Unfold dynamics of generalized Gaussian structures

PHYSICAL REVIEW E

VOLUME 57, NUMBER 5

MAY 1998

Unfold dynamics of generalized Gaussian structures H. Schiessel Materials Research Laboratory, University of California, Santa Barbara, California 93106-5130 ~Received 24 October 1997; revised manuscript received 20 January 1998! We consider the unfold dynamics of generalized Gaussian structures ~GGSs! exposed to different kinds of external forces. A GGS consists of N monomers connected by harmonic springs into a network; when its spectral dimension d s exceeds the critical value of 2 the GGS is in a collapsed state. Sommer and Blumen @J. Phys. A 28, 6669 ~1995!# showed that collapsed structures can be unfolded under external forces; they demonstrate this for the case where each monomer is exposed to a force with a random direction: Then networks with a spectral dimension up to 4 become unfolded. In the present paper we focus on the dynamics of such unfold processes. We investigate GGSs exposed to different kinds of external forces ~pulling one monomer, uncorrelated forces, long-range correlated forces, and diblocklike forces!. We show that external perturbations that act only on a few monomers are not able to unfold a collapsed structure; on the other hand, more general kinds of forces lead to a stretching of GGSs even for d s .2 as long as d s ,d c , where d c depends on the kind of the force field. In general, during the unfold process the size R of the GGS grows via a power law R}t a (0< a 2 the integral in Eq. ~12! diverges at the lower bound; note, however, that for very short times t! z /K 5 z b 2 /T the behavior of the GGS is governed by its local properties. Thus let us introduce z /K as a lower cutoff. For d s .2 this leads to

^^ x ~ t ! && .x 0 1 5x 0 1

SD E

f z z K

d s /2

t

z /K

~13!

where x 0 denotes the bead’s displacement at the time z /K, i.e., x 0 5 ^^ x( z /K) && . For d s 52 one finds a logarithmic time dependence. We note that in the force-free case the meansquared displacement of a single bead shows similar behavior patterns @24#. Before we provide a physical interpretation of these results, we calculate the short- and long-time behavior of ^^ x(t) && . As mentioned above, the continuous description of the spectrum in Eq. ~8! is not valid for t, z /K and t. t G and we have to go back to the exact formula, Eq. ~7!. We consider here only the case of connected objects so that only one eigenvalue vanishes, say l 1 ; thus

^^ x ~ t ! && 5

f f t1 zN NK

N

(

k52

12e 2 ~ K/ z ! l k t . lk

L.

.

d t t 2d s /2

f b ds f b2 2 12d s /2 d s /2 t 12d s /2, T z T

FIG. 1. Response of a GGS with ~a! d s ,2 and ~b! d s .2 to an external force that acts on one given monomer. Schematically depicted is the displacement of the tagged monomer as a function of time ~see the text for details!.

~14!

For t! z /K this leads to ^^ x(t) && .( f / z )t. On the other hand, for long times t@ t G Eq. ~14! takes the form ^^ x(t) && .( f / z N)t1L, where the time-independent term is given by

f b2 T

H

E

l max

l min

dl l d s /222

f b 2 d s /221 f b 2 ~ 22d ! /d s s l N . for d s ,2 T min T f b 2 d s /221 f b 2 l for d s .2 . . T max T

~15!

Note that depending on the value of d s , the lower or the upper bound of the integral in Eq. ~15! determines the behavior of L. Let us now survey and discuss the results. In Fig. 1~a! for the case d s ,2 the displacement of the tagged bead is plotted logarithmically against time. For very small times t! z /K the bead does not ‘‘feel’’ any constraints that arise from the connection to neighboring beads. It moves ballistically with the constant velocity v 5 f / z ; see below Eq. ~14!. Then, in the intermediate-time regime more and more beads are involved in a collective motion; consequently, the velocity of the tagged monomer decreases with time, following a t 2d s /2 behavior; cf. Eq. ~12!. Note that this time domain is very broad for large N. Finally, at t5 t G the motion of the tagged bead crosses over into a small drift motion @see above Eq. ~15!#. The velocity obeys v 5 f / z N, which follows from the balance between the external force f and the friction z N of the whole GGS, i.e., for t. t G the whole network moves ballistically and the tagged bead mirrors the motion of the center of mass. Note further the time-independent term ~15!. It represents the equilibrium value of the stretching of the

