Quantifying Nanoparticle Layer Topography: Theoretical Modeling and Atomic Force Microscopy Investigations

A comprehensive method consisting of theoretical modeling and experimental atomic force microscopy (AFM) measurements was developed for the quantitative analysis of nanoparticle layer topography. Analytical results were derived for particles of various shapes such as cylinders (rods), disks, ellipsoids, hemispheres (caps), etc. It was shown that for all particles, their root-mean-square (rms) parameter exhibited a maximum at the coverage about 0.5, whereas the skewness was a monotonically decreasing function of the coverage. This enabled a facile determination of the particle coverage in the layer, even if the shape and size were not known. The validity of the analytical results was confirmed by computer modeling and experimental data acquired by AFM measurements for polymer nanoparticle deposition on mica and silica. The topographical analysis developed in this work can be exploited for a quantitative characterization of self-assembled layers of nano- and bioparticles, e.g., carbon nanotubes, silica and noble metal particles, DNA fragments, proteins, vesicles, viruses, and bacteria at solid surfaces. The acquired results also enabled a proper calibration, in particular the determination of the measurement precision, of various electron and scanning probe microscopies, such as AFM.


Table of contents:
1. Calculations of the topographical parameters for particle covered surfaces 2. Modeling of Particle Layers -the Monte-Carlo RSA Approach 3. AFM Measurements -Determination of the Topographical Parameters 4. APPENDIX A (Supplement to Section 1)

Calculations of the topographical parameters for particle covered surfaces
The constitutive expression for the average height h of a rough surface is the following ( ) ( ) where S is the projection area of the surface, h(rs) is the local height of the surface profile measured relatively to a reference planar surface located at h0, rs is the surface position vector and ( ) It is evident from eq S1 that the definition of the average height is not unique because it depends on the position of the reference plane.
In contrast, all the central moments of the surface q  can be uniquely calculated from the general formula ( ) where q = 2,3,4… It should be noted that 0 , thus the moments are independent of the location of the reference plane.
Explicitly, the second, third and fourths moments can be expressed as ( ) h is defined by eq S2 and the remaining surface integrals

=  rr
Consequently, the basic topographical parameters such as the root mean square (rms), the skewness (sk) and the kurtosis (ku) of a rough surface are defined as follows If the roughness stems from the presence of surface features (particles) attached to a planar surface located at h = 0, one can express the topographical parameters in a useful analytical form.
Let us, therefore, assume that there are p N features of equal size and the same shape, whose characteristic cross-section area is p S (see Figure S1). .In consequence the average height and other topographical parameters of the particle layer can be expressed as ( ) is the absolute (thermodynamic) particle coverage.
It should be underlined that eq S7 is valid for arbitrary coverage and distribution of particles within the layer.It is also applicable for any particle shape given that the double integrals defined by eq S8 can be evaluated either analytically or by numerical techniques.Obviously, the analytical solution are of a special significance because they can be used for the formulation of S5 general laws governing topography of particle layers and for estimation of the precision of numerical solutions.In this respect, simple expression can be obtained for the category of particles characterized by a cross-section area independent of the coordinate perpendicular to the planar surface.This comprises cylinders of arbitrary, e.g., elliptic, cross-section area, disks, parallelepipeds (cubes), etc.For all these particles all the double integrals defined by eq S8 can be evaluated because ( ) s h r is independent of the position vector.In consequence the and 1 p h to 4 p h coefficients become where p d is the characteristic particle dimension perpendicular to the surface (see Figure S1).Thus, according to eq S7 the average height, the root mean square (rms), the skewness (sk) and the kurtosis (ku) of the particle layer are given by p hd Θ = ( ) Eq S11 indicates that the average layer height and its rms scale up linearly with the particle dimension p d , whereas the skewness and kurtosis are independent of the particle dimension.The h coefficients were also derived for the particles in the form of elliptic caps, sphero-cylinders and cylinders aligned parallel to the interface, by evaluating the corresponding double-integrals in the Cartesian coordinate system (see Appendix A).For sake of convenience these coefficients were normalized as follows Accordingly, the topographical parameters of surfaces covered by such particle class are given by 1 in the main manuscript where also the maximum values of the normalized rms/ p d are given as well as the coverage where the maximum occurs.
As can be inferred from eq S13, for particle shapes, the average layer and the rms of the layers scale up linearly with the characteristic particle dimension p d (perpendicular to the surface), whereas the skewness and kurtosis only depend on the coverage.The latter property has interesting practical repercussions enabling the determination of the particle layer coverage using the data acquired from experiments, e.g., AFM.
Moreover, using eq S13 one can formulate important limiting expression valid low coverage range where Θ << 1, which assume the following form Thus, under this regime, the average layer height linearly increases with the particle coverage, the rms is proportional to the square root of the coverage and the skewness is proportional to Θ − .This indicates that the sensitivity of the latter parameter to the particle coverage becomes exceptionally large in this limit increasing measurement sensitivity.

