  We describe the first X-ray observations of binary millisecond pulsars PSRs
J0023+0923, J1810+1744, J2215+5135, and J2256-1024. All four are Fermi
gamma-ray sources and three are 'black-widow' pulsars, with companions of mass
< 0.1 solar masses. Data were taken using the Chandra X-Ray Observatory and
covered a full binary orbit for each pulsar. Two pulsars, PSRs J2215+5135 and
J2256-1024, show significant orbital variability and X-ray flux minima at the
times of eclipses observed at radio wavelengths. This phenomenon is consistent
with intrabinary shock emission characteristic of black-widow pulsars. The
other two pulsars, PSRs J0023+0923 and J1810+1744, do not demonstrate
significant variability, but are fainter than the other two sources. Spectral
fits yield power-law indices that range from 1.4 to 2.3 and blackbody
temperatures in the hundreds of eV. The spectrum for PSR J2215+5135 shows a
significant hard X-ray component (41% of counts are above 2 keV), which is
additional evidence for the presence of intrabinary shock emission.
