The conformation of linear and star polymers in solution

Prog. Polym. Sci., Vol. 16, 463-508, 1991 Printed in Great Britain. All rights reserved.

THE

CONFORMATION POLYMERS

0079-6700/91 $0.00 + .50 © 1991 Pergamon Press plc

OF LINEAR AND IN SOLUTION*

STAR

G1USEPPE ALLEGRAt and FABIOGANAZZOLI

Dipartimento di Chimica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 1-20133 Milano, Italy

1. I N T R O D U C T I O N

After dissolution in appropriate solvents, polymer chains may attain an enormous number of conformations. As a simple example, a polyethylene chain with N chain bonds develops roughly 3Nconformations, since the rotation angle around each bond may be trans (T), gauche-plus (G÷), or gauche-minus (G-), corresponding to the minima of the torsional potential regarded as rotational states.' Of course, any existing side groups would add to the total number of conformations through the statistical combination of their own rotational states. Such a large number of accessible conformations naturally favours the use of statistical-mechanical methods where single polymer chains, if sufficiently long, may even take the role of canonical systems. In this paper we shall address some aspects of the conformational statistics of polymer chains in solution. Rather than in the classical thermodynamic properties, such as the free energy of mixing or the osmotic pressure, we shall be particularly interested in such conformational properties as the interatomic distances and the overall chain size, typically probed by elastic scattering techniques. Before examining the basic conformational aspects in solution, it is convenient to consider briefly a polymer chain in a typical melt. Specifically, we analyze the statistical behaviour of its chemical bonds, regarded as a sequence. We see that the relative spatial orientation of any two consecutive chain bonds is dictated by Boltzmann's statistical law applied to its conformational states. The energy of these states is basically determined by the local conformation of the chain; packing with the surrounding chains provides an energy contribution that is usually assumed as conformation-invariant, so that it may be effectively *In memory of Prof. Piero Pino. tAuthor to whom correspondence should be addressed. Editors' note: It was our editorial policy from the beginning, in putting together this issue, to refer to the metal complexes used in Ziegler-Natta polymerization uniformly as "initiators" or "initiator systems" rather than as "catalysts", and so the former term has been used in this paper and in the others in this issue. 463

464

G. A L L E G R A and F. G A N A Z Z O L I

neglected. In other words, the chain conformation is statistically influenced by the action of local forces, arising from bond-length, bond-angle and rotationangle potentials, in addition to interactions between atoms separated by no more than ,-, 10-15 bonds. If this behaviour dictated by local forces is taken to fully determine the overall conformation of the chain, no matter how long, we have the so-called phantom chain. 2 For this model, interactions between topologically distant atoms are completely ignored. However, after following its natural trajectory for many bonds, the chain will have wandered enough to come back to some region of space already visited. Unlike the phantom chain, the real chain will avoid the space conflict. In principle, we have a statistical dilemma: (i) either the chain makes what can be regarded as a local detour, without substantially changing its rotational-state statistics overall, or (ii) the chain coil undergoes a substantial expansion to avoid most space overlaps, the remainder being taken care of through local detours. We see at once that alternative (ii) strongly reduces the chain entropy because it implies discarding most conformations. Furthermore, although the chain avoids some space conflict with itself, it generates plenty of new conflict with the surrounding chains, with a further entropy reduction if these conflicts are to be avoided in turn. 3 In conclusion, alternative (i) is much to be preferred, and we have what shall be denoted as the unperturbed chain. It is important to maintain the distinction between this physical state and the phantom chain, wherein admissible forces between chain atoms are limited to short-range interactions, confined to a topological neighbourhood of ,~ 10-15 bonds; we stress that, in spite of the inherent approximation, the phantom chain is a useful and frequently adopted model.~ Although the unperturbed chain accounts for some volume exclusion, it does so through the action of forces falling off rather quickly with increasing interatomic topological separation (or distance along the chain contour). These forces will be denoted as medium-range interactions; 2"4 as we shall see in the solution state longer-range interatomic forces may also appear. Both the phantom and the unperturbed chain obey the important rule that their mean-square size is proportional to the chain length, if this is large enough. Indicating with r(N) and S respectively the end-to-end distance and the radius of gyration of the N-bond chain, we have ($2)o = (r2(N))0/6 oc N,

N >> l,

(1)

where the angular brackets stand for "conformational average" and the subscript zero refers to the unperturbed state of the chain. The same equation holds for the phantom chain, in which case the subscript ph will be adopted. Actually, scattering experiments may yield ($2)0, whereas the mean-square end-to-end distance (r2(N))0 is not usually an observable quantity, though it is easily obtained from computer simulations. Moreover, the relatively simple theoretical evaluation of (r2(N))0 directly gives ($2)0 through eq. (1).

LINEAR AND STAR POLYMERS 1N SOLUTION

465

w(r)

1 0

w.-

/

r

r,~

i

/

',

~e(r)

I i Ii I I

I

i! tl

F[6. I. Schematic drawing of the interatomic potential w(r) (solid line) in arbitrary units and o f the corresponding effective potential e(r) = 1 - e x p ( - w ( r ) / k B T ) (dashed line) as a function of the separation r. The distance r at the minimum of w(r). rvdw, and the separation r* at which it changes sign are also indicated.

Let us now consider a polymer solution. Within the present paper we shall be concerned with dilute solutions, where the polymer coils behave as independent entities. At a quantitative level, if the distance r between any two chain atoms is smaller than their van der Waals' distance, the interaction energy is large and positive (i.e. repulsive), otherwise changing to negative, that is, attractive, for larger r's and going asymptotically to zero for still larger distances (see Fig. 1). The resulting energy plot resembles the well-known Lennard-Jones function, but two considerations must be borne in mind in this regard. First, for any given distance r the plot is assumed to represent the average local energy w(r) between two chain atoms, considering all possible conformational states of their side groups. Second, w(r) is actually to be regarded as a free energy difference between the state under consideration and a reference state wherein the chain atoms are surrounded by the solvent only. In conclusion, the energy reported in Fig. 1, (i) contains an entropic contribution, and (ii) is affected by the solvent quality. In a dilute solution we may assume that the energy contribution prevails over the entropic one. Since the two interacting atoms of Fig. 1 in principle may be separated by all possible

466

G. ALLEGRA and F. GANAZZOLI

distances, it can be shown that the cluster expansion coefficient effectively associated with their interaction 5 is

fl(T) = fv [1 - exp ( - w ( r ) / k , T ) ] d V ,

(2)

where the integral is extended to all space. Since w(r) is mainly energetic, the so-called binary cluster integral fl increases with increasing temperature, at least within the temperature range of interest to us. We shall consider in the following the alternative case where w(r) is mainly entropic. Although we are confining our attention to two-body forces, it must be pointed out that they are to be counted properly. For example, it is possible to see that, if three non-bonded chain atoms come close at the same time, their contact energy is not reducible to a mere sum of two-body contributions o f the type given in Fig. 1. Through the cluster-expansion method it can be shown that the coefficient of each two-body contact needs modification to account for the three-atom contact. We have

Belt(T)

=

(3)

f l ( T ) + a3,

where fleeris the effective factor and 0"3 ( > 0) is the three-body repulsive correction. Although a3, unlike fl, may sometimes be considered as effectively temperature

independent, it is generally dependent on the location of the two atoms along the chain (mainly on their topological separation). Of course, higher-order corrections are also needed, in principle, but their importance quickly decreases with increasing order. Hence we shall retain 0"3 as the only important correction to ft. It may be possible to select a temperature ® (depending on the polymersolvent pair) such that fl is slightly negative, 0"3 turns out to be the same for all the atom pairs and we hence have 2'6 flerr(O) =

fl(O) + 0"3 =

0.

(4)

Under these conditions we essentially reproduce the same unperturbed state existing in the melt. But why is it that in a melt the unperturbed state persists over a range of temperatures instead of appearing at a single temperature, as in solution? Let us first consider that in a melt the potential w(r) (see eq. (2)), cannot contain an important energy contribution. In fact, if the distance between any two atoms is changed, other identical atoms will intervene to keep the density locally uniform, the number of the interatomic contacts will stay the same and so will the energy. Being therefore (mostly) of an entropic nature, w(r) is proportional to T and the integral giving fl is temperature independent. The fact that its value is exactly the one satisfying eq. (4) is not obvious but is suggested by the initial considerations (in fact, fleff :/: 0 would lead to either expansion or contraction of the chain). Going back to the solution state, if T > ®, in the light of the preceding discussion we have fl(T) > fl(O), and consequently flefr(T) > 0, see eq. (4). As

LINEAR AND STAR POLYMERS IN SOLUTION

467

we shall see, there is a predominance of intramolecular repulsions and we are in a good solvent. Conversely, for T < ®, flefr(T) < 0, the intramolecular attractions predominate and we have a poor solvent. As chain length increases, the change of solvent behaviour is usually manifested within a progressively narrower temperature range. We shall be interested in chains long enough that this temperature range is much smaller than O. It is now possible to show that in the unperturbed, or theta, state, where fl¢~ = 0 (see eq. (4)) intramolecular medium-range interactions between different chain atoms still exist. Let a chain comprise N atoms indexed sequentially from 1 to N, and consider two of them with index numbers i a n d j and a mean-square distance . We show that if their van der Waals distance rvow(see Fig. 1) is not negligible with respect to m, the effective value of fl applying to them is more positive than the one given by eq. (2). In fact, the integral must be weighted by the a-priori probability Wi.j(r) of finding the two atoms at a distance r. We shall assume a Gaussian distribution for the probability, i.e.

[3/(2zt(r2(i, j)>)]3/2 exp [ - 3r2/2]

Wi.j(r) =

[3/(2n(r2(i,j)>]3n[l

- 3r2/2(r2(i,j)> + . . . ] .

(5)

Referring to temperature T for generality, the interaction free energy between the two atoms is the average weighted with Wij (r), or (in kB T units, see also eqs (2-4)) a(i,j)

Iv Wi.j(r)[1 - exp ( - w(r)/kB T) + f3(r)]dV

=

(6)

where f3(r) is the overall interaction among atoms i, j and a third atom, the distance r(i,j) being equal to r. Obviously, we have Iv t"3(r) d V =

0.3.

(6A)

Remembering eq. (5), it is convenient to split the integral in eq. (6) as follows a(i,

J)

[3/(2rc(r2(i, j)>)]3/2 {Iv[ 1 _ exp ( - w(r)/kB T) + f3(r)]dV

-3/(2(r2(i'J)>) Iv fl[1 - exp ( - w(r)/ks T) + f3(r)]dV t . (6B) The first integral within curled parentheses is merely/3o~(T), see eq. (4), and we shall write 2'4 a(i,j) ==_ a2(i,j) + ~ a3(i,j, k) + a2s(i,j)

(7)

k

where a2(i,j)

=

[3/(2g(r2(i,j)>)]3n~(T);

X a3(i' j' k) = k

[3/(2rr)]3/2 0.3;

(7A)

468

a2ss(i, j )

G. ALLEGRA and F. GANAZZOLI

=

- zr[3/(2n)]5/2 fv r2[1 - exp ( - w(r)/ka T) + fa(r)] d V.

(7B) At T = 19 the sum of the first two terms in eq. (7) is zero, since fl(19) + 0"3 = 0, see eq. (4) and (7A). The integral in eq. (7B) is negative - in fact, the integrand resembles the function reported in Fig. 1 - and approximately constant in the vicinity of T = 19. Defining ~v r2[l -- exp ( - w(r)/kB T) + f3(r)]dV =

-v¢,

(8)

We have a2s(i,j)

=

n[3/(2n)]5/2v¢,

(9)

that is, a positive (repulsive) contribution sharply decreasing with increasing . While vc is the effective volume per chain atom, may be regarded as the lower limit of the squared distance between the axes of two contacting chain segments; for shortness, it will be referred to as the meansquare chain thickness. 4 In conclusion, at T = 19 the interaction energy between atoms i and j may be written as a ( i , j ) ~ a2s(i,j) oc -5/2.

