Derivation of temperature variation rule and variation rules of temperature lag, heat flows, internal energy and effective specific heat of platelike sample in temperature modulated differential scanning calorimetry

Article abstract of DOI:10.1016/s0040-6031(99)00143-4

In this paper, the temperature variation rule of platelike sample in temperature modulated differential scanning calorimetry (TMDSC) has been drawn with strict mathematical derivation. The obtained analytical result reveals the total variation rule of the platelike sample from its initial equilibrium state to its steady state. With this temperature variation rule, the variation rules of both reversible and irreversible heat flows, temperature lag, internal energy and effective specific heat of the platelike sample have been derived and studied as well. If the thermal conductivity of the sample is so great that the temperature gradients within the sample can be neglected, in this case the temperature variation rule derived from the fundamental equation of the temperature distribution of our TMDSC is the same as the current TMDSC theories. If the modulated amplitude A_Ts or modulated frequency ω equals zero, it reverts to the conventional DSC situation, so all the results in the TMDSC model derived in this article are automatically suitable for the conventional DSC situation and current TMDSC theories.

Full text of DOI:10.1016/s0040-6031(99)00143-4

Thermochimica Acta 336 (1999) 1±15
Derivation of temperature variation rule and variation rules of
temperature lag, heat ¯ows, internal energy and effective
speci®c heat of platelike sample in temperature
modulated differential scanning calorimetry$
a,b,*
En-Yong Ding , Rong-Shi Cheng , Yu-Hui Huang
a
b
a
College of Material Science and Engineering, South China University of Technology, Guangzhou 510641, China
b
Laboratory of Cellulose and Lignocellulosics Chemistry, Guangzhou Institute of Chemistry,
Chinese Academy of Sciences, Guangzhou 510650, China
Received 4 January 1999; received in revised form 2 June 1999; accepted 7 June 1999
Abstract
In this paper, the temperature variation rule of platelike sample in temperature modulated differential scanning calorimetry
TMDSC) has been drawn with strict mathematical derivation. The obtained analytical result reveals the total variation rule of the
(
platelike sample from its initial equilibrium state to its steady state. With this temperature variation rule, the variation rules of both
reversible and irreversible heat ¯ows, temperature lag, internal energy and effective speci®c heat of the platelike sample have been
derived andstudied aswell. Ifthe thermal conductivity ofthesample issogreat that thetemperaturegradients within the sample can
beneglected,inthiscase the temperaturevariationrulederivedfromthe fundamental equation ofthe temperaturedistributionofour
TMDSCisthesameasthecurrentTMDSCtheories.IfthemodulatedamplitudeAT ormodulatedfrequency! equalszero,itreverts
to the conventional DSC situation, so all the results in the TMDSC model derived in this article are automatically suitable for the
s
conventional DSC situation and current TMDSC theories. # 1999 Elsevier Science B.V. All rights reserved.
Keywords: Temperature modulated differential scanning calorimetry; Platelike sample; Temperature distribution; Heat ¯ow; Effective speci®c
heat
1
.
Introduction
nationally. Since Reading [1,18] invented the
TMDSC, the apparatus of TMDSC has been success-
fully commercialized [2±6]. The non-linear heating
rate in TMDSC causes many dif®culties in handling
the data. How to use TMDSC to measure the char-
acteristics of matter effectively, how to solve the
dif®cult problems in dealing data and how to expand
the application of TMDSC are the focuses of studying
in thermal analysis theoretical society.
Temperature modulated differential scanning
calorimetry (TMDSC) is a powerful thermal analysis
tool, which ®nds much use in various regions. The
study and manufacture of new and effective thermal
analysis apparatus have drawn much attention inter-
$Presented at the Ninth Chinese Conference on Chemical
Thermodynamics and Thermal Analysis (CTTA), Beijing,
China, August 1998.
If the real temperature gradients within the sample
are omitted, the measured sample can be taken as the
sample with a uniform temperature. Under this
*
Corresponding author.
0040-6031/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 0 4 0 - 6 0 3 1 ( 9 9 ) 0 0 1 4 3 - 4

