Increasing particle concentration enhances particle penetration depth but slows down liquid imbibition in thin fibrous filters

The transport of particles within thin, porous media is a complex process which received growing attention due to its applications in filtration, printing and microfluidics devices. The effect of particles on liquid imbibition and particle clogging can reduce the performance and lifetime of these applications. However, these processes are still not clearly understood and are challenging to investigate. The goal of this study is to increase our understanding about the effect of particle concentration on the imbibition process in thin fibrous membrane filters. In this study, an Ultra-Fast Imaging NMR method is used to study the particle penetration inside nylon membrane filters for particle suspensions with varying particle concentrations ( C 0 ). The measurements revealed that increasing the particle concentration increases the particle penetration depth S ( t ) as governed by a Langmuir


Keywords: Iron oxide nanoparticles Capillarity Filter membrane NMR SEM A B S T R A C T
The transport of particles within thin, porous media is a complex process which received growing attention due to its applications in filtration, printing and microfluidics devices.The effect of particles on liquid imbibition and particle clogging can reduce the performance and lifetime of these applications.However, these processes are still not clearly understood and are challenging to investigate.The goal of this study is to increase our understanding about the effect of particle concentration on the imbibition process in thin fibrous membrane filters.In this study, an Ultra-Fast Imaging NMR method is used to study the particle penetration inside nylon membrane filters for particle suspensions with varying particle concentrations (C 0 ).The measurements revealed that increasing the particle concentration increases the particle penetration depth S(t) as governed by a Langmuir isotherm given by S(t) = l(t)(1 + κC 0 )/1 + κ(C 0 + C b,m ), with C b,m the bound particles and κ the binding constant.Secondly, in droplet penetration, particles slow down liquid penetration in a Darcy like manner where effect on viscosity (η) and surface tension (σ) determine the penetration speed rather than changes within permeability (K 0 ).The final liquid front (l), scaled according to l 2 ∝σt/η.The particle penetration depths were verified using scanning electron microscopy images.