(10)

This term embodies the medium-range interactions mentioned in the description of the polymer melt, and is responsible for the local adjustments, or detours, of the chain to avoid itself in the unperturbed state, without altering the proportionality between the mean-square size and the molecular weight of the polymer (see eq. (1)). They are also denoted as screened interactions, in that they derive from hard-core repulsions which are screened by attractions at larger interatomic distances. At T ¢: 19, remembering flefr = fl + 0-3, eq. (7) may be written as a(i,j)

=

[3/(21r)]3/2fleff(T ) -~ a2s(i,j)

(11)

where a2s(i,j) is still given by eq. (9). We see that the term comprising fl~fris oC <1"2(i, j)>-3/2, SO that it decreases more slowly than a2s(i, j ) with increasing average distance between the two atoms. It embodies the so-called long-range interactions. In the long-chain limit these interactions produce a very substantial effect going well beyond a mere change of the proportionality constants in eq. ( l ) . 3'7 In fact, new power laws are set up. For T > 19 we have oc oc N 2v ( T > 19, N ~ o0)

(12)

where 2v is close to 1.2, instead of 1 as in the unperturbed state. Conversely, for T < 19, fl~fr(T) < 0. In the infinite-Nlimit the chain shrinks towards a compact globule, wherein the vast majority of the mean-square distances between atom

LINEAR AND STAR POLYMERS IN SOLUTION

469

pairs have the same value, equal to twice the mean-square radius of gyration. 2'6 The exponent v appearing in eq. (12) is equal to 1/3. In the following we shall discuss some significant conformational properties of linear and star polymer chains, in the light o f recent developments. The theta-solution, the poor-solvent ( T < O) and the good-solvent case ( T > ®) will be considered in this order. We apologize in advance for unduly emphasizing some of our own results. 2. T H E G A U S S I A N S E L F - C O N S I S T E N T APPROACH TO CHAIN CONFORMATION

The existence of medium- and long-range interactions makes it useful to adopt chain coordinates of a collective type, 2'8'9 each resulting from contribution of all chain atoms or chain bonds. It is also useful that the conformational averages over the products of different collective coordinates reduce to zero (orthogonal coordinates), which simplifies the results quite effectively. These collective coordinates are conveniently identified with suitable linear combinations of the chain-bond vectors, the coefficients being taken from Fouriertransformation along the chain contour. At first we shall consider a linear chain with N(>> l) chemical bonds along the main skeleton, its conformation being described by the ensemble o f N bond vectors I(1), !(2). . . . . I ( h ) , . . . , l(N). For simplicity, here we shall assume that the chain behaves as if it were the periodically repeating portion of an infinite superchain with a conformation [ . . . I(1), !(2). . . . . l(N), l(l), 1(2). . . . . I(N), l(1), !(2) . . . . ] (see Fig. 2). This assumption enables us to regard all chain bonds as conformationally equivalent, no reference to chain ends being necessary. The collective coordinates, or Fourier components, are then given by 8 N

l(q) =

~ I(h)e iqh

(13)

h=l

where, assuming the number of bonds N to be even, the Fourier coordinate q belongs to the set of values

{q} = O,

+_2~r/N, ___47r/N, . . .

2~(N/2)/N =

~.

(13A)

t(2)j~-...... ~ i ) /~(I)

-I~N-I)

I(N)

FIG. 2. Sketch of the chain as a periodically repeating portion of an infinite superchain (see text).

470

G. ALLEGRA and F. GANAZZOLI

It is important to remark that there are as many different values of q as there are chain bonds, which is in keeping with the equivalence between the two sets of coordinates l(q) and l(h). In particular, from the two previous equations, it is easy to show that l(h) is obtainable from the T(q) set, i.e. I(h) =

N - ' ~ i(q) e-iqh

(14)

and that the Fourier components, or modes, are indeed an orthogonal set. ~° It should be stressed that this conclusion derives from the periodicity assumption that makes the average scalar products (l(h). I(h + k)) independent from h. Therefore it is applicable to any chain model, namely, phantom chain, (real) unperturbed chain at T = O, expanded chain at T > 19, or collapsed chain at T < 19. In the specific case of the phantom chain we put (i(q) "i(q'))ph =

NI2f(q)A(q + q')

(15)

where A is the Kronecker delta, I is the length of the chemical bond along the chain skeleton and C(q) will be defined as the generalized characteristic ratio. 8 Its general expression is C(q) =

1 + 2[(l(h).l(h + 1))ph cos q

(16)

+ (l(h). I(h + 2))ph COS 2q + . . . ] / l 2 all the averages being evaluated for the phantom chain. For q = 0 we have the classical characteristic ratio (of the phantom chain), i.e. C(0) =

Co~,ph.

(16A)

Equation (16) suggests that the infinite sum in square brackets may be performed through an appropriate average of the rotational matrices carrying the intrinsic reference frame of any chain bond into that of the next bond, ~thus obtaining a closed matrix expression. 8 This is indeed possible either within the rotational-isomeric state (ris) scheme, 1"8 or within the alternative scheme that performs the full configurational integral through the correlation matrix carrying the Fourier components of the statistical weight (all-skeletal-rotations, or asr, scheme)) ~ A fairly good and general analytical representation of C(q) turns out to be the following9 C(q) = a/(1 - b cos q)

(16B)

where a > 0, 0 ~< b < 1. The closer b is to unity, the sharper will be the resulting peak at q = 0 and the stiffer will be the chain, meaning that the chain bonds tend to maintain the same local alignment. Typical plots of C(q) are shown in Fig. 3. In many cases we may be interested in long-range conformational properties, namely those involving relatively long chain strands. Considering that a Fourier

LINEAR AND STAR POLYMERS IN SOLUTION

471

10

8

PS

6

S 42

P

E

~

0 0

I 0.05

I 0.10

I 0.15

I 0.20

0.25

q/~t FIG. 3. The generalized characteristic ratio C(q) plotted as a function of q/~ for polyethylene (PE) at 400 K, atactic polystyrene (PS) at 300 K and poly(dimethylsiloxane) (PDMS) at 350 K (see Ref. 2 and references therein).

component i(q) comprises 2rr/q chain atoms within its wavelength, this means that we are interested only in a very small q-range of Fig. 1 near q = 0, where C(q) is basically constant. We may write in this case C(q) ==_ C(0)

=

C~,r,h

(16C)

and we recover the so-called bead-and-spring model, i.e. a chain model where representative beads are connected by perfectly elastic springs. In the real chain model, whether in the 19 state or not, eq. (15) changes into

(l(q).l(q'))

= Nl2~t2(q)f(q)A(q + q')

(17)

where ~2(q) is the strain ratio of the q Fourier mode, namely the ratio between the conformational average of the real chain and of the p h a n t o m chain, as is apparent upon comparison with eq. (15). I f we are in the 19 state and denote with Coo the corresponding characteristic ratio, it is easy to show that C~

=

C(0)a2(0)

(T =

19)

(17A)

F r o m the full set of the strain ratios ~2(q) and the p h a n t o m chain function

472

G. ALLEGRA and F. GANAZZOLI

C(q), all the mean-square interatomic distances may be obtained: (r2(h, h + k))

=

(r2(k))

=

J(k)C(0)l 2

12

sin2(qk/2)

(18)

= ~ Z 82(q)C(q) sinZ(q/2) {q} k being the number of chain bonds separating the two atoms under consideration. Obviously, the phantom chain is also encompassed by this formula if all ~2(q)'s are set equal to 1. If the phantom chain is described by the bead-and-spring model (see eq. (16C)), we have J(k) = k and (r2(k)) oc k. We see that the ~2(q)'s are the actual variables of our conformational problem in that they completely specify the conformational strain if we do not need higher-order geometrical averages beyond the mean-square distances, as is the case within the Gaussian approximation. In turn, the ~2(q)'s are derivable from a self-consistent minimization of the chain free energy, as seen in the foUowing, x6 Expressed in kB T units, the chain free energy A is given by the sum of a conformational, or elastic, contribution Ae~, and of a contribution due to the intramolecular interaction Aintra. We have 6 A = Ael -q- Aintra Ae~ =

~2 [ ~ 2 ( q ) {q}

1 - ln~2(q)]

(19)

(19A)

whereas mintra is the sum of the contributions a(i, j) given in eq. (7) over all the pairs of chain atoms labelled i a n d j (1 ~< i,j <~ N). We have Aintr a

=

A 2 + A 3 + A2s

(20)

where A2 =

~Z

a2(i,j)

I <~i
m3 =

XX~

a3(i,j, k)

I <~i
A2s =

ZX

a2s(i,j).

(21)

l <~i
In these summations, the differences ( j - i) and (k - j ) indicate the number of chain bonds separating atoms that interact through a two- or a three-body contact. In the Gaussian approximation, the probability of a pairwise contact is proportional to (r2(i, j ) ) - m , and so is the corresponding free energy a2(i, j ) (see eq. (7A)). However, for small values of ( j - i) (e.g. < 100 in a flexible hydrocarbon chain) this approximate evaluation is grossly overestimated, as shown by the investigations of Yoon and Flory, ~2 and Elory and Chang ~3 on polyethylene (PE) and on polydimethylsiloxane (PDMS), respectively (phantom chain

L I N E A R A N D S T A R P O L Y M E R S IN S O L U T I O N

473

models). Their results give us a chance of improving the accuracy of eq. (21) by imposing a suitable lower cutoff/~ to the differences (j - i), with no other sacrifice of the Gaussian approximation. 4 Namely, the value of E is derivable from the ratios between the "real" probability of an end-to-end contact for short chains of different length, and the Gaussian probability. In turn, the real probability is evaluated from classical rotational-isomeric-states calculations, as reported in the two quoted papers. 12'13Labelling such ratios as ,ok, where k is the number of chain bonds, we have p, = WkR(0)/W,c(0); Wk6 (0) =

[3/(2g(r2(k)))] 3/2,

(22)

W,(0) being the probability of a zero distance between the chain ends, whereas the suffixes R and G stand for "real" and "Gaussian", respectively. The values of p, are reported in Ref. 12, Fig. 21 for PE and in Ref. 13, Fig. 14 for PDMS, under the form of plots from analytical expressions with different degrees of approximation. In the following we shall refer to the best-approximation plot. The cutoff value h7will be determined through the simple criterion that, if the pairwise free energy of a very long chain is evaluated in the Gaussian approximation by imposing the cutoff, the result must be the same as that obtained from the real chain with no cutoff. We have [ ~ , a 2 ( i , / + k)] k=l

= R

[~,

a2(i,i+

k=E

k)]

(23) G"

Identifying for simplicity a chain portion comprising k bonds with a small k-bond chain, we know that a2(i , i + k) is proportional to W,(0 ). Considering further that W,(0) is about the same for the real as for the Gaussian chain if k > 200, and that WkR(0) may be equated to zero 12'j3 for k < 20, replacing Wk (0) for a2(i, i + k ) w e may set from (23) 200

200

W~R(0) = k=20

k ~ Wkc(0).

(24)

=

Since Wkc(0 ) oc (r2(k)) 3/2 and W~R(0) = Wk~(0)pk (see eq. (22)) we may substitute in eq. (24) if we know (r2(k)), in addition to P,. Taking these values from the ris calculations on the PE phantom chain at ~ 410 K, we solve the resulting equation for k- to yield k7 ~ 42. A very close figure, within a few units, is obtained for PDMS. In the preceding paragraphs we have elaborated at some length on the issue of the meaning and of the evaluation of the cutoff limit kT.Although it is obvious that no interatomic self-contact may exist, so that some non-zero limit E must be invoked, it is frequently believed that its precise value is either irrelevant ~4or derivable on phenomenological grounds) 5 However, we shall see that some conformational effects do depend on /~, which makes the above derivation

474

G. ALLEGRA and F. GANAZZOLI

useful. To avoid unnecessarily precise, and perhaps overly tedious computations, in the case of polymers other than PE and PDMS we may assume with sufficient accuracy that/~ is proportional to the unperturbed characteristic ratio Coo .4

One may question whether the value of ~r could'be modified by the degree of expansion or of compression of the chain, as induced by a good or a bad solvent (i.e. T > 19 or T < 19), respectively. However, we shall be mainly concerned with temperatures within a few degrees of the 19-value. This usually implies very modest changes of the chain conformation at the scale of/~, while leading to substantial expansion or compression for sufficiently long chains (N > 103). Consequently, we shall assume/~ to be independent of temperature. We are able now to write the formal expression of the chain free energy. From eqs (7-9) and (19-21) we have [ 3 \3/2 A = ~X[~2(q) _ 1 - ln~2(q)] + ~ ) ilk=, ~ (N{q} +

{ 3

vc(A r)

N

k)(r2(k)} -3/:

(N - k ) & ( k ) )

N--f,( N - k

+ 33 k~, , ~

( N - - k - k.)Wk.,,(0, 0)

(25)

l

where p is a parameter for the three-body interactions, proportional to v~z, and Wk.k~(0, 0) is the probability of a contact among three atoms labelled i < j < /, with ( j - i) = k, (l - j ) = kl. In general, this quantity is a rather complex function of (r~(k)), (r2(k~)) and (r2(k + kl));6 it simplifies in the particular case T = 19 where Wk.k.(0, 0) --~ W~G(0)WkIc(0)

(26)

the Gaussian contact probability Wkc (0) being defined in eq. (22). In this case eq. (25) may be written as (N ~> /~) A = 3X[fiZ(q) _ 1 - lnfi2(q)] + {q}

U2nC(0) l j

I;(N-

k)

rN dk + K j~ ( N -- k) J(k) 5/2.

dk IUr-k ( N + Kl Jg~JvJ(k) 3/2 J~

dkl J(kl) 3/2

(27)

K = n[3/(2nC(O)12)]5/2vc(Nr)

(27A)

K, = js[3/(2r~C(0)lZ)] 3.

