Testing the Equivalence between Spatial Averaging and Temporal

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Testing the Equivalence between Spatial Averaging and Temporal Averaging in Highly Dilute Solutions Keith M. Carroll, Colin Rawlings, Yadong Zhang, Armin W Knoll, Seth R. Marder, Heiko Wolf, and Urs Duerig Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02730 • Publication Date (Web): 05 Dec 2017 Downloaded from http://pubs.acs.org on December 8, 2017

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Figure 1. a) Experimental schematic showing how we measure the assembly process with time-lapsed videos taken on an inverted microscope. Inset: i) Schematic showing top down view of a patch with a radius α, and radial distance, ρ. b) Example of a measure of the image difference between two frames (1881, left and 1880, right), yielding a particle that is difficult to even notice by eye. Arrows act as indicators where the particle appears. Scale bar 2 µm. c) Example of the drift correction algorithm; left shows uncorrected data, middle shows the corrected data, and the right shows the final fluorescence image of the assembled particles. Scales are the same for all 3 images, scale bar 5 µm. 177x129mm (300 x 300 DPI)

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Figure 2. a) Positions of the particles acquired through the fluorescence tracking. b) SEM image of the fluorescent particles along with their tracked positions. Note, the fingerprint of the particles’ positions measured in both techniques are extremely similar; which suggests that once the particles are down, they do not move, even with the removal of water. This contrasts with larger sized particles which are believed to succumb to capillary forces21. c) Overlay between the fluorescence tracking and the SEM tracking. Red dots indicated correlated points, while the green are missed SEM particles and blue are invalid fluorescence measured spots. On average the fluorescence imaging picks up on about 80% of the particles and 15% are invalid measurements. Note, both blue and green points are fairly uniformly spaced over the square, indicative that errors are not likely to bias a spatial measurement. d) 2-D Histogram of the probability density of the particle error between the fluorescence tracking and the SEM measurements. Fitting against a Gaussian of the form 〖1/(2πσ_x σ_y ) e〗^(-x^2/(2σ_x^2 )-y^2/(2σ_y^2 )) gives deviations σ_x~20 nm and σ_y~17 nm. 161x138mm (300 x 300 DPI)


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Figure 3 a) Time stamps for particles assembling onto a 5 µm patch. Scale bar is 2 µm. b) Plot of the assembly kinetics. Based on our observations, we decompose the assembly process into 3 regimes: transient (marked in red), static (green), and blocking (blue). The regime is determined by which assembly process dominates the deposition rate. 80x103mm (300 x 300 DPI)


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Figure 4. Plot showing for large patterns the transient effect enhances; the transient domain creates an additional time dependent deposition of particles that scales ~α^2 √Dt. Inset: fluorescence image showing the deposition process for early times on a large pattern (25 µm). Scale bar 5 µm. 79x58mm (300 x 300 DPI)


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Figure 5. a) Schematic showing the boundary conditions for the static profile along with the corresponding solution from a finite element simulation. b) Measurement of the total flux rate as a function of different radii, measured over the static regime from 50 to 200 s. As predicted in equation 3, the total flux is linearly dependent on the radius. The solid red line represents a least-squares fit to the data with a slope of 0.4405 particles/(s·µm) (intercept equal to zero), while the dashed line is the theoretical fit with a slope of 0.4059 particles/(s·µm). In the supplementary information, we explain why the measured slope could exceed the value derived from theory. 78x105mm (300 x 300 DPI)


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Figure 6. a) Measurement of the position dependent radial flux, j_r, as measured from the patch’s center. The data is averaged over 8 patches with a 5 µm radius. The curve comes from the theoretical curve predicted in equation multiplied by radial component. The exact functional form is not easy to predict from individual particle motion, but instead must be calculated from the ensemble picture through the diffusion equation. Inset: Final fluorescence image of the pattern after assembly is completed. b) Measurement of the position dependent areal flux for a shape with a changing radius of curvature. For regions of positive radius of curvature (light blue/gray) the divergence is strong, while for regions with a negative radius of curvature (red/green) the divergence is muted. The points are the experimental data, while the lines are the theoretical curve (extracted from a finite element simulation). This experiment provides evidence that the divergence measured in a is not an edge effect, but instead is caused by diffusion. Inset: i) Legend showing which colors correspond to which polar angles. ii) Final fluorescence image of the pattern after particles have finished assembling. 151x61mm (300 x 300 DPI)


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Figure 7. a) Queueing distribution for the assembling particles. As expected for non-interacting particles the distribution is exponentially decaying with a rate determined by the total flux value. b) Distribution showing how the number of particles deposited on the surface varies for different time intervals. The longer the time stamp, the broader the distribution. This fluctuation in the total number of particles is a representation of the flux fluctuation theorem. 77x120mm (300 x 300 DPI)


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Testing the Equivalence between Spatial Averaging and Temporal Averaging in Highly Dilute Solutions Keith M. Carroll1, Colin Rawlings1†, Yadong Zhang2, Armin W. Knoll1, Seth R. Marder2, Heiko Wolf1, Urs Duerig1†*. 1

2

IBM Research – Zurich. Saumerstrasse 4, 8803 Ruschlikon, Switzerland

School of Chemistry and Biochemistry, Georgia Institute of Technology. 901 Atlantic Drive, Atlanta, GA, USA 30332-0400

Diffusion, surface assembly, Smoluchowski equation, ergodicity.

