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Entropy of Mixing: Rigid vs Flexible Molecules: Effect of Varying Solvent on Dissolution Temperature E. B. Sirota,*,† H. Rangwalla,†,‡ and P. Peczak† †

Corporate Strategic Research, ExxonMobil Research and Engineering Company, Route 22 East, Annandale, New Jersey 08801, United States ‡ ExxonMobil Chemical Company, Baytown, Texas 77520, United States ABSTRACT: We report a study of the dissolution temperature of nhexacontane, as a function of concentration, with 52 different solvents, aimed at understanding the effect of molecular flexibility on the entropy-of-mixing. The entropy-of-mixing of rigid molecules may be expected to follow ideal mixing combinatorics and go as ln(mole fraction), while for flexible polymers it is expected to follow Flory−Huggins ln(volume fraction). By isolating the entropy-of-mixing, we have experimentally found that rigidity, through ring structures, causes deviations from the Flory−Huggins behavior; and we have proposed a crossover form for the entropy-of-mixing which varies between ln(volume fraction) and ln(mole fraction) according to molecular rigidity.

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INTRODUCTION The Flory−Huggins form for the entropy-of-mixing (ΔSmixing) “x ln ϕ” is well-known by those who study flexible polymers, as is the ideal “x ln x” form, used by those who consider small rigid, nonchain-like molecules. (Here x is mole fraction and ϕ is volume fraction.) Two natural questions are as follows: How large do molecules have to be, before you need to use “x ln ϕ”, and how should larger molecules with varying degrees of rigidity be treated? The origins and applicability of these simplifications and limiting behaviors, and deviations from them in more complex systems are a subject of a great deal of investigation.1−15 There is practical interest in predicting the phase behavior of systems which contain mixtures of, not only polymers and solvents, but also molecules from petroleum and petroleum residua, which appear in many products including lubricants, fuel oil and asphalt. Such molecules range in molecular weights up to a few thousand. They range from flexible paraffins (i.e., short polyethylene), to relatively rigid polynuclear aromatics; and include molecules combining both types of functionalities, including molecules with multiple rigid multi-ring cores connected by more flexible aliphatic chains.16−18 If we assume the two above-mentioned approaches to the entropy-of-mixing are valid in their respective limits, it is then apparent that we could not use either the “x ln x” or “x ln ϕ” forms for our problem, since the molecules of interest span both extremes. In mixtures of hydrocarbons, especially complex mixtures, the excess volume of mixing is usually quite small, much smaller than for hard spheres of differing size. Because of the soft interaction between hydrocarbons, and their wide variety of shapes, especially in multicomponent complex mixtures, they tend to fill space and therefore do not exhibit large changes of volume on mixing. It is in this context and within this approximation that we are trying to understand the behavior. In © XXXX American Chemical Society

addition, in such complex mixtures, the specifics of the packing of individual species tend to smear out and become less important. On the question of the entropy-of-mixing for more rigid (i.e., nonflexible chain-like) objects, Flory1,2 did point out that the behavior appeared to be better described by the ideal entropyof-mixing, “x ln x”, than by “x ln ϕ” when the larger molecule was more globular, rather than linear (i.e., where the solute was not a coil which was penetrateable by the solvent). “If they are homologous chain-molecules then polymer solution theory should hold. If their diameters, or cross sections differ notably, then a mixing entropy between the ideal and the polymer formulations may be expected”.1 A large, 2-dimensional rigid structure would not behave like the coil either, although differences with a 3D globular structure would also be present, especially with regards to the enthalpic interactions which scale as the surface area. There have been many approaches used to calculate the entropy-of-mixing. These include continuum approaches; for example, computing the entropy in the liquid, as one would in a gas, employing the idea of “free volume” in the liquid state, and making assumptions of how that “free volume” scales with molecular size.3,4 There are also more sophisticated lattice models6,7 with higher-order corrections, including lattice cluster theory;8 and equation-of-state approaches9,10 which were used to calculate the case of hard spheres exactly. Each approach has its limitations, especially when being applied to our nonspecific liquid system with the assumptions about packing and volume described above. Calculations of the hard spheres of different sizes did indeed suggest that the ideal Received: September 26, 2011 Revised: April 27, 2012

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dx.doi.org/10.1021/ma202151g | Macromolecules XXXX, XXX, XXX−XXX

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Entropy of Mixing. First we look at ΔSmixing of identical objects and provide arguments showing that the “x ln(x)” form is indeed reasonable for nonflexible condensed molecules of different sizes. It is convenient and useful to consider a lattice, and ask how many ways can one populate N sites with n solute molecules, and N − n identically sized solvent molecules (see Figure 1a). (Even though drawn as such, we should not think of

entropy-of-mixing was a closer approximation than the Flory− Huggins form.9 However, hard spheres of different sizes have non-negligible excess volume of mixing at constant pressure, making it not a definitive model for the type of system of interest here. The application of the continuum theory making use of free volume gives results which totally depend on how the volume scales with molecular size. Recognizing that reality lies between the two limiting forms with bulky molecules tending to behave more like “x ln x”, Prausnitz had proposed a crossover form,11 different from the one proposed here. We do not intend to give a full review of the subject here, or to prove that our assumptions are true; we only suggest that for they may be reasonable. The question of the applicability of the ideal or Flory−Huggins entropy to rigid bulky molecules in a general sense, is thus not simply resolved; and will not be, with this contribution. In statistical mechanics, the derivation for ΔSmixing for identically sized particles (on a lattice or by a continuum treatment) exactly gives the “x ln x” form, and we will assume (and give arguments) that it should be reasonable for liquid mixtures of different sized, rigid particles. For the case of polymer chains, Flory derived a form, where it is the volume fraction, rather than the mole fraction, inside the logarithm.19 By generalizing Flory’s statistical mechanics derivation of “x ln ϕ”, we have obtained a form for ΔSmixing which behaves as “x ln x” for fully rigid molecules, “x ln ϕ” for fully flexible polymers, and has a crossover, associated with the number of flexible groups. Simply, our proposed form behaves as x ln(β), where β is the “blob fraction”;20 Specifically, on a per unit mole basis: ΔS“xlnβ” = -R∑ixi ln βi, where βi = xiBi/∑jxjBj. Here we associate a “blob number” (Bi) with every molecular species, analogous to the molecular volume. Bi represents the number of freely placeable groups in a molecule. Rigid polynuclear aromatics move as a unit, while a long aliphatic chain has many degrees of freedom in this respect. Where both species in the mixture have the same functionalities, but differ only in degree of polymerization, the blob-fraction equals the volume-fraction, and the Flory−Huggins form is recovered. It is useful to express our proposed form, generally for multiple components, written in terms of molecular mole fractions (xi), and the number of flexible segments (“blobs”) per molecule (Bi). ΔSmixing /R = −∑ xi ln xi + ln(∑ xiBi ) i

i

Figure 1. (a) Schematic showing populating a lattice with two species of identical sized particles. (b) Following the same combinatorics, and trying to populate the crystalline lattice with different sized objects. (c) Following the same combinatorics, but relaxing the requirement of crystalline order in the lattice, as is appropriate for liquids.

