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Title Thermoreversible gelation is a Bose-Einstein condensation
Author(s) Tanaka, F
Citation PHYSICAL REVIEW E (2006), 73(6)
Issue Date 2006-06
URL http://hdl.handle.net/2433/39915
Right Copyright 2006 American Physical Society
Type Journal Article
Textversion publisher; none
Kyoto University PHYSICAL REVIEW E 73, 061405 ͑2006͒
Thermoreversible gelation is a Bose-Einstein condensation
Fumihiko Tanaka* Department of Polymer Chemistry, Graduate School of Engineering, Kyoto University, Katsura, Kyoto 615-8510, Japan ͑Received 23 July 2005; revised manuscript received 27 December 2005; published 13 June 2006͒ On the basis of the classical tree statistics, being combined with the lattice theory of a polymer solution, we prove that within the mean-field approximation thermoreversible gelation of polyfunctional molecules is a third order phase transition analogous to the Bose-Einstein condensation of ideal Bose gases. The derivative of the osmotic compressibility with respect to the concentration of primary functional molecules is shown to reveal a discontinuity at the sol/gel transition point, whose width is directly related to the amplitude of the divergent term in the weight-average molecular weight of clusters. An inclusion of cycle formation, as well as concen- tration fluctuations, may, however, change the nature of the phase transition.
DOI: 10.1103/PhysRevE.73.061405 PACS number͑s͒: 82.70.Gg, 64.60.Ϫi, 64.70.Ϫp, 64.75.ϩg
I. INTRODUCTION into the condensed phase with zero momentum, although there is no phase separation in space. Our argument ͓10,11͔ Recently, thermoreversible gelation in solutions of poly- was based on the fact such that each connected cluster made mers, as well as low molecular weight molecules, has been up of the primary molecules in the solution has a center of ͓ ͔ attracting researchers’ interest 1,2 . Typical examples of the mass translational degree of freedom ͑momentum͒ contribut- phase diagrams with sol/gel transition lines have been re- ing to the entropy of mixing. Clusters of finite size ͑sol͒ have ported in the literature. These include atactic polystyrene in finite momenta. At the gel point, the largest cluster grows to ͓ ͔ ͑ ͒ various organic solvents 3,4 , poly n-butylmethacrylate in a macroscopic network ͑gel͒. Since the network spans the ͓ ͔ ͑ ͒ 2-propanol 5,6 , poly vinyl chloride in dimethyl malonate entire solution, its center of mass ceases translational motion, ͓ ͔ 7 , etc. Many other examples can be found in the reviews thus leading to a vanishing of the mixing entropy originating ͓ ͔ 1,2 . However, the thermodynamic nature of the transition in this cluster. ͑ from a sol state to a gel state, and vice versa sol/gel transi- The gel point to be studied here is defined, in accordance ͒ tion , has not yet been clarified. Hence, there have been in- with the classical gelation theory ͓13–15͔, by the point where tense argument and serious confusion on whether or not the the weight-average molecular weight of the cluster becomes thermoreversible gelation is a real phase transition that is infinite. In other words, it is defined by the percolation point accompanied by a singularity in the physical properties. One where a connected cluster grows to the macroscopic dimen- ͓ ͔ group of researchers 8 insists that gelation is related only to sions to percolate over the entire solution. Such a definition the connectivity properties of the system, and hence should of the gel point on the basis of the connectivity of the system be a continuous change. Their argument is based on a simple is, in principle, different from the rheological gel point that is assumption such that gelation is a percolation, and since a defined by the point where the solution loses its fluidity. For percolation is a