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Journal of Chemistry, Vol. 44 (5), Tr. 632 - 637, 2006 Internal Interaction Factor in Thermodynamic Model for Heterogeneous Mixture Received 25 August 2004 Nguyen Xuan Quang, Le Anh Tuan, Quach Dang Trieu Institute of Chemistry, Vietnam Academy of Science & Technology SUMMARY Based on irreversible thermodynamics, a new approach to study sorption in polymer has been proposed [1]. This treatment gives a standpoint different from the existing sorption models [2 - 9] not only in interpretation and nature, but also in analytical expression. Moreover, the model also accounts for diffusion and the other internal processes stimulated by sorption such as swelling, plasticization, crystallization etc. Some experimental results of fluid sorption in polymer have been interpreted by this model. Sorption species may be created by weak physical interaction between penetrant molecules and various molecular groups in molecular chain of polymer and sorption cluster may be created by weak physical interaction of free fluid molecules with fluid molecules in sorption species. Sorption center which are able to create sorption species and sorption clusters are proposed to be a finite number. Based on these arguments, the fluid sorption in polymer matrix can be considered as a heterogeneous mixture, where there exist processes of internal state changes such as creations of different sorption species, sorption cluster, plasticization etc... Generally, the polymer molecular groups are able to be considered as a sorption species or sorption clusters, with fluid molecules can be called as a sorption center. In the neighborhood of such polymer molecular groups there exist interaction forces. Under influence of this field fluid molecules may get in an excited state. If the distance between the excited molecules and the sorption center is small enough and this sorption center is not occupied, then their interaction may lead to create sorption species. Anomalous processes are stimulated by interaction of fluid molecules and polymer molecular chains, these processes spread out from all sorption center. The such case heterogeneous mixture can be defined by following constitutive equation [10]: {fi ,si ,T ,ki ,q,l,d }= (Fs ,j ,Gs ,hj ,uj ,T,g,) (1) where fi is partial free energy of i-th constituent in the mixture, si is partial entropy of i-th constituent, Ti is partial stress tensor of i-th constituent, ki is partial mixture interaction on i-th constituent (index i takes for all constituent in mixture), q is heating flux, l are kinetics of internal processes such as sorption, anomalous processes (l = lk, where index k takes for all 632 internal processes, such as creation of sorption species, sorption cluster, swelling, consolidation, etc.), d are internal forces (d = dk, index k takes for all internal processes, are constitutive function of (Fs, j, Gs, hj, uj, T, g, ) representing for {fi, si, Ti, ki, q, l, d}, where Fs is deformation tensor of s-th solid constituent, j is mass density of j-th constituent, Gs is second order deformation of s-th solid constituent (index s takes for all solid constituents), hj is mass density gradient of j-th constituent, uj is diffusion velocity of j-th constituent (the index j takes for all fluid constituent and solid constituent taking part in internal processes), T is absolute temperature, g is temperature gradient, is internal variables characterizing for internal processes such as creation of sorption species, swelling, plasticization, consolidation etc. ( = k, where index k takes for all internal processes). D = grad (2) are so-called internal forces of creation sorption species, swelling, plasticization, consolidation etc. (d = dk, index k takes for all internal processes). They certainly exist in the neighborhood of sorption center determined by the nature and order structure of polymer molecular chains. As a fluid molecules gets nearly to a sorption center under influence of internal force field d the internal state of system P/f begins to change (become to be excited) and if thermodynamic condition is favorable then a sorption species or sorption clusters may be formed. The existence of sorption species, sorption clusters is based on the weak bonds of polar, hydrogen, or polar induced interaction of fluid molecules with certain molecular groups on polymer molecular chains or sorption species [8, 11, 12]. These processes lead the present equilibrium state of system to be broken. Up to a certain degree these interaction may provoke a change of molecular chain structure (such as a swelling, consolidation etc.) in order to restore a new equilibrium state). By apply of thermodynamic balances and entropy inequality [10] on the constitutive Eqn. (2) one can obtain following equations: F+Lklk J =ad (3) k where F is total interaction force, Lk is a proportion coefficient with respect k (k are indexes including sorption, swelling, consolidation, …), lk are kinetics of internal process k, J is fluid diffusion flux density in polymer, ad = akdk, ak is an energy of internal state change due to process k. dg = -SdT + Pd + Ks.dF + gidWi + ad s i (4) where g is specific Gibb’s energy of mixture, S is mixture entropy, P is mixture pressure, is specific mixture volume, Ks is partial chemical potential tensor of elastic deformation of solid constituents, gi is partial Gibb’s energy, wi is partial mass fraction. Eqn. (3) expresses a complex relationship between diffusion, kinetics of internal processes including creation sorption species, creation of sorption clusters and anomalous processes (such as swelling, consolidation, plasticization, etc.), internal forces, total interaction forces and internal state change energy. This equation enables to consider not only the sorption itself but all processes parallel proceeding together sorption in a more complex relation. From this equation one can see for example, diffusion flux is inverse proportional to kinetics of sorption and other anomalous (internal) processes. While chemical affinity and interaction force of sorption are proportional to its kinetics etc., equation (4) is a differential thermodynamic state equation. From Eqn. (4) one can find relationship between entropy, mechanical deformation, polymer specific volume, chemical affinity of internal processes and sorption. However, based on Eqns. (3), (4) this is only abstract theory. The best way to understand Eqns. (3), (4) must be concrete applications and illustration to predict experimental data. Some application in diffusion study already had been carried in the previous paper [13]. In the present work, an application of the model in sorption isotherm will be presented. II. It is well known that membrane per-separation very strong depends on sorption properties of polymers. Therefore, many studies of sorption isotherm have been made to characterize polymer sorption properties. Sorption isotherm is a relationship between penetrant pressure and penetrant sorption amount in polymers at steady state. Sorption steady state is a thermodynamic state, at which 633 the sorption kinetics are annul i.e., kinetics of creations sorption species, sorption clusters: lk = 0. From Eqns. (1, 3, 4) one can obtain: ni C =Kiln i +1 (5) i s where C is a fluid sorption concentration in polymer, Ki is a coefficient inversely proportional to energy creating sorption species Ei for i-th type (1/Ki = kiEi/RT), P is fluid pressure, ks is a sorption kinetic interaction coefficient for i-th sorption type, ni is a number of penetrant molecules associated with sorption centers for i-th type. Eqn. (5) shows that, there are three main kinds of sorption parameters that can influence the behavior of fluid sorption in polymer. Depending on the thermodynamic conditions e.g., higher or lower than glass transition temperature Tg, properties and structure of polymer and penetrant the parameter Ki, ki and ni would have different value, and due to the different in values of sorption parameters different type of sorption isotherm such as Henry’s, dual-mode, Flory-Huggins’s or sigmoidal can be observed. These coefficients Ki, ks and ni are strongly dependent not only on physico-chemical properties and structure of polymer and sorption fluid but also physical state of polymer and penetrant in a 40 given thermodynamic condition. First application of this thermodynamic model can apply to analysis of gases sorption in polymer. It is well known that, the interaction between gas and polymer is usually very weak and in this case sorption species are very unstable. For gases, typically observed experimental sorption isotherms are Henry’ type for rubbery polymers and dual-mode type for glassy polymers. So it can be proposed that sorption site (center) usually could associate with only one penetrant molecule i.e., in polymer network there exists only one type of sorption center: mono-molecular center. Mono-molecular sorption would be understood that each sorption center is able to associate with only one penetrant molecule and interaction of this sorption species with next fluid molecule can not leads to any form of molecular species even very unstable. By this restriction, Eqn. (5) is reduced to: C= Kln P +1 (6) s 1) First checking the validity of Eqn. (6) can be made from the experimental sorption data of various gases NO2, CO2, and C2H4 in polymethyl methacrylate (PMMA) reported by Sanders and Koros [4]. Fig. 1 shows the comparision of the measured solubility of the various gases depending on pressure (fugacity) and the calculated curves by Eqn. (6). 35 30 25 20 15 10 5 0 NO2 CO C2H2 0 5 10 15 20 Fugacity, bar Fig. 1. Comparison between curves calculated by Eqn. (6) and experimental data for various gases sorption in polymethylmethacrylate at 30ºC This figure shows that the equation give a very good fitting experiment data. From 634 calculation of K and ks in Tab. 1 one can see that the parameter 1/K for CO2 and NO2 smaller than that for C2H4. This suggests that the energy creating sorption species for C2H4 may be greater than for CO2 and NO2. It is understandable from viewpoint of polar structure of these gases. To create a sorption species with C2H2 should be more difficult than with CO2 and NO2. From table 1 one can also see that sorption kinetics ks between CO2, NO2 and PMMA greater than between C2H2 and PMMA. It is understandable from viewpoint of geometric structure (molecular size). 