Journal of Modern Physics, 2013, 4, 16-21
http://dx.doi.org/10.4236/jmp.2013.47A2003 Published Online July 2013 (http://www.scirp.org/journal/jmp)
The Determination of Surface Thermodynamic Properties
of Nanoparticles by Thermal Analysis
George O. Piloyan, Nikolay S. Bortnikov, Natalia M. Boeva
Institute of Geology of Ore Deposits, Petrography, Mineralogy and Geochemistry,
Russian Academy of Sciences (IGEM RAS), Moscow, Russia
Received April 23, 2013; revised May 26, 2013; accepted June 29, 2013
Copyright © 2013 George O. Piloyan et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The effect of dispersivity on thermodynamic and kinetic parameters of chemical reactions in nanodispersed systems is
theoretically investigated. On the basis of the established theoretical dependences the new method of determination of
surface thermodynamic properties of nanoparticles (surface enthalpy, surface entropy and surface energy) by thermal
analysis (DTA or DSC) was developed. Three examples of calculation of surface properties of nanoparticles were pre-
sented to prove the feasibility of this method.
Keywords: Surface Enthalpy; Surface Entropy; Surface Energy; DTA; DSC; Nanoparticle
1. Introduction
Surface thermodynamic properties of nanoparticles take
a distinct effect on thermodynamic and kinetic parame-
ters of chemical reactions (so-called size effect) in nano-
dyspersed systems. However, this problem is poorly in-
vestigated because of complexity of experiment and ab-
sence of a database on surface thermodynamic properties
of solid. The present article attempts to fill an existing
gap somewhat. In the article, the new method of determi-
nation of surface thermodynamic properties of nanopar-
ticles by thermal analysis is described and its influence
on thermodynamic and kinetic parameters of chemical
reaction is shown.
Owing to its great scientific and practical significance,
the influence of solid body dispersibility on the chemical
reaction velocity has attracted the attention of researchers
for a long time. Three different kinetic models, which
take into account substance dispersibility, were proposed
for different heterogeneous chemical reactions [1]. At the
same time, all the models assume that changes in dis-
persibility affect only the size of the reactive surface in
the solid body. It was shown that the substance dispersi-
bility growth is accompanied by a shift of the reaction
regime from the diffusion-kinetic area toward the kinetic
one [2]. The processes at the phase interface form the
limiting stage of kinetics.
The transition to nanoparticles is characterized by size
effects. It is established that the thermal effect of the re-
action Qr for nanoparticles depends on the surface en-
ergy σ [3].
2. About Terminology
Let’s define some terms which will be used in the fur-
ther.
2.1. Nanoparticle
As this term we shall understand particles of substance
which size even in one dimension lay in an interval 10
nm ≤ r 100 nm [4]. In other words, nanoparticles are lar-
ger clusters, but it is less than microcrystals. Under this
definition get also a number of natural substances (some
clay minerals, some oxides, etc.). The main characteristic
property of nanopartiles is the appreciable contribution of
surface energy to their total energy.
2.2. Nanodispersed System
We shall understand as this term the system which con-
sists of nanoparticles.
2.3. Surface Energy, Surface Tension
Energetic properties of a surface of solid is usually char-
acterized value of a surface tension γ (by analogy to liq-
uids) (J/m2) or surface energy σ (J/m2).
The surface tension is caused by unbalanced field of
C
opyright © 2013 SciRes. JMP

G. O. PILOYAN ET AL. 17
intermolecular forces, i.e., by definition, it is value iso-
tropic. The condition is satisfied always for liquids. An
alternative pattern is observed for solid. Stress tensors in
volume and in surface layer of solid are not isotropic in
general so also the surface tension should be not obliga-
tory isotropic, that contradicts its definition. Thus, the
concept of a surface tension for a solid has concrete phy-
sical sense only within the range of melting temperature
of a solid.
More universal concept is surface energy. There is a
simple thermodynamic relationship between a surface
tension and surface energy for liquid [1]:

,
F
F     (1)
where
F
is area of surface of liquid.
For pure liquids (one-component system) 0,F


so numerically

 
r
G
,
r
GG F
.
According to Gibbs, thermodynamic properties of na-
nopartiles is described the same thermodynamic func-
tions, as microcrystals. For example, Gibbs free energy
of nanoparticle is stated as follows:



