Viscosity of Dilute Polymer Solutions For dilute polymer solutions, one is normally interested not in the value of itself but in specific viscosity s= (-0)/0 (0 is the viscosity of pure solvent) and characteristic viscosity  = (-0)/0 , where is the density of monomer units in the solution. For the solution of impenetrable spheres of radius R Einstein derived 0 1 2.5 where is the volume fraction occupied by the spheres in the solution. If each sphere consists of N particles (monomer units) of mass m, and their density is , we have N A 4 3 N A 4 3 R R mN 3 M 3 where M is molecular mass of the polymer chain, M=mN, and NA is Avogadro number . 4 R3N A 2.5 2.5 3 M For dense spheres N~R3 and  is independent of the size of the particles viscosity measurements are not informative in this limit. E.g. for globular proteins we always obtain  4 cm3/g independently of the size of the globule. However, polymer coils are very loose objects with N R3 N 3 R03 1 3 N 1 2 a 3. If they still move as a whole together with the solvent inside the coil (non-draining assumption), the Einstein formula remains valid. Then 1 4 R3N A 3 2.5 N 2a3 3 mN therefore, for polymer coils there is an N dependence. So by measuring  it is possible to get the information on the size of polymer coils. Conclusions 1. Indeed, if the measurements are performed at the -point we have 6 0 S 2 3 2 0 M (Flory-Fox law), where universal constant 0 2.84 1021 if [ ] is expressed in dl/g. From this relation, if we know M (elastic light scattering, chromatography), it is possible to determine <S2>0, and therefore to obtain the length of the Kuhn segment. 2 l 6S 0 L On the other hand, if we know l we can determine M. 2. By determining [ ] and [ ] in the good solvent, we can calculate the expansion coefficient of the coil. 1 3 3. Another important characteristics of polymer solutions which can be determined from the value of [ ] is the overlap concentration of polymer coils c* c < c* c = c* c > c* Dilute polymer solutions Overlap concentration Semidilute polymer solutions The average polymer concentration in the solution is equal to that inside one polymer coil at the overlap concentration c*. Thus, N 1 c 3 3 12 3 R N a Since c* N/R3 , and  R3/N , we have c* 1. For the practical estimations it is normally assumed c 1 4. At the -point ~ R3 M ~ M 3 2 M . Thus, we can write KM 1 2, where K is some proportionality coefficient. In the good solvent ~ 3 M 1 2 ~ M 3 10 M 1 2 ~ M 4 5 , i.e. KM 4 5, where K is another proportionality coefficient. In the general case KM a It is called a Mark-Kuhn-Houwink equation. Its experimental significance is connected with the fact that by performing measurements for some unknown polymer for different values of M and by determining the value of a it is possible to judge on the quality of solvent for this polymer. 5. Assumption of non-draining coils. Analysis shows that this assumption is always valid for long chains. Let us consider the system of obstacles of concentration c moving through a liquid with the velocity . Inside the upper half-space the liquid will move together with the obstacles, inside the lower half-space it will mainly remain at rest. System of obstacles in the upper half-space move through a liquid with the velocity . L The characteristic distance L connected with the draining is 1 2 L c where is the viscosity of the liquid and is the friction coefficient of each obstacle. In the application of these results to polymer coil, we identify obstacles with monomer units. Then 1 c 3 12 3 N a Thus, we have 1 3 12 3 12 2 N a L c For -solvent 3 12 a N 1 4 R aN 1 2 L For good solvent 3 12 a N 2 5 R aN 3 5 L In both case the value of L is much smaller than the coil size R (for large value of N), thus the non-draining assumption is valid. Analysis shows that the opposite limit (free draining) can be realized only for short and stiff enough chains. Light Scattering from Polymer Solutions It is well-known that all media (e.g. pure solvent) scatter light. This is the case even for macroscopically homogeneous media due to the density fluctuations. If polymer coils are dissolved in the solvent, another type of scattering appears - scattering on the polymer concentration fluctuations. This is called excess scattering; it is this component which is normally investigated for the analysis of the properties of the coils. In this section we will consider elastic (or Rayleigh) light scattering (without the change of the frequency of the scattered light) and the scattering from dilute solutions of coils. Let us assume that the incident beam of light (wavelength 0 , intensity J0 ) passes through a dilute polymer solution. The detector is located at a distance r from the scattering cell in the direction of the scattering angle . The quantity which is measured is the intensity of excess scattering J( ). 