Developing a Macroscopic Mechanistic Model for Low Molecular

Article pubs.acs.org/IECR

Developing a Macroscopic Mechanistic Model for Low Molecular Weight Diffusion through Polymers in the Rubbery State B. Martinez-Lopez,*,†,‡ P. Huguet,§ N. Gontard,† and S. Peyron† †

UMR IATE, CIRAD, INRA, Montpellier SupAgro, Université de Montpellier, 2 Pl. Viala, F-34060 Montpellier, France National Food Institute, Danmarks Tekniske Universitet, 2800 Kgs. Lyngby, Denmark § Institut Européen des Membranes, Université de Montpellier, 34095 Montpellier, France ‡

ABSTRACT: Raman microspectroscopy was used to determine the Fickian diffusivity of two families of low molecular weight molecules through amorphous polystyrene in the rubbery state. Different effects of the temperature on diffusivity for each of the families suggested that molecular mobility is controlled by both the volume and flexibility of the diffusing substance when the movement of polymer chains can generate stress induced deformation of molecules. The diffusing molecules were represented as Newtonian spring−bead systems, which allowed us to quantify their flexibility, in function of the vibration frequency of their bonds by reconstructing their theoretical spectra. Results showed that the use of molecular descriptors that take into account flexibility rather than the most stable conformation of the diffusing molecules may improve the description of the diffusion behavior caused by variations in shape and size of the free volumes of the polymeric matrix in the rubbery state.

1. INTRODUCTION Accurate evaluation of diffusivity remains indispensable for reliable prediction of migrant diffusion within polymeric matrices. Significant advances have been made in the understanding of the fundamentals of molecular mobility and prediction of diffusion coefficients, but the identification of the mechanisms by which diffusion occurs in polymeric systems remains a scientific challenge. The main factors that are known to influence the diffusion comprise (1) the size or bulkiness of the diffusing molecule, (2) the morphology of the polymer that determines the segmental mobility of the polymer chains, and (3) interactions between the diffusing molecule and the polymeric matrix, as a result of polarity or the presence of different functional groups. One of the oldest theories accepted to accurately describe the diffusion of solutes in solid-state polymers is the so-called free-volume theory.1 This theory is based on the assumption that the amorphous fraction of a semicrystalline polymer may be regarded as a network of polymer chains containing free spaces between them of different sizes and shapes. The so-called free volumes are redistributed continuously, since the thermal agitation induces movements of the polymer leading to change in the distribution and location of the spaces. Inside these local holes, the diffusing molecules vibrate at much higher frequencies than the polymer chains. According to the free volume theory, a molecule will diffuse by “jumping” through these holes, only if the hole reaches a volume equal to or larger than the volume of the diffusing molecule. On the other hand, the idea of the “jump” © 2016 American Chemical Society

was refined including the hypothesis that molecules not only did jump but also had a slithering-like movement between the polymer chains.2 There is a number of models based on the free volume theory.3 The most famous is, by far, the model of Vrentas and Duda,4−7 which was refined nearly two decades after its first appearance in the literature.8−11 These so-called macroscopic models usually describe diffusion in polymer solutions instead of solid polymeric matrices. They also often rely on several parameters of very abstract nature, resulting in the models being too complex for a direct application. Since it is well established that the diffusion coefficient of a molecule depends on both its size and shape, there are also models available in literature that rely solely on descriptors of the bulkiness of the diffusing substance. Many of them empirically relate D to the molecular weight of the diffusing substance because of the latter being a very easy descriptor to calculate. Furthermore, the scaling laws that relate diffusivity and molecular weight, initially introduced by Einstein to describe the self-diffusion behavior of macromolecules (1905), then refined by Langevin (1908), proved to be applicable to other diffusion phenomena considering that they are based on the simple fact that the driving force responsible for the mobility of a diffusing substance is offset by the frictional force of a viscous Received: Revised: Accepted: Published: 5078

January 18, 2016 March 24, 2016 April 5, 2016 April 26, 2016 DOI: 10.1021/acs.iecr.6b00233 Ind. Eng. Chem. Res. 2016, 55, 5078−5089

Article

Industrial & Engineering Chemistry Research

2.1.2. Diffusing Molecules. Biphenyl (CAS no. 92-54-4, molecular weight 156.2 g mol−1), p-terphenyl (CAS no. 92-944, molecular weight 230.3 g mol−1), trans-stilbene (CAS no. 103-30-0, molecular weight 180.3 g mol−1), diphenylbutadiene (CAS no. 538-81-8, molecular weight 206.28 g mol−1), and diphenylhexatriene (CAS no. 1720-32-7, molecular weight 232.32 g mol−1) were purchased from Sigma-Aldrich (France). p-Quaterphenyl (CAS no. 135-70-6, molecular weight 306.4 g mol−1), p-quinquephenyl (CAS no. 3073-05-0, molecular weight 382.5 g mol−1), and p-sexiphenyl (CAS no. 4499-83-6, molecular weight 458.6 g mol−1) were purchased from TCIEurope (Belgium). These molecules were chosen in order to relate the differences of the values of D with their geometries. This is represented by using two homologous series of surrogate molecules that share the phenyl group as primary unit. The first of the homologous series, from now on referred to as the oligophenyl series, is formed by adding phenyl groups in the para- position in relation to the others at each step. It consists of biphenyl (or phenylbenzene, two phenyl units), pterphenyl (or p-diphenylbenzene, three phenyl units), pquaterphenyl (four phenyl units), p-quinquepheyl (five phenyl units), and p-sexiphenyl (six phenyl units). The other series, from now on the diphenylalkene series, also starts with biphenyl, but instead of adding more phenyl units, it increases the distance between them by adding an ethylene group at each step. This series then consists of biphenyl, trans-stilbene (or 1,trans-2-diphenylethylene, one ethylene group between the phenyl units), diphenylbutadiene (or trans,trans-1,4-diphenyl1,3-butadiene, two ethylene groups), and diphenylhexatriene (or 1,6-diphenyl-1,3,5-hexatriene, three ethylene groups). 2.2. Fabrication of Films and Sources. Virgin polystyrene films were made by thermoforming PS pellets (hot press) at 200 bar for 5 min at 165 °C. The actual PS film thickness was measured by using a micrometer (Braive Instruments, Chécy, Fr) in quintuplicate, resulting in 200 ± 23 μm. Vestoplast 891 sources (from now on, polypropylene or PP sources) were made by thermoforming vestoplast pellets (hot press) for 5 min at 165 °C in order to obtain a solid vestoplast surface of approximately 5 mm of thickness. This vestoplast surface was cut into squares of approximately 2 cm2. Then, for each surrogate, one of these 2 cm2 pieces was placed on a hot plate in order to raise its temperature above its softening point again. While in its soft or liquid form, 20 mg of surrogate was homogeneously distributed on top of the vestoplast piece by using a spatula. Immediately after that, the vestoplast piece was removed from the hot plate, which causes its almost instantaneous solidification, trapping the surrogate on it. This ensures that the surrogate is fairly evenly distributed on the region of the source close to the surface that will serve as interface with the PS. For molecules that are originally in crystal form (p-quaterphenyl, p-quinquephenyl, and p-sexiphenyl), the sources were placed under a controlled temperature of 100 °C, so its concentration homogenizes. An exception is made for biphenyl and trans-stilbene which, because of being very volatile, were homogeneously distributed with a spatula on top of the 2 cm2 vestoplast surface at room temperature, right before the diffusion assay. A new source was fabricated for each of the studied temperatures. The nominal concentration of the sources is of approximately 20 000 ppm. 2.3. Diffusivity Assay. PP source and PS virgin film were put into contact at 105 (Tg), 115, 125, 135, 145 °C. After contact and before each Raman measurement, the virgin film

