You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


1. Introduction {#sec1}
===============

Planetary ball mills provide high energy density due to the superimposed effect of two centrifugal fields produced by the rotation of the supporting disc and the rotation of the vials around its own axis in the opposite direction [@bib1]. During operation, the grinding balls execute motion paths that result in frictional and impact effects. These high-energy ball mills are suitable to induce mechanochemical reactions and develop a wide range of new technological materials from powder mixtures [@bib2]. Due to their scientific relevance, an increasing number of works are trying to model their operational mechanism, i.e., how the mechanical energy is transferred to powders and its quantification and dependence on the operating parameters. This is essential to propose the optimal milling conditions and compare the results obtained from different experimental devices.

Early models were based on kinematic equations describing the trajectory and velocity of balls inside the vial. For these models, the impacts between the balls and the inner walls of vials were the primary event in the energy transfer \[[@bib3], [@bib4], [@bib5], [@bib6]\]. However, in-situ observations have not shown signs of balls leaving the vial surface to make clear single collisions; the actual situation is much more complex with multiple types of ball-ball and ball-vial interactions \[[@bib7], [@bib8], [@bib9]\]. Nowadays, most of the studies use numerical simulations of the collective three-dimensional ball motion based on the Discrete Element Model (DEM) \[[@bib10], [@bib11], [@bib12]\], since it is possible to quantify in a more precise way the effects of successive small collisions between balls, the lack of single ball-wall collisions and the slipping and rolling of balls, situations that occur in real milling conditions \[[@bib11], [@bib13]\].

In all these studies, the key role played by the rotational speed of the planetary mill and the number and nature of balls (and indirectly the ball-to-powder ratio, BPR) have always been highlighted since these parameters determine greatly the rate of the energy transfer to the powder material. Nevertheless, the influence of the ratio (*k*) between the rotational speed of the supporting disc (*ω*~*d*~) and vials (*ω*~*v*~), (*k = -ω*~*v*~*/ω*~*d*~), on the milling efficiency has not been frequently addressed \[[@bib5], [@bib6], [@bib7], [@bib8], [@bib9], [@bib10], [@bib11], [@bib14], [@bib15], [@bib16], [@bib17]\]. The main reason is that most commercial planetary mills, especially at a laboratory scale, are characterised by a fixed transmission ratio that sets this *k* parameter to a value of approximately 1. However, it has been shown that the motion path of balls is greatly influenced by *k* and, therefore, this parameter also has a significant influence on the energy transferred to the powders.

Based on kinematic models, Le Brun et al [@bib7] determined three different types of ball trajectories in function of *k* that can be defined as cascading, cataracting and rolling. The critical *k* values defining the cataracting domain, the optimum regime, were approximately between 1.3 and 3.1. Chattopadhyay et al [@bib5] calculated that the effectiveness of impacts was enhanced for *k* ≥ 1 and that the total impact power increased with increasing *k*, but not any optimal value was proposed. Kakuk et al [@bib6] determined that the greatest impact energy is achieved at approximately *k* = 2.5, but the greatest milling power (that includes also the impact frequency) is reached around *k* = 2.96.

Based on DEM simulation, Mio et al [@bib10] also distinguished between the three abovementioned ball movement regimes, but with really close transitional *k* values. They determined that the specific impact energy of balls increases with increasing *k* up to a critical value, which depends on the size of the planetary mill, from which the energy began to fall. Zhang et al [@bib14] determined also by DEM that the highest energy utilisation in a horizontal planetary mill was acquired for *k* = 1.5. However, Rosenkranz et al [@bib8] showed by both DEM and in situ-observations that *k* had no considerable influence on the ball motion pattern in the range 1 \< *k* \< 3. A cascading motion pattern was always observed in the absence of mill feed and with a typical ball filling ratio of 0.3. Nevertheless, by increasing the ball filling ratio or the friction conditions, due to the presence of the mill feed, the dynamics of balls changed to a cataracting regime.

