Contradictions and new aspects of the bubble model of doped tungsten wires

International Journal of Refractory Metals & Hard Materials 24 (2006) 311–320 www.elsevier.com/locate/ijrmhm

Contradictions and new aspects of the bubble model of doped tungsten wires I. Gaal a

a,*

, P. Schade b, P. Harmat a, O. Horacsek a, L. Bartha

a

Research Institute for Technical Physics and Materials Sciences, P.O. Box 49, 1525 Budapest, Hungary b HTM Consulting, Berlin, Germany Received 15 November 2005; accepted 9 January 2006

Abstract The interpretation of a series of small angle neutron scattering studies on lamp grade tungsten rods and wires is in contradiction with an extended body of the results that were obtained by fractographic methods and thin foils studies by means of scanning microscopy and transmission electron microscopy. In order to facilitate the re-interpretation of the scattering studies, the state of agglomeration of potassium in the dope phase has to be clarified, due to the peculiar scattering properties of the two-phase inclusions. These efforts yielded to the following results: (1) There are independent pieces of experimental evidence that the dope inclusion are filled with liquid potassium in heavily drawn lamp grade tungsten wires and rods in various stages of the processing. (2) A slight volume compression of liquid inclusion upon wire drawing is compatible with the scaling laws of the Taylor model in a good approximation, because the expected deviations are less than the accuracy of the required measurements. (3) The size increase of the inclusion cannot be explained as the expansion of individual bubbles that do not interact with each other. One has to take into account also various sorts Ostwald type ripening processes.  2006 Elsevier Ltd. All rights reserved. Keywords: Non-sag tungsten; Coarsening of the dope phase; Liquid potassium inclusions

1. Introduction The performance of lamp grade tungsten is governed by a highly dispersed potassium phase. It is well known that it consists of axially aligned, few micrometers long rows of ultra-fine sized inclusions, if heavily drawn wires are stress relieved before the onset of the heterogeneous grain growth. The famous, coarse, sufficiently elongated and interlocking grain structure evolves also in the actual course of filament fabrication in a previously stress relieved microstructure. The interlocking structure is of primary importance, as it prevails also upon service life [1]. There are good pieces of evidence that measures can be devised for the numerical characterisation of the extent of required interlocking grain structure in order to predict not only the excellent high temperature creep performance, *

Corresponding author. E-mail address: [email protected] (I. Gaal).

0263-4368/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmhm.2006.01.002

but also the required limited plasticity of coils and wires at the ambient [2–7]. One has to stress that these measures are governed not only by the mean transversal intercept of the bubble rows, but they are also sensitive on the distribution of their lengths [2]. It is also well documented that the size distribution of the dope inclusions is especially broad on the boundaries of the interlocking grain structure [8–13]. This size distribution is crucial with respect to the excellent creep performance, because the largest intergranular inclusion should not surpass a critical size that depends on the service conditions [8,9]. The required microstructure of the filaments is achieved by close control of the complete manufacturing process from powder preparation through sintering, swaging, wire drawing, coiling and carefully scheduled annealing of the coils [1,7]. The present model on the microstructural evolution of the dope phase has its origin in the classical paper of Moon and Koo [14]. The principal scenario of the dope evolution upon thermo-mechanical processing is quite clear today [1],