medium. Later on and by the means of statistical mechanics, Rouse,12,13 suggested that diffusivity and molecular weight are related by a relationship of the kind D ∼ M−α with a scaling parameter α that can be related to the transport mechanism.14 For example, De Gennes15 stated that a value of α = 2 means that the polymer chain diffuses through the matrix by crawling through a tube consisting of the neighboring chains. Since the self-diffusion of a polymeric chain in a semidilute regime and the diffusion of a small molecule through a solid polymeric matrix are equivalent from a statistical mechanics point of view, some authors have tried to extrapolate this scaling relationship to the latter. This is why it is possible to find studies giving the coefficient α a more meaningful sense.16 Although most of the authors showed a strong relationship between D and the molecular weight of the migrant or the actual volume occupied by the molecule, like the van der Waals volume, bulkiness as represented by M does not appear to be sufficient to fully describe the diffusion behavior through polymeric matrices. It may be assumed that the flexibility of the diffusing substance influences the ease with which it passes through the free volume of polymers.17 The molecular flexibility that translates the intramolecular degree of freedom is a difficult descriptor to evaluate. In the specific case of molecular structure composed of repeating units, flexibility proved to be related to the timeaveraged molecular conformations18 or the end-to-end distance distribution.19 On this basis, current work aims to contribute to the clarification of the influence of the molecular structure on diffusion phenomenon, from a macroscopic point of view, which might result in a model readily available for direct industrial applications at the expense of not providing the atomistic level detail of molecular dynamics. The studies carried out on the self-diffusion in polymer melts showed the influence of the contour length fluctuation of long-chain flexible macromolecules on their motion,20,21 but such a theory cannot be accurately applied to more compact molecules. By use of a method based on local measuring with Raman microspectroscopy that has been proved successful for polymer in rubbery22 and glassy states,23 diffusivity data of two homologous series of molecules that share the phenyl group as the main unit were evaluated in amorphous polystyrene above its glass transition temperature. The homologous series were selected to display a broad range of size and flexibility while remaining linear. Several trends in the diffusivity data were identified by using the classical descriptors molecular weight and volume and by following a new strategy based on the evaluation of compressibility of molecules in amorphous polymers based on a mechanical approach that considers molecules as spring− bead systems: each phenyl group being equated with a rigid bead, and the bond or series of bonds between them with a spring.

2. MATERIALS AND METHODS 2.1. Chemicals. 2.1.1. Polymer and Sources. Amorphous polystyrene with a molecular weight Mw of approximately 285 000 g mol−1 and a glass transition temperature (case II transition) of approximately 105 °C was purchased from Polyone France. Vestoplast 891, an amorphous poly-α-olefin rich in propene, generally used as hot melt adhesive with a glass transition temperature of −33 °C, a softening point of 162 °C, and a molecular weight Mw of 85 000 g mol−1, was purchased from TER France. 5079

DOI: 10.1021/acs.iecr.6b00233 Ind. Eng. Chem. Res. 2016, 55, 5078−5089

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Figure 1. Experimental procedure leading to the Raman cross-section analysis of the PS film.

Figure 2. Raman normalized spectral fingerprint of PS + trans-stilbene, showing the decrease on the peak at 1295 cm−1 used to follow the evolution of the concentration of trans-stilbene through the thickness of the PS film.

PS specific band at 622 cm−1, indicative of the substituted benzene ring24. 2.5. Estimation of Diffusivity. The internal diffusion of a migrant in a plane sheet is given by the eq 125 where x is the distance (m), C is the concentration in diffusing substance (mass diffusing substance/mass of sheet), and D is the diffusivity of the molecule in the sheet material (m−2 s−1). D is assumed independent of the concentration of the diffusing substance, so the system is said to follow Fickean kinetics. Equation 1 can be solved with the initial and boundary conditions that apply to the case, in order to obtain an expression for the concentration distribution.

was removed and wiped with ethanol. Measurements were carried out once after a certain time of contact, which was fixed in relation to the size of the molecule and the temperature of the test. 2.4. Raman Measurement. Surrogate concentration profiles were determined as follows. Thin slices of PS were prepared using a razor blade and stuck on a microscope slide, as shown in Figure 1. Raman spectra were recorded between 800 and 3500 cm−1 Raman shift wavenumber using a confocal Raman microspectrometer Almega (Thermo-Electron) with the following configuration: excitation He−Ne laser, 633 nm, grating 500 grooves/mm, pinhole 25 μm, objective ×50. The resultant spectra were the mean of two acquisitions of 25 s each. Measurements were carried out in the cross section of the sample with different spacings, ranging between 1 and 15 μm depending on the temperature of the assay. All spectra pretreatments were performed with Omnic 7.3 (ThermoElectron). Processing included (i) a multipoint linear baseline correction and (ii) normalization according to the area of the

