Berat Molekul Polimer

MOLECULAR WEIGHT (MW) • Molecular weight, M: Mass of a mole of chains. Low M high M Polymers can have various lengths depending on the number of repeat units. During the polymerization process not all chains in a polymer grow to the same length, so there is a distribution of molecular weights. There are several ways of defining an average molecular weight. The molecular weight distribution in a polymer describes the relationship between the number of moles of each polymer species and the molar mass of that species.

MOLECULAR WEIGHT DISTRIBUTION Mn = the number average molecular weight (mass) __ Mi = mean (middle) molecular weight of size range i xi = number fraction of chains in size range i wi = weight fraction of chains in size range i

The simplest, most common molecular weight is the number-average molecular weight (n) end-group analysis or colligative properties (b.p. elevation, osmotic pressure, etc) others commonly used are weight-average molecular weight (w), z-average molecular weight (z) and viscosity-average molecular weight (u) light scattering (w), sedimentation equilibrium (z) and solution viscosity (u) The simplest, most commonly used, and most correct molecular weight is the number-average molecular weight that is based on methods of counting the number of molecules in a given weight of polymer. The methods that have been used for counting are reducing value measurements, osmometry, and cryoscopy Another type of molecular weight is weight-average molecular weight that is obtained from light scattering techniques. Others include z-average molecular weight and viscosity-average molecular weight.

Number-average molecular weight (n) based on methods of counting the number of molecules in a given weight of polymer the total weight of a polymer sample, w, is the sum of the weights of each molecular species present N = number of molecules M = molecular weight Molecular weights value obtained depend in large measure on the method of measurement. Mn is based on methods in which the number of molecules of each weight in the sample are counted. The total weight of a polymer sample (w) is the sum of the weights of each molecular species present (wi). (Where N and M are the number of molecules and molecular weight of each species i ) Number-average molecular weight (Mn) is the weight of sample per molecule, that is the sum of samples with molecular weight (Mi) multiplied by the number of molecules of the ith weight (Ni) divided by the sum of the number of molecules of the ith weight (Ni).

Weight-average molecular weight (w) determination of molecular weight based on size rather than the number of molecules the greater the mass, the greater the contribution to the measurement w = weight fraction M = molecular weight N = number of molecules Light scattering or ultracentrafugation on the other hand would produce a molecular weight based on mass or polarizability of the species present. The greater the mass, the greater the contribution to the measurement. In contrast to Mn (which is the sum of the number of molecules times its molecular weight). These methods sum the weight fraction of each species times its molecular weight. The value obtained is the weight-average molecular weight (Mn) and is expressed mathematically as The sum of the sampls with molecular weight average Mi squared multiplied times the number of molecules of the ith molecular weight divided by the sum of samples with molecular weight Mi times the number of molecules of the ith molecular weight.

Z-average molecular weight (z) some molecular weight determination methods (e.g. sedimentation equilibrium) yield higher molecular weight averages - z w = weight fraction M = molecular weight N = number of molecules In addition to weight and number-averaged molecular weight, higher molecular weight averages are often reported. z-average molecular weight (Mz) which is the sum of the weights of each molecule present (wi) multiplied by the square of the molecular weight of each molecule (Mi)2 divided by the sum of the weight of each molecule present (wi) multiplied by the molecular weight of each molecule (Mi).

Number-average molecular weight (n) Example - a polymer sample consists of 9 molecules of mw 30,000 and 5 molecules of mw 50,000 Suppose for example, we have a polymer sample consisting of 9 moles of molecular weight 30,000 Da, and 5 moles of molecular weight 50,000 Da. Thus the number average molecular weight is the sum of 9 molecules of MW 30000 and 5 molecules of 50000 divided by the sum of molecules (9+5). If instead, our sample consists of 9 grams of MW 30,000 Da and 5 grams of 50,000 Da. Thus Mn is the sum of grams of polymer divided by the sum of 9 grams divided by MW of 30,000 Da and 5 grams divided by MW of 50,000 Da. (g / g mol-1 = mol)

Weight-average molecular weight (w) Consider the previous example - 9 molecules of molecular weight 30,000 and 5 molecules of molecular weight 50,000 If we consider the same two previous examples, 9 moles of molecular weight 30,000 Da, and 5 moles of molecular weight 50,000 Da. The weight-average molecular weight is the sum of 9 molecules of 30000 MW squared and 5 molecules of 50000 squared divided by the sum of molecules 9 molecules of 30000 MW plus 5 molecules of 50000 MW. Again, if instead our sample consists of 9 grams of MW 30,000 Da and 5 grams of 50,000 Da. Thus Mw is the sum of grams of polymer times the MW divided by the sum of grams of polymer that is 9 grams multiplied by 30,000 Da, plus 5 grams multiplied by 50,000 Da. (g x g mol-1 = g) divided by 9 grams plus 5 grams. Note that in each instance, Mw is greater than Mn

