REVIEW INVITATION : Awakening the Sleeping Giant: Rediscovering Archimedes' Density Method for Fingerprinting of Multicomponent Alloys

[JaiHind15]

11 hours ago, Mordred said:

May be helpful to include the peer review article which includes the related mathematics and methodology.

In case you wish to post those related mathematics here (recommended) the latex structure uses the

\[ latex\*] tag for new line inline \(latex\*) the * is simply there to prevent activation.

 

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

[Mordred]

Yes you already stated that above 

[studiot]

11 hours ago, sethoflagos said:

If integer values are used for component mass percentages in calculating this 'Archimedean' density function, then performing a brute force scan of the inverse function for integer solutions can recover those input integer values. 

 

  

 

SG alone not going to be enough information.

 

For instance looking at random in Lange

Invar, Cyclops 17 and Ferro Nickel valve steel all have sgs of exactly 8.0.

Yet they have very different compositions.

Many others cluster in the region 7.8 - 8.2

[Mordred]

So from the peer review article mentioned above which only has summation equations one must know what elements are likely included in the allow and the density of each alloy.

As that peer review article is extremely short It didn't indicate anything beyond that 

I didn't see any methodology of detection 

Am I correct on the above ?

Edited October 19 by Mordred

[sethoflagos]

3 hours ago, studiot said:

SG alone not going to be enough information.

It can be

IFF: - You know all the components beforehand

        - The components have significantly different elemental densities

        - Elemental densities have no significantly large common denominator

        - The alloying process does not induce a volume change

        - Compositions are precise integer percentages by mass (or moles, volume etc)

        - Alloy lattice is flawless

        - Density measurements are accurate to ~ 4 significant figures +   

[rathorebc]

On 10/18/2024 at 2:30 AM, exchemist said:

I see a bit of problem with this. The density of an alloy will in generral not be a simple linear interpolation between the densities of the components. There will be a degree of interaction between the different elements present, according to their mutual chemical affinity or otherwise, and effects due to the packing of atoms of dissimilar size in the metal lattice. This is addressed for example in this piece of work: https://www.sciencedirect.com/science/article/pii/S0364591619302524. in which the enthalpy of mixing is used as a way to estimate these effects.

I am also rather confused by the following sentence in your post: " 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."  

What is meant by chromosomal structure of probability distributions?

What is meant by a butterfly effect in this context?

What is meant by principle of vernier caliper in multi-dimensions?

 

 

On 10/18/2024 at 7:34 AM, sethoflagos said:

+1

The referenced discussion document makes no allowance that I can find for alloy density being a function of a lattice structure specific to that alloy. Rather, as you seem to suspect, alloy densities are simply assumed to be mass weighted averages of elemental densities.

As far as I can tell, such a weighted average is calculated from an alloy composition constructed from integer component percentages, and of the infinite potential compositions that match that density, the composition that most closely yields integer percentage values is picked as the 'Most Probable Composition'. It's a few years since I studied statistical analysis techniques and I think I must have missed the lecture on the Hogwarts Sorting Hat.

Best guess:- deadcatting.

Dear @exchemist and @sethflagos,

Thank you for your interest in our work and for raising important questions about the concepts we mentioned in our previous post. We appreciate the opportunity to provide more detailed explanations and clarify any confusion.

1. Chromosomal structure of probability distributions: In our analysis of the Probable Iso-density Compositions (PICs) generated by the Density Decoding System (DDS), we observed a striking similarity between the probability distributions of PICs across different series and the structure of chromosomes. Just as chromosomes contain genetic information organized into distinct regions, such as centromeres and telomeres, the probability distributions of PICs exhibit a non-random, structured pattern.

Specifically, we found that the true composition of an alloy consistently appears as a highly probable point in each PIC series, reminiscent of the role of centromeres in chromosomes. Centromeres are crucial for the proper segregation of genetic material during cell division, and similarly, the true composition acts as a focal point that connects and aligns the PIC series. This chromosomal analogy provides a framework for understanding the underlying organization and information content of the alloy composition space.

