Thermodynamic Rarity Assessment of Mobile Phone PCBs: A Physical Criticality Indicator in Times of Shortage

Rising prices in energy, raw materials, and shortages of critical raw materials (CRMs) for renewable energies or electric vehicles are jeopardizing the transition to a low-carbon economy. Therefore, managing scarce resources must be a priority for governments. To that end, appropriate indicators that can identify the criticality of raw materials and products is key. Thermodynamic rarity (TR) is an exergy-based indicator that measures the scarcity of elements in the earth’s crust and the energy intensity to extract and refine them. This paper uses TR to study 70 Mobile Phone (MP) Printed Circuit Boards (PCBs) samples. Results show that an average MP PCB has a TR of 88 MJ per unit, indicating their intensive use of valuable materials. Every year the embedded TR increases by 36,250 GWh worldwide -similar to the electricity consumed by Denmark in 2019- due to annual production of MP. Pd, Ta and Au embedded in MP PCBs worldwide between 2007 and 2021 contribute to 90% of the overall TR, which account for 75, 600 and 250 tones, respectively, and increasing by 11% annually. This, coupled with the short lifespan of MP, makes PCBs an important potential source of secondary resources.


Introduction
The whole world is experiencing soaring energy and raw material costs. Europe is particularly vulnerable to this situation which must import a large part of the raw material domestically consumed by industry and households [1]. Rising energy prices -driven by fossil fuel prices [2]-(electricity [3][4][5][6], natural gas [7] and gasoline and diesel [8]), food [9,10] (fertilizers [11]) and livestock feed [12]), shipping [13,14] and even the lack of microchips for factories [15,16], are examples of this. These supply issues occur when the transformation to a low-carbon economy driven by renewables, electric vehicles, and digitization is beginning to accelerate. This situation could jeopardize the transition since a low-carbon economy requires a large quantity and variety of raw materials. For example, to produce one gigawatt (GW) of electrical power equivalent to that which a natural gas-fired power plant could supply would imply the use of approximately 160,000 tons of steel, 2000 of copper, 780 of aluminum, 110 of nickel, 85 of neodymium and 7 of dysprosium for its construction [17]. These are not negligible amounts if it is estimated that in the future, the power provided by wind turbines in 2050 could be around 2200 GW [18].
Another example is that demand for some minerals for batteries could increase dramatically by 2040 -with respect to 2020-lithium 42 times, cobalt 21 times, nickel 19 or Rare Earth Elements (REE) 7, as the International Energy Agency (IEA) warns [19]. Thus, the use of scarce minerals -needed in a low-carbon economy-could pose a problem for future generations due to their eventual depletion and unavailability in the future [20]. Furthermore, these raw materials are mainly extracted from mines that need fossil fuels to operate. The IEA's World Energy Outlook 2021 indicates that oil and natural gas production could the TR [42] and that many high TR elements end up downcycled as part of alloys or in landfills. Downcycled elements represent 4.5% of the vehicles, while in TR terms, it would be 27% [43]. Currently, EC legislation for End of Life Vehicles requires the recovery of 95% of the vehicle by weight. This can be met by recovering major metals, yet the minor ones become lost or downcycled, losing their functionality. Horta Arduin et al., 2020 [44] has also highlighted this problem in the case of display waste. They state that there is a contradiction between the EC criteria, which on the one hand is concerned with the criticality of CRMs through the publication of lists, but on the other hand, the WEEE recycling regulations focus on weight, causing many critical elements to be lost due to their low contribution in weight. This makes new indicators necessary to reinforce current regulations.
This paper is structured as follows. First, TR indicator is explained. Second, the sources used to calculate the composition of the MP PCBs, the assumptions for calculating the TR and the estimation of resources embedded worldwide are shown. Third, the results of the mass composition, TR and resources worldwide are presented. Finally, the main conclusions are discussed.

