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Mack Weldon advertises that their underwear uses Silver XT2™ technology:

Made possible with Silver XT2™ technology, silver naturally destroys odor-causing bacteria. We added Pima cotton to keep you cool and comfortable all day long.

NanoSilver provides a similar promise:

Compared to most garments Nanosilver offers a unique properties- it conducts sweat and is antibacterial.

Is there misleading marketing or does silver really provide advantages at fighting odor-causing bacteria?

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    @user568458 The article you linked to earlier is still on archive.org but isn't that specific about the study either. However, it seems they were referencing The Behavior of Silver Nanotextiles during Washing [compatible with the findings below and referenced by one of the later studies…] – LangLangC Sep 2 '18 at 16:15
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Yes, this is providing some 'benefits', and yes this is misleading marketing as well.

Unless we see a study that compares body odour effects of silver treated textiles against plain old wool, showing a big effect: It's just not worth it. That is, polyester aficionados might depend on it to not stink that fast and long lasting.

Silver treated textiles do fight bacteria. All bacteria, not just odour causing bacteria. That is one big problem, as your skin has its own, necessary microbiome. Silver is an antibiotic that kills indiscriminately. In that view it may be not a good idea. Silver does not kill bacteria by being in or on the fabric, it does so by being released from it!

Another problem is that too much contact with or ingestion of silver can cause quite a funny effect: The case of argyria or argyrosis, but that does not mean you cannot still run for the Senate, like Stan Jones, a True-blue bids for Senate:

enter image description here

In the case of textiles:

Textiles interact with the skin in a very intensive manner. Therefore, the microorganisms of the skin can influence the skin itself, the textiles as well as the interaction between skin and textiles. During the last few years, the materials for manufacturing textiles show positive tendencies towards a higher functionality. The market has been enriched with innovative antimicrobial products, especially with silver fibers or materials with enclosed silver ions. These textiles could not only find a domain in the wellness sector, but the goal is to use textile fabrics with antimicrobial finishing sufficient for prophylaxis and therapy. On the other hand wearing these new textiles can generate problems, unknown till now. Potential health risks can occur. To minimize such risks, careful and reliable in vitro as well as in vivo test systems should be established, which is, by the way, one of the most important requirements of the European Conference on Textiles and Skin. Standards are necessary for the effectiveness of antimicrobial textiles as well as for the evaluation of their undesirable side effects, like cytotoxicity, allergenic and irritative potentials.
Hipler, below.

You can further conceded that:

Silver has a long and intriguing history as an antibiotic in human health care. It has been developed for use in water purification, wound care, bone prostheses, reconstructive orthopaedic surgery, cardiac devices, catheters and surgical appliances. Advancing biotechnology has enabled incorporation of ionizable silver into fabrics for clinical use to reduce the risk of nosocomial infections and for personal hygiene. The antimicrobial action of silver or silver compounds is proportional to the bioactive silver ion (Ag) released and its availability to interact with bacterial or fungal cell membranes. Silver metal and inorganic silver compounds ionize in the presence of water, body fluids or tissue exudates. The silver ion is biologically active and readily interacts with proteins, amino acid residues, free anions and receptors on mammalian and eukaryotic cell membranes. Bacterial (and probably fungal) sensitivity to silver is genetically determined and relates to the levels of intracellular silver uptake and its ability to interact and irreversibly denature key enzyme systems. Silver exhibits low toxicity in the human body, and minimal risk is expected due to clinical exposure by inhalation, ingestion, dermal application or through the urological or haematogenous route. Chronic ingestion or inhalation of silver preparations (especially colloidal silver) can lead to deposition of silver metal/silver sulphide particles in the skin (argyria), eye (argyrosis) and other organs. These are not life-threatening conditions but cosmetically undesirable. Silver is absorbed into the human body and enters the systemic circulation as a protein complex to be eliminated by the liver and kidneys. Silver metabolism is modulated by induction and binding to metallothioneins. This complex mitigates the cellular toxicity of silver and contributes to tissue repair. Silver allergy is a known contra-indication for using silver in medical devices or antibiotic textiles.
in U.-C. Hipler & Jena P. Elsner (Eds.): "Biofunctional Textiles and the Skin", Karger: Basel, Freiburg, 2006. (PDF)

