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Kinetic dissolution measurement of polymers by solution viscosity recording
Polymer Testing 8 (1989) 339-351
Kinetic Dissolution Measurement of Polymers by Solution Viscosity Recordingt M. M i k u l a , M . ( ~ e p p a n , J. B l e c h a , L. L a p ~ i k Faculty of Chemical Technology, Slovak Technical University, 812 37 Bratislava, Czechoslovakia & V. Kaligek University of Chemical Technology, 532 10 Pardubice, Czechoslovakia (Received 24 February 1989; accepted 18 April 1989)
ABSTRACT A laboratory device for continuous dissolution measurement of polymers by solution viscosity recording is described. The nonlinear viscosity dependence on the polymer content in the solution is utilized to measure the dissolution by recording the driving motor power of the discs rotating within the solution. A short theoretical background of the diffusion process of dissolution is introduced, with comments. The diffusion coefficient of the solvent into the polymer, velocity of dissolution, thickness of the surface swollen layer and other kinetic and activation parameters can be calculated by treating the recalibrated viscosity data. Some results of the dissolution in water of carboxymethylceUulose and some dissolution tests of the degree of fixation of textile materials are given together with suggestions for possible alternative applications.
INTRODUCTION The dissolution of polymers has s o m e typical features when c o m p a r e d with the dissolution of low-molecular-weight solids. The transport of polymer into the solvent does not take place immediately after both fractions come into contact. O n e of the characteristic kinetic parat Dedicated to Professor Josef Schurz, DSc, Carl-Franzens University of Graz, in honour of his 65th birthday. 339 Polymer Testing 0142-9418/89/$03.50 (~) 1989 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland
M. Mikula, M. Ceppan, J. Blecha, L. Lap6ik, V, Kalfgek
340
meters of the process is an induction period to which depends on temperature, molecular weight of the polymer, kinetic and thermodynamic properties of the solvent and also on the method of polymer sample preparation. 1-~ During the induction period the solvent penetrates into the polymer by diffusion and a swollen surface layer is formed. Stationary dissolution starts after the induction period, provided that there is a constant mixing. The thickness of the swollen surface layer becomes invariant in time (Fig. 1). The hydrodynamic pressure of the solvent on the swollen layer surface, caused by mixing the solvent, is the dominant factor determining the swollen layer thickness. 3 Such important technological parameters as the mean diffusion coefficient, the thickness of the swollen layer and the velocity of dissolution can be calculated directly from the dissolution kinetic curve. Moreover, from the temperature dependences various activation parameters can be obtained according to Eyring's theory of reaction rates. 4"5 Apart from the essential influences of the temperature and the type of both the chosen polymer and the solvent exerted on all dissolution parameters, there exists a strong dependence of these parameters on the physical structure of the polymer, on its crystallinity and surface quality and on the whole temperature history of the polymer sample and its preparation. Earlier publications 1-4 have dealt with the use of a refractometry technique to measure the dissolution kinetic curve that is unsuitable for composites or polymers with low-molecular-weight components. The present paper describes the continuous automated measurement of the dissolution process based on registration of the solution viscosity changes being indicated by mechanical rotating elements. The equipment is simple, cheap and easily connectable to a microcomputer. The two disadvantages of the method are the nonlinear calibration curve of an applied rheoviscometer and the undefined shear rate. /
SATURATION/-'
"
B DISSOLUT N
to
Fig. 1.
TIME
Dissolution kinetic curve; to = time of swelling (induction period).
Kinetic dissolution measurement of polymers
341
C O M M E N T ON T H E T H E O R Y OF D I F F U S I O N CONTROLLED DISSOLUTION In the case when the diffusion of the solvent into the polymer is the slowest mechanism in the dissolution process, Fick's first law for every component i can be used:
ji = - ( c i D i / R T ) grad/~i
(1) where j is the flow density, c is the concentration, D is the diffusion coefficient, R is the gas constant, T is the temperature and /~ is the chemical potential. For the steady state of dissolution (the stationary regime--Fig. 1), providing that the hydrodynamics are constant, a simple relation among kinetic parameters can be derived from Fick's first law in one dimension and in ideal solution c o n d i t i o n s ? : The diffusion problem is first solved in the moving coordinate system firmly bound with an advancing phase boundary. Then the transformation of variables from the moving coordinate system to the laboratory one with a fixed solid polymer is performed carefully to obtain eqn (2). -D~(1 - qos)
ac~ = ~. c~ cox
(2)
where $ is the moving velocity of the interfacial swollen layer into the polymer [m/s] (i.e. the dissolution rate, better designated as Uoc4); Ds, q)s, c~ are the diffusion coefficient, the volume fraction and the concentration of the solvent at a given distance x inside the swollen layer, respectively. Provided that D~ does not d e p e n d on c~ (D~ is the constant within the x coordinate) and that (p~ = cJp~, where p~ is the specific mass of the solvent, we can carry out the integration of eqn (2):
f
(I
10c~ = f Uoc Ox
-
(3)
u,,c
Inc~-cJp~-
-- x + K D~
Let us designate the ratio DJuoc by 6o[m], which represents the swollen layer thickness (see below). The value of the integration constant K can be obtained from the boundary condition: c~ = cs: for x = 0 (see Fig. 2), where c,.~ is the concentration of the solvent in the liquid phase. Thus K = In cs: - c~:/p~ and
c~=c~,,exp
(x t -~,
exp
P~
/
(4)
M. Mikula, M. Ceppan, J. Blecha, L. Lap~fk, V. KalEek
342
Cs, t = Ps
I . . . . . . . . .
"% ."1
APPROXlM. \ x ",J LIQUID
~
\\
%~..x SOLVENT JSWOLLEN ~ POLYMER -LAYER -
~
o Fig. 2.
