Determination of Molecular-Weight Distribution of ... - ACS Publications

Chapter 18

Determination of Molecular-Weight Distribution of Water-Soluble Macromolecules by Dynamic Light Scattering

Downloaded by EMORY UNIV on April 21, 2015 | http://pubs.acs.org Publication Date: July 18, 1991 | doi: 10.1021/bk-1991-0467.ch018

Michael J. Mettille and Roger D. Hester Department of Polymer Science, The University of Southern Mississippi, Hattiesburg, MS 39406-0076 An algorithm for the extraction of molecular weight distributions of typical high molecular weight, water-soluble polymers from dynamic light scattering has been developed. A function fit method was used to determine the best distribution parameters of a generalized exponential function. The algorithm employs a Marquardt-Levenburg constrained nonlinear least squares regression. A series approximation of the integral equation which is generated was attempted. This approach was abandoned because of difficulties with series convergence. A modified Rhomberg numerical integration was utilized with limited success. Two water-soluble polymer molecular weight distributions were obtained. The distributions of a polyethyleneoxide and a polyacrylamide were determined and compared with distributions obtained using gel permeation chromatography. The two methods were in fair agreement. Molecular weight characterization of macromolecular systems is extremely important. Many solution properties are functions of the macromolecular size (and hence the molecular weight). An example of one property of major importance is viscosity. High viscosity solutions are useful as polymer flooding agents in enhanced oil recovery applications. A high molecular weight system exhibits higher viscosity than one of lower molecular weight for a single polymer at a specified concentration. In general, high molecular weight water-soluble polymer systems are polydisperse. In other words, the macromolecules are not of a single molecular weight but rather have a distribution of molecular weights. The fraction of polymer molecules having lower molecular weight is much less effective in producing high viscosity solutions. Thus, knowledge of this distribution is very important in designing an economical, effective EOR system. At the present there is no adequate tool for the determination of molecular weight distributions of the large watersoluble macromolecular systems. Various methods have been employed to obtain molecular weight information. Vapor phase osmometry, membrane osmometry, and end group analysis have all been used to obtain number average molecular weights. These techniques are unable to measure molecular weights in the high 0097-6156/91/0467-0276$06.00/0 © 1991 American Chemical Society

In Water-Soluble Polymers; Shalaby, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.

18.

M E T T I L L E & HESTER

Molecular-Weight Distribution Determination

277


molecular weight regime and also suffer from an inability to determine information on the molecular weight distribution (MWD). Classical light scattering has also been used to acquire weight average molecular weights. Although this technique can determine molecular weights of high molecular weight systems, it does not provide information on the distribution of molecular weights. Size exclusion chromatography (SEC) is the classical method of obtaining MWDs. SEC is very difficult with aqueous solutions of very high molecular weight macromolecules because packing materials are not efficient at separating very large polymers. SEC requires molecular weight standards that span the appropriate molecular weight range. Water-soluble standards do not exist of the required high molecular weight. Thus, the utility of this method is limited. One method for determining the molecular weight distribution which does not rely on the existence of high molecular weight water-soluble standards is dynamic light scattering (DLS). Background and Theory DLS is also known as photon correlation spectroscopy, quasi-elastic light scattering, intensity fluctuation spectroscopy, and several other names. Thorough discussions of the theory and mathematics of D L S may be found elsewhere (1). Briefly, the phenomenon that is observed i n D L S experiments is intensity fluctuations in the light scattered from a dilute solution of macromolecules. As the solute molecules which are scattering light undergo brownian diffusion, the relative phase of the scattered fight changes. These changes i n phase cause fluctuations i n the detected scattering intensity. The rate at which these fluctuations occur can be, i n turn, related to the rate of diffusion of the scattering molecules through a time autocorrelation of the scattered intensity (2). The intensity time autocorrelation function can be defined as shown i n Equation 1. (1) In this equation I(t) and Kt+τ) are the intensity of scattered fight at time t and time t+τ. The integral equation shown is normally approximated as a discrete sum i n most instrumentation. For most applications, the measured autocorrelation function G(x) is converted to what is termed the first order normalized autocorrelation function. The transformation is shown i n Equation 2. (2)

Where : Gfc) = Measured autocorrelation function g(x) = First order normalize autocorrelation function Ko = Baseline value, equal to < I> and GK°°) 2

= Square of the average value of the intensity = Average value of the intensity squared 2

2


278

WATER-SOLUBLE P O L Y M E R S

K is measured during the experiment and KQ is determined during the data analysis. In systems that consist of only a single size diffusing species, the relationship between the translational diffusion coefficient of the species i n dilute solution and the first order normalized autocorrelation function is simply a single decaying exponential as shown i n Equation 3. B

g(x) = exp(- q D x)

