applications to swelling of ionic microgels (Sec. IV B), it is useful to relate the normal component of the electrostatic pressure profile [Eq. (11)] to the gel pressure,

which follows from differentiating and then integrating Eq. (39) with respect to r and identifying the gel pressure

the diagonal element $(Pg)rr$ of the gel pressure tensor.

To further elucidate the significance of Eq. (39), we now show that this expression also follows from the condition of mechanical equilibrium. In fact, Eq. (39) can be equivalently expressed in the form

where

is the normal component and

is the transverse component of the total pressure tensor. Here, P_e,∥(r) and P_e,⊥(r) are the normal and transverse components of the electrostatic pressure tensor [Eqs. (11) and (12)], respectively, and we assume uniform swelling and P_g(R) = 0. As illustrated in Fig. 2, the left side of Eq. (43) is proportional to the magnitude of the net force acting on the cylindrical surface of a cylindrical sector (pie slice) of the cell whose cross section is enclosed by a circular arc of radius R subtending a central angle ϕ and two straight radial segments. By symmetry, the net force points in the radial direction and is proportional to R sin(ϕ/2). The first term on the right side of Eq. (43) is similarly proportional to the magnitude of the net force acting on the cylindrical surface of a cylindrical sector of radius r subtending the same central angle, which is proportional to r sin(ϕ/2). The second term on the right side is proportional to the magnitude of the net force acting on the flat sides of the sector. Integration along the radial coordinate is required, as the transverse pressure varies with r. By symmetry, this net force also points in the radial direction and is proportional to sin(ϕ/2). Thus, Eq. (43) simply expresses the condition that the net force on any sector of the cell vanishes. Since the size and location of the sector are arbitrary, it follows that mechanical equilibrium prevails at every point in the cylinder.

The fact that precisely the same expression for the bulk pressure results from two seemingly independent approaches—a variational minimization of the free energy and a force balance condition involving the pressure tensor—provides a strong consistency check on our calculations. In passing, we note that our expressions for P_∥(r) and P_⊥(r) (neither of which is spatially constant) satisfy the condition for mechanical equilibrium,

which follows from substituting Eq. (13) into the general expression for the divergence of a second-order tensor in cylindrical polar coordinates.^71,73 Furthermore, since P_∥(r) and P_⊥(r) are everywhere finite, Eq. (46) implies that the equilibrium normal pressure is continuous.

In a spherically symmetric spherical cell of radius R, the Gibbs free energy functional now takes the form

where r is the radial distance from the center and

The geometric factor of r², from spherical symmetry in spherical geometry, further modifies the results derived in the planar and cylindrical cell models. Substituting g from Eq. (48) into Eqs. (31)–(33) recovers the PB equation [Eq. (7)] in spherical polar coordinates, the electroneutrality condition, ψ′(R) = 0, and the cell theorem for the pressure [Eq. (36)]. Integrating Eq. (37) from r to r = R and using Eq. (48) to evaluate ∂g/∂r, we have

Substituting Eq. (48) for g on both sides of Eq. (49) and integrating by parts, we obtain a general expression for the bulk pressure in the spherical cell model,

Evaluating Eq. (50) at r = 0 yields

Once again, the bulk pressure P(R) can be expressed as an integral over the radial coordinate but is not obtained by simply substituting r for x in Eq. (24). Applications to swelling of ionic microgels (Sec. IV C) are facilitated by relating P_e,∥(r) [Eq. (11)] to the gel pressure,

which follows from differentiating and then integrating Eq. (50). The normal component of the total pressure in the spherical cell is then

