Concentration Dependence of Excluded Volume Effects
aa r X i v : . [ c ond - m a t . s o f t ] J u l Concentration Dependence of Excluded Volume Effects
Concentration Dependence of Excluded Volume Effects
Polymer Solution as Inhomogeneous System
Kazumi Suematsu
Institute of Mathematical ScienceOhkadai 2-31-9, Yokkaichi, Mie 512-1216, JAPANE-Mail: [email protected], Tel/Fax: +81 (0) 593 26 8052
Abstract
The concentration dependence of the excluded volume effects in polymer solutions is investigated. Through ther-modynamic arguments for the interpenetration of polymer segments and the free energy change, we show that thedisappearance of the excluded volume effects should occur at medium concentration. The result is in accord withthe recent experimental observations.
Key Words : Inhomogeneity of Polymer Solutions/ Excluded Volume Effects/ Concentration Dependence/
Concentration dependence of the excluded volume effects is an unsolved problem in polymer physics. To date,only a few theoretical works have been developed on this issue. Quite recently, very reliable experiments with thesmall angle neutron scattering (SANS) have been performed by two research groups [1, 2]; they showed that theunperturbed coil dimensions are realized at medium concentration far below the melt state; the unperturbed statebeing retained over wide concentration range.In this paper we reexamine the concentration dependence of the excluded volume effects on the basis of theclassic thermodynamic theory. The starting point of our investigation is motivated by the suggestion of the scalingformula by Issacson and Lubensky [3] that a critical concentration may occur for d = 3 somewhere [4] between thedilution limit and the non-solvent state above which the excluded volume effects vanish.A polymer solution is intrinsically an inhomogeneous system of segment concentration, which is due to the factthat monomers are joined by chemical bonds (see Fig. 1). A theory must therefore take this fact into consideration.Along this line, first we will make minor amendment of the classic theory of the local free energy in order to applythe theory in a more rigorous manner to the disappearance problem. Then, on this basis, we will show that theexcluded volume effects should really vanish at medium concentration. Excluded volume effects [5–15] of polymer molecules have two different facets: One is the expansion of the polymerdimensions and the other is the repulsion between two polymer molecules. These two phenomena, however, can beunderstood by a single thermodynamic property, the osmosis, namely the spontaneous flow of solvent from a moredilute region to a more concentrated region. The solvent flowing into from the dilute region will expand polymercoils. If all segments are joined by covalent bonds, the osmosis simply leads to the expansion of the polymer coil,while if the segments consist of some polymer molecules, the expansion will necessarily lead to the separationof those molecules, which will be phenomenologically observed like the core-core repulsion between hard spheres.Thus the coil expansion of one molecule and the repulsive interaction between two molecules are the differentmanifestations of the same phenomenon.In order for the osmosis to occur, there must exists concentration fluctuation in the system. Without theconcentration gradient between the inside and the outside of the coil, no excluded volume effects can occur. Our1ask is then to estimate this gradient and calculate the magnitude of the exclude volume effects as a function ofpolymer concentration.In order to investigate whether the transformation of the excluded volume coil to the ideal one occurs at acertain concentration, one must begin by evaluating the potential energy change ∆ F of the system under theconcentration fluctuation. More specifically, one must investigate whether the condition ∆ F ≪ kT can be fulfilledat certain finite concentration. If this inequality is satisfied, the excluded volume effect will be hidden behindthe thermal fluctuation and never be detected experimentally. Our question is then “To what extent must theconcentration fluctuation diminish in order for the excluded volume effect to be screened out?” [7, 9]. In this section we make minor amendment of the classic theory of