312

I. Gaal et al. / International Journal of Refractory Metals & Hard Materials 24 (2006) 311–320

although important features of this evolution are not well understood. These are as follows: (1) What kind of processes govern the length of the inclusion rows during swaging, wire drawing and stress relieve [2]? (2) What kinds of competitive processes make the interlocking grain structure sensitive to the annealing schedule [7–16]? (3) What kinds of processes make the size distribution of potassium inclusion sensitive to the details of the annealing schedule [7–16]? (4) Why is the size distribution of the intergranular and intragranular inclusions markedly different [11]? The study of these questions is often limited by the systematic errors of the usually applied experimental methods [1]. Therefore, the morphology of the dope phase was studied also by means of small angle neutron scattering [17–19]. The present interpretation of these results is in contradiction with the results of the topographic methods in various aspects. Because the neutron scattering is sensitive on the state of aggregation of the studied dispersed phase [19], the present work deals with the phase relations in the dope phase and points out the importance of the inclusions filled with liquid potassium in various stages of processing. 2. State of aggregation of potassium in the dope phase It is usual to consider the dope phase of lamp grade tungsten as an ensemble of potassium bubbles. Of course, this picture is absolutely correct, as long as one deals solely with processes that take place above the critical temperature of potassium (T K C ¼ 2178 K [20]). However, the term bubble may become ambiguous, when the thermo-mechanical processing is analysed, since it applies a great variety of temperatures that are below T K C [21–24]. Fig. 1 shows the T–q phase diagram of potassium according to the data of [20,25–28], in order to visualise the domain, where one evidently has to consider three types of inclusions: (i) voids filled with vapour, (ii) voids filled with liquid and (iii) voids containing liquid and vapour in two-phase equilibrium (Fig. 1). Partly because the behaviour of two-phase inclusions may be quite peculiar, and partly because the twophase region of potassium is broad (Fig. 1), the state of aggregation of potassium may affect also the evolution of the dope phase. This topic has already been considered also by Nagy [29]. To this end, he used a van der Waals equation, the parameters of which were fitted to the critical temperature and critical pressure. Unfortunately, these efforts cannot give reliably support at the clarification of the phase relations, because the applied equation of state results in a limiting liquid density amounting merely to 0.27 g/cm3 [29], while the density of liquid potassium is much higher than this value in a very broad temperature region (Fig. 1). Let us spend few words on the construction of the T–q phase diagram (Fig. 1). The pressures at vapour–liquid

Fig. 1. The temperature-density phase diagram of potassium according to the data of [20,25–28]. The insert depicts two sorts of two-phase inclusion.

equilibrium were inserted into this diagram according to the data of [28] for K. The density and temperature data with asterisk are the data from [26] for K, while the data with triangles were recalculated from the T/TC–q/qC diagram of [20]. The reduced T–q diagram of [20] was constructed with the reduced co-ordinates of K, Rb and Cs, in order to visualizes the validity of the T/TC, P/PC and q/qC scaling law. For the sake of clarity, the insert of Fig. 1 depicts two sorts of two-phase inclusion. Their construction is based on the fact that the tension of the W(solid)/K(liquid) interface is much higher that one of the K(vapour)/K(liquid) interface, while the tension of the W(solid)/K(liquid) and W(solid)/K(gas) interfaces are not very different at liquid–vapour equilibrium, due to the strong adsorption of K on W. Unfortunately, one has to realise that even recent papers consider vapour as the sole state of aggregation of potassium also in cases, where the processes take place below TK C . Because the origin of this habit might be connected with the story of ‘‘potassium bubbles,’’ let us remember that a ‘‘stringer of spherical voids’’ was the first morphological feature that was recognised as a typical microstructural element of doped tungsten (e.g. [30]). This feature was revealed on transmission electron micrographs and on scanning electron micrographs taken on intergranular fracture surfaces (e.g. [30]), and the new finding was rationalised by means of two assumptions: (i) the voids are stabilised through a fill-gas at high temperatures and (ii) the fill-gas should not condense even at the ambient, since extraction replicas fail to identify particles that could be

