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Membrane crossflow ultrafiltration (UF) is widely used for the enrichment and purification of colloids and proteins. In this continuous process, a feed dispersion is steadily pumped through a channel consisting of solvent-permeable membrane walls. The applied transmembrane pressure (TMP) causes the solvent to flow out of the membrane channel. The TMP-induced permeate flux of a colloidal dispersion is commonly smaller than that of pure solvent. This unwarranted permeate flux reduction is due to the following effects. First, the permeate flux transports colloidal particles towards the inner membrane surface where they form a particle-enriched diffuse layer. This so-called concentration polarization (CP) layer increases the osmotic particle pressure which counteracts the applied TMP. Second, membrane-fouling mechanisms are operative, including the formation of adsorbed particle layers partially blocking the membrane pores, and the formation of a cake multi-layer of immobilized particles. Fouling mechanisms decrease the hydraulic permeability of the membrane and possibly cause irreversible damage to the membrane. We have developed a semi-analytic modified boundary layer approximation (mBLA) method to calculate concentration and flow profiles in the UF of colloidal dispersions from knowledge of the concentration-dependent bulk suspension properties [1, 2]. The mBLA is an accurate and fast method, producing results in full agreement with numerically expensive finite-element calculations [1]. Moreover, the semi-analytic mBLA expressions provide the important insight into the influence of dispersion properties such as the collective diffusion coefficient and shear viscosity, which helps to improve the UF efficiency [1, 2]. We present mBLA results for the UF of dispersions of solvent-permeable rigid, and of charge-stabilized colloidal particles in a broad range of particle sizes. A simple cake layer model is used to unravel the relations between critical and limiting permeate fluxes and applied TMP [3]. References[1] G. W. Park and G. Nägele, Journal of Chemical Physics, 2020, 153, 204110.[2] G. W. Park and G. Nägele, Membranes, 2021, 11, 960.[3] G. W. Park, M. Brito, and G. Nägele, manuscript in preparation.

Cross-flow ultrafiltration (UF) is a pressure-driven separation process used, e.g., for water purification, blood treatment by (artificial) kidneys, and protein enrichment. In this process, a feed dispersion is steadily pumped through a channel consisting of solvent-permeable membrane walls. The applied transmembrane pressure (TMP) causes solvent to flow out of channel so that a particle-enriched diffuse layer is formed near the membrane walls. This so- called concentration-polarization (CP) reduces the permeate flux and hence the filtration efficiency, due to osmotic pressure built-up counter-acting the TMP. The particle concentration inside the CP layer increases with the applied TMP. When it reaches a freezing concentration where the particles are immobilized, an unwarranted cake layer is formed at the membrane walls. The permeate flux value signaling the onset of cake formation is called the critical flux. Predictions of the critical flux are commonly made using standard film theory which, however, applies only to colloidal particles and proteins of a diameter smaller than 10 nm. When the diameter roughly exceeds 100 nm, the predicted critical flux is an order-of- magnitude smaller than the experimental one. This is referred to as the critical flux paradox.We present theoretical results for the UF concentration and flow profiles, and the critical flux for dispersions of various size of particles. The results are obtained using a recently published modified boundary layer approximation (mBLA) method of cross-flow UF. The semi-analytic mBLA method provides an accurate description of UF concentration and flow profiles, on accounting for the concentration dependence of dispersion transport properties and osmotic pressure. The considered model dispersions encompass impermeable and permeable hard spheres and charge-stabilized particles. For hard-sphere dispersions, the mBLA method is extended to the microfiltration regime where shear-induced migration becomes important.

Membrane crossflow filtration is widely used for the enrichment and purification of colloidal and protein dispersions. In this pressure-driven process, a feed dispersion is steadily pumped through a channel consisting of solvent-permeable membrane walls. The applied transmembrane pressure (TMP) causes advection of the dispersion toward the membrane, and formation of a particle-enriched diffuse layer near its surface. This so-called concentration polarization (CP) layer increases the osmotic particle pressure counteracting the TMP. When the particle concentration reaches a solidification limit, an immobilized particulate layer, termed cake layer, is formed next to the membrane surface. The cake layer adds to the hydraulic membrane resistance and lowers thus the filtration efficiency. A key problem is to understand quantitatively how cake formation is related to filtration operating conditions, and to dispersion and membrane properties. In this study, we theoretically analyze how permeate flux and cake layer formation are influenced by the size, charge and feed concentration of dispersed particles [1]. We consider dispersions of neutral and charge-stabilized colloidal particles. Under conditions where shear-induced migration matters, empirical expressions for shear-rate dependent transport properties are used. Our results for concentration and flow profiles under filtration are obtained using a recently developed modified boundary layer approximation (mBLA) method [2, 3]. The mBLA is a numerically efficient and accurate method for predicting filtration properties. A thorough dicussion of the so-called critical permeate flux related to the onset of cake layer formation is presented. Moreover, an analytic expression for the critical flux is derived and compared with standard predictions by film theory and mass transfer coefficient calculations. References[1] G. W. Park, J. K. G. Dhont, and G. Nägele, manuscript in preparation.[2] G. W. Park and G. Nägele, Journal of Chemical Physics, 2020, 153, 204110[3] G. W. Park and G. Nägele, Membranes, 2021, 11, 960

We present a boundary layer approximation (BLA) method for calculating the concentration-polarization (CP) layer and dispersion flow in crossflow ultrafiltration with a cylindrical membrane [1]. Results are discussed for various feed dispersions of colloidal particles, namely hard spheres, charge-stabilized particles [2], solvent-permeable non-ionic microgels [3, 4], and deswelling ionic microgels [5]. State-of-the-art methods are used for the gradient diffusion coefficient and viscosity entering the calculation of filtration profiles [3, 5]. We show that the BLA profiles are in quantitative agreement with predictions by a finite-element method [1]. The BLA results are compared with predictions by a similarity solution [4]. The BLA method is extended to conditions where a filter cake is formed at the membrane, with permeate flux results presented for a hard-sphere system [6].