G
(2)
where —Gibbs free energy of microcrystal,
F
—the
area of a surface of nanoparticle,

is surface energy of
nanoparticle:
,
F
F
H
TS

 (3)
where
F
H
—surface enthalpy,
F
S
T
0,
—surface entropy,
is temperature. For pure substances (one-component
systems) 0,
F
FF
H
SH and
F
S do not depend on
temperature as a first approximation [1].
F
H
is often
named surface energy or total surface energy to suppose
approximately the equality
F
H


for nanoparticles at
first approximation.
3. The Offered Model
Let investigated substance represents ensemble indepen-
dent nanoparticles, contacting, but not cooperating with
each other. We shall admit, that everyone nanoparticle
represents pure substance (one-component system). The
total area of a surface of ensemble nanoparticles we shall
characterize the specific area F carried to 1 mole sub-
stances (m2/mol):

,
F
Sc r


Sc 3Sc
2Sc2.5Sc
(4)
where is shape coefficient ( for spherical
particles, for lamellar particles,

for
particles of the complex or uncertain form, etc.),

—
mole density of a particle, r—the characteristic size of a
particle.
Let chemical reaction occur in the system:
,
SD gas
C D
SB gasC
AB

where i

are stoichiometric coefficients, A1,


i.e.
all calculations are related to a mole of the initial sub-
stance.
The assumption, that the transformation process in a
system can be described by one chemical reaction, is
equivalent to an assumption, which this process depends
on one independent variable. The number of moles of
any reaction component can be selected as such a vari-
able, but is more convenient to introduce a new variable
α which is known as the degree of transformation or frac-
tional extent of reaction.
According to the definition:


  (5)
00
,
iiii
nn n   (6)
is dimensionless quantity, 01.



If we assume, that all components of system mutually
insoluble, reaction in system should go up to the end


1 

, unless one of the components is exhausted.
Hence, nanopartile
A
is changed into nanoparticle
as a result of reaction (5) according to our model.
N

1
n
ri
i
GG
4. Some Features of Thermodynamics of
Chemical Reactions
For our model Gibbs free energy of reaction ∆G1 is de-
termined as:
i




G
,i
(7)
where i is thermodynamic potential of component
i

, are stoichiometric coefficients, 0
i


0
for reagents
and i


0
r
G
for the reaction products.
In the equilibrium state, we have
(8)


Let’s consider separate types of transformations, de-
scribed by Equation (5).

Phase transitions

0,1.
BD C
   
.
rCCAA
GGF F
With account for (2) and (5), we can write the follow-
ing equation:




 
,GHTS

(9)
Taking into account equality constancy
of mass and (4), (8), we get (for spherical particles):
2
3
0
3,
C
AC
A
T
QTr












 

Q
(10)
where

is heat of phase transition related to the mi-
crocrystalline state,


TT T
00


 is equilibrium tem-
perature difference between temperatures of phase transi-
tions for the microcrystalline and nanodispersed state of
the substance, ρ is the mole density of nanoparticles and

G. O. PILOYAN ET AL. 18
r is the size of nanoparticles.
Formula (10) for the first time has been derived by
Hill [5]. This formula can be considered as the general-
ized analogue of known Gibbs—Thomson formula.
For heterogeneous endothermic (or exothermic) reac-
tions the equation is obtained similar to Equation (9):

rr
GG 

 .
CCCAA
FF  
0
r
G
(11)
In the equilibrium state, we have
(12)
A little manipulation yields as follows (for spherical
particles):