the local free energy [5]; the amendment isnecessary to apply the theory in a more rigorous manner to real polymer solutions. As is well known, the classictheory for the local free energy does not take into consideration correctly the interaction between segments onthe same chain, so that the classic theory is constructed on the basis of the pseudo self-avoiding chains whosebehavior is virtually equivalent to that of the ideal chain [11]. This feature has been frequently criticized so far asa fundamental deficiency of the Flory theory.Let us consider the three dimensional lattice where a site may be occupied by a segment or a solvent molecule.Following the standard lattice representation, the random occupation of sites is assumed. Let a site be surroundedby z neighboring sites, and let there be δx segments from different molecules in the volume element δV . Thenumber of arrangements of those segments isΩ δV = δx − Y i =0 ( z i − − f i ) (1)where f i is the probability that a given cell is occupied by the polymer segments when i segments are already putin δV , so that f i = i/δn where δn = δx + δn , δn denotes the total number of sites and δn the number ofsolvent molecules in δV . z i is a special number introduced in this amendment and is defined by 0 ≤ z i ≤ z . Thephysical meaning of z i is as follows:There is finite probability that a given segment overlaps with the other segments on the same chain (themultioccupation problem). Such unphysical conformations must properly be removed by subtracting fromthe total number Q i ( z −
1) of feasible conformations. This is possible, because the number of conformationsis enumerable in principle. The subtraction can be achieved simply by reducing z to z i , so that Q i ( z i − z i is unfortunatelyunknown [10], which however is not essential for the present purpose, as is verified below.By eq. (1) the local entropy becomes δS = δS mixing + δS = k log Ω δV = k ( δx − X i =0 log ( z i −
1) + log δn ! δn δx ( δn − δx )! ) (2)Applying the Stirling formula to the above equation, we have δS mixing + δS ∼ = − k ( δn log v + δx − δx − X i =0 log ( z i − ) (3)where δS = δS ( δx = 0) + δS ( δn = 0) represents entropy for respective pure components and v = ( δn − δx ) /δn is the volume fraction of solvents. It is clear by eq. (3) that δS ( δx = 0) = 0, and δS ( δn = 0) = − k n δx − P δx − i =0 log ( z i − o . Hence we have δS mixing = − k δn log v (4)which is exactly the Flory result. It turns out that all the self-avoiding terms are absorbed into the melting entropy, δS ( δn = 0). Noteworthy is the fact that the mixing entropy δS mixing of the pseudo self-avoiding chains andsolvent is exactly equal to that of the genuine self-avoiding chains and solvent; only the standard state must bealtered as pure pseudo self-avoiding chains ⇒ pure genuine self-avoiding chains . δS mixing is a function of thesolvent fraction alone, but independent of the conformational properties of chains.Adding the enthalpy term, δH mixing = kT χδn v to the above equation, we have the formula of the local freeenergy in the volume element δVδF mixing = δH mixing − T δS mixing = kT { log (1 − v ) + χv } δn (5)where v = 1 − v is the volume fraction of the segments and χ the enthalpy parameter. Eq. (5) represents the freeenergy difference between the mixture of the self-avoiding chains and solvent, and the respective pure components.Eq. (5) has already been derived by Flory [5].We realize that eq. (5) has deeper generality along with sound physical basis, and hence is applicable equallyto the excluded volume problem in concentrated systems. α in Concentrated Solution Fig. 1: A snapshot of a polymer solution.
