Polymers and scaling

POLYMERS AND SCALING

D. S. McKENZIE Department of Physics, Queen Elizabeth College, University of London, Campden Hill Road, London W8 7AH, England

(~E NORTH-HOLLAND PUBLISHING COMPANY

—

AMSTERDAM

PHYSICS REPORTS (Section C of Physics Letters) 27, No. 2 (1976) 35 88. NORTH-HOLLAND PUBLISHING COMPANY

POLYMERS AND SCALING

D. S. McKENZIE Department of Physics, Queen Elizabeth College, University of London, Campden Hill Road, London, W8 7AH, England Received December 1975

Contents: 1. 2. 3. 4.

Introduction Polymer models and relation to critical phenomena The scaling laws Calculation of exponents. Numerical methods

37 47 57 65

5. Analytic theories 6. Experimental work 7. Conclusions References Note added in proof

73 82 84 85 88

Abstract: We review the statistical mechanics of polymer solutions with reference to the theories of scaling, current in the theory of phase transitions. Topics include the theoretical background, the relation of the polymer problem to magnetic systems, numerical calculations, Monte-Carlo work, self-consistent field theories, recent field-theory work and experimental work.

Single orders for this issue PHYSICS REPORTS (Section C of PHYSICS LETTERS) 27, No. 2 (1976) 35—88. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 19.—, postage included.

D. S. McKenzie, Polymers and scaling

37

1. Introduction In this review we shall consider the statistical mechanics and thermodynamics of dilute solutions of synthetic polymers. The statistical mechanics of polymer solutions has been of increasing interest to theoretical physicists in recent years especially since it has been shown that polymers have many properties analogous to those of the Ising and related models of ferromagnetism. In particular, the osmotic pressure of a polymer solution is analogous to the magnetisation of the Ising model and the end-to-end length distribution is analogous to the spin—spin correlation function. The scaling law theory of ferromagnetism first proposed by Kadanoff [1] can be easily adapted to apply to polymer solutions. We shall review the results of this work and some of the relevant experimental results. For an earlier review along these lines see Domb [129]. Since the work to be described falls neither wholly in the province of the physicist interested in critical phenomena nor in that of the physical chemist interested in polymer solutions we have thought it fit to give a more extensive review of the behaviour of polymer solutions and of the basic properties of the Ising model than would be necessary in addressing a more homogeneous group of readers. In this introduction therefore, we shall review the basic properties of polymer solutions with emphasis on osmotic pressure and light-scattering. In section 2 we shall show in detail the analogy between the Ising model and the polymer problem. We shall continue in section 3 with statements of the scaling laws which are applicable to the polymer problem and derive scaling relations between the critical exponents introduced in section 2. Section 4 covers numerical calculations of the properties of polymer solutions. Section 5 reviews the theoretical attempts to understand polymer solutions particularly in so far as they provide estimates for the critical exponents. Finally in section 6 we shall give a brief review of experimental work. Polymers are long chain molecules formed by the repetition of a basic unit or segment. Examples of synthetic linear polymers are H

polymethylene

.

.

.

andpolystyrene

.

whose repeat unit is

~

.

.

H

C— C— C I I I HH H

—

whose repeat unit is

H

H

H —

H

C —. I H

H

—~—c—~—c—c—~— .

.

C

A polymer chain may contain any number of segments. The distinguishing characteristic of a synthetic polymer is that it has a very large molecular weight, which is propor~tionalto the number

38

D. S. McKenzie, Polymers and scaling

of segments, and no unique molecular formula. Thus any polymer sample contains a distribution of molecular weights, a property known as polydispersity. Polymers are stable molecules and may exist in the solid or liquid (melt) states and in solution. In polymers containing carbon—carbon covalent bonds in the backbone of the chain the position of each carbon atom is not fixed in space relative to the positions of the preceding pair of carbon atoms, but may assume a position on the circle defined by the fixed bond length (1 .54A) and the tetrahedral angle between successive bonds (fig. 1). This property of internal rotation is true all along the chain so that in the melt or in solution, where the molecule is free to move, a very large number of configurations of the molecule are possible. In solution therefore synthetic polymers are flexible.

Fig. 1. A schematic representation of the relative positions of three successive carbon atoms in the backbone of a synthetic polymer. The z axis is defined by the bond joining the first two atoms; the x andy axes are fixed arbitrarily in the plane perpendicular to the z axis and containing the third carbon atom. The angle ~l = 70.5°(IT — the tetrahedral bond angle). The third carbon atom is free to lie anywhere on the circle shown, in the x, y plane.

However different polymers have differing degrees of flexibility. This arises because the internal rotation about carbon—carbon bonds is not free but to a greater or lesser extent hindered by interactions between the side-groups attached to the carbon atoms. In general these interactions become more important in polymers with larger side-groups. The effect is that the carbon atoms prefer certain positions on the circle of free rotation. This effect can be described by a potential of hindered internal rotation v(~p)where ~pis the azimuthal angle. The potential v(~p)is not known in detail for most polymers, though the position and depth of the minima in v(~p)can be estimated. It has become customary to use the rotational isomeric model developed by Ptitsyn and Volkenstein [2] in which v(~p)is approximated by the statement that the third carbon atom prefers those values of p corresponding to the minima in v(~p)with a probability proportional to the depth of the minimum. Thus each atom possesses a discrete spectrum of ~p-states.The states of adjacent atoms can be related by a transition matrix. But this is just the definition of a Markov process. In other words the polymer chain can be likened to a Brownian motion or random walk in which the step length corresponds to the bond length of the chain and the time variable is analogous to the length

D. S. McKenzie, Polymers and scaling

39

of the chain (or degree of polymerisation). This result is still true if the model is generalised to take account of interactions between any finite number of neighbouring atoms. The added complication is that the transition matrix becomes much more complex but the detailed chemical structure of the chain can be taken into account. This approach to the theory of polymers has been treated at length by Flory [3]. The major quantity of interest in these calculations, indeed in most studies of polymer solutions, is the shape and size of the polymer molecule. This is determined theoretically by segment—segment distribution functions though of these the end-to-end length distribution is most studied because it is usually the simplest to calculate and is assumed typical of the remainder. If the end-to-end length distribution is p,, (r) where n is the number of steps and r is the position vector of one end relative to the other end as origin then it is well known that for random walks, in the limit of large n and small r, p,, (r) is Gaussian. That is we have Pnfr)

=

b2/3)312 exp(—3r2/2nb2)

(2

(1.1)

where b is a length parameter related to the bond length and the flexibility of the chain. The second moment of the distribution, the mean square end-to-end length (r~), is given by (r~)—nb2.

(1.2)

The calculations of Flory and Volkenstein show that the effect of increasing the flexibility of the chain is to decrease the parameter b. Though theory is predominantly concerned with the distribution of end-to-end lengths and in particular with the calculation of the mean square end-to-end length, neither of these quantities is accessible experimentally. The most direct way of determining the size and shape of polymer molecules in solution is by light scattering. The amount of light scattered by a polymer molecule in a particular direction relative to the incident beam, is determined fundamentally by the segment— segment distribution functions of which the end-to-end length distribution is a particular case. Light scattering therefore gives information on the radius of gyration rather than the end-to-end length. The theory of light scattering from polymer solutions is well known (for an introduction see [4] and a review see [5]). Here a brief sketch of the theory will be sufficient. The excess scattering, of a dilute polymer solution over the scattering from a pure solvent can be written as m2

n

5(kjp})~ 7

~

$g~)(r)exp(ik r)dr+~nP~~} .

ft

(1.3)

j

where interference between photons scattered from different molecules has been neglected. Here m is the molecular weight per segment, Xis Avogadro’s number, K is a constant which contains the optical and mechanical constants of the experimental situation and k is the scattering vector. The solution is polydisperse so that p,, is the number density of polymer of length n and (p} represents tI’e set fPi P2, p,,, The distribution function g~)(r) is the probability that, for a molecule of length n, segment i is a distance r from segment when the segments are numbered sequentially from one end. The second term in (1.3) represents the self-scattering term from each segment whereas the first term takes account of the inteference between photons scattered from different .

.

.

.

. }.

j

40

D. S. McKenzie, Polymers and scaling

segments of the same molecule. The distribution functions are normalized so that fg~(r)dr= 1.

(1.4)

In using (1 .3) we make the assumption that all the distribution functions are spherically symmetric. We then obtain 2K sin kr J(k, fp}) m —k— p,~4irfg~(r) kr r2 dr+ ~

{~

and the term sin kr can be expanded so that f(k,fp})

m2K =

—k—

p~

~ 4ir fg~(r) tl

k2r2

—

—a--] r2dr+ ~

flP 0)

2K k2 {~n~n_l)~t~~—-g~Pn ~r~i+~n~n} m~ -~ —

m2K I X knPn_

~ 3

15

2(2)

n

SnPn

where (st,) is the mean square radius of gyration of a polymer of length n. If we now define the weight-average molecular weight by 2n2p,,/~mnp~ (1.6) = ~ m and the total weight concentration of polymer by c~mnp~/X

(1.7)

we obtain .f(k,c)/K°°M~c— 4k2(s~)~M~c,

(1.8)

where =

~

m2n2s~p

2n2p 0/~m

0.

The experimental data are usually expressed in terms of the ratio kc/5(k, c) given by Kc J(k,c)

=~_

M~ L

(1.9)

~

3

2 and extrapolating to k = 0 one can obtain M~and (s~ whence by plotting k a polydisperse system, and all samples of synthetic polymers are to It should be notedagainst that with some extent polydisperse, one measures properties averaged over the distribution of molecular weights and that in light scattering the molecular weight one obtains is the weight-average as defined in (1.6) above. Although the theory of hindered internal rotation can account for the detailed chemistry of the polymer, it neglects interactions between segments far distant from one another along the chain, interactions between segments of different molecules and solvent effects. In particular, the random walk model allows the polymer chain to intersect itself so that the excluded volume effect due to )~.

D. S. McKenzie, Polymers and scaling

41

the repulsive potential between segments on close approach is neglected. Let us therefore for the moment calculate the thermodynamic properties of a solution containing N, polymer molecules of length i (1 ~ I ~ Nm ax) and N0 solvent molecules for the non-interacting case. If Z( {N }, V, T) is the partition function for such a system where {N} = f N0, N1, Nm ax }, the grand partition function can be written as ~(fz}, V, fl ~ H z~Z(f~, V, fl (1.10) fN}\s~o / .

.

.

(

where the summation is over all possible sets fN~and z~is the absolute activity of the ith species. Here V and T are the volume and temperature of the system respectively. For the random walk model the partition function factorises so that Z({N}, V fl11(qsV)NSIN5!

(1.11)

where q5 is the number of configurations of a molecule of length s calculated with respect to a segment fixed in space. The q5 may be calculated using the model of hindered internal rotation and will in general be temperature dependent. The grand partition function becomes Z((z}, V. T)

~

=

H (q5z5V)”~~N5! = H exp(q5z5V)

(N} s~0

(1.12)

s~0

where we have allowed the upper limit on s to become infinite. The osmotic pressure H of a solution is the difference between the internal pressure P1 of a solution in which the solvent is at activity z0 and the internal pressure P0 of the pure solvent at the same activity. We obtain the following relations for P0 and F1, P1V —lnZ((z}, V, Tj= ~ (q5z5V) kT P0V

(1.13)

V. T)q0z0V

so that kT

~

(1.14)

q5z5.

~i

But thc mean number of molecules of length s is given by

ralnzi L

az5

z5q5V.

(1.15)

J

Hence the activity is related to the number density of species s by z5q5N3/Vp5,

(1.16)

so that osmotic pressure becomes H kT

~ Ps. s~’1

Thus for a system of non-interacting polymer molecules we obtain an equation in every way

(1.17)

42

D. S. McKenzie, Polymers and scaling

analogous to that of an ideal gas. It should be noted that from osmotic pressure measurements we obtain no information about the solvent nor about individual polymers apart from their average molecule weight. The latter is obtained by changing variables to the total weight concentration c defined in (1.7)so that H/3? T~C/Mn

(1.18)

where =

~M5N5/~NS.

(1.19)

is the so-called number average molecular weight and 3? is the gas constant. Osmotic pressure measurements therefore give a different average molecular weight than light scattering experiments. The ratio M~,/M0 is used to give an indication of the degree of polydispersity of the solution. One expects eq. (1.17) to be correct in the limit of very low polymer concentration when interactions between different polymer molecules are negligible. At higher concentrations the effect of interactions on the osmotic pressure may be included by expanding H in terms of p. Thus, for a monodisperse system, eq. (1 17) becomes 2 +A 3 (1.20) H/kTp+A2p 3p which is analogous to the virial expansion of a gas. The rth virial coefficient accounts for the mutual interaction between r polymer molecules so that for the random walk model all virial coefficients are identically zero. Of course, the properties of polymers are influenced by the fact that the molecules interact. The most important interaction is the hard-core repulsion between the segments at close contact. Since all intermolecular potentials have a hard-core, its effect is always felt in any solution under all conditions. This effect is known as the excluded volume effect. The easiest way to include the excluded volume effect in the thermodynamic properties is to construct a theory analogous to Van der Waal’s theory of real gases. For a solution containing a single solute species consisting of Nmolecules of length n, the osmotic pressure in the absence of excluded volume is given by (1.17) which we shall write as

M~

.

+.

..

H/kT=N/V.

(1.21)

If the excluded volume per molecule is b~,that is, the volume excluded by one molecule to occupation by a second, eq. (1.21) can be modified to ~

VNb~~1

—pb~)’ =p+b~p2+b~p3+

(1.22)

Since bn has the dimension of volume, the rth virial coefficient has the dimensions of volume to the (r 1 )th power. Eq. (1.22) can be compared with the result of Flory [6], namely —

Hv

2u~p2] (1.23) 1/3?T—°—[ln(1 —nv1p)+(l l/n)nv1p+~n where v 1 is the molar volume of the solvent and we have replaced Flory’s 02 by no1 p/X. On expanding the logarithmic term (1.23) becomes 2P2/X + o~n3p3/3X2 (1.24) H/kTp +(~ X)V1fl —

+...

.

D. S. McKenzie, Polymers and scaling

43

The difference between (1.22) and (1.23) in that the term (1 b~p)’ has become —ln(l b~p)is an artefact of the lattice system used by Flory to derive (1.23). Furthermore, a heat of mixing term xn2 o~p2 has been added so as to maintain approximate agreement with experiment at least as far as the second virial coefficient. To account for polydisperse systems, eq. (1.22) should be further modified. We obtain —

—

N~ kT

(1.25)

~ (V~brsNr)

where N 5 is the number of molecules of polymer species s (s = 1, 2, and brs is the volume excluded by a molecule of species r to occupation by one of species s. Eq. (1.25) may be rewritten as .

.

.

)

1

H

~ p5(l =

~

p 5

s

+

—

~

b,.5pr)

~ b,.5prP5 r,s

+ ~

brsbstprPspt +...