these as necessarily spherical, or hard.) The mole fraction is x = n/N. With N possible locations for the first molecule and N − (n − 1) for the nth, and accounting for the fact that all the molecules of each species are identical, one obtains Ω=

N! (N − n) ! n!

and applying Stirling’s formula ΔSmixing /R =

ln(Ω) = −x ln x − (1 − x) ln(1 − x) N (2)

which is the familiar “x ln x” form. What if the particles are not identical in size? In one dimension, it is obvious that the size of the objects never makes a difference, since the number of configurations depends only on the order in which they are put down, without any regard to their relative size; and there are no packing problems. For three dimensions, imagine that they are initially equal sized balloons on a lattice and we inflate some and deflate others. One would still have the same combinatorics regarding which balloons were inflated and deflated. If different sized objects are placed randomly on a uniform lattice, then one runs into excluded volume (overlap) problems, as in Figure 1b. But if they can be allowed to rearrange from their original lattice positions, then they will fit, as in Figure 1c. With the lattice sites changing volume as necessary and moving around to pack, the combinatorics of the above equations regarding the population of the sites is the same as for particles of the same size, and ΔSmixing for different sized rigid objects would still be given by eq 2. The varying of cell volumes might, at first, suggest a potential problem (if taking the lattice too seriously): The long-range order of the lattice will be destroyed; The coordination number of the lattice will not be constant (i.e., some species will have different number of nearest-neighbors than others). In addition, one may be concerned about the entropy associated with the destruction of that lattice order when going from the pure state of identical molecules on a perfect lattice, to the mixture where the order is destroyed.

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This is discussed in detail below and in the Appendix. In many formulations and approaches to solution behavior, an entropy-of-mixing form is assumed, and all nonidealities or any unaccounted-for effects are lumped into the “activity coefficient”, which is then often empirically determined.21,22 These effects may be a combination of both enthalpic interaction effects and entropic effects. While it is useful to use such approaches to fit data and make practical predictions, it does not let us look carefully at the underlying physics. By attempting to isolate or limit the impact of all effects but one, in the present case, ΔSmixing, we can hope to develop a more complete understanding. We carried out experiments on the solubility of nhexacontane in a large number of solvents to determine the effect of the molecular flexibility on the phase behavior, specifically ΔSmixing, and to what extent such a crossover form for ΔSmixing is warranted. B


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Lattice models are often used in studying systems where the lattice is really crystalline, however in this system it is not. Here we only used the lattice for convenience. So far we never accounted for the fact that we are attempting to calculate ΔSmixing in liquids, not crystalline alloys. Here is where the liquid state comes in: We are attempting to calculate the free energy of mixing components which are already in their liquid state, so long-range lattice order does not exist in either the mixed or demixed state. The constraint of a positionally ordered lattice, need not and indeed should not be enforced. The liquid state of almost all pure liquids has a density lower than that of the crystal, meaning that there is “free volume” available to allow the many configurations that distinguish the liquid from the ordered crystal. There is no extended positional order, and all molecules have a spread in the number of nearest neighbors. While higher-order corrections to the statistics are certainly present, we aim to capture the essential features that enable thinking about this crossover, in an intuitive, visual way, to try to understand the underlying physics. If we consider Flory’s derivation of ΔSmixing for chains on a lattice, but where each lattice site is not necessarily occupied by a monomer of a constant volume, but rather a rigid element or “blob” of arbitrary size, then the procedure and combinatorics of Flory’s derivation are unchanged, leading to ΔS = −R∑ixi ln βi, and expanded as eq 1. These calculations are shown in the Appendix. This crossover expression for the entropy-of-cmixing can also be obtained23 using either a method similar to Huggins’ derivation of the entropy of random mixing5 or the free-volume approach of Hildebrand.4

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Figure 2. Schematic showing the experimental apparatus used to measure the dissolution temperature. The transmitted and scattered light were monitored. The temperature setting and stir-bar on/off were under computer control.

EXPERIMENT

In these experiments, we measured the equilibrium dissolution temperature of a crystalline n-alkane, as a function of concentration, in different solvents, varying both molecular weight and structure of the molecules. The n-alkanes crystalline component was nC60 (nhexacontane), as obtained from Aldrich. The experimental setup is shown schematically in Figure 2, and involves monitoring light transmission and scattering while scanning temperature, to determine the dissolution temperature of the nC60. The solutions were contained in a quartz cell with a square cross section and a path length of 1 cm. The solute and solvent were brought well above the dissolution temperature and homogenized with a computer controlled magnetic stir-bar. The temperature was then rapidly dropped below the crystallization temperature, by opening a valve for a cool circulating bath. The intensity of the transmitted light through the sample was monitored using a fiber optic connected to a photodiode. In addition, the scattered intensity at 45° was monitored using a photomultiplier tube. However, it was the transmitted intensity which was used for the data analysis reported here. With the stir-bar turned on, the sample temperature was ramped up slowly at 0.1 °C/min, while monitoring the transmitted intensity. The dissolution temperature ideally is the temperature at which the sample becomes clear. When analyzing the data, we called the dissolution temperature the intersection of the extrapolation of the fast rising portion of the transmitted intensity curve, and the transmission baseline value when the samples were clear. This determination was consistent with the measurements of the disappearance of scattered intensity, and easier to determine reliably. The melting profile was typically repeated 4 times on each sample. For each solvent−solute pair, four mixtures were characterized, spanning about a decade in concentration, and in the vicinity of mole fraction x = 0.001. A typical series of transmission−temperature data at different dilutions is shown in Figure 3. The solvents used were n-alkanes from nC9 to nC18, n-alkyl cyclohexanes, from cyclohexane to nC13-cyclohexane, alkyl-benzenes,

