**Encyclopedia of Engineering**

Editors

**Raffaele Barretta Ramesh Agarwal Krzysztof Kamil Zur ˙ Giuseppe Ruta**

MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin

*Editors* Raffaele Barretta University of Naples Federico II Italy

Giuseppe Ruta University "La Sapienza", & National Group for Mathematical Physics Italy

Ramesh Agarwal Washington University in St. Louis USA

Krzysztof Kamil Zur ˙ Bialystok University of Technology Poland

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Topical Collection published online in the open access journal *Encyclopedia* (ISSN 2673-8392) (available at: https://www.mdpi.com/journal/encyclopedia/topical collections/encyclopedia engineering).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. *Journal Name* **Year**, *Volume Number*, Page Range.

**ISBN 978-3-0365-7000-6 (Hbk) ISBN 978-3-0365-7001-3 (PDF)**

Cover image courtesy of Raffaele Barretta

© 2023 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications.

The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND.

## **Contents**

## **About the Editors**

#### **Raffaele Barretta**

Professor Raffaele Barretta is a Full Professor of Solid and Structural Mechanics in the Department of Structures for Engineering and Architecture at the University of Naples Federico II. He was born in Naples (Italy) March 20, 1980. He received his Master Degree (5 years) in Civil Engineering with magna laude at the University of Naples Federico II, 2003—Title of thesis: Polar Models of Beams and Shells in Large Deformations. He received his Ph.D. in Structural Mechanics at the University of Naples Federico II, 2007—Title of thesis: Mixed Variational Methods in Elasticity. He was an Assistant Professor of Solid and Structural Mechanics at the University of Naples Federico II (2010–2015). He was an Associate Professor of Solid and Structural Mechanics at the University of Naples Federico II (2015–2021). His research interests mainly focus on Continuum Mechanics; Beams, Plates, Shells; Nano-Materials; Non-Local Elasticity; Functionally Graded Materials; MEMS/NEMS; Nanoscience and Nanotechnology. Awards: Top Italian Scientist in the Engineering Area from Single Year Career 2017 (Source paper: Ioannidis J.P.A., Baas J., Klavans R., Boyack K.W. A standardized citation metrics author database annotated for scientific field. PLoS Biology 17(8): e3000384 (2019) https://bit.ly/2LITT4P); One of the top 90 scholars of the University of Naples Federico II in the database of 100,000 top scientists published by PloS Biology: http://bit.ly/37ox6Uk. He is ranked #532 in the world and #9 in Italy among Top Scientists for 2022. The full world ranking can be found here: https: //research.com/scientists-rankings/mechanical-and-aerospace-engineering. The entire ranking for Italy can be dound here: https://research.com/scientists-rankings/mechanical-and-aerospace-engineering/it.

#### **Ramesh Agarwal**

Professor Ramesh K. Agarwal is the William Palm Professor of Engineering in the Department of Mechanical Engineering and Materials Science at Washington University in St. Louis. From 1994 to 2001, he was the Sam Bloomfield Distinguished Professor and Executive Director of the National Institute for Aviation Research at Wichita State University in Kansas. From 1978 to 1994, he was the Program Director and McDonnell Douglas Fellow at McDonnell Douglas Research Laboratories in St. Louis. Dr. Agarwal received PhD in Aeronautical Sciences from Stanford University in 1975, M.S. in Aeronautical Engineering from the University of Minnesota in 1969 and B.S. in Mechanical Engineering from Indian Institute of Technology, Kharagpur, India in 1968. Over a period of 45 years, he has worked in several disciplines within mechanical and aerospace engineering, and energy and environment which include computational fluid dynamics, computational electromagnetics and acoustics, control theory, multidisciplinary design and optimization, turbomachinery and pumps, chemical looping combustion, carbon capture and sequestration, and wind energy. He is the author and co-author of over 600 publications. He has given many plenary, keynote and invited lectures at various national and international conferences in over sixty countries. He is a Fellow of 26 professional societies including the American Institute of Aeronautics and Astronautics (AIAA), American Society of Mechanical Engineers (ASME), Institute of Electrical and Electronics Engineers (IEEE), Society of Automotive Engineers (SAE), American Association for Advancement of Science (AAAS), American Physical Society (APS) and American Society for Engineering Education (ASEE). He has received many prestigious honors and national/international awards from various professional societies and organizations for his research contributions including the AIAA Reeds Aeronautics Award, SAE Medal of Honor, ASME Honorary Membership and Honorary Fellowship from Royal Aeronautical Society.

#### **Krzysztof Kamil Zur ˙**

Professor Krzysztof Kamil Zur is a Researcher at the Faculty of Mechanical Engineering, Bialystok ˙ University of Technology. He received his Ph.D. in Theoretical and Applied Mechanics. His research is concerned with the applications of meshless methods to the dynamics of discrete–continuous composite structures with a non-linear distribution of parameters. Professor Zur is an independent ˙ European scientist who is the head of many international scientific groups working on the mechanical problems of structures and composite materials at diverse scales (from nano to macro). He is also an expert in analytical, mesh, and meshless numerical methods applied to solve different challenging linear and non-linear boundary value problems. His research is complex and strongly interdisciplinary, connecting research areas such as material science, mechanics and nanomechanics, physics (multi-field), and numerical methods. To go beyond the current state of knowledge and present new insights into the mechanics of investigated multi-scaled structures and materials, Professor Zur connects people ˙ of different abilities from around the world (US, Australia, EU, Asia, Africa) to perform high-quality interdisciplinary investigations and publish convincing and reliable results in reputable journals. He is the "Highly Cited Researcher" from the publications that rank in the top 2% by citations for field and publication year in the Web of Science™ citation index.

Professor Zur is a well-known and respected scientist within international scientific communities. ˙ He serves as one of the main editors of two reputable journals: *Engineering Analysis with Boundary Elements* (ELSEVIER) and *Applied Physics A* (Springer), and an Associate Editor, Subject Editor, and Editorial Board Member in more than 20 reputable and internationally recognized journals. He has worked for several international grants carrying out complex tasks. He works as a referee to support grant proposal assessments for a few national and international funding agencies as well as governments. He plays the role of reviewer in more than 150 journals and works as a referee for different publishers.

#### **Giuseppe Ruta**

Born in 1967, Giuseppe Ruta graduated in Mechanical Engineering from the University "La Sapienza" of Rome in 1992, receiving a PhD in Theoretical and Applied Mechanics from the same university in 1996. He received postdoc positions at the University Roma Tre and became an Assistant Professor at "La Sapienza" in 2000, then an Associate Professor in 2012, and since 2018 he is eligible for a full Professorship. He serves in the Mechanical Engineering courses and in the Ph.D. School of Theoretical and Applied Mechanics of "La Sapienza", and as a didactic expert in the Italian National Agency for University Evaluation. He is author and co-author of 150 scientific contributions, of which more than 70 are international journals. His research interests include: one-, two-, and three-dimensional continuum models; flexural-torsional instabilities for thin-walled slender elastic beams, the presence of damage and stiffeners; perturbation methods for non-linear elasticity in moderately thick cylinders; fluid–structure interactions; static and dynamic instability of beams on various elastic soils; functionally graded materials; one-dimensional continua for the dynamics of circular and parabolic arches; local damages in arches via static and dynamic measures in view of structural identification and monitoring; non-local elasticity in beams; historical–epistemological study of the works of Gabrio Piola, Enrico Betti, Luigi Federico Menabrea; and elastic models of Cauchy, Voigt, Poincare, Born-Von K ´ arm´ an, Gazis, Hrennikoff, Eringen. He is regular reviewer for several ´ international journals (*Meccanica*, *Journal of Sound and Vibration*, *Thin-Walled Structures*, *Acta Mechanica*, *Mathematical Reviews*, *Journal of Vibration and Control*, a.o.); he is Academic Editor for the journal *Shock and Vibration* and an active collaborator of MDPI publishing house. He was part of the organizing committee of national and international congresses at "La Sapienza" in 2004, 2012, 2019, and 2021.

## *Entry* **Vibration-Assisted Ball Burnishing**

**Ram ón Jerez-Mesa 1,\*, Jordi Llumà 2 and J. Antonio Travieso-Rodríguez 1**

**Definition:** Vibration-Assisted Ball Burnishing is a finishing processed based on plastic deformation by means of a preloaded ball on a certain surface that rolls over it following a certain trajectory previously programmed while vibrating vertically. The dynamics of the process are based on the activation of the acoustoplastic effect on the material by means of the vibratory signal transmitted through the material lattice as a consequence of the mentioned oscillation of the ball. Materials processed by VABB show a modified surface in terms of topology distribution and scale, superior if compared to the results of the non-assisted process. Subgrain formation one of the main drivers that explain the change in hardness and residual stress resulting from the process.

**Keywords:** ball burnishing; acoustoplasticity; vibration-assistance; surface integrity; surface topology

**Citation:** Jerez-Mesa, R.; Llumà, J.; Travieso-Rodriguez, J.A. Vibration-Assisted Ball Burnishing. *Encyclopedia* **2021**, *1*, 460–471. https://doi.org/10.3390/ encyclopedia1020038

Academic Editors: Krzysztof Kamil Zur, Raffaele Barretta, Ramesh ˙ Agarwal and Giuseppe Ruta

Received: 17 May 2021 Accepted: 8 June 2021 Published: 11 June 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. History: From Ball Burnishing to the Vibration Assisted Version of the Process**

This Encyclopedia entry deals with the main aspects and details related to the ball burnishing process assisted with a vibratory signal (namely, vibration-assisted ball burnishing or VABB henceforth). Ball burnishing is based on deforming plastically with a preloaded sphere the irregularities of a surface that has been previously machined, so that its roughness or texture features are reduced while hardness is increased due to cold deformation (Figure 1a). However, this interaction is three-dimensional and is very much influenced by the friction between the ball and the material. The main physical vector to achieve that deformation is the preload force with which the ball is preloaded and the number of passes by which the target surface is covered. By assisting the process with vibrations, a vibratory component of the force *Fv* is overlapped to the preload *Fp*, resulting in the overall vibratory burnishing force *Fb*, as shows Figure 1b.

The process must be understood as the upgrade of a classical operation complemented with an extra layer that introduces new dynamics and modifies the way the tool interacts with the material of the target surface. The oldest references related to ball burnishing itself refer to the processing of certain parts of the automobilistic industry in the sixties [1]. The process was described simply as a means whereby the motion of a ball or roller displaces the peaks of the surface roughness profile into the valleys. Today, we know that this apparently simple description does not account for the very complex mechanisms that are put at stake when this kind of process is deployed to provide a certain workpiece with a desired finishing state. The phenomenon whereby the material surface is modified is more likely to be compared to how the wavy surface of calm waters on the sea are smoothly moved by the effect of the wind, changing their direction, but keeping a very similar pattern all the way through.

Ball burnishing has been often cited because of its direct effects on the surface texture. The actual description of this modification can be described as a triplet:

As the last of the described effects is only observed if the proper force and number of passes are combined to obtain the desired surface finishing, it could be said that the original explanation of ball burnishing in which material peaks were introduced in the valleys is not totally accurate.

Besides the topological effects of the process, the material also embodies other transformations that define its state after ball burnishing. Specifically, by experiencing cold deformation, the material is ultimately cold-hardened, providing the final workpiece with a reinforced outer layer with enhanced performance. Furthermore, a higher compressive residual stress profile is formed in the subsurface layers of the material. This change of mechanical state of the material is also often observed in the change of the microstructural state of the outer layers of the material itself, if a cross-section of the processed surface is observed.

The assistance of ball burnishing in the mid-twentieth century responded at the time to an extended trend in the manufacturing innovation ecosystem based on providing classical operations with extra functions that enhanced the outputs of these processes. This is how hybrid processes such as vibration-assisted machining [2] or laser-assisted ball burnishing [3] were born and are still today used in many manufacturing companies. Specifically, VABB was brought into play into the finishing operations industry, incorporating a vibratory movement to the burnishing ball simultaneous to its rolling over the surface irregularities while it runs the programmed trajectory. VABB was for the first time reported during the 1970s, designed as ultrasonic burnishing. It was assisted by 41.5 kHz vibrations and a variable amplitude from 5 to 10 μm [4]. The first detailed academic bibliography dealing with VABB dates from the 1980s [5], although some references could be found in previous years focusing on the comparison of the friction coefficient, wear rate or load bearing capacity of VABB-treated surfaces with regards to surfaces finished through other processes such as boring, grinding or even simple ball burnishing. However, these references did not focus on the phenomenology behind the results or their relation with the descriptive parameters of the surfaces themselves.