(1.26)

.

r,S,t

To proceed further, we may estimate the molecular weight dependence of the excluded volume in (1 .26). Let us treat the excluded volume effect as a hard-core interaction between the segments of different molecules. Thus, if v is the volume excluded by one segment of a molecule of length r to occupation by a segment of a molecule of length s we may estimate b,.~= ursS~5since we have rs possible interactions between the r segments of species r and the s segments of species s. Here 6rs equals ~ if r = s and is unity otherwise. Remembering that the second virial coefficient is determined by the interaction of two molecules, we can represent the above result diagramatically as in (i) below brs

Ii)

(ii)

The result b~3 vrsfSrs is an upper bound since configurations with two or more interactions of the form (ii) and higher are neglected. It is shown by McKenzie [7] that a lower bound to b,.5 ~5 bry = o(r + 5)~rs. For a solution containing a single polymer species of length n, the above results may be written as 2. (1.27) fl~ bnn/0 ~ n One is therefore led to expect that, at least for large n, the molecular weight dependence of b~,~is determined by a power law of the form b~~/vczn2

(1.28)

with 0 < e < 1. As will be shown later such is indeed the case and the value of e is about According to this theory, by substituting bnn for b~in (1.22), the molecular weight dependence of the rth virial coefficient will be proportional to ~(‘~ 1 )(2 ~ compared with n’~according to the Flory theory. Qualitatively, the effect of excluded volume on the end-to-end length distribution is easy to ~.

—

44

D. S. McKenzie, Polymers and scaling

assess. The excluded volume implies that less space is available to the molecule so that the molecule spreads over a larger volume and the mean end-to-end length is increased. All attempts to calculate the mean shape of the molecule or the second moment (r~) have disadvantages, which shall become more apparent as we proceed. However it is generally agreed that in the presence of excluded volume, the distribution is no longer Gaussian and that (r,~) n~‘~for large n where v > 4. Note that ~= 4 corresponds to the Gaussian model. Furthermore, there is an upper bound of i-’ = 1 which occurs when only completely straight configurations of the molecule are possible. To this point, we have discussed polymer solutions with respect to concentration, molecular weight, distribution of molecular weights, and chemical structure as independent variables. The only interaction we have introduced is that caused by the excluded volume effect. However, the properties of polymer solutions are also dependent on temperature and solvent. Let us consider first the effect of temperature and solvent on the shape of a polymer molecule in solution. We assume now that there are attractive forces acting between the polymer segments and the solvent molecules. The detailed shape of the intermolecular potential is unnecessary to the subsequent argument so long as the potential is short-ranged. We are therefore excluding discussion on polyelectrolytes which contain charged groups. We have therefore polymer—polymer, polymer—solvent and solvent—solvent forces acting in the solution. The important effect in discussing solution behaviour is whether the polymer—solvent interactions are of greater or lesser strength than the mean of the polymer—polymer and solvent— solvent interactions. If polymer—polymer interactions are energetically weak whereas polymer— solvent interactions are favoured, the polymer will tend to avoid itself and seek regions of the solution rich in solvent. The effect therefore is to enhance the excluded volume effect. However, when polymer—polymer interactions are energetically favoured, the polymer tends to shun the solvent and the molecule will retreat into itself. In extreme circumstances, the polymer molecule will collapse into a ball. If we denote the excess of polymer—solvent interactions over the mean of polymer—polymer and solvent—solvent interactions by w, then it will be shown later that the effect of solvent and temperature will enter the partition function of the system through a factor of the form exp(2w/kT). When w = 0, the net interaction is zero and only the excluded volume is present; when w is negative, polymer—solvent interactions are favoured. Under these circumstances, the polymer is said to be dissolved in a good solvent. The case when w = 0 is known as an athermal solution since there is then no heat of dilution. The effect of temperature on a good solvent is weak; temperature has no effect on an athermal solution, though real solutions are never completely athermal over very wide temperature ranges. On the other hand, when w is positive and polymer—polymer interactions are favoured, lowering the temperature favours the more condensed polymer configurations so that a transition at a finite temperature to a collapsed state is favoured. A transition temperature is only possible in poor solvents; in good solvents no transition is possible. Flory defines the transition temperature, the e temperature, as the point at which the molecule has been sufficiently contracted that it can be described by a Gaussian distribution function. That is, at the e point the strength of the attractive forces exactly balances the dispersive nature of the exluded volume effect. Hence, if we write the mean square length as (r~) = R~n2I) then in general R0 is a function of temperature and at the ® point t.’ = The behaviour of ~ above the ® point is unclear. However, as T oo, 2w/kT—* 0 and thus the solution becomes athermal for which it is accepted that ~>4. With the introduction of attractive forces between the molecules of a solution, the thermodynamic properties become temperature dependent. The important experimental quantity is the ~.

-~

D. S. McKenzie, Polymers and scaling

45

differential heat of dilution Mid. The heat of dilution in the limit of infinite dilution can be used to characterise a polymer—solvent system though a systematic study of heats of dilution has not been attempted. The heat of dilution is related to the other excess thermodynamic properties of the solution through the equation =

—k1~ —ln(z1/z?)

(1.29)

where z1 is the absolute activity of the solvent in solution and Z? is the absolute activity of the pure solvent. Furthermore, we have ~ln(zl/z?)~Hvl/3?T

(1.30)

so that the heat of dilution is related to the rate of change of the osmotic pressure with temperature. A positive heat of dilution is characteristic of a poor solvent, athermal solutions have zero heat of dilution, a negative heat of dilution characterises good solvents. The addition of attractive forces introduces a variable into the polymer problem which is absent from the analogous problems of magnetism and simple fluids. In the bulk of this review therefore we shall not be concerned with this added complication. However, in this introduction we shall briefly explore some of the interesting consequences of the interplay of attractive and repulsive forces in polymer solutions. First, the presence of attractive forces raises the possibility of a phase transition in solution. Such is indeed the case. For a solution containing a single polymeric species there is a transition between two phases, one relatively rich in polymer and the other relatively dilute. The transition can be followed as one changes the concentration or temperature by noting the sharp increase in turbidity. This phase transition is analogous to the liquid-gas phase transition. In particular, the transition is second order and has a critical point characterised by a critical concentration and a critical temperature. The behaviour of polymer solutions near the critical point is explained theoretically only by the Van der Waal’s theory proposed by Flory [6] (eq. (1.23)). The critical point is defined by ~ and x~such that 2 = 0. (1.31) OH/Op = 02 H/Op The critical density is given by no

2 1p~ ~

(1.32)

1~n”

for large n, and the critical value Xc is given by /

Xc

1

~k~’ ~ç~) \2

or X~—4

=

l/~/~

for large n. Flory assumes that X

x—4—%,D(l—eIT)

(1.33) —

4 may be replaced by (1.34)

46

D. S. McKenzie, Polymers and scaling

where (1

—

~i

is a constant and

e is a characteristic temperature. The critical temperature T~is given by

e/T~)= —l/i~1iV’~

(1.35)

with T~
—

—

®.

—

—

D. S. McKenzie, Polymers and scaling

47

properties of the solution will become independent of molecular weight. Since p is the number of molecules per unit volume and bn is the effective volume per molecule, the condition for the transition to the more concentrated region may be expressed by p> l/bn. This may be compared with (1.22) where it is seen that the osmotic pressure is singular at p = 1/b,,. Edwards [11] has proposed that there is a further region of moderately concentrated solution, which he shows using a self-consistent field argument to be described by an equation of state with the form ~p+bnp

~

2

312

/b~p\

(1.36)

where P is the partial vapour pressure of the solution. The region of applicability of this equation is l/R~
(1.37)

where Rn is the characteristic or scaling length describing the size of a single polymer chain and d is the dimension. However, a result of the scaling theory to be developed later is that bn Rd,, (3.43). Since it is the dependence on n which is important in (1.37), this result reduces the region of applicability of the moderately concentrated regime to zero. Osmotic pressure measurements of polymer solutions over a large concentration range are lacking though the transition to the concentrated regime is noticed in measurements of viscosity [1281. It is clear that the properties of polymer solutions depend on the distribution of molecular weights. All samples of synthetic polymers are polydisperse, the distribution of molecular weights depending on the nature of the polymerisation process. In the subsequent development we shall often use a molecular weight distribution of the form =

(.Kc/m)(l ~p)2p~—1

(1.38)

where m is the molecular weight of a segment and c is the total weight concentration of polymer. Here p is a parameter related to the probability per unit time of attaching a new segment to the end of a growing chain. Numerically p is close to but slightly less than unity. Many different forms of molecular weight distribution are encountered in the literature. The form chosen here has the advantage of being the simplest and it describes some of the distributions encountered in practice. Using (1.38) we obtain W and (1.39) I—p —

so that MW/MR

1

+p

which is approximately 2 when p is nearly unity.

2. Polymer models and relation to critical phenomena The thermodynamic properties of a monodisperse dilute polymer solution depend on four variables; the temperature, the solvent, the concentration, and the molecular weight assuming that the pressure is always constant which is the normal experimental condition. Of these variables the effect of the solvent and temperature are closely linked, and it is suggested that the relevant variable is x = exp(2w/kl) 1 where w > 0 and x > 0 for good solvents whereas w <0 and x <0 for poor solvents. When w 0 the solution is athermal. The equation of state for a dilute polymer solution —

48

D. S. McKenzie, Polymers and scaling

can be written formally in terms of the osmotic pressure H namely H~H(x,p,M)

(2.1)

where p is the number density and M the molecular weight. For magnetic systems the magnetisation .~#is a function of temperature T and magnetic field ~° so that the equation analogous to (2.1) is .~=J(T,~’).

(2.2)

Hence polymer solutions are analogous to magnetic systems only for a particular value of x. The value of x usually chosen is x = 0 at which the solution is athermal. The other variables are related by .$° p and (1 T~/T) l/M where T~is the critical temperature. Hence the limit )~° ~ 0 corresponds to infinite dilution in the polymer case and the limit T -÷ T~+ corresponds to infinite molecular weight. The reason for these analogies will become apparent as we proceed. In this section I shall be concerned with the details of the similarities and differences between polymer and magnetic systems. To this end it is necessary to introduce a model of a polymer solution. Models fall into two general categories discrete and continuum. In this section only one discrete model, namely, the representation of a polymer molecule as a self-avoiding walk on a regular crystal lattice, will be dealt with. A discussion of other discrete models and continuum models will be encountered in later sections. We therefore consider a system consisting of N1 self-avoiding walks of length 1, N2 self-avoiding —

—

walks of length 2, etc. placed on a lattice of 3? sites and coordination number q so that multiple

occupation of a lattice site is excluded. The self-avoiding condition takes account of the excluded volume. The remaining lattice sites are occupied by solvent molecules. Moreover, nearest-neighbour forces act along the bonds of the lattice between the segments of the polymer molecules and between the solvent molecules. The statistical mechanics of this model can be found in McKenzie and Domb [7]. Here we shall compare this work with the Ising model of ferromagnetism. Details of the Ising model can be found in Domb [12] and Hill [13]. Domb [14]. and Bowers and McKerrell [151 have explored the connection between the polymer problem and the Ising and related models of ferromagnetism. In what follows we have attempted to be as brief as possible without sacrificing intelligibility. However, we have found the introduction of some rather cumbersome expressions unavoidable and we hope readers will appreciate that the complications might have been worse. We are considering the lattice model of a polymer solution as defined above and we suppose that the interaction energy for polymer—polymer interactions is CA A’ for polymer—solvent interactions CA B~and for solvent—solvent interactions CB B~Then, using the methods of nearest-neighbour lattice statistics, the partition function for the system can be written as ~ N,,

Q(fIV},N0, 3?, fl°°e~ H

(J~) n~0 U,,1

~ NAA>O

g’(~N},3?,NAA)YNAA

(2.3)

where X=qeB~/2kT 2AB

a,,

=

exp{[qn(CAB

—

EBB)

—

(n

—

—

EBB)]

/kT}

(2.4) (2.5)

l)(

(2.6)

49

D. S. McKenzie, Polymers and scaling

with WEAB

(2.7)

4(AA + EBB).

—

Here fN} represents the set ~N 1, N2 } and N0 is the number of solvent molecules; /~~ (T) takes account of the vibrational and translational part of the partition function for the molecules of species n; NAA is the number of polymer—polymer nearest-neighbour contacts; and g’(~N},.3?, NAA) is the number of ways of placing the set of { N} molecules on a lattice with 3? sites so that there are NAA nearest-neighbour polymer—polymer interactions. By defining . .

z,,

.

/,, exp(p~/kfl/[j0 exp(p0/kT)]”,

(2.8)

where ji,, is the chemical potential of the nth species (n ~ 0) and Z({N}, 3?,

fl =

H (Nn?.) ~

n~1

n

~

(2.9)

g’({N}, 3? flyNAA

NAA~O

we may relate the statistical mechanics to the osmotic pressure through the equation /

exp(H 3?/kT) = ~

z~”n\

H ~—fl—) Z(~N},3?, T).

(N}n~1

(2.10)

N,,!i

Eq. (2.10) may be used as the starting point for the development of the virial series. In the usual way one obtains H

kT

+

n

~ A(fN}, flHp~n

(2.11)

{N}

where the virial coefficients A({N}, T) are independent of 3?. In (2.10) the symbol

~N}

represents

summation over all sets of (N} including the empty set whereas in (2.11) the summation is over all sets ~N} with the restriction that ~ 1N~~ 2. The first few terms of (2.10) may be written as exp(H3?/kT) = 1

+

~ Z((n), P4, T)z,,

+

4~Z((n, m),

.3?,

T)ZnZm

+...

(2.12)

where there is an obvious change of notation. We now define g((n), 3?, NAA) as the number of ways of placing a self-avoiding walk of length n on a lattice with 3? sites so that there are at least NAA nearest-neighbour polymer—polymer interactions and g((n, m), 3?, NAA) as the number of ways of placing a self-avoiding walk of length n and a self-avoiding walk of length m on a lattice with 3? sites so that there are at least NAA nearest-neighbour polymer—polymer interactions. If we define xy—l,

(2.13)

we obtain Z((n), 3?,

~ U,,

Z((n, m), 3?

g((n),

3?,NAA)XNAA

(2.14)

NAA~O

1

UnGm

~

g((n, m),3?,NAA)xNAA.

(2.15)

NAA~O

This change of variable from y to x is non-trivial. Furthermore, it should be noted that in deriving

50

D. S.

McKenzie, Polymers and scaling

the virial series (2.11) from (2.10) the virial coefficients are normalized so that for example the second virial coefficient [ A((n, rn) T) =

Z((n, rn), 3?, 1)

3?önm I

LZ((n), 3?, T) Z((rn), 3?, T)

—

1 ii j

.

(2.16)

Thus the factors cr,, cancel out and H/kT becomes dependent on temperature only through x as expressed in (2.1). Because of this normalization, which expresses the fact mentioned in the introduction that the osmotic pressure is measured with respect to an infinitely dilute solution, it is the activity series (2.1 2) which can be compared directly with the Ising model rather than the virial series (2.11). To facilitate this comparison it should be noted that g((n), 3?, 0) and g((n, m), 3?, 0) are the number of self-avoiding walks and the number of two separated self-avoiding walks respectively. Thus g((n), 3?, 0) is the number of embeddings on the lattice of a walk of n steps such that intersections are disallowed. We may interpret this result graphically: the segments of the polymer chain are represented by the vertices of the graph and the steps in the walk by the edges of the graph, for example

,

,

,

We represent the number of embeddings of a graph G on the lattice 2 with the restriction that no vertices intersect by (G; 2) [16]. (These embeddings are sometimes known as weak or high temperature embeddings.) When no particular lattice is understood this notation may be abbreviated so that for example we may write g((3), P4,0)(

/\/).

