indentation stress relaxation of sol–gel-derived organic/inorganic hybrid coating

Acta Materialia 53 (2005) 4455–4462

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Indentation stress relaxation of sol–gel-derivedorganic/inorganic hybrid coating

M. Sakai *, M. Sasaki, A. Matsuda

Department of Materials Science, Toyohashi University of Technology, Tempaku-cho, Toyohashi, Aichi-ken 441-8580, Japan

Received 18 April 2005; received in revised form 1 June 2005; accepted 1 June 2005Available online 3 August 2005

Abstract

The indentation stress relaxation is examined of sol–gel-derived phenylsilsesquioxane film coated on a soda-lime glass plate. Therheological transition associated with the condensation reactions from the sol (liquid) to the gel (solid) is elucidated in indentationload relaxation tests. The effects of the heat-treatment-temperature and the heat-treatment-time on the process of gelation (the solid-ification process) are studied. The change in the chemical structures and the evolution of siloxane networks are discussed in terms ofthe Fourier transform infrared-spectra (FTIR-spectra) combined with the corresponding rheological transition of the coatings dur-ing the condensation processes.� 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Sol–gel hybrid coating; Microindentation; Viscoelasticity; Stress relaxation; Elastic modulus

1. Introduction

Polyorganosilsesquioxane, termed an organic/inor-ganic hybrid, which is derived in sol–gel processes of sil-icon alkoxides, has considerable potential in the scienceand engineering of photofunctional materials, coatingfilms and in applications in the surface modification ofengineering materials [1]. The advantage of employingthese hybrid materials is in the capacity for controllingthe characteristic material properties of the hybridthrough prescribing the sol–gel chemical processing. Infact, the material properties of organic/inorganic hy-brids are significantly modified by varying the ratioand the characteristic of the organic group chemicallybonded to siloxane networks. The conditions for thecondensation reactions in sol–gel processing, includingthe drying temperature and time, rate of hydrolysis,and the pH of the solution, are also very critical in

1359-6454/$30.00 � 2005 Acta Materialia Inc. Published by Elsevier Ltd. A

doi:10.1016/j.actamat.2005.06.005

* Corresponding author. Tel.: +81 532 44 6798; fax: +81 532 485833.

E-mail address: [email protected] (M. Sakai).

controlling the properties of the hybrid materialssynthesized [1].

A number of recent developments have been reportedfor these organic/inorganic hybrid coatings [2], althoughfew studies have been conducted in a systematic manneron the mechanical properties of hybrid coatings. Malzb-ender and de With [3] first intensively studied, using thenano-indentation technique, the elastic modulus, hard-ness, fracture toughness and the adhesion characteristicof methylsilsesquioxane (MeSiO3/2) coatings filled withcolloidal silica. They argued the contact mechanics ap-plied to the thin films, and then considered the effectof sol–gel processing on the properties of the resultanthybrid coatings. The role of organic modifier groupsin the mechanical properties (Young�s modulus, hard-ness, creep, and adhesion characteristic) of organ-osilsesquioxane was examined by Atanacio et al. [4].They deposited hybrid sol–gel coatings on silicone andcopper substrates in terms of the various organicmodifier groups (including methyl-, ethyl-, vinyl-, andglycidoxypropyl-groups). They demonstrated that theaddition of organic substitutes to inorganic sol–gel

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4456 M. Sakai et al. / Acta Materialia 53 (2005) 4455–4462

networks dramatically changes the indentation responseand the adhesion characteristic. A significant decreasewas observed in the elastic modulus and the hardnessof the resulting films, and there is a transition from elas-tic/brittle to viscoelastic behaviors, when the dimensionof the organic modifier group is increased. A similarbehavior has also been reported by the present authorsfor methyl- and phenyl-groups; the former is very elas-tic/brittle even for lower temperatures and shorter timesof drying, whereas the latter is very viscoelastic even forlonger times and higher temperatures of drying [5–7].

In this work, instrumented microindentation tests areconducted to probe the viscoelastic behavior during thecondensation reactions in the sol–gel processing ofphenylsilsesquioxane (PhSiO3/2) film coated on a soda-lime glass (SL-glass) plate. The rheological transitionfrom a viscoelastic sol in shorter heat-treatment-time(HT-time) to a brittle solid in longer HT-time is tracedthrough measuring the stress relaxation characteristics.Along with this rheological analysis, the Fourier trans-form infrared-spectra (FTIR-spectra) are utilized toexamine the change in the chemical structures of silox-ane clusters/networks that evolved during the condensa-tion reactions.

