Diffusion and Chemical Potential in Polymer Solutions
Kazumi Suematsu, Haruo Ogura, Seiichi Inayama, Toshihiko Okamoto
aa r X i v : . [ c ond - m a t . s o f t ] M a r Theory of Excluded Volume Effect
Diffusion and Chemical Potential in Polymer Solutions
Kazumi Suematsu † , Haruo Ogura † , Seiichi Inayama † and Toshihiko Okamoto † Institute of Mathematical ScienceOhkadai 2-31-9, Yokkaichi, Mie 512-1216, JAPANE-Mail: [email protected], Tel/Fax: +81 (0) 59 326 8052
Abstract
We give a mathematical proof for the preceding derivation of the excluded volume theory on the basisof the concept of diffusion, chemical potential, and the theory of total differential.
Key Words : Diffusion/ Chemical Potential/ Total Differential/ Excluded Volume Effects/
Generally, diffusion occurs from a concentrated region to a dilute region so as to reduce the concen-tration gradient. As has been well understood, however, this is not always true. The excluded volumeproblem of polymers is a good example in which this empirical rule is not applicable. Since monomersare joined by chemical bonds, there is retraction force due to the rubber elasticity, so that the systemmust maintain the balance between the forward force due to diffusion and the backward force due toelastic retraction. It thus makes possible the reverse flow of segments from the dilute region to theconcentrated region against the concentration gradient. For this reason, the complete homogeneity ofsegment concentration can not always be expected for polymer solutions. In this short paper, we willinvestigate this problem on the basis of the concept of diffusion and chemical potential, and then givea mathematical proof for the equation derived in the preceding paper [4].
Let n i be the number of molecules of species i . The basic thermodynamic equation is dG = V dP − SdT + X i (cid:18) ∂G∂n i (cid:19) T,P,n j dn i (1)Consider a two-component system consisting of n solvent molecules and n solute molecules. Supposethat there are two phases in the system, phase I and phase II which have an equal volume size( V I = V II ); for instance, the phase I is rich in n and II is rich in n . Let n and n be able to diffusefreely between these phases. In this paper, we use the word, phase, in a broader meaning; e.g., it † The author takes full responsibility for this article. † Kitasato University † Keio University † Tokyo University II Fig. 1: Segment migration between the phase I and the phase II. does not necessarily mean the abrupt change between two phases such as the
Gas - Liquid transition.Under constant T and P , we have dG = dG + dG = µ n I dn I + µ n II dn II + µ n I dn I + µ n II dn II (2)The constraint conditions are n I + n II = n = const. and n I + n II = n = const. , so that dn I = − dn II dn I = − dn II (3)Substituting Eq. (3) into Eq. (2), we have dG = ( µ n II − µ n I ) dn II + ( µ n II − µ n I ) dn II (4)If the diffusion occurs spontaneously, we must have dG = ( µ n II − µ n I ) dn II + ( µ n II − µ n I ) dn II ≤ dG = 0. If dn II and dn II are independent of each other (such as in the case of a mixingprocess of two ideal gases partitioned by a certain membrane), the equal sign holds for arbitrary dn II and dn II , so that each term in Eq. (5) must be equal to zero. Note that the differential operators, dn II and dn II , are quantities that approach to zero without limit, but not numbers in the ordinarysense, so these cannot be equated with zero. The solution is thus µ n I = µ n II µ n I = µ n II (6)This is the familiar conclusion for the phase equilibria. However, the differential operators dn I and dn I are, in general, not independent. Let us consider the simplest case in which a solvent and a solute have the same molecular volume,and that a substitutional diffusion process occurs so that the flow of one solvent molecule from thephase I to the phase II is concurrently accompanied by the displacement of one solute molecule fromthe phase II to the phase I − the substitutional diffusion, of course, does not exclude the displacementbetween the same molecular species: solute-solute or solvent-solvent in which the Gibbs potentials ofrespective phases do not change. The excluded