A Study of Molecular Diffusion in Polymer Solutions by a Microinterferometric Method

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Masters Theses Student Theses and Dissertations

1966

A study of molecular diffusion in polymer solutions by a microinterferometric method

Girish T. Dalal

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Recommended Citation Dalal, Girish T., "A study of molecular diffusion in polymer solutions by a microinterferometric method" (1966). Masters Theses. 5757. https://scholarsmine.mst.edu/masters_theses/5757

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ABSTRACT

A microinterferometric method was used to study the effect of solute concentration on the diffusion coefficient in polymer solutions and also to d etermine the effect of polymer concentration on the dif fusion coefficient. The polyacrylonitrile-dimethylforruamide system was used as a means of dete rmining the accuracy of the experimental apparatus and procedure. The non-ionic, water soluble polymer, hydrox~thyl cellulose (commercially known as Na.trosol), was used to study this effect with urea and D-glucose as the solutes.

The results obtained for the polyacrylonitrile-dimethylform amide s ystem were in close agreement with the results obtained by

Secor, which meant that the experimental techniqu.e is accurate enou.gh to give r eproducible data. The differential diffusion coefficient in creased with the increase in solute concentration in all cases. The effect of solute concentration on the differential diffusion coefficient was found to be s imilar for various polymer concentrations and dif ferent solutes. The integral diffusion coefficient temained almost constant for the concentration range of the solutes used in this work.

No effect of polymer concentration on the integral diffusion coefficient could be deduced from the experimental data. iii

TABLE OF CONTENTS

Page

LIST OF FIGURES v LIST OF TABLES viii NOMENCLATURE ix

I. INTRODUCTION 1

II. LITERATURE REVIEW 4

Theory Of Diffusion 4 Laws of Diffusion 5 Techniques for the Measurement of Diffusivity • 10 Steady State Diffusion. • • • •••• 1 1 Free Diffusion ..•••.•• 13 Restricted Diffusion . .••• . . . . 17 Optical Techniques .. 17 Schlieren Methods ...•••• 17 Interferometric Methods ...•• ...... 19 The Optical Wedge Technique ••• 21

III. EXPERIMENTAL 25

Mate rials . ... 25 Expe rimental Apparatus ..... 2 5 Preparation Of Optical Wedge 28 Microscope ...... · . · · · · · · · · · 31 Concentration Measurement Apparatus . . . . 31 Expe rimental Procedure...... 34 Procedure For Obtaining Experimental Interference Pattern.• ...... 34 Measurement Of Concentration ...... 35 Computation Technique ...... 39 Evaluation Of The Integral ...... 40 Evaluation Of The Concentration Gradient 44 Data and Results . 46

IV. DISCUSSION ...... 48

Effect Of Solute Concentration .•. 48 Effect Of Polymer Concentration On Integral Diffusi vities • • •••.•• . ••••••..•.••• . . . 63 Experimental Equipment • •••••••••• • ••• 65 Prediction of Diffusi vity of a Solute in Very Dilute Solutions • • • • • • • • • • • • • • • • • • • • • • 6 7 iv

Page

v. CONCLUSIONS...••• . . . . 68 VI. RECOMMENDATIONS . . . . 70

VII. APPENDICES ...... 71

Appendix A - Derivation Of Formula For Diffusion Coefficient

Appendix B - Materials .....•. 77

Appendix C - Data and Results •. 78

Appendix D - Computer Programs 92

Appendix E - Integration of Concentration distance Curves •••.•... 107

Appendix F - Evaluation Of Concentration Gradient ...... 112

Appendix G - Measurement of Interference Pattern ••. •.• .• 11 5

VIII. BIBLIOGRAPHY .•.• ...... 117

IX. ACKNOWLEDGMENT • ...... 119

X. VITA .•...... 120 v

LIST OF FIGURES

Figure Page

2. 1 Schematic Diffusion Cell for Steady-state Diffusion 12

2. 2 Free Diffusion. The Initial Arrangement of Solutions in the Cell at t = 0, and Concentration Gradie nt Curves at a later time , t1 14

2.3 Restricted Diffusion. Concentration Curve for a time, t2, after the concentrations have changed appreciably at the ends of the Cell 18

2.4 The Optical Wedge 22

3. 1 Schematic Drawing of Experimental Apparatus 27

Experimental Apparatus Assembly 29

3.3 Preparation of the Optical Wedge for Diffusion Measurements 30

3.4 Microscope used for the Experiment 32

3. 5 Abbe-Spencer Refrac tomete r use d for M easure ment of Refractive Inde x as a Function of Con centration 33

3. 6 Light-inte rference P atterns Observed During a D iffu sion Experiment 36

3 .7 Experimental Interference Patte rn at t = 11 seconds, for System 4. 37

3.8 Experimental Interference Patte rn at t = 15 seconds , fo r System 8. 38

3.9 Refractive Inde x (Sodium D-line ) of Solutions of Polyacrylonitrile in Dimethylformamid e at 25°C 41

3. 10 Expe rimental Concentration Profile for Poly acrylonitrile-Dimethylformamide System at t = 90 seconds 42 vi

Figure Page c 3. 11 Evaluation of .Jo(' x dC 45 3. 12 Experimental Concentration Gradient as a Function of Distance for Polyacrylonitrile Dimethylformarnide System at t = 90 Seconds 47

4. 1 Effect of Concentration on the Diffusion Coefficient for Solutions of Polyacrylonitrile in Dimethyl formamide at 77°F, with t = 90 seconds 49

4. 2 Diffu~ivity versus concentration for 0. 8% Natrosol in Water with Urea as Solute, at 77°F, with t = 12 seconds. 51

4. 3 Diffusivity versus Concentration for 0. 8% Natrosol in Water with D-Glucose as Solute, at 770F, with t = 15 seconds. 52

4.4 Diffusivity versus Concentration for 1. 0% Natrosol in Wa ter with Urea as Solute, at 77°F, with t = 15 54

4.5 Diffusivity versus Concentration for 1. 0% Natrosol in wate r with D-Glucose as Solute, at 77°F, with t = 15 s e conds 55

4. 6 Diffusivity versus Concentration for 1. 88% Natrosol in Wate r with Urea as Solute, at 77°F, with t = 10 seconds 56

4.7 Diffusivity versus Concentration for 1. 88% Natrosol in Water with D-G1uc ose as Solute , at 77°F, with t = 30 seconds 57

4.8 Diffusivity versus Concentration for 2. 5% Natrosol in Water with Urea as Solute, at 77°F, with t = 11 Seconds 58

4.9 Diffusivity versus Concentration for 2. 5% Natrosol in Wa t e r with D-Glucose as Solute, at 77°F, with t = 1 7 S e conds 59

4. 10 Diffusivity versus Concentration for 3 . 51% Natrosol in Water with Urea as Solute, at 77°F, with t = 10 seconds 60 vii

Figure Page

4. 11 Diffusivity versus Concentration for 3. 51% Natrosol in Water with D-Glucose as Solute, at 77°F, with t = 16 seconds. 61

4. lZ Integral Diffusivity as a Function of Polymer Concentration for Natrosol-Water Systems. 63

E. 1 Evaluation of De from a Concentration-distance Curve Using Equation (Z. ZZ). 109

E. Z Evaluation of the .Jor.~ X dC. 110 F. 1 Concentration Profile for System 3, at t = 16 Seconds. 114b

G. 1 A Sketch of Experimental Interference Pattern, showing how to measure the Fringe Pattern. 116 viii

LIST OF TABLES

Table Page

3. 1 Polymeric Systems Used for the Study. 26 c. 1 Polymeric Systems Used for the Study. 79 c. 2 Data and Results for System 1. 80 c. 3 Data and Results for System 2. 81

C.4 Data and Results for System 3. 82 c. 5 Data and Results for System 4 . 83 c. 6 Data and Results for System 5. 84 c. 7 Data and Results for System 6. 85 c . 8 Data and Results for System 7. 86 c. 9 Data and Results for System 8. 87 c. 10 Data and Results for System 9. 88 c. 11 Data and Results for System 10. 89

c. 12 Data and Resulty for System 11. 90 c. 13 Values of Integrated Diffusivities, D, for the Systems Studied. 91

F. 1 Comparison of Concentration Gradients Obtained by Two Different Methods. 114

F. 2 Average Absolute Percentage Deviation of Observed Concentrations from Those Predicted by Gompertz Equation and Davis' Method. 114a ix

NOMENCLATURE

A = area under the curve of concentration versus distance, (gm./100 cc. soln. )em.

a = intercept of straight line that results when (x - x 1 ) / § is plotted against x.

a' = cross-sectional area, sq. em. a 1, a 2 , a 3 = constants in Gompertz equation.

b = slope of straight line that results when (x - x 1 )/ § is plotted against x .

C = concentration of solute, gm. Icc.

C 0 = initial concentration of solute, gm. Icc.

c' = concentration of solute at original interface, gm. Icc.

c = concentration of solute, gm./100 cc.

D = molecular diffusion coefficient of a solute, sq. em. /sec.

D = integral {average) diffusion coefficient of a solute, sq. em. I sec.

Dnn = diffusion coefficient of a solute in a non-Ne wtonian fluid, sq. cm. / sec.

d = distance between adjacent bright fringes, em.

h = h eight, em.

M = molecular weight.

N = diffusion flux, gm. I sq. em. sec.

-' N = diffusion flux in vector notation, gm. /sq. em. sec.

n = refractive index of solution.

no = refractive index of solvent. X

r = x/2 ~~ Boltzmann's variable, ern. I (sec. ) 112

T = absolute temperature, °K.

t = time, seconds.

V = volume, cu. ern.

w = weight fraction of solute.

X = fractional area occupied, sq. ern.

x = coordinate, corresponds to distance, ern.

y = coordinate, corresponds to concentr ation of solute, gm. / c c .

y I = small angle of deflection, rad.

z = coordinate, corresponds to depth of the medium, em.

Subscripts

A = solute A.

A' = rese r vo1r. A ' .

B = solvent B. B ' = reservo1r. B' . c = at some concentration C.

Co = at initial concentration C 0 • cp = prope rties or parameters of the continuous phase portion of the non-Newtonian fluid.

i = i th sol ute.

t = at time t.

x = direction x.

y = direction y. xi

z = direction z.

1, 2 = positions 1 and 2 in a system.

Greek Letters and Other Symbols

a = intercept, Gompertz equation.

A = change in property.-

9 = wedge angle, minutes or rad.

~ = wave length, mi.J. or em.

~0 = wave length in vacuum, m.,a. or em ..

1..1. = viscosity of fluid, cps.

; = number of neighbors of the diffusing molecule which are sheared during its advancing a distance equal to one lattice parameter.

V = vector notation.

~ = an association factor for solvent •

.~ = symbol used for sigmoid function. I. INTRODUCTION

Many operations in the chemical process industry involve the transfer of mass from one phase to another. The rate of mass trans fer must be considered in the design of equipment where two phases are in contact and mass is interchanged between the phases. There fore, an under standing of the diffusion process in various cases is vital in the design of many types of chemical process equipment, particularly in the fields of extraction, absorption and reactor design.

However, the knowledge in the area of diffusion process and the pre diction of the diffusion coefficients is not adequate at present.

The spontaneous migration of various substances that leads to uniform concentration in a single phase or a multiphase system is the phenomenon of diffusion. This spontaneous approach to uniformity on a molecular scale in a stagnant liquid, and that which may be superimposed upon the process as a result of bulk motion of a fluid, is called the .molecular diffusion. In spite of a sound explanation of the mechanism of molecular diffusion (that is, gross transport equations), theories of the structures of liquids and their molecular transport char acteristics are not yet adequately developed to permit rigorous treatment in predicting molecular diffusivity.

The Wilke and Chang (29, 30) correlation has been recommended for predicting the molecular diffusivity in dilute solutions of nonelec trolyte&. Powell, Roserveare and Eyring (18) have shown that in 2

concentrated solutions the diffusivity will be a function of the activity of the solute in solution.

For high viscosity liquids, correlations for predicting diffusion coefficients are not yet available. Among the high viscosity liquids, polymer solutions are often encountered in the chemical process industry. Mass transfer operations involving polymer solutions are frequently controlled by molecular diffusion. However, the values of the diffusion coefficients in polymeric systems are usually lacking.

The purpose of this investigation is to study the effect of solute concentration on the diffusion coefficient in polymer solutions and also to determine the effect of polymer concentration on the dif fusion coefficient. The non-ionic, water soluble polymer, hydroxy ethyl cellulose (commerically known as Natrosol), was used to study this effect with urea or D-glucose as the solute.

Diffusion coefficients of the polymeric systems were deter mined by a microinterferometric method. Refractive index method was used to measure concentrations. The experimental technique used was similar to the one used by Secor and others ( 1, 3, 7, 16, 17,

20, 22). This technique is very well suited to the low diffusion rates encountered in high viscosity liquids. According to the theory of

Brownian movement (9, 10) the average of the square of the distance over which a particle is randomly wandering is proportional to the 3

time during which it was traveling. In other words, the time varies inversely with the square of the distance traveled. Therefore, if the diffusion process is observed over a small distance, it shouJd be possible to reduce the time required for the observation by the square of the magnification factor. Thus, if the diffusion measurement is carried out under a microscope with a magnification factor of 10, the time scale is reduced by 100.

Another feature of the micro-diffusion technique is that dif fusional processes which normally take pl ace over microscopic dis t a nces, e. g., those associate d with synthetic fiber formation and with living cells, can be readily observed. Other advantages of the microinterferometric method over conventional and other interfero metric methods are that the former equipment is less expensive, and

only micr0gram amounts of solution are required. It is estimated that the method employed has a probable error of from + 5 to + 10

per cent (22). 4

II. LITERATURE REVIEW

The literature published on the subject of mass transfer is voluminous and much of it is not pertinent to this work. Therefore, only a brief discussion of the basic fundamentals of diffusion will be reviewed. This discussion is divided into two parts: first, the theor y of molecular diffusion, and second, the techniques used in the experi mental work.

Theory of Diffusion

Considering a single or multiphase system not at equilibrium, it is observed that it is spontaneously altered, ultimately reaching a state of equilibr ium where there no longer exists thermal or concen tration gradients within the system (28). This phenomenon, by which uniformity of concentration is achieved, is known as "diffusion. "

There are two types of diffusion processes, namely "molec ular diffusion" and "eddy diffusion. "

Molecular diffusion is prevalent in cases in which the fluids are stagnant and also in the case of fluids moving in streamline or laminar flow. Molecular diffusion is governed by the nature of the fluids involved and not by the type of flow patte rn. At higher ·veloc ities, the flow of fluids becomes turbulent. This t ype of flow is characterized by swirls of eddies. In this type of flow, mass is 5

transferred not only by molecular diffusion, but also by the eddies; hence the name, "eddy'' diffusion.

Eddy diffusion is treated in detail elsewhere (28). Only

"molecular diffusion" will be treated in detail in this discussion.