This entry is divided in three sections. The first describes the overall results observed on different materials after VABB. The second offers an insight into the physical origins of the vibratory assistance. Furthermore, finally, the hardware and physical systems reported in literature are explained.

#### **2. Effects of VABB on Materials**

In general, the affectation of surfaces after VABB can be described very similarly to the one resulting from the conventional process, namely as a comprehensive effect on the material at the surface on the topology or roughness, microhardness and residual stress with a higher affectation of the surface. The references that are included in this section show that VABB does not necessarily enhance all properties simulatenously, hence the importance and need to know the process and decide whether to use it or not and to distinguish its adequateness according to the desired surface characteristics.

The first results reported after VABB by Marakov (1973) [4] found a relevant interaction between the vibration amplitude and the obtained surface roughness. Indeed, the anticipated positive effect of higher force values on the resulting surface roughness was only observed on those mild steel specimens treated with a 2 μm amplitude. A reduction in the friction coefficient between the burnishing ball and the recipient material was also reported in those conditions. Later on, Pande and Patel (1984) [5] reported results on low-frequency (10 to 70 Hz) vibratory burnishing in contrast to the high-frequency assited process. Results provided evidence for an inverse interaction between the preload and the amplitude of the assistance, obtaining lower surface roughness values for amplitudes lower than 0.5 μm. They also found that the vibration-assistance was remarkably positive with regards to the residual hardness obtained after the tests, as an assistance with 60 Hz vibration allows the lowest preload to be successful in increasing these values compared to the unassisted process.

No other relevant research sources about ultrasonic burnishing can be found until the 2000's, when new references to the process start to be found on different materials. Bozdana et al. (2005) [6] applied the process on Ti-6Al-4V specimens assisted with 20 kHz and 6.75 μm vibration on a milling machine. It was proved that there is a critical value from which surface roughness is harmed when the VABB process is applied, and should be defined. For the ultrasonic process, that point is at a much lower preload level, probably because transient softening due to the transmission of the vibratory signal through the material favours the in situ plastic deformation. Consequently, the effects of vibration-assistance can fire back by deforming excessively the material at the surface, and should be carefully selected [7]. However, residual stress and hardness results were much more promising, as the ultrasonic process resulted in higher values with half the preload required for the non-assisted process. Therefore, it can also be stated that the process seems not to be neatly positive in affecting the surface under different perspectives, i.e., some aspects might be improved while others are harmed.

The references related to VABB since the 2010s have increased considerably, including few references applied on a lathe on different materials [8,9]. The VABB applied on milling machines clearly dominates the state of the art in this sense. A 2-kHz assistance was reported to improve the surface roughness of AISI 1038 [10] and EN AW 7078 [11] with regards to the non-assisted process (NVABB), although the results in terms of microhardness were questionable. Extensive experimental research has been performed to analyze the impact of VABB on different ball-end milled surfaces of AISI 1038 [12], Ti-6Al-4V [13], nickel-based Udimet 720 alloy [14] and AISI 306 [15]. In all cases, following the same research pattern by applying Taguchi experimental design, the authors conclude that VABB proves its effectiveness to modify effectively the surface topology of all surfaces, and redistributing the material to a Gaussian state. Other works on the AISI 316L steel has also been conducted [16]. Furthermore, the threshold value of the preload from which the surface is harmed was identified, being different for each of the tested materials. It was also noted that it seems that in applying the VABB process, the effect of the original surface is of upmost importance, as it defines the improvement potential of the surface itself. The authors conclude that the vibration-assistance should only be selected to improve surface topology if the original Sq descriptor of the surface is 5 μm or less; but on the other hand that topological improvement can be accompanied with a lower level of compressive residual stress [13].

The nanoscopical level also shows information about how the material at the surface is modified after VABB. It has been found that the process succeeds in refining the grain structure at the subsurface on many alloys such as aluminum 6061 [17] or AISI 1045 [18], and is even able to promote the phase transformation in materials such as Ti-6Al-4V [19], or austenitic metastable AISI 306 [15] (by forcing the generation of martensite by cold plastic strain) [20]. This translates into a higher residual hardening and stress. In biocompatible materials, investigators have succeeded in generating subsurfaces that favour cell adhesion, and can, therefore, increase the biocompatibility of materials [21]. Grain refinement on 17-4PH stainless steel surfaces also resulted in a wear and corrosion resistance of the material, compared to the conventional unassisted version of the process [22]. In all cases, new research seems to show that the correct direction to continue with investigations about VABB is to consider how the microstructure is changed.

#### *Pros, Cons and Capabilities of VABB*

The development of the VABB throughout the years and the very positive results obtained in research have positioned VABB as a potential process to be implemented in many kinds of industries. Although no works comparing VABB with other finishing processes have been reported, its non-assisted counterpart has proved to be superior in terms of residual stress and topological improvement if compared to it direct competitors, such as laser shock peening or shot peening [23]. If it is assumed that VABB is an upgrade of NVABB, the general superiority of VABB with regards to other competitive finishing processes can be inferred by extension.

The capabilities of the process do not only result from the effects on the material itself, but also the ease with which it can be introduced in a manufacturing routine through numerical control has to be highlighted. The authors work, for instance, with a company that is substituting their manual polishing for moulds for the automotive industry by this automatised process. The introduction of VABB in their routine not only has reduced the processing time of each part but has also allowed the owners of the company to exploit their machine tools overnight with the automatised process.

Companies from the aeronautical industry are also eligible to implement VABB in their routines. This industry is on the search of processes that can help them improve the conditions of selective surfaces that are subjected to fatigue stress. The target of VABB does not have to be a whole surface but specific sectors of the part that engineers have identified as critical, what makes the process still more interesting in comparison with other ones that cannot be so selective, such as sand blasting or laser shock peening.

The process also demonstrates disadvantages, as the equipment required to execute it is based on an external circuit that has to be branched to the VABB tool so that it can work. This wiring could be a handicap to automatise the process, or at least could be a conundrum for production engineers willing to guarantee the security of the process itself. Furthermore, it would require more space to install the external power circuit.

Theoretical models of VABB are scarce but have arisen the fact that the simultaneous improvement of texture and residual stress cannot always be possible with VABB [24]. The solution to that is to adjust very thoroughly the burnishing parameters to achieve this simultaneous effect. Therefore higher preprocessing and preparation time can also be cited as a drawback of the process and introduces a new challenge for industries willing to implement it in their routines.

The authors consider that these disadvantages cannot overshadow the evident advantages of the process itself, as it has been presented above, not only in practical terms but also in material modification and performance after being finished through VABB. It is true that the time required to define a correct selection of VABB parameters (explained in subsequent subsections) is long, but it can definitely pay off later on once the process is effectively implemented in the manufacturing routine of the adopters.

#### **3. Physical Principles behind VABB**

Originally, the introduction of vibration assistance was brought into the industry just by following the hypothesis that a vertical movement of the burnishing ball, simultaneous to its longitudinal feed movement, could have a similar effect on the material as if successive impacts were applied on the surface, i.e., due to a hammering effect on the material that composed the interface of the workpiece. The results obtained after VABB, and that shall be described in the next section, evidence that the process leads to different results compared to its original counterpart. However, it is not clear today what the phenomenological explanation is that accounts for the process results. This is caused by the fact that it is impossible to visualize how the actual engagement of the ball with the surface material is modified due to the vibratory movement, and how stress is transmitted into the material subsurface layers.

Regardless of this limitation, research during the last few decades has allowed the scientific community to obtain new insights of the technology itself. The evidence found demonstrate that there are two main causes whereby VABB offers different results with regards to it conventional counterpart, namely:

#### *3.1. The Acoustoplastic Effect*

Acoustoplasticity consists of the decrease in the quasi-static stress to which a material must be subjected to be plastically deformed by means of overlapping a vibratory signal over the physical force that causes that strain. As ball burnishing is based on plastic deformation, it is, therefore, eligible to be enhanced by this effect. Acoustoplasticity was reported for the first time by Blaha and Langenecker in 1955 [25] on pure zinc crystals radiated by a 800-kHz ultrasonic wave. Hence its alternative designation as Blaha effect. It was proved later on that it can be universally observed in metals [26], although the degree of affectation varies according to the properties of the materials. For instance, the higher the acoustic impedance and the higher elastic modulus of a material, the higher its sensitivity to be affected by acosutoplasticity [27]. The acoustoplastic effect has proven to be independent of the vibration frequency [28] but its effects vary according to the vibration amplitude although the source of this influence is not clear [29,30].

The consequences of acoustoplasticity are dual because it can cause residual softening but also residual hardening [31]. Gindin et al. (1972) [32] concluded that the residual hardening is only present if an intensity threshold is surpassed. This observation is interesting to justify the assistance of ball burnishing with vibrations because it could facilitate plastic strain during the process while provoking a residual hardening of the target surface.

Despite the fact that acoustoplasticity has been experimentally observed on a high variety of materials, its actual physical source is still controversial. The conundrum is based on the fact that there is no agreement on whether acoustoplasticity has an intrinsic or extrinsic causes, i.e., whether it is provoked by a reaction inside the material's microstructure to the external source of vibrations, or by the increase of the power deployed into the system without change of the material behaviour. Ultimately, the reader shall find that acoustoplasticity has a polyhedral nature.

The intrinsic approach to the issue is based on the hypothesis that ultrasonic energy was preferentially absorbed by defects in the metal lattice (e.g., dislocations or grain boundaries). The intrinsic approach is based on the idea that these defects are actual responsible for the mechanisms of plastic deformation, hence its potential to explain why acoustoplasticity works and can be observed at a macro level. This idea was deffended by Blaha and Langenecker [25,33]. Later on, Mason (1955) [34] argued that as the lattice defects absorb the vibratory energy, dislocation mobility is enhanced and this effect allowes the metal to deform under lower loads. Based on this theory, Gindin et al. (1972) [32] justified the residual hardening observed on materials deformed through acoustoplasticity because of dislocation loops on the material lattice and the new stable vacancies accumulated in it. Pohlman and Lechfeldt (1966) [35] reinforced this intrinsic approach by observing that the force drop during ultrasonic strain was only observed during the plastic strain phase, and not in the elastic one, as plastic deformation mechanisms are related to metal lattice dynamics. Langenecker (1966) [31] proposed that the ultrasonic energy at lattice defects caused a microheating effect, facilitating the material strain. Imperfections in the metal lattice tend to look for minimal energy positions, provided that a certain threshold value of energy is exceeded. Therefore, they defended that the increase in dislocation mobility must be a thermally activated process, meaning that acoustoplasticity would only happen if a certain activation energy was surpassed. Although they contributed to the understanding of acoustoplasticity, intrinsic theories could explain why other mechanisms related to energy absorption by lattice defects such as resonance or hysteresis based on ultrasonic do not have the same effect as acoustoplasticity, and, therefore, evidenced limitations.

Chronologically simultaneous were the works undertaken by Nevill and Brotzen (1957) [36], who defended extrinsic theories. They proposed that the observed stress decrease through acoustoplasticity was independent of the temperature. Therefore, they explained the acoustoplasticity phenomenon as a result of macroscopic superposition of steady and oscillatory stresses. Kirchner et al. (1985) [37] developed an extrinsic model by a set of experiments performed with a universal testing machine on aluminium specimens, by programming different overlapped sinusoidal forces on the deforming forces at low, medium and high frequencies. However, as this model lacked total correspondence with experimental observations, the role of internal friction was introduced eventually in the system, assuming its responsibility for the equilibrium of forces that need to be satisfied during the quasi-static deformation of a material [38,39]. Still, results were not exactly consistent with experimental observations. Therefore, a purely extrinsic approach to the topic did not seem to be sufficient to explain satisfactorily the sources of acoustoplasticity.