∂C ∂ 2C =D 2 ∂t ∂x

(1)

The analytical solutions of eq 1 applied in this work are given by eqs 2 and 3.26 Equation 2 represents a thickness that is several orders of magnitude greater than the region of the system in which diffusion occurs or can be detected. This kind of solution (called semi-infinite or short-time solution) is 5080


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Table 1. Summary of Geometric Magnitudes of Both Homologous Series of Tracers, Including Molecular Weight (M), Molecular Volume (Vm), Increase of Molecular Volume with Respect to the First Molecule in the Series (Always Biphenyl), Number of Phenyl Units (n), Length of the Equivalent Spring (Lseq), Elasticity Constant of the Equivalent Spring (keq), Area of the Minimal Projection (Amin)

two homologous series of oligomers that have the phenyl group as the primary unit in common. These series are the oligophenyl series, consisting of biphenyl, p-terphenyl, pquaterphenyl, p-quinquepheyl, and p-sexiphenyl, and the diphenylalkene series, consisting of biphenyl, trans-stilbene, diphenylbutadiene, and diphenylhexatriene. By use of homologous series of molecules, the differences in diffusivity can be related to their geometry. The selection of homologous series of molecules was strategically established in order to be able to relate the differences in their diffusion coefficients to their molecular structures. Most authors describe the diffusivity differences using the molecular weight as the main molecular descriptor, which might not be sufficiently representative of the steric hindrance. This way, other geometric descriptors are proposed, such as the molecular or van der Waals volume (Vm) which corresponds to the volume enclosed by the van der Waals surface, the latter being a fictitious surface of the union of spherical atom surfaces defined by the van der Waals radius,27 or the minimum area that can be projected by the molecule, also based on the van der Waals radius (Amin). The values of these descriptors, gathered on the free access Web site www. chemicalize.org (ChemAxon), are reported in Table 1. While there is an increase in the molecular weight and van der Waals volume in both series, it is more noticeable in the case of the oligophenyl series, since the chain enlongation of the oligophenyl series is made by adding one phenyl ring, while in the case of the diphenylalkene series, it is made by the addition of one ethylene group, thus increasing the distance between the phenyl groups. In terms of molecular weight or volume, the addition of one phenyl ring is equivalent to the addition of three ethylene groups, which means that the difference between the two smallest molecules of the oligophenyl series is equivalent to the difference between the

recognizable because of the use of the error function and the absence of the thickness of the film L as a result of the integration of the original differential equation. In some cases, the molecules have shown greater diffusing speeds than expected, leading to diffusing penetrations that almost reach the thickness of the film. This results in the system being considered as finite instead of semi-infinite. For those cases, the analytical solution of eq 1 is given by eq 3. Different from eq 2, eq 3 takes into account the sheet thickness, represented by L. Both solutions allow the evolution of a local concentration profile to be formalized over time. It is to be pointed out that in both eqs 2 and 3, the concentration of the diffusing substance at equilibrium or C∞ is required. ⎛ x ⎞ C ⎟ = erfc⎜ ⎝ 2 Dt ⎠ C∞

C x 2 = + π C∞ L

∞

∑ n=1

(2)

−1 ⎛⎜ nπx ⎞⎟ (−n2π 2 / L2)Dt sin e ⎝ L ⎠ n

(3)

23

As stated in previous work, for this kind of system, the parameter C∞ can be obtained from the Raman peak area ratio of the region immediately adjacent to the source/film interface. Diffusivity was identified from experimental data by minimizing the sum of the squared residuals between experimental (Figure 2) and predicted profiles and by using an optimization method (Levenberg−Marquardt algorithm, optimization routine predefined from Matlab software).

3. RESULTS 3.1. Geometrical Descriptors of Molecules. The diffusing molecules or surrogates used in this study belong to 5081


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Industrial & Engineering Chemistry Research first and last molecule of the diphenylalkene series To illustrate this, p-terphenyl is 1.46 times bigger than biphenyl but psexiphenyl is only 1.19 times bigger than p-quinquephenyl. In the case of diphenylalkenes, trans-stilbene is 1.17 times bigger than biphenyl while diphenylhexatriene is 1.13 times bigger than diphenylbutadiene. Also, if only van der Waals volume and molecular weight are taken into account, both series have a common point, represented by p-terphenyl and diphenylhexatriene, which have fairly comparable values of both properties. 3.2. Spring−Bead Representation of the System. Both molecule series used in this work can be considered as spring− bead systems. Each phenyl group is represented as a rigid bead, and the bond or series of bonds between them are represented as a spring. This way, three types of springs can be defined, the first represented by the Ph−Ph bond between two phenyl groups, a second one represented by the C−C single bond between a phenyl group and a CC double bond, and a third one represented by the CC double bond between two C−C single bonds. The first one is present in the oligophenyl series, while the last two are present in the diphenylalkene series. Each kind of individual spring has an associated force or elasticity constant k, which will be denominated kPh for the Ph−Ph bond, k‑ for the C−C single bond between the phenyl group and the double bond, and k for the CC double bond between two single bonds. These single constants can be calculated with Hooke’s law, according to Newtonian mechanics. This way, the spring is assumed to be attached to the center of mass of two point masses, represented by the phenyl rings at the extremities. The elasticity constant of the single spring can then be calculated according to eq 4: f=

1 2π

k ⇒ k = μ(2πf )2 μ

made under different assumptions and simplifications depending on the geometry of the molecule, which is shared by the molecules of the same family. The oligophenyls are series combinations of rigid spheres of a ratio equivalent to that of the van der Waals volume of a phenyl ring, attached to each other with phenyl−phenyl springs. The equivalent elasticity constant of a spring series combinations in the same axis is given in eq 6:

keq =

k Ph m

(6)

Ph

where k is the elasticity constant of the phenyl−phenyl bond and m is the number of phenyl−phenyl bonds (1 for biphenyl, 2 for terphenyl, etc.). The length of the equivalent spring (Lseq) can be expressed by eq 7: Lseq = mL Ph − Ph + 2rPh(n − 1)

