Fatigue crack growth in ferroelectrics under electrical loading

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Fatigue crack growth in ferroelectrics under electrical loading

Transcript Of Fatigue crack growth in ferroelectrics under electrical loading

Journal of the European Ceramic Society 26 (2006) 95–109

Fatigue crack growth in ferroelectrics under electrical loading
Jay Shieha, John E. Huberb,∗, Norman A. Fleckb
a Department of Materials Science and Engineering, National Taiwan University, 1 Roosevelt Road, Sec. 4, Taipei, Taiwan b Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK Received 8 August 2004; received in revised form 12 October 2004; accepted 16 October 2004 Available online 8 December 2004
Abstract
Fatigue crack growth is investigated in the polycrystalline ferroelectrics PZT-5H and PLZT 8/65/35 under electrical loading. The fatigue cracks exhibit features such as bifurcation and tunnelling, followed by crack arrest. The nature of the resulting damage differs in the two materials: in PZT-5H a narrow zone of intergranular cracks propagates across the specimen, wedging the crack surfaces apart. In PLZT 8/65/35 a broad microcracked band spreads across the specimen. The rate of crack growth is found to correlate well with the amplitude of electric displacement. © 2004 Elsevier Ltd. All rights reserved.
Keywords: Electroceramics; Ferroelectricity; Fatigue; PZT; PLZT

1. Introduction
Ferroelectric ceramics experience fatigue cracking under the influence of alternating electric fields, which degrades their electrical and mechanical properties and thus reduces their reliability. The growth in applications of ferroelectrics as electromechanical sensors, electronic filters, micropositioners and displacement actuators1,2 has made fatigue a critical issue faced by manufacturers of ferroelectric components.
In the ferroelectrics literature, the term fatigue generally refers to the gradual degradation of bulk material properties, such as the saturation remanent polarization, in a cyclically loaded specimen. Most studies in this area have focused on the electrical degradation of ferroelectrics, i.e. the cycle dependence of remanent polarization, strain hysteresis, coercive field and dielectric constant of ferroelectric ceramics upon electrical fatiguing.18–26 In contrast, the present experimental study is concerned with the distinct phenomenon of fatigue cracking—the nucleation and growth of cracks or microcrack damage due to cyclic electrical loading.
∗ Corresponding author. Tel.: +44 1223 332781; fax: +44 1223 332662. E-mail address: [email protected] (J.E. Huber).
0955-2219/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jeurceramsoc.2004.10.002

In several previous experimental studies of electric field-
induced fatigue crack growth of ferroelectric ceramics,
researchers have grown fatigue cracks from Vicker’s inden-
tations, and have shown that the predominant direction of
fatigue crack growth is perpendicular to the applied electric field.3–5 The phenomenon of fatigue crack growth in ferro-
electrics under cyclic electric fields above the coercive field (Ec) is well documented.4,6 Electric field-induced fatigue crack growth in PZT under an alternating electric field below the coercive field has also been observed.7 Weitzing et al.8 ex-
plored fatigue cracking in three different PZT materials (one
rhombohedral composition and two morphotropic composi-
tions with different grain sizes) under cyclic electric fields
up to 1.5Ec; they found that the fatigue crack growth rate decreases with increasing cycle number, and a saturation point is reached after approximately 105 cycles. Several authors
have used in situ field-emission scanning electron microscopy
(SEM) and transmission electron microscopy (TEM) to study
crack tip deformation and microcracking from fine pores due to electrical loading.9–11 By altering the relative orientation
between the poling and electrical loading directions, it has
been demonstrated that an indentation crack could be re-
directed to propagate on planes which are inclined to the original crack plane.12,13