(27B)

k--

k~)

where

LINEAR A N D STAR POLYMERS IN SOLUTION

475

It should be noted that the independent variables ~2(q) affect the final result, (25) or (27), both by appearing explicitly and by influencing (r2(k)) or J(k) (see eq. (18)). Consequently, minimization of A with respect to the full set of the ~2(q)'s may be regarded as a self-consistent process. The resulting set of equations is highly non-linear and may be solved through numerical iteration. 6 Hitherto we focussed our attention on linear chains in the periodic assumption (see Fig. 2, eqs (13-14)). Actually, this assumption is possible only when N is large and specific end effects are not investigated. Otherwise the periodic constraint may induce some artificial behaviour in the chain ends. To avoid this inconvenience we may use the so-called "open-chain" Fourier transform, whereby sine functions replace the complex exponentials e ~qh in the equations (13-14). Under these circumstances, unless we adopt the phantom chain model, we find a two-fold deviation from the periodic assumption: (i) the mean-square distance (r2(h, h + k)) depends on both k and h (unlike in eq. (18)); (ii) the Fourier components l(q) do not constitute an orthogonal set (as in eq. (17)). While both deviations are relatively minor in the linear chain case even for large overall expansion or contraction, this is not so for the regular star polymers, for which appreciable deviations from mode orthogonality were checked by us even for an overall mean-square expansion around 2 over the phantom chain. ~6 However, corrections for non-orthogonality effects are possible, to any degree of accuracy. 16 In any case, the periodic model does not lend itself to a natural treatment of the branch-point "atom". For all these reasons, use of the open chain transform becomes unavoidable in the star case. From the early analysis of Zimm and Kilb ~7 it turns out that the Fourier components may be distinguished into an even-order and an odd-order set. While the former set has a unit multiplicity, the latter has a multiplicity equal to ( f 1), w h e r e f i s the number of equallength arms in the star molecule. However, the actual expression of the Fourier components reported in Ref. (18a) is correct only in the phantom case, which entails more erroneous consequences. ~SbThe correct expression will be given in Section 4.2. 3. THE THETA TEMPERATURE The ® temperature was defined in the introduction as that producing equilibrium between the long-range two-body and three-body intramolecular interactions (see eq. (4)). However, the original, although equivalent, definition is given in terms of an effective cancellation of intermolecular interactions. 3 Specifically, O is the temperature at which the second virial coefficient .42 (measured, e.g., by light scattering) vanishes. A2 may be expressed as 5

`42

~ NA~ ~(2 V M 2 ) f [W(l, 2) -- W(1)W(2)]d(l, 2),

(28)

476

G. ALLEGRA and F. GANAZZOLI

where NAy, V and M are Avogadro's number, the volume of the solution and the polymer molar mass, respectively, whereas W(I), W(2) and W(1, 2) indicate the respective Boltzmann statistical weights of chain 1, of chain 2 and of the two interacting chains. The differential indicates that the integral must be performed over all the configurational coordinates of the two chains. It may also be expressed in terms of the Helmholtz free energies A, conveniently expressed in kB T units: ~9 I [W(1, 2) - W(l)W(2)]d(1, 2) =

V2{exp [ - A ( I , 2)] - exp [ - A ( I )

- A(2)]}

(29)

the V 2 term deriving from integration over the solution volume of the two centre-of-mass coordinates. From the above, we have A2 oc 1 - exp [-Ainte,(1, 2)]; A~nter(1, 2) =

A(1, 2) - A(I) - A(2).

(30) (30A)

We see that Ainter(l, 2) is the interaction free energy, that is, the excess free energy of the ensemble of two chains over that of two isolated chains. At the ® temperature, A2 = 0, and from eq. (30) we have Ainter(1, 2) = 0. It may be shown that the interaction free energy Aimer(l, 2) is derivable from the intramolecular contributions given in the previous section, although the medium-range terms a2s(i, j ) (i.e. the screened interactions) are ineffective and may be ignored as far as the interaction between different chains is concernedfl This conclusion is consistent with eq. (11) for the overall interaction energy a(i, j ) between two atoms, which shows that the vanishing of fl~trat T = 19 does not modify the value of a2s(i, j). Therefore Ainter(1, 2) depends only on the classical long-range two- and three-body interactions and for T ~ 19 it is given by: Aint~r(1, 2) oc fl(T)[3/(2~C(O)I2)] 3/2 + 2K~ f~ (1 - k/N)J(k)-3/2dk

(31)

where the symbols were already defined in eq. (27). Let us put Zph =

(T-

19ph)/T,

19oh being the temperature at which the binary cluster integral vanishes (see eq. (2). It may be referred to as the 19 temperature of the phantom chain. It is possible to show that for sufficiently small values of Irphl we have

fl(T) ~- arph,

(33)

where the constant a is roughly equal to the volume per chain atom vc. The linear dependence is merely a truncated series development, considering that 13 = 0 for Zph = 0. Concerning the constant a, we shall make recourse to an argument similar to that used in the van der Waals theory of real gases. 2° Let us denote by r* the distance of two approaching atoms below which their

L I N E A R A N D S T A R P O L Y M E R S IN S O L U T I O N

477

interaction potential w(r) becomes positive (see Fig. 1). Since w(r) is extremely large for r < r* and relatively small and negative for r > r* (the argument would obviously fail if there were some strong interaction like a hydrogen bond) we may put: /~(r)

4~ fr
=

--

exp ( - w ( r ) / k B T)]r2dr

+ 47r f,>,. [1 - exp ( - w(r)/kB T)]r2dr

~- 4~ fro* r2dr + 4~/(kBT) f f w(r)r2dr = a - b/T,

(34)

where a =

4re(r*)3/3 ~- vc

(34A)

Vc being the covolume per chain atom, and

b = --4zc/kB f f w(r)r2dr

(34B)

where b is positive since w(r) is negative for r > r*. In eq. (34), the series expansion of exp ( - w(r)/kB T) is truncated at the linear term since we assume w(r) < kB T, which is a reasonable approximation, especially for many synthetic nonpolar polymers. From eqs (34) and fl(Ooh ) = 0, we have fl(T) a(1 -- Oph/T) = a%h, i.e. eq. (33), where Oph = b/a. The Q-temperature of the real chain may be obtained as a function of N from the requirement A2 = 0, hence Ai,tcr(l, 2) = 0 (see eq. (31)). Defining ~ph =

rob( T = O) =

(0

--

Oph)/O

,

(35)

and assuming I~hl ~ 1, we have O ( N ) - Oph[l + ~oh(N)]-

(35)

Proceeding to explicit results, let us assume J(k) - k in eq. (31), which is true for large k in the phantom chain (k > 100). We get 639'21 ~ph(N) =

(O -- Oph)/O =

-4K,/(B(E)~/2)[1 -- (I~/N)'/2] 2

(36)

where (see eqs (33) and (34))

B = a[3/(2rcC(O)12)] 3/2.

(36A)

From eq. (36) we have, 21 for N >> /~, O ( N ) = O~ + 2qg/x/N

(36B)

with O~

=

Oph(1-

4K~/B(E)'/2);

q~ = 4K, Oph/B.

(36C)

We see that, with increasing N, the O temperature decreases to a limit O~, which is smaller than Oph because the residual three-body repulsive interactions

478

G. ALLEGRA and F. GANAZZOLI

(terms with coefficient Kt) need to be compensated by two-body attractive interactions, thus requiring a temperature lower than ®vh- Moreover, ®oo is reached from the above, since the three-body interactions are relatively less important for shorter chains, hence requiring weaker two-body interactions, that is, a smaller temperature decrease from ®ph. More specifically, in a threebody interaction occurring between two chains, the two atoms belonging to the same chain must be separated by at least/~ bonds to produce a contact, whereas no such stereochemical constraint applies to the two-body interactions involving two chains. Therefore, due to the cutoff, comparatively fewer atoms are involved in the three-body repulsive interactions when the chain is shorter. This may also be seen from the explicit evaluation of the three body repulsive parameter 0 3 (see eqs (3) and (4)) in the present case. Considering that fl = fl(®), from eqs (33), (36) and (36A) we have 0-3 =

4Kl/(~)l/2127tC(O)12

/313/2[ 1 -- (~]N)l/2] 2

(36D)

showing 0-3 to decrease with decreasing N. It should be added that more sophisticated models of chain conformations in the O state (see, e.g. the following section) do not basically alter the above results, at least in the limit N >> /~ considered in eq. (36). 19 The procedure outlined above for linear polymers may be easily extended to regular star polymers with f arms and N + 1 atoms, hence N / f atoms per arm if one atom is at the branch point. Since we must take into account that the three-body interactions among two stars involve two atoms on the same star which, in turn, may be either on the same or on two different arms, let us rewrite eq. (31) in the following, more general form:

li - j/ >/ k-

(37)

where the double sum runs over all the star atoms and J(i, j ) is defined in analogy with J(k). Due to the molecular symmetry, we may simply evaluate the above sum for the two atoms belonging to the same arm with multiplicity f, or on two different arms with multiplicity f ( f 1)/2. We take again J(i, j ) = li - jl, where the atoms are numbered from 1 at the free end of one arm to 2N/f + 1 at the free end of the other arm, the branch point being labelled as N / f + 1. (The detailed evaluation of the double sum in eq. (37) is in Ref. 21.) Neglecting terms of order N -l , we get ®(N)

= 0o~ - ~oz(f)N -~/2

(38)

®o~ and ~o being defined in eq. (36C), and z(f)

= f ' / 2 [ f ( 2 - 2 ~n) + 21/2 - 4].

(38A)

LINEAR AND STAR POLYMERS 1N SOLUTION

479

28

/ 20102

/ 103

10~

105

106 N/f

FIG. 4. "]'he tbeta temperature of linear and star polymers with farms as a function of the number of atoms per arm N/f (the linear chain corresponds to f = 2). The calculated curves are from eq. (38), the experimental points are for polyisoprene in dioxane 22 (@o~ = 33.4°C, ~ = 34K, see text).

Of course, the linear chain is recovered by setting f = 1 or 2, so that X(1) -- Z(2) = - 2, hence eq. (36B). The present results are reported in Fig. 4 in comparison with some experimental data from polyisoprene in dioxane. 22 Let us note, incidentally, that both O~ and ¢p depend on the given monomer/solvent pair, whereas the effect of topology is embodied in x ( f ) and the molecularweight dependence is explicitly given in eq. (38). The opposite way of approaching ®~ with increasing N, for f larger or smaller than 4, should be notedfl t With the experimental data reported in Fig. 4, the plots of Fig. 5 were also obtained for a comparison with eq. (38). Thus, O(N) is reported vs (N/f) 1/2for b o t h f = 8 a n d f = 12. The common intercept ®~ was set at the experimental value 22 of 306.4 K, whereas the two interpolating straight lines were obtained through a linear regression. The resulting value of ~o from the two sets of data is ~ - (34 + 2)K, denoting a substantial consistency. It is instructive to compare the results given above with those from the approach of Candau et aL,23 who extended to branched polymers the OrofinoFlory theory. 24 Here, the polymer is described as a gas of monomers, with an interaction free energy depending on the monomer density. Their result may be cast in a form equivalent to eq. (38) but, due to the neglect of chain connectivity, two incorrect conclusions are derived: (i) ®~o coinciding with ®ph (in other words, within this model, for large molecular weights, the probability of threebody collisions is negligible compared to that of two-body interactions) and (ii) for linear chains ® increasing with N towards the asymptotic value (or, the probability of a three-body interaction appears to become larger, as the chain

480

G. ALLEGRA and F. GANAZZOLI 0 36

~" 32-

28

\ 24

•

f = 12

÷\ 20

I 0

J 2

I

J 4

i

i 6

,

ooI¢

8

FIG. 5. The experimental theta temperature for regular stars w i t h f = 8 and 12 arms (from the previous figure) plotted as a function of ( N / f ) - 1t2according to eq. (38). The best-fit lines through the common intercept O~ are shown.

becomes shorter). Both results depend on the assumption of a uniform monomer density inside the coil, which is d oc N/V; taking V oc ( $ 2 ) 3/2 oc N 3/2, one gets d oc N-1/2. This would make the probability of a three-body encounter more and more unlikely with increasing molecular weight, hence with increasing N. This conclusion is in conflict with the consideration that the numbers of twoand of three-body interactions increase with N in (about) the same way. This is because any interaction between two chain atoms must be accompanied by a proportionate amount of three-body interactions involving a third atom close to either of them because of the chain connectivity, as first suggested by Khokhlov. 25 The same conclusion would apply to those higher order interactions involving two groups of neighbouring atoms. It should be added here that eq. (38) was first obtained by Cherayil et al. ~5 through a different approach. Treating the chain as a continuous curve with a suitable lower cutoff analogous to E to avoid physically impossible selfinteractions, they employed a perturbative analysis to get the temperature O at which the second virial coefficient A2 vanishes. Their resulting equation is identical to eq. (38), provided that we identify their cutoff with/~, their threebody parameter z° with K~ (which is taken to be the same both for branched and for linear polymers) and their two-body parameter z°e with ~phB(N[f) ~/2, ~ph = (O - O~h)/O.