Fundamental force of the coil expansion is the osmosis of solventmolecules from a more dilute region, C valley , to a more concen-trated region, C hill around the center of gravity of a coil, where C denotes segment concentration. In order for the osmosis to occurfor a given polymer coil, there must exists non-zero free energydifference between the inside of the coil ( C hill ) and the outside( C valley ). If this circumstance is realized, positive molecular forcefrom the outside region inevitably arises, creating the coil expan-sion.The size of the expansion factor α is determined by the forcebalance between the osmosis and the retraction force due to rub-ber elasticity [5] (see an alternative derivation in Appendix). Theequilibrium condition is ∂F/∂α = ∂F osmotic /∂α + ∂F elastic /∂α = 0 (6)By eq. (5) the free energy of mixing polymer segments and solvent molecules is written in the form:∆ F M = kT Z { log (1 − v ) + χv } δn (7)where the subscript 1 denotes the solvent, the subscript 2 the solute (polymer segment), and χ is the enthalpyparameter as mentioned above. Note that δn = (1 − v ) δV /V with V denoting the molecular volume of a solventmolecule. Substituting this into eq. (7), we have∆ F M = kTV Z (1 − v ) { log (1 − v ) + χv } δV (8)Expanding the logarithmic term, we have∆ F M = kTV Z (cid:26) − (1 − χ ) v + (1 / − χ ) v + 16 v + · · · (cid:27) δV (9)Since we want to calculate the local free energy difference, ∆ F , between the hill and the valley areas, let us write∆ F in the form:∆ F =∆ F M,hill − ∆ F M,valley = kTV Z (cid:26) − (1 − χ ) ( v hill − v valley ) + (1 / − χ ) ( v hill − v valley ) + 16 (cid:0) v hill − v valley (cid:1) + · · · (cid:27) δV (10)For v ≪ F ∼ = kTV Z (cid:8) − (1 − χ ) ( v hill − v valley ) + (1 / − χ ) ( v hill − v valley ) (cid:9) δV (11)3et V denote the volume of a polymer segment and we have v = V C . For the Gaussian chain, with the equality δV = α d ( x − a ) d ( y − b ) d ( z − c ) = α dxdydz in mind, we can write generally the segment concentration at a givenpoint ( x, y, z ) in the form C ( x, y, z ) = N X { a,b,c } (cid:18) βα π (cid:19) / e − β { ( x − a ) +( y − b ) +( z − c ) } (12)where N is the number of the segments on a molecule, β = 3 / R g with R g denoting the radius of gyration of anunperturbed chain and ( a, b, c ) the coordinate of the center of gravity of a polymer molecule; so the summationrepresents the accumulation of segments emanating from a number of different polymers and therefore reflectsthe segment concentration at the point ( x, y, z ). A graphical representation is given in Fig. 1 which shows thata polymer solution is the typical inhomogeneous system. In contrast to the homogeneous system of a monomersolution, there is wild fluctuation ∆ C = C hill − C valley in the solution. ∆ C is a function of α ; it rapidly decreaseswith increasing α , because segments pervade more deeply the whole system as α increases.It is useful to extract the pre-factor from eq. (12) and recast it in the form C ( x, y, z ) = N (cid:18) βα π (cid:19) / G ( x, y, z ) (12 ′ )so that the quantity G ( x, y, z ) = X { a,b,c } e − β { ( x − a ) +( y − b ) +( z − c ) } (13)is a function independent of α . Now eq. (11) may be rewritten in the form∆ F ∼ = N kT V V Z Z Z ( − (1 − χ ) (cid:18) βπ (cid:19) / ( G hill − G valley ) + V N (1 / − χ ) (cid:18) βα π (cid:19) (cid:0) G hill − G valley (cid:1)) dxdydz (14)The first term of eq. (14) is, by eq. (13), independent of α , so it vanishes by the differentiation with respect to α .We have then ∂ ∆ F osmotic /∂α = − N kT V α V (1 / − χ ) (cid:18) βπ (cid:19) Z Z Z (cid:0) G hill − G valley (cid:1) dxdydz (15)Note that for any fluctuation model of polymer solutions, we can establish one-to-one correspondence between the C hill and the C valley by properly setting the defined spaces. Thus it is sufficient to take account of the elasticforce per one molecule (this point will be discussed in Sec. 4). The classic work [16] showed that ∂ ∆ F elastic /∂α =3 kT ( α − /α ). The force balance between the osmosis and the rubber elasticity is obtained by substituting theseresults in eq. (6), namely3 N kT V α V (1 / − χ ) (cid:18) βπ (cid:19) Z Z Z (cid:0) G hill − G valley (cid:1) dxdydz = 3 kT ( α − /α ) (16)By rearrangement, we arrive at the expression (see an alternative derivation in Appendix): α − α = N V V (1 / − χ ) (cid:18) βπ (cid:19) Z Z Z (cid:0) G hill − G valley (cid:1) dxdydz (17)For the dilution limit, we have G hill → e − β ( x + y + z ) and G valley → α = 1) at the theta point ( χ = 1 / V limit, the coil is nearly ideal, and (iii) the fifth power rule of α isstill valid, but (iv) the coil must also be ideal at the point of G hill = G valley in which the concentration fluctuationdisappears. u in Concentrated Solution Prior to the derivation of the excluded volume u in the concentrated solution, let us review briefly the classic work.Let a system contain