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A substantial literature now exists concerning electric field induced fatigue crack growth. However, there is still a lack of agreement on the fundamental mechanism. Cao and Evans6 presented a crack wedging mechanism which explains steady-state electric field-induced fatigue crack growth as a result of the actuation of asperities along the fracture surfaces, giving active wedging. Zhu and Yang14–17 argued that the non-uniform domain switching near the crack tip leads to strain mismatches and produces a cyclic stress field characterised by a cyclically varying stress intensity factor near the crack tip. There is a lack of data for fatigue crack growth over 106 or more cycles; moreover, the available data are limited in scope in terms of material type, and loading amplitude.
The current study gives a systematic set of measurements of fatigue crack growth in ferroelectrics under purely electrical loading and assesses various possible correlating parameters. Some of the fatigue crack growth data have been previously presented;27 the present work extends and develops this previous study by reporting additional measurements, detailed morphology of fatigue cracks, and by calculating each of a set of loading parameters in order to find a parameter that correlates with the rate of crack growth. The correlating parameters considered are the range of electric field,
E; the range of applied voltage, V; the range of electric displacement, D; the cyclic J-integral, J and the cyclic dissipation, W. The evaluation of J from readily available experimental measurements is discussed in Section 2. The focus of the experimental part of this study in Sections 3 and 4 is on the characterisation and observed mechanism of crack propagation in two ferroelectric compositions: soft lead zirconate titanate (PZT-5H) and lanthanum-doped lead zirconate titanate (PLZT) of composition La/Zr/Ti = 8/65/35. Electric field-induced fatigue crack growth measurements, in the form of crack extension as a function of the number of cycles, are presented. The dielectric hysteresis of cracked specimens of the two materials during fatigue, and the morphology of the fatigue crack itself, are described. Critical information required for the evaluation of potential correlating parameters is obtained from the experimental data.

2. Evaluation of J

Consider a non-linear ferroelectric solid in which the state of stress σij, electric field Ei, strain εij and electric displacement Di are known at each material point. Assume the existence of the electric enthalpy h defined by

h = h(εij, Di) = σij dεij − Ei dDi

(1)

The J-integral is defined on a closed path Γ with unit outward normal ni that surrounds a crack tip C in the solid, as shown in Fig. 1a. In a linear piezoelectric solid, the integral

J = (hn1 − σijnjui,1 + DjnjE1) dΓ

(2)

Γ

is path independent and is equal to the energy release rate G
for a crack parallel to the x1 direction, propagating in the x1 direction.28 Care is needed in evaluating J because closure of
the path within the body requires the path to include a por-
tion of the crack faces. There is an electrical contribution to
J from the crack surface when DjnjE1 is non-zero. However, under some circumstances the contribution to Eq. (2) from
DjnjE1 vanishes on the crack surface. The “impermeable” crack model assumes a charge free crack surface and a crack
medium with zero permittivity; in this case, Djnj = 0 everywhere on the crack surface and hence the contribution from
DjnjE1 vanishes. Similarly, in a conducting crack, E1 = 0, and again DjnjE1 vanishes.
The concept of a cyclic J-integral, J has been used
to characterise fatigue crack growth where bulk yielding occurs.29 Let the minimum load point in a fatigue cycle correspond to loading (σ0, E0) and a strain and electric displacement state of (ε0, D0). Define deviations from the minimum load state during the loading part of the cycle by σ = σ − σ0
and so forth. A cyclic J-integral can be defined by

J = (hcn1 − σijnj ui,1 + Djnj E1) dΓ (3)
Γ
where hc is a cyclic analogue of the electric enthalpy, defined by

hc =

σijd εij −

Did Ei

(4)

In the present work, the electrical fatigue of a rectangular specimen with a thin crack growing parallel to the surface electrodes is considered (see Fig. 1b). The specimen has zero traction on its surface, and zero voltage along BC, while a voltage V is applied along the surface ED. For the integration contour ABCDEFGA of Fig. 1b, the cyclic J-integral becomes

J=

hcn1 dΓ + Djnj E1 dΓ

(5)

AB+CD+EF

FGA

In writing Eq. (5), terms hcn1 vanish along FGA because n1 vanishes on the surface of a thin crack. Also, the approximation DiniE1 = 0 is made along the external surface ABCDEF of the specimen. This is justified on BC and DE because the presence of conducting electrodes ensures that E1 = 0. On AB, CD and EF the external medium has a much smaller permittivity than that of the specimen, and the surfaces carry no free charge, such that Dini ∼ 0. If the crack is thin and conductive, then the contribution to J from segment FGA vanishes, giving