2. Butterfly effect: The butterfly effect is a concept from chaos theory that describes how small changes in initial conditions can lead to large-scale, unpredictable consequences in complex systems. In the context of our work, we observed a phenomenon analogous to the butterfly effect when analyzing the sensitivity of the matched composition to slight variations in the input density.

As we incrementally changed the input density, we noticed that the matched composition—the point at which PICs from different series converge—exhibited abrupt shifts at certain critical density values. These sudden changes in the matched composition, despite the small changes in density, are reminiscent of the butterfly effect, where a minor perturbation can trigger a significant alteration in the system's behavior.

This finding highlights the intricate and nonlinear nature of the relationship between alloy density and composition, and it underscores the importance of high-precision density measurements for accurate composition determination. Understanding the butterfly effect in this context can guide the development of more robust and reliable methods for alloy characterization and design.

3. Vernier caliper principle in multi-dimensions: The Vernier caliper is a precision measuring tool that uses two scales with slightly different spacings to achieve high accuracy. The principle behind the Vernier caliper relies on the alignment of the two scales at specific points, allowing for precise measurements that exceed the resolution of either scale alone.

In our work, we discovered a multi-dimensional analog of the Vernier caliper principle in the convergence of PICs from different series to the true composition. Each PIC series can be thought of as a scale, with the individual PICs representing the markings on the scale. When multiple PIC series are combined, they form a multi-dimensional space where the true composition is located at the point of convergence, similar to the alignment of scales in a Vernier caliper.

This multi-dimensional Vernier caliper principle allows us to pinpoint the true composition with high accuracy by leveraging the collective information from multiple PIC series. The convergence of PICs from different series acts as a self-reinforcing mechanism, increasing the confidence in the determined composition. This principle underscores the power of the DDS in navigating the vast composition space and identifying the true composition among numerous possibilities.

We hope these detailed explanations provide a clearer understanding of the concepts we introduced and their relevance to our work. We encourage you to explore our recent paper for a more comprehensive discussion of these findings and their implications for alloy characterization and design.

Thank you again for your engagement and thought-provoking questions. We value the input and expertise of the scientific community in refining our methodology and advancing our understanding of these fascinating phenomena.

Sincerely,

Dr. B. C. Rathore and Research Team

[studiot]

11 minutes ago, sethoflagos said:

It can be

IFF: - You know all the components beforehand

        - The components have significantly different elemental densities

        - Elemental densities have no significantly large common denominator

        - The alloying process does not induce a volume change

        - Compositions are precise integer percentages by mass (or moles, volume etc)

        - Alloy lattice is flawless

        - Density measurements are accurate to ~ 4 significant figures +   

You are misreading my point.

Yes,  but if you are told what its constituents are then you have more information than just the overall density.

If the only information you have on a sample is its density it is impossible to determine the alloy.

This was why I asked earlier about the spectroscopy part of the original OP picture.

To date I have received no answer to this question or the very natural one of

What else do you need ?

 

 

 

 

[sethoflagos]

46 minutes ago, studiot said:

You are misreading my point.

On the contrary, your point is abundantly clear. But I think that bird has flown a while back. The mystery is not in the mineralogy, it's in the mathematical method.  

1 hour ago, rathorebc said:

we found that the true composition of an alloy consistently appears as a highly probable point in each PIC series, reminiscent of the role of centromeres in chromosomes.

 

1 hour ago, rathorebc said:

These sudden changes in the matched composition, despite the small changes in density, are reminiscent of the butterfly effect,

Please refrain from tabling dead cats. The topic is metallurgy.

What we have here is effectively a closed box containing material of a precise given weight and monetary value.

Given no further information I have no idea what the contents are.

However, if I'm told the contents consist solely of nickles, quarters and dimes, I think I might be able to work it out. 

Restricting a search solely to integer roots of equations provides a massive simplification.

[JaiHind15]

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:

 

On 10/19/2024 at 11:31 AM, sethoflagos said:

On the contrary, your point is abundantly clear. But I think that bird has flown a while back. The mystery is not in the mineralogy, it's in the mathematical method.  