2) The next verifying validity of Eqn. (6) is comparable between calculated curves and experimental solubility of CO2, CH4, Ar and N2 in polyphenylene oxide (PPO) [14]. Table 1: Sorption parameters of various gases in polymethylmethacrylate Sorption parameters according to Dual-mode sorption Thermodynamic sorption K Gas atm-1 NO2 1,340 CO2 1.173 C2H4 0.586 b cm3(STP) cm3.atm 0.234 0.186 0.217 C`H 1/K cm3 (STP) cm3.atm 14.782 0.044 16.079 0.042 10.631 0.103 ks bar 5.219 6.079 3.240 60 50 40 30 20 10 0 CO2 CH4 Ar N2 0 5 10 15 20 25 Pressure, atm Fig. 2: Comparison of curves calculated by Eqn. (6) and experimental sorption isotherm of polyphenylene oxide and various gases at 30ºC From Fig. 2, one can also see a very good agreement between experimental data and theoretical calculation for CO2, CH4, Ar and N2 solubility in PPO. Calculated values of K and ks for these systems are shown in table 2. Table 2: Sorption parameters of various gases in polyphenylene oxide at 35ºC cm3(STP) cm3.atm ks, atm N2 Ar CH4 CO2 0.150 0.116 0.063 0.050 14.631 10.460 7.855 2.835 635 From this table one can see that the value of 1/K of sorption species increases from CO2 to N2, sorption kinetics interaction ks also increases from CO2 to N2. From viewpoint of molecular structure of these gases this order also would be understandable. Solubility of penetrant depends not only on structure of fluid molecules and polymers but also strongly on thermodynamic condition [3]. For temperature higher than glass transition temperature Tg, gas solubility in polymer usually described by Henry’s Law while for temperature lower than glass transition temperature it would be described by dual-mode theory[3, 4, 5]. With increasing temperature, gas solubility in polymer decrease. The different behavior of gas solubility in rubbery polymer and glassy polymer can be explained by the existence of microvoids in glassy polymer. However, from thermodynamic point of view, the existence of microvoids in polymers is implausible [14 - 16]. 18 25ºC 35ºC 45ºC 55ºC 65ºC 75ºC 8 95ºC 6 4 2 0 0 5 10 15 20 25 Pressure, atm Fig. 3: CO2 solubility change with temperature for poly(ethylene terephtalate), Tg = 85ºC 3) Fig. 3. shows a change of CO2 solubility in poly(ethylene terephtalate) (PETP) with temperature. This figure shows that not only CO2 solubility in PETP decreases with temperature, but also the curvature of sorption isotherm is reduced and it becomes a line as temperature is higher than glassy transition temperature (for PETP, Tg = 85ºC). By dual-mode sorption theory in glassy state due to thermal mobility and flexibility of polymer molecular chains the amount of microvoid in polymer decreases with temperature, overcome Tg, at rubbery state the amount of microvoid in polymer equal to zero and sorption can be describes by Henry’s Law. A direct consequence of thermal mobility and flexibility 636 of polymer molecular chains may be increasing of sorption kinetics interaction ks between penetrant molecules and polymer molecules with temperature. Due to these thermal effects the energy creating sorption species (i.e., T/K) may be also increased with temperature by addition a certain kinetics energy. The change of polymer such as from glassy state to rubbery state by overcoming Tg usually lead to a drastical change in all chemico-physical parameters including the sorption parameters K and ks due to change in solid structure in polymer, as it is shown in table 3. Table 3: Sorption parameters of CO2 in poly(ethylene terephtalate) at various temperature Temperature, T/Kcm3(STP) cm .atm 25 45,424 2,682 35 49,383 3,840 45 62,544 3,946 55 65,652 5,593 65 69,782 7,348 75 73,319 9,766 95 4,458 421,585 From this table one can see that sorption kinetic parameter ks in rubbery state are larger than that in glassy state, however, energy creating sorption species (i.e., T/K) species in rubbery state is smaller than that in glassy state of polymer. This may cause by the difference of mobility, flexibility and local orientation of polymer molecular chains in both state [14]. In rubbery state penetrant solubility also decreased (i.e., ks increased) with temperature as shown in figure 4. As one can see above, from thermodynamic point of view, the difference between Henry’s sorption type and dual-mode sorption type, there is only the difference between the values of sorption kinetic parameters ks for rubbery and glassy state of polymer. It should not be a result of difference in sorption mechanism between the both polymer state such as consideration of dual mode sorption theory. Using Taylor’s expansion, ln(x+1) x for small x, Eqn. (6) can be rewritten as C = KP/ks, for small x = P/ks ... - tailieumienphi.vn
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