..:' ~ \
.'"
.: P
SOLID
\
INFILTRATION
\
Uoc/ r////~.9c//
&o
'~
Steady-state situation at solvent-polymer interface,
o the
~o= Ds/uoc
x
Fig. 3. The solvent volume fraction inside the swollen layer: - - , calculated from eqn (4); . . . . , linear approximation. Polymer volume fraction, qgp: . . . , obtained experimentally by Ueberreiter. 3
The variation of the concentration of the solvent in the swollen layer according to this equation at the beginning of the dissolution process (when cs,,- p~) is illustrated in Fig. 3. Any polymer property was not considered until eqn (4) was derived, except for the assumption of the existence of the swollen layer. The shape can be compared with that of the polymer volume concentration q0p in the swollen layer taken from the experiment by Ueberreiter 3 (the dotted line in Fig. 3) for polymers at temperatures below Tg. In order to calculate the diffusion coefficient from the experimental data, usually including the induction period to and the stationary dissolution rate U o c - d c / d t , eqn (2) has to be averaged within the swollen layer. First of all, the expression Ds(1 - qg~) in eqn (2) can be substituted by the mutual diffusion coefficient D which depends on the volume fractions of both components: D = Ds(1 - ~s) + Dp(1 - tpp)
(5)
where Dp is the diffusion coefficient of the polymer in the solvent. For macromolecules Dp---->0 and eqn (2) becomes ¢9Cs - D ~ x = UocCs(X)
(6)
This equation can be integrated in a different manner than was used
Kinetic dissolution measurement of polymers
343
previously [eqn (3)] to obtain averaged variables:
- f D aCs= Uocf cs(x)dx
(7)
After the integration through the swollen layer, supposing that there exists 6 for which Cs(6) = 0, eqn (7) becomes
D = Uoc6~s/ACs
(8)
where 6 represents the real thickness of the swollen layer a n d / ) is the mean mutual diffusion coefficient: /) _
1 mCs
f
D acs = 1 swollen layer)
•
fc(O)D aCs
Cs,I Jc(~)
where Acs = Cs(0) - Cs(6) = cs,i (see Fig. 3). Using the constant gradient approximation of a concentration profile in the swollen layer, eqn (8) becomes: 15 = Uoc6~ (9) when 6s=C~.l/2, ?s/Acs=½ and / ) = D s ( 1 - q 0 s ) = D s / 2 . In this approximation 6 equals 6o because eqn (9) is identical to the definition of 6o. Ueberreiter 3 and Lap~ik and Valko 4 did not perform the averaging rigorously. They replaced the ratio ?-.JAcs in eqn (8) by 1. The parameter c~,l can be kept constant during the dissolution process (Cs,l = p~) to maintain the stationary dissolution, for instance by adding fresh solvent to the dissolved surface continuously, but in the case when the proper dissolution is measured by polymer concentration in the solution Co.l, the increase of the Cp.l(t) value is unavoidable and evident because of the finite and constant volume of the solvent. The increase of Cp,i is remarkable mainly in terms of long-term dissolution in the small volume of the solvent comparable with the solid polymer volume. Macromolecules solvated in the solution bond a great number of solvent molecules and so the effective volume fraction of the free solvent qgs,~reedecreases rapidly. Because only free solvent molecules are able to penetrate through the swollen layer, the rate of dissolution decreases, the thickness of the swollen layer is no longer constant and the process is not stationary, which is a serious condition for the validity of the essential eqn (2). However, the dissolution can be described in the first approximation as quasi-stationary in a time region when Cp.l(t)(
D
Uoc -- 61-~s Cs.Lfree(t )
(10)
344
M. Mikula, M. Ceppan, J. Blecha, L. LapEik, V. KaHgek
] ~o
/
'
C_pA,_sQ~ur
~
,
/~NON-S TAT. STATE
§ :E
~
y UASl- rA // STATIONARY/
STATE
S A 'IT~
0 Jf 0 TIME t" Fig. 4. The dissolution curve following the mathematical model used. Realizing that
Cs,,,free=/Os~gs,,,free=Os. [1
~P" ] t~0p,l,satur _l
where qvp,, and qgp,l.~at.,are volume fractions of the polymer dissolved in the solution, eqn (10) becomes;
c°"(t)]
U o c - 6 _ ~ Os[1 •
(11)
Cp,l,satur J
During the stationary state the dissolution rate Uoc is constant; applying the law of mass conservation, one can write Uoc
AtApp
= Acp, I Vs
where A is the area of the interfacial constant, ~ is the volume of the solution and pp, Cp.i are mass densities of the polymer in the solid state and in the solution, respectively. This means that the velocity of dissolution is directly proportional to the measured change of polymer concentration in the solution:
6% Uoc =
dt
According to Daynes, 6 who has derived a mathematical expression for the time-lag to in diffusion phenomena associated with the approach to the stationary state, the following equation can be written by analogy: to
(13)
= 62/6/)
Combining eqns (9), (12) and (13), one immediately obtains, for steady-state dissolution,
/)
3 [- Vs dcp,i
2t°LApo'--dt
]2 ,_,,, /
(14)
Kinetic dissolution measurement of polymers
345
where /) is a function of two measured kinetic parameters t o and d%,/dt at the beginning of the dissolution (Fig. 1). Temperature behaviour of the kinetic parameters D, 6, Uoc, to was extensively studied 3~4 on the basis of Eyring's transition-state theory where the typical temperature dependences were confirmed: D=Doexp
(
AHn_ RT/
6 = &oexp -
RT/
(15)
where AH n is the activation energy of diffusion in connection with the dissolution and RAo is the work of expansion of the macromolecular coil coming into solution from the solid phase, both per mole. Similar expressions can be written for Uoc and t o.