Introduction
A wide range of science and engineering applications depend on transport, deposition and clogging of nanoparticles within porous media.This includes multiple filtration applications such as waste water treatment [1][2][3][4][5][6] and industrial separation (food [7], beverage [8,9] and pharmaceutical industry [10,11]), ink penetration [12,13] and microfluidic devices [14,15].In most applications, particle transport behaviour is directly linked to the performance and quality of the application such as filter efficiency, print quality and reliability of your microfluidic device.A major challenge in filtration processes is filter fouling.In most cases, membrane fouling is caused by particle deposition.Particles such as contaminants will tend to clog pores or form a filter cake.This will significantly lower the flux through your membrane and in some cases even terminate the filtration process, which directly impacts the efficiency of the application.Therefore, a proper understanding of the particle penetration behaviour and clogging is of great importance.
Multiple studies are devoted to understanding phenomena involved in particle capture within porous media.In most research, four different capture mechanisms are distinguished [16][17][18]: (1) sieving, where the particle radius exceeds the pore radius and the particles are blocked due to size exclusion, (2) arching/cake formation, where multiple particles form an arch or cake over a pore or the complete porous media which prevents other particles to enter, (3) flow-induced aggregation, where multiple particles will form an aggregate that will increase in size until it completely blocks a pore and (4) depth filtration, where particles are captured by the porous medium due to physicochemical interactions with the pore walls.The effect of each phenomenon depends on parameters such as the particle size, porosity, initial particle concentration, density of surface pores and physicochemical interactions.
Multiple studies have been performed to improve our understanding about particle transport and the effect of multiple physicochemical parameters on the transport processes.These studies showed that the particle retention and clogging depends on parameters such as the flow rate [14,19], porosity [20], effect of particle size and the formation of a filter cake [21,22], and matrix-particle interactions [14,23].Also, the effect of particle deposition on the porous matrix could be investigated, showing the change in porosity and permeability during particle capture [24,25].In a study performed by Derekx et al. [1] the effect of particle size was investigated, which showed that when the particle size was larger than half the pore radius, the particles were unable to penetrate within the porous media.In the same study, it was shown that smaller particles were able to penetrate deeper within the porous membrane.Commonly used particles to study particle transport and characterize filter and membrane performance are latex particles.Studies with latex particles also provided information about parameters such as the retention value [16,21,26], the effect of particle size [1] [16,26], flow velocity [19] and particle-particle interaction [14] on the transport processes.While there is already quite some information about the particle transport behaviour and effect of the porous media and particle size.There still exist a few gabs such as the extreme small-and fast-time scales and the effect of particle concentration upon the penetration behaviour.
The main problem with current research studies is that no experimental studies allow to measure the particle distribution throughout a porous membrane with sufficient temporal and spatial resolution.In applications such as printing or filtration, media thicknesses and transport time are often below 100 µm and 100 ms, putting huge constraints on the experimental resolutions.Most of the current experimental studies on particle transport are performed on columns, micro-fluidics devices or membrane filters.In these experiments, a particle suspension is pushed through the porous media and data is gathered by analysing the residue [26].These measurements only provide indirect information about the particle behaviour.This severely limits the amount of information and conclusions which can be obtained.As a results, current experimental results are unable to verify and provide input for theoretical models.To improve our understanding of the varying processes involved in particle transport, particle distribution profiles could provide substantial amount of information.
In a recent study, Nicasy et al. showed how an Ultra-Fast Imaging (UFI) NMR method is able to measure spatial resolved information about the transport of latex nanoparticles within a thin, porous media [31].The UFI-method was able to capture the particles penetration depth and particle concentration profiles with a spatial and temporal resolution of 18 µm and 10 ms respectively.The UFI method provides a way to obtain particle concentration profiles for processes such as printing or filtration which was unavailable before.Having access to the particle concentration profiles can substantially improve our understanding of these fast processes.
The aim of this work was to understand how the particle concentration impacts the penetration depth of particles and the speed of fluid imbibition.To accomplish this, UFI-NMR profiles are taken during the penetration of particle suspensions with varying particle concentrations within a nylon membrane filter.From these profiles, the particle penetration depth and particle distributions can be extracted.The particle penetration depths are then compared to Scanning Electron Microscopy (SEM) images.Finally, the profiles are used to extract liquid imbibition speed for all particle concentrations which will be compared to theoretical models such as Darcy [27][28][29] which showed to hold in similar membranes using water-glycerol mixtures [30].Therefore, it will be investigated if Darcy's Law still holds when using particle suspensions.

Liquid solutions
Table 1 shows the liquid mixtures used in this study together with their physical parameters.The mixtures are prepared by first taking a fixed weight of demineralized water (type I), glycerol, Clariscan (C 16 H 25 GdN 4 O 87 ) and Fe 3 O 4 -latex particles which depend on the required wt%.Thereafter, all components are mixed together.The solutions are coded by GxPyFz, where x represents the amount of glycerol in wt%, y the number of particles in wt% and z the amount of iron oxide incorporated within the latex particles.The latex particles mainly consist of styrene and contain 0.5 wt% of iron oxide.For more details about the synthesis and characteristics of the particles we refer to earlier work [31].The iron oxide is incorporated to make the particles detectable for NMR imaging.The exact particle composition, along with an extensive description and NMR calibration can be found in earlier work [31].Clariscan is added to change the NMR characteristics of the liquid solution and improve the signal-to-noise ratio of the measurement [30,32].Clariscan is a gadolinium based contrast agent commonly used in medical MRI [33].It is provided in a water like solution with a concentration of 279.3 mg/ml (0.5 M), a pH between 6.5 and 8, a density of 1.349 g/ml and a viscosity of 3.0 mPa.s at room temperature [34].The concentration of Clariscan in all solutions was 0.005 mol/l, based on earlier work [32].The glycerol (> 99.0% purity) has a density of 1.26 g/ml and is used to tune the viscosity of the liquid solutions.
To quantify the effect of the latex particle concentration on the physical properties of the liquid suspensions, measurements of the viscosity and surface tension were performed with an Anton Paar MCR302 rheometer (20 • C) and Wilhelmy plate method.The measured data shows that increasing the particle concentration leads to an increased viscosity and decreased surface tension, see Table 1.