Laws of Diffusion: A basic law for one dimensional diffusion was proposed by Fick (11). It is the simplest of the experimental mass flux relations and serves to define the diffusion coefficient, D.*

For diffusion at constant temperature and pressure in two-component systems which show no change in volume on mixing, Fick' s first law for one dimensional transport of the solute is

N · =- D --a C·)1 ( 2. 1) 1x ( a X t

where subscript i denotes the solute.

This equation shows that at any time, t, and position, x, in

the x direction, the flux, N, of solute is directly proportional to the

first power of solute concentration gradient (a Ci/ 8x)t. · From here

on, the su?script, t, will not be written, but it should be understood

that all partial derivatives of concentration, Ci, with respect to

distance, x, are always taken a t some time, t. The flux Nix, is

positive ~n the direction of increasing x and can be defined as the

amount of solute crossing a unit area perpendicular to the direction

of the flow per unit time. The negative sign in the equation (2. 1)

*The symbols are explained in the nomenclature. 6

arises because diffusion occurs in the direction opposite to that of increasing concentration (6).

For liquid diffusion in a three dimensional system, equation

(2. 1) may be rewritten as

_,.

N·l =

ac = n ( ac + + ac) ax 8y az

= - D ( V C) (2. 2)

Fick's second law for three-dimensional system may be

written as,

ac = _ ( aNix + 8Niz) 8t ax + az (2. 3)

The subscript i will now be dropped for convenience. In some cases,

e. g. diffusion in dilute solutions, D, can be taken as being reasonably

constant. For such cases Nx, NY, and Nz are given by equation (2.1) ·

in each direction, and equation {2. 3) becomes

(2. 4)

reducing simply to

a·c (2. 5) at = D [::~ J 7

if there is a concentration gradient only in the x direction. Equations

(2. 4) and (2. 5) are known as Fick's second law.

Several empirical methods for estimating the value of D in dilute Newtonian solutions of non-electrolytes are presented by Reid and Sherwood (19), and the corr~lation of Wilkie and Chang (29, 30) is recommended for general purposes. From the theoretical indi- cation that the quantity DAB P.BIT should be a function of molal volume, Wilke and Chang correlated the available data to within about

10 per cent by the relation

0 AB = (2. 6)

where DAB = diffusivity of A in dilute solution in solvent B, sq. cm./sec.

MB = molecular weight of the solvent.

T ::: temperature, oK.

I p. ::: viscosity of the solution, centipoises.

VA = solute molal volume at the normal boiling point, eel g. mole.

Cf ::: an association factor for the solvent.

::: z. 6 for water as solvent.

This correlation fails to handle systems which are very viscous, for example, solutions of high polymers, and also systems where com- plexes are formed (28). At infinite dilution the diffusion coefficient 8

of an electrolyte can be related to the ionic mobilities as shown by

Nernst (15 ). These have been summarized by Treybal (28).

Clough, Read and Metzner and Behn (5), have developed a

theoretical approach for the prediction of diffusivities in viscous

and non-Newtonian fluids. The equation they obtained is

0 nn xcp __l.l._ = -~- (2. 7) D ~cp l-'-cp

where D = diffusivity of a solute in a Newtonian fluid, em 2 / sec. ' 2 D diffusivity of a solute in a non-Newtonian fluid, cm /sec. nn = subscript cp = properties or parameters of the continuous phase portion of the non- Newtonian fluid.

Xcp = fractional area occupied by the continuous phase.

lJ. = viscosity of the fluid.

~ = number of neighbors of the diffusing molecule which are sheared during its advancing a distance equal to one lattice parameter.

The Eyring rate equation for mass and momentum transfer was ex-

tended to include diffusion in slurries and in non- Newtonian fluids.

Both the Wilke-Chang and the Metzner relations are for systems in

which the solute concentration is very low and hence both D and D nn are assumed to be independent of solute concentration.

In many systems, e. g . in high polymer solutions, Dis

frequently very dependent on solution concentration. In such a case,

D varies from point to point in the solution and equation (Z. 3) 9

then becomes

ac a 0 ac ) + ~ ( 0 ac) + a (2. 8) at = ax ( ax ay ay az where D may be a function of x, y, z, and C. Thus for a case of one dimensional diffusion when the diffusion coefficient, D, is a function

of concentration, C, equation (2. 8) may be written as

ac a (2. 9) at = ax

In 1894 Boltzmann (4) showed that for certain boundary con-

ditions, provided Dis a function of C only, C may be expressed in 1/2 ten:ns of a new single variable, x/2t • Equation (2. 9) may, there-

fore, be reduced to an ordinary differential equation by the introduction

of a new variable, r, where

X (2. 10) r = 2,..fi

Thus we have

ac 1 = (2. 11) ax 2,.ft ( :~)

and

ac X = (2. 12) at - 4 t3/2 ( :~) 10

and hence

a a ( D ;;) = dC ) ax ax ( 2~ dr

1 d = {2 . 13 ) 4t dr ( ri :~ ) so that finally (2. 9) becomes

-2 r dC = d ( 0 dC) (2. 14) dr ar dr

The appl ication of this equation for the determination of D as a function of concentration will be described later in this thesis.

Techniques for the Meas urement of Diffusivity

Though Fick' s first law, equation (2. 1), defines the diffusion ceofficient in terms of the flux of solute and its concentration g r adient, it is not possible to calc ulate D by direct measurement of these two quantities. It is possible to measure concentration gradient, but the flux of solute can not usually be measured. The steady state method discussed below, is the closest approac h to a direct deter - mination of the flux. All other methods of determining D utilize integrated forms of Fick' s second law, equation (2. 5). For systems in which Dis very concentr ation dependent, e. g. in high polymeric systems, integrated forms of equation (2. 9) are generally used to calculate D. 11

Steady-State Diffusion: ( 13 ). This is probably the simplest experimental arrangement for determining the value of D. Figure

2. 1 illustrates the essential characteristics of this method. Diffusion occurs in a narrow vertical tube of uniform cross-sectional area, ~.

1 and length, h, between two reservoirs of solution, A and B • The

I . solute concentration CB' in B is greater than the solute concentration

I CA. in A. VA. and VB', the volumes of the two reservoirs, are known and are both equal to V. After sufficient time has elapsed, the concentration gradient becomes constant throughout the tube, and we get

( 2. 15)

provided D is independent of concentration. It is assumed in equation

(2. 15) that CA_ changes very slightly and can be considered constant for purposes of calculating the gradient. The flux N, may be calculated by measuring the concentration change ~C , in C A occurring in a small time interval At. For this hypothetical apparatus, the diffusion coefficient D is

N (2. 16) D = -Wx)

A practical form of steady-state diffusion apparatus is the diaphragm cell designed by Stokes (23, 24). A procedure for calculating D for concentration dependent systems was obtained by Gordon

(12) and extended by Stokes (24). 12

r I A

h Tx 'Y

Figure 2. 1. Schemati c Diffusion Cell for Stead y state Diffusion. 13

Free Diffusion (13): Free diffusion will occur when two phases having different concentrations are brought in direct contact with each other. The physical arrangement of a free diffusion ex periment is schematically illustrated in Figure 2. 2. Figure 2. 2 a illustrates a typical rectangular diffusion cell of uniform cross- section at the start of the experiment with only one solute in the system.

1 Solution B is more concentrated than solution X (CB•> CP!_). At time t = 0, .a sharp initial boundary is formed at level x = 0 between the two phases. After diffusion has proceeded for a time t = t 1 the original boundary no longer exists, and the curve of C versus x look~ similar to the one shown in Figure 2. 2 . b . The corresponding con centration gradient, (aC/ ax) versus x, curve is illustrated in Figure

2 . 2 c. Throughout free diffusion, the curve continues to flatten and spread with time, while keeping the same area. This continues until the concentrations begin to change measurably at the ends of the cell.

The experiment essentially consists of the measurement of either the concentration distribution and/ or the concentration gradient distribution, as a function of time. The diffusion coefficient, D, can be calculated as a function of concentration from the shape of either or both curves at a given time.

In free diffusion for two-component systems, the equation for calculating D from measurements of solute concentration, C, and/ or concentration gradient, (8C/ ax), is derived by integrating t = 0 t = t 1 t = t 1 ~------. -- I

C_A.

CB• Tx

CA_ CB'

ac ---'>- c~ ax (a) (b ) (c)

Figure 2. 2. Free Diffusion . The initial arrange m e nt of solutions in the cell at t 0 , and concentration gradie nt curves a t a late r time, t = 1

II>- 15

F i ck' s second law (equations 2. 5 or 2. 9) subject to proper boundary conditions. To simplify this integration, Boltzmann' s (4 ) variable is used. Boltzmann pointed out that in free diffusion of th e variabl es x and t always occur in the ratio x f t l / 2• This rel ation is true wheth er or not D depends on concentr ation.

If two infinite media are brought together at t = 0 , t he d iffusion

coefficient and its dependence on concentration can readily be deduced from the concentrati on distribution observed at some s u bsequent time. The conditions of the experiment are

c = 0, t = 0, X< 0 (2. 17)

0 ( 2. 18) c = c 0, t = o, x>

c = 0 t = t x -- 00 (2. 19)

C = C t = t x - oo ( 2. 20) 0

covco c = t - 00 all x' s ( 2. 21) +v vo co

where C is the conce ntration of the component in which we are

intere ste d, and x = 0 is the position of the initial interface between

the two phases at time t = 0. C 0 is the initial c.oncentration of the

V and V represent the volume s of the solute in the solution. 0 co

solvent and of the solution with initial solute concentration, r espectively.

A s suming that ther e is no overall change of volume on m ixing a nd that

D i s a funct ion of C only, w h ere C i s measur e d i n m a ss per unit v ol ume, 16

we can use the Boltzl?ann variable r (equation 2 . 10) to obtain

_ 2 r dC = ..i_ ( D dC) {2. 14) dr · dr dr

On applying boundary conditions and subsequent rearrangement, a method for determining D as a function of C for two component sys- terns was devised by Boltzmann (4), who derived the relation

(Beckmann and Rosenburg {2) )

xdC {2. 22)

Complete derivation of the equation {2. 22) is shown in Appendix

The c' axis, i.e. locus of X= o, is defined by the requirement that c' xdC (2. 23) -s X dC 0

This approach has the advantage that no functional dependence of Don Cis assumed. Numerical values giving this dependence are obtained directly from equation (2. 22).

Free diffusion is commonly studied by using optical methods.

Either the refractive index and/ or refractive index gradient of the medium in the diffusion cell is measured as a function of distance.

Hence, an accurate graphical or analytic representation of refractive index as a function of concentration is needed. 17

Restricted Diffusion ( 13): When a free diffusion experiment is continued for sufficient time, the solute concentration, C, begins to change appreciably at the ends of the cell. At this point free diffusion ceases, and the experiment is said to have entered the stage of "restricted diffusion. " The curve of C versus x for some time, t , during restricted diffusion is shown in Figure 2. 3. For 2 restricted diffusion the concentration is no longer a function of a

1 2 single variable x/t 1 .

Optical Techniques: There are several optical methods for the measurement of refractive index distribution and/ or refractive index gradient, for the subsequent determination of the diffusion

coefficient as a function of concentration (refractive index being a function of concentration). All the methods, except the optical wedge

technique which has been used in the presen,t work, will be mentioned

very briefly.

Schlieren Methods: "Schliere" means optical inhomogeneity.

The methods are based on curved light paths due to a refractive index

gradient. The resulting small angle of light deflection can be increased

in various manners, resulting in different techniques.

I 1 an y~ (2. 24) n ay 18

t = t 2

Figure 2. 3 . Res.tricte d Diffusion. Concentration Curve for t ime, tz, after the Concentrations have changed appreciably at the ends of the Cell. 19

where

y = small angle of deflection.

n = refractive index of medium.

x and y = coordinates.

Other Schlieren methods are the Shadow-graph, the Toepler and the Philpot-Svenson methods.

The Lamm-scale method measures the refractive index gradient as a function of height in the cell. This method has been used less as interferometric methods have been developed.

The Interferometric methods: These methods are based on a phase difference due to different propagation rates. Phase difference is detected by interference of one wave with another wave, which may be a reference beam or another part of the same beam. Phase change

Az (numbe r of wave lengths) for a one-dimensional index field is given by the relation

d' Az = An (2. 25) "-o where

d' = cell dimension.

"-0 = wave l ength in vacuum.

An= change in refractive index. zo

The Gouy interfere nce method is particularly useful for the study of fr~e diffusion. The interference fringes formed by the optical system of this method permit a precise determination of the shape of the refractive index gradient curve. However, the curve should be symmetrical and have only one maximum.

The Rayleigh interference method produces fringes which have a shape directly proportional to n versus x in the cell. There fore , from a single experiment on a two-component system, one may obtain the diffusion coefficient and also obtain information about its dependence on concentration. All analyses of data from this method have to be done in terms of the n versus x curve, since the refractive index gradient cannot be calculated from the integral curve without appreciable loss of accuracy. However, Svensson (25, Z6) and

Svensson, Forsberg and Lindstrom (Z7) have developed a modification of this optical system which automatically performs the equivalent of a numerical differentiation of the n versus x curve. Its accuracy is considerably greater than that of the Schlieren and Lamm scale methods but not quite as much as the Gouy and Rayleigh methods.

The Jamin interference method also provides data for the n versus x curve. There are several other optical methods, each having its own advantage and disadvantage; however, it is not possible to review all of them here. A review of different techniques used in the determination of diffusion coefficients, which are of potential interest for the study of heterogeneous mass transfer in solutions, is 21

given by Muller ( 14). An extensive list of references is also provided.

The Optical Wedge Technique: The microinterferometric method used in this work was adapted from the method used for study ing concentration profiles around growing crystals (1, 3 ), for measure ments of local viscosities (17), and for the study of diffusion {7, 16,

20). It has also b een used successfully by Secor (22) to study the effect of concentrations on diffusion coefficients in polymer solutions.

The diffusion cell consists of a wedge made from two partially metallized, plate glass microscope slides separated by a thin spacer at one end. In order to produce sharper fringes, the slides are partially metallized on one side so that they become partly trans mitting and partly reflecting. The monochromatic light passes through the wedge, producing interference fringes that were viewed and photographed through a microscope.

The principle on which the optical w edge works is illustra ted in Figure 2. 4 . A ray of monochromatic light AB, enters the wedge at point A and is partly transmitted and partly r eflected at point B.

The r eflected ray travels along the path BCD. When the difference in the lengths of the optical paths of the reflected and transmitted

rays is an integral numbe r of wave lengths, reinforcem ent occurs, and ~ bright fringe is observed. Between the bright fringes formed

by this reinforcement, where the paths of the two rays differ by an 22

d = 2n9 d = Distance between adjace nt bright fringes. n = Refractive Index of medium in wedge. A. = Wavelength of light. e = Wedge angle.

