ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978
1997
Effect of Thermal Lag and Measurement Precision in Differential Scanning Calorimetry: Theoretical Guidelines for Enzyme-Substrate Reactions by the Method of Orthogonal Collocation L. F. Whiting' and P. W. Carr" Department of Chemistry, Kolthoff and Smith Halls, University of Minnesota, Minneapolis, Minnesota 55455
A simplified model of a differential scanning calorimeter (DSC) with large (40-120 pL) aqueous enzyme sample was simulated digitally by the mathematical technique called orthogonal collocation in order to observe the errors due to thermal lag (temperature and concentration gradients) in calculating the first-order Arrhenius kinetic parameters 2 and A€. Only two dimensionless parameters were found to influence the determinate errors in the data, one related to the scan rate and the other related to the rate of diffusion relative to the rate of chemical reaction. The errors were independent of A H , C,, and D, and also the parameters 2,A € , and T o as long as the initial rate constant, k,, remained unchanged. Simulations also indicate that In Z and A € can be obtained to within 5 % for an 80-pL sample at a scan rate of 5 K/min. An additional study was carried out involving the indeterminate errors incurred when one scans slowly relative to the rate of chemical reaction. Results indicate that random errors can easily be several times greater than the determinate errors related to thermal lag.
Differential thermal analysis (DTA) and differential scanning calorimetry (DSC) are uniquely suited to t h e measurement of the temperature dependence of physical and chemical processes. Both techniques have been employed in the study of rates and mechanism of chemical reaction in liquids and solids ( 1 ) . In this context, one of the earliest studies was that of Borchardt and Daniels who measured the activation energy for the decomposition of benzenediazonium chloride by use of DTA in stirred solutions (2). Very few studies after the initial work of Borchardt and Daniels have been concerned with solution phase measurements of kinetic parameters. However, during the past few years DSC has been applied to helix-coil transitions in DNA ( 3 )and poly(A) ( 4 ) ,conversion of ovalbumin to S-ovalbumin ( 5 ) ,shrinkage of collagen fibers ( 6 ) ,thermal stability toward denaturation of soluble and immobilized nicotinamide adenine dinucleotide dependent dehydrogenases (71, complex transitions in whole blood and serum involving different molecular forms of hemoglobin ( 8 ) ,and investigations of phase transitions and stability of lipids, lipoproteins, and biomembranes (9,101. In spite of this great interest in biological applications, the study of enzyme catalyzed reactions by DSC has not been pursued in depth (11). Because of the advantages inherent in the high sensitivity and small sample sizes provided by DSC, we have initiated both experimental and theoretical studies on such reactions by DSC ( 1 2 ) . This paper will deal exclusively with the theoretical aspect of the studies while a later paper will develop the experimental studies. 'Present address. Analytical Laboratories. 574 Building, Dow Chemical USA, Midland, Mich. 48640. 0003-2700/78/0350-1997$01 O O i O
One of the major problems in any temperature scanning technique is the presence of temperature gradients (thermal lag) in the sample. Detailed studies of the effect of thermal lag on the measurement of reaction enthalpy, heat capacities, and phase transition temperatures have been carried out (13-16). The chief objective of the present work was to study the consequences of slou. heat transfer in t h e sample. Unfortunately there have been very few studies of the effect of slow heat transfer on the measurement of chemical kinetics. One of the reasons for this is the cost of computer simulation of the partial differential equations by classical finite difference techniques. During the past several years the technique of orthogonal collocation has received considerable attention from chemical engineers in the simulation of heat and mass transfer processes in reactors ( I 7) because this numerical analysis scheme can provide very accurate results more efficiently than classical methods. In a recent study we were able to simulate chronoamperometric systems to 0.1% accuracy in less than 10 seconds (18). We therefore studied the applications of orthogonal collocation to the simulation of the effect of slow heat and mass transfer with coupled chemical reaction on DSC curves. T h e only previous theoretical study (19) outlined the effect of physical properties of the sample and reference, heating rat.e, sample and reference diameters, and heat leakage along the thermocouple leads on the peak area, peak temperature, and general peak shape in DTA. Although DSC has the advantages of high sensitivity and small sample size, it has several limitations as far as enzyme-substrate reactions are concerned. For the majority of enzyme systems of interest, one cannot "freeze-out" the