(7)

where m is the number of phenyl−phenyl bonds, LPh−Ph is the length of the phenyl−phenyl bond, rPh is the radius of a sphere of a volume equivalent to that of a phenyl ring, and n is the number of phenyl rings present in the molecule. The use of (n − 1) instead of (n − 2) is made to consider that the equivalent spring is attached to the center of masses and not the surface of the phenyl groups situated at the extremes extremities. The case of the diphenylalkenes is slightly more complex. Due to the geometry of the phenyl−phenyl bonds, the stretching was assumed to occur in the same axis for all bonds, but that is not the case of the C−C single and the CC double bonds. In fact, considering that the center of masses of the left phenyl ring is in the origin of the system, the carbons forming the C−C single and the CC double bonds are bent approximately 55° with respect to the horizontal reference axis. All single bonds are assumed to be aligned parallel to the longitudinal reference axis, while all angles of the carbons between single and double bonds were considered the same for each molecule of the series. Also, the C−C single bonds between the CC double bonds of diphenylbutadiene and diphenylhexatriene were considered to vibrate at the same frequency as the double bonds, according to the simulation results, which might be due to the electron delocalization due to their placement between two double bonds. Distance between atoms and the angles were measured after geometry optimization with the free, cross-platform, open-source molecular editor package Avogadro.28 This way, two force constants may be calculated for each molecule, a series combination of the longitudinal contributions keqH and a series combination of the radial contributions keqV, which are respectively given by eqs 8 and 9:

(4)

where μ is the reduced mass of the phenyl rings and f is the frequency of the stretching vibration mode of the atomic bonding. The reduced mass allows the relative motion of two objects to be described that are acted upon by a central force as if they were a single mass and can be calculated according to eq 5: m1m2 μ= m1 + m2 (5) where m1 and m2 are the values of both masses. Considering that in this case both masses have a value equal to that of the phenyl ring, the reduced mass constitutes half of the latter. The stretching vibration frequencies of the bonds were determined by reconstructing the theoretical spectra of the molecules with the molecular modeling software Gaussian (Gaussian Inc., USA). Resulting from the elasticity constant of all three kinds of single springs, the stiffness tensor of the whole molecule can be deduced by calculating the elasticity constant of the associated equivalent spring (keq). The equivalent spring is a simplification of the backbone of the molecule, derived from the series association of all the single springs attached to the center of masses of two point masses μ. This calculation allows the compressibility of all molecules to be evaluated and compared. In the same way, the length of the associated equivalent spring (Lseq), defined as the length of the straight line binding the center of masses of the phenyl groups situated at the extremities of the molecules, can be easily deducted just with simple trigonometrical relationships. These calculations are

keqH =

k−k cos α k [p + (p − 1) cos α] + 2k cos α

(8)

keqV =

k sin α p

(9)

−

where k− is the elasticity constant for the C−C single bond between the phenyl group and the CC double bond, k is the elasticity constant for the double bond between two single bonds, p is the number of nominal double bonds present in the molecule (1 for trans-stilbene, 2 for diphenylbutadiene, 3 for diphenylhexatriene), and α is the angle of the carbon between the single and the nominal double bond. This way, the elasticity constant can be deduced from the expression, which gives the total force of the system FT 5082


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Figure 3. Representation of the molecules as springs−beads systems: p-terphenyl (top), diphenylbutadiene (middle), and their simplification into the equivalent spring (bottom). Each kind of bond has its single force constant: kPh−Ph for the phenyl−phenyl, k− for the single bond between a phenyl group and a double bond, and k for double bonds. The equivalent spring is defined by the distance between the center of masses of the extreme phenyl groups, called length of the equivalent spring (Lseq), and the force constant of the equivalent spring (keq), which takes into account the series association of the individual springs. In the case of the diphenylalkenes, the calculations of keq and Lseq are made taking into account the angles α and β, since, different from the oligophenyls, the springs are not in the same axis.

Figure 4. Evolution of the concentration of diphenylhexatriene in the thickness of the PS film: (left) at 105 °C after 48 h of contact; (right) at 135 °C after 2 h of contact. The scatter represents the experimental points, and the black solid line represents the best fit from which diffusivity was determined. 5083


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Figure 5. Plot of the diphenylalkene series (a, left) and the dependence of diffusivity with temperature of the oligophenyl series (b, right). Since biphenyl is the starting molecule for both series, it is represented in both plots.

according to its longitudinal FH and radial FV compounds, given by eq 10:

activation energy increases with the molecule length. What is more, the influence of temperature on the diffusion coefficient does not appear to be related to the molecular weight or volume: at a given temperature, the diffusivity difference between biphenyl and trans-stilbene is comparable to that of diphenylbutadiene and diphenylhexatriene. This is also shown by the shapes of the concentration profiles: concentration profiles of biphenyl or trans-stilbene are almost straight lines, while diphenylhexatriene shows a decreasing exponential curve. The apparent activation energies appear normal for amorphous PS, going from 86 to 195 kJ mol−1. 3.4. Diffusivity of Oligophenyls. The relative content of oligophenyl molecules was assessed according to the area of the specific band at 1290 cm−1 assigned to the inter-ring C−C stretching band (νC4−C7, C10−C13). The intensity of this signal, which increases with the number of phenyl groups in the chain, was used to establish the concentration profile through the thickness of the PS film. Diffusivity was identified using eq 2 or 3, on the basis of three concentration profiles per molecule and temperature. Figure 5b shows the evolution of diffusivity with temperature, which can be described as linear if biphenyl is excluded. In contrast with the results observed in the diphenylalkene series, the molecule size appears to impact only biphenyl and p-terphenyl, which show a difference of around 3 or 4 orders of magnitude between them and of up to 2 orders of magnitude between pterphenyl and the bulkiest molecules, p-quaterphenyl, pquinquephenyl, and p-sexiphenyl, which present low diffusivity (reaching 10−18 m2 s−1) regardless of the temperature. More specifically, p-quaterphenyl and p-terphenyl displayed a size variation of only ×1.3, while the difference between their diffusivity is ×5 at 105 °C and reaches ×100 at 135 °C. Between p-quaterphenyl and p-quinquephenyl, the latter being 1.24 times bulkier, diffusivity differences do not generally reach ×10, and between p-quinquephenyl and p-sexiphenyl, the latter