geometric transition with no thermodynamic chemical gels with covalent cross links, these two definitions singularity associated, sol/gel transition should not have any give the same point, but we do not study chemical gels here ͓ ͔ singularity. Another extremity 9 insists that the transition because they are thermodynamically irreversible. In what becomes a first order phase transition if the formation of follows we will focus on physical gels with cross links that excessive large-scale cycles in the gel part is taken into con- may break and recombine to reach thermal equilibrium. Al- sideration. Therefore, it seems to be an urgent issue to re- though solutions are percolated by the gel networks, they solve this problem. may flow by repeating association-dissociation of the cross ͓ ͔ In our previous study 10,11 , we showed that the transi- links when external stress is given. Such fluidity of the sys- tion is a third order phase transition analogous to the Bose- tem does not affect the nature of the Bose-Einstein singular- ͑ ͒ Einstein condensation referred to as BEC of ideal Bose ity, although the amplitude of the singularity depends upon gases. It is well known that an ideal Bose-Einstein gas real- the strength of the cross-link bonds described by the associa- izes a characteristic state of order when approarching to the tion constant ͑T͒ to be studied below. ͓ ͔ absolute zero temperature 12 . The transition is accompa- In the previous study, we showed ͓10,11͔ that the second nied by a thermal discontinuity. At a finite temperature given order derivative of the mixing entropy ͑third order derivative in terms of the number density of Bose molecules, the spe- of the free energy͒ with respect to the temperature or con- cific heat at a constant volume has a peak with discontinuous centration exhibits a discontinuity under the mean-field as- slope. Below this transition temperature, a finite fraction of sumptions such that ͑i͒ the free energy of clusters in the all molecules goes into the lowest state with zero momen- solution is given by the Flory-Huggins theory of polymer tum. The process is characterized as a gradual condensation solutions, ͑ii͒ the clusters take Cayley tree forms, and ͑iii͒ in momentum space from one phase with finite momentum reactivity of the functional groups is not affected by the full effect of self-avoidance on the lattice. The calculation for proving this, however, involved complicate combinatorial *Electronic address: [email protected] consideration since we treated multiple associations. There-
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⌬ ⌬ ͑ ͒ fore, the paper seemed to be difficult to access for the ma- l = l 1 2.5 jority of researchers. In what follows, we attempt to draw the same conclusion more clearly on the basis of a simpler pair- for association and dissociation. Then, we find the volume wise cross linking, and stress the analogy between thermor- fraction of l clusters to be given by eversible gelation and BEC. Since our proof of the parallel to = K l, ͑2.6͒ BEC is based on the above mean-field assumptions ͑i͒–͑iii͒, l l 1 it may fail when any of these assumptions is eliminated. where 1 is the volume fraction of the unassociated molecules, and the equilibrium constant is given by Kl II. THEORETICAL MODEL OF GELLING POLYMER ͑ ⌬ ͒ =exp l−1− l . SOLUTIONS III. APPLICATION OF THE CLASSICAL TREE We start from a model solution in which primary mol- STATISTICS FOR GELATION ecules of the molecular weight n ͑in terms of the number of ͒ ⌬ ⌬comb statistical units on them carrying the number f of functional We now split the free energy into three parts l = l ͓ ͔ ⌬conf ⌬bond groups are dissolved in a solvent 16 . For simplicity, we + l + l . To find the combinatorial part, all clusters are assume that the functionality of the primary