2. Experimental details

2.1. Sol–gel processing

The condensation reaction is very fast in the sol–gelprocessing of methyltriethoxysilane (MTES) [5,7]. Theresultant film coated on a solid substrate becomes verybrittle even at the rather mild treatment conditions ofthe HT-temperature of 50 �C and the HT-time of30 min [5]. Accordingly, the viscoelastic study of thechange in the mechanical properties associated withthe evolution of MeSiO3/2-networks in the sol–gel pro-cessing is very difficult, as the evolution of siloxanechains and networks takes place during rheological test-ing. The equilibrium rheological parameters and func-tions, therefore, cannot be obtained experimentally.

As suggested by Atanacio et al. [4] and Matsuda et al.[7], the rate of condensation reaction becomes progres-sively and significantly more sluggish when methylgroup is replaced with a larger (bulkier) organic modi-fier. In this study, we adopted phenyltriethoxysilane asan organic precursor. The replacement from methyl tophenyl in organosilane-structure, as demonstrated inthe following sections, results in a highly prolonged con-densation rate; in fact, the resultant phenylsilsesquiox-ane (PhSiO3/2) coating on a SL-glass substrate made atthe HT-temperature of 50 �C behaves as a viscoelasticliquid even at the HT-time of 10 h.

The sol–gel coating solution was prepared by addinghydrochloric acid (0.1 wt.% HCl aqueous solution

used as a hydrolysis agent) to phenyltriethoxysilane,PhSi(OEt)3 (PTES) in dry ethanol (EtOH); all of thesereagent chemicals were supplied from Wako PureChemical Industries, Ltd., Osaka, Japan. The molar ra-tio of PhSi(OEt)3:EtOH:H2O (0.1 wt.% HCl) was 1:3:4.First, 0.05 mol PTES was mixed with 0.15 mol EtOH,and then stirred for 30 min at room temperature.Hydrochloric acid was then added to the solution dropby drop. After stirring the mixture for 1.5 h, the hydro-lysis and the condensation reactions resulted in a clearPhSiO3/2 sol. SL-glass plate was used as the substrateafter ultrasonic cleaning in pure water and then in 2-pro-panol. A method of dropping the sol on the substratewas adopted to obtain a thick coating with its thicknessof 20 ± 2 lm; using a micropipette, an appropriateamount of the sol was dropped and spread spontane-ously onto the substrate, the edges of which wereshielded by coating with a hydrophobic agent. The filmcoated on the substrate was put into an oven and heat-treated at 50, 80, and 100 �C for various values of HT-time for soaking, ranging from 30 min to 8 h at 50 �C,from 10 min to 4 h at 80 �C, and from 3 min to 1 h at100 �C. Microindentation tests were conducted rightafter soaking. The thickness of the heat-treated filmswas measured by a profilometer (SURFTEST SV-3000M4, Mitutoyo Co. Ltd., Kawasaki, Japan). Changein the chemical structure during the condensation reac-tions was examined in a transmission mode for the coat-ings on a silicon wafer by the use of a FTIRspectrometer (FT/IR-7300, JASCO, Tokyo, Japan) inthe range of wave numbers from 400 to 4600 cm�1.

2.2. Indentation load relaxation test

Viscoelastic indentation tests were conducted, usinga micro/nano-indentation apparatus, schematicallydepicted in Fig. 1. The apparatus was specially designedto be a compact test system having the height of 0.25 mconstructed on an aluminium base (0.3 · 0.3 · 0.05 m3)with a honeycomb structure. This compact system is effi-cient in reducing the thermal drift and fluctuation, whichresult in undesirable variations of the observed mechan-ical responses in indentation tests. The penetrationdepth was controlled by piezo actuators (PiezomechanikGmbH, Munich, Germany) with a computer-assisteddriver. The indentation load was monitored by a minia-ture load cell (LCFA-50, Omega Engineering Inc., Stan-ford, CT, USA) with the precision of ±0.5 lN. Therelative displacement between the indenter tip and thespecimen surface was measured as the penetration depthby a linear transducer (K4-20242-A, Shinko Denki Co.Ltd., Osaka, Japan) with the precision of ±0.5 nm.The temperature of the test specimen was held at23 ± 0.1 �C using a Peltie thermo-controller (DPC-100,Daitron Technology, Osaka, Japan). The outputs fromthe load cell, linear transducer, and the temperature

