For example, researchers at Argonne National Laboratory and the University of Illinois Chicago have been working on what can be viewed as an "artificial leaf" for taking carbon dioxide out of the atmosphere. A press release from Argonne described it this way: "To make carbon dioxide into something that could be a usable fuel, Curtiss and his colleagues needed to find a catalyst — a particular compound that could make carbon dioxide react more readily. When converting carbon dioxide from the atmosphere into a sugar, plants use an organic catalyst called an enzyme; the researchers used a metal compound called tungsten diselenide, which they fashioned into nanosized flakes to maximize the surface area and to expose its reactive edges. While plants use their catalysts to make sugar, the Argonne researchers used theirs to convert carbon dioxide to carbon monoxide. Although carbon monoxide is also a greenhouse gas, it is much more reactive than carbon dioxide and scientists already have ways of converting carbon monoxide into usable fuel, such as methanol." The research was just published in the July 29 issue of Science magazine, in "Nanostructured transition metal dichalcogenide electrocatalysts for CO2 reduction in ionic liquid," by a long list of co-authors headed by Mohammad Asadi (vol. 353, issue 6298, pp. 467-470).
As another example, researchers at the USC Loker Hydrocarbon Research Institute at the University of Southern California "have directly converted carbon dioxide from the air into methanol at relatively low temperatures," according to a summary of the research by Robert Perkins. "The researchers bubbled air through an aqueous solution of pentaethylenehexamine (or PEHA), adding a catalyst to encourage hydrogen to latch onto the carbon dioxide under pressure. They then heated the solution, converting 79 percent of the carbon dioxide into methanol." The researchers hope that the method might be viable at industrial scale in 5-10 years. The research was published earlier this year in the Journal of the American Chemical Society, "Conversion of CO2 from Air into Methanol Using a Polyamine and a Homogeneous Ruthenium Catalyst," by Jotheeswari Kothandaraman, Alain Goeppert, Miklos Czaun, George A. Olah, and G. K. Surya Prakash (2016, 138:3, pp 778–781).
One of the co-authors of the USC study, Alain Goeppert, points out in an article in the Milken Institute Review by Lawrence M. Fisher (Third Quarter, 2016, pp. 3-13) that a company in Iceland is already recycling carbon to make methanol and exporting it to Europe.
“A company in Iceland is already doing that: Carbon Recycling International,” Goeppert said. “There, they are recycling CO2 with hydrogen they obtain from water. They use geothermal energy, which is relatively cheap. They have been producing methanol that way for five years, exporting it to Europe, to use as a fuel. It’s still relatively small scale, but it’s a start.”Methanol can easily be mixed into gasoline, as ethanol is today, or cars can be adapted fairly cheaply to run on 100% methanol. Diesel engines can run on methanol, too.
Of course, I don't know if carbon-dioxide-to-methanol can put a real dent into atmospheric carbon in any cost-effective way. But again, I'm a consider-everything kind of guy. And before I get too skeptical about how fields of artificial leaves might work for this purpose, it's worth remembering that fields of solar collectors didn't look very practical as a method of generating electricity a couple of decades ago, either.
Addendum: One more study along these lines, this one involving experimental results for converting carbon dioxide to ethanol done by researchers at Oak Ridge National Laboratory, reached the trade news in October 2016.