The recent advances in microstructural analysis has allowed researchers to revive the discussion about the roots of acoustoplasticity. Vickers microindentation tests performed with a vertical 30-kHz vibrating indenter were conducted in 2011 by Siu et al. on pure aluminium [40], copper and molybdenum [41] proved that the diamond-shaped indentations for the ultrasonic-indented prints were bigger. This confirms a decrease in the hardness experienced by the material during the application of the ultrasonic plastic deformation. It was confirmed later on that this is due to dislocation annihilation [42] promoted by acoustoplasticity, as the positive vibratory cycle promotes dislocation travel to further places, and the negative cycle slows them down to favour that annihilation. Furthermore, SEM observations evidenced that subgrains are formed after indentations performed with a vibration assistance unlike the results evidenced by specimens indented quasi-statically. That explains residual hardening, as subgrains act as secondary boundaries which increase the required energy to move the dislocations because of the increase in heterogeneity in the direction of slipping planes inside the material lattice. That is associated to hardness increase. On the other hand, that strain hardening is highly unbalanced, what derives in higher residual stress [43].

This explanation that combines dislocation annihilation and subgrain formation by means of acoustoplasticity is actually a fusion of extrinsic and intrinsic theories and is so far the explanation that accounts more accurately for acoustoplasticity. Indeed, neither is acoustoplasticity just a stress addition effect, nor the preferential absorption of vibratory energy by lattice defects. It also supports the non-dependence of the softening results on the frequency [44]. In contrast to that, it seems tha the vibratory amplitude does influence the residual hardening results, as observed in 2015 by Cheng et al. (2015). In fact, this author remarked that the effect of acoustoplasticity is only conspicuous if an amplitude threshold value is surpassed [45]. This result is in line with the mid-20th century acoustoplasticity experiments explained above.

#### *3.2. Modification of the Engagement Dynamics Ball-Material*

The previous subsection has shown that the usefulness of vibration-assisted ball burnishing can be justified with the resources that material science is able to deliver. However, focusing on how the material is modified during deformation assisted with a vibration is not enough to explain why ball burnishing happens. Indeed, there is a second relevant mechanism that explains the change of the effects of ball burnishing due to vibrations related to the fact that the interaction of both solids changes as the ball moves or is moved by a dynamic mechanism. That is, the frictional behaviour of the ball and the material must be of great importance to the results, because it is that contact that enables vibratory transmission.

The described effect has been formulated guided by the extensive experimental observation and out of intuition, as it is not possible to actually see what is the interaction between the ball and the surface during the process. It is still more complicated to visualise what the impact of the vibratiory movement is. For this reason, researchers are incipiently working on finite element model that can show in detail what are these interactions and predict eventual results of the process [46]. Shen et al. (2019) have developed a 3D FE model of VABB that shoes that a forced vibration overlapped on a static force (preload), the alternative force can be understood as a dynamic hammering that derives in a higher penetration of the residual stress. They also highlight that the compressed layer will be saturated at a certain static load and that, therefore, the room for improvement after VABB is not infinite.

This line should be higher explored in the future, to better understand the dynamics of the process. Lacking the possibility of actually observing the interphase between ball and surface, numerical models are a clear alternative to understand the phenomenology of the process and know how the engagement of the ball and the material occurs during VABB.

#### **4. Equipment to Deploy of Vibration-Assisted Ball Burnishing**

To date, in this text, VABB has been explained as a single process. However, there are numerous ways whereby the vibrations can be introduced in the system and how they are technically deployed:

**Figure 2.** Schematic representations of VABB systems. (**a**). Systems based on deflective plates. (**b**). Systems based on sonotrode deformation.

The readers must take into account that all the described systems are based on the general idea of precharging the burnishing tool on the surface and then activating the vibratory system. As a consequence, the free oscillatory movement of the plate or sonotrode are restricted. That is, their normal free movement when they are excited without contact with the workpiece is dampened according to the elastic properties of the material that is being treated. As a consequence, the actual mechanical system that represents VABB is evidently complex and must be understood as a version of how the tool moves and vibrates when it is not constricted. For this reason, the correct functioning of VABB equipment should be checked to confirm that the vibratory signal originated by the vibrating tool is successfully transmitted though the material lattice. Direct dynamometric measurements or acoustic emission sensors could do the job if installed properly [48]. The challenge in this case would be to have accessibility to acquisition systems that have a high enough sampling frequency to reconstruct the signal, if the assistance is ultrasonic.

#### *VABB Conditions*

The numerous factors that can be chosen to apply the process makes it very easy to design a particular application of VABB for a certain material. Of all these parameters, some of them are directly related to the productivity of the process and the other are responsible for its technical implementation and the actual effectiveness of VABB on the target material. The reader shall find a description of all of the in the next paragraphs:

From the explained parameters, the combination of preload, number of passes, trajectories and lateral offset must be defined in the NC routine implemented to apply VABB on the target surface. The former is actually the linear coordinate the ball has to be positioned at to guarantee a certain pressure on the material surface before starting the routine, whereas the three others are programmed through interpolation functions in the ISO code.

The need to define all these parameters before implementing the process is a challenge for those willing to use VABB to improve the finishing routines inside their industries. For this reason, it is necessary to follow a certain strategy to define Jerez-Mesa (2018) [50] defined after extensive work with different materials that a certain protocol has to be defined, and it depends on the alloy that has to be treated and its original surface state. Figure 3 is an extension of what can be consulted on the referenced Thesis Dissertation and it summarizes that protocol. The frequency and amplitude of the system are normally fixed by the VABB tool. Therefore, to apply the process, the user has to take into consideration what is the target material and what is its current topological state. That defines the actual preload and number of passes to be chosen to modify the topology, residual stress and hardness of the surface. On the other hand, the definition of the trajectories and the lateral offset between passes must be decided to define the desired directionality of the surface texture and preferential residual stress component. By defining these parameters, the user shall be able to master the conditions under which VABB must be executed to maximise its results. It should also be note that, in case the VABB process leads to stress relaxation, then the non-assisted process should be considered instead, although probably the adjustment of the processing conditions could lead to an eventual improvement of the effects of VABB.

**Figure 3.** Recommended protocol to be followed to design the VABB processing conditions.

#### **5. Conclusions and Prospects**

The VABB process featured in this Encyclopedia entry has proved to be a procedure of interest for the industry and researchers during many decades and is starting to be more prominent now with the development of new research activities and the proliferation of practical tooling systems that are easy to manufacture. The best way to technologically implement the process in an actual environment requires a previous testing phase in which the most convenient parameters should be fixed to increase the potential of the process as much as possible and achieve simultaneous effects on texture, residual stress and hardening of the material.

VABB has all the ingredients to be the first option to be adopted as finishing technology in numerous manufacturing environments. However, the prospects of the technology are associated with certain challenges to be tackled that are related to understand the phenomenology behind the physical driver of plastic deformation under the acoustoplastic effect. Extensive experimental research has proved that the results of the process are highly dependant on the interaction of the ball and the original texture. Indeed, the interaction of the vibratory deforming body and the target material is micrometrical, and has proven to be highly influential on the actual results of the process. Increasing the theoretical knowledge of the process not only would lead to the deeper understanding of VABB utterly important in the academic field– but could also reduce the timespan of the previous assessment phase referred to previously to plan the actual implementation of the process.

The innovation of the tooling systems to apply the process are also another innovation line to be explored in the future. The most extended systems reported in the literature to apply the actual process have been explained in this entry. However, the high frequency that these instruments vibrate at, and the dynamics of the mechanical system composed by the long tool pressed on the target surface, makes it difficult to monitor de process with conventional acquisition systems. VABB tooling must, therefore, be explored with different techniques so that the effectiveness of its systems are ratified.

All in all, VABB has proven to be an interesting process with some drawbacks in comparison with other direct competitors such as shot peening, but its capability of being applied selectively on specific areas of industrial parts and the easiness of integration in a manufacturing routine have earned it a prominent position in the present and future of finishing techniques.

**Funding:** Financial support for this study was provided by the Ministry of Science, Innovation and Universities of Spain, through grant RTI2018-101653-B-I00, which is greatly appreciated. Furthermore, by the regional government of Catalonia and FEDER funds for regional development through grant 2019PROD00036.

**Acknowledgments:** The main author Ramón Jerez-Mesa acknowledges the Serra Hunter programme of the Generalitat de Catalunya.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Entry Link on the Encyclopedia Platform:** https://encyclopedia.pub/11771.

#### **Abbreviations**

The following abbreviations are used in this manuscript:

NVABB Non-vibration-assisted ball burnishing VABB Vibration-assisted ball burnishing

#### **References**

## *Entry* **The Foundation of Classical Mechanics**

**Danilo Capecchi**

Department of Structural and Geotechnical Engineering, Sapienza University of Rome, 00185 Roma, Italy<EMAIL_ADDRESS>
**Definition:** Mechanics is the science of the equilibrium and motion of bodies subject to forces. The adjective classical, hence Classical Mechanics, was added in the 20th century to distinguish it from relativistic mechanics which studies motion with speed close to light speed and quantum mechanics which studies motion at a subatomic level.

**Keywords:** classical mechanics; fundaments; history; epistemology; analytical mechanics

#### **1. Introduction**

*Classical mechanics*, or more commonly *Mechanics*, is the discipline devoted to the study of the equilibrium and motion of bodies subject to forces; the adjective *classical* sets it apart from *Relativistic mechanics* which studies motion with speed close to light speed and *Quantum mechanics* which studies motion at a subatomic level. From a historical perspective *Classical mechanics* is the form of mechanics after Newton; for this reason it is often referred to, though quite improperly, as *Newtonian mechanics*. For the 16th and 17th century one speaks of *Early classical mechanics* while for previous periods the locution *Ancient mechanics* is often used.

What is now called mechanics has played a preponderant role in the development of science since ancient Greece [1]. To explain its influence it is not enough, as done just above, to define mechanics as the reasoned study, that is a science, of the phenomena of equilibrium and motion. First of all, mechanics knows how to measure the phenomena of motion: in other words, however complex their appearances may be, whatever qualitative aspects they reveal, whether it is the changing figure of a cloud, of a waterfall, deformations and resistances of an elastic solid, mechanics knows how to define entirely with the help of numbers these motions, these resistances, these deformations. It is a quantitative discipline.

As for the influence of mechanics on the development of other sciences, it is not difficult to see the reasons for this. First of all, the phenomena of motion or of equilibrium occur constantly and everywhere, whether they appear alone or whether they are accompanied by other more complex phenomena (electrical, chemical, etc.). Mechanics was therefore the necessary basis for other sciences, at least as soon as they wanted to be sufficiently precise.

There was not a historical accident which gave mechanics its preponderance. Among all the phenomena, those which are the least difficult to measure, that is to define completely with numbers, are the phenomena of motion: it is therefore mechanics which became the first of all sciences to take a quantitative form. However, by virtue of the more abstract and geometric nature of the phenomena it studies, it had to vegetate as long as it was not in a condition to assume a quantitative form in all its aspects. Let us follow, for instance, a projectile launched into the air; what will we be able to say precisely about its motion if there is no way to register it? It is therefore understandable that the development of mechanics, or better of that part which studies motion, was at the same time so late and so prodigiously rapid once it had begun, and it immediately took precedence over other sciences, even over those which, like alchemy in the Middle Ages, seemed to precede it ([2], pp. 1–4).

Classical mechanics is today generally considered a mathematical physics discipline and as such of little interest to physics researchers. It is a mathematical theory whose

**Citation:** Capecchi, D. The Foundation of Classical Mechanics. *Encyclopedia* **2021**, *1*, 482–495. https://doi.org/10.3390/ encyclopedia1020040

Academic Editors: Ramesh Agarwal, Raffaele Barretta, Krzysztof Kamil Zur and Giuseppe Ruta ˙

Received: 17 May 2021 Accepted: 16 June 2021 Published: 19 June 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

axioms, although they draw inspiration from the physical world, are considered as given; the only developments that can be expected are discoveries of new theorems more or less interesting for the applications of the discipline. It must be said that this point of view is not unassailable and the historical analysis of the discipline provides many elements to counter it.

Despite the unusually distinguished and successful role Newtonian [classical] mechanics has played in the history of modern science, its foundations have been under vigorous debate since Newton first formulated his laws of motion. Moreover, although the axioms have received much more than two centuries of critical attention from outstanding physicists and philosophers, there still is wide disagreement about what they assert and what their logical status is. The axioms (or their logical equivalents) have been claimed to be either a priori truths, which can be asserted with apodictic certainty; [or] to be necessary presuppositions of experimental science though incapable either of demonstration by logic or refutation by observation; or to be empirical generalizations, "collected by induction from phenomena" ([3], p. 174).