FT = FH + FV = keqLseq = keqHLseq cos(β) + keqVLseq sin(β)

(10)

where β is the angle between the center of masses of the phenyl bead set at the origin and the horizontal reference axis. In this case, given that all bond lengths and angles were known, β as well as Lseq could be calculated with trigonometric relationships. For the sake of clearness, all values of the molecular descriptors are included in Table 1, while the spring−bead systems are represented in Figure 3. It can be noticed that because of the torsions and the nature of the bonds of both series, longer molecules are not always more flexible. 3.3. Diffusivity of Diphenylalkenes. Raman microspectroscopy was applied to establish the concentration profiles of biphenyl, trans-stilbene, diphenylbutadiene, and diphenylhexatriene through the thickness of PS films. The Raman fingerprint of diphenylalkenes exhibits a specific peak of high intensity at 1250 cm−1 which was well suited to follow the sorption of the molecule (Figure 2). This signal increases according to the number of double bonds in the chain. Three concentration profiles per molecule and temperature were separately plotted, as the ones shown in Figure 4, and used to evaluate the diffusivity using eq 2 or 3. As shown in Figure 5a, the evolution of D in relation to temperature for each molecule can be described as linear. It can be pointed out that the molecule length influences diffusivity to a large extent, leading to diffusivity differences greater than ×10 between the different molecules of the series. The variation in temperature highly impacts diffusivity with, in general, a 10 °C increase in temperature resulting in an increase of diffusivity of 5×. In addition, the slope of D vs T curves appears noticeably different according to the size of the molecules, suggesting that the 5084


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Figure 6. Representation of the log10 of diffusivity versus the log10 of the rigid descriptors M (a) and and

Lseq keq

Lseq Vm

(b), as well as the flexible descriptors

Lseq keq

(c)

A min (d), issued from the springs−beads model. The black line represents the best fit of the form D = BFα, where F is the descriptor and

where B and α are the fitting coefficients. The dashed line in the cases of (a) and (b) represents the best fit admitting a different diffusion behavior for each of the homologous series. The symbols correspond to compounds as follows: (○) biphenyl, (×) trans-stilbene, (◇) DPDB, (▲) pterphenyl, (▼) DPHT, (■) p-terphenyl, (★) p-quinquephenyl, and (six-point star) p-sexiphenyl.

At first glance, comparison of the results obtained on both series seems to confirm the major influence of the bulkiness, since diphenylhexatriene and p-terphenyl, which display similar molecular weights/volumes, show diffusivities of the same order of magnitude. However, the impact of temperature appears slightly different with, in particular, a slope of the D vs T curve that is slighty steeper for p-terphenyl. Added to the fact that this translates into a higher activation energy, this statement could additionally suggest a change in the diffusion mode depending on the temperature. Nevertheless, and as it was the case with the diphenylalkene series, activation energies also lay in a normal range for amorphous PS, going from 120 to 212 kJ mol−1. Comparative analysis of these diffusivity values with literature data is difficult due to the lack of investigation performed on similar amorphous polystyrene at the same temperature range. Bernardo29−31 reports values at the same range of temperature for homologous series composed of linear alkanes, alcohols, and carboxylic acids ranging from 32 to 256 g mol−1, while Pinte et al.32 used polycyclic aromatic compounds as fluorescent probes for the determination of diffusion coefficient using FRAP

being 1.19 times bigger, diffusivity differences are not significant taking into account the measurement accuracy. In consequence, plotting D as a function of the molecular weight or molecular volume would result in a decreasing exponential curve at any temperature. What is more, the differences found in the values of the diffusion coefficient are well reflected in the shape of the concentration profile since, for example, at 125 °C, biphenyl manages to go through the whole film thickness in 10 min (D = 4.32 × 10−11 m2 s−1), while p-sexiphenyl penetrates just 6 μm after 9 h of contact. The difference in the diffusing speeds affects the accuracy of the diffusivity determination, reflected in the two slowest cases exhibiting standard deviations above those expected. The average standard deviation, for any molecule belonging to either the diphenylalkene or the oligophenyl series, at any temperature is 0.47 times the value of the measured coefficient, with two exceptions: quinquephenyl and sexiphenyl at 115 °C, which show standard deviations of 13 and 17 times the value of their measured diffusion coefficient. While high, these values are below what could have been expected if such slow kinetics were monitored with other classical methods.23 5085


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results with, in particular, inaccurate predictions for molecules such as alkenes, carboxylic acids, and alcohols of small molecular weight, while the prediction of diffusion coefficient of rubrene (532.7 g·mol−1) based on such a relation is fairly in agreement with the value found experimentally.35 Coming back to Figure 6a and b, it can be observed that despite the a priori good values of the RMSE, a logarithmic scale in the diffusivity axis does not result in linear regression, a sign of deviation of the power law, particularly on the low molecular weight range. In-depth analysis of the results suggests distinct constitutive behavior for both homologous series. In particular, the shape of the oligophenyl D vs M or Vm regression is a decreasing exponential, while the diphenylalkene series one is a straight line. Another interpretation of the plot would be to consider two straight lines with a slope break at diphenylhexatriene/pterphenyl, p-terphenyl being slightly smaller and lighter. Such interpretation of the results leads to the most accurate regression of diffusivity data (with R2 > 0.9) and consequently confirms that a description solely based on M (and in this particular case, on Vm) is not enough to explain the diffusion behavior of low molecular weight molecules. L The use of seq as descriptor (Figure 6b) yields slightly

