Thermal Analysis of Polymeric Materials Part 13 ppt

7 Multiple Component Materials ___________________________________________________________________ 706 Fig. 7.1 7.1 Macromolecular Phase Diagrams 7.1.1 Phase Diagrams The thermodynamics of the phase diagrams of multi-component systems of small molecules are described in Sects. 2.2.5 and 2.2.6. The key equations and the eutectic phase diagram are reviewed in Figs. 2.24–27. Applications of these equations are found in the molar-mass determinations described in Sect. 1.4. One-component pVT- phase diagrams are discussed in Sect. 4.1.7. To review a general phase diagram in the condensed state with two components, a series of DSC heating-traces, labeled A–F, are shown in Fig. 7.1 with a vertical temperature axis and a horizontal heat-flow-rate axis. The phase diagram derived from these DSC curves is indicated by the dotted line using the same temperature axis and a horizontal concentration axis [1]. The DSC trace A is for a sample with a concentration somewhat beyond the pure component, x 1 . The single, broad, melting peak suggests a solution of the two components. The beginning and end of melting indicate the positions of the solidus and liquidus, as represented by the dotted lines. Both change with variations in the overall concentration. When analyzing run B, a second endotherm can be seen, a peritectic transition. It is best identified together with run C at a higher mole fraction of component 2. At the peritectic temperature, a second solid solution that exists at larger x 2 turns unstable. This becomes obvious when completing the full phase diagram, as is shown in area 6 of Fig. 7.2, below. The DSC runs D, E, and F reveal a simple eutectic phase diagram of crystals of component 2 with an intermediate compound (point 5 in Fig. 7.2). The phase diagram is interpreted in Fig. 7.2. Proof of the assumed phases usually needs a detailed X-ray structure analysis of all the indicated phase areas. 7.1 Macromolecular Phase Diagrams ___________________________________________________________________ 707 Fig. 7.2 Only the eutectic portion on the far right of Fig. 7.2 fulfills the conditions of complete solubility in the liquid phase and complete separation in the crystals, as was assumed in Fig. 2.27. The pure crystals, marked by the concentration x 1 =1,andthe compound “5,” participating in the eutectic “3,” have a small solubility in their crystals, the areas “9” and “8,” respectively. All the solid solutions indicated in the diagram need additional specification of the concentration-dependence of thechemical potential of the components in the single phase areas (I) of limited solubility (“8,” “6,” and “9”). The solid solution “6” decomposes above the peritectic temperature “7” into the solid solution “9” and the liquid solution. All two-phase areas (II) consist of the two phase-separated compositions given at any chosen temperature by the points of intersections of a horizontal with the boundaries of the phase-areas. At equilibrium, the possible numbers of phases are governed by the phase rule: P + F = C + 2 (see Sect. 2.5.7). In Fig. 7.2, three of the condensed phases are in contact at the eutectic temperatures “3” and “4,” and the peritectic temperature “7.” Adding the gas phase, omitted in the present discussion, leads to four phases (P) and no degrees of freedom (F). These quadruple points are thus the points of sharp transitions in the two- component phase diagram of Fig. 7.2 (C = 2). The next step towards the description of phase diagrams that include macro- molecules is to change from the just discussed ideal solution to the real solutions, mentioned in Sect. 2.2.5. For this purpose one can look at the phase equilibrium between a solution and the corresponding vapor. The simplest case has a negligible vapor pressure for the second component, 2. The chemical potentials for the first component, 1, in solution and in the pure gas phase are written in Fig. 7.3 as Eqs. (1–3), following the discussion given in Fig. 2.26. For component 1, the chemical potential of the solvent,  1 o , is defined in Eq. (1) for its pure state, x 1 =1.0, but at its vapor pressure, p 1 o , not at atmospheric pressure. It must then be written as 7 Multiple Component Materials ___________________________________________________________________ 708 Fig. 7.3 