If we look at just cotton, as the claimant seems to do, things in vitro 'look good':

Nano silver can be used effectively as an antimicrobial agent for cotton. The higher the concentration of antimicrobial agent, the larger the zone of inhibition in the cases of both Gram-positive and Gram-negative bacteria. SEM study of antimicrobial-finished fabric reveals that a continuous polymer film has been formed on the fabric. The concentration of PVOH controls the bending length and crease recovery angle. The higher the concentration of PVOH, the greater the bending length and crease recovery angle. Curing temperature and time have profound impacts on the tensile strength. The higher the curing temperature and time, the lower the tensile strength. In the case of commercial Product A (Sanitized® T 27-22 silver) treated cotton fabric, the zone of inhibition of Gram-positive bacteria was a minimum of 24 mm and a maximum of 29 mm, while for Gram-negative bacteria the minimum was 14 mm and the maximum 18 mm. In the case of commercial Product B (Sanitized® T 25-25 silver) treated cotton fabric, the zone of inhibition of Gram-positive bacteria was a minimum of 24 mm and a maximum of 29.5 mm, while for Gram-negative bacteria the minimum was 14 mm and the maximum 18.6 mm.

A.I. Wasif and S.K. Laga: "Use Of Nano Silver As An Antimicrobial Agent For Cotton", AUTEX Research Journal, Vol. 9, No1, March 2009. (PDF)

But we are interested solely in the odour prevention?

Abstract:
The purpose of this study was to determine whether polyester textiles treated with bioactive concentrations of an antimicrobial silver chloride (SC) compound were effective in reducing axillary odour and axillary bacterial populations before and after multiple washes. A polyester knit fabric was treated with two concentrations of a SC formulation (resulting in 30 and 60ppm of silver) and evaluated at two levels of wash treatments (unwashed and washed 30 times). Treated fabrics were matched with an untreated control fabric and worn against the axillae of male participants (n = 8). A sensory panel evaluated odour intensity using two different methods (paired comparison and line scale method). Overall, results showed that the treated fabrics did not lower odour intensity compared with the untreated fabrics. Bacterial populations extracted from the treated fabrics were also not significantly lower, despite there being evidence of antimicrobial activity in in vitro testing. The paired comparison method was found to be more sensitive in detecting small differences between treated and untreated fabrics. However, the line scale method was deemed to be a more appropriate method for evaluating odour intensity on fabrics because the magnitude of the difference could be assessed. It is recommended that as in vitro efficacy does not necessarily predict in vivo efficacy of an antimicrobial treatment that sensory evaluation and in vivo testing should be conducted when examining the odour reducing properties of an antimicrobial. Conclusion:
The primary purpose of this research was to carry out an in vivo trial to determine whether one silver-based antimicrobial-treated textile could noticeably reduce odour intensity emanating from polyester fabrics following wear against the underarm. As only one type of antimicrobial compound was evaluated in this study, the results cannot be extrapolated to other anti-microbials, or even to the same compound at higher levels of concentration. However, the significance of this study is that it highlights an important problem, that is, in vitro tests of antimicrobial efficacy do not necessarily correspond to antimicrobial efficacy in vivo. Published studies evaluating the odour-reducing properties of antimicrobial-treated textiles are minimal. Not only may in vitro efficacy tests not necessarily correspond with efficacy in vivo, but odourous compounds produced in the axillary region may transfer and affix to the fabric itself. Regardless of evidence of antimicrobial activity in vivo, it is essential that sensory measurement of odour be conducted if the purpose of the antimicrobial is to reduce the build-up of body odour in a textile.
It is clear that appropriate methods for assessing antimicrobial efficacy in vitro need to consider the end-use application for the fabric. The effect of components in sweat on antimicrobial efficacy of silver-based products during real-use is unclear and requires more attention, given the use of these products in general clothing.