(fluorescence recovery after photobleaching). Regarding the results obtained with surrogates of high molecular weight, diffusivity values obtained on the quater-, quinque-, and sexiphenyl are in agreement with the value reported on fluorescent probes (10−14−10−15 m2 s−1) displaying a molecular weight of 236 and 426 g·mol−1. For molecules of lower molecular weight, diffusivity of biphenyl (156.2 g·mol−1) above the glass transition temperature of PS corresponds relatively well to trend values, which is measured around 10−11 m2 s−1 at 105 °C for the aforementioned linear alkanes, alcohols, and carboxylic acids. However, diffusivity of trans-stilbene (around 10−13 m2 s−1), is of the same order of magnitude as that of alkanes or alcohols of comparable molecular weight at temperatures of 75 °C and lower. This comparison suggests that bulkiness may not be sufficiently described by the molecular weight, despite its major influence on diffusivity. 3.5. Description of D Using a Rigid Descriptor. Most of the comparisons between the diffusion coefficients of different molecules found in literature are made using molecular weight as the main descriptor, or molecular weight and temperature. This is possibly due to the fact that molecular weight is very easy to calculate, while other parameters that would yield more accurate descriptions, like the aforementioned molecular volume, require more complex calculations, sometimes even with molecular modeling software that was not generally available. Figure 6a and b reports the experimental values of diffusivity of all of the surrogates at 125 °C as a function of two descriptors: M (Figure 6a) and the ratio between the length of the equivalent spring (Lseq) and Vm (Figure 6b), indicative of the degree of resemblance between the shape of the molecule and a cylinder. The denomination “rigid descriptor” comes from the fact that they do not take into account any form of flexibility of the molecules, although Lseq has been calculated as a part of the spring−bead model. Diffusivity data of both series were plotted together and resulted in a curve apparently described by a decreasing power law of the type D = BFα. Biphenyl was used as the starting molecule, i.e., the most elementary molecule for both homologous series. These projections show strong correlations, with an RMSE of 0.64 L and 0.67 for M and seq , respectively. Regarding the evolution as

Vm

different results. The RMSE is not significantly different and the points align better around the prediction, but the deviation from the power law is still present, as it is shown by the underestimation of the biphenyl and the overestimation of the diphenylhexatriene. If with M the oligophenyl series seemed to follow an exponential decreasing trend and the diphenylalkene series a linear one, in this case, each of the series appears to follow linear trends with different slopes and the same intercept, as indicated by the dashed lines. The improvement of the prediction is probably due to the use of two parameters that describe bulkiness/geometry better (Lseq and Vm), instead of a generic one (M). This view has already been shared by several authors, who pointed out the influence of the shape of a molecule on its diffusion mode.17 The higher probability for a long molecule of having many degrees of freedom should facilitate its displacements by crawling, while nonlinear molecules are supposed to diffuse by jumping. This statement has led to the emergence of an alternative concept according to which the molecular mobility is investigated as a function of a proposed fragmentation of the molecule volume by considering the mobility of each part of the molecule separately. This way, each molecule is considered as an association of linear and flexible chains that can take several conformations making the displacements easy, and the rigid parts (generally constituted of cycles) for which the conformation is locked. The fractionated volume of the molecule corresponds to the sum of the different partial volumes of each part, the jump displacements being facilitated by the easier or harder relaxation of the flexible parts of the molecule. Considering the satisfactory correlation between the fractionated volume and the diffusivity, this concept has clearly allowed progress to be made on understanding the mechanism of displacement through polymeric matrices. However, as usual size parameter, fractionated volume was assessed on the basis of the most stable conformation of the molecules (or of each part of it). As a consequence, this approach did not take into account the variation in shape of the surrogate or the energy necessary to induce the modification of the molecule conformation. 3.6. Description of D by a Spring−Bead Model. With the aim of proposing alternative molecular parameters that

Vm

a function of the molecular weight, this behavior is not surprising taking into account the base postulates of variation described by the relation D ∝ M−α. This scaling law has its origins in the use of statistical mechanics to describe the selfdiffusion of polymeric chains in dilute solutions12 or more concentrated regimes13,15 and has been the subject of research for many years. As a matter of fact, the exponential increase of computing power has allowed these scaling relationships to be checked, via complex molecular dynamics simulations.33,34 According to these theories, the parameter α would be indicative of the transport mechanism. For example, de Gennes postulated that a value of α = 2 indicates that the polymeric chain reptates through a tube formed by the neighboring ones. While these postulates have been proved accurate to describe the self-diffusion of polymer chains in polymeric networks, they might not be generalizable to any other kind of diffusing molecule. This way, other theories, studying the influence of parameters other than M and with a more meaningful concept of the coefficient α, have been recently developed,16 aiming to find a general model to predict diffusivity in polymeric matrices. However, the use of the molecular weight to predict diffusivity of the molecules found in literature generates contradictory 5086


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Table 2. Coefficients B0, B1, α0, and α1 for the Shape Factors That Take into Account Compressibility, as Well as the R2 Coefficients, Representative of the Goodness of Their Correlation with Temperature B0

shape factor −41

2.9 × 10

Lseq

R2

B1 −1

−1

[N·m·s ·Å ]

−1

α0 −1

α1

R2

0.37 [°C ]

0.9999

−12.9 [°C ]

0.07 [−]

0.9986

18.1 [−]

0.9988

−11.9 [°C−1]

0.06 [−]

0.9928

keq Lseq keq

A min

7.5 × 10−53 [N·m·s−1·Å−3]

Figure 7. Prediction of the models at the three studied temperatures. The plots have been made with the coefficients shown in Table 2.

of the diphenylalkenes and p-terphenyl lay on both sides of the fitted curve and not systematically below it, as was the case of the rigid descriptors, which means that the diffusivity decreases in the range between 150 and 230 g mol−1 using these shape factors rather than solely using molecular weight. This means that, while yielding slightly lower RMSE, M or Vm proved to be poorer descriptors of the diffusivity/bulkiness trend, thus reinforcing the hypothesis according to which splitting up generic molecular descriptors into others richer in structural information results in an increase in accuracy, which can be done with simple geometric considerations. Since the polymeric chains have more mobility in the rubbery than in the glassy state, these statements should not be verified in the glassy state. While the results presented here are specific to the homologous series of molecules that have been studied, the geometric concepts on which they are based are not difficult to understand, and the calculation of the parameters is pretty straightforward and does not require an intensive amount of computing power. It might be said that the spring−bead concept could lay the foundations of a more general model of diffusion in the rubbery state in which the polymeric chains have more mobility. 3.7. Possibilities of Generalization for the Rubbery State. As previously stated and shown in Figure 6, the relationship between diffusivity and bulkiness can be described by a power law of the type D = BFα, where F is any of the

could further explain the mobility of diffusing molecules within a polymeric matrix, it can be pointed out that polymer chains in a rubber-like state exhibit a degree of mobility that can influence the diffusivity of molecules. On this basis, molecules endowed with flexibility should demonstrate a higher ability to diffuse in a moving polymer network, since they would be able to overcome the compressing stress induced by this movement. The goal of this approach being not to try to predict diffusivity but to progress in the description of the diffusion behavior, new and less generic molecular descriptors have been considered to translate the flexibility of molecules. Considering the studied class of linear and rigid molecules, the flexibility was drawn up by the assessment of their compressibility which was achieved by using the magnitudes issued from the spring−bead consideration of the system (Lseq and keq) and combining them into shape factors. This way, two shape factors were L defined: (1) the ratio seq or linear compressibility of the keq

molecule; (2) the linear compressibility times the parameter Amin, which transforms the linear compressibility into

