Experimental Study of Sorption and Diffusion of n-Pentane in

Article pubs.acs.org/jced

Experimental Study of Sorption and Diffusion of n‑Pentane in Polystyrene Hana Hajova,† Josef Chmelar,† Andra Nistor,† Tomas Gregor,‡ and Juraj Kosek*,† †

Department of Chemical Engineering, Institute of Chemical Technology Prague, Technicka 5, 166 28 Prague 6, Czech Republic Research Centre New Technologies, University of West Bohemia, Univerzitni 8, 306 14 Pilsen, Czech Republic

‡

ABSTRACT: Polystyrene (PS) foams are widespread heat insulation and impact absorption materials, but many blowing agents used in their production are harmful to the environment. Pentane is a less harmful alternative and is commonly used in the manufacturing of expandable PS, but there are inadequate diffusion data of pentane in PS available in the literature at conditions relevant to PS impregnation and foaming. We conducted gravimetric measurements of sorption dynamics of n-pentane in PS particles and thin films at temperatures from 45 to 110 °C and pressures from 0 to 4.8 bar. The measured data were fitted using the Fick’s and case II diffusion models and the obtained transport parameters are presented together with an analysis of the dominant diffusion mechanism. We also present the Flory−Huggins interaction parameter for the PS + n-pentane system and show that it can be used to accurately predict the sorption equilibria. This paper thus provides important parameters for the optimization of PS foam production.

1. INTRODUCTION The knowledge of diffusion dynamics and sorption equilibria of organic solvents in polystyrene (PS) and its copolymers is important for numerous applications, for example, for the design of food packaging materials and degassing equipment in PS manufacturing. Here we stress out the relevance of pentane diffusion in PS for processes related to PS foams: (i) impregnation of PS by blowing agents, (ii) storage of impregnated PS beads, (iii) foaming of PS, and (iv) diffusion of blowing agents from PS foams into the environment. The two frequently used processes in PS foam manufacturing are expansion of blowing-agent impregnated PS beads1,2 and extrusion of polymer melt with dissolved blowing-agent.3,4 Pentane is typically used in the former of these processes and the impregnated PS beads are referred to as expandable PS. The selection of a suitable blowing agent is important, because it significantly affects the mechanical and heat-insulation properties of the produced PS foam. Most classical blowing agents are greenhouse gases and some cause the depletion of the ozone layer (e.g., fluorinated hydrocarbons). It is thus important to find environmentally friendly blowing agents or, at least, to reduce the used amount of the classical agents. Despite their industrial importance, sorption equilibrium data of pentane in PS at conditions relevant to the manufacturing of expandable PS have been reported only recently.5 Diffusion affects the length of the PS bead impregnation period and the uniformity of pentane concentration profiles in the PS beads. In industrial suspension polymerization processes, pentane is added at the final stage of polymerization when polymer particles still contain a few percent of unreacted styrene. The © 2013 American Chemical Society

unreacted monomer increases the rate of pentane diffusion and the pentane dissolved in the polymer beads can reduce the Norrish−Trommsdorff effect.6−8 At the end of the polymerization run, the suspension of impregnated PS beads is cooled to a temperature below their glass-transition temperature (Tg) so that a vitrified structure of pentane in PS is obtained. The storage of impregnated PS beads before further processing, that is, before their foaming, can last several days to several weeks, and pentane can partially desorb from the beads during this period. Pentane diffusion also plays a role in the process of foaming as it affects the rate of bubble growth and as pentane acts as a polymer plasticizer. Slow diffusion of blowing agents out of the manufactured PS foam and counter-diffusion of air affect the heat-insulation properties of the foam and result in the release of blowing agents into the environment. In this paper, we continue in the publication of our measurements previously reported by Chmelar et al.,5 who presented only sorption equilibrium data and their fitting by the PC-SAFT equation of state. Here we study the sorption dynamics of n-pentane in both PS particles and thin PS films at condition relevant to the production and processing of PS foams. The dynamic data were evaluated by the Fick’s and case II diffusion models both with and without the effect of PS swelling. The resulting transport parameters are presented together with conclusions on the dominant diffusion mechanism. Furthermore, we present previously unpublished Received: August 19, 2012 Accepted: February 20, 2013 Published: March 5, 2013 851

dx.doi.org/10.1021/je300916f | J. Chem. Eng. Data 2013, 58, 851−865

Journal of Chemical & Engineering Data

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sorption equilibrium data of n-pentane in thin PS films. We also evaluated the temperature dependence of the Flory−Huggins interaction parameter for the PS + n-pentane pair and successfully tested the quality of predictions calculated with this parameter. Survey of Equilibrium Sorption and Diffusion Measurements in PS. Articles studying the sorption equilibria and diffusion dynamics of various relevant solvents in PS are summarized in Table 1. In most cases, the experimental

adoption of the SAFT EOS led to many modifications of the original work, for example, the perturbed chain SAFT (PCSAFT),26 the variable range SAFT (VR-SAFT)27 or the simplified PC-SAFT.28 A significantly different approach is the clustering theory of Zimm and Lundberg,29 which uses statistical mechanics to account for fluctuations. This theory does not predict sorption isotherms, but enables their interpretation in molecular terms. In the case of gas sorption into glassy polymers deep below Tg, the well-known S-shaped isotherms were observed.30,31 These S-shaped isotherms can be explained by the dual-mode sorption model.32 The Flory−Huggins Theory. The Flory−Huggins model was used for the fitting of sorption equilibrium data in this work. This theory is based on the concept of a microscopic organization of solvent molecules and polymer macromolecules which are arranged in a lattice. We employ the Flory−Huggins equation in the form

Table 1. Studies of Sorption Equilibria and Diffusion Dynamics of Relevant Solvents in PS authors Vieth and Sladek (1965)32 Hopfenberg et al. (1969)41 Holley et al. (1970)55 Baird et al. (1971)43 Stewart et al. (1987)70 Chmelar et al. (2011)5 Hopfenberg et al. (1969)41 Enscore et al. (1980)71 Hopfenberg et al. (1969)41 Kwei (1973)37 Krüger and Sadowski (2005)30 Grolier and Randzio (2011)9 Grolier and Randzio (2011)9 Sato et al. (1999)17 Grolier and Randzio (2011)9

T/°C

p/bar

methane n-pentane

25 to 45 25 to 50

0 to 32 0 to 0.62

n-pentane n-pentane n-pentane n-pentane, i-pentane n-hexane

25 to 50 30 to 40 30 45 to 110

0 to 0.62 0 to 0.62 0 to 0.70 0.9 to 5.4

30 to 45

0 to 0.62

n-hexane n-heptane

30 to 50 35 to 50

0 to 0.055 0 to 0.62

benzene toluene

30 to 70 30 to 115

0.016 to 0.11 0 to 0.40

carbon dioxide

65 to 130

0 to 450

nitrogen

40 to 80

0 to 800

nitrogen HFCs

40 to 80 112 to 140

0 to 170 0 to 200

solvent

⎛ 1⎞ ln a1 = ln φ1 + ⎜1 − ⎟(1 − φ1) + χ (1 − φ1)2 ⎝ x⎠

(1)

where a1 is the penetrant activity, ϕ1 is the volume fraction of the penetrant, x is the scaled polymer chain length, and χ is the polymer−solvent interaction parameter. Under the simplifying assumption of volume additivity, the volume fraction of the penetrant ϕ1 can be expressed as φ1 =