Fig. 1. Schematic of the micro/nano-indentation apparatus. The piezoactuator (L) is used for positioning the tip of indenter, and the actuator(S) is for the computer-controlled penetration. The test system isthermo-controlled at 25 ± 0.5 �C. A Peltier element coupled with athermister is utilized for controlling the temperature of the testspecimen with the precision of ±0.1 �C.

M. Sakai et al. / Acta Materialia 53 (2005) 4455–4462 4457

controlling thermo-system were recorded by a dataacquisition system (NR-2000, Keyence Corp., Osaka,Japan).

Using the piezo actuator, a stepwise penetration h0 ofabout 1.4 lm (about 7% of the film thickness) was

Fig. 2. Indentation load relaxations P(t) measured at 23.0 ± 0.1 �C for vari100 �C. The time-dependent quasi-stepwise penetration h(t) is also shown.

imposed within a period of 1.0 s, and then the resultantviscoelastic relaxation of indentation load was mea-sured, using a Berkovich trihedral indenter. Preliminaryrelaxation test results confirmed that: (1) the viscoelasticrelaxation behavior is linear, where the stress relaxationmodulus E(t) is independent of the imposed penetrationsfor h0 6 1.5 lm; (2) the film-only relaxation behavior isobserved when the imposed penetration depth is lessthan about 2 lm (less than about 10% of the filmthickness).

3. Results

Examples of the indentation load relaxation areshown in Figs. 2(a)–(c) for the coatings with variousHT-temperatures and HT-times. As readily expected,the indentation load relaxation is very fast for the lowerHT-temperatures and/or the shorter HT-times, i.e., forinsufficient condensation reactions. However, the relax-ation times are progressively increased during the con-densation reaction, and eventually become infinite(transition from liquid sol to solid gel) for the HT-timeexceeding about 2 h at the HT-temperature of 100 �C,by way of example.

It is worth noting in Figs. 2(a)–(c) that the penetra-tion depth is not ideally stepwise; it gradually increaseswith the increase in time, although the displacement ofthe piezo actuator is held to be a fixed value. The releaseof elastic strains of the test system (the release of frame

ous HT-times at the HT-temperatures of (a) 50 �C, (b) 80 �C, and (c)


compliance of test system) that occurs during the relax-ation processes of the viscoelastic test specimen (film) re-sults in this time-dependent penetration. The lineartransducer in the present instrumented indentation testsystem is designed to eliminate any effect of framecompliance on the observed penetration depth in time-independent elastic/elastoplastic contact. However, intime-dependent viscoelastic contact, the frame compli-ance (about 2.5 · 10�5 m/N) results in a quasi-creepingpenetration (about 0.25 lm for the applied indenta-tion load of 10 mN, being equivalent to the framecompliance).

As pointed out in the literature [8,9], this type of quasi-stepwise displacement imposed on the test materialalways overestimates the stress relaxation times, andsignificantly so. In order to circumvent this difficultyin viscoelastic relaxation tests, it has been common tointroduce an electromechanical feedback system to con-trol the displacement (penetration depth) to be fixed,giving an ideally stepwise displacement [9]. In the pres-ent study, instead of introducing an electromechanicalfeedback system, a mathematical algorithm was appliedto the quasi-stepwise penetration using Laplace trans-form inversion, the details being given in Appendix.

The FTIR-spectra are shown in Fig. 3 for the films atthe HT-temperature of 80 �C for various HT-timesranging from 5 to 240 min. The absorption spectra atthe specific wave numbers for the O–H, Si–Ph, Si–OH,and the H–Ph vibration modes [10] are illustrated.

40080012001600

Wave number (cm-1

)

30004000

5min

10min

20min

40min

60min

120min

HT-time:240minO-H

Si-Ph

Si-OHH-Ph

Abs

orba

nce

HT-temperature: 80˚C

Fig. 3. FTIR-spectra of the coating for various HT-times (HT-temperature of 80 �C) with the respective assignments for the stretch-ing, bending, and the wagging modes [10].

