JaiHind15

[latex]\frac{100}{\frac{1}{2}+\frac{3}{4}+\frac{5}{6}}=7[/latex] well... that was painful

Hi @John Cuthber, For 50% Au11.5Ag57.4Cu31.1 + 50% Ag100, the resultant density of the alloy is 10.49499847 which is different than ‘10.49 exact’ as provided before. Please see the calculations below: 50% Au11.5Ag57.4Cu31.1 + 50% Ag100 Au5.75Ag28.7Cu15.55 + Ag50 Au5.75Ag78.7Cu15.55 [latex]\frac{100}{\frac{5.75}{19.32}+\frac{78.7}{10.5}+\frac{15.55}{8.96}}=10.49499847[/latex] Please see the details and the reasoning of pure silver computations 10.5 in the previous response: The performance of an algorithm that identifies a unique solution for an underdetermined system (mathematically impossible task) is being evaluated here. To ascertain that the result produced is correct or not, the composition of the alloy must be known beforehand for testing purposes, so that after identification the result obtained from DDS can be matched, verified and authenticated accordingly. For instance, A pure metal or an alloy: Ag100 has density of 10.5 (in this case taken as standard density for the DDS). On subjecting this density (10.5) in DDS as input density with Au=19.32, Ag=10.5 and Cu=8.96 as constituents, it produces numerous compositions from pure metal to ternaries and conclusively determines the only concordant composition i.e. Ag100 as the unique solution confirming that the algorithm or the methodology of finding concordance between series effectively works accurately. If you wish to evaluate the functioning efficiency of this algorithm by testing the alloys of you own compositions, you may access our DDS platform (www.densityfingerprinting.com). We have evaluated many alloys (from pure metals up to 8-metal alloys) using DDS to understand the functioning and those results are also presented in both papers. Sincerely, Jai

We decoded density = 10.49 for Au-Ag-Cu and the algorithm found Au11.5Ag57.4Cu31.1 as the composition of the alloy for i=0.1. The results are provided below. 10.49.pdf

Surely. This should be available without paywall. Kindly refer the following paragraph, Table 1&2 and Figures 3&4 in the paper given below: "3.1.1 Decoding Au75 Ag18 Cu7 and Au90 Ag5 Cu3 Zn2 Alloys: Proof of Concept The Density Decoding System (DDS) was initially put to the test by decoding the densities of imaginary ternary and quaternary alloys into their elemental percent compositions. The alloy Au75Ag18Cu7 (D=15.68) was subjected as an input parameter in 3-Metals System, selected for Au, Ag, and Cu, with their respective standard densities of 19.32, 10.5, and 8.96, at a 1% scale of discretization (i.e., iterative step, i=1). The system generated three distinct series containing 21, 29, and 8 Probable Iso-density Compositions (PICs), with each series invisibly holding the Most Probable Composition (MPC) as Concordant Compositions (CCs) at serial numbers 8, 19, and 3, respectively, as shown in Table 1. Similarly, the quaternary alloy Au90Ag5Cu3Zn2 (D=17.3928) was tested in 4-Metals System, selected for Au, Ag, Cu, and Zn (d=7.14), at the same scale of discretization. The system yielded six series with 42, 56, 17, 43, 30, 8 and 86 PICs, each series containing MPC as CCs at serial numbers 23, 32, 10, 23, 17, and 46, respectively, as shown in Table 2" Reference: Rathore JH, Rathore P, Rathore B, Rathore BC. Awakening the Sleeping Giant: Rediscovering Archimedes’ Density Method for Fingerprinting of Multicomponent Alloys. ChemRxiv. 2024; doi:10.26434/chemrxiv-2024-wxzt9

Dear @exchemist and @John Cuthber Kindly refer the following paragraph and Table 2 in the paper given below: "On comparing the two Series, values seem to intersect each other at a point where the mass fractions of constituent metals in both compositions become exactly equal to each other (Fig. 2). Thus, only single composition emerges as common (concordant) composition within the two series, which stands as the ‘best probability’ among such a large number of probabilities. Its numerical values have also been found exactly equal to the actual composition of alloy for which the theoretical density was originally calculated. This peculiar pattern constitutes the finger-prints of alloy for a given density. It has also been observed that high precision and accuracy in densities are essentially required because a fractional variation may lead to incorrect results." Reference: B.C. Rathore, Pratibha and Bharati, “Theoretical optimisation of constitution of alloys by decoding their densities”, Materials Letters, Volume 61 (June 2007), 2956-2960. https://doi.org/10.1016/j.matlet.2006.10.052