Figure 3. Transmission versus temperature data for a series of concentrations of nC60 in nC11-cyclohexane. and various other compounds including decalin. All were commercially obtained from TCI or Aldrich. Table 1 lists the compounds, their molecular weight (MW), molecular volume and a nominal solubility parameter discussed below. In addition, we list the carbon number (C#), and the number of those carbons which are in, or directly connected to ring structures (raC#) which we call “ring-associated” carbons. This table also contains tabulations of the experimental results which are discussed below. Since the main focus and intent of this study was to look at entropyof-mixing effects, we chose mostly solvents which are not strongly polar or aromatic, so the enthalpy of mixing with the alkane solutes would be small. However, there are still finite contributions to the enthalpy-of-mixing which will affect the dissolution temperature. To account for these consistently, we have employed solubility parameters obtained by group additivity.24 The values we used are included in Table 1, thus facilitating the recalculation of derived quantities using any other measure of the enthalpic interactions. We are well aware that the solubility parameter approach is only an approximation for the enthalpic interactions. In this work we are using the simplest approximation where we treat each molecule as spatially homogeneous in its interactions, and calculate the total enthalpy in mean-field theory, approximating the molecules as being randomly located. In the case where the interactions are not too strong, this is not an unreasonable approximation. However we are aware that, in general, molecules will be in nonrandom configurations such that the local composition in the vicinity of a given molecule is different from the bulk average composition, causing ephemeral clustering of likemolecules. In addition, molecules with different spatially separated functional groups, will be expected to have different populations of neighbors in the vicinity of the different functionalities. We have, C


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Table 1. Listing of the Solvents Used, Including the Short Acronym and the Chemical Namea

a

We also list the MW, carbon number (C#), number of ring carbons (rC#), the solubility parameter [in (J/mol)1/2] as described in the text, the molar volume and the enthalpic interaction (V2χ). Based on the parameterization of the experimental results, we list Td at x = 0.001 and the slope of Td(ln(x)) curve and the values obtained for the parameter Z, as described in the text.

D


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The first term is the difference between the entropy-ofmixing where some of the solute is in the liquid phase and, where all of the solute is in the liquid phase. The solute in the solid phase gives no contribution to the entropy-of-mixing, since the solid phase fully excludes the solvent. The second term is the free energy of melting, for the solute. Here, ΔG is the free energy difference between the crystalline and melted state of the alkane, at a given temperature (T). In the simplest case this is

therefore, chosen molecules for this study which tend to minimize these interactions, focusing on saturated ring compounds rather than aromatics, and avoiding highly polar solvents. We, however, limited ourselves in this first study, to compounds which were commercially available in sufficient quantities.

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DATA ANALYSIS Solution Theory. We analyzed our data in the context of the regular solution theory approach given below. We define as follows: x  solute mole fraction (total in mixture) θ  fraction of solute which is crystallized Ψ  liquid-phase solute mole fraction Thus, Ψ = x(1 − θ)/(1 − θx). In addition to mole fractions, we also consider the volume fraction, and the “blob fraction”, as we had discussed earlier. The “blob fraction” is like the volume fraction, but weighted by how many independently positionable units there are in a molecule, rather than the molecular volume. ϕ0  solute volume fraction (total in mixture) β0  solute blob fraction (total in mixture) ϕ  liquid-phase solute volume fraction β  liquid-phase solute blob fraction For subscripts, we define the following convention: 1 = solvent 2 = solute with V1, V2  molar volume B1, B2  blob number We define the following ratios:

ΔG = ΔSm(T − Tm) Here Tm is the melting temperature of the pure solute, and ΔSm is its melting entropy. For the thermodynamics of the crystallization of the solute: we need to assume a single consistent value of an effective Tm, ΔSm, and ΔCp. Since the pure nC60 actually crystallizes with multiple phase transitions, the values used here is will not be exactly the bulk measured values.27 As confirmed by differential scanning calorimetry (DSC), there is no phase transition occurring in nC60 in the temperature range studied, justifying the use of single “effective” values for those parameters. When there is a solid−solid transition occurring in the pure solute at temperatures above where dissolution is being studied, then the driving free energy is ΔG = ΔSm1(T − Tm1) + ΔSm2(T − Tm2) = ΔStot (T − Tmeff )

where ΔStot = ΔSm1 + ΔSm2

ρ ≡ V1/V2

and

Tmeff = (ΔSm1Tm1 + ΔSm2Tm2)/ΔStot

ζ ≡ B1/B2 ≡ 1/Z

Here the subscript “m1” is the main melting transition and “m2” refers to the solid−solid transition. We can also look at the effect of a heat capacity difference between the liquid and solid phase. Here the driving free energy has the following temperature dependence where ΔT  T − Tm:

The following relations hold: φ=

Ψ 1 = Ψ + (1 − Ψ)(V1/V2) 1 + [(1/Ψ) − 1]ρ

β=

Ψ 1 = Ψ + (1 − Ψ)(B1/B2 ) 1 + [(1/Ψ) − 1]ζ 1 1 + [(1/x) − 1]ρ

expanded for T near Tm gives

φ0 = β0 =

1 1 + [(1/x) − 1]ζ

Therefore, we can replace ΔSm with ΔS’ which will incorporate the heat capacity term:

ΔG = ΔSmΔT − ΔCp[(T − Tm) + T ln(Tm/T )]

ΔG ≅ ΔSmΔT + (ΔCp/2)(ΔT 2/T )

⎡ ⎛ T ΔT ⎟⎞⎤ ΔS′ = ΔSm − ΔCp⎢1 + ln⎜1 − ⎥ ⎣ ΔT ⎝ T ⎠⎦

We also write the enthalpic interaction parameter, in terms of solubility parameters:25 χ = (δ2 − δ1)2

and for T near Tm

We compute the difference in the free energy (ΔF) between the state with a finite fraction of the solute in the crystallized form, and the state with all the material dissolved in a single liquid phase. The simplest case has no enthalpy-of-mixing and the entropy-of-mixing (per unit mole) is given by the “ln(mole fraction)” form:26

ΔS′ ≅ ΔSm +

ΔT ΔCp 2T

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To write the free-energy difference for the general case, we include the enthalpy-of-mixing in terms of the mean-field interaction parameter (χ), and write the entropy-of-mixing with the proposed “ln(blob fraction)” form (The Flory−Huggins form is mathematically the same, but with the volume fraction, ϕ, instead of β, inside the logarithm.)