S(p , T ) V1 V1 = = ρ Vmix Vpol + V1 S(p , T ) + 1,LIQ ρ pol

(2)

where ρpol and ρ1,LIQ are the temperature-dependent densities of the polymer33 and penetrant,33,34 respectively, Vmix is the volume of the polymer−solvent mixture, Vpol is the volume of the polymer, V1 is the volume of the penetrant, and S is the penetrant solubility per unit mass of polymer obtained from gravimetric measurements

conditions are out of the range used in PS foam manufacturing. Industrially interesting are temperatures from 80 to 110 °C and higher pentane pressures from 4 to 5 bar. The diffusion and sorption equilibrium data are usually estimated and presented together (most studies in Table 1). Only few papers report the sorption equilibrium data separately.5,9 The classical blowing agents are mixtures of chlorofluorocarbons (CFCs) and are still used in the industry.9 Because CFCs are harmful to the environment, it is necessary to eliminate their use in industrial processes. Hydrochlorofluorocarbons (HCFCs) and alkanes are considered as less harmful alternative blowing agents. However, even these compounds are not completely safe. Researches are thus currently attempting to find effective environmentally neutral blowing agents, for example, carbon dioxide,5,10 supercritical carbon dioxide,11 moisture from wood fibers4, or water.12

S(p , T ) =

m(p , T ) − m(0, T ) mpol

(3)

where m(p,T) is the gravimetrically evaluated mass (eq 9) at temperature T and pressure p (or at vacuum), and mpol is the mass of the pure polymer. In this work, we study the sorption of n-pentane in PS samples with molecular weights (Mw) above 100 kg/mol, that is, with scaled chain length x > 1000. Because the sorption equilibria of n-pentane in PS are independent of Mw for Mw values high above 10 kg/mol,35,36 we arbitrarily set the scaled chain length to x = 1000. Krüger and Sadowski30 reported that for the PS + toluene system the Flory−Huggins model was not able to predict the curvature of the slightly S-shaped sorption isotherm at low penetrant concentrations, but at higher concentrations (rubbery polymer) it described the isotherms correctly. In this paper, we focus on the industrially important range of higher pentane pressures; therefore we do not consider the possible deviations of the Flory−Huggins model at low pressures. Complex relations were proposed for the temperature dependence of the interaction parameter χ,24 but we employ the simple reciprocal temperature dependence

2. THEORY OF EQUILIBRIUM SORPTION AND DIFFUSION 2.1. Theory of Equilibrium Sorption. There is a large number of models developed for the description of phase equilibria in polymer−solvent mixtures, for example, the Flory−Huggins equation,13,14 the Sanchez−Lacombe equation of state (EOS),9,15−18 UNIFAC-FV models,19 and the statistical associating fluid theory (SAFT) EOS.20,21 The Flory−Huggins model and its modifications are widely used in practical applications due to their simplicity.22−25 The widespread

χ= 852

α +β T

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annealing on the orientation of PS films decreases with increasing Mw. Baird et al.43 investigated the effect of Mw on the relaxation-controlled diffusion of n-pentane in glassy PS and found the diffusion rate to be independent of Mw over a broad range of molecular weights (from 9.95 × 103 g/mol to 1880 × 103 g/mol). Hassan and Durning52 and Vrentas and Vrentas53 also concluded that the effect of Mw on the diffusion mechanism is not significant for higher values of Mw. Polymer orientation significantly influences diffusion, especially in the case of non-Fickian diffusion.54 Bray and Hopfenberg50 found that slight differences in the residual orientation of PS films can result in significantly different diffusion rates. Such residual orientation can, for example, arise during the solution casting preparation of thin PS films on rough surfaces.41,50,53 Also Baird et al.43 observed that pentane diffusion in uniaxially oriented PS films was up to 10 times faster than in annealed (unoriented) films. However, Holley et al.55 concluded from their experiments of pentane sorption in PS that the polymer orientation does not affect the transport processes in the case of very low penetrant activities. De Francesco and Duckett51 observed decreases in the orientation of PS films upon annealing at constant temperature and found that the annealing time influenced these changes in orientation. The time−temperature history of a PS sample can thus affect the orientation of PS chains. Cross-links are not present in PS, but it is known that they influence the diffusion of penetrants in polymers. For more information on this topic, we recommend the detailed analysis of cross-linking effects on penetrant transport presented by Ekenseair and Peppas.44 2.3. Models of Diffusion. 2.3.1. Transport in Polymer of Constant Volume. In this work, two simple limiting models of transport are used: Fick’s diffusion and case II diffusion. The simplest form of these models assumes the PS particles and films as perfect spheres (radius R) and planar films (thickness d), respectively. The n-pentane material balances and the equations for the molar flux intensities N are presented in Table 2, where c(r,t) and c(x,t) are the n-pentane molar

with parameters α and β evaluated from experimental solubility data. This simple approach enables good fitting of the measured data as shown in section 4. Studies investigating the effect of polymer orientation on sorption equilibria are scarce in the literature. Kwei37 found slightly higher solubilities of benzene in unoriented than in biaxially oriented PS films and discussed this difference in solubility using the clustering theory. The clustering tendency of benzene in oriented PS is greater than in unoriented PS. It is known that polymer chains are oriented in PS foams and that this orientation is a result of the expansion of the blowing agent bubbles.38 However, it is still not clear as to which molecular orientation arises during this process. Further research is thus required to better understand this topic. 2.2. Theory of Diffusion. The diffusion of pentane in PS is both practically important and theoretically interesting. The properties of amorphous polymers above and below Tg have been studied for a long time, but the structure of the glassy state is still a subject of interest.39 The glassy state is a nonequilibrium one, the polymer chains and the free-volume among them are “frozen” and polymer chains relax very slowly. Polymer relaxation is thus rate limiting for the diffusion in glassy polymers, which is then referred to as relaxationcontrolled. During PS impregnation by pentane, the relaxation is enhanced by swelling and osmotic stresses and depends on the pentane concentration.40 It is also necessary to consider the dependence of Tg on the composition of the PS + pentane mixture.25,37 As a result, the diffusion in glassy polymers is often difficult to interpret. At temperatures deep below Tg, the diffusion into glassy PS can be described as a moving front of pentane with constant velocity causing the phase-transition from glassy to plastic state. This diffusion mechanism is called the case II diffusion.41 The case II diffusion can be characterized by a single parameter: the velocity of the moving diffusion front.42 According to Baird et al.,43 the principal characteristic of the case II diffusion is the presence of a distinct boundary between the outer layers, which are in a state close to sorption equilibrium, and the still unrelaxed and unswollen glassy inner core. There should also be a step change in penetrant concentration at this boundary. At temperatures above Tg, the diffusion mechanism is Fickian. However, in the case of nonswelling vapors, the Fickian diffusion model can correctly describe even the diffusion in glassy polymers.48 Both the Fickian diffusion and the case II diffusion are only limiting cases as there exist a number of combined diffusion mechanisms collectively named nonFickian diffusion. Examples of these combined mechanisms are sigmoid, super case II, anomalous, or pseudo-Fickian diffusion.30,44 The dependence of pentane diffusion rate in PS on temperature and pentane concentration can also provide information on the plasticizing effect.45 Cohen and Turnbull46 presented a concept of the free-volume theory that enables the evaluation of the effects of solvent concentration, polymer Tg, mixture viscosity, and other parameters on the solvent diffusion coefficient. But the high number of parameters and high sensitivity to these parameters make the application of this concept to polymer systems practically difficult.47 Rudd48,49 and Bray and Hopfenberg50 studied the effect of polymer molecular weight Mw on relaxation-controlled diffusion. They concluded that the relaxation rate decreases with increasing Mw for PS with Mw below 6 × 103 g/mol. De Francesco and Duckett51 also observed that the effect of

Table 2. Equations for n-Pentane Transport in PS for the Case without Polymer Swelling particle

thin film

material balance

∂c 1 ∂ = − 2 (r 2N ) ∂t r ∂r

∂c ∂N =− ∂t ∂x

molar flux intensity (Fick’s diffusion)