As in the cylindrical cell model, we can validate our expression for the bulk pressure [Eq. (50)] by expressing it as a force balance condition on a spherical sector,

where the left side is proportional to the magnitude of the net force acting on the spherical surface of a spherical sector of the cell whose cross section is enclosed by a spherical cap of radius R and a cone with the apex at the center of the sphere. By symmetry, the net force is in the radial direction and is proportional to R². The first term on the right side is similarly proportional to the magnitude of the net force acting on the spherical surface of a spherical sector of radius r subtending the same solid angle, which is proportional to r². The second term on the right side is proportional to the magnitude of the net force acting on the lateral surface of the cone, which by symmetry also points in the radial direction. A radial integration is required, as the transverse pressure varies with r. Finally, we note that our relations for P_∥(r) and P_⊥(r) satisfy the condition for mechanical equilibrium,

obtained by substituting Eq. (13) into the general expression for the divergence of a tensor in spherical polar coordinates.^71,73 Note the factor of 2 [cf. Eq. (46)]. As in cylindrical geometry, neither component of the pressure is spatially constant and P_∥(r) is continuous.

The approach described above can be implemented for any fixed charge distribution. Next, we illustrate applications to idealized distributions that model microgels.

The planar cell model has been widely used to model the structure of electrolytes near charged surfaces,^2,3 e.g., metallic electrodes, charged colloids, and lipid membranes.^15–17 If the fixed charge uniformly coats a flat, impenetrable wall at x = a with valence/unit area σ, then n_f = σδ(x − a) and the electrostatic pressure is uniform throughout the available volume of the cell (a < x < L). According to Eqs. (24) and (25), precisely at the surface (x = a), P_e,∥(x) and P_g(x) are discontinuous, jumping by equal but opposite values of magnitude σψ′(a) such that the total pressure remains continuous. Evaluating Eq. (27) at x = a, and invoking the boundary condition ψ′(a) = 4πλ_Bσ, yields the exact contact value theorem,^24,25,74,75

which is valid only for planar charged surfaces.

The planar cell model can also be used, however, to model a flat, permeable membrane or a slab of ionic gel. The latter we refer to as a planar microgel, since one dimension of the gel is not macroscopic. If the fixed charge coats the surface of a permeable membrane or gel at x = a, then P_e is discontinuous, jumping between different values on the left and right sides of the membrane. Evaluating Eq. (27) at x = 0 and x = L, we have (with n ≡ n₊ + n₋)

Evaluating Eq. (27) as x → a from both sides,

which implies a discontinuity (osmotic pressure) of

Combining this result with the boundary condition

yields an alternative expression for the difference in electrostatic pressure across the membrane or gel surface,

Because the electric field points away from the surface, the two terms on the right side are of opposite sign. In the symmetric case, a = L/2, the terms cancel and ΔP_e = 0. Equations (59) and (61) are equivalent forms of a new “contact value theorem” for surface-charged permeable membranes or planar microgels.

If the fixed charge is instead evenly distributed over the volume of a microgel in the region 0 ≤ x ≤ a, then

where θ(x) is the unit step function. While fixed charges are discretely distributed within a gel, thermal motion of polymer chains makes a continuum model reasonable. The electrostatic pressure is then given by Eq. (28) as

implying an internal electrostatic osmotic pressure

In contrast to the case of a surface-charged microgel, the electrostatic pressure of a volume-charged microgel does not make a jump at the surface but instead rises continuously with increasing distance into the gel.

The cylindrical cell model has been extensively used to model polyelectrolyte solutions and suspensions of charged rodlike colloidal particles.^23–25 For reference, we first consider an impermeable cylinder of cross-sectional radius a and fixed charge per unit length −λe uniformly spread over the surface with number density,

The bulk pressure is given by Eq. (39) evaluated at r = a,

where σ ≡ λ/(2πa) is the valence/unit area and we have applied the boundary condition for the electric field at the cylinder surface,

Equation (66), derived here within PB theory, represents a contact value theorem for charged cylinders.²⁴ As noted in Sec. III C, this theorem can be interpreted as a force balance condition on a cylindrical sector of the cell.