2
3
,
CC A
A C
M
M










i
0
3A
AC
AA
MT
QTr











(13)
where
M
are the molecular masses of nanoparticles A
and i ,C

are their densities. Complex

MM 
CC A
is known as Pilling—Bedward coefficient in the litera-
ture [6].
5. Some Features of Kinetics of Chemical
Reactions in the Nanodispersed Systems
The effect dispersivity of a solid on the chemical reac-
tions rate has attracted the attention of researches for a
long time, owing to its great scientific and practical sig-
nificance. The various kinetic models, which take into
account dispersivity, were proposed for different chemi-
cal reactions [2].
However in all models it is supposed, that the change
of dispersivity changes only the area of a reactionary
surface of a solid. The dispersivity growth has been shown
to be accompanied by a shift of the reaction regime from
the diffusion—kinetic area toward the kinetic one [7]. A
limiting kinetic stage becomes the processes going on
surface of the interface. In this case, reaction rate de-
pends on value of the surface and can be presented Equa-
tion [2]:
d,
d
t
t
VkF
t
V
(14)
where t and t
F
are accordingly volume and a surface
of the particle, not reacted by the moment , is the
rate constant.
tk
By introducing degree of transformation α (5) into
Equation (14), we obtain following equation:
 
01n
kr
d
dt


 (15)
where n is the order of reaction (for particles of the
spherical form 23,n for flat 1n2, etc.0 r), is
the size of the particle.
The rate constant k is usually described by the Ar-
rhenius equation:


expkA ERT (16)
where A is a pre-exponential term, Е is empirical (ap-
parent) energy of activation, R is a gas constant, Т is
temperature in Kelvin.
In the theory of the activated complex (one of the ba-
sic theories of chemical kinetics) [8], the rate constant is
defined by the equation:


10
exp ,kA GRT (17)


, where 1B
A
kT h  
B
k
G
is Boltzmann’s constant, h
is Planck’s constant, 0 is activation free energy,

is transmission coefficient.

defines a probability that
the system to jump activation barrier. It usually is as-
sumed that 1


.
Equation (17) for nanoparticle must be changed by
analogy with Equation (9). According to work [9] the rate
constant of reaction of the activated complex which has
already formed on a surface nanoparticle, should not de-
pends on dispersivity. According to this assumption Equ-
ation (9) may be transformed as follows:
00 0rArAAA AA
GGG GGFGF




 
G,
(18)
A
A where Ar is free energy initial nanoparticle G

is Gibbs free energy for microcrystals of an initial com-
ponent ,
A
0 G

is free energy of formation of the ac-
tivated complex.
Thus Formula (18) may be written as follows:




10
expkAGFRT   (19)
where

is surface energy,
F
is the mole area of
surface.
Formula (19) can be transformed taking into account
equality (3):




20
expkAHF RT   (20)

where 0
H
is activation enthalpy,

21 0
exp ,
A
ASR
0
S
0
is activation entropy.
The following equation has been derived in the the-
ory of the activated complex [7]:
H
ERT (21)


where E is activation energy of reaction for microcrys-
talls.
Having substituted (21) in (20) we shall receive:




0expkAEF RT
 
,
(22)
where 02
A
eA

Substitution in (15) Formulas (20) and (21) gives:
 


00
dexp 1
d
n
ArEFRT
t


  (23)
From Equation (23) we may deduce that rate of

G. O. PILOYAN ET AL. 19
chemical reaction should increase with growth of disper-
sivity.
Of course, this is an idealized situation, which takes
into account neither defects nor the covering degree of
active centers on the reactive surface of the substance.
A more general example of heterogeneous catalytic re-
actions is considered in [9], where it is shown that the
dispersivity growth under stationary filling of active cen-
ters of the catalyzing agent 1,


1,
the activation energy
should decrease, while under


T
m
TT
the latter should
grow.
Below it will be shown, what even the simplified
model considered in the article, leads to satisfactory re-
sults.
6. Some Formulas from the Theory of the
Thermal Analysis
The thermal analysis experiments are known to run in
non-isothermal conditions. According to the theory of the
thermal analysis [10], parameters of thermal curves (DTA,
DSC, TG, DTG) contains the usefulness information on
investigated substance and processes, in it proceeding.
For the decision of our problem it is enough to use
only one parameter—peak temperature of the thermal ef-
fect (temperature of the maximal deviation of thermal
curves from a base line in an interval of thermal effect).
The peak temperature of curve DTA, DTG or DSC is
usually assumed to correspond to the temperature of ma-
ximal rate of chemical reaction. This assumption can be
accepted only in the case where inertia of balance for
DTG curves or thermal inertia of substance for curves
DTA or DSC may neglect.
It has theoretically been shown in work [11], that the
peak temperature on curves DTG should advance peak
temperature on curves DTA for endothermic reactions.
However, practice shows, that this difference is usu-
ally insignificant, so as a first approximation it is possi-
ble to accept, that for curves DTA or DSC the peak tem-
perature m of endothermic effect is approximately
equal to temperature of maximal rate reaction. Thus, at
the equation is accomplished:
2
2
d0
dt