2
E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
approximate condition, there are some articles dealing
with sample heat capacities, glass-transition tempera-
ture, etc. in TMDSC [7±12]. Just as in the conven-
tional DSC, if the temperature gradients within the
sample are omitted, the measuring errors in TMDSC
will occur unavoidably, which are sometimes rather
large in some case [13]. To minimize the measuring
errors, and to explain the physical meanings of each
eigen point and each eigen curve correctly, it is
necessary to consider the temperature gradients within
the sample [14]. Thus, the variation rule can be
revealed correctly between the temperature distribu-
tion within the sample and the heating form of stove,
and much valuable information can be effectively
obtained from the real thermal analysis curve.
thermal transference equation
2
@
Tꢀx; t†
@ Tꢀx; t†
2
ˆ a
where T(x,t) is the sample temperature at the depth x
;
(1)
2
@
t
@x
2
and at the time t, a ˆ ꢀ=ꢁc , ꢀ is the thermal con-
p
ductivity of the sample at temperature T, ꢁ the mass
density of sample at temperature T, c is the specific
p
heat capacity of sample at temperature T. Here, for
simplicity, the value of ꢀ, ꢁ and c are assumed as
constants in the studied temperature interval.
Almost all the existing popular theories of TMDSC
are based on the approximate assumptions that the
p
values of ꢀ, ꢁ and c of the sample are assumed as
p
constants in the studied temperature interval, and the
temperature gradients within the sample are omitted
which actually implies that the thermal conductivity of
the sample is in®nite. In most situations, the assump-
For simplicity, in this paper we assume that the
pan's thermal resistor are so small that can be
neglected. This assumption will not in¯uence the
universality of our following theory.
tions that the ꢀ, ꢁ and c of the sample are constant in
p
the studied temperature interval can be rationally
accepted which may cause little error, but the assump-
tion that the temperature gradients within the sample
are omitted may cause obvious errors, and may cause
ratherbig errors in some cases. Although therearesome
thermal analysis theories dealing with conventional
DSC [15] and TMDSC [16,17] in which the tempera-
ture gradients in the sample are considered, there are
also some obvious approximations in these theories. In
our TMDSC theory, by considering the temperature
gradients within the sample we will try to obtain the
exact temperature distribution within the sample and
its variation rule. Using the obtained temperature
variation rule we can obtain the variation rule of some
physical quantities, so we can improve the existing
analytical theory of TMDSC and obtain more precise
values of physical quantities by TMDSC experiments.
The sample can be taken as a total depth 2l with two
surfaces exposed to the heating surrounding, or
equivalently a total depth l with one adiabatic surface
and another surface exposed to the heating surround-
ing, so we get boundary condition:
To enhance the measuring precision and decrease
the measuring error caused by the temperature gra-
dients within the sample, the sample is generally made
in the platelike form and the quantity of sample is as
small as possible within the sensitivity of the appara-
tus. Because in the general thermal apparatus, tem-
perature detector (e.g., thermocouple) is placed in the
central position under the sample box, the measured
temperature actually is the sample temperature in the
center of outer surface, so the real sample can be taken
as a plate, and the boundary effect caused by the ®nite
sample size can be rationally omitted.
In this article, we will use strict mathematical tools
to solve the temperature distribution rule of platelike
sample in TMDSC model. Then, we will use this
temperature variation rule to obtain the strict mathe-
matical expressions of reversible and irreversible heat
¯ows, temperature lag, internal energy and effective
speci®c heat of the platelike sample.
2
.
Mathematical derivation of temperature
variation rule of platelike sample in
TMDSC model
ꢀ
ꢁꢂ
ꢂ
ꢀ @T
ꢂ
T� K @x
ˆ Ts;
ꢂ
0
ꢂ
ꢂ
ꢂ
ꢂ
@
T
Assume the sample shape studied in TMDSC is ¯at,
ˆ 0:
l
(2)
@
x
it can be taken as a plate. For a platelike sample, we
only need to study the temperature distribution in the
plate depth direction. In this condition, there is a
where Ts ˆ T0 ‡ qt ‡ AT sin !t, T
is the program-
the
s
s
controlled stove temperature in TMDSC model, T
0

E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
3
initial temperature of stove, q the linear heating rate of
stove, and AT and ! are, respectively, amplitude and
frequency of modulated heating rate.
Sample's initial condition is
The left side of Eq. (11) is the function of time t, but
the right side is the function of place x. Because the
equation is valid, both side of the equation must be
s
2
2
equal to a constant. De®ning this constant � ꢃ a , we
get:
Tꢀx; 0† ˆ T0; (3)
i.e., at the initial time, whole sample's temperature is
2
2
1
dU a d X
ˆ
2
2
ˆ � ꢃ a :
X dx2
(12)
U dt
T .
0
De®ne
From these, the obtained solutions are
2
2
Tꢀx; t† ˆ T0 ‡ qt ‡ AT sin !t ‡ Dꢀx; t†;
(4)
� ꢃ a t
;
s
U ˆ Uꢀ0†e
Xꢀx† ˆ A sin ꢃx ‡ B cos ꢃx:
(13)
(14)
where Á(x,t) is a correction function. Only in the
extremely ideal situation that the thermal conductivity
of the sample is infinite, the correction function Á(x,t)
is equal to zero. In the general situation Á(x,t) is not
equal to zero, so Eq. (1) can be changed into the
following form:
�
ꢃ
ꢀ @X
K @x
From a recent boundary condition X�
j0 ˆ 0,
we can obtain
ꢀ
B ˆ Aꢃ:
K
(15)
2
dXꢀx†
dx
@
Dꢀx; t†
@ Dꢀx; t†
From another boundary condition
get
jl ˆ 0, we
2
�
a
ˆ � q� A ! cos !t: (5)
Ts
2
@
t
@x
The boundary condition (2) becomes
Aꢃ cos ꢃl� Bꢃ sin ꢃl ˆ 0:
Combining Eq. (15) with Eq. (16), if follows
(16)
ꢀ
ꢁꢂ
ꢂ
ꢀ
@D
K @x
ꢂ
D�
ˆ 0;
ꢂ
0
K
ꢂ
ꢂ
ꢂ
ꢂ
ꢃn ˆ ctgꢃnl; n ˆ 0; 1; 2; . . .;
(17)
@
D
ꢀ
ˆ
0;
(6)
(7)
@
x
l
i.e., to satisfy the boundary conditions, ꢃ must be the
roots of Eq. (17).
So we have a general solution
and the initial condition (3) becomes
Dꢀx; 0† ˆ 0:
‡
1
X
By using the method of impulse theorem and de®n-
2
2
R
� ꢃna ꢀt� ꢂ†
t
vꢀx; t; ꢂ† ˆ
Cnꢀꢂ†e
ing Dꢀx; t† ˆ vꢀx; t; ꢂ† dꢂ, Eq. (5) becomes
0
nˆ0
ꢄ
ꢅ
2
@
vꢀx; t†
@ vꢀx; t†
ꢀꢃn
2
�
a
ˆ 0:
(8)
 sinꢃnx ‡
cosꢃnx :
(18)
2
@
t
@x
K
P
‡
1
The boundary condition (6) becomes:
From
ꢀ
initial
condition
Cnꢀꢂ†‰sinꢃnx
nˆ0
ꢀ
ꢁꢂ
ꢃn
K
ꢂ
‡
cosꢃ xŠ ˆ � q� A ! cos !ꢂ, we get
ꢀ
@v
n
Ts
ꢂ
v�
ˆ 0;
ꢂ
K @x
2
0
2K
ꢂ
ꢂ
ꢂ
ꢂ
Cnꢀꢂ† ˆ �
@
v
2
2
2
ꢃnꢀK l ‡ lꢀ ꢃn ‡ ꢀK†
 ꢀq ‡ A ! cos !ꢂ†;
ˆ
0;
(9)
@
x
l
(19)
Ts
and the initial condition (7) becomes
where we have used orthogonal relation of the intrinsic
function
vꢀx; t ˆ ꢂ ‡ 0† ˆ � q� AT ! cos !ꢂ:
By using variable separation method and de®ning
(10)
s
Z
l
2
Cnꢀꢂ†
jXnꢀx†j dx ˆ � ꢀq ‡ AT ! cos !ꢂ†
s
v(x,t)=U(t)X(x), we have
0
Z
l
2
2
1
dU a d X
ˆ
Xnꢀx† dx;
(20)
:
(11)
U dt
X dx2
0

