Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

M.
Iatalese
^{a},
M. L.
Coluccio
^{b},
V.
Onesto
^{b},
F.
Amato
^{b},
E.
Di Fabrizio
^{c} and
F.
Gentile
*^{d}
^{a}Akka Technologies, Via Giacomo Leopardi 6, 40122 Bologna, Italy
^{b}Department of Experimental and Clinical Medicine, University Magna Graecia, 88100 Catanzaro, Italy
^{c}Physical Science & Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia
^{d}Department of Electrical Engineering and Information Technology, University Federico II, 80125 Naples, Italy. E-mail: francesco.gentile2@unina.it

Received
14th June 2018
, Accepted 23rd July 2018

First published on 17th August 2018

Electroless deposition on patterned silicon substrates enables the formation of metal nanomaterials with tight control over their size and shape. In the technique, metal ions are transported by diffusion from a solution to the active sites of an autocatalytic substrate where they are reduced as metals upon contact. Here, using diffusion limited aggregation models and numerical simulations, we derived relationships that correlate the cluster size distribution to the total mass of deposited particles. We found that the ratio ξ between the rates of growth of two different metals depends on the ratio γ between the rates of growth of clusters formed by those metals through the linearity law ξ = 14(γ − 1). We then validated the model using experiments. Different from other methods, the model derives k using as input the geometry of metal nanoparticle clusters, decoded by SEM or AFM images of samples, and a known reference.

The method enables the fabrication of nanostructures by incrementally depositing smaller building blocks on a flat substrate. Pre-patterning of the substrate by optical or electron beam lithography allows site selective formation of metal nanoparticles with tight control over the final aspects of the nanoparticles.^{5} On the active sites of the silicon surface exposed to growth, metal ions aggregate into clusters where the geometrical characteristics of the aggregate will depend on a fine-tuning of size and shape of preexisting patterns on the substrate, and the parameters of electroless growth including temperature, pH, and concentration of metal ions in solution.^{5}

In the absence of stirring or other convective flows, ions are transported towards the active sites of the substrate by pure diffusion. Since the mechanisms of diffusion are very well understood,^{10,11} electroless deposition can be simulated using discrete models of transport in cellular spaces. In the models, cells of a grid can take definite 0 or 1 states – 1 indicates the presence of a particle (ion). A particle will move with time in a continuous time probabilistic Brownian motion (Fig. 1a), which is discretized in the space as a random walk (Fig. 1b). Particles aggregate upon contact. Displacement of a great many particles in a lattice (Fig. 1c) enables the formation of numerical aggregates with a dendritic appearance that is typical of fractals (Fig. 1d). Simulation of particle growth using iterative arrays is called Diffusion Limited Aggregation (DLA).^{12–17} In previously reported analyses, we have used DLA models to examine the effects of pattern size^{4,5} and pattern distance^{18} on the characteristics of metal nanoparticles in electroless growth. Here, we revise DLA models of particle growth to consider the effects of the sticking probability of an ion to the aggregate – p.

On changing p between 0 and 1, one can modulate the packing fraction and superficial aspect of the aggregates. Low values of p generate aggregates with high atomic packing densities (Fig. 1e), in contrast to low-density aggregates obtained from high values of p (Fig. 1f). p regulates the adhesion of ions to the substrate. If p is low (p → 0) the probability of adhesion of a particle to the aggregate is low, meaning that a particle may not immediately stick to the aggregate upon contact. Under these conditions, a particle may penetrate more deeply in the structure without being captured by the external shells of the aggregate. Higher penetration depths imply, in turn, higher final densities of the numerical clusters of particles. Notice though that, even if for small p the surface density of the aggregates is high, the growth of the aggregate is slow compared to systems with larger sticking probabilities, as demonstrated in the following sections of the paper. If p = 0, no adhesion can occur on the surface.