I. Gaal et al. / International Journal of Refractory Metals & Hard Materials 24 (2006) 311–320

correlated with the dopant stringers. Believe (ii) was supported also by fracture replicas [14], as the shadow contrast of them proves that the ‘‘void-like’’ features on the fracture surfaces are really empty. Therefore, it was a crucial achievement, when Snow proved by thin film selected area diffraction that potassium condensed into an epitaxial crystalline layer on the tungsten surface of dope voids in samples that are annealed at supercritical temperatures and afterwards were quenched to the ambient [31]. One may, of course, ask two questions. Why are the dope voids empty at the fracture surfaces? Is eventually the crystalline potassium layer extremely thin in the ‘‘dope voids’’? In order to answer these question samples having a diameter of 900 lm were preannealed at 2200 K and studied by Auger electron spectroscopy [32,33]. When longitudinal grain boundaries of these samples were opened up by in situ fracture in the AES chamber at 77 K, no K signal was detected in the used set up. Consequently, the average K coverage on the measured area (having a diameter of 5 lm) was less than 0.05 monolayer. However, a marked potassium signal appeared, when the samples were heated up to 300 K in the AES chamber. The detected signal could be ascribed to 0.7–1.5 monolayer potassium. Since the Auger studies were not supported by micrographs, we cannot give reliable estimates for the potassium content of the opened up inclusions. In spite of this, this study is of importance: it revealed a mechanism by means of which potassium is able to leave the intergranular dope inclusions upon fracture. The essential points are as follows. The binding energy of solid potassium is 21 kcal/mol at 300 K with respect to a dilute potassium gas source, while the binding energy for potassium adatoms on the free tungsten surface amounts to about 40 kcal/mol at 1 monolayer coverage with respect to the same source. Consequently, it is an energetically favourable process, when K adatoms spread on a free W surface upon leaving a solid potassium source that is attached to it. Since the diffusion rate of K adatoms on the W surface is high at the ambient, the final conclusion is [32,33]: solid potassium can leave a fine intergranular dope inclusion (d < 100 nm) within few minutes, if is opened up by fracture. The potassium content of the individual inclusions is one of the crucial parameters, whenever a model has to be fitted to experimental data [1]. In spite of this, reliable experimental data are scarce for this quantity. Let us quote, therefore, only the data concerning the average potassium density of the dope phase. Wright [34] measured the volume fraction of the dope phase, VV, in two sorts of high quality wires having a diameter of 173 lm and annealed at 2800 K. Wright evaluated VV in three independent thin foils for each wire type. The volume fractions amounted to 0.17 ± 0.05% and 0.33 ± 0.05%, respectively. The corresponding potassium density amounted to 0.65 and 0.4 g/ cm3, respectively, as the mass fraction of potassium was 60 lg/g. Small angle neutron scattering should be also an adequate tool to reveal the two-phase character of dope inclu-

313

sions [19]. The reason is as follows. The neutron scattering cross section of K atoms is much higher than that one of W atoms, and consequently the scattering of a two-phase inclusion will markedly depend on the volume ratio of the gas phase and the liquid phase. Since the potassium phase in as-drawn wires give rise to anisotropic small angle neutron scattering [17–19], the samples (d = 180 lm) had to be preannealed at 2200 K, in order to create rows of spherical inclusions that cause isotropic small angle neutron scattering [18]. In order to detect the expected temperature variation in the ratio of potassium phases, the neutron scattering was measured at various temperatures [19]. It turned out that the intensity as a function of the scattering angle significantly depend on the measuring temperature up to 1000 K, and the changes are especially marked between 800 and 1000 K. This finding has to be considered as a piece of evidence for a marked change in the volume fraction of the two potassium phases below 1000 K. The re-evaluation of the experimental data is in progress by means of an improved model. Preliminary results suggest that the inclusions will be entirely filled with liquid around 1400 or 1600 K. When we insert these results into the phase diagram, then we can conclude: the twophase inclusions turn to liquid inclusions somewhere around 1400 and 1600 K, and consequently the average potassium density in them should range from 0.6 to 0.5 g/cm3 (see Fig. 1). These preliminary data are in agreement with the results of Wright [34]. The final conclusion is: about one half of the volume of the dope inclusions is filled with solid potassium in heavily drawn and annealed doped tungsten wires, when the annealing temperature is not too high. Snow arrived to a similar conclusion along other lines [31]. 3. Bubble growth at supercritical temperatures The usual treatment takes the potassium content of the individual bubbles as a fitting parameter that has to be derived from the ‘‘equilibrium’’ size that a bubble may attain at high temperatures, where diffusion processes are able to change also its volume [1]. This kind of ‘‘equilibrium’’ is actually a less common type of phase equilibrium, in which the equilibrium is attained between a stress-free matrix and a single spherical bubble that has a fixed mass of potassium, m. In this case the balance of the fluid pressure, PF, and the Laplace pressure, PL, represents the sole isothermal equilibrium condition. For the sake of simplicity, it is also assumed that the surface tension of the W–K interface, c, is isotropic. In order to evaluate the equilibrium radius from a single condition, PF and PL, must depend on a common variable. To this end, the fluid density is expressed by the relation q = 4pmr3/3, where r is the bubble radius [1,24,29]. Thus, the equilibrium bubble radius, a, obeys the relation   4pm 2c ð1Þ P F T o; 3 ¼ PL ¼ ; a a