Abstract: Diffusion relates the flux of particles to the local gradient of the particle density in a deterministic way. The question arises what happens when the particle density is so low that the local gradient becomes an ill-defined concept. The dilemma was resolved early last century by analyzing the average motion of particles subject to random forces whose magnitude is such that the particles are always in thermal equilibrium with their environment. The diffusion dynamics are now described in terms of a probability density of finding a particle at some position and time and a probabilistic flux density which is proportional to the gradient of the probability density. In a time average sense the system thus behaves exactly like the ensemble average. Here we report on an experimental method and test of this fundamental equivalence principle in statistical physics. In the experiment we study the flux distribution of 20 nm radius polystyrene


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particles impinging on a circular sink of micrometer dimension. The particle concentration in the water suspension is approximately 1 particle in a volume element of the dimension of the sink. We demonstrate that the measured flux density is exactly described by the solution of the diffusion equation of an infinite system and the flux statistics obeys a Poissonian distribution as expected for a Markov process governing the random walk of non-interacting particles. We also rigorously show that a finite system behaves like an infinite system for very long times despite of the fact that a finite system converges to a zero flux empty state.

Main Text: Transport phenomena, such as an assay assembly or material translocation through pores and channels, depend on the diffusive motion of particles or molecules1,2. Models, which describe these diffusive processes, are continuously evolving as new knowledge about the spatial and temporal dynamics provides insights to improve and understand such diffusive systems3,4. In particular, simulations of the assembly process rely heavily on the diffusion equation, which provides spatial and temporal information through the flux5. The derivation of the diffusion equation, however, is based on there being a continuous material, which for moderate to high analyte concentration is a valid assumption6. However, for dilute mixtures, the continuity breaks down, and concepts such as a flux become ill-defined. To overcome this limitation, the diffusion equation is perceived as a probabilistic one, an interpretation justified through centuries old work examining the action of individual molecules culminating in an ensemble description with probabilistic equations, which rely on the system existing in a local thermal equilibrium7–10.

Transforming to the probabilistic interpretation, however,

fundamentally shifts deterministic quantities, such as the flux, into stochastic variables, and this newly imparted “randomness” leads to natural fluctuations which act as hard limits on the accuracy. An implicit assumption in dilute cases is the validity of the probabilistic interpretation


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of the diffusion equation - particularly in non-equilibrium. Validity must show not only that the total measurable effects between probabilistic interpretation and measurement are equivalent (for example rates), but also that the spatial arrangement must also be fully equivalent. Moreover, there must be an effect from the stochastic nature that imparts itself on the deterministic results in the form of the aforementioned fluctuations. In this work, we develop a method to verify the probabilistic interpretation by quantitatively measuring the rates, spatial arrangements, and fluctuations for a system in non-equilibrium. To test the validity of the probabilistic interpretation we measure the diffusion controlled assembly of nanoparticles onto a surface. We create positively charged patches surrounded by a passivating material to assemble negatively charged particles.

While the electrostatic field

provides a driving force for assembly, the guiding force for assembly is diffusion. Assembly experiments are attractive because with our method, they allow us to measure the spatial arrangement of deposited particles with a high level of resolution (~20 nm) as a function of time (resolution ~50 ms); moreover, using high resolution techniques such as scanning electron microscopy (SEM), we can confirm the spatial arrangement remains fixed, suggesting the particles, once assembled, do not move.

With the ability to detect the spatial-temporal

deposition of individual particles, we can quantify the assembly process with excellent accuracy, something which would be difficult to do with molecules. As part of this work, we demonstrate a quantitative measurement of a spatial-dependent flux, and we also measure the fluctuations caused by the inherent stochastic nature of highly dilute diffusive assembly. In particular, we show that the measured time averaged particle flux corresponds to the deterministic result derived from the diffusion equation for an ensemble average thus experimentally verifying the Ergodic hypothesis under non-equilibrium conditions.


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Methods and Materials: Polymer Preparation: In this work, we fabricate a polymer surface composed of two materials: polyphthalaldehyde (PPA) and a cross-linker polymer with a protected amine group. The 7 nm top layer, PPA, prevents negatively charged particles from depositing through an electrostatic repulsion caused by the negatively charged PPA due to the preferential adsorption of anions11. The bottom layer, once activated produces primary amine groups; the activated amine groups, when placed in water are protonated to make a positively charged patch, which attracts the particles11. To simultaneously activate the amine protected polymer and remove the top layer of PPA, we use thermochemical scanning probe lithography (tc-SPL) combined with thermal scanning probe lithography (tSPL). The chemical structure, reactions, and example pattern are shown in the supplementary information (materials and methods, Fig. S1-S3, Table S1-S2). Particle Preparation: 40 nm Fluorescent nano-particles (NPs) (Orange Carboxylate-modified FluoSpheres 40 nm) were purchased from Thermo-Fischer Scientific.

