Nov 2, 1998

PHY 711 - Notes on derivation of surface energy for membranes - (Eq. 46.8 in Fetter & Walecka)

Following the discussion in Chapter 8 of Fetter and Walecka, we argue that the potential energy per unit area associated with stretching the membrane is given by

V = t æ
ç
è 		dS
dA
 -1 ö
÷
ø 		,
(1)
where dS/dA represents the rate at which the membrane stretches per unit area of the original membrane. According to Fig. 46.1 and Eq. 46.3 in the text, we can estimate this quantity according to:
dS
dA
 = 1
^
z
 
 · ^
n
 
,
(2)
where [^(n)] is the normal to the stretched surface and [^(z)] is the normal to the original (unstretched) surface. Arguing that the surface of the stretched membrane is described by the equation (Eq. 46.4):
F(x,y,z,t) º z - u(x,y,t) = 0,
(3)
we then find that a vector pointing normal to that surface is given by the gradient ÑF so that a unit vector pointing in this direction is given by
^
n
 
 = ÑF
|ÑF|
 .
(4)

Since ÑF = [^(z)] - Ñu, and [^(z)] · Ñu = 0, it follows from Eqs. 46.6 and 46.7 that

1
^
z
 
 · ^
n
 
=   æ
Ö
1 + |Ñu|2
 
 ,
(5)
and Eq. 46.8a and 46.8b follow.

File translated from TEX by TTH, version 1.92.
On 2 Nov 1998, 20:16.