314

I. Gaal et al. / International Journal of Refractory Metals & Hard Materials 24 (2006) 311–320

where To is the temperature of equilibration. The equilibrium radius can be evaluated also graphically [24,29]. To this end one has to plot the fluid pressure and the Laplace pressure as function of the bubble diameter, r, at constant m, To and c, and the point of intersection of curves PF  r and PL  r gives the equilibrium fluid pressure and the equilibrium radius. It may be of some advantage to use directly the P–q isotherms at the evaluation of the equilibrium size. To this end, we devised an algorithm that uses the potassium density as variable. Since the inclusions preserve their potassium mass, their individuality and they are not in diffusional interaction with other non-equilibrium inclusions in the considered model m¼

4p 3 4p 3 ri qi ¼ rq 3 3

ð2Þ

and  1=3 2c 2c ri i q PL ¼ ¼ ¼ PL . r ri r qi

(a)

ð3Þ

Here, the index i refers to an initial condition that may be either an equilibrium state at a reference temperature or an arbitrary non-equilibrium initial condition. Consequently, the initial state has two independent parameters in this algorithm. One has two equivalent choices for them: (i) potassium mass and initial diameter 2ri, or (ii) initial density qi. and initial diameter 2ri. We shall characterise the initial state with qi and 2ri. When PL and PF is plotted as a function of q for a given initial condition (P iL , qi), then the point of intersection of the curves PL(q) and PF(q) gives equ the equilibrium parameters (qequ, P equ F and P L ), while the equilibrium radius is given by the relation  i 1=3 P iL q a ¼ ri equ ¼ ri equ . ð4Þ PL q When one plots several PF–q isotherms on the same graph, one gets a proper insight into the evolution of potassium density and bubble radius (1/PL) as a function of the equilibration temperature also for bubbles that start from a non-equilibrium initial state. Fig. 2a depicts the proposed algorithm for bubbles filled with (a hypothetical) ideal potassium gas, although the selected density range is compatible with the expected potassium density of the dope phase. Fig. 2b visualises the same algorithm with potassium isotherms that were derived from experimental data. To this end, we have made use of the fact that the T–P–q relations obey the law of corresponding states for K, Rb and Cs, and the three critical parameters are the scaling parameters [20]. The data points for potassium below 20 MPa and 0.3 g/cm3 were evaluated with this scaling on the basis of the Cs data of [25]. The other data points were obtained by means of the rule of straight isocores P ðT ; qÞ ¼ cðqÞðT  T o Þ þ P ðT o ; qÞ.

ð5Þ

(b) Fig. 2. Depicts the PF(q) fluid pressure isotherms and the PL(q) Laplace pressures at various temperatures and initial conditions, respectively. The upper diagram is constructed for an ideal gas, and the lower one is constructed from experimental data points. The numbers at the PL(q) curves give the initial diameters of the spherical inclusions at an average potassium density of 0.6 g/cm3. The tension of the K/W interface was taken as 2 J/m2, as it was recommended in [24]. (The choice of the symbols for the experimental data is explained in the text.)

This rule is valid in a quite broad parameter region at high temperatures [35], because according to general thermodynamic relations ocðT ; M=qÞ 1 oC V ðT ; V Þ ¼ oT T oV

ð6Þ

and the heat capacity at constant density, CV, depends only very slightly on the molar density (or on the molar volume V) in any elementary phase above the Debye temperature [35]. The pressure at densities higher than 0.3 g/cm3 was evaluated according to Eq. (6) from the Rb data of [27].