The reported

excitation/emission wavelengths are 540/560 nm respectively. To track the individual assembly of the NPs, we dilute, in a solutions of 0.01 mM and 0.1 mM KCl, the particles down to a concentration of 32 pM (~10-5 % by weight) just prior to running an experiment. Experiments were also run at 1 mM KCl, but at these concentrations and higher, we report the particles were visibly aggregating, making these salt concentrations measurements unreliable. We measured the diffusion constant as ~9·10-12 m2/s (see SI for full details). Chemical Patterning of Polymer Substrates: ITO samples coated with the layered amine and PPA materials were chemically patterned using an in house tc-SPL/t-SPL instrument. tc-SPL and tSPL are well described techniques12–14, which rely on thermal probes to create a localized temperature profile in a surface; this temperature gradient can induce a local surface


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transformation, such as the amine protected polymer’s activation or the decomposition and removal of PPA. By tuning different parameters, such as the force or the temperature15–17, the reactions extent can be tuned. For the verification of our tracking method, we work at fixed speeds and loads, allowing the temperature to remain fixed for a given square.

For the

quantitative diffusion measurements, we fix the speed, loads and temperatures to reduce variations caused by heterogeneous patterns. Assembly Measurements: The experimental setup is the same for every experiment: the patterned ITO samples are mounted onto an epi-fluorescence microscope (TE Nikon microscope).

For the tracking and measurement, we use a commercially available TRITC

fluorescence cube and a digital camera (Hamamatsu CMOS Digital Camera ORCA-Flash 4.0). First, a droplet of solution (with either 10 or 100 µM of KCl, depending on the salt conditions we measured) was placed over the patterned areas. We initialize the assembly process with a drop with no particles for two reasons: to equilibrate against possible electrostatic interactions that could interfere with the assembly process and to allow for accurate focusing with a perfect focus system. After approximately 5 minutes, we add an equal volume of particle solution (total particle concentration of 16 pM). We avoid issues with mixing by using small droplets (20-50 µL) and pipetting directly into the previous droplet. For figures 1-6, we initially watch the assembly at 2 frames per second (FPS) for the first 50 s and then we switch to 1 FPS. For figure 7, we increase the frame rate to 20 FPS; this allows us to increase the temporal resolution to 50 ms to measure the queueing distribution. SEM Measurements: SEM measurements were made with in-lens detections on a Leo 1550 SEM. SEM images shown here have been contrast enhanced to facilitate particle visualization.


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Particle Tracking: We track the assembly process with a time-lapse video recording the deposition of particles. We circumvent the diffraction limit by taking the difference between two consecutive frames along with centroid based particle tracking positions18 (see figure 1b). To ensure that particles actually stayed on the pattern, we also took the image difference between every other frame (referred as the second image difference). Particle positions on the second image difference were then compared to the consecutive image difference and if the particles overlapped within 40 nm, i.e. the size of the particle, the particle was assumed to have permanently adsorbed. To protect against double counting particles we impose the requirement that the position separations between detected particles in subsequent frames are larger than 40 nm.

During the time lapsed video, drift, inherent to the microscope over long durations, was measured through a reference pattern. The reference pattern is either a previously assembled pattern or the patterns themselves. From the reference pattern, a reference point was extracted by finding the maximum of a correlation function between the reference pattern and the ideal reference pattern (computerized). To expedite the drift measurement the reference points were extracted every 10-100 frames; we assume the drift to be linear over these short times, and comparisons with drift points extracted from every frame showed minor differences. Particle Tracking and SEM comparison: We employed a previously developed program which optimizes alignment between two images19; this optimization programs allowed us to compare the SEM tracked positions with the fluorescence tracked positions. An image was constructed from the fluorescence tracked positions convolved with a Gaussian (standard deviation of 40 nm). This image was scaled to the approximate length and size of the SEM image; the two


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images were then input into the optimization program which searched for the affine transformation (translation, shear, rotation and scaling) which minimised the difference between the normalised copies of the SEM and optical images. The extracted transformation matrix was applied to the fluorescence tracked positions. An algorithm was developed to match the pairs of the fluorescence positions with the SEM positions based off nearest neighbor and a threshold condition of 100 nm (~4-5 the standard deviation of distance separations). Separation distances between the matched pairs were also extracted and binned for statistical comparison. Particle Tracking To test our method for measuring particle assembly, we show in figure 1 an example experiment. Figure 1a shows the experimental setup for measuring the assembly process on an inverted microscope with time lapsed videos. Since the particles act independently, the emitted signals should on average be incoherent from one another; this incoherence makes it possible for us to faithfully map out the particle assembly with the help of image differences since each particle will effectively superpose its signal when adsorbed20. In figure 1b, we demonstrate an example of measuring a particle position with image differences. Specifically, we show that by subtracting frame 1880 from 1881 a single adsorbed particle can be seen and measured. The detected particle (highlighted in red) is difficult to pick out by eye; however, the image difference removes the previous assembled particles making it possible to track the assembly process at higher densities. Since these positions are extracted from fluorescence signal, we refer to them as the fluorescence tracked positions or the fluorescence tracking interchangeably.