Using data from the simulations, we developed models that correlate the cluster size distributions and the mean cluster size with p. Since p is related, in turn, to the kinetics of metal deposition, these models enable extraction of the rate of growth of nanoparticles from topographical maps of samples, obtained by SEM, AFM or other similar techniques of imaging. We tested the performance of the model analyzing the growth of gold and silver aggregates in patterned silicon substrates. The model predicted a difference of growth rate between the two metals of a factor of 10. The growth kinetics of metal nanoparticles measured by UV spectrophotometric techniques matched the predictions of the models with a good level of accuracy.

Coupled to experimental SEM or AFM data, this scheme can be used to derive the growth rate kinetics of metals plated on a substrate by electroless methods.

The density–density correlation function c(r), also called pair correlation function, is a distribution routinely used in statistical mechanics to describe how density varies as a function of distance from a reference point. In the present configuration, we used as a reference point the lines of nucleation sites of the aggregate, i.e. the base of the aggregate. c(r) is a measure of the probability of finding a particle at a distance r from the reference. It describes the internal structure of an aggregate as a function of continuous, smoothly varying spatial coordinates – for this it can be used to link the microscopic to the macroscopic characteristics of a system: c(r) delivers the information content of an aggregate as a function of the scale of the aggregate.

In a log–log plot, c(r) is a line with a slope α (Fig. 2). Since the fractal dimension D_{f} of an aggregate is an index that quantifies the change in detail to a change of scale, D_{f} may be derived from α as D_{f} = 2 − α (Methods). The fractal dimension varies with the sticking probability and rapidly undergoes transition from an initial value D_{f} ∼ 1.9 for p = 0 to D_{f} ∼ 1.625 for p = 1. Notice that for sufficiently high values of p, p > 0.2, the best fit of D_{f} attains the steady state value D_{f} ∼ 1.625, similar to the theoretical limit 5/3 ∼ 1.667.^{14} The purpose of calculating the fractal dimension is deriving the cluster size distribution of an aggregate. Clusters are defined as naturally separated trees – a collection of particles connected to the same nucleation site through nearest neighbours forming the deposit^{16,17} (Fig. 2d). From ref. 16, the average cluster size 〈S〉 of an aggregate with fractal dimension D_{f} and mass N is

〈S〉 ∼ N^{Df/(Df−1)} | (1) |

Q = N/〈S〉 | (2) |

The cluster-size distribution I(S,N), i.e. the number of clusters in an N-aggregate as a function of cluster size S, is:^{19}

(3) |

The probability of finding an S-site tree on a nucleation site is limited by N.

(4) |

is a measure of the number of separate particles that are deposited on a substrate over time. This assertion is substantiated by the following argument. A cluster is a structure, in the aggregate, with some internal correlation. An operational definition of a cluster is that it is a structure clearly distinguishable as a subsystem – a self-contained system within the larger system. For this, it is the analogue of an isolated particle in a real process of chemical deposition. Therefore, can be considered a good estimate of the number of separate particles that are deposited on a substrate in the unit time.

k is a key parameter in materials science and nanotechnology and is relevant for the rational design of processes or structures that imply the deposition, growth and self-assembly of metal nano-materials. On the other hand, is experimentally observable and can be readily determined through conventional imaging techniques, including SEM and AFM. It may therefore be convenient to express p as a function of .

Since numerical simulations and Fig. 4a reflect metal deposition less quantitatively than qualitatively – calibration standards may be required for matching the predictions of the model with experimental data (see comments in the Discussion) – in the following we will use ratios between variables rather than the absolute values of variables to describe particle growth.

We then define the non-dimensional parameters ξ = p_{2}/p_{1}, γ = (p_{2})/(p_{1}), the subscript i = 1, 2 indicates two different states or materials with p_{2} > p_{1}. Then, we derive from the diagram in Fig. 4aξ as a function of γ: ξ(γ). The best fit of data and graphical representation of ξ(γ) (Fig. 4b) indicate that ξ varies linearly with γ as

ξ = 14.34γ − 13.7 | (5) |

In separate ESI 1,† we provide a full statistical report of the linear regression, including values for the parameter table, r^{2}, adjusted r^{2}, estimated variance and ANOVA table. We used at least 25 simulations for each p to fit data and evaluate the model.