molecules is assumed to take tree forms. The cycle formation within a monodisperse and the functional groups form pairwise bonds cluster is neglected. We consider the entropy change on com- that can break and recombine during thermal processes. We bining l identical f-functional molecules to form a single ͓ ͔ ⌬ comb employ a lattice theoretical description of the Flory-Huggins Cayley tree. The classical tree statistics 14 gives Sl ͓ ͔ ͓ l ͔ type 17–19 , and incorporate the cluster formation by using =kB ln f l , where the conventional tree statistics of the gelation reaction ͓ ͔ ͑fl − l͒! 13–15 . Our starting free energy is given by ϵ ͑3.1͒ l l!͑fl −2l +2͒! ⌬ ͑ ͒ ͑ ͒⍀ ⌬ F = ͚ Nl ln l + N0 ln 1− + 1− + ͚ lNl lജ1 lജ1 is Stockmayer’s combinatorial factor. The free energy is given by ⌬comb=−⌬Scomb/k . ␦͑͒ G ͑ ͒ l l B + N , 2.1 For the conformational free energy, we employ the lattice ͓ ͔ where Nl is the number of clusters formed by l primary mol- theoretical entropy of disorientation 19 , ecules ͑l-clusters͒, their volume fraction, N the number of l 0 n͑ −1͒n−2 solvent molecules, the total volume fraction of the primary S ͑n͒ = k lnͩ ͪ ͑3.2͒ dis B n−1 molecules, ⍀ the total number of the lattice cells, is Flory’s e ͒ ⌬ ϵ͑ؠ ؠ parameter, and l l −l 1 , the free energy change ac- for a chain consisting of n statistical units, where is the companying the formation of an l-cluster from the separate lattice coordination number, and the symmetry number of primary molecules in their standard reference state ͑super- the chain. We then find script circle͒. The last term is necessary only in the postgel regime where a gel network formed by a macroscopic num- ͑ −1͒2 l−1 ⌬Sconf = S ͑ln͒ − lS ͑n͒ = k lnͫͩ ͪ lͬ, ber NG of primary molecules exists. The free energy to bind l dis dis B en a molecule onto the gel is given by ␦͑͒ϵ͑ؠG −ؠ ͒.By 1 ͑3.3͒ differentiation, we find for chemical potentials with most probably =1. ⌬ ⌬ ͑ ͒2 l = l +1+ln l − nl + nl 1− Finally, the free energy of bonding is given by ␦Ј͑͒G͑ ͒ ͑ ͒ + nl 1− , 2.2a ⌬bond ͑ ͒⌬ ͑ ͒ l = l −1 f0, 3.4 ⌬ ͑ ͒ 2 ␦ ͑͒G ͑ ͒ 0 =1+ln 1− − + − Ј , 2.2b because there are l−1 bonds in a tree of l molecules, where ⌬ f0 is the free energy change on forming one bond. for the l cluster and the solvent molecule, where Combining all results together, we find ϵ ͑ ͒ ͚ l +1− 2.3 f l−1 lജ1 K = fl ͩ ͪ ͑3.5͒ l l n gives the total number of molecules and clusters possessing the translational degree of freedom. The summation for the equilibrium constant, where ͑ ͒ϵ͓͑ ͒2 ͔ ͑ ⌬ ͒͑͒ ϵ ͑ ͒ T −1 / e exp − f0 3.6 ͚ l 2.4 lജ1 is the association constant. gives the number density of finite clusters in the solution, We first consider the pregel regime where all clusters are and G ϵNG /⍀ is the number density of polymer chains con- finite. The total volume fraction and the total number of clus- ͑ ͒ tained in the gel network. ters in the solution are then given by using Eq. 2.6 as In thermal equilibrium, the solution has a distribution of ϱ l ͑ ͒ clusters with a population distribution fixed by the equilib- /n = ͚ l lx , 3.7a rium condition l=1
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ϱ ͑,T͒ϵ͑,T͒/n +1/͑1−͒ −2. ͑4.2͒ = ͚ xl. ͑3.7b͒ ͒ץ ץ͑ l ϵ l=1 Here, a new function is defined by n / T. This result was previously derived in the case of multiple cross ϵ ͑ Here, the parameter x is defined by x 1 /n 1 being the links ͓10͔. The singularity in the osmotic pressure originates volume fraction of primary molecules that remain unassoci- in this function: the translational entropy of clusters. From ated in the solution͒. The osmotic pressure of the solution the fundamental two relations ͑3.7a͒ and ͑3.7b͒ given above, is given by the chemical potential of the solvent through the the function is given by 3 ⌬ thermodynamic relation a =− 0. Explicitly, we have l 2 l ¯ ͑ ͒ = ͚ l lx ր͚ l lx =1/lw 4.3 3 ͑ ͒ 2 ͑ ͒ a /kBT = − −ln 1− − 3.8 ¯ in terms of the weight-average aggregation number lw of in the pregel regime. It is basically proportional to the total clusters. In what follows, we show that is continuous number of clusters since all molecules with a translational across the gel point concentration, but its derivative .