4
E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
Z
l
2
2
2
n
K l ‡ lꢀ ꢃ ‡ ꢀK
Finally, we have
2
jXnꢀx†j dx ˆ
;
2
2
K
0
Z
l
Tꢀx; t† ˆ T ‡ qt ‡ A sin !t ‡ Dꢀx; t†
0
Ts
1
Xnꢀx†dx ˆ
:
(21)
‡
1
X
ꢃn
0
ˆ
T ‡ qt ‡ A sin !t�
0
Ts
nˆ0
So we have
2
K2
‡
1
Â
X
2
2
2
n
ꢃnꢀK l ‡ lꢀ ꢃ ‡ ꢀK†
vꢀx; t; ꢂ† ˆ � ꢀq ‡ A ! cos !ꢂ†
Ts
ꢀ
ꢁ
nˆ0
ꢀꢃ
n
Â
Â
sin ꢃ x ‡ K cos ꢃnx
K2
n
2
2
n
2
�
ꢃ a ꢀt� ꢂ†
Â
e
Xnꢀx†:
(
2
2
2
n
ꢃnꢀK l ‡ lꢀ ꢃ ‡ ꢀK†
q
2
2
2
A !
Ts
ꢀ1� e�
ꢃ a t
† ‡
n
2
4
n
4
2
(22)
ꢃ a
ꢃ a ‡ !
n
)
h
i
2
2
Thus, we can obtain the correction function
2
n
2
� ꢃna t
 ꢃ a ꢀcos!t� e
† ‡ !sin !t
;
Z
t
X‡1
(
25)
Dꢀx; t† ˆ
vꢀx; t; ꢂ†dꢂ ˆ
DnꢀA; !; x; t†
0
nˆ0
Tꢀx; t† ˆ T0 ‡ qt ‡ AT sin !t ‡ Dꢀx; t†
s
‡
1
2
X
2
K
‡
1
X
ˆ
�
Â
Â
2
2
2
n
ꢃnꢀK l ‡ lꢀ ꢃ ‡ ꢀK†
ˆ T0 ‡ qt ‡ AT sin !t�
nˆ0
s
ꢀ
ꢁ
nˆ0
ꢀ
ꢃn
2K2
sin ꢃ x ‡
cos ꢃnx
n
K
Â
2
2
2
n
(
h
ꢃnꢀK l ‡ lꢀ ꢃ ‡ ꢀK†
2
n
2
A !
Ts
ꢀ
ꢁ
q
ꢀ1� e�
ꢃ a t
† ‡
ꢀꢃ
n
2
n
2
4
n
4
2
Â
sin ꢃnx ‡
cos ꢃnx
(25A)
ꢃ a
ꢃ a ‡ !
K
)
i
(23)
8
2
n
2
2
n
2
� ꢃ a t
 ꢃ a ꢀcos !t� e
† ‡ !sin !t
;
>
<
q
2
n
2
AT !
s
1� e�
ꢃ a t
ꢀ
† ‡ q sinꢀ!t ‡ ꢄn†
2
2
>
:
ꢃ a
4
n
4
2
ꢀ
0ꢁxꢁl; tꢂ0†:
n
ꢃ a ‡ !
)
2
n
2
2
n
2
� ꢃ a t
AT !ꢃ a e
Á is defined as follows:
s
n
 �
;
ꢀ0ꢁxꢁl; tꢂ0†;
4
4
2
ꢃ a ‡ !
n
2
K2
D ꢀA; !; x; t† ˆ �
n
where ꢄ is defined as
n
2
2
2
n
ꢃ ꢀK l ‡ lꢀ ꢃ ‡ ꢀK†
n
ꢀ
ꢁ
2
2
ꢀ
ꢃ
n
ꢃ a
n
Â
Â
sin ꢃnx ‡
cos ꢃnx
ꢄ ꢃarcsin q :
(26)
n
K
4
n
4
2
(
ꢃ a ‡ !
q
2
2
2
AT !
4
ꢀ1� e�
ꢃ a t
† ‡
s
n
2
4
2
ꢃ a
ꢃ a ‡ !
Eqs. (25) and (25A) are the temperature distribution
rule within the platelike sample in TMDSC model, and
this is the fundamental and most important equation of
our TMDSC theory.
n
n
)
h
i
2
n
2
2
n
2
� ꢃ a t
 ꢃ a ꢀcos !t� e
† ‡ !sin !t
;
2
n
2
�
ꢃ a t
If the time is long enough, the item e
becomes
ꢀ
0ꢁxꢁl; tꢂ0†;
(24)
so small that it can be neglected. In this case the
sample is in the steady state, and Eqs. (25) and (25A)
can be rewritten as
where Á is the nth temperature distribution correc-
tion item of the sample.
n

