Okay, WSP -- you have asked a good question, so don't give up. I'm not entirely sure I have a good answer, because I've been searching for info on this issue for some time without a lot of luck. Mostly what I've found is the "common wisdom" that we read on soapcalc and other popular soaping sides. While this info is helpful in an empirical way, I have not been able to uncover much, if any, of the science to back up this common understanding.
Here's my first stab at a preliminary answer. Bear in mind this is very much a work in progress, so don't hold my feet to the fire if my point of view evolves with time.
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I'm ambivalent about describing soap as moisturizing, emollient, conditioning, etc. All those words are used to describe something that soothes, protects, and softens the skin. My current opinion is that the
superfat in a soap can be moisturizing, emollient, conditioning, etc., but I question how the
soap itself can be.
What I do know is that soap dissolved in water has the ability to emulsify fats and thus dissolve greasy soil. Call it "cleaning" for short.
The ability to clean is related to the soap's ability to dissolve in water (solubility) and to its ability to emulsify fats.
Solubility is partly related to the type of ion that is attached to the fatty acid to make the soap. These ions include ammonium (NH3), potassium (K), sodium (Na), lithium (Li), magnesium (Mg), calcium (Ca), barium (B), lead (Pb), mercury (Hg), and others. I've listed them in order of how soluble their soap is in water -- from high solubility (ammonium, potassium) to mostly or completely insoluble (magnesium, calcium, barium, lead, mercury).
Add the word "hydroxide" after the words ammonium, potassium, and sodium and you will get the name of the most common soaping lyes. Any particular soap recipe can be changed just by using a different lye. A soap made with sodium hydroxide (NaOH) will be a firm bar soap and the least soluble and lowest lathering. If you make the same recipe except use either potassium hydroxide (KOH) or ammonium hydroxide (NH3OH), the resulting soap will be softer, ranging from a paste to liquid in form. The soap will also be more soluble, and lather more freely. By using a mix of two or more of these lyes, you can adjust the relative solubility, hardness, and lathering ability of the finished soap.
* Side note: Concentrated ammonium hydroxide cannot be safely used as a lye for making handcrafted soap, because it is extremely hazardous to work with -- far more dangerous than KOH or NaOH. Some soap recipes include "household ammonia" which is a dilute solution (5%, if I remember correctly) of ammonia in water. That is just not concentrated enough to use as 100% of the lye, so recipes with household ammonia must also enough KOH or NaOH for complete saponification. *
In addition to the choice of lye, the fatty acids in the recipe will affect the solubility and the cleansing ability of a soap. Some soaps are very soluble and are also very efficient emulsifiers -- cleaning to the point of leaving the skin dry and uncomfortable. These would be soaps made with short chain, saturated fatty acids such as myristic, lauric, capric, butyric, etc.
Other soaps are also relatively soluble, but they are not quite so efficient at cleaning because they are less efficient at emulsifying fats. These soaps tend to leave more of the skin's oils intact, so they are considered more "emollient" or "conditioning". These would be the soaps with mono- and poly-unsaturated fatty acids -- oleic, linoleic, and linoleic.
Still more soaps are relatively INsoluble and moderate to low on cleansing efficiency -- these would be soaps with longer chain, saturated fatty acids such as stearic and palmitic.
It's my guess at this point that the soaps that are considered more "conditioning" are the ones that are more INsoluble and/or less efficient emulsifiers. They simply leave more of the skin's oils intact. Whether they actually leave a film of "conditioning stuff" behind on the skin after the suds are rinsed off (not counting any superfat) -- that I cannot say. I'm still looking into this.
See also:
http://www.soapmakingforum.com/showthread.php?t=48845