Modeling of Particle Layers -the Monte-Carlo RSA Approach
Computer experiments aimed at the determination of the above define topographical parameters were carried out according to the algorithm comprising the following steps: (i) creation of the particle layer of a desired coverage applying a Monte-Carlo type random sequential adsorption (RSA) approach [1][2][3][4][5][6][7] for various physicochemical parameters related to experimental conditions, (ii) quantitative characteristic of the layer structure in terms of the radial distribution function, (iii) superimposition of a regular net over the particle layer characterized by the mesh size corresponding to the range of atomic force microscopy scans, (iv) calculation of the topographical parameters for the discrete set of points resulting from the mesh point distribution.
For the creation of the particle layer, the extended RSA modeling (referred to as soft-RSA) was applied considering the electrostatic interactions of the incoming particle with all previously attached to the surface.The calculation algorithm was based on the following rules [1][2][3]5] a virtual molecule of a fixed size was generated whose position within the adsorption domain was selected at random; if the particle fulfilled defined adsorption criteria it was treated as firmly attached to the surface, thus its position was unchanged during the entire process; if the adsorption criteria were not fulfilled, a new adsorption attempt was made uncorrelated with previous ones.Two necessary deposition criteria corresponding to experimental measurements were adopted: no overlapping of the virtual particle with others attached to the surface and (ii) the availability of uncovered surface area on the substrate surface large enough to accommodate the virtual particle.
Because of the simplicity of governing rules, this RSA based algorithm enabled to generate large populations of deposited particles.In this way, the information about the deposition S8 kinetics, the maximum coverage, the blocking function and various correlation functions characterizing the distribution of particles in the layers can be extracted with a sufficient precision [7].
The net interaction energy of the virtual particle v  was calculated by summing the pair potentials within the interaction zone expressed using the linear superposition approach [6]. where and vi  is the interaction energy of the virtual particle with the i-th particle attached to the substrate, 0  is the pair energy on contact, vi r is the distances between their centers, ap = dp/2 is the radius of the particle, is the electrical double-layer thickness, ε is the permittivity of the medium, k is the Boltzmann constant, T is the absolute temperature, e is the elementary charge and I is the ionic strength of the electrolyte solution.
Upon evaluating the interaction energy, the deposition probability density at a given point at the surface was calculated from the Boltzmann formula: The number of particles in the layer generated in a single modeling run Np was typically equal to 2×10 3 .In order to increase the precision of the modeling, a few independent runs were performed.
The absolute coverage of particles is define as before

S9
The particle distribution within the layer can be conveniently characterized in terms of the radial distribution function g(r), referred also to as the pair distribution function).This function reflects the probability (averaged over the entire population) to find a particle pair separated by the distance r normalized to the uniform probability at large distances.If a sufficiently large particle population is available from the modeling this function can be calculated in a discrete form using the finite-difference equation [7] ( ) / 2 where pi N is the number of particles within the ring 2 rΔr  drawn around the central particle and < > means the ensemble average.
The topographical parameters for the particle layers generated in the modeling were calculated as ( ) N is the number of mesh points (pixels) corresponding to the net superimposed on the particle layer.