I. Gaal et al. / International Journal of Refractory Metals & Hard Materials 24 (2006) 311–320

In Fig. 2a the data points of each isotherm were connected by spline interpolation, although various fitting formulae for the equation of state of dense alkali fluids have been recently published [36,37]. The first message of Fig. 2 is as follows. When one seeks to get equilibrium bubble sizes that are larger then the initial, non-equilibrium bubble size, then both the initial average potassium density and the initial Laplace pressure must be sufficiently high. With respect to the initial conditions we have to distinguish between two scenarios. • When an equilibrium bubble size is attained at a certain temperature, then the subsequent equilibrium bubble size increases or decreases with increasing or decreasing equilibration temperature, as it is described in any treatment that considers solely equilibrium bubbles [1,24,29]. • Fig. 2 draws, however, attention also an other scenario. When the initial potassium density is low, then the bubble will markedly shrink even at the ‘‘common’’ bubble sizes (i.e. Laplace pressures), if the equilibration temperature is relatively low (e.g. at 2200 K). The reason is evident: the fluid pressure must be higher than the Laplace pressure already in a non-equilibrium initial state at the start of the equilibration, in order to obtain bubble growth. (The opposite situation evidently leads to bubble shrinkage, when the rates of the suitable diffusion processes are not too low.) In addition, the bubble size would grow continuously upon successively higher and higher equilibration temperatures solely in such a case, in which the equilibrium is really attained at the first temperature of equilibration. This process may be very time consuming, when the applied equilibration temperature is low, and the non-equilibrium bubbles may shrink and grow also at the expense of each other due to their diffusional interaction. In an extreme case, the total volume may remain even constant in this Ostwald ripening type coarsening of a bubble population (see Section 4.4). The second message of Fig. 2 concerns the quantitative predictions. The ideal gas model and the dense fluid model predict quite similar equilibrium bubble sizes for the fine bubbles (ri = 33 nm), while they predict very different equilibrium sizes, when the initial size is coarser (ri = 100 nm). This difference relies on a quite general behaviour of the compressibility factor ZðT ; qÞ ¼

PM ; qRB T

ð7Þ

where RB is the universal gas constant and M denotes the molar mass of potassium. The crucial points are as follows [35]. Z is much lower than 1 along a supercriticial isotherm at moderate densities in the critical density region (Fig. 2b). Thereafter, it increases continuously with increasing density and attains values close to 1 at a limited range of intermediate densities. This region is the well-known Boyle region, where the ideal gas law has an apparent transient

315

validity. Of course, Z increases to higher and higher values with increasing density above the Boyle region. 4. Potassium inclusions at subcritical temperatures 4.1. Width increase of elongated inclusions in the preheating period It is well established that the spherical dope inclusions get elongated and narrower upon swaging and wire drawing. There is a broad range of processing parameters at which the inclusion width, w, decreases proportionally to the actual diameter, d, of rods or wires [1,21,24] wðdÞ=wðDÞ ¼ d=D;

ð8Þ

where D denotes a reference diameter. However, this scaling law is markedly violated, when too high working temperatures are applied [1,2,14,21,24]. In order to clarify the reasons of this important discrepancy, Briant devised and conducted a series of well-controlled swaging, rolling and wire drawing experiments and published also the crucial process parameters [1,21,24]. Let us restrict our discussion to drawing experiments [1,21,24], in which swaged rods having a diameter of 3.4 mm were drawn at first to a diameter of 2.67 mm, while a final size of 1.19 mm was achieved in several consecutive drawing steps. The drawing speed varied from 6.1 to 7.1 m/ min, and the temperatures of preheating ranged from 1323 to 1073 K in the three variants of the applied drawing schedules [22]. The average width of the elongated intergranular potassium inclusions was 20 nm at the final rod sizes and the width of individual inclusions ranged from 10 nm to 30 nm. The nature of deviation from the scaling law can be characterised by the following figures: d=D ¼ 2:67 mm=1:19 mm ¼ 2:24; waverage ðdÞ=waverage ðDÞ ¼ 29 nm=21 nm ¼ 1:38; wmax ðdÞ=wmax ðDÞ ¼ 47 nm=29 nm ¼ 1:62; wmin ðdÞ=wmin ðDÞ ¼ 11 nm=13 nm ¼ 0:85. These figures have an important message: the observed deviation should depend also on the width of the individual inclusions. What should be the reason of this effect? Since our inclusions are far from each other with respect to their size, the observed size effect cannot find any explanation in the framework of the classical continuum theory of plasticity, as this theory does not have any internal length scale [39]. Since diffusion processes and capillary effects have natural length scales also in their common continuum description, it is at hand to seek an explanation in this domain at first [1,24]. In this context, it is of primary importance that the rod passes the drawing die in a very short time with respect to the preheating period that the rod has to spend before the entry into the die. Therefore, one may suspect that a cumulative effect governs the