Eqn (5) puts in relation the growth kinetics of different materials expressed in different forms. ξ is the ratio between the kinetics of growth of two materials, γ is the ratio between the rates of growth of clusters emerging from the deposited volume of those materials. The proportion between ξ and γ is ∼14. Say that we have two different metals A and B, the kinetics of deposition of A is known, being k_{A}. Then k_{B} can be readily derived using an approximate form of eqn (5):

(6) |

On comparing the experiments with the predictions of the model, we propose that the aspect ratio of the patterns exposed to growth is the same in the experiments and in the numerical DLA scheme. In this configuration, the patterns are short systems, with d > h and aspect ratio lower than one . Since the model is a comparative method of analysis, it does not necessitate a one-to-one correspondence between the geometry of the real physical prototype and the model. More sophisticated evolutions of the model that will be developed over time will enable direct simulation of real systems and the determination of absolute values of growth. As regarding spacing between patterns, δ, we have used in the experiments spacing between patterns three times larger than the pattern size, which guarantees non-interference between patterns^{18} and justifies the use of a single-well numerical scheme and an isolated system to simulate particle formation.

Then, we examined the structure of metal clusters at different times t from immersion in the electroless solution, t = 5, 20, 60, 120 s, using scanning electron microscopy (SEM) and atomic force microscopy (AFM) imaging techniques. We fabricated at least 5 different samples per time of deposition, and acquired more than 25 SEM images and 2 AFM images for the samples. We used SEM images for determining the rate of growth of the nanoparticles, and AFM imaging to extract the topographical details of the clusters.

For fixed values of time, we observe that the number and density of isolated particles on the silicon surface are greater for silver compared to gold deposition (Fig. 5a). This trend is maintained for all considered time frames of growth (Fig. 5b). Using standard image analysis algorithms, we extracted the number Q of isolated particles formed during silver and gold deposition over time (Fig. 5c). Q varies linearly with time (r^{2} = 0.989 for gold, r^{2} = 0.984 for silver; the statistical significance of the linear regression is reported in separate ESI 2†), in agreement with the theoretical model and the simulations. We elaborated information contained in Fig. 5c to derive how rapidly the number of clusters in an image changes with time: at any time deposition of silver is more rapid than that observed for gold. Numerical analysis of data yields the values for silver and for gold, and a ratio between the two γ^{Ag/Au} ∼ 3.45. This value of γ will be used in the model to determine k.

From AFM imaging (Fig. 6a) one can observe that the number of particles deposited during electroless growth of silver is larger than the number of gold nanoparticles deposited in the same time (60 s), in accordance with the results of the model and SEM inspection of samples. Few clusters of gold nanoparticles are particularly large, this is deceptive and may suggest that the cluster growth rate of gold is larger than silver – that it is not as proved by image analysis algorithms applied to AFM images that yield an estimate of Q_{Ag} ∼ 106 clusters in a square pattern of 2 μm for silver, and Q_{Au} ∼ 29 clusters for gold, with a ratio . While AFM imaging may be as accurate as SEM in deriving the topographical characteristics of a sample surface, we use here SEM results to benchmark the model because SEM imaging of samples is faster compared to AFM. SEM images are more in number, have larger formats, include more particles, are more informative and statistically significant than AFM images. While AFM can achieve ultra-high resolution and can examine sample topography at the sub nanometer level, it is not high-throughput.

Post-processing of AFM topographic data enabled us to derive the power spectrum density function for both silver and gold (Fig. 6b). From this, we derived the values of fractal dimensions for the clusters of Ag (D_{f} = 2.7) and Au (D_{f} = 2.65). Notice that, remarkably, the experimental values of fractal dimension derived in the space are about one dimension higher than the corresponding numerical values derived in the plane, D^{3d}_{f} ∼ D^{2d}_{f} + 1.