Eq. (2) for the gas phase, but with its chemical potential  1 o ' defined at atmospheric pressure p = 1.0 atm. Equation (3) expresses then the chemical potential of the (pure) solvent vapor over the solution. Note, that in these expressions one always uses atm as unit of pressure, a non-SI unit of pressure (see Fig. 2.3, for conversion factors see Sect 4.5.1). At equilibrium, the two logarithmic terms in Eqs. (1) and (3) must be equal,  1 s =  1 v , and produce Raoult’s law. Raoult’s law is indicated by the diagonal in the graph of Fig. 7.3. At concentrations approaching the pure solvent, x 1  1.0, this equation must always hold. At higher solute concentrations (x 2 =1 x 1 ), however, deviations occur for real solutions and must be evaluated in detail. The deviations from the ideal solution are often treated by introduction of an activity a 1 replacing the concentration x 1 . A well- known interpretation of the activity for the strong interactions, as are found for ions dissolved in water, is given by the Debye-Hückel theory [2]. Clusters of water are bound to the solute and reduce p 1 . Repulsions, on the other hand, reject the solvent from the solution into the pure gas phase and increase p 1 , as is shown in the example of Fig. 7.3. At high solute concentration x 2 , the indicated tangent can approximate the vapor pressure and is called Henry’s law. For macromolecules the vapor pressure of a the low molar solvent is decreased so much that the Henry’s law activity, a 1 , remains practically zero for a wide concentra- tion range. Only close to x 1 = 1.0 does the vapor pressure approach Raoult’s law, i.e., a large mole fraction of low-molar-mass solvent dissolves in a polymer melt without producing a noticeable vapor pressure. Perhaps, this should not be surprising if one remembers that x 1 must be large compared to x 2 , the mole fraction of the polymeric solute, to achieve comparable masses for the two components. The Flory-Huggins equation, to be described next, will resolve this problem and describes the vapor pressure in Fig.7.3 for cases such as natural rubber dissolved in benzene. 7.1 Macromolecular Phase Diagrams ___________________________________________________________________ 709 1 Fraction of empty space, for the related concepts of packing fractions, see Figs. 4.24 and 5.17, and for the hole model, see Sects. 6.13 and 6.23, as well as Fig. 6.5. Fig. 7.4 7.1.2. Flory-Huggins Equation To assess the influence of size on chemical potentials, one first searches, as in Sect. 2.2.5, for an ideal entropy of mixing, but now for macromolecules and small molecules. In Fig. 7.4, the description for a eutectic equilibrium between a pure polymer crystal and an amorphous solution is attempted as in Fig. 2.26 with Eqs. (1) and (2). The large-molecule component is 2, the small-molecule solvent, 1. Next, one tries tofind expressions for the RHS and LHS of the differences in chemical potential, as before. The RHS is identical to the prior case and written on the right in Eq. (2). One needs to remember that H f refers to one mole of large molecules, i.e., it has a much bigger effect than for small molecules, perhaps by a factor of 1,000! The left- hand side, the mixing expression, will be given by the Flory-Huggins equation. The free enthalpy of mixing, G mix , will first be treated by considering a new ideal entropy of mixing that depends on molecular size. Then, it will be expanded to a real solution by adding the interaction term, G*, seen in Eq. (3). Easiest is the Hildebrand derivation [3]. One assumes that ideal liquids have a universal free volume fraction, 1 v f . On mixing, the molecules expand into the available free volume, as in an ideal gas. They change from the free volumes in the pure states, V f1 =V 1 v f and V f2 =V 2 v f , to the total free volume V f1,2 =V total v f with a change in the entropy of S ideal = n 1 Rln(V f1,2 /V f1 )  n 2 Rln(V f1,2 /V f2 ), where n 1 and n 2 are the moles of the respective molecules. Since the free volume fraction is assumed to be constant during the mixing, this leads to Eq. (4) in Fig. 7.4 with the definition of the volume fractions 7 Multiple Component Materials ___________________________________________________________________ 710 Fig. 7.5 written as Eq. (5) and Eq. (6). An identical expression can be derived by placing the