 (23)
More complex picture, especially for fast reactions
with greater heat effect, is observed for exothermic reac-
tions.
Let’s consider endothermic reaction of the first order.
Twice differentiating (15) with respect to t we shall re-
ceive:
 
0
ERT r
m
TT
^2 expERTdTdt A (24)
It is known [10], that for curves DTA or DSC at
dd ,Tt b
TT
(25)
where b is heating rate of the furnace or reference sam-
ple.
For curves DTG at m the derivative
dd .Ttb

For this reason the use of curves DTA or
DSC is more preferable, as they contain less unknown
parameters than curves DTG.
In [12] it has been shown, that at as a first
approximation:
m
TT


1
11
n
n 
.



(26)
Hence, Equation (24) can be used and for reactions of
the n-order.
If instead of k from (16) to use k from (22) we shall
receive:

 

00
^2 exp.rbEF RTAEF RT   (27)
For convenience, Formula (27) transforms more con-
venient form:
 




00^2 exp.rART bEFEFRT
  

(28)
Denote the term in a square brackets by B. Equation
(28) may be rewritten as follows:


 
0exprBEFRT
 
const.B
. (29)
At change Т or r term B changes essentially more
slowly, than the exponential term. As a first approxima-
tion the term B may be assumed to be constant:
(30)

Write down (29) in form more convenient for calcula-
tion, having substituted instead of
F
and

their va-
lues from Formulas (3) and (4):





ln ln
,
mFm
F
rBERTScHRrT
Sc SRr


 

rE
T
R
(31)
where is the size of a particle, is empirical acti-
vation energy for microcrystals, m is peak temperature
of thermal effects on curves DTA or DSC, is a gas
constant,


Sc —shape factor,
F
H
is surface enthalpy,
ρ is a mole density of substance, S
F
is surface entropy.

Thus, the value

ln r depends on three independent
variables—1,1TrT
mm
and 1 Coefficients at these
variables from Equation (31) is calculated by method of
multiple regression. The admissible calculation accuracy
can be achieved if to use not less than 6 experimental
points.
.r
7. The Calculations
Let’s consider on the several examples, how much well
offered model describes experimental data.

G. O. PILOYAN ET AL. 20
The published experimental data of different authors
(see Table 1) have been used as initial data.
The first example is related to dehydration of boehmite.
(α—AlOOH). This reaction is known to belong to the
class of topochemical reactions; i.e., their limiting stage
is represented by processes at the phase interface. The
rate reaction is described by Equation (23).
The authors [14] synthesized boehmite nanoparticles 1
to 26 nm across and described their thermal curves ob-
tained on the differential scanning calorimeter (DSC).
The size of particles was defined by X-ray method. Their
experimental data are presented in Table 1. Using the
boehmite data from Table 1, the multiple regression equa-
tion was calculated:




ln18.675112366.1 1
0.0049445 1
rT
r


3.37732 1
mm
rT
The coefficient R2 = 99.68%.
Using the coefficients from Equation (31), we obtain
(see Table 2 that the activation energy of the boehmite
dehydration reaction is E = 102.8 kJ/mole, the surface
enthalpy is 0.58 J/m2, surface entropy is 0.000844 J/
(m2·K), surface energy is calculated at 298 K σ298 =
0.329 J/m2, using Formula (3). At calculation shape coef-
ficient has been put (Sc) = 2.5 as synthesized boehmite
nanoparticles have the uncertain forms. In the literature
there are data on the surface enthalpy of boehmite, ob-
tained by a method high-temperature calorimetry [15]:
НF = 0.52 J/m2. Our data are fairly consistent with liter-
ary data.
The second example is related to the oxidation of mag-
netite.
Table 1. Initial experimental data.
 -Al2O3 Magnetite Fe3O4 Boehmite α-AlOOH Sample
r Tm r Tm r Tm
2.65 1463 9.5 358 1.13 653
2.69 1471 16 378 1.56 676
3 1476 30 388 2.04 686
3.3 1479 44 398 2.42 701
4.5 1522 48 418 6.9 744
6.2 1562 60 433 14.2 781
6.6 1563 80 433 26.3 801
- - 95 438 - -
Phase tansformation
 -Al2O3  α-Al2O3
Oxidation, the
formation
 -Fe2O3
Dehydration, the
formation of  -Al2O3 The reaction
[13] [14] [13] References
The note: Tm is taken in К, r is taken in nm.
Table 2. The results of calculation of surface thermody-
namic properties of investigated substances.
Sample Е,
kJ/mol
HF,
J/m2 SF, J/m2K σ298, J/m2
ρ,
g/cm3(Sc)
Boehmite102.80.58 0.0008437 0.329 3.082.5
Magnetite16.1 1.85 0.00567 0.16 5.2. 3
γ-Al2O3 95.1 1.90.00144 1.474 3.6 3
The note: Е is activation energy of chemical reaction for microcrystals,
is surface enthalpy, is surface entropy, σ298—surface energy at Т
= 298 К, ρ—density of substance, (Sc)—factor of the form.
F
HF S
Several kinetic models of oxidation reaction were
proposed for many classes of solid substances [6,15]. All
of them describe, however, processes for microcrystal-
line substances. The change-over to nanoparticles alters
the picture of reaction. The limiting stage of reaction, as
well as in case of with boehmite, becomes the process
going on at the phase interface. Reaction rate is described
by Equation (22). The authors of [14] synthesized mag-
netite nanoparticles 9.5 to 95 nm across and investi-
gated oxidation reaction of magnetite and formation γ-
Fe2O3 by DSC. The size of particles was determined by
X-ray method. Their experimental data are listed in Ta-
ble 1. Using a calculation procedure similar to that in the
situation with boehmite, we obtain the following multiple
regression equation:

 



ln9.392711937.36 129.7373 1
0.091178 1
mm
rTrT
r


0
0.01 ,
Fy
The coefficient R2 = 99.53%.
Using coefficients from Equation (31), we obtain that
the activation energy of the magnetite oxidation reaction
is E = 16.1 kJ/mole, the surface enthalpy is HF = 1.85
J/m2, surface entropy is SF = 0.00567 J/(m2·K). The sur-
face energy under 298 K calculated in line with Equation
(3) is σ = 0.16 J/m2 (Table 2).
No data on magnetite surface properties are available
in the literature. Let us use for the theoretical assessment
of the surface enthalpy the Orovan equation [16], which
allows the surface enthalpy for metals and some oxides
to be calculated at first approximation:
H
Ea (32)

where
y
E is Young module and a0 is the parameter of
the lattice. Using the available published data Ey = 231.3
gPа [17], a0 = 0.8394 nm [18], we obtain НF = 1.941
J/m2. Quite satisfactory agreement to our result is ob-
served.
The third example is related to the phase transforma-
tion γ-Al2O3  α-Al2O3.
In work [13] γ-Al2O3 has been synthesized at dehydra-
tion of boehmite nanoparticles. The size of particles was
determined by X-ray method. Thermal curves DSC have

G. O. PILOYAN ET AL. 21
REFERENCES been written down. Necessary for the further calculations
the initial data are placed in Table 1. Using already de-
scribed procedure of calculation we has been obtained
the following multiple regression equation:
[1] Yu. G. Frolov, “Course of Colloid Chemistry,” Khimiya,
Moscow, 1982.