In the 18th century, a century in which classical mechanics had reached a mature status, many important scientists like Euler and d'Alembert believed that mechanics were an *a priori* science, that needed no recourse to experience to be established. Otherwise other scientists, such as Daniel Bernoulli for instance, supported the idea of mechanics as an empiric discipline.

The possibility of this long dispute depended on the essence of mechanics. Even those who declared mechanics as an experimental science referred to the experience of everyday life and not to complicated experiments carried out in the laboratory, as was the case, even in the 18th century, for electricity and magnetism or thermology. For instance the law of lever was based on the observation that the more the heavier a body, the more it acts on an arm; the law of motion was based on the constant effect of a cause and that heavy bodies felt downward. The principle of inertia was stated based on simple thought experiments and the principle of action and reaction seemed to be self-evident. Of course no one denied the role of experience when a mechanical theory was be applied. For instance, to evaluate the law of falling of a heavy body, it is not enough to say that its speed varies proportionally to time. It is also necessary to know the value of the constant of proportionality, which is the acceleration of gravity. This can be known only by means of accurate experimentations in a laboratory. Equally the evaluation of masses, forces, times, distances may require sophisticated experiments and instruments.

Up to now no shared formulations of a completely axiomatized formulation of classical mechanics have been provided, even though some attempts have been made [4–7]. Commonly incomplete axiomatized versions circulate in which in some cases the difference between axioms and theorems is not completely clear. For example, Newton's second law of motion, *f* = *ma*, can be treated as a principle, a relationship between force *f* and acceleration *a*, in which both members of the relationship are considered as primitive terms of the theory, or it can be treated as a simple definition of force. This matter of fact strongly suggests that mechanics needs some more studies by physicists in the attempt to improve its comprehension.

The difference between the two kinds of axiomatic formulations, the complete and the incomplete one, is not only of a quantitative but also epistemological nature. A complete axiomatic theory is by definition a closed system, straightforwardly formalizable with the sufficiently sophisticated language of mathematical logic so that it can contain most of mathematics. It has by definition a hypothetical deductive nature; that is, no truth value is given to the axioms of the theory. What matters is that the theory is coherent, even if this coherence as argued by Gödel's theorems cannot be proved but only assumed in the absence of evident inconsistencies. Primitive terms, definitions, axioms, have no definite meaning in themselves, even if they are given names used in everyday life, such as force, mass, inertial system. These quantities take on values through some rules of correspondence, connecting the real world with the theory, which include all the conceptual difficulties of the physical theory that are avoided in the formalized theory. If the theory and the correspondence rules provide results corresponding to the empirical measurements, the theory is said to be validated (see the next sections).

In incomplete axiomatized theories, one starts from some axioms, generally treated as empirical laws and therefore considered true as such. From these axioms one derives conclusions that are true in themselves because they are deduced by true axioms with the indubitable laws of logic. If it were to be verified that the conclusions were not empirically true, the blame would not be attributed to the theory but to those who applied it, not having been able to take into account the precise manner of the different parameters involved. The incomplete axiomatic approach of contemporary treatments is from a logical point of view perfectly equivalent to the approach of Hellenistic Greece scholars, of Archimedes in particular, as will be clear from the next section. From certain points of view, therefore, it can be said that the essence of mechanics has remained unchanged over a period of more than two thousand years.

There are various formulations of classical mechanics. Perhaps the most important distinction is between a vector formulation that emphasizes the notion of force and refers to Newton and Euler and an analytic formulation that emphasizes the related notions of work and energy and refers to Leibniz and Lagrange. Up to now the equivalence of the two formulations has not been proven in a shared way, even if most scholars are convinced of it, or if they are not convinced they do not consider the fact particularly interesting. It must be said that many textbooks, even of a high level, try to prove this equivalence, unfortunately with some logical leaps. The few supporters of the non-equivalence between analytic and vector theories point out that within the analytic formulation, work and energy are concepts that can be easily derived from thermodynamics, which is not true for the vector formulation.

#### **2. Ancient Mechanics**

In ancient Greece mechanics was the name of the science dealing primarily with the study of equipments or machines (in Greek *μηκανη*), to transport and lift weights. The search for equilibrium was not of practical interest and mechanics, at least at the beginning, did not care for it. From this point of view mechanics was different from modern statics which is instead seen as the science of equilibrium.

Mechanics with astronomy, music and optics was considered as a mixed mathematical discipline (Renaissance nomenclature), because it dealt with both mathematics and physics. Astronomy which studied phenomena of which only a limited direct experience could be shown, was a hypothetical deductive theory (modern terminology). That is a theory based on a principle assumed as hypothetical—for instance the diurnal rotation of the fixed stars sphere—that is not true for sure. The theory, but not the principle, was validated by its adherence to empirical observations. Different theories based on a different principle could save the same phenomenon. Mechanics, with music and optics, was instead based on indubitable axioms derived either from natural philosophy, or metaphysics or experience.

Certainly there have been very ancient reflections about mechanics; hardly any trace remains of them, however. An ancient mathematician credited with some theorizations was Archytas of Tarentum (fl. 400 BC) but the first known writing in the West is the *Mechanica problemata* attributed to Aristotle [8,9]. The attribution is doubtful, sometimes accepted, sometimes rejected by historians. Certainly a modern reader is surprised to see how a philosopher not particularly interested in mathematics, Aristotle, had taken an interest in mechanics.

The style of the *Mechanica problemata* is not the sober one of mathematicians and the fact that some scholars have seen in it the demonstration of the law of the lever while others had not, testifies to the ambiguity of the text. The first proofs of the law of the lever truly accepted by ancient mathematicians were those of Euclid and Archimedes. That of the latter is still considered by many scholars as the most convincing even if some controversial points remain, as happens in any proof of the 'axioms' of a science.

Euclid is credited as the author of a text known in the West as *The book of the balance* ([10], pp. 24–30); Archimedes was the author of the *De planorum aequlibriis* [11]. Both the authors developed a theory of a deductive kind based on some simple (empirical?) axioms. Euclid, or better the pseudo-Euclid, assumed:

Archimedes on his side assumed:

All the rest of the theory was presented by Euclid and Archimedes as a pure deduction. From the law of the lever, the Hellenistic mathematicians derived the laws of all the machines for lifting weights, or rather of their fundamental components: the pulley, wedge, winch, called simple machines, with the exception of the inclined plane. The solution of the problem posed by this last machine had to wait for the treatises of the Arabs in which a convincing demonstration of the law of the lever was resumed closer to that suggested by the *Mechanica* than by the *De planorum aequlibriis*, and lent itself naturally to the demonstration of the law of the inclined plan. Jordanus de Nemore in the 13th century was the first to exhibit a correct law for the inclined plane. Although not all historians agree, he is credited to have applied the rule: what lifts a weight *np* to a height *h* lifts a weight *p* to a height *nh*, which is a form of the principle of virtual works [12].

The ancient mechanics that in the Middle Ages was called the *Science of weights* remained substantially unchanged until the beginning of the 17th century. It was the discipline developed by mathematicians; therefore, it was of a quantitative nature that studies the equilibrium and displacement of heavy bodies.

#### **3. Early Classical Mechanics**

With the 17th century, a new perspective opened up, which at the beginning was seen as completely separate from that offered by ancient mechanics. Mathematicians began to be interested in the motion of bodies not only in that occurring with the use of lifting machines but also in the more general case such as the natural motion of falling bodies and the violent motion of an object thrown from a slingshot, whose study since then was carried out by philosophers like Plato, Aristotle in an almost exclusively descriptive way.

The novelty was made possible because mathematicians began to conceive of time as a physical magnitude that could be measured with precision; conversely, one could probably say that the need to treat time as a physical magnitude derived from the need to study the motion of bodies. The main protagonist of this development was Galileo Galilei, who in the 17th century obtained the temporal law of the free fall of a heavy body according to which the space passed downward is in the square ratio of the time.

This law seemed too complex to Galileo to be taken as a principle and he tried to deduce it from a simpler principle. He obtained it by verifying that the law of falling bodies could be derived by assuming the speed of the body increasing linearly with time. Galileo's procedure was quite elaborate—a primordial use of infinitesimal calculus indeed—and it is not completely clear, at least to me, whether it should be considered as a hypothetical deductive approach, for which the law of proportionality of speed and time is tentatively fixed and, if the passed space derived from it is in accordance with the square ratio of the time, the hypothetically assumed law is valid, or if it followed the analytical method of mathematicians, which derived one or more axioms from a theorem with purely deductive procedures. A modern mathematician would opt for this second solution because the two laws, speed proportional to time and space proportional to the square ratio of time, are obtained from each other with derivation/integration operations.

No less important and perhaps more basic was the formulation of the law of inertia and the principle of relativity (modern terminology) strictly related to it—based no longer on metaphysical concepts as it had been in the Greek and medieval impetus theory but on thought and real experiments. Galileo's results were motivated by the defense of the heliocentric system and would not have been possible without the publication by Copernicus of the *De revolutionibus orbium coelestium* of 1543 [13].

A very clear exposition of Galileo's principle of inertia is that of the *Dialogo sopra i due massimi sistemi* of 1632, where its experimental—though idealized—nature is shown. In the following exchange, Salviati-Galileo made Simplicius to say that the motion of a body on a horizontal plane, once it has received an initial impetus, would move forever, on the condition that there are not impediments; the motion should occurs without acceleration or delay; it would thus be uniform.

#### SIMP. *I cannot tell how to discover any cause of acceleration, or retardation, there being no declivity or acclivity.* [emphasis added]

SALV. Well: but if there be no cause of retardation, much less ought there to be any cause of rest. How long therefore would you have the moveable to move? SIMP. *As long as that superficies, neither inclined nor declined shall last* [emphasis added]. SALV. Therefore if such a space were interminate, the motion upon the same would likewise have no termination, that is, would be perpetual. SIMP. I think so, if so be the moveable be of a matter durable.

SALV. That hath been already supposed, when it was said, that all external and accidental impediments were removed, and the brittleness of the moveable in this our case, is one of those impediments accidental ([14], p. 173. English translation in [15]).

In the *Discorsi e dimostrazioni matematiche sopra due nuove scienze* of 1638, Galileo added that, in truth, even though all the above mentioned resistances were eliminated, a further resistance would arise to contrast the motion, due to the heaviness of the body. Indeed the horizontal plane is supposed to be represented by a straight line and each point on this line equally distant from the center; this is not the case, for as one starts from the middle of the line and goes toward either end, he departs farther and farther from the center of the earth and is therefore constantly going uphill. Whence it follows that the motion cannot remain uniform through any distance whatever, but must continually diminish ([16], p. 274).

This explanation of Galileo has led many historians to affirm that Galileo did not have a clear idea of the principle of inertia and that his inertia was in fact just a circular inertia ([17,18], p. 352). It seems clear to me that in the affirmation that if resisting forces did not act on a moving body it would continue to move indefinitely, was the purest expression of the principle of inertia, even though it must be said that it is always difficult to establish a correlation between a modern concept and its older formulation. That there are in fact no bodies in nature that can move without resistance, which is another matter. Compared with Newton's formulation, Galileo's differs in one, perhaps non-trivial, way: Galileo did not admit the possibility that there were bodies on which resistant forces did not act, for example weightless bodies; Newton did. Galileo's was a terrestrial view, Newton's a cosmological one. In any case, when Galileo studied the motion of projectiles, he considered a horizontal inertia, without specifying whether he ignored the change in the direction of gravity when moving in a limited space because in this case the direction varies little, or he actually ignored the effect of gravity, thus separating the effect of weight from that of mass.

Cavalieri investigated the motion of projectiles in his *Lo specchio ustorio, overo trattato delle settioni coniche* of 1632 [19]. In this text he had paused to consider the nature of the motions to be combined.

Moreover I say, that considering that the motion [of the body] is a straight line toward any part, if it had not other motive virtue that would push it towards another direction, [the body] should go in the place indicated by the projector along a straight line, because of the virtue impressed to it was in the straight line, from which direction it is not reasonable that the mobile deviates, as long as there is no other motive virtue that deflects it, and that when between the two terminal points there is not impediment. ([19], p. 154. My translation).

This is an expression of the principle of inertia, closer to Newton than to Galileo. Cavalieri said clearly, in a few lines, that a projectile would move in a straight line when not impeded by something (gravity for instance). The main difference compared to the sentence of Galileo is its explicitness.