solution on a lattice where the solvent molecules occupy one unit cell each, and the macromolecules use x unit cells each (x = V 2 /V 1 ) [4,5]. Note that with x = 1, i.e., for equal sizes of both components, the equation describing the mixing reverts back to the result for the ideal gas in Fig. 2.24 (v 1 =x 1 and v 2 =x 2 ). A simple calculation clarifies the effect of molecular size. Assuming two types of molecules mixed to a fixed volume V with fractions v 1 =v 2 = 0.5. Allowing large and small molecules of ratio 1,000:1, Eq. (4) yields a S = 693 R when both molecules are small (x = 1.0, and n 1 =n 2 = 500), when one molecule is large and the other one small, x = 1,000, and now S = 347 R (n 1 = 500 and n 2 = 0.5), and when both molecules are large, x is again 1.0, but this time S=0.7R(n 1 =n 2 = 0.5). One concludes that for small molecules the positive entropy strongly drives the dissolution and may overcome the endothermic heat of solution which usually exists in the absence of specific dipole or H-bond interactions between solvent and solute. Large molecules still may dissolve in a small molecular solvent for entropic reasons. Macromolecules, in contrast, will not dissolve other macromolecules, unless the overall free enthalpy is negative, i.e., the heat-of-dissolution is exothermic (see Sect. 2.2.3). The last step in the derivation involves the evaluation of G * , the interaction term, which may help the dissolution of a macromolecule by becoming exothermic. Equation (7) in Fig. 7.4 and the sketch in Fig. 7.5 show how one can assess G * from the number of contacts a macromolecule can make with its low-molar-massneighbors. If the interaction parameter, w 1,2 per contact pair, is taken to be an enthalpy, the resulting mixing expression applies to a regular solution (ideal entropy and a non-zero enthalpy of mixing). Making it a free enthalpy by including a correction to the ideal entropy into w 1,2 , the mixing expression for a real solution of unequal molecular sizes results. The equivalence in Eq. (8) allows one to eliminate x from Eq. (7), and 7.1 Macromolecular Phase Diagrams ___________________________________________________________________ 711 Fig. 7.6 the introduction of the interaction parameter 3, expressed in units of RT, yields Eq. (9) and allows, finally, to change Eq. (7) in Fig. 7.4 to Eq. (10) of Fig. 7.5, and yields with Eq.(3) the Eq. (11) in Fig. 7.6. The new interaction parameter 3 is central to the description of the polymer solutions. The bottom equations in Fig. 7.5 help in the derivation of the operation suggested by Eq. (2). The result is the Flory-Huggins equation, Eq. (12). The boxed Eq. (13) is the equation for T m , which describes the freezing-point lowering of polymer crystals in the presence of a solvent under equilibrium conditions, with the solvent being rejected from the polymer crystal (eutectic phase diagram). Examples are given in Sects. 7.1.5 and 7.1.6. At equal sizes of the two components and with a small v 1 , the ideal T m of Fig. 2.26 is recovered, as is indicated by Eq. (14) of Fig. 7.6. With a macromolecular component 2, i.e., a very large x and also for a small concentration v 1 , the heat of fusion of the macromolecule in Eq. (15) of Fig. 7.6 can be replaced by H f /x, the heat of fusion of the macromolecule per reference volume of the low-molar-mass component 1, which is often written with Eq. (16) as H u . Quite analogous to the phase equilibrium of the macromolecular component 2 in Eq. (13), one can also write the Flory-Huggins expression for the small-molecule component 1 (see Fig. 2.26) to complete a eutectic phase diagram (see Fig. 2.27): with the RHS representing the H f of the small solvent molecules, which have a much smaller effect than the macromolecules (factor of 1/x). The left-hand sides of the equations, representing the partial molar mixing, which are available for both components from the overall expression for G mix by differentiation with respect to the component under consideration, n 1 . The result is used in Fig. 7.7, below. 