Thermodynamic Rarity Indicator
TR is an indicator, based on exergy, used to measure the thermodynamic criticality of raw materials, depending on their scarcity in the earth's crust and the energy intensity associated with mining, beneficiation, and refining processes. Exergy is a property of a system relative to an associated reference state. It is the maximum work a system can deliver as it interacts with another large, but real, system, namely, a reservoir. Such a reservoir attracts the system toward degradation or entropy creation. The reference state selected for the exergy assessment of minerals is a planet, called "Thanatia" (from Greek Thanatos, meaning "death") with the following characteristics [ This imaginary state of the planet does not need to be "reachable", but it is a baseline to assess the quality of any resource physically. It further allows us to objectively identify which resource is closer to depletion in the race to exhaustion. Any mineral resource with a concentration higher than that found in Thanatia has exergy, and therefore, its quality can be quantified in energy terms [39]. TR incorporates two aspects. The first is the embedded exergy cost (kJ), i.e., the useful energy required to extract and process a given mineral from the cradle to the gate (i.e., until it becomes a raw material for the manufacturing industry). The second is, in fact, an avoided cost for having minerals concentrated in mines and not dispersed throughout the crust (i.e., it can be seen as a natural bonus). As mines become depleted, it becomes exponentially harder to obtain commodities (embedded costs increase), whereas the bonus reduces. This bonus is calculated as a hypothetical exergy cost required if the given mineral would be restored to its initial composition conditions and concentration in the original mines from an utterly dispersed state, i.e., its state in Thanatia. This is the exergy replacement cost (ERC) (kJ) and can be seen as a grave-tocradle-approach [36] or as a natural avoided exergy cost, i.e., as a natural bonus. Thus, the TR is presented as a physical indicator, stable over time, based on thermodynamic fundamentals. However, it is conditioned by mining technology, as it could reduce the exergy costs of mineral extraction and the knowledge of the earth's crust that would modify the composition established for Thanatia. Another advantage is that it allows classifying the elements in order of criticality since each element has a unique value, measured in exergy terms.
TR values (R i ) of the analyzed elements, measured in GJ ton i are shown in Table 1 [28]. Nevertheless, TR values could be higher than those used. As an example, Palacios et al. [45] obtained TR values 2 to 3 orders of magnitude higher than previous values for Cu and Au, using metallurgical process simulation, more specifically the HSC Chemistry software.  The TR of a mobile phone (MP) PCB has been calculated through Equation (1). First, the TR of a kg of PCB is calculated (in parentheses). To do this, the product of the TR of an element (R i ) by its concentration in the PCB is done and then the units of kg of PCB are transformed into units of MP.
Therefore, it is necessary to know its average weight and the percentage of PCBs it contains in relation to its weight. In this paper, as indicated in Equation (1), an average phone weight of 100 g and a PCB percentage by weight of 20% have been used to obtain conservative results. Table 2 shows the percentage of PCBs in phones according to different references. Equation (1) is also used to calculate the contribution of each element to the total TR to analyze the thermodynamic criticality of each element. To estimate the mass of elements embedded in MP PCBs worldwide, the annual sales of 2020 -around 1.5 billion-and the cumulative sales between 2007 and 2021 -around 14.8 billion units-are taken [22].
Finally, the ratio between the amount of elements embedded in MP PCBs worldwide and the annual extraction of the elements is calculated. For this purpose, the quantity of each element is divided by its extraction. Thus, two percentages are obtained depending on the number of MP considered. On the one hand, the cumulative quantity is considered, i.e., 14.8 billion units between 2007 and 2021; and, on the other hand, the annual sales are considered, i.e., 1.5 billion units. Thus, the first percentage represents the annual production that could be provided if that element were recovered from all the accumulated PCBs. In addition, the second percentage of annual production could be covered with the PCBs of one year. In other words, it would be approximately the percentage of the annual production that is used to produce MP PCBs. The extraction data for the elements were obtained from the U.S. Geological Survey 2021 commodity summaries [58]. In 2020, 210 tons of Pd, 1700 tons of Ta, 3200 tons of Au, 300 tons of Ga, 20,000,000 tons of Cu, 170 tons of Pt and 900 tons of In were mined.

Composition and Thermodynamic Rarity of Mobile Phone PCBs
The 70 MP PCBs samples reviewed are composed of 55 different chemical elements, of which 31 are considered as CRMs by the EC (Figure 1). Although the EC list contains Entropy 2022, 24, 100 6 of 21 30 commodities, some of them are groups of elements such as light REE or platinum group metals (PGM), so the number obtained is greater than 30. Taking this into account, 25 elements in the MP PCB are considered critical by the EC. Nevertheless, the contribution by weight of these elements to the total PCBs is very different. Figure 2a,b shows the results of the mass contribution of each element. As can be seen, more than 90% of the weight of PCBs is made up of 8 elements: Cu, Si, Fe, Br, Sn, Ni, Al and Zn, being two CRMs according to the EC: Si and Al. Using the CE criterion, the remaining 47 elements constitute 10% of the overall weight, concentrating up to 23 CRMs. Therefore, most of the critical elements are characterized by low mass concentrations.
of one year. In other words, it would be approximately the percentage of the annual production that is used to produce MP PCBs. The extraction data for the elements were obtained from the U.S. Geological Survey 2021 commodity summaries [58]. In 2020, 210 tons of Pd, 1700 tons of Ta, 3200 tons of Au, 300 tons of Ga, 20,000,000 tons of Cu, 170 tons of Pt and 900 tons of In were mined.