Figure 2. 4. The Optical Wedge 23

odd number of half wave lengths, destructive interference occurs and a dark fringe is observed. When a material of constant refractive index is in the wedge, the fringes are paralle l and are equally spaced at a distance, D, given by

d = 'f.../2 n 9 (2. 26)

The wedge angle shown in Figure 2-4 is greatly exaggerated. Actually, it is very small - about 20 to 40 minutes of an arc. The theory of the optical wedge has been treated in greater detail by Searle {21).

The interference pattern has two important characteristics upon which the experimental technique depend:

1. Along any line drawn parallel to the original interface, the dista nce between any two adjacent fringes is constant; and

2. Along any line ~rawn perpendicular to the original inter face, the change in refractive index between any two adjacent fringes is constant.

From the interference pattern photographed at some time, t, and knowing the r efractive index as a function of concentration, the curve of C versus x can be obtained. From this curve, the concen tration gradient curve can be obtained and subsequently the diffusion coefficient, D, can be calculated as a function of concentration, C, 24

by the relation c -So x dC zt (:;)c (2. 22}

The experimental apparatus, procedure a~d computation technique are explained in details in the next section.

The major advantages of using the microinterferometric method (as is done in this thesis} rather than the interferometric method are (1) the former equipment is less expensive; (2) the time required to obtain the data is less; (3) only microgram amounts of solution are required; (4) the diffusion cell is very thin and so convection currents are not important and hence good temperature con- trol as required in macroscopic apparatus is not essential here.

This is also true as the time required to obtain the data is very small. 25

III. EXPERIMENTAL

The purpose of this investigation was to study the effect of solute concentration on the diffusion coefficient in polymer solutions and also to determine the effect of polymer concentration on the diffusion coefficient. The microinterferometric method was used to measure the diffusion coefficient as a function of concentration.

Concentrations were measured ·as a function of refractive index.

This section is essentially divided into five parts: The first includes information about the material used. The second describes the apparatus used for the experiment. The procedure followed is ex plained in the third. The fourth deals with the computation technique used. Data and results are included in the fifth.

Materials

The polymeric systems used to conduct the study are shown in Table 3. 1 on the next page. A list and detailed description of these materials are given in Appendix B.

Expe rimental Apparatus

Diffusion coefficients for the polymeric systems were deter mined at constant temperature by the microinterferometric method.

The experimental apparatus is schematically shown in Figure 3. 1.

The light from a sodium lamp is passed through a collimating lens to make the rays parallel. This beam of parallel, monochromatic light Table 3. 1. Polymeric Systems Used for the Study.

System Polymer Solvent Polymer Solute Solute No. cone. wto/o cone. gm 100 cc soln. 1 Polyac r ylonitrile Dimethylformamide 17.72

2 Natrosol Water 3. 51 Urea 10

3 Natrosol Water 3. 51 D-Glucose 10

4 Natrosol Water 2.50 Urea 10

5 Natrosol Water 2.50 D-G1ucose 10

6 Natrosol Water 1. 88 Urea 10

7 Natrosol Water 1. 88 D-Glucose 10

8 Natrosol Water 1. 00 Urea 10

9 Natrosol Water 1. 00 D-Glucose 10

10 Natrosol Water 0.88 Urea 10

11 Natrosol Water 0.88 D - Glucose 10

c:r-N 27

Camera

Microscope

Objective Lens

Stage

Diffuser Lens

Light Source

Figure 3. 1. Schematic Drawing of Experimental Apparatus 28

if reflected by an optically flat mirror at the base of the microscope and passes upward through the diffusion cell into the objective lens of the microscope. The diffusion cell consists of a wedge formed by two partially metallized, glass microscope slides separated by a spacer at one end. Light rays passing thr<;mgh the liquid in this wedge produce interference fringes. These fringes were parallel to each other if there were only one liquid in the wedge; but whenever there were two liquids in the wedge, and if a concentration gradient existed between them, the fringes are distorted. Measurement of the fringe distortion provided a means for calculating the diffusion coefficient,

D, and for determining its dependence upon the concentration, C, of the solute. A photograph of the apparatus assembly is shown in

Figure 3. 2.

Preparation of the Optical Wedge: The diffusion cell con sisted of two partially metallized, glass microscope slides separated by a spacer at one end to form a wedge. The slides were 3 inchs x

1 inch and were coated on one side with aluminum by vacuum evaporation.

The amount of aluminum coating was such that the per cent transmit tance of light through the slide was reduced by approximately 70% to

75o/o. This metallization of the slide rendered it partly transmitting and partly reflecting. The polymer solution was placed in the wedge formed by the two slides whose metallized surfaces faced each other.

Details of the optical wedge are shown in Figure 3. 3. 29

Figure 3. 2 . Experimental Apparatus Assembly 30

Transmitting metallize d surfaces

Ple>.te glass Spacer mh:roscope slides Polymer (3 - in. x l-in. ) solution

(B)

Figure 3. 3. Preparation of the Optical Wedge for Diffusion Measurements. 31

Microscope: The microscope used for this purpose was an

American Optical, Microstar Series 4 microscope. Figure 3. 4 is

a picture of the microscope. The microscope had a Kodak 35 mm

camera attached to it. Kodak Tri-X Pan, fast black and white film

was used with an exposure time between 1/25 to 1/50 second. Kodak

Poly-contrast paper was used to develop prints. The condenser on

the microscope was removed as a parallel beam of light was required

for the experiment. There were four nose pieces (objective lenses)

having 3. 5 X, 10 X, 43 X, and 97 X magnification power, respectively.

Nose pieces having 3. 5 X and 10 X magnifying power were used for

the present work, as higher magnification reduced the intensity of

t~e light passing through the microscope. The eye piece on the microscope had a 10 X magnifying power.

Concentration Measurement Apparatus: An Abbe- Spencer refractometer was used for measurement of the refractive index.

Measurements were taken at constant temperature. The Abbe

Spencer refractometer and its constant temperature bath are shown in Figure 3. 5. The refractive index readings taken had a deviation of + 0. 0002. A Gate 1 s sodium lamp was used as an external light source for both the refractometer and the microscope. The average wave length of the light was 589. 3 m~ (21). 32

Figure 3. 4. Microscope Used for the Experiment

,. ,.• _,.

• •• • Jl; 33

Figure 3. 5. Abbe-Spencer Refractometer Used for Measurement of Refractive Index as a Function of Concentration. 34

Experimental Procedure

This section has been divided into two sub-sections. The first sub-section describes the experimental pro~edure followed to obtain photographs of the interference pattern for the polymeric system. The technique for measurement of the concentration of solute by refractive index measurement constitutes the second sub section.

Procedure for obtaining Experimental Interference

Pattern: A polymer solution, of known concentration and refractive index, was placed on the lower slide, which was fixed on the microscope stage. Two cover classes were placed as a spacer on one end of the slide. The upper slide of the wedge was then lowered to obtain contact with the polymer solution. The metallized surfaces of both the slides were then in contact with the polymer solution. A set of parallel vertical interference fringes were observed through the eyepiece. The interface between the polymer solution and the surrounding air was then scanned through the microscope to find a region where the interface was perpendicular to the interference fringes. In case such a r egion could not be found a new set of slides was prepared, and the whole procedure r epeated. In practice, it is advisable to apply the polymer solution to the lower slide in the form of an elongated strip, approximately parallel to the long dimension 35

of the slide. Usually, one or two attempts will yield a suitable

interface.

After a suitable interface is obtained and brought into focus,

a drop of solvent is placed in contact with the side of the wedge.

The drop of solvent is immediately drawn in by surface tension

forces and as it comes into contact with the polymer solution diffusion

begins. The time at the moment the solution and the solvent come

into contact is t = O. The interference pattern at t = 0 is shown in

Figure 3. 6 A. After some time, t, has elapsed ~nd sufficient dif

fusion has occurred the interference fringes were curved producing

an interference pattern like that shown in Figure 3. 6 B. If the dif

fusion process were allowed to proceed for a long time, the material

in the wedge approached uniform concentration, and the fringes

became parallel, straight and uniformly spaced, as shown in

Figure 3. 6 C.

By photographing the interference pattern at some known time,

t, after sufficient diffusion had occurred, it was possible to obtain

the concentration - distance profile. A ruled microscale was also

photographed through the microscope at the same magnification in

order to provide a distance scale for measuring the interference pattern.

Such photographs for systems 4 and 8 are shown in Figures 3. 7 and 3 . 8.

Measurement of Concentration: The refractive index method was used for measurement of concentrations. The refractive 36

(A)

Time• = 0

(B)

Time• = t

(C)

Time• = oo ·

Figure 3. 6. Light- interference Patterns Observed During a Diffusion Experiment. 37

Figure 3. 7. Experimental Interference Pattern at t = 11 seconds, for System • · the solvent is located at the top of the photograph. 38

Figure 3. 8. Experimental Interference Pattern at t = 15 seconds, for System 8, the solvent is located at the top of the photograph. 39

index - con~entration relationship can be obtained by making measure - mente of a series of polymer solutions with known solute concent ration using the Abbe-Spencer refractometer. The refractive index of the pure sol vent was obtained by refractometer measu rem ent.

Computation Technique

From Fick' s second law, the differential equation describing

the molecular diffusion process for a one dimensional case is

ac at ( 2. 9 )

The initial and boundary conditions for this case a r e

c = 0, t = 0, X < 0 (2. 17)

c c t = 0, X > 0 (2. 18) = o ' c = 0, t = t , X- • 00 (2. 19)

c = co, t = t, x - oo (2. 20)

( 2. 2 1) c = Co Yeo t-oo all x ' s Vo +vco

Boltzman (4) showed that for a case of molecular diffusion depending

on concentration, c may be expressed by a new single variable, r, where

X r = (2. 10 ) 40

By· substituting the new variable, r, into equation (Z. 9) the following

equation is obtained:

(Z. ZZ)

The detailed derivation of equation (Z. ZZ) is given in Appendix A.

Evaluation of the Integral: A photograph of the interference pattern is taken after some time, t, after sufficient diffusion has taken place, but before the concentrations of the initial materials in the regions remote from the original interface have been affected.

The difference in the refractive index between the polymer solution and the solvent is known by previous refractometer measurements.

By photographing a ruled microscale through the microscope, (using the same magnification as that used for photographing the interference pattern) the change in refractive index between any two adjacent fringes in the direction of diffusion can be measured. The refractive index-concentration r elationship can also be obtained from this measurement. A curve of refractive index as a function of concentration (see Figure 3. 9), may be obtained from the refractive index measurements of solutions of known concentrations. A concentration- distance curve obtained in this manner is shown in Figure 3.10.. The 41

0.020

0. 016

0 . 012

s::0 -s:: 0. 008

h-· • ~ ,....,..,.. . '

0. 004 ..

T. 0 . 000 . 0. 0 4.0 8.0 12. 0 16. 0 20.0 Concentration, g. polyacrylonitrile/ em. 3 solution

Figure 3. 9. Refractive index (sodium D-l ine) of solutions of polyacrylonitrile in dimethylformamide at

25. 0°C . (where (n - n 0 } is the difference between the r efr active indices of the solution and the solvent ). 42

s:: ... . ._ ...... ,0 ...... ::l 0 (/) 16 (V") E u 0 0 ...... 12 cu 0 Exfterimental E va ue ...... >- 8 Calculated value 0 0..

bO s:: 8 0 ...... , rd .._,... s:: cu u s:: - 0 4 0

0 0.00 0.02 0.04 0.06 0. 08 0 . 10

Distance, em.

Figure 3. 10. Experimental Concentration Profile for Polyacrylonitrile-Dimethylformamide System at t = 90 seconds. 43

I G axis (locus of x =0) is defined by t~e requirement that

c' co - So x de = t X dG (2. 23) •C~

which requires equal areas under the concentration-distance curve

above and below the c'axis (6).

The nature of the concentration-distance curve (Figure 3.1 0)

is sigmoid. Equations for such curves are usually desirable,

particularly equations that represent close fits and permit easy

differentiation so as to provide accurate slopes.

The Gompertz equation

has been found to be satisfactory in representing such data (8). The

modified Gompertz equation is

aX y = a· + a1 a2 3 (3. 2)

where a 1• a2, and a 3 are constants, and ct is an in~ercept. A c?rn ..

puter program for evaluating these constants and obtaining the

Gompertz equation for the various experimentally obtained concen ..

tration-distance curves is shown in Appe ndix D.

After the Gompertz equation was obtained the line of the original interface, i. e. locus of x =0, was obtained. A distance,

1 say x =x , is · specified and Simpson's nwnerical method was used 44

to evaluate the areas Al and A 2 as shown in Figure 3. 11 a. Then area A 3 was evaluated and compared with area A 2. If these are unequal, then another value of x is specified, depending on which area is large or small. This trial and error procedure was followed until the areas A 2 and A 3 were obtained equal. Th1s satisfied the requirement that c' co x de = l x dC -.Jor .Jc' (2. 23)

and the line of original interface was obtained (6). The computer

program for evaluating the line of original interface is shown in

Appendix D.

The integral in equation (2. 22) was evaluated by numerical integration of the concentration-distance relation given by the

Gompertz equation fit of the experimental data. Simpson's numerical method was used for the purpose. In Figure 3. 11 b, the area repre- c1 senting x dC is shown shaded. The computer program for this .S 0 numerical integration is shown in Appendix D.

Evaluation of the Concentration Gradient: The derivative

(dC/ dx) is obtained by fitting an empirical equation as shown by

Davis (8). Sigmoid curves, both normal and skewed, can be fitted satisfactorily by the equation

.§ = X- X} (3 . 3) a+ bx 45

~---- u

...... 0

......

u 46

20 where k - log ( Y ) J log(lOO- y) (3. 4)

x 1 corresponds to y = 0. 1, and a and bare the intercept and slope, respectively, of the straight line that results when (x- x 1 )/~ is plotted against x. On differentiation of equations (3. 3, 3. 4), we get

(3. 5)

where

4343 s = o. 4343 ( yl + o. ) (3. 6) (1 00-y) log( 1 00-y)

Details of this m e thod are explained by Davis (8). The computer program for computing the concentration gradient is shown in

Appendix D. The relationship between concentration gradient and dist~nce obtained in this way is shown in Figure 3. 12 for the polyacrylonitrile-dimethylformamide system.

After the above calculations were p e rformed, the diffusion coefficient D, was obtained as a function of concentration, C, by appropriate substitution in equation ( 2. 22).

