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COUPLING OF COLLOIDAL PARTICLES AND
RECOILLESS FRACTION
H. Übelhack, F. Wittmann
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 12, Tome 37, De'cembre 1976, page C6-273
COUPLING OF COLLOIDAL PARTICLES AND RECOILLESS FRACTION
H. J. UBELHACK and F. H. WITTMANNAbteilung fiir Werkstoffphysik, Technical University of Munich, Germany
Resume. - Dans les systkmes colloldause la fraction sans recul est reduite si les cristaux micros- copiques sont lies seulement lbgkrement 2 une position d'equilibre. On demontre que le facteur de Debye Waller peut 6tre factorise. Pour illuminer l'influence de la liaison des particules colloldales nous avons fait des mesures relatives de la fraction sans recul f vers la tempkature. Les valeurs de f
d'un systkme lie 16gkrement sont comparees avec les valeurs d'un systkme lie fortement. Le spectre de vibration des cristaux lies 16gkrement a 6tB fait plus vigide par une reduction de la teneur en eau intracristalline cornme il a et6 observk par effet Doppler quadratique. La reduction de la fraction sans recul, qui Btait observee, pourrait &tre attribuh seulement 2 la liaison 16g6re. Le problkme est analyse quantitativement et l'application de l'effet Mossbauer aux etudes des interactions des particules collo'idales a Bte demontree.
Abstract. - The recoilless fraction in colloidal systems is reduced, if microcrystals are held in their equilibrium position by weak coupling only. It is shown, that the Debye Waller Factor of colloidal systems may be factorized.
In order to elucidate the role of the coupling of colloidal particles we have measured the recoilless fraction of a xerogel in the temperature range between liquid N Z and room temperature. The related
f fraction of a weakly coupled colloidal system is compared to the corresponding value of a well coupled system. The phonon spectrum of the microcrystals in the weakly coupled system has been stiffened artificially by the reduction of the crystal water content. The stiffening of the lattice has been detected by the temperature dependent second order Doppler shift. Therefore the observed reduction of the recoilless fraction can be attributed to a change of the coupling only. A quantita- tive analysis of the problem is given. This approach allows us to apply Mossbauer spectroscopy to study the interactions of colloidal particles. The influence of coupling forces within a colloidal structure on the mechanical behaviour is discussed.
1. Introduction. - In previous experiments the recoilless fraction in colloidal systems has been found to differ from the value in the bulk [l-71. Possible modifications of the phonon spectrum of microcrystals have been discussed : (1) a cut off of the phonon spec- trum at long wave lengths should lead to an increase of f for small particles [l], (2) a contraction of the lattice due to surface forces as observed by electron- and x-ray diffraction (e. g. [g]) should rise the Debye temperature by about 5-10
%
[9] and (3) the mean square displamment of surface atoms is increased, as observed by low energy electron diffraction 1101, and as a consequence the recoilless fraction for atoms near the surface is reduced [2]. Modifications of type (3) seem to be dominating and there is clear experimental evidence from measurements with surface enriched samples [6]. The experimental evidencz of modifica- tions of type (1) and (2) is doubtful as a repetition of the experiment of Marshal1 and Wilenzick gave contradicting results [ l l].Van Wieringen has described a n experiment which indicates that the recoilless fraction of colloidal sys- tems is reduced if weak coupling of individual particles predominates [12]. We have been able to point out that the
f
fraction varies in hydro- and xerogels having one given particle size distribution under different states ofcompaction and/or moisture content and exhibits a characteristic temperature behaviour [13].
I n this paper we shall indicate a quantitative analysis of the influence of the coupling of colloidal particles t o their surrounding matrix on the recoilless fraction. Modifications of the phonon spectrum can be excluded in these experiments to be the origin of the observed reduction o f f as the phonon spectrum can be artifi- cially varied by changing the amount of crystal water bound in the gel particles.
2. Theoretical considerations. - Typical colloidal interactions between small particles can be supposed to be weak compared to the intracrystalline interactions of atoms or molecules. Within the lattice relative coordinates xL may be introduced with respect t o the center of mass X,,, of the colloidal particles.