E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
5
‡
1
X
dTꢀx; t†
dx
ꢀ
s†
T
ꢀx; t† ˆ T0 ‡ qt ‡ AT sin !t�
s
nˆ0
ꢀ
ꢁ
‡
1
K2
X
2
ꢀꢃn
2
K2
ˆ�
cos ꢃ x� K sin ꢃnx
n
2
2
2
n
Â
Â
K l ‡ lꢀ ꢃ ‡ ꢀK
2
2
2
n
nˆ0
ꢃ ꢀK l ‡ lꢀ ꢃ ‡ ꢀK†
n
(
ꢀ
ꢁ
ꢀ
ꢃ
q
2
� ꢃ a t
2
2
A !
Ts
n
n
Â
q sinꢀ!t ‡ ꢄ †
n
sinꢃnx ‡
cos ꢃnx
(25B)
ꢀ1� e
†‡
2
K
ꢃ a
4
n
4
2
n
ꢃ a ‡ !
(
)
Ã
)
Â
2
n
2
n
2
q
AT !
2
� ꢃ a t
s
2
n
2
A !ꢃ a e
Ts
‡
ꢃ a cos !t ‡ ! sin !t
;
2
n
2
4
n
4
2
�
:
(27A)
ꢃ a
ꢃ a ‡ !
4
n
4
2
ꢃ a ‡ !
‡
1
X
ꢀs†
T
ꢀx; t† ˆ T ‡ qt ‡ A sin !t�
0
Ts
3
.
Discussion on the temperature distribution rule
nˆ0
2
K2
To verify Eqs. (25) and (27), now we study a very
special example.
Â
2
2
2
n
ꢃnꢀK l ‡ lꢀ ꢃ ‡ ꢀK†
ꢀ
ꢁ
ꢀ
ꢃn
Â
sinꢃnx ‡
cos ꢃnx
Example 1. Assume that in the studied temperature
interval the sample's thermal conductivity ꢀ is 1, its
mass density ꢁ is 1, its specific heat c is 1.5, so we get
K
2
3
7
6
q
2
AT !
s
p
 4
‡ q sinꢀ!t ‡ ꢄ †5;
2
n
2
a ˆ ꢀ=ꢁc ˆ 0:6667. We also assume that sample'
p
ꢃ a
4
n
4
2
n
ꢃ a ‡ !
depth l is 0.1. From Eq. (17) we can get the roots,
l =8.6033357, l =34.25618387, l =64.3729811, l =
ꢀ
0ꢁxꢁl; tꢂ0†:
(25C)
0
1
2
3
9
5.29334263,l =126.45287025,l =157.71284669,etc.
4
5
From Eqs. (25) and (25A) it can be known that the
temperature distribution function T(x,t) is related to
the experimental conditions, such as heating rate of
surrounding, Newton's Law Constant of the sample
box, sample's initial temperature, its speci®c heat, its
Assume the Newton's constant K is 10, the modu-
lated amplitude and frequency, ATs and ! are 1 and ꢅ,
respectively. We can get a relation between the tem-
perature of stove T and the time t as shown in Fig. 1.
From Fig. 2, we can know that Á is the dominant
item, the sum of other items is just only the 2% of the
value of Á . For simplicity, in this special example we
s
0
thermal conductivity and other factors. When AT or !
s
equals zero, it reverts to the conventional DSC situa-
tion, so the temperature distribution rule within the
sample in the TMDSC model derived in this article
automatically suits the conventional DSC situation.
The temperature gradients within the sample can be
derived from the basic Eqs (25) and (25A):
0
onlyneedtoconsiderthein¯uenceoftheÁ andtheerror
0
caused by this approximation will be less than 2%.
In this special situation, Eqs. (25) and (25A) can be
rewritten as follows:
Tꢀx; t†ꢄT0 ‡ qt ‡ AT sin !t ‡ D0ꢀA; !; x; t†
2K
s
‡
1
2K2
X
2
dTꢀx; t†
dx
ˆ
�
Â
Â
ˆ T ‡qt‡A sin !t�
2
n
0
Ts
2
0
2
2
2
2
K l ‡ lꢀ ꢃ ‡ ꢀK
ꢃ0ꢀK l‡lꢀ ꢃ ‡ꢀK†
nˆ0
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢃn
ꢀꢃ0
cos ꢃ x�
sin ꢃnx
Â
Â
sinꢃ x ‡ K cos ꢃ0x
n
0
K
(
h
(
h
q
2
n
2
A !
Ts
q
2
2
A !
Ts
ꢀ1� e�
ꢃ a t
† ‡
ꢀ1� e
� ꢃ a t
† ‡
0
2
2
4
n
4
2
2
0
2
4
0
4
2
ꢃ a
ꢃ a ‡ !
ꢃ a
ꢃ a ‡ !
n
)
)
i
27)
i
2
n
2
2
2
2
n
2
� ꢃ a t
2
0
2
� ꢃ a t
0
 ꢃ a ꢀcos!t� e
†‡! sin !t
;
 ꢃ a ꢀcos !t� e
† ‡ ! sin !t
;
(
(25D)