reaction by reducing the temperature to 0 "C. This is primarily due to the low activation energies (10-20 kcal/mol) of most enzyme reactions. An important consequence of the above property of enzyme reactions is that an ideal DSC trace for the enzyme reaction will resemble an exponential decay rather than the usual peaked DSC trace. Although the DSC trace will appear to be exponential, kinetic information is still present in the data and can be extracted by any of the data analysis methods available which apply to the enzyme kinetic system (20-24). Another problem inherent to enzyme-substrate reactions is the low initial substrate Concentration needed to ensure that one is in the first-order kinetic region of the Michaelis-Menten rate law. Since the initial substrate concentration must be below the K , (Michaelis constant) which is typically to M, the enthalpies of reaction (AH]are usually less than 20 kcal/mol, and sample sizes are limited to less than 100 pL; the reaction generated heat will be small (-2 mcal). This could pose a sensitivity problem for enzymes whose K , are M. less than Finally, enzymes can be studied only over a somewhat limited temperature range, a t best from about 0 to 70 "C, because of denaturation, whereas solid-state reactions can C 1978 American Chemical Society
1998
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978
often be studied over a much larger temperature range. Obviously, the larger is the temperature range, the more temperature-dependent kinetic information will be contained in the data. Problems are also found in the reaction cell design, sample handling, and particularly in mixing the solutions containing the enzyme and substrate (25). These experimental aspects of this study will be given in a later paper since they are not germane to the present theoretical discussion. I t should be noted here that, although the above deals in particular with DSC studies of enzyme-substrate systems, much of it is, in general, also applicable to any solution phase DSC kinetic study. All of the commonly employed methods for the analysis of thermoanalytical kinetic data tacitly assume that the sample is at all times homogeneous in temperature and composition (concentration). On this count DSC has been routinely criticized since t h e dynamic nature of the technique cannot satisfy these requirements in an absolute sense ( 1 ) ;however, in the limit, as the temperature programming rate and sample thickness decrease, the two assumptions hold. Since sample volumes of 25 to 100 1L are desirable in terms of sensitivity and these volumes would be many times thicker than typical solid-state samples, t h e homogeneity in temperature and concentration of an aqueous enzyme sample (un-stirred) might be difficult to maintain during a temperature scan. I n a physical sense, only one boundary of the liquid sample is in intimate contact with t h e programmed temperature. If the temperature of the programmer increases a t a rapid rate, the unstirred solution which is farthest from the programmer may not “see” a change in temperature for several seconds after the scan is initiated. This introduces a temperature gradient throughout the aqueous sample. Naturally, since the sample temperature is not homogeneous, the reaction rate will be different throughout the sample and a concentration gradient will develop. The resultant DSC kinetic trace will be distorted and subsequent analysis of these data will yield incorrect kinetic parameters. T h e degree of determinate error incurred will depend on the severity of the temperature and concentration gradients. Our theoretical model allows the simulation of the effect of scan rate, sample thickness and reaction rate parameter on the DSC curve for a first-order chemical reaction. T h e model contains a number of other simplifying, but we believe conservative, assumptions. This work develops “worst-case” guidelines for t h e DSC study of a solution phase first-order reaction. I t should be emphasized here that this theoretical error analysis applies only to the effect of slow heat and mass transfer in the sample and does not take into account possible slow instrument response or interfacial effects as described previously by Flynn (26, 27). These instrumental and interfacial effects are coupled to the sample effects, but, with proper experimental design, they can presumably be made negligible compared to the slow heat and mass transfer effects in the sample. I n any case, we wish to focus on slow heat transfer in the sample and have decided to exclude the effects described by Flynn from the present treatment.
THEORY Under the assumptions that the sample is homogeneous in temperature and concentration at all times, one can mathematically describe an ideal DSC signal, i.e., q,, the heat flow due t o reaction (cal/s) a t any time. The instantaneous temperature is that of the temperature programmer, T = To + rst, where To,rs, and t are the initial temperature (K), temperature scan rate (K/s), and time (s), respectively.
