AFML-TR-67-121PART I

DfAR (aDo •5,3 _ PFFLCIAL FILE tLo ,

EVALUATION OF MOLECULAR WEIGHT FROMEQUILIBRIUM SEDIMENTATION

PART 1. COMPUTATION BASED ON EXPERIMENTALSCHLIEREN PLOT

MATATIAHU GEHATIA

TECHNICAL REPORT AFML-TR-67-121, PART I

JUNE 1967

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AFML-TR-67-121PART I

EVALUATION OF MOLECULAR WEIGHT FROMEQUILIBRIUM SEDIMENTATION

PART I. COMPUTATION BASED ON EXPERIMENTALSCHLIEREN PLOT

MATATIAHU GEHATIA

Distribution of this document is unlimited.

AFML-TR-67-121Part I

FOREWORD

This report was prepared by the Polymer Branch of the Nonmetallic Materials Division.The work was initiated under Project No. 7342, "Fundamental Research on MacromolecularMaterials and Lubrication Phenomena," Task No. 734203, "Fundamental Principles Deter-mining the Behavior of Macromolecules," and the researchwas conducted by Dr. M. T. Gehatia(MANP). The work was administered under the direction of the Air Force Materials Labora-tory, Research and Technology Division, Air Force Systems Command, Wright-PattersonAir Force Base, Ohio.

The report covers research conducted from January 1966 to February 1967. The manuscriptwas released by the author in February 1967 for publication as a technical report.

This technical report has been reviewed and is approved.

WILLIAM E. GIBBSChief, Polymer BranchNonmetallic Materials DivisionAir Force Materials Laboratory

ii

AFML-TR-67-121Part I

ABSTRACT

A new method has been developed to determine molecular weight (specifically, ratio2gsw /D, where w = angular velocity, s = sedimentation coefficient, D = diffusion coeffi-cient) from an experimental schlieren plot, dc/dr vs r, taken at equilibrium. Computation ofthe quantity g can be accomplished by applying the newly derived formula:

-L (I -C .. (g k) + I 2,r dcr 2

whereI2k(c) exp(- --•-gr )

is a constant, and c is concentration at distance r from center of rotation.

For a case of apolydisperse system, the following method of computation has been developed.The heterogeneous system was replaced by an equivalent N-component paucidisperse solute.

A set of N g(i) values was arbitrarily chosen, where each g(i) is related to component, i,present in the system. The following expressions were defined:

2 2 ,y. = 2/r. (rb - r

is a quantity measured at point j on the experimental plot; b and m denote bottom and menis-cus, respectively; and the function

a. = 1 M 2(i)2)/ exp(-1 - (g) r2 exp (i)-L)m 219 exp (2g r 2 rb )2xp -Mg

Since it is shown that

N 0)y]= a. cOi=a ji 0

where c is the initial concentration of component i, evaluation of aji and y. experimentallyleads directly to the determination of c(0 i. Therefore, by applying this method, one can

determine the entire distribution of molecular weight from equilibrium sedimentation, aswell as the average values.

iii

AFML-TR-67-121Part I

TABLE OF CONTENTS

SECTION PAGE

I INTRODUCTION 1

1. Equilibrium Sedimentation 1

2. Limitations of Equilibrium Sedimentation 2

3. Prior Attempts to Improve Equilibrium Sedimentation 2

II MATHEMATICAL ANALYSIS OF EQUILIBRIUM SEDIMENTATION 3

III POLYDISPERSE IDEAL SYSTEM 5

REFERENCES 10

V

AFML-TR-67-121

Part I

SECTION I

INTRODUCTION

1. EQUILIBRIUM SEDIMENTATION

Equilibrium sedimentation was first applied by Svedberg in his early works (Reference 1).Since, at equilibrium, concentration of polymers in solution does not depend upon time, theequation of ultracentrifuge reduces to an ordinary differential equation:

ac- 5 divJ =, (I)

where

c = polymer concentration

J = flux of solute.

Hence

IJ = IDcr + sW 2crt = const. (2)

D and s are coefficients of diffusion and sedimentation, respectively.

w = angular velocity of ultracentrifuge;

r = distance from center of rotation

c r= dc/dr.

Since at equilibrium no flux occurs, J = 0, the following formula holds:

dcd--r" -w s cr (3)

By introducing a new constant, g = sw 2/D, the Differential Equation (3) can be written in ashort form:

dc grc (4)dr

or:

d c r2 (5)2

Formula (5) leads to a well known method of computation (Reference 1) which determines gfrom any two points i and j of an experimental plot c vs r:

I 2 2A",(Ci/C = -g(r -r2 ). (6)

S2 i

AFML-TR-67-121Part I

By substituting g into the Svedberg Equation (Reference 2), molecular weight can be easilyevaluated:

R TgM " ~ (7)

( I -Vp )w2

where

M = molecular weight

R = gas constant

T = absolute temperature in °K

V = partial specific volume of solute

p = density of solvent.