Hi @studiot Thank you for sharing the table of specific gravities and the link showcasing the details of specific gravity. In the link shared: Since specific gravity is a comparison of two densities (material X and water), any sort of error in measurement of any of the two has a cascade effect and in turn diminishes the quality of the density measurement obtained from specific gravity. Apart from Invar, Ferronickel and cyclops 17, we noticed a couple others sharing same specific gravities such as Misco C, Durimet A, Pyrasteel, Durimet B sharing the same specific gravities: 7.89. Durimet A and Durimet B is an interesting case as it shares the same constituents except the inclusion of Chromium in Durimet B yet yields the same specific gravities. Please note: Durimet A Percent composition does not add up to 100: 75+20+5+0.25 =100.25 >100 Pyrasteel Percent composition does not add up to 100: 57+25+15+0.3 =97.3 <100 Durimet B Percent composition does not add up to 100: 48+35+12+5+0.25 =100.25 >100 Cyclops 17 Percent composition does not add up to 100: 70.9+20+8+0.75+0.4 =100.5 >100 Based on Standard Densities of metals and our density equation, the densities of the following alloys are as follows: Ferro Nickel Valve Steel Fe67.8Ni32C0.2: 8.13916 g/cc Invar Fe63.8Ni36C0.2: 8.178813 g/cc Feeding these into DDS: (the results are enclosed) Please note that the accuracy of measurements of the specific gravities of different alloys are from 1-2 decimal places (based on the number of decimal places reported in the table) and there are going to be errors in measurements. This raises an important question: Is this measured density the exact density of the alloy? Well, no, the exactness of the density depends on the measuring capability of the instrument, errors propagate in form of noise, etc. that plague the data to be utilized in further analysis. It is a similar situation as our assumption of conservation of volume, is it the truth? well, no, but it is a good enough assumption that enables us to perform analysis. To be clear, I am not saying that the measured specific gravities are wrong, but rather, they are not up to spec to be able to be decoded by DDS. Luckily in DDS, there is a limitation to the accuracy needed to decode a density correctly. In other words, any additional increment in accuracy of density does not affect the output from DDS once a critical threshold is reached (another finding mentioned in the preprint). I apologize for missing the first part of the question. I believe I have answered the inputs to the algorithm early on with an example. In regards to determining which metals to input, please see the following: And in the preprint, we mention that the type of alloy can be determined with various methods such as XRF, LIBS etc. to identify the alloying metals. These conventional radiation based nondestructive technologies are limited to surface analysis upto ~100 microns depth. So, they are not efficient to quantify the composition of bulk material. The integration of the two technologies shall enable us to decode the composition of bulk alloy/material. Since we assume conservation of volume, we theorize that the conditions for measuring the density of the alloy must be identical to the conditions of measuring the density of constituents to reduce the errors caused by temperature differences and other parameters at play such as pressure. This is the most crucial part as our system is sensitive to the accuracy of density measurements, in other words, “The ‘decodability’ of density depends on the quality of density obtained” @Mordred We are not using any experimental data in our paper except standard densities of metals. These standard densities are replaceable with measured densities or densities derived from atomic weight and lattice structure of constituents. As mentioned previously, we have considered an ideal case to understand the underdetermined system stripped off of any measurement errors. Thank you for suggesting the Hume-Rothery rules for narrowing down the composition of alloy from a tremendous amount produced after discretization. We appreciate your kind suggestions and we are looking into it very seriously. Introducing another parameter, in theory, should bring down the number of Probable Iso-density Compositions to decrease the computational load. In DDS, we have observed that regardless of the number of Probable Iso-density Compositions obtained, between two series, there is one and only one common composition (Concordant Composition or CC) that happens to be the correct composition (True Composition or TC). This has been the basis of identification of a unique composition in DDS. To understand this further, lets see the case of coins: @sethoflagos Thank you for the calculations you put forth. Believe it or not, we struggled with the same problem. Btw, the correct number of coins we chose for the problem is: 2100 nickels!!!, 1500 dimes and 2000 quarters I see you got the number of nickels correctly!!! but in this underdetermined system, although integer solutions brought down the number of solutions that work for the system of equations, it fails to converge to a unique solution. The equation used in coins case: Cost (C) =ac1+bc2+cc3 Weight (W) =aw1+bw2+cw3 Coefficients: c1,c2,c3 are terminating decimals w1,w2,w3 are terminating decimals Bounds: 0 ≤ a,b,c < ∞ The equation used in rule of mixtures: Density (D) =m1d1+m2d2+m3d3 Mass (M) =m1+m2+m3 Coefficients: d1,d2,d3 are terminating decimals 0 ≤ m1,m2,m3 ≤ M The equation we have considered: 100Density (D) =m1d1+m2d2+m3d3 Edited due to Latex issues: 100/Density(D) = m1/d1 +m2/d2+m3/d3 100=m1+m2+m3 Coefficients: 1/d1,1/d2,1/d3 are all non-terminating decimals (not irrational numbers as they can be represented in p/q form) And m1, m2 and m3 are limited to [0,100] These non-terminating coefficients and bounds make sure that there is no repetition of the common composition in the alloy space! Also, if you notice, the relationship between the density and mass percent is no longer linear but an inverse relationship as is observed experimentally as opposed to rule of mixtures. Thus, it is abundantly and explicitly clear that we are NOT following the rule of mixtures in any manner. @KJW Thank you very much for your kind suggestion. Really it is worth appreciable. We wish to know and learn more about this problem statement in detail and your kind help and guidance is anticipated positively. I hope I was able to address all the questions raised so far. Please let me know if anything is missed or still unclear. Sincerely, -Jai on behalf of the research team PS: This is my first time working with latex in this forum, so if the equations don't turn out readable, i'll repost. Ferro Nickel Valve Steel.pdf Invar.pdf