316

I. Gaal et al. / International Journal of Refractory Metals & Hard Materials 24 (2006) 311–320

observed deviations and the essential processes evolve in the preheating periods. In other words: the inclusion width will evolve according to the following scenario. The deformation process itself obeys the scaling law. However, the inclusions expand in the preheating period before entering into the die of the next deformation step. The extent of expansion depends on the actual width of the individual inclusions according to the nature of the capillary effects and the required diffusion processes. Consequently, there are two effects that effect the drawing in the next die: (i) the actual inclusion width will be wider at the start of the next deformation step, than it would correspond to the previous deformation process alone and (ii) this widening is size dependent. When the effect of the preheating period is not negligible with respect to the deformation induced narrowing in the next drawing die, then a marked deviation from the scaling law may evolve in the course of a series of consecutive drawing steps. In short terms: the observed size effect has nothing to do with the plastic deformation itself. It can be considered as a peculiar effect of annealing, since the extent of plastic strain is virtually zero in the preheating section. Consequently, it would be somewhat misleading to speak on size sensitive inclusion deformation in context of the described size effect. In order to prove the described explanation, Briant conducted also annealing experiments [1,24]. As-drawn rods produced by the experimental drawing procedure were heat treated at 1523 and 1693 K. The average width of the ellipsoid was 30 nm in the starting condition, and it increased in two minutes to 64 nm and 62 nm at 1523 and at 1693 K, respectively. One may, therefore, conclude: the processes leading to the observed deviation from the scaling law [24,38] can evolve also in the preheating periods [1,24]. Briant explained the observed annealing effect as follows [1,24]. The inclusions are filled with potassium vapour at the temperature of the studied processes. This vapour will exert a pressure on the K/W interface, when the temperature is sufficiently high. When one assumes that the Laplace pressure is low, and it cannot balance the vapour pressure, the width of the inclusion will increase, provided that the rate of a suitable diffusion process is high enough. This idea is quite general and can be applied also for two-phase inclusions, since the fluid pressure in them is equal to the equilibrium vapour pressure. Unfortunately, the Laplace pressure in the present case turns out to be higher than the vapour pressure of the liquid–vapour equilibrium, that is also the upper limit for any vapour pressure at a prescribed temperature. Therefore, we have to conclude that the assumed driving force for an expected expansion does not exist. In contradiction with the expectation, the inclusions should get narrower under the effect of the high Laplace pressure until an equilibrium size is attained in the liquid region. This scenario has a single crucial point: how high is the Laplace pressure with respect to the equilibrium pressure of the two-phase region. The actual figures are as follows. The vapour–liquid equilibrium pressure of potassium vapour is 2.2 and 4.0 MPa at 1523 and

