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Wet conditions led to a lengthy delay to the start of proceedings on the second day of the tournament, but the overnight leader continued to set the pace, finishing three-under for the day as he paused his round after 13 holes.
The American, who has made just one bogey at the event so far, will be one of 42 golfers to return from 7am tomorrow to complete their second rounds ahead of the third round.
Danny Willett remains the clubhouse leader, having carded a three-under 69 on day two, leaving the Englishman just one shot behind Johnson after going around in a six-under-par 66 in the first round.
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Jordan Spieth, who is bidding for a third straight win in a Major, sits five shots off the lead on five-under-par, having gone through the first 13 holes on level par with three birdies and three bogeys.
Meanwhile, veterans Tom Watson and Sir Nick Faldo, who are playing in The Open for the final time in their careers, are likely to miss the cut, with a bogey on the 18th leaving the American last on 12-over, while the three-time champion is just two shots better off.
Former world number one Tiger Woods, who has also won three times before, is in danger of missing the cut on five-over, but still has seven holes to play tomorrow, bogeying twice in the first six holes before getting a shot back shortly before play was suspended.
The cut is currently projected to be level par.