Hi everybody, Apologies for a delayed response. Thank you for all the great comments and questions. I shall try to address all of them, so please let me know if I missed or misinterpreted something. The questions and comments I gathered are as follows: 1. Specific Gravities of Invar, Cyclops 17 and Ferro Nickel are exactly 8.0 yet they have different compositions so how can Specific Gravity be an identifying factor? The density equation considers Density not Specific Gravity. We have not worked with the mentioned alloys but we found the following densities of the alloys: Invar 36: 8.055 g/cc https://www.hightempmetals.com/techdata/hitempInvar36data.php Cyclops 17: --- Not sure, I could not find any references for this alloy Ferronikel FeNi 25: ~8.1 g/cc https://a.storyblok.com/f/94542/x/154109402a/ferronickel-data-sheet.pdf FeNi 55: ~8.4 g/cc Please note the differences in the alloy densities. This difference, however minuscule, is important for DDS to detect the different alloys. 2. The research paper is missing the Methodology of alloy detection. What is the methodology? In the paper the Normal (or Forward) and Reverse equations (Summation equations) are mentioned that are literally rewritten Alloy Density equation. From computing the mass percents of two metals by iterating the remaining, produces two lists of compositions, a Forward Series and a Reverse Series is obtained. The iterative step can be i=1,0.1,0.01,0.001…. so, not just integer compositions but fractional compositions can also be identified. The paper mentions the two series computed in tabulated form to showcase the following: a. The desired composition is present in both the series. b. The desired composition is the Only Common Composition (a.k.a. Concordant Composition) between the two series and is thus shown as the detected alloy. This summarizes the alloy detection algorithm. 3. The alloy has to be a perfect lattice. The Alloy density equation demands an ideal condition, thus this point is valid. In real world, there are going to be some deviations from grain boundaries, lattice defects, lamellar microstructures etc. Since the topic of this research is developing a method to attempt at solving the underdetermined system produced for 3 or more metal alloys, a simple equation to model an ideal condition is used to prevent deviations and experimental errors. So, no volume change during alloying process or porous structure is to be considered. Because of this ideal nature of the alloy, any inclusion of new metal or fluctuation in composition brings a change in density and that change drives the algorithm to find the new correct composition. 4. The Constituents are required to use density to identify alloy. So how is alloy density the only parameter that can identify the composition? This method is extendable to any number of metals in an alloy. Since not every combination of metals produce an alloy, the maximum number of metals to be considered can be limited to max 7 or 8. This would include trace metals as well. This does however increase the computational load. In case, we select an 8-Metals system, it may effectively decode each single density from the lowest to highest densities of alloy bearing any conceivable combination of the preselected metals. For instance, 8-Metals system selected for Pt(d=21.45), Au (19.32), Ag (10.5), Cu (8.96), Co (8.9), Sn (7.31), Zn (7.14) and Al (2.7) may correctly characterize all conceivable potential alloys i.e., from binary to octonary alloys of preselected metals including constituents within seconds. A list of some binary to octonary alloys of preselected metals including constituents is tabulated in Table 3 of our preprint paper. 5. If integer compositions are being detected to be the alloy composition then the alloy identification problem is obviously solvable with brute force composition calculation. I like the analogy by @sethoflagos: Will you kindly allow me to most respectfully ask a simple question based on your analogy: Suppose, I am having a box of coins worth $755.00 weighing 25.242kg bearing coins of denominations 5 (nickel), 10 (dime) and 25 (quarter) cents, weighing 5g, 2.268g and 5.67g respectively. We do not understand how to compute the number of coins of each denomination in the box. Will you kindly help us in solving this problem? Coin weights: https://uscode.house.gov/view.xhtml?req=granuleid:USC-prelim-title31-section5112&num=0&edition=prelim 6. The density accuracy needs to be 4+ decimal places Not always, it can be seen that alloy densities that lie towards the least and most dense constituent metal require as low as 2 decimal place accuracies. Towards the center of the spectrum, the accuracy increases to 4+ decimal places. Please note that the cheap densitometers currently available have density measurement accuracy of 0.001 g/cc with repeatability of 0.002 g/cc. This is bordering the range required for Density Decoding System. @sethoflagos I am not sure what you meant by the “large common denominator” in this: Will you please elaborate? Sincerely, Jai on behalf of Research Team