LINEAR AND STAR POLYMERS IN SOLUTION 4. T H E

POLYMER

EXPANSION

AT

T =

481 O

4.1. Linear polymers We now turn to the question of the actual conformation of a linear polymer at the O temperature. It was earlier pointed out in Section 3 that the shift of O from the phantom chain value (see eq. (36)) results from the need to compensate the long-range interactions through an equilibrium between three-body repulsions and two-body attractions. Actually, we recently showed that such equilibrium may be rigorously attained only within internal portions of a linear chain.19 However, for simplicity we shall proceed by assuming that at T = O the compensation is perfect, so that the only intramolecular interactions effectively surviving are due to the medium-range screened interactions. As discussed in Section 2, the average conformation may be determined by minimization of the chain free energy with respect to the full set of the strain ratios ~2(q), defined in eq. (17). In the present case, cancellation between the long-range intramolecular contributions A2 and A 3 reduces the free energy A to the sum (Ae~ + A2s), Ae~ being the configurational, elastic contribution whereas A2s is due to the screened interactions (see eqs (7B) and (19-21)). Linkage between the ~2(q)'s and A2s is provided through eq. (18), where (r2(k)) = , J(k)C(O)l 2 is the mean-square distance between atoms separated by k bonds. Free energy minimization gives ~2(q) =

{1

~ [(1 - k/N) sin2(qk/2)/J(k)7/2]dk } l . (39) 5K sina(q/2) 3

The resulting system of equations may be solved numerically through an iterative procedure starting from the approximate phantom-chain result J(k) = k (see eq. (18)). It may be convenient, however, to use a perturbative approach which provides an explicit analytical expression. To this purpose, let us note that if we express k and N in/~ units, that is, if we replace them by k/Eand N/F, and disregard the local conformational aspects adopting an appropriate bead-and-spring model with as many beads as chain atoms (i.e., C(q) = C(0)), the equation is independent of K r if we use K/~/--~ as the universal variable. For a typical polymer like atactic polystyrene w e 4 estimated K = 0.65 and/~ = 50, hence K/,,//-~ = 0.092, so that a perturbative approach appears to be permitted. In agreement with the full numerical results, we get, to the second order of perturbation: ~9 (r2(k))

=

kC(0)12~2(1 - Al~/k)

(40)

where both ~2 and A are of order unity, ~2 > I. Since (rZ(k))ph = kC(O)l 2 in the bead-and-spring model, we may write 0¢2(k)

=

(r2(k))ol(r2(k))ph

=

a2(1-

A/x/k)

(40A)

482

G. A L L E G R A and F. G A N A Z Z O L |

and since fi2 and A/x/~ are a function of K / x / ~ only, the expansion factor ~2(k) depends on the reduced topological separation k/E. Equation (40), valid for relatively large k values ( > 50-100), indicates a finite expansion with respect to the phantom chain, in keeping with the medium-range character of the screened-interaction potential. According to our estimates of the parameters K and/~ for various polymers in Ref. 4, the asymptotic quadratic expansion should amount to more than 30% and 20% respectively for (relatively bulky) isotactic polypropylene and atactic polystyrene, and to about 5% for (relatively thin) polyethylene. Let us point out here that we expect a somewhat larger expansion for a strand of k bonds if it is part of a much larger chain than if it is a smaller chain itself, due in the former case to the presence of outer atoms producing additional repulsive interaction. In fact, this expectation is in agreement with our numerical results, although the additional expansion turns out to be quite small. By assuming eq. (40) to hold for any k ~< N, we obtain for the mean-square radius of gyration:

(S250 = N-2 fo (N - k)(r2(k))odk = N(C(0)12/6)~2(1 -

~A/~/N). (41)

The procedure previously described rests explicitly on the assumption that the long-range interactions are effectively absent at the ® temperature. Actually, in the absence of the medium-range screened interactions, for an interatomic separation k close to N the chain would contract somehow because cancellation of long-range two- and three-body interactions is incomplete (in fact, the end atoms are involved in a smaller number of three-body interactions due to the absence of outer atoms). Accordingly, we performed recently a perturbative calculation along the same lines as described before, but including all the intramolecular interactions, that is, A:, A3 and A:s (see eq. (27)); fl(O) was derived from Ai,ter(1, 2) = 0 (see eq. (31)). As a result, we obtained again eq. (40), although in this case both ~: and A also depend on the three-body interaction parameter K~ .26 However, since with our estimate K~ is smaller than K / , J ~ by two orders of magnitude, the numerical values of ~: and A are not affected appreciably. Conversely, if we switch off the screened-interaction term by setting K to zero (see eq. (27A)), the resulting contraction of the asymptotic (i.e. N ~ ~ ) mean-square end-to-end distance j5'27 is very small ( ~ 0.2% with our estimate K~ ~ 10-3), with an opposite sign of the molecular-weight dependence of aZ(N) (i.e. A < 0, see eq. (40A) with k replaced by Ntg). The results of our calculations accounting for the screened interactions appear to explain quite satisfactorily some recent Monte Carlo simulations performed by Bruns on self-avoiding chains placed on a lattice, z8 The ® state was obtained through the condition A2 = 0 by applying an attractive potential

LINEAR A N D STAR POLYMERS IN SOLUTION

483

<52~ NI 2 2 t

1.5

0.5

0

I

I

0.1

0.2

I

0.3 N-92

FIG. 6. The mean-square radius of gyration of atactic polystyrene29 plotted as (S2)o/NI 2 vs N ,n. Filled circles: small-angleX-ray scattering. Open circles: smallangle light scattering. The solid line is the best-fit line through the points, the dashed line the ris calculation (see Ref. 29 and references therein).

to non-bonded a t o m pairs on adjacent lattice sites. The end-to-end distance closely follows eq. (40) (k -+ N), the plot of ( r 2 ( N ) ) / N I 2 being linear in N -1/2 with a negative slope both for a cubic and for a diamond lattice, as expected by us. Moreover, both the ~2 ( > 1) and the A value ( > 0) are in fair agreement with our estimates. Direct experimental results are more difficult to obtain, requiring carefully chosen polymers of small molecular weight, whose size cannot be studied by conventional light scattering techniques. However, very recent small angle X-ray scattering from polystyrene samples 29 having 578 ~< Mw ~< 97,300 do show the same N -j/2 dependence as predicted by eq. (41) for the radius of gyration, see Fig. 6, thus suggesting a procedure to obtain A from experiment. The value of 1.49 obtained from Fig. 6 falls nicely between our calculated figures, 1.I for internal portions, and 2.4 for the end-to-end distance. '9 It should be added here that Martin 3° independently of us considered a finite-range interatomic potential in the (9 state, obtaining an asymptotic expansion of the chain (N -~ oo) analogous to that discussed here.

484

G. A L L E G R A and F. G A N A Z Z O L I

4.2. Regular star p o l y m e r s In this case the molecular topology requires all the arms to join at the branch point, thus enhancing the amount of intramolecular interaction. As done before with the linear chain, we minimize the polymer free energy with respect to all the degrees of freedom ~2(q). We take into account the star symmetry, which also dictates different multiplicities for even- and odd-order normal modes l(q)'s, 17 see the following. For simplicity, we adopt the bead-and-spring model, disregarding conformational strains at the scale of 10-20 chain bonds; this is equivalent to the procedure followed in Section 4.1 for linear polymers, whereby the characteristic ratio C(q) was identified with C(0)) 9 We may still designate as "atoms" the centres of force along the chain, therefore keeping the same interatomic cutoff E as done previously. If the star comprises N + 1 atoms and f arms, thus having N / f atoms per arm and one atom at the branch point, we number sequentially the atoms on each arm starting from zero at the branch point to N / f a t the free end. Furthermore, the h-th bond vector on arm j, I~J)(h), is chosen to join atoms h - 1 and h pointing towards the latter atom. (Here we cannot use the periodic chain transformation given in eq. (13), for the atoms are in no way equivalent among themselves due to the presence of the branch point.) The normal modes of a regular star turn out to be 31 i(q) = (2/N) 1/2 ~ h=l

l(J)(h) sin [q(h -

even modes (nq even) (42)

l(J)(h) e -ij~

( 2 I N ) I/2 ~ h=l

1/2)1

j=l

j

cos [q(h -

1/2)]

odd modes (nq odd)

I

where ~k =

2~z/f',

q

=

rrnq/m,

(nq = 1, 2 . . . . .

m m -

=

2N/f+

1).

1 (42A)

There a r e f - 1 different odd normal modes with factors e -i~ . . . . . e -i(i- ])J~ in square brackets. However, these symmetry-related modes have the same mean-square value, hence we may consider only one set of odd modes with multiplicity f 1. Of course, for f = 2 the above expressions reduce to the usual transforms for an open chain, 2 provided we keep in mind the bond vector orientation chosen here. The elastic free energy Aet is still given by eq. (19A), but each term within the sum has a multiplicity factor q~ defined as following: {If ~0 =

for even modes -

1

(43)

for odd modes.

Concerning the screened interaction term A2s, the simple sum within eq. (21)

LINEAR AND STAR POLYMERS IN SOLUTION

485

must be rewritten as a sum over all the atom pairs due to the lack of periodicity: A2s

---'=

(K/2)[C(O)lz]5/2 ~,X (r2(i'J))-5/2'

(44)

i :~j

where (r2 (i, j ) ) is the mean-square distance between atoms i andj. Adopting the numbering scheme described above, (r2(i, j ) ) is given by

(r2(i,j)) = (412C(O)/mf) ~ {fia(q)u~(q) + ( f -

l)~2(q)w~j(q)

lq}

+ 2f~)~t2(q)woi(q)Woj(q)}

(45)

where q and m are reported in eq. (42A),

i)q/2] sin [(i + j)q/2]/sin (q/2) (even modes) w0.(q) = sin [(j - i)q/2] cos [(i + j)q/2]/sin (q/2) (odd modes) u0(q ) =

sin [(j -

(45A)

and 7 is equal to either 0 or 1, according to whether the two atoms belong to the same arm or they are on different arms. Note that for the phantom chain ~2(q) = 1 and from eq. (45)we get (r2(i,j))ph = C(0)12Ii - Jl i f i a n d j b e l o n g to the same arm, or C(0)12(i + j ) if they are on different arms, whereas in general (r2(i, j ) ) depends on the location of both atoms and not only on their topological separation. It should be noted that, due to the star symmetry, we need only consider two arms in eq. (44) (and in any other double sum over atom pairs) by grouping together the equivalent terms with the appropriate multiplicity. By free-energy minimization we get an integral equation for fiZ(q) analogous to that reported in eq. (39) for the linear chain and depending on the whole set of the mean-square distances (r2(i, j)). Since these depend in turn on the {~2(q)} set through eq. (45), we have again a coupled set of equations which can be solved numerically in a self-consistent way. The perturbative approach may also provide useful analytical results, although in the limit of very long arms where the influence of the star centre plays a relatively minor role. The interatomic distances are very closely given by the equation that applies to the linear chain (see eq. (40), k being the topological separation). Concerning the mean-square radius of gyration ( S 2)0, it is convenient to express it as the difference between the mean-square distance (X2)0 of the atoms from the branch point, and the mean-square distance ( Z 2)0 of the latter from the centre of mass. 32 The perturbative results are 3~ (S2)0 =

C(0) { x

12 N 3f - 2 6f

f

lO K [ (~r) 1 + ~-(~)1/-----5 1 - p

'/2f(23/2- 1) + 2 - 23/2]} 3f--2-

. (46)

We point out incidentally that in a recent paper, ~8a due to incorrect representation of the normal modes, ~Sbwe reported the wrong conclusion that the

486

G. A L L E G R A and F. G A N A Z Z O L I

unperturbed star chain undergoes a non-affine deformation with respect to the phantom state even in the limit N ---, oo. l2 N ( X 2>0 = C(0) 2 f x

(z

1 + -~-(k-)l/----~ 1 -- p

= c(o)

(~,~)-l/2f(23n -

1) + 4 3f 23/21}

(47)

12N1 { 7Y

-{- T l/'-'-~ (k-)

1 -- p

(48)

where p =

5

(

(49)

and ((p) is Riemann's zeta function. For comparison, let us first consider the phantom chain, obtained by setting K = 0 in the above equations. If we connect to the branch point an increasing number of arms of equal length N/f, the mean-square distance of the atoms from the branch point (X2)ph does not change w i t h f ( s e e eq. (47), K = 0), since the arms do not interact and can intersect freely. (For the same reason, the meansquare distance between any two atoms is also independent from f, being simply proportional to their topological separation.) On the other hand, each additional arm makes the centre of mass closer to the branch point; as a consequence, the mean-square distance (Z2)ph is inversely proportional to f. Otherwise said, the fluctuations of the branch point around the centre of mass decrease w i t h f i. Therefore, for a given arm length N/f, the mean square radius of gyration ( S 2)ph = (.e~ 2)ph -- ( 2 2 ) p h increases with f towards a finite limit equal to ( X 2)ph ( f -+ 00). In the presence of the ® expansion due to the screened interactions - i.e. to the intrinsic chain thickness - ~2x = (X2)o/(X 2)ph, O~2z= (Z:)o/( Z2)ph and ~2s = (S2)o/(S2)ph are larger than unity and for N -+ oo tend to the finite common limit ~2 independent from f :

~2 = 1 + (lO/3)K/(k") '/2.