Z-average molecular weight (z) Consider the previous example - 9 molecules of molecular weight 30,000 and 5 molecules of molecular weight 50,000 Using the same two previous examples in the calculations of Mn and Mw, 9 moles of molecular weight 30,000 Da, and 5 moles of molecular weight 50,000 Da. The z-average molecular weight is the sum of 9 molecules multiplied by 30000 MW cubed and 5 molecules of 50000 cubed divided by the sum of molecules 9 molecules of 30000 MW squared plus 5 molecules of 50000 MW squared. Again, using the sample consisting of 9 grams of MW 30,000 Da and 5 grams of 50,000 Da. The the sum of grams of polymer times the MW squarred divided by the sum of grams of polymer multiplied by their molecular weight. That is 9 grams multiplied by 30,000 Da cubed, plus 5 grams multiplied by 50,000 Da cubed [g x (g mol-1)3 = g4 mol-3 ) divided by 9 grams multiplied by 30,000 Da plus 5 grams multiplied by 50,000 Da. In all cases, Mz (42,136) is greater than both Mw (40,000) and Mn (37,000)

A Typical Molecular Weight Distribution Curve 104 wi n = 100 000 g mol-1 4.0 w = 199 900 g mol-1 3.0 2.0 z = 299 850 g mol-1 With very few exceptions, polymers consist of macromolecules with a range of molecular weights. Since the molecular weight changes in intervals equal to the repeat unit (or weight of the smallest repeating structure in the polymer chain) the distribution of molecular weights is discontinuous. However, for most polymers these intervals are extremely small in comparison to the total range of molecular weight and the distribution can be asumed to be continuous, as exemplified above. 1.0 200 000 400 000 600 000 800 000 1 000 000 Mi (g mol-1)

The narrower the molecular weight range, the closer are the values of w and n , and the ratio w / n may thus be used as an indication of the breadth of the molecular weight range in a polymer sample. The ratio is called the polydispersity index, and any system having a range of molecular weights is said to be polydispersed The narrower the molecular weight range, the closer are the values of Mw and Mn, and the rates of Mw / Mn, may thus be used as an indication of the breadth of the molecular weight range in the polymer sample. This ratio of Mw / Mn, is called the “polydispersity index” and any system having a range of molecular weights is said to be “polydispersed” Using our previous examples, the polydispersity of our 5 moles of 50,000 MW and 9 moles of 30,000 MW polymer would be 40,000/37,000 = 1.08

Polymer Solution Viscosity When a polymer is dissolved in a solvent and then subjected to flow through a narrow capillary it exerts a resistance to that flow. This resistance is very informative. It provides information on the size of the polymer Its Flexibility and shape in solution Its interactions with the solvent it is disolved in. A characteristic feature of a dilute polymer solution is that its viscosity is considerably higher than that of either the pure solvent or similar dilute solutions of small molecules. This arises due to the large difference in size between polymer and solvent molecules, and the magnitude of the viscosity increase is related to the dimensions of the polymer molecules in solution. Therefore, measurements of the viscosities of dilute polymer solutions can be used to provide information are polymer structures, molecular shapes, D.P. and polymer solvent interactions. However, it is mainly used to determine molecular weight. For a dilute polymer solution, the ratio between flow time of a polymer solution (t) to that of the pure solvent (to) is effectively equal to the ratio of their viscosities - or “relative viscosity”. As this has a limiting value of unity, i.e.  = o , a more useful quantity is specific viscosity (sp ) such that

For dilute solutions the ratio between flow time of a polymer solution (t) to that of the pure solvent (to) is effectively equal to the ratio of their viscosity (h / ho) As this has a limiting value of unity, a more useful quantity is specific viscosity (hsp) A characteristic feature of a dilute polymer solution is that its viscosity is considerably higher than that of either the pure solvent or similar dilute solutions of small molecules. This arises due to the large difference in size between polymer and solvent molecules, and the magnitude of the viscosity increase is related to the dimensions of the polymer molecules in solution. Therefore, measurements of the viscosities of dilute polymer solutions can be used to provide information are polymer structures, molecular shapes, D.P. and polymer solvent interactions. However, it is mainly used to determine molecular weight. For a dilute polymer solution, the ratio between flow time of a polymer solution (t) to that of the pure solvent (to) is effectively equal to the ratio of their viscosities - or “relative viscosity”. As this has a limiting value of unity, i.e.  = o , a more useful quantity is specific viscosity (sp ) such that

