Commercial sodium stearate is a powder for ease of processing, but I suspect the powder form isn't a strict necessity. I'd try grating home-made sodium stearate into fine flakes/shreds.
There's a hint on formulating the deo base in this file: http://www.duponttateandlyle.com/sites/default/files/Water Based Deodorant_GelStick_TechnicalBulletin_Zemea_CPC.pdf
Here's a paraphrase of the key info -- "...Sodium stearate is dissolved in propylene glycol [and water] at 75-80 C, followed by the addition of any remaining ingredients...."
Here are 2 generic formulations from the same PDF --
51% water and 40% propylene glycol and 6.5% sodium stearate and solubilizing agent (PEG 400 monolaurate) and active ingredients
OR
31% water and 64.5% propylene glycol and 2% sodium stearate and solubilizing agent and active ingredients
A higher % of stearic soap is going to increase the melt temperature and add hardness to the mixture. If I'm looking at the correct lush deo, I see it's more of a flattish shape like a soap bar rather than an upright stick like a lip-balm style. The flatter shape will be helpful -- you won't have to make the product so firm that it's hard to apply.
There's more hits that look productive, but I'm going to let you check them out. Search google using this keyword phrase: water, propylene glycol, sodium stearate
There's a hint on formulating the deo base in this file: http://www.duponttateandlyle.com/sites/default/files/Water Based Deodorant_GelStick_TechnicalBulletin_Zemea_CPC.pdf
Here's a paraphrase of the key info -- "...Sodium stearate is dissolved in propylene glycol [and water] at 75-80 C, followed by the addition of any remaining ingredients...."
Here are 2 generic formulations from the same PDF --
51% water and 40% propylene glycol and 6.5% sodium stearate and solubilizing agent (PEG 400 monolaurate) and active ingredients
OR
31% water and 64.5% propylene glycol and 2% sodium stearate and solubilizing agent and active ingredients
A higher % of stearic soap is going to increase the melt temperature and add hardness to the mixture. If I'm looking at the correct lush deo, I see it's more of a flattish shape like a soap bar rather than an upright stick like a lip-balm style. The flatter shape will be helpful -- you won't have to make the product so firm that it's hard to apply.
There's more hits that look productive, but I'm going to let you check them out. Search google using this keyword phrase: water, propylene glycol, sodium stearate
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