One may note that in this context the lattice 2 is being considered as a graph. In particular no metric properties of the lattice are being used. Indeed, the number of embeddings of a graph G1 on a second graph G2 with the restriction that no vertices intersect, that is (G1 G2), is just the number of subgraphs of G2 isomorphic to G1. Thus no other properties of the lattice are necessary other than those which can be derived from the representation of the lattice as a graph. Equation (2.12) may be compared with the high temperature series expansion of the free energy of the Ising model in a magnetic field. The Hamiltonian of the Ising model is mIC

J

a,

(2.17)

where a, (= ±1) is the z component of the spin on site i; J is the interaction energy between spins on neighbouring lattice sites; the term in a1u1 is summed over all the nearest-neighbour -bonds of the lattice; in is the magnetic moment per spin and 1C is the magnetic field in the z-direction. The partition function of the system is An ~ exp[H/ktj lie) = ~ [ H exp(Ja,a1/4kT)][H exp(mlCa,/2kfll.

li

n.n.

(2.18)

D. S.

McKenzie, Polymers and scaling

51

In the usual way we introduce the high temperature variables o

tanh(J/4kT) = tanh K

(2.19)

and z = tanh(m~C/2kT)

(2.20)

whence 1 (2.21) cosh~’K[l + ~(a1o1)o + ~~(G,Gj)(UkUl)V2 where = qP4/2 is the number of nearest-neighbour bonds of the lattice and the first sum on the right-hand side of (2.21) is over ft terms, the second over ~ ( !t — 1)/2 terms and so on. That is, in the contribution to the coefficient of 0” in (2.21) each bond of the lattice occurs once and once only. The contributions to each term may be interpreted graphically so that the term in o corresponds to the diagram +...

~°

ii

and the term in v2 corresponds to two diagrams

/\~.

and

Because of the contribution from the magnetic field these graphs have different weights and must be kept distinct. The graphs which contribute to any term have no double bonds and no vertex of a graph is allowed to intersect another vertex since this forms a different graph with, in general, a different weight. The sums in (2.21) over the nearest-neighbour bonds of the lattice imply that for any graph one must calculate the number of ways of embedding the graph on the lattice so that each vertex of the graph lies on a distinct lattice site and each edge of the graph coincides with one of the bonds of the lattice. Because of the restriction that the vertices occupy distinct lattice sites, with no overlaps, the contribution of a graph is its number of weak embe.ddings. In other words, this restriction is the same as that applied to the self-avoiding walk model of a polymer chain. Substituting (2.21) into (2.18) the partition function becomes A = cosh~K ~ [ii exp

(‘‘)]

[1

+

~(u,a

2

+...].

(2.22)

1)o ÷~(UjGj)(GkGl)U The summation over licr} associates a weight factor with each graph in the expansion in terms of v. The constant term gives a weight ~

[exp(m~C/2kfl

+

exp(—m~C/2kfl]~ .

(2.23)

The weights of graphs contributing to higher order terms depend on the valence of the vertices of each graph. The valence of the vertex of a graph is the number of edges incident at the vertex. Vertices of even valence, including vertices of valence zero, give rise to a weight exp(m~C/2kI)+ exp(—m~C/2kfl whereas vertices of odd valence give rise to weight exp(m~W/2kT) exp(—m~C/2kT). —

52

D. S.

McKenzie, Polymers and scaling

Let ~ (1, e) be the set of all graphs with 1 edges and e odd vertices, let G represent any graph and let z = tanh(rn~lC/2kT).The partition function can now be written as (2.24) A = cosh~Kt~[1 + ~ ~ (G; 2 )UlZ2e 1. l~0e~0

Gec4(l,e)

The terms in eq. (2.24) may be interpreted graphically as follows. The coefficient of z0 includes contributions from graphs all of whose vertices are of even valence, thus we may write K 0=

~

2)v’

(G:

Ge ~(l,0)

= ~

[(Q)÷(~)+(~~~)]v6+....

(2.25)

2 produces the following expression The term in z K 2= ~

(G:

2)v’

Ge ~c(l,2) = ()o+(~

+

)~2 +

[(/~/\/)

(~)v3

+

[(~)+ (/\~/)+

(~)]

~

~

(2.26)

4 gives while, finally, the term in z K 4= ~ (G; 2)o’ Ge c#(l,4)

+[(/,~)+ ~~)]o3

(//)~2

+[(/~~/)+(~,~)+ (2.27)

From the above, it can be seen that a “first approximation” to K0 involves only the self-avoiding polygons whereas K2 and K4 can be approximated by the self-avoiding walks and two separated self-avoiding walks respectively. If we use the notation u~to represent the number of self-avoiding polygons with n edges on a particular lattice, c,, to represent the number of self-avoiding walks of n steps and c,,,,, to represent the number of separated pairs of self-avoiding walks of n and rn steps then, numerically, the first few coefficients of o in (2.25), (2.26) and (2.27) are dominated by u,,, c~and c,,,,, respectively and one may assume that they continue to dominate the higher coefficients. Thus we may write, approximately, K0 K2 ~ K4

~ u,,u” n~’3

(2.28)

~ c,,v”

(2.29)

n~a1

~

~

n~1 m~n

Cnm0~m.

(2.30)

D.

S.

McKenzie, Polymers and scaling

53

With a small change in notation (2.24) becomes (2.31) Acosh~’Kt~[1+ e~0 E K2eZ2e]. This equation may be compared with the analogous equation in the polymer case, namely (2.12). Using (2.14) with x 0 it can be seen that the term linear in the z,, in (2.12) involves the number of self-avoiding walks and so is analogous to the term in z2 in (2.3 1). Similarly, the term in Zn Zm in (2.12) involves the number of two separated self-avoiding walks and thus is analogous to the term z4 in (2.31). There is no term in (2.12) analogous to the term in z0 in (2.31). To pursue the analogy further it is assumed in the Ising model that in zero field the specific heat diverges at the critical point T~as C~~(1 — 7~/fl°’ (l —v/u~)~”

(2.32)

the zero field susceptibility diverges as

x~

kT(02 in A/O~C2)~( 0 (1 ~

—

o/u~)~’

and higher derivatives of the free energy diverge, in zero field, as 2k in A/O~C2~~ kT(0 0 (I v/o~Y~. —

(2.33)

(2.34)

In (2.32)—(2.34) we are assuming that Tis always greater than T~; the change from the usual Ising model notation in that primes have been added to a, ~yand ~k is because we wish to use these symbols later in a slightly different context. Equations (2.32) and (2.33) imply that K0 ~

~ na’2(v/vc)n

(2.35)

n~’3

since the specific heat in zero field (z = 0) is given essentially by the second derivative of K0 with respect to v, and K2 ~

~ n~~t’_1(v/oc)P2.

(2.36)

n>1

Now it is known from the work of Hammersley [17, 18] that u,, and c,, behave as =

~i(n)~/’

(2.37)

=

Ø(n)j.t”

(2.38)

and

where ~i is known as the effective coordination number or the connective constant and [i~Li(n)] 1/n ~ 1 and [0(n)] ~ 1 as n —~oo~This result has been derived for the hypercubical lattices. If we assume that 2 and 0(n)con~’ n)u0n’~ then the generating functions ‘u,, v” and ~Cn 0” behave, for large n, as —~

~—2

~ u,,vn = u 0 ~ n n

n

~

(pu) ~

0

(i—pu)

~

1

(2.39)

54

D. S. McKenzie, Polymers and scaling

and ~

CnVn =

c0

~

n

~

‘(po)”

C0

(2.40)

-

(1 —pu)~

Hence we can identify p with 1 /v~and a and ‘y with the specific heat exponent a’ and the susceptibility exponent ‘y’ respectively. Unfortunately, this identification of the critical point and the exponents between the two problems does not extend so far as complete equality. Although Temperley [191 conjectured that I/i-i = v~,Fisher and Sykes [201 constructed a rigorous lower bound to p for the plane square lattice which was greater than the exact result o~= 1 + ~J2 found by Onsager [211. The relation between the self-avoiding walks and the Ising and Heisenberg models has been explored in great detail by Domb [141 who has put forward reasons why the values of the critical point and the exponents change from model to model. Bowers and McKerreli [1 5] give precisely the nature of the selfavoiding walk approximation. It is not the place here to go into these details, but the underlying reason for the change is that in the polymer problem one is concerned with only one class of graph, say, self-avoiding walks or self-avoiding polygons, whereas in the Ising model an indefinite number of new types of graph must be taken into account as one proceeds to higher terms in the series expansion of, say, the susceptibility or specific heat. Thus, for example, the assumption that the u,, continue to dominate the coefficients of the series for K0 (2.25) is incorrect. It is unfortunate that the analogy between the Ising model and the polymer problem is not closer in that the activities z,~are proportional to some parameter so that ~ would be analogous v. Furthermore, the variables z,~are never measured experimentally, activities being notoriously difficult to measure for non-electrolyte solutions. The normal experimental procedure is to measure the osmotic pressure as a function of the total weight conceiltration c of polymer in solution. This formulation also has some theoretical advantages. Returning to (2.11) we substitute the molecular weight distribution (1.38) for the p,, to obtain 1 (2.41) H 1 3?Tc M,, 1~2 ~ B,c

r

where Mm the number average molecule weight, is defined in (1.39) and B,

=

X’’(l _p)2’ m1

~

A({~,x) Hp(n_t)Nn

where the summation is restricted so that ~ A((i,/,.

. . , k), 0)

(2.42)

(N)

= 61,/,...kAl(ii..

. k)~’

= 1.

We now make the assumption (2.43)

where A, is a constant and öi,j,...k is a symmetry factor which, for example, equals 1/2! when two of the chains are of equal length and 1/3! when three of the chains are of equal length and so on. The assumption (2.43) is plausible for the second virial coefficient as explained in the introduction and it also allows us to recover the analogue in the polymer case of (2.34) for the Ising model. Substituting (2.43) into (2.42) we obtain for x = 0 BlX,~~[l+~,n~t1pfl1

m

1, (l—p)

,=C,(M~)’~~’~ ‘

(2.44)

D.

S. McKenzie, Polymers and scaling

_~55

where C, is a constant. Thus the exponent l(~, 1) is analogous to the exponent ~k introduced in (2.34). It is also interesting that the parameter p behaves in a similar fashion to v/va in (2.34). Hence the limit u v,, -‘-or T —~T~+ corresponds to p 1. Through (1 .39) it can be seen that this implies taking the limit M,, 00• We turn now to the analogy between the end-to-end length distribution function p,, (r) and the zero field spin—spin correlation function K °aGb ) of the Ising model, where (Ga Gb ) represents the correlation between a spin on site a with one on site b. The correlation function is defined by —

-~

—~

-+

AK

GaGb)

cosh~K ~ [ H (a)

OaGb(l +

a,a1o)] [II exp(m.~a1/2kfl].

n.n.

(2.45)

I

The term involving the product over nearest-neighbours can be expanded in a series in o as before and each term represented graphically. Because of the extra factor GaUb the only terms in the expansion which contribute in zero field are graphs whose vertices are all of even valence and which have an odd vertex at site a and an odd vertex at site b. If ‘~‘ab(G;2) represents the number of weak embeddings of the graph G on the lattice 2 with the restriction that a vertex of G occupies site a and a vertex of G occupies site b then we may write AK GaGb>

2~cosh~K~

l~1

~

‘~‘at(

2)v’

(2.46)

GE~g(1,2)

where the odd vertices of G must coincide with a and b. Suppose the sites a and b are nearestneighbours then we may interpret the above equation graphically as follows AKUaUb)

2~

cosh~K[(~) ~+(/\)u2

+(fl)03

+[(<>)

+(~,~)+(~)]~v4÷[(~)+(e) J~(

)] ~

+ . .

.]

(2.47)

where the open circles represent the vertices on sites a and b. The contributions to the term in v~ always include the chain of 1 bonds between a and b so that in the same sense as previously the selfavoidiiw walks provide a ‘first approximation’ to the spin—spin correlation function. If c,, (r) is the number of ways of placing a self-avoiding walk of n steps starting at the origin and ending at r then KGaUb)~

~ c,,(r)u”

(2.48)

n~1

where a is at the origin and r is the vector from a to b. It should be noted that here again there is a ëhange in normalisation between the Ising model and the polymer problem. Thus the end-to-end length distribution function is normalised by Pn(r) so

=

c,,(r)/c~

(2.49)

that

fp,,(r)dr=

1.

(2.50)

56

D. S. McKenzie, Polymers and scaling

The spin—spin correlation function is related to the zero field susceptibility Xo by rn2 (2.51)

~

where the sum is over all lattice sites. The corresponding equation in the polymer case is Cn =

fcn(r) dr.

(2.52)

Furthermore, the polygons are related to the end-to-end length distribution close to the origin; thus Ufl~ 1

2(n+ 1)Cn(l).

(2.53)

This can be related to the internal energy E of the Ising model by E= q3?J

(2.54)

(UaGa+i).

Although the end-to-end length arises naturally when considering the analogy between the Ising and polymer problems, it is interesting that a generating function similar to (2.47) is produced in the polymer problem when one considers the segment—segment distribution functions. This generating function is also of importance since it comes about quite naturally in the theory of lightscattering from polymer solutions. We obtain this result merely by substituting (1 .38) for the p,, in (1.3), whence 2 [~pn 1 ~ fg~1)(r)exp(ik r) dr + ~ np” ~1]~ (2.55) j(k, {p}) = cmK(l p) .

—

We now make the assumption that the g~7~(r)depend only on I i / I and r for all n. There is numerical and theoretical evidence, which will be mentioned later, in favour of this assumption. Let us therefore write —

g~(r)-°g~(r)

where

rn

=

li—il.

(2.56)

Equation (2.55) may now be written as

f(k,(p})cmK(l

—p)2[2 E n~2

p~

~

(n—rn) fgm(r)exp(~r)dr+

~ np”~].

(2.57)

n>i

m1

This equation is simplified by defining the generating function G(r,p)

~ g,,(r)p”

(2.58)

n)’l

so that f(k, ~)

[~—p)2 JG(r, p) exp(~ r) dr

=

cmK(1

=

cmK[2Ô(k, p)

—

~)2

+

1]

+

(1 1

p)2]

(2.59)

where O(k, p) is the Fourier transform of G(r, p). Thus the behaviour of the experimentally measur-

D. S. McKenzie, Polymers and scaling

57

able scattering function Kc/fl (k, p) is determined fundamentally by the behaviour of the function G(k, p). It should be noted that the turbidity is given by f(O~P)[

2p +1] =M;. 1—p

(2.60)

3. The scaling laws In the preceding discussion it has been shown that the theory of athermal polymer solutions, in which the excluded volume effect is dominant, is analogous to the Ising model of ferromagnetism. The limit of large molecular weight corresponds to the approach to the critical point. By analogy with the Ising model one can define exponents which describe the asymptotic behaviour for large molecular weight of the virial coefficients in the expansion of the osmotic pressure, and the root-mean-square end-to-end length of the chain. In addition one may define formally exponents to describe the asymptotic behaviour of the total number of self-avoiding walks and the total number of self-avoiding polygons though these quantities are not experimentally measurable. Suppose for a monodisperse athermal polymer solution of concentration c and molecular weight M we express the osmotic pressure H H(0, c, M) (see (2.1)) as a virial series in c by H/3?Tc-~-+ ~ A,(0,M)c~ M

(3.1)

l~2

where A,(0, M) is the /th virial coefficient when x the exponents e, v, ~yand a as follows:

= 0.

