Theory of the shape relaxation of a rouse chain

CHEMICAL PHYSICS6 (1974) 124-129.8 NORTH-HOLLAND PUULISHINC COMPANY

THEORYOFTHESHAPERELAXATIONOFA ROUSECHAIN MASAODOI Departmenr of Physics, Faculty of Science, Tokyo Merropoliran University. Seragaya&u, Tokyo, .Iawn

and HARUHIKONAKAJIMA Depurtmen! of Applied Physics, Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo, Iawn

Received16 March1974 Revisedmanuscriptreceived10 July 1974

An analyticalinterpretationis made for the Monte Carlo results on the shape of an unrestricted random Llight chain. Boththe average shape and the time correlation function of the shape fluctuation are dircusscd. The Monte Carlo results are well reproduced by armming that the shape OFthe chain is primarilydetermined by the first few normalmodes.

I. lntroductioa

Recently Sole and Stockmayer [ 1,2] have shcwn through Monte Carlo technique that the shape of a random flight chain is quite asymmetric. To characterize the shape of the chain, they introduced the tensor ld = (Nb)-*

c

a

raa rap

,

(1)

where ra,(u = I,2 ,... N, Q,P = x, y, z) is the @cornponent of the position vector of the uth bead relative to the center of mass and b is the bond length. The tensor I3 is related to the inertia tensor Jcrlpas

(2) (m: mass of the bead), and the principal valuesof I,, are proportional lo the squared length of the principal axes of an equivalent ellipsoid with the same moment of inertia as that of the chain. We shall call Id the shape tensor and denote its principal values by X,, X2 and h3 (xl > Xz> 1,). Sole zindStockmayer have

investigatedthe distribution of Xiin detail and found tit the ratio between averagesof I is rather high: (A$ : t12>: (A+ = 1: 0.23 : 0.085 and the probability of spherical shape is very small. Kranbuebl et al. [3] have investigated autocorrelation functions, p(r) = (Q,(f) hA0))- Q)/(ch;>

- (h#J2>

(3)

by a computer experiment for the stochastic jump model [4] of the random flight chain, and found that the ratio among the relaxation times for pi(l) is aboutl:i:k. Herawe shall show that these results are well e* plained in terms of the conventional Rouse model (bead spring model). As is well known, the random flight model and the stochastic jump model reduce to the Rouse model when the chain is sufficiently long. Therefore in what follows we assumethat N is sufficiently

large and neglect terms of order l/N.

2. Approximate pticipal values In principle, by solvingthe eigenvalueequation,

M.

det IMs,-f,l = 0, we can express the hi as a function of ta and calculate their mean values by use of the equilibrium distribution function,

P&g)

= i-l ($rby’*

function of 4 at time t, is givenby [S]

fl; =(12/a’)

X exp [-3(rti1 - fa)2/2b2] .

(4)

However this procedure is almost impracticable, because the 4 thus obtained are very complicated functions of r,. Therefore here we adopt an approximate expression ftir pi. Let us rewrite the shape tensor in terms of the normal coordinates defined by Rk = (2/N)“* c cos [@I (k= 1,2,3,..., NY 1)or

r, = (2/N)‘1z c cos [(2Q- OWN&. a

(6)

Substituting eq.(6) into es.(l) weobtain

.

(7)

Eq. (7) has the same form as eq. (1). Thus we can discuss the shape of the chain in terms of Rk. For the sake of intuitive understanding,let us imagineN particles located at 4. The shape of thisN normal mode system is equal to that of the original polymer chain. The advantagein using the normal coordinates is that their motion is independent of eacS other; Rk

represents an overdamped oscillator bound to the origin with force constant $kgT(see fig. 1). The diftkion

125

with

a

fas=(Nb)-2T RkJQo

of a Rouse chain

Doi and H. Nakajima. Theory of rhe shaperelation

sin2(ki$X’) ,

(91

where 5 is the friction constant of the bead; and the equilibruim distribution function of 4 is P,&&l)

= v (27#)-“‘*

exp(-iP$#

-

00)

‘l’hc basic idea of our approximation is as follows: The fluctuation amplitude U?$ is equal to pi2 a km2 (for k 4 N) and decreasesrapidly as k increases.(Note that the sum Z& kD2 is well approximated by the first few terms.) Therefore the shape of the N oscillator system is primarily determined by the fmt few ma&s. As is understood from fig. 1, the longest principal axis is almost always in the direction of R,, and the second longest principal axis is peqendicular to the fti. maybe in the plane determined by R, and R2. Thus we may assume that the first principal axis is parallel to R 1, and the second lies in the plane containing Rt and R2, perpendicularly to R, ; and the third is per-

pendicular to these two. Let r+ (i = 1,2,3) be the unit vector parallel to these principal axes: I+ = R,/IR,I

,

u2 = [R2 - u1(R2. q )I /lR, - u1(R2. u,)l .

