2.2 Polymer Solutions 13 volatile compound.For the solvent,61 can be calculated directly from the latent heat of vaporization (AHp)using the relationship of Eq.2.11: △E=△Hap-RT (2.11) R is the gas constant,and T is the temperature in kelvins.Thus,the cohesive energy of solvent is shown in Eg.2.12: d1= △Hap-l R /2 (2.12） Since polymers have negligible vapor pressure,the most convenient method of determining 82 is to use group molar attraction constants.These are constants derived from studies of low-molecular-weight compounds that lead to numerical values for various molecular groupings on the basis of intermolecular forces.Two sets of values (designated G)have been suggested,one by Small [3],derived from heats of vaporization and the other by Hoy [4],based on vapor pressure mea- surements.Typical G values are given in Table 2.1.Clearly there are significant differences between the Small and Hoy values.The use of which set is normally determined by the method used to determine o for the solvent. G values are additive for a given structure,and are related to 6 by i=d>G (2.13) M where d is density and M is molecular weight.For polystyrene -[CH2- CH(CHs)],for example,which has a density of 1.05,a repeating unit mass of 104,and 6 is calculated,using Small's G values,as Table 2.1 Representative group molar attraction constants [3,4] Chemical group G[(cal cm)mol-] Small Hoy H3C— 214 147.3 —CH2— 133 131.5 CH一 28 86.0 < -93 32.0 -CH2 190 126.0 =CH一 19 84.5 一CsHs(phenyl 735 -CH =(aromatic) 117.1 C=O(ketone) 字 262.7 C02一(ester) 310 326.6
volatile compound. For the solvent, d1 can be calculated directly from the latent heat of vaporization DHvap using the relationship of Eq. 2.11: DE ¼ DHvap RT ð2:11Þ R is the gas constant, and T is the temperature in kelvins. Thus, the cohesive energy of solvent is shown in Eq. 2.12: d1 ¼ DHvap RT V 1=2 ð2:12Þ Since polymers have negligible vapor pressure, the most convenient method of determining d2 is to use group molar attraction constants. These are constants derived from studies of low-molecular-weight compounds that lead to numerical values for various molecular groupings on the basis of intermolecular forces. Two sets of values (designated G) have been suggested, one by Small [3], derived from heats of vaporization and the other by Hoy [4], based on vapor pressure measurements. Typical G values are given in Table 2.1. Clearly there are significant differences between the Small and Hoy values. The use of which set is normally determined by the method used to determine d1 for the solvent. G values are additive for a given structure, and are related to d by d ¼ d R G M ð2:13Þ where d is density and M is molecular weight. For polystyrene –[CH2– CH(C6H5)]n–, for example, which has a density of 1.05, a repeating unit mass of 104, and d is calculated, using Small’s G values, as Table 2.1 Representative group molar attraction constants [3, 4] Chemical group G[(cal cm3 ) 1/2mol-1 ] Small Hoy H3C 214 147.3 CH2 133 131.5 CH 28 86.0 C -93 32.0 CH2 190 126.0 CH 19 84.5 C6H5 (phenyl) 735 – CH (aromatic) – 117.1 C O (ketone) 275 262.7 CO2 (ester) 310 326.6 2.2 Polymer Solutions 13
14 2 Polymer Size and Polymer Solutions 6=105133+28+735)=9.0 104 or using Hoy's data, 6=1.051315+85.9+6117.1】=9.3 104 Both data give similar solubility parameter.However,there is limitation of solubility parameter.They do not consider the strong dipolar forces such as hydrogen bonding,dipole-dipole attraction,etc.Modifications have been done by many researchers and available in literature [5,6]. Once a polymer-solvent system has been selected,another consideration is how the polymer molecules behave in that solvent.Particularly important from the standpoint of molecular weight determinations is the resultant size,or hydrody- namic volume,of the polymer molecules in solution.Assuming polymer molecules of a given molecular weight are fully separated from one another by solvent,the hydrodynamic volume will depend on a variety of factors,including interactions between solvent and polymer molecules,chain branching,conformational effects arising from the polarity,and steric bulkiness of the substituent groups,and restricted rotation caused by resonance,for example,polyamide can exhibit res- onance structure between neutral molecule and ionic molecule. 0 -C-NH- Because of Brownian motion,molecules are changing shape continuously. Hence,the prediction of molecular size must base on statistical considerations and average dimensions.If a molecule was fully extended,its size could easily be computed from the knowledge of bond lengths and bond angles.Such is not the case,however,with most common polymers;therefore,size is generally expressed Fig.2.3 Schematic representation of a molecular coil,r end to end distance, s radius of gyration [2]