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Figure 1. a) Experimental schematic showing how we measure the assembly process with timelapsed videos taken on an inverted microscope. Inset: i) Schematic showing top down view of a patch with a radius , and radial distance, . b) Example of the image difference between two frames (1881, left and 1880, right), yielding a particle that is difficult to even notice by eye. Arrows act as indicators where the particle appears. Scale bar 2 µm. c) Example of the drift correction algorithm; left shows uncorrected data, middle shows the corrected data, and the right shows the final fluorescence image of the assembled particles. Scales are the same for all 3 images, scale bar 5 µm.


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In addition to measuring the frame difference, time lapsed videos allow us to track the drift of the sample over the period of the assembly experiment. We track the drift with a marker of particles that were previously assembled or with the patterns themselves. The left-hand side of figure 1c shows the raw positions of assembled particles.

Without drift correction the patterns are

indiscernible; once the drift is compensated (middle of figure 1c), the tracked positions show the correct assembled patterns shown on the right hand side of figure 1c. Armed with the drift compensated fluorescence tracked positions, we aim to test the spatial resolution by comparing against a high resolution image of the assembled particles. Figure 2 shows an example of comparing the two measurements; figure 2a shows the fluorescence tracked positions, while figure 2b shows a SEM measurement along with the extracted positions. There is a striking similarity between the particle fingerprints seen in the fluorescence and SEM tracked positions.

As an example we have circled the same respective areas on the SEM and

fluorescence tracked positions. The fingerprint in the fluorescence tracked circle is that of five particles in a ring; by eye each particle can be paired to a corresponding partner in the SEM. The fact that we see, without any optimization for comparison purposes, the same fingerprint between the SEM and fluorescence measurements is indicative that the particles do no move once they have assembled.


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Figure 2. a) Positions of the particles acquired through the fluorescence tracking. b) SEM image of the fluorescent particles along with their tracked positions. Note, the fingerprint of the particles’ positions measured in both techniques are extremely similar; which suggests that once the particles are down, they do not move, even with the removal of water. This contrasts with larger sized particles which are believed to succumb to capillary forces21. c) Overlay between the fluorescence tracking and the SEM tracking. Red dots indicated correlated points, while the green are missed SEM particles and blue are invalid fluorescence measured spots. On average the fluorescence imaging picks up on about 80% of the particles and 15% are invalid measurements.

Note, both blue and green points are fairly uniformly spaced over the square,


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indicative that errors are not likely to bias a spatial measurement. d) 2-D Histogram of the probability density of the particle error between the fluorescence tracking and the SEM measurements. Fitting against a Gaussian of the form

gives deviations ~20

nm and ~17 nm. In figure 2c, we show the overlap between the two tracked results after optimizing the overlap with a previously developed program. This optimization program corrects for offsets, rotations, shear, and scale differences between the two tracked results. The red points in figure 2c are the correlated pairs between the fluorescence and SEM measurements; the blue points are particles measured with the SEM but missing from the fluorescence measurements; and the green points are particles measured with fluorescence but not connected to any extracted from SEM measurements. We report that on average we extract 80% of the particles measured from SEM images; in addition, we over-count our particle number by 20%. As a result of these two opposite effects, we effectively measure the correct number of particles accurately. Finally to quantify our method, we show in figure 2d, the 2-D histogram probability density for the positional differences between the paired SEM and fluorescence measurements. The histogram can be approximated by a Gaussian profile and a fitting showed an x/y deviation of 20 nm/17 nm respectively. Coupling this narrow distribution with the similar fingerprint observation, we confidently declare that once the NPs are down, most particles stay down, even with the removal of water. These observations and measurements contrast with the observation and results observed for larger particles which seem to succumb to capillary forces21.


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Results and Discussion To set the stage we first discuss some generic properties of the diffusion equation (, ) = ∇ ∙ (, ) ∇(, )" (1) where (, ) denotes the particle density and

(, ) is the diffusion coefficient which

henceforth will be assumed to be independent of positon and particle concentration. Traditionally, the diffusion equation is derived for continuous media. For high particle densities the continuum approximation makes perfect sense. Here we may assume that the number of particles in a small volume element # $ is sufficiently large such that these particles constitute a good approximation to an ensemble in the sense of statistical physics. The particle flux density %(, ) = − ∇(, ) (2) is fully deterministic and given by the local gradient of the particle concentration. For a highly dilute particle suspension the ensemble concept underlying the derivation of the diffusion equation becomes invalid, i.e. there might not be even one particle in a volume element of the characteristic size of the problem as is the case in our experiments. The resolution of this dilemma comes from viewing the diffusion process through an ensemble of individual particles. Each particle follows a trajectory dictated by Langevin dynamics, which incorporates not only external deterministic forces but also includes uncorrelated stochastic forces. The strength of the stochastic forces is tuned such that the particles are in thermal equilibrium with their environment, viz. Boltzmann statistics is obeyed22. In this individual particle picture a single particle’s trajectory through phase space is unpredictable, but a probabilistic distribution can be computed through the Fokker-Planck equation. In particular, in the high friction limit (low