Rachel H. McQueen et al.: "In vivo assessment of odour retention in an antimicrobial silver chloride-treated polyester textile", The Journal of The Textile Institute, 104:1, 108-117, 2012. DOI

So the real life effect of silver treated textiles seems quite small. It seems that in comparing the performance to wool, cotton is quite good naturally, but polyester just the worst. this effect seems to be old folk knowledge, because it is just that much bigger than treating coton with silver:

enter image description here The relationship among textile fibres/fabrics, axillary bacteria and odour is highly complex. Through the detection of volatiles using PTR-MS, compounds likely to be associated with axillary malodour were found to increase over 7 days in polyester fabrics. This increase was not evident for either the wool or cotton fabrics. The intensity of axillary odour emanating from fabrics was inversely related to fibre hygroscopicity. Therefore, differences in the chemical structure and physical morphology of the fibres, and hence the availability of reactive sites for absorption of volatile compounds, may also account for differences in odour retention. Both cotton and wool have many hydroxyl groups in which to form hydrogen bonds with polar molecules, and wool has many reactive amino acid sites for which volatile compounds can bind to within the fibre. Polyester on the other hand has no reactive groups, and the olephilic tendency of polyester makes it likely to attract oily ‘soils’ present in human apocrine and sebaceous secretions. As bacterial numbers per se cannot be a predictor of odour intensity emanating from fabrics that differ in fibre content, the difference among fabrics may be a result of metabolic versatility of some resident microbial strains.

R. H. McQueen et al.: "Retention of axillary odour on apparel fabrics", The Journal of The Textile Institute, 99:6, 515-523, 2008. DOI

Applying silver to general use clothing is a nice idea that has problems of "obviously!" levels like washing out of the fabric:

The initial Ag contents ranged from 254 to 350 μg Ag/g of the product in lab-prepared fabrics and from 1.2 to 44 μg Ag/g of the product in consumer products. After 20 wash cycles, 48 to 72 % of Ag was lost from the prepared fabrics washed with Milli-Q water, while a greater loss of 84-94 % of Ag occurred after washing the prepared fabrics with commercial detergent. The Ag released during the washing process is present dominantly in particulate form. In the consumer products after 20 washes with Milli-Q water, the percent Ag remaining was found to be around 46 to 70 %. Statistical analysis of the Ag release rate between consumer products and lab-prepared fabrics in Milli-Q washing water by independent t test showed no significant difference after 20 washing cycles (p > 0.05).
P Limpiteeprakan: "Release of silver nanoparticles from fabrics during the course of sequential washing.", Environ Sci Pollut Res Int. 2016 Nov;23(22):22810-22818. Epub 2016 Aug 26.

As usual in the commercial sphere some fabrics do not contain any silver from the start, less than necessary for odour control or in wholly unsuitable form, which then is rapidly lost during normal washing and rinsing, or just inactivated if not washed out:

As found in several other studies, not all commercial products contain what is promised on their packages (Benn and Westerhoff, 2008; Kulthong et al., 2010). In this study, one textile contained no Ag at all in the fabric and another one contained only very low amounts. Only one label explicitly stated nano-Ag (Textile 6), although two more products were also based on nanoparticles according to information obtained directly from the manufacturer (Textiles 4 and 7). Many so-called ‘‘conventional’’ silver forms are indeed based on nanosilver but labeled in a different way (Nowack et al., 2011). Based on manufacturer labeling alone it is impossible to know which textiles contain what form of silver or if a given textile contains silver at all.
The three textiles that contained silver in the form of nanoparticles (according to the manufacturers) all exhibited a very high antimicrobial activity and clearly outperformed all the other textiles. The use of nanoparticles is therefore coupled to a superior functionality. Many forms of silver can be applied to textiles, but not all show antibacterial activity, which in fact is related to the respective release rates of Ag+. This is especially apparent for the textile with the silver wire, which had no antimicrobial activity, although its Ag content was much higher than that of two of the nanotextiles. Presumably, the silver ion releasing surface was too small to attain potent concentrations, indicating again the much better functionality of nanobased silver-formulations.
The release of Ag from the textiles depends on the one hand on the form and amount of Ag in the textiles but also on the medium that is used for washing. Previous studies have used distilled water (Benn and Westerhoff, 2008), tap water (Benn et al., 2010), and washing solution (Geranio et al., 2009). The main difference caused by different media is clearly the amount of Ag that is released in ionic form. Whereas Benn and Westerhoff (2008) reported up to 86% of the Ag released as dissolved Ag+, the amount found in this work and by Geranio et al. (2009) in washing solution are much lower and the particulate fraction was much more important. A common washing solution has a high pH (10 and above) and contains a variety of other components that may chemically interact with Ag, e.g. chloride ions. For an assessment reflecting the washing of textiles in households, distilled water is surely not the suitable medium. It is also important to note that the detergent used in our study did not contain any bleaching agents, which have been shown to rapidly oxidize nano-Ag and result in a fast release of dissolved Ag+ (Geranio et al., 2009). Tap water as used by Benn et al. (2010) and in this study is a suitable extractant because of which the rinsing step during washing also uses tap water.
Of the four textiles that released measurable Ag, three released AgCl particles (Textiles 4, 5 and 7) and only one released metallic Ag nanoparticles (Textile 6). The released AgCl was found in three very different morphological forms: nanocomposites of Ti/Si and nanoparticles, large AgCl particles for a textile labeled to contain ‘‘silver ions’’ and agglomerates of AgCl nanoparticles from a textile where we had the information that it contained AgCl particles. Speciation calculations show that the 8.7 mg L 1 chloride in the washing solution would be able to precipitate a majority of the total Ag released from textiles 4 and 6 (94 and 70%) if released in ionic form. Secondary formation of AgCl is therefore a possibility and we cannot rule out the possibility that Ag was released as dissolved Ag+ that precipitated in the washing solution as AgCl.
In the washing solution of one sample (textile 5) Ag-sulfide containing nanoparticles were detected. Ag2S is the prevalent silver species found in wastewater and is the result of rapid transformation of all forms of silver including nano-Ag (Kim et al., 2010; Kaegi et al., 2011). Because this Ag-sulfide species was almost never found in any of the other samples, it is unlikely that it was formed in the washing solution. The washing powder used contains 21% of Na2SO4, so at least there would be enough sulfur around that could be reduced, although this is rather unlikely given that all experiments were conducted under oxygenated conditions. This textile also did not retain its antimicrobial activity after washing, although more than 85% of the silver was still present. If the released particles are representative of the particles in or on the textile, the low activity could be explained by low bioavailability of Ag2S or of large AgCl particles, which have a much slower dissolution rate than small nanoparticles of AgCl.
Only agglomerated nanoparticles were found in the washing solutions during the microscopic investigations except for the silver sulfide particles released from textile 5, showing that even in washing solution with its high concentration of detergents that are known to stabilize Ag-NPs, aggregation is an important process.
Our results show that the restriction of the nano-Ag discussion to metallic nano-Ag may miss an important aspect of nano-Ag use and release: of the eight studied silver textiles, only one released metallic nano-Ag. Two of them instead released nano-AgCl and one nanosilver sulfide. Additionally from one textile larger AgCl particles were released. Because washwater is directly released into wastewater, the further fate of these different forms of Ag during treatment will determine the possible release of nano-Ag to the environment (Nowack, 2010). The research on the behavior and effects of nano-Ag should therefore not focus on metallic nano-Ag alone but should also include studies on nano-AgCl and nano-Ag- sulfides.
Ag-sulfides have a very low bioavailability (Blaser et al., 2008) whereas AgCl and nano-Ag both release silver ions that are known to have antimicrobial activity (Marambio-Jones and Hoek, 2010). In contrast to AgCl, which can release Ag+ ions by dissolution, nano-Ag particles first need to be oxidized before Ag+ can be released into solution. The knowledge on the actually released form of silver is therefore crucial for predicting the further environmental effects of silver.

C.Lorenz et al.: "Characterization of silver release from commercially available functional (nano)textiles", Chemosphere, Volume 89, Issue 7, October 2012, Pages 817-824. DOI

An 'independent' review of the exact product in question confirms:

Swamp ass is real, and will sneak up on you. The second you turn around. There is damn swamp ass ready to ruin your day. Making you all self-conscious. Stupid swamp ass.
On a light note these have excellent odor control. After 18000+ these actually smelled better than the undershirt that I was wearing for the day. However, it wasn’t odorless much like the manufacture of XT2 talks about. The fabric manufacture claims that you won’t stink at all, and they still had a little odor to them at the end of the day.

This is in brand new underwear of course. These features cannot improve as the fabric ages.

In conclusion:

Silver based textiles show some promise, and they may well be a real boon in medical applications. But they are currently showing only small, short time effects in consumer clothing, less effective than natural fibres alone and already a potential threat for the environment.

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