d ¼ 1:05ð133 þ 28 þ 735Þ 104 ¼ 9:0 or using Hoy’s data, d ¼ 1:05 131 ½ :5 þ 85:99 þ 6ð117:1Þ 104 ¼ 9:3 Both data give similar solubility parameter. However, there is limitation of solubility parameter. They do not consider the strong dipolar forces such as hydrogen bonding, dipole–dipole attraction, etc. Modifications have been done by many researchers and available in literature [5, 6]. Once a polymer–solvent system has been selected, another consideration is how the polymer molecules behave in that solvent. Particularly important from the standpoint of molecular weight determinations is the resultant size, or hydrodynamic volume, of the polymer molecules in solution. Assuming polymer molecules of a given molecular weight are fully separated from one another by solvent, the hydrodynamic volume will depend on a variety of factors, including interactions between solvent and polymer molecules, chain branching, conformational effects arising from the polarity, and steric bulkiness of the substituent groups, and restricted rotation caused by resonance, for example, polyamide can exhibit resonance structure between neutral molecule and ionic molecule. Because of Brownian motion, molecules are changing shape continuously. Hence, the prediction of molecular size must base on statistical considerations and average dimensions. If a molecule was fully extended, its size could easily be computed from the knowledge of bond lengths and bond angles. Such is not the case, however, with most common polymers; therefore, size is generally expressed Fig. 2.3 Schematic representation of a molecular coil, r = end to end distance, s = radius of gyration [2] C O NH C O- NH+ 14 2 Polymer Size and Polymer Solutions
2.2 Polymer Solutions 15 in terms of the mean-square average distance between chain ends,72,for a linear polymer,or the square average radius of gyration about the center of gravity,32, for a branched polymer.Figure 2.3 illustrates the meaning of r and s from the perspective of a coiled structure of an individual polymer molecule having its center of gravity at the origin. The average shape of the coiled molecule is spherical.The greater the affinity of solvent for polymer,the larger will be the sphere,that is,the hydrodynamic volume.As the solvent-polymer interaction decreases,intramolecular interactions become more important,leading to a contraction of the hydrodynamic volume.It is convenient to express r and s in terms of two factors:an unperturbed dimension (ro or so)and an expansion factor (a).Thus, 2=x2 (2.14) 32=x2 (2.15) 3)B 义= ( (2.16) The unperturbed dimension refers to the size of the macromolecule exclusive of solvent effects.It arises from a combination of free rotation and intramolecular interactions such as steric and polar interactions.The expansion factor,on the other hand,arises from interactions between solvent and polymer.For a linear polymer,2=632.The a will be greater than unity in a"good"solvent,thus the actual(perturbed)dimensions will exceed the unperturbed dimensions.The greater the value of a is,the "better"the solvent is.For the special case where a=1,the polymer assumes its unperturbed dimensions and behaves as an "ideal"statistical coil. Because solubility properties vary with temperature in a given solvent,a is temperature dependent.For a given polymer in a given solvent,the lowest tem- perature at which =1 is called the theta(0)temperature (or Flory temperature), and the solvent is then called a theta solvent.Additionally,the polymer is said to be in a theta state.In the theta state,the polymer is on the brink of becoming insoluble;in other words,the solvent is having a minimal solvation effect on the dissolved molecules.Any further diminish of this effect causes the attractive forces among polymer molecules to predominate,and the polymer precipitates. From the standpoint of molecular weight determinations,the significance of solution viscosity is expressed according to the Flory-Fox equation [7], 外 (2.17) where n is the intrinsic viscosity (to be defined later),M is the average molecular weight,and is a proportionality constant (called the Flory constant)equal to