We may now write the defining equations for

~

A2(0,M)

a2M_e,

(3.2) (3.3)

where the a1 are constants and the exponent ~, is introduced in this manner instead of the ~, in (2.44) because of a simplification which will occur later; furthermore, if K r~)1/2 is the root-meansquare end-to-end length, e,, is the total number of self-avoiding walks, with n = M/m, and u,, is the number of self-avoiding polygons then 112—r (r1~> 0M°, (3.4) ‘°,,

and

c0n~~p”,

(3.5)

2pn. (3.6) u,, uona The above equations have been written for a single component system. They can be generalised to polydisperse systems by, for example, using the molecular weight distribution (1.38) and defining generating functions as in (2.39), (2.40), (2.44) and (2.58). The variable M is then replaced by 1/(l p) or, through (1.39), bYMn. The generating functions for c,, and u,, are functions of 1 /( 1 pp) but since we shall not be concerned directly with e,, and u,, the net effect of the polydispersity is to replace M by Mn. In this section we shall deduce various relations between these exponents and others to be introduced later. The treatment is the standard scaling law argument and we have followed the notation — —

D. S. McKenzie, Polymers and scaling

58

of Stanley [221. Some of the ideas presented here can also be found in Domb [23]. The scaling argument is to assume that the osmotic pressure is a homogeneous function of the independent variables C and M, that is, H(0, Xac, AIIM) =

XH(0,

e, M),

(3.7)

where A is an arbitrary parameter. Taking successive derivatives of H with respect to first OH(0 XaC AIIM) O(Xae)

011(0

C

M)

C

we obtain

=Xfl~)(O,c,M);

that is, H~’~(0, 0,M) = X~1H~1~(0,0, AIIM). But Xis arbitrary so we may choose). ~M1/” so that fl~’~(0, 0,M) =M~’—”)/’~’fl~1~(0, 0, 1).

(3.8)

(3.9)

From (3.l)it can be seen that HW(0, 0,M)M’

whence

ba—l.

(3.10)

In a similar way we obtain H(2)(0, 0,M)

(3.11)

0, X’~’M)

= X2”’H~2)(0,

so that H~2~(0, 0,M) =M~’_2a)/bH(2)(O

0, 1);

(3.12)

whence, using (3.2) cb2a—l.

(3.13)

Equations (3.10) and (3.13) may be solved fora and b; we obtain a

= 1

—C

and

2—c

b

=

—l

(3.14)

2—c

The higher derivatives of H with respect to c give us [J(r)(0 0,M) =M~1_~~)/l~H(r)(0, 0, 1)

(3.15)

and H~~’~(0, 0,M) ~

0, 1).

(3.16)

Dividing (3.15) by (3.16) and using (3.3) we obtain A,.(0,M)/A,._

(3.17)

1I~. 1(0,M) c~M—a

Hence we may define an exponent =

—a/b =

1

—

(3.18)

~.

such that if Ar_i (0, M) ~ M~then Ar(0, M) ~ M~

D. S. McKenzie, Polymers and scaling

59

Thus, the scaling assumption implies the existence of a gap index which relates the molecular weight dependence of successive virial coefficients. Furthermore, when x = 0, it should be noticed that only one exponent e is necessary to describe the molecular weight dependence of the osmotic pressure. This result may be compared with the Van der Waal’s theory developed in the introduction. To do this note that from (3.15) with (3.14) the molecular weight dependence of the rth virial coefficient is given by (3.19) Changing the variable in (3.1) from e to p using c = Mp/X (see (1 .7)) we obtain ll/kTp+ ~ CrM_2~pr

(3.20)

r~2

where the Cr are constants. This equation may be compared with (1.22) whence it is seen that the molecular weight dependence of the excluded volume per molecule b~is given by n2 ~ and, with the assumption (1.28), the Van der Waal’s theory gives correctly the molecular weight dependence of the higher virial coefficients. To complete the scaling relations for the osmotic pressure, one should include the dependence of H on the temperature and solvent variable x. The arguments that follow are tentative and we include them only for completeness. Let us consider the function K(x,c,M)

0311/OxOc2.

(3.21)

In the limit e -~0,K(x,0, M) is the derivative with respect to x of the second virial coefficient. Using the scaling argument as above we obtain K(x,e,M)X2~e_lK(Xex,X~~C,XLIM), so that

(3.22)

K(o,0,M)=M(t_e_2a)/bK(0,0, 1).

(3.23)

We now assume that K(0, 0,M)

K

=

0M~~

(3.24)

where ~ is an exponent which describes the molecular weight dependence of the temperature derivative of the second virial coefficient close to the athermal point. Thus we obtain e=

~/(2

—

e).

(3.25)

With (3.25) it can be seen that the osmotic pressure scales as 2_e H(M~x,M’ H(x, c,M)

M

C,

~

M 2_~Mx~M ~c).

(3.26)

Unpublished work suggests that ~ is small and negative, perhaps of the order of The interesting question which remains is whether the osmotic pressure is a homogeneous function of M~xand M’ ~c as the system approaches the 8 point. This is unlikely to be the case. For fixed M the point of critical mixing x,, (M) a’~de~ (M) is defined by the equations (see (1 .31)), —~.

(OH/Oc)X,M =

0

(3.27)

60

D. S. McKenzie, Polymers and scaling

and (02H!0e2)X,M = 0. The 0 point x®, c® is then given by x®

=

(3.28)

lim x~(M)

(3.29)

M-°=

and =

lim e~(M)

(3.30)

M -°

if these limits exist. Experimental evidence suggests that xe is finite whereas C® is zero. This behaviour is predicted by Flory (see equations (1.32) and (1.35)) who also suggests that near the® point the appropriate variables are 114”2(x — x®) and M”2c. This suggests that near 8, the independent variables are t = x x 0, q = C C® and 1 /M, which is the finite on taking the limit M —~oo. One can now make the scaling hypothesis that near 0 the osmotic pressure can be considered as a homogeneous function of these variables, that is, —

—

H(Xa’t, X!~’q,A~~’M_1)= XH(t, q,M’) where a’, b’ and C’ are constants and A is an arbitrary parameter. Scaling with respect to M’ by choosing A = Mi/c we obtain H(Ma’/c’t, M”’~’q, 1)

=

Mh/c’H(t,

q,

M’ ),

that is (3.31) 1 at q = 0, which with We canimplies obtain that values a’, b’= and if we the predictions fact that (OH/Oq)~,~ M = 4 whence a’b’ = (3.31) (b for 1)/C —1, c’ and the use Flory that a’/c’ ==b’/C’ and c’ = The above results apply at the 8 point, t = 0, which for finite M is not identical with the point of critical mixing so that the requirement that (OH/Oq) = 0 at x = x,, and c = e~,is not violated. The second requirement that limM. (02 H/0q2 )o,M = 0 and the assumption that C® = 0 necessarily imply that at the 8 point the second virial coefficient vanishes. This point has been made previously by Kennedy [471. The Flory predictions for the exponents a’, b’ and c’ above are obtained from what is essentially a mean field theory. A more advanced theory may suggest changes in the values of the exponents. In particular, the scaling variable M1’ Ic q near 8 may be identical with the scaling variable M’ ‘C appropriate far from 0. However a corresponding identication for the x variable cannot be made because of the finite, non-zero value of x®. This suggests that one obtains one type of scaling behaviour close to 0, and another type far from 0 and that there is a cross-over between the two types of behaviour. To investigate the cross-over let us define a region about the line of critical points x~,Cc for given 1k! in the space defined by the variables x, c and M. The appropriate phase diagram can be found in [47]. Since the line must end at the 8 point the more sensible variables are t, q and M1. Furthermore, using the scaling hypothesis (3.31) near®, we can transform to the scaled variables y 1M” Ict H(t,q,M_1)~M_h/c’g(Ma’/c’t,MlI’/c’q). —

61

D. S. McKenzie, Polymers and scaling

andy2 =M~~’/c’q. In the space defined byy1 andy2 the line of critical points reduces to a point = k, and Y2 = k2 where k, and k2 are constants. The region near 0 is then bounded by a curve h(y1 Y2) = 0 in the Yi y~space. The exact form of h, that is the shape of the curve, is unimportant. In t, q, M’ space this curve becomes a cone whose apex is the 0 point. The 0 region can be defined by the inequalities k1
(3.32)

k~c
(3.33)

The tth moment of the distribution about the origin, p(t, M) is given by p(t, M)

fP(r, M) rtdr.

=

(3.34)

Using (3.33) and (3.34) we obtain p(t, AIIM)

=

(3.35)

A(d+t)~~p(t,M)

where d is the number of dimensions. As before we put A p(t,M) =M_L(+t)a+1 I

/bp(~

=

M1

/“

to obtain

1).

But the distribution function is normalised so that p(O, M) =

(3.36) 1

whence

a—l/d.

(3.37)

Furthermore, the second moment, from (3.4), behaves as M2 whence “

(d

+

2)a

+

1

= —2vb

(3.38)

which, with (3.37) gives b—l/dv.

(3.39)

Substituting for a and b in (3.33) we find that P(r,M)

M_c~~~P(r/Mv, 1).

(3.40)

62

D. S. McKenzie, Polymers and scaling

Hence the end-to-end length distribution can be described by a single scaling length R,,

=

R0n’~such

that -~F(r/R~).

pnfr)

(3.41)

The results of numerical calculations, to be mentioned later, of the mean square radius of gyration (s~) suggest that the molecular weight dependence of K s~) is determined byrepeats the same 2 This suggests that, if one theexposcaling nent v as for lengths, distribution that is K s~) functions, x n argument forthe the end-to-end segment—segment g, 1(r), or g,,, (r) with rn = I i I I where we make the same assumption as in (2.56), then each distribution function is determined by a single scaling length Rm which is proportional to rn°. Thus the scaling hypothesis implies that the size of a molecule in solution is determined by a single length. But the second virial coefficient is merely the volume excluded by one molecule to occupation by another. We can nowmake the strong scaling hypothesis namely that this volume is proportional to the cube of the characteristic length which determines the size of the molecule, that is, in d dimensions 2. (3.42) A2(0,M)~M~’° (In (3.42) we have divided by Al2 because the A,. are defined in (3.1) with reference to the virial series in the concentration c rather than the number density p.) Referring to (3.2) we see that the strong scaling hypothesis implies the scaling relation “.

—

dv =

2

—

e.

(3.43)

This relation is the most important Of the results derived in this section; much of the discussion in subsequent sections is devoted to the question of whether (3.43) is correct. Lastly, the higher virial coefficients have the dimensions of volume to the (r — 1 )th power so that the strong scaling hypothesis implies A,(0,M) xM(~’)”~~.

(3.44)

Substituting for dv in (3.44) from (3.43) we see that the strong scaling hypothesis is consistent with the result of our earlier discussion of the higher virial coefficients, namely (3.19). To proceed further with the end-to-end length distribution additional assumptions besides homogeneity must be introduced. We shall now describe the derivation of an analytical form for Pn (r) from the Ornstein—Zernicke hypothesis. This theory is described by McKenzie and Moore[241 and is based on the earlier work of Fisher [251 and Fisher and Burford [261. The end product of this work has been to show that p,,(r)~(L)texp[_(~~],

(3.45)

when(r/Rn)~°1, with t

= (1

—

d/2

—

+

dv)/(l — v)

(3.46)

and = i/(l

—

ii).

(3.47)

63

D. S. McKenzie, Polymers and scaling

Qualitatively, a distribution with the form (3.45) is sensible since the distribution function shows a dip near the origin caused by the excluded volume and a steep decay far from the origin caused by the finite length of the walk. The above form for Pn (r) includes the Gaussian case for which v = 4 and’y 1. Fisher [251 first derived the relation between 6 and zA by considering the generating function (3.48) F(O, r) = ~ p,,(r) e”0. Using the facts that Pn (r) = 0 for r> na where a is the step length, Pn (r) < 1 and p,, (r) ~ can be shown rigorously that F(O, r) decays exponentially with r, so that one may write F(O, r) = B(r, 0) e”

1/c,,, it

(3.49)

where B(r, 0) does not vary exponentially in either r or 0. Equation (3.49) may be inverted using Cauchy’s theorem to obtain 1 p,,(r)=— 2in

J

c+i~r

F(0,r)e”°dO

(3.50)

c—hr

where c is chosen larger than the real part of any singularity of F(0, r). Fisher makes the additional assumption that K=Ko0v’ for small 0 so that (3.50) becomes p,,(r)

=

1 —

f

2iri

c+i~rB(0,r) exp(nO

(3.51)

—

i~0~’ r) dO.

(3.52)

c—iir

For large n this integral may be approximated using the method of steepest descents. We obtain p,,(r)

=

C(r, n) exp[—(r/R,,)~]

(3.53)

with

6

= I/(l

—

v’)

(3.54)

and R,,

=

(i4’(l

—

v’)

(1

T~’)v~v’)nP.

(3.55)

In (3.53) the function C(r, n) does not vary exponentially in either r or n. On making plausible assumptions concerning the behaviour of B(0, r) for small 0, Fisher shows that one recovers the scaling form forp,,(r) namely (3.41) and thence one can make the identification v’ = v. Thus Fisher’s argument shows that the shape of a molecule is changed by the excluded volume effect, and in particular the shape exponent 6 becomes greater than 2 when v is greater than 4. McKenzie and Moore extended Fisher’s treatment by using an Ornstein—Zernicke argument starting from the generating function ~ cnpn(r)u”.

n= 1

(3.56)

64

D. S. McKenzie, Polymers and scaling

This function is directly analogous to the spin—spin correlation function of the Ising model (see (2.48)). Subsequent development relies on the assumption, first made by Fisher in the context of the Ising model, that l~(k,o), the Fourier transform of F(r, v) with respect to r is given by (3.57) l~(k,u)=AK~/(K2 +k2) for small k, where A is a constant. The parameter i~is assumed to depend on v so that KK 0(l —v/v~)~ where

~

(3.58)

is a constant and o is close to o~.Hence, when k

= 0,

we obtain (3.59)

But if we substitute (3.5) for c,, in (3.56) we find 1(pu)” fpn(r) exp(ik

r)dr,

(3.60)

1(k, v) = ~2n~ so that, using the normalisation of p,,(r), we obtain 1 [‘(0, v) = ~ n~l(j~~)~~ n=i (1 —pv)~ 00

.

(3.61)

Comparing (3.61) with (3.59) we can identify l/p with o~and we obtain the relation (3.62) between the exponents involved. The remainder of the argument of McKenzie and Moore is to change variable from v to U where v = e0 and take the Fourier—Laplace transform of (3.57) using Cauchy’s theorem and the method of steepest descents to obtain the scaling relations (3.54) and (3.55). This derivation holds for small k and large ~r. These conditions imply that n is large and r ~. R,,. The other assumptions are the Ornstein—Zernicke form (3.57) for F(k, v) and that ,~~ for small 0. The introduction of the term l~’~ in (3.57) is of no fundamental significance; its purpose is to avoid the relation ‘y = 2v which the most reliable values of’y and v appear to preclude. The theory of McKenzie and Moore was developed by forcing identification between the Ising spin—spin correlation function and the unnorrnalised distribution functions ~ (r) (eq. (3.56)). However the distribution function which arises naturally from considering light-scattering from polymers is G(r,p)

~ g,,(r)p”

n~1

(3.63)

(eq. (2.58)), where the g,~(r) are the normalised segment—segment distribution functions. As seen from (2.59), the dependence on scattering vector and molecular weight of the light scattering from a polydisperse polymer solution comes through G(k, p), the Fourier transform of G(r, p). By the same argument as above one can assume that Ô(k, ~)

=

(3.64)

K2+k2

where now i<

~

(1

—

p)P• However, because of the change in normalisation,

D. S. McKenzie, Polymers and scaling

65

p 1 —p so that to obtain agreement between (3.64) and (3.65) we must have (2

—

~i’)~=

(3.65)

1

(3.66)

instead of (3.62). This implies that i~’is not equal to i~.One may continue the argument by taking the inverse transforms as before and obtain g,, (r)

(r/R,, )t’exp[ —(r/R,, )II]

=

(3.67)

with 6

=

(3.68)

1 —v

but now t’

d(v— 1/2)

1—v

.