 

Please refrain from tabling dead cats. The topic is metallurgy.

What we have here is effectively a closed box containing material of a precise given weight and monetary value.

Given no further information I have no idea what the contents are.

However, if I'm told the contents consist solely of nickles, quarters and dimes, I think I might be able to work it out. 

Restricting a search solely to integer roots of equations provides a massive simplification.

 

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:

 

On 10/19/2024 at 10:16 AM, sethoflagos said:

It can be

IFF: - You know all the components beforehand

        - The components have significantly different elemental densities

        - Elemental densities have no significantly large common denominator

        - The alloying process does not induce a volume change

        - Compositions are precise integer percentages by mass (or moles, volume etc)

        - Alloy lattice is flawless

        - Density measurements are accurate to ~ 4 significant figures +   

Will you please elaborate?

 

Sincerely,

Jai on behalf of Research Team

[Mordred]

So your using a densitomer for detection correct ?

I take it your algorithm isn't applying any other algorithm other than what you described briefly above. Have you considered incorporating Hume-Hothery rules to help narrow down possible alloy configurations ?

Or other methods of narrowing down possible configurations such as alloy conductivity? This may prove useful in narrowing down computations 

Edited October 21 by Mordred

[sethoflagos]

19 hours ago, JaiHind15 said:

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?

Of course, though $755.00 is a lot of coins, and it is a highly instructive question for a reason I didn't anticipate (or at least bother to check in advance). 

It turns out that I'd wrongly assumed that the standard coin masses have no large common denominator whereas your reference indicates that five dimes and two quarters actually have indentical mass and value. This guarantees that if there is one solution, there are many: just substitute five dimes for two quarters as many times as you like.

Let us start by guessing there are no quarters. So nickels and dimes sum to $755.00 and 25.242 kg. Two variables; two equations which simply solve to 2,100 nickels: 6,500 dimes. Yay!! Integers!! So we have a solution!!

Do the same for one quarter and we get 2,100 nickels again but a non-integer number of dimes (6,497.5) so we can reject this one.

But for two quarters we get another solution 2,100 nickels: 6,495 dimes - ie we've taken the first solution and exchanged 5 dimes for 2 quarters. And so on...

So we can do no better than state that the box contains $105 worth of nickels and $650 dollars of mixed dimes and quarters...

... because:

19 hours ago, JaiHind15 said:

I am not sure what you meant by the “large common denominator” in this:

 

On 10/19/2024 at 3:16 PM, sethoflagos said:

It can be

IFF: - You know all the components beforehand

        - The components have significantly different elemental densities

        - Elemental densities have no significantly large common denominator

        - The alloying process does not induce a volume change

        - Compositions are precise integer percentages by mass (or moles, volume etc)

        - Alloy lattice is flawless

        - Density measurements are accurate to ~ 4 significant figures +   

Expand  

 

 

Will you please elaborate?

The above example illustrates this very well. If instead of 5.67g we set the mass of the quarter to 5.669g, we retain the previous solution of 2,100:6,500:0 however, the slight deviation from a large common denominator introduces increasingly large deviations from integer values which invalidates all other potential candidates. This is easily demonstrated with a simple Excel spreadsheet (I've omitted lines 16 - 3,200 for sanity's sake)

Your methodology requires significant common denominantors in alloy composition figures to keep the number of permutations of composition down to a manageable finite number (to facilitate a brute force computational sieve), but component densities should ideally be irrational numbers (which in actuality we'd expect them to be) to prevent the existence of multiple integer solutions.

19 hours ago, JaiHind15 said:

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.

For purposes of my argument, any equal incremental step process is essentially based on stepping through integers.

I'd often wondered in the past why banks etc. went to the trouble of counting coins individually rather than just weighing them in batches and exploiting the limited possible combinations to compute the value. Now I've a clearer picture. Thank you for that.  

Edited October 21 by sethoflagos

[studiot]

1 hour ago, sethoflagos said:

I'd often wondered in the past why banks etc. went to the trouble of counting coins individually rather than just weighing them in batches and exploiting the limited possible combinations to compute the value. Now I've a clearer picture. Thank you for that.