volume problem of a polymer will correspond to thiscase. In such a case, additional restraint conditions should occur: n I + n I = const.n II + n II = const. (7)2o that dn I = − dn I dn II = − dn II (8)Then, G can be expressed as a one-component-function of n or n , and we have µ n I = (cid:18) ∂G∂n I (cid:19) T,P = (cid:18) ∂G∂n I (cid:19) T,P dn I dn I = − (cid:18) ∂G∂n I (cid:19) T,P = − µ n I µ n II = (cid:18) ∂G∂n II (cid:19) T,P = (cid:18) ∂G∂n II (cid:19) T,P dn II dn II = − (cid:18) ∂G∂n II (cid:19) T,P = − µ n II (9)which result in the equality: ( µ n II − µ n I ) dn II = ( µ n II − µ n I ) dn II (10)If dG is a function of n or n alone, then we have µ n = 0 or µ n = 0. This means that either of theterms of the right-hand side in Eq. (5) must be abolished, which yields, for instance, dG = ( µ n II − µ n I ) dn II (11)This is a natural consequence, because the two opposite flows of n ( I → II ) and n ( II → I ) concur-rently occur. The above discussion must be modified for the excluded volume problem in polymer solutions, since,in addition to the diffusive migration of segments, there is retraction force due to the rubber elasticityof a polymer. The total free energy of a polymer must be written as G = G diffusion + G elasticity , and itmust satisfy dG = dG diffusion + dG elasticity ≤ dG = dG diffusion + dG elasticity = ( µ n II − µ n I ) dn II + dG elasticity ≤ dG must take the minimum value. dG diffusion The fundamental equation [1] of the Gibbs potential of mixing pure solvent and pure polymer melt is:∆ G mixing = kTV Z (cid:26) − (1 − χ ) ˆ v + (1 / − χ ) ˆ v + 16 ˆ v + · · · (cid:27) δV (14)where V denotes the system volume and V the volume of a solvent molecule, and ˆ v representsthe volume fraction of polymer segments in the local area δV . Let N be the number of segmentsconstituting a polymer. Assuming the Gaussian distribution of segments around the center of gravity,ˆ v can be expressed in the form:ˆ v = V ˆ C = V N (cid:18) βπα (cid:19) / X { a,b,c } exp (cid:26) − βα (cid:2) ( x − a ) + ( y − b ) + ( z − c ) (cid:3)(cid:27) ≡ V N (cid:18) βπα (cid:19) / ˆ G ( x, y, z ) (15)3here V denotes the volume of a segment, and β = 3 / h s N i has the usual meaning (the subscript0 denotes the unperturbed dimensions); { a, b, c } signifies the location of individual polymers, so thatˆ C represents the number concentration of segments at the coordinate ( x, y, z ) with the symbol ˆdenoting the Gaussian approximation specified by Eq. (15). In this problem, n corresponds tosolvent molecules, and n corresponds to segments. Note that ˆ C is a function of ˆ v ; so ∆ G mixing isa one-variable-function of n . As we can see from Eq. (15), a polymer solution is an inhomogeneoussystem of the segment concentration. The inhomogeneity causes physical instability of the solution.To acquire the stability, a polymer must reduce the inhomogeneity by expanding the mean radius andmaking its segments overlap with those of other molecules.Because of the wild inhomogeneity of segment concentration, we may regard a polymer solution asconsisting of two phases, a concentrated region (phase II) and a dilute region (phase I) which is putin contact with each other.To proceed with our discussion, it is useful to modify the basic thermodynamic equation [4].Multiply Eq. (1) by V /V , and we have dG = V dP − SdT + X i (cid:18) ∂G∂c i (cid:19) T,P dc i (16)where c i represents the number concentration of molecular species i . Thus we can introduce a newdefinition of the chemical potential as a measure of the rate of the change of Gibbs potential as againstthe change of solute concentration [3] under constant T and P : µ c i = (cid:18) ∂G∂c i (cid:19) T,P (17)Using Eq. (17), Eq. (13) may be recast in the form: dG = dG diffusion + dG elasticity = ( µ c II − µ c I ) dc II + dG elasticity (18)The equilibrium point is determined by the force balance between the expansion of a molecule dueto the diffusive force of segments and the retraction force due to the rubber elasticity; it correspondsto the minimum point of G . To find the equilibrium point, differentiate Eq. (18) with respect to theexpansion factor, α , to yield dGdα = ( µ c II − µ c I ) dc II dα + dG elasticity dα = 0 (19)Eq. (19) is a basic equation for the excluded volume effects of polymer solutions.To apply Eq. (19) to real problems, what we must do is only to express this equality in terms ofpolymer solutions, making one-to-one correspondence of the notations between the thermodynamicformula (Table 1), namely µ c = (cid:18) ∂G∂ ˆ v (cid:19) T,P d ˆ v d ˆ C (20)where ˆ v is given by Eq. (15), and ˆ C = N (cid:18) βπα (cid:19) / ˆ G ( x, y, z ) (21)Eq. (20) is common to both the phases I and II, since the specification of the phases is determined onlythrough the integral operation with respect to dV = dxdydz . By Eqs. (14) and (15), d ˆ v /d ˆ C = V and µ c = (cid:18) ∂G∂ ˆ v (cid:19) T,P d ˆ v d ˆ C = kT V V Z Z Z (cid:26) − (1 − χ ) + (1 − χ ) ˆ v + 12 ˆ v + · · · (cid:27) dxdydz (22)4 ˆ C/∂α can be calculated directly using Eq. (21) to yield ∂ ˆ C∂α ! = − α ˆ v V + 2 βNα (cid:18) βπα (cid:19) / X { a,b,c } s ( x, y, z, a, b, c ) exp (cid:26) − βα s ( x, y, z, a, b, c ) (cid:27) (23)where s ( x, y, z, a, b, c ) = ( x − a ) + ( y − b ) + ( z − c ) .Table 1: One-to-one Correspondence of NotationsThermodynamic equation Excluded volume problem n Number of solvent molecules n Number of segments ( N ) c Segment concentration ( ˆ C ) v Fraction of solvent ( v ) v Fraction of segments (ˆ v )Phase I valley : Outside of a polymerPhase II hill : Inside of a polymer dG elasticity The expression for the elastic potential is already given in the previous works [1, 2, 4]: (cid:18) ∂ ∆ G elasticity ∂α (cid:19) T,P = − T (cid:18) ∂ ∆ S∂α (cid:19)
T,P = 3 kT ( α − /α ) (24) Substituting Eqs. (22) and (24) into Eq. (19), and taking difference between the phase I and thephase II, we have α − /α = − V V (Z Z Z (cid:18) (1 − χ ) ˆ v II + 12 ˆ v II + · · · (cid:19) ∂ ˆ C∂α ! II dxdydz − Z Z Z (cid:18) (1 − χ ) ˆ v I + 12 ˆ v I + · · · (cid:19) ∂ ˆ C∂α ! I dxdydz ) (25)In Eq. (25), we have made use of the aforementioned assumption that the phases I and II have anequal volume ( V I = V II ). Since the change in order of the integral and the differentiation does notchange the result, Eq. (25) may be recast in the form: α − /α = − V ∂∂α (cid:26)Z Z Z (cid:18)(cid:0) − χ (cid:1) ˆ v II + 16 ˆ v II + · · · (cid:19) dxdydz − Z Z Z (cid:18)(cid:0) − χ (cid:1) ˆ v I + 16 ˆ v I + · · · (cid:19) dxdydz (cid:27) (26)which is exactly the result in the preceding paper [4], if we alter the notations as I → valley and II → hill . As a consequence, the preceding formula derived intuitively has gained a mathematicalbasis. 5 eferences [1] (a) P. J. Flory. Principles of Polymer Chemistry. Cornell University Press, Ithaca and London(1953).(b) P. J. Flory, Statistical Mechanics of Chain Molecules. Interscience Publishers, John Wiley &Sons, New York (1969).[2] L. R. G. Treloar. The Physics of Rubber Elasticity: Third Edition. Oxford University Press,Oxford (1975).[3] (a) P. G. de Gennes. Statistics of branching and hairpin helices for the dAT copolymer. Biopoly-mers, , 715 (1968).(b) P. G. de Gennes. Collapse of a polymer chain in poor solvents. Journal de Physique Lettres,
36 (3) , pp.55-57 (1975).(b) P. G. de Gennes. Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca andLondon (1979).[4] (a) K. Suematsu. Minor Amendment of the Local Free Energy. arXiv:1012.2505 [cond-mat.soft]12 Dec 2010.(b) K. Suematsu. Concentration Dependence of Excluded Volume Effects. arXiv:1106.5488[cond-mat.soft] 3 Jul 2011; Colloid Polym. Sci. , 481 (2012).(c) K. Suematsu. Coil Dimensions as a Function of Concentration. arXiv:1208.0097 [cond-mat.soft] 1 Aug 2012.(d) K. Suematsu. Molecular Weight Dependence of Excluded Volume Effects. arXiv:1310.6135[cond-mat.soft] 26 Jan 2014.(e) K. Suematsu. Radius of Gyration of Randomly Branched Molecules. arXiv:1402.6408[cond-mat.soft] 26 Feb 2014.(f) K. Suematsu. Excluded Volume Effects of Branched Molecules. arXiv:1606.03929v3 [cond-mat.soft] 29 Dec 2016.(g) K. Suematsu. Volume Expansion of Branched Polymers. arXiv:1709.08883 [cond-mat.soft] 26Sep 2017.(h) K. Suematsu, Haruo Ogura, Seiichi Inayama, and Toshihiko Okamoto. Alternative Approachto the Excluded Volume Problem: The Critical Behavior of the Exponent νν