Membrane
The membrane filter used in this study is a Whatman nylon-6,6 membrane with an average pore radius of 0.53 µm.The membrane has a thickness of 170 µm and a porosity of 65%.Both the porosity and pore radius were determined via Mercury Intrusion Porosometry (MIP) in an earlier study [30].In that study, the penetration of liquid suspensions was well characterized and could be explained using Darcy's Law.For this reason, the same membranes are used within this study.

Nuclear magnetic resonance
Nuclear magnetic resonance is a non-destructive method capable of imaging liquid and particle distributions in porous media.The first part of this section will introduce the principles behind NMR and how it can be used to measure 1D-moisture profiles.Secondly, the UFI-NMR method is introduced, which is used to study the penetration of varying particle suspensions within a nylon membrane.Thirdly, it is shown how the UFI signal intensity is linked to Fe 3 O 4 -latex particle concentration.Lastly, the NMR-setup and experimental procedure will be introduced.A detailed description of the UFI-method and how it can measure both liquids [32] and Fe 3 O 4 -latex particles [31] was the topic of earlier research.Therefore, this section will only give a brief overview of the method, setup, and experimental procedure, which is required to understand the experimental results shown in Section 3.

Basic theory
NMR can be used to image nuclear spins, such as the 1 H atoms.To record a signal profile, the NMR uses a main magnetic field with magnitude B 0 [T].Due to the magnetic field magnitude, the spins experience a torque that makes them resonate at a frequency f[Hz] = γB/2π, also called the Larmor frequency.To record spatially dependent information, a magnetic field gradient with magnitude G[T/m] is used to make the magnetic field vary in space according to B = B 0 + Gx.After introducing this gradient, the Larmor frequency becomes: where γ[MHz/T] is the gyromagnetic ratio, which for hydrogen is 42.58 MHz/T and x[m] the position.By using a Fourier transformation, the frequency signal can be turned into a position dependent signal.
To measure signal intensity, spins need to be excited by a radio frequency (RF) pulse, sent by an RF-coil.After sending the RF pulse, the same RF-coil is used to record the signal intensity.The measured signal intensity is proportional to the density of hydrogen atoms ρ[kg/m − 3 ] and decreases over time due to spin-spin or T 2 -relaxation and spin-lattice or T 1 -relaxation.The characteristic time scales of these processes are T 2 [s] and T 1 [s] and depend on the liquid properties.The signal at a certain location is given by, where t r [s] is the repetition time, the time between two subsequent pulsed experiments, and t e [s] is the echo time, the time at which the signal is recorded.
To measure signal profiles, different NMR pulse sequences can be used such as the Hahn-spin echo [35] or CPMG sequence [36].Detailed descriptions of these and other possible pulse sequences can be found in dedicated textbooks [37].In our study, profiles are measured using the UFI-method, which uses an adaptation to the Ostroff-Waugh (OW) [38] pulse sequence and is explained in the next section.

UFI-method
To record the signal profiles, a recently introduced UFI method is used, which is able to measure signal profiles with a spatial resolution of 18 µm and a temporal resolution of 10 ms [30,32].To achieve this temporal resolution, the method requires two main ingredients: the decrease of the T 1 relaxation time by the addition of a contrast agent (Clariscan) and the summation of multiple echoes by using a variant of the OW pulse sequence [38].
The pulse sequence used by the UFI method is given by 90 where N is the amount of repetitions and 2τ = t e , the echo time.The final signal intensity is built up by a summation over all the echoes.Because there are 2 echoes within one repetition, the signal is built up by 2N echoes.
In all experiments, the echo time was set to 50 µs and the number of repetitions N, is set to 16.The pulses length of the excitation pulse was set to 1 µs.The relative short pulse length allows to have a large excitation bandwidth.The final Field-Of-View of our setup is around 400 µm and is determined by the decrease in sensitivity when moving away from the RF-coil.