Torricelli studied the motion of projectiles in Book II of the *De motu gravium*, written before 1644. For him the composition of motion became just a matter of mathematics, the composition of two motions highly idealized: a downward uniformly accelerated motion and a uniform rectilinear motion ([20], *De motu proiectorum*, p. 156).

The evolution of the principle of inertia due to Cavalieri and Torricelli is an example of the growth of scientific knowledge. A scientist (Galileo) developed a form of the principle of inertia, which he did not elaborate completely. His internal troubles transpired in his work, signaled by some uncertainties in its use. Scientists after him (Cavalieri and Torricelli) who had read his writings were not fully aware of the troubles and applied with no doubt the theory to any possible case (the angled launch of a bullet for instance). The process is similar to what Kuhn refers as gestalt switch [21].

Cavalieri and Torricelli moved ahead of Galileo because their mathematics had moved ahead too. Cavalieri and Torricelli, more inclined to mathematics, did make use of the actual infinite in mathematics; Galileo did not. For them the mathematical physical theory (broad meaning) had its reality, maintaining contact with the external world. For this reason the quasi-rectilinear but still slightly curved trajectories of bullets, moving over the earth sphere, can actually be assumed to be rectilinear and of infinite length, identifying the approximation with reality. However, Cavalieri and Torricelli were more cautious than Newton; for instance, they hardly used the quantifiers *for all* and *for ever*, as Newton did.

The concept of the principle of inertia was substantially revolutionary. It overturned completely the traditional approach to motion according to which it was caused by a force, either internal or external. From now on the force acting on a body is responsible for the change of velocity, and not displacement.

After Galilei, the attention of scientists who studied motion focused on the phenomenon of collision, with important contributions by Descartes and Huygens (with Wren and Wallis). The goal of these scholars, classifiable as mechanistic, was to provide general laws of motion that could explain all phenomena. This project of mechanical philosophy was too ambitious and was essentially aborted. However, new concepts arise from these studies especially thanks to Descartes and Huygens that will prove fundamental at the beginning of the 19th century; those of work and kinetic and potential energy.

Descartes contributed to the development of mechanics mainly as a promoter of a mechanistic philosophy (with Gassendi and Hobbes), simpler than the older philosophy which allowed many people to devote themselves to a quantitative study of nature. He contributed to spread the concept of energy/work in statics (the product of weight and vertical displacement) and in dynamics (the product *m*|*v*| of mass and speed), but mainly he was the supporter of a rationalistic view of mechanics, based on evident and clear notions, with no recourse to experience. Huygens played a different but fundamental role, that for the sake of space is not commented upon here. For instance he, with Leibniz (see below), contributed to the development of the ideas of work and energy and formulated

in nuce the concept of conservation of mechanical energy, developed then by Johann and Daniel Bernoulli [22,23].

John Wallis published in 1669–1671 the text *Mechanica sive de motu, tractatus geometricus* [24], where the term mechanics, until then restricted to statics, was also applied to the new science of motion. Aside from a greater wealth of content, mechanics took on a different function. It was no longer a discipline destined to solve a particular problem, of mainly technological interest, but became a new philosophy of nature with which the mathematical philosophers (that is mathematicians) and not the generalist philosophers intended to explain nature.

#### **4. Classical Mechanics**

The transformation of mechanics into a new philosophy of nature was particularly evident with Newton's *Philosophiae naturalis principia mathematica* of 1687 [25], which took up the work of Archimedes and Galileo by unifying statics and dynamics with few very simple axioms: the three famous laws of motion, considered certain and inductively derivable from experience. The approach was exactly that of the mixed mathematics of the ancient Greece, nothing substantially new after two thousand years. However, the mathematics was more refined, using modern categories it can be said that the *Principia* was a treatise of differential geometry.

Below are the first two laws of motion as reported in the 1726 edition.

Law I.

*Everybody perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed*. Law II.

*A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed*. ([26], p. 13. English translation in [27]).

Notice that Newton's second law is quite different from what is known today under this name. The modern relation, force equals mass by acceleration, is due mainly to Euler (see below).

Modern philosophers of science have clearly shown that Newton's laws/axioms cannot be derived from experimental data by induction, not least because many of them do not believe that induction exists. However, Newton and his followers were convinced of the empirical derivation of the laws and that all natural phenomena could be derived from them with certainty.

The comment of Ernest Nagel on the nature of Law I is worth reporting. According to him, it is clear that the first law of motion, taken in isolation, is seriously incomplete as a statement intended to have empirical content. To say that a body will persevere in its state of rest or uniform motion in a straight line, unless compelled to change its state by forces impressed on it, is to say nothing definite, if nothing further is stated as to:

Assumed that the first point will in some way be solved, continued Nagel, what is the criterion for asserting that a body is under the action of no forces? What defines the perseverance of the body in its state of rest or uniform motion in a straight line? However, in such a case, the law is a concealed definition, a convention which specifies the conditions under which one will say that there are no impressed forces acting on a body. Moreover, in addition to assuming a definite spatial frame of reference, in its usual formulation the law takes for granted a definite system of chronometry. If, then, some method not involving the explicit or tacit use of the law were available for identifying the absence of forces, the law could be construed as an implicit definition either of uniform motion or equal time intervals ([3], pp. 175–178).

The exposition of the three laws was followed by some examples that made them plausible even though there was no reference to their empirical nature. Only in the scholium did Newton say that his laws were accepted by mathematicians and confirmed by numerous experimental results. They are attributed to Galileo

Scholium The principles I have set forth are accepted by mathematicians and confirmed by experiments of many kinds. By means of the first two laws and the first two corollaries [the law of composition of forces] Galileo found that the descent of heavy bodies is in the squared ratio of the time and that the motion of projectiles occurs in a parabola, as experiment confirms, except insofar as these motions are somewhat retarded by the resistance of the air ([26], p. 21. English translation in [27]).

This attribution is generally considered too generous on Newton's part. In fact, Galileo had not possessed the concept of impressed force and, if he had read the *Principia*, most likely he would not have understood them. However, it should be noted that, in the 18th century, mathematicians who took up the theory, such as Varignon and Euler for instance, referred to Newton's laws as the laws of motion of Galileo.

Newton considered only the third law as original of him:

Law III.

*To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction* ([26], p. 14. English translation in [27]).

In addition to its explanations, thought experiments were reported; the simplest of them is described below.

I demonstrate the third law of motion for attractions briefly as follows. Suppose that between any two bodies A and B that attract each other any obstacle is interposed so as to impede their coming together. If one body A is more attracted toward the other body B than that other body B is attracted toward the first body A, then the obstacle will be more strongly pressed by body A than by body B and accordingly will not remain in equilibrium. The stronger pressure will prevail and will make the system of the two bodies and the obstacle move straight forward in the direction from A toward B and, in empty space, go on indefinitely with a motion that is always accelerated, which is absurd and contrary to the first law of motion ([26], p. 25. English translation in [27]).

However, the results of a real devised experiment were also reported.

I have tested this with a lodestone and iron. If these are placed in separate vessels that touch each other and float side by side in still water, neither one will drive the other forward, but because of the equality of the attraction in both directions they will sustain their mutual endeavors toward each other, and at last, having attained equilibrium, they will be at rest ([26], p. 25. English translation in [27]).

With his laws, Newton was able to prove old results and to find new ones. The proof of the ellipticity of the orbits of the planets about the sun, as predicted by Kepler, and the formulation of the law of universal gravitations are very interesting.

#### **5. Post Newtonian Mechanics**

Newton's mechanics proved to be incomplete as it was substantially limited to the motion of free mass points. This was particularly clear to the handful of brilliant mathematicians of the 18th century, of which the most known are Leibniz, Johann Bernoulli, Varignon, Hermann, Euler, d'Alembert, Lagrange that were able to fill the gap [28]. In the following, for the sake of space, a short hint to the contributions of Leibniz, Euler and Lagrange only is given.

Leibniz played a role similar to that of Descartes—both were great philosophers and mathematicians. His main contribution to mechanics was a critical discussion about Newton's concepts: force, time and space, the introduction of the term *dynamics* to indicate that

part of mechanics devoted to the study of the motion of bodies, and, mainly, the development of a form of infinitesimal calculus more efficient in mechanics than that proposed by Newton. It is worthy signaling however that Leibniz's name, though his role is well recognized by historians, is hardly found in modern textbooks of mechanics.

Euler is usually considered as Newton's most important heir. His main merit is of having transformed mechanics from a geometric discipline into an analytical one, considered as a purely rational discipline. The incipit of his early treatise *Mechanica sive motus scientia analytice exposita* of 1636 are quite famous:

Thus, I always have the same trouble, when I might chance to glance through Newton's *Principia* or Hermann's *Phoronomia*, that comes about in using these [synthetic methods], that whenever the solutions of problems seem to be sufficiently well understood by me, that yet by making only a small change, I might not be able to solve the new problem using this method. Thus I have endeavored a long time now, to use the old synthetic method to elicit the same propositions that are more readily handled by my own analytical method, and so by working with this latter method I have gained a perceptible increase in my understanding. Then in like manner also, everything regarding the writings about this science that I have pursued, is scattered everywhere, whereas I have set out my own method in a plain and well-ordered manner, and with everything arranged in a suitable order. Being engaged in this business, not only have I fallen upon many questions not to be found in previous tracts, to which I have been happy to provide solutions, but also I have increased our knowledge of the science by providing it with many unusual methods, by which it must be admitted that *both mechanics and analysis are evidently augmented more than just a little* ([29], Preface. English translation by Bruce I).

Notice that the title of Euler's treatise paraphrased that of Wallis, by changing *geometric* with *analytic*.

Apart from the algebraization of mechanics Euler improved much the 'primeval' Newton's development. For instance he contributed to give the second law of motion its modern form [30,31]. Moreover he applied the laws of motion to the case of extended and constrained bodies, fluid included. Euler was an extremely prolific and methodical mathematician. With him, mechanics became the discipline we know. And even today it makes sense to read some of his writings on mechanics, finding interesting topics for current research.

After Euler mechanics had become an algebraic theory; in some parts however geometry was still present, for example in the use of vectors. A concept, that of vector, today possibly introduced without any reference to classical geometry but that in the 18th century could not. Lagrange with his *Méchanique analitique* of 1788 [32] (Since the second edition of 1811 the title of Lagrange's treatise changed to *Mécanique analytique*.) undertook an important and big step toward a full algebraization of mechanics. His treatise of 1788 added little new to the development of mechanics and collected most results obtained by Lagrange himself since the 1760s. However, it was completely new because of its way of presentation and its logical–epistemological conception. Mechanics then became analytical.

The locution analytical mechanics had/has different meanings. Today its more diffuse acceptation is that of a (mathematical) theory based on algebra and Calculus, whose axioms are presented with scalar relations; this definition substantially covers *Méchanique analitique* use. According to Truesdell, analytic mechanics is limited to discrete systems (excluding thus continua, solid and fluid), and this limitation of the term is quite diffuse also [33]. Vectors could be still present, but they were introduced only through their components and mainly there could be many different local frames in the same mechanical assembly. According to the meaning assumed beforehand, that of Euler is not analytical mechanics, even though he himself referred to it as such.

In the preface of his masterpiece, Lagrange stated that his mechanical theory had become a branch of analysis. The axioms of this branch were represented by general formulas, the principle of virtual work and the principle of d'Alembert. They were enough to solve any practical problem, both in statics and dynamics. The axioms were considered to be given, and no serious interest was addressed to their derivation or proof. Accordingly the *Méchanique analitique* should be considered one of the first texts of modern mathematical physics.

In the following a large quotation of the preface of Lagrange's treatise is reported, even though well known in the literature, because of its relevance.

I propose to condense the theory of this science and the method of solving the related problems to general formulas whose simple application produces all the necessary equations for the solution of each problem. I hope that my presentation achieves this purpose and leaves nothing lacking. In addition, this work will have another use. The various axioms presently available will be assembled and presented from a single point of view in order to facilitate the solution of the problems of mechanics. Moreover, it will also show their interdependence and mutual dependence and will permit the evaluation of their validity and scope. I have divided this work into two parts: statics or the theory of equilibrium, and dynamics or the theory of motion. In each part, I treat solid bodies and fluids separately. *No figures will be found in this work* [emphasis added]. The methods I present require neither constructions nor geometrical or mechanical arguments, but solely algebraic operations subject to a regular and uniform procedure. *Those who appreciate mathematical analysis will see with pleasure mechanics becoming a new branch of it* [emphasis added] and hence, will recognize that I have enlarged its domain ([32], Preface. English translation in [34]).