4, = -AHV-
dC dt
SCV””LE
RE‘EPENCE
T - T O t fst
T:-O+rSt
Figure 1. simplified model of differential scanning calorimeter for sample thickness L
For a first-order reaction with Arrhenius dependence of the rate constant on temperature: dC dt =
-
-zexp( -AE m)c ZAE
p(z) =
:(
-
l=:du)
-
AE
= -;zo
RT
(3)
(F lom -
AE
= -
R To
:du)
(4) (5)
where AH,V, C, 2,SE,R , T a r e the heat of reaction (cal/mol), total sample volume (cm3), concentration (mol/cm3), preexponential factor (s-’), activation energy (cal/mol), gas constant (cal/K-mol), and temperature (K), respectively. Equations 3-5 are obtained by integrating Equation 2 over temperature. The exponential integrals present in the function p ( z ) are not integrable in closed form but can be approximated by any of several polynomial functions, one of the more accurate of which involves an expansion in series of Bernoulli numbers ( 2 1 ) . I t was found that for the values of the parameters required to describe an enzyme system, Equation 4 must be available to a very high degree of accuracy and the second half of the equation cannot in general be neglected, especially a t low scan rates. This is in contrast to the case with solid-state systems where the degree of accuracy is not critical and high scan rates ( > 5 K/min) are the rule rather than the exception. In reality the sample will not be homogeneous in temperature and concentration during a temperature scan. When heat and mass transfer within the sample are not infinitely fast, then gradients will develop causing distortion of the data. In order to account for these possible gradients and their effects on the resultant DSC data, the following simplified model has been assumed for a differential scanning calorimeter (Figure 1). T h e top of the sample is approximated by an adiabatic boundary (physically this is a water-air boundary). T h e bottom of the sample is assumed to be held a t the instantaneous scanning temperature, Le., all of the effects by Flynn are ignored. As a first approximation, we have assumed that side wall effects are negligible, thus reducing the problem from two dimensions (cylindrical coordinates) to one dimension (planar coordinates). These approximations are justified as a “worst-case” since the usual encapsulated aqueous sample would be surrounded by an aluminum sample pan of relatively high thermal conductivity (compared to water). Thus, the sample should experience a heat input from both the sides and top which will tend to decrease temperature gradients in the solution. Clearly, the assumption that the upper boundary is adiabatic is extremely conservative. Our mathematical calculations become simpler when the coordinate is chosen such that x = 0 a t the adiabatic boundary
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978
and x = L a t the temperature scanning boundary. Both the sample and the reference cells are assumed to have the same configuration. This “worst-case” model does not include effects due to convective currents which are difficult to account for. If thermally or mechanically induced convective currents occur in t h e aqueous sample, they will aid only in reducing the temperature and concentration gradients (thermal lag) which develop in the cell. T h e above simplified DSC model allows one to write a boundary value problem in temperature and concentration which will simulate t h e temperature and concentration gradients produced during a temperature scan. As in the ideal case, one also assumes a first-order reaction with Arrhenius dependence of the rate constant on temperature. A separate boundary value problem must be written for both the sample and t h e reference.
# = D,/ZL2, rs* = r , / Z T o T h e problem becomes:
Sample Cell
aC* at*
a”*
- = 8--
C*exp(-y/T*)
ay?
Boundary Conditions
dT*
(O,t*) = 0
~
CIS’
T* ( l , t * ) = 1 + r,*t* dC*
Sample Cell
1999
-(O,t*) a?’
OC*
(22)
(23)
(l,t*)= 0
(24)
T* b,O) = 1, C * ( j , O ) = 1
(25)
=
~
ay
Initial Conditions
Reference Cell
05x I L, 0 < t
aT*
d2T*
z = 4 2ay .
Boundary Conditions
aT
Boundary Conditions
- ( 0 , t ) = 0 (conservation of energy) ax
dT* (O,t*) = 0 a?’ T* ( l , t * ) = 1 + r,*t*
T(L,t)= To + r,t
__
(27) (28)
Initial Condition Initial Conditions
T(x,O) =
r,
C(x,O) =
co
T* (y,O) = 1
T h e calculated heat flux, q,., or DSC signal obtained by numerical solution of the above boundary value problems can be compared quantitatively to the ideal heat flux, qi, in order to ascertain what range of parameter values, scan rate, sample thickness, reaction rate, etc. are acceptable in terms of the corresponding determinate errors in the data. Although direct comparison of 4, to 4, is meaningful, a more important comparison can be made after the data have been put in a form suited for the calculation of the kinetic parameter Z and AE. T h e method of Achar e t al. (23) is exact for first-order reactions and is applicable to our problem. The first-order mechanism described by Equation 2 can be transformed into the following:
Reference Cell
Boundary Conditions
dT
-
ilX
(29)
(0,t) = 0 (conservation of energy) T ( L , t ) = To + r,t
Initial Condition
T(x,O) = To
(30)
T h e heat flux a t x = L is given by where a is the fraction reacted at t . Linearization of Equation 30 yields Equation 31.
So for a true DSC scan, the output in cal,/s will be given by: 4DSC = qsamp -4ref (18) where K, p , c , D,, C , A , are the thermal conductivity (cal/ cm.K.s), density (g/cm3), heat capacity (cal/g.K), mass diffusion coefficient (cm2/s),reactant concentration (moI/cm’) and surface area (cm2) of the sample a t x = L , respectively, and all other symbols remain as previously defined. By making the following substitutions the entire problem can be reduced to dimensionless form. T* = T/To, C* = C/Co, y = x / L
t* 4
=
=
Zt, @
DT/ZL2,
=
(-1H)D&o/KTO
= 1E/RTU, DT = K / p c
(19)
In h = In Z
s
-
--
RT
where k dn/dt/’(l - a ) . When In k is ploted against l / T , the slope of the resultant straight line is - S I R while the intercept is In 2.Both q, and q, can be analyzed in this manner and compared.