Finally, by in-situ observations and using an image processing program, Rogachev et al [@bib9] observed different types of ball movement depending on *k*. At *k* \< 1.7, the rolling of the balls primarily took place and some free fly trajectories were detected. An important proportion of stagnant balls (making small oscillations around a steady position) were also observed. At *k* = 1.8--1.9, the free fly type of ball motion became dominant and at *k* ≥ 2.0, a circular motion of balls formed a rotating layer on the wall of the vial.

Some other works have also tried to study the effects of *k* on the milling efficiency from experimental observations. For example, Le Brun et al [@bib7] observed based on microhardness values that a higher milling efficiency was achieved during milling Cu powders for a *k* value of 1. However, Schilz et al [@bib15] found during the formation of the Si-Ge solid solution and the Mg~2~Si intermetallic that the shortest alloying time (based on X-ray diffraction (XRD) results) was observed for *k* = 2.5 and 3, respectively. Ghayour et al [@bib16], using the ignition time (t~ig~) of the mechanically induced self-sustaining reaction (MSR) process involving Al, B~2~O~3~ and TiO~2~, determined that the optimum *k* value was between 1.2 and 1.4 depending on the BPR value. Rogachev et al [@bib9] determined from the increase of the reactivity of highly exothermic mixtures (Ni-Al, Ti-Si and Si-C) that the most efficient milling regime was achieved when *k* = 1.5. Boytsov et al [@bib17], studying the microstructure evolution of Cu powders by XRD, claimed that it is rather difficult to select an optimal value due to a non-linear dependence of the milling parameters with *k*.

The contradictory results of these experimental studies, which often also contradict the theoretical predictions, can originate from the way of measuring the milling efficiency, which is carried out indirectly from changes in composition, morphology, structure or crystallinity in the powder charge at increasing milling time. Frequently, it is difficult to correlate the sample property analysed with the mechanical energy involved in the process and its dependence on the milling conditions. However, in MSR processes, a direct relationship between 1/t~ig~ and the mechanical dose rate provided by a mill has been established [@bib18], since ignition occurs when the milling device has provided a specific amount of energy (per unit mass) to the reactant mixture. Several studies have already shown that variations of t~ig~ with different milling parameters can be used as a fingerprint of the milling efficiency \[[@bib18], [@bib19], [@bib20], [@bib21], [@bib22], [@bib23], [@bib24], [@bib25],[@bib25],[@bib26]\]. Moreover, t~ig~ is a reproducible parameter that can be measured in real-time during milling by recording the temperature and/or the pressure inside the vial, since both signals show a sudden increase at ignition due to the heat released by the highly exothermic reaction. It has been observed in solid-solid reactions, in which no gaseous reactants or products are involved, that the pressure signal is more sensitive to self-sustaining processes than temperature [@bib27].

In this work, the MSR process involving the formation of TiB~2~ from the stoichiometric elemental mixture was selected as a model system to study the influence of *k* on t~ig~ and, therefore, on the milling efficiency. TiB~2~ is a stoichiometric compound with high adiabatic temperature (T~ad~), which has already been obtained by MSR [@bib28]. The high T~ad~ ensures the occurrence of the MSR process in a wide range of different milling conditions without the interference of competitive side reactions. With this study, we tried to shed some light on the contradictory results found in the literature, both from experimental and theoretical approaches.

2. Experimental {#sec2}
===============

Titanium powder (99% purity, \< 325 mesh, Strem Chemicals), boron powder (95--97% purity, amorphous, Fluka) and high-purity argon gas (H~2~O and O~2~ \< 3 ppm, Linde) were used in this work. All milling experiments were conducted under an Ar atmosphere using a Ti/B mixture with an atomic ratio of 1/2, a 60 mL tempered steel vial (inner diameter = 45 mm) and WC-Co (d = 15 mm and m = 26.4 g) balls. The powder mixtures were ball-milled in a modified Pulverisette 4 (Fritsch) planetary mill able to detect t~ig~ in MSR processes by measuring the pressure inside the vial during milling. This was possible by means of a vial with a special lid that was continuously connected to the gas cylinder at a pressure of 5 bar by a rotating union (model 1005-163-038, Deublin) and a semi-rigid polyamide tube (Legris). During milling, the gas pressure was monitored by a pressure transducer (AKS, Danfoss) connected to a paperless recorder (Ecograph T RSG35, Endress + Hauser). When ignition occurred, the heat released produced a sharp peak in the pressure-time record, from which t~ig~ was accurately determined.

