X-ray computed tomography (CT) is a powerful technique for imaging three-dimensional objects revealing the interior of a sample. X-ray CT is routinely used in medical imaging and industrial inspection, for example. Currently, contrast in X-ray CT images is largely based upon how strongly X-rays are attenuated by the sample. For example, bones in the human body strongly attenuate X-rays incident upon them, which is why bones, which are highly attenuating, appear with relatively high contrast in X-ray CT images. There is much interest in extending X-ray CT to be able to identify particular materials in a sample. For example, personalised diagnosis of complex diseases increasingly involves the use of imaging agents. Our method would provide the ability to pinpoint multiple agents simultaneously in a single scan, allowing for more effective diagnostic tests to be performed in a simple and speedy manner At present, material identification using X-ray CT can only be performed using two separate CT scans which therefore takes twice the amount of time of a single CT, thus delivering double the radiation does to the sample. Furthermore, this approach only works for monochromatic X-ray source, such as are available at synchrotrons, which are expensive to operate and have very limited capacity. Even if synchrotron access is possible, in many applications, such as medical imaging, performing two scans is not feasible due to the dose they can safely receive. More recently, methods which make use of expensive detectors can achieve this using a single scan, however severely limiting the sample size. We propose a method of performing material identification which uses a single phase-sensitive CT scan and is based on the edge illumination x-ray phase imaging technique which employs standard, therefore cost effective, x-ray imaging equipment. Our technique works by recognizing that three-dimensional images of electron density and effective atomic number, reconstructed from a single edge-illumination CT data set will only have sharp interfaces between two materials when both of these materials have been correctly identified. Thus, the unknown material can be determined by varying a reconstruction parameter and inspecting an interface between the known and unknown materials. The parameter which leads to this interface becoming sharp essentially reveals the unknown material. This phenomenon can be used sequentially to identify all materials in a sample, resulting in complete material identification using a single CT scan.