͒ reveals a discontinuityץ/ץ͑ degree of freedom equally contribute to the pressure within ͑ ͒ T the ideal gas approximation. By solving Eq. 3.7a with re- To simplify theoretical analysis, let us introduce the kth ͑ ͒ spect to x, and substituting the result into Eq. 3.8 , we obtain moment of Stockmayer’s distribution by the osmotic pressure as a function of the temperature and ϱ volume fraction of the primary molecules. ͑ ͒ϵ k l ͑ ͒ ͑ ͒ Sk x ͚ l lx k = 0,1,2, ... . 4.4 l=1 IV. SIMILARITY TO BOSE-EINSTEIN CONDENSATION Then, the number- and weight-average cluster sizes are given At this stage, we readily realize that our equations are by mathematically parallel to those we encounter in the study of ¯ ͑ ͒ ͑ ͒ ͑ ͒ BEC of ideal Bose gases ͓12,20͔. The number density N/V ln = S1 x /S0 x , 4.5a and the pressure p of an ideal Bose gas consisting of N molecules confined in the volume V is given by ¯ ͑ ͒ ͑ ͒ ͑ ͒ lw = S2 x /S1 x . 4.5b ϱ All moments are monotonically increasing functions of x and 3 l 3/2 ͑ ͒ TN/V = ͚ x /l , 4.1a have a common radius of convergence l=1 x = x* ϵ͑f −2͒f−2/͑f −1͒f−1. ͑4.6͒ ϱ For xϾx* all moments diverge. Exactly on the radius of 3 l 5/2 ͑ ͒ * p T/kBT = ͚ x /l , 4.1b convergence x , S0 and S1 take finite values, but all moments l=1 with kജ2 are infinite. Since the weight-average cluster size becomes infinite at this point, x=x* gives the gel point. The where x is the activity of the molecule, and T volume fraction at the gel point is therefore given by ϵh/͑2mk T͒1/2 the thermal de Broglie wavelength. The B * /n=S ͑x*͒=͑f −1͒/ f͑f −2͒2. With further increase in the coefficient of the infinite series on the right-hand side is re- 1 volume fraction of primary molecules, the excess molecules placed from Stockmayer’s combinatorial factor to 1/l5/2, l in the part −* condense into the macroscopic cluster ͑in- but other parts are completely analogous. The infinite sum- finitely extended branched network͒ through a cascade pro- mations on the right-hand side of these equations are known ͓ ͔ 3 5 cess, and will not contribute to the number density, so that it as Truesdell’s function 21 of order 2 and 2 . Their singu- will remain a constant value * =2͑f −1͒/͑f −2͒. The larity appearing at the convergence radius x=1 was studied ͓ ͔ weight-average cluster size also remains infinite. We have, in detail 21 . Since the internal energy of the Bose gas is therefore, a kind of condensation phenomenon in momentum U pV related to the pressure as =3 /2, the singularity in the space. compressibility and that in the specific heat has the same The analogy of BEC can be seen more clearly if we re- nature and reveal discontinuity in their derivatives ͓12͔. ͑ place Stockmayer’s combinatorial factor l by its asymptotic Hence the transition condensation of macroscopic number Ӎ *−1 5/2 ͒ form l x /l for large l. This form is derived by apply- of molecules into a single quantum state turns out to be a ing Stirling’s formula to Eq. ͑3.1͒. We find third order phase transition ͓12,20,22͔. ϱ We now show that a similar picture holds for our gelling 1 x l solution; a finite fraction of the total number of primary mol- /n = ͚ ͩ ͪ , ͑4.7a͒ l3/2 x* ecules condenses into a single state ͑gel network͒ with no l=1 center of mass translational degree of freedom ͑no momen- ϱ tum͒, although we have no quantum effect. We first calculate 1 x l ͑ ͒ = ͚ ͩ ͪ . ͑4.7b͒ the dimensionless osmotic pressure defined by KT l5/2 x* ͒ l=1ץ ץϵ͑ 3͒͑ kBT/a / T / as a function of the temperature and the volume fraction. By taking the concentration derivative Thus, we can see that the singularity at x=x* is perfectly ͑ ͒ −1 2͑ ͒ of Eq. 3.8 ,wefindKT = ,T , where identical to those in Truesdell’s functions at x=1.Inour
