Question

for problems where a compuster is used to generate a plot or table , example handwritten calculation must be shown

Consider the van der Waals and EDL interaction energies per unit area between two infinite planar surfaces to be Upw 1200 and UzDL = 64","? z' exp(-xD) respectively, where D is the separation distance between the surfaces. In the above expression, Z = tanh(zew/k). The conditions for the potential energy barrier to just disappear are U duro = Uypr +UzD2 = 0 and dU DETO. 0 dD simultaneously. Show that this occurs at D = 2.

Answer 1

We are given that :

Uduto = Uy DW + UERL = 0

------------ 1.

64nikt UOLTO = 12D2 Ez-.e-KD = 0

Hence, ------------- 2.

64nykt K . 127 D2

DUDLTOO dD

Hence differentiating 1. wrt D and equating to 0.

-A -2 , 6ᏎnᎥᎢ 12Ꭲ Ꭰ8 * Ꮶ . £z.e-ᏦᎠ(-Ꮶ) = 0

A 2 64nykT 72e-KD (K) 127 D3 K2

By 2. we simplify above expression :

A 2 A 12π D3-12-D2-(A

Simplifying above expression :

KD= 2

Please feel free to comment for any doubts. :)