DESIGN AND OPERATION
Dissolution equipment Our method of measuring polymer dissolution kinetics is based on continual detection of polymer concentration rise in the solution through the solution viscosity changes. The viscosity is measured by a rotary disc viscometer that works also as a stirrer and a pump. Rotating discs are driven by a DC motor, the power of which is a function of solution viscosity. We applied two models of rotary viscometers: a one-disc type with a higher disc diameter and lower speed (q~ = 58 mm; ~2--1200rain-'; gap between rotary and static discs d = 1.5 mm) and a two-disc type, smaller and quicker ( q ) - - 3 2 m m ; Q = 1 5 0 0 m i n '; d = 0 . 5 m m ) with a different design of static components. The mixing and the rectified motion of a liquid are given, first of all, by a centrifugal force from the disc rotation acting on the liquid and also by pumping and rectifying holes (Fig. 5). The pumping holes for the passage of liquid through the rotary discs are drilled obliquely to achieve downward pumping. The two-disc device [Fig. 5(b)] cannot be self-centring by rotation as the single disc device can [Fig. 5(a)] because of a longer axis relative to the disc diameter. For this reason the Teflon bearing is used to centre the needle-end axis. The compact disc system is immersed in the vessel with a defined temperature (Fig. 6). The diameter of the vessel is just a little bigger than the diameter of the static discs to minimize the solvent volume (V~ = 60ml) and thus to achieve good sensitivity. The height of the
346
M. Mikula, M. Ceppan, J. Blecha, L. Lap6fk, V. Kalfgek
IDC MOTOR o
I DC MOTOR
i
Fig. 5. The single-disc (a) and the two-disc (b) viscometers used in our experiments. 1, polymer sample; 2, holder; 3, static discs; 4, rotary discs; 5, pumping holes; 6, rectifying holes; 7, Teflon needle bearing.
liquid column above the upper rectifying holes ought to be sufficient to prevent the creation of bubbles by pumping the air. In order to reduce an intense liquid rotation the edges of the static discs were shaped (or grooved) in a suitable manner. The vessel containing a viscometric system was placed directly in the common thermostat (Fig. 6). The discs, the vessel and the other mechanical elements that are in contact with the liquid phase were made from stainless steel or nickel-plated brass. Principle of operation The solution viscosity is recorded continually in time using a personal computer. During the dissolution the solution viscosity increases and so PE CONNECTINGSLEEVE ELECTRIC UNIT
11 PERSONAL1 COMPUTER THERMOSTAT
I, VESSEL
Fig. 6.
SOLVENT SURFACE
The position of the dissolution viscometer in the whole measuring system.
Kinetic dissolution measurement of polymers
347
the power of the DC motor that drives the rotary discs changes too. The computer is recording a numerical quantity proportional to the power, which is a complicated complex function of the polymer solution concentration Cv.~ that is required. The measured data can be transformed by a computer through a calibration function to the required physical dimension. So the dissolution curve, Cp.~(t), obtained by this procedure is used to calculate all the kinetic parameters. The requirement of steady-state hydrodynamics (e.g. a constant disc speed) was satisfied by means of the electronic unit (Fig. 6), but generally it can be controlled by the computer. Keeping the rotary disc speed and the bath temperature constant at the beginning of the experimental procedure, the stabilization of the hydrodynamic and electronic state in a pure solvent can be watched on the screen of the computer (line recorder mode). At time to (= 0) the polymer sample is immersed in the solvent and the computer starts to record the viscometric data in the memory in equidistant time steps chosen in advance (measurement mode). After completion of the dissolution process (usually less than 60 min), when a set of about 100 viscometric values have been recorded, the data treatment m o d e follows. Moreover, one recorded value represents the average of many measured samples from the whole time interval. The dispersion variance of samples in the interval is used to determine the error and thus the statistical weight of the recorded value for the interval. The software developed allows the transformation of the measured data through the nonlinear calibration function. This function is constructed for a particular polymer and temperature in a similar manner to the actual dissolution measurement. The actual dissolution is replaced in the calibration measurements by the known increase of the polymer concentration in the measured solution under the same measurement conditions. The statistical treatment of calibration data is performed by a separate calibration program (with the calibration function as a result). After the transformation of the dissolution data to the reai physical scale, a cubic spline is used to obtain the dissolution kinetic curve (Fig. 1) that gives the two required kinetic dissolution p a r : ~ e t e r s to, d c / d t from which the others (D, Uoc, 6) can be easily calculated through eqns (12)-(14). EXPERIENCE AND APPLICATIONS The rheoviscometer has frequently been used for dissolution measurements of water-insoluble cellulose derivatives, particularly for the
348
M. Mikula, M. Ceppan, J. Blecha, L. Lap6ik, V. Kaffgek v
[mV] 60
T:I
30 ~
0
50
0
1
2
Cp,I [kg m-3 ]
Calibration curves for C M C - N a (~7/= 1.1 x 105, degree of substitution= 0-95) in water at different temperatures using the two-disc (nickel-coated) viscometer. The polymer sample dimensions were 2 x (35 m m x 16 m m x 0-5 mm). Fig. 7a.