5778

H. SCHIESSEL

GGS due to the external force and follows from the summation over the 1/l k values. It can also be obtained by setting t5 t G in Eq. ~12!. The stretching leads to an N (22d s )/d s dependence of the size of the GGS, which is significantly larger than the typical size in the force-free case, which is proportional to N (22d s )/2d s @cf. Eq. ~1!#. In Fig. 1~b! we depict the case d s .2. The behavior for very short times t! z /K and long times t@ t G is again governed by the ballistic motion of the tagged bead and of the whole network, respectively. On the other hand, in the intermediate-time regime z /K!t! t G one has a very small displacement of the bead: At t5 z /K one finds ^^ x(t) && . f /K; see below Eq. ~14!; then, during the following intermediate regime the tagged monomer moves only a distance f /K @cf. Eq. ~13!#, which is of the same order as the elongation at t5 z /K. Thus the conformation at t5 t G is still governed by the local properties of the network. Now, as noted above, GGS with d s .2 are collapsed in the force-free case. Our considerations show that it is not sufficient to pull at one monomer in order to unfold the collapsed structure. We give now a scaling argument from which Eq. ~12! follows directly ~cf. also Ref. @26#!. As already discussed above, the generalized Rouse equation is formally equivalent to a diffusion equation on the corresponding fractal lattice. For d s ,2 the number S(t) of different sites visited by a random walker grows with time following Eq. ~10!. In the framework of the Rouse dynamics, S(t) can be interpreted as being the number of monomers that move collectively with the tagged bead. The domain that moves collectively has the velocity v (t)5 @ z S(t) # 21 f . The average displacement of the single bead can be estimated from the average displacement of the S(t) monomers, i.e., x(t). * t0 d t v ( t ), which ~for d s ,2! leads again to Eq. ~12!. Note that Eq. ~10! is valid as long as the network does not move as a whole. At later times t. t G , g(t)5N holds and we recover the drift motion of the c.m. with v 5 f / z N. The crossover time t G can be interpreted as being the largest internal relaxation time of the GGS. We close this section by noting that the above-given considerations also hold for the case when one applies external forces to a group of g monomers as long as g!N ~a situation that occurs in a class of rheological experiments @27#!.

57

^ x k~ t ! & 5

E

1 z

t

0

^ f n f m & 5 f 2 d nm ,

~16!

where the angular brackets denote the average with respect to different realizations of $ f k % . Polyampholytic networks in external electrical fields and copolymers or networks of copolymers at interfaces are examples of such kinds of external perturbations @29#. Following the preceding section @cf. Eqs. ~2!–~6!#, we find for the displacement of the kth bead after switching on the force field at t50

(m ~ e 2~ K/ z ! M t ! km f m ,

~17!

where the angular brackets denote the thermal average. Here we are interested in the mean-squared displacement. Taking the square of Eq. ~17!, then averaging over k, and, finally, averaging over the different realizations of $ f n % leads to

^^^ x ~ t ! & 2 && 5

1 z N 2

3

E tE t

d

0

1

t

0

dt2

~ e 2 ~ K/ z ! M ~ t 1 t ! ! mn ^ f n f m & ( m,n 1

f2 5 2 z N

E tE t

d

0

1

t

0

2

N

dt2

(

e 2 ~ K/ z ! l k ~ t 1 1 t 2 ! .

k51

~18! Similarly to Sec. II, we calculate now the intermediatetime behavior z /K!t! t G by going to a continuous distribution of the eigenvalues. For d s ,2 this leads to

^^^ x ~ t ! & 2 && . 5

SD E E

f2 z z2 K

d s /2

t

0

f 2b ds

z 22d s /2T d s /2

dt1

t

0

d t 2 ~ t 1 1 t 2 ! 2d s /2 ~19!

t 22ds/2.

For d s 51 one recovers the t 3/2 dependence that was found in Ref. @5#. Now Eq. ~19! is also valid for 24, however, the displacement shows a strong dependence on «; here we introduce, similarly to Eq. ~13!, z /K as a lower cutoff for both integrations in Eq. ~19!. This leads to

III. UNCORRELATED FORCE FIELD

In this section we assume that each monomer is exposed to a constant force with a random direction independently from the force directions of the neighboring monomers. Denote by f k the force on the kth monomer; the force on each monomer is taken to be directed either in the positive or in the negative X direction, i.e., f k 56 f . Then one has

dt

^^^ x ~ t ! & 2 && .

F S D G

K f2 t 2 12 K z

22d s /2

.

~20!

The behavior for t! z /K and t@ t G can be calculated similarly to Sec. II by treating in Eq. ~18! the contribution of the vanishing eigenvalue separately. For t! z /K one finds a mean-squared displacement that goes as ( f 2 / z 2 )t 2 ; this corresponds to the single-bead drift that we already discussed below Eq. ~14!. Furthermore, the long-time behavior t@ t R obeys ^^^ x(t) & 2 && .( f 2 / z 2 N)t 2 1L 2 . We find, similarly to Sec. II, that the long-time behavior has two contributions: a drift motion ~here, however, by a factor N 1/2 larger, which follows from the fact that the typical total force is of the order N 1/2! and a time-independent stretching term, which represents the typical equilibrium size L of the stretched network:

UNFOLD DYNAMICS OF GENERALIZED GAUSSIAN . . .

57

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Let us try now to derive the power law ~19! through a scaling argument similar to that presented at the end of Sec. II. For a given monomer the number S(t) of neighboring beads that move collectively with it is given by Eq. ~10! for d s f 2N2g, with 0 < g