ΔF = RT[(1 − θx){Ψ ln(Ψ) + (1 − Ψ) ln(1 − Ψ)} − {x ln(x) + (1 − x) ln(1 − x)}] + ΔGθx E


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ΔF = RT[(1 − θx){Ψ ln(β ) + (1 − Ψ) ln(1 − β)} − {x ln(β0) + (1 − x) ln(1 − β0)}] + V2χ {(1 − θx)φ(1 − φ)[Ψ + ρ(1 − Ψ)] + φ0(1 − φ0)(x + ρ(1 − x))} + ΔGθx

The dissolution temperature, Td, is the temperature at which dΔF/dθ = 0 at θ = 0. For the case where the blob and mole fractions are equal, representing rigid molecules, i.e. Ψ = β: Td = Tm + +

Vuχρ2 (1 − x)2 [ρ(1 − x) + x]2 [ΔSm − R ln(x)]

RTm ln(x) ΔSm − R ln(x)

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More generally Td = Tm + {V2χρ2 [ζ(1 − x) + x](1 − x)2 − RTm[ρ(1 − x) + x]2 {(1 − ζ )(1 − x) − [ζ(1 − x) + x][ln(x) − ln(ζ(1 − x) + x)]}} /{[ρ(1 − x) + x]2 {ΔSm(ζ(1 − x) + x) + R {(1 − ζ )(1 − x) − [ζ(1 − x) + x] [ln(x) − ln(ζ(1 − x) + x)]}}}

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In Figure 4, we show how the expression for Td(x) (eq 5) behaves with variation of the different parameters. It is particularly insightful to look at the x≪1 approximation for eq 5 shown in eq 6. We define for convenience: Z  1/ζ, the solute/solvent ratio of the blob numbers. Expanding for dilute solutions where x ≪ 1 we get:

Figure 4. Computed dissolution temperature from eq 5, illustrating the effect of variation of different parameters: (a) varying ΔSm; (b) varying ΔCp; (c) varying V2χ; (d) varying Z; (e, f) simultaneously varying Z and V2χ.

fewer “blobs” than the solute) always acts to reduce the dissolution temperature, as a downward shift (Figure 4d). We describe these last two effects as “shifts” to Td; however, we must understand that this is in the small-x approximation. At x = 1, Td = Tm. Thus, the crossover to the “dilute behavior” takes place over the first order-of-magnitude or so in dilution, according to the full equation (eq 5), and as can be seen in Figure 4. It is also useful to consider the slope of the dissolution curve. The derivative of Td respect to ln x, is:

Tm ln x Td ≅ Tm + ΔSm/R − ln x + {Z − 1 − ln Z} +

V2χ /R − Tm{Z − 1 − ln Z} ΔSm/R − ln x + {Z − 1 − ln Z}

Td ≅ Tm +

Tm ln x + V2χ /R − Tm{Z − 1 − ln Z} ΔSm/R − ln x + {Z − 1 − ln Z}

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Solving for Z, we obtain: ⎡T ⎤ ΔS Vχ {Z − 1 − ln Z} ≅ ln x + ⎢ m − 1⎥ m + 2 RTd ⎣ Td ⎦ R

dTd Tm ≅ ΔSm/R − ln x + {Z − 1 − ln Z} d(ln x) ⎧ ln φ + (V2χ /TmR ) − {Z − 1 − ln Z} ⎫ ⎬ ⎨1 + [ΔSm/R − ln x + {Z − 1 − ln Z}] ⎭ ⎩

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which we will use below. We see from the numerator of eq 6, that there are three basic contributions. The main one is the dilution effect (x) which reduces the dissolution temperature logarithmically with dilution, the slope being dominated by ΔSm. (Figure 4a) (We note that such straight logarithmic decrease in Td would cause it to drop below absolute 0 K at very high dilutions; but this is prevented by the “−ln x” in the denominator which sends Td → 0 K as x → 0.) A heat capacity jump changes the curves, based on eq 3, and causing the slope to change with decreasing temperature as shown in Figure 4b. Then there is the enthalpic interaction effect (χ) which acts to raise the dissolution temperature, essentially as an upward shift (Figure 4c). The last term is the conformational entropy contribution, which when Z > 1 (the case where the solvent has

In the dilute regime, if the interaction term is not excessive, i.e. if (V2χ/TmR) ≪ −ln x + {Z − 1 − ln Z}, its contribution is negligible and the slope can be approximated as follows: dTd TmΔSm/R ≅ d(ln x) [ΔSm/R − ln x + {Z − 1 − ln Z}]2

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Solving for Z: {Z − 1 − ln Z} ≅

TmΔSm/R + ln x − ΔSm/R slope

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As we can see from eq 9, the slope depends on Z, and to firstorder, it is not affected by the interaction enthalpy. So, as we F


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can see in Figure 4, parts e and f, while increased Z and increased χ will offset each other as far as the value of Td at a given concentration, the slope will not be the same. In the analysis of the data, we used eq 7, to obtain values for Z, based on Td at x = 0.001, using the effective ΔSm, obtained from eq 3. Because the heat capacity effect causes curvature in the Td(x) curve, (see Figure 4b) one could not use the closed form (eq 8a) to obtain Z from the slope; we therefore obtained the slope from the numerical derivative of eq 5, and numerically determined Z. While eq 5 is numerically solvable, eqs 6 and 8 give us physical insight into the effects of the different physical parameters on the phase behavior. In our data we can consider both the slope and the offset. The slope is not significantly affected by the solubility parameter or the enthalpic interaction, but rather only by Z, or the ratio of the two components’ molecular volumes (in Flory−Huggins), or “blob numbers” (as we propose). The offset, or shift in Td at a given (dilute) ϕ, however, has a term which will give a shift to the dissolution temperature due to the enthalpic interaction, which will, in general, raise the dissolution temperature; and a contribution which will decrease the dissolution temperature due to the inequality of molecular volumes (or blob numbers). Connection to Activity Coefficient Description. The “activity coefficient” is often used to describe or tabulate the deviation of the phase behavior from “ideal”.22,28 For the simple case this is typically written: ln x =