J=

hcn1 dΓ

(6)

AB+CD+EF

Eq. (6) allows J to be evaluated as the net flux of electric enthalpy into the specimen during the loading part of the cycle. The assumption of a conductive crack may in practice be more realistic than the assumption of an impermeable crack,

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Fig. 1. Evaluation of the J-integral. (a) General path ABA CA enclosing a crack tip C. (b) Path ABCDEFGA following the surface of a rectangular electrical fatigue specimen. (c) Path BCDEB enclosing a specimen with a growing damaged zone. (d) Calculation of J from the charge Q vs. voltage V response of the whole specimen.

given that electrical discharge has been observed within fatigue cracks during cyclic loading (see Section 4). This is consistent with the observation30 that the measured potential drop across a crack in a ferroelectric is much less than that predicted by assuming an impermeable, insulating medium.
In the set of measurements discussed in this paper, detailed information about the electro-mechanical state at each point on the integration path is not readily available, and a steady state approximation of J is used. For this interpretation, consider a developing damage zone of finite height, spreading into the material specimen in the x1 direction, as shown in Fig. 1c. The damaged zone behaves as an inhomogeneity whose mechanical and dielectric behaviour differ from the undamaged bulk. By following the contour of integration around the outer surface ABCDEFA of the specimen shown in Fig. 1c, Eq. (6) becomes

J=

hcn1 dΓ

(7)

CD+EB

In steady state, with purely electrical loading of a specimen of width BC CD, the stress σij is zero everywhere on the integration path, whilst Di is uniform and parallel to the x3

direction on each of the segments CD and EB. Then, labelling the upstream state on CD by a superscript ‘u’ and the downstream state on EB by a superscript ‘d’, Eq. (7) can be rewritten as

J=

D3u(V ) dV −

D

d 3

(V

)

dV

(8)

where V = CD − E3 dx3 is the voltage on the upper electrode DE. Thus J can be evaluated if the electric displacement is known as a function of applied voltage both upstream and downstream of the zone of damage growth. In practice, the quantities that are easily measured are V and the total charge Q delivered to the upper electrode. In steady state, let the damage zone advance over a small area δA projected onto the upper electrode. This has the effect of converting a region of material of projected area δA from the upstream state to the downstream state. Consequently, the change in the total charge Q delivered to the upper electrode is δ( Q) = δA( Dd − Du) and the expression for
J reads

J = − ( Q|A+δA − Q|A) dV

(9)

δA

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Fig. 2. Schematic of the test specimen and electric field loading apparatus.

as illustrated in Fig. 1d. Eq. (9) is used to estimate the cyclic J-integral in this study.
3. Experimental procedure
The general procedures used were as previously reported by Shieh et al.27 A brief description of the key features of experimental procedure is given here for clarity.
3.1. Choice of materials
The two ceramics PZT-5H and PLZT 8/65/35 were chosen for this study because they are widely used in engineering devices, exhibit nonlinearity at relatively low field strengths, and have strong electromechanical coupling. The PZT-5H was obtained in bulk polycrystalline form1 with grain size 5–10 ␮m, a composition close to the morphotropic phase boundary, and a Curie temperature of approximately 195 ◦C. The PLZT 8/65/35 was also obtained in bulk polycrystalline form,2 with grain size 3–6 ␮m, a predominately rhombohedral crystal structure, and a Curie temperature of approximately 110 ◦C. PLZT 8/65/35 exhibits memory type electro-optical characteristics (birefringence hysteresis). It is transparent in the unpoled and minor-loop states, translucent in the poled ferroelectric state, and birefringent under strain or electric field. Both the PZT-5H and PLZT
1 Morgan Matroc Ltd., Transducer Products Division, Thornhill, Southhampton SO9 5QF, UK.
2 Alpha Ceramics Inc., 5121 Winnetka Avenue North, Minneapolis, MN 55428, USA.

