Engineering Water and Solute Dynamics and Maximal Use of CNT Surface Area for Efficient Water Desalination.

While polymer-based membranes and the consistent plants and elements have long been considered and optimized, there are only few studies on optimization of the new generation of carbon-based porous membranes for water desalination. By modeling the elements and their corresponding parameters in a vertical configuration via COMSOL Multiphysics software, an experimental setup was modified that contained various bare and carbon nanotube (CNT)-covered microprocessed porous membranes in parallel and in series. Several design parameters such as inlet pressure, length of outlet, vertical distance of the parallel membranes, and horizontal distances of the series membranes were optimized. Taking advantage of the uttermost surface area of CNTs and the engineered particle trajectory, almost 90% NaCl rejection and 97% Allura red rejection were obtained with very high permeation values. Considering microsized outlets, the results of particle rejections are outstanding owing to the smart design of the setup. The results of this work can be extended to larger and smaller scales up to the point where the governing equations still hold.

Introduction

Carbon-based materials, including carbon nanotubes (CNTs),[1,2] graphene,[3−5] graphene oxide,[6−9] and porous carbon,[10,11] have been the center of interest for possible replacements of state-of-the-art polyamide membranes for water desalination. Carbon-based material candidates have properties superior to those of the current membranes, including lower surface roughness, higher porosity, hydrophilicity, and more environment friendly, which can overall improve the efficiency of the membranes. Among these membranes, CNTs have attracted a lot of interest in separating water and salt ions due to their high mechanical strength, high chemical stability, and advanced functionality.[12−17] Although there are two main categories of CNT prototypes that have been realized for separation processes and specifically for water desalination: mixed matrix CNT membranes[18−22] and vertically aligned CNT membranes,[23−26] these are not the only ones. Decorating the surface of a porous membrane with grown CNTs can modify the surface properties and the large surface area of the CNTs improves the process of rejecting salt ions and transporting water.[27] It has also been found that the presence of carbon nanotubes can favorably offer the effective antimicrobial properties to diminish the biofouling during extended period of filtration.[28]

In an early report, it was surprisingly found that CNT-decorated microporous membranes contributed to salt rejection when exposed to vertical flow of salty water.[27] The underlying mechanism for this rejection was mainly related to the high bar/hole ratio of the samples and the high surface area of the CNTs.[27] But the other aspects of the mechanism may be related to the dynamics of water and solute transport across such a system. To understand the details of the rejection mechanism, it was necessary to model such a system and study theoretically the dynamics of water flow and particle trajectory across these membranes at millimeter scale. There are numerous reports on the dynamics of water when passing through polyamide membranes,[29−31] CNTs,[32−34] carbon composites,[35] nanoporous graphene,[36−38] and graphene oxide[39,40] at nanoscales using nonequilibrium, equilibrium, or classical molecular dynamics. Computational fluid dynamics is a promising tool that has been employed for optimization of reverse osmosis (RO) modules[41] and desalination membranes[42−44] at submicron scales. To have a general sense of the dynamics of water and particles as they flow in the whole assembly of new generation of desalination membrane setups, i.e., from the inlet all the way to the outlets, millimeter scale calculation is essential. This large scale of modeling is rarely found in the literature while polymer-based desalination membranes and the entire consistent plants that allow reverse osmosis (RO) processes have long been studied and optimized.

Here, using COMSOL Multiphysics modeling software, we modeled velocity field and magnitude of water flow (without particle) approaching several microporous membranes with varied outlet lengths. The different membranes were then mounted in parallel with different geometries, akin to application of spacers in polymer-based desalination membranes. Velocity field, velocity magnitude, and particle trajectory of water flow (with particles) were modeled. Equivalent experimental measurements were accompanied in most cases. From the significant variations in the dynamics of water and particles in the models, we designed a modified experimental setup to envisage the improvement in particle rejection across the membranes. To take the most advantage of the optimized geometry parameters and the surface area of the membranes, we coupled the two aspects by using CNT-decorated microporous membranes in an improved experimental configuration. High salt rejection, high dye rejection, and high water permeation were obtained in the final configuration and via the uttermost area of CNTs. The model specifications are not unique to the current experimental setup and can be extended to a general model to be envisioned in designing a variety of separation-based apparatus. The scale of the whole model can be reduced to 2 orders of magnitude while the results hold for these scales unchangeably.

Results and Discussion

Bare Individual Supports, Bare Parallel Supports

Experimental Results

The schematics of the setup and a tilted view of all individual porous supports used in particle rejection and water permeation measurements are presented in Figure a. From G300 to G1500, the size of the holes decreases and the density of the holes varies, as presented in Table . Si25 and Si49 also show a significant difference in the number and size of the holes. The grid’s outer diameter is a fixed value of 3.05 mm as they are provided commercially. The Si porous supports are squares with side length of 10 mm that were fabricated as discussed in Experimental Section. The water measurement setup (Figure b) including the sample holder, the spacers, and vertical Plexiglas cylinder (with height H2) was designed to accommodate the larger Si samples and to ensure comparable measurements for both the grids and the Si supports. This explains why two different bar/hole ratios are calculated in Table ; one ratio describes the bar area per hole area in each individual sample regardless of the size of sample holder and the other is related to the bar area of the sample regarding the area of the sample holder (100 mm2) per hole area of the sample. These two ratios were identical for Si-based supports and significantly different for the grids.

Figure 1

(a) Schematics of tilted view of G300, G400, G850, G1500, Si25, and Si49 showing significant differences in hole size and hole distribution at comparable scales in each support. (b) Schematics of the experimental setup for water permeation and salt rejection measurements in two-dimensional (2D) with six different inlets demonstrating details in the dot box. (i, ii) Individual grid and porous Si support, respectively, as the outlet of salty water flow. (iii, iv) One and two G850 held in parallel with Si49 and at height h1, respectively. The volume and purity of the flow after going through the four arrangements in (i)–(iv) are significantly different. (v, vi) Three-dimensional (3D) schematics of (iii) and (iv) where one and six G850 were held in parallel with Si49. h1 denotes the vertical distance at which grids were mounted above Si49, and D is the horizontal distance between the grids in series. H2 for all measurements was fixed at 10 mm, but H1 was 100 mm for 103 Pa and 1000 mm for 104 Pa measurements.

Table 1

Comparison between the Geometric Characteristics of Different Porous Supportsa

porous support	hole length (μm)	no. of holes in 3D	area of holes (μm2)	area of bars (μm2)	bar/hole per support	bar/hole in 100 mm2	no. of holes in 2D
G300	70	6.5 × 102	3.18 × 106	9.74 × 105	0.3	3.04 × 101	24
G400	45	1.37 × 103	2.78 × 106	1.37 × 106	0.49	3.49 × 101	33
G850	20	1.45 × 103	4.56 × 105	3.29 × 105	0.72	2.18 × 102	40
G1500	10.5	2.65 × 104	1.89 × 106	1.25 × 106	0.66	5.19 × 102	118
Si25	30	25	1.76 × 104	9.99 × 107	5.66 × 103	5.66 × 103	5
Si49	7	49	2.40 × 103	9.99 × 107	4.16 × 104	4.16 × 104	7

G850 among the grids and Si49 among the microfabricated Si supports have the highest bar/hole ratio.