To explore the dependence of t~ig~ on the milling conditions, different milling experiments were carried out by modifying the following parameters: *ω*~*d*~ from 250 to 400 rpm, *k* from 1 to 3 and BPR from 24 to 46. These values were within the range normally used by the milling practitioners and in the studies mentioned in the introduction section. In this work, the absolute *k* ratio (*k = -ω*~*v*~*/ω*~*d*~) was employed, i.e., if the supporting disc rotates at 400 rpm and the vial rotates around its own axis at the same speed but in the opposite direction (-400 rpm), the *k* value is 1. Note that some manufacturers use the relative speed ratio in their technical data. For the example mentioned, the relative ratio would be -2. Moreover, the BPR value was adjusted by changing both the powder charge and/or the number of balls to ensure the correct filling of the vial. A BPR value of 46 was achieved with 7 WC-Co balls and 4 g of charge, a value of 33 with 5 WC-Co balls and 4 g of charge and, finally, a value of 24 with 5 WC-Co balls and 5.5 g of charge. The conversion of the elemental mixtures into TiB~2~ after the MSR process was ascertained by XRD. The XRD patterns were obtained on a X\'Pert Pro MPD diffractometer (PANalytical) equipped with a graphite-diffracted beam monochromator and a solid-state detector (X\'Cellerator) using Cu Kα radiation over a 2θ-range of 20° to 80° with a step size of 0.05° and a counting time of 300 s/step.

3. Results and discussion {#sec3}
=========================

[Fig. 1](#fig1){ref-type="fig"} shows as an example the pressure-time record corresponding to a milling experiment performed using the following experimental conditions: *ω*~*d*~ = 250 rpm, *k* = 1.5 and BPR = 24, from which a t~ig~ value of 110 min and 22 s was determined. As can be seen, the pressure spike at ignition is very intense and the t~ig~ value can be determined accurately. Similar records were always obtained for the different milling experiments performed in this work. Each milling experiment was repeated at least three times and the t~ig~ value obtained was always found within a range of ± 5 %. [Fig. 2](#fig2){ref-type="fig"} presents the XRD pattern of this sample collected a few minutes after ignition that clearly demonstrates the conversion of reactants into TiB~2~ during the MSR process. In this same figure, the XRD pattern corresponding to another milling experiment with a significantly shorter t~ig~ value is also shown (experimental conditions: *ω*~*d*~ = 350 rpm, *k* = 1 and BPR = 46). For all milling experiments, the final phase obtained after ignition was always TiB~2~ regardless of the experimental conditions used and t~ig~ values found. These results confirmed that the Ti/2B system chosen is appropriate to study the influence of milling parameters on the energy supplied by the planetary mill.Fig. 1A plot of the pressure-time record during the MSR process, from which t~ig~ can be properly determined. Milling experimental conditions: *ω*~*d*~ = 250 rpm, *k* = 1.5 and BPR = 24.Fig. 1Fig. 2XRD patterns of products after two MSR processes with significant different t~ig~ values, showing the formation of TiB~2~. Milling experimental conditions: (a) *ω*~*d*~ = 250 rpm, *k* = 1.5 and BPR = 24; (b) *ω*~*d*~ = 350 rpm, *k* = 1 and BPR = 46.Fig. 2

[Fig. 3](#fig3){ref-type="fig"} shows the results of the full set of experiments from which the dependence of t~ig~ on *k* can be inferred for different *ω*~*d*~ and BPR values. For a given *k* value, [Fig. 3](#fig3){ref-type="fig"} shows the expected behaviour (frequently observed in the literature) of an increase in the milling efficiency, i.e., reduction of t~ig~, as *ω*~*d*~ and BPR increase. It is clear that when *ω*~*d*~ is larger, both the frequency and the energy of impacts increase, which produce the shortening of t~ig~. On the other hand, for the milling experiments with a BPR of 46, seven balls were employed instead of five and, hence, the number of collisions per unit of time was greater, inducing a shorter t~ig~. In the case of the milling experiments with a BPR of 24, the amount of powder sample was larger (5.5 g instead of 4 g) and, therefore, the energy transferred per gram of powder and unit of time due to collisions and friction was lower, which reduces the activation of the reactant mixture and induces a longer t~ig~.Fig. 3Dependence of t~ig~ on *k* for different *ω*~*d*~ and BPR values: (a) BPR = 46, (b) BPR = 33 and (c) BPR = 24.Fig. 3

