It should become standard for married chassidim to wear silk kapotes on Shabbos and Yom Tov, and I wish to thank in advance those who will do so. The idea that one’s garments on Shabbos and Yom Tov should be silken is mentioned in a Ma’amar of the Rebbe Rashab .
In general, the behavior [of chassidei Chabad] should be distinctive, and this should express itself in the fact that on Shabbos and Yom Tov one wears a silk kapote.
I am surprised at those chassidim who do not wear silk kapotes on Shabbos and Yom Tov It appears that the reason for this [failure to observe this custom] is the poverty that prevailed in Russia, which meant that chassidim were not able to wear silk kapotes.
In any case, the time has come that people should start wearing silk kapotes, and over time all married chassidim will come to follow this practice.
Toras Menachem 5711, Vol. 1, p. 53.
 The following sources are cited in the footnote to this sicha:In addition to the point the Rebbe makes that on Shabbos chassidim ought to wear a silk kapoteh, it appears clear and self-evident that the Rebbe is also making an implied statement that ... on Shabbos chassidim ought to davka wear a kapoteh, and not suffice with a suit.
In Sefer HaMa’amarim 5660-1-2, p. 231 the Rebbe Rashab cites from the works of Kabbalah that on Shabbos one’s garments are connected with the world of Beriah, which parallels chai, the animal world; thus, one’s garments should be made of silk, which comes from the category of chai. (To explain, the four spiritual worlds correspond to the four levels of creation: Atzilus—medaber, mankind; Beriah—chai, the animal kingdom; Yetzirah—tzomeach, plant life; Asiyah—domem, inorganic matter. Thus, since on Shabbos there is a revelation of the world of Beriah, it is fitting to wear garments made from animal kingdom.)
Likewise, in Sefer HaMa’amarim 5671, p. 247 the Rebbe Rashab mentions the same practice, and also explains that since on Shabbos the world receives from the level of Hashem’s thought, so to speak (the worlds of Beriah, Yetzirah, and Asiyah corresponding in Kabbalistic thought to thought, speech, and action respectively), on Shabbos one should wear garments made from chai, while during the week one should wear garments made from tzomeach, plant life.
In Sefer HaMa’amarim 5672, Vol. 2, pp. 872-873, the Rebbe Rashab cites this in the name of the Mikdash Melech (a classic commentary on the Zohar written by Rabbi Shlomo Bazuglo).