Skip to main content
SpringerLink
Log in

A mathematical model for Modulated Differential Scanning Calorimetry

  • Published:
Journal of thermal analysis Aims and scope Submit manuscript

Abstract

Reading and co-workers introduced a new technique a few years ago called Modulated Differential Scanning Calorimetry or MDSC. Here the first part of a theoretical analysis for this technique is given. A simple mathematical model for modulated differential scanning calorimetry in the form of an ordinary differential equation is derived. The model is analysed to find the effect of a kinetic event in the form of a chemical reaction. Some possible sources of error are discussed. A more sophisticated version of the model allowing for spatial variation in a calorimeter is developed and it is seen how it can be reduced to the earlier model. Some preliminary work on a phase change is also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a :

(remaining) mass fraction of reactant or crystalline material

a 0 :

initial mass fraction

b :

temperature ramp

c :

specific heat capacity

f :

rate of heat intake by a kinetic process within the sample

g :

temperature-dependence function of a reaction rate

h :

distribution function for molecular weights

k :

thermal conductivity

K a :

function appearing in the Avrami equation

l :

length scale of a (one-dimensional) calorimeter

l s :

half-width of a sample

m :

complex spatial-decay rate of modulated temperature

m j :

coefficients in a power series for α1 (j=1,2,...)

n :

outward normal

n j :

coefficients in a power series for α2 (j=1, 2,...)

p :

order of a reaction

q, q j :

coefficients appearing in expressions for\(\mathop \Delta \limits^ \sim \) T(j=C, S, D, 1, 2, 3, 4)

r :

reaction rate

r′ :

temperature derivative of reaction rate

s :

time used in integrations

s d :

dimensionless position of a free boundary

s f :

position of a free boundary

t :

time

t 2 :

time scale of melting

u :

real part ofw

v :

imaginary part ofw

w=wR, ws,W F :

functions of position determiningT

W j :

coefficients in power seriesw (j=0, 1, 2...)

x :

position within calorimeter or sample

y :

dimensionless position

zo,zR,zs:

functions of position determining¯T

A :

pre-exponentional rate constant

A * :

adjusted pre-exponential rate constant

A + :

scaled pre-exponential rate constant

A j :

coefficients appearing in expressions of cyclic temperature in regionj (j=1, 2, 3)

B :

amplitude of modulation of temperature

B :

complex amplitude of temperature (a complex multiple ofB)

B j :

complex amplitude ofT j (j=F, R, S, RM or SM)

C(T):

heat capacity of the sample when it depends up on temperature

CR,CS:

heat capacities of pan and sample

C j :

coefficients appearing in expressions of cyclic temperature in regionj(j=1, 2, 3)

¯C s :

value of sample's heat capacity given by underlying “measurement”

C s :

value of sample's heat capacity given by cyclic “measurement”

D :

heat capacity of sample regarding it as a function ofσ

E :

activation energy

F, G :

functions of time appearing in a nearly periodic signal

FM,¯GM,GM:

“measured” versions ofF, G

G a :

function appearing in the Avrami equation

H :

heat of reaction

I :

function of time appearing in a nearly periodic signal

ĪM,ĪM:

“measured” versions ofI

K :

effective heat-transfer coefficients

Kt,K0,K1,K2:

internal heat-transfer coefficients

L :

latent heat

M :

mass of sample

M j :

coefficients in expressions for\(\mathop \Delta \limits^ \sim \) T(j=1,2,3,4)

N :

exponent appearing in the Avrami equation

P :

coefficient appearing an\(\mathop \Delta \limits^ \sim \) T

QR,Qs:

heat contents of reference pan, sample pan plus sample

R :

gas constant

S :

a surface

S b :

cross-sectional area of a bar

S c :

cross-sectional area of a sample

S F :

external surfaces of calorimeter

SR,Ss:

surfaces of reference and sample pans

T :

temperature

T :

scaled complex amplitude of modulated temperature

Ti(x):

temperature on initiation of melting

T c :

melting temperature

T e :

external temperature

TF,T R ,TS:

temperatures of furnace, reference, sample

TRM,TSM:

temperatures measured for reference and sample

T o :

initial temperature

Ta,T1,T2:

typical temperatures

T oj :

initial values ofT j(j=F, R, S, RM, SM)

Toj,Tlj:

coefficients appearing in expressions for underlying temperature in regionj of a one-dimensional calorimeter (j=1,2,3)

ΔT :

temperature difference,T R-T S

ΔT j :

coefficients in a power series ofΔT (j= 0,1, 2,...)