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͓ ͔ 5 previous study 10 , we referred to this important similarity pointed out that the case c= 2 turns out to be analogous to in the case of multiple association. BEC. The Stockmayer factor in the combinatorial tree statis- A simple calculation gives tics falls also on this category. The effect of cycle formation in the sol, as well as in the gel, beyond the level of tree ¯ Ӎ ͑* ͒͑͒ lw A/ − 4.8 statistics, is however important. It is well known that the gel with a constant amplitude point shifts to a higher conversion, and modifies the critical index of divergence in ¯l . It may also modify the nature of 3 w A ϵ fn/͑f −2͒ ͑T͒͑4.9͒ the phase transition as was discussed in Ref. ͓9͔, although their argument was not plausible. ͑In the case of the first Ͻ* ¯ ϱ Ͼ* for , and lw = for . We thus find the disconti- ͒ order transition, there should be a finite concentration gapץ ץ͑⌬ nuity in the slope of the function is given by / T between the sol phase and gel phase, called “chimney,” on =1/A. This leads to a discontinuity in the osmotic compress- the temperature-concentration phase plane that comes out of ibility of the form the macroscopic phase separation region, but this has never been observed.͒ What we presented here is a proof for theץ * ץ KT ⌬ͩ ͪ =−K2ͩ ͪ⌬ͩ ͪ =−B/͑*,T͒2, existence of a third order singularity in thermoreversible gels ץ T nץ T T within the conventional mean-field description of gelation. ͑4.10͒ where VI. TREATMENT OF THE POSTGEL REGIME B ϵ f2͑f −2͒9͑T͒4/͑f −1͒3n5 ͑4.11͒ The theoretical argument given here is based on Stock- mayer’s picture ͓14͔ for treating the gelling reaction in the is a constant depending only on the temperature, functional- regime after passing the gel point. Theoretically, this is not ity, and the number of statistical units on a chain. For large the only consistent way to treat the postgel regime. There are molecular weight polymers, the amplitude B is small. This is other ways of treating it that fulfill the fundamental thermo- the main reason why an experimental detection of the singu- dynamic laws ͑Gibbs-Duhem relation͒. In fact, Flory had larity has so far been difficult. However, there is a good proposed a different picture in his old work on a gelation chance for the observation to be feasible. Quite often, the reaction of trifunctional molecules ͓13͔. In his treatment, ͑ sol/gel transition line crosses the phase separation line bin- molecules in the sol part react with those in the gel part with ͒ odal and spinodal at the top, or on the shoulder, of the phase an increase in the concentration, and as a result, formation of separation region. Such a crossing point is called either a cyclic linkages within the gel part is allowed. The weight- tricritical point or critical endpoint depending upon the rela- average cluster size ¯l in the sol goes back to a finite de- tive position of the transition curves. As we approach the w creasing function of the concentration because large clusters spinodal point ͑by changing the temperature under a fixed in the sol part are connected into the gel network as reaction concentration, for instance͒ where the condition ͑* ,T͒ proceeds. Upon adopting this treatment, we readily find =0 is satisfied, the discontinuity is enhanced by critical fluc- ͑ ͒ tuations as seen from Eq. 4.10 , and there may be a chance ¯l Ӎ A/͉ − *͉͑6.1͒ to observe the singularity. w near the gel concentration, and hence the discontinuity in the ͒ץ ץ͑⌬ slope of is doubled as / T =2/A. The third order V. BOSE-EINSTEIN