N = −D

molar flux intensity (case II diffusion)

N = cvCII

equation

∂c ∂r

N = −D

∂c ∂x

N = cvCII

concentrations in the particles and films, respectively, D is the diffusion coefficient, vCII is the case II diffusion front velocity, r is the radial coordinate (for particles), x is the spatial coordinate (for thin films) and t is the time coordinate. The initial and boundary conditions are summarized in Table 3, where c0 is the equilibrium n-pentane concentration in the sample before the pressure-step (at t = 0) and cS(t) is the timedependent surface concentration, which is assumed to be in equilibrium with the surrounding gas phase. The timedependent boundary conditions (for r = R and x = d/2) were introduced because it is not technically possible to realize ideal stepwise pressure changes and also because slight pressure changes may occur during long experiments. Thus cS(t) was 853



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Table 3. Initial and Boundary Conditions for the Case without Polymer Swelling condition

description

thin film

particle

initial

equilibrium before the pressure-step

t = 0,

0 < r < R,

boundary

symmetry

t ≥ 0,

r = 0,

boundary

instantaneous sorption equilibrium on the surface

t ≥ 0,

r = R,

∫0

R

∫0

m(t ) = 2M1

4πr 2c(r , t ) dr

(5)

d /2

wlc(x , t ) dx

(6)

1 n

(7)

i=1

where n is the number of points in the given data set and miexp and micalc are the experimental and calculated mass of the sorbed n-pentane at the ith point, respectively. The fitting parameters were D (for Fick’s diffusion) and vCII (for case II diffusion), that is, only one fitting parameter per model. 2.3.2. Transport in Swelling Polymer. According to Alsoy and Duda,57 the mathematical models of pentane diffusion in PS should account for the swelling of the PS matrix by pentane. The equations used for transport in swelling polymers are analogous to those presented in Table 2, but incorporate swelling terms, and the polymer dimensions become time dependent (R(t) for particles and d(t) for thin films). The resulting set of equations is presented in Table 4 and the modified initial and boundary conditions are presented in Table Table 4. Equations for n-Pentane Transport in PS Including the Effects of Polymer Swelling equation

particle

thin film

material balance I

∂c 1 ∂ = − 2 (r 2N + r 2uc) ∂t r ∂r

∂c ∂ = − (N + uc) ∂t ∂x

material balance II

∂ρ 1 ∂ = − 2 (r 2M1N + r 2uρ) ∂t r ∂r

∂ρ ∂ = − (M1N + uρ) ∂t ∂x

molar flux intensity (Fick’s diffusion) molar flux intensity (case II diffusion)

N = −D

∂c ∂r

N = cvCII

N = −D

∂c =0 ∂r

t ≥ 0,

x = 0,

c = cS(t )

t ≥ 0,

x = d /2,

c = c0

∂c =0 ∂x

c = cS(t )

(8)

where ρPS is the pure PS density at the relevant temperature. We also evaluated the dependence of ρ on W from the PCSAFT equation of state, but the calculated densities did not agree with the values estimated from the video-microscopic experiments. This was not surprising, because it is well-known that condensed phase densities are calculated by most equations of state with high uncertainties or deviations. 2.3.3. Combined Diffusion Model. An alternative to the limiting case II diffusion model is the “combined diffusion” model. This model is composed of a Fickian precursor, a case II diffusion-type concentration front and a Fickian foot. The presence of the Fickian precursor was first predicted from theory and later experimentally verified.58,59 The precursor represents the initial Fickian diffusion into the glassy polymer and is responsible for the plasticization of the polymer (plasticizing effect), that is, for initiating the very rapid expansion of the polymer network.44 This expansion is accompanied by a step change in penetrant concentration (concentration front), which is similar to the case II diffusion front. The difference is that the penetrant concentration does not instantaneously reach the equilibrium value. The subsequent establishing of equilibrium again follows the Fick’s diffusion mechanism and is called the Fickian foot.60 This combined diffusion model seems more rigorous, but has four fitting parameters: the diffusion coefficients in the glassy polymer (Fickian precursor) and plasticized polymer (Fickian foot), the diffusion front velocity vCII and the concentration at the diffusion front. Apart from the diffusion coefficient in the plasticized polymer, these parameters are hard to estimate and the fitting of the data is complicated by many local minima of the optimization error function. This would make the interpretation of the obtained results complicated. As a result, we decided to avoid the use of the combined diffusion model.

n

∑ (miexp − micalc)2

0 < x < d /2,

⎛ W ⎞⎟ ρ = ρPS ⎜1 + ⎝ 2⎠

where w and l are the PS film width and length, respectively. The fitting of experimental data by both models was based on the minimization of the sum of error squares S2 defined by S2 =

t = 0,

5. Note that ρ is the polymer phase density, M1 is the molar weight of n-pentane and u(r,t) and u(x,t) are the polymer growth rates due to swelling in the particles and thin films, respectively. The time-dependent surface concentration cS(t) was again assumed and calculated from the sorption isotherms. The equations were processed in the same way as in the case of the model without swelling and S2 (eq 7) was used for the fitting of experimental data with only one fitting parameter per model (D and vCII). If swelling is considered, an equation for the dependence of the density ρ on the n-pentane concentration is needed. In this work, we used an empirical equation relating ρ to the n-pentane relative weight fraction W (eq 8) based on our videomicroscopic experiments (see Salejova and Kosek2 for information about the experimental technique).

calculated from the pressure at each integration step using the experimentally measured sorption isotherms. The finite volume method,56 which discretizes the polymer into a large number of elements (shells), was used to process the equations presented in Table 2 together with the appropriate initial and boundary conditions. The resulting system of ordinary differential equations was simulated in Matlab. Once the concentration field c(r,t) or c(x,t) in the polymer was evaluated, the evolution of the mass of sorbed n-pentane with time m(t) was calculated from eq 5 in the case of particles or from eq 6 in the case of films m(t ) = M1

c = c0

∂c ∂x

N = cvCII

854



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Table 5. Initial and Boundary Conditions for the Case with Polymer Swelling condition

description

thin film

particle

initial

equilibrium before the pressure-step

t = 0,

0 < r < R(t ),

boundary

symmetry

t ≥ 0,

r = 0,

boundary

instantaneous sorption equilibrium on the surface

t ≥ 0,

r = R(t ),

c = c0

∂c =0 ∂r

c = cS(t )

t = 0,

0 < x < d(t )/2,

t ≥ 0,

x = 0,

t ≥ 0,

x = d(t )/2,

c = c0

∂c =0 ∂x c = cS(t )

3. MATERIALS AND METHODS 3.1. Experimental Apparatus and Procedure. For the determination of sorption equilibria and transport parameters, a gravimetric apparatus based on a magnetic suspension balance was used (Figure 1). The apparatus consists of (i) a heated

Figure 2. The course of a typical sorption experiment with three pressure-steps at 90 °C.