1 Normally the size of the coil R aN 2 is less than 100nm and is, therefore, much smaller than the wavelength of light 0 . In this case the coil can be regarded as point scatterer. Scattering of normal nonpolarized light by point scatterers has been considered by Rayleigh. The result is : 16 4 2 1 cos 2 J 4 2 c0VJ 0 0 r 2 where c0 is the concentration of coils (scatterers), V is the scattering volume, while is the polarizability of the coil ( defined according to P E ; P being a dipole moment acquired by the coil in the external field E ). Experimental results are normally expressed in terms of reduced scattering intensity J r 2 16 4 2 1 cos2 I 4 c0 J0 V 2 0 The value of I does not depend on the geometry of experimental setup. Traditionally for polymer scatterers the value of polymer mass per unit volume, , is used instead of c0 : c0 N A M where M is the molecular mass of a polymer, and NA is Avogadro number. Therefore J r 2 16 4 2 1 cos2 I 4 NA J0 V 2 0 M The polarizability can be directly expressed in terms of the change of the refractive index of the solution n upon addition of polymer coils to the solvent. n M n , 0 2N A where n0 is the refractive index of the pure solvent. The value of n is called refractive index increment; it can be directly experimentally measured for a given polymersolvent system. Thus, 2 4 n n 1 cos 2 I 4 M 0 N A 2 2 2 0 2 1 cos , We have I HM 2 2 2 4 2 n0 n where H is so-called optical 4 N A 0 constant of the solution. It depends only on the type of the polymer-solvent system, but not on the molecular weight or concentration of the dissolved polymer. So, by measuring I( ) we can determine the molecular mass of the dissolved polymer. For example, if I( 900 ) is the scattering intensity at the angle 900, 2 I 900 M H The physical reason for the possibility of the determination of M from the light scattering experiments can be explained as follows. The value of I is proportional to the concentration of scatterers c0 ( ~ 1/M) and to the square of polarizability ( ~ M2 ) , thus I ~ M. Whether it is possible to obtain from the same experiment the size of the coil, R , in addition to M ? The answer is yes, and this can be explained as follows. It should be emphasized that the coil actually can not be regarded as point scatterer, as long as R > /20. In this case it is necessary to take into account the destructive interference of light scattered by different monomer units. The waves scattered from the monomer units A and B in the direction of the unit vector u are shifted in phase with respect to each other, because of the excess distance l . This phase shift is small, as soon as l << , but still it is responsible for the partially destructive interference which leads to the decrease in I. This effect should be larger for higher values of . From the scattering theory we know that 1 I I 0 N2 1 I 0 N2 N exp i k rj j 1 2 N N exp i k rj rl j 1 l 1 2 u 4 k sin 0 u For the light scattering always k r 1 (where r is the distance between two monomer units), since k 1 0 . Thus, we can expand the expression for I in the powers of k. Since linear terms vanishes after averaging, this gives I 1 1 S 2 k 2 I 0 3 where is the mean square S 2 radius of gyration of polymeric coil. Thus : 1. By measuring the intensity at any specified angle it is possible to obtain the molecular mass of a polymer chain, M. 2. By measuring the angular dependence of scattered light it is possible to obtain the mean square radius of gyration of a polymer coil, S 2 . Inelastic light scattering from dilute polymer solutions In the method of inelastic light scattering we measure not only the intensity, but also the frequency spectrum of scattered light. For this method the incident beam should be obligatory a monochromatic light from laser ( reduced intensity I0 , frequency w0 , wavelength 0 ). The light scattered at the angle is actually not monochromatic, since the scattering objects are moving. If I(w0 + w) is the intensity of the light of frequency w0 + w scattered at the angle , then from the general scattering theory it follows I 0 iwt 3 ikr I w0 w dt e d r e δc0,0 δcr , t 2 kwhere is the scattering wavevector ( k 4 sin ) , and δс 0,0 δc r , t 2 0 is so-called dynamic structure factor of polymer solution; δc r , t c r , t c is the deviation from the average polymer concentration at the point r and the time moment t . Thus: (i) scattering is connected with the dynamics of concentration fluctuations; (ii) intensity of scattering is given by the Fourier-transformation ( vs. time and spatial coordinates ) of the dynamic structure factor. By studying the scattering at a given angle (or k ), we investigate the dynamics of polymer chain motions with the wavelength 1 k . For dilute solutions at k R 1 the wavelength 1 k R and for this case we can study the internal motions with the coil. These condition can be realized for the scattering of X-rays or neutrons. But for the light scattering normally even at 1 ; ( k 1 0 ), k R R 0 1 . In this limit the method of dynamic light scattering probes the motion of the coils as a whole. Coils move as point scatterers with the diffusion coefficient D. The concentration of coils (scatterers) с r , t obeys the diffusion equation cr , t Dcr , t t where 2 x 2 2 y 2 2 z 2 . For the Fourier transform ( vs. time and spatial coordinates ) of the dynamic structure factor the diffusion equation is known to give 2 Dk I w0 w I 0 2 2 Dk w2 The dependence I(w) is called a Lorenz curve. I The spectrum of the light scattered at some angle w w0 w0+ w The characteristic width of the Lorenz curve is w Dk 2 (16 2 D 20 ) sin 2 . 2 Thus, by measuring the spectrum of the scattered light it is possible to determine the diffusion coefficient of the coils, D. What should be expected for the value of D? If coils are considered as impenetrable spheres of radius R ( we will see below that this is the case in many situations ), then according to Stokes and Einstein kT D 6R where is the viscosity of the solvent. Thus, by measuring D it is possible to determine R. This is a more precise method for the determination of the size of a polymer coil than elastic light scattering. Detailed analysis shows that the value thus determined is the socalled hydrodynamic radius of the coil 1 1 RH 2 N 2 1 i 1,i j , j 1 rij N N 2 12 The values of SR2 H1 ,2 Rand are of the same order of magnitude; the difference is in only numerical coefficients.