2. LIMITATIONS OF EQUILIBRIUM SEDIMENTATION

The following are limiting factors in the application of equilibrium sedimentation.

a. The time of the experiment becomes exceedingly long if M > 250,000. If M > 1,000,000this method is not feasible. Therefore, equilibrium sedimentation can only be advantageouslyapplied if molecular weight is below 250,000.

b. Schlieren optics is the simplest and most generally used experimental technique. How-ever, this method provides a plot of dc/dr vs r. Since the conventional methods of computationrequire a plot of c vs r (see Equation (6)), the experimental schlieren curve has to be trans-formed into an integral one. Such a transformation is very laborious and not sufficientlyprecise.

c. A plot of c vs r can be obtained by applying interference optics. Unfortunately, thismethod cannot be used where high speed, elevated temperature, or corrosive solvents areinvolved.

d. Absorption optics also provide a plot c vs r. However, this technique requires a tediousprocedure of calibration and evaluation, and very good conditions for precise photographicprocessing.

e. In recent years a new technique has been under development (Reference 3) which alsoleads to an experimental plot of c vs r. This method is based upon direct scanning of anultracentrifugal cell by photosensitive elements. Unfortunately, this optical system is not yetcomplete and, in addition, necessary equipment is very expensive.

3. PRIOR ATTEMPTS TO IMPROVE EQUILIBRIUM SEDIMENTATION

Some attempts have been made to improve the method of equilibrium sedimentation and torelax its limitations.

Archibald (References 4 and 5) applied Equation (4) to evaluate g at the meniscus at anytime; e.g., before equilibrium. Since no transport can exist at the meniscus and bottom of theultracentrifugal cell, an equilibrium equation must always hold at this particular point. Someof his evaluations are of general use, and are mentioned subsequently. Experimental difficul-ties, however, arose which limited the application of Archibald's method.

2

AFML-TR-67-121Part I

Another method, "sedimentation at constant density gradient," was successfully developedby Meselson, et al. (Reference 6). This technique, which significantly shortens the time of anexperiment, requires the presence of a special binary or multicomponent solvent. Such asystem is supposed to give a constant density gradient and to cause sedimentation and flo-tation of polymer toward the middle of the cell. It is evident that each family of polymerscan be investigated in this manner only if an appropriate solvent system has already beendeveloped. Unfortunately, at present there are only two successful systems -- one for deoxy-ribonucleic acid and one for polystyrene.

Since molecular weights of polymers under current investigation in the Air Force MaterialsLaboratory do not exceed 250,000, conventional equilibrium sedimentation can be successfullyapplied. Therefore, the objective of this work is to improve the methods of computation anddetermine g directly from an experimental plot of cr vs r.

SECTION II

MATHEMATICAL ANALYSIS OF EQUILIBRIUM SEDIMENTATION

Integration of Differential Equation (4) leads to the following solution:

I 2

c = ke . (8)

where k is a constant

Hence:1 2 1 2 1 2---- gr2 -- yr ! __1(rb

k = c= Cm 1 - = Cb" (9)

m denotes meniscus and b bottom of the cell.

Following Equation (9) the concentration at any point can be given as a function of any chosenreference point (c q, r q):

Cq O L g ( r 2 - r 2 )

C q= C q ger2 q (10)

The first derivative of c is given by Equation (11):

Tg(r 2 -. rq)

Cr = grcq_ (01)

3

AFML-TR-67-121Part I

Further differentiation of Equation (11) leads to the following relationship:I g(r2_r 2 2I _g r2_q 2

Crr= gCq 9G 2 + g r Cq (9 (12)

By combining Equations (10), (11), and (12), one will obtain:

r + + gr)cr (13)Crr= rr

or:d QitiCr

-r gr (14)dr r

Equations (8) and (14) lead to the following expression:

tvCr = n (kgr) + - gr 2 (15)

and finally:

I 2S(cr/r) z v (gk) + - gr ,(6

Equation (16) can be applied for the computation of g. By plotting L-1 (cr/r) vs r 2 , one can

obtain the constant values g/2 and 124(gk). With the help of g and k the concentration c can beevaluated at any r; cm and cb (concentrations at meniscus and bottom respectively) can be

determined in the same manner.