(3)

2

Where : q = Scattering vector =

4

*

( λο

n s i n

e / 2

>


D = Diffusion coefficient η = Refractive index of solvent λο = Radiation wavelength i n vacuum However, polymers have a distribution of molecular weights and this must be accounted for i n the relationship. The above equation can be modified to become a summation over the various sizes of polymer molecules and is shown below. oo

g(x) = XF(Di)exp(q Dix) 2

W)

i=0

Where : F(Di) = The discrete distribution function of diffusion coefficients Di = The diffusion coefficient of the i sizemolecule th

If one assumes that the discrete distribution can be approximated as a continuous distribution, the summation equation may be transformed into an integral equation. It is assumed that the diffusion coefficient may be related to molecular weight as shown i n Equation 5. Di = a M f

(5)

p

The α and β parameters are tabulated for many polymer-solvent systems. Those values not appearing directly i n the literature may be estimated from the Mark-Houwink constants. One may then transform the integral equation from decay constant space to molecular weight space (3). This transform is shown i n Equation 6. j°°Ci F(M) M gW-

7 p _ 1

exp(- o q M " Ρχ) d M 2

— CiF(M)M * dM

J

(6)

0

Where : F(M) = The number distribution of molecular weights is an intraparticle interference factor. This factor arises from the interference of light that is scattered from a single particle. As the size of scattering particles approach the wavelength of the incident radiation, the light scattered from one portion of a particle may interfere with the light scattered from another portion of the same particle. For monodisperse systems the result


18.



279

is simply a change i n overall intensity without changing the shape of the correlation function. For polydisperse systems the result is markedly different. Particles of a size such that Ci is high (or the intensity of scatter is low) contribute little to the integral equation and particles where Cj is small contribute greatly. C acts to appropriately weight the scattering intensity from each molecular size. If this factor were not applied the M W D may show anomalous results. In its most general form for spherical particles, takes a form known as the Mie scattering equation and is dependent upon the size of the scattering particle, the refractive indices of both the particle and solvent, the angle of scattered light, and the wavelength of the scattered radiation (4). This form, however is very complicated and cumbersome. Fortunately, this very general form is necessary only i n cases of extremely large particles. For many macromolecular systems the Rayleigh Debye approximation to the Mie equation is adequate. This approximation of the intraparticle interference factor is shown i n Equation 7 (5). Downloaded by EMORY UNIV on April 21, 2015 | http://pubs.acs.org Publication Date: July 18, 1991 | doi: 10.1021/bk-1991-0467.ch018

x

C i

=

9[sin(qR)-qRcos(qR)] (qR)

2

6

As shown, the Rayleigh Debye approximation is dependent upon the scattering vector, q, and the particle radius, R. The hydrodynamic radius, R, is related to the diffusion coefficient through the Stokes-Einstein relationship (6). Equation 5 can be used to represent the diffusion coefficients i n terms of molecular weight. R

,_

kbT

kbTMf

βπηϋί

6πηα

^)

Where : Τ = Absolute temperature kb = Boltzman constant η = Solvent viscosity Theoretically with knowledge of g(t), α, β, and the scattering vector, q, the M W D may be extracted by inverting Integral Equation 6. The inversion of this equation to find the molecular weight distribution is not a simple task. Equations of this type appear frequently i n science and are known as Fredholm Integral Equations. The mathematical properties of this class of equations are ill-conditioned. Thus, without making any priori assumptions about the properties of the distribution there is no guarantee that a solution for the distribution, F(M), exists. In addition, i f there does exist a solution, the uniqueness of that solution is not assured. Algorithm Development Various methods have been used i n the past to extract distribution information (7-11). The approach we have chosen is a type of function fit method. This approach assumes a distribution type for the M W D and then attempts to find the best set of distribution parameters consistant with the scattering data. These "best" set of parameters consistant with th scattering data define the "best" F(M). The distribution type chosen was the generalized exponential (GEX) distribution. The G E X distribution function is shown i n Equation 8.