Since our modeling results were in 2D, the last column in Table shows the number of holes of each porous support in 2D, i.e., the number of holes across the diameter of the grids or across the straight line, which connects the two sides of the square and the center of the square in Si supports. H1 + H2 was the height of salty water used as the driving force for water transport across the membranes (Figure b). The valve was situated directly above H2 and was opened when measurements started. Before the measurements, a homogeneous solution of salty water occupied the whole length of H1. H2 = 10 mm was kept constant and H1 produced a hydrodynamic pressure of 103 Pa if H1 ∼ 100 mm and 104 Pa if H1 ∼ 1000 mm. Although 104 Pa is smaller than the required osmotic pressure for water desalination (5–8 MPa), using gravitational pressure-driven setup in laboratory did not allow much higher pressures.

The inlets (i) and (ii) in Figure b schematically demonstrate a grid or a Si porous support used as the separation membrane in the measurement setup. Inlets (iii) and (iv) show a single or double grid in parallel with a Si porous support. These schematics show that from (i) to (iv), water becomes more purified after passing through the membranes. The 3D schematic image (v) shows a single G850 directly over Si49 and in parallel to it, where h1 varies to see the effectiveness of the vertical spacing between the two membranes. In inlet (vi), there are six G850 mounted symmetrically on a plane with no holes in its center, in parallel to and above Si49 at h1 vertical distance. D in this figure represents the distance between the outer diameters of every two G850 in series.

We performed water permeation and NaCl rejection measurements on all of the bare porous supports at 104 Pa. To relate our experimental results with the results from the modeling, velocity values were obtained in addition to permeation values through the following relationsIn relation , v (m/s) is the velocity of the flow at the outlet, V (m3) is the volume of the flow passing though the outlets, A (m2) is the area of the outlet, and t (s) is the time the flow takes to pass through the outlets. For relation , there are slight changes in the units of the quantities since permeation p is usually reported in the literature with L/(m2 h bar) units and P, the pressure of the inlet, is expressed in bars (bar), V is expressed in liters (L), and t is expressed in hours (h). Figure summarizes the experimental results of NaCl rejection, and velocity and permeation of the flow across several porous supports. The velocity obtained from experiments is the average velocity of the fluid, and compared to modeling, it is slightly lower than the maximum velocity in the center of the outlets. It is clear from relation that although v increases with increase in V, it decreases with the product of A and t. From fluid dynamics,[45]t changes inversely with A and V changes inversely with t. The larger (smaller) the outlet area of the fluid, the faster (slower) the fluid passes through with smaller (larger) t and the higher (lower) is the volume V of the fluid that crosses over the outlets.

Figure 2

Experimental results of bare individual supports and parallel supports. (a–c) Velocity, permeation, and NaCl rejection of individual bare supports G300, G400, G850, G1500, Si25, Si49. (d–f) Velocity, permeation, and NaCl rejection of parallel sets G850-Si49, D6_G850-Si49, and D3_G850-Si49 at h1 = 0.5, 1.5, 4.5, and 9.5 mm. h1 is the vertical distance between G850 and Si49, and D is the in-plane spacing between G850s, whose values are 6 and 3 mm, respectively. (g, h) Three- and two-dimensional views of the parallel setups with one and six G850 over Si49 and velocity field by streamlines (obtained from 3D modeling), which presents significant differences in the setups. (i) Images of mounted one and six G850 on Plexiglas that are laser-drilled with one and six holes, respectively.

For individual porous supports from G300 to Si49, the velocity of the fluid decreases with decrease of outlet area (Figure a). Permeation of the flow follows the same trend as the velocity. The data are discrete and connected via β-spline line to demonstrate the trend of changes more clearly. NaCl rejections of these supports are around 5–8% in the grids and 20–25% in the microfabricated Si hole array supports. We explored experimentally what would happen to the velocity and permeation of the flow and to the NaCl rejection if we combined grid and Si hole supports.

We knew from the modeling that beyond 5 mm above the outlets, the flow was laminar all the way up to the serum that contained the feed solution. Hence, all of the significant variations in velocity field and magnitude occur in the vicinity of the outlets. This was the reason we chose a window of 10 mm × 10 mm for all models. With this regard, the vertical distance of the grids with Si hole support, h1 (Figure g), changed from 0.5 to 9.5 mm, which lies in the extent of the model window. We had three different parallel setups: the first was with just one G850 above Si49 with zero offset and named as G850-Si49 (Figure g); the second was with six G850 in series with D = 6 mm and named as D6_G850-Si49; and the third with six G850 in series with D = 3 mm and named as D3_G850-Si49 (Figure h). Three-dimensional velocity field demonstrated by streamlines obtained from 3D modeling shows significant differences between the parallel setups having just one G850 and six G850 above an Si49 outlet. The details of these differences will be discussed in the next subsection. At this stage, we only mount two G850 in series at D values, to keep the experiments simple and comparable to the results from the models in 2D. In Section , we mount all six G850 as described in Figure h and perform measurements. While permeation of the water flow remains high in parallel setups, NaCl rejections are slightly improved with respect to individual supports. Interestingly, the parallel setup with six G850 and D = 6 mm reaches around 38% when h1 = 4.5 mm. The velocity in parallel setups varies with h1, but all of the values lie between the velocity of individual G850 and Si49.

A brief comparison between calculated streamlines in 2D and 3D models of the two main parallel setups with one and six G850 over an Si49 (Figure g,h) gives some indication of the differences in how the water flows from the two setups and reaches the outlets. To gain more insight into the underlying reasons for the improvements in the desalination parameters in parallel supports, we will discuss the modeling results for bare individual and bare parallel supports with two approaches: steady-state and time-dependent. By evaluating the parameters that affect the desalination performance in the next two subsections, we will demonstrate how we envisaged these understandings into designing a setup with specific water dynamics that will take advantage of the optimized geometrical values for much better NaCl rejection than individual supports and still high permeations.

Steady-State Modeling Results

It is known that during steady flow, the fluid properties can change point to point within a geometry, but at any fixed point, they remain constant. One of the main properties of a fluid is its velocity. From the definition of incompressible fluid, the variation of volume with time at both the inlet and outlet of the geometry, where the fluid is flowing, should remain constant. This means that the velocity at the inlet multiplied by the area of the inlet should be equal to the velocity at the outlet multiplied by the area of the outlet. Since the area of the inlet is much larger than the area of the outlet, velocity increases vastly at the outlets.

Figure compares 2D (x, z) velocity magnitude and velocity field of water for G300, G400, G850, G1500, Si25, and Si49 as outlets at 103 Pa. The inlet of all models is the entire 10 mm length at the top of the window (where z = 10 mm). The velocity magnitude has been demonstrated in two styles: surface (first column) and contour (second column). Due to nonslip conditions at the nonporous walls and from the surface representation, the velocity magnitude has a minimum value of 0 alongside all of the vertical walls, attains higher values near the pores in all outlets, and reaches its maximum value inside the holes. Contour plots show curves of constant values and reveal regions of high or low values of velocity magnitude with color legend same as the surface. It is evident that the highest values of velocity lie in the center of the outlets regardless of the length of the outlet in all six membranes.

Figure 3

Two-dimensional (x, z) velocity magnitude in surface (first column) and contour (second column) representation and velocity field streamlines in uniform density style (third column) and magnitude-controlled style (fourth column) of G300, G400, G850, G1500, Si25, and Si49 at 103 Pa.