Thanks for that clearer picture.  +1

 

20 hours ago, JaiHind15 said:

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.

A really disappointing non scientific response.

Density and specific gravity are interconvertible and in the correct circumstances numerically equal.

 

Quote

https://www.riccachemical.com/pages/tech-tips/density-and-specific-gravity

 

Density and Specific Gravity are never the same, because Density has units and Specific Gravity is dimensionless, but they are numerically equal when three conditions are met: (1) Density is measured in grams per cubic centimeter, grams per milliliter, or kilograms per liter; (2) Density and Specific Gravity are measured at the same temperature; and (3) the Specific Gravity is referenced to water at 4°C, where its Density is very close to 1 gram per cubic centimeter. The Density of a substance may be calculated by multiplying the Specific Gravity by the Density of water at the reference temperature.

Of course temperature also plays a part but we are then talking of several decimal places and you have yet to mention the effect of sample temperature on your measurements.

 

Since you could not find all my alloys, my mid 1970s copy of Lange gives the following information, including wt %

 

I no longer have access to up to date handbooks to compare with.

 

 

Finally this is the fourth and last time I ask my most important question, which has been steadfastly ignored.

 

On 10/18/2024 at 5:17 PM, studiot said:

What would be most helpful would be to state plainly what measurements are needed to identify an unknow metal specimen.

Ie what do you need to know to input into your 'algorithm' ?

 

 

 

[KJW]

On 10/21/2024 at 6:05 AM, JaiHind15 said:

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?

Related to this is the problem of determining the molecular formula of a compound from the molecular mass obtained using a high-resolution mass spectrometer.

 

[JaiHind15]

Hi @studiot

Thank you for sharing the table of specific gravities and the link showcasing the details of specific gravity.

In the link shared:

On 10/21/2024 at 12:09 PM, studiot said:

https://www.riccachemical.com/pages/tech-tips/density-and-specific-gravity

 

Density and Specific Gravity are never the same, because Density has units and Specific Gravity is dimensionless, but they are numerically equal when three conditions are met: (1) Density is measured in grams per cubic centimeter, grams per milliliter, or kilograms per liter; (2) Density and Specific Gravity are measured at the same temperature; and (3) the Specific Gravity is referenced to water at 4°C, where its Density is very close to 1 gram per cubic centimeter. The Density of a substance may be calculated by multiplying the Specific Gravity by the Density of water at the reference temperature.

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.

On 10/21/2024 at 12:09 PM, studiot said:

 

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 Fe_67.8Ni₃₂C_0.2: 8.13916 g/cc
Invar Fe_63.8Ni₃₆C_0.2: 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.

On 10/16/2024 at 9:23 PM, JaiHind15 said:

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 regards to determining which metals to input, please see the following:

On 10/20/2024 at 4:05 PM, JaiHind15 said:

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.

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

On 10/21/2024 at 11:06 AM, sethoflagos said:

Let us start by guessing there are no quarters. So nickels and dimes sum to $755.00 and 25.242 kg. Two variables; two equations which simply solve to 2,100 nickels: 6,500 dimes. Yay!! Integers!! So we have a solution!!

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

Edited October 29 by JaiHind15
Latex issues in the equations mentioned. Preview was good but after posting, it changed the equation.

[sethoflagos]

3 hours ago, JaiHind15 said:

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!!!

Your input combination was one of the 1,300 other possible solutions also given on my spreadsheet. There is no unique 'correct' solution.

3 hours ago, JaiHind15 said:

The equation used in rule of mixtures:

 

Density (D) =m₁d₁+m₂d₂+m₃d₃

Mass (M) =m₁+m₂+m₃

If D is volume, and d is specific volume, then yes.

3 hours ago, JaiHind15 said:

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!

If any of the various q values share a common factor, there will likely be multiple solutions. This is readily apparent in the coins problem as the common factors are of a similar order of magnitude to q. The problem does not go away even with irrational numbers.

3 hours ago, JaiHind15 said:

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.

