# W r of the three dimensional space see fig 1 it is

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+ w ( r ) of the three- dimensional space (see Fig. 1). It is assumed that the container is much larger than the colloidal particles so that w ( r ) is slowly varying on colloidal length scales. Due to the incompressibility of the solvent, i.e., ∇· v ( r , t ) = 0, ∇ · w ( r ) = O ( bardbl w bardbl 2 ) (17) holds for small deformations bardbl w bardbl → 0. Since the con- sidered deformations conserve the volume as well as the number of colloidal particles of the fluid elements (see Fig. 1), the number densities are not changed by the de- formation map: ̺ w ( r w ) = ̺ ( r ). Upon deformation of φ ( r ) = ̺ ( r ) ¯ ̺ with the displace- ment field w ( r ), the excess free energy in Eq. (4) leads to βF ex [ φ w ] βF ex [0] ≃ − 1 2 integraldisplay d 3 r w integraldisplay d 3 r w ¯ c ( | r w r w | ) φ w ( r w ) φ w ( r w ) = 1 2 integraldisplay d 3 r integraldisplay d 3 r ¯ c ( | r r + w ( r ) w ( r ) | ) φ ( r ) φ ( r ) . (18) Writing c (2) ( r ) := ¯ c ( | r | ), the direct correlation function in Eq. (18) can be expanded for small deformations: c (2) ( r r + w ( r ) w ( r )) = c (2) ( r r ) + ( w k ( r ) w k ( r )) ∂c (2) ∂r k ( r r ) + O ( bardbl w bardbl 2 ) , (19) where an implicit summation over the vector components k ∈ { x, y, z } is performed. Since ∂c (2) ∂r k ( r r ) vanishes for distances | r r | large compared to colloidal length scales whereas w ( r ) varies smoothly on colloidal length scales, the term w k ( r ) w k ( r ) in Eq. (19) can be approximated by w k ( r ) w k ( r ) ( r r ) ∂w k ∂r parenleftbigg r + r 2 parenrightbigg (20) with ∈ { x, y, z } . Therefore Eq. (19) can be rewritten as c (2) ( r r + w ( r ) w ( r )) = c (2) ( r r ) + B kℓ parenleftbigg r + r 2 parenrightbigg ∂c (2) ∂r k ( r r )( r r ) + O ( bardbl w bardbl 2 ) , (21) where the gradient B kℓ ( r ) := ∂w k ∂r ( r ) (22) has been introduced. Using the relation i ( r r ) exp( i q · ( r r )) = ∂q exp( i q · ( r r )) (23) an integration by parts leads to ∂c (2) ∂r k ( r r )( r r ) = integraldisplay d 3 q (2 π ) 3 iq k hatwide ¯ c ( | q | )( r r ) exp( i q · ( r r )) = integraldisplay d 3 q (2 π ) 3 ∂q ( q k hatwide ¯ c ( | q | ) ) exp( i q · ( r r )) = integraldisplay d 3 q (2 π ) 3 parenleftbigg δ kℓ hatwide ¯ c ( | q | ) + q k q | q | hatwide ¯ c ( | q | ) parenrightbigg exp( i q · ( r r )) . (24) Noting B kk ( r ) = ∇ · w ( r ) = O ( bardbl w bardbl 2 ) (see Eqs. (17) and (22)), one obtains from Eq. (21) ¯ c ( | r r + w ( r ) w ( r ) | ) = ¯ c ( | r r | ) integraldisplay d 3 q (2 π ) 3 q k q | q | B kℓ parenleftbigg r + r 2 parenrightbigg hatwide ¯ c ( | q | ) exp( i q · ( r r )) + O ( bardbl w bardbl 2 ) . (25)
5 Using Eqs. (10) and (25) in Eq. (18) yields βF ex [ φ w ] = βF ex [ φ ] + 1 2 integraldisplay d q z 2 π integraldisplay d p z 2 π | q z | hatwide B zz (0 , 0 , p z ) hatwide ¯ c ( | q z | ) hatwide φ parenleftBig q z p z 2 parenrightBig hatwide φ parenleftBig q z p z 2 parenrightBig + O ( bardbl w bardbl 2 ) (26) with hatwide B zz ( p ) being the Fourier transform of B zz ( r ). Whereas hatwide B zz (0 , 0 , p z ) in Eq. (26) contributes only for small wave numbers p z corresponding to those large length scales on which the displacement field w ( r ) varies, hatwide ¯ c ( | q z | ) contributes only for wave numbers q z correspond- ing to colloidal length scales. Consequently | p z | ≪ | q z | holds for the dominant contributions to the integrals in Eq. (26) so that the approximations hatwide φ parenleftBig ± q z p z 2 parenrightBig hatwide φ ( ± q z ) (27) apply, thus βF ex [ φ w ] = βF ex [ φ ] + 1 2