While quantitative study of fluid dynamics requires advanced mathematics, much can be learned from flow visualization. The velocity field is demonstrated by streamlines (curves that are everywhere tangent to the instantaneous direction of fluid velocity vector), indicating the direction of fluid flow. Streamlines here have been demonstrated in two styles: uniform density (third column) and magnitude-controlled (fourth column). The first style was used to show an even distribution of the flow in the entire geometry and also the small turbulence of the flow produced in the bottom right-angled corners of the setup. The dimension of this turbulent flow extends more in (x, z) for outlets that are smaller in length. The second style demonstrates high- and low-density flows distributed unevenly in the entire geometry of the setup; a large number of streamlines are densely packed directly above the outlet and are much less dense near vertical walls. For Si49, with smallest outlets, the streamlines above the outlets are less dense in comparison to all other outlets. Both styles show in common that at z = 5 mm and above, the flow is completely laminar and for z = 0–5 mm, the streamlines converge into the outlets. It is evident that the streamlines in the vicinity of outlets (in the fourth column) are perpendicular to contours (in the second column) for identical regions as expected by the definition of streamlines. We note that the turbulence in a region forces the water to flow in cycles in that region until all of the laminar flow nearby drains completely through the outlets. As soon as the laminar flow volume is fully drained, the volume of water from turbulent flow will start exiting the outlets. A comparison between velocity field and magnitude in the region where we have turbulence demonstrates that the velocity magnitude of turbulent flow is lower than that of the laminar flow. The larger the volume of the turbulence region, the larger the amount of water with lower velocity. This slowing down of the flow in turbulence regions could result in longer interaction of salty water with a surface that is in contact with and can impact the particle rejection rate. However, according to our models, the volume of water from turbulent flow is usually much smaller than that from the laminar flow. Generally, some percentage of the energy of the flow is lost in turbulence; therefore, it is better to avoid these turbulent flows in optimizing design parameters.

The variation of velocity magnitude and field with increase in pressure from 103 to 106 Pa for G850 as outlet is demonstrated in Figure . A similar study is done for all other five outlets and presented in the Supporting Information (Figures S1–S5). The trend of variations is typically the same in all six types of outlets, and therefore we only describe G850. The main difference in streamlines of uniform density (Figure a) at different inlet pressures is the distribution of the turbulent flow in the bottom corners of the setup. With increase in pressure from 103 to 105 Pa, the turbulence becomes smaller, and at 106 Pa, it becomes slightly larger but being pushed up along the vertical walls. Streamline of magnitude-controlled style (Figure b) shows that increase in pressure increases the extent of laminar flow closer down to outlets. Contour representation of the velocity magnitude (Figure c) and the respective magnified images show how the area of higher velocity (from light blue to red) is hammered on the outlets as the pressure increases. A comparison of maximum velocity value from the legend shows that with linear increase in pressure, the maximum velocity (which is distributed inside the outlets) increases linearly.

Figure 4

Two-dimensional (x, z) velocity field streamlines in magnitude-controlled style (first row) and uniform density style (second row); velocity magnitude in contour style (third to fifth row) and in surface style (sixth row) of G850 as outlet at 103, 104, 105, and 106 Pa. The fourth row is a magnification of selected region in the third row, and the fifth row is the magnification of selected region in the fourth row.

Velocity field and velocity magnitude behavior for G850-Si49 is presented in Figure at vertical distances h1 = 0.5, 1.5, 4.5, and 9.5 mm and at inlet pressure 103 Pa. Results of similar comparison at other pressures for this geometry are depicted in the Supporting Information (Figures S6 and S7). At a fixed pressure of 103 Pa and at different h1 values, magnitude-controlled streamlines show no difference in the distribution of turbulent flow when approaching G850 outlet (indicated in red triangles), but there is a significant change in the size and distribution of the turbulent flow in the region between G850 and Si49 (indicated in green dotted contours). At h1 = 0.5–4.5 mm, the turbulence grows larger and fully fills the corners of the region between G850 and Si49 with converging flow directly between the holes in G850 and Si49. In h1 = 9.5 mm, the turbulence distribution is only limited to the four corners of the region with a size similar to turbulence highlighted in red triangles. As the inlet pressure increases to 104 and 105 Pa, there are no apparent changes in velocity field streamlines. When the inlet pressure is 106 Pa at all h1 values, turbulent flows in the right-angled corners of the setup approaching G850 (indicated in red triangles) become very small in comparison to 103 Pa (Figures S6 and S7). At h1 = 4.5, 9.5 mm, the turbulent flow in the region between G850 and Si49 becomes denser and larger and widens the area of converging flow between the holes of G850 and Si49 (Figure b). However, for h1 = 0.5, 1.5 mm, there is no significant difference in turbulent flow in the region between G850 and Si49. At all h1 values and inlet pressures, uniform density streamlines are closely packed and have very high density in the vicinity of G850 and Si49 holes. As h1 increases, the very dense streamlines get separated by the vertical distance between G850 holes and Si49 outlet. The turbulent flow (with 300 levels) is usually not apparent in magnitude-controlled style for other geometries, but it can be seen at some h1 values in this particular geometry. Velocity magnitude in contour style is in agreement with velocity field uniform density style. Magnified images of the contours near the outlets show less curvature for h1 = 0.5 mm but negligible differences at other h1 values (Figure ). With the increase of inlet pressure, the values of maximum velocities inside outlet holes increase linearly with increase in pressure (Figure b). When the grids are in series together and in parallel with Si49, D6_G850-Si49, a major difference with the single-grid case is the vanishing of turbulent flow almost everywhere in the magnitude-controlled style streamlines (Figure a, first row). In this geometry, the flow is directed to the grids that are spaced with D = 6 mm and are very close to the vertical walls. In D6_G850-Si49, the flow is directed from G850s and at h1 = 0.5, 1.5 mm, it almost reaches the outlet in Si49 horizontally, and therefore, minor turbulence forms (indicated in green contours). At h1 = 4.5 mm, the flow between G850s and Si49 is not horizontal but it converges from G850s to Si49 with almost no turbulent flow. At h1 = 9.5 mm, turbulence flow similar to G850-Si49 forms in the bottom corners of the setup. The above description holds for all inlet pressures with an exception. At 106 Pa and h1 = 4.5, 9.5 mm, turbulent flow forms directly beneath the G850–G850 region (Figure b), while no turbulence is present in this area at lower pressures (Figure a).

Figure 5

(a) Two-dimensional (x, z) velocity field streamlines in magnitude-controlled style (first row) and uniform density style (second row); velocity magnitude in contour style (third and fourth rows) of G850-Si49 at 103 Pa. The fourth row is a magnification of the selected region in the third row. (b) Two-dimensional (x, z) velocity field streamlines in magnitude-controlled style and uniform density style of G850-Si49 at 106 Pa at h1 = 4.5 and 9.5 mm.

Figure 7

Data analysis of 2D (x, z) models on velocity magnitude. (a, b) Variation of maximum velocity per individual porous supports and four parallel porous supports. (c) Variation of maximum velocity in four different parallel supports at h1 = 0.5, 1.5, 4.5, and 9.5 mm and at four different inlet pressures: 103, 104, 105, and 106 Pa.

Figure 6

(a) Two-dimensional (x, z) velocity field streamlines in magnitude-controlled style (first row) and uniform density style (second row); velocity magnitude in contour style (third and fourth rows) of D6_G850-Si49 at 103 Pa. The fourth row is a magnification of the selected region in the third row. (b) Two-dimensional (x, z) velocity field streamlines in magnitude-controlled style and uniform density style of D6_G850-Si49 at 106 Pa at h1 = 4.5 and 9.5 mm.