Processing of Gravimetric Data. During a gravimetric experiment the time, pressure, temperature, and sample mass are automatically recorded. The measured mass mmeasured has to be corrected because of the buoyant force exerted by the surrounding vapor. This correction is done according to the following equation

Figure 1. Scheme of the magnetic suspension balance gravimetric apparatus.

m(p , T ) = mmeasured (p , T ) + ρgas (p , T )VDV(0, T )

(9)

where m(p,T) and mmeasured(p,T) are the corrected and measured mass of the weighed object at pressure p and temperature T, respectively, ρgas is the gas phase density at pressure p and temperature T calculated by the Lee−Kessler’s equation of state34 and VDV(0,T) is the displaced volume of the weighed object at vacuum. To estimate VDV(0,T), the mass of the weighed object is measured at vacuum and at several pressures of He. Because He has a negligible solubility in PS,5 changes of the measured mass at different He pressures are caused only by the buoyant force. Therefore, the displaced volume can be estimated from the obtained buoyancies. The penetrant solubility per unit mass of the polymer is then evaluated from eq 3. Two types of pressure-steps employed in our experiments have to be distinguished. The first type (referred to as the initial pressure-step) is characterized by the step increase from vacuum to a higher level (e.g., 2 bar). The second type (referred to as the subsequent pressure-step) corresponds to the increase from a nonvacuum level (at least 1 bar) to a higher level. Our original plan was to also apply small initial pressuresteps, for example, from 0 to 0.5 bar. However, the establishing of sorption equilibria as a response to such steps is very slow and the magnetic suspension balance is not stable enough during very long measurements. Furthermore, the low pressure region is not industrially relevant. We have thus performed only larger initial pressure-steps. The equilibrium sorption and diffusion characteristics are evaluated from the weight response to each pressure-step. That means that each pressure-step is processed separately. For the sake of simplicity, the time axis of each pressure-step is shifted

sorption cell with inlets, outlets, and temperature and pressure sensors, (ii) a magnetic suspension balance, (iii) an industrial PC for data acquisition, and (iv) other auxiliary systems. The control and data acquisition program was made in the LabView software. The magnetic coupling of the balance transmits the measured weight outside of the sorption cell, which protects the weighing equipment and allows measurements in a wide range of conditions. The technical specifications of the balance are pressure range (from vacuum to 50 bar), temperature range (from −196 °C to 150 °C), resolution (from 10 μg to 1 μg), reproducibility (± 20 to ± 2 μg), relative error (≤ 0.002 % of the measured value) and maximum load (30 g). Sorption experiments were conducted at constant temperature in the range from 45 to 110 °C. The PS sample was placed inside a stainless steel basket in the case of PS particles or coiled and fixed on a hook in the case of thin PS film. The mass of the PS samples ranged from 0.5 to 0.65 g. The sorption cell was then evacuated and the constancy of sample weight was checked. The sample was subsequently subjected to several different pressures of helium (He) at the desired temperature in order to estimate the volume of the weighed object (polymer sample, sample container, hook and wire), which is required for the buoyancy correction. After the second evacuation, a stepwise increase in n-pentane pressure was realized by introducing n-pentane vapor into the sorption cell and the weight response of the sample was recorded. Series of such stepwise pressure increases were conducted for several different initial and final pressures at various temperatures. An example of the sample weight evolution measured at 90 °C and stepwise pressure rise from 0 to 4 bar is presented in Figure 2. 855



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Table 6. List of PS Samples sample

commercial name

type

form

particle size/mm

A B C

noncommercial Koplen 0207 F Koplen 0513 F

with nucleation agent with nucleation agent with nucleation agent

particles particles film

0.60 to 0.63 0.40 to 0.63 film

approximately 30 cm long. Various film thicknesses (from 45 to 70 μm) were achieved by changing the slit height of the film applicator. Note that films prepared by solution casting are not oriented. The prepared films were degassed at room temperature for 48 h and then placed in vacuum at 100 °C for another 48 h to completely remove any residual toluene. The degassed films were wrapped into small coils and placed in the gravimetric apparatus. Complete solvent removal was then verified by the stabilization of the gravimetrically measured sample weight. Two independent methods were used to measure the film thicknesses d. The first method was based on a laser measuring instrument constructed in our laboratory, which found the relative deviation of film thickness to be smaller than 10 %. The second method was based on the measurement of the film mass mf in the gravimetric apparatus at vacuum. The film thickness was calculated according to eq 10 from the measured mf, the temperature dependent PS density ρPS, and the manually measured film length l and width w.

to zero value. Note that the sorption isotherm is obtained from the steady part of the weight response, whereas the diffusion coefficient is determined from the dynamic parts of the weight response (see Figure 2). The swelling of polymers during sorption changes their volume and thus influences the buoyant forces. Studies presented in the literature suggest that this effect may be neglected for the evaluation of sorption equilibria of n-pentane in PS. Alsoy and Duda57 performed a theoretical analysis of the effects of (i) swelling, (ii) volume changes upon mixing, and (iii) diffusion-induced convection on both sorption equilibria and diffusion rates in thin film polymer−solvent systems. They concluded that the above-mentioned phenomena can be neglected in the analysis of equilibrium sorption for most polymer−solvent systems. In contrast, the correction for the swelling of the polymer phase can be significant in the analysis of sorption dynamics, particularly when a large step change in solvent concentration is considered. In a previous paper,5 we suggested that the error due to neglecting PS swelling in the evaluation of n-pentane solubilities is below 1 % and thus negligible. In the studied range of temperatures and pressures, the changes in the PS sample volume were always safely below 10 % and the polymer comprised less than half of the weighed object volume, resulting in volume changes of the weighed object smaller than 5 %. As the buoyant forces acting on the weighed object change the measured solubilities by less than 20 %, the maximum error due to neglecting swelling is below 1 %. Changes of the gas phase volume due to sample swelling and decrease of n-pentane concentration (pressure) due to sorption uptake can influence the gravimetric measurements. The volume of the measuring cell is 125 cm3, while the PS sample volume was always below 0.7 cm3 and changed by less than 10 % due to swelling. The changes of the gas phase volume were thus below 0.06 % and safely negligible. In extreme cases (large pressure steps at low temperatures), changes of the gas phase pressure by up to 8.5 % are possible due to sorption uptake. However, we recorded the pressure in the gas phase continuously during the experiments and used a pressuredependent boundary condition for diffusion coefficient calculations. Any pressure changes were thus accounted. Note that equilibrium solubilities were evaluated from steady-state data and thus not influenced by these phenomena. 3.2. Materials and Sample Preparation. Three different PS samples were obtained from SYNTHOS Kralupy a.s. (Table 6). The Mw of the PS samples were well above 100 kg/mol, that is, in the region where the sorption processes are independent of Mw (see Section 2.2). The Tg of pure PS should be approximately 105 °C and this value was confirmed by DSC measurements for sample A (TgA = 104.5 °C) and sample C (TgC = 106.5 °C). Toluene in 99.9 % grade was obtained from Merck KGaA and n-pentane in p.a. grade from Penta s.r.o. Thin PS film samples were prepared from particles of sample C by the solution casting method.50 A homogeneous solution of toluene (50 wt %) and PS (50 wt %) was prepared, cast on a flat glass surface by a film applicator and the solvent was evaporated. The produced films were 3 cm wide and

d=

mf wlρPS

(10)

4. RESULTS AND DISCUSSION In this section, the evaluation of the measured sorption and diffusion data is presented. Experiments with the industrially important PS + n-pentane system were carried out in a broad temperature and pressure interval covering the region of expandable PS processing conditions. Sorption was measured both in PS particles and PS films, because the measurement of particles is relevant to industrial applications, while the use of PS films is advantageous for the studies of diffusion mechanisms. It is not possible to exactly calculate the errors of transport parameter estimation, because a single sample cannot be measured repeatedly due to foaming upon degassing and two samples are never identical, because the particles are not monodisperse and the irregularity of the individual films varies. However, it is possible to estimate the experimental error based on uncertainties. There are three sources of uncertainty in the evaluation of transport parameters: (i) inaccuracy of the fitting, (ii) uncertainties in the measurement and evaluation of the sorption uptake, and (iii) nonuniformity of sample dimensions and errors in their measurement (film thickness is not homogeneous and the particles are not monodisperse perfect spheres). Note that the inaccuracy of the fitting can be easily calculated (eq 7). The uncertainty in the mass measured by the gravimetric apparatus is below 0.1 % of the sorption uptake. More significant errors may result from the buoyancy corrections, where (i) errors up to 0.2 % of the sorption uptake may result from the inaccuracy of the weighed object volume estimation, (ii) errors up to 1.5 % of the measured mass may result from pentane density calculation, and (iii) errors up to 1 % of the sorption uptake may arise due to neglecting swelling (see section 3.1). Even if all these errors were at their maximum and 856