(50)

In particular, ~,2 is still given by fi2(1 ~(Ai/x/~)) as for the linear chain (see eq. (41)), although Ai ( > 0 ) increases with f. It should be stressed that eqs (46-50) were obtained by a perturbative approach, assuming that the variable K/(~) 'n is small compared with unity and neglecting terms of order (N/f E)-' or smaller. In this context, it should be added that when the number of arms gets very large, correlations across the branch point become very strong, due to the requirement of space filling around the branch point, while preserving the connectivity of the star. In this case, a simple scaling approach fulfilling this requirement appears to be promising. 33 A quantity often reported in the literature is the topological index g, which gives a measure of the polymer compactness and is defined as the ratio between

LINEAR A N D STAR P O L Y M E R S IN SOLUTION

487

the mean-square radius of gyration of the star polymer and that of the linear chain having the same molecular weight, therefore having the same total number of atoms N + 1. From eq. (46), the perturbative result for g is given by g = gph [ 1 + - - ~lOp h ( f ) N

'/21

(51)

where p and fi2 were defined in eqs (49) and (50), and gph = h(f)

=

(3f-

1 -fl/2[f(23/2 -

2)/f2;

(52)

1) + 2 - 23/2]/(3f- 2).

(53)

h ( f ) vanishes for the linear chain [h(l) = h(2) = 0], while being negative for star polymers ( f ~> 3). Therefore, according to these perturbative results, g tends from below to the asymptotic value gph for increasing molecular weight. Recent experimental results on 12-arm atactic polystyrene suggest an opposite behaviour. 34 However, let us point out that numerical results obtained by solving the full set of self-consistent equations show that for short chains (i.e. N/f of the order of a few tens) higher-order terms are non-negligible and such as to make g larger than gph by a few percent. 3. It is interesting to note that a somewhat similar result was obtained by Burchard et al., 35 who could get a decreasing value o f g in their computer simulations only by assuming a specific star centre to keep apart the arms in the core region. As a final remark in this section, it must be recalled that g should be derived from a regular star and from a linear polymer of the same molecular weight at their own 19 temperature, which is not the same temperature. In fact, the stars have in general a lower 19 temperature than the linear chains, the more so the larger the number of arms and the smaller the molecular weight (see eq. (38) and Figs 4 and 5). 5. T H E

POOR-SOLVENT

CONTRACTION:

T <

O

If the temperature is lower than t9, the binary cluster integral fl(T) of eq. (2) becomes so negative that fl~fr(T) = fl(T) + a3 is also negative. Under these conditions, the attractive long-range two-body interactions overcome the repulsive terms, and the chain contracts. Although the interatomic attraction becomes stronger and stronger as the contraction progresses - in fact the interatomic free energy between atoms separated by k bonds changes as (r2(k)) ~/2 _ we do not have a catastrophic collapse to a point-like globule because the repulsive energy increases as well, and an equilibrium collapse stage is eventually reached, dictated by the density requirementfl '6~7 The different free-energy contributions were already defined in Sections 1 and 2. In short, referring to eqs (7) and (19-20), the non-attractive terms are: (i) The elastic (entropic) configurational potential Ac~, opposing any departure from the set of the most-probable phantom-chain configurations. It turns

488

G. A L L E G R A and F. GANAZZOLI

out to be very important only in the close vicinity of T - 19, where the two-body and the three-body interactions effectively cancel. (ii) The screened-interaction repulsion A2s, related to the chain thickness. This is a medium-range interaction insofar as its general term changes as (r2(k)) -5/2 instead of (r~(k)) 3/2 as the attractive potential, and is especially effective in resisting the contraction of relatively short chains. 6 (iii) The three-body repulsion potential A3, with a long-range character, that becomes dominant for long chains. The chain conformation at T < 19 may be obtained again by minimization of the chain free energy A = Ael + A 2 + A 3 + A2s with respect to the strain ratios 82(q). Consistent with our primary interest in long chains and departures of only a few degrees from the 19;temperature, which may be sufficient to produce a significant overall contraction or expansion, we shall disregard any deformation effect at the local scale (i.e. k < 100-200 chain bonds). Within these limits, our reference state is conveniently identified with the unperturbed, or ®-chain instead of the phantom chain, the generalized characteristic ratio C(q) being replaced by the characteristic ratio of the 19-chain C® = C(0)~ 2 (see eq. (40). With this procedure we neglect any q-dependence of the characteristic ratio, consistent with our disregard of localized effects, enabling us to assume q ~ 0. Denoting as 0t2(q) without the tilde the strain ratio of the general Fourier component with respect to the unperturbed chain, we have the following simpler equations for 0t2(q) and (r 2(k)), replacing eqs (17) and (18) (periodic chain model): ~2(q) = (r2(k))

=

(i(q) • i*(q))/(NC~ l 2)

C~l 2 N ~ ~2(q) sina(qk/2)/sin2(q/2).

(54) (55)

M It should be borne in mind that assuming the realistic unperturbed chain as our reference state, instead of the phantom chain, implies both the above reported geometrical rescaling and the temperature shift below Ooh, as seen in Section 3, eqs (35) and (36). The elastic free energy Ael is given again by eq. (19A) with the mere replacement of fi2(q) with ~2(q). In analogy with the O expansion, the optimization equations dA/3~2(q) = 0, together with eq. (55), connect the two sets of variables {ct2(q)} and {(r2(k))} in a self-consistent system. This was solved through a numerical iteration procedure, but the results turn out to be virtually identical with those obtainable in a much more compact way through the approximate procedure described a s f o l l o w s . 2'6'36 We assume that for T < 19 the free energy mintr a = A2 + A3 --I- A2s due to the interatomic interactions (see eq. (20)) is a single valued function of the mean-square radius of gyration (SZ), independent from the actual values of the ~2(q)'s. The rationale behind this assumption is that at T < 19 we have an essentially compact, uniformly filled globule, whose interaction energy is deter-

LINEAR A N D STAR P O L Y M E R S IN SOLUTION

489

mined by its size rather than by the detailed configuration of the chain strands. In turn, ( S 2) is univocally determined by the actual temperature or, more conveniently, by the degree of undercooling z = (T - ®)IT. Therefore, the equilibrium conformation may be obtained by minimization of Art + f ( ( S 2)), Am,r, = f((SZ)) being an unspecified function linking ( S z) to the intramolecular free energy. Both Ae~ and ( S 2) may be expressed as sums over the Fourier modes, therefore depending on the set {ct2(q)}. The result of this constrained minimization may be expressed as 2'6'36 ~2(q)

=

j2q2/( i + j2q2),

(56)

where the constant J includes the first derivative of f ( ( S 2)). In the periodic model, q = 21rnq/N, see eq. (13A), and for J/N >~ 1 we recover the unperturbed state, wherein ~t2(q) = 1 for any q. Conversely, for J/N .~ 1 we are in a strongly contracted state; note, however, that while we may have a strong contraction for collective modes (i.e. ~2(q) ~ 1, nq small, that is, q ,~ 1), the chain is usually unperturbed on a more localized scale (that is ~2(q) ~ I for nq >> 1). It is worth pointing out that the above result also applies to the more accurate open-chain model 37where the permissible values of q are q = 7rnu/N. With either model in the limit J >> N the unperturbed geometry is recovered, whereas for J ~ N the mean-square interatomic distances (r2(k)) equal the unperturbed values if the topological separation k is ~ J , otherwise attaining a constant plateau if

N~k>>

J: (r2(k))

=

JC~I 2.

(57)

In view of the periodic constraint (see Fig. 2), interatomic distances involving atoms close to opposite chain ends are incorrectly evaluated within the periodic model. With the more accurate open-chain model, the mean-square distance between atoms i a n d j is, 37 (1i - Jl >> J): (r2(i,j))

=

wJC~l 2

(58)

where w = 1 for a central portion (see eq. (57)), w = 3/2 for a terminal portion and w = 2 for the end-to-end distance. Particularly relevant results are the strain ratio of the mean-square radius of gyration 37 ~ and the elastic free energy 38 Ae~, both given as a function of J/N:

~

= ($2)/($2)o Ae, -

=

3(p coth p - ' - p2);

p =

J/N;

~ 3 4,o2 + ~ In (p sinh p ').

(59) (60)

For J ,¢ N we get the strong-contraction results

ot~ ~ 3J/U Ae~ ~ 9/(4~), where it should be remembered that J depends on both z and N.

(61)

490

G. A L L E G R A and F. G A N A Z Z O L I

The resulting, significantly contracted or collapsed chain (supposing p = J / N < 1/2) was denoted as the Random Gaussian Globule. 6"36 The term

implies very quick loss of correlation among chain bond vectors separated by a number of bonds on the order of J or more. It may be shown that this correlation falls off exponentially2'6 (l(h). l(h + k))

-

1 C~l 2 2 J exp ( - k / J )

(62)

(k >> 1), which gives theoretical support to the use of the Gaussian distribution for the interatomic distances. The negative value of the average should also be noted: it implies that the angle formed by distant bond vectors is larger than 90 ° , on average, in keeping with the idea of a tightly folded structure. An interesting description of the Random Gaussian Globule may be given in terms of the familiar blob model. 7 Each blob comprises about J atoms and its mean-square diameter is just 2JC~/2; within each blob the chain is unperturbed, but all the blobs have a common centre o f mass, thus pervading the same space. To get a connection between ( S 2) and temperature, or better, between the quantities J/N and z = ( T - ®)/T, let us note that: (i) ~2(q) depends only on J / N and on the mode number nu; (ii) (rZ(k)) depends only on J / N (through the function cc2(q)) and on the topological separation k; (iii) the free-energy terms depend only on J / N either directly through ~x2(q) (see Ael, eqs (19A) and (56)) or indirectly through (r/(k)) (see A2, Aa, Azs, eqs (7), (21) and (55)). Therefore, both the overall free-energy A = Aen + A2 + A3 + A2s and its derivative with respect to ~2(q) will depend only on J/N and on the parameters characterizing each intramolecular free energy contribution. In conclusion, the equilibrium condition may be directly obtained by minimization of A with respect to ~ , leading to the equation 2'36 Ael -- r B x ~ A 2 + (K/x/N)A2s + KIA 3 =

O,

(63)

where the A's are proportional to the derivatives of the corresponding freeenergy terms and depend only on ~ through J/N. It turns out that the dependence from E disappears because we are taking as the reference state the unperturbed chain at T = O, that is, the chain with the screened interactions expansion. The A's of eq. (63) are evaluated once and for all as a function of ~ and the connection with r is established through the same equation. In the limit of strong collapse (~s '~ l) the temperature dependence of ~s may be given explicitly: -~s/x/3

=

z B x / N + (16/3)K,/~ g + 5 K ] ( ~ x / m ) ,

(64)

to be compared with the Flory-like affine result 39'4° ~ -- ~

= z B x / U + y/ct 3.