The streamlines in uniform density style are closely packed in the holes of G850s and Si49 (Figure a, second row). However, there is an evident horizontal dense streamline region between the grids and Si49 when h1 is small (e.g., h1 = 0.5, 1.5 mm). This region is of interest for the experimental measurements where the water flow is in close and probably longer contact with the surface of Si49. The velocity magnitude presented by contours in the third row shows the same aspects of the flow as in the uniform density streamlines. At h1 = 0.5 mm, the curvature of the contours in the vicinity of the outlet is very different from that at higher h1 values. A full comparison of velocity magnitude in contour style and velocity field streamlines in magnitude-controlled style and uniform density style of D6_G850-Si49 at 104 and 106 Pa is presented in Figures S8 and S9 where h1 = 0.5, 1.5, 4.5, and 9.5 mm. With the increase of inlet pressure, the values of maximum velocities inside outlet holes increase linearly with the increase in pressure (Figure b). Similar calculations were performed on D3_G850-Si49 at 104–106 Pa (Figures S10–S12) and Ext_G850-Si49 at 103–106 Pa (Figures S13–S15) both at h1 = 0.5, 1.5, 4.5, and 9.5 mm. D3_G850-Si49 is very similar to D6_G850-Si49, but the spacing between series G850s is 3 mm and each G850 has a distance of 2.5 mm from the vertical walls, which causes a larger distribution of turbulent flow. Ext_G850-Si49 geometry is similar to G850-Si49, but the holes of G850 extend across the 10 mm length. We do not have an experimental analogy to this geometry, but it was essential to see the effects of extended holes on the velocity field and magnitude at least in the models.

The maximum velocity of all of the different geometries modeled in this section is plotted in Figure .

This value is obtained from velocity field magnitude graphs in contour style where the maximum refers to the center of the main outlet hole. For parallel supports, the outlet holes in all cases correspond to Si49. The maximum value of velocity is slightly larger than the average velocity from experimental data, but the trend of their variation is very similar. For most individual porous supports, the maximum velocity increases almost an order of magnitude with the stepwise increase of inlet pressure from 103 to 106 Pa. However, for the supports with the largest holes (e.g., G300 and G400), this increase is less than an order of magnitude (Figure a). For all parallel geometries, the increase of maximum velocity with pressure is linear (Figure b). Considering the effect of the vertical distance between G850s and Si49, h1, there are some minor variations in the maximum velocity of different parallel supports at each inlet pressure (Figure c). The pattern of these variations for every parallel support is barely changed with the increase in inlet pressure. h1 = 1.5 mm seems to be a characteristic vertical distance since at most of the inlet pressures, it has maximum velocity values. We will later see that optimum velocity is not the only parameter that is critical for better performance of the membranes. Particle trajectory (in the models) and particle rejection (in the experiments) are other important parameters that need to be optimum when engineering the dynamics of water-containing particles across porous membranes.

Time-Dependent Modeling Results

In the previous section, the effect of geometry was examined on the velocity field and magnitude of pure water flow and some optimum values were obtained in parallel porous supports. Once the water is not pure and contains particles, it is important to understand how these particles find their way from the inlet to the outlets of the porous supports and how do inlet pressure, size of the pores, and vertical distances of the supports in parallel setups impact their transport time and pathways. For this reason, we performed particle trajectory on a limited number of particles in optimal geometries from the previous section. The number of the particles in all geometries was 300, and the diameter of the particles was 1 μm. Since an extremely fine physics-controlled mesh was used with a minimum element size of 0.24 μm, this diameter was reasonable. Although the diameter of the salt ions and the dyes in our experiments were several times smaller than the finest pores in the outlet membranes, the diameter of the particles in the model was only 7 times smaller than the finest pore size in the porous supports, which was inevitable due to the minimum element size in the mesh. The choice of the number of particles was based on test runs and was reasonable because it was enough to follow the trace of the particles. Furthermore, the calculation time for each geometry was not too long with this number. As described in Simulation Model and Method, we modeled another set of porous supports with three parallel membranes, where an outlet O is in parallel with G850-Si49, D6_G850_Si49, and D3_G850-Si49. We summarized the velocity field and magnitude results obtained from the steady-state models for these new geometries and particle trajectory results for better comparison. To keep the calculations simple, we selected a single hole with diameter d ranging from 5 μm to 1 mm with a varied vertical distance h2 from Si49. For h1 = 1.5 mm, h2 was taken as 0.5, 1, 1.5, 4.5, and 9.5 mm, and for h2 = 1.5, h1 was taken as 0.5, 1.5, 4.5, and 9.5 mm. The inlet pressure was varied from 103 to 106 Pa for minimum values of h1, h2, and d. Particle trajectory was studied with time variation where a timeline was produced as the result. A timeline is a set of adjacent fluid particles that are marked at the same (earlier) instant in time. Because of friction at the vertical walls and the no-slip condition, the fluid velocity there is zero. Hence, the left and right of the timeline are anchored at their starting locations. In regions of flow away from the walls, the marked fluid particles move at the local fluid velocity, deforming the timeline. For all calculations, velocity field (mainly with uniform density style and some with magnitude-controlled style) obtained by steady-state calculations was superimposed over the particle trajectory at each snapshot of the timeline to distinguish the difference between water streamlines and particle pathlines. The full timeline of each calculation contained too much data; therefore, it was impossible to show all of the timelines. We chose main snapshots of the timelines, and in specific cases, a video of the full timeline is presented in the Supporting Information. However, the full result of the timeline for each calculation is given in a graph demonstrating the variation of total number of particles with time evolution. For simplicity and reducing calculation time, we modeled particle trajectory on G850-Si49-O and examined the effect of various parameters, and from the obtained results, we performed further calculations with optimum parameters on D6_G850-Si49-O and D3_G850-Si49-O.

The first column in Figure shows velocity magnitude (in contour style) for G850-Si49 mounted over the third outlet with diameter d (5, 10, 20, 50, 500 μm) and at vertical distance h2 = 1 mm with h1 = 1.5 mm and at 103 Pa. As d increases, the contours become denser around G850, Si49, and the final outlet O, which is depicted clearly in the magnified inset image. The pathlines of the particles in the second and third columns are shown for T = 0 and 60 s superimposed on velocity field streamlines. At T = 0 s, the spectrum of particles at the top of the inlet demonstrates variation in initial velocity depending on the location of the particles. The legend of the figure shows the velocity magnitude of particles. With the increase in time, the velocity of the particles increases and therefore the particles gain acceleration. The time at which the particles reach the first holes (in G850) is dependent on the diameter of the final outlet O. It takes 53 s for the particles in the geometry with d = 5 μm to reach G850, and at 60 s, about 75 particles out of 300 reach the final outlet (fourth column). A more detailed timeline for d = 5 μm is presented in Figure S16, Supporting Information. This figure compares time evolution of the particles in long enough time, where they reach stability and their pathline almost fits the velocity field streamline. Velocity field streamlines are represented in two styles, magnitude-controlled and uniform density. From a detailed study on the particles time evolution, it is clear that the particles do not fall into the turbulent flow. For d = 500 μm, it takes only 2 s for the particles to reach G850 and about 260 of them reach the final outlet at 60 s. This is also reflected in velocity field streamlines where at larger d values, the streamlines are much denser even at the inlet compared to streamlines in smaller d values. It is clear that the pathlines for the particles deviate from the streamline of water in all geometries until a stable situation is reached in the time-dependent picture (see Figure S16). For instance, when d = 10 μm and for T ≥ 18 s, a stability is reached and the pathlines from unsteady-state modeling almost fit the streamlines from steady-state modeling. When inlet pressure increases from 103 to 106 Pa with d = 5 μm, h1 = 1.5 mm, and h2 = 1 mm, the main impact on time evolution of the particles is the significant drop of onset time of particles in reaching G850 and the slight increase in the number of particles that reach the final outlet at 60 s. For the inlet pressure of 106 Pa, it only takes 0.2 s for the particles to reach G850 and by 60 s, almost 200 of them reach the final outlet. Details are presented in Figure S17, Supporting Information. For d = 5 μm and h1 = 1.5 mm, at 103 Pa and h2 = 0.5, 1.5 mm, the onset of reaching G850 are 52 and 53 s, respectively, but for h2 = 4.5, 9.5 mm, the onset does not fall in the 0–60 s range, and therefore an extra 30 s was added to the calculations (Figure S18). When h2 is fixed at 1.5 mm and h1 = 0.5, 1.5, 4.5, and 9.5 mm, the onset remains in the 0–60 s range (Figure S19). The most characteristic variations for h1 and h2 are presented briefly in Figure . A minimum number of particles (∼10) reach final outlet at 60 s when h1 = 9.5 mm and h2 = 1.5 mm.