ΔSm ⎛ T ⎞ ⎜1 − m ⎟ − ln γ R ⎝ Td ⎠

Figure 5. Plots of measured and fitted nC60 dissolution temperature (Td) versus concentration (x) for various solvents: (a) alkanes, (b) alkylbenzenes, (c) alkylcyclohexanes, and (d) compounds with 2-fused rings.

line, and parametrized with its value at x = 0.001 and its slope. These results are shown in Table 1. On the basis of the theoretical curves in Figure 4, we know that our data should be smooth and straight. Furthermore, for homologous series, as a function of chain-length, the fitted parameters should be smooth and continuous. When there were Td(x) data points which were not colinear (which we attribute to experimental error), we identified the errant points using the smoothness of the homologous series dependence. For the analysis of the data shown here, we chose the parameters for our nC60 material, based on best representing the data: Tm = 96 °C, ΔSm = 600 J/mol-K, and ΔCp = −250 J/ mol-K, not inconsistent with a DSC measurement on the sample. It turned out that this hexacontane sample from Aldrich was not extremely pure, and thus the values used were not simply the pure component values for nC60. However, all the solutions reported here are using the same batch of nC60, and thus a single set of parameters describing it are used. We also point out that the conclusions of this paper do not depend on the exact choice in these values, since the solute is not varied here, only the solvents. In Table 1, we have two columns where we list the two computed values of Z, determined independently from both the slope of the Td(x) data, and from the value of Td(x = 0.001). We also list the ratio of the molecular volumes of nC60 and solvent (VnC60/Vsolvent), and the ratio of the computed Z’s to that volume ratio. There are a few data points where the Z derived from the slope is listed as “13”. A smaller slope translates to a larger value of Z, and it is very sensitive to the slope. When the measured slope becomes just a little too small, the calculated value for Z becomes undefined, in that eq 5 gives no solution for it. Thus, in the range of larger Z, the experimental error on the slope can result in a large/undefined calculated value of Z. For these, we list Z = 13, which is greater than the largest of the other values.

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If we start with eq 4 and write it in the form of eq 10 above, we obtain ln x =

ΔSm ⎛ T ⎞ V2χρ2 (1 − x)2 ⎜1 − m ⎟ − R ⎝ Td ⎠ [ρ(1 − x) + x]2 RTd

Thus, the activity coefficient is a function of concentration. ln γ = −

V2χρ2 (1 − x)2 [ρ(1 − x) + x]2 RTd

However, using our dilute concentration expansion, eq 6, we can write it in the form: ⎛ T ⎞ ln x ≅ ⎜1 − m ⎟(ΔSm/R ) + {Z − 1 − ln Z} − V2χ /TdR Td ⎠ ⎝

so ln γ ≅ V2χ /TdR − {Z − 1 − ln Z}

There are two terms here: the enthalpy term (which is itself a function of Td), and the configurational entropy term. This may be useful in the relating the present results to other data tabulations.

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RESULTS Typical raw data of transmitted intensity versus temperature for nC60 solutions in nC11-cyclohexane is shown in Figure 3. From these, plots of dissolution temperature versus concentration are made for each solvent. These are shown for a few series of our solvents in Figure 5a−d respectively for alkanes, alkylbenzenes, alkylcyclohexanes, and compounds with 2-fused rings. Then from these, the data has been fit to a logarithmic G


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DISCUSSION We now recall what Z represents. In eq 5, Z  1/ζ, which in the context of our proposed form of the entropy-of-mixing, is Z = BnC60/Bsolvent or the ratio of the “blob numbers”. However, if the Flory−Huggins form (“ln(ϕ)”) were to be generally valid, then Z = VnC60/Vsolvent should hold, meaning that Z will only depend on the molecular volume of the solvent, not its flexibility. Likewise, if the ideal-solution form (“ln(x)”) was generally valid, then Z would be constant, not even depending on molecular size; i.e. Z = 1 (Table 2).

ones. This independently confirms the results in the previous figures. We now discuss some of the solvent molecules which have some confounding physics that causes deviations from the simple analysis above. The interaction we compute between molecules treats the molecules as compositionally homogeneous, and thus the solubility parameter of the molecule is an average of that attributable to the various functionalities. We call this the “mean-field molecule” approximation. However the reality is that some molecules have different functionalities spatially separated. Consider molecule A with two parts, onehalf contributing a high solubility parameter, and the other half contributing a low solubility parameter, the average being the same solubility parameter as molecule B, which is homogeneous (see Figure 7). In the mean-field molecule approach, the interaction enthalpy between molecules A and B would be 0; however it is obvious that such molecules would not have a zero interaction enthalpy, because molecule B would not be happy in contact with either part of molecule A. The nC60 solute molecule is compositionally homogeneous. For the solvents studied here, the saturated rings do not have a very high solubility parameter, while the aromatics do. Thus, the aromatics with aliphatic chains attached would tend to exhibit this effect, which is that the effective χ will be greater than that computed in the mean-field molecule approximation. The effect of this would be a lower calculated Z when we are using Td(0.001). However, since using the Td(ln(x)) slope to calculate Z does not depend on χ, this effect is not manifest in that data. We identify such molecules in Table 1 with an asterisk (∗) next to their solubility parameter and in Figure 6d− f with different symbols. These molecules include hexadecylnaphthalene and the alkylbenzenes, as well as phenylcyclohexane. This also highlights the value in doing both the slope and offset analysis, since the enthalpic interaction does not affect the slope. Quantitatively handing this effect is beyond the scope of this paper; however, we are developing a new formalism which will account not only self-consistently for the fact that molecules are not situated randomly, but also that like-molecules will favor being in contact with one another and also independently for the functionalities on the molecule.29 The other interesting feature is that Z appears to increase with decreasing solvent size. This is because the ratio of the blob numbers (Bsolute/Bsolvent) or molecule volume (Vsolute/ Vsolvent) increases with decreasing solvent size. But as we get to small enough molecules, we see that Z levels off, especially apparent in the data from Td (0.001). (Figure 6d) This occurs at a molecular volume of about 140 Å3. We can understand this in the following manner. Up until here we have qualitatively discussed, looked for and found evidence for the entropic effect of the relative rigidity of molecules. While we do not believe the present data is accurate enough to unambiguously quantitatively assign Bi values based on structure, we offer a suggestion about how this may be assigned, which can be tested with further studies. If we are considering the case of mixing the linear aliphatic nC60 with other paraffins, then we would be in the Flory−Huggins limit where the blob-fraction (βi) and the volume fraction (ϕi) would be the same. The two molecules are made of the same building blocks with the longer one having more of them. We would like to know, not just what is the ratio of the Bi of these molecules, but what are their actual values. This will be important once we are interested in mixing them with more rigid molecules. We