Imaging particles with UFI
The pulse sequence introduced in the previous section, allows to detect hydrogen atoms inside the liquid solutions.Due to the short T 2time of the polystyrene, the hydrogen atoms within the latex particles are not detected, see earlier work [31].In the same study it was shown that by incorporating iron oxide within the latex-particles, their concentration profiles could be measured.Due to the magnetic properties, the Fe 3 O 4 within the latex particles will locally suppress the signal intensity of the hydrogen atoms of the solvent [39,40].In a previous study, it was shown that the iron oxide latex particles lower the signal intensity by two effects: (1) decreasing the T 2 -time and (2) reducing the hydrogen density by replacing water molecules with latex particles.After incorporating both effects, the final UFI signal was given by, where ρ max (1 − aC) is used for the hydrogen density.ρ max is the density of

Table 1
The compositions and physical properties of the liquids used in the imbibition.visible hydrogen nuclei in a pure liquid and a[wt% − 1 ] is a constant that represents the decrease in liquid hydrogen nuclei per wt% of latex particles.
To link the signal with the particle concentration and simultaneously remove the coil-profile, every measurement is normalized with the UFI signal profile of a G70P0 solution which contains no particles, given by S = ρ max ∑ 2N n=1 exp( − nt e /T 2bulk ).After normalizing, the relative signal intensity could be linked to the particle concentration C[wt%] by the following equation, where R 2 [s.wt% − 1 ] is the relaxivity of the iron oxide latex particles.Using this formula, the measured 1D-signal profiles can be used to extract 1D-concentration profiles of the iron oxide latex particles.Within a porous media, the signal intensity decreases due to the porosity of the porous membrane ϕ[ − ].When incorporating this effect into Eq.( 4), we end up with,

Setup and experimental procedure
For the NMR measurements, a GARField NMR is used [41].This GARField NMR has specially designed curved poles that produce a high magnetic gradient of 41.5 T/m [42].Because the magnetic gradient is directly linked to spatial resolution, the GARField NMR achieves resolutions as high as 5 µm [42].Fig. 1 shows a schematic representation of the setup where also the curved magnet poles of the GARField NMR can be seen.The figure also shows a membrane sample during the penetration of a particle suspension.The measurement area is located above the RF-coil, which has a diameter of 4 mm.
During an experiment, a syringe was used to jet droplets on top of the nylon membrane filter (brown speckled).The droplet was typically around 8 -12 µl.In all experiments, the droplet covered the entire measurement area.A droplet sensor was used to trigger the NMR and start the UFI pulse sequence.Samples were made by gluing a membrane filter on top of a glass plate (dark grey) using double sided tape (light grey).
In Fig. 1, a typical particle penetration experiment is shown where a liquid-particle mixture with 5 wt% of particles penetrates within a nylon membrane.Within this example, the liquid has partly penetrated the sample and the particle front p(t)(orange, 50 µm) lags the liquid front l(t)(blue, 100 µm).On the right, the corresponding UFI-signal profile is shown.The signal is shown within the droplet (x < 0 µm), membrane (0 µm < x < 170 µm), tape (170 µm < x < 195 µm) and glass plate (195 µm < x).The signal inside the droplet was calculated based on Eq. ( 4), which for a particle concentration of 5 wt% results in 0.625.Within the membrane, the signal intensity can be split into three regions: region I, with both liquid and particles, region II, with only liquid, and region III without particles and liquid.In region I, the signal intensity drops compared to the droplet because of the porosity ϕ of the nylon membrane (Eq.( 5)).In region II, the signal starts to increase again because of the absence of particles.Finally in region III, the signal intensity drops to zero because no liquid is found.After the membrane, the tape and glass plate are located which never have signal, because the mixtures are unable to penetrate these regions.