Express in terms of specific volume rather than density and the form is identical to your stated rule of mixtures.

Edited October 29 by sethoflagos
small clarification

[John Cuthber]

Copper is less dense than silver.
Gold is more dense than silver.
Therefore there is a binary alloy of copper and gold which has exactly the same density as silver. We can call it "match" because it has a density that matches that of silver.

An alloy of mainly copper with a little gold will be slightly more dense than copper. We can refer to this alloy as "light" and similarly we can have an alloy, which we can call "heavy" made from mainly gold with a little copper.

If we start with pure silver we can add a small amount of heavy which will raise the density, and then we can add light to reduce the density.
The overall effect will be to produce an alloy which has the same density as silver and which is a ternary alloy.
We can repeat that process and get a second alloy- again containing all 3 metals and which has exactly the same density as silver. It will have a different composition from the first one.

And we can , add further amounts of light and compensating amounts of heavy until we get arbitrarily close to "match".

All these alloys have the same density as silver.
How does your method distinguish among them?

(The option of extending this to have an alloy with, for example, a density half way between that of gold and silver or whatever also exists)




 

Edited November 8 by John Cuthber

[rathorebc]

Hi @sethoflagos

On 10/29/2024 at 5:14 AM, sethoflagos said:

If any of the various q values share a common factor, there will likely be multiple solutions. This is readily apparent in the coins problem as the common factors are of a similar order of magnitude to q. The problem does not go away even with irrational numbers.

It is the combination of both; Bounds and Non-terminating decimal expansions that ascertain unique solutions.

On 10/29/2024 at 5:14 AM, sethoflagos said:

Express in terms of specific volume rather than density and the form is identical to your stated rule of mixtures.

Of course, it is absolutely correct. Nevertheless, we have not considered this term (specific volume), instead we have emphasized the importance of Density being in denominator as a consequence of the volume additions. As a result, we obtained the correct compositions associated with input densities of non-binary alloys, which would have not been possible.

 

@John Cuthber

On 10/29/2024 at 10:14 AM, John Cuthber said:

Copper is less dense than silver.
Gold is more dense than silver.
Therefore there is a binary alloy of copper and gold which has exactly the same density as silver. We can call it "match" because it has a density that matches that of silver.

An alloy of mainly copper with a little gold will be slightly more dense than copper. We can refer to this alloy as "light" and similarly we can have an alloy, which we can call "heavy" made from mainly gold with a little copper.

If we start with pure silver we can add a small amount of heavy which will raise the density, and then we can add light to reduce the density.
The overall effect will be to produce an alloy which has the same density as silver and which is a ternary alloy.
We can repeat that process and get a second alloy- again containing all 3 metals and which has exactly the same density as silver. It will have a different composition from the first one.

And we can , add further amounts of light and compensating amounts of heavy until we get arbitrarily close to "match".

All these alloys have the same density as silver.
How does your method distinguish among them?

(The option of extending this to have an alloy with, for example, a density half way between that of gold and silver or whatever also exists)




 

Thank you for this wonderful question.

It is counterintuitive, the way DDS performs, to what is expected from an underdetermined system.

Initially, we have defined “match” as a binary alloy of Au (19.32) and Cu (8.96) having density equal to pure Ag (10.5). We have also defined “Light alloy” comprising Cu with little Au and “heavy alloy” comprising Au with little Cu respectively.

As per question, we began mixing these “light” and “heavy” alloys with pure silver ‘bit by bit’ in such a manner, so that the overall density of resultant alloy remains intact equal to the density of pure silver (10.5). We wanted to know about the composition of ternary alloys so obtained.

We further kept on repeating the mixing of “light” and “heavy” alloys continuously with resultant ternary alloy until its composition reached close to “match” bearing density equal to pure silver (i.e., 10.5).

We wished to know the compositions of various alloys obtained during the entire course of mixing process from “pure silver” to the “match”, particularly when all these alloys have same density as that of silver (10.5).