Nevertheless, [Fig. 3](#fig3){ref-type="fig"} suggests that the exact evolution of t~ig~ with *ω*~*d*~ and BPR also depends on *k*. Note that at 400 rpm and independently of the BPR value, t~ig~ was progressively reduced as *k* increased. The shortest t~ig~ was observed for *k* = 3. However, for the other *ω*~*d*~ values, the same order of reducing t~ig~ with *k* was not followed and this order was also dependent on BPR. This fact should be related to how the frequency and the energy of impacts depend on *ω*~*d*~ and BPR for the different *k* values. Besides, the proportion of friction effects associated with the time spent by the balls sliding on the vial wall should be also considered, especially at the lower *ω*~*d*~ values.

Although results in [Fig. 3](#fig3){ref-type="fig"} do not allow finding a single optimal *k* value independently of the experimental conditions, some clear trends can be extracted from this figure. First, the lowest milling efficiency (longer t~ig~ values) was always found for *k* = 1, which is the usual value employed in routine milling works and also the only available in many commercial planetary mills with a fixed gear ratio between the disc and vial movements. Second, differences in t~ig~ for different *k* values are reduced as *ω*~*d*~ increases. And third, best efficiencies were found for the larger *k* values (2.5 or 3). In this sense, at lower *ω*~*d*~, the shortest t~ig~ was obtained for *k* = 2.5 and at higher ω~d~ for *k* = 3, independently of BPR.

These results are only in agreement with the predictions of Kakuk et al [@bib6] based on kinematic equations and the experimental observations of Schilz et al [@bib15] based on XRD measurements. Note that Kakuk et al [@bib6] considered in their calculations the characteristic geometrical parameters of a Pulverisette 4 planetary mill, as employed in the present work, and Schilz et al [@bib15] used a Retsch PM400 planetary mill with a similar geometry as the Pulverisette 4 mill. These considerations outline the fact that the optimal *k* value should be dependent on the size of the planetary mill. In this sense, Schilz [@bib29] determined on the basis of classical mechanics that the impact power exhibits a maximum value for a certain *k* for every mill geometry, characterised by the ratio between the vial (*r*) and the revolution (*R*) radii. He claimed that for a typical planetary mill with *r*/*R* ratio between 0.1 and 0.3, the optimal *k* is between 3.25 and 2.25. For our set of experiments, the r/R ratio was 0.18 and the optimal *k* value should be approximately 2.5 in agreement with our results. This good accord is observed despite the fact that in their calculations, the balls were considered as single mass point, without any interaction or influence between them. Also no rolling or slip was taken into account that is responsible for the friction effects in real milling experiments and can have more influence as *ω*~*d*~ is reduced. Nevertheless, as observed in [Fig. 3](#fig3){ref-type="fig"}, the optimal *k* value is not only dependent on the planetary mill geometry, but also on the specific milling conditions (*ω*~*d*~ and BPR).

In contrast, our results surprisingly do not agree with any of the few works that exist in the literature dealing with *k* based on DEM simulations. For example, Zhang et al [@bib14] determined an optimal *k* value of 1.5. Although Mio et al [@bib10] determined on the basis of DEM simulation that the optimal *k* value was dependent on the revolution radius of the mill, they estimated for a revolution radius of 200 mm a low optimal *k* value of 1.8. Note that the Pulverisette 4 mill has a radius of 120 mm and the optimal *k* value should be even smaller according to their calculations. Moreover, Rosenkranz et al [@bib8] claimed that their results contradicted prior considerations and theoretical calculations that identified *k* as a key influencing parameter on the grinding ball motion pattern.

