Counterion condensation and fluctuation-induced attraction

COUNTERION CONDENSATION AND FLUCTUATION- ...1AF Bd 1b2 22 B R d2 q B2 2 1 B R24b 2 1b 2 2 d R1b 2 2q Gd/22Jd/2 d2 q2 2 B2q Gd/2Jd/2 21b2 Jd/22q R 2 1 2 /q 2 2 2Mqsinhqdq R 1 2 /q 2 1M q,2where J(d/2) is defined by the expressionJd/2 1 2 1M 2 q 2Mqcosh qd1M 2 q. A261M 2 qFinally, we turn to the last term in Eq. A1, Eq. A22. Withthe help of the integral G3D z,z;qdzd/2 zd/2 B 1b 2 23 22q 2 R 1 2 Gd/2 1 2 1 2 Lq ,Eq. A22 can be written asA271AF Cd 1b2 22 B R d2 q 2 2qB Gd/21Lq2 24b2 1b 2 d2 q2 2 B2q2 d 1b 2 R Gd/21Lq2, A28which can be combined with the expression for F B /dabove note that G(d/2) cancels nicely to yield1AF BC 1b2 22d B RPHYSICAL REVIEW E 66, 041501 2002 4b2 1b 2 B4 RI 3 1 B R22 d R1b 2 B4 RI 2 I 3 ,A29where we have defined the following dimensionless integrals:I 2 d2 q2 2 4 R2q 21b2 Jd/2q R 2 1 2 /q 2 2 2Mqsinhqdq R 1 2 /q 2 1M q,2A30I 3 d2 q2 2 4 R2q 1 2 1 2 LqJd/2 . A31With some straightforward but tedious algebra, they can becast into a more explicit form,I 2 d/ R 0dx2x1b 2 x 2 1b 2 x1b 2 xb 2 4b 2 x 2 4b 2 x 2 1M 2 x1b 2 2 1xx1b 2 x1b 2 x0dx2xM 2 x1b 2 x1b 2 x1b 2 xb 2 x4b 2 x 2 4b 2 x 2 1M 2 x1b 2 2 1b 2 x1b 2 xA32andI 3 d/ R 0dx2xb 21M 2 x1b 2 2 1xx1b 2 x1b 2 x 2xM 2 xxb 2 xdx0 1M 2 x1b 2 2 1b 2 x1b 2 x ,A33041501-13