Results and discussion
In this section, the effect of particle concentration on the penetration process will be studied by performing penetration experiments on a nylon membrane with suspensions containing 6 different particle concentrations, see Table 1.First, the time evolution of UFI signal profiles will be discussed in Section 3.1.Secondly, in Section 3.2, these UFI signal profiles will be used to extract particle concentration profiles and particle penetration depth which will be compared to Scanning Electron Microscopy images.Finally, Section 3.3 studies the effect of particle concentration on liquid imbibition speed and compare it to Darcy's model.Profiles are marked with varying colours to indicate the initial particle concentration within the droplet: 0 wt% (black), 1 wt% (orange), 2 wt% Fig. 1.Experimental setup with a typical sample situation and the corresponding 1D -UFI-signal profile as calculated by Eq. ( 5).Shown are the curved magnetic poles of the GARField NMR, the RF-coil, the syringe and droplet sensor.The sample consists of a droplet, nylon membrane (brown), double sided tape (light grey) and glass plate (dark grey).The liquid within the droplet and membrane is shown in blue and particles are shown with brown circles.The corresponding liquid front l(t) and particle front p(t) are marked with orange and blue line respectively.The same markings are used within the 1D-signal profiles.(light blue), 3 wt% (green), 5 wt% yellow and 8 wt% (dark blue).The UFI-profiles mainly consist of three regions: (1) the droplet (x < 0μm), (2) the membrane ( 0μm < x < 170μm) and (3) the double side tape and glass plate (x > 170μm).Inside a profile, the signal intensity changes within every region.In the droplet, where the measurement area is fully saturated with liquid, the signal is given by Eq. ( 4).Inside the membrane, the signal drops, due to the porosity ϕ of the nylon membrane, see Eq. ( 5).In final part of the profile x > 170μm, the tape and glass plate, the signal intensity drops to zero, because no liquid is present.

UFI Signal profiles
Based on the profiles, the process could be split into several phases: (I), before penetration, (II) early phase liquid penetration (homogeneous), (III) late phase of liquid uptake (front splitting, particles lag behind) and (IV) the end state (lateral liquid penetration with vertical particle penetration).Fig. 3, shows a schematic representation of these four phases.In this figure, the droplet, membrane and tape are shown, and a blue colour is used to indicate water, while brown circles indicate particles.
Phase I is shown in Fig. 2a, which shows profiles before liquid penetration.In these profiles, there is only signal within the droplet because no liquid has entered the porous media, as shown in Fig. 3.The signal intensity within the droplet varied for every profile and depended on the initial particle concentration of the suspension, as given by Eq. ( 4).
In phase II where t = 0.5 s-0.75 s, shown in Fig. 2b and c, the liquid started to penetrate the membrane.As explained before, the signal intensity of liquid is lower in porous media due to the porosity ϕ.The profiles show that liquids with higher particle concentrations imbibed slower into the porous media.Furthermore, the signal intensities within the membrane were rather homogeneous.Because the signal is directly linked to the particle concentration, see Eq. ( 5), a homogeneous signal profile indicated a constant particle concentration.Therefore, at the early stage of penetration, the liquid and particles penetrated as a homogeneous suspension, see Fig. 3 phase II.Another possibility was that the particle-liquid front splitting does not exceed the NMR-resolution (18 µm), see [31].
Fig. 2d-e show phase III of the penetration process.During this phase, the signal profiles started to become inhomogeneous and developed a signal increase near the liquid front.This was the case for the lower particle's concentrations (1, 2 and 3 wt%) in Fig. 2d and for all liquids in Fig. 2e.We know from Eq. ( 5) that a higher signal intensity corresponds to lower particle concentrations.Therefore, it could be concluded that the particle concentration started to decrease near the liquid front.A similar situation was already shown in earlier work [31].In that study, it was shown that this signal increase indicated the splitting between particle and liquid front, see Fig. 3 phase III.In Fig. 2e, at a time of 4.0 s, most of the liquid had reached the membrane-tape interface.A closer look at the signal intensities revealed that signal intensities of the lower particle concentrations (1, 2 and 3 wt%) became equal to the signal of the reference solution containing no particles (black).Because the signal intensity was similar to the reference sample, there were no particles within this particular region.For the higher concentrations.5 wt% (yellow) and 8 wt% (dark blue), the signal did not completely recover.
This either meant that some particles were able to penetrate within this area or that the resolution of the setup was unable to resolve the splitting between the particles and liquid front.
Profiles of phase IV are shown in Fig. 2f.At this stage, the liquid suspension had reached the membrane-tape interface and vertical penetration of liquid was stopped.However, liquid from the droplet was still entering the membrane due to lateral penetration.Due to this penetration, particles were still penetrating the membrane which further increased the particle concentration.This situation is schematically shown in phase IV of Fig. 3.The increase in particle concentration within the measurement area continued to affect the signal intensity, as given by Eq. ( 5).When comparing Fig. 2f and e, the influence of the particles could be seen by a further decrease in the signal intensity near the droplet-membrane interface.In Fig. 2f, the position of the particle front can be clearly observed by the increase in signal intensity within the membrane.These positions are marked with circles.