Intrinsic Viscosity [η] To eliminate concentration effects, the specific viscosity (hsp ) is divided by concentration and extrapolated to zero concentration to give intrinsic viscosity [h] To eliminate concentration effects (although minimal if existent in dilute solution), the specific viscosity is divided by concentration and extrapolation to zero concentration to yield Intrinsic viscosity [ ]. Thus plotting solution viscosity () minus the solvent viscosity (o) divided by the solvent viscosity (o) multiplied by polymer concentration will provide Intrinsic viscosity [ ] (y-intercept) and KH (Huggins constant) from the slope. The following example demonstrates this. Thus plotting hsp/c vs c, the intercept is the intrinsic viscosity [h] and from the slope, KH (Huggins constant, typically between 0.3 - 0.9) can be determined

Intrinsic Viscosity Determination 3.5 KH[η2]  3.0  2.5   From the flow times in a capillary viscometer for a dilute polymer-solvent system, the intrinsic viscosity [] and Huggins constant KH can be determined. The y-intercept (2.18) is the corresponding intrinsic viscosity [], thus enabling the determination of the Huggins constant by dividing the slope by the square of the intrinsic viscosity ( ). 2.0 [h] 0.2 0.4 0.6 0.8 1.0 C (g dl-1)

Viscosity-Molecular Weight Relations Intrinsic viscosity [h] can be related to molecular weight by the Mark-Houwink-Sakurada Equation Applicable for a given polymer-solvent system at a given temperature Log [h] vs log M (w or n) for a series of fractionated polymers produces log K (intercept) and a (slope) A wide range of values have been published a ~ 0.5 (randomly coiled polymers) ~ 0.8 (rod-like, extended chain polymers) K between 10-3 and 0.5 Intrinsic Viscosity [] is the most useful of the various viscosity designations because it can be related to molecular weight by the Mark-Houwink-Sakarada equation where Mv is the viscosity average molecular weight. The constants K and a are the intercept and slope respectively, of a plot of log [] versus log Mw or Mn of a series of fractionated polymer samples. Such plots are linear (except at low molecular weight) for linear polymers. As a result, a wide range of constants K and a have been published.

Typical Mark-Houwink-Sakurada Equation Constants for Several Polysaccharides Table 1. Shows several K and a constants for various polysaccharides. Included are the molecular weight ranges of the polymer standards and the method of molecular weight determination. SD - sedimentation equilibrium OS - osmometry LS - light scattering Note K and to a lesser extent a is very much dependent on solvent and temperature of the system used and will vary according to these parameters

Typical Intrinsic Viscosities, a and K values for Several Naturally Occurring Polymeric Materials Table 2. shows typical intrinsic viscosities [], K and a constants for kraft lignin, cellulose and xylan of equal molecular weight (50,000 Da). Note from a values the shape of each macromolecule can be determined Kraft Lignin - Newtonian sphere Cellulose - non-freedraining coil Xylan - freedraining coil (more rigid) Thus [] can provide information not only in molecular weight, but also on molecular shape in solution. The degree of expansion or shape of the molecular coils of a polymer can be ascertained from its a values (Table 2) lignin (Newtonian sphere), cellulose (non-freedraining coil) and xylan (freedraining coil)

Viscosity-average molecular weight (u) viscosity, like light scattering, is greater for the larger-sized polymer molecules than the smaller ones, and is much closer to Mw than Mn w = weight fraction N = number of moles M = molecular weight a = A constant Thus from the Mark-Houwink-Sakarada equation the Mv can be determined, and is defined as - the sum of weight fraction of each species (wi) multiplied by its molecular weight (Mi) to the power a (a constant) all to the power of 1/a, or - the sum of the number of molecules of the ith weight (Ni) multiplied by its molecular weight to the power a+1 (Mia+1) divided by the sum of the number of molecules of the ith weight (Ni) multiplied by its molecular weight (Mi) all to the power of 1/a. For most polymers, a varies between 0.5 (for a randomly coiled polymer) and 0.8; for a more rod-like polymer. a may be as high as a=1.0, when Mv=Mw. Factors may complicate the application of the Mark-Houwink-Sakarada equation - such as chain branching, too broad a molecular weight distribution used to determine K and a, solvation of polymer molecules, and the presence of alternating block sequences in the polymer backbone. When a = 1, u= w , usually a ~ 0.5-0.9 a is a measure of the the hydrodynamic volume of the polymer varies with polymer, solvent and temperature