Using the ratio between cluster growth dynamics (γ) determined from experiments in the model of eqn (6), we estimate that the ratio between the rates of growth of silver (k_{Ag}) and gold (k_{Au}) is about ξ(γ) = ξ^{model} ∼ 34.

The last point implies that, while in nanotechnology the geometrical characteristics of metal nanomaterials are relevant for their functions,^{29–33} traditional methods for the determination of k ignore the geometrical form of the material assembling into nanostructures.

Here, we used mathematical modelling and numerical simulations to correlate the topological properties () of a system grown by electroless deposition with the variation of its volume (Ṅ → p) to determine k. Different from other existing methods, our approach centers on geometry. The geometry of target structures, in turn, can be easily reconstructed using SEM, AFM, or other techniques of imaging that can be easily found in a nanotechnology laboratory and that are, on the other hand, routinely used to examine the aspects of the structures and the efficiency of a method of fabrication. Thus, incidentally, much of the SEM, AFM, TEM material produced during ordinary inspection of samples and reports can be redirected as the input of the model – generating high volumes of data (big data) that can contribute to reducing noise, reducing uncertainty, and increasing the accuracy of the estimate of k. Moreover

(i) Our model uses as input the ratio between cluster multiplication velocity of two species, . In the more general case, _{A} and _{B} should be determined as the variation of Q over time, which in turn implies determining the number of clusters Q at different times through multiple measurements. Nevertheless, assuming linearity, γ can be more simply determined as the ratio between one of the possible couples Q_{A}(t_{a}) and Q_{B}(t_{a}), , measured at any time t_{a} comprised between zero and the final time of growth, 0 < t_{a} < t_{fin}. This reduces the search for k to the estimate of the number of clusters (i.e. isolated metal nanoparticles) in two different SEM or AFM images, and the division between the two – that dramatically reduces the number of measurements and the time necessary to find k compared to other experimental techniques of analysis. In determining the number of clusters Q from SEM or AFM imaging of samples, it is preferable that the nanoparticles do not overlap (or overlap partially) on the substrate to assure accuracy. Images may be analyzed using algorithms developed over time,^{5,34,35} which deconvolute information using maximum likelihood methods and Fourier transform decomposition/reconstruction of images. Image analysis is accurate under sparse and sub-confluent conditions. When particles on the substrate overgrow, non-dense assumption breaks down and the number of clusters/particles in the field of view can be under-estimated. To avoid miscalculations, one should limit the analysis to the early time of particle deposition, or in any case to the linear regime of particle growth, which is the case of Fig. 5d and results presented in this work. When Q(t) deviates from linearity, results of the analysis can be inaccurate.

(ii) Our model derives k using direct optical, SEM or AFM inspection of samples. The method does not require sample treatment, preparation or modification, it is not destructive, does not hamper and has no adverse loading effects in the process of growth. The measure of k is carried on the real physical prototype and not on a simplified version of it. The size and shape of the patterns on the silicon exposed to growth are preserved and their effects are correctly incorporated in k, which must not be therefore further adjusted.

(iii) Our model is general in nature. It simulates the assembly of ions or atoms into supramolecular structures and can be used to estimate the ratio between the rates of growth of several materials, including non-metallic materials, provided that the process of deposition is limited by diffusion and the process of chemical reaction at the interface is fast compared to diffusion. Under similar assumptions, a material can be examined using DLA. As a rule of thumb, if a material exhibits a dendritic structure, with some level of order and recursive patterns in the structure – like crystals – it can be analyzed using this model.

In its current form, the model makes the following assumptions and has the following limitations:

(1) In electroless deposition, motion of metal ions in a solution is much slower than the chemical reaction of reduction of those ions into metal on a patterned silicon substrate (DLA assumption).

(2) The diffusion lengths of different metals is the same – assuming that the thermodynamics conditions of growth are held constant, using the Stokes–Einstein relationship^{10} this is equivalent to state that the sizes of different metals is the same.

(3) The differences between different species are lumped in the sole chemical reaction of deposition, i.e. in p.