sodium salt of carboxymethylcellulose (CMC-Na). Typical calibration and dissolution curves for C M C - N a are shown in Figs 7a and 7b. It is necessary to use a thick foil as the polymer sample for dissolution (~>0.5 mm for one-sided dissolution) in order to attain independence of measured kinetic parameters t o, d%,l/dt from the sample thickness. In the case of thin foils the dissolution is finished before the steady-state swollen layer is formed. The linear dependence of In D on the inverse temperature 1/T (Fig. 8) confirms the assumption about the diffusion character of dissolution, governed by the production of free vacancies. The value RA~, which represents the work of expansion done under the internal pressure by
Opt [kgl m3] 2 T=5
0
Fig. 7b. The dissolution
3
6
9
12 t {rain] Fig. 7a) in water
curves of C M C - N a (as in nickel-coated viscometer.
using the two-disc
Kinetic dissolution measurement of polymers
In D [m2s 1]
349
ln~i[m ]
-27
-10
-28
so'c I,
3.0
M ,
,
3.2
3.4
,?
cI 1 /
i/T[1(~3~1]
Fig. 8. The temperature dependence of the diffusion coefficient of water in CMC-Na and of the swollen layer thickness. AEt~ = 39 kJ mol-t; RA6 = 14 kJ mo1-1. macromolecular coils coming from the solid phase into the swollen layer and into the solution, can be used to estimate the internal pressure Pi in the system studied: 7"9 Pi = R A ~ / A V *
( - 8 x 103 atm for C M C - N a )
where A V * is an apparent molar volume difference of p o l y m e r segments in the different surroundings mentioned. O f course, swelling and dissolution rates are considerably influenced by the chemical constitution of the solvent and polymer. E v e n a small content of electrolytes, e.g. in the industrial cellulose derivatives present, m o d e r a t e s the dissolution in water. There exists also a strong d e p e n d e n c e b e t w e e n the dissolution kinetic parameters and the physical structure (supermolecular structure, degree
013, i TdissoLu,cton =40"C [kgr~]Ja// ~/"C 104 rflx 0.2 [ 0 t
0
....
300
" 600 tIs]
Fig. 9. The dissolution curves of poly(ethylene-terephthalate) fibres, fixed 5 win at different temperatures, using phenol/chlorobenzene (1 : 1) at 40 °C as the solvent.
350
M. Mikula, M. Ceppan, J. Blecha, L. Lap6ik, V. KalEek
uiv 1
T
0.5
~
o/
-0.5
w a l e r , 20"C o
1
- rI [lO'3po.s]
Fig. 10. Sensitivity curves of the single-disc (1) and the two-disc (2) viscometers. U, measured electric quantity. The common measured range for water dissolution problems is illustrated.
of crystallinity, orientation, temperature history, etc.) of the polymer. As can be seen in Fig. 9, the swelling time (induction period to) of textile fibres depends very sensitively on the degree of thermal fixation used in fibre technology. Our viscometric device is inconvenient for the measurement of the absolute viscosity value because of a limited viscosity range, the nonlinear sensitivity response (Fig. 10) and the undefined shear rate within the rotating disc. The device is, however, suitable for any measurement of the liquid viscosity changes relative to the initial state. As well as the dissolution, it can measure polymerization kinetics or polymer degradation in the solution. In a typical measurement we usually use just a small part of the sensitivity response illustrated in Fig. 10. In this small region the linear approximation U - U0-- kp(~/- rio) can be applied, enabling simple and fast measurement of the limiting viscosity number; where U is the measured quantity (voltage) and q is the viscosity of the solution (U0 and r/0 are the values for the pure solvent). A definite amount of the concentrated polymer solution is added stepwise to the pure solvent and the computer records the value ( U - U o ) / ( k p ~ h , C p , , ) = (fl - ~lo)/(~loCp.,) as a function of Cp.~(Fig. 11). The constant kp has to be exactly determined for the polymer from a single measurement using another calibrated viscometer (e.g. capillary). The speed of the DC-driven motor (Fig. 6) must be chosen not to overload (to avoid overheating) the motor winding. This leads to a lowering of the winding resistance and thus to an increase in power
Kinetic dissolution measurement of polymers
351
U-Uo Cp,t
%c~. [dt/g]
l'
3
I
i
0.02
0.04 •
I
I
0.06 Cp, t [ g / d t l
Fig. 11. Intrinsic viscosity of CMC-Na (purified 'Lovosa', Czechoslovakia) and HEC250H (from Hercules, The Netherlands) at 25 °C, measured by single-disc viscometer. (The intrinsic viscosity increase for low concentrations of CMC-Na is caused by the properties of the polyelectrolyte, x)
consumption. If overloading occurs the stabilization time b e f o r e the m e a s u r e m e n t is prolonged and also the m e a s u r e d data are d e f o r m e d . The two-disc system [Fig. 5(b)] needs relatively less liquid than does the single-disc system and, in addition, its stabilization if quicker, but it requires a more precise mechanical w o r k s h o p for construction.
REFERENCES 1. Ueberreiter, K. & Asmussen, F., J. Polym. Sci., 23 (1957) 78. 2. Ueberreiter, K. & Asmussen, F., J. Polym. Sci., 57 (1962) 187, 199. 3. Ueberreiter, K., The solution process. In Diffusion in Polymers, ed. J. Crank & G. S. Park. Academic Press, London, 1968. 4. Lap~ik, L. & Valko, L., J. Polym. Sci. A-2, 9 (1971) 633. 5. Glasstone, S., Laidler, K. J. & Eyring, H,, The Theory of Rate Processes. McGraw-Hill, New York, 1941, 6. Daynes, H., Proc. R. Soc., A97 (1920) 286. 7. Valko, L. & Lap~ik, L., Zbornik Chem. Fak. SVST (Czechoslovakia), Slovak Technical University, Bratislava, 1967, p. 47. 8. Morawetz, H., Macromolecules in Solution. Interscience, New York, 1975. 9. Lap(:ik, L., Valko, L., Mikula, M., Jan6ovi~ov~i, V. & Pan~ik, J., Progr. Colloid Polym. Sci., 77 (in press).