X = X,,,
+
X, ( 1 4P =
+
PL.
(lb)Then the quantum state of a colloidal particle,
I
C P>
can be separated into two factors with COM- or L coordinates only :I C P > = I L > I C O M > . (2) The probability, that
I
C P>
remains unchanged18
AND F. H. WI'ITMANN during the emission or absorption of a gamma quan-
tum with wave vector k, can be written as follows : w(CP ; CP) =
=
I
<
CPI
expC-
i(k, xcoM+
xL)lI
CP>
1'
=
<
LIexp[- i(k,xL)]I L >1'
XX
I
<
COMI
exp[- i(k, xcoM)]I
COM>
1'
= w(L ; L). w(C0M ; COM)
.
(3)Hence the recoilless fraction may be factorized similar as in the case of molecular crystals 1141.
The secondary Debye Waller Factor fcoM will be of importance, if the states
I
COM,>
correspond to differences in energy much greater than the natural width of the Mossbauer line.For a colloidal particle, which is fixed to an equili- brium position with a force constant 5, fco, can be derived simply by discussing the mass dependence of the Einstein expression for the usual Debye Waller Factor.
The Einstein temperature 8, of a colloidal particle of N times the mass m, of an identically coupled Mossbauer atom is reduced by a factor N-'/, (-- 10',- 10-4).
In the cases investigated here 8, will be of the order of 8,
5
10 K. Using Fe57 a stronger coupling would give values of fcoM > 0.99 at T = 300 K and therefore the influence of the secondary f-value may be neglected. The high temperature limit of the Einstein formula may be used and one obtainsFrom eq. (6) a critical value
tCr
can be estimated.(
fCOM (300 K, tcr) = 0.95). Below this value the secon- dary Debye Waller Factor becomes increasingly important. Some typical examples forg,,
are listed in table I.TABLE I
Three typical Mossbauer transitions with the critical coupling strength Isotope
-
5,r
(105 dynlcm) -Fe57 (14.4 keV) 4
Sn119 (23.8 keV) 11
Auig7 (77.3 keV) 115
The temperature dependence of the second order Doppler shift (SOD) can be derived by using the Debye model of a solid [15]. Assuming constant source tem- perature the resonance lines at a temperature T are
shifted relative to the position at T = To by the second
order Doppler effect as much as
In this equation c =
-
2.7 X 10-4 mm/sK if Fe57 isused as Mossbauer isotope and f (y) has the usual meaning :
A possible change of the Debye Temperature 8, of the absorber material can be estimated by extrapolation for T -t 0. Absolute values of 8, may be derived from the temperature dependence of the SOD.
3. Experiments and results.
-
The system under investigation consists of calcium-ferrite-hydrate par- ticles (CaOFe,O, .(H,O),) with a mean diameter of about 200A
incorporated in a xerogel of hydrated cement. These particles already present in hardened cement paste are used as marker particles allowing Fe57 Mossbauer spectroscopy. Drying in water vapour pressures above P,O, reduces the content of crystal water from X = 7 to X = 5.5 [16].Two types of samples have been specially prepared. In sample A the particles hydrated in excess water have been dried in an atmosphere of about 50
%
relative humidity and then compacted with a pressure of 1.3 kbar. The xerogel reached in this way has a porosity of 0.3. Sample B in contrast has not been compacted. Therefore the porosity was about 0.6. But sample B has been stored in a dessicator with 10-3 torr for 5 days. In sample A the marker particles are well coupled to the xerogel structure, whereas in sample B the coupling is weak. The Debye temperature of the microcrystals in sample B has been increased by removal of crystal water.Mossbauer spectra have been recorded with a constant acceleration drive system using CoS7 in platinum as a source. The effective thickness of the absorber was t, = 2. The spectrum consists of two doublets with an intensity ratio of about 5 : 1 (IS, = 0.62 mm/s ; AEQ, = 0.72 mm/s ; IS, = 1.23 mm/s ; AEQ, = 0.8 mm/s ; isomershifts are given against Na,(Fe(CN), .NO). 2 H,O). An example of a spectrum is shown in figure 1. The side lines can be attributed to unhydrated cement clinker material i. e. wustite. The parameters of the resonance lines have been determined by a fit with Lorentzian lines. The isomershift and the area of the dominating lines of the colloidal consti- tuents is shown in figures 2 and 3 for both samples as a function of temperature.