(11)

u3 = q x ll2 .

Then the principal valuesare given by

equation for p({&}; f), the distribution

hi = (Nn)-2 Z;: (Ui’Rk)2 -

02)

Since there is no smallnessparameter in our problem, we cannot answer questions such as in what condition eq. (12) is justified. We can only verify eq.(12) by comparing the numerical results with those of the Monte Carlo calculations. However it is worthwhile to note that eq. (12) satisfiesthe exact relation

Fig. 1.

which is derived Fromthe relation between the roots and the coefficients of the eigenvalueequation.

126

M. Dai and Ii. Nakajima,Theoryof fheshaperelaxation of a Rouse chain

3. static propertie!

Let us fmt calculate (xl). From eqs. (11) and (12),

(A@ = ,;=, I

(Ls(uI. Rk)‘A(ul . R,,)2) ,

(22)

where

we have

(x,1= (Nlr)_2 (@

+ kFz (@I - K33)

*

(14)

A@, 4)’ l

!he

By use of eq. (lo), we have

= (rq mh)2-

the 4’s

(23)

((u, i’Q2> . l

are statistically independent of each

other, we have

(R$ = 3&* .

(19

&)2A(~,*

C[A(z+* l$)*]

2, = 3(@

Rk)2] 2, (24)

- uZ;J*)

=@3+

Hence eq.(14) is written as cXt)=(N?J)-* (3fli2tkg2 oi2).

l

(25)

(16)

l

Q)2> = &([A@

From eq. (lo), it follows

If we take z+ as the temporary z-axis, the average C(U~*R#) is calculated to be ((11, R#> =_u& = pi2 .

W,*

= tf&

(17)

2(NZ$/3d ,

- (R&j2

(k = 1)

(26)

= 2fli4 = 2(h%)4/9k4n4 , (k 2 2)

.Irr the sum of eq.(17), the dominant terms are those with smallk. Since we are consideringthe case of NSbl,& canbeexpandedas

Hence we obtain

/3; = 3k2g/N2b2

(A$} = 2,3x4 t kg2 2,9k4n4 = l/405 + 16/9n4 , (27) =

(18)

and the summation fork can be taken to +m. Thus we obtain (x1)= I,rr2tk$

1/3k2$=

1/18t2/3n2 .

(19)

and by use of eq. (19)

a;,=

1~180+2/27a2+8/9rr4.

(28)

The other second moments a$) and A$ can be cal-

culated just iu the same way. The result is In a similarmauner we can calculate (A,) and (x3): $1 = (MI)-*((R;) -((u, * R2)2) tkFS ((~~9 Rk)2>)

= (Nb)-*(za?_* t kG pi*)

= 2,3n2t ks3 1/3kzn2 = I/18- I/4n2 ,

(20)

a$ = (A@)-* kq3 <(q- R&)3

1~5 .

1/3k2n2 = I/18-5/12rr*.

(21)

Calculation of the second moments cX$ is not dif-

ficult. As an example let us calculate @, It is convenient to consider the deviation AX, = X, - ch,) instead of A, itself. From eq. (12),

(h;,= 1]180- I/36n2-7/48n4 (A$)= l/180-5/108n2-

,

1/16rr4 .

(2%

(30)

In table I, the values 03 and (x,2) obtained in eqs.(19)-(21) and (28)-(30) are compared with the results of the Monte Carlo studies by Sole [2] for the random flight chain with N = 100. The agreement is quite satisfactory for the first and second principal axes, but not so good for the third axis. This is because the absolute values of (A,>and (h$ are so small that small errors in the direction of the first and second principal axes seriously affect Q$ and (h$. lf absolute differences between the analytical and Monte Carlo results are compared, they turn out to be of almost the same order of magnitude.

The sums of @I and can be calculated exactly by an analytical method (21 and are also listed in table 1. These exact values of the sums indicate that the Monte Carlo result itself includes small statistical

M. Doi and H. NakqYma,

Theory of rhe shape relaxarion of a Rouse chain

127

Table 1 The tirst and second moments of the squaredlengths ofthe principalaxe3

qx

10

($, x 10’

i=3

i=2

i= 1

Sum

An&Itical

Analytical

Monte Carlo

Analytical

Monte cad0

AnaJytical

Maate

Exact

Carl0

Cd0

vaIue

1.231

1.2356

0.3023

0.2916

0.1334

0.1077

1.667

1.655

1.667

2.219

2.235

0.1244

0.1070

0.0223 1

0.01361

2.366

2.356

2.437

Monte

errors. The sum of &) for the present calculation is equal to the exact value l/6. This is evident because the sum of our approximate principal values satisfies the exact relation (13).

is known to be [4] ~(n}>= ~~n>(U$,) = (Q!)-“~

&k$WkJ

s (36)

NW) = 2 “k&ok f 4.