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inertia), the Fokker-Planck equation reduces to the Smoluchowski equation. In absence of external forces the Smoluchowski equation is functionally equivalent to the diffusion equation (1) where the particle density is now interpreted in terms of a probability density of finding a particle at some position and time. Likewise the particle flux is a statistical quantity subject to random fluctuations. Similar to quantum mechanics one cannot simultaneously know the position of a particle and the flux associated with the particle since the particle motion is an entirely random diffusive process. In our experiment we observe the arrival of a particle on a sticky surface, i.e. the particle position at a given time. Therefore, we need to integrate over time until enough events have been collected such that an estimate of the particle in-flux can be made. Ergodicity then says that our time averaged observation should yield the same result as for an ensemble average derived from the diffusion equation. We now introduce the concepts of static and transient diffusion. To avoid mathematical complexity we consider a spherical sink with radius ' in an infinite medium. At < 0 the particle concentration is ) throughout the whole space. At = 0 the sink is turned on forcing the particle density to drop to a zero value at the surface of the sphere. This problem was solved by Smoluchovski23 and the concentration profile can be written as 3 4

2' √56 1 ' (*, ) = ) +1 − , + ) 0 #2 * * √/ ) (3) where * = √ ∙ is the radial coordinate as measured from the sphere center. The first term in eqn (3) does not depend on time and it constitutes the asymptotic solution for → ∞. Correspondingly it is called static solution. The second term, called transient solution, describes how the system evolves towards the static solution. As explained in the SI the static profile propagates from the sink into the free space over a distance on the order of 2√ . The flux ACS Paragon Plus Environment

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density at the sphere is constant and the total particle flux integrated over the surface of the sphere is given by 9 = 4/ ) ' + 4/ )

'

√/ (4)

The important point here is that the static flux scales with ' whereas the transient flux scales with ' and diverges for → 0. This divergence is a consequence of neglecting the finite size of the diffusing particles which leads to a finite velocity relaxation time. The latter is on the order of 100 ps for our 20 nm polystyrene particles in water and it also constitutes a lower bound for in the flux equation. As a summary conclusion we can say that the transient flux dominates at small times < ' /

and it entirely prevails as ' → ∞ corresponding to a purely 1-dimensional

diffusion problem of a flat surface sink. On the other hand, the total flux is dominated by the static term for > ' / . The real boundary conditions for our finite system demand a zero flux at the droplet-air interface (since particles cannot escape and no particles are added). Therefore we need to study what happens if the spherical sink is embedded in a finite size system with zero flux across the outer boundary. It is immediately clear that the static solution is (*) ≡ 0 in this case (since all particles in the system will eventually reach the sink). In the SI we rigorously solve this problem for a spherical sink embedded in a finite spherical environment with outer radius >. The solution can be written in the form of an infinite series of eigenfunctions: J

(*, ) = ? @A B5CDE6 AK

sin BIA (* − ')E * (5)


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where

IL

are

normal

modes

given

by

the

transcendental

equation

>IA = tan BIA (> − ')E and the coefficients @L are given by: @L =

' 20 IL > − ' − > cos2 BIL (> − ')E (6)

The key difference between the infinite and finite case is that the former conditions support a time-independent static solution while for the latter (and real) conditions the only non-trivial solution is strictly time dependent. However, for times < > / the solution of the finite system is virtually identical to the one of the infinite system. To see the analogy more clearly we rewrite eqn (5) in the following way: (*, ) = (*, ) − 63 (*, )" + 63 (*, ) = QR (*, ) + 63 (*, )

(7)

with J

(63)

63 (*, ) = ? @A B5SDE6 AK

sin BTA (* − ')E * (8)

where (63)

@A =

' 2 TA > − ' (9)

and TA = (2L + 1)

/ 2 ∙ (> − ') (10)

The term QR (*, ), referred to as pseudostatic solution since it is not a static solution in the strict sense, corresponds exactly to the static term in eqn (3) and 63 (*, ) refers to the transient solution describing ACS Paragon Plus Environment

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how the system evolves to the pseudostatic regime (see Figs S12 – S14). For > > /

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the transient

diffusion front reaches the finite boundary leading to an overall depletion of particles and correspondingly the solutions of the finite and infinite systems become substantially different.