(3.69)

Thus one expects a change in the value of t for the segment—segment distribution functions. Such a change is not unsurprising if one remembers that t determines the shape of the distribution near the origin. The dip near the origin is likely to be more pronounced for the segment—segment distribution functions compared with the end-to-end length because the free ends of the chain enhance the excluded volume effect, particularly when the distance between the segments is small. So far these results hold in the limit r ~ Rn. However, the homogeneity argument concludes that p,, (r) has the scaling form (1 /R~)F(r/R,,) for all values of r/Rn. This is an equivalent form of the strong scaling hypothesis in that only one scaling length Rn characterises all regimes. McKenzie and Moore use this hypothesis to deduce a scaling relation valid in the limit r 0. They suppose that when r/R~is small we may assume (3.70) F(y) = B 5 —~

0y

where B

0 is a constant. When the walk returns after n steps to within one step of the origin we have (3.71)

pn(l)=10(n~~)_l:=Blna_i_7

for large n. Comparing (3.71) with (3.70) withy g =

(~y+ 1 — dv

—

= 1/Rn

we obtain the scaling relation

a)/v.

(3.72)

The interest of this equation is that it relates the “short-range” behaviour of p~(r) characterised by the exponent a to the “long-range” behaviour characterised by the exponent v. 4. Calculation of exponents. Numerical methods In the preceding section we have described how the theory of dilute polymer solutions is related to the theory of critical phenomena. In particular we have described how the scaling theory of

66

D. S. McKenzie, Polymers and scaling

critical phenomena is adapted to the theory of polymers. In so doing we have defined various exponents which describe the molecular weight dependence of quantities of interest and various scaling relations between the exponents. It is of obvious interest to obtain numerical values for the exponents not only to make it possible to test theory against experiment but also to test the validity of the scaling relations and by implication the assumptions used in their derivation. The theoretical methods employed to obtain numerical values for the exponents can be classed either as numerical or analytical. In this section we shall describe the results of numerical work; the next section concentrates on analytical studies. As we shall describe in this section, numerical work appears to confirm the assumption of the homogeneous form for the osmotic pressure and the longrange behaviour of the end-to-end length distribution but doubts are cast on the validity of the strong-scaling hypothesis and its application to the short-range behaviour of the end-to-end length distribution. The exponents in which we shall be interested are y, which describes the number of self-avoiding walks, a, which describes the number of self-avoiding polygons, g, t and 6 which describe the shape

Table 1 The symbols ~, ~‘, a, a, v, 6, t and g are defined in table 2. q is the coordination number of the lattice. The nomenclature of most of the lattices is self-explanatory, but by q-tree we mean a regular tree of coordination number q, for example, a finite section of the 3-tree is

and by dD-hypercubic we mean the generalisation of the plane square and simple cubic lattices to d dimensions. The row labelled ‘Random walk’ gives the values of the exponents for a random walk on a regular lattice of coordination number q. The entries NA stand for ‘not applicable’. Lattice

q

~s

q-tree Honeycomb Plane square Triangular Diamond Simple cubic Body-centred cubic Face-centred cubic 4D-hypercubic 5D-hypercubic 6D-hypercubic Random walk

q

(q — 1) 1.8478 2.6385 4.1520 2.8792 4.6838 6.5295 10.0355 6.7680 8.8313 10.8720 q

3 4 6 4 6 8 12 8 10 12 q

‘y

a

1 [29] [29] [29] (29] [29] [29] [29] [30] [30] [30]

4/3 4/3 4/3 7/6 7/6 7/6 7/6 15/14 33/32 73/72 1

[36] [29] [29] [29] [29] [29] [29] [30] [30] [301

a

NA 1/2 1/2 1/2

1 [36] [33] [31]

—

1/4 1/4 1/4 1/11 1/27 1/63 0

v 1 — —

1/2 [7] —

[33] [33] [31] [30] [30] [30]

6

1/4 [7] 1/4 [7] 1/4 [7]

00 —

3/4 3/4 3/5 3/5 3/5 3/5

—

— —

—

—

—

NA

1/2

g

t 00

[34] [32] [35] [34] [34] [32]

4 [37] 4 [38]

2/3 [39] 2/3 [38]

—

5/2 [37] 5/2 [39] 5/2 [38]

0 [39] 0 [381

—

1/3 [39] 1/3 [39] 1/3 [38]

—

—

—

—

—

2

NA

—

—

—

0 [39] 0 [391 0 [38]

—

0

—

0

D. S. McKenzie, Polymers and scaling

67

of the end-to-end length distribution. No calculations leading to the gap exponent ~ have been made so that we do not discuss ~ further in this section. The numerical work can be broadly subdivided into those using exact enumeration methods and those using Monte Carlo methods. Let us first consider the exact enumeration results. (Application of the method and the information which can be obtained by it with reference to ferromagnetic models are reviewed by Domb [27].) Exact enumeration methods are based on the model of a polymer molecule as a self-avoiding walk on a regular crystal lattice. A regular lattice is one for which every point is equivalent to any other point. Embedding the walk on a regular lattice makes the enumeration problem much more tractable but in principle a random network would be just as satisfactory. The basis of the method is to calculate exactly the quantity of interest the total number of walks, end-to-end length or whatever for short walks up to a maximum number of steps which is determined by the practical difficulty involved in calculating the next term. The discreteness of the lattice means that for any number of steps the total number of configurations of the walk, per lattice site, is finite and an integer. For example, the total number of self-avoiding walks per lattice site on the simple quadratic (plane square) lattice with from zero to six steps are 1,4, 12, 36, 100, 284 and 780. At 24 steps the number of configurations is 46 146 397 316 [19]. In the exact enumeration method one must then analyse the numbers using extrapolation techniques to find the asymptotic behaviour when the number of steps is large. Various extrapolation techniques are available, and are reviewed by Gaunt and Guttman [281 , but how they are used depends to quite an extent on the ingenuity of the individual worker. A list of the exponents and the effective coordination number p found by the exact enumeration method for various lattices is shown in table 1. For the convenience of readers a list of the defining equations for the exponents is given in table 2. Because of the extrapolation, the exact enumeration method does not produce precise estimates for the exponents. Furthermore, the extrapolation procedure is to some extent subjective so that —

—

Table 2 Defining equations. The equations are those applicable to an n-step self-avoiding walk in the limit when n is large. The osmotic pressure is written in terms of the weight concentration c for a monodisperse system. Quantity

Equation

Total number of walks

c,, c0n12” 2p.” ~ u0n°

Total number of polygons Osmotic pressure

=

_~i~~_ =

=

Second virial coefficient

2

~‘

Rn)

n~

4R,,)

(3.2)

1 p~(r)

=

t exp[—(r/R,,)6]

(3.45)

—~(r/Rn)

R~= R,n°

Distribution of end-to-end lengths

(r

(3.1)

k~2

A

Distribution of end-to-end lengths

(r

~ Aj~c~’~

+

G~Tc M

(3.5) (3.6)

1 Pn~ ~

/ ~

r

g

(3.70)

68

D. S.

McKenzie, Polymers and scaling

the possible error in the preferred value of an exponent is not easy to determine. The values given in the table should be considered as mnemonics, the extrapolation results being close to these values. However, this caution does not imply that the results are completely arbitrary. First, one’s confidence in the reliability of the results is strengthened by the stability of the values as more data has become available over the years through improved numerical techniques, and second the internal consistency of the values when they are employed in different situations. To give some quantitative feeling for the probable errors we shall consider some particular cases. Watts [29] discusses the values of p and ‘y for the most-common lattices. The values he gives for p are close to those previously known, for example Sykes et al. [33] give p = 4.15 17 for the triangular lattice whereas Watts finds 4.1 520 and 4.1 516 depending on the extrapolation technique employed. Values of y are in the range 1 .162 to 1 .167 for the three-dimensional lattices and in the range 1.33 to 1 .34 for the two-dimensional lattices. It should be noted that in three dimensions the value of is accepted now as being 7/6 which is significantly different from the value 8/7 deduced earlier from more limited data by Fisher and Sykes [201. The values of v and e are of importance in testing the scaling relation (3.43). The values of e given by McKenzie and Domb [7]may be slightly greater than ~ perhaps as high as 0.27, but are unlikely to be less than Concerning the values of v McKenzie [38] and hoe [36] find v for the triangular lattice to be closer to 0.74 than ~, whereas Watts [34] using a different extrapolation technique finds v = 0.75 for two-dimensional lattices, but the same technique applied to the data for three-dimensional lattices lowers the value of v from ~ to. 0.59. McKenzie [38, 391 finds 6 about 3.84, that is, slightly less than 4, for the plane square and triangular lattices. Some general features can be immediately deduced from table 1. Most important, the values of the exponents appear to depend solely on the dimensionality of the lattice. Furthermore, the values of ‘y, a, c, 6 and t appear to decrease as dimensionality increases and to be bounded below by the corresponding values for the random walk. Thus as the dimensionality is increased the difference between the random walk and the self-avoiding walk, that is, the excluded volume effect, becomes less pronounced. Also of importance is the apparent agreement between the values of a and c. Finally, the obvious upper bound p (q 1) to p is satisfied in all cases, apart of course from the random walk. Let us use the data of table 1 to test first the scaling relation ‘~‘

~.

~

—

(4.1)

dv2—e.

It is easily seen that (4.1) is obeyed for two-dimensional lattices whereas for three-dimensional lattices we have dv = = 2 e. For the random walk e is undefined since there is no excluded volume. The breakdown of the scaling relation in three dimensions appears to be outside the error limits in e and v. The numerical data suggest that if anything c is greater than ~ which would imply, in three dimensions, that v <~ ( 0.583). This value is lower than any value it is reasonable to deduce from the numerical data. Furthermore, the interesting result of the numerical work that a and e appear to be equal, suggests that the possibility exists of a relation between the polygons and the second virial coefficient. The identification of a with is important in that it unities the tneory of polymer solutions by relating the single chain properties through a to the osmotic pressure through e. Thus one of the critical exponents may be redundant. The results summarised in table 1 can be used to check the scaling involving the shape parameters 6, t and g, for the end-to-end length distribution. These relations are —

69

D. S. McKenzie, Polymers and scaling Table 3 The exponents 6, t and g calculated from equations (4.2), (4.3) and (4.4).

t

= (1 —d/2

6

= l/(l

—

+dv

—

Lattice

6

t

q-tree Two-dimensional lattices Three-dimensional lattices Random walk

00

00

4 5/2 2

2/3 1/3 0

‘y)/(l

—

v)

v)

g N.A. 1/3 7/36 4 —d

(4.2) (4.3)

and g =

(~y+ 1

—

a — dv)/v.

(4.4)

Substituting for ‘y, v, a and d from table 1 we obtain the results which are summarised in table 3. Comparing with the observed values we can see that the relations (4.2) and (4.3) are confirmed in all cases whereas the relation (4.4) can only be true for the random walk in four dimensions. Table 3 rather over-dramatises the agreement with the data for the relations (4.2) and (4.3) and the disagreement in the case of the relation (4.4). In fact the data are difficult to analyse but McKenzie [38, 39] showed that consistent results were obtained with the relations (4.2) and (4.3) for r ~ R,, whereas the best values of g were significantly lower than the expected values 7/36 for all three-dimensional lattices or ~ for all two-dimensional lattices. The breakdown of relation (4.4) is significant since it was derived on the assumption of the strong scaling hypothesis that for all values of r, Pn (r) is given by p~(r)

=

~-~F(r/R,,)

(4.5)

where R~is the scaling length. McKenzie suggests that this assumption is incorrect and that, though the distribution function shows a long-range behaviour which obeys the scaling laws, for small r and large n, the end-to-end length distribution function is better fitted to the form Pnfr)

=

(a

+

b ln r)/c0n~ i

—a

(4.6)

where a and b are constants. This result cannot be reconciled with the strong scaling hypothesis. It implies that the short-range behaviour of the end-to-end length distribution is characterised by a second scaling length L, of the order of the lattice bond length, such that L = exp(—a/b).

(4.7)

One may suspect that the problems of the breakdown of the scaling relation (4.1), the indentification of a with e and the breakdown of the scaling relation involving g are interrelated. Although these results are reasonably clear in the three-dimensions, the apparent agreement of the strong scaling relation (4.1) in two-dimensions contradicts the result of the above analysis of the shortrange behaviour of the end-to-end length distribution in two dimensions. To summarise the results of the exact enumeration method, the polymer problem has been

70

D. S.

McKenzie, Polymers and scaling

studied on a wide variety of regular lattices. The results show that the values of the critical exponents are independent of the coordination number of the lattice, differ from the random walk values but vary according to the dimensionality of the lattice. Furthermore, the strong scaling hypothesis appears to be incorrect, at least in three dimensions. Let us now turn to the Monte Carlo work. In many ways Monte Carlo studies and exact enumeration work complement each other. In contrast to the exact enumeration method which computes every configuration of the polymer molecule, the Monte Carlo method enumerates merely a sample. By this means it is possible to study a greater variety of models. The exact enumeration method is, for practical purposes, confined to lattice models so that for a finite number of steps there is a finite and integral number of configurations. In contrast, the Monte Carlo method can be applied to situations for which the number of configurations is not countable. Moreover, the Monte Carlo calculations are carried to considerably longer walks than is possible by the exact enumeration method. The application of the Monte Carlo method to polymers was pioneered by Wall and his collaborators [40]. The major interest has been the behaviour of the mean-square end-to-end length and the mean-square radius of gyration. In the early papers lattice models were used exclusively. The main problem with the Monte Carlo work is to generate a sufficiently large sample of long walks. It was soon noticed that the number of successful self-avoiding walks remaining after n steps, decayed exponentially with n. Thus jfNn is the number of walks remaining at n steps and N0 the number of walks started, we have Nn —N0 e’~

(4.8)

where A is a parameter, the attrition constant, which depends on the lattice. The attrition constant can, for each situation, be related to the effective coordination number p, defined earlier. Agreement can be established between the Monte Carlo work and the exact enumeration work through the values found for p. For example, for the plane square lattice Wall and Erpenbeck [411 find A = 0.128 which corresponds to p = 2.64. This agrees well with the result of the exact enumeration work namely p = 2.6385 (table 1). In the early work the generation of walks longer than 100 steps was rarely attempted. For the diamond and simple cubic lattices [42] and for the four-choice simple cubic lattice, the body-centred cubic lattice and the face-centred cubic lattice [43] 2v was found equal to 1.22. For the 6-choice four-dimensional hypercubic lattice Wall et al. [43] find 2v = I .11. For the plane square, triangular, honeycomb and two-choice plane square lattices Wall et a!. [44] find 1 .42 ~ 2v ~ 1 .49. Thus the agreement with the exact enumeration values of v = in three dimensions and v = ~ in two dimensions is not unsatisfactory. Later Monte Carlo work has been based on efforts to overcome the attrition problem. That is, to devise techniques so that significant samples of much longer walks, up to several thousand steps, may be generated. Techniques are described by Wall et al. [41], Gans [45] and Alexandrowicz [46]. In general, authors have not repeated the earlier lattice calculations using these methods but have studied more complex problems. However, Wall and Erpenbeck [41] report 2v = 1.18 for the diamond lattice and 2z.’ = 1.50 for the plane square lattice. Two problems that have been fairly extensively studied are first the effect on v of including nearest-neighbour interactions and second the behaviour of self-avoiding walks which are not constrained to a lattice. The first problem has been studied by Mark and Windwer [49], Mazur [50], Mazur and McCrackin [51], and Wall eta!. [52, 531. The results suggest that when the polymer— polymer interactions are sufficiently strong and attractive, in three dimensions, 2v is less than unity