There is another way to evaluate g by approximation in the vicinity of a chosen reference

point (c , r ). Since under these circumstances 1/2 g(r 2- r ) is a small number, Equa-q q q

tion (10) reduces to the following series:

Cr + 2 g2C( 2 2q1

r =Cq 2 C r- ) + q

A plot of (cr/r) vs (1/2) (r2 - r 2) provides values of the constants g, cq, and g2 c q, and makes

it possible to determine g and c q. Since no flux exists at the boundaries of an ultracentrifugal

cell, any equation describing equilibrium always holds at the meniscus. Therefore, the methodof computation expressed by Equation (16) can be applied in the vicinity of the meniscus, andsuch an application can be made at any time; e.g., even before equilibrium is established.Under these conditions g (and cm) have to be determined as limiting values for r ---- rm

4

AFML-TR-67-121

Part I

SECTION III

POLYDISPERSE IDEAL SYSTEM

Let us consider an ideal dilute polydisperse system composed of N-noninteracting com-ponents. At equilibrium each component attains its own equilibrium state. Therefore, thereexists a set of the following N equations:

(i) (i) (i) 2 (i)

D cr s w rc (18)

where (i) denotes component i.

Or in terms of g(i) = I(i) w' 2 /D(i):

(i) (i) (i)Cr g rc (19)

Solutions of the differential equations given by Formula (19) can be expressed either by

constants k(i) or by any reference concentration c(i)q

~ 12 (i) 2 M 2 2 .) i)2 r2)c k a 'g c -W e ( ) ( c)e g (r - rm) =i... (20)

Another relationship can be derived from the law of conservation of matter. Since the amount

of solute enclosed in a cell remains constant, the following equation holds at any time:

rb

A f cdr, (21)rm

where A is the amount of solute in the cell, and *j (r) is the area of the cylindrical surfacewithin the cell at the distance r from the center of rotation. In the case of a sectorial cellwith angle 4, and depth h:

* = 4rh (22)

and Equation (21) goes into the following formula:

rb

A : f h(Acrdr (23)rm

where A, h and 0 are constant.

If co denotes the initial homogeneous concentration, one can derive from Equation (23) thefollowing relationship:

Srb20 (,r- rm f 2 d 14

Mr

5

AFML-TR-67-121

Part I

Equations (21 through 24) were first derived by Archibald (References 4 and 5).

By substituting Equation (8) into (24) one can obtain:

2

2 k b I 2 2 '2co drr -b( -r e (25)

rmHence:

Co(r -rm2 ) = -2 -(cb-c) (26)

Equation (26) was derived in a different way by Van Holde and Baldwin (Reference 9).

The value (cb - cm) which expresses the increase of concentration between the meniscus

and bottom of the cell can be easily determined from experiment by integrating cr betweenall boundaries:

rb

Cb- f Crdr (27)rm

By substituting the value of (cb - cmr) into Equation (26) one can easily determine g.

In the following, Equation (26) will be applied to evaluate the distribution of a polydispersesystem. Since Equation (26) holds for any component of a heterogeneous system, one can write:

(0) 2[c - cC b cm (28)

Equations (20) and (11), if applied at the meniscus for component i, go into the followingexpressions:

M ~ ~ (i) - )(r 2 r2

b = C m (29)

and:Mi) M~ ~ -'M ( r 2 M icr = 9 r m Cm (30)

First, the quantity cb(i) will be eliminated. By combining Equations (28) and (29) the following

expression will be obtained:

M (i)2 [ ( 2 20i g'( 2 - ) [--g- (rb -,m)i ] c(i)31Co = 49 -- b mcm (31)b rm )

6

AFML-TR-67-121

By eliminating the quantity Cm I) from Equations (30) and (31), one may express Crr M as a

function of gi) and co(i)"

2 2 r 21 M gi2 1 g r1

r( rb -r c 2) (32)= 2 2~r 1_ ,(i)r2 0r 2 1. _ 92 rm

Formula (32) can be applied to any one of the N components present in the system underinvestigation. Therefore, this formula represents a set of N equations. By carrying out thesummation of all N equations of Formula (32) one will obtain the following relationship:

r gi 0L(i)2 0)r

I 2 2 N __M_ 2__M(i)C r ( r -r ) .. . . C (33)Sbm I (i) 2 I ]r2

49d' 2 rb - 4949 M

Nwhere c. C r can be directly measured experimentally.

i=1

The objective of this section is to evaluate the molecular weight distribution. Since theratio, P:

RT

( I - Vp) W 2

which appears in Equation (7) is an experimental constant, identical for all componentspresent in the system:

(i)M Pg , (35)

the distribution of molecular weight, M(i), can be replaced by a distribution of quantities gIt should be mentioned that evaluation of a distribution means an appropriate adjustment of the(gi) o(i))pairs (g, c 0 ) Such a definition gives one the freedom to arbitrarily choose sets of

quantities mentioned above. This freedom is utilized in the following way:

First, a set of N g(i) values will be chosen arbitrarily. However, one has to bear in mind,that the experimental average g values cannot be outside of this set. In addition such a setcannot be very narrow, and it must be extended over a wide range of possible g values.

