I went through the 2007 thread and cleaned this post up
Purdey Gun & Rifle Makers, The Definitive History by Donald Dallas
There was very little discussion of the barrels, and no mention of any specific named pattern nor, after the mid-1820s, if any barrels were out-sourced. I was able to establish the progression of barrels used from the excellent pictures.
In summary: prior to the 1840s all barrels were twist. In the 1840s-1860s Laminated steel and 2- and 3- Iron “Oxford” Damascus appear - asymmetric large oval scrolls

About 1870, a distinctly finer 3- and 4- iron English Best “Turkish” (smaller annular symmetric scroll) patterns appear

c. 1890, most of the barrels were Whitworth steel
p.15
Purdey began producing his own barrels in the mid-1820s.
Prior to that time, most were obtained from Charles Lancaster (‘
CL’ marked) or William Fullerd (‘
WF’ marked)
p. 27
Barrels made by Purdey were stamped ‘
JP’ after 1826
Also Thomas Evans ‘
TE’ and Thomas Parkin ‘
TP’ in mid-1820s.
No further mention of any out-sourced barrels thereafter.
Aston Snr. began with Purdey's in the 1850s; his son, Harry, was also a barrel maker until 1930.
Barrel patterns in images1818: 6b Flintlock #14 - Twist
1821: 4b Dbl percussion shotgun #287 - Twist
1844: 16b Dbl percussion rifle #3845 - Laminated
1863: 12b Dbl Bastin/Purdey slide action pinfire #6424 - Damascus ?2 rod
1864: First pattern thumb-hole pin-fire #6829 - Laminated Damascus?
1866: 12b Dbl First pattern thumb-hole #7225 - Laminated steel
1867: 40 cal. Dbl Express rifle #7464 - Large scroll 2 Iron “Oxford” Damascus
1868: 12b Second pattern thumb-hole #7745 - Large scroll 3 Iron Damascus
1869: 12b muzzle-loading shotgun #7902 - Large scroll 3 Iron Damascus
1872: 10b bar-in-wood hammer gun #8623 - First appearance of Fine small scroll “Turkish” 4 Iron Damascus
1873: 12b bar-in-wood hammer guns pair #8998 & 9000 - same Damascus as above.
c. late 1870s: Introduction of Quality A - E
Jan. 1, 1880 - Pair #10614/10615 were delivered to Mr Hunt by James Purdey 'The Younger' with The "New Whitworth Fluid Pressed Steel".
1881: Bar-in-wood hammer gun #11090 - same 4 Iron “Turkish”.
1885 Catalog - Quality A- Best Gun “The barrels of best Guns can if desired be made of Sir Joseph Whitworth & Co.s fluid pressed steel at an extra cost...”
Quality B- “Fine Damascus barrels.” No other mention of barrel material.
Crolle pattern illustrated is the same 4 Iron “Turkish”.
1884: Beesley gun #11837 - 3 Iron “Turkish”
1886: 8b hammer rifle #12371 - 3 Iron “Turkish”
c. 1890s: Guns illustrated had Whitworth steel.