Data and Results

The data taken and the results obtained by performing the calculations shown above are presented in Appendix C. 47

I H·t-"-1+++< - lt<~l~ I:tjj:!. ~ flTf 1-'- at il~ fJ+H •• . -,.. ~ E ~i: ~ .. tt ht Iff I+ ft!i1~ .. +t u 5 '14 + ~' -r . ~- -fl. s:: • ltli •tl tt:r •i 0 I ' ...... 1., . :;j """"0 (/) • ('() ,1-i- E 4 ..u u ~-~--!-; ...... H .. Q) -

~ f+- ~ -IE .-i Ft:ii 0 l . d- ±; 0. :tf.t= 3 ~ 1--H+ b.O.. :'I~ ...... r!:t H ~ t;:~ ~~I:Jl·~ ...... Q) ,T 'U 11-4-l-14-U - ~\ -4++ rd ~+·. -,-..I_ ~ ~ ;!: ~r:tt-;: ::t""t-t ~ .. .X · () w. - ~ 2 .... ···:_r ~ 1-- ~. J::._ ;- 1- -=li+JI .... 1:.: !;:" l FI1F. ·1+-4 .... • •• f • • • ~- :J .0.... :t--'-1 . -7"" 17; ..... ft;#l± - rd ·r+ .....H ~ - s:: rf Q) .. u s:: 1 0 (.) ' ' I t

~ r. /1 ~- <::+ : / ~ l .. ~ -:p::::;:f /{ 0 . 0.00 0. 02 0. 04 0. 06 0. 08 o. 10

Distance, em.

Figure 3. 12. Experimental Concentration Gradient as a Function of Distance for Polyacrylonitrile Dimethylformamide System at t = 90 seconds. 48

IV. DISCUSSION

In the microinterferometric study, the water soluble, non ionic polymer, hydroxyethyl cellulose (commerically known as

Natrosol) was used in the preparation of the polymeric solutions used. Urea and D-glucose were used as solutes. The rate of molec-

·ular diffusion of these two solutes in polymer solutions of various polymer concentrations was quantitatively measured in terms of the molecular diffusivity. From the Wilke-Chang correlation (30), it can be easily seen that the diffusion coefficient is an inverse function of the viscosity of the solution and the solute molal volume, and a direct function of the molecular weight of the solvent in an isothermal system. To facilitate the study of the effect of the molal volume of the solutes, the solutes selected differed substantially in molal volume (Molal volume of urea = 48 cu. em. I g. mole and that of

D-glucose = 177. 6 cu. em./ g. mole). Polymer concentrations were used which would give solutions which were viscous enough to allow observation of the diffusion process.

Effect of Solute Concentrations: The plots of diffusion coefficients versus solu.te concentrations are shown in Figures 4. 1 to

4. 11. The plot of diffusivity, D, versus concentration, C, for poly acrylonitrile-dimethylformamide is shown in Figure 4. 1. This system was studied by Secor (ZZ) and wa~ used as test of the e~perimental -7 · - , · :· . . :-· 1 · ~- =:·-=-· • , ... -- -·· . -:-·· ..... l 0-·-. ·----:- :-·-. . :::-:·-· .. :-:. -- I. . ·:··-- H q· . ·I . . ·•++. t '' t!f .. • +.. 1 .t i • ..H ~ ... -" . · 't • • - -~,.. .. - i' • '- 1 ~>- - I f l' l! . • ' 4 • - +-+ •-t t- ~ .,. • t t 1 J • I ·t;l +J ; • !-1-l ~ · • t t!t; ~+r e> t ) t ' • • 1' .. ·- r-' + t ' t ~ ·! • t • ~ i·+t • • , • "• • .. 5 :I , 11= I+ ITff t tt ttl ~ lffi f-t 1-t t 1± It- til It· IJ:El - %t tfl!;;:~ l!ffi j!!_t '" ·~ t , I . ~ 1t '' ;.r ,. :. 1~~ t:trt· ft l!h ·~· ·~~ p' 't lk. ~r . ~ I I I r ~n mru ~ t; . : 1 a±. tr 1:i!.t hh ~ H L-t- i-t:f ...... - .,._~ ~ ~ . p + rl 1/ tit .• , •• 'tT ll 11· , ,., .. ~~ ~~·:: ~u if im• ··+ f+ il ~t :J:i=;+ . ~~r.: t ~r + -· '' JHt +! ' "--'. r.t:;!:. rl.""'~ ·'"r--- ;~ · ~"li .: ·U~ . :r:l .. ~ ... t ' . -~ r.h::::: 4 -tt . < •• -f---+-< • t-·. ;.t '/, :'f. ~H ~1 1·;::: 1 ~:-- £$- *~ ' ~R -t . + ~ -~ -1 0 1..... 1-l- v . ; Ql T ...... '. I+ f" N ... +1.:.. cl . e 3 . 0 ±= t:a: ~$*...... c T ~ -~ ffi t tT :1 j .• i ~-zq: H- ... ·, 0 H: :r . . tlt I± H , .... j:d' t Ff ! ...... t , ;~ , ,; ..,.. Jm ...... ~ ; . H- :~~ r.-- •. 1-t- d'',. tt v ~·"!"t..,. ~ -- "-r-.. >< t+ 1 / 1-:+t-:- Q _ I '-~- tiff c- -rm ·i~ • ±. I~ .w~~- . ~'Tl:pt ;!-:- *-..... ,+ ~ >...... , 2 + ...... ~7: ~ ·tt;..:- "- 11= ..jr.m .;. · ll-1:;- m± .'5!... . trt . -· Ql ±W: 0 -4-4- ~ :! '+-1 . ++t+ '+-1 . l .... • ~H- Q 1-H '·:rt .. . , • •.... l=F! ,. tt~ ~ tt± ~ ~ £±!: 1 ""':": IT • ! · :.hr . - + . - '~r1 rrrrtrt! ffi. ~ ±!_-:+ t: t bt ~ ·rt . .,. .. ~ ·. 1..r t 1 tP.-- H ;m rit i± · 1. . tr rrr! ~ -tr.l t+t ±7=- lh ,. f 8tH: t~t:=I::- 1 :I:.: h- ::tt :1 ·li : · rf r~ T -+I- : ,f ~ tt H~ . fl't u:ttltrt-t cr :-- ~ :j:j: ~ · - ~~ -rt flttt t I tt :r 1H1 ~-tt 111 - ! It ' l·l·d .• UW-1 ~!1 +H+ rt+t 0 i l {l~ ilt 2 4 6 8 10 12 14 16 Concentrati on, g. solute polymer I 100 cm3 solution tf:o. ..c Figur e 4 . 1. Effect of Conc entration on the Diffusion Coefficient for Solutions oi Polyacrylonitrile i n Dimc thylformamide at 77°F, with t = 9 0 s econds. 50

equipment and procedure. Experimental results obtained were in complete agreement with those obtained by Secor. He used 60 seconds as a contact time for diffusivity measurements, while 90 seconds w ere used in this work. Initially, a 60 seconds contact time was used for photographing the interference pattern; however, this photograph was damaged. Subsequently a 90 seconds contact time was used, which was found to give satisfactory results. Likewise, the

curve plotted was identical with that obtained by Secor. When the diffusion coefficient was integrated over the solute concentration

range studied, it was found that the integrated diffu sivities were also in close agreement with the work of Secor ~ The integrated diffusivity

obtained in this work was 2. 06 x 10-6 S'q. em. I sec. over the concentration range of 0 to 14 gms. polymer per 100 cc. solution. The 6 value obtained for Secor's data was 1. 98 x 10- sq. em. I sec.

Figures 4. 2 and 4. 3, show marked similarity though the dif-

fusion coefficient versus solute concentration curve (Figure 4. 2) for

the urea system is relatively steeper than the one for the D-glucose

system (Figure 4. 3 ). The, integrated (average) diffusion coefficient was obtained by integrating these curves over the range of solute

The integral diffusion coefficient for urea concentration studied. 5 in 0. 8% Natrosol solution in water was found to be 3. 38 x 10-

sq. em. 1sec. , while the integral diffusion coefficient of D-glucose 5 . in the same solution of polymer was found to be 3. 0 x 10- sq. em. / sec. 51

7 - :. i J::tflt~ l !:t:

.... rt+ I 4 :!:1+1tf-t .+1 f+ri 1+1-!- '-!.fJ t· ::t;± 6 !· rrrltti• I :.: t ~ ~-i: ±ti

u ' Q) 1/J N- 4 uE 1.{) -t '' 0 ± 1-1 .. . + ->: - Cl .. 3 r >. -1-' • rl ...... > 1/J ' ~ '+-< '+-<..... Q 2 I A Wilke-Chang . Value . • .

1 - .

0 2 4 6 8 10 Concentration, gm. solute/ 100 cc. solution

Figure 4. 2. Diffusivity versus concentration for 0. Bo/o Natrosol in Water wit h Urea as Solute, at 77°F, with t = 12 Seconds. 52

7 ± r:t:' H+ ~rn ~ ~ WIYf~ l~~ ,· / Tf£ . ~ $ .ff!.! ·l=!¥1t :m·+ 6 Ll: :t: ri .411, 14: ·t .; oi' J--' J-,~ H-'-1 5± i1f I+H l.t H+ ~ ~++-t .--.. 5 +-:-r ~ ~ ... ±;: f+ -l-1- f* .X: -H ' J:. I t+ ·& llj

() v *- (/) •t+- ...... • N 4 E () f -rr +I lJ) ~ -f # 4- -+I 1E 0 '-'• ...... ~ +I-' ·++: ..,t~ ... ~ ... X H .. +i-h- ·t ±:: .~ lt1=R 1-H J-H+ H-1-i 1±1·+1 t+ lli~ .. Cl ~ .r Jl 1-1-1 ,~ .. !l :X r-r,...,. 3 i::t::. ., ~ t· '-'- . ~ ~ --- ~ 1"1 '" -~ : .. t-'- f-'- :): .: ?&::. b ::F:-r-·'1 o::: ,.... P::·t 4-'>- ...... tr ''!; T • t+ ...... > :q (/) ...... ::1 ...... t+ i'~ I H-1 Q 2 . A Wilke - Chang Value 11-h- I i: t c!:f 'I : !::+ f4t 1-H::!= .. ~ ti:::::i ;=t: -·-~- + ;...-..-;+;~ -14

·I"~ - ~ h· 1 i-t t+r+ p t • "1 . - m m 11 - =F -~· 0 2 4 6 8 10 Concentration, gm. solu te/ 100 cc. solution

Figure 4. 3. Diffusivity Ve rsus Concentration for 0. 8% Natrosol in Water with D-Glucose as Solute, at 77°F, with t = 15 Seconds. 53

This is consistent with the expectation, since the molal volume of

D-glucose is larger than that of urea, and the diffusion coefficient is an inverse function of the molal volume.

Figures 4 . 4 and 4. 5 indicate that D-glucose system curve

(Figure 4. 5) is flat between solute concentration of 4 gms. /100 cc. to 7 gms. I 100 cc. This flattening of a curve does not exist in the case of urea (Figure 4. 8) in the same polymer solution. However, in general, both the solute systems have similar diffusivity versus solute concentration curves.

For the 1. 88% concentration of Natrosol in water, the effect

,of the solute concentration is shown in Figures 4. 6 and 4. 7. There is also a flat region noted in the curve of the D-glucose system

(Figure 4. 7) as was observed in the 1% Natrosol concentration results.

Contrary to the 1% Natro~ol concentration - urea system (Figure 4. 4 ), in this case there is a flat region for the urea system (Figure 4. 6).

However, the flat region for the urea system is very short compared with that in the D-glucose system (Figure 4. 7). Diffusion coefficients for urea were found to be higher than for the D-glucose.

The diffusivity versus solute concentration curves obtained are very much the same · for the 2. 5% and 3. 51% Natrosol systems

(Figures 4. 8 to 4. 11). There was no flattening of the curve noted, however, in the 3. 5% Natrosol solution (Figures 4. 10 and 4. 11) such as was obser ved in other cases (e. g. 1% and 1. 88o/o Natrosol syst ems). 54

7 ~ :t:::th: 1¥ I ff ~H+ l- I+ Fi=

H m ~ wt ++++H+H fH, -±:< < .. !+ . . . .. :i. .. 11:1' : : .. .: : it q: -1.t1 ,.;-._ .. ;:t 6 t1ti l~ r mlr

~Wt t I 5 ...... ,.. :;:: ; .. t+ I:J: R I 1-t u (l) (IJ :t . • ...... N 4 6 u ll) 0 ~... ..; t r.rt- R:f· f'l+t+ : ·t+~ ,. -~"'· 3 ...... • 4 t+ .• 1-f:.~';l=~ .' . ;~ H-t-t 1-H I IJJ 4-1-l- ..1.. ~:It :;; t±~ -~ . ;::;:l t · +H- J: .-f-'-'- ·~ +- . : .·•· +hi It -1001!f-H :t!:t.j 2 IF - ~i 1 A Wilke-Chang I 1/ H ·· Value i+ / !l++- ..... i+H.l. H_ l:tt~ Hl. !=t;:;:l#tt ~ ._._..t t±:l1~ $-;;r(td1 1:r4 ·I- f;i:Lt- t'!:li ....._.._ 1-'-t t+. : : t:ft-'-t- ··tr rl-t f-r- ,...,_. 'it- ~ lft:E ~:-.- 1 ~·l+r I=Ht-~ 1-r--· ~--~ IH:;=

H+l-llH +Hl !=H= # 0 4 6 8 10 Conc entration, gm. solute/ 100 cc. solution

Figure 4. 4. Diffusi vity versus Concentration for 1. O% Natrosol in Water with Urea as Sol ute, at 77°F, with t = 15 Seconds. 55

7 Jl$1: FHH- I+ j:i • I IE ~ :~~ !=h. H ~ H-1+ • ...... - .- II 6 It -

H+t r 5

--1 ·n u tr . v + Cl) ...... ~ N 8 4 u I'· l() 0 -I· t! ~ ~ ~ ltl=l rt 0 1....:. f!j~ !- ;. 3 :E;z:::-+ ..... ' ,:2 •.-i

•.-i> {/) I ::I . :__+ '+-4 ~ '+-4 •.-i h rt- l:f± 0 Ill 2 t+ A Wilke- Chang H· @: -~ Value 4 ~ k±~-~ M+r--:- .~ M+- r$± Tt -,-~ '"'"+- 1-+t: ~ . ~.,_ H± ~ 1

~- ~

i-t

f-1+ If Ht Lli n 0 2 4 6 8 10 Concentration, gm. solute/ 100 cc. solution

Figure 4. 5 . Diffusivity versus Concentration for 1. Oo/o Natrosol in Water with D - Glucose as Solute, a t 77°F, with t = 15 seconds. 56

7 ~fJlr. ttl=! ~-'-~s,1:1.· .n:Eitm~t 81t±i ~ -~~--r- ..... :I++ i-f :::ttf •"-H er;. ~ ~· ·- 1-f-'-l..L, T." :: t ..__ +·+ -I.. f-.-+ -~ ft:H: .._.- fl- T :j:: ·'-l-i4++ 6 mt -~ f-+ ti .... T- t 't.L liil: .. T ,:;.. ~.:-:-~- .. ~ -~

,} ..,-r:t= :J.:~ rt ifF+ 5 I+ +l-f r· ~

if +FF+ 1-c- _gr. ~ . u <1> ... . 1/) ' t:t:;.~ N 4 t-o-+ ... H•· H- 'iT '1 "' .... :.~ ··+> . uE _:t- -:: ,...... , IJ"'' ,. . 0 .... -.... ,"1.:: ...... i ~~ ::n >< :t f+ Q . ~ 3:~ 3 -h-l+r._. r<- ~ Ff IF -~tfH t'l- ~ ... >- H+ t H+ .,...... +- .....> H+ 1/) rf~ ' ::l ,'...... ,... LJ..1..U. ~ 2 I+ A Wilke-Chang tt ~t Valu e . r;~ • rtF . 1 ~ :f 1-t 1 ·'+t;·