Kinetic Dissolution Measurement of Polymers by Solution Viscosity Recordingt M. M i k u l a , M . ( ~ e p p a n , J. B l e c h a , L. L a p ~ i k Faculty of Chemical Technology, Slovak Technical University, 812 37 Bratislava, Czechoslovakia & V. Kaligek University of Chemical Technology, 532 10 Pardubice, Czechoslovakia (Received 24 February 1989; accepted 18 April 1989)
ABSTRACT A laboratory device for continuous dissolution measurement of polymers by solution viscosity recording is described. The nonlinear viscosity dependence on the polymer content in the solution is utilized to measure the dissolution by recording the driving motor power of the discs rotating within the solution. A short theoretical background of the diffusion process of dissolution is introduced, with comments. The diffusion coefficient of the solvent into the polymer, velocity of dissolution, thickness of the surface swollen layer and other kinetic and activation parameters can be calculated by treating the recalibrated viscosity data. Some results of the dissolution in water of carboxymethylceUulose and some dissolution tests of the degree of fixation of textile materials are given together with suggestions for possible alternative applications.
INTRODUCTION The dissolution of polymers has s o m e typical features when c o m p a r e d with the dissolution of low-molecular-weight solids. The transport of polymer into the solvent does not take place immediately after both fractions come into contact. O n e of the characteristic kinetic parat Dedicated to Professor Josef Schurz, DSc, Carl-Franzens University of Graz, in honour of his 65th birthday. 339 Polymer Testing 0142-9418/89/$03.50 (~) 1989 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland
M. Mikula, M. Ceppan, J. Blecha, L. Lap6ik, V, Kalfgek
340
meters of the process is an induction period to which depends on temperature, molecular weight of the polymer, kinetic and thermodynamic properties of the solvent and also on the method of polymer sample preparation. 1-~ During the induction period the solvent penetrates into the polymer by diffusion and a swollen surface layer is formed. Stationary dissolution starts after the induction period, provided that there is a constant mixing. The thickness of the swollen surface layer becomes invariant in time (Fig. 1). The hydrodynamic pressure of the solvent on the swollen layer surface, caused by mixing the solvent, is the dominant factor determining the swollen layer thickness. 3 Such important technological parameters as the mean diffusion coefficient, the thickness of the swollen layer and the velocity of dissolution can be calculated directly from the dissolution kinetic curve. Moreover, from the temperature dependences various activation parameters can be obtained according to Eyring's theory of reaction rates. 4"5 Apart from the essential influences of the temperature and the type of both the chosen polymer and the solvent exerted on all dissolution parameters, there exists a strong dependence of these parameters on the physical structure of the polymer, on its crystallinity and surface quality and on the whole temperature history of the polymer sample and its preparation. Earlier publications 1-4 have dealt with the use of a refractometry technique to measure the dissolution kinetic curve that is unsuitable for composites or polymers with low-molecular-weight components. The present paper describes the continuous automated measurement of the dissolution process based on registration of the solution viscosity changes being indicated by mechanical rotating elements. The equipment is simple, cheap and easily connectable to a microcomputer. The two disadvantages of the method are the nonlinear calibration curve of an applied rheoviscometer and the undefined shear rate. /
SATURATION/-'
"
B DISSOLUT N
to
Fig. 1.
TIME
Dissolution kinetic curve; to = time of swelling (induction period).
Kinetic dissolution measurement of polymers
341
C O M M E N T ON T H E T H E O R Y OF D I F F U S I O N CONTROLLED DISSOLUTION In the case when the diffusion of the solvent into the polymer is the slowest mechanism in the dissolution process, Fick's first law for every component i can be used:
ji = - ( c i D i / R T ) grad/~i
(1) where j is the flow density, c is the concentration, D is the diffusion coefficient, R is the gas constant, T is the temperature and /~ is the chemical potential. For the steady state of dissolution (the stationary regime--Fig. 1), providing that the hydrodynamics are constant, a simple relation among kinetic parameters can be derived from Fick's first law in one dimension and in ideal solution c o n d i t i o n s ? : The diffusion problem is first solved in the moving coordinate system firmly bound with an advancing phase boundary. Then the transformation of variables from the moving coordinate system to the laboratory one with a fixed solid polymer is performed carefully to obtain eqn (2). -D~(1 - qos)
ac~ = ~. c~ cox
(2)
where $ is the moving velocity of the interfacial swollen layer into the polymer [m/s] (i.e. the dissolution rate, better designated as Uoc4); Ds, q)s, c~ are the diffusion coefficient, the volume fraction and the concentration of the solvent at a given distance x inside the swollen layer, respectively. Provided that D~ does not d e p e n d on c~ (D~ is the constant within the x coordinate) and that (p~ = cJp~, where p~ is the specific mass of the solvent, we can carry out the integration of eqn (2):
f
(I
10c~ = f Uoc Ox
-
(3)
u,,c
Inc~-cJp~-
-- x + K D~
Let us designate the ratio DJuoc by 6o[m], which represents the swollen layer thickness (see below). The value of the integration constant K can be obtained from the boundary condition: c~ = cs: for x = 0 (see Fig. 2), where c,.~ is the concentration of the solvent in the liquid phase. Thus K = In cs: - c~:/p~ and
c~=c~,,exp
(x t -~,
exp
P~
/
(4)
M. Mikula, M. Ceppan, J. Blecha, L. Lap~fk, V. KalEek
342
Cs, t = Ps
I . . . . . . . . .
"% ."1
APPROXlM. \ x ",J LIQUID
~
\\
%~..x SOLVENT JSWOLLEN ~ POLYMER -LAYER -
~
o Fig. 2.