Composition and Thermodynamic Rarity of Mobile Phone PCBs
The 70 MP PCBs samples reviewed are composed of 55 different chemical elements, of which 31 are considered as CRMs by the EC (Figure 1). Although the EC list contains 30 commodities, some of them are groups of elements such as light REE or platinum group metals (PGM), so the number obtained is greater than 30. Taking this into account, 25 elements in the MP PCB are considered critical by the EC. Nevertheless, the contribution by weight of these elements to the total PCBs is very different. Figure 2a,b shows the results of the mass contribution of each element. As can be seen, more than 90% of the weight of PCBs is made up of 8 elements: Cu, Si, Fe, Br, Sn, Ni, Al and Zn, being two CRMs according to the EC: Si and Al. Using the CE criterion, the remaining 47 elements constitute 10% of the overall weight, concentrating up to 23 CRMs. Therefore, most of the critical elements are characterized by low mass concentrations. To measure criticality, this paper uses the TR indicator. Thus, Figure 2c shows the results of the TR contribution of each element in kJ per unit of MP and Figure 2d the results in percentage, according to the data and assumptions outlined in Section 2.2. If the TR of an element is unknown, it has been counted as 0, as for Te or Br (see Table 1). Taking TR To measure criticality, this paper uses the TR indicator. Thus, Figure 2c shows the results of the TR contribution of each element in kJ per unit of MP and Figure 2d the results in percentage, according to the data and assumptions outlined in Section 2.2. If the TR of an element is unknown, it has been counted as 0, as for Te or Br (see Table 1). Taking TR as a criterion, the results are radically different. There are now 3 elements that contribute to almost 90% of the TR: Pd, Ta and Au, 4 others that account for 8%: Ga, Cu, Pt, and In, and remaining 48 for only 2%. Thus, seven elements account for 98% of the TR, being all of them CRMs according to the EC except Au and Cu -the most abundant in PCBs-.
Summing the contribution of each element as shown in Equation (1), the results indicate that the TR of a PCB is 88 MJ per MP unit. This result does not include other parts of the MP, such as the display, camera, or battery, so the TR of the complete MP is higher than obtained. Considering that between 2016 and 2020 mobile sales stagnate at around 1.5 billion mobiles per year (Figure 3b), the TR embedded in MP PCBs worldwide would increase by 1.305·10 11 MJ o 36,250 GWh per year, an amount comparable to the electricity consumed by Denmark in 2019 [59].

Resources Embedded in Mobile Phone PCB worldwide
In order to estimate the amount of resources embedded in the PCBs of MP, two sources of information have been taken. On the one hand, the number of MP sold between 2007 and 2021 is 14.8 billion, doubling the world population (Figure 3a). On the other hand, the number of MP put on sale annually considered is 1.5 billion units, which since 2016 has stagnated as shown in Figure 3b. Figure A1 (appendix A) shows the results for each element, and Table 3 shows the results for the highest TR (rows A and B). It indicates that these elements' quantity embedded in MP PCBs increases by approximately 11% each year.
This strong annual increase and the short lifespan of the MP -of around four years [23]-make such devices an interesting source of valuable raw materials. Accordingly, we now explore how much of the annual production could -theoretically-be covered by the resources embedded in the MP PCBs. Table 3 shows the annual primary extraction of each element in row C. In the last two rows, the ratios between row A and C, and, B and C are calculated. These ratios indicate the percentage of a year's global extraction that could be replaced if all of the embedded mass between 2007 and 2021 (A/C) could be recovered, or if all of the mass produced in one year could be recovered (B/C). It is important to emphasize that recovering the entire PCBs from MP is impossible. For example, in Reuter et al. 2018 [60], they only recover 22% of the metals from a MP in the best case. However, they achieve very high recovery rates for some elements such as Au (90-100%), Pd (10-100%) or Ga (80-90%), but much lower for others -Ta (0-10%). Another example is found in Valero-Navazo et al. 2014 [61], in which Pd, Au, Ag, Cu, Ni, Pb and Sn are recovered with recovery rates between 80 and 95%. As can be observed, the elements with higher TR are not always recovered, for example, Valero-Navazo et al. 2014 does not recover Ta or Ga, while in Reuter et al. 2018, the recovery efficiencies of Pd and Ta are 10% and Ga 80% in the worst cases. Therefore, the percentages in Table 3 should be interpreted as a theoretical maximum -unreachable-or, from another perspective, as the percentage of the extraction hoarded by the MP PCBs. In addition to the physical limitations, separate collection rates are very low, ranging from 2 to 16% [61], so high recovery rates are still far from being achieved.