Lagrange's and Euler approaches were different in two main aspects. First, Euler's mechanics had a substantially axiomatic structure, in which concepts were explained before being used. Lagrange's mechanics did not have such an axiomatic structure. It generally used concepts and axioms already accepted by experts of mechanics. Moreover the general formulas Lagrange talked about were not true formulas but rather rules that needed interpretation. Second. Lagrange's mechanics made no reference to natural philosophy to introduce its assumptions, differently from Euler's. 'Axioms' were justified with mathematical reasoning. Where it would be necessary to exit from this ambit, Lagrange was elusory. Instead of considerations drawn from natural philosophy or metaphysics, he referred to a historical account [35]. History gave a justification of the theory and replaced the justification based on natural philosophy. Lagrange thought that mechanics were nothing but the results gained in its history, by Archimedes *in primis*. In his history there was no room for the philosophy of nature.

#### **6. The Crisis of Classical Mechanics**

In the 19th century, mechanics was a mature discipline, or at least it was considered as such by its experts. There were no substantial innovations in the understanding of the relevant phenomena. One of the main 'discoveries' was the Coriolis effect, that is the fact that in a reference system in rotation with respect to the fixed stars, apparent forces orthogonal to the motion are noted. The effort of mathematicians and physicists was concentrated on perfecting the formal aspects of the theory and in the criticism of its axioms.

At the turn of the century, after the publication of Lagrange's *Méchanique anlitique*, there was a heated debate on the logical status of the principle of virtual works, which was at the basis of Lagrange's analytical mechanics. The protagonists of this discussion were the French mathematicians and physicists, including Fourier, Ampère, Coriolis, Laplace and Poinsot ([12], pp. 317–351).The discussion showed how difficult it was to 'prove' in a rigorous way, that is according to the stringent standard of rigor of the 19th century, axioms that appeared nearly self-evident to the common people, as the composition of forces, the law of lever, the principle of virtual works. For this purpose the words by Lazare Carnot at the very end of his *Essai sur les machines en général* of 1782/86 are illuminating.

Sciences are as a beautiful river whose course is easy to follow, when it has acquired a certain regularity; but if one wants to sail to the source one cannot find it anywhere, because it is far and near; it is diffuse somehow in the whole earth surface. The same if one wants to sail to the origin of science, one finds nothing but darkness and vague ideas, vicious circles; and one loses himself in the primitive ideas ([36], p. 107).

In the second half of the century, the analysis of the foundations concerned the whole mechanics; in particular, the Newtonian force was the subject of criticism. The protagonists were mathematicians, physicists and philosophers, such as: Saint-Venant, Kirchhoff, Hertz, Helmholtz, Reech, Mach, Poincaré, Duhem ([37], pp. 417–443).Particularly effective were the contributions of the latter three scholars. Mach in his *Die Mechanik in ihrer Entwicklung historisch-kritisch dargestellt* of 1883 scrutinized all the ideas of Newtonian mechanics, starting from the ideas of absolute space, force and mass [38]. Poincaré in the *Science et hypotèses* of 1902, commented upon the epistemology of the axioms of mechanics. According to him, mechanics is neither fully empiric nor fully a priory. For instance let us consider the principle of inertia:

*The Principle of Inertia—A body under the action of no force can only move uniformly in a straight line*. Is this a truth imposed on the mind à priori? If this be so, how is it that the Greeks ignored it? How could they have believed that motion ceases with the cause of motion? or, again, that every body, if there is nothing to prevent it, will move in a circle, the noblest of all forms of motion. [. . . ] Is, then, the principle of inertia, which is not an à priori truth, an experimental fact? Have there ever been experiments on bodies acted on by no forces? and, if so, how did we know that no forces were acting? ([39], pp. 112–113).

Beside the attempts to clarify the axioms, there was also the problem of the nature of mechanics in itself. Until then mechanics had been considered as the mother of the other disciplines; even electricity could be framed into a mechanical context. However, at the end of the 19th century it was clear that this was not a tenable position. Thermodynamics dealt with concepts hardly reducible to mechanics, and a similar consideration held good for electromagnetism.

Duhem championed 'energetics', holding generalized thermodynamics as foundational for physical theories; that is, thinking that all of chemistry and physics, including mechanics, electricity, and magnetism, should be derivable from first axioms of thermodynamics. His *Traité d'énergétique ou de thermodynamique générale* of 1911 [40] represented one of the first attempts to establish a physical theory on modern axiomatic basis. The author believed that the axioms of a physical theory did not require any justification apart from the assessment of their internal consistency. He considered, however, that the axioms could not be chosen by chance, but that they should benefit from formulations of past similar axioms; in this way the history of science becomes an integral part of science ([40], pp. 183–246).

The attempt by Duhem to save classical mechanics was frustrated by the emergence of serious paradoxes in the electromagnetic theory, still framed into classical mechanics for some aspects, evidenced at the end of the 19th century. They pushed to consider classical mechanics as a simple approximation of a true mechanics, the relativistic one. From then on, physicists ceased to be interested in classical mechanics, considering it scarcely interesting, or worse a dead discipline. However, this is another story.

Together with the intense work on foundation there was also an effort to clean up the theory. Hamilton and Jacobi moved in this direction, giving analytical mechanics its modern structure, based on a functional, well-known as the Hamiltonian, which allowed the treatment of statics and dynamics, rigid and continuous deformable solid bodies and fluids in a unitary way. With them the formal aspects, the elegance, the clarification of the spheres of validity of the theorems had reached the apex [41,42]. The rearrangement of mechanics also made the solutions of some problems easier and also suggested the solution of others; it is so restrictive to speak of formal perfecting only. Mechanics was now ready to be framed into the broad field of modern mathematical physics.

The expression mathematical physics had substantially two different meanings. On the one hand, it simply indicated modern physics, strongly rooted in mathematics; in this sense, the great mathematical physicists were Galileo, Kepler, Newton, Euler, etc. On the other hand, it meant that branch of science born in the 19th century which afforded some problems regulated by partial differential equations, e.g., the propagation of heat, the theory of potential, the theory of elasticity; in this sense, great mathematical physicists were Fourier, Lamé, Gauss, Piola, Beltrami, etc. Today the expression indicates an academic discipline cultivated by mathematicians which has as its basis some physical axioms whose merit was not questioned.

Examining epistemological aspects of mathematical physics gives the opportunity to stipulate an appropriate conventional meaning. As a first step, it is useful to explain the essence of physical theories. They are made up of three parts:

When the correspondence rules are missing, the physical theory turns to a mathematical physical theory as its axioms speaks about the physical world and not only of pure mathematical entities. The boundary between pure mathematics and mathematical physics is not exactly specified and various positions could be assumed. Clifford Ambrose Truesdell, for instance, does not see any difference and claims mathematical physics to be simply a branch of pure mathematics. He actually refers to his field of competence, rational mechanics, but his considerations apply to any mathematical physical theory:

Is rational mechanics part of applied mathematics? Most certainly not. While in some cases known mathematical techniques can be used to solve new problems in rational mechanics, in other cases new mathematics must be invented. It would be misleading to claim that each achievement in rational mechanics has brought new light in mathematics as a whole as to claim the opposite, that rational mechanics is a mere reflection from known parts of pure mathematics ([43], p. 337).

Other people sustain that a theory can be called mathematical only if it is addressed to purely mathematical objects.

Quite interesting in the development of modern mathematical physics was the contribution by Carl Neumann [44]. He recognized that the theorems of mathematical physics should be confirmed by experimental data ([45], p. 130), but as a mathematician (or mathematical physicists) he found that he should focus his attention on the mathematical description of axioms and on the improvement of mathematical tools ([45], p. 127). In his time there was a substantially Aristotelian–Euclidean vision of mathematics, according to which a mathematical theory should be based on indubitable axioms so that the resulting theorems could not be disputable. According to Neumann, a mathematical physical theory was different from a pure mathematical theory because in it some 'axioms' might be hypothetical, and consequently its theorems could not necessarily be true. This possibility made mathematical physical theories very interesting for a mathematician because of their greater potential of invention (hypothetical deductive theories). These considerations by Neumann, shared by Mathieu ([46], pp. 110–111), contributed to the development of the modern concept of a mathematical theory based on arbitrary premises.

The approach by Neumann and Mathieu, not very different indeed from that of Duhem, paralleled a similar process emerging in mathematics, that will continue in the 20th century. After moving away from the Euclidean rigor for a long time, mathematics had returned to it. Therefore, demonstrations of many properties that were previously considered evident were required; indeed, this was in many cases the only way to discover the limits of their validity. The concepts of function, continuity, limit, infinity had revealed the need for a more precise determination; the negative number and the irrational number, which have long since become part of mathematics, have had to be subjected to a more precise examination of their justification. Thus, there was the tendency to give rigorous proofs everywhere ([47], p. 1).

In geometry, the attempts to prove the postulate of parallels (fifth Euclid's postulate) from the other postulates proved to the contrary its independence from them. This suggested to Lobachevsky and Bolyai the idea to replace it with a different assertion on the parallels: they could meet. This gave rise to the so called non Euclidean geometry. This form of geometry furnished results in contrast with many points with the Euclidean geometry, even though the forms of reasoning were similar and suggested the possibility of creating a mathematical theory in which the assumption of the truth of axioms was not required [48,49].

The new theories of mathematics or mathematical physics were hypothetical deductive. It is true that since the ancient Greeks there had been theories based on hypotheses and at large to be classified as hypothetical deductive, but there the hypothesis could not be fully arbitrary; they were suggested in some way by observations or deductions and were considered to assume a value of truth if the theory gave true results. Modern mathematical physics has released this requirement.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

**Entry Link on the Encyclopedia Platform:** https://encyclopedia.pub/12056.

#### **References**

## *Entry* **Metal Nanoparticles as Free-Floating Electrodes**

**Johann Michael Köhler 1,2,\*, Jonas Jakobus Kluitmann 1,2 and Peter Mike Günther 1,2**

**Definition:** Colloidal metal nanoparticles in an electrolyte environment are not only electrically charged but also electrochemically active objects. They have the typical character of metal electrodes with ongoing charge transfer processes on the metal/liquid interface. This picture is valid for the equilibrium state and also during the formation, growth, aggregation or dissolution of nanoparticles. This behavior can be understood in analogy to macroscopic mixed-electrode systems with a freefloating potential, which is determined by the competition between anodic and cathodic partial processes. In contrast to macroscopic electrodes, the small size of nanoparticles is responsible for significant effects of low numbers of elementary charges and for self-polarization effects as they are known from molecular systems, for example. The electrical properties of nanoparticles can be estimated by basic electrochemical equations. Reconsidering these fundamentals, the assembly behavior, the formation of nonspherical assemblies of nanoparticles and the growth and the corrosion behavior of metal nanoparticles, as well as the formation of core/shell particles, branched structures and particle networks, can be understood. The consequences of electrochemical behavior, charging and self-polarization for particle growth, shape formation and particle/particle interaction are discussed.

**Keywords:** nanoparticles; colloidal solutions; electrical charging; self-polarization; mixed-electrode; particle growth; particle interaction

#### **1. Introduction**

Metal nanoparticles have attracted a lot of scientific interest in recent years. The most important practical motivations come from their interesting electronic and optical properties [1–4], their applicability for nanolabeling [5,6] and sensing [7–10] and their catalytic properties [11,12]. In addition, they are fascinating targets for basic research for understanding the nature of nano-objects and the interaction with biomolecules and living cells [13–16] and for designing new materials, as well as micro- and nanosized tools [17,18].

An important field of nanoparticle generation and handling is liquid-phase synthesis resulting in colloidal solutions of metal nanoparticles [19,20]. The existence of nanoparticles in the form of a thermodynamically stable dispersion in a liquid was firstly explained by Michael Faraday about one and a half centuries ago. Already at this time, the importance of the electrical properties of colloidal particles was recognized.