6
E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
Fig. 1. Heating rate in a TMDSC experiment.
Tꢀx; t†ꢄT ‡ qt ‡ A sin !t ‡ D ꢀA; !; x; t†
where ꢄ is defined as
0
Ts
0
ˆ
T ‡ qt ‡ A sin !t
0
Ts
2
0
2
ꢃ a
2
ꢄꢃarcsin q :
Now, we study the relationship between the modu-
2
K
�
4
0
4
2
2
0
ꢃ a ‡ !
2
2
ꢃ ꢀK l ‡ lꢀ ꢃ ‡ ꢀK†
0
ꢀ
ꢁ
ꢀ
ꢃ0
Â
sin ꢃ0x ‡
cos ꢃ0x
K
lated temperature of the stove and the 0th temperature
distribution correction item of the sample Á . From
Fig. 3, we know there is a phase shift, ꢅ+ꢄ, between
them. In this example, we have ꢄ = 1.507. The main
cause of this phase shift is the sample's limited
thermal conductivity.
Now, we study the temperature gradients within the
sample. Fig. 4(a) shows us the temperature distribu-
tions within the sample at the different time. To
obviously observe the temperature gradients within
the sample and its variation rules, at ®rst, we derive the
(
0
q
2
2
ꢀ1� e�
ꢃ a t
†
0
Â
2
0
2
ꢃ a
AT !
s
‡
�
q sinꢀ!t ‡ ꢄ†
4
0
4
2
ꢃ a ‡ !
)
2
0
2
2
0
2
� ꢃ a t
AT !ꢃ a e
s
;
ꢀ0ꢁxꢁl; tꢂ0†;
4
0
4
2
ꢃ a ‡ !
(25E)

E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
7
Fig. 2. Compare between 0th and 1st correction items.
dTꢀx; t†
dx
2K2
relationship between the temperature gradient and the
depth x. From Eq. (27A), it is easy that the temperature
gradients within the sample can be derived
ˆ
�
Â
Â
2
2
2
0
K l ‡ lꢀ ꢃ ‡ ꢀK
ꢀ
ꢁ
ꢀ
ꢃ0 sinꢃ0x
2K2
2
0
cos ꢃ x�
dTꢀx; t†
dx
K
ˆ
�
2
2
0
K l ‡ lꢀ ꢃ ‡ ꢀK
(
q
2
2
0
2
2
ꢀ
ꢁ
� ꢃ a t
0
ꢀ1� e
†
ꢀ
ꢃ0
ꢃ a
Â
cos ꢃ0x�
sin ꢃ0x
K
(
AT !
s
‡
q sin !ꢀt ‡ ꢂ†
q
2
2
2
ꢀ1� e�
ꢃ a t
†
4
0
4
2
0
Â
ꢃ a ‡ !
2
ꢃ a
0
)
2
0
2
2
0
2
� ꢃ a t
A !ꢃ a e
4
AT !
Ts
s
�
;
‡
q sinꢀ!t ‡ ꢄ†
4
2
ꢃ a ‡ !
4
0
4
2
0
ꢃ a ‡ !
)
ꢀ0ꢁxꢁl; tꢂ0†;
where ꢂꢃꢄ=!, in this example, t=0.48.
(27C)
2
0
2
2
0
2
� ꢃ a t
A !ꢃ a e
4
Ts
 �
;
(27B)
4
2
ꢃ a ‡ !
0

8
E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
Fig. 3. Variation of modulated temperature and sample's surface temperature.
From Fig. 4(b) we can obviously know that
If we draw a ®gure about the relationship bet-
ween the sample's surface temperature and the time,
we can get an almost the same curve as shown in
Fig. 1. But if we amplify the curve, we can discover
the temperature difference (i.e. temperature lag)
between the sample's surface and the stove (as shown
in Fig. 6).
From above brief discussion we ®nd out that the
temperature variation rule of platelike sample is self-
satis®ed, and the obtained results in all these ®gures
conform to our previous experiences about the thermal
analysis.
there is really temperature gradient within the sample
which varies periodically with the time. In Fig. 4(b),
we can discover the temperature gradient in depth l is
zero, which is the inexorable result obeying the
boundary condition. We also can ®nd out another
interesting phenomenon, i.e., the temperature gradi-
ents variation in a periodic time is asymmetric to the
abscissa. This phenomenon is reasonable, in the
heating process the equivalent heat ¯owing direction
is from stove to sample in a periodic time so the total
temperature gradients within the sample must be
negative.
Now, we discuss another representative example.
In Fig. 5, we can ®nd out the in¯uences of the
modulated amplitude and frequency on the 0th tem-
perature distribution correction item of the sample.
From these calculation results, it can be easily known
that bigger the modulated amplitude or the frequency,
the bigger is the variation extent of the 0th temperature
distribution correction item.
Example 2. From above discussion, we know that in
the general situation if the thermal conductivity of the
sample is big enough, that is ꢀ! ‡ 1, in Eq. (25), we
only need to study the dominant item Á and neglect
0
other correction items. So Eq. (25) can be simplified as
follows:

E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
9
Fig. 4. Temperature distribution within the sample: (a) temperature distribution within the sample at different time; (b) temperature gradients
within in a periodic time.