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would all be either positive or negative, the overall error in the evaluated sorption uptake would still be below 3 %. Uncertainties in sample dimensions have to be discussed separately for the individual samples. The relative errors of the film thickness measurements were below 4 %. The maximum errors arising from the width of the particle size fractions of the particle samples are 2.5 % and 22 % for samples PS A and PS B, respectively. These errors correspond to a hypothetical situation when all of the particles would be at the edge of the fraction, for example, if all particles in sample PS B (0.40 mm to 0.63 mm) would have a diameter of 0.63 mm. However, such a situation is highly unlikely and, therefore, the maximum errors in the particle diameters are estimated to be 1 % and 10 % for samples PS A and PS B, respectively. To sum up, we conclude based on the previous discussion and our long-term experience that the errors in the presented transport parameters are in the range from 5 % to 10 % (up to 20 % for sample PS B) plus the fitting error. 4.1. Sorption Equilibria of n-Pentane in Polystyrene. Sorption equilibrium data were evaluated as described in section 3.1. The obtained sorption isotherms were fitted with the Flory−Huggins model to evaluate the polymer−solvent interaction parameter χPS,n‑pentane. Both the experimental data and the fitted Flory−Huggins curves are shown in Figure 3 and

Figure 4. Dependence of the Flory−Huggins interaction parameter χPS,n‑pentane on reciprocal temperature. The symbols represent experimental data and the solid curve represents the evaluated temperature dependence (eq 11).

Figure 5. Dependence of n-pentane solubility in PS on the activity a defined by eq 12. Data for all samples and temperatures are plotted simultaneously. The symbols represent experimental data and the solid line represents the polynomial activity correlation (eq 13).

and activity in the whole studied range of temperatures and pressures. This correlation was fitted by a third order polynomial (eq 13) that provides good fitting accuracy and a reasonable representation of the nonlinear increase of solubility with activity.

Figure 3. Sorption isotherms of n-pentane in PS at various temperatures. The symbols represent experimental data and the solid lines are Flory−Huggins calculations (with χPS,n‑pentane defined by eq 11).

S = 0.2033a3 − 0.1237a 2 + 0.0935a

the obtained χPS,n‑pentane parameters are presented in Figure 4. The temperature dependence of the χPS,n‑pentane parameter was estimated in the interval from 45 °C to 110 °C and is presented in eq 11 and Figure 4 (solid line). Although the data exhibit non-negligible scatter, they clearly show a linear dependence on (1/T) in agreement with theory.25,61 572.12 χPS, n ‐ pentane = + 0.50 (11) T The sorption equilibrium data were also plotted against the penetrant activity at the individual temperatures and pressures a(T,p): p a(T , p) = sat p (T ) (12)

(13)

To compare the accuracy of the Flory−Huggins calculations and the activity correlation, we need a measure of the deviations between the experimental and calculated data. The average relative deviation in terms of percent (ARD) was chosen as a suitable quantity: ARD = 100

1 n

n

∑ i=1

|Siexp − SiF − H| Siexp

(14)

where n is the number of experimental points in the given F−H isotherm and Sexp are the experimental and calculated i and Si solubilities at the ith point, respectively. The ARD values obtained for the Flory−Huggins calculations with the temperature dependent χPS,n‑pentane (eq 11) were in the range from 0.49 % to 14.39 % with an average value of 5.80 %. The average ARD of the activity correlation (eq 13) was 8.29 % with the errors for the individual sorption isotherms ranging from 4.13 % to 14.52 %. The average ARD of the

where p is the pressure and psat(T) is the saturated vapor pressure at temperature T. The data are shown in Figure 5 and one can see a reasonable correlation between the solubility data 857



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where z is the co-ordination number, R is the universal gas constant, ΔCp is the difference between the heat capacity of a supercooled liquid and glass, Mp and Ms are the molecular weights of the monomer unit and solvent, respectively, and ω is the solvent weight fraction. Note that z serves more as a fitting parameter66 because of several assumptions of the Chow’s model, such as monolayer absorption, temperature-independent ΔCp, or equal lattice size for the polymer repeat unit and solvent. The parameters used for estimating the mixture Tg are listed in Table 7.

Flory−Huggins calculations was significantly lower than the value for the activity correlation, but the maximum ARD values were similar. However, an even lower average ARD of 3.95 % was obtained with PC-SAFT.5 We would suggest that even the activity correlation can be used for first estimates of n-pentane solubility in PS, but the Flory−Huggins model with the χPS,n‑pentane parameter defined by eq 11 is more suitable for predicting the sorption equilibria of n-pentane in PS. Its slightly lower accuracy compared to PC-SAFT is compensated by its simplicity and widespread use in the industry. However, if high accuracy is desired, one has to use advanced models. 4.2. Diffusion of n-Pentane in Polystyrene. In this section, the gravimetrically measured sorption dynamics of n-pentane in PS are fitted using two principally different models: (i) Ficks diffusion and (ii) case II diffusion. For Fick’s diffusion, models both with and without PS swelling were used. Only the model without swelling was used for case II diffusion, because this mechanism does not consider any transport in the swollen volume (i.e., behind the concentration front) and the case II velocities are thus not influenced by swelling. The Fick’s diffusion fitting errors decreased in average by 30 % when swelling was considered, and the diffusivities evaluated with and without swelling differed on average by 9 % (maximum difference was 33 %). The difference between the diffusivities evaluated with and without swelling generally increased with increasing pressure, that is, with increasing degree of swelling. However, the general trends and the error ratios of the case II and Fick’s diffusion models were similar for both model types (with and without swelling). At conditions studied in this paper, swelling should change the particle diameters or film thicknesses at most by 3 %.5 As the diffusion path comes in a square term, the effect of swelling is estimated on the order of 6 %, which is similar to the observed differences in diffusivities. The differences in diffusivities are also of the same order of magnitude or smaller than the errors of transport parameter estimation (5 % to 20 % plus the fitting error). Further in this paper, we present the result obtained with the models accounting for polymer swelling, as they provide more accurate diffusion coefficients and lower fitting errors. However, reasonable estimations can also be obtained with the models without swelling, which are simpler and need less input parameters. The Tg values of our PS samples were approximately 105 °C. However, when a polymer contains a solvent, the Tg of the polymer + solvent mixture is lower than the Tg of the pure polymer. The Tg of the polymer + solvent mixtures was estimated by Chow’s model,62,63 which is based on the work of DiMarzio and Gibbs64 and uses the Bragg−Williams approximation.65 The model combines classical and statistical thermodynamics to explicitly express the Tg of polymer− solvent systems as Tg = Tg0 exp[β {(1 − θ ) ln(1 − θ ) + θ ln θ }]

Table 7. Parameters of PS Used in the Chow Model Calculations

(15)

θ=

zR ΔC P

(16)