only two polymer molecules. Consider the interpenetration of the two polymer molecules.4et a distance be L between their centers of gravity. The chemical potential change (∆ F = ∆ H − T ∆ S ) bringingthese molecules close from L = ∞ to L is∆ F ( L ) = 2 kT (1 / − χ ) (cid:0) V /V (cid:1) Z ρ k ρ ℓ dV (18)where ρ k and ρ ℓ are the respective segment density of the two polymer molecules k and ℓ within the small volumeelement δV , which has the form: ρ k = N (cid:18) βα π (cid:19) / e − β ( x k + y k + z k ) (19)Then the excluded volume u can be calculated by the equation u = Z ∞ πL (cid:16) − e − ∆ F ( L ) /kT (cid:17) dL (20)The parameter χ is, according to the definition, positive for poor solvents but negative for good solvents. It followsthat the interpenetration of the two polymer coils is strongly hindered in good solvents resulting in the molecularrepulsion. We derive in the following the corresponding free energy change in the concentrated polymer solution. Suppose that a single polymer molecule k is added to the concentrated polymer solution. There are two cases forthe molecule to be located in the solution: one is the case where the molecule is put within the C hill region and theother is the case where it is put within the C valley region. The free energy difference between these two locationswill correspond to the free energy difference between overlapping state and non-overlapping state in concentratedsolutions. Now we can generalize the Flory excluded volume theory to include the concentrated system. Thequantity in the small volume element δV is δ (∆ F k,C ) = δ (∆ F k,C hill ) − δ (∆ F k,C valley ) = kT ( δV /V ) n (1 − ρ k V − C hill V ) log (1 − ρ k V − C hill V ) − (1 − C hill V ) log (1 − C hill V ) − (1 − ρ k V − C valley V ) log (1 − ρ k V − C valley V )+ (1 − C valley V ) log (1 − C valley V ) − χρ k V ( C hill − C valley ) o (21)Eq. (21) can be written in the series form: δ (∆ F k,C ) = 2 kT { (1 / − χ ) + O} ρ k ( C hill − C valley ) (cid:0) V /V (cid:1) δV (22)where the symbol O represents the higher terms of the series and a function of C hill , C valley and ρ k . Below themedium concentration, these terms are negligible and eq. (22) reduces to δ (∆ F k,C ) ∼ = 2 kT (1 / − χ ) ρ k ( C hill − C valley ) (cid:0) V /V (cid:1) δV (23)Let L be a distance between the centers of ρ k and C hill . Then C valley is also a function of L , because the distancebetween C hill and C valley is fixed, on average, for the solution of a given concentration. Applying the equality δV = α dxdydz to eq. (23) and integrating over the defined spaces, we have the free energy difference as a functionof L ∆ F k,C ( L ) = 2 kT (1 / − χ ) (cid:0) V /V (cid:1) α Z Z Z ρ k ( C hill − C valley ) dxdydz (24)In the limit of the infinite dilution, we have C hill → ρ ℓ and C valley →
0, and we recover the classic equation (18).Using the Gaussian approximation, we may recast the above equation in the form:∆ F k,C ( L ) = 2 N kT (1 / − χ ) (cid:0) V /V (cid:1) (cid:18) βαπ (cid:19) Z Z Z e − β ( x k + y k + z k ) ( G hill − G valley ) dxdydz (25)where G = P { a,b,c } e − β { ( x ℓ − a ) +( y ℓ − b ) +( z ℓ − c ) } . As the concentration fluctuation decreases, namely, G hill − G valley →
0, then ∆ F k,C →
0. And it follows from eq. (20) that u →
0. The excluded volume disappears inparallel with the disappearance of the fluctuation. 5
Solution According to Lattice Model
By the equations (17) and (25), we have a general statement that the excluded volume effects are a strong functionof the concentration fluctuation. This is a very natural conclusion, because the osmosis can occur only in thepresence of the concentration gradient. Our task is then to evaluate the local concentration gradient around agiven polymer coil.Eqs. (17) and (25) are formal solutions not easy to solve, since the relative coordinates ( a, b, c ) of individualmolecules can not be specified in real solutions; moreover the boundary conditions of the integral terms in eqs.(17) and (25) are not clear. In this sense, no rigorous calculation seems possible. However it is possible to extractessential features of the equations by making use of the lattice model. In this paper we show the solution for eq.(17) only, since eq. (25) is more complicated and requires heavy calculation that seems to exceed the ability ofcomputers available. p pp xy z