(65)

Apart from the absence of the screened-interactions term (last term on the right

LINEAR AND STAR POLYMERS 1N SOLUTION

491

(1. s

1.0

0.8

0.6

0.4

0.2

L

-20

I

i

-10

i

I

0

i

/

10

i

i

0 i

20 T('C)

FIG. 7. The experimental43 and calculated cts for polystyrene in dioctylphthalate (® = 22°C) as a function of T. The parameters employed in eq. (63) are: N = 75,000 (matching the experimental molecular weight), K = 0.37, K t = 1.9 x 10 3, B = 0.027, while the transition temperature is T* = 4°C. In eq. (63) the free-energy derivatives are taken from Ref. 36.

side of eq. (64), the affine model has a different term on the left side. This difference arises from the difference in the elastic free energy at high compression, the affine model (Ael = In ( e ~ ) - j ) underestimating it compared to the R a n d o m Gaussian Globule model (Ae~ = 9/(4~), see eq. (61)). However, in both cases, we get ~2 = ( - z ) 2/3N-I/3 in the mathematical limit as--* 0 (remember that z < 0), which ensures that the density of the collapsed globule is preserved for N - ~ ~ . In fact, since ( $ 2 ) o oc N, one gets ( S 2) = ~ 2 ( $ 2 ) 0 oc N 2/3 and, taking the globule volume V oc ($2) 3/2, its density is d oc N / V = constantN. The predictions of the model outlined above were compared 6 with the experimental results of T a n a k a et al. 4~ from atactic polystyrene (PS) in cyclohexane. Unfortunately, the polydispersity of that sample ( ~ 1.3) was not as small as one could wish, which proved to be an important point. 6'4z In Fig. 7 we show a similar fit carried out on the data of ~t~pfinek et al. 43 from PS with a very small polydispersity (1.05) in dioctylphthalate. A good agreement is obtained with the parameters reported in the figure caption: N was chosen to match the actual molecular weight, while K and K~ are the same as for PS in cyclohexane. The difference in solvent is reflected only in the B parameter -- apart, of course, from the O temperature - in qualitative agreement with theoretical predictions concerning the molar volume of the solvent. 6'24

492

G. ALLEGRA and F. GANAZZOLI

/i ' i / i '

i

-.!i !/

(~*BvrNJ)I

-1

(~*8~rN)21 0

0

~B~ FIG. 8. A schematic drawing of ~ts plotted vs the reduced variable TBx/Nin the poor-solvent region (T < ®) for two different values of K~ (see text).

Concerning the transition from the unperturbed coil in the O state to the collapsed globule with decreasing temperature, let us first point out that, unlike the good solvent expansion, no single universal variable like zBx/N is sufficient, since K/x/N and Ki are also needed (see eq. (63)). Considering very long chains (N ~ ~ ) and neglecting for simplicity the screened-interaction term with coefficient K/x/N, the plot of as vs zBx/N takes two different aspects depending on the value of Kt (see Fig. 8): 36.40curve 1, with a positive slope throughout if K~ is larger than a critical value K*, or curve 2 if K~ < K*. (We note incidentally that K~ increases with an increasing volume/length ratio of the statistical segment. 4°) The basic difference between the two curves is that only one free-energy minimum exists for any value of zBx/N in curve 1, whereas two such minima and one maximum are present in curve 2 if zB x/N is within a suitable range. In curve 2 the states corresponding to the absolute minima (the stable states) are shown by a solid line, those corresponding to the relative minima - i.e. the metastable states - by a dashed line and those corresponding to a maximum (the unstable states) by a dotted line. The analogy with the curves of a van der Waals fluid in the pV plane is quite evident, K* playing the role of the critical temperature. The dash-and-dot line connects the absolute minima in the upper and the lower branch. Due to this discontinuity, we are tempted to denote the collapse as a first-order transition for curve 2 and a second-order transition for curve 1, in the thermodynamic limit N --* ~ . However, it should be noted that in the same limit the transition

LINEAR AND STAR POLYMERS IN SOLUTION

493

takes place in an infinitesimal temperature interval below 0 in both cases because the product z*~/N tends to a constant for N ~ ~ , if z* = (T* - O)/T*, T* being the transition temperature. Furthermore, a peculiarity o f the transition in the case of curve 2 is that it is quasi-athermal, since the change in internal energy per chain atom at the transition, AU*[N, is proportional to N-1/2, thus tending to zero for N --+ ~ . This unusual result is an effect o f the chain connectivity.2"6"a4 Studies are being carried out in our laboratory to extend the above results to the collapse of regular star polymers. So far, only the collapse o f a flexible polymer chain has been described. Things are a bit more involved when stiff chains are considered, wherein the number o f chain bonds N is not much larger than Co~ or, in the language of the worm-like chain, 45"46the contour length L is not much larger than the persistence length lp~r~. Fairly good examples o f such polymers in the collapsed state are provided by the globular proteins, whose detailed molecular structure is known in many cases from X-ray single crystal studies. The crystallographic results enable us to easily obtain the interatomic distances between amino acid residues, which can be compared with theoretical predictions. 47Protein chains are characterized by a rather high bending stiffness, i.e. their chain axes are unlikely to make sharp turns especially because o f the large amounts o f rigid ~-helical and r-sheet strands. On the other hand, from its very definition the worm-like chain is subject to the constraint that the unit vector tangential to the chain axis is continuous in space, so that it lends itself as a model for polymer chains with bending stiffness. The persistence length is the usual parameter to account for such a stiffness, and is related with the generalized characteristic ratio through the equation 47'4s C(q) =

2(l[l~r~)/(q 2 + (l[locrs)2)

(66)

1 ~ 0 being the " b o n d " length (the worm-like chain is a smooth model with no chemical bond); considering that q = 2rCnq/N, with nq = 0, -Jr 1, _ 2 . . . . and that N = L/I --+ ~ , C(q) is extremely sharp-peaked and may be considered as a limiting form ofeq. (16B) with b very close to unity. The chain free energy may be written again as a sum of the elastic term and o f the intramolecular contribution, the latter being taken as a single-valued function o f the mean-square radius of gyration. A constraint ensuring that the overall contour length is constant must also be introduced, otherwise the collapsing chain would bend and shorten at the same time. While bending may be achieved at a few specific sites along the amino acid sequence, shortening along the chain axis is most unlikely, since it involves compression o f bond angles and in general disruption of stabilizing intramolecular hydrogen bonds. Our results are reported in full in Ref. 47. Here we show graphically in Fig. 9 a few typical example calculations from a flexible chain (L/lp¢~ = 103, curve a) and from a relatively stiff chain ( L / I ~ = 10, curve c) undergoing large compression ( ~ = 0.05 in both cases). Defining as A the curvilinear coordinate

494

G. ALLEGRA and F. GANAZZOLI

~oo 3.0' 2.5 a

2.0

~

~

1.5 1.0 0.5

0"00,0

0,2

0.4

0.6

0.8

1.0

^/L FIG. 9. (r2(A))/2Ll~rs plotted as a function of AlL for a worm-like chain with different degrees of rigidity L/l~,~ at various degrees of contraction: (a) collapse of a flexible chain with L/l~r~ = l0 s (very similar to a bead-and-spring chain) and a t = 0.05; (b) unperturbed rigid chain ( ~ = 1) with L/l~,~ = I0 (for clarity, only the initial part of the plot is shown); (c) collapse o f a rigid chain with L/l~ = 10 and ~t~ = 0.05. 47

separating two points on the chain axis, the existence of one or more maxima in the plot of (r2(A)) vs A is typical of stiff polymers and somewhat resembles what is observed from the crystallographic data of most globular proteins taken into account by US° 47 A fit was carried out between a calculated curve of the same type and a suitably weighted average of the experimental results, thus permitting us to estimate l~rs, inter alia. The resulting diagram is shown in Fig. 10, where (r2(k)) is shown as a function of k, or the number of amino acid units separating two chain points; k is obviously proportional to A. Both k and (r2(k)) are normalized by the coordinates of the first peak. The agreement is satisfactory for a small topological separation k, where the statistical sampling is more accurate, and becomes only qualitative at large k where the number of conformational possibilities enormously outweighs the number of experimental data under consideration. The suggestion deriving from this statistical-mechanical approach is that the gross, average features of protein folding follow a very general pattern dictated by the collapse of stiff polymers, while only the details of folding depend on the specific amino acid sequence. Otherwise said, even in the case of protein folding nature seems to obey criteria of highest probability, and differences among different proteins may be merely regarded as statistical fluctuations.

LINEAR AND STAR POLYMERS IN SOLUTION

Y=

495

2.0

1.6

-

1.2 -

-

~

0.8 0.4 0.0

t

t 2

I

I z,

1

I 6

I

x=k/kmax FIG. 10. Comparison between the calculated (dashed curve) and observed (solid curve) mean-square distance between aminoacid units separated by k units. Both the abscissa and the ordinate were normalized to the coordinate of the first peak. The observed curve reports the experimental values averaged over 31 globular fragments of 21 typical non-membrane proteins as detailed in Ref. 47.

6. GOOD SOLVENT E X P A N S I O N T > O

At T > O the effective value of the interaction parameter fle~(T) + 03 is positive (see eqs (3) and (4)). Besides, from eq. (36D) the three-body contribution 0"3 may be considered as independent of temperature, of the location o f the interacting atoms within the chain and of the molecular length N, if it is large enough. This enables us to adopt the truncated series expansion fle~ = a(T - ®)IT (see eq. (33)); the higher the temperature, the more pronounced is the expansion effect. Since the inequality fl¢fr > 0 implies that the polymersolvent interactions are favoured over the average of the polymer-polymer and solvent-solvent interactions, it is commonly said that at T > ® we are in the presence of a good solvent. 3 Considering eqs ( 7 - 9 ) , we see that the overall interatomic potential a(i, j ) may be written as the sum of two contributions, the first one being proportional to fle¢(T)(r2(k)) -3/2 (where k = li - Jl) and the second one to vc (A 2r ) ( r 2(k)) 5/2. These contributions are respectively denoted as the long-range and the medium-range (or screened) interactions. At T = 19, where only the screened interactions are effective, both (r2(N)~/N and ( S 2)/N tend to a constant value for N ---, ~ . At T > 19, however, the long-range interactions are dominant, making both ratios increasingly large for N --* ~ . We will here consider primarily the large-scale properties o f polymers, so that only the smaU-q Fourier components (i.e. q ,~ 1) will be considered. Accordingly, as done in the previous section, the phantom-chain characteristic ratio C(q) will be replaced throughout by C~ = C(0)~ 2 (see eq- (40)), effectively

496

G. ALLEGRA and F. GANAZZOLI

embodying the screened-interaction expansion ~2 in the characteristic ratio Coo of the chain at the ® point. Therefore, from eq. (7) the free-energy of interaction between atoms separated by k(>> l) bonds is simply given by 2 a(i,j)

= a(k = li - J l ) J(k) =

~ fle~(T)[3/(EnC®12J(k))] 3/2

(r2(k))/(C®12).

(67) (67A)

It is useful at this point to comment briefly on the approximations that are implicit in eqs (67) and (67A). First of all, we adopt the Gaussian approximation. In essence, the probability distribution Wi.j(r) between any two chain atoms is taken as a single three-dimensional Gaussian function, instead of a weighted average of the Gaussian probability distributions pertaining to all possible chain contact graphs: As was previously said, we are using a mean-field approach whose basic inaccuracy consists of replacing a constant for Wi.j(r) in the limit of small r's, instead of a function ocre, 0 < ~ < 1, as suggested by statistical investigations of self-avoiding walks placed on regular lattices. 7"49"5°However, (i) the true distribution, through the Renormalization Group Approach, 7 leads to an exponent 2v ~ 1.176 for (r2(k)) oc k2', k ~ ~ , whereas the Gaussian approximation leads to 2v = 1.2 with a relatively small error; 5~and (ii) for finite values of k, or for finite chains where the asymptotic power law is not yet reached (i.e. in the intermediate range of expansion), the Gaussian approximation does lead to results very similar to those obtained with more sophisticated, and complicated, approaches (see below)) 2 As a second assumption, the root-mean-square distance between interacting atoms is taken as much larger than the van der Waals distance rvdw (see Fig. 1), so that the effective potential function is fleff(T)Wi.j(O)t~3(r) = a(i, j ) ~ 3 ( r ) s e e eq. (67)). As a further assumption adopted in our simple periodic model, the overall potential a ( i , j ) is taken to depend only on the topological separation k = li - J l between the interacting atoms and not on their location within the chain, as specified by the indices i andj. Actually, one expects a larger expansion for a strand of k atoms if they are placed in the centre of the chain rather than at one end. In fact, in the former case there are more repulsive interactions involving outer atoms rather than in the latter (see Scheme 1 below, where the

SCHEME I.