Influence of particle concentration on particle penetration depth
In the previous section, the UFI profiles revealed that during penetration, a particle front developed that lagged the liquid front.Furthermore, it was observed that the particles were unable to penetrate the full thickness of the membrane.In this section, particle penetration depth and concentration profiles will be extracted with the goal of studying the influence of particle concentration on the particle penetration behaviour.This data will then be compared to SEM images of the cross-sections of the samples.
The profiles shown in Fig. 2 gave a good idea about the particles position within the membrane.However, due to limits in resolution and the small thickness of the membrane, only few positions could be extracted to gain information about the particle front through time.With a larger separation between the particle and liquid front or with larger membranes (> 250 µm), more particle fronts could be extracted, making this technique valuable in multiple applications.In this study, particle concentrations and fronts were extracted at later timescales.In a previous study, it was shown that by using Eq. ( 4), the UFI profiles shown in Fig. 2 can be turned into particle concentrations [31].Fig. 4 shows the particle concentration profiles that were determined from the UFI-profiles shown in Fig. 2f.
A first observation of the particle concentration profiles revealed that no jump in concentration was observed at the droplet-membrane interface, therefore the particles were not hindered by the interface.It seemed that the difference between the pore radius of the membrane (540 nm) and the particle radius (50-100 nm) was large enough for the particles to move freely inside the membrane.This is in line with a study performed by Derekx et al. [1] were the effect of particle size was investigated, which showed particles were hindered if the particle size was larger than half the pore radius.
Furthermore, the data showed again that the membrane region could be divided into two regions.A region, near the droplet-membrane interface, containing the same particle concentration as inside the droplet, and a region near the membrane-tape interface, containing no Fig. 3. Schematic representation of the penetration process.R.J.K. Nicasy et al. or almost no particles.Particle liquid splitting and a final particle penetration depth are also observed in other porous systems found in literature [43].The interface between both regions could be identified as the particle penetration depth and was marked in Fig. 2f and Fig. 4 with coloured dots.These particle penetration depths revealed that with increasing particle concentration, particles were able to penetrate deeper within the nylon membrane.
In the final part of this section, the UFI measurements are compared to SEM images.Because SEM images of the cross-section could only be taken after the penetration was finished, the SEM images were compared to profiles measured just before the droplet disappeared and the penetration process stopped.Fig. 5 shows UFI -profiles (a) and the corresponding particle concentration profiles (b) after 230 s, just before the penetration process finished.In two profiles, the one of 1 wt% (orange) and the one of 2 wt% (light blue), the disappearance of the droplet can be observed by the decrease in signal intensity around − 75 µm, indicating that no liquid is present.The exact time at which the profiles were measured varied slightly for all liquids due to differences in penetration and evaporation rates.
When comparing the particle concentration profiles of Fig. 4 and Fig. 5b, it can be seen that the particle concentrations within the membrane further increased, but the particle penetration depth was unchanged.The increase in particle concentration was however not homogenous but the particles seemed to build up from the inside towards the droplet-membrane interface.
The particle penetration depths and profiles of Fig. 5 could be verified using SEM images.Fig. 6 shows SEM-images of the cross-section of the samples used for the UFI measurements shown in Fig. 5 for all particle concentrations: a (0 wt%), b (1 wt%), c (2 wt%), d (3 wt%), e (5 wt%) and d (8 wt%).In these figures, the position of the particles was marked with a white line and the particle penetration depth was given by a white arrow.In Fig. 6f, a filter cake of particles could be seen on top of the membrane.This particle layer was not observed within the UFIsignal profiles, probably because the measurement was stopped too early, and the thickness of the filter cake formed during penetration was too small to be picked up with the experimental resolution of 18 µm.
Similar to the NMR results, a final particle penetration depth was observed, that increased with the particle concentration.Furthermore, a similar concentration gradient was observed where a dense particle layer had built up from the inside towards the droplet-membrane interface, similar to what was observed in Fig. 5b.This is most visible in Fig. 6e, where a dense particle concentration can be seen near the white line.
To investigate the effect of particle concentration on the particle penetration depth and compare both the UFI and SEM results, Fig. 7 shows the particle penetration depths as determined by UFI (black) and SEM (red).The particle penetration depths for UFI were taken from Fig. 4. Clearly, a good correlation exists between both measurements.
In order to explain the final particle penetration depth, a formula for the particle front S(t) [m] as a function of the liquid front l(t)[m] was taken from Kuijpers et al. [44], given by, where κ[m 3 ]is the binding constant,C 0 [m − 3 ] the initial particle concentration and C b,m [m − 3 ] the maximum concentration of bound particles.In this model, it was assumed that the particles attached on the membrane following a Langmuir model, C b = C b,m κC 0 /(1 + κC 0 ).The use of Langmuir models for the attachment of small molecules in thin fibrous membranes has already been used in several other research studies [45,46].When κand C b,m were used as fitting parameters, this formula could be used to fit the data found for the particle penetration depths by   6) is able to describe the final particle front positions quite well.Therefore, it was reasonable to think that the particle's during migration adsorb to the membrane in a Langmuir fashion.This means that for low particle concentrations, increasing the particle concentration will result in more surface coverage.However, for larger concentrations, the binding of particles to the surface becomes harder and the total surface coverage stays similar.Therefore, with higher particles concentrations, the particles will on average move faster throughout the medium because they spend less time near the surface, and it becomes less likely to bind to the surface.The increased penetration depth when increasing particle concentration was also observed in a few other research studies on modelling of deep bed filtration [47] and on column experiments [44].