8/65/35 materials were thermally depoled to an isotropic state.
3.2. Specimen preparation
Cuboidal specimens measuring 5 mm × 2 mm × 3 mm (width × thickness × height) were cut from the bulk materials. The general arrangement is shown in Fig. 2. Silverbased paints were applied onto the 5 mm × 2 mm faces of the specimens to form electrodes, and all other surfaces were polished to a 1-␮m finish. (Unpolished specimens were found to develop numerous cracks rapidly during high amplitude cyclic loading.) A V-notch of depth approximately 0.1 mm was then scribed mid-way between the electrodes on one of the 2 mm × 3 mm side faces with a diamond-tipped scriber; the V-notch was parallel to the electrodes and ran across the entire 2 mm thickness of each specimen. The fatigue cracks were initiated from a notch to produce, as nearly as possible, a through-thickness crack. Conductive epoxy was used to connect wires to the electroded surfaces of the specimens for electric field excitation.
3.3. Equipment and testing
Electric field loading was applied using a high-voltage amplifier, driven by a signal generator. The net charge on the electrodes of each specimen was monitored by a 2.96 ␮F metering capacitor connected in series between the specimen and ground. From the high voltage source, a bipolar cyclic voltage of sinusoidal waveform at a frequency of 20 Hz was applied to each specimen, to produce electric field parallel to the 3 mm height direction of the specimen. Fully reversed

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Fig. 3. Measured electric displacement D vs. electric field E hysteresis curves after 10 cycles for PZT-5H and PLZT 8/65/35; loading frequency = 0.2 Hz.

electric loading was applied, with the amplitude of the cyclic field set to selected values in the range 0.9Ec to 3.0Ec (where Ec is the coercive field of the test material). Electrical breakdown outside the specimen was prevented by immersion in oil, which was filtered regularly to remove any conductive contaminant particles. (Preliminary trials showed that contamination in the oil could contribute to the nucleation of microcracks on the specimen surface.) A long-focal-length video camera was used to observe and record fatigue crack

growth and a digital cycle counter was used to track the cycle number. All tests were carried out at laboratory temperatures in the range 17–21 ◦C. Some temperature rise in the specimen was expected during high amplitude electric field testing due to dissipation associated with ferroelectric switching. Periodic checks were made on the specimen and oil temperature using remote infrared sensing; it was found that the oil temperature in the vicinity of the specimen remained stable and close to the laboratory temperature throughout the tests.

Fig. 4. Comparative a vs. N curves for V-notched PZT-5H, at 20 Hz. (a) Results for the entire test duration, shown on log-linear axes. (b) The first 105 cycles of the same tests shown on linear axes.

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3.4. Preliminary measurement of hysteresis
The stable cyclic dielectric hysteresis (electric displacement D versus electric field E) of each material was measured before commencing the fatigue tests. A cyclic electric field of amplitude 1.4 MV m−1, frequency of 0.2 Hz, and sinusoidal waveform was applied to an un-notched specimen of each material. The measured D–E curves for PZT-5H and PLZT 8/65/35 are shown in Fig. 3, and the resulting values of coercive field Ec and saturation remanent polarization Pr of both materials are listed in Table 1, along with manufacturers’ data for dielectric permittivity K3T3, piezoelectric coefficient d33 and Curie temperature Tc.

Table 1
Properties of PZT-5H and PLZT 8/65/35 (K3T3, d33 and Tc are manufacturers’ data)

Quantity
Coercive field, Ec (MV m−1) Remanent polarization, Pr (C m−2) Relative dielectric constant, K3T3 Piezoelectric strain coefficient, d33
(×10−12 m V−1) Curie temperature, Tc (◦C)

PZT-5H 0.70 0.26
3400 593
195

PLZT 8/65/35 0.48 0.30
3350 640
110

Fig. 5. (a) SEM images of a band of damaged material on crack flanks (PZT-5H). (b) Damaged band height H vs. distance behind the crack tip, x, for specimens loaded at various amplitudes of electric field. All specimens were V-notched PZT-5H; loading frequency = 20 Hz.
Fig. 6. (a) Electrically fatigue-induced intergranular fracture surface (PZT-5H). (b) Mechanically separated transgranular fracture surface (PZT-5H).