7

AFML-TR-67-121Part I

Let us now denote by j a specifically chosen point (rj, c r.) on the experimental curve. For

J (i)any chosen pair (rJ, c r) and for any arbitrarily chosen value of , the following quantitiescan be determined: 2

y. 2C.j/ r (r r -rm (36)ri b m

and:

2 (1 ) 2

a.. : (37)ii _I (i)r2 (i) 2

20 rb ' g rm9 -0-

By introducing the definitions given in Equations (36) and (37), the expression cited in Equa-tion (33) can be transformed into summation:

NY a. ji. C , (38)

where yj and aji are known quantities.

Let us choose L pairs (r., c r) where L> N. Since Equation (38) holds for any one of such3 r.

defined L pairs, there exists a system of L simultaneous linear equations, with N unknown

quantities ci)N

: O .. C 0lYl i=I a..

(39)N

YL= a Li )0i=lI

The following definitions of two vectors and one matrix:

Y = (Y Y (40)

(I) (N)

0 0 ) , (41)

A ' (42)

8LI a L N

AFML-TR-67-121

Part I

make it possible to express the System indicated in Equation (39) in a compact form:

Y = AC (43)

If L = N one can immediately write the solution

C = B Y (44)

where B = A is the inverse matrix of A.

If L > N a method of least squares can be applied to evaluate vector C.

Since the elements of C are correlated to the appropriate set of g(i) values, the evaluationof vector C accomplishes the determination of the molecular weight distribution.

The theory derived in this work is an approximate one. It disregards any influence of virialcoefficient, interactions within a multicomponent system, compressibility in the ultracentrifuge,and other nonspecified factors.

9

AFML-TR-67-121Part I

REFERENCES

1. Svedberg, T. and Pedersen, K. 0., Die Ultrazentrifuge, Edited by T. Steinkopff, Drezden

and Leipzig, 1939, pp. 5-9.

2. Ibid, p. 6.

3. Williams, J. W., Ultracentrifugal Analysis in Theory and Experiment, Academic Press,New York and London (1963), pp. 185-200.

4. Archibald, W. J., J. Appl. Phys, 18, 362 (1947).

5. Archibald, W. J., J. Phys. and Colloid Chem, 51, 1204 (1947).

6. Meselson, M., Stahl, F. W., and Vinograd, J.,Proc. Natl. Acad. Sci. U.S., 43, 581 (1957).

7. Lansing, W. D. and Kraemer, E. 0., J. Am. Chem. Soc., 57, 1369 (1935).

8. Yphantis, D. A., Biochemistry, 3,297 (1964).

9. Van Holde, K. E. and Baldwin, R. L., J. Phys. Chem, 62, 734 (1958).

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3. REPORT TITLE

EVALUATION OF MOLECULAR WEIGHT FROM EQUILIBRIUM SEDIMENTATIONPART I. COMPUTATION BASED ON EXPERIMENTAL SCHUIEREN PLOT

4. DESCRIPTIVE NOTES (Type of report and inclusive dates)

Summary Report (January 1966 to February 1967)S. AUTHOR(S) (Loot name. first name, initial)

Gehatia, M. T.

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May 1967 15[ 98a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S)

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13. ABSTRACT

A new method has been developed to determine molecular weight (specifically, ratiog= sW 2 /D, where w = angular velocity, s = sedimentation coefficient, D = diffusioncoefficient) from an experimental schlieren plot, dc/dr vs r, taken at equilibrium. Com-putation of the quantity g can be accomplished by applying the newly derived formula:

-L(!~ JL 2'&v (gk) + _Ig 2r dr 2

wh reI 2k (c)exp(---Igr )2

is a constant, and c is concentration at distance r from center of rotation.

For a case of a polydisperse system, the following method of computation has beendeveloped. The heterogeneous system was replaced by an equivalent N-componentpaucidisperse solute. A set of N g(i) values was arbitrarily chosen, where each g(i) isrelated to component, i, present in the system. The following expressions were defined:

2 2

y. = 2/r (r 2 r 2

b m

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is a quantity measured at point j on the experimental plot; b and m denote bottom andmeniscus, respectively; and the function

aj, [ g~i)2 (i2) [exp ex(-gg r,)

a.. U ) 2 exp (Lg 9)r 2j / (- )rb2)- 1i)2)

Since it is shown that

Ny. a..c

i I ii 0 i

where c 0 (i) is the initial concentration of component i, evaluation of a.. and y. experimen-

tally leads directly to the determination of c 0(M. Therefore, by applying this method, onecan determine the entire distribution of molecular weight from equilibrium sedimentation,as well as the average values.