Fig. 2: Representation of the simple cubic lattice: Edgesare occupied by polymer molecules having the Gaussiandistribution. The C maximum and the minimum pointslie on the z = x plane. Fig. 3: Representation of the concentration fluctuationin the solution of Gaussian polymers arranged on thesimple cubic lattice. A peak represents the C hill and abottom the C valley : the center of a peak corresponds tothe center of gravity of a chain. Consider the simple cubic lattice with the unit lengths ( p × p × p ). Polymer molecules are arranged on everysites with the Gaussian segment distribution. The relative coordinates of the molecules have then integer valuesof the form: ( a, b, c ) = ( ip, jp, kp ) ( i, j, k = −∞ , · · · , − , , , · · · , + ∞ ). The segment number density ( / ˚A ) at agiven coordinate ( x, y, z ) can be calculated by the simple summation of equal steps: C ( x, y, z ) = N + ∞ X ( i,j,k )= −∞ (cid:18) βα π (cid:19) / e − β { ( x − ip ) +( y − jp ) +( z − kp ) } = N (cid:18) βα π (cid:19) / G ( x, y, z ) (26)To find the maximum and the minimum points, differentiate eq. (26) to yield dC = (cid:18) ∂C∂x (cid:19) y,z dx + (cid:18) ∂C∂y (cid:19) z,x dy + (cid:18) ∂C∂z (cid:19) x,y dz (27)Since x , y and z are independent of each other, the solution for the equality dC = 0 must satisfy (cid:18) ∂C∂x (cid:19) y,z = (cid:18) ∂C∂y (cid:19) z,x = (cid:18) ∂C∂z (cid:19) x,y = 0 (28)Here (cid:18) ∂C∂x (cid:19) y,z = const + ∞ X i = −∞ ( x − i p ) e − β { ( x − ip ) +( y − jp ) +( z − kp ) } = 0 (29)6bviously x = i p satisfies eq. (29). The other solutions are x = (1 / ℓ ) p ( ℓ = −∞ , · · · , − , , , · · · , + ∞ ). Fromthese, we have C max = { x = i p, y = j p, z = k p } C min = { x = (1 / ℓ ) p, y = (1 / m ) p, z = (1 / n ) p } (30)where ℓ, m, n = −∞ , · · · , − , , , · · · , + ∞ . Thus C max and C min lie on z = x + np planes. In Fig. 2, an example( z = x plane) of those planes is illustrated. The mean number density of segments is directly calculated by theequation: ¯ C = Np ( / ˚A ) (31)so that the mean polymer volume fraction ¯ φ is ¯ φ = Np V (32)An important quantity to be evaluated is the integral term appearing in the final expressions (17); this measuresthe 3-dimensional density fluctuation in the system. We put J α = Z Z Z (cid:0) G hill − G valley (cid:1) dxdydz (33)As can be seen from Figs. 1-3, the G hill and the G valley areas are discontinuous; its domains can be partitionedinto a number of intervals. By the lattice symmetry, we may define the intervals as [ − p/ , p/
4] for G hill for eachaxis and [ p/ , p/
4] for G valley . This choice is simply a matter of convenience for numerical calculation. Then theabove integral may be specified as J α = Z Z Z p/ − p/ G dxdydz − Z Z Z p/ p/ G dxdydz (34)We show in Fig. 4-a the concentration fluctuation on the x = y = z line that goes through the centers of the C hill and C valley . The curves were calculated according to eq. (26) for α = 1 as a function of the mean polymervolume fraction ¯ φ , modeling polymethyl methacrylate (PMMA: N = 1000) solutions. We have assumed the sizeof the segment to be equal to that of the repeating unit so that V = 140 ˚A. The light-brown peak with ¯ φ = 0represents the isolated polymer molecule in the dilution limit. As one can see, the concentration fluctuation decaysrapidly with increasing ¯ φ and vanishes for ¯ φ & . J α /N is plotted as a function of ¯ φ and N . The dotted line shows the result for N = 1000 and the solid line the result for N = 5000. The function J α /N also decreases strongly with increasingconcentration and vanishes at medium concentration. ( a ) ( b ) Fig. 4: Concentration fluctuation in polymethyl methacrylate solutions ( V = 140). (a) Fluctuation of C on the x = y = z line (see also Fig. 2) as a function of the volume fraction ¯ φ of the polymer ( N = 1000). (b) Numericalsolution of J α /N as a function of ¯ φ ; J α was calculated according to eq. (34) together with eq. (32); dotted line ( · · · ): N = 1000 and solid line ( − ): N = 5000. .2 Expansion Factor α as a Function of Concentration With the help of the estimation of J α , we can now solve eq. (17) as a function of the segment concentration.Employed parameters are listed in Table 1. The