(4) It is a comparative method of analysis. This depends on the fact that, in this current form, the model shows a very high sensitivity to the geometry of the system. Since it may be impractical and computationally intractable to reproduce the entire physical system and simulate a real electroless process of growth, we use ratios between variables rather than the absolute values of variables to describe particle growth. In doing so, we can examine exclusively and focus on the parameter of interest, i.e. k, with the remaining conditions held constant. The effects of an inaccurate representation of the systems would cancel each other out when we compare two different metals. This in turn reduces uncertainty and increases the precision of the model. The cost of an augmented precision is the use of the model as a comparative rather than absolute method of analysis.

More sophisticated formulations of the model that will be developed over time may potentially relax the above constraints and be used for a more complete and precise description of metal growth for applications in materials science.

Brownian motion is a mechanism of ion transportation implemented at the atomic level – it describes the process of particle deposition using a discrete sequence of events in time, and guarantees maximum accuracy in reproducing the structure of an aggregate. Fick's laws of diffusion are derived from Brownian motion and molecular diffusion: they represent a generalization of Brownian motion at the continuum limit. Thus, direct numerical simulation using DLA is more appropriate for micro- and nano-scale systems, for which the continuity assumption breaks down, while continuous Fick diffusion is more efficient when one considers macro-scale systems.

In what follows, the word particle is used interchangeably with ion, they both indicate the smallest building blocks that assemble together to form the final aggregates. Ions drift in the domain (grid) due to pure diffusion. Therefore, displacement of ions in the grid obeys a random uniform distribution; at each time of the process particles move from a position of the grid to one of its nearest neighbors, each of those positions being equally probable. We call Δx the size of the cells (pixels) of the grid. It is the size of the smallest features in the grid and corresponds to the resolution of the system. Since the motion of a particle in a grid takes discrete steps, in a macroscopic interpretation Δx is also the mean path length of the particles, i.e. the average distance traveled by a particle between successive collisions. Cases in which the mean path length is different from the resolution of the systems are discussed below. During motion, particles have a fixed velocity v. The mean time interval between collisions is thus . The nature of Δx, v and τ is determined by the energy of the system and its temperature.^{10} Consider the scheme in Fig. 1c.

5.1.1 Geometry.
The portion of the silicon substrate (well) exposed to electroless deposition is represented at the bottom of the diagram as a line of nucleation sites Λ. Its length is w. The height of resist walls that delimit the well is h. The ratio determines the aspect ratio of the well. If the aspect ratio is lower than one, then we have a short geometry. In contrast, tall geometries have aspect ratios greater than one. Short geometry reproduces more accurately the real physical prototype and the experimental set up, where the dimension of lithographed patterns (10 μm) is larger than the thickness of the resist (∼1 μm). Here, we chose w and h as w = 100 pixels, h = 50 pixels, such that . Dimensions of the entire domain are 400 pixels (length) and 500 pixels (height).

5.1.2 Initial and boundary conditions.
At the left and right boundaries of the domain, periodic boundary conditions are imposed. At the upper boundary of the domain, we enforce a bouncing boundary condition. Moreover, we assume that particle deposition occurs in an excess of solute, i.e. the number n of metal ions in the domain is generally high and it is maintained constant as n = 1000. At the initial time of growth, metal ions are positioned in a region of the domain at a distance l from the well.

5.1.3 Aggregation rules.
Then, the system is left free to evolve. At any iteration, particles move in the grid by one pixel. After an iteration, a particle may find itself in two separate, mutually exclusive states: (i) isolated and (ii) contact state. In the isolated state (s = I) all sites surrounding the particle are 0 (the particle misses the aggregate). In the contact state (s = C) the minimum distance between the particle and the elements of the seed Λ is 0 (the particle hits the aggregate). The algorithm starts with the condition s = I and proceeds until the test s = C for at least one particle in the domain yields true. When a particle hits the aggregate, it is incorporated into the seed to form an aggregate of particles β if a randomly generated number q is such that q < p. In the simulations, the sticking probability p can be varied between 0 and 1. If p = 0, particles would indefinitely migrate in the domain never accumulating to the aggregate. If p = 1, particle deposition is deterministic, and particles would be captured by the aggregate anytime that they hit it. After a certain number of repetitions aggregates shall have the aspect depicted in Fig. 1d. The multi-branched arrangement of particles recalls the dendrite, fractal nature that electroless systems reveal under certain growth conditions. On changing p, one can modulate the size, shape and density of the aggregates.

