A LNDG PA34 EXPERIENCED GREATLY REDUCED BRAKE EFFECTIVENESS CAUSING A RWY EXCURSION DURING A TWR REQUESTED EXPEDITED TAXI.

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AFTER TOUCHDOWN; THE TWR REQUESTED THAT THE ACFT EXPEDITE ITS TAXI FROM THE RWY TO ALLOW FOR A JET DEP PRIOR TO A SINGLE ENG ACFT LNDG. BOTH THE PLT AND FLT INSTRUCTOR ATTEMPTED TO EFFECTIVELY REDUCE SPD SO THE ACFT COULD CLR ONTO AN APCHING TAXI AREA. DUE TO THE SPD OF THE ACFT IT WAS AGREED THAT THE FIRST EXIT WAS UNATTAINABLE AND THAT A CONTINUED TAXI ON THE RWY TO THE NEXT EXIT WAS REQUIRED. AS THE STUDENT BEGAN TO TAXI CLR OF THE RWY AT THE NEXT INTXN; ALMOST ALL EFFECTIVE BRAKING WAS LOST. THE TURN REQUIRED TO REMAIN ON THE TXWY WAS NOT POSSIBLE WITHOUT BRAKING. THE STUDENT CONTINUED STEERING WITH MINIMAL RESULTS AND DIRECTED THE ACFT BTWN TXWY EDGE LIGHTING WHILE THE INSTRUCTOR SHUT OFF BOTH ENGS BY STOPPING FUEL FLOW BY WAY OF THE MIXTURE CTLS SO AS TO PREVENT A PROP STRIKE TO THE TXWY LIGHTING OR THE UNEVEN SURFACE OF THE GRASS AREA ADJACENT TO THE TXWY. THE ACFT WAS THEN SAFELY BROUGHT TO A STOP IN THE GRASS AREA NEXT TO THE TXWY. THE ACFT WAS LATER TOWED BACK TO A HANGAR WHERE IT WAS DETERMINED THAT AIR BUBBLES IN THE BRAKE LINE WERE THE CAUSE OF THE BRAKE FAILURE. IT WAS ALSO FOUND THAT THE R BRAKE HAD THE LEAST AMOUNT OF FORCE OF THE 2 MAIN BRAKES WHICH WOULD MAKE A TURN TO THE R VERY DIFFICULT; IF NOT IMPOSSIBLE. THE ACFT IN THIS CASE EXITED THE TXWY WHILE ATTEMPTING A R TURN. I BELIEVE THE EVENT WAS A RESULT OF EQUIP FAILURE COMPOUNDED BY THE ENVIRONMENT AT THE TIME THAT IT OCCURRED. THE FOCUS AND STRESS ASSOCIATED WITH A SINGLE ENG INST APCH AND THE RUSHED OR HURRIED ATTITUDE PRESENTED BY THE CTLR PRODUCED AN ENVIRONMENT THAT WAS ABNORMAL.