enthalpy parameter χ is hard to estimate. Here we use the valuesof χ = 0 . − chloroform and χ = 0 . − n -nonadecane which is an assumed value at 150 ◦ C by theextrapolation of the HandBook data [17].The calculation results are illustrated in Figs. 5 and 6 to be compared with the observed points by Cheng,Graessley and Melnichenko [1] and those by Westermann, Willner, Richter and Fetters [2]. As one can see, theexpansion factor α drops strongly with increasing polymer concentration and falls to the unperturbed value at themedium concentration in accordance with the experimental observations [1, 2]; the results are also consistent withthe prediction of the Issacson-Lubensky scaling formula [3, 4].According to the present calculation (Figs. 5 and 6), it is found that the essential features of the two systemsof PMMA and PE are very alike: both of the systems show the swollen-to-unperturbed coil transition at themedium concentration. Comparing the two systems, it is found also that the coil expansion is less pronouncedin the PE − n -nonadecane system than in the PMMA − chloroform system. The reason can be found by inspectingeq. (17); it simply comes from the special combination of the polymer and the solvent, namely n -nonadecane withthe large molecular volume V and polyethylene with the small segment volume V . According to eq. (17), such acombination necessarily depresses the expansion factor α to a lower level.Table 1: Basic parameters of polymer solutions [1, 2]parameters † notations valuespolymethyl methacrylate (PMMA) volume of a solvent (CHCl ) V
134 ˚A volume of a segment (C O H ) V
140 ˚A degree of polymerization N F ℓ . ◦ C) χ . n -C H ) V
569 ˚A volume of a segment (C H ) V
49 ˚A degree of polymerization N F ℓ .
54 ˚Aenthalpy parameter (150 ◦ C) χ . The lattice model is a strong approximation where the dynamic aspect of real solutions is completely neglected.However, there are two cases in which this static model is expected to be a good representation of real solutions:(i) the lattice model will predict correctly the behavior in dilute solutions where polymer coils are, on average, farapart from each other so that the interaction among chains is weak; (ii) the model will predict correctly the locationof the swollen-to-unperturbed coil transition at which the concentration fluctuation disappears. The agreementwith the recent observations supports this reasoning.One of the important questions throughout the present work was whether one can put the equality between thel.h.s and the r.h.s. terms in eq. (16); i.e., whether one can equate the osmotic pressure due to the density gradient † The unperturbed size is calculated by the equation: R g = C F Nξ ¯ ℓ , where ξ is the bond number per one repeating unit. Thenumerical estimation of R g is in accord with the observed values [18]. ig. 5: Expansion factor vs ¯ φ plot for PMMA − CHCl .Solid line ( − ): theoretical line by eq. (17) for χ = 0 . · · · ): theoretical line by eq. (17) for χ = 0.Open circles ( ◦ ): observed points by Cheng, Graessleyand Melnichenko. Fig. 6: Expansion factor vs ¯ φ plot for PE − n -C H .Solid line ( − ): theoretical line by eq. (17) for χ = 0 . ◦ ): observed points by Westermann, Will-ner, Richter, and Fetters. exactly with the retractive force of one whole chain. To examine this problem, we must take the following fact intoconsideration: the osmotic phenomena within polymer coils are very different from the ordinary osmosis [19]. Theordinary osmosis occurs between two systems separated by the semipermeable membrane, in which solvent moleculesdiffuse from a dilute system to a concentrated system, thus lowering the solute concentration in the concentratedsystem, whereas raising that in the dilute system. This process of the change of the solute concentrations on eachside of the membrane takes place as a result of the volume change of the respective systems.In contrast, there exists no semipermeable membrane in polymer solutions so that no volume change of therespective systems can occur. The change of the concentrations in the two regions (dilute and concentrated) canoccur only through the interchange of the space coordinates between polymer segments and solvent molecules,which leads to the coil expansion. The expansion of the polymer dimensions is