Influence of particle concentration on liquid imbibition speed
Increasing the particle concentration will influence the imbibition behaviour of the liquid mixtures.While the previous section focused on the particle's, this section will investigate the effect of the particle concentration on the liquid imbibition behaviour.
From the liquid profiles shown in Fig. 2, it was concluded that liquid mixtures containing more particles, move slower inside the nylon membrane.For a more in-depth discussion of the imbibition behaviour, liquid front positions were extracted from the UFI-signal profiles at half the maximum signal intensity, in a similar way as explained in a previous study [24].
Fig. 8a shows the liquid front position as a function of time for all particle concentrations.Every experiment was performed three times, from which an average liquid front position and deviation could be extracted.
The data clearly showed that increasing the particle concentration slowed down liquid penetration.Explanations for this observation can be searched in two directions: (1) modification of the porous media leading to permeability reduction, commonly reported in literature [48,49], or a change in the properties of the liquid (i.e.viscosity, surface tension, …).
To investigate if the change in viscosity and surface tension was responsible for the slower penetration, the data was compared to a particular form of Darcy's Law within 1D, l 2 = 4K 0 cos(θ)σrt/η [27,28].This form of Darcy's Law predicts the fluid front position in function of the viscosity and surface tension.In a previous study, it was shown that this equation was valid for water-glycerol mixtures within the same nylon membrane [24].When replotting the liquid fronts of Fig. 8a on a rescaled time axis given by , the effects of surface tension and viscosity, as given by Darcy, can be accounted for.Fig. 8b shows the liquid front position of Fig. 8a as a function of . Because all liquid fronts fell onto one master curve, it could be concluded that the predicted scaling with surface tension, viscosity and square root of time was valid for particle suspension within the nylon membrane.Therefore, it could also be concluded that the particles changed the penetration speed due to their influence on the liquid properties rather than altering the porous media structure.
This indicates that on short timescales, particle attachment did not influence the liquid penetration speed or permeability.This was probably due to the relative low increase in particle concentration in the early stages of penetration which can be seen in Fig. 4 and Fig. 5.These profiles showed that no significant particle concentration increase was observed before 95 s and that only on later timescales (t > 230 s) significant clogging was observed, where the particle concentrations doubled.Within our experiment, vertical penetration happened on timescales two orders of magnitude faster (t < 3 s) which could explain that particle attachment did not influence the measured liquid fronts.
On longer times scales (t > 230 s) shown in Fig. 5, a clear increase in particle concentration was observed.Therefore, on timescales exceeding 230 s particle attachment and clogging will probably effect the liquid imbibition speed similar to what is observed in several studies were particle concentration increases will reduce the porous media permeability [48][49][50].However, on short timescales, permeability reduction will not play a significant role.