..:' ~ \
.'"
.: P
SOLID
\
INFILTRATION
\
Uoc/ r////~.9c//
&o
'~
Steady-state situation at solvent-polymer interface,
o the
~o= Ds/uoc
x
Fig. 3. The solvent volume fraction inside the swollen layer: - - , calculated from eqn (4); . . . . , linear approximation. Polymer volume fraction, qgp: . . . , obtained experimentally by Ueberreiter. 3
The variation of the concentration of the solvent in the swollen layer according to this equation at the beginning of the dissolution process (when cs,,- p~) is illustrated in Fig. 3. Any polymer property was not considered until eqn (4) was derived, except for the assumption of the existence of the swollen layer. The shape can be compared with that of the polymer volume concentration q0p in the swollen layer taken from the experiment by Ueberreiter 3 (the dotted line in Fig. 3) for polymers at temperatures below Tg. In order to calculate the diffusion coefficient from the experimental data, usually including the induction period to and the stationary dissolution rate U o c - d c / d t , eqn (2) has to be averaged within the swollen layer. First of all, the expression Ds(1 - qg~) in eqn (2) can be substituted by the mutual diffusion coefficient D which depends on the volume fractions of both components: D = Ds(1 - ~s) + Dp(1 - tpp)
(5)
where Dp is the diffusion coefficient of the polymer in the solvent. For macromolecules Dp---->0 and eqn (2) becomes ¢9Cs - D ~ x = UocCs(X)
(6)
This equation can be integrated in a different manner than was used
Kinetic dissolution measurement of polymers
343
previously [eqn (3)] to obtain averaged variables:
- f D aCs= Uocf cs(x)dx
(7)
After the integration through the swollen layer, supposing that there exists 6 for which Cs(6) = 0, eqn (7) becomes
D = Uoc6~s/ACs
(8)
where 6 represents the real thickness of the swollen layer a n d / ) is the mean mutual diffusion coefficient: /) _
1 mCs
f
D acs = 1 swollen layer)
•
fc(O)D aCs
Cs,I Jc(~)
where Acs = Cs(0) - Cs(6) = cs,i (see Fig. 3). Using the constant gradient approximation of a concentration profile in the swollen layer, eqn (8) becomes: 15 = Uoc6~ (9) when 6s=C~.l/2, ?s/Acs=½ and / ) = D s ( 1 - q 0 s ) = D s / 2 . In this approximation 6 equals 6o because eqn (9) is identical to the definition of 6o. Ueberreiter 3 and Lap~ik and Valko 4 did not perform the averaging rigorously. They replaced the ratio ?-.JAcs in eqn (8) by 1. The parameter c~,l can be kept constant during the dissolution process (Cs,l = p~) to maintain the stationary dissolution, for instance by adding fresh solvent to the dissolved surface continuously, but in the case when the proper dissolution is measured by polymer concentration in the solution Co.l, the increase of the Cp.l(t) value is unavoidable and evident because of the finite and constant volume of the solvent. The increase of Cp,i is remarkable mainly in terms of long-term dissolution in the small volume of the solvent comparable with the solid polymer volume. Macromolecules solvated in the solution bond a great number of solvent molecules and so the effective volume fraction of the free solvent qgs,~reedecreases rapidly. Because only free solvent molecules are able to penetrate through the swollen layer, the rate of dissolution decreases, the thickness of the swollen layer is no longer constant and the process is not stationary, which is a serious condition for the validity of the essential eqn (2). However, the dissolution can be described in the first approximation as quasi-stationary in a time region when Cp.l(t)(
D
Uoc -- 61-~s Cs.Lfree(t )
(10)
344
M. Mikula, M. Ceppan, J. Blecha, L. LapEik, V. KaHgek
] ~o
/
'
C_pA,_sQ~ur
~
,
/~NON-S TAT. STATE
§ :E
~
y UASl- rA // STATIONARY/
STATE
S A 'IT~
0 Jf 0 TIME t" Fig. 4. The dissolution curve following the mathematical model used. Realizing that
Cs,,,free=/Os~gs,,,free=Os. [1
~P" ] t~0p,l,satur _l
where qvp,, and qgp,l.~at.,are volume fractions of the polymer dissolved in the solution, eqn (10) becomes;
c°"(t)]
U o c - 6 _ ~ Os[1 •
(11)
Cp,l,satur J
During the stationary state the dissolution rate Uoc is constant; applying the law of mass conservation, one can write Uoc
AtApp
= Acp, I Vs
where A is the area of the interfacial constant, ~ is the volume of the solution and pp, Cp.i are mass densities of the polymer in the solid state and in the solution, respectively. This means that the velocity of dissolution is directly proportional to the measured change of polymer concentration in the solution:
6% Uoc =
dt
According to Daynes, 6 who has derived a mathematical expression for the time-lag to in diffusion phenomena associated with the approach to the stationary state, the following equation can be written by analogy: to
(13)
= 62/6/)
Combining eqns (9), (12) and (13), one immediately obtains, for steady-state dissolution,
/)
3 [- Vs dcp,i
2t°LApo'--dt
]2 ,_,,, /
(14)
Kinetic dissolution measurement of polymers
345
where /) is a function of two measured kinetic parameters t o and d%,/dt at the beginning of the dissolution (Fig. 1). Temperature behaviour of the kinetic parameters D, 6, Uoc, to was extensively studied 3~4 on the basis of Eyring's transition-state theory where the typical temperature dependences were confirmed: D=Doexp
(
AHn_ RT/
6 = &oexp -
RT/
(15)
where AH n is the activation energy of diffusion in connection with the dissolution and RAo is the work of expansion of the macromolecular coil coming into solution from the solid phase, both per mole. Similar expressions can be written for Uoc and t o.