Turning to a cylindrical microgel with uniform surface charge per unit length −λe, the fixed charge density is the same as for an impermeable cylinder [Eq. (65)], but the surface is now permeable to microions. Substituting n_f(r) into Eqs. (41) and (42), we find that P_e,∥(r) jumps discontinuously at r = a. The magnitude of this jump is the electrostatic osmotic pressure of the microgel,

Equation (68) can be viewed as a contact value theorem for cylindrical surface-charged ionic microgels.

It is important to note that, in contrast to planar geometry, the normal and transverse pressures are now not spatially uniform. Thus, whereas the electrostatic osmotic pressure of a planar surface-charged microgel is proportional to the difference in microion density between the inner (x = 0) and outer (x = L) boundaries of the cell [Eq. (61)], the same relation does not hold for a cylindrical microgel. The electrostatic osmotic pressure of a cylindrical surface-charged microgel is not simply proportional to the difference in microion density between the cylinder axis (r = 0) and the edge of the cell (r = R) but rather equals the jump in electrostatic pressure at the microgel surface (r = a).

Next, we consider a cylindrical microgel with fixed charge per unit length −λe uniformly distributed over its volume with number density

Substituting this form of n_f(r) into Eq. (42) yields an internal electrostatic osmotic pressure (for r < a)

which is continuous at the surface of the gel and increases from the gel surface inward.

The spherical cell model is a well-established model of charge-stabilized colloidal suspensions.^23–25 For comparison, we first consider an impermeable sphere, modeling a colloidal particle, of radius a and uniform fixed surface charge −Ze with number density,

The bulk pressure is given by Eq. (50) evaluated at r = a,

where σ = Z/(4πa²) is the valence/unit area and we have applied the boundary condition for the electric field at the particle surface,

Equation (72) represents a contact value theorem for spherical charged colloids.²⁴ As discussed in Sec. III D, this theorem can be interpreted as a force balance condition on a spherical sector of the cell.

Turning to a spherical microgel with uniform surface charge −Ze, the fixed charge density is the same as for an impermeable spherical colloid [Eq. (71)]. Substituting n_f(r) into Eqs. (52) and (42), we find that P_e,∥(r) jumps discontinuously at r = a. The magnitude of this jump is the electrostatic osmotic pressure of the microgel,

Equation (74) can be regarded as a contact value theorem for spherical surface-charged ionic microgels.

As with cylindrical geometry, the normal and transverse pressures are not spatially uniform. Thus, the electrostatic osmotic pressure of a spherical surface-charged microgel is not simply proportional to the difference in microion density between the center (r = 0) and edge (r = R) of the cell but rather equals the jump in electrostatic pressure at the microgel surface (r = a). Our approach thus differs sharply from that of Ref. 76, in which the pressure inside the microgel is approximated as uniform, and the electrostatic osmotic pressure is claimed to depend only on the microion density at the center.

A spherical microgel with fixed charge −Ze uniformly distributed over its volume has charge number density,

Substituting this form of n_f(r) into Eq. (42) yields an internal electrostatic osmotic pressure (for r < a),

which, as in planar and cylindrical geometries, rises continuously from the surface inward. To check and compare with the above results, obtained within PB theory, we next derive theorems that provide exact relations for the electrostatic component of the osmotic pressure in the cell model, which are not tied to PB theory.

For comparison with the predictions of Poisson-Boltzmann theory, we next derive some exact statistical mechanical relations for the electrostatic contribution to the osmotic pressure of an ionic microgel in the cell model, following the approach developed in previous work³⁴ and inspired by Wennerström et al.²⁴ The derivations start from a decomposition of the electrostatic part of the Hamiltonian,

into the microgel self-energy U_m(a) and microgel-microion and microion-microion interaction energies, U_mμ({r}; a) and U_μμ({r}), respectively, which depend on the coordinates of all N microions, {r} ≡ {r₁, …, r_N}. Only the first two terms in Eq. (77) depend on the microgel size. The microgel-microion interaction energy depends on the microgel-microion pair potential $vmμ$(r; a),