D. S. McKenzie, Polymers and scaling

71

and that 2v increases as the interaction is made increasingly repulsive so that when occupation of the nearest-neighbour site is excluded 2v 1 .3. These conclusions are supported by the exact enumeration studies made by Fisher and Hiley [541, who had very limited data, and Kumbar and Windwer [551.The values of p thus range from below 4, the value expected for a random walk, to greater than ~ the accepted value for self-avoiding walks as the strength of the interaction increases from negative values. The value v = 4 occurs for a particular value of the interaction parameter x (cf. (2.13) and (2.6)) and thence one may define the 0 temperature of the system. The variation of v with interaction strength has been criticised by Mazur and McCrackin [51] and by Domb [81, who suggest that for infinitely long walks v remains equal to its self-avoiding walk value until the 0 point is reached at which point v becomes equal to 4. This argument is based on the ‘smoothness’ hypothesis propounded by Griffiths [56]. Confirmation by numerical studies requires considerable data collection and the results would probably always be rather unconvincing. To continue with studies of polymers with nearest-neighbour interactions both Domb [81 and Rapaport [57] argue, on the basis of exact enumeration work, that the definition of the 0 point given above is not unique. Considering the collapse of a single chain, many definitions of the 0 point are possible, for example, the value of x for which p = q or the value of x for which the rth moment of the end-to-end length distribution function behaves as ~r/2 The numerical data suggest that each definition gives a different value of x and thus Domb and Rapaport are led to postulate the existence of a 0 region rather than a 0 point. This problem should not be confused with the definition of the 0 point as the point of critical mixing of a polymer solution which is a real second order phase transition. The behaviour of a single chain near the point of critical mixing may well not be closely connected to the behaviour of the solution as a whole. The second problem, namely the behaviour of self-avoiding walks which are not constrained to a lattice is more directly relevant to the subject of this paper. The models provide a link between the lattice models and models studied by analytical methods. The details of the models studied are: freely-linked chains with variable excluded volume (Fleming [58, 59]); chains with tetrahedral bond angles but random and variable deviations of the position of each third atom in any triplet from the position occupied in the diamond lattice (Loftus and Gans [60]); and successive atoms occupy positions according to a tetrahedral lattice or a four-choice simple cubic lattice respectively (Windwer [61]). Windwer finds that 2v = 1.29 for the mean square end-to-end length and 2v = 1.32 for the radius of gyration. Fleming [58] finds that 2v varies from 1.015 (±0.025)for zero excluded volume to 1 .161 (±0.017)when the range of the repulsive potential equals half the bond length. In the latter case Fleming [59] finds 2v = 1.18 on pushing his computations from 20 to 100 step walks. Finally Loftus and Gans find that v is very sensitive to deviations of the walk from perfect lattice regularity. Even a 10 deviation is sufficient to change 2v from 1.200 to 1.2297 (±0.333).The value of v increases less rapidly as the size of the allowable deviation is further increased. The maximum value, 2v = 1.2450 (±0.022),is observed in the case of completely free rotation. Similar behaviour is observed for the mean-square radius of gyration except that the values of v are slightly larger. It is plain that the results of these investigations are not compatible with each other. A clue to the discrepancies may lie in the fact that all authors report an increase in the attrition constant. For example, for Windwer’s model A ~ 0.4 compared with A ~ 0.04 for the tetrahedral lattice. Hence it is much more difficult to generate large samples and the authors may have underestimated the spread in their results. One suspects, with recourse to the smoothness hypothesis, that v would remain unchanged between on-lattice and off-lattice walks. The available Monte Carlo work does not rule out this hypothesis.

72

D. S. McKenzie, Polymers and scaling

On studies of the mean-square radius of gyration K s~) it is agreed both by Monte Carlo studies and exact enumeration work that (4.9) Ks~)cn2v, for large n, where the values of the exponent v, to all intents and purposes, equal the values of v for the end-to-end length deduced by the same author using the same model. More significantly, the ratio (s~ )/KR~) is found to be about 0.155 in three dimensions and about 0.14 in two dimensions (Wall and Erpenbeck [62] , Windwer [611, Hioe [361, Domb and Hioe [63]). This ratio may be predicted using the relation between the mean-square radius of gyration and the segment—segment distribution functions introduced in (1.3), namely, Ks,~

,~

ln—1 )=—~

~ 4irfg~(r)r~idr.

~

n j=1

(4.10)

j=i+i

If it is assumed that, as in (2.56), the g,(7 ~(r) depend on I i 4ir Jg~7)(r)r4 drAI i~i

2u

—

I I and r so that (4.11)

where A is a constant, then (4.1 0) may be written as A

where m = I I K s~) =

n--i

—

(4.12) j I. For large n this sum may be approximated by an integral whence we obtain

An2 ~‘/(2v+ 1 )(2v + 2).

(4.13)

But, in the above notation K R~) = An2 L’ so that the ratio K s~ )/K R~) isgiven by 1 / f(2v + 1 )(2v + 2) Substituting v = and v = one predicts that the ratio should be 0.114 in two dimensions and 0.142 in three dimensions. Thus the results of the numerical work differ in both two and three dimensions from the predicted values and also the value ~ expected for the Gaussian model. The reason for this discrepancy is not clear. There have been a number of Monte Carlo calculations of the distribution of end-to-end lengths. Of these, that of Mazur and McCrackin [51] is by far the most thorough though contributions have been made by Mazur [64, 651, Schatzki [66], Alexandrovicz and Accad [67], Verdier and Stockmayer [681 and Wall eta!. [52]. Apart from Mazur and McCrackin none of these studies is particularly helpful in determining the exponents t, g and 6 either because of the crudeness of the data or because the authors have interpreted their data in terms of analytical theories incompatible with the approach followed here. However, all show that the distribution of end-to-end lengths for selfavoiding walks is significantly non-Gaussian. Mazur and McCrackin find for self-avoiding walks on the face-centred cubic and simple cubic lattices that 6 = 2.85. This disagrees with the value 2.5 found by McKenzie for the face-centred cubic lattice and by Domb et a!. for the simple cubic lattice. On the other hand, Mazur and McCrackin have assumed g = 0, an assumption which is likely to increase the estimated value of 6. Mazur and McCrackin take account of nearest-neighbour interactions. They show that for increasingly attractive interactions 6 decreases through the value 2 expected for a Gaussian distribution. Finally, Bellemans and Janssens [69] have used the Monte Carlo method to calculate the second }.

j

D. S. McKenzie, Polymers and scaling

73

virial coefficient for self-avoiding walks on the simple cubic lattice. They estimate e = 0.28, in fair agreement with the exact enumeration result. Recent Monte Carlo work on ring closure and tadpoles has been done by Whittington et al. [1321, on off-lattice models by Smith and Fleming [133] and on the end-to-end length distribution by Wall and White [134]. Rapaport [135] has recently studied the segment—segment distribution functions using the method of exact enumeration. The method has also been used by Torrie and Whittington [136] to calculate values for v for the diamond and body-centred cubic lattices.

5. Analytic theories In this section we shall consider those theories of polymers which are analytical and which are of relevance to scaling theories. The theories have a common background in that they treat the polymer problem as being analogous to the Brownian motion of a particle. Thus the path of the particle is analogous to a chain configuration and the time variable is analogous to the length of the chain. Moreover, the excluded volume effect is introduced by means of an external repulsive potential under whose influence the Brownian particle moves. Differences arise in the manner in which the problem is treated mathematically. Three main groups of theories can be discerned: straightforward perturbation expansions as first proposed by Zimm [70] ; the self-consistent field theories pioneered by Edwards [71,72]; and theories based on the quantum field theory techniques associated with Wilson [73]. The subject matter of this section is closer to the main-stream of research in polymers and consequently has been more extensively reviewed. In particular, the review of Yamakawa [74] discusses much of the material of the following section and has an extensive bibliography. However, the stand-point of this paper differs from that of Yamakawa in that we shall concentrate more on the fundamental aspects which connect the various theories. We shall be quite brief since in some ways the analytical theories, particularly the perturbation theories, do not bear directly on the application of the scaling laws to polymer solutions. Moreover, the renormahisation group approach, in which scaling is explicitly assumed and to which we shall give some space, is in a period of rapid development. Let us first derive the basic equations which are the starting point of the analytical theories. We use here the notation of Freed [751. Suppose the probability, that segment / 1 of a polymer chain and segment / are separated by a distance r1 — r~ and that a contour length along the chain L~s~ is —

associated with segment j, is p(r1 — r1 ; i.~s1).Then, in the absence of excluded volume, the probability that, for a chain of n segments and of contour length L, segment r0 is at the origin and segment rn at R is given by P(R,L)

J6(ro)t5(R _rn)[H~(rj_ri_i;~sj)] (fEd?~)

(5.1)

with L = ~ ~s1. It is now normal to make the assumption that p(rj

—

r~1 ;

~s1) = (2~

)3/2

exp

[

2l~s1~ — r1

1 )2]

where / is a characteristic intersegment length. This assumption replaces the stiff bonds with

(5.2)

74

D. S.

McKenzie, Polymers and scaling

Gaussian equivalent bonds. By substitution and elementary integration it can be shown that 21 / 3 ~ 3/2 r 3R P(R, L) = ~~—) exp L (5.3) 2irL/ 2L1 so that with the Gaussian equivalent bonds we recover the well-known Gaussian distribution for a random walk. The excluded volume effect is introduced by a potential energy of interaction w 11 between the segments so that the total potential energy is —

W

=

2!

W11 ~

—J

(5.4)

J~S1.

i~j

If we define 2

(5.5)

V11 = (1 — 6,~)w11/kTl

where 6q is the Kronecker delta, then P(R, L) becomes P(R,L)

f6(ro)6(R —r,,) ~

(

2ls~~

312~~[ 2~~1)

x exp

11

un

— ~ V,

1

[j=0

.

1 i~s,z~S1] H di~]

2 ~,j

(5.6)

This equation is now transformed to a path integral by taking the limits n -~ such that ~ ~s1 = L. Thus we obtain 2~

j=1 ~s ‘\(~‘—r1_1~ 1~&S 1 /

and max i~s1 0 -+

2 ds 1L

(5.7)

(dr(s)/ds)

0

and V~~

~SJ

~

I

ds f

ds’ V[r(s)

—

(5.8)

r(s’)].

Whence (5.6) becomes 3

r(L)R

P(R, L) =A 5,’ r(0)0

~[r(s)] exp

—

2!

1

L

P(s)ds

f 0

—

-

2

L

ds

f 0

5

L

ds’ V[r(s)

—

r(s’)])

(5.9)

0

where A is a normalisation constant, t(s) represents differentiation of r with respect to s and the functional integration is performed over all paths r(s) with the given restrictions on the end points. The most direct way of handling this equation is by a perturbation expansion. That is, one expands in series form the exponential involving V in (5.9). The rth term in the resultant series for P(R, L) then takes account of r interactions between points on the chain. The terms in the expansion may be interpreted quantum mechanically, in which case the diagramatic expansion is in terms of Feynman diagrams, or classically as shown below. The terms may be written diagramatically as

D. S. McKenzie, Polymers and scaling

75

follows ~P(R,L)=K0(R,L)

s~s’

---

+

+

~

+

+

~

}

+

(5.10)

L

0

3/2 exp(—3R2 /2L) is the unperturbed distribution function, and the first where K0(R, L)be= (3/2irL) order term can written as 1L5

55 K0 (r(s), s) K0 (r(s)

—

r(s’), s

—

s’)

K0 (r(s’)

—

R, L

—

s’)

V[r(s)

—

r(s’)] dr(s) dr(s’) ds’ ds.

That is, each dashed line corresponds to a term V[r(s) r(s’)] and each solid line corresponds to a random walk, represented by K0, between the appropriate limits. As formulated above, the integrals overs s and s’ diverge. The singular part may be removed by changing the limits to f~—b f~÷b ds’ ds where b is assumed small. It is interesting that, on normalisation, the term linear in L vanishes along with the singular part of the integral. The classical approach has a long history, being first formulated in terms of Mayer theory by Zimm in 1946. The subsequent development, although very important in the theory of polymers is not directly applicable to subject of this paper. The field has been reviewed recently by Yamakawa [76].Here we give merely a critique of the method. A fundamental difficulty with the perturbation method is that the ground state from which the system is perturbed is the random walk for which interactions are completely neglected. But this state is physically unrealisable. Even the argument that at the 0 point attractive forces between the segments of the molecule counterbalance the repulsion due to the excluded volume effect in such a way that intersegmental distances are distributed Gaussianly, an argument which has not been justified mathematically and which disagrees with the numerical work of Domb [8] and Rapaport [571, cannot justify the assumption that the molecule there behaves as a random coil in the sense that interactions are negligible. On the contrary, interactions are required to offset the excluded volume effect. Furthermore, one expects that, foi the most probable state of the system, the number of interactions to be proportional to the number of segments in the chain and thus for long chains to be correspondingly large. Hence, the perturbation method which assumes that the standard state of the system is that in which no interactions, including the excluded volume effect, occur appears on physical grounds to be unpromising. Rather, if one considers formally a perturbation in the strength of each interaction, as opposed to the number of interactions, then as soon as the interaction strength deviates from zero, one would expect the full effect to become apparent. This idea has been explored by Domb and Joyce [77] who refer to the smoothness hypothesis of Griffiths. Moreover, Domb and Joyce point that the the coefficients variable z which occursonly in the expansion is 112outwhile are correct to perturbation O(L~12).These conditions proportional to (1 0/T)L are contradictory since the coefficients are true constants only in the limit L —i’ oo which implies that the region of applicability of the series about T = 0 is neghible if the series has a finite radius of convergence. Recently Edwards [137]has shown that the series (5.10) is asymptotic, that is, its radius of convergence is zero. To this criticism one may add that three terms only of the series are known and at best the theory is applicable only to a small region close to the 0 point. In this region the solution is close to the point of critical mixing and its properties are varying rapidly. The second method of handling (5.9) has been the self-consistent field method pioneered by Edwards. De Gennes [78]has given a short review of this approach. Further details can be found in the review of Wiegel [79].Edward’s model of a polymer molecule is of a random walk starting at —

—

76

D. S.

McKenzie, Polymers and scaling

the origin and moving in a spherically symmetric repulsive field which takes account of the excluded volume. Given a field the average configuration of the molecule is calculated. The field is then determined by a consistency condition dependent on the average configuration. There are a number of variations of this basic approach, for example, one can take both ends of the walk as being fixed points, in which case the field should have ellipsoidal symmetry. Apart from destroying the isotropy of space, the self-consistent field is introduced in all cases in a way such that the essentially nonMarkovian behaviour of (5.9) is replaced by a Markovian model. Thus if P(R, L) and V[r(s) r(s’)J are replaced by FscF(R, L) and VSCF [r(s)1 respectively, we obtain —

PSCF(R,L)

r(L)R

Af

r(0)0

~[r(s)] exp

3

~ ~

—

—2

f L0

5L

[~(s)]2ds —

VSCF [r(s)] ds

0

It is well-known that a functional integral of this form can be expressed in differential form (Gel’fand and Yaglom [751)so that

[~/~ Hence

—*

1v2

+

VSCF(r)] GSCF(r,s)6(r)ä(s—L).