....._ ~·

L-llJ. 0 2 4 6 8 10 Concentration, gm. solute/ 100 cc. solution

Figure 4 . 6. Diffusivity versus Concentration for 1. 88% Natrosol in Water with Urea as Solute, at 77°F , with t = 10 Seconds. 57

7 JH Hf . ....;.;: ftrrl ~~- ~!tt :~ 1·:;::. ~ P·... if. 6 q~ '

ltt.t:,.._ FH+ It• •+- · · · + a~ £ Wilke- Chang .L ~ :t -+t- t Value 11-r- ~ +t.!-l- U-:! ~ -~ ~ FF ... '+l- :~I:+ u ,: ::} v .. 00 F-~-::;: "t (/) I"H"n" ...... ,.~.. :·· ....t+ N 4 6 u l() 0 tt~ 1-+:- Ti-4 ~;_,.-,'T • ~

~ Iff: -~ Ittl 3 ~ I tt I +I= ..I·t~ft . -H£1 =* ~ ; ..,l ~ . . . . ::t ,..._ H-r+ ~ -- ~ ~t:T ·4-1- . ·- q. +t H ·z h-t H+ • :1. lffEt 1 ~·1-t- . :~~;±!-+- ':'' 1-t+t ~ ·t ~.1--+t+ :t; . ,.fi: ~ -H+ ~- _--;:;;: . ~ . :·L ~+.. . ~ - H- ':::: hP.~ ;)..,...(:>.1:!: • •+IJ..J-1-+- • ~ Ht + ~ . H- ± • '+· j I~ r# 1 ..":: · + ::+ !l: =::-T~ ~ --- -- f!i ~(.1 ~ i-f1 ~

' 0 2 4 6 8 10

Concentration, gm. solute/ 100 cc. solution

Figure 4. 7. Diffusivity versus Concentration for 1. 88o/o Natrosol in Water with D - Glucose as Solute, at 77°F, ·with t = 30 seconds. 58

7 i. 1+1- ~}: li:tW:G± t ·HH- mHI ..

"1-'-1- + It'++ j:l ·-. f-t~+- '!.i- .• ':-:-;~;t:H=tt"~ 6 H- .. .._ p ~ I ¢ : ~.+ • . ~ r-F 5 f+ n;: I~ - Rf /- . It',. rt u ~· "" t (1) It' ...... (/) . ll- I+ ' N 4 ~- E ' u lfl 0.....

X ~ I A 3 - .. rtf;~ >- I*~ ...... 1-'-!-'- .....> fll ::l "H ....."H A 2

,;.. .._1 A Wilke- Chang . f ~l r-1- ,. Value =>+ ,- f-i: ~ · 1 j

ll-

0 2 4 6 8 10 Concentration, gm. solute/ 100 cc. solution

Figure 4. 8. Diffusivity versus Concentration for 2. 5% Natrosol in Wa t er with Urea as Solute, at 77°F, with t = 11 Seconds. 59

7 ~·Ifp.. -~ H= I+ rrl-rt ~ H+ >>h-1- f+ *"_,. F t-ffi.c.ft 1-f rY+II 6 +'+ H+

1::1::

II tt ~ H f:t 5 t:1 lffl F! + : t+ :Ji I~ H ·- 1-'-'+ 1+'- 1-:-r- ...... ~t+j::l: ... It ~ Ff :ru r• r~ltt1 tf u I~· ,...... I+t++ ++-- . ~ flt!L

1 t $

IH+T 8 10 0 2 4 6 Concentration, gm. Solute/ 100 cc. Solution

Figure 4. 9. Diffusivity versus Concentration for 2. So/o Natrosol in Water with D-Glucose as Solute, at 77°F , with t = 17 Seconds. 60

7 H l:t II H± if ~rnit~ ~ TI:f :a: +

5 ,'-r- H-rl-lt+ '-'+- . i-t- + u v IT+ II) l:± ...... N 4 11= e ,c . + u . +r- I it 1.0 0

3 14

m::;• 8. Wilke-Chang ++- Value

1 ... Ft# ~~:l " '+ -:-If ± ~ ~~:=! :~~~ ...... I+ - 111 ~t ~ +tt tfti I·HI 1t: ~:tt. -1+ 1 ~:t i :mt IL 6 8 10 0 2 4 Concentration, gm. solute/100 cc. solution Figure 4. 10. Diffusivity versus Concentration for 3. 51% Natrosol in Water with Urea as Solute, at 77°F, with t = 10 seconds. 61

!11 l fH

6 :H.,. ~li f!-o-t r j• . ;-;..t Y, .t-1 H t ·.,. f ~ tw~~~ ~, . :·.,. '-+-

h IT . 5 !::::...... u v (/) ...... \.) E 4 u V"' 0

q .. -- l.u.. 2

~ !-

,•....::;- :r.+ ~ ~· lt ~:R ·~~ ~~. 1 ~- u- i+ 1-i-' ..

. ijLl:;.i : ·~ IR-1-:r t 0 2 4 6 8 10 Concentration, gm. solute/ 100 cc. Solution

Figure 4. 11. Diffusivity versus Concentration for 3. 51% Natrosol in Water with D-Glucose as Solute, at 77°F, with t = 16 Seconds. · 62

This rna y mean that a relatively more viscous solution exhibits greater solute concentration dependence than do the less viscous

solutions. This change of behavior may have been caused by the

change of internal structure of the polymer solution. The integrated

diffusivities obtained for all systems were approximately equal to the differential diffusivities obtained in the flat region of the curve of differential diffusivity versus solute concentration.

Effect of Polymer Concentration on Integral

Diffusivities: In Figure 4. 12, the integrated diffusion

coefficients are plotted against the polymer concentrations. From

the figure it .can be observed that there is no set pattern relating the

integral diffusivity and polymer concentration. While the integral

d~ffusion coefficient does change, this change is more of a fluctuation

in value. This fluctuation makes it impossible to reach any conclusion

concerning the effect of the polymer concentration upon the integral diffusivity ·of a solute in a polymer solution. The results for D-glucose are similar to the urea sys.tems, as shown in Figure 4. 12. Therefore in the case of these two solutes, no specific conclusion can be made.

When Nishijima and Oster ( 17} studied diffusivities of sucrose m polyvinyl pyrrolidone, they found that beyond the critical concentration of the polymer in the solvent, the diffusivity of the solute decreases as the polymer concentration increases. Similar conclusions, however, cannot be drawn from this work in view of the fluctuating results obtained. 5 F:Tili: Ll:d_:_ it ' rtdtT~ L ~~~L+f Jr~lliii l"r 1iiiH i,I~~~~ ;rw~ mi Il Ll~, ~H -- ;I U rea as Solute

D - Glucose as Solu te

rl ~l~~~...... ---~~ ..... ll) tl:tt 0 ...... , ..... ->< m 0 l litl~dl v 1-1- It) 3 ~T ...... s I 0 u . m m=r:_· ,~ :·· ·:.JI._. '_. T . o< _:__1__~ ~·~tD·-- ...... - ••·-·• (/) tn- illt: ~tT ti-t I t-H IQ 2 ttL~ tt >- IL±t ~~m ·.tr ...... t; CP.' ' ' . ' ... =1 .....> - ,,,r, h;~~.l.·--.l.m ·_ (/) J! 1-' ~' .....~ Q I+ ...... 1 I'd ~ b{) I H+ v ...... ~ f!:H. If 0 1 2 3 3 .5

Polymer Con c e ntration, w o/o

Figure 4. 12. Integr al Diffusivity as a Function of P ol ymer Concent r a tion :~r : ~a trosol "'w Water Systems. 64

Gosting and Akeley (13 . a) obtained diffusivity data for urea

in water at 77°F. Their diffusivity value was 1. 363 x 10-S sq. cm./sec.

The value of diffusivity obtained by using the Wilke-Chang correlation 5 was 1.46 x 10- sq . c m. I sec. The value of diffusivity obtained for

D-glucose using the Wilke-Chang correlation was 0. 67 x 10-5

sq. em. I sec. All the integrated diffusivities obtained in the polymer

solutions studied, were found to be higher than those obtained in a

pure solvent, i.e. water. As the Natrosol concentration was increased,

the viscosity of the solution rose very rapidly. However, this increased

viscosity of the solution did not show any pronounced effect on the

diffusivities. In general, this may be expected with some polymers.

In this work, only the molal volume of the solutes, appears to

have materially affected the diffusion coefficient. The diffusion

coefficients for urea were found to be higher than the corresponding

diffusion coefficients for D-glucose in the particular Natrosol concen-

tration solutions used. This result may be expected theoretically

f rom the Wilke- Chang correlation.

Attempts were made to measure the viscosities of the polymer

solutions. I t was not possible to measure these viscosities since the

solu~ions were very viscous and the equipment available was not suitable for the purpose. Hence, it was not possible to obtain predicted values of diffusion coefficient using equation {2. 7), obtained by Clough,

Read, Metzner and Behn {5). Thus the comparison of experimentally 65

obtained values of integral diffusion coefficients with those predicted

by theory was not possible.

Experimental Equipment: The microscope shown in Figure

3. 4, page32, has a built-in tungsten light source in its base. As this

ordinary light would give rainbow colored fringes, it would be hard

to differentiate a bright fringe from a dark one. This difficulty can

be avoided by using monochromatic light. The monochromatic light

used in this work was an external sodium lamp. The light passes

through a diffuser, and is then reflected upwards into the diffusion

cell by an optically flat mirror at the base of the microscope. Several

lens combinations were tried in order to obtain a bright beam of parallel,

monochromatic light. The lens used for this work has a focal length of

10 ems.

Considerable difficulty was experienced in metallizing the glass

microscope slide one side only. This was required, however, in order

to make the slides partly transmitting and 'partly reflecting. The slides

should have a per cent transmittance of 25 to 30 per cent (22. a ). After

several attempts were made, such slides were successfully prepared

by depositing a metallized surface on the slide from evaporating

aluminum under vacuum. If the coating were too thin, the per cent

transmittance would be too great and it would not be possible to get bright fringes. On the other hand, a thicker coat of aluminum tends to :make the slide opaque. 66

This method of investigation is dependent on the existence of a relatively large difference in refractive index between the solution and the solvent initially contacted. If the difference in refractive index were small, little or no distortion of the fringes would be observed. The minimum required difference in refractive index depends on severa) factors. In general, a minimum difference of

0. 01 is usually adequate (22). The viscosity of the solution initially contacted should also be considered. If the visc<;>sity of both the solutions were low, considerable convection may be observed when the drops come in contact of each other. This sort of behavior may be due to surface tension forces. When the drops of low viscosity come in contact with each other due to surface tension, they may not be capable of dampening the initial convection currents set up as the fluid is drawn into the wedge. Convectional movement could be seen in the low viscosity solutions due to the presence of some fibrous dirt. In these systems, the concentrations became uniform in about one second; and hence, after this short period of time no distortion of the fringes could be observed. This type of behavior was not noted with drops of higher viscosity, as might be expected. However, at very high viscosities, the solution became gelatinous, and it was not physically possible to handle these solutions. In general, it can be said that either both the solutions should be viscous or either one of the solutions should be viscous. The higher volatility of the solution 67

may cause a problem due to evaporation on the slide. Therefore, the volatility of the sol vent should be considered.

The photographs were taken with a Kodak Tri-X Pan (ASA speed 400), fast, black and white film. An exposure time of from

1/25 to 1/50 second was found to be satisfactory. The developing of prints using Kodak Kodabromide paper did not give satisfactory results. Kodak Polycontrast paper was used satisfactorily for this work.

Prediction of Diffusivity of a Solute in Very Dilute Solutions :

From the results obtained (Figures 4. 1 to 4. 11), it appears that the

Wilke- Chang correlation may approximately predict the differential diffusivity of the solutes at very low concentrations of the solutes in the P

V. CONCLUSIONS

The following conclusions have been drawn from the data and

results obtained for the polyacrylonitrile-dimethylformamide system

and 10 gm. of solute (either urea or D - glucose) in 90. 0 cc. of 3. 51%,

2. 5%, 1. 88%, 1. Oo/o and 0. 8% solutions of Natrosol in water.

1. Since the results ~btained for the polyacrylonitrile-dimethyl formamide system are in close agreement with the results obtained by

Secor, the experimental technique is considered to be sufficiently accurate to give reproducible data.

2. The microinterferometric method is a rapid technique for determining diffusivity in viscous solutions.

3. The effect of solute concentration on the differential dif fusion coefficient is similar for various polymer concentrations and different solutes. The differential diffusion coefficient increases with the increase in solute concentration in all cases.

4. The integral diffusion coefficient remains almost constant within the range of solute concentration used in this work.

5. The effect of polymer concentration on the integral diffusion coefficient cannot be concluded from the available information. Addi tional work will be required to determine the effect of polymer concen tration on the integral diffusion coefficient.

6. The differential diffusivity of a solute decreases with an increase in solute molal volume. 69

7. Low viscosity solutions may not give satisfactory fringe distortion. Therefore, for these solutions, the interference technique rna y not be used to determine the diffusion coefficient. However, the range of viscosity in which it may be applicable, can be quickly de

termined by trial and error.

8 . The Wilke-Chang correlation may be used to approximately

predict the diffusivity of a solute at its very low concentrations in

polymeric solutions. 70

VI. RECOMMENDATIONS

The following are the recommendations for further work in

this area:

1. Different polymers should be used to study the effect of

polyn1er concentration on the molecular diffusion process.

2. If possible, more concentrations of polymer solutions

should be used in an effort to determine the effect of varying polymer

concentration on the diffusion coefficient.

3. It appears that it is not as much significant to use various

solutes, as much as it is impor tant to use various polymer concen

trations. It is therefore recommended that one solute should be used.

4. Non- aqueous polymeric systems should also be studied,

since considerable information about other properties {e. g. viscosity,

transition temperature, solution structure, other rheological properties,

etc. ) of these polymers are available.

5. Limitations of the refractive index, volatility and viscosity

of the solution should be established for both solute and polymer sys

tems when the microinterferometric method of analysis is to be used.

If molecular diffusion coefficients are required as a part of liquid

extraction studies, it may be necessary to have several t~chniques and apparatus available in order to cover the wide range of polymer con centrations which may be encountered in the extraction studies. 70a

6. A viscometer capable of measuring the very high viscosities is necessary if the Metzner relation is to be tested. 71

VII. APPENDICES 72

APPENDIX A

Derivation of equation (2. 22) for evaluation of the diffusion coefficient, D, as a function of concentration, C, is shown in this

section (6).

Fick' s first law for molecular diffusion obeys the relation

~ N = (- D V" C) (A. 1)

where

~ N = flux of the solute,

D = the molecular diffusion coefficient,

'1 C = the vector gradient of solute concentration.