Resources Embedded in Mobile Phone PCB Worldwide
In order to estimate the amount of resources embedded in the PCBs of MP, two sources of information have been taken. On the one hand, the number of MP sold between 2007 and 2021 is 14.8 billion, doubling the world population (Figure 3a). On the other hand, the number of MP put on sale annually considered is 1.5 billion units, which since 2016 has stagnated as shown in Figure 3b. Figure A1 (Appendix A) shows the results for each element, and Table 3 shows the results for the highest TR (rows A and B). It indicates that these elements' quantity embedded in MP PCBs increases by approximately 11% each year. Table 3. Annual increase in resources embedded in MP PCBs worldwide. Comparison between annual element production and quantity embedded in Mobile Phone PCBs. Extraction data from reference [58].

Elements
Pd Ta  This strong annual increase and the short lifespan of the MP -of around four years [23]make such devices an interesting source of valuable raw materials. Accordingly, we now explore how much of the annual production could -theoretically-be covered by the resources embedded in the MP PCBs. Table 3 shows the annual primary extraction of each element in row C. In the last two rows, the ratios between row A and C, and, B and C are calculated. These ratios indicate the percentage of a year's global extraction that could be replaced if all of the embedded mass between 2007 and 2021 (A/C) could be recovered, or if all of the mass produced in one year could be recovered (B/C). It is important to emphasize that recovering the entire PCBs from MP is impossible. For example, in Reuter et al., 2018 [60], they only recover 22% of the metals from a MP in the best case. However, they achieve very high recovery rates for some elements such as Au (90-100%), Pd (10-100%) or Ga (80-90%), but much lower for others -Ta (0-10%). Another example is found in Valero-Navazo et al., 2014 [61], in which Pd, Au, Ag, Cu, Ni, Pb and Sn are recovered with recovery rates between 80 and 95%. As can be observed, the elements with higher TR are not always recovered, for example, Valero-Navazo et al., 2014 does not recover Ta or Ga, while in Reuter et al., 2018, the recovery efficiencies of Pd and Ta are 10% and Ga 80% in the worst cases. Therefore, the percentages in Table 3 should be interpreted as Entropy 2022, 24, 100 9 of 21 a theoretical maximum -unreachable-or, from another perspective, as the percentage of the extraction hoarded by the MP PCBs. In addition to the physical limitations, separate collection rates are very low, ranging from 2 to 16% [61], so high recovery rates are still far from being achieved.
Coincidentally, most elements with the highest ratios are those with the highest TR, i.e., Pd, Ta, Ga, Au, Pt and In, except for Cu. This may be due to the relationship between geological scarcity and low extraction rates. However, this should not necessarily be the case, as it is a result that depends on the composition of the devices to be analyzed. What is important to note is that the recovery of these elements should be prioritized, as they are not only the most critical from the point of view of TR, but if they were recovered, they could make an important contribution to world production. For example, in the case of Pd and Ta, their contribution to world production could theoretically reach 35% if the tons incorporated between 2007 and 2021 could be fully recovered. This figure would be 15% and 8% for Ga and Au, respectively. Considering only the tons embedded in a year, the contribution would drop to 3.8% for Pd and Ta; and to 1.7% and 0.8% for Ga and Au, respectively.

Conclusions
The volatility and increase in raw material prices and even the unavailability of some components may jeopardize the energy transition. The search for secondary raw materials and their recovery becomes necessary to alleviate this shortage situation, which could worsen in the future due to ore grade decline, among other factors. In addition, reducing primary extraction would provide other benefits such as less environmental deterioration and greater availability of resources for future generations. To this end, identifying new sources of secondary resources is essential.
This article analyzes the PCBs of the MPs, through the TR. These devices are promising candidates due to their large sales and their short useful life. The use of TR-a physical indicator based on thermodynamics allows obtaining stable values of material criticality in the medium to long term, which can only be influenced by mining technology and knowledge of the earth's crust. This physical point of view is an essential reinforcement of the criticality assessment of any government, based on the importance of the elements for the given economy's region and the supply risks. Being decoupled from these time-varying factors, the TR can help establish long-term policies. Another advantage of the TR is that it allows to classify and quantify the elements in order of criticality, as each element has a unique value. This helps identify products and parts with a high content of critical and valuable materials and is helpful for eco-design.
The results show that Pd, Ta, Au, Ga, Cu, Pt and In are the highest contribution to TR in MP. All are considered critical by the EC, except for Cu and Au. In addition, a considerable percentage of the world's production of Pd, Ta, Ga and Au is hoarded in MP PCBs. These results show the need for the recovery of these elements, not only for the conservation of TR, (i.e., of the exergy embedded in the most geologically scarce elements) but also for their significant contribution to the world's commodity production. However, 100% recovery of the resources embedded in the equipment is impossible, so to achieve the maximum recovery rate, it is necessary to develop and promote recycling processes that allow it. However, these processes are energy-intensive and require further thermodynamic analysis. This will be analyzed in a forthcoming paper.