MP ω zMS 1 − ω

(17)

value

unit

reference

Tg0 ΔCp z

105 27.03 4.272

°C J mol−1 K−1 -

This work 62 66, 72

At the start of each experiment, a pure PS sample was placed in the gravimetric apparatus. This means that sorption into glassy PS was measured in all initial pressure-steps except for the experiments at 105 and 110 °C, when the temperatures were equal to or above Tg,PS. Subsequent pressure-steps took place into the polymer already swollen by n-pentane. The Tg values of these swollen polymers estimated from eq 15 were all below the experimental temperature, except for the first subsequent pressure-step carried out at 70 °C with the thin film (calculated Tg was 76.7 °C). However, this pressure-step exhibited Fickian behavior. Because the calculated Tg is close to the experimental temperature and the models for estimating mixture Tg have higher inaccuracies, we will consider all subsequent pressure-steps as sorption into plasticized PS. According to theory, a significant difference between the diffusion in glassy and plasticized polymers is expected.45,58 If we strongly generalize, then the diffusion into plasticized PS should be Fickian and the diffusion into glassy PS should be non-Fickian or case II (see section 2.2 for detailed discussion of the diffusion mechanisms). The difference between these two limiting diffusion mechanisms is most apparent in thin polymer films, whereas in spherical polymer particles these two diffusion mechanisms nearly coincide and are hardly distinguishable (see section 4.2.2). 4.2.1. Diffusion in Thin Polystyrene Films. The measured sorption dynamics of n-pentane in thin PS films were fitted with the Fick’s and case II diffusion models. The estimated transport parameters are presented in Table 8 and Figure 6. The fitting errors are presented in the form of the sum of error squares S2 (eq 7). The transport parameters are listed together with the relevant pressure-steps. Concentration steps can be deduced from these pressure steps and the equilibrium data (e.g., from eq 13). The average activity amid (eq 18) is used to represent the individual pressure-steps for further discussion. This variable was chosen for three reasons: (i) diffusion of n-pentane in PS is concentration dependent, so the starting pressure and the end pressure do not correctly characterize the pressure-step, (ii) the use of activity enables better comparison of the individual experiments, and (iii) the activity was used by other authors studying the diffusion of n-pentane in PS.41,55,67,68 The value of amid is calculated as

where Tg and Tg0 are the glass-transition temperatures of the swollen and pure polymer, respectively, and β and θ are nondimensional parameters expressed as β=

parameter

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Table 8. Pressure-Steps Carried out for Thin Film Samples Together with the Evaluated Diffusion Coefficients D and Case II Diffusion Velocities vCIIa T/°C 110 110 90 90 90 90 90 90 90 90 90 90 90 80 80 80 70 70 70 70 70

d/μm 52 52 55 55 55 55 55 63 63 63 67 67 67 50 50 50 48 48 48 48 48

pst/bar 0 4.30 0 1.98 2.37 2.89 3.34 0 2.44 3.08 0 1.97 2.83 0 1.54 1.92 0 1.30 1.56 1.96 2.19

pend/bar 4.31 4.80 1.98 2.37 2.87 3.35 3.77 2.44 3.08 3.82 1.97 2.37 3.81 1.54 1.92 2.26 1.30 1.56 1.94 2.18 2.44

amid/− 0.29 0.62 0.21 0.46 0.56 0.66 0.76 0.26 0.59 0.73 0.21 0.46 0.71 0.21 0.47 0.57 0.23 0.51 0.62 0.73 0.82

S0.7/(g gPS‑1) 0.039 0.072 0.023 0.042 0.054 0.071 0.089 0.029 0.060 0.087 0.022 0.042 0.083 0.021 0.039 0.051 0.022 0.042 0.058 0.074 0.093

D/(m2 s‑1) 3.86 6.67 3.05 1.55 2.53 3.43 3.96 6.37 2.92 4.38 2.74 1.55 5.47 9.38 6.49 1.43 1.58 4.68 1.34 2.06 2.07

× × × × × × × × × × × × × × × × × × × × ×

S2Fick/−

−12

10 10−12 10−13 10−12 10−12 10−12 10−12 10−13 10−12 10−12 10−13 10−12 10−12 10−14 10−13 10−12 10−14 10−13 10−12 10−12 10−12

6.5 4.7 3.8 6.9 1.3 6.1 7.4 5.4 1.7 3.1 3.4 8.3 2.6 7.0 6.1 1.6 2.2 6.4 3.1 6.5 3.8

× × × × × × × × × × × × × × × × × × × × ×

−04

10 10−04 10−03 10−04 10−04 10−05 10−04 10−03 10−04 10−05 10−03 10−04 10−04 10−03 10−04 10−04 10−02 10−04 10−04 10−04 10−04

vCII/(m s‑1) 1.1 1.2 1.5 2.1 2.8 3.9 9.1 2.6 4.1 6.2 1.1 4.5 1.2 5.5 3.0 5.4 1.2 2.8 4.8 6.3 5.6

× × × × × × × × × × × × × × × × × × × × ×

−07

10 10−07 10−08 10−08 10−08 10−08 10−08 10−08 10−08 10−08 10−08 10−08 10−07 10−09 10−08 10−08 10−09 10−08 10−08 10−08 10−08

S2CII/− 4.6 1.5 2.1 1.9 3.3 2.8 1.4 1.3 5.9 5.1 2.7 4.6 9.1 2.0 5.1 5.6 7.0 8.1 4.9 7.8 1.0

× × × × × × × × × × × × × × × × × × × × ×

10−03 10−02 10−03 10−03 10−03 10−03 10−02 10−03 10−03 10−03 10−03 10−03 10−03 10−03 10−03 10−03 10−04 10−03 10−03 10−03 10−02

a Columns pst and pend are the pressures before and after the pressure-step, respectively, amid is the average activity (eq 18), d is the film thickness and S2Fick and S2CII are the sums of error squares of the two limiting diffusion models.

Our results showed that both sorption equilibria and diffusion coefficients of n-pentane in PS films of various thicknesses vary only within the experimental error (see results obtained at 90 °C in Table 8). The determination of the dominant diffusion mechanism is an important outcome of the processing of experimental results. To estimate the dominant mechanism of transport, one has to compare the fitting errors of both limiting diffusion models. For this purpose, the ratio of S2 values for both limiting diffusion models defined by eq 19 was chosen

ratio =

p ⎞ 1 ⎛ pst ⎟ ⎜ sat + end 2⎝p psat ⎠

SFick 2

(19)

If this ratio is well above 1, then Fick’s diffusion is the dominant mechanism (Figure 7a), whereas for this ratio deep below 1 the case II diffusion mechanism prevails (Figure 7b). If the ratio is close to 1 then the diffusion process can be described by both diffusion mechanisms with similar accuracy. In the case of thin PS films, this means that none of the two limiting mechanisms described the measured data satisfactory (Figure 7c). The S2 ratios calculated for all experiments conducted with thin PS films are shown in Figure 8. On the basis of the calculated Tg values, the theory predicts that the case II diffusion mechanism should be dominant for the initial pressure-steps at 70 °C, 80 °C, and 90 °C, while Fick’s diffusion should be the dominant mechanism for all other pressure-steps. Let us first discuss the initial pressure-steps. The calculated S2 ratios are below 1 for all initial pressure-steps except for the one at 110 °C, as predicted by theory. However, only the pressure-step at 70 °C is accurately described by the case II model (Figure 7b). The initial pressure-steps at 80 °C and 90 °C were closer to the case II mechanism, but the deviations of the calculated and measured curves were relatively high (e.g., Figure 7c). This suggests that the diffusion mechanism was non-Fickian, but also not purely case II. This

Figure 6. Fickian diffusion coefficients of n-pentane in thin PS film samples at various temperatures. Note that the vertical axis is in logarithmic scale and that amid is the average activity defined by eq 18.

amid =

SCII 2

(18)

where pst and pend are the pressures before and after the pressure-step, respectively, and psat is the saturated vapor pressure at the relevant temperature. Figure 6 shows diffusion coefficients increasing with T and amid in agreement with theoretical expectations and previous studies.37,41,43,55 The increase of D with activity can be clearly seen at all temperatures and also the increase of D with temperature is evident. Owing to the highly nonlinear effects of T and amid, it was not possible to propose a simple empirical correlation between D, T, and amid. We thus suggest conducting interpolation between the presented diffusivities to obtain the values of D at various temperatures and activities. We also verified that various thicknesses of our PS films practically did not affect the sorption and diffusion processes. 859