LINEAR AND STAR POLYMERS IN SOLUTION

497

thick segments stand for k-bond strands and the dotted lines indicate the repulsive two-body interactions involving outer atoms). This expectation is indeed verified if we adopt the more accurate open-chain model, whereby the chain atoms are not equivalent. 52 However, even in the periodic model we find that a k-bond strand is more expanded than a free chain with k bonds, due to the same reasons pointed out in the Scheme. 53"54 The equilibrium conformation may be obtained by the usual minimization of the free energy A = Ae~ + Ai,tr,. In turn, from eqs (67) and (67A), the intramolecular interaction term of the periodic chain is given by Aintra =

k_~k.(N -

(68)

k)a(k)

where /~ is the lower cutoff, discussed in Section 2, obtainable from stereochemical considerations. We recall that typical values of £ are ~ 40 for polyethylene and ~ 50 for atactic polystyrene.4 Concerning Ae~, it is given by eq. (19A) where the strain ratio ~2(q) relative to the phantom chain is substituted by ct2(q), relative to the unperturbed chain (see eq. (54)), as done in Section 5 for the collapse in poor solvent. Again an analogy with Section 5 is that is given by eq. (55) and non-localized Fourier modes will be considered (i.e. [ql '~ 1). The dimensionless function J(k) defined in eq. (67A) will also be used. After minimization of the chain free-energy, we get ct2(q) =

1

zB sin2(q/2 ) f ; dk(l -

}-l

k/N) sin2(qk/2)/J(k) 5/2

(69)

a set of equations to be solved consistently with eq. (55). We shall now focus our attention on the asymptotic expansion of infinitely long chains (N ~ oo), considering very long chain portions with k ,> E. We shall start by assuming o?(q) oc IqJ-r, or, equivalently, oc J(k) oc k I+r, with the exponent y(> 0) to be determined from analysis of eq. (69). It turns out 5mthat 7 can be neither > !/5, nor < 1/5: in fact, 7 < 1/5 leads to a nonphysical divergence of the quantity in curled brackets, eq. (69), whereas 7 > 1/5 implies that ot2(q) depends on the cutoff value/~ even in the limit q ~ 0, contrary to what is expected from consideration of long-range interactions. We are therefore left with the only choice 7 = 1/5, which gives the exponent 2v = 1 + 7 = 1.2. However, the substitution cannot satisfy eq. (69), since the right side contains a logarithmic divergence of the type In (1/q£). If the simple power-law dependence is modified into ~2(q) oc [q1-1/5 (1 + a In (1/q~)) 21~, the divergence still exists but is reduced to the milder form In (1 + a In (1/q/~)). This suggests that ot2(q) must contain an infinite product of nested logarithmic terms which render the divergence milder and milder for each added factor. It may actually be shown that no divergence exists if we accept a self-consistent conjecture that relies on continuity considerations. 51The final result may be cast

498

G. ALLEGRA and F. GANAZZOLI

in the form s

~ C~I2(TB)2/Sk2,tk),

(70)

where the effective exponent 2v(k) takes into account the logarithmic terms; for k --+ oo, 2v(k) tends to 1.2, being somewhat larger (---1.25-1.30) for finite k. This result is consistent with the numerical solution of eq. (69)52 and, approxirnately, with Monte Carlo simulations: 9 W e note in passing that some recent experimental results do indeed show an exponent larger than 1.20.55The correct asymptoticexponent I.176... from the renormalization approach, is likely to be reached only for extremely large polymer chains,56 usually beyond the available range of molecular weights. As hinted above, a somewhat different approach to solve the set of coupled eqs (69) and (55) is through numerical methods. Although the results are not easy to translate into analytical terms and cannot be used to investigate the large-expansion limit due to numerical problems, this procedure has the advantage that finite chains can be easily handled. Furthermore, the different conformational properties of real polymers can be taken into account through the function C(q) (we would have a prefactor C(q)/Co~ in front of the integral ofeq. (69)). Also, the procedure can be easily extended to the open chain, as done in Ref. 52. Renouncing the periodic approximation, let it suffice here to say that the periodic approximation only affects seriously the first two or three modes, which turn out to be slightly more expanded than in the open chain: in fact, the periodic assumption is somewhat akin to considering the chain as embedded in an infinite superchain, whose outer atoms increase the chain expansion through their own repulsive effect. As a consequence, the expansion factor of the end-toend distance is somewhat enhanced. However, there is an almost negligible effect on the expansion factor of the radius of gyration, which depends more substantially on the internal distances, and hence on the internal Fourier modes. The results of the numerical, self-consistent solution both for the periodic and for the open chain are shown in Fig. 11 in terms ofa~ = /o and of ~2 = /o as a function of the universal variable z = zBx/N in the intermediate range, which includes many cases of practical interest. Analytical equations fitting quite accurately the numerical results a r e : 2 0t2R =

[1 + 10Z + 25Z2 + 72Z3]2/15

Ct2 =

[1 + (67/7)Z + 18Z2 + 55z3] 2/15.

(71) (71A)

As an indirect check of the approximations made, we show for comparison the results of other approaches, including Monte Carlo simulations (curve DB) 57 and renormalization group results (curves D F and MN), 58'59 the latter being obtained without the Gaussian approximation. The agreement may be considered as satisfactory, suggesting that the approximations involved in the present approach are not too serious, at least in the range taken into consideration. We

L I N E A R A N D STAR POLYMERS IN SOLUTION

499

! 2.5

Per.--/

2.0 ]. .i i J .: • • ./

/

1.5

1"010-2

2

~ 1 5

I a 10-1

I 2

"" ..¢"....

.......

MN

I 100

5

2sl

PeK

Op.,.~.t 2.0

DF~,;":

-

4 js','/....' /"

1.5-

a s2

1.0 10 "2

~

- ;~"~" 4~"

: 2

5

10-1

2

5

100

FIG. 1I. ct~tand at~as a function ofthe reduced variable z = "rBx/Nin the intermediate range of the good-solvent expansion. The results from both the open- and the periodic-chain model are shown. The other theoretical curves reported are: Flory modified6° (Fm), Douglas and Fred 5~ (DF), Muthukumar and Nickel59 (MN), and Domb and Barrett 57 (DB). s h o u l d stress h e r e t h a t k n o w l e d g e o f the w h o l e f u n c t i o n ~2(q) also p e r m i t s d e r i v a t i o n o f t h e r e l a x a t i o n t i m e s - h e n c e t h e c h a i n d y n a m i c s - as d o n e for i n s t a n c e in R e f . 61 to c a l c u l a t e t h e i n t r i n s i c viscosity. W e believe the g o o d agreement of our calculations with intrinsic viscosity measurements - much

500

G. ALLEGRAand F. GANAZZOL1

better than attained with the renormalization group approach at least in the present state 62 - is due to our rather accurate description of the intramolecular degrees of freedom, especially in the intermediate range. Concerning the assumption of orthogonality of the normal modes for the open chain, a recent analysis, still carried out within the Gaussian approximation, ~6 shows that in a linear chain with ~ ~ 3 the largest non-diagonal component expressed as (I(ql) • I(q2))[(NCo~12), is less than 0.02, to be compared with the diagonal components comprised between 1 and 3, which gives a n excellent support to the orthogonality assumption in this expansion range. Conversely, a star chain with 6 equal branches, under a comparable overall expansion, has non-diagonal components about 10 times as large. Further investigation of the expansion of regular star polymers is presently in progress in our laboratory. 7. THE STRUCTURE FACTOR The conformational properties of a polymer chain are naturally suited to investigation by elastic scattering techniques. Depending on the observation distance, inversely proportional to the scattering coordinate Q ( = 4~ sin (~/2)/g, is the scattering angle and g the wavelength), both the overall size and the statistical distribution of the intramolecular distances can be probed. According to the degree of resolution required, proportional to the wavelength, either light scattering (2 ~ 103-104A) or neutron and X-ray scattering (g ~ 1/~) are typically employed. The experimental quantity is the structure factor, given by S(Q) = N -2 ~.~ (exp ( - i Q - r(h,j)))

(72)

where Q is the scattering vector and r(h, j ) the distance between atoms h and j. Within the Gaussian approximation, the equation reduces to S(Q) = N -2 ~

exp [-Q2(rE(h,j))/6]

(73).

h,j

which may be calculated knowing the set of the mean-square distances {(r~(h, j))}. In turn, these are obtained from the generalized characteristic ratio C(q) and the strain ratios ~2(q), see eqs (16-18). Let us first note that for Q 2 ( $ 2 ) ~ 1, which implies Q2(r~(h,j)) ,~ 1 for any h or j , we have, from series development of eq. (73): S(Q) --- 1

--

Q2($2)/3

(74)

independently of the model adopted and of the solvent quality. Therefore, the mean-square radius of gyration is derived from the initial slope of the smallangle scattering intensity in the region Q2 ($2) ,~ 1. The influence of chain expansion or contraction and the effect of chain topology on the shape of the structure factor are especially apparent for

LINEARAND STARPOLYMERSIN SOLUTION

501

p2S(O) 2.0 f=2

1.5

f=3

f=4

--

1.0 =

0.5

0.0

I

I

I

1

10

0

I

20

30 p2

FIG. 12. The structure factor S(Q) plotted as/~2S(Q) vs/~2 = Q2<$2) for unperturbed linear ( f = 2) and star polymers in the limit N --* oo.

Q2 ( s 2 ) > 1, where relatively local properties of the polymer are probed. A convenient way to analyze the results within this range is through the so-called K r a t k y plot. F o r a better comparison o f different models, we will use in the following the plot o f #2S(Q) vs #2, where /~2 = Q2($2) is a useful nondimensional variable. In Fig. 12 we report the calculated structure factor for unperturbed regular star polymers in the infinite molecular weight limit. Using the variables mentioned above, the curves coincide with those calculated for the phantom-chain model, 63 tJ2 S(Q) 2.5 c)

2.0 a) 1.5

1.o

0.5

0.0

I

0

I

4

I

I

8

I

I

12

I

I

16

I

20

p2 FIG. 13. Same as Fig. 12 for the linear chain at different temperatures: (a) unperturbed chain; (b) collapsed chain (T < O) with e~ = 0.03; (c) expanded chain (T > O) with • g = 2.l.

502

G. ALLEGRA and F. GANAZZOLI

showing again that in this limit the screened interactions merely produce a uniform affine expansion. The effect of polymer topology is evident: the very nature of a star polymer implies a clustering of atoms near the branch point, which gives rise to a peak centred at # values of order unity f o r f i> 4. Furthermore, the peak sharpness increases with the number of arms. This is a direct consequence of the more uniform space filling within a sphere of radius 0~/2 in a star polymer compared to the linear chain, the more uniform the larger isf. The influence of the solvent quality on the scattering plot of a linear polymer is shown in Fig. 13. In comparison with the unperturbed chain (curve a), the opposite cases of chain contraction (T < ®, curve b) and the chain expansion (T > 0 , curve c) show an opposite trend. In the contraction, or collapse, case the most significant feature is provided by the maximum at about Q ~- <$2> -~/2, which is due to the large and uniform density within the collapsed globule. At larger Q's the plot is fiat, indicating that in this region S(Q) oc Q-2 as in the unperturbed chain: we recall, in fact, that at the local scale the chain is indeed unperturbed. Conversely, in the presence of chain expansion the plot does not show any levelling off; rather, it increases monotonously. In fact, it may be shown that for Q2<$2> >> 1 the structure factor is S(Q) oc Q-l/v, where v is defined through oc /fly(see eq. (70)). In the ® state, v = 1/2, whence we get Q2S(Q) = constant, the constant being equal to 2/<$2>0. It is remarkable that the same value is attained by a collapsed chain; therefore, since I~2 = Q2<$2>, the horizontal asymptote in Fig. 13 is 2ct2 both in the ® state (~2 = 1) and in the collapsed state ( ~ ,~ 1). On the other hand, for T > 0 , v > 1/2, whence Q2S(Q) increases as Q" with r/ = 2 - 1Iv (in the limit of good solvent expansion v = 3/5, which implies that r/ = 1/3). 8. S U M M A R Y

AND CONCLUDING

REMARKS

In a dilute solution, for a given polymer-solvent pair there may be a temperature - denoted as the O, or ideal temperature - at which the net interaction energy between different chains is zero. 2'3 Actually, the effective energy between two interacting atoms placed on different chains may be regarded to consist of an attractive and of a repulsive part, both varying as -3/2, where r is the interatomic distance. While the attractive component is proportional to the binary-cluster integral fl(T = O), the repulsive component is a term correcting the cluster expansion to take into account the three-body contacts, and may be considered as temperature-independent in a reasonable vicinity of ®. Of course, within each macromolecule long-range interactions of the same sort as observed between the macromolecules exist as well, and they also reach equilibrium at T = 0 . For this reason, the macromolecule is said to be in its unperturbed state at this temperature, frequently denoted as the unperturbed temperature. In the unperturbedstate, if observed at a scale coarse enough that the smallest details comprise several bonds, the conformation of a linear chain looks like a

LINEAR AND STAR POLYMERS IN SOLUTION

503

random walk, because any chain strand is conformationaUy independent of any other. Related properties are: (i) The mean-square distance between any two chain atoms is proportional to the number of bonds separating them; (ii) The mean-square radius of gyration is also proportional to the total number N (assumed to be very large) of chain bonds, or ( S 2) oc N

(75)

This equation also applies to a regular star chain with f arms, although the proportionality coefficient is different and depends on f In this case, some geometrical correlation between the arms may also exist. Let us now suppose the temperature T to be slightly larger than O, say by a few degrees. If the effective interatomic potential is of the usual Lennard-Jones type (see Fig. 1), it is possible to show that the attractive interactions will decrease in absolute value, being less negative, and the repulsive interactions will prevail, albeit slightly. However, if N ~ oo the overall repulsive effect will be inevitably very large and ( S 2) will increase so much that a new type of N-dependence is established, namely ( S z ) oc N zv