in terms of the mean-square average distance between chain ends, r2, for a linear polymer, or the square average radius of gyration about the center of gravity, s2, for a branched polymer. Figure 2.3 illustrates the meaning of r and s from the perspective of a coiled structure of an individual polymer molecule having its center of gravity at the origin. The average shape of the coiled molecule is spherical. The greater the affinity of solvent for polymer, the larger will be the sphere, that is, the hydrodynamic volume. As the solvent–polymer interaction decreases, intramolecular interactions become more important, leading to a contraction of the hydrodynamic volume. It is convenient to express r and s in terms of two factors: an unperturbed dimension (r0 or s0) and an expansion factor ð Þa . Thus, r2 ¼ r2 0a2 ð2:14Þ s 2 ¼ s 2 0a2 ð2:15Þ a ¼ r2 ð Þ1=2 r2 0 1=2 ð2:16Þ The unperturbed dimension refers to the size of the macromolecule exclusive of solvent effects. It arises from a combination of free rotation and intramolecular interactions such as steric and polar interactions. The expansion factor, on the other hand, arises from interactions between solvent and polymer. For a linear polymer, r2 ¼ 6s2. The a will be greater than unity in a ‘‘good’’ solvent, thus the actual (perturbed) dimensions will exceed the unperturbed dimensions. The greater the value of a is, the ‘‘better’’ the solvent is. For the special case where a ¼ 1, the polymer assumes its unperturbed dimensions and behaves as an ‘‘ideal’’ statistical coil. Because solubility properties vary with temperature in a given solvent, a is temperature dependent. For a given polymer in a given solvent, the lowest temperature at which a ¼ 1 is called the theta ð Þh temperature (or Flory temperature), and the solvent is then called a theta solvent. Additionally, the polymer is said to be in a theta state. In the theta state, the polymer is on the brink of becoming insoluble; in other words, the solvent is having a minimal solvation effect on the dissolved molecules. Any further diminish of this effect causes the attractive forces among polymer molecules to predominate, and the polymer precipitates. From the standpoint of molecular weight determinations, the significance of solution viscosity is expressed according to the Flory-Fox equation [7], ½ ¼ g U r2 ð Þ3=2 M ð2:17Þ where ½ g is the intrinsic viscosity (to be defined later), M is the average molecular weight, and U is a proportionality constant (called the Flory constant) equal to 2.2 Polymer Solutions 15
16 2 Polymer Size and Polymer Solutions approximately 3x 1024.Substituting for we obtain Mark-Houwink- Sakurada equation: (哈x2)32 (2.18) M Equation 2.18 can be rearranged to M=(M)mPx (2.19) Since T and are constants,we can set then =KM2 (2.20) At the theta temperature,=1 and =KM2 (2.21) For conditions other than the theta temperature,the equation is expressed by In]KM (2.22) Apart from molecular weight determinations,many important practical con- siderations are arisen from solubility effects.For instance,one moves in the direction of "good"solvent to "poor",and intramolecular forces become more important,the polymer molecules shrink in volume.This increasing compactness leads to reduced "drag"and hence a lower viscosity which has been used to control the viscosity of polymer for ease of processing. 2.3 Measurement of Molecular Weight Many techniques have been developed to determine the molecular weight of polymer [8].Which technique to use is dependent on many factors such as the size of the polymer,the ease of access and operation of the equipment,the cost of the analysis,and so on. For polymer molecular weight is less than 50,000,its molecular weight can be determined by the end group analysis.The methods for end group analysis include titration,elemental analysis,radio active tagging,and spectroscopy.Infrared spectroscopy (IR),nuclear magnetic resonance spectroscopy (NMR),and mass spectroscopy (MS)are commonly used spectroscopic technique.The IR and NMR are usually less sensitive than that of MS due to the detection limit. Rules of end group analysis for Mn are:(1)the method cannot be applied to branched polymers unless the number of branches is known with certainty;thus it is practically limited to linear polymers,(2)in a linear polymer there are twice as