In addition to the presence of an electrical charge on metal nanoparticles, the exchange of charges and the interaction with ions are important for the generation and behavior of metal nanoparticles. Charge transfer processes can include the release of ions from the metal or the conversion of adsorbed metal cations into metal atoms. These processes, as well as oxidation and reduction reactions of other species, can be regarded as local electrochemical processes [21]. In the following, important examples of such processes will be regarded and discussed from the point of view of the electrode character of colloidal metal nanoparticles.

**Citation:** Köhler, J.M.; Kluitmann, J.J.; Günther, P.M. Metal Nanoparticles as Free-Floating Electrodes. *Encyclopedia* **2021**, *1*, 551–565. https://doi.org/10.3390/ encyclopedia1030046

Academic Editors: Raffaele Barretta, Ramesh Agarwal, Krzysztof Kamil Zur and Giuseppe Ruta ˙

Received: 10 June 2021 Accepted: 28 June 2021 Published: 12 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **2. Formation of Colloidal Metal Nanoparticles by Reduction of Solution Precursors**

In the past, the general model of LaMer [22] was frequently used for explaining the formation of nanoparticles in liquid media. This model postulates a two-step process, whereby in a first step, single atoms or molecules aggregate to form nuclei, and, in a second step, these nuclei grow by the attachment of further atoms or molecules, as long as the concentration of particle-forming species is above the critical saturation concentration. The concept of LaMer is based on the assumption that the critical concentration threshold for nucleation is significantly above the saturation limit. Thus, nucleation proceeds only at very high concentrations, which means in a short start phase, whereas particle growth can occur over a longer time until the exhaustion of particle-forming species above the saturation limit.

The LaMer model was not developed for the specific conditions for the reactive formation of metal nanoparticles by the reduction of a precursor. For the example of colloidal gold, it was shown by Polte et al. that the formation of spherical metal nanoparticles was caused by an aggregation and growth mechanism, which could be proved by in situ-X-ray scattering measurements [23]. The proposed mechanism of Polte et al. is very conclusive and also gives a convincing explanation for the polycrystalline character of spherical gold nanoparticles [24].

The start of reductive metal nanoparticle formation is a molecular process. The primarily formed products are clusters, and their nature is closer to small molecules than to a classical metallic solid. The simplest description is given by the following reaction equation:

$$\text{M}^{\text{n}+} + \text{n} \,\text{e}^- \to \text{M}\_0 \tag{1}$$

The electrons are supplied from the oxidation of the reducing agent (RA):

$$\text{RA} \rightarrow \text{RA}^{\text{m}+} + \text{m} \,\text{e}^- \tag{2}$$

Both processes are coupled and can be written formally as a redox reaction:

$$\rm{M}^{n+} + \rm{n/m}\rm{RA} \rightarrow \rm{M}\_0 + \rm{n/m}\rm{RA}^{m+} \tag{3}$$

Single free metal atoms are usually not stable in a liquid environment. Thus, Equation (1) should be written better as forward and backward processes or as equilibrium:

$$\text{M}^{\text{n}+} + \text{n} \text{ e}^- \rightarrow \text{M}\_0 \rightarrow \text{M}^{\text{n}+} + \text{n} \text{ e}^- \tag{4}$$

Irreversibility is achieved if a stable cluster is formed consisting of a small number x of metal atoms; for example, z = 4:

$$\text{z}\,\text{M}^{n+} + \text{n} \times \text{z}\,\text{e}^- \to \text{(M}\_\text{z}) \tag{5}$$

The above-written equations are a very formal description of the processes during the reaction of a metal precursor species in a polar solvent. It is to be assumed that the formation of clusters does not result in a neutral particle without any ligands. Instead, it has to be expected that the clusters are electrically charged and connected with ligands by analogy with the coordinatively bound or solvated metal ions of the precursor.

Metal ions are solvated or connected with ligands L in coordination compounds or complex ions. The primary formed cluster (Equation (5)) is also connected with ligands X, which can differ from the original ligands of the metal ion complex. The following equations give an example of how the formation of a stable cluster could proceed, whereby the electrons come from direct reduction by interacting molecules of the reducing agent, whereby noncharged ligands L and X and a single positive charge per metal ion are assumed, for simplification:

$$\text{a } 2[\text{ML}\_{\text{n}}]^{+} + \text{e}^{-} \rightarrow [\text{M}\_{2}\text{L}\_{(2\text{n-a})}]^{+} + \text{a L (reduction in solution)}\tag{6}$$

$$\left[\mathrm{M\_2L\_{\left(2n\cdot a\right)}}\right] + \mathrm{b} \times \rightarrow \left[\mathrm{M\_2L\_{\left(2n\cdot a\cdot b\right)}}\mathbb{X}\_b\right] + \mathrm{b}\text{ L}\text{ (partial ligand exchange)}\tag{7}$$

or

$$\begin{array}{c} \text{[M $\_2$ L\_{(2n\text{-a})}] + (2n\text{-a}) \times \rightarrow [M $\_2$ X}\_{\text{(2n\text{-a})}}]^{+} + (2n\text{-a}) \text{ L} \\ \text{(complete ligand exchange)} \end{array} \tag{8}$$

The following steps are written for the example without ligand (L by X) exchange:

$$\text{[M}\_2\text{L}\_{\text{(2n-a)}}\text{]}^+ + \text{[ML}\_\text{n}\text{]}^+ \rightarrow \text{[M}\_3\text{L}\_{\text{(3n-a-b)}}\text{]}^{2+} + \text{b L (metal ion adsorption)}\tag{9}$$

$$\left[\mathrm{M\_3L\_{\left(\text{3n-a-b}\right)}I}\right]^{2+} + \text{e}^- \rightarrow \left[\mathrm{M\_3L\_{\left(\text{3n-a-b}\right)}}\right]^{+} \text{ (reduction step)}\tag{10}$$

$$\left[\mathrm{M\_3L\_{\left(2n\text{-}a\text{-}b\right)}}\right]^++\left[\mathrm{ML\_{\left(2n\text{-}b\text{-}c\right)}}\right]^+ \rightarrow \left[\mathrm{M\_4L\_{\left(4n\text{-}a\text{-}b\text{-}c\right)}}\right]^{2+}+\mathrm{c L}\text{ (metal ion adsorption)}\tag{11}$$

$$\left[\mathrm{M}\_{4}\mathrm{L}\_{\mathrm{(4n\cdot a\cdot b\cdot c\cdot)}}\right]^{2+} + \mathrm{e}^{-} \rightarrow \left[\mathrm{M}\_{4}\mathrm{L}\_{\mathrm{(4n\cdot a\cdot b\cdot c\cdot)}}\right]^{+} \text{ (reduction step)}\tag{12}$$

and so on.

The equations above illustrate the alternation of steps of metal ion or complex, respectively, adsorption and reduction. It is not necessary to assume the formation of metal(0) atoms that have to aggregate spontaneously to form a nucleus that can subsequently grow. It is much more probable that there is a direct way from the formation of a coordinated cluster of very few atoms to a continuous growth process. This growth process is always composed of the reduction of adsorbed metal ions into metal(0) and the electron supplying oxidation of the reducing agent. From this point of view, there is no classical nucleation process, as postulated by the LaMer model. Instead of this, the reductive formation of metal nanoparticles can be described by the coupling of reduction and oxidation from the very beginning of the first metal–metal interaction.

After completion of the particle growth, the particles in a stable colloidal solution are in a thermodynamic equilibrium state. This means that cathodic and anodic elementary processes are further running but with equal rates in both directions. As a result, the surfaces of metal nanoparticles experience a fluctuation in the adsorption and release of metal ions, as well as in the oxidation and reduction of residual active redox species in solution. Each elongation of the thermodynamic equilibrium state—for example by changing temperature, pH, metal precursor, ligand or reducing agent concentrations induces shifts in the intensity of the electrochemical processes on the particle surfaces and, therefore, changes in the size or shape of particles.

#### **3. Forming Colloidal Metal Nanoparticles as Mixed-Electrode Objects**

The reaction system of reductive metal nanoparticle formation consists of two different object types: the ion-containing solution with the character of an electrolyte and the metallic clusters and nanoparticles. Bulk metals and metal films and metal clusters are marked by high internal charge carrier mobility. This is the essential feature of the metallic state. Together with the electrolyte, an electrode is formed.

Each electrode is marked by its electrode potential. This potential can be measured, in principle, by the voltage between the electrode and a reference electrode. However, such a direct measurement is impossible in the case of nanoparticles.

The electrochemical potential of a metal particle during its formation results at least from the metal deposition and the oxidation of the reducing agent (RA). The metal reduction represents the cathodic partial process as formulated by Equation (1). The anodic partial process can be written as:

$$\text{RA} \rightarrow \text{RA}^+ + \text{e}^- \tag{13}$$

Due to the, at least, two electrode processes, the particle is a mixed-electrode object (Figure 1). The absence of any outside current source means that this potential is only controlled by the ongoing electrode processes. Metal nanoparticles have a free-floating potential.

**Figure 1.** Scheme of the formation of an electrochemical mixed potential of a growing nanoparticle by the coupling of a cathodic and an anodic partial process.

The potential of clusters and nanoparticles of z atoms Mz is modified, in addition, by the adsorption of ions, ligands and other reaction partners, for example by following ligand exchange reaction:

$$(\mathbf{M\_z L\_n}) + \mathbf{X}^- \to (\mathbf{M\_z L\_n} \text{-1}\mathbf{X})^- + \mathbf{L} \tag{14}$$

The small size of nanoparticles causes a high sensitivity of the particle potential against the exchange of single charges. The clusters and the growing nanoparticles behave as electron confinements with a fluctuating potential. Each adsorption or desorption of an ion and each uptake or release of an electron changes the electrochemical potential. An increase in potential enhances the probability of oxidation of an adsorbed or colliding molecule of the reducing agent. A decrease in potential enhances the probability of metal deposition.

The order of magnitude of potential change by the transfer of one elementary charge qe can be estimated by regarding the particle surface as an electrical capacitor with the capacitance C, an electrode distance in the order of magnitude of the electrochemical double-layer thickness d and the surface area A:

$$\mathbf{C} = \varepsilon\_0 \times \varepsilon\_\mathbf{r} \times \mathbf{A} / \mathbf{d} \tag{15}$$

The stored charge on such a capacitor Q depends on the capacitance and the voltage:

$$\mathbf{Q} = \mathbf{C} \times \mathbf{U} \tag{16}$$

From these equations, the potential U can be estimated:

$$\mathbf{U} = \mathbf{Q} \times \mathbf{d} / (\mathbf{A} \times \boldsymbol{\varepsilon}\_0 \times \boldsymbol{\varepsilon}\_r) \tag{17}$$

For the nearly atomic distances, εr can be approximated for the vacuum (=1), and the expected double-layer thickness is in molecular dimensions (about 0.5 nm). Then, the change of nanoparticle potential UNP by one elementary charge qe can be approximated by:

$$\mathbf{U\_{NP}} \approx \mathbf{q\_{e}} \times 0.5 \text{ nm} / (\mathbf{A} \times \varepsilon\_{0}) \tag{18}$$

Significant shifts in electrochemical potential should be expected for small nanoparticles, which means in the early stage of growth. In this phase, the reacting sites of the particle surface and their stochastic interactions with species from the solution have a large effect on the electrochemical state of a particle (Figure 2). It has to be kept in mind that changes of some tens of millivolts typically have a significant effect on the intensity of electrochemical processes, and changes of several hundred millivolts are related to strong changes in the electrochemical process intensities, often connected with drastic changes in the general chemical behavior. The estimation shows that such strong effects caused by fluctuations of single elementary charges must be expected for small particles with sizes below about 2 to 4 nm.

**Figure 2.** Estimated contribution of a single elementary charge on the potential of a single metal nanoparticle in dependence on the particle size for values of dielectric constant (electrical permittivity) of 1, 2, 4 and 10 and an assumed thickness of the electrochemical double layer of 0.5 nm.

In a later phase, when the particle consists already of about thousands or more atoms, these stochastic effects are less important for the state of the particle, and its electrochemical potential is mainly controlled by the concentration of the electrochemically active components of the solution and the general particle surface state.

