Biomedical Engineering Reference
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hydrogen bonds and appear in zigzag structure instead of a line (see Fig. 1.7 ), the
distance between the two water molecules along the z direction is about 2.6 A.
The total potential U ( z ) of water molecules inside the CNT is given by
i DC X
U. z / D
u . z C id/f. z i /;
(1.1)
i D1
where z is the coordinate of a water molecule (denoted by P 0 ). i > 0 indicates that
the water molecule locates only in the right side of the P 0, vice versa. f. z i / is a
truncation function, which is given by
z max ;
0if z i < z min ; z i > z max ;
1if z min
6
z i
6
f. z i / D
where z min and z max are the z -coordinates of the left and right ends of the CNT,
respectively.
The
probability density
of finding water
molecules at
position z
can be
assumed as
. z / D A expΠU. z /=k B T;
(1.2)
where k B is the Boltzmann constant, T is the temperature of the system, and A can
be determined by the average number of water molecules in the CNT,
Z
z max
<N> D
. z /d z :
(1.3)
z min
According to ( 1.1 )and( 1.2 ), we can calculate the probability density of the
representative systems.
Comparing between the theoretical results and the simulation results for ı D 0 A
(see Fig. 1.18 ), we can find that the main structure of water probability density
distribution is consistent. Similar to the simulation results, the curves of the
theoretical result in the case of N D 5, 6 have five and six troughs, respectively.
The peaks of the curves of the theoretical results are a little sharper than those of
the simulation results, mainly results from that in the above model, the distance
between water molecules inside the CNT is assumed to be a constant. In reality,
the distance between neighboring water molecules fluctuates according to the
interaction potential.
When the CNT is deformed by external force, the water-CNT interaction energy
will increase correspondingly (see Fig. 1.17 ). According to ( 1.2 ), the probability
of finding water molecules in the narrow portion of the CNT decreases. The
deformation greatly changes the water density distribution especially for N D 5,
while slightly for N D 6. The errors in the theoretical predictions increase with the
deformation of CNT. The main reason is that the distance between neighboring
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