(5.11)

is the Green’s function for a diffusion process in a spherically symmetric field Vs C F~Gs C F is normalised so that lim 5~0 G~C F (r, s) = 6(r). Equation (5.11) is a Fokker—Plank equation for the diffusion of a particle in an external field and has been derived before in the context of the theory of polymers [80,81]. To complete the self-consistent field argument, the selfconsistency condition can be taken as GSCF(r, s)

VSCF(r) = fVscF(r

—

r’) p(r’) dr’

(5.12)

where p(r’), the density of walks at r’, can be found from GSCF. However, it has been pointed out by Freed [75]and Whittington [82]that one may define a hierarchy of two-point, three-point and so on, distribution functions so that the n-point distribution function can be related to the (n 1)-point function by an integral equation in the same function space as (5.9). The existence of this hierarchy is strictly analogous to the existence of the KirkwoodBorn—Green hierarchy ‘of integro-differential equations among the distribution functions describing the behaviour of fluids. This approach was first applied to polymers by Naghizadeh [83]and has subsequently been related to the self-consistent field approach by Kyselka [84]. The self-consistent field approximation may be introduced at any level in the hierarchy so that one obtains an approximation to the n-point distribution function. Freed has shown that Edwards introduces the approximation at the lowest level that is for the two-point function whereas Reiss [85]introduces the approximation at the level of the three-point 413 function. Edwards has shown that, in his approximation, the self-consistent field behaves as r whereas Reiss has obtained r’. These results imply that, in three dimensions, the mean square length of the chain behaves as L615 according to Edwards and L413 according to Reiss, that is v = and v = respectively. Edward’s result has been confirmed by Gillis and Freed [86]who also show that for large r the field satisfies the self-consistency condition. Furthermore, Gillis and Freed report that in d dimensions the field behaves as r —2 (d 1) / ~, which implies v = 3/(d + 2), when d is less than or equal to 4. In higher dimensions, the self-consistent field decays faster than r2. The random walk terms then dominate the solution so that v = 4 for all d> 4. The simple relation v = 3/(d + 2) between v and the dimensionality d was first proposed by —

-

4

—

D. S. McKenzie, Polymers and scaling

77

Fisher [25]based on the early work of Flory [6]and the exact enumeration work of Domb [351. Flory’s argument is essentially a mean field theory though it is not thus formulated explicitly. Fisher [871has recast Flory’s argument in a form which emphasises the dependence on dimensionality. We reproduce Fisher’s exposition here because his formulation is relatively simple, the results are interesting and because the original is published in a rather out-of-way place. Briefly, Fisher writes the partition function Z(h) for a chain of n segments for which the end-to-end length has fixed modulus h, as (5.13) Z(h) = Cdh”~exp(—dh2/2N)Kexp(—I3UN))h where Cd is a constant; ~3represents 1/kT; the term h”~ exp(—h2/2N) is the Gaussian term which takes account of the connectivity of the chain; and the term involving UN, the total pair interaction between segments, is averaged over all configurations such that I rN r 0 I = h. The free energy of the chain on this model can be written as —

F=i3hiZ(h)=U0+U1

(5.14)

where U0 takes account of the free energy due to the diffusive part of the problem and U1 accounts for the interactive term. The main assumption of the Flory theory is to let U1 =4B(T)N(N/Adh”),

(5.15)

that is, the free energy due to interactions is proportional to the effective strength of each interaction B(T) multiplied by the number of segments, times the probability (N/Adh”) that a segment will interact with another. This probability is given basically by a mean field approximation, being the number of segments N divided by the effective volume Adh” of the chain. The procedure is now to maximise F with respect to h. One easily obtains 2 d—1 1 B N2 hN d (5.16) Now in any dimension h must increase at least as fast as N 1/2 the random walk result, so that for d>4, the constant term on the right-hand side of (5.16) dominates and we obtain hccN’/2

(N-+oo,d>4).

(5.17)

For c/ < 4 it is assumed that the interaction term dominates so that hcxN3/(’~2)

(N—*oo,d<4).

(5.18)

(For a repulsive interaction B is negative.) The case d = 4 is a borderline. It should be noted that the relation v = 3/(d + 2) is supported by the numerical work in one, two and three dimensions (table 1). The above derivation has been criticised by des Cloiseaux [88,89] who argues that the leading order terms in both U 2 /N and N2 /h” respec0 and U1 are should be proportional to N rathershould than hbe considered as correctively. Both these terms which proportional toM4 —d)/(d+2), tions to the leading order terms whose contribution has been neglected. Before leaving the Flory treatment it should be noticed that, defining a parameter a~-to take account of the swelling of the chain due to the excluded volume by h2

a~h~

(5.19)

78

D. S. McKenzie, Polymers and scaling

withh~defined from (5.16) by putting B = 0 whence h~= (d form — ~ = CM4 ~)/2,

—

l)N/d,

we may recast (5.16) in the (5.20)

where C is a constant. With d = 3, (5.20) reduces to the well-known result first derived by Flory [61.Domb [130,138], by combining the Flory treatment with the results of exact enumeration work, has shown the superiority of (5.20) over its many rivals (see [74]). Domb further suggests that the variable z (1 ®/T)N”2 is sufficient to characterise all regimes of polymer solutions, an idea which agrees with the work of Berry [481. The above results suggest that in four dimensions the excluded volume problem becomes trivial since one recovers the critical exponents for a random walk, particularly v = 4. This recalls the studies of Brownian motion in d-dimensional Euclidean space by Dvoretzky, Erdos, and Kakutani [901who have shown, that in four or higher dimensions almost all paths of a particle undergoing Brownian motion have no intersections, whereas in three or fewer dimensions almost all paths intersect. Thus, in four or more dimensions there is no essential difference between a random walk and a self-avoiding walk since the probability of an intersection is vanishingly small. This interesting result was first noticed to be applicable to the polymer problem by Rubin [911.Hence, for the polymer problem four dimensional space represents the point at which solutions of the problem exhibit a qualitative change, the problem being non-trivia! in less than four dimensions and trivial in four or more dimensions. The third method of tackling equation (5.9) uses the theory of the renormalisation group which has been adapted to the study of critical phenomena by Wilson. The method has been extensively reviewed by Wilson and Kogut [73]and Fisher [140]. The method is based on a generahisation of the derivation of scaling by Kadanoff [1].Kadanoff’s argument is that near the critical point, correlations in a magnetic or fluid system become very long ranged. Thus, in any small finite region of a magnetic system the spins are highly correlated. This region will have a mean magnetic moment. By dividing the system into cells of linear size characterised by 1 each containing several spins, the problem is transformed to a similar problem in which the cells interact with one another though with a new Hamiltonian. Kadanoff assumes that this new Hamiltonian has a similar form to the old except that the magnetic field and the temperature are re-scaled by some power of 1. This procedure relates the change in the linear dimensions of the system to the thermodynamic variables and thence one can derive scaling relations between the critical exponents. In particular one obtains the relation dv = 2 a characteristic of strong scaling. Kadanoff’s argument is over-simplified because he assumes that the resultant magnetic moment of the cells have only two states as in the original Ising model. This objection has been removed by Wilson by studying~theHamiltonian in momentum space. That is, the spins are replaced by Fourier series formed over the first Brillouin zone of the lattice. One makes the assumption that near the critical point the short-range properties of the system are slowly varying so that one can transform to a new Hamiltonian by integrating over the high spatial frequencies. Furthermore, this transformation can be iterated so that one obtains a group of transformations — the renormalisation group. As far as scaling is concerned the renormahisation group approach reproduces the scaling relations of Kadanoff. It should be noticed that strong scaling is built into the method. It has been shown by Coniglio [139]that the existence of a renormalisation group implies strong scaling and vice versa. The iterative re-scaling removes information about the short-range structure of the lattice so that the discreteness of the lattice is lost and the model becomes a continuum model. It is con~

—

—

D. S. McKenzie, Polymers and scaling

79

venient to introduce the continuum in the original Hamiltonian by writing H —

=

kT

1

n

1~

5 dx 4[~vs1(x)j

1

+ 5ro ~ s~(x) +

,

121

uo[Es~(x)j J.

(5.21)

j=1

Here s~(x)is a spin field with n components (1 ~ I ~ n) and r0 and u0 are constants. One obtains with n = 1 the Ising model, with n = 2 the planar Heisenberg model, with n = 3 the usual Heisenberg model and as n -~ cc one recovers the spherical model. With u0 = 0 one obtains the Gaussian model of Berlin and Kac [141]. The properties of this model are well-known and are similar to those derived by mean field theory, namely ‘y = 1, i~= 0 and v = 4 in any dimension. Furthermore, scaling is obeyed and a = 2 d/2 for d ~ 4. If one assumes the Gaussian model to be a trivial case, then the addition of the term involving u0 to the Hamiltonian makes the problem non-trivial, and more interesting. It should be noted that the extra term involves fourth powers of the fields s,(x). The renormahisation group procedure applied to the above Hamiltonian renormahises the coupling constants r0 and u0 to r1 and u1 respectively so that r1~ and u1÷1 are defined recursively in terms of r1 and u,. The set of all such transformations forms a group. From the fixed point theorem this group has at least one fixed point. In fact, Wilson’s group has two fixed points, one of which is trivial in that it gives Gaussian behaviour and the other is non-trivial. The fixed points correspond to the critical points. In the Wilson theory four dimensions has special significance. For d < 4, iteration of the recursion relations converges on to the non-trivial (Ising) fixed point whereas for d> 4, the Gaussian fixed point is picked out. The case d = 4 requires special treatment. It is not clear that the special significance of four dimensions is a necessary consequence of the theory or whether it is accidental in that the result depends on the details of the Hamiltonian, in particular, the introduction of the term quartic in the spin fields. However this result is not unsurprising because of the results mentioned above on Brownian motion in d-dimensional Euclidean space, and the fact that the known properties of the spherical model show similar behaviour. The critical exponents can be calculated by studying the behaviour of the renormalisation group close to the Ising fixed point. One assumes that near the fixed point the coupling constant u1 is of the order 1C where c = 4 d. When e is small we may write 1c ~ 1 + e ln 1 and expand the partition function as a pertubation series in c. Results for the critical exponents can then be found, also as a series in e, using the homogeneity of the free energy which is implied by the scaling properties of the method. The renormalisation group method has been applied to the polymer problem by des Cloiseaux [126], de Gennes [93,131] and Burch and Moore [138].De Gennes [93]was first to notice that one could, formally, write down the c-expansion for a system with zero degrees of freedom and that the diagrams which then occurred in the expansion contained no closed loops, merely chains. De Gennes was therefore led to identify the Wilson theory with zero degrees of freedom as equivalent to the polymer problem. The details of this identification are best followed by using the notation of des Cloiseaux. Des Cloiseaux generalises (5.21) by introducing a magnetic field H and by replacing u0 by a more general pair-potential V(x — y) so that the Hamiltonian becomes —

—

= fdx4’2

+

E s~(x)

—

~

H1s~(x)}+ ~ fdx fdy ~ [s~(x)] [s7(x)] V(x —y). (5.22)

In momentum space defined by s1(k)

=

fcix exp(ik

.

x)s~(x) and

c’(q)

=

1

(2ir)

~ jdx exp(ik x) V(x) .

80

D. S. McKenzie, Polymers and scaling

we can expand the partition function as a series of Feynman diagrams, examples of which are shown below

for which a factor (k2 + m~)‘ is associated with each part of a solid line, a factor V(q) is associated with each dashed line, a factor of the form 6(k + k’ + q) is associated with each vertex and each solid line corresponds to a component / of the field. Each diagram is integrated over the external momenta and one sums the contribution over all values of/. However, because of the summation over the components of the field, a closed loop in a diagram contributes a factor proportional to n. By setting n = 0 all diagrams with loops vanish and the remaining diagrams contain only chains. Des Cloiseaux’s object is to investigate the scaling properties of the osmotic pressure. This is the reason for introducing the magnetic field which is analogous to the activity of the polymer species. The number density of the polymer molecules, p, is given by Ha p

~-j~~ ln Z(H)

=

where the grand partition function Z(H) is given by Z(H)

r

1 = 1 +

~

M=12

~

M.

[H2l~2)12(2ir)°’’2

]2MJ___ 1

. . JIdLM __~_et(~~/l)Z(Li . . . LM) /

(5.23)

and 1

M

Z(L 1

. . . LM)

=

f H ~ [rm (pm)] exp mi 12

2m,m’ ~

Lm

Lm’

5 0 dpm 5 0

2 1Lm

—

— j m41

rarm(12m)1

dpm I

L

0

(5.24) is the generalisation to a many component system of (5.9). Des Cloiseaux makes the connection between (5.23) and the partition function using the Hamiltonian (5.22)by ordering the contributions to the diagrams in the perturbation expansion of the partition function so that each solid line is identified with a polymer molecule. This is not equivalent to identifying each molecule with a particular component of the spin fields. The number of components n is put to zero as before to remove closed loops from the diagrams. Burch and Moore in their treatment of this problem define the Hamiltonian in terms of an m x n component field and subsequently let n -~ 0. The important point of this work is that the renormalisation group approach is not applied directly to the polymer problem represented by (5.23) or (5.9) but the polymer problem is first shown to be formally equivalent to the magnetic problem defined by the Hamiltonian (5.22), or its equivalent, by putting n = 0 in the perturbation expansion. The relation between the Ising model and the polymer problem is thus similar to that reviewed in section 2, in that for the Ising model new diagrams must be taken into account which do not contribute in the polymer case. In three dimensions de Gennes obtains the estimates ‘y = I .176 and i~= 0.03 2 which imply, using —

dpm’ V[rm (prn)

ap,,,

—

rm’(pm’)]

D. S. McKenzie, Polymers and scaling

81

the relation y = (2 i~)v,v = 0.598. These results are close to the results of numerical work, particularly so for v. Des Cloiseaux finds that at low concentrations the osmotic pressure defined from (5.23) by —

ill = — ln Z(H) kT V —

has the scaling form H/kTp —F(l”M’~p)

where M is the number average molecular weight. This scaling form is identical with (3.20) if one puts dv = 2 , which is the strong scaling relation (3.43). Thus des Cloiseaux’s treatment produces the strong scaling form for the osmotic pressure. This result is not unexpected if one remembers that strong scaling is an integral part of the renormalisation group technique. Des Cloiseaux also produces the estimate 0.362 for the ratio of the third virial coefficient to the square of the second virial coefficient. Furthermore, des Cloiseaux predicts that, at high concentrations, H/kTc c1 /(dv— 1) —

where c is the total weight concentration of polymer. An important feature of the renormahisation group method is its ability to cope with tricritical behaviour. If the Hamiltonian is made more complex by the introduction of extra field variables, the renormalisation group becomes more complex and in particular, one obtains more fixed points for the transformation. Variation about one of the fixed points will define a domain within which iteration of the transformation will converge on that fixed point, whereas outside the domain one of the other fixed points dominates. The 0 point has been identified as a tricritical point by de Gennes [13 1] who has studied the collapse of a single polymer molecule while Burch and Moore have studied the behaviour of polymer solutions close to the 0 point. The Wilson estimates for critical exponents are not exact since the series for y and i~are asymptotic. However the measure of agreement with numerical work is gratifying. A further satisfactory feature of the method is the choice, in the polymer problem, of the random walk in four dimensions as the ground state from which to perturb the system. This appears to be much more sensible than the choice of the random walk in three dimensions as ground state which is the approach of the earlier studies of the problem using series expansions. In three dimensions the random walk corresponds to no physically realisable state of the system whereas in four dimensions the random walk is the expected behaviour. One suspects that the self-consistent field theory and the Wilson method should be more closely related than is apparent from the preceding discussion since both are field theory methods and both pick out four dimensions as a special transitional case. It has not been possible to ascertain the precise nature of this relation since the details of the self-consistent field theory of Gillis and Freed [86] have not yet been published. It is interesting that while the self-consistent field method, according to Gillis and Freed, gives the explicit relation v = 3/(d + 2) the Wilson approach gives only approximate values for the critical exponents. Before leaving this section, it should be remembered that the self-consistent field theories Sand the Wilson theory are derived solely in terms of d-dimensional Euclidean space. Furthermore, the discreteness of the individual segments of the chain is removed and replaced by a smeared-out, continuous repulsive potential. Among other things this obviously introduces singularities into the problem

