TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Method: In the method (M1), the surface modification compound is cross-linking compound having at least two amine groups. During the exposing step, maintaining the conditions to form an amide bond between imide groups of the polyimide and amine groups of the surface modification compound. The exposing step involves maintaining the temperature at 50-212 degrees F (10-100 degrees C) for 1-60 minutes. The method (M1) involves vaporizing a solution of the surface modification compound to generate the vapor phase; and agitating the solution of the surface modification compound during the vaporization step. The agitating step involves passing a gas through the solution of the surface modification compound that is inert to the surface modification compound, where the surface modification compound comprises an aliphatic hydrocarbon chemically coupled to the amine group. The cross-linking compound comprises two amine groups chemically coupled by a hydrocarbon linker (R'), the cross-linking compound is of formula H2N-R'-NH2 (I). The method (M1) involves selecting an aromatic polyimide as the polyimide of formula (II). The exposing step involves modifying the polyimide membrane that is present in the outer layer of the hollow fiber; and maintaining the temperature of 50-212 degrees F (10-100 degrees C) 1-5 minutes.R'=hydrocarbon linker which is a saturated or unsaturated, branched or straight chain aliphatic hydrocarbon or a saturated or unsaturated aliphatic ring hydrocarbon, where the aliphatic group is 1-8C alkyl;Ar1=quadrivalent organic group (preferably group of formulae (i)-(ix));Ar2=divalent organic group (preferably group of formulae (x)-(xiii));n=number of monomer units in the polyimide such that the polyimide has an inherent viscosity of at least 0.3 as measured at 25 degrees C on 0.5 wt.% solution in N-methylpyrrolidinone;Z'=group of formula (xiv), -C(CF3)2-, -C(=O)-, -O-, -S-, -S(=O)2-, -C(CH3)2- or -CH2-;andX', X1-X3=H, 1-5C alkyl, 1-5C alkoxy, phenyl or phenoxy. EXAMPLE - No suitable example given.