Finally we address the diffusion of particles towards a circular sink embedded in a perfectly reflecting planar surface, see Fig. 5a. Starting with a spherical sink we note that for symmetry reasons the reflecting surface has no influence on the solutions. The transition from a spherical to a planar sink may be thought of a conformal mapping procedure which is mathematically rather involved but the general structure of the solution entailing transient and (pseudo)static diffusion and the corresponding temporal and size scalings will still be valid. Intuitively we can visualize the process in terms of a simple projection operation (see SI for details) leading to a highly nonuniform flux density at the circular sink. Rigorous solutions have only been worked out for the static case of an infinite system29,30. We are only interested in the functional form of the flux density which for the static case is given by %() = −

2 U 1 = 2 / V − (11)

where is the polar radial distance measured from the center of the sink, is the particle’s concentration profile, 2 is the direction normal to the sink and the embedding surface, and is the radius of the sink, see Fig. 5a. As outlined above, eqn (11) describes the expected flux density in the pseudo static regime. Correspondingly, we obtain for the total pseudostatic flux integrated over the patch area 9QR = 4DU α (12) which up to a factor of π is equivalent to the total flux for a sphere, see eqn (4).


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In addition, there is a transient contribution to the flux density with a similar radial distribution as for the pseudostatic flux density but with a relative strength scaling as /√/ , see eqn (4). In analogy with eqn (12) we obtain for the total integrated transient flux 963 = 4 )

'

√/ (13)

Note that the above equation is an approximation based on the projection argument. The exact expression may differ by a factor of order unity. The transient contribution will be most noticeable at short times < /

and for large sink radii. As the patch size increases ( → ∞),

the static solution is an increasingly negligible term to the transient term, and the rate determining deposition is always given by the transient solution. This limit ( → ∞) reduces to a 1-D adsorbing plane5,31,32, for which the only time dependence is inversely proportional to √ (i.e. fully transient).

The main purpose of the experimental work is to test the validity of the probabilistic interpretation of the diffusion equation in non-equilibrium. Specifically we demonstrate that ensemble and time averaging yield within experimental errors the same results. To this end we measure the spatial distribution and the time evolution of the in-flux of particles onto a perfectly absorbing circular patch. We work with particle concentrations of about 1 particle per 125 µm3 (5 µm x 5 µm x 5 µm) to measure the flux dynamics faithfully. Thus we test the applicability of the diffusion equation when the length scales being probed are on the order of (or smaller) than the average particle spacing. Figure 3a shows time-lapsed results from a typical experiment for a patch with a 5 µm radius.

The fluorescence images captured at different times show the


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evolution of the assembly process. We observe at early times that the deposition is directed and is preferential to the patch’s rim exactly as expected from eqn (11) which stipulates a diverging flux density at the rim. This can be seen in the time stamps for the first 500 s in which there are more particles deposited at the rim versus those in the center. The strong bias towards assembly of particles at the rim implies that despite the fact that particles are moving randomly, particle assembly is not a random process; instead, the assembly process is governed by the laws of the diffusion equation. The second effect, which is well documented and studied, is the saturation of particles on the patch; this saturation arises from the steric (and electrostatic) hindrance from previously assembled particles24–26.

We refer to this hindrance as blocking. Examples of

blocking dynamics are the renowned Langmuir adsorption model and the modified blocking functions discussed previously27,28.


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Figure 3. a) Time stamps for particles assembling onto a 5 µm radius patch. Scale bar is 2 µm. b) Plot of the assembly kinetics. Based on our observations, we decompose the assembly process into 3 regimes: transient (marked in red), static (green), and blocking (blue). The regime is determined by which assembly process dominates the deposition rate. For a quantitative comparison we show in Figure 3b the total number of particles that have adsorbed on the patch as a function of time. Based on the diffusion equation the number of particles is given by integration of eqs (12) and (13) Y = 4 ) α + 4) '

2√ √/

(14) The pseudo-static regime is marked by a linear adsorption of particles versus time as expected from eqn(14) whereas the transient regime exhibits a √ time dependence which dominates at times less than ) ~ Z . For a 5 µm radius patch, this corresponds to a time ) ~2.5 s, too short

to do a statistically meaningful measurement. In working with a finite system, with outer dimensions, ~5 mm radius of the suspension droplet, much larger than inner, 5 µm patch radius, the time scales required for the finite size effects to be noticeable on our patterns are huge (~10] s) in comparison to the time scale of our experiment such that we can faithfully approximate the pseudo-static assembly process by the static case. For assembly times > 600 s the number of particles starts to level off due to blocking. To enhance the transient effect, we have in figure 4 made early time assembly measurements on a large patch (25 µm). Given the larger size of the patch, the transient regime should last ~25 times longer, long enough for us to faithfully measure the effect. Figure 4 shows that the √ dependence is indeed enhanced as expected and measureable.


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Figure 4. Plot showing for large patterns the transient effect enhances; the transient domain creates an additional time dependent deposition of particles that scales ~ √ .