Figure 8

Two-dimensional (x, z) velocity magnitude in contour style (first column); particle trajectory (pathline) and timeline (second and third columns) in a G850-Si49-O system where the length of the hole varies from 0.005 to 1 mm at 103 Pa. Variation of the total number of particles with time (fourth column) shows the onset of the time where particles reach G850 and the number of particles that reach O at 60 s.

Figure 9

Two-dimensional (x, z) variation of the total number of particles with time (first row) in a G850-Si49-O system where d = 0.005 mm for different h1 and h2 values at 103 Pa. Corresponding particle trajectory (pathline) and timeline (second to fifth row).

In parallel supports, when a G850 is added in series with separation D to the other G850, the particle trajectory becomes more interesting. Figure compares the time evolution of particles passing through three geometries: G850/Si49/O, D6_G850-Si49/O, and D3_G850-Si49/O. In the 0–90 s range, it takes 84 s for the particles in D6_G850-Si49/O to reach G850s, while less than 40 particles manage to reach the final outlet at 90 s. In D3_G850-Si49/O, the onset of reaching G850s is 74 s and almost 130 particles reach the final outlet at 90 s. The timeline comparison clearly shows that when G850 is directly above Si49, the particles have the shortest path to travel to reach the final outlet. But when there are two G850 separated by D, the particles travel a longer path. When the separation D is 6 mm, this path is the longest. This is briefly demonstrated in Figure e, where the time range was 0–90 s. Figure summarizes the results on particle trajectory and maximum velocity modeling in different geometries and for variations in d, h1, h2, and inlet pressure. The number depicted (per second) against the second y data is the onset time when particles reach G850.

Figure 10

Two-dimensional (x, z) variation of the total number of particles with time (first row) in G850-Si49-O, D6_G850-Si49-O, and D3_G850-Si49-O, where d = 0.005 mm, h1 = 1.5 mm, and h2 = 1.5 mm at 103 Pa. Corresponding particle trajectory (pathline) and timeline (second to fourth row).

Figure 11

Data analysis from 2D (x, z) models on particle trajectory and velocity magnitude of different geometries. (a) d variation in G850-Si49-O, (b) P variation in G850-Si49-O, (c) h1 variation in G850-Si49-O, (d) h2 variation in G850-Si49-O, (e) comparison between G850-Si49-O, D3_G850-Si49-O, and D6_G850-Si49-O, and (f) comparison between G850, Si49, G850-Si49, D6_G850-Si49, O (final outlet with d = 0.005 mm), and D6_G850-Si49-O.

All of the data presented here are discrete and the dotted lines only connect the data to show the trend of changes. The general result from data analysis in Figure a–e shows an increase in maximum velocity with decrease in the onset time of particles reaching G850 and increase in the maximum number of particles that reach the final outlet in any particular time range and vice versa. Figure f compares maximum velocity and maximum number of particles reaching the final outlet between six different geometries. This comparison highlights the advantages of using parallel porous supports over individual ones in optimizing the membrane performances. When two G850 are put in parallel with Si49 and a single outlet O, the number of particles that reach the final outlet is 38 at 90 s. At high enough times (t ≥ 300 s), this number increases to 76 and stabilizes. This is the lowest number of particles reaching the final outlet compared to other geometries. The ratio of this number over 300 subtracted from 1 gives a value that can be a measure of particle rejection in the parallel geometry. This means that particles with smaller diameter than the membrane holes can be rejected through other mechanisms than size rejection. While this rejection is negligible in individual porous supports or even parallel supports without O, it is quite substantial in parallel supports with O.

Furthermore, the particles reach G850s at 84 s and the maximum velocity inside the final outlet O is 6.2 × 10–3 m/s in D6_G850-Si49-O. In any given time range, the longer it takes for the particles to reach G850, the lower the acceleration of the particles. From Newtonian mechanics, acceleration of a particle has a direct relationship with the path it has traveled and an inverse relationship with the square of the time its travel has taken. With the optimum values found for h1, h2, d, P, and also the more ideal geometries in the models, the possibilities of our experimental setup in the next section narrow down to a few cases. We only use the results of the models and the trends in parameter variations to elevate our experimental results, and due to the differences in particle diameters, we will not attempt to compare the exact particle rejections from the models to particle rejections from experiments.

Individual and Parallel CNT/Supports for Desalination

From the results of Section and the results from the modeling, we can draw some lines: bare individual supports have very low salt rejections and bare parallel supports without O have improved but still low rejection rates. However, parallel supports with very small O demonstrate candidates with well improved salt rejection, but they are not all quite the same. For a geometry where G850 is directly above Si49, lower acceleration of the particles due to the presence of very small O has some improvement in particle rejection. In geometries with two G850 in series being farthest apart, we expect higher improvement because the acceleration of the particles decreases more as they reach G850 and the particles eventually flow to the final outlet O. On the other hand, in the region between G850s and Si49, and for small h1 values, the flow is horizontal and almost laminar. Hence, we expect the particles to have more interaction with the surface of Si49 especially away from its holes. We will see in Section that salt rejection in the geometries with horizontal flow improves quite significantly.

It is shown that some types of CNTs can reject salt ions such as NaCl and MgSO4 to an extent using their surface properties[27] rather than size rejection through the inner diameter of the tubes.[23,24] Among them, spaghetti CNT had better performance. In the next subsection, we use CNTs to enhance the surface properties of the membranes along with decreased particle acceleration and horizontal flow to vastly improve salt rejection while retaining water permeation higher than the state-of-the-art desalination membranes.

CNT Growth and Characterization on Porous Supports

Spaghetti CNT was grown on all porous supports mentioned in Section . We followed a similar protocol for the growth of CNTs to that in ref (46), where a tip-growth mechanism was proposed. Therein, (110) planes of the nickel were catalytically active surfaces during the growth. Mechanisms of CNT growth have been discussed by Chhowalla et al.,[47] where both tip and root growth conditions are possible, depending on the adhesion of the seed layer to the substrate. As observed in the magnified field emission scanning electron microscopy (FESEM) images, the Ni seed has been lifted upward during the growth of the CNTs and positioned at the top of each tube, confirming a tip-growth mechanism. While the growth process on Si supports was very reproducible, it was not very successful on the grids. Some of the successful growths on G850 and on G1500 are shown in Figure .