Conclusions
In this study, an Ultra-Fast Imaging NMR method (UFI) was used to study the effect of particle concentration on the imbibition process within a nylon filter membrane.The effect was investigated by performing imbibition experiments with particle suspensions containing varying particle concentrations.Compared to existing techniques, UFI allows to collect spatially resolved information on timescales going below 1 s and with temporal resolutions below 15 µm.This spatially resolved approach enabled to examine particle and liquid transport throughout the membrane rather than measuring the contribution of the whole membrane as in most studies performed.
Due to the limited studies on the effect of particle concentration on penetration in thin, porous media, the primary aim of this study was to investigate the influence of particle concentration on the penetration dynamics.This study had two main findings related to the particle penetration depth and the imbibition speed as function of the particle concentration.
First, increasing the particle concentration led to an increase in the particle's penetration depth.The dependence of the particle penetration depth on the particle concentration was verified by scanning electron microscopy images of the samples cross-sections.The dependence could be linked to a theoretical model based on a Langmuir adsorption model.This aligns with other studies performed on particle suction in porous media that report changes in particle penetration depths [44,47] or report attachment based upon a Langmuir Isotherm within thin porous media [45,46,51].
Secondly, the study demonstrated that increasing the particle concentration slowed down the imbibition speed of the particle suspensions.This change in penetration speed could be linked to an increase in viscosity and decrease in surface tension, as described by Darcy's Law.This

Fig. 2
Fig. 2 shows the UFI signal profiles measured during the penetration of the particle suspensions outlined in Table 1 within the nylon-6,6 membrane.The profiles are presented for 6 different time steps: a (0.05-0.1 s), b (0.50 s), c (0.75 s), d (1.25 s), e (4.00 s) and f (90-95 s).Profiles are marked with varying colours to indicate the initial particle concentration within the droplet: 0 wt% (black), 1 wt% (orange), 2 wt%

Fig. 4 .
Fig. 4. Particle concentration profiles as determined from the UFI-profiles shown in Fig. 2e.Marked with coloured dots are the particle penetration depths.

Fig. 5 .
Fig.5.a) Measured UFI-NMR signal profiles for the penetration of liquids containing 70 wt% of glycerol with varying particle concentration: 0 wt% (black), 1 wt% (orange), 2 wt% (light blue), 3 wt% (green), 5 wt% yellow and 8 wt% (dark blue).The profiles are measured at timescales beyond 230 s, just before the droplet disappears.The exact time is slightly different for all concentrations because the penetration and evaporation rate are different for all solutions.The particle penetration depths for all signal profiles are marked with circles.b) particle concentration profiles corresponding to figure a.A similar colour code is used to mark the varying initial particle concentrations.

Fig. 6 .
Fig. 6.Scanning electron microscopy images of the cross-section of nylon membrane after the penetration of a liquid containing 70 wt% glycerol and varying particle concentrations: a: 0 wt%, b: 1 wt%, c: 2 wt%, d: 3 wt%, e:5 wt% and f: 8 wt%.These samples correspond the samples used for measuring the UFI signals in Fig. 2 and Fig. 5. Marked with white arrows are the particles penetration depths.

Fig. 7 .
Fig. 7. Final particle penetration depth as determined by UFI-NMR (black) and Scanning Electron Microscopy (red).Shown in dashed lines are the fitted penetration depths based on equation.

Fig. 8 .
Fig. 8. Liquid front positions (l) as a function of a) time and b) rescaled time (