As indicated earlier in the introduction section, it has previously been shown that 1/t~ig~ and the energy milling power provided by a mill are directly linked [@bib18]. In [Fig. 4](#fig4){ref-type="fig"}, 1/t~ig~ is plotted as a function of *k* for the different *ω*~*d*~ and BPR, from which the multi dependence of *k*, *ω*~*d*~ and BPR on the milling efficiency can be observed. In this figure, it is more clearly visible as the best milling efficiency was obtained for *k* = 3 at the highest *ω*~*d*~, and it is highly enhanced when BPR is increased. As *ω*~*d*~ is reduced, differences are less important, but milling efficiency is always better for large *k* values. The reason for the fall in efficiency at *k* = 2.5 that increases again at *k* = 3 is related to the fact that the milling efficiency is dependent on the product of the impact energy multiplied by the frequency of impacts, and both parameters do not correlate with *k* in the same way. It has been shown in [@bib6] that at *k* = 2.5 the impact energy (of a single ball) is greater than at *k* = 3, but the impact frequency is lower. Therefore, the higher impact frequency at *k* = 3 induces the best milling efficiency only for high *ω*~*d*~ values. Note that the impact frequency increases with *ω*~*d*~.Fig. 4Dependence of 1/t~ig~ on *ω*~*d*~ for different *k* and BPR values: (a) BPR = 46, (b) BPR = 33 and (c) BPR = 24.Fig. 4

A great resemblance was observed between [Fig. 4](#fig4){ref-type="fig"}, especially that with a BPR of 24, and [Fig. 5](#fig5){ref-type="fig"} (taken from reference 6) that corresponds to the calculations by Kakuk et al [@bib6] (based on kinematic equations) for the milling power provided by the Pulverisette 4 mill as a function of *k*, at different *ω*~*d*~ values. The similarity between both figures confirms the direct relationship between 1/t~ig~ and the milling power and, therefore, t~ig~ of MSR processes can be a valuable milling parameter that could be used to validate the models developed to describe the operation of planetary mills.Fig. 5Changes of the milling power (*P*) as a function of the ratio (*i≡k*) at different *ω*~*d*~, reprinted from reference [@bib6]. This figure was obtained from kinematic equations using the geometrical parameters of a Pulverisette 4 planetary mill.Fig. 5

4. Conclusions {#sec4}
==============

The influence of the ratio (*k*) between the rotational speed of the supporting disc (*ω*~*d*~) and vials (*ω*~*v*~) at different *ω*~*d*~ and BPR on the milling efficiency of a Pulverisette 4 planetary mill was studied. The milling efficiency was determined indirectly by measuring the ignition time (t~ig~) for the formation of TiB~2~ from its elemental mixture via a mechanically induced self-sustaining reaction (MSR); longer t~ig~ values imply lower milling efficiencies. Results showed that the optimal *k* value is not only dependent on the geometric characteristics of the planetary mill (size of the supporting disc and vials), but also on the specific milling conditions (*ω*~*d*~ and BPR), which can explain the differing optimal values that can be found in the bibliography for this parameter. For the Pulverisette 4 mill, the lowest milling efficiency was always found for *k* = 1. The best efficiencies were found for the larger *k* values (2.5 or 3). At lower ω~d~, the shortest t~ig~ was obtained for *k* = 2.5 and at higher ω~d~ for *k* = 3. Surprisingly, a better agreement was found with predictions made on the basis of kinematic equations than on DEM, which in principle must be able to reproduce in a more realistic way the ball trajectories and the forces involved during a milling process.

Declarations {#sec5}
============

Author contribution statement {#sec5.1}
-----------------------------

Concepcion Real, Francisco J. Gotor: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Funding statement {#sec5.2}
-----------------

This work was supported by the Secretaría de Estado de Universidades, Investigación, Desarrollo e Innovación, Spain, under Grant MAT2014-52407-R, financed, in part, by the European Regional Development Fund.

Competing interest statement {#sec5.3}
----------------------------

The authors declare no conflict of interest.

Additional information {#sec5.4}
----------------------

No additional information is available for this paper.