Figure 12

(a) FESEM images of spaghetti grown on G850, G1500, Si25, and Si49. (b) Experimental mounting of G850 over CNT/Si49 and in parallel with a third outlet O for water measurements. (c) Schematic image of spaghetti CNTs grown on Si49 showing the length and density of the CNTs. (d) Raman spectrum of CNT/Si supports representing D, G, and G′ peaks, which are signatures of CNT formation.

Thin nickel grids with thickness ≤30 μm tended to curve beyond 600 °C, where the required temperature for the growth of spaghetti CNT was above 650 °C. It was very hard to mount bended grids over tiny holes in Plexiglas and avoid water leakage from the sides. CNT/microporous Si supports (Si25 and Si49) were either taken further for water measurements individually or used in parallel with bare G850 and O supports (Figure b). Raman spectroscopy (Figure d) demonstrates the formation of CNTs. D-band (1348 cm–1), G-band (1594 cm–1), and G′-band (2697 cm–1) confirm the typical characteristics of multiwall carbon nanotubes.[48] Raman scattering is highly sensitive to the electronic structure and is an essential tool to characterize carbonaceous materials. The G-band corresponds to the tangential stretching mode of an ordered graphite structure with sp2 hybridization and the D-band relates to the disorder-induced phonon mode due to finite-size crystals and defects.[49] G′-band is an important band in carbon nanotubes (CNTs), which gives information about the degree of nanotube crystallinity.[50]

The sharp G′-band in Figure d suggests metallic nanotubes rather than semiconducting ones. The ratio of D to G band intensities (ID/IG) is known to associate with the in-plane crystal domain size and has been used to estimate the degree of disorder in graphitic carbon.[51] While an ID/IG ratio near zero indicates high crystallinity (order), a ratio close to or greater than 1 demonstrates high disorder due to abundant defects in the graphitic structure; ID/IG = 0.791 in Figure d suggests somewhat defective surface or disorder of the grown CNTs.

As mentioned, Ni remained at the end of the tips of the tubes. The remaining Ni at the tip of the tubes (and mainly inside the tubes) has a minor chance to escape from the rigid crystalline structure of the CNTs and flow across the apertures of the membrane into the permeate solution and contaminate the solution. For large-scale fabrication of CNT membranes, removing Ni may be an issue to avoid water contamination and health hazards if the membranes are to be used for water desalination. To remove the Ni from the end of the tubes, there is a protocol that can be followed.[27]

NaCl and Dye Rejection and Water Permeation

Figure a compares NaCl rejection and water permeation in porous supports with grown spaghetti CNT. The rejection rates for the grids with CNT improves slightly with respect to their rejection without CNT (in Figure c). For all of the rejection measurements in this section, we used the pressure of 103 Pa, due to optimizations in the models and a constant permeate solution of 20 cc. The rejection rate in G850-CNT/Si49-O has been significantly improved with respect to G850-Si49-O due to the presence of CNT. The permeations in samples with final smaller outlets are higher. Permeation as described by relation has an inverse relation with the area of the outlet and the time it takes for a certain volume of water to pass the outlet. When the area of the outlet is small, it takes more time for the flow to pass through, which overall results in high permeation. In a comparison with ultrafiltration membranes, the values we obtained for water permeation were a few orders of magnitude higher. One main reason is that our experimental applied pressures were about 3 orders of magnitude lower than the applied pressure on ultrafiltration membranes and permeation has an inverse relation with pressure. Also, the permeation values reported for ultrafiltration membranes is related to large area samples that fit into industrial plants but the overall area for our samples here are orders of magnitude smaller.

Figure 13

(a, b) NaCl rejection and water permeation in various individual, parallel, and CNT-covered porous supports. (c) SEM image of a typical O outlet, with d ∼ 0.005 mm fabricated on SiN/Si/SiN membrane. The topmost circle layer is the thin nitride remained from the physical etching process. (d) Dye rejection (Allura red and Indigo) for three parallel supports. (e, f) UV–vis absorbance spectra of 0.1–1 mM Allura red and Indigo dyes.

D6-G850-Si49-O (Figure b) shows better NaCl rejection with respect to G850-Si49-O, which demonstrates the effectiveness of having two G850s spaced as far as possible rather than having just one G850 directly above Si49. The horizontal flow that forms between G850s and Si49 and the decrease in particle acceleration are the underlying reasons for this improvement. Strikingly, D6-G850-CNT/Si49-O (Figure b) shows much better NaCl rejection with respect to G850-CNT/Si49-O, where both Si49 are covered with CNT. In the former, the horizontal flow of water and the particles directly above CNTs and the decrease in particle acceleration in this region play an important role. Experimental sample D6-G850-CNT/Si49-O had two G850s equivalent to the models. But to take advantage of the whole surface area of CNT/Si49, we used six G850s (as described in Figure ) in a symmetric manner and we name it 3D6-G850-CNT/Si49-O. This allowed water and salt flow to reach all of the surface area of CNT before draining out of Si49 holes. NaCl rejection in this sample reached almost 90%, which is amazingly high. Using a smart configuration of parallel supports with effective distances at low pressure, microporous membranes covered with spaghetti CNT managed to reject ions of nanometer scale; 1 mM Allura red and Indigo dyes were also passed through the three samples with highest NaCl rejections to measure the effectiveness of their filtration further. UV–vis spectra were taken on 0.1–1 mM dyes, as shown in Figure e,f. The permeate solution of the dyes was compared to the calibration spectra, and the rejection was calculated (Figure d). The rejection of Allura red was higher than Indigo in all three samples, since Allura red is a longer chain of atoms and also has a double molecular mass than the Indigo. In the measurements involving CNTs, the error was slightly higher than that without CNT. This could be related to the random distribution of the tubes in different directions.

Effect of Wettability and Porosimetry on Desalination Performance

Wetting is the process of water interacting with a surface, and wettability studies usually implicate the measurement of contact angles. The contact angle of a liquid drop on an ideal solid surface is defined by the mechanical equilibrium of the drop under the action of three interfacial tensions: liquid–vapor, solid–vapor, and solid–liquid interfacial tensions, described first by Thomas Young.[52,53] The contact angle is geometrically obtained by applying a tangent line from the contact point along the liquid–vapor interface in the droplet profile. Usually a small contact angle is observed when the liquid spreads on the surface, while a large contact angle is observed when the liquid beads on the surface and forms a compact liquid droplet. The most widely used technique of contact angle measurement is a direct measurement of the tangent angle at the three-phase contact point on a sessile drop profile.[54] Here, the wettability was evaluated using dynamic water contact angle tests on the CNT/SiN and CNT/Si49 surfaces (Figure S21). Since the substrates were relatively large, contact angles were measured at multiple points and an average value, representative of the entire surface, was recorded. As the water dropped onto the CNT/SiN surface, the contact angle declined slowly with time from 134.75 to 121° in 210 s while the values suggest a hydrophobic nature when there is no pore on the surface and gravity is the only external force. The contact angle on CNT/Si49 had values below 90°, suggesting a hydrophilic nature with the pores sucking the water through themselves. When a drop of 0.1 M NaCl was released on the same surface, the initial contact angle was lower than the initial contact angle with pure water and it declined faster with time (Figure e).