Crossflow filtration is a pressure-driven separation and enrichment process for colloidal dispersions where the feed dispersion is continuously pumped through a membrane pipe. The transmembrane pressure (TMP) causes solvent to flow out of the membrane, while the colloidal particles are retained inside the pipe. Consequently, a particles-enriched diffuse layer is formed near the membrane wall which reduces the filtration efficiency. This so-called concentration-polarization (CP) layer is due to the balance of flow advection of particles towards and gradient diffusion away from the membrane. In the ultrafiltration regime where Brownian motion dominates flow convection, the collective diffusion coefficient and the dispersion viscosity are the key transport properties determining the CP layer in conjunction with the TMP and transmembrane osmotic pressure. In this study, we present a finite-element (FEM) method for calculating the CP layer and suspension flow profiles in a crossflow ultrafiltration setup [1]. We show that the FEM-calculated CP layer profiles are in quantitative agreement with results obtained from a simplifying boundary layer analysis [1]. Results for the filtration and flow profiles are discussed for different globular dispersion particles including colloidal hard spheres (as a reference system), solvent-permeable particles such as non-ionic microgels [2], impermeable charge-stabilized particles [3], and ionic microgels with concentration-dependent size [4]. We pay particular attention to the effect of the different transport properties and osmotic pressure dependencies on the ultrafiltration profiles.References[1] G. W. Park and G. Nägele, “Analytic and numerical approaches to concentration-polarization layers in cross-flow ultrafiltration with a cylindrical membrane pipe” (in preparation)[2] R. Roa et al., “Ultrafiltration modeling of non-ionic microgels,” Soft Matter 11, 4106-4122 (2015) [3] R. Roa et al., “Ultrafiltration of charge-stabilized dispersions at low salinity,” Soft Matter 12,4638-4653 (2016)[4] M. Brito et al., “Deswelling effects on structural and transport properties of ionic microgelsuspensions” (in preparation)

Cross-flow inside-out filtration is a pressure-driven separation and enrichment process of microgel and colloidal suspensions where the feed suspension is continuously pumped tangentially through a membrane pipe. The transmembrane pressure (TMP) causes permeate (solvent) to flow out through the membrane, while the microgel particles are retained inside the pipe. Consequently, a particle-enriched diffuse layer is formed near the membrane wall which reduces the filtration efficiency. This so-called concentration-polarization (CP) layer is due to the balance of flow advection of particles towards and gradient diffusion away from the membrane. The resulting CP layer osmotic pressure operates against the TMP and counteracts the permeate flow.To model the CP layer with its physical effects in the ultrafiltration regime where Brownian motion dominates flow convection, it is crucial to determine the gradient diffusion coefficient, suspension viscosity, and osmotic pressure of concentrated microgel suspensions. In this study, we calculate these properties for non-ionic microgels treated as solvent-permeable and softly interacting colloidal particles. The suspension flow is described by the effective Stokes equation, and the permeate flux by Darcy’s law. Pore-blocking of the membrane is assumed to occur uniformly throughout its surface, with the sole result of increasing the overall hydraulic membrane resistance. We solve the coupled advection-diffusion equations quantifying of our ultrafiltration model by using a finite element method (FEM). The FEM results are compared with the standard similarity solution in boundary layer approximation [1]. As a reference, we also compare with simplifying analytic solutions for pure solvent flow, and for suspension flow where the microgel transport properties are taken as concentration independent [2]. Furthermore, we generalize the FEM analysis to a segmented membrane with impermeable rings to observe the weakening of the CP layer [3].References[1] R. Roa, E. Zholkovskiy, and G. Nägele, “Ultrafiltration modeling of non-ionic microgels,” Soft Matter 11, 4106–4122 (2015).[2] G. W. Park and G. Nägele, work in progress.[3] G. W. Park, M. Brito, E. Zholkovskiy, and G. Nägele, work in progress.

Hydrophobically modified exthoxylated urethane (HEUR) is one of the most popular examples of telechelic associating polymer. In a certain range of HEUR concentration in aqueous solution, (i) linear viscoelasticity follows a single-Maxwellian behavior with a characteristic time controlled by the network relaxation due to association/dissociation kinetics. In simple shear flows, HEUR systems often (ii) exhibit strain hardening in the transient startup when the shear rate exceeds the reciprocal relaxation time, and (iii) at steady state deviate from the Cox-Merz rule, deviations including (iv) a shear-thickening phenomenon. Recently, Park and Ianniruberto [1] suggested a new stochastic simulation to describe the complex rheological behavior listed in (i-iv). The results indicate that finite extensibility effects are mostly responsible for the strain hardening, while deviations from the Cox-Merz rule are due to the persistence of bridge chains. However, the detailed mechanism behind the shear thickening is still unclear. In this study, we show additional observables by using the same simulation method for a better understanding of the shear-thickening phenomenon. Furthermore, we extend the range of the parameter space explored in [1] by examining larger (and more reasonable) values of the ratio between the loop dissociation time and the Brownian diffusion time of flower-like micelles. [1] G. W. Park and G. Ianniruberto, J. Rheol. 61, 1293 (2017)