To solve this problem, we subjected the density of pure Ag (10.5) as input parameter in Density Decoding System (DDS), initially selected for constituent metals Au, Ag and Cu along with their standard densities 19.32, 10.5 and 8.96 respectively. It was operated at initial (the lowest or ground state) accuracy level i.e., Iterative step (i)=1. This accuracy level visualizes only the whole number compositions of alloys along with their allied compositions of next higher accuracy levels.

The DDS, effectively displayed the presence of Ag₁₀₀ in result along with three discrete PIC series, each one showing presence of Ag₁₀₀ as Concordant Compositions (CCs), the density spectrum revealing fingerprints of Ag₁₀₀, the triangular plot of PICs showing the Isopycnic Region (IR) of Ag₁₀₀ and 2D projection of two PIC series. The detailed results are enclosed as Figure-1.

 

https://densityfingerprinting.github.io/Discussions/Ternary%20Plot%2010.5

The Isopycnic Region (IR) or Alloy Space (AS) of Ag₁₀₀ obtained at lowest accuracy level (i=1) shows the following trends:

(i)     It initially shows the presence of pure silver (Ag₁₀₀) at the vertex of Ag of trigonal plot which extends further in straight line obliquely intersecting the Au-Cu axis and finally terminates constituting a binary alloy of Au_27.41Cu_72.59. 

(ii)   Just after Ag₁₀₀, the IR shows presence of very first ternary alloy Au_0.3Ag₉₉Cu_0.7 and the last (penultimate) ternary alloy with composition Au_27.1Ag_0.9Cu_72.0 before binary alloy.

In compositions, the Ag content exhibits a decreasing trend across the IR from pure Ag to binary alloy with increasing Au and Cu contents, which ultimately becomes significantly low and practically vanishes to become untraceable. At higher accuracy levels, i= 0.1 and 0.01, the Ag contents in corresponding penultimate ternary compositions have been recorded being Au_27.334Ag_0.067Cu_72.599 and Au_27.35Ag_0.005Cu_72.645 respectively.

(iii)      Interestingly, in IR of Ag₁₀₀, we recorded ternary alloys containing equal amount of Au & Ag (such as Au_21.5Ag_21.5Cu₅₇) as well as similar amount of Ag & Cu (such as Au_15.9Ag₄₂Cu_42.1) at density 10.5, equal to pure Ag.

(iv)     There are, however, infinite number of unique ternary compositions in entire Alloy Space of Ag₁₀₀ between pure Ag and Au-Cu binary alloy, but only finite number of alloys are visualized at particular iterative step (i).

 

As already stated, that the iterative step (i) intelligently slices the infinite composition space for a given density at its particularly selected accuracy level. For instance, the iterative step (i=1) visualizes only the PIC series associated with whole number compositions, whereas the reducing iterative steps such as i=0.1, 0.01, 0.001, and so on, show the PICs of corresponding higher accuracy levels i.e., accuracy of compositions up to respective decimal places 0.1, 0.01, 0.001, and so forth. 

 

At each accuracy level, the iterative step shows finite number of compositions in PIC series without redacting a single composition. The number of PICs at each accuracy level shows exponential increase resulting in higher and higher computational load making the problem NP-Hard.

 

Now, the question arises, how does DDS decode the different alloys of given Isopycnic Region (IR) or Alloy Space (AS) despite identical densities?

 

The Isopycnic Region (IR) or Alloy Space (AS) of a density is actually the composition space, where densities of all unique infinite compositions are identical.

 

In order to experimentally showcase the functioning of DDS, how does it determine the different compositions in an Isopycnic Region (IR), we selected the following compositions of alloys obtained from the IR or Alloy Space of Ag₁₀₀ and determined their compositions using DDS, initially selected for Au, Ag and Cu constituents along with their standard densities:

 

1.      Au_27.41Cu_72.59 (D=10.5038712); ∆D = +0.0038712

The composition of Au_27.41Cu_72.59 for binary alloy may be determined conventionally by Archimedes Density equation being Au= 27.41% and Cu=72.59%. It may also be calculated through DDS for its varied density 10.5038712, which shows a difference ∆D = +0.00387 from density of pure Ag (10.5).