1
0
E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
Fig. 5. Inference of the modulated amplitude and frequency: (a) the relation between the modulated amplitude and sample's surface
temperature; (b) the relation between the modulated amplitude and sample's temperature distribution; (c) the relation between the modulated
frequency and sample's surface temperature; (d) the relation between the modulated frequency and sample's temperature distribution.

E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
11
Fig. 5. (Continued )

1
2
E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
Fig. 6. Temperature of stove and sample's surface temperature.
Tꢀx; t†ꢄT ‡ qt ‡ A sin !t
If the sample's thermal conductivity is suf®ciently
great, the temperature gradients within the sample can
0
Ts
2
2
K
�
be rationally omitted. In the situation ꢀ! ‡ 1, from
2
2
2
0
ꢃ ꢀK l ‡ lꢀ ꢃ ‡ ꢀK†
0
K
ꢀ
2
0
ꢀ
ꢁ
ꢃ0 ˆ ctgꢃ0l, we can get ꢃ ˆ K=ꢀl and
ꢀ
ꢃ0
ꢂ ˆ ꢀꢁc l=K† ˆ ꢀC=KS†, so we obtain ꢆ = 1. Thus,
p
Â
sin ꢃ0x ‡
cos ꢃ0x
K
we get
(
h
q
2
0
2
AT !
ꢀ1� e�
ꢃ a t
† ‡
s
Â
� t=ꢂ
2
0
2
4
0
4
2
Tꢀ0; t† ˆ T ‡ qt� qꢂꢀ1� e
†
0
ꢃ a
ꢃ a ‡ !
h
i
)
A
Ts
sin !t� !ꢂꢀcos !t� e� t=ꢂ
A !ꢂ
i
2
2
‡
†
2
0
2
� ꢃ a t
2
2
0
1 ‡ ! ꢂ
 ꢃ a ꢀcos !t� e
† ‡ ! sin!t
�
t=ꢂ
Ts
� t=ꢂ
ꢆ
ꢇ
ˆ T0 ‡ qt� qꢂꢀ1� e
† ‡
e
2
2
�
t=ꢂ
1 ‡ ! ꢂ
ˆ
T ‡ qt ‡ A sin!t� ꢆqꢂ 1� e
0
Ts
AT
s
h
i
‡ p sinꢀ!t� !ꢂ†; ꢀtꢂ0†
ꢆA !ꢂ
2
Ts
ꢀcos !t� e� t=ꢂ † ‡ !ꢂ sin !t :
2
2
1
‡ ! ꢂ
� 1
2
‡ ! ꢂ
(
30)
(
25F)
This equation is exactly the same as the basic
equation of the current TMDSC theories in which
the temperature gradients are omitted [7]. So it can be
seen that the current TMDSC theories in which tem-
perature gradients are omitted are only special exam-
ples of our general theory.
where some de®nitions are used
2
0
2
ꢂꢃꢃ a ;
(28)
(29)
2
Kꢀ
2
ꢆꢃ
;
2
2
0
K l ‡ lꢀ ꢃ ‡ ꢀK