U :

dimensionless temperature based on reaction rate

W :

molecular weight

Z(W):

mass fraction with molecular weight ≤W

α:

scaled amount of reactant

αj :

surface integrals of normal derivatives ofw's (j=1,2,3)

β:

spatial decay rate of modulated temperature

γj :

surface integrals of normal derivatives of z's (j=1,2,3)

δ:

dimensionless temperature over whichC(T) varies

ɛ:

1/dimensionless activation energy

ɛm :

Stefan number

η:

error in measurement of temperatures due to location of thermo-couples

λ:

degree of bias

ρ:

density

σ:

rescaled dimensionless temperature

τ:

dimensionless time

ϕ:

phase lag

ϕb, ϕc :

baseline and corrected phase lags

ϕs :

phase of alternative “reference” signal, e.g. furnace modulation

ψ:

ratio indicating importance of ωC R compared withK (“departure from perfection”)

θ:

dimensionless temperature

θi :

dimensionless temperature on initiation of melting

ω:

angular frequency of modulation of temperature

Ω:

rescaled angular frequency

-:

underlying part (notC's,F's,G's,I's)

∼:

cyclic part (notC's,F's,G's,I's)

':

derivative with respect to argument (notr)

References

  1. M. Reading, D. Elliot, and V. L. Hill, “Some aspects of the theory and practice of modulated differential scanning calorimetry”, Proc. 21st North American Thermal Analysis Society Conference, 1992, p. 145–150.

  2. M. Reading, Trends Polymer Sci., 1 (1993) 248.

    CAS  Google Scholar 

  3. M. Reading, A. Luget, and R. Wilson, Thermochim. Acta, 238 (1994).

  4. M. Reading, R. Wilson and H. M. Pollock, in Proceedings of the 23rd North American Thermal Analysis Society Conference, 1994, p. 2–10.

  5. B. Wunderlich, Y. Jin and A. Boiler, Thermochim. Acta, 238 (1994) 277.

    Article  CAS  Google Scholar 

  6. J. Hatta, Jpn. J. Appl. Phys., 33 (1994) 686.

    Article  Google Scholar 

  7. B. Wunderlich and A. Boiler, Abstract, 24th NATAS Conference, 1995.

  8. J. E. K. Schawe, Thermochim. Acta, 261 (1995) 183.

    Article  CAS  Google Scholar 

  9. S. M. Marcus and M. Reading, “Method and apparatus for thermal conductivity measurements”, U.S. Patent 5335993, 1994.

  10. J. D. Murray, Asymptotic Analysis, Oxford University Press, 1973.

  11. E. J. Hinch, Perturbation Methods, Cambridge University Press, 1991.

  12. B. Wunderlich, A. Boller, I. Okazaki and S. Kreitmeier, Thermochim. Acta, 283 (1996) 143.

    Article  Google Scholar 

  13. K. J. Jones, I. Kinshott, M. Reading, A. A. Lacey, C. Nikolopoulos and H. M. Pollock, “The origin and interpretation of the signals of MTDSC”, preprint.

  14. S. Marcus, Private communication

  15. M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliff, N. J., 1967.

    Google Scholar 

  16. K. Nakamura, T. Watanebe, K. Katayama and T. Amano, J. Appl. Polym. Sci., 16 (1972) 1077.

    Article  CAS  Google Scholar 

  17. K. Nakamura, K. Katayama and T. Amano, J. Appl. Polym. Sci., 17 (1973) 1031.

    Article  CAS  Google Scholar 

  18. K. A. Q. O'Reilly and B. Cantor, Proc. Roy. Soc. Lon. A, 452 (1996) 2141.

    Article  Google Scholar 

  19. J. Crank, Free and moving boundary problems, Oxford University Press, 1984.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Heriot-Watt University, EH14 4AS, Riccarton, Edinburgh

    A. A. Lacey & C. Nikolopoulos

  2. ICI Paints, Wexham Road, SL2 5DS, Slough, Berks, UK

    M. Reading

Authors
  1. A. A. Lacey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Nikolopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Reading
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Prof. Menachem Steinberg on the occasion of his 65th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lacey, A.A., Nikolopoulos, C. & Reading, M. A mathematical model for Modulated Differential Scanning Calorimetry. Journal of Thermal Analysis 50, 279–333 (1997). https://doi.org/10.1007/BF01979568

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01979568

Keywords

Search

Navigation