Fick' s second law, in vector notation is,

ac + V N = o (A. 2) at

ac. _ - V "N at

= - '\l• (- D \l C) = V .(D v C) (A. 2- a) a ac a (DoC/ox) + - (D8C/8y) at = ax ay .

a (nac/az) (A. 2-b) + az 73

The following assumptions are made:

1. Equation (A. 1) is valid only for an isotropic medium.

2. The structure and diffusion properties of the medium

in the neighborhood of any point are the same relative

to all directions.

3. Assumption (2) implies that the flow of the diffusing

substance at any point is along the line of constant con-

centration through the point and normal to the surface

contacting the diffusing substance.

If the diffusion is in the x direction only, then equation (A. 2-b)

reduces to

ac a (A. 3) at = ax

The following initial and boundary conditions apply:

c = 0, t = 0, X < 0 (A. 4)

c = co, t = 0, . X > 0 (A. 5) c = 0, t = t, X - - 00 (A. 6) c = Co, t = t , X -oo (A. 7)

c v 0 co c = t - 00 all x' s (A. 8) Yo +vco

It is also assumed that Dis a function of C only. 74

Therefore equation (A. 3) can be reduced to an ordinary differ ential equation by the introduction of a new variable, r ,

{Boltzmann's variable) where

1 X r = -- {A. 9) 2 ..Jt dC ac --1 (A. 10) ax = dr 2..ft and

ac = - X dC (A. 11) at 4 t3/2. dr

Substituting equations (A. 10) and (A. 11) in equation (A. 3) we get

D X dC = a dC) 3/2 dr ax ( 2,Jt dr 4t

a D ar = dC ) - (A. 12) ar ( 2 t1/2 dr ax

From .equation (A. 9):

ar = 1 (A. 13) ax 2,Jt

X 2r (A. 13a) = -1/2 t

X dC d (A. 14) . . = ( - 4t3/2 dr dr (2~ ~; ) 2~ ) 75

X dC d -- = D dC) (A. 15) tJt dr dr ( dr

Therefore, we get:

dC d - 2r = D dC) (A. 16) dr dr ( dr

On integrating equation {A. 16) with respect to r, we get

dC ) dr dr

(A. 1 7)

c D dC) _ (o dC) - 2 S r dC = ( dr C= C dr C = 0 0 since (D dC/dr) = 0 when C = 0, we get

2 (' C r dC = ( D -dC ) (A. 18) Jo dr C=C substituting for r, we get

(A. 19)

where D indicates that D is a function of concentration. c Therefore,

(2. 22) 76

Equation (2. 22} is the required equation for evaluating the diffusion coefficient, D, as a function of concentration C . 77

APPENDIX B

Materials

The following is a. complete list of the materials used in this investigation. A detailed analysis of the chemicals may be obtained from the chemical catalogue of the respective supplier.

1. Dimethylformamide. Reagent Gracie . Fisher Scientific

Company, Fair Lawn, N.J. {used as solvent for polyacrylonitrile).

2. D-Glucose. Reagent Grade., Molecular weight 198. 18.

Eastman Organic Chemicals, Distillation Products Industries,

Rochest e r , N . Y . (u. sed as solute for Natrosol-water systems).

3. Hydroxyethyl Cellulose. (Commercial name - Natrosol).

High molecular weight. Lot No. 12039, Type 250 HR. Cellulose and

Protein Products Department, Hercules Powder Company, Inc. ,

Wil~i ..... crton. Delaware.

4. Polyacrylonitrile. Unfractionated polymer. Number

average molecular weight: 50, 600. Weight average molecular weight:

122, 200. E. I. du Pont de Nemours and Company, Wilmington,

Delaware.

5. Urea (Carbamide). Reagent Grade. M.P. 132-134°C.

Matheson Coleman and Bell, Division of the Matheson Company, Inc. ,

Norwood (Cincinnati), Ohio. (used as solute for Natrosol- water systems).

6. Water, Distilled. Distilled water was obtained from the

Nuclear Reactor, University of Missouri at Rolla, Rolla, Missouri. 78

APPENDIX C

The experimental data taken during the investigation and the results obtained are included in this appendix. The systems used for the study are tabulated on the next page, in Table (C. 1 ). All the data were obtained at 77°F. Table C. 1. Polymeric Systems Used for the Study

System P olymer Solvent Polyme r Solute Solute No. cone. wt. % cone. gm . 100 cc soh'l . 1 P olyacrylonitrile Dirnethylformamide 17.72

2 Natrosol Water 3.51 Urea 10

3 Natrosol Water . 3. 51 D-Glucose 10

4 Natrosol Water 2 . 50 Urea 10

5 Natrosol Water 2.50 D-Glucose 10

6 Natrosol Water 1. 88 Urea 10

7 Natrosol Water 1. 88 D-Glucose 10

8 Natrosol Water 1. 00 Urea 10

9 Natrosol Water 1. 00 D-Glucose 10

10 Natrosol Water 0.88 Urea 10

11 Natroso1 Water 0.88 D-Glucose 10

~ ...0 80

Table C. 2 . Data and Results for System 1

R efractive c dC Distance Index Concentration dC Sox dx D em. (n - n 0 ) gm.polymer per 100 cc. sol. gin/sq. em. grn/cdcm sq. em/ sec

\) • oo u~ 1 . l(J() • Qt· R ?· ft i \ .

I ?- . ';')0 ,, • 7 7 (~ 1 o I ~ ·~ l . 007.7 • ' , " r'

•7 4 . 6S 0 1. 1 . ~s • 00':-:3 ) • ~ 10 . , . '7 . uo. u • 00 • 1 () ~- !• . r 1 '· .

10 • 1 J. 7 ? • 9f'll-:- ) • 1 . I J l . 00(>9 ..

/ ) • ! lQ • J.~i) ?.',')(", •.. ~ .o o:;~ 1

10. :. ()(} .1 ?.0 ·~ • 5? s .' . I I . 0085

:: ) . 0097 lL >O . ] /../:· ?. • l 7 ".) .

,,, (, 1 . 621 =' • ()· ,._,, • 0 110 13 ._) ')l) ... "

• 1 R2 } l 1, . 0127 15. 6 10 . ..., 17. ~50 • 21r0 - - • • 0 1 '~1

Time t = 90 seconds. 81

Table C. 3. Data and Results for System 2

c Refracti ve Concentration SO x dC dC Distance Index gm. solute dx D em (n - n 0 ) per 100 cc. sol. gm/ sq.cm. gm/cc/cm sq. em/sec

, ' • (J I . • orl l5 1. ()50 • 02 7 . s.~ .I e • • i'

. ) ' . ant:. 3 . 'J:)\) . 0$?8 1 . 5t)~, ' .

• 0. o lJ I} (• 1 4 . 0 ~)(I . I.)" 7 1 . 5 71 • 1 l

.) o L 5 • () ') ) • 1 n?. 1 • 5 l:.fl -

. o li I ·, • r) 0 . ~ 5 5 . 6~)0 • 1• n ·.•:.1 1. 530 . 7

• ll· . 009 1 j ) • I I I;\) .103 1. 511 .'«) • 3

'1 , G .. 1 iJ 5 7 . () 0 () • 1 nr~ 1 . '~ 7 7 •

} · •t'• 0 (; . OJ. ?d • ( J . • 113 J. · '~ 17 ..

I· '.0 • r • ' J 13 t:) '.1 • :1l" . ).'27 1. 35fl ' . .

I ) • 1 ...) ]..052 .79 I (J 0 (J 0 I ~ .1 l l ') .

Time t = 10 seconds. I 82

Table C. 4. Data and Results for System 3

('c Refractive Concentration Jo x dC dC Distance Index gm. solute dx D

em (n - n0 ) per 100 cc. sol. gm/sq.cm. gm/cc/cm sq.cm/sec

• \.I ~ . 0~ 11 3 e ':• 30 e \130 . Sll ]_ e ' :>

l • .>:)CJ 1- e oJ ) .(, 7 . 750 . 0 7 9 .

j_ • -1 <"; oll. • oos 1 i . ~u . ong 1 . 56'1

• 09(, 1. e :> L,.';- ... o :I • i' n '1':· I~ • ? 7 .)

5 . (l:)r) . :1C>9 • (I I .\ }7 ')

I I -, t) .~ .. ) 100 J. • ': :; () !:. • o (I • .( ' 0:! . :... •

-I.;. 1:· 1. 393 ;, . ll , .. • 0100 ·1 • J. ..J C1 • 1 0 •

,,.., 1 0 3? I • v . 0 11 5 . 11 2

... . 1.39 1. 1:37 'J • 70 • t I • l) l ?. " 9 . 10u

.179 (',I s . :--1 • t. o I J :~ '·.I 10 . 0 00

Time t = 16 seconds

- 83

Table C. 5. Data and Results for System 4 c .Refractive Concentration Sox dC dC Distance Index gm. solute dx D em (n - n ) per 100 cc.sol. 0 gm/sq.cm. gmlcdcm sq. em/ sec

1/ I. . 0012 . 300 . IJ 2 7 J• • 21 R l.Ol

,c:- . 0025 1. j.J • l)t~!:J l • 6 ~· 6 1. - J

• ' • .> . uo::;r. 2 o :HI L) . ()'),, 1 • \ {> :'

• ' -I . OQC)o) 3 . ~ j () • or) 7 ?. • 15 3 j •'!· /:

•) ·. l r' • u:' I • 00 () 3 l~ . l :i 0 • 0 75 ,_ • :J -- ~ • L•. 7

r· l . /. ) . 007A '" · ';50 • 0 79 ~ . ':-3:..:

l I· 7 ~ () . oo n 5 • ():) 0 . oeo 2 · '~II 9 . '

( ~. I 7 • 0 11 /~ '1 . '~:10 . on7 . oc.7 • J

. 776 :i . 77 • • f ~; 4 . 01/.7 P . ~90 . 098

Time t = 11 seconds 84

Table C. 6. Data and Results for System 5 c Refractive Concentration Sox dC dC Distance Index gm. solute dx D

em (n - n 0 ) per 100 cc. sol. gm/sq. cm. gm/cdcm sq. em/ s e c

C 1 I 1 . G• ll • : no . o~r. . o .

'' • 00?.5 • j

,. • on? , ? • ? I 1 . -

I ' :~ / , 3. 7:-iO

• 00 ,r,,) . 0 7 () . . .

1 • - -, . (. /. . 007 0 • 0 7 ° . '

I o • 0 \ / /. . OOB3 . nno

7. I· I I)

1 ? Q ' ) / ' 'I . "·!)0 ·~ ,_ . n ,..' ~\ ( • . 0107 • 100 ......

1 r' ·, ...... ' I - 0 J. ), / 9 . 200 . . -

Time t = 17 seconds

.. 85

Table C. 7. Data and Results for System 6

c Refractive Concentration Sox dC dC Distance Index gm. solute dx D em (n - n 0 ) per 100 cc.sol. gm/sq.cm. gm/cc/cm sq.cm/sec

t O"J. . 001/. . 'l J . -,.,:... ]_ • L·

., . • . '? ) • 0 r 7 1 ...LU'' 9 '-

I • GU ':~) .:J.: • I"'Q J. • ~;... ·.: . ?I:

I, I ~~ • I; :, d j/~ 1 . 5~ - ,. "':I ' !;7 • O(J • J.

, -( :· 0 J ., ·) • 1;. .it) 1 n ·1 ) • 5 P, l J. • .

'! . . ,. -;} ( I % " .' J J . '

,. l . :) J l -. ~ ) -·~= • Q;) ~) 0! (l • .:; . 10 ..

-1 • ;, 1. ;,~ . I .. , .I . ~JJ.) 1 t l . 110

1 • : .. ,. 'I) J.l ., l . 57..) . 0 I • () ll t ' . . -

l' ';· l. ~I.., • : 0 . t ( • (j 12 '( . - J

~ • lr L;. . J~ t • 5 J 1 • I • ~· 1..~9 • s

Time t = 10 seconds 86

Table C. 8. Data and Results for System 7

c Refractive Concentration Sox dC dC Distance Index gm. solute dx D em ) (n - n 0 per 100 cc. sol. gm/sq.cm gm/cc/c m sq.cm/ s e c

,. ... , , .. ,) ~ . {) • 00 13 . Y90 . 028 • ( •' 0 • 5l'

' • 00 1r 1 3. ono • no~ 1 . 35L;. 1. 17

1 ?0' • \;1; 7 . 0055 .~ o J IJ'J . 106 ·- I .%7

r L;·/ :-) ,_ ?'_, ,._ 0 (, .;. .r,u69 5. GOO . 113 1 • .

1 ~ ., I 1 • ~-2?, :) ., . . . 0077 5. l)ll 0 • 113 -.

1 ~ L:·2/. • •• ~j <) . 00(33 6 . i) )\) • 111· .....

3\ ,, /. . 0097 \) . <.:9 0 • 118 1. 412 1 . .

I t::. 1.. '). c, 1.3 H5 • - t • ; -, J. . 0 111 !) . 05[) .

. 159 1 . 32 3 ~~ . on .(, ·2 . 0125 9 . 000

r r ; 1 . 233 • • J ) J ! '· . 0139 ~ .

Time t : 30 seconds 87

Table C. 9. Data and Results for System 8

c Refractive Concentration Sox dC dC Dista nce Index gm. solute dx D

em (n - n 0 ) per 100 cc. sol. gm/sq.cm gm/cc/cm sq. cm/sec

.I "{ . 11()19 l • l ;. )() • ()td J • u ·r ' J • • 2(

• on :>9 ( • c iO • 1. ()I. 1 .-· 'I

I . 7·• • OOTI _, • ·' r J ) • J. D c J. . :nn

1 ?; '1 i" 1 • ((l7G 5. ·roq • 1 ()0 - . ..

• t:. • {'" a 1.1 7. 1 10 • 1 )_ ·~ l. 20) . ' IJ • I\) _,

1 1 • r, ', l ..I ;l • ' ?, ,) . 110 . · ·r 2 • n 119

• (' 2 9 . ?? • Q ].II() J. 1. 0()0 • 2 P4·

Time t = 15 seconds 88

Table C. 10. Data and Results for System 9

c Refractive Concentration Sox dC dC Distance Index gm. solute dx D

em (n - n 0 ) per 100 cc. sol. gm/sq.cm gm/cc/cm sq. em/sec

. ) . 'I I . • OOL~ J. • .) ~~· ) oO~ E1 0 f)~ l l a 15

1-. ') ~ ., 003C1 3.000 . 0°5 l.L..3P. '· . ;; 2

1: • J '0 o Fl5 1.:5 ff) ~ . :; 5

n /.1 j :; i) o ('I .1 (: •.; :-> • 1 'II) .llO l. 5?? "

,) • I l (lt") J. 1. ') ) .51·." o I J • 0 :'f) . l107 .': .

- ; ].. r, . ') / .' . ft3 0 0:1 ,_. 1 7. ;J • 1 J. ::·.