7 Multiple Component Materials ___________________________________________________________________ 712 Fig. 7.7 7.1.3. Upper and Lower Critical Temperatures Liquid-liquid phase separations are treated in this section. The top equation in Fig. 7.7 repeats the Flory-Huggins equation for the chemical potential of a macromolecular component 2 dissolved in a low-molar-mass solvent 1, as derived in Fig. 7.6, Eq. (12). By analogy, the chemical potential of component 1 is written in the second equation. From Fig. 7.4, one can, furthermore, state that for the phase separation into solutions I and II, the equations in the center of Fig. 7.7 must hold. The condition for a phase separation is thus that the system must have two concentrations, x 2 I and x 2 II , with the same slopes 0G/0x 2 , as shown. The equation in the center of Fig. 7.7 is analogous to the evaluation of the partial molar quantities in Fig. 2.25 for volume and leads to the constructions shown. The limiting two concentrations share the same tangent to the G mix curve. Between the two areas of stable solutions lies the two-phase area. A system in the two-phase range separates into two different concentrations, set by the limits of the two-phase area. Its total G mix is less than the heavy line in Fig. 7.7 (more negative). The sum of the free enthalpies of mixing of the two solutions is given by the tangent to the two concentra- tions of phase separation. On gradually entering the region of phase separation by changing the concentration, one notes that the free enthalpy of mixing still decreases as long as 0 2 G mix /0x 2 2  0, i.e., starting the phase separation needs a nucleation event to overcome an interfacial free energy. The single phase may remain metastable. The limiting value 0 2 G mix /0x 2 2 = 0 is reached at the point of inflection. From this point on, phase separation is spontaneous, one has reached the spinodal decomposition concentration. Changing the concentration along the tangent of inflection causes no change in G mix for concentration fluctuations, and the two phases given by the phase limits (the binodals) can grow continuously without nucleation. 7.1 Macromolecular Phase Diagrams ___________________________________________________________________ 713 Fig. 7.8 The phase diagrams expressed in temperature versus volume fraction v 2 are shown in Fig. 7.8 for a series of different molar masses of polystyrene dissolved in diisobutyl ketone (the lower curves correspond to lower molar masses). Lower molar masses dissolve at lower temperature and yield a narrower two-phase area. To find the upper critical temperature of phase separation, the UCST, one can insert for low polymer concentrations v 2 the first few terms of the series-expansion for ln v 1 =ln(1 v 2 ) given in Fig. 7.8 into the expression for 0G mix /0n 1 (given in Fig. 7.7). With the first two terms of the expansion this leads to the equation listed for ln a 1 , the activity of the component 1 ( 0G mix /0n 1 =  1 , see also Fig. 7.3). First, one notices that when the interaction parameter 3 reaches a value of 0.5, the term in parentheses, the second virial coefficient, becomes zero and the term in v 2 2 vanishes. The temperature where this condition is met is called the -temperature. It is easy to see that, in analogy to the non-ideal gases, this is the temperature where the repulsive effect of the interaction between the large and small molecules overcomes the excluded volume and the system behaves as if it were ideal. For macromolecules and small molecules this condition is discussed inFigs. 1.33 and1.34 and methods to measure the second virial coefficient are found in Sect. 1.4.2. The second observation is that a horizontal tangent and tangent of inflection for incipient phase separation (UCST) is obtained at 0 1 /0v 2 = 0 2  1 /0v 2 2 = 0. Carrying out the differentiations with the appropriate equation of Fig. 7.7, leads first to the result (1  v 2c ) 1  (1  x 1 )  23v 2c = 0 and then to (1  v 2c ) 2 23 = 0. The solution for the UCST is thus v 2c =(1+x 1/2 ) 1 , i.e., v 2c is close to zero for large molar masses. Similarly, the critical 3 c is  0.5 +x 1/2 . Figure 7.9 shows the phase diagramfortwopolystyrenes of different molar masses with acetone. It illustrates that the value of 3 can be separated into an enthalpy effect, 7 Multiple Component Materials ___________________________________________________________________ 714 Fig. 7.9 3 h , and an entropy effect, 3 s .If3 has a minimum in its temperature-dependence, an LCST, lower critical solution temperature, is possible in addition to the UCST, as exhibited by the lower-molar-mass