E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
13
4
.
Derivation of the variation rules of
temperature lag, heat flows, internal energy
and effective specific heat of platelike sample
This is the generalized temperature lag rule of the
platelike sample in TMDSC.
Under the steady-state condition, i.e. the term
2
2
�
ꢃ a t
n
e
can be omitted, we have
Further discussion of Eqs. (25) and (25B) will
give rise to many new subjects, such as the varia-
tion rules of sample's internal energy, the energy
q
AT !
4
ꢀs†ꢀt†ꢃ
s
2
2
n
f
‡
ꢀꢃ a cos !t‡! sin !t†:
n
2
4
ꢃna2 ꢃna ‡ !
2
¯
ow within the sample, the temperature lag, the
(34A)
reversible heat ¯ow and the irreversible heat ¯ow
within the sample, etc., so we will discuss them in
the following.
Thus, under the steady-state condition, there is
temperature lag rule of the platelike sample
‡
1
X
ꢀ
s†
hnfꢀs†ꢀt†:
ꢇT ˆ �
(35A)
n
4
.1. Temperature lag rule of the platelike sample
nˆ0
First of all, we study the temperature lag dT of
the sample surface temperature with the variation of
surrounding temperature. dT is de®ned as follows:
4
.2. Variation rule of reversible and irreversible
heat flows in the platelike sample
Under the general-state condition, the ideal rever-
ꢇTꢃTꢀ0; t†� T ꢀt† ˆ Tꢀ0; t†� T � qt� A sin !t:
s
0
Ts
sible Newton's heat ¯ow HFrev which is ¯owing into
the sample through the boundary of sample within a
unit area and in a unit time is
(31)
According to Eq. (25), Eq. (31) has the form
‡
1
‡
1
X
X
dQs
1
HFrev ˆ dt ˆ � KꢇT ˆ K
h f ꢀt†;
n n
(36)
ꢇT ˆ � 2Kꢀ
2
2
2
n
K l ‡ lꢀ ꢃ ‡ ꢀK
nˆ0
nˆ0
(
q
2
n
2
AT !
where Q is the heat energy absorbed by the sample.
s
ꢀ1� e�
ꢃ a t
† ‡
s
Â
2
n
2
4
n
4
2
Irreversible heat ¯ow is
ꢃ a
ꢃ a ‡ !
)
h
i
2
2
HFn:r: ˆ hHFi� HFrev;
(37)
2
n
2
� ꢃ a t
n
 ꢃ a ꢀcos !t� e
† ‡ ! sin!t
:
where hHFi is the standard heat flow detected practi-
cally with the experiment.
Under the steady-state condition, the ideal reversi-
(
32)
To simplify the form of Eq. (32), we make the
following de®nitions:
ꢀs†
ble Newton's heat ¯ow HF which is ¯owing into the
sample through the boundary of sample within a unit
area and in a unit time is
rev
2
Kꢀ
2
hnꢃ
;
(33)
2
2
n
K l ‡ lꢀ ꢃ ‡ ꢀK
ꢀs†
‡1
X
dQs
ꢀ
s†
ˆ � KꢇTꢀs† ˆ K
hnfꢀs†ꢀt†:
q
2
2
AT !
HF
ˆ
ꢀ1� e�
ꢃ a t
s
rev
n
n
dt
fnꢀt†ꢃ
† ‡
2
2
4
n
2
n
4
2
nˆ0
ꢃ a
ꢃ a ‡ !
n
(
36A)
2
2
n
2
� ꢃ a t
 ‰ꢃ a ꢀcos !t� e
† ‡ ! sin !tŠ:
Irreversible heat ¯ow is
(
34)
D
E� HFꢀs†
ꢀ
s†
ꢀs†
Thus Eq. (32) can be written as
HF ˆ HF
;
rev
(37A)
n:r:
‡
1
X
ꢈ
ꢉ
ꢀ
S†
ꢇT ˆ �
hnfnꢀt†:
(35)
where HF is the standard heat flow detected prac-
tically in the steady state in the experiment.
nˆ0

1
4
E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
4
.3. Variation rule of internal energy of platelike
sample
The de®nition of effective speci®c heat is
1
dEꢀt† :
ceffꢃ
(39)
(40)
ꢁ
dTꢀ0; t†
Eq. (39) can also be written
In the general situation, there is an equation about
internal energy of sample at time t in a unit volume
Z
l
1 dEꢀt†
1
ꢁ
cp
l
ceff ˆ
:
Eꢀt†ꢃ
Tꢀx; t† dx
ꢁ
dt dTꢀ0; t†=dt
0
"
In the general situation, from Eq. (38) , there is a
following relation:
ˆ
ꢁcp ꢀT ‡ qt ‡ A sin !t†
0
Ts
"
#
#
‡
1
X
‡
1
K2
dEꢀt†
dt
K
X
2
ˆ
ꢁcp ꢀq‡AT ! cos !t†�
hngnꢀt† ;
�
fnꢀt†
s
2
2
n
2
2
2
n
ꢀꢃ l
ꢃ lꢀK l ‡ lꢀ ꢃ ‡ ꢀK†
nˆ0
n
nˆ0
"
(
41)
ˆ
ꢁcp ꢀT ‡ qt ‡ A sin !t†
0
Ts
where g (t) is defined as follows:
n
#
dfnꢀt†
dt
2
n
2
A !
Ts
‡
1
� ꢃ a t
X
K
gnꢀt†ꢃ
ˆ qe
‡
4
n
4
2
�
hnfnꢀt† :
(38)
ꢃ a ‡ !
2
ꢀ
ꢃ l
nˆ0
n
2
n
2
2
n
2
2
n
2
� ꢃ a t
 ‰ꢃ a ꢀꢃ a e
� ! sin !t†
In the steady-state situation, there is an equation
about internal energy of sample at time t in a unit
volume
2
‡
! cos !tŠ:
Because there is a relation
(42)
Z
‡1
l
X
from Eq. (40), there is a relation as follows:
ꢁ
cp
dTꢀ0; t†
ꢀ
s†
ꢀs†ꢀx; t† dx
E ꢀt†ꢃ
T
ˆ q ‡ AT ! cos !t�
hngnꢀt†; (43)
s
l
dt
0
nˆ0
"
ˆ
ꢁcp ꢀT ‡ qt ‡ A sin !t†
0
Ts
P
‡
1
K
q ‡ AT ! cos !t�
2
hngnꢀt†
#
s
nˆ0
ꢀ
ꢃ l
n
‡
1
X
ceff ˆ cp
P
Under steady-state condition, there is a similar
:
K
‡1
�
hnf ꢀs†ꢀt†
(38A)
q ‡ ATs ! cos !t�
hngnꢀt†
2
n
n
nˆ0
ꢀ
ꢃ l
nˆ0
(
44)
4
.4. Variation rule of effective specific heat of
platelike sample
expression of sample's effective speci®c heat
P
‡
1
K
ꢀꢃnl
ꢀs†
q ‡ A ! cos !t�
2
h gn ꢀt†
Ts
nˆ0
n
ꢀ
s†
c
ˆ cp
;
Because of the thermal resistance of the sample and
eff
P
‡
1
ꢀs†
q ‡ AT ! cos !t�
hngn ꢀt†
s
nˆ0
the non-linear heating or cooling rate in the TMDSC
model, there is a sample's temperature lag effect that
takes place with the variation of surrounding environ-
mental temperature. Because the temperature mea-
sured in the real TMDSC experiment is the
temperature of the sample's outer surface, the mea-
sured speci®c heat in TMDSC is not the same as the
real speci®c heat of the sample. We use equivalent
speci®c heat or effective speci®c heat to obtain the
exact value of the measured sample's speci®c heat in
TMDSC.
(
44A)
ꢀs†
where the used definition gn ꢀt† is as follows:
ꢀ
s†
dfn ꢀt†
dt
AT !2
ꢀ
s†
s
g ꢀt†ꢃ
ˆ
n
4
4
2
ꢃ a ‡ !
n
2
2
 ꢀ! cos !t� ꢃ a sin !t†:
(42A)
n
The expression in Eq. (44) is the effective speci®c
heat of sample in TMDSC. With this expression, the
exact value of real speci®c heat of sample can be