Dynamic

properties

Time correlation functions of any quantity expressed as a function of Rk for the free draining Rouse model are most conveniently calculated by employing the eigenfunctions for the diffusion operator. This was first suggested by Zimm [S] , and later discussed in detail by Fixman [6,7] who deviseda method of boson representation. Here we summarizehis method in a form convenient for our calculation. Consider a time correlation function of any function A of the normal coordinates:

(37)

where nh is a nonnegative integer, InI stands for the set of nka (k = I ,2,3,...; a = x,y, z), H,(x) is the Hermite polynomial of degree n: Ho(x) = I ,

H,(x) = x ,

H*(x) =x2-

(37)

I,...

and ok is given by o&= 12kB7j{b2 sin’(ka/uv) = k3at . (forkeN).

(38)

The eigenfunctions are normalized in such a way as (MlCmI) =s v d3R, ‘Q,)({Rk))

(A(r)A(O))=Iy

d3RLA exp(-Dr)APq

.

(31) x *{~](W)peJ

If we introduce the operator L defined by (32)

(39)

‘$)+?I}

where I+,,}{~) is Kronecker’s del!a which is equal to unity when nka= ,,,&, for allk and a, and zero otherwise. If we expand A in terms of the eigenfunctions:

which satisfies A({RkZk))= $ (OIAIbd) Ibd) ,

DA% = PqLA ; eq. (3 1) is rewritten as (A(t)A(OY=s v d3Rk Pes A exp(-Lt)A

with .

(34)

The complete set of the eigenfunctions and eigenvalues for L satisfying

Ll{nI) = oM)lInI)

,

(40)

(33)

(35)

(OIAl{n)) =fn

L

d3R,A+{d Pq ;

eq. (34) reduces to a simple form: (A(r) A(O))= 5 I(OlAl{n))12 exp[-a({n~)t]

.

(42)

M. DoiandH. Nakajimu.Theoryof the shaperekxation of a Rouse chain

128

This equation offers a convenient way to calculate the relaxatjon spectrum for the time correlation func. tions. If the matrix element (OIAI{n))does not vanish, Ol(r)A(O)>has a component rclting with the relaxa. tion time l/Q({n}). The longest relaxation time of (A(t is determined t;y the I{n]) which minimdzes -d(n)) with nonvanishing~OIAl{n]).Since a~({n])takes discrete values,we can find the longest relaxation time by examination of the vanishingof (OIAl(n})for all In) with increasinglylarger valuesof a({n}). As an example, let us consider $(t) Al(O)>. The smallesteigenvalueis q, to which belong three eigenktions, lnlx= l), III,~ = 1)and )nlz = 1). Here IQ = 1) stands for the eigenfunction with nlr = 1 and II&~0 0 for other k. cu;i.e., the nka of zero value areomitted from the expression. The matrix elements (Ol~,ln,,= I) (a = x. y. z), however, vanish because lq== 1) is an odd function of RI,. The next smallest eigenvalueis 241, which includes six eigenfunctions, Inl,=2~andInl,=nla= I,(~~,~=x,y,z;a#~). The matrix element (O(hllnla = nlB = 1) is found to vanish but (OlhlInl, = 2>has a finite value, which is calculated as follows. Since In Ix = 2) =

2-l” (R;&

I)lO)

Let us denote the pth relaxation time and corresponding relaxation strength for (A#)A&O>>by rip) and E?, respectively: C+(f)Aj(O)>= (Ai> + C E~p)exp(-f/r~)) , (r?) > Tj2) > ...) , (47) P

Then the above calculation yields r\‘) = l/2&, and E\‘)= 213~~.

The next longest relaxation time ri2) can be found by sequentially examining(OlhtIin)) with increasingly larger valuesof ct({tz}).Since Al is a quadratic functionofR1,termsof(OIXtlntx=m,,nty=my, nlz = mz; mXtm,,+m, > 3) vanish,and furthermore ~OJh~lntx=m,,nly=my,~lz=m,,n201=1~vanishes because the latter is an odd function of R,. Thus the second longest relaxation time is found to be 1/2a (nz, = 2). The corresponding relaxation strength is calculated in a similar manner mentioned above. The result is 42’ =

(49)

E~*)=~)(OIA~~~~=~)I*=11126~~.