82

D. S. McKenzie, Polymers and scaling

and means, in the Wilson theory, the use of renormalisation techniques as in quantum field theory. Hubbard [95] has related the partition function of the Ising model on a lattice to the partition function in functional form in terms of suitable fields which is the starting point for the Wilson theory, and has deduced that the critical indices are independent of lattice, spin and interactions provided the latter are short-range and ferromagnetic. Though it is not easy to criticise Hubbard because his letter is rather sketchy, in our opinion a one-to-one correspondence between the lattice, or discrete models, and the continuum models has not been satisfactorily established. In support of this assertion, one notes that the total number of walks, the total number of polygons and the endto-end length distribution, though not the moments of the distribution, for the lattice models depend solely on those properties of the lattice which are deductible from the definition of the lattice as a graph and are in no way connected with how, or even if, the lattice is embedded in Eucidean space. In particular, none of the metric properties of the Euchidean space are used. This point is rather hard to put across since the regularity and symmetry of the lattices as normally illustrated are seductive. The problems of the lattice models are therefore purely algebraic and not analytical. Even the mean-square end-to-end length, where obviously some metric properties are introduced, may not be an exception to the above argument. Martin [961 suggests that the function ~rPnfr) ln p~(r),which is independent of the metric, behaves as ln n’~ so that v can be defined in a manner independent of the space in which the lattice is embedded. There are some further mathematical results which suggest that discrete and continuum theories are not interchangeable. Of note are the studies of Montroll and Weiss [97] and Erdös and Taylor [98] on the number of intersections of a random walk on the d-dimensional hypercubical lattices. It has been shown that the probability of an intersection is finite for any dimensional lattice. In one and two dimensions, the probability of an intersection at any particular lattice site is unity, a result first proved by Polya [99] and the probability, though still finite, falls steadily as the dimension is increased from three upwards. As Erdös and Taylor point out, this result is in complete contrast to the continuum results where the probability of an intersection is effectively zero in four or higher dimensions. For lattice systems the difference in behaviour comes in moving from two to three dimensions whereas for continuum theories the difference occurs in moving from three to four dimensions. The result of Polya has been generalised to locally finite infinite graphs by NashWilliams [100]. Finally, Fisher and Gaunt [30], using numerical data, find a smooth decrease in ‘y and a towards their random walk values for the 4, 5 and 6 dimensional hypercubical lattices. Though this result supports the hypothesis that for the discrete models the random walk behaviour is not expected in four dimensions, their results can be criticised on the grounds that their numerical data was limited and that with longer series the results may come closer to random walk behaviour.

6. Experimental work As far as we are aware, no experiments on polymers have been performed with the expressed purpose of obtaining estimates of critical exponents. Of the critical exponents introduced in this review only two are readily accessible experimentally. These are the molecular weight dependence of the mean-square radius of gyration which behaves as (6.1) Ks,~)ccM2”

D.

S. McKenzie, Polymers and scaling

83

and the molecular weight dependence of the second osmotic virial coefficient which behaves as A2 xM~.

(6.2)

The exponents y and a which describe the behaviour of a single molecule are not in principle measurable by thermodynamic means because, as explained in the introduction, the thermodynamics of polymer solutions are measured with respect to the standard state of infinite dilution, that is, a solution so dilute that the polymer molecules do not interact. The properties of single polymer chains in solution may be accessible experimentally by labelling a molecule in some fashion and following its behaviour by a suitable scattering experiment. Neutron scattering experiments along these lines are being performed at Strasbourg. The ‘gap’ exponent ~ could be obtained by measuring the third virial coefficient but this probably requires greater accuracy in the experiments than is presently available. The relevant experimental techniques are measurements of osmotic pressure and light-scattering. Both sets of data are necessary since from the osmotic pressure one obtains the number-average molecular weight M~whereas light-scattering gives the weight-average molecular weightM~.The ratio c1 = M/M~gives an indication of the degree of the polydisperity of the sample. However lightscattering is the more powerful technique since from the same series of measurements one obtains the weight-average molecular weight, the mean-square radius of gyration and, though this has not been mentioned previously, the second virial coefficient. The thermodynamic and hydrodynamic behaviour of dilute polymer solutions have been reviewed by Berry and Casassa [1011 who give numerous references to experimental work. Unfortunately, most experimental work has been interpreted in terms of those theories of polymer solutions which rely on series expansions away from the 0 point at which point the molecules are assumed to obey Gaussian statistics. Consequently most experimenters work with systems close to the 0 point rather than in the athermal solutions (solutions with ‘good’ solvents) which are more relevant to determining v and c. However, numerous values for i’ and c can be found in the literature almost invariably being quoted as a by-product of some other investigation. The work described in references [102—121] has been examined in some detail in that we have replotted the data points so as to obtain some idea of the likely errors involved. The work covers a wide variety of polymers, solvents and temperatures. For good solvents the results indicate values of v about 0.6 and values of e about 0.25. Though these values agree with the theoretical values little significance can be attached to this agreement since the errors in a single experiment appear to be of the order of 10% and the variation between different experimenters and different systems is much larger than 10%. It would be interesting to test the strong scaling hypothesis that 3v = 2 — e but the present experimental estimates of v and e are too inaccurate to give any indication of the validity or invalidity of the hypothesis. Some of the difficulties of the experimental work and the errors involved can be gained from the report by Strazielle and Benoit [122] of a project in which the same samples of polymer were distributed around a large number of laboratories across the world with wide variations in the results in such quantities as the number and weight-average molecular weights, among others. McIntyre, Mazur and Wims [123, 1241 have tested experimentally the hypothesis that in good solvents the function describing the distribution of inter-segmental distances is non-Gaussian, by examining the shape of the scattering function KcM~/J(k,0) (see (1.9)) found from light-scattering. They deduce that the shape of the molecule in good solvents is non-Gaussian. From their data it may be possible to deduce a value for the shape exponent 6.

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McKenzie, Polymers and scaling

One of the major sources of systematic error in the experimental work is the degree of polydispersity of the sample. All samples of synthetic polymer are polydisperse. Although the degree of polydispersity can be reduced by means of suitable fractionation techniques, fractionation is in fact changing the shape of the distribution of molecular weights in the direction, one hopes, of making the distribution more sharply peaked at particular molecular weights. However, one is still left with a distribution of molecular weights. The conventional method of determining the degree of polydispersity, namely the ratio c~,must be used with caution. The rule of thumb that the closer this ratio is to unity the less polydisperse is the sample can be misleading; for example, a long tail to the distribution at low molecular weights may have little effect on ‘I but would have an appreciable effect on the second virial coefficient which, through (6.2) with e = ~, is dominated by the lower molecular weights. The theory reviewed in this paper has the advantage that it can quite easily deal with polydispersity because of the simple power law relationships involved. Consequently, it may be an experimental advantage not to fractionate samples bu.t to rely on one’s knowledge of the molecular weight distribution from details of the polymerisation process used to prepare the sample. Finally, the work described by Krigbaum and Flory [121]involved, among other things, measuring the second viria! coefficient A2 of a mixture of two fractionated samples of polymer with different molecular weights as a function of the relative concentration x of the two species. They obtained a maximum in A2 as a function of x a result which contradicts the theory presented here (2.43) and in ref. [7] unless the samples used by Krigbaum and Flory were not as sharply fractionated as they believed.

7. Conclusions In this review we have attempted to trace the various strands in the theory of polymers which find counterparts in the theory of critical phenomena. We hope to have given a unified and logical account of the field so that the relations and differences between the component parts are made clear. It remains here to point out those aspects which are not yet settled and which merit further study. First it has been shown that there are reasonable grounds for belief that, for athermal polymer solutions, the osmotic pressure is a homogeneous function of concentration and molecular weight and that the long-range behaviour of the end-to-end length distribution is determined by a single scaling length. However, evidence is rather against the strong scaling hypothesis that this scaling length is sufficient either to characterise the short-range behaviour of the end-to-end length distribution or to determine the behaviour of the equation of state. These conclusions are supported by numerical work using three-dimensional models; the work with two-dimensional models appears to support the strong scaling hypothesis though the evidence is somewhat contradictory. The difference in behaviour, or rather, the incompatibility between short-range and long-range effects, if it exists, is worth serious study. Second, wehave shown that work using discrete models and work using continuum models may not be so closely related as appears at first sight. Indeed, one might not be overstating the case if one says that the models based on a discrete space are qualitatively different from models based on a continuum space and that discrimination between the two lies in experimental work. Third, little theoretical work from the point of view of work on critical phenomena has been done on real phase transitions in polymer solutions. The fact that most work has been concerned

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85

with athermal solutions only, is a limitation. The further study of temperature and solvent effects would greatly extend the range of applicability of the theories discussed herein. Fourth, the theory of dynamical scaling has not yet been applied to the polymer problem though a recent paper by Kapral et a!. [142] discusses some of the relevant features. Finally, polymer systems offer useful additional experimental material for study of critical phenomena. Although the behaviour of the single polymer molecules in solution is not easy to measure and impossible by normal thermodynamic means because of the standard state to which the properties of the solution are referred, the latter fact can be an advantage because it also removes the necessity of finding the critical point, the major source of error in experiments on the critical behaviour of simple fluids and magnetic systems. Balanced against this advantage are the complications caused by the polydispersity of polymer samples. We wish to acknowledge with thanks the comments of C. Domb, V. Bloomfield, M. F. Sykes, D. S. Gaunt, and S. McKenzie.

References [1] L. P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V. Palciauskas, M. Rayl, J. Swift, D. Aspnes and J. Kane, Rev. Mod. Phys. 39 (1967) 395. [2] M. V. Volkenstein, Configurational statistics of polymeric chains (1963, New York, Interscience). [3] P. J. Flory, Statistical mechanics of chain molecules (1969, New York, Interscience). [4] C. Tanford, Physical chemistry of macromolecules (1961, New York, Wiley). [5] V. E. Eskin, Soviet Phys. Usp. 7 (1964) 270. [6] P. J. Flory, Principles of polymer chemistry (1953, Ithaca, New York, Cornell University Press). [7] D. S. McKenzie and C. Domb, Proc. Phys. Soc. 92 (1967) 632. [81 C. Domb, Polymer 15 (1974) 259. [9] R. Finsy, M. Janssens and A. Bellemans, J. Phys. A8 (1975) L106. [10) A. R. Massih and M. A. Moore, J. Phys. A8 (1975) 237. [11] S. F. Edwards, Proc. Phys. Soc. 88 (1966) 265. [12] C. Domb, Adv. in Phys. 9 (1960) 149. [13) T. L. Hill, Statistical mechanics (1956, New York, McGraw-Hill). [14] C. Domb, J. Phys. CS (1972) 1399. [15] R. G. Bowers and A. McKerrel, J. Phys. C6 (1973) 2721. [16] M. F. Sykes, J. W. Essam, B. R. Heap and B. J. Hiley, J. Math. Phys. 7 (1966) 1557. [17] J. M. Hammersley, Proc. Cam. Phil. Soc. 53 (1957) 642. [18] J. M. Hammersley, Proc. Cam. Phil. Soc. 57 (1961) 516. [19] H. N.Y. Temperley, Phys. Rev. 103 (1956) 1. [20] M. E. Fisher and M. F. Sykes, Phys. Rev. 114 (1959) 45. [21] L. Onsager, Phys. Rev. 65(1944)117. [22] H. E. Stanley, Introduction to phase transitions and critical phenomena (1971, Oxford, England, Clarendon Press). [23] C. Domb, in: Critical phenomena in alloys, magnets and superconductors, eds. R. E. Mills, E. Ascher and R. I. Jaffec (1971, New York, McGraw-Hill). [24) D. S. McKenzie and M. A. Moore, J. Phys. A4 (1971) L82. [25] M. E. Fisher, J. Chem. Phys. 44 (1966) 616. [26] M. E. Fisher and R. J. Burford, Phys. Rev. 156 (1967) 583. [27] C. Domb, Adv. in Phys. 19 (1970) 339. [28) D. S. Gaunt and A. J. Guttman, in: Phase transitions and critical phenomena, Vol. 3, eds. C. Domb and M. S. Green (1974) p. 181. [29] M. G. Watts, J. Phys. A8 (1975) 61. [30) M. E. Fisher and D. S. Gaunt, Phys. Rev. 133 (1964) 224. [31] J. L. Martin, M. F. Sykes and F. T. Hioe, J. Chem. Phys. 46 (1967) 3478. [32] J. L. Martin and M. G. Watts, J. Phys. A4 (1971) 456. [33) M. F. Sykes, A. J. Guttman, M. G. Watts and P. D. Roberts, J. Phys. A5 (1972) 653.

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Note added in proof We apologise to des Cloiseaux for inadvertently omitting his paper [1431 on the Lagrangian theory of single polymer chains. His model is the same as that described for multi-component systems in section 5. He confirms, in the context of the renormalisation group theory, the scaling relations (3.46), (3.47) and (3.72) and also the strong-scaling relation dv = 2 a. Recently, Morita [144] has extended Massih and Moore’s [101 exact calculations to a tree of hexagons. Barrett [1451 has considered the relation of the model of Domb and Joyce [771 to the Monte Carlo calculations of Smith and Fleming [1331. The second osmotic virial coefficient with nearest-neighbour attractive forces has been studied by the Monte Carlo method by Janssens and Bellemans [1461. Monte Carlo calculations have also been made on the osmotic pressure of short chains on the plane square lattice by Okamoto [147]. Jasnow and Fisher [1481 and Hilhorst [149] have applied to the lattice formulation of the Wilson theory by Niemeijer and van Leeuwen [150] to the self-avoiding walk problem. Jasnow and Fisher are concerned with the relation to the Ising model and show that the relation can be made without recourse to diagrammatic techniques. Hilhorst also calculates values for v. —

References [142] R. Kapral, D. Ng and S. G. Whittington, J. Chem. Phys. 64 (1976) 539. [143] J. des Cloiseaux, Phys. Rev. AlO (1974) 1665. [144]T. Morita, J. Phys. A9 (1976) 169. [145] A. J. Barrett, J. Phys. A9 (1976) L33. [146] M. Janssens and.A. Bellemans, Macromol. 9 (1976) 169. [147] H. Okamoto, J. Chem. Phys. 64 (1976) 2686. [148] D. Jasnow and M. E. Fisher, Phys Rev. B13 (1976) 1112. [149] H. J. Hithorst, Phys. Lett. 56A (1976) 153. [150] Th. Niemeijer and J. M. J. van Leeuwen, Phys. Rev. Lett. 31(1973)1411.