CONDENSATION IN CLASSICAL nature of the phase transition remains the same as in Stock- STATISTICAL MECHANICS mayer’s picture, although the molecular mechanism of con- There are several other condensation phenomena in clas- densation and hence the amplitude of the discontinuity are sical statistical mechanics that are similar to BEC. Jacobson different. These two treatments do not contradict fundamen- and Stockmayer ͓23͔, for example, pointed out that, for lin- tal laws of thermodynamics, and both are equally acceptable. ear polycondensation systems with chains and rings, it is Their comparison and application to phase diagrams were ͓ ͔ possible to obtain 100% yield of rings beyond a certain criti- detailed in our preceding paper 11 . cal dilution by driving the polycondensation to completion. To study the postgel regime in more detail, let us intro- ␣ The origin of this singularity comes from the entropy of duce a new parameter through the positive root of the rings. They showed that, within Gaussian chain statistics, the equation l 5/2 population of rings of size l is proportional to x /l , where x x ϵ ␣͑1−␣͒f−2. ͑6.2͒ is the molar concentration of the primary molecules that re- main unassociated in the solution. Similar singularity, and Then, the first three moments of Stockmayer’s distribution occurrence of a phase transition, was reported in the study of are explicitly calculated as ͓14͔ the melting of nucleic acids ͓24͔. In the melting of duplex S ͑x͒ ␣͑ f␣ ͒ f͑ ␣͒2 ͑ ͒ DNA, the entropy of loops formed by the complementary 0 = 1− /2 / 1− , 6.3a chains due to partial dissociation of the hydrogen bonds ͑ ͒ ␣ ͑ ␣͒2 ͑ ͒ gives rise to a singularity. The nature of the phase transition S1 x = /f 1− , 6.3b depends upon the index c characterizing the loop entropy in c ͑ ͒ ␣͑ ␣͒ ͓ ͑ ͒␣͔͑ ␣͒2 ͑ ͒ the form 1/l for a large number of monomer units l.Itwas S2 x = 1+ /f 1− f −1 1− . 6.3c
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To see the physical meaning of ␣, let us calculate the proposed a third model which takes the reaction between sol extent of the reaction, i.e., the probability for a randomly and gel network into account, while the cycle formation in chosen functional group to be associated. Since an l cluster the gel is forbidden as in Stockmayer’s model. This third includes the total of fl groups, among which 2͑l−1͒ are as- treatment, however, turned out not to fulfill Gibbs-Duhem ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ ␣ ͓ ͔ sociated, it is given by 2 S1 x −S0 x / fS1 x = . Thus, it relation when applied to thermoreversible gelation 11 . turns out that ␣ introduced by the formal relation ͑6.2͒ actu- Later, to ensure the equilibrium distribution, additional terms ally gives the extent of reaction. By using ␣, the average describing reversible reaction ͑fragmentation͒ were intro- cluster sizes are given by duced to the kinetic equation by van Dongen and Ernst ͓27͔. Since their study was limited only to Flory’s and Stockmay- ¯ ͑ ␣ ͒ ͑ ͒ ln =1/ 1−f /2 , 6.4a er’s model, the possibility of another new treatment within the classical tree statistics remains unexcluded. From the ¯ ͑ ␣͒ ͓ ͑ ͒␣͔ ͑ ͒ mathematical analysis given in this paper, however, it is lw = 1+ / 1− f −1 . 6.4b highly probable that a new thermodynamically consistent Since the weight-average diverges at ␣=␣* ϵ1/͑f −1͒, the treatment, even if it exists, leads to the third order singularity extent of the reaction at the gel point is given by ␣*. This lying somewhere between Stockmayer’s one and Flory’s one. value corresponds to the maximum of x=x* in Eq. ͑6.2͒.At low concentration with the extent of reaction smaller than the VII. CONCLUSIONS AND DISCUSSION critical value ␣*, Eq. ͑6.2͒ gives a unique value of ␣ for a given x. On passing the gel point, ␣ becomes larger than ␣*, We