approximately 3 9 1024. Substituting r2 0‘a2 for r2, we obtain Mark-HouwinkSakurada equation: ½ ¼ g U r2 0a2 3=2 M ð2:18Þ Equation 2.18 can be rearranged to ½ ¼ g U r2 0M 1 3=2 M 1=2 a3 ð2:19Þ Since r0 and M are constants, we can set K ¼ U r2 0M 1 3=2 , then ½ ¼ g KM 1=2 a3 ð2:20Þ At the theta temperature, a ¼ 1 and ½ ¼ g KM 1=2 ð2:21Þ For conditions other than the theta temperature, the equation is expressed by ½ ¼ g KM a ð2:22Þ Apart from molecular weight determinations, many important practical considerations are arisen from solubility effects. For instance, one moves in the direction of ‘‘good’’ solvent to ‘‘poor’’, and intramolecular forces become more important, the polymer molecules shrink in volume. This increasing compactness leads to reduced ‘‘drag’’ and hence a lower viscosity which has been used to control the viscosity of polymer for ease of processing. 2.3 Measurement of Molecular Weight Many techniques have been developed to determine the molecular weight of polymer [8]. Which technique to use is dependent on many factors such as the size of the polymer, the ease of access and operation of the equipment, the cost of the analysis, and so on. For polymer molecular weight is less than 50,000, its molecular weight can be determined by the end group analysis. The methods for end group analysis include titration, elemental analysis, radio active tagging, and spectroscopy. Infrared spectroscopy (IR), nuclear magnetic resonance spectroscopy (NMR), and mass spectroscopy (MS) are commonly used spectroscopic technique. The IR and NMR are usually less sensitive than that of MS due to the detection limit. Rules of end group analysis for M n are: (1) the method cannot be applied to branched polymers unless the number of branches is known with certainty; thus it is practically limited to linear polymers, (2) in a linear polymer there are twice as 16 2 Polymer Size and Polymer Solutions
2.3 Measurement of Molecular Weight 17 many end groups as polymer molecules,(3)if the polymer contains different groups at each end of the chain and only one characteristic end group is being measured,the number of this type is equal to the number of polymer molecules, (4)measurement of molecular weight by end-group analysis is only meaningful when the mechanisms of initiation and termination are well understood.To determine the number average molecular weight of the linear polyester before cross-linking,one can titrate the carboxyl and hydroxyl end groups by standard acid-base titration methods.In the case of carboxyl,a weighed sample of polymer is dissolved in an appropriate solvent such as acetone and titrated with standard base to a phenolphthalein end point.For hydroxyl,a sample is acetylated with excess acetic anhydride,and liberated acetic acid,together with carboxyl end groups,is similarly titrated.From the two titrations,one obtains the number of mini-equivalents of carboxyl and hydroxyl in the sample.The number average molecular weight(i.e.,the number of grams per mole)is then given by Eg.2.23: Mn=2×1000×sample wt. (2.23) meqCOOH meqOH The 2 in the numerator takes into account that two end groups are being counted per molecule.The acid number is defined as the number of milligrams of base required to neutralize 1 g of polyester which is used to monitor the progress of polyester synthesis in industry. Of the various methods of number average molecular weight determination based on colligative properties,membrane osmometry is most useful.When pure solvent is Fig.2.4 Schematic Measuring representation of a membrane tubes osmometer [2] △h Solution Solvent Semipermeable membrane
many end groups as polymer molecules, (3) if the polymer contains different groups at each end of the chain and only one characteristic end group is being measured, the number of this type is equal to the number of polymer molecules, (4) measurement of molecular weight by end-group analysis is only meaningful when the mechanisms of initiation and termination are well understood. To determine the number average molecular weight of the linear polyester before cross-linking, one can titrate the carboxyl and hydroxyl end groups by standard acid–base titration methods. In the case of carboxyl, a weighed sample of polymer is dissolved in an appropriate solvent such as acetone and titrated with standard base to a phenolphthalein end point. For hydroxyl, a sample is acetylated with excess acetic anhydride, and liberated acetic acid, together with carboxyl end groups, is similarly titrated. From the two titrations, one obtains the number of mini-equivalents of carboxyl and hydroxyl in the sample. The number average molecular weight (i.e., the number of grams per mole) is then given by Eq. 2.23: M n ¼ 2 1000 sample wt: meqCOOH þ meqOH ð2:23Þ The 2 in the numerator takes into account that two end groups are being counted per molecule. The acid number is defined as the number of milligrams of base required to neutralize 1 g of polyester which is used to monitor the progress of polyester synthesis in industry. Of the various methods of number average molecular weight determination based on colligative properties, membrane osmometry is most useful. When pure solvent is Fig. 2.4 Schematic representation of a membrane osmometer [2] 2.3 Measurement of Molecular Weight 17