On subjecting density D=10.5038712 in DDS as input parameter, selected at iterative step i=0.01, we obtained correct composition of binary alloy of Au_27.41Cu_72.59 as shown in Figure-2_. The interactive fingerprint of this alloy is shown as follows:

 

https://densityfingerprinting.github.io/Discussions/Spectrum%20Plot%20AuCu%20Binary

 

2.      Au_27.1Ag_0.9Cu_72.0 (D = 10.49966) ∆D = -0.00034

On subjecting density 10.49966 in DDS as input parameter, selected at i=0.1, we obtained correct composition of ternary alloy of Au_27.1Ag_0.9Cu_72.0 as shown in Figure-3_. The interactive fingerprint of this alloy is shown as follows:

https://densityfingerprinting.github.io/Discussions/Spectrum%20Plot%20Last%20Ternary

 

3.      Au_0.3Ag₉₉Cu_0.7 (D=10.501747) ∆D = +0.001747

On subjecting density 10.501747 in DDS as input parameter, selected at i=0.1, we obtained correct composition of ternary alloy of Au_0.3Ag₉₉Cu_0.7 as shown in Figure-4_. The interactive fingerprint of this alloy is shown as follows:

 

https://densityfingerprinting.github.io/Discussions/Spectrum%20Plot%20of%20first%20ternary%20from%20Ag

 

4.      Au_21.5Ag_21.5Cu₅₇ (D=10.501926) ∆D = +0.001926

On subjecting density 10.501926 in DDS as input parameter, selected at i=0.1, we obtained correct composition of ternary alloy of Au_21.5Ag_21.5Cu₅₇ as shown in Figure-5_. The interactive fingerprint of this alloy is shown as follows:

https://densityfingerprinting.github.io/Discussions/Spectrum%20Plot%20of%20Au21.5Ag21.5Cu57

 

5.      Au_15.9Ag₄₂Cu_42.1 (D=10.50239) ∆D = +0.00239

On subjecting density 10.50239 in DDS as input parameter, selected at i=0.1, we obtained correct composition of ternary alloy of Au_15.9Ag₄₂Cu_42.1 as shown in Figure-6_. The interactive fingerprint of this alloy is shown as follows:

https://densityfingerprinting.github.io/Discussions/Spectrum%20Plot%20of%20Au15.9Ag42Cu42.1

 

In the last, you have asked the option of extending this (mixing “light” and “heavy” alloys with silver) to have an alloy with, for example, a density half way between that of gold and silver or whatever also exists.

 

When we continued mixing “light” and “heavy” alloys with pure silver, the density lying between Au and Ag (i.e., D=14.91) produced a ternary alloy Au₆₇Ag₂₇Cu₆; density between Cu and Ag (i.e., D=9.55) produced Au₁₁Ag₂Cu₈₇; whereas the density lying between Au and Cu (i.e., D=14.14) produced Au₅₉Ag₃₄Cu₇. The results are enclosed as Figure-7, 8 and 9 respectively.

 

The Isopycnic Regions/Alloy Spaces (IRs/ASs) for alloy compositions obtained from decoding densities D=14.91, 14.14, and 9.55 were found to be parallel to each other and to that of Ag₁₀₀ with density D=10.5. The main difference was that densities D=14.91 and 14.14 showed Au-Ag and Au-Cu binaries on their corresponding axes, while density 9.55 showed an Ag-Cu binary on the extreme left and an Au-Cu binary on the extreme right. However, all IRs exhibited similar trends to the IR/AS of Ag₁₀₀ with density D=10.5.

 

The veracity of these results and findings can be conveniently inspected and observed in the interactive graphics by clicking the corresponding links embedded in each image. Comprehensive results obtained from the DDS for the corresponding alloys are enclosed in pdfs from Figures-1 to 9.

 

Remarkably, the DDS effectively exploits differences in alloy densities (∆D) to correctly identify the composition associated with each density, producing a tremendous number of fractional compositions with non-terminating mass percents.

