Most of the article is in fact a standard rehash of ID arguments. The only part that relates specifically to Sewell's Second Law argument is
I take a closer look at the equation for entropy change, which applies not only to thermal entropy but also to the entropy associated with anything else that diffuses, and show that it does not simply say that order cannot increase in a closed system. It also says that in an open system, order cannot increase faster than it is imported through the boundary. According to this equation, the thermal order in an open system can decrease in two different ways -- it can be converted to disorder, or it can be exported through the boundary. It can increase in only one way: by importation through the boundary. Similarly, the increase in "carbon order" in an open system cannot be greater than the carbon order imported through the boundary, and the increase in "chromium order" cannot be greater than the chromium order imported through the boundary, and so on.
Sewell appears to be making a fundamental and astonishingly simple error here (for a math. professor at UTEP). The second law states merely that the total entropy must increase in a spontaneous process. If you break the total entropy into component parts - 'carbon entropy', 'chromium entropy', 'thermal entropy', etc., it is simply false to say that the increase in "carbon order" in an open system cannot be greater than the carbon order imported through the boundary
The 'carbon order' can most definitely increase as long as the total order decreases, if a microscopic process exists that couples carbon entropy to some other type of entropy. So, for example, if we supercool some water to -5 C, and then isolate the system, part of the water can freeze, liberating heat, which will raise the temperature of the system to 0 C, increasing the thermal entropy, but decreasing the 'water entropy'. The water molecules become more ordered, even though the total order of the system has decreased.
While I'm often unpleasantly surprised by how little science pure mathematicians know, I am genuinely astonished Sewell would not run this by a physicist or physical chemist before embarrassing himself in this very basic way.