Polymer Properties Intrinsic viscosity [h] is a measure of the effective hydrodynamic volume of the molecule. For a hard non-swelling sphere, Einsteins equation is valid Where V is the specific volume of the material in the sphere. In a linear solvent-swollen polymer like cellulose, V and thus also [h] is much larger. A low [h] value means that the molecule is compact and thus occupies a relatively small volume Intrinsic viscosity [h] is a measure of the effective hydrodynamic volume of the molecule. For a hard non-swelling sphere, Einsteins equation is valid, such that Where V is the specific volume of the material in the sphere. In a linear solvent-swollen polymer like cellulose, V and thus also [h] is much larger. Recall that the intrinsic viscosity values for cellulose, 1.81 dL g-1 where as for kraft lignin it is 0.06 dL g-1. A low [h] value means that the molecule is compact and thus occupies a relatively small volume.

Polymer Morphology The ultimate properties of any polymer (plastic, fiber, or rubber) result from a combination of molecular weight and chemical structure. Polymers require a particular MW, which depends largely on the chemical structure, to have desirable mechanical properties. Molecular Weight Mechanical Property The ultimate properties of any polymer, whether plastic, rubber or fiber, results from a combination of molecular weight and chemical structure. Polymers have to be of a certain molecular weight before they have useful mechanical properties, and the particular molecular weight necessary depends largely on chemical structure.

The mechanical properties result from attractive forces between molecules dipole-dipole interactions, H-bonding, induction forces, London forces or ionic bonding, ion-dipole interactions H-bonding dipole-dipole The mechanical properties result from attractive forces between molecules: dipole-dipole interactions, (including hydrogen-bonding), induction forces, dispersion or London forces between nonpolar molecules, or they may result from ionic bonding and ion-dipole interactions with polymers containing ionic groups. Induction (induced dipole) interatcions and London forces are weak forces, but become increasingly significant as molecular weight increases. In particular it is London forces that give rise to the molecular cohesion and resulting mechanical properties of nonpolar polymers like polyethylene. Secondary bonding forces, i.e. hydrogen-bonding vs dipole-dipole and their relationship to molecular weight can be seen in comparing a polyester (dipole-dipole) with a polyamide (hydrogen-bonding). In comparing the physical properties of both polymers a higher molecular weight polyester is required to obtain the fiber properties comparable to those of a much lower molecular weight polyamide. A lower MW polyamide will produce good fiber properties as compared to the polyester  H-bonding

Hydrogen Bonding A dipole-dipole interaction for hydrogens bonded to electronegative elements Electrostatic Interaction very important in cellulose Hydrogen-bonding plays an extremely important role in wood chemistry, particularly cellulose, and thus deserves further elaboration. It is a dipole-dipole interaction involving heteroatom containing molecules. In such systems weak electrostatic interactions will exist between the heteroatom hydrogens and lone-pair electrons on an adjacent heteroatom. It is a weak bond ~ 5 kcal mol-1 (as compared to ~ 81 kcal mol-1 for covalent c-c bonds) and requires a short bond distance ~2.5Å (however, slightly longer than a c-c covalent bond ~1.46Å). Although a weak bond, it is extremely important in cellulose due to the large number of hydrogen bonds present. Weak bond ~ 5 kcal mol-1 (c-c ~ 81 kcal mol-1 ) Require short bond distance ~ 2.5Å (c-c ~ 1.46Å)

Hydrogen Bonding A dipole-dipole interaction for hydrogens bonded to electronegative elements Electrostatic Interaction very important in cellulose Hydrogen-bonding plays an extremely important role in wood chemistry, particularly cellulose, and thus deserves further elaboration. It is a dipole-dipole interaction involving heteroatom containing molecules. In such systems weak electrostatic interactions will exist between the heteroatom hydrogens and lone-pair electrons on an adjacent heteroatom. It is a weak bond ~ 5 kcal mol-1 (as compared to ~ 81 kcal mol-1 for covalent c-c bonds) and requires a short bond distance ~2.5Å (however, slightly longer than a c-c covalent bond ~1.46Å). Although a weak bond, it is extremely important in cellulose due to the large number of hydrogen bonds present. Weak bond ~ 5 kcal mol-1 (c-c ~ 81 kcal mol-1 ) Require short bond distance ~ 2.5Å (c-c ~ 1.46Å)