, ! II 121 1.54:"' / .. uC) • l) ].04- '1 0 0 . . "' -. 1. 5 (1

. 203 l.l·lL~ :·. (C o I \; 5 • 0130 J. o. r;n•)

Time t = 15 seconds 89

T a ble C. 11. Data and Results for System 10

c R efractive Concentration Sox dC dC Distance I ndex gm. solute dx D

em (n - n 0 ) per 100 cc. sol. gm/sq.cm gm/cc/cm sq.cm/sec

I 1 f, ,1\.. . 0016 l . DOO o i/31 l o It:

- • 00 6l~ Lr- • J 1} 0 . 1 () 1 l • :·~ 7(") . .r

. .'·.(_; . OOGO 5.000 . 1n1:> J • .:.# .) r; :-~ • 2 .,.

r, ' Qfjr,~) _, " .. , • 1 1J6 1 • ~· :- 0 :.. • . !. 0 :u

-; I I! '> 0 :·J Q . 10 7 l. 31Z. .. ' 0 ' .. \':· . lJ!Jf...J6

•l O I) ('1 () 1 . 2 5? ":1 . 7',_ . ,· • I I J 1.2 . 11?.

• 1 ;~ rJ 1.176 'r • l ' .' ,•, . lJl2E :. uUO

1tS () 1 • ,, /~':- ' • :J •L . 0 l 'A 9 . ,JOO .

. 19C • 92 3 ~ . r:q .. ' :.:> . 0 1 60 l0 . 'J00

Time t = 12 seconds 90

. Table C . 12. Data and Results for System 11

c Refractive Concentration Sx dC dC Di :~ t ttn c e Index gm. solute · · 0 dx D per 100 cc. sol. gm/sq .cm gm/cc/cm sq.cm/sec. e m (n - n 0 )

Time t : 15 seconds Table C. 13. Values of Integrated Diffusivities, D, for the Systems Studied

System Solute 15 No. Polymer Solvent 10 gm. /100 cc. soln. sq. em. I sec. -6 1 Polyacrylonitrile Dimethylformamide 2.06xl0 5 2 Natrosol 3. 51 o/o Water Urea 3.42 X 10- 5 3 Natrosol 3. 51 o/o Water D-Glucose 2.31xlo- 5 4 Natrosol 2. 5o/o Water Urea 1. 83 X 10- 5 5 Natrosol 2. 5o/o Water D-Glucose 1. 52 X 10-

6 Natrosol 1. 88% Water Urea 3. 38 x 1o-S

1 Natrosol 1. 88o/o Water D-Glucose 1. 42 x to-s 5 8 Natrosol 1. Oo/o Water Urea 3.28x1o-

9 .. Natrosol 1. Oo/o Water D-Glucose 2. Sx1o-5

10 Natrosol 0. 8o/o Water Urea 3.38x 10-S 5 11 Natrosol 0. 8o/o Water D-Glucose 3. 0 X 10-

..4)- 92

APPENDIX D

Computer Programs

The programs used for the computations described in this thesis are given in this appendix. All programs are written in the

Fortran II language. The programs were tun on an IBM 1620 Model

II equipped with auxiliary disk storage. Program No. 1

C C***47980:NX032 DALAL GIRISH T 05/02/66 FORTRA N 2 0015 0 18 0 C NON-LINEAR CURVE FITTI NG BY HOD IFIED GQ,\lPER TZ EQUATIONS , FOR E0UAL C I NTERVALS OF THE I ND EPE NDANT VA~IABLE . DIMENSION X(20) , Y(2J) , S(3) , G(20) READ 905 , KK DO 900 I JK=1 7 KK PRINT 901 PRINT 902 .I JK PRINT 903 PRINT 904 READ 100 ,( S(J) ,J=1,3) READ 101,N,A READ 109,(G(I) 7 Y(l),I=1,N) DO 1 l=l,N 1 X(I)=(G(l)/0 . 01- 1. 0 ) SUM=O . O K=N/3 L=1 K1= K DO 10 J=1 , 3 DO 11 l=L, K 11 S (J) =S(J)+LOGF(Y(I) - A) L=K+1 10 K=K + K1 P=N/3 P=l./P CN=( $ ( 2 )-5 ( 3) )/( S( 1)-$(2)) C=CN **P AL =( S ( 1 ) - ( ( S ( 1)-S ( 2 ) ) I ( 1. -C N) ) ) * P AL=EXPF( AU BL=(S( 1)-$ ( 2 ) ) * ( 1.-C)/( ( l. - CN >**2l BL=EXPF(8L)

SR =0 . 0 -D w Continued PRINT 103,A,AL,BL,: PUNCH 10 2,A,AL,BL ,C PRINT 105 PRINT 104 DO 12 I=1, N YC =A+ AL >:q BL ~:: ;'< ( C:;: :::X { I ) ) ) D=YC -Y {I ) DV = {D /Y { I ) ) * 10 0 • 0 SUM=SUM+{A9SF{DVll SR=SR+(D**2l PRINT 107,X{!),Y(I),YC,D,DV 12 CONTINUE PRINT 105 PRINT 106, SR E=N DEV=SUM/E PRINT 105 PRINT 108,DEV 900 CONTINUE CALL EXIT 100 FORMAT(3E18.8) 101 FORMAT(l5,E18.8) 102 FORMAT(4E18.8) 103 FORMAT(7H ALPHA=,E1B.8,4X,2HA=,E18.8,4X2HB=,E18.8,4X2HC=,E18.8) 104 FORMAT(9X4HX(l),14X4HY(!),12X8HY(I)CALC,12X4HDIFr,12X8HPER DEVIl 105 FORMAT{///) 106 FORMAT{25H SUM OF RESIDUAL SQUA~E =,E18.8) 107 FORMAT(5E18.8) 108 FORMAT(25H ABS. AVE. PERCENT DEVI.=,E18.8) 109 FORMATC2El8.8) 901 FORMAT ( 1H 1) 902 FORMAT(1X10HSYSTEM NO.,I3l 903 FORMAT(//) 904 FORMAT(/) 905 FORMAT (I 5) ...c END ~ G(I) = Distance, em.

Y(I) = Concentration, gm. I 100 cc. soln.

N = Total number of data points.

A = Alpha (intercept).

AL, BL, CL = Constants for Gompertz Equation.

-.J:I U'l Program No. 2

*FANDK1604 C C***4 8582CNX032 DALAL , G. T. 05/03/66 FORTRA N II 00 10 015 0 C CALCULATI ON OF THE DERIVATIVES OF SIGMO I D CURVES 0 I t~ EN S I 0 f\! , YT ( 4 0 0 ) , P T ( 40 0 ) , Y( 2 0 } , X ( 2 0 ) , F ( 2 0 ) , P ( 2 0 ) , PC: ( 20 ) , S ( 15 , 16 ) OIHE NS I ON YC(20) READ 100 , (YT (I l ,PT( I l, I=1, 389 ) READ 905 , KK DO 900 IJK=l , KK PRINT 90t PRINT 902 , IJK PUNCH 905 , I JK PRINT 903 PRINT 904 READ 10l,Sl,S2,S3,S4,S5 READ 102,K READ 109,(X(I) , Y(l),I=l,K) DO 1 I = 2, K P(l)~L OGF(20 . 0*Y (l)/(L OGF(l00 .0-Y(l))/ 2 . 303 )) /2 . 303 F(l )=(X(l)-X(l))/P(l) S1=S1+X(I l S2=S2+X(I l *X( l) ~ S3=S3+F( l ) 1 S 4 =$4 +X ( I ) :::~ F ( I ) S (1,U =K - 1 S <1, 2 l =S1 S ( 1, 3)=S3 $(2 , ll=Sl S(2 , 2l=S2 S ( 2 , 3 l =S4 PRINT 100,($( l ,I ) ,I =l, 3) PRINT 100 ,( $(2 , Il,I=l,3) B= ( S ( 1 , 2 ) * S ( 1 , 3 ) - S ( l , l ) ::~ S ( 2 , 3 ) ) I ( S ( l , 2 ) * S ( 2 , l ) - S ( l , l ) ::~ S ( 2 , 2 l ) A= ( S ( 1 , 3 l - B:::: S { 2 , l > l IS { 1 , l l -c PRINT 10 6 , A, B a- Continued

· 'l DO 2 I =2 ,-\ P C I I ) = I X I I l - :·: ( 1 l l I ( :. + : ..:: X ( I l l J = l 3 J= J+ l PS2 =PT ( J) - ?C (I ) ------·- -

IFIP S~ l 4 , t.. , 5 4 PSl= P T( J ) ------... c-1 o 3 ------5 PS2=PT(J) YC ( I ) = ( PC ( I ) - P S 1 ) ::'( ·( T ( J l - YT ( J- 1 l l I ( P S 2 - P S l ) + YT ( J - l ) O=Y(ll-YC

Y = Concentration, gm. / 100 cc. soln.

DV = Concentration Gradient, gm. I 100 cc. soln. I em.

...0 (X) Program No. 3

C C***48581CNX032 DALAL, G.T. 05/03/66 FORTRAN II 0010 015 0 C CNX032 DALAL, GIRISH T. 05311 2/18/66 C LOCATING THE LINE OF OR IGI NAL INTERFACE DIMENSION YA(25) ,YB(25) READ 905, KK DO 900 IJK=1,KK PRINT 901 PRINT 902.IJK PRINT 903 PRINT 904 READ 100 , A, B, f>_, N READ 10l,ALP,AL,BL, CL PRINT 102 DP=0.0001 F=N-1 M=F/2.0+1.0 DA= ( 8-A) /F X=A 00 10 I=l,N YA(Il=ALP+AL*( BL**(CL**(X/O.Ol-1.0))) X=X+OA 10 CONTINUE Sl=O.O 52=0.0 MA=N -1 DO 20 1=2 2 Sl=S1+YA(l) 20 CONTINUE MB=N-2 0 0 2 l I =3 , r-1 B , 2 S 2= S 2 + YA ( I ) 21 CONTINUE SIM=(YA(l)+YA(N)+4 . J*Sl+2.0*S2)*DA/3.0 ._o ._o Continued Al=SH-1 15 DB=(P-Al/(F/2.0) X=A DO ll I= l, ~1 YB ( I ) =ALP+ Al ~q BVP'.: ( Cl ::: * ( XI 0 • 0 1-1 • 0 ) ) ) X=X+Db 11 CONTINUE Sl=O.O 52=0.0 MA=t-1-1 DO 22 I=2,MA 1 2 Sl=Sl+YB(I) 22 CONTINUE MB=M-2 QQ 22 I=3,MB,2 S2=S2+YB( I) 23 CONTINUE SIM=(YB(l)+YB(M)+4.0*S1+2.0*S2l*DB/3. 0 A2= SIN A3=(B-P)*YA(N) - (Al- A2l O=A2-A3 OV=(A2-A3)/A2 PD=DV*lOO.O PRINT l03,A1,A2,A3,D,PD,P IF ( AB S F ( DV ) - 5. 0 E- 0 3 ) 6 ,- 6-; 7 7 IF(A2-A3) 12,6,14 12 P =P+DP PRINT 10 5 'p GO TO 15 14 P=P -OP PRINT 105 'p GO TO 15 6 PRINT 104 PRINT 102 PRINT 103,Al,A2,A3,0,PO,P PUNCH 10ltALP_,. ALtBL,CL .... 0 PUNCH 101,A,B,P 0 900 CONTI NUE CALL EXIT 100 FORMAT(3El8 . 8 ,I l8) 101 FORMAT( 4El8 . 8 ) 102 FORMAT(llX , 2HA1 ,15X,2HA 2,15X, 2HA3 ,13X,S HA2 - A3 ,11 X,5HP . DEV ,l6X,lHP ) 10 3 FORt-I ATToFTT. 8 ) 104 FORMAT ( I I I) 105 FO RMA T(5X,2HP= , F l 8 . 8 ) 901 FO R~I AT ( lH 1) 902 FORMAT(lXlOHSYSTEM NO ., I3) 903 FORMAT (I!) 904 FORMAT(/) 905 FORMAT (I 5) END

A = Lower limit for Distance, em.

B = Upper limit for Distance, em.

P = Position of the line of original interface.

ALP, AL, BL, CL = Constants for Gompertz Equation.

A ,A ,A =Areas, asshowninFigure3.lla, page45. 1 2 3

-0 Problem No. 4

c C***485B2:NX032 DALAL , G.T. 05/03/66 FORTRAN II 00 10 0 15 0 c CNX032 DALAL , GIRISH T. 0 5311 3/5/66 c EVALUATIO N OF Ii'ITEGRAL DIMENSION X(50}, C(50} 7 X0(50} 7 YD(50l READ 905, KK DO 900 IJK=l,KK PRINT 901 _____..P_..R,_,.I...... ,Nw.T_9D2 ._1 J K PRINT 903 PRINT 904 READ lOO,ALP,AL,BL,CL READ lOl,A,B,P READ l03,N,K,L,NT F=N READ 104, (X 0 ( I } , YD ( I l , I= 1, NT l JJ= 1 DO 10 I=1,K GO T 0 ( 30,31 l , J J 30 M=I*N GO TO 32 31 i'l=(I*Nl+l JJ=JJ-2 GO TO 32 32 X(ll=A C(1)=ALP+AL*(BL**(CL**(X(l)/0.01-l.Ol ll MA=M-1 t'vtB=fJ\-2 At'vi=!'-tA X(Ml=XO(!} OX= (X ( H l -X ( ll ) I ( Ml l Sl=O.O s 2=0 .o DO 11 J=l,~· I A

Continue 0 -N X(J+1) =X (J) +OX C(J+1) =ALP+A L*(BL**CCL** (X(J+1)/) . 01- 1. 0))) 11 CONTI NUE D0 12 J = 2 , 1·1A , 2 S1=S1+C(J) 12 C O N Ti i~ UE D 0 1 3 J = 3 , l'i 8 , 2 S2 = S2 +C ( J ) 13 CONTI NUE SIM=CCC1)+C(N)+4. 0*S1+2.0*S2) *DX / 3 . 0 SIM=SIM+(C(M) * ( P-X( M))) PRINT 10 2 , X( M),C( M),SIM,I PUNCH 100,SIM JJ=JJ+1 10 CONTINUE A2=SIM JJ= 1 DO 20 I=1,L GO TO (40 ,4l),JJ 40 M=I*N GO T O 4 2 41 M=CI *N )+1 JJ=JJ-2 GO TO 42 42 XCU=P C(1)= ALP +AL*(BL** lCL** (Xl1) /0 . 0l-1. 0 ))) MA=M-1 MB=M- 2 AM=MA KJ=K +I IF ( KJ - NT) 2 4 , 2 4 , 9 00 24 X( M)=XD( KJ) OX= (X ( i'-l ) -X ( 1) ) I l Ai'\ ) S1=0.0 -----~$2=0 .o Continue w0- DO 21 J=l, MA X(J+1l=X(Jl+DX C(J+ll= ALP +AL * ( BL** (CL**(X(J+1l/1 . 0 1-1. 0 l)) 21 CONT HJUE DO 22 J=2,t·1A , 2 S1=Sl+C(Jl 22 CONT1 NUE D 0 2 3 J = 3 t ~-I 8 t 2 S2= S2+C ( J l 23 CO NT INU E SIM=(C(ll+C(Ml+4. 0*S1+2 .0*S2l*DX/3. 0 SIM=C(Ml * ( X(M ) - P)-SIN SIM=SIM+A2 PRINT l02 1 X(N),C(Ml,SIM,I PUNCH 100 , SI M JJ=JJ+1 20 CONTI NUE 900 CONT INU E - CATLEXTT 100 FOR t-I AT (4E 1a . a l 101 FORHAT ( 3E 1a . a) 102 FORHAT (3Fla . a , Ila) 103 FORMAT(4 110l 104 FO RNAT( 2E l 8 . 8 l 901 FORMAT(lHl) c CONTI NUED 902 FORMAT (lXlOHSYSTEM NO ., I3l 903 FORMAT(//) 904 FOR~·\ AT (I l 905 FORMAT(I 5 l ENO

Same as in Pr-ogram No. 3

Sim = Value of x dC. 0 Soc -~ 105

'·· !('. ~· \.