(Ol~lln,,=2)=2-“2(X,(R:,P:-I)),

W)

because (OIAW is equal to oi) [see eq. (41)]. The averagein eq. (44) can be evaluated in the same way discussedin the foregoing section. Substituting eq. (12) into eq. (44). (OlhtIqX = 2) = 2-1’2 (MJ)-~[(R;(R$+ + C(q* &)2(R;X&l))]

Wecalculated the first and the second longest relaxation times and corresponding relaxation strengths also for (X2(t)X2(O))and (x,(r) X3(0)>.The results are summarizedas follows: @= 1/8al ,

1))

.

Ey) = 1/547r4,

@=x,y,z);

ILJ2,=2) (45)

,f)= l/lQt ,

The second term in eq. (45) vanishesbecause l

1/2a2= l/Sat ,

(43)

we have

<(ut Rk)‘(&

(48)

l”l*=“*/j=

ep = 1/3oOil4, nza=nw=

1)

(a,p=x,u,r;aZ/3):

0; -I)) = uZ$U$~~ - 1) = 0 ; (46)

and the firsttermin eq. (45) is evaluated by use of cq. (I 0). yielding, tOI? lq* = 2) = 2l’t(Nb)-2&2

= d/37?

.

1/18aI ,

E$‘)= l/2 I 87n4 ,

lFIh,=2)

(47)

Ftysymmetry, (Olhllnly = 2 and (01X,IQ = 2) have the same value. Thus the relaxation strength for this rekation time is equalto 31Qh~lnt,=2)12=2/3n’.

7$‘)1)=

7p=

1/2oal ,

(50)

(a=x,y,zh

ep = 4f5467n4 ,

lq4=“,~=n3,=“3p

=

1) (Q, fl=x,y,z;afj);

where the relatingeigenfhctions arealso listed. Table2 shows the numericalvaluesfor #-‘) and

hi. Doi and H. Nakajima.

Theoa

of the shape rekuation

Table 2

Relaxation time and relaxation strength for the autocodation furlction. where p = 1 denotes the component with longest relaxation ’ !&$“i:“c;c~~;;$;$)!~~ one. The value of c, tally observed$3 and ($1 i=2

i= 1 p=2

p=l rmax/r;P) ,ip’

T

x lo4

68.44

i=3

p=l

p=l

p=2

p=2

8

8

10

18

20

0.475

1.901

0.342

0.208

0.017

,iP)

il.973

0.007

0.576

0.104

0.208

0.017

$P)’

1.041 0.007

0.866

0.156

0.467

0.037

ep) calculated io eqs. (48)-(50).

The

relative relaxa-

tion strength #P’ = E;P’/((x?) - (hd2)

(51)

describes the relaxation strength for the normalized autocorrelation function Pi(t in eq. (3). and is also tabulated in table 2. Since 3 i’) z.1, the relaxation of PI(t) can be almost described by the first relaxation process. For p2(t) and p3(t), however, the second and other higher order relaxation processes have a considerable contribution to the total relaxation strenghts. Thus the apparent longest relaxation times observed for pa(t) and p3(r) will be shorter than their longest relaxation times $) and TV), respectively. This explains why the ratio observed between the longest relaxation times, 1 : f : h, deviates from 71’) : $) : #I= 1:;:;.

5. DiWussion The

entire result of the present calculations is based

of a Rouse chain

129

upon eq.(l 1). This assumption enables us to treat *he problem analytically. ConcemQ the nature of ap proximation (11) we would like to make the following remark. Assumption (: 1) is not an approximation which holds accurately for every conformation of ti 4 pO[yTt’Ier chain. There are conformations where eq. (11) breaks down completely, e.g., the case where R2 is longer than R, and the fmt principal axis is directed parallel to R,. However this is highly improbable and does not affect the results when the statistical average is taken. For this reason our approximation is quite similar to Kuhn’s [S] . He assumed that the polymer chain in elongated in the direction of the endto-end vector. Kuhn’s treatment, however, cannot predict the difference between (x2) and (x3). In order to explain this difference, the introduction of the normal coordinates is necessary. Finally it should be mentioned that from the viewpoint of application, it is more favorable to know the mean density of beads around the center of mass in coordinates fixed to the principal axes of the chain. This problem is now being investigated and will be reported on in the near future.

References [l] K. Sole and W.H. Stockmayer, J. Chem. Fhys. 54 (1971) 2756. 121 K. Sole. J. Chem. Phys. 55 (1971) 335. (31 D.E. Kranbuehl, P.H. Verdier and J.M. Spencer, J. Chem. Phys. 59 (1973) 3861. [4] P.H. Verdicr, J. Chem. Phys. 52 (1970) 5512. [S] B.H. Zimm, I. Chem. Phys. 24 (1955) 85. [6] hf. Fii. J. Chem. Phys.42 (1965) 3831. [7] C.W. Pyun and M. Fii, J. Chem. Phys. 42 (1965) 3838. [S] W. Kuhn, Kolloid-Z. 68 (1934) 2.