E.-Y. Ding et al. / Thermochimica Acta 336 (1999) 1±15
15
obtained from the effective speci®c heat measured in
TMDSC. From the expression of effective speci®c
heat of sample, it is obvious that the effective speci®c
heat of sample can vary with the variation of the
modulate amplitude and modulated frequency.
Now, let us reconsider the example two. Consider-
ing an imaginary situation in which the thermal con-
ductivity of the sample is in®nite, i.e. ꢀ! ‡ 1. In this
ultimate situation, the temperature gradients within
the sample can be neglected. From the Eqs. (33) and
variation rule. The obtained results show that rever-
sible and irreversible heat ¯ows, temperature lag,
internal energy and effective speci®c heat of the plate-
like sample are functions of experimental conditions,
such as modulated amplitude and modulated fre-
quency. So if we use TMDSC to obtain the character-
istics of the matter we must deal with the experimental
data according to corresponding physical rules. In the
general situation, the sample's thermal conductivity is
not great enough, so the effects caused by temperature
gradients cannot be omitted. All the experimental data
obtained in the TMDSC must be dealt carefully with
appropriate calibration methods such as considering
the temperature gradients within the sample.
(
tends to zero and the value of g (t) is a ®nite quantity,
42) we know when ꢀ tends to in®nite, the value of hn
n
so we can easily obtain the relation ceff ˆ cp from Eq.
(
44). Only in the ultimate situation that the sample's
thermal conductivity is so high that the temperature
gradients within the sample can be neglected, the
value of sample's effective speci®c heat in TMDSC
equals to that of real speci®c heat.
References
[
[
1] M. Reading, Trend Polym. Sci. 8 (1993) 248.
2] C. Tomasi, P. Mustarelli, N.A. Hawkins, V. Hill, Thermo-
chim. Acta 278 (1996) 9.
From the above derivation, it is not dif®cult to know
that if the sample's thermal conductivity is not so high
that the temperature gradients within the sample can-
not be neglected. In this general situation, the tem-
perature gradients within the sample must be
considered. The analytical theory here is far more
complex in form than the current theories and we can
anticipate that our analytical theory of TMDSC is
more precise.
[
3] J.E.K. Schawe, G.W.H. H oÈ hne, Thermochim. Acta 287
(
1996) 213.
[4] T. Ozawa, K. Kanari, Thermochim. Acta 288 (1996) 39.
[
[
5] Y. Saruyama, Thermochim. Acta 282/283 (1996) 157.
6] D.J. Houston, M. Song, A. Hammiche, H.M. Pollock, M.
Reading, Polymer 38(1) (1997) 1.
[
7] B. Wunderlich, Y. Jin, A. Boller, Thermochim. Acta 238
(
1994) 277.
[8] A. Boller, Y. Jin, B. Wunderlich, J. Therm Anal. 42 (1994)
07.
9] En-Yong Ding, Xue-Hai Liang, J. Instrumental Anal. 17(6)
1998) 25.
All these equations derived here will be veri®ed by
both TMDSC experiments and computer simulations.
The detailed results will be offered in other articles.
3
[
(
[
[
10] A. Zanier, H.W. Jackle, Thermochim. Acta 287 (1996) 203.
11] B. Wunderlich, A. Boller, I. Okazaki, S. Kreitmeier,
Thermochim. Acta 282/283 (1996) 143.
5
.
Conclusion
[
[
[
12] A. Boller, I. Okazaki, B. Wunderlich, Thermochim. Acta 284
(
1996) 1.
13] En-Yong Ding, Xue-Hai Liang, Chinese J. Anal. Chem. 21(6)
1993) 660.
The strict analytical temperature variation rule of
platelike sample in TMDSC model derived here
reveals the total variation rule of the platelike sample
from its initial equilibrium state to its steady state.
This theory is more general than the current TMDSC
theories. Both current TMDSC theories and conven-
tional DSC theories are included in our theory. The
variation rules of some sample's physical quantities
can be derived from this fundamental temperature
(
14] En-Yong Ding, Xue-Hai Liang, J. Instrumental Anal. 12(5)
(1993) 50.
[
[
15] H. Hoff, Thermochim. Acta 187 (1991) 293.
16] A.A. Lacey, C. Nikolopoulos, M. Reading, J. Thermal Anal.
50 (1997) 279.
[
[
17] F. Stager, Thermochim. Acta 304/305 (1997) 219.
18] Reading, B. Hahn, B.S. Crows, US Patent 3 224 755, 6 July
1993.