NUMERICAL METHODS T h e above set of dimensionless boundary value problems was solved by a numerical analysis technique called orthogonal collocation ( I 7,28)which is relatively simple to apply and is much more efficient and more stable than the common finite difference techniques. Orthogonal collocation has been applied in several heat and mass transfer problems in chemical engineering with excellent results. Previously, we have utilized
2000
ANALYTICAL CHEMISTRY, VOL.
50, NO. 14, DECEMBER 1978
Table I. Typical Parameter Values Chosen for Study T,>= 273.0 K C,, = 1.0 X 10.' mol/cm3 A H = 10.0 kcal/mol K = 1.43 x cal/cm,K.s 2 = 4.0 X 10' S C ' A E = 12.0 kcalimol D , = 1.43 X c m 2 / s D , = 1.0 X cm2/s rs = 5 K/min L = 0.2 cm c = 1 . 0 ca1ig.K p = 1.0 g/cm3
c-
g - k
D
L
a_
.
8
'@.IC
_.__~ :.:2
3.25
C.?T
2.EC
3.52
:.-5
:,*-
~~~-
:.0c
Z-Pri
Figure 3. Plot of relative error in dnldt, (A): 1 - a , (B); d a l d t l ( 1 a ) , (C); and In k , (D) vs. alpha. Run number 1 , r, = 0.3 Klmin, L = 0.105 cm. All other parameter values are listed in Table I1 r-
Figure 2. Calculated theoretical DSC output (calls) for first-order reaction: GI, or exponential integral method, (A): and QC,orthogonal collocation method, (B). r, = 5 Klmin, L = 0.3 cm, 2 = 3.2 X 10' s-', If = 12.0 kcallmol, T o = 273.0 K , k , = 0.08 s-', C , = 1.0 X lo4 mol/cm3,A H = 10.0 kcal/mol. All other parameters are as listed in Table I
orthogonal collocation in the simulation of several electrochemical diffusion processes and obtained very accurate (Q2
1.3;
2,'s
2.87
7.12
Figure 12. Plot of relative error in d a l d t , (A): 1 - N , (B); d n / d t / ( l o ) ,(C); and In k , (D) vs. alpha. Run number 9, r, = 0.3 K/min, L = 0.105 cm, Z = 1.0 X l o ' s - ' , A € = 12.0 kcalimol, T, = 273.0 K, k, = 24.86 X s-'. All other parameter values are listed in Table I1
--
C.30
,
0.12
2.25
0.37
2.51
1.62
0,'s
3.87
1.00
:-PrlP
Figure 13. Plot of relative error in dtuldt, (A); 1 - a , (6); d n / d t l ( l a ) ,(C); and In k , (D) vs. alpha. Run number 10, r, = 0.3 K/min, L = 0.105 cm, Z = 4.0 X l o 8 s-', A € = 12.0 kcal/mol, T o = 273.0 K, k, = 99.44 X ss'. All other parameter values are listed in Table
I1
-i
C. Comparison of Scan Rate and Sample Thickness Effects. A comparison of In k for Figures 3 through 9 is shown in Figure 10. Once the maximum allowable relative error in AE and In Z has been selected, one can decide from Figure 10 the maximum limits on sample size and scan rate. For example, if one desired an accuracy of 5?&for both AE and In 2 with minimum of 50% usable data, then only the conditions of runs 1, 2, 4, and 5 are acceptable. T h u s a maximum sample thickness of 0.2 cm and scan rate of 5 K / m i n are allowable for the 5% error criterion. Obviously, other criteria can be employed.
D. Effect of Pre-exponential Factor, Z, Activation Energy, A E , and Initial Temperature, To. Figures 11 through 14 illustrate the effect of the pre-exponential factor, 2,on the Achar plots. When the rate of reaction is slow
Figure 14. Composite plot of relative error in In k vs. alpha for run number 8, (A), Z = 2.0 X l o 7 s-', k, = 4.97 X s-'; run number s-'; run number IO, (C), 9, (B), Z = 1.0 X lo's-', k , = 24.86 X Z = 4.0 X l o 8 s-', k, = 99.44 X s-'. All other parameters are constant and are listed in Table I1
compared to heat transfer in the sample, then the calculated
DSC trace follows the ideal DSC trace faithfully except a t small a (