Lseq keq

A min

or volumetric compressibility and makes it potentially applicable to molecules other than linear. These compressibility evaluators might indicate how easy a molecule can get through free spaces between the polymeric chains. As well as in the case L of M and seq , the best fit was achieved with a decreasing power Vm

law of the kind D = BFα, where F is the studied shape factor and B and α are coefficients issued from the fitting (Table 2). The fitting yielded RMSE of 0.78 and 0.60, respectively (Figure 6c,d), which would a priori indicate similar but slightly worse L fits than the curves against M or seq . However, taking a more

studied descriptors, either rigid (M, Vm, Lseq

Lseq

eq

keq

(k ,

Lseq Vm

) or flexible

× A min ). Figure 6 only displays data and predictions

at 125 °C, but the same kind of law has also been found to successfully describe the trend at 115 and 135 °C. The law D = BFα remains the same, while the value of the fitting variables B

Vm

detailed look at the plots, it can be noticed that the diffusivities 5087


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Industrial & Engineering Chemistry Research and α varies as a function of T. At any of these three temperatures, the descriptors can be ranked according to the accuracy of the fit they are able to provide. This way, if only the RMSE is taken into account, the ranking is the same, regardless L of the temperature: the rigid descriptors M and seq would be

135 than at 125 °C, resulting in the rupture of the trend coefficient-temperature. As it can be seen, the models developed on the basis of the rigid descriptors, besides providing poorer predictions at each of the single temperatures, are especially sensitive to the scatter of the data, due to experimental issues, which is more noticeable at 135 °C. Consequently, the models derived from this work can be easily formalized into eqs 11 and 12 for the prediction of diffusivity on polystyrene in the rubbery state as a function of the linear and volumetric compressibility of the molecules.

Vm

better descriptors, while the flexible shape factors Lseq keq

Lseq keq

and

A min would be a priori worse descriptors, again only

considering the RMSE as an indicator of the accuracy of the fit and without taking into account the biases of each model described in section 3.6. Keeping in mind that the coefficients B and α change along with temperature, the relationship between them and temperature can be explored in order to determine which is the descriptor that models the data the best. This exercise has led to interesting results and shows a clear difference between the rigid and the flexible descriptors. Results show that the coefficients B or α obtained from the modeling L with the rigid descriptors M and seq are not related to

D = B0 e

⎡ L ⎤[α0 − α1T ] ⎥ ⎢⎣ keq ⎥⎦

B1T ⎢

⎡ L ⎤[α0 − α1T ] A min ⎥ D = B0 T ⎢⎣ keq ⎥⎦

(11)

B1⎢

(12)

Vm

temperature in any way whatsoever. The exact opposite happens for the descriptors that take into account the flexibility of the molecule: not only both coefficients B and α follow a trend in relation to temperature, but the correlation is so strong that it is possible to derive a mathematical relationship with R2 coefficients that vary from 0.99 to 0.9999. Particularly, B follows L an exponential law of the kind B = B0 eB1T for the seq descriptor

4. GENERAL CONCLUSIONS Considering the influence of the physicochemical characteristics of low molecular weight molecules, results indicated the influence of the geometry of the molecule on its diffusion behavior. In practical terms, the classical comparison between molecules of different nature solely based on the molecular weight can induce a perfectible estimation, especially for small molecules of up to approximately 300 g mol −1 . To quantitatively improve the estimation, a new approach to describe the diffusion behavior by splitting the most generic molecular descriptors like M or Vm into more refined ones is proposed on the basis of relatively simple geometric considerations that describe the shape of the molecules and their flexibility more accurately. This way, new descriptions based on the ratio length/volume, as well as on the linear or volumetric compressibility, were proposed. The variables translating the molecular compressibility are based on the inclusion of the elasticity constant issued from the consideration of the molecules as springs−beads systems, according to Newtonian mechanics. While still far from a general model to describe diffusion behavior in the rubbery state, this work shows how a description based on geometric considerations is feasible with simple calculations that do not require an extensive amount of computing power, which is the case of the models based on molecular dynamics or based on too abstract concepts, which is usually the case of the models based on the free-volume theory. The constants present in the developed equations are a priori pertinent in the unique case of amorphous polymers in a rubbery state exhibiting a potential segmental mobility. In other words, since this model is based on principles like the conformation of free volumes, a continuous change in the surrounding environment is required for the model to be applicable. It may be expected that reduced mobility of the polymeric chains would result in less stress on the diffusing molecules, leading to diffusivity and compressibility not being related (e.g., diffusion through polymers on the glassy state). On the other hand, diffusion through looser and continuously changing media (e.g., diffusion in the gas phase) may be under the scope of the model, provided the fitting parameters α and B are found. Nevertheless, since this work is simply a first step toward a general understanding of the relationship through the compressibility principle, these

keq

and a power law of the kind B = B0 eB1 for the

Lseq keq

A min ,

descriptor. Regarding the coefficient α, it follows a linear trend of the kind α = α0 − α1T in any case. Resuming, only the flexible descriptors allow diffusivity to be predicted at any temperature in the rubbery state, by expressing B and α as a function of the four temperature dependent coefficients B0, B1, α0, and α1 listed in Table 2. A plausible explanation for the lack of correlation between B and α in relation to temperature for the rigid descriptors comes from (as stated in section 3.6) trying to derive a general power law from data that do not actually follow it. Figure 7 displays the power law for each of the descriptors at each of the studied temperatures: 115, 125, and 135. For the sake of clearness, both axes are shown in logarithmic scale, which means that a power law of the kind D = BFα is transformed into a linear polynomial of the kind log10 D = log10 B + α log10 F, where log10 B is the intercept and α is the slope. Several facts can be deduced from this figure: • The slope becomes more gradual in relation to temperature; i.e., the diffusivity difference between the lighter (biphenyl) and the bulkier (sexiphenyl) molecules becomes smaller in relation to temperature. • The intercept hardly varies, which is quite realistic, taking into account the minimal variations on the diffusivity of biphenyl (which acts as first point of the series) between 115 and 135 °C. • The same figure using the rigid descriptors derives in systematic underestimatation the diffusivity of the lighter molecules (biphenyl and stylbene) at any temperature (this can be noticed in Figure 6, parts a and b). Consequently, the intercept (log10 B) varies randomly, and the slope (log10 α) becomes a lot more gradual at 5088