Figure 14

(a, b) Two-dimensional (x, z) velocity magnitude in surface for G850 and one of the holes in G850-Si49 at different pressures. As the pressure increases, the flow of high velocity water gets suppressed. (c) Dynamic water contact angle measurement with time for CNT/SiN and CNT/Si49. (d) Contact angle vs pore diameter considering the surface tension of water and CNTs at 104 Pa. (e) Pore diameter vs contact angle variations at different pressures.

It was not straightforward to assess the contact angle of water in the pressure-driven setup we used for water permeation and salt rejection measurements. From the literature,[55] at high contact angles, very high pressures are required to allow water to pass through pores. The simulations on various porous supports under applied pressure presented in Section showed that with increase in pressure, the velocity of the water flow increased linearly, and as a result, water permeation increased with the same trend. Although it is not possible to give quantitative values for water contact angle, the simulations provide some insight into the effect of pressure on the wettability of the surface. Figure a shows the velocity magnitude variation of G850 with pressure. The suppression of high velocity water onto the hole area is evident as the pressure increases (comparing the vertical distance of the horizontal dashed line with the top of the high velocity area). At the lowest pressure (103 Pa), on the edges of the hole cross section, velocity makes angle Ψ < π/2 (depicted with red arrows) with the horizontal hole cross section. It suggests that water only goes through the holes if it is directly above the holes, and at all other regions, velocity has almost zero value. At higher pressures, velocity attains tangent components with Ψ > π/2 and nonzero velocity is extended beyond the edges of the hole cross section. Note that Ψ should not be confused with θ, whereas the former corresponds to velocity and the latter to the topography of water. The suppression of high-velocity water above holes cross section at higher pressures in Figure b is less significant but still present. In this figure, the depicted hole is the central hole of Si49 in the G850-Si49 structure. It is clear that pore size and density have a direct impact on wettability of the surface. These analyses on the velocity variation of water flow with pressure across a given bare porous support can be applied to a CNT-covered surface. With CNT on top, surface tension of a porous support is nonzero relative to a bare porous support, but its magnitude does not change with the increase in inlet pressure. Therefore, the inlet pressure still works against the surface tension and increases the wettability at elevated pressures. For the desalination purposes, higher wettability helps in the increase in water permeation, but it may not be favorable to salt rejection since water permeation and salt rejection always have a trade-off. However, our results show that it is possible to engineer the measurement setup to take advantage of the low hydrophilicity of CNT-covered supports to improve salt rejection and keep water permeation high enough for efficient desalination.

To investigate how contact angle and pore diameter relate, we used the following relationwhere P is the liquid pressure, γ is the surface tension, θ is the contact angle, and D is the pore diameter. Since all our samples were either microfabricated or commercially purchased, the geometry and size of the pores were determined and checked with FESEM. The minimum diameter of the pores in the samples was 5 μm (Table ). When Si49 was covered with CNT, the diameter of the holes decreased by 10–15% according to FESEM image analyses, which can, in principle, increase the contact angle by a small amount. From the literature, the surface tension between water and multiwall CNTs is about 72 mN/m.[56,57] We calculated the contact angle of a porous surface at 104 Pa and at different pore diameters by substituting these values into (relation ). Figure d shows the variation of contact angle with pore diameter. As the pore diameter increases from 0 to 15 μm, the contact angle decreases linearly from 90 to 58°, but beyond this point, it drops exponentially to 0° for pore diameter 28.8 μm. At 103 Pa, the contact angle starts with 90° for zero pore diameter but decreases much faster with increase in pore diameter. At this low pressure, a pore must be 288 μm wide to have zero contact angle. For higher pressures of 105 and 106 Pa, the maximum value of pore diameter must be 2.88 μm and 288 nm, respectively, to have zero contact angle. Overall, at all pressures, hydrophilicity increases with the increase in pore diameter, but the rate of the increase is higher at higher pressures.

Generic Applications and Future Perspective

The sizes of the different elements in the models proposed in this work are not limited to the current values, they can be larger and even smaller. The trend of variations in the parameters is also extendable to other scales. For example, the overall scale of the model G850-Si49-O with h1 = h2 = 1.5 mm, d = 0.005 mm, and P = 103 Pa was reduced to 1:10 and another time to 1:100 (Figure S20 in the Supporting Information). There was almost no change in the particle trajectory and timeline when the scales were lowered. We did not look into scale 1:1000 since the size of the elements would drop to nanometer, and in this range, the governing equations describing the physics of the system would presumably change. To have a sense of the size of the model elements in these reduced scales, let us consider scale 1:100. In this scale, the vertical and horizontal size of the model reduces from 10 mm to 100 μm, the typical size of the outlets reduces from 5–7 μm to 50–70 nm, the typical vertical distances between the parallel supports reduce from 1.5 mm to 15 μm, the horizontal distance between supports reduces from 6 mm to 60 μm, and the diameter of the particles reduces from 1 μm to 10 nm. These values are typical in microfiltration and nanofiltration processes.

We could not model the CNTs directly in the current scale of this work, due to the minimum element size of the mesh. The overall size of the setup was important in this work and we did not want to eliminate the effects caused by millimeter scale. Now that we have learned those aspects, as part of future work, we can zoom in to the area where parallel supports provide horizontal flow and include the CNTs to directly see the effect of their presence on particle trajectory. We could also charge the particles and the CNTs by applying an electric field and study the interaction of CNTs with the particles and the modified particle trajectory. There are so many other aspects that can be part of the future work such as the variation of the number of particles and the shape and orientation of the tubes if the scale of the measurement is lowered.

Polymer-based desalination membranes and the entire plant that holds the membranes along with all other parts that allow RO processes have long been studied and optimized. Now using a new generation of porous membranes requires optimization at all levels. Engineering the measurement setup including parallel porous supports decorated with carbon-based materials in a smart way to increase particle rejection and maintain high permeation is a must in widening the future perspective for water desalination.

Conclusions

Motivated by the potential of unprocessed CNTs in water desalination through their surface area, we aimed to increase the efficiency of these materials by developing an engineered measurement setup with optimized design parameters such as inlet pressure, outlet length, and vertical and horizontal distances between the membranes. Using a set of microporous supports in parallel and series, we took the uttermost advantage of the surface area of CNTs to enhance salt rejection and maintain water permeation at high levels. In the most optimized configuration, almost 90% NaCl rejection was obtained while the lengths of the outlets were at submicron level. For larger particles such as Allura red, almost 97% rejection was obtained. Water permeation values for all configurations were much higher than polymer-based desalination membranes.

Experimental Methods

Preparation of Holes in Si3N4/Si/Si3N4 Substrates

A combination of wet and dry etching processes was carried out on single-side-polished Si3N4/Si/Si3N4 wafers 10 mm × 10 mm × 500 μm, with the nitride layer as a hard mask, following the protocol in ref (5). Two sets of array patterns were produced, one with 5 × 5 array consisting of 25 holes and the other with 7 × 7 array consisting of 49 holes, denoted as Si25 and Si49 hereafter, respectively. The lengths of the holes in Si25 and Si49 were 30 and 7 μm, respectively. The distance between hole centers in Si25 was 400 μm, and in Si49, it was 50 μm. Using the same protocol, another array was produced with 5 μm hole length and 400 μm distance between hole centers. All of the holes were blocked using photoresist apart from the middle hole before physical etching in reactive ion etcher. This membrane with only one hole is named O, hereafter.