Table 2 model

Z

ideal Flory−Huggins “blob”

1 Vsolute/Vsolvent Bsolute/Bsolvent

Our data gives two independent measures of Z: one with greater scatter, but independent of χ, and one with smaller uncertainty, but dependent on accounting for the enthalpy of mixing. We therefore will plot our experimentally measured Z versus Vsolvent, and determine whether it best behaves as • a constant, unity in particular, suggesting the “ln(x)” form • a function only of Vsolvent, VnC60/Vsolvent in particular, suggesting the “ln(ϕ)” form, or • a function not only of size but also rigidity where for molecules of the same Vsolvent, the less flexible (smaller Bi) species will have a higher Z, suggesting the “ln(β)” form. In Figures 6a−c, we plot using Z derived from the slope: (a) Z as a function of the solvent’s molecular volume, Vsolvent, (b) Z divided by the volume ratio VnC60/Vsolvent, and (c) Z as a function of VnC60/Vsolvent. The points are labeled with the solvent, and color coded, grouped by the number of ringassociated carbon atoms; with red/diamonds representing paraffins, purple/circles representing molecules with 5−8 ring-associated carbons, and blue/squares representing molecules with >9 ring-associated carbons. Here, a ring-associated carbon refers to the carbons in the ring or on a directly connected carbon. We consider this as a rough measure of the molecule’s rigidity for the purpose of the present discussions. What is first clear from these plots is that, first Z is not constant, but very roughly inversely proportional to Vsolvent. This shows us specifically that even for alkyl-chain molecules of this size, the “rigid molecule” or ideal “ln(x)” assumption is not valid. We then ask whether Z is only a function of Vsolvent, or whether molecules with the same Vsolvent have a higher Z if they are more rigid. Looking at Figures 6a−c, we see clearly that, for the same x-coordinate, the blue/squares are well above the red/ diamonds, with the purple/circles being intermediate. Thus, while there is a degree of scatter due to the uncertainty in the smaller differences in the slope, there is no interference from the enthalpy and a clear deviation from the Flory−Huggins entropy-of-mixing associated with rigidity is apparent. Similarly, we plot in Figures 6d−f, Z derived from Td(0.001). (Points from solutes of compositionally mixed character are shown with open symbols and will be discussed below.) Here too, we see that there is an inverse dependence on molecular volume for the flexible molecules, and the more rigid molecules (blue) have a systematically higher Z than the more flexible H


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Figure 6. continued

I


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Figure 6. continued

J


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Figure 6. (a, d) Z vs Vsolvent; (b, e) Z/(VnC60/Vsolvent) vs Vsolvent ; (c, f) Z vs (VnC60/Vsolvent). (a−c) Z is determined from the slope. (d−f) Z is determined from Td(0.001). The points are labeled with the solvent. Red/diamonds = paraffins, purple/circles = 5−8 ring-associated carbons, and blue/squares = >9 ring-associated carbons. The points with open symbols in parts d−f are molecules with highly different spatially separated functionalities, as discussed in the text.

K


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to reduce the uncertainty in the Td(ln(x)) slope value. The current study was accomplished with off-the-shelf commercial materials. These results suggest that synthesis or purification of additional compounds or highly monodisperse polymers might be of value to confirm and further quantify this effect.

■

APPENDIX We use the following variables: z = coordination number of lattice in lattice model x = solute mole fraction ϕ = solute volume fraction β = solute “blob fraction” r = volume of a solute molecule (in solvent volume units) B = number of blobs in a solute molecule v = volume of a blob of solute (in solvent volume units) Thus, r = vB. N = number of lattice sites (solvent molecules occupy one site) n = number of solute molecules The number of molecules is

Figure 7. Schematic representing interactions between two molecules, one homogeneous with intermediate solubility parameter (purple), and one with two parts: a high solubility parameter part (red) and a low solubility-parameter part (blue). A mean-field approach to the molecules would describe these molecules as being identical, and would incorrectly result in predicting no interaction enthalpy between such molecules, where finite enthalpy would clearly be present.

therefore are interested in how many carbons in an aliphatic chain constitute a blob. We suggest that it should near the persistence length of the chain, for alkanes (polyethylene) being about 7 carbons.30 In the lattice derivation of the Flory− Huggins entropy-of-mixing, we allow the chain to change directions at every lattice point. In reality, there is a persistence length (of multiple carbons) associated with direction change. A ring structure, aromatic, saturated or mixed, along with pendent methyls, would act as a single entity in a chain structure. When we look at the values for Z as the solvent’s molecular volume gets small, we see that it appears to level off (near Z ∼ 7.5, when looking at the data from Td(0.001)). This leveling off is consistent with the fact that Bi will never drop below unity. Even if a molecule is smaller than a persistence length, (or even a methane molecule), it will have Bi = 1. Thus, the limiting value of Z would be the Bi of the solute. Thus, for a 60 carbon chain if Z is as high as 7.5, this would correspond to 60/7.5−8 carbons per blob, roughly consistent with the persistence length. We reiterate that we do not believe the present data is accurate enough to unambiguously assign Bi based on structure; however it does show the clear trend associated with the molecules’ rigidity. On the basis of the present results and methods of analysis, future experiments can be carried out to further quantify this phenomenon.

(N − nr ) + n

The mole volume and blob fractions in terms of n are therefore x=

n nr nB φ= β= N − nr + n N N − nr + nB

Conversions between β and ϕ: β=

φ vβ φ= (1 − φ)v + φ (1 − β) + vβ

Conversions between β and x: β=

β Bx x= (1 − x) + Bx (1 − β)B + β

Conversions between ϕ and x: φ rx φ= x= (1 − x) + rx (1 − φ)r + φ

■

With this nomenclature, we consider the two “familiar” forms for ΔSmixing, x ln(x) and x ln(ϕ) and our own proposition x ln(β), and write them in terms of x only.