COUPLING OF COLLOIDAL PARTICLES AND RECOILLESS FRACTION C6-275
RG. l.
-
Mossbauer spectrum of the iron containing compounds present in fully hydrated high alumina cement.(See for further details description in the text.)
FIG. 2. - Isomershift measured on samples A and B as a function of temperature. The theoretical lines represent the thermal shift due to the second order Doppler effect calculated by using the Debye model and inserting values of the Debye
temperature as indicated in table I1 column 2.
FIG. 3. -Related f-fraction as function of temperature. Theoretical lines are calculated according to eq. (4) and (6) using the values for the Debye temperature and coupling cons- tant given in table I1 columns 2 and 3. The deviation of experi- mental data of sample A from the calculated line at higher temperatures is caused by a decrease of mutual coupling due
to the gradual liquifying of adsorbed waterfilms.
pendent way from the temperature dependent related f fraction with the help of the usual Debye formula. The results are listed in table 11. For sample B the two values for 8, obtained were found to differ largely.
Debye temperature 0, and coupling constant
5
estimated as indicated in the text5 (dynlcm) Sample
2
&(K) (fD(~)) BD(K) (SOD) (eq. (4, (6))-
A-
-
A
-
500-
500 > 4 X 10sB
--
300-
800--
3 X 104To explain this discrepancy the recoilless fraction has been computed according to eq. (4). The lattice contribution
f,
has been calculated using the Debye expression with the Debye temperature as estimated from the SOD. The obtained values have been corrected by taking into consideration the secondary Debye Waller Factor f,,, with the help of eq. (6). The coupling constant5
has been varied as a parameter. The optimal value found in this way is also shown in table 11.4. Discussion.
-
Modifications of the phonon spec- trum of microcrystals can be excluded to be the origin of the observed reduction of the recoilless fraction for sample B. Measurements of the SOD indicate, that the phonon spectrum has been stiffened substantially. It may be concluded, that there really exists a secondary Debye Waller Factor for a colloidal particle as a whole. The decrease off is caused by the weak coupling of the microcrystals to the surrounding xerogel structure used as a matrix.Quantitative analysis shows, that even in relatively strongly coupled colloidal systems, as for instance hardened cement paste
(5
--
4 X 105 dynlcm) a reduc-tion off of the order of some per cent is to be expected at room temperature. If corresponding measurements are performed by using Sn119 instead of Fe5' the influence of the coupling should be even more pro- nounced. In the case of Aulg7 it should not even be negligible in strongly coupled systems (see Table I). It should be realized, that modifications of the phonon spectrum would cause a decrease of the recoilless frac- tion even at T = 0, whereas the secondary Debye
Waller Factor .fco,,, will approach unity at low tem- peratures.
loidal systems as obtained with the help of Mossbauer spectroscopy cannot be achieved by any other method so far.
Using the concept described in this contribution it is possible to explain mechanical properties of colloidal systems such as hardened cement paste as function of porosity. The determination of coupling forces between individual colloidal hydrate particles allows us for instance to predict the corresponding elastic modules [17]. Moreover, the influence of absorbed water films on the mutual coupling of gel particles
can be studied on samples having different moisture contents and different temperatures. Results are in fair agreement with the mechanical properties observed at the corresponding temperatures. Based on these results the deviation of the recoilless fraction measured on sample B from the computed theoretical line in figure 2 at temperatures above
-
60 OC thus can be explained by a diminution of the coupling forces caused by the gradual liquifying of the absorbed structured water films between opposite surfaces as may be observedby other methods too [18].
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