polystyrene. The same equations as derivedabove for the UCST apply for the case of an LCST. 7.1.4. Phase Diagrams with Low-mass Compounds In this section, phase diagrams of systems with two components are discussed, one of low and one of high molar mass. Only the case is treated where both components crystallize in a eutectic-like manner. The top equation in Fig. 7.10 describes the liquidus line for the macromolecule located on the right side of a eutectic phase diagram. It applies to equilibrium, and represents Eq. (13) of Fig. 7.6, the Flory- Huggins equation. The second equation applies to the left, low-molar-mass side and can similarly be derived from the appropriate equations of Fig. 7.4–7. The phase diagram for polyethylene and 1,2,4,5-tetrachlorobenzene (TCB) at the bottom of Fig. 7.10 looks, indeed, like a eutectic phase diagram. The data points were obtained by DSC, as outlined in Sect. 7.1.1. To minimize nonequilibrium effects, the results were extrapolated to a heating rate of zero. The volume fraction is assumed to be close to the weight fraction. The low-molar-mass side is close to equilibrium, but polyethylene should show a T m o of 414.6 K at w 2 = 1.0, i.e., the phase diagram follows at best, a zero-entropy-production curve, paralleling equilibrium (see Fig. 6.79). The equations at the lower right in Fig. 7.10 express the freezing point lowering for the pure solvents on adding small amounts of the solute, i.e., the tangents to zero concentrations. The top equation is the ideal solution limit (a 1 =x 1 =1 x 2 , Raoult’s law, see Fig. 7.3). It arises from the second of the top equations for the low-molar- mass solvent in Fig. 7.10 if the polymer solution is so dilute that the macromolecules 7.1 Macromolecular Phase Diagrams ___________________________________________________________________ 715 Fig. 7.10 are isolated from each other (compare to Fig. 7.6). This equation is used to determine molar masses of polymers in Fig. 1.63 and 6.64 (see Sect. 1.4.3). The data points from the right-hand side of the eutectic phase diagram can be used with the second equation which is identical to Eq. (16), derived in Fig. 7.6. The method of adding small amounts of a low-molar-mass compound to the polymer can be used to determine H u from the freezing point lowering, measured by DSC or dilatometry, and is known as the diluent method. The slope 0T/0v 1 allows the evaluation of H u . Some typical experimental values for the heat of fusion of polyethylene with the diluent method are 3.89, 3.91, 4.14, and 4.06 kJ mol 1 for the solvents ethyl benzoate, o-nitrotoluene, 1,2,3,4-tetrahydronaphthalene (tetralin), and -chloronaphthalene, respectively [6,7]. The method is not absolute, since polymer crystals are usually not in equilibrium. The measurements rely, thus, on analyzing crystals of identical lamellar thickness (identical metastability) for the different concentrations. A distinct advantage of the diluent method is that the crystallinities of the samples needs not to be known. The reasonable agreement with the established equilibrium heat of fusion (4.11 kJ mol 1 ) suggests, furthermore, that the free enthalpy curves of crystals of different metastabilityare oftenparallel,as assumed,for example, for the schematic in Figs. 2.86–88 and Fig. 6.79. Figure 7.11 illustrates the irreversiblebehavior of the same TCB and polyethylene, but under nonequilibrium conditions. The DSC data were collected after quenching to 354 and 388 K for crystallization. Comparing with Fig. 7.10 shows reasonable agreement for TCB. For polyethylene, crystallization at the higher temperature, 378 K, raises the eutectic and melting temperatures relative to crystallization at 354 K. The cooling traces, in contrast, show even lower crystallization temperatures for the polymer and at higher concentration, the TCB phase diagram is also affected. It is of particular interest that both the eutectic temperatures and the eutectic concentrations [...]