4. Discussion

4.1. Viscoelastic constitutive equation for pyramidal

indentation

Consider the mechanical behavior when a conical ora pyramidal indenter is pressed into contact with a linearviscoelastic body. During the viscoelastic growing depthof penetration h(t) with time t, its contact area alsogrows, and the distribution of contact pressure changeswith time, resulting in the change of the indentation loadP(t) with time. Sneddon�s elastic solution for a conicalindenter with the inclined face angle b is simply rewrit-ten into the following hereditary integral by replacingthe elastic modulus E with the stress relaxation operatorE(t � t 0) [11–13];

P ðtÞ ¼ gk2c2

� �1

1� m2

� �Z t

0

Eðt � t0Þ dh2ðt0Þdt0

� �dt0; ð1Þ

where m is Poisson�s ratio, c is the geometrical factor forcharacterizing the surface profile of the indentationimpression, having been approximated to be 1.0 for sim-plicity in the viscoelastic analysis. The geometrical fac-tor g of a cone/pyramid indenter, which relates thecontact depth hc to the projected contact area Ac

through Ac ¼ gh2c , is 24.5 for both trihedral Berkovichand tetrahedral Vickers indenters. The indenter�s geom-etry-dependent parameter k is simply approximated ask = tanb using the inclined face angle b of a cone/pyra-midal indenter.

A constant depth of penetration h0 imposed at t 0 = 0in a stepwise manner, i.e., h(t 0) = h0u(t

0) [u(t 0): Heavisideunit function] in Eq. (1) affords the following indenta-tion load relaxation P(t) in terms of the stress relaxationmodulus E(t) [14];

P ðtÞ ¼ Ah20EðtÞ; ð2Þwhere the frontal factor A in the right-hand-side ofEq. (2) is defined by A = (gk/2c2)/(1 � m2). Accordingly,the measurement of the relaxation load of P(t) in inden-tation test directly yields the stress relaxation modulusE(t), one of the important rheological functions. Oncewe determine E(t), various rheological parameters areobtained, such as the glassy modulus Eg (equivalent toYoung�s modulus), equilibrium modulus Ee for visco-elastic solid, steady-state viscosity g for viscoelasticliquids, relaxation time spectrum H(s) [15].

The simple constitutive relation of Eq. (2) is onlyapplicable to indentation relaxation tests with a con-stant depth of penetration h0 imposed in an ideally step-wise manner. However, as is easily understood fromFigs. 2(a)–(c), the elastic frame compliance of the pres-ent indentation test apparatus results in undesirablecreeping penetration (quasi-stepwise penetration) withtime, resulting in a significant overestimate of the relax-ation times. To circumvent this difficulty, in the present


study, the stress relaxation modulus E(t) was determinedusing a Laplace transform and its inversion. We applythe convolution theorem to Eq. (1) [16], and then makethe Laplace transform, affording

P ðsÞ ¼ Ah20EðsÞ � F ðsÞ ð3Þor

EðsÞ ¼ 1

Ah20

P ðsÞF ðsÞ

; ð30Þ

where PðsÞ, EðsÞ, and F ðsÞ are the Laplace transforms ofP(t), E(t), and F(t). The function F(t) is defined in thefollowing formula:

F ðtÞ ¼ 1

h20

dh2ðtÞdt

� �. ð4Þ

The function F(t) reduces to du(t)/dt (the derivative ofthe Heaviside unit function with respect to time t, equiv-alent to the delta function d(t)), when the imposed pen-etration is ideally stepwise, and then the Laplacetransform inversion of Eq. (3) yields Eq. (2). The stressrelaxation modulus E(t), therefore, can be obtainedthrough the Laplace transform inversion of Eq. (3 0),when the imposed penetration is not stepwise. The ana-lytical/numerical details for solving Eq. (3) or Eq. (3 0)are given in Appendix.