,...... ,_ <. I - ' ..

i -

· r: t- t"'·.. , ' ( \.

··-. ' )

1 -'- - U i

u c

v , 0 IJI

::.> :::: _.I tll .. ~ · v: ..-:..,

I!_ 1!1 .-I '.L II ~ 0 ~I) 1-< ..,_. ::: Ill ., .. .. lr) .. V l .-I .-I : I I ~---< II II V) c. ) 0 1-- • c-: >-I ( "\J 0 IC\ Q 0 1! 1 _,....- Q ... r-l ' '- 1- n '-.::: c . (/) ( C) ( '> ...... -· n. 1- -::- ~ ...... ('_ : ·:< ·.~· - ~ >- ~ .:::: ..-l ,_, - ._... 0 ' r·"\ Vl -"' ~ II .. 1-- X W W - · Y. t--c ...... ~- . ,., 0 ...... j· u .. -,.-IN rC\ ... c; ('(') ,-t N ..j· .-- ...... M ..., -;:-: .-I - (<'\ I- z OO Q C>N.-tNN II - w 0 I '.> 1. .> o 0 UJ lU - 1- ..... c:~ lC\ ..... 0 C) 000 v 0 ' N N 0 ' 0 0 0 0 ,_. lU :::.::: C! t- 0 N 0' :::> .::. > ~ ~- :· ·· · ..... c; O' 0' <2.' 0 ("\j E .-l...-1.-1...-l Vl II II "J II II I.U g nl ....; - · ··; 1/) lJ" 0 ,...... f 0 II ...... , CJ ~I-1-,_,,_, ,_. 0 0 I- 1- I- I- I- I- I- 1-- ...... -c-._...... -.....,.,_~.7z t-t- _J ..;j · U>- 11! 0()"' _,._ o __ ,;:.. ·"- oooo b.O 1-4 - - ..,_... ,_.. ,.:_ . ..: . .-J i:: 0 CJ ~u lL <{ 0 ,_. 1~(\ .~~,, CJ ,_, U

uuu 1 oo r= o.::: ; ;.·.- ~- < r s ; := 1·5 :-rr--·-· 1 0 1 F G ~ : . ::~ T ( 2 ;:: 1 :: • .-, ) 10 2 F r :~.i i·. l < E ! -: • ?. ) 2 0 0 := C' ?. . ;_, T ( ::> :~ -· :-iT ! .. c T = , r 5 • l ) 2 o 1 F o~:.; . ;;.--. -r < s xo. r J r s T , .. :: ::: :: , 2 :< ::> '-i : o:!:: • , ? >:-= LiJ : ~ T i:: ..; ~ :\ L , 2 ·~ 1 1 r-· >,: ·: • ,: :- ; o: • , 2 :· z~ . !' 1 > 2 o2 FoR : · ;:,. T < 5 x3 :-1c .• , s ~< 9 H G. · • 1 1 : L , 5 x 1·.., • 'r., · · • 1 s :-: • c: ,: . , 5 :n oH G , : • 1 · L 1 :: · • , :::; >: ?. : I ~ :- • l 2 0 3 F 0 ~~;:, ;\T ( ? ); , f l ~.l • 3 , :. lJ • '.· , r 1 1 • 3 , F ·, • 3 , ? F l 0 • 3 l 90 1 rD f'?.i ·i..cJ. T( 1'-i 1 l 902 FCRi·:t.·.T ( l X 10HSY Sn:·; ;;·) ., I 3 ) 903-FORi:-;;T ( I I) 9 0 £:- rOP,i:ii·.T (I) 905 FC' f<.:·.t\1 (! 5 ) E i\~ D

X(I) = Distance, em .

R(I) = R efractive Index.

C(I) = C o n centrat i on, gm. /100 c c. so ln. EN(I) = Va lue of Jorc X dC. ED(I) = Vo lue of Concentration Gradient.

DC(I) = Diff e r e ntial Diffusiv ity, sq. em. /sec.

T = Time , s e c.

-0 0' 107

APPENDIX E

Integration of Concentration-distance curves: The diffusion coefficient and its dependence on concentration can be observed by contacting two media at time t =O, and then studying the c oncentration distribution at some known subsequent t ime t. The conditions of the experiment are

c = 0, t = o, x < 0 (2. 17)

c = co, t = o, X > 0 (2. 18)

c = 0, t = t, X -- oo (2. 19)

c = Co, t = t , x-oo (2. 20)

covco all x ' s (2.21) c = ' yo+Vco wher e C is the concentration of the component we are interested in;

Vis the volume of the phases; and x = 0 is the position of the initial

interface between the two phases at timet= 0. Assuming that there

is no change of volume on mixing, that the concentration C is measured

as mass per unit volume of the system, and as a result of boundar y

conditions (2. 19) and (2. 20), the Boltzmann variable

112 (2. 1 O) r = x/2t

and as explained in Appendix A, the relati on below can be obtained.

x dC •s; . (2. 22) = Zt (~) ~ 108

In order that the boundary condition be satisfied, the origin

from which x is measured must be such that

I c reo So x dC = Jc' x dC (2. 23)

is satisfied. This means that the plane, x = 0, must be selected so

that the two shaded areas in Figure E. 1 (a}, may be equal. It is

physically impossible to know where the plane, x = 0, is located.

Equation (2. 23) can be derived as follows:

Consider a two dimensional system as shown in Figure E-2,

with the third dimension (depth of the fluid layer, not indicated on the

diagram) being unity. From a material balance we know that the

solute transferred from one phase to the other is equal to the solute

present in the latter phase, if it had no solute at the beginning. Thus, from Figure E-2, on considering the element dC, the solute present in it is g iven by

solute in element dC = (- x) (dC) ( 1. 0} (E. 1)

Therefore, the total amount of solute present in area A 2 from C = 0

I to C = C · is given by cl cl

A 2 =So -x dC .- - So x dC (E. 2)

1 Similarly, for area A 3 , from C = C to C = C 0 , the solute present is given by (E. 3) c c

----- X : 0 X X= 0 X

(A) (B)

Figure E. 1. E valuation of De from a Concentration-distance Curve Using Equation {2. 22).

...... 0 -.!) 110

X A

0 c

-x

c Figure E. 2. Evaluation of the SO x dC 111

But from the material balance, as explained above, we know that

Solute in Az = Solute in A 3 c' co · s X dC = X dC (Z. 23) 0 Sc'

We have assumed the system to be a constant volume system.

In a constant volume system,· equation {2. 23) represents a conser-

vation of mass condition. It can be satisfied if x is measured from

the initial position of the interface between the two solutions at time

t = 0. The negative sign on the left hand side of equation (2. 23) is

necessary to take account of the negative area and slope of the portion.

of the curve to the l eft of the locus of x = 0 .. Thus in this case, a plot of concentration versus distance is prepared for a known time t, as shown in Figure E. 1 (b). A plane x = 0, i.e. the position of the initial interface is located and then De is evaluated at various concentrations

C, using equation (2. 22). The value of the integral can be calculated by using a numerical integration method such as Simpson's rule, and the gradient is calculated by taking the slope at that concentration.

The shaded area in Figure E. I (b), represents the value of SoC x dC. 112

APPENDIX F

Some diffusion data take the form of an elongated S when

plotted. Such curves are characterized initially by a very small

slope followed in order by a rapidly increasing slope, an interval

of nearly constant slope, and finally a rapidly decreasing slope which ultimately approaches zero. The modified Gompertz equation has

been found to be satisfactory in representing such data. The modified

Con1pertz equation is

( 3. 2)

wbere a , a and a are constants, and a is the value of y at x 0. 1 2 3 = Tbe concentration- distance curve can be represented by equation

(3. 2). In order to get the concentration gradient-distance curve, it

is necessary to differentiate equation (3. 2).

du d au au ln a - (F. 1) dx = dx

Therefore, substituting

(F. 2) X :;;: p a3

we get p (F. 3) y = a l az 113

Therefore,

~= dx (F. 4)

Now, from equation (F. 2), we get

dP = (F. 5) dx

Substituting in equation (F. 4), we get

(F. 6)

Substituting equation (F. 2) for P, we get

~= dx (F. 7)

Equation (F. 7) may be used for evaluation of the concentration

gradient. However, this method was not found satisfactory when com-

pared with Davis' method (8). A comparison of the concentration

gradients obtained by both the methods is given in Table F -1, for

the test system polyacrylonitrile-dimethylformamide. Davis' method

for evaluation of the concentration gradient is applicable even for

cases when the Gompertz method is not found to be satisfactory (8).

Hence Davis' method was used for evaluating the concentration

gradients.

The Gompertz relation was, however, used to evaluate the integral in equation (2. 22). The average absolute percentage de.viation 113a

of observed concentrations from those predicted by the Gompertz relation for each system is given in Table F . z.

An example of the accuracy of the Gompertz equation is shown in Figure F. 1 on page 114b for System 3. 114

TABLE F. 1. Comparison of Concentration Gradients Obtained by Two Different Methods

c gm/100 cc. sol. (~)I (:~) z

. 75 1 ..48 1. 55 2. 85 3.80 4.00 5. 30 4.35 4.70 7. 05 4.41 4.80 8.05 4. 34 4.60 11. 80 3. 31 2. 17 14. 10 2.05 1. 55 15.50 1. 15 1. 10 16.50 . 61 . 65 17. 15 . 31 . 32 17.30 . 16 . 10

= Concentration gradient evaluated using derivative of Gompertz equation {Equation F. 7)

= gm. Icc. sol. em.

= Concentration gradient evaluated using Davis' method {8).

= gm. Icc. sol. em. 114a

TABLE F. 2. Average Absolute Percentage Deviation of Observed Concentrations from Those Predicted by Gompertz Equation and Davis' Method

System No. Average Absolute Average Absolute % Deviation Using % Deviation Using Gompertz Method Davis' Method

1 4 . 00 4.26

2 7. 13 4.02

3 5.08 2.94

4 2.47 2.48

5 8.01 3.91

6 5.25 1. 98

7 3. 74 1. 59

8 2. 17 3.85

9 5.30 4. 41

10 3.47 0. 51

11 0. 96 6 .84 114b

l+t+t +i=FIH-l It-! 1-H IH ltf:tt m ~ n ~l i nt i:J :q I jt t 11 I!~ • • ;:. :!f Iii! il'i IiIt t ! :iI ii r I! I l 10 I I M1 11H 1iP! :.lHi' ~ I -t I :E t•• I:! ...... ,0 :;j • )(] ...... 8 0 VI i Cl). u u ~ 0 i 0 ...... -..,v 6 ...... :;j 0 Cl). b.O.. I:! 0 Experimental ...... 0 4 .., - •' value cU f-4 & Calculated value 4 L ~v u ~ ~ I:! 0 u 2 .

- 0 0. 00 o. 02 0.04 0.06 0.08 0,10 Distance, ern.

Figure F. 1. Concentration Profile for System 3, at t = 16 Seconds. 115

APPENDIX G

Procedure for Measuring Experimental Interference

Pattern: A sketch of an experimental interference pattern is shown in Figure G. 1. M easurement of the fringe pattern provided a means for obtaining a concentration versus distance curve.

A line PQ (parallel to the fringes and perpendicular to the interface) was drawn, as shown in Figure G . 1. Several fringes were intersected by this line at various points. The number of the points of intersection were counted. The difference between the refractive index of the solution and of the solvent was divided by the above number of points in orde r to obtain an increment of r efractive index represented by the distance between two points. Along any line per pendi'cular to the original interface, the change in refractive index between any two adjacent fringes is constant. A microscale,

(magnified by. the sa~e amount as the interference pattern) was thE:m used to measure the distance between the points of intersection. The refractive index-concentration relationship was known fr om previous measurements. Therefore , a curve of concentration versus distance could be prepared from the measurement of distances along the line

PQ as described in this procedure. 116

' ! l I 1 I I

Fig ure G. 1. A sketch of Experimen tal Interference Pattern, showing how to measure the Fringe Pattern. 117

VIII. BIBLIOGRAPHY

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~ . ACKNOWLEDGMENT

The author wishes to thank Dr. R. M. Wellek, Associate

Professor of Chemical Engineering, who suggested this investigation

and served as research advisor. His help, guidance and encourage

ment are sincerely appreciated.

Thanks are extended to Dr. M. R. Strunk, Chairman of the

Department of Chemical Engineering, University of Missouri at Rolla, for his aid in providing financial assistance.

The author is also grateful to Dr. R. M. Secor, Research

Engineer, E. I. du Pont de Nemours and Co., (Inc.); .Wilmington,

Delaware, for his advice during the course of this investigation and for supplying the author with a polymer sample.

Thanks are a lso extended to Mr. J. J. Carr of the Chemical

Engineering D eP,artment, and Professor R. Gerson of the Depart ment of Physics, for their help and guidance in the photographic work and metallization of microscope slides, respectively. The assistance of Don Byrd and other individuals of the Computer Science

Center, University of Miss~uri at Rolla, is gratefully acknowledged. lZO

X. VITA

The author, Girish Trikamlal Dalal, was born June z, 1941,

in Ahmedabad, India. His elementary and high school education were

obtained at the St. Xavier's High School, Ahmedabad, graduating in

1957. After high school, he attended the Gujarat University,

Ahmedabad, graduating in 1961 with a degree of Bachelor of Science in Chemistry. He then served as a Chemist with the Atul Products

Pvt. Ltd., Bulsar, for a year. In September 196Z, the author entered the University of Missouri at Rolla and received a degree of

Bachelor of Science in Chemical Engineering in August 1964. In

September 1964, he enrolled as a candidate for the Master of Science degree in ~h e mical Engineering. He was employed as a part-time

Graduate Assistant in the Department of Chemical Engineering during the period from September 1964 to May 1966.