DESIGN AND OPERATION
Dissolution equipment Our method of measuring polymer dissolution kinetics is based on continual detection of polymer concentration rise in the solution through the solution viscosity changes. The viscosity is measured by a rotary disc viscometer that works also as a stirrer and a pump. Rotating discs are driven by a DC motor, the power of which is a function of solution viscosity. We applied two models of rotary viscometers: a one-disc type with a higher disc diameter and lower speed (q~ = 58 mm; ~2--1200rain-'; gap between rotary and static discs d = 1.5 mm) and a two-disc type, smaller and quicker ( q ) - - 3 2 m m ; Q = 1 5 0 0 m i n '; d = 0 . 5 m m ) with a different design of static components. The mixing and the rectified motion of a liquid are given, first of all, by a centrifugal force from the disc rotation acting on the liquid and also by pumping and rectifying holes (Fig. 5). The pumping holes for the passage of liquid through the rotary discs are drilled obliquely to achieve downward pumping. The two-disc device [Fig. 5(b)] cannot be self-centring by rotation as the single disc device can [Fig. 5(a)] because of a longer axis relative to the disc diameter. For this reason the Teflon bearing is used to centre the needle-end axis. The compact disc system is immersed in the vessel with a defined temperature (Fig. 6). The diameter of the vessel is just a little bigger than the diameter of the static discs to minimize the solvent volume (V~ = 60ml) and thus to achieve good sensitivity. The height of the
346
M. Mikula, M. Ceppan, J. Blecha, L. Lap6fk, V. Kalfgek
IDC MOTOR o
I DC MOTOR
i
Fig. 5. The single-disc (a) and the two-disc (b) viscometers used in our experiments. 1, polymer sample; 2, holder; 3, static discs; 4, rotary discs; 5, pumping holes; 6, rectifying holes; 7, Teflon needle bearing.
liquid column above the upper rectifying holes ought to be sufficient to prevent the creation of bubbles by pumping the air. In order to reduce an intense liquid rotation the edges of the static discs were shaped (or grooved) in a suitable manner. The vessel containing a viscometric system was placed directly in the common thermostat (Fig. 6). The discs, the vessel and the other mechanical elements that are in contact with the liquid phase were made from stainless steel or nickel-plated brass. Principle of operation The solution viscosity is recorded continually in time using a personal computer. During the dissolution the solution viscosity increases and so PE CONNECTINGSLEEVE ELECTRIC UNIT
11 PERSONAL1 COMPUTER THERMOSTAT
I, VESSEL
Fig. 6.
SOLVENT SURFACE
The position of the dissolution viscometer in the whole measuring system.
Kinetic dissolution measurement of polymers
347
the power of the DC motor that drives the rotary discs changes too. The computer is recording a numerical quantity proportional to the power, which is a complicated complex function of the polymer solution concentration Cv.~ that is required. The measured data can be transformed by a computer through a calibration function to the required physical dimension. So the dissolution curve, Cp.~(t), obtained by this procedure is used to calculate all the kinetic parameters. The requirement of steady-state hydrodynamics (e.g. a constant disc speed) was satisfied by means of the electronic unit (Fig. 6), but generally it can be controlled by the computer. Keeping the rotary disc speed and the bath temperature constant at the beginning of the experimental procedure, the stabilization of the hydrodynamic and electronic state in a pure solvent can be watched on the screen of the computer (line recorder mode). At time to (= 0) the polymer sample is immersed in the solvent and the computer starts to record the viscometric data in the memory in equidistant time steps chosen in advance (measurement mode). After completion of the dissolution process (usually less than 60 min), when a set of about 100 viscometric values have been recorded, the data treatment m o d e follows. Moreover, one recorded value represents the average of many measured samples from the whole time interval. The dispersion variance of samples in the interval is used to determine the error and thus the statistical weight of the recorded value for the interval. The software developed allows the transformation of the measured data through the nonlinear calibration function. This function is constructed for a particular polymer and temperature in a similar manner to the actual dissolution measurement. The actual dissolution is replaced in the calibration measurements by the known increase of the polymer concentration in the measured solution under the same measurement conditions. The statistical treatment of calibration data is performed by a separate calibration program (with the calibration function as a result). After the transformation of the dissolution data to the reai physical scale, a cubic spline is used to obtain the dissolution kinetic curve (Fig. 1) that gives the two required kinetic dissolution p a r : ~ e t e r s to, d c / d t from which the others (D, Uoc, 6) can be easily calculated through eqns (12)-(14). EXPERIENCE AND APPLICATIONS The rheoviscometer has frequently been used for dissolution measurements of water-insoluble cellulose derivatives, particularly for the
348
M. Mikula, M. Ceppan, J. Blecha, L. Lap6ik, V. Kaffgek v
[mV] 60
T:I
30 ~
0
50
0
1
2
Cp,I [kg m-3 ]
Calibration curves for C M C - N a (~7/= 1.1 x 105, degree of substitution= 0-95) in water at different temperatures using the two-disc (nickel-coated) viscometer. The polymer sample dimensions were 2 x (35 m m x 16 m m x 0-5 mm). Fig. 7a.
sodium salt of carboxymethylcellulose (CMC-Na). Typical calibration and dissolution curves for C M C - N a are shown in Figs 7a and 7b. It is necessary to use a thick foil as the polymer sample for dissolution (~>0.5 mm for one-sided dissolution) in order to attain independence of measured kinetic parameters t o, d%,l/dt from the sample thickness. In the case of thin foils the dissolution is finished before the steady-state swollen layer is formed. The linear dependence of In D on the inverse temperature 1/T (Fig. 8) confirms the assumption about the diffusion character of dissolution, governed by the production of free vacancies. The value RA~, which represents the work of expansion done under the internal pressure by
Opt [kgl m3] 2 T=5
0
Fig. 7b. The dissolution
3
6
9
12 t {rain] Fig. 7a) in water
curves of C M C - N a (as in nickel-coated viscometer.