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models, while the subsequent pressure-steps can be described by the Fick’s diffusion model with sufficient accuracy. 4.2.2. Diffusion in Polystyrene Particles. The results obtained by fitting the measured sorption dynamics of n-pentane in PS particles with the Fick’s diffusion model are presented in Table 9 and Figure 9. The fitting errors are presented in the form of S2 values defined by eq 7, while amid calculated from eq 18 is used to represent the individual pressure-steps for further discussion. The diffusion coefficients are listed together with the relevant pressure-steps. Concentration steps can be deduced from these pressure steps and the equilibrium data (e.g., from eq 13). According to Figure 9, the diffusion coefficient increases with T and amid in agreement with theoretical expectations. The increase of D with T at constant activity is best seen in the activity region just below 0.4. The dependence on activity is demonstrated at temperatures 70, 80, 90, and 95 °C, at which several pressure-steps were measured. As in the case of thin PS films, the highly nonlinear dependence of D on T and amid does not allow the proposition of a simple empirical correlation D(T, amid). Our suggestion is thus the same as for PS films, that is, to use interpolation between the presented data points to estimate the values of D at different temperatures and activities. All sorption experiments with spherical particles, except for the one at 45 °C, can be described by the Fick’s diffusion model with acceptable error (Table 9). However, to correctly estimate the dominant diffusion mechanism, the ratio (S2CII/S2Fick) has to be evaluated. Figure 10 displays the dependence of this ratio on n-pentane activity and we can conclude that Fick’s diffusion is the dominant transport mechanism for all pressure-steps (except at 45 °C). However, there are significant differences between the results of measurements with thin films and spherical particles. According to the thin film results, the initial pressure-steps carried out with spherical particles at temperatures from 70 to 100 °C (below Tg,PS) should result in nonFickian diffusion, but this was not the case (Figure 10). Moreover, some of the subsequent pressure-steps were described by both limiting mechanisms with similar errors (Figure 11), although the diffusion at these conditions should have been strongly Fickian according to theory. The observed deviations were probably caused by geometrical effects, which hinder the evaluation of the dominant diffusion mechanism for spherical particles. In the case of planar films, the mass uptake during sorption should be linear with time for the case II diffusion and nonlinear with time for Fick’s diffusion, which enables clear differentiation between the two mechanisms. However, in the case of spherical particles, the mass uptake is nonlinear with time also for the case II diffusion due to the nonplanar geometry (Figure 12). This explains the Fickian-like responses to initial pressure-steps carried out at temperatures from 70 to 100 °C (below Tg,PS) and also the fact that both diffusion mechanisms had similar errors for pressuresteps that should have been Fickian. The above-mentioned geometrical effects did not enable a clear evaluation of the dominant diffusion mechanism from the experiments with spherical PS particles. However, it is possible to conclude that Fick’s diffusion describes the transport of n-pentane in spherical PS particles in the region of PS foam processing conditions (80 to 110 °C) with acceptable errors. This conclusion, together with the presented diffusion coefficients, should help in the optimization of PS foam production.

Figure 7. Comparison of experimental data and fitted Fick’s and case II diffusion curves for thin PS films: (a) dominant Ficks diffusion, temperature = 90 °C (d = 67 μm), pressure-step from (2.83 to 3.81) bar; (b) dominant case II diffusion, temperature = 70 °C, pressure-step from (0 to 1.30) bar; (c) neither Fick’s nor case II diffusion provides satisfactory description, temperature = 90 °C (d = 67 μm), pressurestep from (0 to 1.97) bar. Dimensionless mass on the vertical axis is expressed by (mt − m0)/(minf − m0), where mt is the mass of the sample at time t and m0 and minf are the sample mass before and after the pressure-step, respectively.

Figure 8. Ratios of case II and Ficks diffusion fitting errors in terms of S2 (eq 19) for thin PS film samples. Note that the vertical axis is in logarithmic scale and that amid is the average activity defined by eq 18.

is probably caused by the higher experimental temperatures, which are closer to Tg,PS. Fick’s diffusion was the dominant mechanism for all subsequent pressure-steps. This is in agreement with theory, as the experimental temperatures were high above the Tg values of the relevant PS + n-pentane mixtures. Although the Fick’s diffusion fitting errors vary, no clear dependence of the fitting errors or their ratio (S2CII/S2Fick) on T and amid was observed. This suggests that the diffusion was fully Fickian for all subsequent pressure-steps. Otherwise the ratio (S2CII/S2Fick) would increase with T and amid. We can therefore conclude that in the interval of PS foam processing conditions (80 to 110 °C), the initial pressure-steps below 110 °C cannot be described by any of the limiting 860



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Table 9. Pressure-Steps Carried out for Particle Samples Together with the Evaluated Diffusion Coefficients Da T/°C 45 70 70 70 75 80 80 90 90 90 95 95 100 105 110

pst/bar 0 0 2.15 2.46 0 0 2.65 0 0 3.05 0 3.44 0 0 0

pend/bar 1.02 2.15 2.46 2.69 1.97 2.65 3.06 2.12 3.05 3.48 3.44 4.03 3.58 2.11 3.82

p0.7/bar

amid/−

0.72 1.50 2.36 2.62 1.38 1.85 2.94 1.49 2.14 3.35 2.41 3.85 2.51 1.48 2.68

0.38 0.38 0.81 0.91 0.31 0.36 0.78 0.23 0.32 0.69 0.33 0.71 0.30 0.16 0.26

S0.7/(g gPS‑1) 0.040 0.049 0.111 0.149 0.036 0.046 0.101 0.026 0.043 0.085 0.045 0.095 0.040 0.017 0.034

D/(m2 s‑1) 2.69 1.23 7.89 1.08 3.62 1.62 1.02 3.09 2.07 1.12 3.36 1.28 4.45 5.00 5.35

× × × × × × × × × × × × × × ×

−14

10 10−12 10−12 10−11 10−13 10−12 10−11 10−13 10−12 10−11 10−12 10−11 10−12 10−13 10−12

S2/− 2.60 9.94 8.97 5.33 1.09 3.83 9.61 1.79 6.80 1.52 8.04 9.46 4.20 1.29 7.46

× × × × × × × × × × × × × × ×

10−03 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−03 10−04 10−04 10−04 10−04 10−04

a

Columns pst and pend are the pressures before and after the pressure-step, p0.7 and S0.7 are the pressure and solubility at 70 % of the concentration step, amid is the average activity (eq 18) and S2 is the sum of error squares.

Figure 11. Experimental data and fitted Fick’s diffusion curves for PS particles: temperature 70 °C, pressure-step from 2.46 to 2.69 bar. Dimensionless mass on the vertical axis is expressed by (mt − m0)/ (minf − m0), where mt is the mass of the sample at time t and m0 and minf are the sample mass before and after the pressure-step, respectively.

Figure 9. Diffusion coefficients of n-pentane in PS spherical particles at various temperatures. Note that the vertical axis is in logarithmic scale and that amid is the average activity defined by eq 18.

Figure 10. Ratios of case II and Ficks diffusion fitting errors in terms of S2 (eq 19) for PS particle samples. Note that the vertical axis is in logarithmic scale and that amid is the average activity defined by eq 18.

Figure 12. Comparison of sorption dynamics for a spherical particle and planar film; mt is the mass of the sample at time t and m0 is the sample mass before the pressure-step.