4.2. Stress relaxation modulus

As pointed out in Section 3, the elastic frame compli-ance of the indentation apparatus is released with timeduring the relaxation process of viscoelastic test speci-men, resulting in a quasi-stepwise penetration, even ifthe penetration h0 imposed by the piezo actuator is fixedwith time. In such an undesirable situation, the stressrelaxation modulus E(t) estimated in Eq. (2) using theobserved load relaxation P(t) is always and significantly

Fig. 4. An example for the normalized stress relaxation moduli with(solid line) and without (dashed line) the correction for the quasi-stepwise penetration. The test temperature is 23.0 ± 0.1 �C.

overestimated [8,9]. Accordingly, we need to reckon E(t)through the Laplace transform inversion of Eq. (3 0) (re-fer to Appendix). The comparison of E(t) with andwithout the correction for the quasi-stepwise penetra-tion is demonstrated in Fig. 4 for the PhSiO3/2 coating(HT-temperature of 50 �C and HT-time of 5 h). The nor-malized relaxation behaviors are plotted in Fig. 4 with re-spect to the instantaneous modulus E0[=E(0)]. Fig. 4clearly shows that the quasi-stepwise effect is so signifi-cant (see the dashed line without correction in Fig. 4)that the apparent relaxation times as well as the relaxa-tion modulus E(t) directly estimated in Eq. (2) are alwaysand considerably overestimated. Accordingly, in whatfollows, all of the observed load relaxation P(t) are con-verted to the stress relaxation modulus E(t) through theLaplace transform and its inversion (refer to Eqs. (A1)–(A6)).

The instantaneous elastic modulus E0 was estimatedin the present study as E0ð¼ Eð0ÞÞ ¼ Pð0Þ=Ah20 (seeEq. (2)), in which the time-independent plastic penetra-tion hp was implicitly neglected for simplicity. Since theinstantaneous penetration h0 is the sum of the instanta-neous elastic and plastic penetrations of he and hp, i.e.,h0 = he + hp, it must be noticed that the neglect of theplastic contribution to h0 always results in a finite under-estimate of E0 [13,14].

The normalized stress relaxation modulus, E(t)/E0,for various HT-temperatures and HT-times are shownin Fig. 5, readily showing how critically the condensationconditions affect the rheological transition (from liquid-like to solid-like transition) in the sol–gel-processing.Further details for the prolonged relaxation associatedwith the formation and the growth of polymeric siloxanechains and networks can be given in terms of the relaxa-tion time spectrum, H(s) [15,17];

HðsÞ � d2EðtÞdt2

� t2�t¼2s

; ð5Þ

where use has been made of the second-order approxi-mation of Schwarzl and Staverman [17]. SubstitutingEq. (A6) into Eq. (5) leads to the following simpleexpression for the numerical calculation of H(s),

Hðs=2Þ ¼ s2Xn

i¼1

ð1=siÞ2Ei expð�s=siÞ. ð6Þ

Examples of H(s), thus, calculated are demonstrated inFig. 6 for the coatings heat-treated at 50 �C for varioussoaking time (HT-time) of 0.5, 3.0, 5.0, and 8.0 h. In theterminal zone of the spectrum (the region of the relaxa-tion times in which the spectrum terminates), all of thepossible relaxation processes are expired, leading to apurely viscous flow. The relaxation time is the character-istic time scale describing the thermally activatedBrownian motions of the constituent molecules, clustersand the chain networks of viscoelastic materials [15].

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

HT-time 30min1h3h4h5h6h8h

(a) HT-temperature 50˚C

(b) HT-temperature 100˚C

20 40 60 80 1000Time, t (s)

HT-time 3min6min

10min20min30min1h

Nor

mal

ized

str

ess

rela

xatio

n m

odul

us, E

(t)/

E0

Fig. 5. Examples of the normalized stress relaxation moduli measuredat 23.0 ± 0.1 �C for the coatings heat-treated at (a) 50 �C (the HT-times are 30 min–8 h from the bottom to the top of the relaxationcurves) and (b) 100 �C (the HT-times are 3 min-1 h from the bottom tothe top of the relaxation curves).

10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 40

0.1

0.2

0.3

0.4

0.5

Relaxation time, τ (sec)

Rel

axat

ion

time

spec

trum

H (

τ) (

GP

a)

HT-temperature: 50˚C

HT-time0.5 hrs3.05.08.0

Fig. 6. Relaxation time spectra of the coatings (HT-temperature of50 �C) for various HT-times. Each spectrum is estimated at23.0 ± 0.1 �C.