AFM characteristics of particle layers
It should be mentioned that all the above defined topographical parameters correspond to the situation where the entire interface area uncovered by particles is accessible to the scanning probe tip.However, in the case of AFM measurements, the acquired height values will be different because of finite probe (tip) dimension.This deviation should depend on the particle and the tip shape (cone angle), on the particle coverage and their distribution within the layer S10 described by the pair correlation function.Although a precise calculation of this correction requires extensive Monte-Carlo type modeling, it significance can be estimated considering the two-particle geometry schematically shown in Figure S2.One can deduce that that a part of the surface becomes inaccessible to the tip if the average distance between the particle surfaces p h becomes smaller than the critical distance * p h given by * tan 1 cos 22 where  is the cone angle of the tip (see Figure S2).Considering that the average surface to surface distance between particles in the layer is given by the formula [8] this postulate can be formulated as: where mx  is the maximum particle monolayer coverage.
The inequality expressed by eq S23 can be converted to the explicit form: 2 ( 1) where As can be seen the critical distance between particles p h (where the tip convolution effects star to appear) is independent of the particle size.
Considering that the typical cone angle of AFM tips is very small and varies between 15 o and 30 o eq S25 can be simplified to the following form Therefore, for a sharp tip with the cone angle of 15 o where p h = 0.131, on can calculate from eq S24 that the convolution effects should appear if the particle coverage in the layer exceeds 0.43 (for the RSA process where mx Θ = 0.547).For  = 30 o , p h = 0.262, which gives the critical particle coverage of the layer equal to 0.34.
If the normalized distance between particle surfaces becomes smaller than the above estimated values of p h , a part of the surface area becomes inaccessible to the tip, which instead moves around the particles.Considering that for two touching particles the maximum radius of the inaccessible surface area is equal to On can calculated from eq S28 that even for the tip cone angle of 30 o the normalized inaccessible surface area calculated from eq S28 is below 1%.

APPENDIX A (Supplement to Section 1)
In the section the following surface integrals are calculated for particles of various shape ( )

=  rr
Let us first consider the case of a particle in the form of a cap of an elliptic cross-section area (see Fig. A1).where (x,y,z) are local Cartesian coordinates with the origin at (0,0,0).
Analogously, the cap height expressed in this coordinate system is given by Considering this definition of the cap height and its symmetry, the double integrals defined by eq A1 can be expressed as follows 11 a a q q q q q I dx Y dy I x dx bb where q = 1, 2, 3, 4 and Explicitly: ( ) Combining eq A8 with eq A6 one obtains the following integrals


In this way, combining eq A9 with eq A4 one obtains the final result for the cap Additionally, the results obtained for the caps can be generalized in a simple way to the case of a hybrid particle composed from on elliptic cylinder (disk) of the length c with an elliptic cap of the same height c.Thus, for such a particle The topographical parameters were also derived for a hemi-cylinder of an elliptic crosssection area attached side-on to the surface (perpendicular to the surface) (see Figure A2).In this case the q h parameters are given by where L is the cylinder length.
Evaluating the integrals expressed by eq A16 one obtains Analogously as before, one can generalize these results to the hybrid particle composed of a parallelepiped of the rectangular cross-section 2ac and the height L, and the hemi-cylinder (cap) analyzed above.
Combining eq A18 with eq A15 the topographical parameters for such composite particle

Figure S1 .
Figure S1.Schematic view of an interface covered by surface features (particles) of equal size with the characteristic dimension dp.

Figure S2 .
Figure S2.Schematic representation of the two spherical particle/AFM tip geometry pertinent to the large coverage limit for the critical surface to surface distance equal to * p h .

Figure A1 .
Figure A1.Schematic view of the ellipsoidal cap attached to a planar surface.
< a, and -b < y < b.The cap cross-section area is the integrals I1-I4 can be transformed to the following general form S14

4 I
integrals can be evaluated substituting ( ) These results are valid for arbitrary elliptic caps, obviously comprising the hemi-sphere.
cap height.Considering this, the h1, h2, h3, h4 topographical parameters for the hybrid particle can be expressed as

Figure A2 .
Figure A2.Schematic view of the hemi-cylinder attached to a planar surface.