Inset:

fluorescence image showing the deposition process for early times on a large pattern (25 µm). Scale bar 5 µm. We limit the rest of this text to show the equivalence of the spatial distribution and predicted rate equations in the pseudo-static regime to those predicted from the diffusion equation. Part of the difficulty in measuring the pseudo-static dominated assembly correctly relies on understanding that measurements must be done at a time in which effects from both the transient and blocking regimes are negligible. We highlight in the plot in Figure 3b the three rate determining regimes. The distinction between the transient and the pseudo-static is clear and is quantified by which term is expected to dominate the rate. The exact cutoff between the pseudostatic and the blocking regimes is less clear because of the non-uniform particle deposition on the patch: the rim, since it fills faster than the patch’s center, enters the blocking regime earlier. We discuss in the supplementary information (Fig. S6) the careful balance to correctly measure the dynamics in the pseudo-static regime. Eqn (12) tells us that the flux of particles onto the patch is proportional to . We test the validity of the macroscopic rate in Figure 5b in which we compare the total flux for different


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radii. The functional agreement between the experimental measurements and the theoretical prediction is excellent. Our error (order 1) is experimental in nature and we explain in the supplementary information (supplementary text, Fig. S15) that part of the deviation from the theoretical value is likely to come from collective effects from the transient regime. We note that previous experiments showed the proportionality to

33

, but here, we have quantified the rates

with high accuracy. The total flux’s linear dependence measured in Figure 5b and predicted by eqn (12), is significant because it aids in our understanding of surface assembly. After the assembly is complete, the total number of particles (Y^ ) on the patch will be proportional to 2 ; if we assume assembly is random, the total flux (9) would be proportional to 2 as well. The time scales (~

Y6U6 Z9) for assembly would be independent of . The radial independence goes

against physical intuition that particles will fill small patches faster than large patches. By instead having the total flux proportional to (and not random), the assembly time scales are proportional to , implying small patches fill faster than larger ones.


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Figure 5. a) Schematic showing the boundary conditions for the static profile along with the corresponding solution from a finite element simulation. b) Measurement of the total flux rate as a function of different radii, measured over the static regime from 50 to 200 s. As predicted in equation 3, the total flux is linearly dependent on the radius. The solid red line represents a leastsquares fit to the data with a slope of 0.4405 particles/(s·µm) (intercept equal to zero), while the dashed line is the theoretical fit with a slope of 0.4059 particles/(s·µm). In the supplementary information, we explain why the measured slope could exceed the value derived from theory.

The total flux being proportional to , however, is indirect evidence that the assembly process itself is not random, a prediction expected form the probabilistic interpretation of the diffusion equation. Ultimately we need to verify the validity of spatial dependence of the flux density predicted by eqn (11). We plot, in figure 6a, the radial flux as a function of distance from the


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patch’s center for times before blocking effects render eqn (11) ineffective. We define the radial flux, %3 , as the flux density integrated over the polar angle _ :

%3 = 0 #_ ∙ %() = )

ρ 4 U / V − (15)

The rational is to improve the event statistics in the measurement of the radial flux rather than the flux density. The key feature, namely the divergence at the rim is not affected by this procedure. The measurements shown in figure 6a come from the average of 8 patches of the same size (5 µm). To the best of our knowledge, figure 6a is the first quantified measurement of the position dependent radial flux. The theoretical curve comes from the diffusion equation, and the agreement shows not only is the total rate (9) calculable, but the average spatial distribution (%) is nicely predictable as well.


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Figure 6. a) Measurement of the position dependent radial flux, %3 , as measured from the patch’s center. The data is averaged over 8 patches with a 5 µm radius. The curve comes from the theoretical curve predicted in equation multiplied by radial component. The exact functional form is not easy to predict from individual particle motion, but instead must be calculated from the ensemble picture through the diffusion equation. Inset: Final fluorescence image of the pattern after assembly is completed. b) Measurement of the position dependent flux density for a shape with a changing radius of curvature. For regions of positive radius of curvature (light blue/gray) the divergence is strong, while for regions with a negative radius of curvature (red/green) the divergence is muted. The points are the experimental data, while the lines are the theoretical curve (extracted from a finite element simulation).

This experiment provides

evidence that the divergence measured in a is not an edge effect, but instead is caused by diffusion. Inset: i) Legend showing which colors correspond to which polar angles. ii) Final fluorescence image of the pattern after particles have finished assembling. The nature of the divergence derives physically from the rim’s ability to pull from both above and next to the rim for particles, whereas the center can only pull from above. It is not an effect from patterning nor is it an electrostatic effect. To validate this conjecture, we performed the


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exact same experiments using a ten times higher KCl salt concentration in the particle suspension (0.1 mM corresponding to a Debye length of 30 nm versus 10 µM corresponding to a Debye length of 100 nm) and we observed the same flux divergence at the rim also in this case, see Fig. S5. As a further test we exploit the fact that the flux determined from the diffusion equation depends on the shape of the sink. As a result, a change in the radius of curvature (Rc) for a given pattern tunes the strength of divergence: patterns with positive Rc will have a strong divergence, whereas patterns with negative Rc will weaken the divergence. In figure 6b, we show the results of an experiment for a pattern that has a varying Rc as a function of polar angle. We plot the standard

flux

density,

%,

(not

the

radial

flux,

%3 )

to

insure

we

measure

the

enhancement/diminishment of the rim’s divergence and to account for the dependence in the polar angle.