5.1.4 Stop condition.
Simulations are halted after ∼1.2 × 10^{7} iterations.

5.1.5 Considering the thermodynamic state of a system.
Thermodynamic variables are implicitly contained in the model. Super-saturation, thermodynamic potentials and the temperature itself depend on the sole temperature field T of the system, assuming that pressure gradients are vanishingly small everywhere in the domain and that the number of ions n in the system is maintained constant over time (i.e. particle deposition proceeds in an excess of solute). Under these conditions, we can express the spatial variables of the simulation in terms of the sole T. Specifically: the root mean square distance (i.e. the variance) of a bolus of diffusing particles is

where x, y, r are the coordinates of the bolus, and D is the molecular coefficient of diffusion

with k_{b} the Boltzmann constant, T the absolute temperature of the system, μ the viscosity, and a the diameter of the particles. Moreover, using results from the kinetic theory of gases, we can write

where δx is the mean path length of an ion in solution and τ is the time between collisions. In molecular-scale systems, the kinetic energy of a particle with mass m and velocity v is

which is a function of time. Eqn (11) correlates the mean path length of an ion with the thermodynamic variables of a system and the temperature T. Since the mean path length is a variable of the model, one can change the temperature of the system by tuning δx: the thermodynamic variables of the system are lumped in the term δx.

〈r^{2}〉 = 〈x^{2}〉 + 〈y^{2}〉 = 4Dt | (7) |

(8) |

D = δx^{2}/2τ | (9) |

(10) |

Combining these equations, one obtains:

δx = 2D/v = f(T) | (11) |

5.1.6 Additional remarks.
The model is a numerical diffusion limited aggregation simulation model – it reproduces aggregation of smaller ions into larger particles under the assumption of a process that is limited by diffusion, i.e., movement of ions is a random walk, and there are no preferential directions of motion in the domain. The model neglects external driving forces or fields including convective fields, electro-magnetic fields, the generation of mass in the domain through internal chemical or nuclear reactions. Nevertheless, assuming linearity, in a more sophisticated evolution of the model these effects can be easily included by adding a constant term to the analysis and considering superposition of individual stimuli. Since ions add up to the aggregate immediately upon contact, another assumption is that motion of ions is much slower than the chemical reaction of reduction of ions into metals at the liquid–solid interface, which is instantaneous. However, since different materials have different values of reaction kinetics – that in turn influences the rate k of growth of the metal – we introduced in the model an additional variable, p, comprised between 0 and 1, to account for this diversity. The assumption that k depends on the sole p is heuristically supported by the numerical diagram in Fig. 3b, where the mass of the system (i.e. the size of the aggregate N) varies linearly with p. Moreover, the model is 2D. The model accepts as inputs (i) the geometry of the system, (ii) the resolution of the system, i.e. discretization, (iii) the mean path length of ions (that depends on the thermodynamic variables of the system), (iv) the concentration of ions in solution, (v) the total length of the simulation. The output of the simulation is an aggregate of pixels, where the geometrical characteristics of the aggregate, measured through mathematical variables including fractal dimension, are indicative of the characteristics of the true particle assemblies.

(12) |

Me^{n+} + Red = Me^{0} + Ox | (13) |

Si + 2H_{2}O → SiO_{2} + 4H^{+} + 4e^{−} | (14) |

Ag^{+} + e^{−} → Ag^{0} | (15) |

Si + 2H_{2}O → SiO_{2} + 4H^{+} + 4e^{−} | (16) |

Au^{3+} + 3e^{−} → Au^{0} | (17) |

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8na00040a |

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