ln9.899711442.90 119
– 0.0147631
rT
r
 .468 1
mm
rT
[2] Yu. D. Tret’yakov, “Solid Phase Reactions,” Khimiya,
Moscow, 1978.
[3] G. O. Piloyan and N. S. Bortnikov, Dokl. Akad. Nauk,
Vol. 416, 2007, pp. 247-249.
The coefficient R2 = 99.98%. [4] A. I. Gusev and A. A. Rempel, “Nanocrystal Materials,”
Fizmatlit, Moscow, 2001. Using coefficients from Equation (31), we obtain that
the activation energy of the phase transformation E is
95.1 kJ/mole, the surface enthalpy γ-Al2O3 is = 1.90 J/m2,
surface entropy γ-Al2O3 is SF = 0.00144 J/(m2·K). The
surface energy under 298 K calculated in line with Equa-
tion (3) is σ = 1.47 J/m2 (Table 2).
[5] T. L. Hill, “Thermodynamics of Small Systems,” Benja-
min, New York, 1963.
[6] P. Barret, “Reaction Kinetics in Heterogeneous Chemical
Systems,” Elsevier, New York, 1975.
[7] D. A. Frank -Kamenetskii, “Diffusion and Heat Transfer
in Chemical Kinetics,” Nauka, Moscow, 1967.
In the literature there are data on the surface enthalpy
of γ-Al2O3, obtained by the method high-temperature
calorimetry [19]: НF = 1.7 J/m2. Our data are fairly con-
sistent with literary data.
[8] N. M. Emanuel and D. G. Knorre, “Chemical Kinetics,”
Vyshaya Shkola, Moscow, 1969.
[9] V. N. Parmon, Dokl. Akad. Nauk, Vol. 413, 2007, pp.
53-59.
8. Conclusions [10] G. O. Piloyan, “Introduction to the Theory of Thermal Ana-
lysis,” Nauka, Moscow, 1964.
The basic equation of the method has been received:
[11] R. L. Reed, L. Weber and B. S. Gottfried, Industrial & En-
gineering Chemistry Fundamentals, Vol. 4, 1965, pp. 38-
46. doi:10.1021/i160013a006
 

ln lnm
F
rBERT
Sc SRr 
 

Fm
ScHRrT 

[12] H. E. Kissenger, Analytical Chemistry, Vol. 29, 1957, pp.
1702-1706.
where r is the size of a particle, Е is empirical activation
energy for microcrystals, Tm is peak temperature on
curves DTA or DSC, R is a gas constant, (Sc) is shape
coefficient, HF is surface enthalpy, ρ is mole density of
substance, S is surface entropy.
F

2
0
B
A
RT
[13] X. Bokhimi, J. A. T. Antonio, M. L. Guzman-Castillo, et
al., Journal of Solid State Chemistry, Vol. 161, 2001, pp.
319-326. doi:10.1006/jssc.2001.9320
[14] C. Sarda, F. Mathieu, A. Vajpei and A. Rousset, Journal
of Thermal Analysis, Vol. 32, 1987, pp. 865-873.
doi:10.1007/BF01913772
b FE  , b—heating rate of the refer-
ence substance or the furnace, ln(B) ≈ const.
The equation involves one independent variable ln(r)
and three dependent variables 1/Tm, 1/rTm and 1/r. Coef-
ficients in the equations are calculated by the multiple
regression method. Surface energy is calculated by the
known equation
F
F
[15] J. Majlan, A. Navrotsky and W. H. Casey, Clays Clay
Minerals, Vol. 48, 2000, pp. 699-707.
[16] M. I. Gol’dshtein, V. S. Litvinov and B. M. Bronfin, “Me-
tallophysics of High Strength Alloys,” Metallurgiya, Mos-
cow, 1986.
H
TS

. On an example of three
reactions (dehydration of boehmite (α-AlOOH), oxida-
tions of magnetite (Fe3O4  γ-Fe2O3) and phase transi-
tion (γ-Al2O3  α-Al2O3) methods of calculation of sur-
face properties were shown. The obtained results were
compared with literature data. Our results quite well co-
incide with known literary data.
[17] I. N. Frantsevich, F. F. Voronov and S. A. Bakuta, “Hand-
book on Elastic Constants and Moduli of Elasticity for
Metals and Nonmetals,” Naukova Dumka, Kiev, 1982.
[18] S. Clark Jr., “Handbook on Physical Constants,” Geolo-
gical Society of America, New York, 1966 (Mir, Mos-
cow, 1969).
[19] A. Navrotsky, Proceedings of National Academy of Sci-
ence of the USA, Vol. 101, 2004, pp. 12096-12101.
doi:10.1073/pnas.0404778101
Three examples of calculation of surface properties of
nanoparticles for three types of reactions show, what
even for such simplified model which is accepted in pre-
sent paper, it is possible to receive quite satisfactory re-
sults by thermal analysis.