280

WATER-SOLUBLE POLYMERS

h =h Where : L =

B-l

m

L

(8)

exp

M-Mo MM-Mo

In its most general form it has five adjustable parameters. Three of these parameters have easily observed physical meaning, the maximum peak height (h ), the initial point at which the value of the function is defined to be zero (Mo), and the location of the peak maximum (M ). These are diagrammatically illustrated i n Figure 1. The other two parameters are shape parameters (A,B) and are functions of the inflection points of the distribution (12). The G E X distribution function was chosen for two reasons. First, it is an extremely versatile distribution function and can approximate a wide variety of distribution types (13). Also the G E X function has been shown to approximate MWDs fround from chromatagraphic analysis (14). The ill-conditioned nature of the equations which govern the relationship between the M W D and the scattered intensity correlation function can make the inversion of the integral equations very difficult i f not impossible. By placing appropriate restrictions on the search space many of the pitfalls associated with the inversion may be possibly avoided. The choice of the G E X may be viewed as just such a limitation of search space. In this case the choice of a G E X distribution function constrains the polymer M W D to be both positive and unimodal. The positivity constraint is completely valid because it is physically meaningless to discuss a negative number of molecules. The restriction that the distribution be unimodal is more limiting. However, most unfractionated water-soluble polymer systems are unimodal or near unimodal. Two other constraints may also be applied. The first is that MQ is zero. This means that any real polymer M W D has some molecules of very low molecular weight and at this low molecular weight limit the number of molecules approaches zero. This assumption simplifies many of the calculations involved and reduces the number of parameters required to be determined for the M W D function. The weight average molecular weight, M , is easily attainable through the use of low angle laser fight scattering. With the M information and the assumption that Mo is zero, it is possible to define M i n terms only the weight average molecular weight and the two shape parameters, A and B. This eliminates the need to solve for another parameter i n the M W D function. These constraints have reduced a four parameter problem to a two parameter problem. This lessens the complexity and time requirements for any regression technique which is utilized to find the remaining two parameters, A and B. The G E X function was also initially chosen because it has moments which are analytically integrable (15). The advantage of integrability is that it diminishes computer computation time. Numerical integration is a time consuming procedure. Initial attempts to integrate both the numerator and denominator of Equation 6 did not produce a solution i n closed form. Prior to the inclusion of the intrapartical interference factor the denominator was simply the second moment about zero of the G E X function and as such was analytically integrable without having to resort to numerical integration. When one includes C i , the intraparticle interference factor, the resulting integral equation is no longer analytically integrable. Thus a larger amount of computer time is required to calculate the value of g(x). m


M

w

w

M


METTILLE & HESTER



18.


281

282

WATER-SOLUBLE P O L Y M E R S

Series Use for Integration. Attempts were made to solve integral Equation 6 for the shape parameters, A and B , i n a manner that did not require lengthy numerical integration routines. One very powerful technique that is often applied is to approximate the kernel of the integral by a power series. The order of summation and integration may be exchanged. Thus, the integral may now be represented by a summation of integrals. It is also possible that each separate integral i n the series can be expressed i n closed form and that the series is convergent. First let us examine the numerator of the Equation 6. C is dependent upon the radius of the particle. The radius of the particle can be related to the diffusion equation by the Einstein-Stokes relationship (6). As previously explained, from a Mark-Houwink-like relationship and Equation 5, one may relate the particle radius to the molecular weight. By expanding the square into its subsequent terms, the following relationship is obtained. Downloaded by EMORY UNIV on April 21, 2015 | http://pubs.acs.org Publication Date: July 18, 1991 | doi: 10.1021/bk-1991-0467.ch018

t

Ci = — ^ [ s i n ( q R ) - q R S i n ( 2 q R ) + (qR) Coe (qR)] (qR) 2

2

2

(9)

Substituting the expanded Ci into the integral and exchanging the order of integration and summation provides a series of three integrals. Each is comprised of the product of a trigonometric function, an exponential term and the G E X distribution function. These trigonometric functions and the exponential function can be expanded in power series form as shown below. Sin (x) 2

Sin(2x)

yC-l)*^- * * ^ (2i)! i=l 1

^ i=0

(10)

(2i+l)!

Cos (x) = l - S i n ( x ) 2

2

2

(2i)! i=l

Substituting these summations for the appropriate functions i n Equation 9 is the next step i n determining a power series expansion for the numerator of Equation 6. The integral formed by the product of the cosine squared and exponential terms can subsequently be separated into two integrals. The original integral has now been separated into four integrals. Three of these integrals have kernels which are comprised of the product of two power series, the G E X function and the molecular weight raised to a power. The product of two power series may be transformed into a single power series by performing the multiplication on a term-by-term basis. Gathering terms of like power provides the power series coefficient for each term i n the series. The kernel of the remaining integral is composed of the product of only a single power series, the G E X function and the molecular weight raised to a power. When the order of integration and summation is exchanged the resulting equation is obtained for the numerator of Equation 6, I . N


18.

oo

lN=

oo

yC ,l

w N—

283



N

f °° M P ^ " ο

1

" ) F(M) 1

_1

d M - Κι Τ

CN,2 / " M ^ ^ ^ F O V I )

ο

M

H—

dM (

N

)

X B j ( " M W W - ^ d M - XCN,1 rM

Determination of Molecular-Weight Distribution of ... - ACS Publications

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