(76)

where 2v ~ 1.2. Since the intramolecular interactions are mediated by the solvent, and a chain expansion entails a larger amount of polymer-solvent contacts, it is frequently said that at T > ® we have a good solvent. Conversely, for T < ® there will be a prevalence of intramolecular attractions (poor solvent), and the infinitely long chain will collapse no matter how small the degree or undercooling. Unlike in the case of expansion, where the threebody interactions have a minor role and the binary repulsions are dominant, in the course of chain collapse the three-body repulsions are the key factor that resists the binary attractions and ultimately enables the chain globule to attain a reasonable density. The power law given in eq. (76) is still valid, but now we have the exponent 2v = 2/3. An important consequence is that in the asymptotic limit N ~ ~ three parameters are required to describe the chain collapse, instead of two parameters as in the expansion, z'5 With reference to eq. (63), parameters required for collapse are those for the two-body attraction r B ~ / N (T = ( T - ® ) / T ) and for the three-body repulsion K~, in addition to the overall chain dimension in the unperturbed state, like the mean-square radius of gyration. In the expansion case K~ becomes unnecessary. Besides, if we plot a typical size parameter like ~z2 =

($2)/($2)o

(77)

vs z B x / N (the suffix zero stands for the unperturbed chain), we see that the curve is always continuous in the expansion range with • > 0 (see Fig. 11, for example), whereas it may have a first-order discontinuity for r < 0 when collapse takes place. The discontinuity is made possible if the three-body par-

504

G. ALLEGRA and F. GANAZZOLI

ameter K~ is sufficiently small, or, in stereochemical terms, if the segment thickness of the chain is small enough compared with the segment length, given by the characteristic ratio Coo, for example. In other words, if the chain is "slim" enough it may undergo a first-order transition to collapse. Otherwise the transition is of the second order, in the limit N --) oo.4°'6 From the preceding considerations the (9 temperature has a very peculiar phase-transition meaning in the thermodynamic limit N --) oo. Moving toward higher temperature, we have the transition from the unperturbed regime of eq. (75) to the expansion regime of eq. (76) with 2v ~ 1.2, whereas toward lower temperature the transition to the collapsed regime goes with 2v = 2•3. This qualifies the (9-temperature to be denoted as a tricritical temperature. 7 However, we have also seen that a sharp physical dissymmetry exists between the two transitions for the (9-state, which has an interesting conformational counterpart. At T > (9 the power law of eq. (76) is valid at any level of intramolecular distances between atoms separated by k bonds, provided k >> /~, /~ being the cutoff below which no interatomic contact is effectively possible. In other words, we may write (r2(k)) oc k 2v, with 2v ~ 1.2. Conversely, at T < (9 the mean-square distances quickly reach a plateau value for/~ ~ k, J ,~ N (see eq. (57)), according to the law6 (r2(k))

= JCool2[1 - exp ( - k / J ) ] .

(78)

For k ,~ J this equation gives the unperturbed result kCool 2, whereas we get the constant value JCool2 for k >> J. Qualitatively, chain strands whose average extension is less than the globule diameter are basically unperturbed, but as soon as they exceed this dimension they tend to bounce back at the globule's wall. Consequently, their extension does not increase any more over the plateau value proportional to the diameter itself. The resulting statistical model is denoted as the Random Gaussian Globule. 2'6 In conclusion, the intramolecular mean-square distances ( r2(k ) ) do not obey a power law of the same form as in eq. (76), unlike the mean-square radius of gyration, for which 2v = 2•3. Rather, we have (r2(k)) oc ~ " , where v' = 1/2 for k ,~ J and v' = 0 for k >> J. It should be noted that, considering a truly infinitely long chain, the globule diameter also reaches an infinitely large value, owing to the density requirement. In such a case, the unperturbed strands become very long, in agreement with Flory's prediction that in a melt, or in a concentrated solution, a polymer chain is unperturbed; 3 this prediction has been amply confirmed by neutron scattering experiments on partially deuterated polymers. 64 Hitherto we supposed the chain to be extremely long, so that a deviation of a few degrees from the (9 temperature is sufficient to produce large changes in the whole molecule, for example making ~t2 (see eq. (77)) very different from unity. However, the conformational strain decreases as we go to a smaller scale, usually becoming very small for chain strands with a length on the order of 100

LINEAR AND STAR POLYMERS IN SOLUTION

505

chemical bonds. In terms of the strain ratio ~2(q) (see eq. (17)), i.e. the ratio between the mean-square value of the Fourier configurational component l(q) (see eq. (13)) and its unperturbed value, we would have ~2(q) g 1 for [q[ > 2n/100, considering that any component corresponds to a conformational "wave" along the chain contour comprising 2rc/[q[ chain bonds in its wavelength. As otherwise said, for small-to-moderate deviations from the O-temperature, the chain is locally unperturbed, as shown by high-resolution NMR spectra, usually indicating little change within temperature ranges of ,--10-15°C, thus suggesting that the local environment of the chain nuclei essentially stays the same.65 If we have either large temperature shifts IATI = IT - OI or strong interatomic forces such as those due to hydrogen bonding, local perturbation of conformations may accompany chain expansion or collapse. Under these circumstances, even a relatively short macromolecule may undergo a significant strain. As an example, when in their globular state, protein chains may be considered as strongly collapsed from their ideal unperturbed conformation, due to hydrogen bonding as well as to hydrophobic and hydrophilic interactions. We investigated the statistical structure at collapse of such chains, using the worm-like model that enables us to account for the chain rigidity.47 (It may be shown that even more flexible hydrocarbon polymers like polyethylene and polystyrene are describable as worm-like chains if observed at sufficiently short distances of observation. 48) As done for collapse and expansion, the chain free energy (see eqs (19-21)) is expressed as a sum of the configurational, or elastic, contribution, of the long-range two-body and three-body interactions and of the medium-range screened interactions that will be commented upon shortly. According to our previous numerical calculations on chain collapse, 6 the overall interaction free energy may be regarded as a function of the single variable (S 2). Recognizance of this simple dependence simplifies the evaluation of the strain ratios ~2(q) through minimization of the chain free energyY 6 Then, the mean-square distances (r2(k)) between k-th neighbouring amino acid units may be evaluated. The results can then be compared with the corresponding averages determined from crystallographic data on a suitable set of globular proteins. 47 The satisfactory agreement shown by Fig. l0 suggests that the so-called "protein folding" follows a high-probability path if regarded as a whole. Of course, the specific amino acid sequence determines highly selective, and biochemically important, local conformational differences, which may be regarded as statistical fluctuations from the probability viewpoint. Polymer chains in the @ state are free from long-range interactions in that the long-range attractive and repulsive contributions achieve a perfect mutual balance. Nevertheless, chain atoms do still repel one another even at bond separations of ,,~ 100 chemical bonds or more, in view of the medium-range, screened interactions. 4 These repulsions arise from the fact that the attractive range of the effective interatomic potential does not coincide with the repulsive

506

G. ALLEGRA and F. GANAZZOLI

range (see Fig. 1), although the integrated value of the potential itself may be zero. The associated free energy changes as ( r 2(k)) - 5/2, instead o f ( r 2(k)) - 3/2 as for the long-range interactions, and is proportional to the mean-square chain thickness. It is important to stress that these interactions do not contribute to the intermolecular free energy in a dilute solution (see eqs (29-31)) and therefore do not influence the O temperature. ' Free-energy minimization shows that the screened interactions produce an expansion of the chain over its phantom state - i.e. the unperturbed state with no such interactions - that increases with molecular length N as (a - b/x/N), in excellent agreement with recent Monte Carlo studies by Bruns, 28 as well as with scattering results on polystyrene chainsfl However, in the large-N limit this O-expansion is essentially afline and is the same for linear and star-branched polymers with the same monomeric structure. It is also important to remark that the screened interactions are very effective in resisting chain collapse for moderate chain lengths, in which case their effect is larger than that due to the three-body repulsions. 6'36 A few comments on the phantom chain and on its relation with the real chain are in order. Although free of both medium- and long-range interactions, the phantom chain still experiences the short-range interactions arising from the local stereochemical structure. These typically include the gauche-trans statistics o f the rotational states, the syn-diaxial interactions, etc, and develop within very short chain strands, usually no longer than 10-15 chemical bonds, l The chain may be ideally split into many identical statistical segments, that may be regarded as effectively uncorrelated chain strands. Obviously, no short-range interaction may take place between two chain atoms separated by one statistical segment or more. (In the language of the worm-like chain, the persistence length is conceptually analogous to the statistical segment. 45"46) The conformation of the phantom chain may be regarded as a random walk wherein the elementary step consists of a single statistical segment. Steric conflict between topologically distant chain atoms is inevitable in a real chain. In the unperturbed state - i.e. either in a O solvent or in a melt - it may be relieved by relatively modest local detours, produced by the screened interactions whose effect consists mainly of an affine expansion of all the statistical segments at very large observation scale. 4 A larger expansion would produce entropy loss not compensated by interatomic interactions. The present approach is based on the Gaussian approximation for all the interatomic distances. Although it should never be confused with the so-called "Gaussian chain", which is basically a random-walk model, this approximation is of a mean-field type, since the chain contact graphs are not taken into consideration individually. A clue to the type o f inaccuracy entailed by the Gaussian approximation may be given by the following considerations. According to the Central Limit Theorem, the distance vector r(k) between two atoms separated by k chain bonds obeys the Gaussian distribution if it is given by the

LINEAR AND STAR POLYMERS IN SOLUTION

507

sum of many, mostly uncorrelated vectors of about the same size. This requirement may be fulfilled only if (r2(k)) is proportional to k~, where ~ is no larger than unity. This is verified both in the collapsed and in the unperturbed state, unlike in the expanded state, where ct ~ 1.2. 53 We know in fact that under expansion, or excluded volume conditions as frequently said, the interatomic distribution deviates from the Gaussian law. 5° However, we have shown that, within the Gaussian approximation, (i) in the asymptotic limit N ~ oo the exponent 2v of eq. (76) is 1.25' apart from logarithmic factors, in good agreement with the true value of 1.176; (ii) for finite chains, up to an overall expansion ~ around 2.5, the intramolecular expansion through the whole range of topological separation is in good agreement with several results from different approaches 52 (see Fig. 11). These considerations appear to provide a valid support to the conclusion that the Gaussian approximation, within the selfconsistent approach of free-energy minimization, may lead to accurate results in most cases of current interest. ACKNOWLEDGEMENTS

This work was financially supported by the Italian National Research Council (Consiglio Nazionale delle Ricerche) and by the Italian Ministry of Public Instruction (Ministero della Pubblica Istruzione), 40% funds. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. I0. 11. 12. 13. 14. 15. 16. 17. 18. 19.

P.J. FLORY, Statistical Mechanics of Chain Molecules, Interscience, New York (1969). G. ALLEGRA and F. GANAZZOLI,Adv. Chem. Phys. 75, 265 (1989). P.J. FLORY, Principles of Polymer Chemistry, Cornell University Press, Ithaca (1953). G. ALLEGRA, Macromolecules 16, 555 (1983). H. YAMAKAWA,Modern Theory of Polymer Solutions, Harper and Row, New York (1971). G. ALLEGRAand F. GANAZZOLI,J. Chem. Phys. 83, 397 (1985). P . G . DEGENNES, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca (1979). G. ALLEGRA, J. Chem. Phys. 68, 3600 (1978). G. ALLEGRAand F. GANAZZOLI,J. Chem. Phys. 74, 1310 (1981). The Fourier components sometimes represent very closely the independent modes of conformational motion, which justifies their frequent nomination as Fourier modes, or even normal modes? G. ALLEGRA and A. IMMmZl, Makromolek. Chem. 124, 70 (1969); S. BRf3CKNER, Macromolecules 14, 449 (1981). D.Y. YOON and P. J. FLORY, J. Chem. Phys. 61, 5366 (1974). P.J. FLORY and V. W. C. CHANG,Macromolecules 9, 33 (1976). B. DUPLANTIER, J. Chem. Phys. 86, 4233 (1987). B.J. CHERAYIL,J. F. DOUGLASand K. F. FREED, J. Chem. Phys. 83, 5293 (1985); J. Chem. Phys. 87, 3089 (1987). G. ALLEGRA, E. COLOMBOand F. GANAZZOLI,in preparation. B.H. ZIMM and R. W. KILB, J. Polym. Sci. 37, 19 (1959). (a) F. GANAZZOLI,M. A. FONTELOSand G. ALLEGRA,Polymer 32, 170 (1991); (b) Polymer, submitted. G. ALLEGRA and F. GANAZZOLI, Macromolecules, in press.

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