Up to now, we plotted the diffusion coefficients against activity. This approach provides well readable plots and enables easy comparison with theoretical expectations. However, a more complex analysis is needed for comparing the individual diffusivities. In the case of diffusivities increasing with penetrant concentration, the data should be plotted against solvent weight

fractions (solubilities in g/g are used in this work) equal to 70 % of the sorption step.68 The solubility at 70 % of the concentration step S0.7 is calculated from the measured sorption 861



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pressure-steps (starting from nonzero pressure) were accurately described by the Fick’s diffusion model. This finding is in agreement with theoretical expectations. However, only the initial pressure-step at 70 °C was accurately described by the case II diffusion model, whereas none of the limiting diffusion models accurately described the other non-Fickian responses to the initial pressure-steps. Our results suggest that the simple Fick’s diffusion model approximates well the sorption dynamics above Tg,PS and for subsequent pressure-steps also below Tg,PS. For PS particles, sorption responses to all pressure-steps except the one conducted at 45 °C were described satisfactorily by the Fick’s diffusion model. Fick’s diffusivities estimated in this paper can thus be used to predict the diffusion dynamics of n-pentane in PS particles at PS foaming conditions (80 to 110 °C, pressures from 2 bar up to 90 % of the saturated vapor pressure). However, both for PS particles and thin films, it was not possible to propose a simple correlation for the dependence of diffusivity on temperature and n-pentane activity. The consideration of advanced diffusion models is required to obtain such a correlation. Our future work will thus concentrate on obtaining a chemical potential gradient based model of concentration dependent diffusion.

isotherms as the solubility at 70 % of the pressure-step (pressure p0.7) which is defined as p0.7 = pst + 0.7(pend − pst )

(20)

The diffusivities estimated from the thin film and spherical particle data are plotted against S0.7 in Figure 13. One can see

Figure 13. Diffusion coefficients of n-pentane in PS particles and films plotted against the solubility at 70 % of the concentration step S0.7. Note that the vertical axis is in logarithmic scale.

■

AUTHOR INFORMATION

Corresponding Author

that diffusivities in spherical particles seem to be slightly higher. However, these differences are mostly within the range of the experimental error and the scatter of the data. Furthermore, the time−temperature history of the film and particle samples was not identical due to differences in their production and it was previously shown that the PS time−temperature history can influence the sorption of n-pentane in PS.43,54,55,69

*Tel.: +420 220 44 3296. Fax: +420 220 44 4320. E-mail: Juraj. [email protected]. Funding

Financial Support from the Czech Grant Agency (GA CR 106/ 10/1912) and specific university research (MSMT No 21/ 2011) is acknowledged. The result was developed with instruments available in the CENTEM project, Reg. No. CZ.1.05/2.1.00/03.0088, cofunded by the ERDF as part of the Ministry of Education, Youth and Sports’ OP RDI programme.

5. CONCLUSIONS We measured the sorption of n-pentane in polystyrene (PS) for a range of temperatures and pressures covering the region of PS foam manufacturing conditions. Equilibrium and dynamic sorption measurements were carried out on a high-precision gravimetric apparatus. The system PS + n-pentane was the subject of our investigation due to (i) its industrial importance, (ii) interesting theoretical background, and (iii) limited information about the diffusion of n-pentane in PS at industrially relevant conditions available in the literature. The Flory−Huggins model is widely used in the industry for calculating the sorption equilibria in polymers. We thus evaluated the temperature dependence of the Flory−Huggins interaction parameter χPS,n‑pentane, which was, to the best of our knowledge, not previously reported in the literature. We also demonstrated that the evaluated χPS,n‑pentane can be used to accurately calculate the sorption equilibria of n-pentane in PS (average ARD below 6 %). For the estimation of transport characteristics from sorption dynamics data, we used several models based on Fick’s and case II diffusion, including models accounting for polymer swelling. Transport parameters presented in this paper were estimated with the models accounting for swelling, which provide a more rigorous description of the diffusion processes and better fitting accuracy. However, even the constant volume models provide acceptable results and have the advantage of less input parameters (e.g., the polymer phase density is not needed). For the sorption of n-pentane in thin PS films, we observed the initial pressure-steps (starting from vacuum) below Tg,PS to be non-Fickian, while those above Tg,PS and all subsequent

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors would like to thank O. Holecek and M. Frühbauer for their contribution to the gravimetric measurements.

■

862

LIST OF SYMBOLS a(T,p) = activity of the penetrant at pressure p and temperature T, − a1 = activity of the penetrant, − amid = average activity, − c = concentration of n-pentane, mol m−3 cS(t) = time-dependent surface concentration of n-pentane, mol m−3 c0 = initial concentration of n-pentane, mol m−3 ΔCp = heat capacity difference, J mol−1 K−1 d = film thickness, m D = diffusion coefficient, m−2 s l = film length, m m(p,T) = evaluated mass at pressure p and temperature T, g m(t) = mass of sorbed n-pentane at time t, g mf = mass of a PS film sample, g miexp = experimental mass of sorbed n-pentane at the ith point, g micalc = calculated mass of sorbed n-pentane at the ith point, g dx.doi.org/10.1021/je300916f | J. Chem. Eng. Data 2013, 58, 851−865


Article

mmeasured(p,T) = gravimetrically measured mass at pressure p and temp T, g mpol = mass of the pure polymer, g mt = sample mass at time t, g m0 = sample mass before a pressure-step, g minf = sample mass after a pressure-step (at time t→∞), g Ms = molecular weight of the solvent, g mol−1 Mp = molecular weight of the monomer unit, g mol−1 Mw = polymer molecular weight, g mol−1 M1 = molar weight of n-pentane, g mol−1 n = number of experimental points, − N = molar flux intensity, mol m−2 s−1 p = pressure, bar pend = pressure after a pressure-step, bar pst = starting pressure before a pressure-step, bar psat = saturated vapor pressure, bar p0.7 = pressure at 70 % of the pressure-step, bar r = spherical coordinate, m R = radius of polymer particle, m R = universal gas constant, J mol−1 K−1 S2 = sum of errors squares, − S(p,T) = solubility at pressure p and temperature T, (g/gPS) S0.7 = solubility at 70 % of the concentration step, (g/gPS) t = time, s T = temperature, K Tg = glass transition temperature, °C Tg0 = glass transition temperature of a pure polymer, °C Tg,PS = glass transition temperature of pure PS, °C u = polymer growth (swelling) rate, m s−1 vCII = case II diffusion velocity, m s−1 Vmix = volume of the polymer−solvent mixture, m3 Vpol = volume of the polymer, m3 V1 = volume of the penetrant, m3 VSample = volume of the PS sample at vacuum and temperature T, m3 w = film width, m W = relative weight fraction of n-pentane, − x = chain length, − x = spatial coordinate, m z = coordination number, −

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PC-SAFT = perturbed chain statistical associating fluid theory PS = polystyrene SAFT = statistical associating fluid theory VR-SAFT = variable range statistical associating fluid theory

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Greek Letters

α = parameter in the temperature dependence of χ, − β = parameter in the temperature dependence of χ, − β = dimensionless parameter of the Chow model, − θ = dimensionless parameter of the Chow model, − ρ = polymer phase density, kg m−3 ρGAS(p,T) = gas phase density at pressure p and temperature T, kg m−3 ρ1,LIQ = density of liquid pentane, kg m−3 ρpol = density of the polymer, kg m−3 ρPS = density of polystyrene, kg m−3 ϕ1 = volume fraction of the penetrant, − χ = polymer−solvent interaction parameter, − χPS, n‑penatne = polystyrene−pentane interaction parameter, − ω = solvent weight fraction, − Acronyms

ARD = average relative deviation CFCs = chlorofluorocarbons DSC = differential scanning calorimetry EOS = equation of state HCFCs = hydrochlorofluorocarbons HFCs = hydrofluorocarbons 863



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Experimental Study of Sorption and Diffusion of n-Pentane in

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