SUMMARY AND CONCLUSIONS We have experimentally shown that the presence of rigid units requires modification to the Flory−Huggins entropy-of-mixing expression. Z is not constant for all Vmol, therefore “x ln(x)” cannot be used even for such short “oligomeric” or “lubes range” molecules. Comparing Z at constant Vmol, we see that Z is larger for more rigid (less flexible) solvent molecules. The Flory−Huggins ln(volume fraction) form for ΔSmixing cannot explain this, while the ln(blob fraction) form at least qualitatively accounts for the deviations. Therefore, corrections to the Flory−Huggins x ln(ϕ) form for ΔSmixing are needed to account for differences in molecular rigidity. A generalization of Flory’s derivation results in a proposed form for this dependence, where x ln(ϕ) is replaced by x ln(β). Further experimental work should be able to quantify the effective “blob numbers” further. This should be accomplished using, for example, a higher purity and better defined solute, and taking data over wider concentration ranges and with care

ΔS

“x ln x ”

ΔS

/R ≡ −x ln(x) − (1 − x) ln(1 − x)

“xln φ”

(A1)

/R ≡ −x ln(φ) − (1 − x) ln(1 − φ) = −x ln(x) − (1 − x) ln(1 − x) − x ln(r ) + ln[(1 − x) + rx] (A2)

ΔS

“x ln β ”

/R ≡ −x ln(β) − (1 − x) ln(1 − β) = −x ln(x) − (1 − x) ln(1 − x) − x ln B + ln[(1 − x) + Bx] (A3)

Flory Calculation

We now look carefully at the lattice calculation for flexible chains from which the xln(ϕ) form has been derived. L


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Flory Lattice Derivation.19 The number of ways to place a molecule with r contiguous sites is

ΔSFL = ΔSrigid molecules + ΔSconformational

(N − rn)z(z − 1)r − 2 (1 − fn )r − 1

ΔSconformational /R = ln[1 + x(r − 1)] + x(r − 1) [ln(z − 1) − 1]

where z is the coordination number of the lattice, and f n is probability that an adjacent cell is occupied. The 4 factors are as follows: (1) places for first segment: (N-rn) (2) choices for second segment: z (3) choices for third+ segments: (z − 1)r‑2 (4) probability that site is open for all but the first segment: (1 −f n)r‑1 Equivalent configurations are accounted for by dividing by n! Assuming that there is no correlation between the occupancy of different sites:

1 − fn =

(A6)

Within the context of this calculation, (which may not be the best way of computing this, however this shows us what the terms mean) we compute for a polymer melt, by setting x = 1 in eq A6 and obtain: ΔSconformational melt /R = ln(r ) + (r − 1)[ln(z − 1) − 1] (A7)

This is what Flory refers to as ΔSdisorientation. In this framework, eq A7 can be considered the melting entropy of the pure polymer. Clearly, this only represents the melting entropy in this polymer lattice model intended to look at mixing in the liquid phase; Therefore, one could ideally replace this term with better models for melting. However it is insightful to look at the term which represents the excess entropy of a melting polymer in solution, compared to melting in the pure state.

N − rn N

The entropy-of-mixing can be written as “per mole of molecules” (as we do throughout this paper). The above assumptions yield what we call the “Flory lattice” entropy-ofmixing, ΔSFL:

ΔSexcess conformational = ΔSconformational − xΔSconformational melt

ΔSFL /R = −x ln φ − (1 − x) ln(1 − φ) + x{ln r + (r − 1)[ln(z − 1) − 1]}

ΔSexcess conformational /R = ln[1 + x(r − 1)] − x ln r

(A4)

We note that the first two terms in eq A4 comprise the common “x ln ϕ” Flory−Huggins form (i.e., eq A2). The third term is simply proportional to x and is often ignored, being left as part of the chemical potential of the polymer species. It is usually described as the entropy associated with melting a chain19,31 or the “entropy of disorientation”.2 While some might assume that the conformation, and therefore the entropy, associated with the different conformations of a polymer in a melt and in a solvent are the same, in general, we realize that the number of chain configurations available to a polymer in a melt is less than those available in a solution of monomers. As a simple example, one can consider cases where a polymer’s conformation created isolated pockets too small to accommodate another polymer, but large enough for a monomer. Those conformations are available to a polymer in a solvent, but not in a melt. We find it insightful to consider, separately, contributions to the entropy from the different configurations associated with the placement of the polymer chains, as though they were rigid objects, from the entropy associated with the fact that the chains are flexible. Assuming, as discussed above, that for rigid molecules the entropy-of-mixing is

From eq A2, we also see that: ΔS“x ln φ” = ΔSFL − xΔSconformational melt = ΔS“x ln x ” + ΔSexcess conformational

ΔSexcess_conformational (eq A8) is the part not associated with melting, and is in-fact the difference between the entropy as computed with “x ln(ϕ)” and “x ln(x)”. Within the framework presented here, and in the context of our assumptions, it represents the additional entropy afforded by the difference in the size of the flexible molecules. Extend to Different Sized Segments: “Liquid Lattice”. For different size segments, we consider the approach where we allow the lattice sites to have variable volume, not maintaining the perfect lattice registry, in keeping with the liquid nature of the system When we allow the lattice sites to have variable volume, the combinatorics are the same as in Flory’s calculation; and the size of the blob “v” in the solute does not come into play. Since N represents the number of lattice sites, it is also the total number of blobs. And therefore β = nB/N. We can compute for the possible configurations: ln Ω = (N − Bn + n) ln N − (N − Bn) ln(N − Bn)

ΔSrigid molecules/R = −x ln x − (1 − x) ln(1 − x)

− n ln n + n(1 − B) + n(B − 1)[ln(z − 1) − 1]

and then rearranging eq A4 in terms of its difference from the “x ln(x)” form we obtain:

and dividing by the total number of molecules which is N-nB +n, we obtain

ΔSFL /R = −x ln x − (1 − x) ln(1 − x)

ΔSvariable lattice/R = −x ln β − (1 − x) ln(1 − β)

+ ln[1 + x(r − 1)] + x(r − 1) [ln(z − 1) − 1]

(A8)

+ x{ln B + (B − 1)[ln(z − 1) − 1]}

(A5)

Thinking of the “x ln(x”) terms as being the “base case” for rigid molecules, we now have additional terms associated with the chain conformations, and write:

which is identical in form to eq A4 with ϕ replaced by β, and r replaced by B. Rewriting this in terms of mole fraction only M


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which is x ln x for rigid molecules, x ln ϕ for flexible chains, and more generally as x ln β for more complex species. The Flory−Huggins form written in this way has the Bi replaced by ri.