Hi @Mordred, Thank you very much for suggesting the latex structuring for mathematical equations in future responses. The methodology, algorithm, and mathematics used in this research have already undergone rigorous peer review in our previous paper: "Theoretical Optimization of Constitution of Alloys by Decoding Their Densities", published in Materials Letters (Elsevier) in 2007 Link to peer reviewed paper: https://doi.org/10.1016/j.matlet.2006.10.052 This work has already been cited in several research papers. This published work outlines the fundamental principles of our approach, and further explores the dynamics and insights of the Density Decoding System. We acknowledge that our explanations may have been unclear or not easily understandable. We will make effort to present our methodology and findings in a more transparent and understandable manner. We shall focus on the key aspects of our work and their implications. Sincerely, Jai on behalf of research team

Hi all! Archimedes proved that the crown was not pure gold by doing the following: 1. Measuring the volume of the crown using immersion in water (Archimedes's Principle) 2. Compare the density obtained from the volume and mass for the crown with that of pure gold D(crown) ≠ D(pure gold) Therefore the crown can not be pure gold! The events could be subject to speculation but, using density, he was able to prove that the crown was not gold. The density of the alloy is dependent on many factors such as composition, temperature, in some cases even pressure. Archimedes' work was a qualitative analysis based on the density - composition relationship. Any perturbations in the composition of an alloy will affect its density. Since, the mass percents of each constituent affects the density, this means that the information of constituents is already naturally "encoded" in the form of density, all we need to do is "decode" this information by understanding the dynamics of density-constituent relationship. The work of Archimedes as described above has been quantified already for binary alloys which utilizes conservation of mass from constituents to alloy and assumes conservation of volume. We extended this relationship further for ternary, quaternary etc. This produces the governing equation: V=v1+v2+v3... ---(1) M=m1+m2+m3... ---(2) Rewriting into density terms: D=M/(m1/d1 +m2/d2 +m3/d3 +....) In terms of mass percents, this shows a linear relationship. Since there are two equations (1) & (2), only two variables can be uniquely identified, i.e. binary alloys! Extending this for multi-component alloy makes this an underdetermined system. We tackled the problem of underdetermined system by first considering mass percents (M=100), so the alloy space (VAS) constricts to the area in ternary plot (3-metals), tetrahedral plot (4-metals) etc. Then we discretized. This is Density Decoding System (DDS). The results of this are the following: Test alloy: Produce a theoretical alloy density based on the governing equation e.g., Au90Ag5Cu3Zn2 -> 17.3928 [Au:19.32, Ag:10.5, Cu:8.96, Zn:7.14] 1) Calibrate DDS: Metal Densities, Iterative Step (used in convergence, dictates the discretization) e.g., selected: Au(19.32), Ag(10.5), Cu(8.96), Zn(7.14); i=1 2) Input: Alloy Density (Theoretical) e.g., Density: 17.3928 3) Output: Percent Composition for multi-component alloy e.g., Alloy: Au90Ag5Cu3Zn2 We have presented upto 8-metal alloy identification using density in the paper. I hope this clears things as this has been peer reviewed already in 2006. In this paper we delved deeper into the functioning of the algorithm and tried to understand why this method works. In this pursuit, we have found evidences of chromosomal structure of probability distributions of the probable iso-density compositions, butterfly effect stemming from alloy density, principle of vernier caliper in multi-dimensions etc. We wish to share these findings through this paper. Please take a look as we believe it is a fascinating find. Sincerely, Jai on behalf of research team