therefore similar to the dissolutionof a solid material into a solvent. The coil expansion resulting from the interchange of the coordinates tends toreduce the density fluctuation in the system, whereas the coil contraction will augment it. Only one way for thepolymer solution to reduce its own fluctuation is to expand the polymer dimensions. Such adaptation of the coildimensions should occur uniformly over the whole system because of the absence of the semipermeable membrane,thus validating the equality (16). Note that in the alternative derivation shown in Appendix, this problem onthe equality (16) does not arise explicitly because of the delta function approximation for the segment densitydistribution.There exists another question,“what do we mean by the unperturbed (or ideal) chain?” To answer this question,let us return to the basic assumption of the present work. We have calculated the free energy difference ∆ F bysubtracting the free energy in the dilute region from that in the concentrated region. Thus the term “unperturbed”necessarily refers to the configuration under the condition that satisfies C max = C min . By the result of eq. (30)it corresponds to the configuration at p = 0, namely the configuration at infinite concentration ( C → ∞ ) whereall excluded volume effects should vanish rigorously as discussed earlier [4]. Hence the term “unperturbed” definesa standard state for the configuration of a chain composed of mathematical dots and lines (with no volume andno thickness) and immersed in the true athermal solvent . In that hypothetical limit, the segment should take thedistribution of the form [20]: W ( s ) = 1 N N X k =1 (cid:18) πR ω k (cid:19) / exp (cid:18) − R ω k s (cid:19) (35)where ω k = t k + (1 − t k ) , t = k/N and R is the end-to-end distance and R = 6 Rg . Usually this distributionis approximated by the corresponding Gaussian distribution as we have done in the text. Note that the abovedefinition for the unperturbed chain is clearly different from that used in the classic theory [5] where the standardstate has been taken to be a pure polymer (melt state of self-avoiding chains). The two definitions are, however,virtually the same, since a chain obeys in effect the random flight statistics over all concentration range above themedium concentration, as we have seen in Figs. 5 and 6.9 Conclusion
1. We have made a minor amendment for the local free energy, showing that the classic theory has deep generalityand sound physical basis.2. Making use of the thermodynamic arguments, we have developed the extended theory of the excluded volumeeffects, showing that the expansion factor α is a strong function of the concentration fluctuation eq. (17).3. According to the solution of the lattice model, the fluctuation is maximum in the dilution limit, but decreasesstrongly as the polymer concentration increases, and vanishes at the medium concentration Fig. 4.4. In parallel with the behavior of the fluctuation, the excluded volume effects manifest themselves most pro-nouncedly in the dilution limit, but decay rapidly with increasing concentration and vanish at the mediumconcentration.The theoretical results are in good accord with the recent experimental observations [1, 2] Figs. 5 and 6. An Alternative Derivation of Eq. (17)
Let W ( s ) be the unperturbed segment distribution around the center of gravity. We introduce the partition functionfor the excluded volume chain [6, 21]: Z = Z s W ( s ) exp (cid:18) − V ( s ) kT (cid:19) πs ds (A1)where V ( s ) represents a potential function. Then the perturbed segment distribution can be formulated as p ( s ) = 1 Z W ( s ) exp (cid:18) − V ( s ) kT (cid:19) πs (A2)Making use of the Gaussian approximation for W ( s ), we have α = h s i R g = Z s s p ( s ) dsR g Z s p ( s ) ds = Z t exp (cid:18) − t − V ( t ) kT (cid:19) t dt Z t exp (cid:18) − t − V ( t ) kT (cid:19) t dt (A3)In eq. (A3), we have made the variable transformation: t = s /R g . Independently we have another equality [21]: α = Z t t δ ( t − α ) dt Z t t δ ( t − α ) dt (A4)where δ ( t − α ) signifies the delta function peaked at t = α . If we identify eq. (A3) with eq. (A4), this amountsto making the approximation: δ ( t − α ) ≈ exp (cid:16) − t − V ( t ) kT (cid:17) t . Our remaining task is then only