The particle potential remains nearly constant in the latest phase of particle growth if the particle is larger and the changes of concentrations of reacting components in the solution become negligible. In the case of constant electrochemical potential, the sum of all partial currents of electrochemical processes is zero. That means that the currents of all cathodic partial processes I are completely compensated by the electrochemical currents of anodic partial processes I+:

$$\|\|\Gamma^{+}\|\| = \|\|\Gamma^{-}\|\|\tag{19}$$

The electrochemical potential of a growing metal nanoparticle increases when the intensity of cathodic partial process (metal deposition) exceeds the intensity of the anodic partial process:

$$\|\|\Gamma^{+}\|\| < \|\Gamma^{-}\|\| \to \text{increasing particle potential} \tag{20}$$

The electrochemical potential of a growing metal nanoparticle decreases when the intensity of the anodic partial process (oxidation of reducing agent) exceeds the intensity of the cathodic partial process:

$$|I^+| > |I^-| \to \text{decreasing particle potential} \tag{21}$$

#### **4. Self-Polarization Effects of Nonspherical Metal Nanoparticles**

Metal nanoparticles are marked by a high mobility of charges. This matches the electron gas model of metallic solids and is valid for charged nanoparticles (or ion-like clusters), too. The electrical excess charges of metal nanoparticles are fluctuating, but the fluctuation of single elementary charges is coupled with the others due to electrostatic repulsion. Therefore, the charges are concentrated at the surface of the particle. Spherical particles are marked by a regular distribution of the excess charges at their surface due to the tendency to maximize the distances between all single charges.

The symmetry in the charge distribution gets lost if the shape symmetry of a sphere is broken. The distribution of charges then depends on the particle shape and two or more charge centers of gravity appear. Some examples of such geometry-dependent charge distributions are shown in Figure 3.

**Figure 3.** Formation of charge centers of gravity in the case of electrically charged nonspherical metal nanoparticles (blue symbolizing negative excess charge, red symbolizing positive excess charge).

This effect of the self-polarization of charged nanoparticles is in strong contrast to the charge distribution of macroscopic metallic objects. Due to the smaller dimensions of the nanoparticles, its charge distribution is closer to the conditions in molecules with delocalized electrons than to the electrical conditions in a larger metal body, in which the charge distribution is dominated by thermal fluctuations for temperatures near room temperature. This effect can easily be illustrated by the comparison between the change of thermal energy ΔT and electrostatic energy for single elementary charges in dependence on charge distance r:

$$\left[1/(4\varepsilon\pi)\right] \times \left[\text{q}\_{\text{e}}\right]^{2}/\text{r} = \text{k}\_{\text{B}} \times \Delta\text{T} \tag{22}$$

$$
\Delta \mathbf{T} = \left[ \mathbf{1} / (4 \varepsilon \pi \mathbf{t}) \right] \times \left. \mathbf{q}\_{\text{e}} \right|^{2} / \left( \mathbf{r} \times \mathbf{k}\_{\text{B}} \right) \tag{23}
$$

The strength of the effect depends on the relative permittivity εr. This is a dimensionless material-specific parameter that can be assumed to be in the range between about 5 and 100. For this range, significant self-polarization effects that are not leveled by thermal fluctuations should be expected for nonspherical particles between about a few tens and few hundreds of nanometers size depending on the material (Figure 4).

**Figure 4.** Estimated temperature differences T corresponding to the thermal energy equal to the electrostatic energy of a pair of elementary charges in distance r (for permittivity of 5, 10, 30 and 100).

The self-polarization effect has important consequences for the growth and for the corrosion behavior of metal nanoparticles. The simplest case is the self-polarization of a nanorod, and this case can serve as an example to explain the principle effects. Dictated by the repulsion of equal charges, the charges are concentrated on the poles of the nanorod.

In the case of a positive excess charge, the potential is higher on the poles and lower in the center of the particle (Figure 5a). This means that in the case of corrosion, the poles are corroded faster than the center due to the higher intensity of the potential-dependent anodic partial on the pole. In the case of deposition (particle growth), a positive excess charge lowers the deposition on the poles and enhances the deposition in the center region. Thus, it is expected that at a positive excess charge, corrosion and deposition result in the lowering of the aspect ratio.

**Figure 5.** Self-polarization effect: effect of excess charge on the change of particle shape in dependence on charge sign shown for the example of a flat triangular metal nanoprism (schematically); (**a**) enhanced positive charge in the corners of triangular particle, (**b**) enhanced negative charge in the corners of particles.

In the case of a negative excess charge, the potential at the poles is decreased stronger than in the center of a nanorod (Figure 5b). This means that the deposition rate is enhanced on the poles and lowered in the center. This promotes the growth in the axial direction or the forming of "dog bone"-like geometries. In the case of corrosion, the material loss is lowered on the poles but enhanced in the center leading to a thinning of the central region. Both processes result in the tendency of an increase in the aspect ratio or the bone-shaping of nanorods.

#### **5. Factors in Shape Control of Growing Metal Nanoparticles**

Some general mechanisms can be formulated for the evolution of metal nanoparticle shape during particle growth [25]. It has to be kept in mind that surface energy plays a crucial role in metal deposition and etching in addition to the electrochemical potential [26]. Therefore, the lattice structure of the growing or corroding nanorods and the crystallographic planes at the particle surface are further important factors for the resulting geometries. However, considering the above-mentioned size-dependent effects, it can be concluded that a stronger influence of the lattice structure on the geometry evolution in the case of larger particles can be expected, whereas small particles are probably strongly dependent on the electrochemical potentials and self-polarization effects. It is well known that halogen ions determine the character of metal nanoparticle growth by the formation of adsorbates [27,28]. In addition, surfactants and their specific interaction with crystal planes are important for the shape development of metal nanoparticles [29,30]. In the case of the specific adsorption of shape-steering additives on preferred lattice planes and their inhibition of local metal deposition during the nanoparticle formation, it can be assumed that electrochemical effects such as self-polarization modulate the anisotropic particle growth and shaping.

The formation of regular crystals can be expected if the adsorbing metal ions or the formed metal adatoms have a high surface mobility. In this case, they tend to move to a thermodynamically preferred place. The formation of regular crystal lattices is then controlled by the minimization of lattice energy.

In the case of lower surface mobility, there is a mixture between a random adsorption of metal ions and adatoms and energy-controlled translocation taking place. This can result in more or less regular crystal structures. A typical result is the formation of branched crystals and dendritic nanocrystals with local crystallinity and a combination of different lattice orientations in different domains of the nanoparticle. In addition to the random adsorption of metal ions, an accidental local binding of ligands or other molecular effectors can influence the growth behavior of nanoparticles and also the formation of branched or polycrystalline structures.

Random effects in the attachment of adsorbing metal ions on the nanoparticle surface and the tendency of binding on thermodynamically excellent sites are superposed by the electrostatic self-polarization effect. On the one hand, local electrical fields direct approaching ionic species into preferred regions of the growing particle. On the other hand, the distribution of electrical charges on the particle surface modulates the local surface energy. The formation of different aspect ratios or the appearance of dog-bone-like nanorod structures is a typical consequence of this effect.

In addition to the shape control, the size control in the formation of metal nanoparticles is also very important because the electronic and optical properties of metal nanoparticles are size dependent. For the growth of larger nanoparticles from seed particles, the ratio of seed concentration to the concentration of precursor is generally the key parameter for steering the particle size. In the case of the complete consumption of the precursor during a homogeneous nanoparticle growth, this ratio determines the particle size directly. High seed concentrations result in small final product particles, and low seed concentrations cause the formation of large particles.

#### **6. Electrically Controlled Assembly of Metal Nanoparticles**

The use of attractive forces of oppositely charged nanoparticles is a simple strategy for their assembly. However, the mixing of particles of opposite charges involves the possibility of fast coagulation. A particle of one type can act as "particle glue" for connecting the particles of another type. Therefore, it is important to work with suitable particle concentrations (Figure 6).

**Figure 6.** Coagulation and assembly of oppositely charged nanoparticles; (**a**) high probability of coagulation at lower concentration ratios, (**b**) forming of dispersed assemblies at higher concentration ratios between small negative and larger positive particles.

Assembly can also take place in the case of equally charged particles. Therefore, forces are needed for overcoming the barrier of electrostatic repulsion. The electrostatic repulsion of charged nanoparticles is well known for the stabilization of the colloidal state. The domination of kinetic energy and binding over repulsion leads to the aggregation of particles, which can finally result in complete coagulation and precipitation. In other cases, moderate lowering of the particle velocity while maintaining the particle charges can cause the limited assembly and restabilization of colloids. Such a mechanism can cause the formation of polycrystalline metal nanoparticles by an in situ assembly process during Au nanoparticle formation, as discovered by Polte et al. [24], and is also a probable reason for the formation of star-like Au/Ag nanoparticles [31].

Discharging of colloidal particles can be induced by:

In all cases, a restabilization mechanism has to take place in order to avoid a continuation of aggregation, the uncontrolled formation of large aggregates and precipitation. In principle, this restabilization can be induced by controlling the electrochemical particle potential, too. Therefore, a "backward strategy" or a "forward strategy" can be applied. In the "backward strategy", the addition of complementary effectors invert the primary-induced potential shift for the stabilization of colloids after the temporary destabilization. The idea behind the "forward strategy" is to continue the primary-induced potential shift in the

same direction to overcome the zero-potential point quickly and to achieve the colloidal restabilization by inversion of the charge sign. Thus, the addition of adsorbing anions or the stepwise enhancement of the reducing agent concentration can lower the positive potential of colloidal metal nanoparticles in a first step but can lead to restabilized colloids by negatively charged particles. On the other hand, a colloid of negatively charged metal nanoparticles can be destabilized by the addition of oxidizing agents or metal cations but restabilized by their further addition in the positive potential range. It has to be mentioned that the addition of oxidizing agents has to be performed carefully in order to avoid the corrosion or complete dissolution of the colloidal particles. It seems that reductants can also cause a disruption of particles, especially strong ones such as NaBH4, by inducing a Rayleigh instability [32].

Self-polarization or the induced polarization of nonspherical particles causes different conditions for particle–particle interactions dependent on position. This can decide the formation of compact, linear or branched assemblies. Such different behavior was observed on polarizable polymer nanoparticles carrying polyionic macromolecules [33], and it can be, obviously, the reason for the formation of linear branched and network structures in the in situ assembly of metal nanoparticles.

#### **7. Secondary Metal Deposition**

An electrostatic model was proposed for the formation of networks of Au/Ag nanoparticles during the deposition of silver on preformed spherical gold nanoparticles [34]. It describes the preferred bonding of spherical particles on the poles of nanorods due to the induced polarization by the electrostatic interaction of the charged metal nanoparticles. This preferred bonding on the poles leads to the successive prolongation of primary formed nonspherical aggregates. The intrinsic reason for this linear aggregation is the symmetry break by binding of two spherical nanoparticles, which means a transition from a spherical geometry into a body with an enhanced aspect ratio. This initial process is responsible for the symmetry break in the distribution of surface charges. High repulsion forces result in a strict linear growth. Lower charging means a certain probability of occasional binding events on intermediate positions, working as branching points.

In general, higher electrical charges lead to more stable colloids and a separate growth of single nanoparticles, whereas lower electrical charges enhance the tendency of aggregation. The aggregation can end with restabilization after a few binding events after the formation of larger aggregates or networks or lead to complete coagulation and precipitation [35]. These effects can well be observed in the deposition of silver on preformed gold nanoseeds. In the case of a strong dominance of positively charged silver cations, the colloid remains stable at a higher potential. In the case of a reduced charge, particles can join each other. Due to the mobility of electrical charges, self-polarization takes place, and a preferred axial growth results in rod- and astragal-like structures (Figure 7). Therefore, not only are compact aggregates observed but also single particles. Further in situ aggregation during silver deposition can lead to branched astragal-like structures and to the formation of nanoparticle networks (examples in Figure 8).

**Figure 7.** Assembly behavior of charged metal nanoparticles: (**a**) spherical nanoparticles; (**b**) nonspherical metal nanoparticles, for example nanodiscs.

The effect of electrical control and self-polarization is well reflected by the formation of nanoframe structures by galvanic replacement reactions [36,37]. The deposition of gold on flat silver nanotriangles starts on the edges, at the preferred cathodically active sites, whereas the corrosion of silver starts in the center of the particles, at the preferred anodically active area. As a result, a central hole is formed, and the original prismatic particle shape is transformed into a frame structure (Figure 9).

**Figure 8.** In situ formation of aggregate particles during the chemical deposition of silver nanoparticles on gold seeds: (**a**) small aggregate particles; (**b**) enhanced aggregate size; (**c**) network formation.