We find that at regions with positive Rc (light blue/gray) there is a strong

divergence, while for negative Rc (red/green) the divergence is nearly extinguished. The strong agreement between the experiment and the curves, which come from finite element simulations of the diffusion equation, indicates that the assembly is solely governed by diffusion. In Figs 5 and 6 we have measured the flux distribution and magnitude by averaging over time and we compare these measurements against values extracted from the diffusion equation. The agreement between the experimental time average and the spatially computed average is sufficiently strong to have successfully demonstrated the validity of the probabilistic interpretation of the diffusion equation. In addition, we also successfully tested the equivalence between the ensemble space and time average. This equivalence is commonly referred to as the ergodic hypothesis. Experimental verifications of the ergodic hypothesis usually rely on systems in equilibrium, but we have verified the hypothesis in a case of non-equilibrium with a non-zero flux (i.e. particles are removed from solution).


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The probabilistic interpretation, however, only relates the spatial arrangement to the time averaged spatial flux, but it does not include the particles’ time dynamics and fluctuations. By increasing our temporal resolution (from 500 ms to 50 ms), we measure in figure 7, the time distributions between 2 particles assembling (hereto referred to as the queueing distribution). As expected for non-interacting particles34,35, the queueing distribution decays exponentially a6 at a rate b determined by the total macroscopic flux as shown in figure 7a.

Figure 7. a) Queueing distribution for the assembling particles. As expected for non-interacting particles the distribution is exponentially decaying with a rate determined by the total flux value. b) Distribution showing how the number of particles deposited on the surface varies for different


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time intervals. The longer the time stamp, the broader the distribution. This fluctuation in the total number of particles is a representation of the flux fluctuation theorem. The exponential decay of the queueing distribution can, in turn, be used to calculate the fluctuations expected in a measured flux. In figure 7b, we compare the theoretical distribution curves, given by the Poisson distribution a6 (b)c /Y! where Y denotes the number of particles that have absorbed on the patch after a time interval , against the experimentally measured number of particles assembled for different time intervals (5, 10, and 20 s). These distribution curves are a manifestation of the flux-fluctuation theorem36,37 which relates the variation of the measured flux to the finite probability for a particle to move opposite to the concentration gradient. For a large number of particles the significance of this upstream flux diminishes exponentially and the flux becomes perfectly deterministic as expected for a large particle ensemble. By verifying the applicability of the flux-fluctuation theorem to the assembly process, we have introduced a theoretical limit for a single measurement’s accuracy in measuring a particle/molecular concentration in flux based bio-/chemical sensors. This fluctuation implies that there is an inherent error in sensor based measurements caused by natural fluctuations, and any measurement is only valid over the range of the fluctuation. Conclusions: In this work, we have developed a technique which allows us to experimentally test the validity of the probabilistic interpretation of the diffusion equation. As part of our method, we showed strong experimental evidence that nano particles, once assembled, do not move. We then used our technique for a quantitative measurement of a spatially dependent flux. By working with highly dilute particle suspensions the deterministic nature of the flux as derived from the diffusion equation breaks down. We demonstrate that the probabilistic interpretation of


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the diffusion equation, which entails the Ergodic hypothesis, is valid under non-equilibrium conditions. By its very nature, the probabilistic picture leads to effects not predicted from the deterministic diffusion equation; we demonstrated these stochastic residual effects as observable fluctuations in the flux. These fluctuations lead to a fundamental limit on sensitivity for assembly based processes. This transformation from the continuous to the stochastic description of assembly becomes necessary particularly at highly dilute concentrations. The value being that with the deterministic equation we predict average spatial profiles and rates, but the stochastic nature must be taken into account to make an accurate temporal measurement. Both components are necessary because they relay two different types of information: spatial design and temporal fluctuations. Associated Content The Supporting Information is available free of charge on the ….. Complementary details on sample preparation and particle characteristics; a sample set of data of the flux measurement at a high salt concentration; details on flux measurements; analysis of the propagation of the transient solution; details on the derivation of the solution of a spherical sink in a finite spherical environment with zero flux conditions at the outer boundary. Corresponding Author *[email protected] Present Address †now at SwissLitho AG, Technoparkstrasse 1, Zurich 8005. Author Contributions


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The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Acknowledgments: Additional support and methods and materials are provided in the Supplementary Infromation. We acknowledge the support and help provided by Ute Drechsler (IBM Research – Zurich) and Martin Spieser (SwissLitho). We acknowledge N. Mojarad (ETH Zurich) and A. Stemmer (ETH Zurich) for help with Zetasizer measurements.

We also

acknowledge the helpful discussions with E. Riedo (Advanced Research Center, CUNY). This publications is supported by the Swiss National Science Foundation as part of the NCCR Molecular Systems Engineering. A.K. acknowledges support by the European Research Council StG No. 307079.

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ToC Graphic


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Testing the Equivalence between Spatial Averaging and Temporal

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