The evolution of siloxane clusters and its three-dimen-sional networks may lead to the progressive increase inthe dimensions of these components with the increasein HT-time, and then significantly affect the relaxationtime spectrum in the terminal zone. In fact, as shownin Fig. 6, the longest relaxation times, in the respectiveterminal zones, are about 0.1, 10, 200, and 3000 s forthe HT-times of 0.5, 3.0, 5.0, and 8.0 h, respectively.On the other hand, in the region for shorter relaxationtimes of the coatings, the relaxation spectrum havingthe relaxation times smaller than 0.01 s, that dominatesthe coatings with the HT-time of 0.5 h, is extinct, as wellas the spectra having longer relaxation times signifi-cantly increase with the increase in HT-times. In otherwords, the evolution of the siloxane clusters and chainnetworks significantly broadens the relaxation timespectrum. This fact suggests that the small-size mole-cules and clusters convert into lager ones with increasingHT-time, i.e., with the progress in the condensationreactions. The detailed profiles of the relaxation timespectra shown in Fig. 6 cannot be related to the evolvedsiloxane structures in the present study, though it maybe possible once we have detailed information on thechemical structures and the dimensions of the molecules,clusters, and the chain networks in sol–gel processing.This is a very challenging issue in the science and engi-neering of sol–gel-derived hybrid coatings.

4.3. Changes in the instantaneous elastic modulus E0 and

the steady-state shear viscosity g during the evolution ofsiloxane networks in the condensation process

The instantaneous modulus E0(”E(0)) or the glassymodulus Eg (equivalent to the Young�s modulus, E)characterizes the mechanical behavior of viscoelasticmaterials in the shorter relaxation time regime. Inthe terminal zone with long relaxation times, in con-trast, the steady-state viscosity g dominates themechanical responses of viscoelastic ‘‘liquids’’, andthe equilibrium modulus Ee dictates the equilibriumdeformation of viscoelastic ‘‘solids’’. In the intermedi-ate regime of relaxation times, the stress relaxationtime spectrum H(s) represents the viscoelastic deforma-tions and flows.

The instantaneous modulus E0 and the shear viscosityg of the PhSiO3/2-coatings are plotted in Fig. 7 againstthe HT-time. Both of E0 and g significantly increase withthe HT-time, reflecting the evolution of siloxane net-works associated with the condensation reactions. Theserising behaviors are obviously enhanced with the in-crease in the HT-temperature. The shear viscosity g inFig. 7 was determined through the integration of E(t)using the following formula [15]:

g ¼ 1Z 1

EðtÞ dt; ð7Þ
3 0

1

2

Inst

anta

neou

s m

odul

us

E(0

) (G

Pa)

50˚C 80˚C100˚C

50˚C80˚C100˚C

0 2 4 6 8

6

8

10

HT-time, t (h)

She

ar v

isco

sity

log

η (P

a s)

Fig. 7. The dependences of the instantaneous modulus and the shearviscosity on the HT-time for the coatings with various HT-tempera-tures. The shear viscosities are the values at 23.0 ± 0.1 �C.


where the viscoelastic liquid is assumed to be incom-pressible (the Poisson�s ratio is assumed to be 1/2). Asdemonstrated in Fig. 7, the viscosity rises and ap-proaches to the value of �1012 Pa s with the increasein the HT-time, implying that the PhSiO3/2-sol changesto the glassy gel in the final stages of condensation [13].

0 1 20.4

0.5

0.6

0.7

4

6

8

10

Rel

ativ

e ab

sorb

ance

of F

TIR

Abs(Si-OH)/Abs(H-Ph) Shear viscosity

HT-time, t (min)

She

ar v

isco

sity

, log

η (

Pa

s)

Fig. 8. Correlation between the relative intensity of FTIR-absorbanceand the shear viscosity (23.0 ± 0.1 �C) in terms of the HT-time. Therelative intensity of FTIR-absorbance is calculated as the ratio of theintensities for the Si–OH stretching and the out-of-plane wagging ofSi–Ph-bonded H atoms.

An example of the correlation between the changes inthe mechanical property and in the chemical structure ofthe coating at the HT-temperature of 80 �C is shown inFig. 8, in which the relative intensity of the specificabsorbance of FTIR-spectrum (the ratio of the absor-bance for Si–OH stretching and H–Ph wagging modes[10]) is compared with the shear viscosity. The extinctionof Si–OH bond (i.e., the evolution of Si–O–Si chains andnetworks) during the condensation, is well correlatedwith the increase in the shear viscosity, g.