CNT Synthesis and Characterization

Patterned Si25 and Si49 were transferred in physical vapor deposition system, and 20 nm Ni was deposited on the polished patterned side. These samples and commercial bare Ni grids with hole lengths of 70, 45, 20, and 10.5 μm (named as G300, G400, G850, and G1500 hereafter, respectively) were then taken into a homemade direct-current plasma-enhanced chemical vapor deposition system for the growth of CNTs. With hydrogen gas flow, 30 s hydrogen plasma, acetylene gas was injected into the chamber to initialize CNT formation. Meanwhile, the hydrogen flow and temperature remained at 100 sccm and 700 °C, respectively.

Experimental Setup for Salty Water Filtration

Using various flat and cylinder-shaped Plexiglas, epoxies, stainless steel screws, washers and threads, volume expanders, and their tubing kit including a valve and connecting rubber tubing, we designed a simple pressure-driven setup for measurement of water transport and salt rejection of bare grids, Si supports, and CNT-covered samples. For all measurements, the source and tubing (with height H1) was initially full of feed solution with the valve closed, and with the start of measurements, the valve was half-way opened. The density (ρ) of the feed solution, 10–2 M NaCl, is roughly 1000.2 kg/m3 at room temperature. Since salt concentration in brackish water is 500–3000 ppm and in moderate saline water, it is 3000–10 000 ppm, our feed concentrations stand at a higher range of brackish water definition.[58] Permeate solutions were collected in small beakers for salt rejection measurements.

Electrical Conductivity (EC) and UV–Vis Measurements

Electrical conductivities (ECs) of feed and permeate solutions were measured by a portable multirange EC meter (HANNA Instruments HI8733, Romania). The probe of the instrument was held firmly in beakers containing certain volume of feed/permeate solutions until conductivity values were stable and recorded. The conductivity of 10–2 M NaCl solution was 1240 μS/cm. Salt rejection rate (%) was measured as . Each measurement was conducted three times and the resulting rejection rates were averages of the three measurements to reduce the reading/sampling errors. After each measurement, the sample surfaces were washed thoroughly with deionized (DI) water. Red (Allura) and blue (Indigo) dyes with molar masses of 496.42 and 262.27 g/mol, respectively, were also used as feed solution in an identical setup described in previous section. A 1 mM solution of each dye was prepared in distilled water. UV–vis absorbance spectra (using Avantes, λ = 200–800 nm) of the two dyes were taken as feed solutions. After passing through the membranes, similar measurement was carried out on permeate solutions. The difference in absorbance of the two (feed and permeate solutions) was considered as a tool to measure the rejection rate of the dyes.

Water Contact Angle Measurements

Dynamic water contact angle measurements were done on bare SiN surfaces and CNT/SiN surfaces once with DI water droplet and again with 10–1 M NaCl solution. The contact angle was measured with time variation using the sessile drop method by IRASOL (CA-500A) instrument equipped with a digital camera.

Simulation Model and Method

Steady-State Modeling

Navier–Stokes was the governing equation for steady-state and incompressible flow using COMSOL in Cartesian coordinate system[45]where ρ is the density of the fluid (kg/m3), = (u, u, u) is the velocity vector (m/s), P is the pressure (Pa), is the volume force vector (N/m3) representing external force, μ is the dynamic viscosity of the fluid, and T is the absolute temperature (K). Here, the density of the fluid was 1000 kg/m3, the dynamic viscosity of the fluid was 0.001 Pa s, and there was no external force. The boundary conditions in the inlet were: laminar inflow with four different entrance pressures, 103, 104, 105, and 106 Pa, and no-slip boundary condition in walls. In all 2D models, = (u, u) and extremely fine physics-controlled mesh was used with maximum element size of 8.04 × 10–2 mm and minimum element size of 2.4 × 10–4 mm. The geometry of the model was considered according to the geometry of the experimental setup. The inlet of the flow was an opening of 10 mm width (in all models) and the height of the fluid was also considered as 10 mm. Although the maximum height in the experimental setup was 103 mm, fluid mechanics and our test runs disclose that the flow is laminar and the velocity field and magnitude are constant along the tube, well above the outlets. When the height of the fluid was <5 mm, in the vicinity of the outlets, the geometry of the setup starts to change and results in significant variations in the velocity. The lengths of the holes in individual porous supports Si25, Si49, G300, G400, G850, and G1500 were initially used as the outlet of the fluid. Then, in parallel supports, the outlet was the holes of Si49, with G300 in a parallel plane but directly on top (with offset 0) at different heights h1 = 0.5, 1.5, 4.5, and 9.5 mm. In another geometry, two G850s were situated in series at two different spacings (D = 3, 6 mm) in parallel with Si49, at all mentioned h1 heights. In this section, a hole-bar line akin to G850 but extended across 10 mm length was modeled above and in parallel with Si49 at all h1 values. This geometry did not have an equivalent experimental setup. Finally, three parallel supports were used, G850, Si49, and O, while the outlet was just O. The velocity of the flow through all membranes was studied from the solutions represented by surface and contours for velocity magnitude (with magnitude-controlled level: 300) and by streamlines for velocity field (with uniform density of 0.02 mm and magnitude-controlled level: 300).

Time-Dependent Modeling

In addition to the steady-state solutions for the fluid for various porous supports and at different pressures, the time evolution of trajectory of the particles in the fluid was determined for various geometries and pressures. For particle trajectory modeling, the Newtonian formulation was usedwhere mP is the particle mass (kg) and F is the force (N) exerted on the particle in time t. The particle density was 2160 kg/m3, and its diameter was 1 μm with 0 charge number. The initial position of the particles was set at the inlet with the number of particles per release equal to 300 based on some pretests.

The initial particle’s velocity was considered as the fluid velocity field, which was determined by the steady-state modeling. For determining the motion of the particles in the fluid, drag force was used with the Stokes drag law[45]where mP is the particle mass (kg), τP is the particle velocity response time (s), v is the velocity of the particle (m/s), and u is the fluid velocity (m/s). The drag force in 2D models have FD(x) and FD(z) components with (u– v) and (u – v) parts. The particle velocity response time for spherical particles in a laminar flow is defined aswhere μ is the fluid viscosity (Pa s), ρP is the particle density (kg/m3), and dP is the particle diameter (m). The geometries used for outlets in this section were optimized geometries from the steady-state results. Since the variation of velocity in-plane (x, y) were small relative to the variation in vertical direction (z), direction y was ignored with negligible error. The flow was modeled conveniently being two-dimensional (2D) since the average velocities obtained from modeling were very close to the related velocities in experimental measurements (with ±0.5% error). Three-dimensional modeling was also tested in some cases, but the results are not presented here. While extremely fine physics-controlled mesh was used in 2D models with maximum element size of 8.04 × 10–2 mm and minimum element size of 2.4 × 10–4 mm, only predefined element size with maximum of 6.7 × 10–1 mm and minimum of 5 × 10–4 mm was used as the mesh element size in 3D models. Even this control on mesh size in 3D resulted in much larger calculation run times than 2D (more than 10 times). In addition, the 3D calculations did not consider cylindrical coordinate system and solved the Navier–Stokes equations in the Cartesian coordinate system. Therefore, a similar level of accuracy was not obtained in 3D models in comparison to 2D models. We used some of the 3D models for a clearer presentation of the setup and some qualitative discussions but not for quantitative comparison with 2D models or experimental results.