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(14) Lodge, T. P. Reconciliation of the Molecular Weight Dependence of Diffusion and Viscosity in Entangled Polymers. Phys. Rev. Lett. 1999, 83 (16), 3218. (15) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (16) Fang, X.; Domenek, S.; Ducruet, V.; Refregiers, M.; Vitrac, O. Diffusion of Aromatic Solutes in Aliphatic Polymers above Glass Transition Temperature. Macromolecules 2013, 46 (3), 874. (17) Reynier, A.; Dole, P.; Humbel, S.; Feigenbaum, A. Diffusion Coefficients of Additives in Polymers. I. Correlation with Geometric Parameters. J. Appl. Polym. Sci. 2001, 82 (10), 2422. (18) Rouvray, D. H.; Kumazaki, H. Prediction of Molecular Flexibility in Halogenated Alkanes via Fractal Dimensionality. J. Math. Chem. 1991, 7 (1), 169. (19) Jeschke, G.; Sajid, M.; Schulte, M.; Ramezanian, N.; Volkov, A.; Zimmermann, H.; Godt, A. Flexibility of Shape-Persistent Molecular Building Blocks Composed of P-Phenylene and Ethynylene Units. J. Am. Chem. Soc. 2010, 132 (29), 10107. (20) Ozisik, R.; von Meerwall, E. D.; Mattice, W. L. Comparison of the Diffusion Coefficients of Linear and Cyclic Alkanes. Polymer 2002, 43 (2), 629. (21) von Meerwall, E. D.; Dirama, N.; Mattice, W. L. Diffusion in Polyethylene Blends: Constraint Release and Entanglement Dilution. Macromolecules 2007, 40 (11), 3970. (22) Mauricio-Iglesias, M.; Guillard, V.; Gontard, N.; Peyron, S. Application of FTIR and Raman Microspectroscopy to the Study of Food/packaging Interactions. Food Addit. Contam., Part A 2009, 26 (11), 1515. (23) Martinez-Lopez, B.; Chalier, P.; Guillard, V.; Gontard, N.; Peyron, S. Determination of Mass Transport Properties in Food/ packaging Systems by Local Measurement with Raman Microspectroscopy. J. Appl. Polym. Sci. 2014, 131, 40958. (24) Nishikida, K.; Coates, J. Handbook of Plastics Analysis; Lobo, H., Bonilla, J. V., Eds.; Marcel Dekker, 2003; pp 201−340. (25) Fick, A. Ueber Diffusion. Ann. Phys. 1855, 170 (1), 59. (26) Crank, J. The Mathematics of Diffusion; Oxford University Press, 1980. (27) Whitley, D. C. Van Der Waals Surface Graphs and Molecular Shape. J. Math. Chem. 1998, 23 (3−1), 377. (28) Hanwell, M.; Curtis, D.; Lonie, D.; Vandermeersch, T.; Zurek, E.; Hutchison, G. Avogadro: An Advanced Semantic Chemical Editor, Visualization, and Analysis Platform. J. Cheminf. 2012, 4 (1), 17. (29) Bernardo, G. Diffusivity of Alkanes in Polystyrene. J. Polym. Res. 2012, 19 (3), 1. (30) Bernardo, G.; Choudhury, R. P.; Beckham, H. W. Diffusivity of Small Molecules in Polymers: Carboxylic Acids in Polystyrene. Polymer 2012, 53 (4), 976. (31) Bernardo, G. Diffusivity of Alcohols in Amorphous Polystyrene. J. Appl. Polym. Sci. 2013, 127 (3), 1803. (32) Pinte, J.; Joly, C.; Dole, P.; Feigenbaum, A. Diffusion of Homologous Model Migrants in Rubbery Polystyrene: Molar Mass Dependence and Activation Energy of Diffusion. Food Addit. Contam., Part A 2010, 27 (4), 557. (33) Durand, M.; Meyer, H.; Benzerara, O.; Baschnagel, J.; Vitrac, O. Molecular Dynamics Simulations of the Chain Dynamics in Monodisperse Oligomer Melts and of the Oligomer Tracer Diffusion in an Entangled Polymer Matrix. J. Chem. Phys. 2010, 132 (19), 194902. (34) Vitrac, O.; Lezervant, J.; Feigenbaum, A. Decision Trees as Applied to the Robust Estimation of Diffusion Coefficients in Polyolefins. J. Appl. Polym. Sci. 2006, 101 (4), 2167. (35) Tseng, K. C.; Turro, N. J.; Durning, C. J. Tracer Diffusion in Thin Polystyrene Films. Polymer 2000, 41 (12), 4751.

affirmations may be too speculative. It is the purpose of future work to apply the principles of the spring−bead model presented here to other series of molecules available in the literature in order to validate it without the bias of the nature of the diffusing molecules or the methodology used to gather the experimental data. Such a validation might lay the foundations of a general model to predict diffusion in polymers in the rubbery state.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work has been possible because of funding provided by the French funding agency Association Nationale de la Recherche within the framework of the Research Project SafeFoodPack Design. The authors thank Dr. Matthieu Saubanère from Institut Charles Gerhardt (Montpellier, France) for his introduction to the two-body problem, and Eddy Petit, from Institut Européen des Membranes (Montpellier, France) for the theoretical spectra calculations.

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REFERENCES

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Developing a Macroscopic Mechanistic Model for Low Molecular

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