have shown that, within the mean-field treatment of so that Eq. ͑6.2͒ gives another root ␣Ј that lies below ␣*. both gelation reaction and polymer solution, thermorevers- Flory ͓13͔ adopted this lower root as the extent of reaction in ible gelation is a third order phase transition analogous to the the sol part, while ␣ is regarded as that of the total solution. BEC. The gel network corresponds to a Bose condensate In the present context of thermoreversible gelation, the first with no translational motion ͑momentum͒ as a whole. There equation ͑3.7a͒ remains valid in the form is a frequent exchange of molecules between the gel network ͑with no momentum͒ and the sol part ͑with finite momen- /n = S ͑␣͒͑6.5͒ 1 tum͒, but on thermal average a finite fraction ͑gel fraction since it gives the total volume fraction. The second one is wG =1−wS͒ in the number of molecules loses their momenta changed to in the postgel regime. The effect of fluctuations in cluster formation is important S S ͑␣Ј͒ ͑ ͒ = 0 , 6.6 near the transition point as it usually is in many phase tran- because the number of clusters can be counted only for finite sitions. It may change the nature of the transition. In the clusters ͓25͔. ͑Superscript S indicates the sol part.͒ The vol- present gelation problem, however, one has to consider it ume fraction of the sol part is similarly given by from two sides. Firstly, the effect of cycle formation during the gelation reaction beyond the classical tree statistics must S ͑␣Ј͒ ͑ ͒ /n = S1 , 6.7 be considered. Secondly, concentration fluctuations in the S ͑␣ ͒ ͑␣͒ polymer solution must be considered beyond the mean-field leading to the sol fraction w =S1 Ј /S1 . The function that appeared in the compressibility now includes the term Flory-Huggins treatment. These two different aspects of fluc- proportional to tuations should be studied from a unified theoretical point of view without violating the fundamental law of thermody- ͒ ͑ ͒ ¯Sץ Sץ͑ n / T =1/lw, 6.8 namics. This is an open problem. The critical exponent and the nature of the singularity may be changed by these fluc- where tuation effects at extremely near the transition point. The criterion for the mean-field prediction to be valid ͑Ginzburg ¯lS ͑␣Ј͒ = ͑1+␣Ј͒/͓1−͑f −1͒␣Ј͔͑6.9͒ w criterion͒ should also be found in order to characterize non- refers to the weight-average cluster size in the sol part in the classical singularity in the experimeriments. This problem postgel regime. Thus, we find a discontinuity in the slope of also lies beyond the scope of the present study. the osmotic compressibility also in Flory’s treatment. As for the absolute value of the discontinuity in the slope The difference in the above two treatments were later ex- of the osmotic compressibility, closer experimental observa- amined from a kinetic point of view. For an irreversible re- tion near the spinodal point with enhanced singularity by action, Ziff and Stell ͓26͔ clarified the reaction mechanism critical fluctuations of phase separation is necessary since the ͑sol-gel interaction͒ in the two treatments after the gel point amplitude B is small for polymeric gels with nӷ1. ͑For a is passed. They found that in Stockmayer’s treatment reac- trifunctional molecules with f =3, for example, we have B tive groups in the sol do not interact with those in the gel, =94 /8n5.͒ For this purpose, measurements near the tricriti- and the gel grows through a cascade process of the sol into cal point, the point where the sol/gel transition line ͑continu- the gel, while in Flory’s treatment all functional groups are ous transition͒ crosses the binodal ͑discontinuous first order allowed to react. On the basis of such a kinetic study, they transition͒ at the top of the miscibility gap, are suitable.
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