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Chapter 8: On the Modeling of Carbon Nan... > 4. NUMERICAL RESULTS AND DISCUSSION - Pg. 227

On the Modeling of Carbon Nanotubes as Drug Delivery Nanocapsules tot = n R i R o ( 2 + ) d 2 g 2 + n g R i R o ( 2 - ) d 0 - 2 2 0 where H 1 = - A , ( 1 ) G = 3 , 1 8 ( 2 ) G 1 = 63 , 256 ( 1 ) G = 3 , b 4 H 2 = B 1 ( G ) = 4 9 21 2 1 ( 2 ) ( ( , G ) = , G 2 = , G 32 ) = 160 80 128 63 G b ( 2 ) = 128 (13) Note that is an even function in term of and Equation (13) can be rewritten as 2 tot = 2 n g R i R o ( 2 - ) d 0 2 ( G 52 ) = 1 10 (18) (14) and u 1 = Z - L 1 , u 2 = Z + L 1 (19) In Equation (14), ( ) can be obtained as follow ( ) = 1 ( , u ) dvdu 2 * u 4 2 ( Z + L 1 ) - u 1 + * ( , u ) dvdu u 1 u 2 * u 2 * u 1 2 ( Z + L 1 ) - u 2 ( Z - L 1 ) - u (15) Note that is independent of v . Thus, ( ) = * u 1 - * * ( u 4 - u 1 ) ( , u ) du + * u 4 * u 1 * ( - u + u 4 ) ( , u ) du (16) where * * * u 1 = Z - L 1 , u 2 = u 3 = - * and u 4 = Z + L 1 . After an extensive manipulation of Eq.uation (16), the two integrals appearing in this equation can be analytically evaluated and the following analytical expression is obtained for ( ) It can be seen that the quadruple integral of Equation (5) can be easily evaluated by comput- ing the single integral of Equation (14). Also, due to the symmetry of the problem, the axial interac- tion force between the tubes is only considered. Therefore, by differentiating Equations (14) and (17) with respect to Z , the following expression is given for the interaction force: (see Box 2). Finally, the present semi-analytical solution procedure leads to a single integral to obtain the total potential energy and axial interaction force between two concentric single-walled carbon nanotubes. It should be noted that the above nu- merical integration of this integral is very quick and more accurate compared with the higher order integrals. In (Baowan & Hill, 2007; Baowan et al., 2008), the quadruple integral of Equation (5) is reduced to a double integral in terms of the Appell transcendental functions whose numerical evalu- ations demand considerable computational time. 2 2 ( - 1 ) m 1 ( ) = H k 2 2 3 k - 1 m = 1 2 ( 3 k - 1 ) ( u 2 + ) k = 1 m m u 2 2 ( ( - 1 ) + G 1 k ) 6 k - 1 u m 2 tan - 1 m m = 1 2 2 ( n - 3 k ) 2 3 k - 1 u 2 L ( + ( - 1 ) m G nk ) m + G b ( k ) 6 1 - 1 2 ( 2 + u m 2 ) n 2 k m = 1 n = 1 4. NUMERICAL RESULTS AND DISCUSSION In this section, numerical results are obtained for the van der Waals interaction between two con- centric and single-walled carbon nanotubes and the acceptance condition is investigated. The in- ner tube (tube 1) is assumed to be (6,6) with a (17) 227