to evaluate themaximum point of the function, exp (cid:16) − t − V ( t ) kT (cid:17) t , so that ddt log (cid:26) exp (cid:18) − t − V ( t ) kT (cid:19) t (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) t = α = 0 (A5)Here we use ∆ F ( α → t ) in eq. (14) as the potential function V ( t ). After some rearragement, we obtain α − α = N V V (1 / − χ ) (cid:18) βπ (cid:19) Z Z Z (cid:0) G hill − G valley (cid:1) dxdydz (A6)which is just eq. (17) in the text. 10 eferences [1] (a) W. W. Graessley. Polymer chain dimensions and the dependence of viscoelastic properties on concentration,molecular weight and solvent power. Polymer, , 258 (1980).(b) W. W. Graessley and R. C. Hayward. Excluded-Volume Effects in Polymer Solutions. 2. Comparison ofExperimental Results with Numerical Simulation Data. Macromolecules, , 3510 (1999).(c) W. W. Graessley. Scattering by Modestly Concentrated Polymer Solutions. Macromolecules, , 3184(2002).(d) G. Cheng, W. W. Graessley, and Y. B. Melnichenko. Polymer Dimensions in Good Solvents: Crossoverfrom Semidilute to Concentrated Solutions. PRL, , 157801 (2009).[2] S. Westermann, L. Willner, D. Richter and L. J. Fetters. The evaluation of polyethylene chain dimensions asa function of concentration in nonadecane. Macromol. Chem. Phys., , 500 (2000).[3] (a) J. Issacson and T. C. Lubensky. Flory exponents for generalized polymer problems. J. Physique, Letters, , L-469 (1980).(b) T. C. Lubensky and J. Vannimenus. Flory approximation of directed branched polymers and directedpercolation. J. Physique, Letters, , L-377 (1982).[4] K. Suematsu. Recent Progress of Gel Theory: Ring, Excluded Volume, and Dimension. Advances in PolymerScience, , p-213 (Appendix) (2002).[5] P. J. Flory. Principles of Polymer Chemistry. Cornell University Press, Ithaca and London (1953).[6] (a) M. Fixman. Excluded Volume in Polymer Chains. J. Chem. Phys., , 1656 (1955).(b) M. Fixman. Radius of Gyration of Polymer Chains. II. Segment Density and Excluded Volume Effects. J.Chem. Phys., , 3123 (1962).[7] (a) S. F. Edwards. The theory of polymer solutions at intermediate concentration. Proc. Phys, Soc., , 265(1966).(b) S. F. Edwards. The size of a polymer molecule in a strong solution. J. Phys, A: Math. Gen., , 1670 (1975).[8] P. G. de Gennes. Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca and London (1979).[9] M. Muthukumar and S. F. Edwards. Extrapolation formulas for polymer solution properties. J. Chem. Phys., , 2720 (1982).[10] J. des Cloizeaux and G. Jannink. Polymers in Solution: Their modelling and structure. Clarendon Press,Oxford, Chapter 8 (1990).[11] M. Doi. Introduction to Polymer Physics. Clarendon Press, Oxford (1996).[12] F. Tanaka. Osmotic pressure of ring-polymer solutions. J. Chem. Phys., 87, 4201 (1987); Introduction toPhysical Polymer Science, Shokabo Publishing Co., Ltd. (1994).[13] L. Sch¨afer. Excluded Volume Effects in Polymer Solutions: as Explained by the Renormalization Group.Springer-Verlag Berlin Heidelberg (1999).[14] M. Plischke and B. Bergersen. Equilibrium Statistical Physics. World Scientific, New Jersey, Chapter 8 (1999).[15] J. P. Wittmer, P. Beckrich, H. Meyer, A. Cavallo, A. Johner, and J. Baschnagel. Intramolecular long-rangecorrelation in polymer melts: The segmental size distribution and its moments. Physical Review E, , 011803(2007).[16] L. R. G. Treloar. The Physics of Rubber Elasticity. Oxford University Press (1958).[17] A. F. M. Barton. CRC handbook of polymer-liquid interaction parameters and solubility parameters. CRCPress (1990).[18] J. M. O’Reilly, D. M. Teegarden, and G. D. Wignall. Small- and Intermediate-Angle Neutron Scattering fromStereoregular Poly(methyl methacrylate). Macromolecules, , 2747 (1985).[19] Samuel Glasstone. The Elements of Physical Chemistry. Macmillan & Co Ltd, London (1956).1120] (a) A. Ishihara. Probable Distribution of Segments of a Polymer Around the Center of Gravity. J. Phy. Soc.Japan, , 201 (1950).(b) P. Debye, and F. Bueche. Distribution of Segments in a Coiling Polymer Molecule. J. Chem. Phys., ,1337 (1952).[21] (a) J.J. Hermans and J. Th. G. Overbeek. The dimensions of charged long chain molecules in solutionscontaining electrolytes. Rec. trav. chim., , 761 (1948).(b) T. B. Grimley. Equivalent Sphere Models of Real Chains. Trans. Faraday Soc.,55