5. Conclusion

The linear viscoelastic stress relaxation of PhSiO3/2-coatings was examined in microindentation load relaxa-tion tests. In order to eliminate the effect of the substrateon the measured relaxation properties, thick coatings ofabout 20 lm were prepared. A stepwise penetration wasimposed on the coating with the depth of about 1.4 lm(about 7% of the thickness of coating), and then the loadrelaxation was measured with time. The effect of theframe compliance on the relaxation behavior was elimi-nated via a mathematical algorithm using a Laplacetransform and its inversion combined with numericalcollocation.

The rheological transition from the sol (liquid) to thegel (solid) during the condensation reactions was tracedin terms of the changes in the instantaneous modulus E0

(Young�s modulus), steady-state shear viscosity g, stressrelaxation modulus E(t), and the relaxation time spec-trum H(s). The effect of soaking conditions (the heat-treatment temperature and time) on the rheologicaltransitions of the coatings was intensively studied. Theevolution of siloxane clusters and chain networks insol–gel processing significantly affected the rheologicaltransitions. The changes in the FTIR-spectrum of thecoatings during the condensation reactions were wellcorrelated to the results and the findings of the microin-dentation rheology.

In viscoelastic studies of coating/substrate systems,indentation load relaxation test may have a definiteadvantage compared to the conventional indentationcreep test, for the penetration depth is kept constant inthe former (meaning that the separation between thetip of indenter and the surface of substrate is fixed),while the depth progressively increases with time in thelatter (meaning that the tip of indenter approaches thesurface of the substrate with creep time, and the effectof substrate becomes progressively significant). Accord-ingly, the indentation creep test adds undesirablecomplexity to indentation test results. Furthermore,due to the dual nature between the creep complianceC(t) and the relaxation modulus E(t) [11–16], i.e.,CðsÞ � EðsÞ ¼ 1=s2 in the Laplace space, it is easy to


obtain C(t) from the E(t) measured in indentation relax-ation test.

Acknowledgment

Part of the present work was supported by The JapanSociety for The Promotion of Science (The Grant-in-Aidfor Scientific Research).

Appendix. Laplace transform and its inversion for the

indentation load relaxation with quasi-stepwise

penetration

Both of the time-dependent observables, the loadrelaxation P(t) and the penetration-related functionF(t) (see Eq. (4)) are expressible with the followingapproximating functions in exponential series:

P ðtÞ ¼ P e þXn

i¼1

ai expð�t=k1iÞ ðA1Þ

and

F ðtÞ ¼Xn

i¼1

bi expð�t=k2iÞ. ðA2Þ

In Eq. (A1), Pe indicates the equilibrium load after com-plete relaxation, implying that Pe reduces to zero for vis-coelastic ‘‘liquids’’. The adjusting parameters of ai, bi,k1i, and k2i are easily determined by collocation atn-points. Transforming Eqs. (A1) and (A2) into therespective Laplace spaces yields, respectively, the follow-ing formulas:

P ðsÞ ¼ P e

sþXn

i¼1

aisþ 1=k1i

ðA3Þ

and

F ðsÞ ¼Xn

i¼1

bisþ 1=k2i

. ðA4Þ

Substitute Eqs. (A3) and (A4) into Eq. (3 0), and approx-imate the result using the following formula:

EðsÞ ¼ Ee

sþXn

k¼1

Ek

sþ 1=sk. ðA5Þ

The adjusting parameters of Ek and sk are againdetermined by the numerical collocation of n-points.The Laplace transform inversion of Eq. (A5) is straight-forward conducted, resulting in the stress relaxationmodulus,

EðtÞ ¼ Ee þXn

i¼1

Ei expð�t=siÞ. ðA6Þ

The equilibrium modulus Ee is zero for viscoelastic liq-uids. In Eq. (A6), Ei represents the elastic modulus ofthe ith viscoelastic component having the relaxationtime of si. For the experimental curves in the presentrelaxation test results, it is sufficient enough to choosethe collocation points n of 10.

References

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indentation stress relaxation of sol–gel-derived organic/inorganic hybrid coating

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