For a suspension that is free to exchange microions with a salt reservoir, the electrostatic part of the semigrand potential can be expressed as Ω = −ln Ξ, where (continuing to express energies in k_BT units)

is the electrostatic partition function and μ₀ is the chemical potential of microions in the reservoir. The osmotic pressure of the suspension—the difference in pressure between the suspension and reservoir—is defined via the derivative of the free energy with respect to volume V . In a planar cell of constant surface area A (V = AL), the bulk osmotic pressure is given by

In a cylindrical cell of radius R and height h (V = πR²h),

while in a spherical cell of radius R (V = 4πR³/3),

Substituting for Ξ from Eqs. (77)–(79) yields the respective cell theorems, namely, Eqs. (21) and (36).^23,24

Similarly, the osmotic pressure of a permeable microgel is defined as the change in pressure at the surface of the gel. The electrostatic component of the osmotic pressure of an ionic microgel, ΔP_e, can be defined via a derivative of the electrostatic semigrand potential with respect to the gel volume $v$⁠. In a planar cell with $v$ = Aa,

where angular brackets denote an ensemble average over microion configurations. In a cylindrical cell (⁠$v$ = πa²h),

while in a spherical cell (⁠$v$ = 4πa³/3),

We emphasize that Eqs. (83)–(85) yield average osmotic pressures, in contrast to the spatially varying pressure profiles derived from PB theory. Within the planar, cylindrical, and spherical cell models, these theorems for the electrostatic part of the osmotic pressure of a permeable ionic microgel are formally exact. Next, we present explicit results for specific fixed charge distributions.

A slab of permeable ionic gel of thickness a, or a flat membrane at position x = a, with a surface charge/unit area −σe, modeled by fixed charge number density n_f(x) = σδ(x − a), has a self-energy that is independent of a. Since the electric field magnitude of the fixed charge, |E_f| = 2πσe/ϵ, is constant, the fixed charge interacts with the microions with energy (in k_BT units)

where z_i = ±1 is the valence of microion i and the sums are restricted to microions that are inside (x_i ≤ a) or outside (x_i > a) the gel. Substituting into Eq. (83),

where, for microion number density profiles n_±(x),

are mean numbers of counterions/coions per unit area inside the gel and we used the electroneutrality condition

Note that in the symmetric case of a membrane fixed at the center of the cell (a = L/2), $N+−N−=σ/2$ and then ΔP_e = 0.

A slab of ionic gel with the same charge evenly distributed over its volume has a fixed charge number density n_f(x) = (σ/a)θ(x − a), generating an electric field

which stores a self-energy (in k_BT units)

The corresponding interaction energy between the fixed charge and the microions is

plus a constant U₀(a), dependent on a, determined by the choice of reference point. Substituting Eqs. (91) and (92) into Eq. (83) and choosing U₀(a) = πλ_Bσ²a yield

where

are second moments of the microion density profiles inside the gel. Our choice of U₀(a) ensures that Eqs. (87) and (93) agree in the thin-gel (a → 0) and high-charge (σ → ∞) limits, in which case n₊(x) → (σ/a)θ(x − a). Of the six systems considered—surface- and volume-charged planar, cylindrical, and spherical microgels—the volume-charged planar microgel is the only one for which the reference potential affects the electrostatic pressure. In passing, we note that in the limit of a macroscopic gel, much thicker than the Debye screening length (κa ≫ 1), when virtually all counterions are confined to the gel, the gel is electroneutral, n₊(x) = n_f(x), and ΔP_e = 0.

For a cylindrical microgel with fixed charge uniformly distributed over its surface [Eq. (65)], the self-energy and the microgel-microion interaction energy per unit length are (in k_BT units and to within a-independent constants)

and

Substituting Eqs. (95) and (96) into Eq. (84) yields

where, for microion radial number density profiles n_±(r),

are mean numbers of counterions/coions per unit length inside the microgel.

A cylindrical microgel with fixed charge uniformly spread over its volume [Eq. (69)] creates an electric field

storing a self-energy per unit length (in k_BT units)

and interacts with the microions with energy