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Fig. 7. D–E curves at the first cycle and after 0.75 × 105 cycles of loading (arrest of crack growth) in PZT-5H; loading amplitude = 3.0Ec, loading frequency = 20 Hz.

4. Results and discussion
4.1. Fatigue crack growth in PZT-5H
The crack extension, a, as a function of the number of cycles, N, under various amplitudes of bipolar electrical loading for the V-notched PZT-5H specimens is shown in Fig. 4a on log-linear axes. In each specimen, crack growth began

immediately, with no incubation period. The rate of fatigue
crack growth da/dN decreased with increasing number of cy-
cles until crack arrest (see Fig. 4b). For the sample loaded
to a maximum value Emax = 3.0Ec, the observation of crack growth was terminated at about 7 × 104 cycles due to the ini-
tiation of secondary cracking at the electrodes. Crack arrest occurred after approximately 7 × 104 cycles for the applied field of Emax = 2.5Ec, and after approximately 2 × 106 cycles

Fig. 8. Comparative a vs. N curves for V-notched PLZT 8/65/35, at 20 Hz. (a) Results for the entire test duration, on log-linear axes. (b) Results for the first 600 cycles after the incubation period, on linear axes.

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Fig. 9. Progressive change of D–E loop of PLZT 8/65/35 during fatigue testing (load amplitude = 1.4Ec, sample 1, loading frequency = 20 Hz).

for the applied fields of Emax = 2.0Ec and 1.5Ec. Fatigue crack growth under the alternating electric field of amplitudes 1.0Ec and 0.9Ec was undetectable by the video imaging system after 107 cycles. Both the arrested fatigue crack length and the average crack growth rate increased with electric field amplitude in PZT-5H. Here, the average crack growth rate is defined by the total crack extension to crack arrest divided by the total number of cycles to crack arrest.
The fatigue cracks in PZT-5H were observed to emanate from the tip of the V-notch. Some crack branching was observed, but the predominant direction of crack growth was along the mid-plane of the specimen perpendicular to the

applied electric field. With nominal field amplitudes above 2.0Ec, electrical arcing was clearly visible within the fatigue crack during the tests. As the fatigue crack propagated through the ceramic, a band of damaged material formed around the crack (see Fig. 5a) and later examination of the specimen surface by scanning electron microscopy (SEM) indicated fragmentation into discrete grains along the crack flanks. The height H of the damage band is plotted as a function of distance x behind the crack tip in Fig. 5b for selected values of Emax. It is evident that H varies approximately with x1/2 and is independent of load level. This behaviour is characteristic of the near crack-tip region of a crack-like

Fig. 10. (a) The development of a single macroscopic zone of microcracks in V-notched PLZT 8/65/35 specimen after 1.44 × 104 cycles at 1.5Ec, 20 Hz. (b) Cracking zone height vs. distance behind crack tip at various applied electric fields for V-notched PLZT 8/65/35; loading frequency = 20 Hz.

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feature propagating in steady state. After fatigue testing, the specimens were broken open by applying a mechanical load until separation occurred, through the fatigue crack growth zone. Examination of the fracture surfaces by SEM indicated that the fatigue crack in the PZT-5H specimen existed across the section thickness and was not merely a superficial crack. The fatigue-crack surface was predominantly intergranular, whereas the monotonic fracture surface under tensile loading was predominantly transgranular (compare Fig. 6a and b). Micro-cracks, wear debris and pulverised material were apparent on the fatigue-induced intergranular fracture surfaces, giving the appearance of severe wear.
The evolution of the nominal D–E curves for the Emax = 3.0Ec specimen, from that recorded at the start of the fatigue loading to that after crack arrest (a = 3 mm), is shown in Fig. 7. A 15% reduction in the saturation remanent polarization of the specimen was observed during the test, and the switch of polarisation in the vicinity of the coercive field strength became less abrupt with increased cycling.
4.2. Fatigue crack growth in PLZT 8/65/35
The PLZT 8/65/35 ceramic also showed crack growth and arrest under a fully reversed cyclic electric field. The fatigue crack extension, a, is shown in Fig. 8a as a function of the number of cycles, N, at selected amplitudes of bipolar elec-

