A C172 DEPARTED THE RWY FOLLOWING A NORMAL APCH WHEN THE NOSEWHEEL STUCK CAUSING THE ACFT TO DEPART THE RWY. RUDDER INPUT DID NOT CORRECT ACFT.

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ZZZ SKY CONDITION BROKEN; CEILINGS APPROX 2000 FT; TOPS APPROX 4500 FT; SURFACE WINDS AT 170 DEGS AT 5-8 KTS. APPROX 40 MI S OF ZZZ; I CALLED ATC AND REQUESTED IFR CLRNC FOR APCH INTO ZZZ. REQUEST APPROVED; ASSIGNED ALT 5000 FT. RECEIVED 'DIRECT-TO' CLRNC TO IAF ALT 2400 FT; 8-10 MI S OF IAF. CLRED FOR ILS APCH RWY 7 APPROX 5-8 MI S OF IAF; ALT 2400 FT. ESTABLISHED ON APCH AT IAF AT APPROX XA40. APCH UNEVENTFUL TO TOUCHDOWN ON RWY 7. TOUCHDOWN OCCURRED WITHIN LNDG ZONE ON CTRLINE OF RWY 7; TIME APPROX XA45; AT THIS POINT LNDG UNEVENTFUL. AS NOSEWHEEL TOUCHED DOWN; ACFT VEERED SHARPLY TO R. ATTEMPTED TO CORRECT DIRECTION BY APPLYING L RUDDER -- NO DIFFERENTIAL BRAKING WAS APPLIED. ACFT LEFT RWY. NO EFFECT FROM APPLICATION OF L RUDDER WAS NOTED. AFTER LEAVING RWY FULL UP ELEVATOR WAS APPLIED; PWR WAS AT IDLE; L RUDDER CONTINUOUSLY APPLIED WITH NO EFFECT. AS ACFT SLOWED TO A STOP; NOSEWHEEL SANK IN THE MUD WITH PROP STRIKING MUD AND STOPPING ENG. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE REPORTER STATED THAT THE ACFT WAS EXAMINED BY TWO SEPARATE MECHANICS WHO GAVE DIFFERING CAUSES FOR THE FAILURE. THE MECHANIC AT THE STATION WHERE THE ACFT HAD THE EXCURSION SAID THE NOSEWHEEL SYSTEM HAD A MALFUNCTION AND INDEED COULD NOT BE CONTROLLED BY THE PILOT. THE FBO MECHANIC WHO NORMALLY MAINTAINED THE ACFT SAID THAT THE PILOT DID NOT USE DIFFERENTIAL BRAKING AND THEREFORE COULD NOT CONTROL THE ACFT. THE REPORTER STATED THAT INDEED HE DID NOT USE DIFFERENTIAL BRAKING BUT DID NOT BECAUSE HE FEARED GND LOOPING THE ACFT. BECAUSE OF THE SUDDEN STOPPAGE WHEN THE PROP HIT THE MUD THE ENGINE HAD BEEN DISASSEMBLED FOR INSPECTION. THERE WAS NO OTHER DAMAGE TO THE ACFT. THE REPORTER STILL QUESTIONS HIMSELF AND WHETHER THE EVENT COULD HAVE BEEN PREVENTED OR WAS RESULT OF HIS NOT USING HARD DIFFERENTIAL BRAKING TO CORRECT THE DRIFT.