 

The segregation of PICs into different PIC series based on the ⁿC₂ combinatorial notation and the interconnectedness of PICs through Concordant Compositions (CCs) play a crucial role in the correct identification of the True Composition (TC) associated with the input density.

 

In the ternary plots shown for any composition, the number of unique PICs in each series is different and equidistant, but not between series. By considering one series as the main scale and another as the vernier scale, the vernier coincidence produced will always be a unique solution representing the coincidence point between the two scales as the correct value. Interestingly, all PIC series collectively show an identical point of coincidence, authenticating the presence of the True Composition (TC) for the input density.

 

For instance, in the ternary plot shown for Ag₁₀₀, it can be seen that the PICs in each series are equidistant (blue to blue dot or red to red dot or green to green dot) but between the series, they are not (red to next blue etc.).

 

Think about taking any one series as the main scale and the second as vernier scale. The vernier coincidence produced will always be a unique solution representing the coincidence point between the two scales as the correct value.

 

In summary, a subset of underdetermined systems has been found that can be solved in a very particular fashion, as mentioned in the preprint publication. Other concepts discussed include superpositions/superimposition or overlapping of CCs, breaking of asymmetry of composition fractals by perfectly ordered C-bands of CCs, and wave interference patterns shown by PIC series analogous to quantum phenomena.

 

 

I hope, I was able to address the questions put forth. In case, I missed something or am not clear in explanation, please feel free to reach out to us accordingly. Your kind feedback, comments etc. shall be highly appreciated.

 

Sincerely,

 

Dr. B. C. Rathore on behalf of research team

 

Comments from the team members:

 

“We, the authors, wholeheartedly commend and appreciate this question posed in the form of a beautifully crafted brilliant brain-teaser, which elegantly encapsulates the core challenges our research aims to address, such as the unsolvable underdetermined systems in non-binary alloys, the infinite solutions (probabilities) they generate, and the DDS's ability to conclusively determine the True Composition (TC) from these probabilities for a given input density. This insightful inquiry demonstrates a deep understanding of the complexities in multicomponent alloy characterization and the significance of our novel approach. We are grateful for the opportunity granted by this wonderful forum to engage in a meaningful discussion that further elucidates the key aspects of our work.”

 

 

 

Figure-3 AuAgCu (last ternary alloy).pdf Figure-6 Equal parts of Ag-Cu.pdf Figure-1 Pure Silver (10.5).pdf Figure-4 AuAgCu (first ternary alloy).pdf Figure-5 Equal parts of Au-Ag.pdf Figure-2 Au-Cu (Binary alloy).pdf Figure-7 Density between Au & Ag (14.91).pdf

Uploading the remaining Figures 8 & 9

Figure-9 Density between Au & Ag (14.14).pdf Figure-8 Density between Cu & Ag (9.55).pdf

[John Cuthber]

So... if I tell you the density of the alloy is 10.5, you can tell me it has a composition which lies on that Isopycnic  line.

But you can't tell me which composition it is.
In other words, you can't tell me anything that Archimedes couldn't have

Edited November 14 by John Cuthber

[studiot]

6 minutes ago, John Cuthber said:

So... if I tell you the density of the alloy is 10.5, you can tell me it has a composition which lies on that Isopycnic  line.

But you can't tell me which composition it is.
In other words, you can't tell me anything that Archimedes couldn't have

Which comes back to my still ananswered point.

What else is required to complete the analysis ?

+1

[John Cuthber]

1 hour ago, studiot said:

Which comes back to my still ananswered point.

What else is required to complete the analysis ?

+1

In the particular case of the CuAuAg system, I think the colour would give you a hint.
If you measured the reflectance spectrum carefully, it would probably be sufficient.
The electrical conductivity might also work, or the hardness, or the speed of sound, or the melting point or magnetic susceptibility or the electrode potential or melting point.

But the point remains; you need something to tell you where on that line you are.
If you are in the UK and there's a point on the line corresponding to 9  (or 18) carat gold then that's a fair bet, but no proof.
But in the USA they seem to prefer 10 carat (as far as I can tell).