Polymer Physics

For each bead in the middle of the chain, the total elastic force results from the
springs of its two sides, as given by

fi¼

@Eel

@ri

ðriþ 1 riÞðriri 1 Þ¼

@^2 ri

@i^2

(5.21)

In the case of simple fluids, the force above generates a constant velocity of the
bead, as given by

fi¼z

@ri

@t

(5.22)

Therefore, we obtain the so-called Rouse equation in the continuous limit,

@ri

@t



@^2 ri

@i^2

(5.23)

Under the boundary conditions of@ri=@ij 0 ¼@ri=@ijn¼0, the analytical solution
of the equation above is in eigenmodes, as given by

ripðtÞ¼apcosð

ppi

n

Þexpð

t

tp

Þ (5.24)

where the mode numberp¼1, 2, 3...andn.nis the total number of beads on the
chain and is proportional to the total amount of monomers (for the sake of
simplicity in the following scaling analysis, we will directly treatnas the number
of monomers). Equation (5.24) represents any term in Fourier series expansion,
provided that the random-coil conformation of polymers is equally parted into
pwave-lengths of a stochastic vibration holding amplitudeap. The relaxation
timetpis the characteristic time for each sub-molecule to containn/pmonomers
and to diffuse through its end-to-end distanceRp. Accordingly,

tp

R^2 p

Dp

(5.25)

For an ideal chain,

R^2 p

nb^2

p

(5.26)

Fig. 5.2 An illustration of
the bead-spring model of the
Rouse chain

5.2 Short Chains 81