ΔSvariable lattice/R = −x ln x − (1 − x) ln(1 − x) + ln[1 + x(B − 1)] + x(B − 1) [ln(z − 1) − 1]

ΔSFlory − Huggins/R = −∑ xi ln xi + ln(∑ xiri)

which is identical to eq A5 with r replaced by B. We thus see that the proposed “x ln β” expression is consistent with a Flory-style lattice derivation, where the sites can have variable volume. Multiple Component Generalization. By mathematically generalizing, as opposed to a rigorous derivation, we now write an expression for ΔSmixing with multiple components of concentrations xi with different Bi. For two components we have eq A3, which can be rewritten putting in the fact that we implicitly assumed that the blob number of the solvent is unity.

■

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We would like to acknowledge useful discussions with Scott Milner, Bill Olmstead, Hubert King, and Howard Freund who also provided calculations of solubility parameters; as well as calorimetry measurements by Manika Varma-Nair and Diana Smirnova.

− x ln B

■

Since x is the solute concentration, x2  x and x1 = 1 − x. Likewise B2  B for the solute, and the solvent blob-number which was implicitly assumed to be unity is now written more generally as B1.

− x1 ln B1 − x 2 ln B2

For more components we write generally: ΔS“x ln β ”/R = −∑ xi ln βi i

∑ xi ln Bi i

Similarly ΔSmixing = ΔS

“x ln β ”

+ xΔSconformational melt

ΔSmixing /R = −x ln(x) − (1 − x) ln(1 − x) + ln[(1 − x) + Bx] + x(B − 1) [ln(z − 1) − 1]

and for multiple components ΔSmixing /R = −∑ xi ln xi + ln(∑ xiBi ) i

i

+ [ln(z − 1) − 1][∑ xi(Bi − 1)] i

The last term is only proportional to xi for each species and can be treated as part of the chemical potential of that species. This is because the contribution from that term will be the same whether the components are mixed or not. As discussed in the body of the paper, it may be associated with the entropy of melting, but it is not part of the entropy-of-mixing. We can then write our form for the entropy-of-mixing generally as (eq 1): ΔSmixing /R = −∑ xi ln xi + ln(∑ xiBi ) i

REFERENCES

(1) Flory, P. J.; Krigbaum, W. R. Annu. Rev. Phys. Chem. 1951, 2, 383−402. (2) Flory, P. J. Discuss. Faraday Soc. 1970, 49, 7−27. (3) Huggins, M. L. J. Phys. Chem. 1948, 52, 248. (4) Hildebrand, J. H. J. Chem. Phys. 1947, 15, 225. (5) Huggins, M. L. J. Polym. Sci. 1955, 16, 209−219. (6) Krukowski, A. E.; Chan, H. S.; Dill, K. A. J. Chem. Phys. 1995, 103, 10675. (7) Chan, H. S.; Dill, K. A. J. Chem. Phys. 1994, 101, 7007. (8) Freed, K. F.; Dudowicz, J. Adv. Polym. Sci. 2005, 183, 63−126. (9) Meroni, A.; Pimpinelli, A.; Reatto, L. J. Chem. Phys. 1987, 87, 3644−3646. (10) Mansoori, G. A.; Carnahan, N. F.; starling, K. E.; Leland, T. W. J. Chem. Phys. 1971, 54, 1523−1525. (11) Lichtenthaler, R.; Abrams, D. S.; Prausnitz, J. M. Can. J. Chem. 1973, 51, 3071. (12) Giesen, D. J.; Cramer, C. J.; Truhlar, D. G. J. Phys. Chem. 1994, 98, 4141−4147. (13) Shinoda, K.; Hildebrand, J. H. J. Phys. Chem. 1958, 62, 292. (14) Sharp, K. A.; Nicholls, A.; Fine, R. F.; Honig, B. Science 1991, 252, 106−109. (15) Lazaridis, T.; Paulaitis, M. E. J. Phys. Chem. 1992, 96, 3847− 3855. (16) Altgelt, K. H.; Boduszynski, M. M., Composition and Analysis of Heavy Petroleum Fractions. Marcel Dekker: New York, 1994. (17) Quann, R. J.; Jaffe, S. B. Ind. Eng. Chem. Res. 1992, 31, 2483− 2497. (18) Jaffe, S. B.; Freund, H.; Olmstead, W. N. Ind. Eng. Chem. Res. 2005, 44, 9840−9852. (19) Flory, P. J. Principles of Polymer Chemistry; Cornell Univ. Press: New York, 1953; p 497. (20) Peczak, P.; Sirota, E. B. Asphaltene Precipitation And LiquidLiquid Phase Separation: A Statistical-Thermodynamic Model And Experiment, 7th Intl. Conference on Petroleum Phase Behavior and Fouling (Petrophase), Asheville, NC, 2006; Asheville, NC, 2006. (21) Abrams, D. S.; Prausnitz, J. M. AlChE J. 1975, 21, 116. (22) Kniaz, K. Fluid Phase Equilib. 1991, 68, 35−46. (23) Peczak, P.; Sirota, E. B. Manuscript in preparation. (24) Fedors, R. F. Polym. Eng. Sci. 1974, 14, 147−154. (25) Hildebrand, J. H.; Scott, R. L. The solubility of nonelectrolytes; Dover Publications: Mineola, NY, 1964. (26) Prausnitz, J. M., Molecular Thermodynamics of Fluid-Phase Equilibria. Prentice Hall: Englewood Cliffs, NJ, 1969. (27) Sirota, E. B. J. Chem. Phys. 2000, 112, 492.

ΔS“x ln β ”/R = −x1 ln(x1) − x 2 ln(x 2) + ln[B1x1 + B2 x 2]

i

AUTHOR INFORMATION

*E-mail: [email protected].

+ ln[1·(1 − x) + Bx] − (1 − x) ln(1)

i

i

Corresponding Author

ΔS“x ln β ”/R = −x ln(x) − (1 − x)ln(1 − x)

= −∑ xi ln xi + ln(∑ xiBi ) −

i

i

N


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(28) Coutinho, J. A. P.; Andersen, S. I.; Stenby, E. H. Fluid Phase Equilib. 1995, 103, 23−39. (29) Milner, S. C.; Sirota, E. B. Manuscript in preperation. (30) Flory, P. J. Statistical Mechanics of Chain Molecules; Wiley: New York, 1969. (31) Strobl, G. The Physics of Polymers; Springer: Berlin, 1997.

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