trical loading. At field amplitudes below 1.2Ec, fatigue crack growth was negligible. At greater amplitudes, cracks grew until they arrested at a crack length which increased with load amplitude. When the field amplitude was greater than 1.2Ec, three distinct stages were observed: an incubation period of Ni cycles, during which crack growth was negligible, a period of rapid crack growth, and finally, crack arrest. The number of cycles Ni in the incubation period reduced monotonically with increasing electric field amplitude. The rapid, steady growth rate after the incubation period was about 10–40 ␮m/cycle for all specimens, with some sensitivity to Emax. Fig. 8b shows the crack growth after the incubation period for each specimen, on log-linear scales.
A progressive change in shape of the nominal D–E hysteresis curve during fully reversed cyclic loading was observed in all of the specimens. Fig. 9 shows this progression for the specimen loaded at amplitude Emax = 1.4Ec (labelled sample 1 at 1.4Ec in Fig. 8). At the start of the fatigue test (curve 1 in Fig. 9) the D–E curve was a minor-loop with little ferroelectric switching; this corresponded to the incubation period of 180 cycles shown in Fig. 8a. At approximately 180 cycles, the D–E curve expanded into a major-loop (see curve 2 in Fig. 9), corresponding to the onset of rapid fatigue crack growth at about 50 ␮m/cycle. During the period of rapid growth, the D–E curve collapsed gradually from the major-loop state (see curve 3 in Fig. 9) until, at about 500

Fig. 11. Evolution of the cracked zone with cycle number in various V-notched PLZT 8/65/35 specimens. Heavy contours show the extent of the damage zone at the number of cycles indicated.

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cycles, fatigue crack growth arrested. During the following 104 cycles, there was little crack growth, and only a slight change in the D–E curve. The D–E curves for specimens cycled at and below 1.2Ec stayed in the minor-loop (low hysteresis) state throughout the test—these specimens showed negligible crack growth.
Fatigue crack growth in PLZT 8/65/35 was by the development of a single macroscopic zone of microcracks which exhibited features such as bifurcation and tunnelling (see Fig. 10a). The microcracked zone had an approximately constant height along its entire length. However, the zone height increased monotonically with the strength of the applied electric field — ranging from about 0.12 mm for a field strength of Emax = 1.1Ec to about 0.35 mm for Emax = 1.5Ec (see Fig. 10b). Subsequent SEM inspection of sectioned specimens and fracture surfaces showed that the microcracks were isolated intergranular cracks. This suggests that the microcracks form by the failure of grain boundaries due to local conditions in the vicinity of the individual grains. Since PLZT 8/65/35 is transparent, it was possible to observe that the zone bifurcated severely and was not continuous through the whole thickness of the specimen. Electrical arcing within the microcracked zone

during high amplitude testing was also visible in PLZT 8/65/35.
The evolution of the length and shape of the microcracked zone with fatigue cycles for various PLZT 8/65/35 specimens is shown on a side view of the specimen in Fig. 11: the contours give the boundary of the microcracked zone with increasing number of cycles. For each specimen, a gradual broadening of the macroscopic zone of microcracks in the direction of the applied electric field is observed, whilst the predominant direction growth was along the centreline of the specimen, perpendicular to the applied electric field.
4.3. Dielectric hysteresis of microcracked material and uncracked material
Measurements of the dielectric response of uncracked material and material taken from the microcracked zone were made, as follows. A V-notched PLZT 8/65/35 specimen was subjected to a cyclic electric field of Emax = 1.4Ec at 20 Hz, until the macroscopic zone of microcracks grew across the full width of the specimen. The uncracked and microcracked regions of the fatigued V-notched specimen were then cut out as separate specimens, with dimensions

Fig. 12. (a) The un-cracked and cracked regions of the fatigued PLZT 8/65/35 specimen are cut out as two separated specimens. (b) Measured D–E responses for cracked and uncracked regions of the fatigued PLZT 8/65/35 specimen; loading frequency = 20 Hz. The bold lines link the tips of the hysteresis loops for both the uncracked and the cracked material.
SpecimenFigFatigue Crack GrowthCrack GrowthSpecimens