... large x and also for a small concentration v1, the heat of fusion of the macromolecule in Eq (15) of Fig 7.6 can be replaced by Hf/x, the heat of fusion of the macromolecule per reference volume of the low-molar-mass component 1, which is often written with Eq (16) as Hu Quite analogous to the phase equilibrium of the macromolecular component 2 in Eq (13) , one can also write the Flory-Huggins expression... area Its total Gmix is less than the heavy line in Fig 7.7 (more negative) The sum of the free enthalpies of mixing of the two solutions is given by the tangent to the two concentrations of phase separation On gradually entering the region of phase separation by changing the concentration, one notes that the free enthalpy of mixing still decreases as long as 02Gmix/0x22 0, i.e., starting the phase separation... Hf of the small solvent molecules, which have a much smaller effect than the macromolecules (factor of 1/x) The left-hand sides of the equations, representing the partial molar mixing, which are available for both components from the overall expression for Gmix by differentiation with respect to the component under consideration, n1 The result is used in Fig 7.7, below 712 7 Multiple Component Materials. .. equation in the center of Fig 7.7 is analogous to the evaluation of the partial molar quantities in Fig 2.25 for volume and leads to the constructions shown The limiting two concentrations share the same tangent to the Gmix curve Between the two areas of stable solutions lies the two-phase area A system in the two-phase range separates into two different concentrations, set by the limits of the two-phase... 7.4 to Eq (10) of Fig 7.5, and yields with Eq.(3) the Eq (11) in Fig 7.6 The new interaction parameter 3 is central to the description of the polymer solutions The bottom equations in Fig 7.5 help in the derivation of the operation suggested by Eq (2) The result is the Flory-Huggins equation, Eq (12) The boxed Eq (13) is the equation for Tm, which describes the freezing-point lowering of polymer crystals... enthalpy of mixing) Making it a free enthalpy by including a correction to the ideal entropy into w1,2, the mixing expression for a real solution of unequal molecular sizes results The equivalence in Eq (8) allows one to eliminate x from Eq (7), and Fig 7.5 7.1 Macromolecular Phase Diagrams 711 _ the introduction of the interaction parameter 3, expressed in units of RT,... freezing-point lowering of polymer crystals in the presence of a solvent under equilibrium conditions, with the solvent being rejected from the polymer crystal (eutectic phase diagram) Examples are given in Sects 7.1.5 and 7.1.6 Fig 7.6 At equal sizes of the two components and with a small v1, the ideal Tm of Fig 2.26 is recovered, as is indicated by Eq (14) of Fig 7.6 With a macromolecular component 2, i.e.,... shown in Fig 7.8 for a series of different molar masses of polystyrene dissolved in diisobutyl ketone (the lower curves correspond to lower molar masses) Lower molar masses dissolve at lower temperature and yield a narrower two-phase area To find the upper critical temperature of phase separation, the UCST, one can insert for low polymer concentrations v2 the first few terms of the series-expansion for... heat -of- dissolution is exothermic (see Sect 2.2.3) The last step in the derivation involves the evaluation of G*, the interaction term, which may help the dissolution of a macromolecule by becoming exothermic Equation (7) in Fig 7.4 and the sketch in Fig 7.5 show how one can assess G* from the number of contacts a macromolecule can make with its low-molar-mass neighbors If the interaction parameter, w1,2... may overcome the endothermic heat of solution which usually exists in the absence of specific dipole or H-bond interactions between solvent and solute Large molecules still may dissolve in a small molecular solvent for entropic reasons Macromolecules, in contrast, will not dissolve other macromolecules, unless the overall free enthalpy is negative, i.e., the heat -of- dissolution is exothermic (see Sect . heat of fusion of the macromolecule in Eq. (15) of Fig. 7.6 can be replaced by H f /x, the heat of fusion of the macromolecule per reference volume of the low-molar-mass component 1, which is often. negative). The sum of the free enthalpies of mixing of the two solutions is given by the tangent to the two concentra- tions of phase separation. On gradually entering the region of phase separation. respectively ln v 1 , approach infinity because of the high molar entropy of fusion ( H f /T m o ) and a value of x = 1 in Eq. (13) . 7 Multiple Component Materials ___________________________________________________________________ 718 Fig.