using the two-disc
Kinetic dissolution measurement of polymers
In D [m2s 1]
349
ln~i[m ]
-27
-10
-28
so'c I,
3.0
M ,
,
3.2
3.4
,?
cI 1 /
i/T[1(~3~1]
Fig. 8. The temperature dependence of the diffusion coefficient of water in CMC-Na and of the swollen layer thickness. AEt~ = 39 kJ mol-t; RA6 = 14 kJ mo1-1. macromolecular coils coming from the solid phase into the swollen layer and into the solution, can be used to estimate the internal pressure Pi in the system studied: 7"9 Pi = R A ~ / A V *
( - 8 x 103 atm for C M C - N a )
where A V * is an apparent molar volume difference of p o l y m e r segments in the different surroundings mentioned. O f course, swelling and dissolution rates are considerably influenced by the chemical constitution of the solvent and polymer. E v e n a small content of electrolytes, e.g. in the industrial cellulose derivatives present, m o d e r a t e s the dissolution in water. There exists also a strong d e p e n d e n c e b e t w e e n the dissolution kinetic parameters and the physical structure (supermolecular structure, degree
013, i TdissoLu,cton =40"C [kgr~]Ja// ~/"C 104 rflx 0.2 [ 0 t
0
....
300
" 600 tIs]
Fig. 9. The dissolution curves of poly(ethylene-terephthalate) fibres, fixed 5 win at different temperatures, using phenol/chlorobenzene (1 : 1) at 40 °C as the solvent.
350
M. Mikula, M. Ceppan, J. Blecha, L. Lap6ik, V. KalEek
uiv 1
T
0.5
~
o/
-0.5
w a l e r , 20"C o
1
- rI [lO'3po.s]
Fig. 10. Sensitivity curves of the single-disc (1) and the two-disc (2) viscometers. U, measured electric quantity. The common measured range for water dissolution problems is illustrated.
of crystallinity, orientation, temperature history, etc.) of the polymer. As can be seen in Fig. 9, the swelling time (induction period to) of textile fibres depends very sensitively on the degree of thermal fixation used in fibre technology. Our viscometric device is inconvenient for the measurement of the absolute viscosity value because of a limited viscosity range, the nonlinear sensitivity response (Fig. 10) and the undefined shear rate within the rotating disc. The device is, however, suitable for any measurement of the liquid viscosity changes relative to the initial state. As well as the dissolution, it can measure polymerization kinetics or polymer degradation in the solution. In a typical measurement we usually use just a small part of the sensitivity response illustrated in Fig. 10. In this small region the linear approximation U - U0-- kp(~/- rio) can be applied, enabling simple and fast measurement of the limiting viscosity number; where U is the measured quantity (voltage) and q is the viscosity of the solution (U0 and r/0 are the values for the pure solvent). A definite amount of the concentrated polymer solution is added stepwise to the pure solvent and the computer records the value ( U - U o ) / ( k p ~ h , C p , , ) = (fl - ~lo)/(~loCp.,) as a function of Cp.~(Fig. 11). The constant kp has to be exactly determined for the polymer from a single measurement using another calibrated viscometer (e.g. capillary). The speed of the DC-driven motor (Fig. 6) must be chosen not to overload (to avoid overheating) the motor winding. This leads to a lowering of the winding resistance and thus to an increase in power
Kinetic dissolution measurement of polymers
351
U-Uo Cp,t
%c~. [dt/g]
l'
3
I
i
0.02
0.04 •
I
I
0.06 Cp, t [ g / d t l
Fig. 11. Intrinsic viscosity of CMC-Na (purified 'Lovosa', Czechoslovakia) and HEC250H (from Hercules, The Netherlands) at 25 °C, measured by single-disc viscometer. (The intrinsic viscosity increase for low concentrations of CMC-Na is caused by the properties of the polyelectrolyte, x)
consumption. If overloading occurs the stabilization time b e f o r e the m e a s u r e m e n t is prolonged and also the m e a s u r e d data are d e f o r m e d . The two-disc system [Fig. 5(b)] needs relatively less liquid than does the single-disc system and, in addition, its stabilization if quicker, but it requires a more precise mechanical w o r k s h o p for construction.
REFERENCES 1. Ueberreiter, K. & Asmussen, F., J. Polym. Sci., 23 (1957) 78. 2. Ueberreiter, K. & Asmussen, F., J. Polym. Sci., 57 (1962) 187, 199. 3. Ueberreiter, K., The solution process. In Diffusion in Polymers, ed. J. Crank & G. S. Park. Academic Press, London, 1968. 4. Lap~ik, L. & Valko, L., J. Polym. Sci. A-2, 9 (1971) 633. 5. Glasstone, S., Laidler, K. J. & Eyring, H,, The Theory of Rate Processes. McGraw-Hill, New York, 1941, 6. Daynes, H., Proc. R. Soc., A97 (1920) 286. 7. Valko, L. & Lap~ik, L., Zbornik Chem. Fak. SVST (Czechoslovakia), Slovak Technical University, Bratislava, 1967, p. 47. 8. Morawetz, H., Macromolecules in Solution. Interscience, New York, 1975. 9. Lap(:ik, L., Valko, L., Mikula, M., Jan6ovi~ov~i, V. & Pan~ik, J., Progr. Colloid Polym. Sci., 77 (in press).