1693 K, respectively [28]. Let us show that this fluid pressure is an order of magnitude lower than the lowest limit of the expected Laplace pressure. In order to obtain this lower limit, let us consider only the largest width (wmax = 60 nm) that was measured at the start of the heat-treatment, as the lowest Laplace pressure is encountered in the largest bubbles. In order to relate to inclusion width to the radius of curvature, let us consider two limiting cases. At first, let us assume that the interfacial tension of the K/W interface amounts merely to 1 J/m2. (This very low interfacial tension has been measured on the W(solid)/ Sn(liquid) interface [40].) In this case, the shape of an elongated inclusion can be approximated with two cylindrical cups in its middle part. The radius of curvature of these cups is equal to the measured inclusion width, when the surface energy of the grain boundary is 1 J/m2, as the dihedral angle at the intersection of the grain boundary and interface is 120 in this case. Consequently, the Laplace pressure of this model is equal to c/w = 16.7 MPa. We considered also an other limiting case, in which interfacial tension of the K/W interface was high with respect to the surface energy of the grain boundaries. In this case the relevant radius of curvature is w/2, and the Laplace pressure amounts to 66.6 MPa, when c is taken as 2 J/m2, as it was recommended by Briant [24]. On the basis of these estimates we have to conclude: vapour filled bubbles (including two-phase inclusions) should shrink at the inclusion sizes of the quoted experiments. In this context, let us note that the trials with very low interface energies cannot help, because potassium will wet the grain boundaries at them, and one cannot rationalise the spherical shape of the intergranular potassium inclusion observed in the transmission electron micrographs taken on thin foil prepared from swaged rods that were annealed for 45 h in dry hydrogen [30]. 4.2. The fluid pressure in elongated potassium inclusions Of course, one may apply the basic idea of Briant with success to liquid inclusions, provided that their volume is compressed upon swaging and drawing in such an extent that the liquid pressure in them will become higher than the Laplace pressure. We claim that this kind of assumption is in accordance with the predictions of the classical continuum theory of plasticity for the following reasons. First of all, this theory is able to predict the volume shrinkage of empty voids in suitable modes of plastic deformation in accordance with the experimental findings. For example, the volume of empty voids decreases upon uniaxial compression at 1300 K in molybdenum in accordance with numerical predictions of a suitable continuum model even at void volume factions below 1% [41–43]. In this context, one has, however, to admit that no size limits or volume fraction limits have been devised that should stop the application of these models to empty voids, although such limits should exist in the dislocation mediated plasticity for example for vacancy clusters. In spite of this, one may assume

I. Gaal et al. / International Journal of Refractory Metals & Hard Materials 24 (2006) 311–320

that such limits can be devised for vapour filled inclusions without the explicit use of definite dislocation models, because energy balance argument will lead also to appropriate limits for the volume shrinkage. The idea is as follows. In order to decrease the volume of a vapour filled inclusion ensemble, work has to be done against the vapour pressure. The inclusion volume should shrink, as long as this work is sufficiently low with respect to the work that is dissipated in the course of plastic deformation. Therefore, we may expect that the volume of the two-phase inclusions will shrink as long as the total amount of potassium is transformed to liquid. This believe is based on the following figures. The volume fraction of the dope phase is low, and the highest possible vapour pressure in the twophase region (the critical pressure of potassium (15 MPa)) is also low with respect to the expected yield stress, because the material is strain hardened, and the strain rates are high at drawing and swaging. Therefore, one may suspect that two-phase inclusion will be filled with liquid potassium at a critical plastic strain, whenever compressive deformation modes (like swaging or wire drawing) are applied. Of course, the energy balance argument will allow also an additional decrease in the inclusion volume upon which the liquid pressure should markedly increase. For example, a relative decrease of 10% in the inclusion volume, will increase to liquid pressure to 60 MPa at 1590 K, while a similar change of 6.8% will increase the pressure to 66 MPa at 1260 K, when one uses the Rb data of [27]. These figures may appear as strange with respect to the compressibility of more common liquid phases (like oxides) in more common matrixes. The figures given for the W–K system concern a less-common situation for the following reasons. A given pressure increase requires much lower volume change in a liquid, the melting point of which is much higher than that one of potassium, because the compressibility scales with the melting point of the corresponding solid. This effect should be also stronger, when the temperature of the process would not be so far from the melting point of the fluid as it is in the case of tungsten and potassium. In addition, the yield stress of the matrix is also quite high, partly because the elastic constants of tungsten are high due to its high melting point, and partly because also the effect of strain hardening is marked at the rod diameters of the quoted experiments [30]. 4.3. Shape and volume change of liquid inclusions upon drawing Of course, one has to test the relations between the Taylor model of the globally homogeneous plastic deformation and the extent of the expected volume compression of a liquid inclusion upon the plastic deformation of its metallic matrix. To this end let us give a short description of the classical Taylor model. Taylor assumed that the grains of a polycrystals or plastic inclusions of a matrix will follow the global plastic shape change of the body at least in a first approximation. This approximation relies on the following

317

principle. The stress field in the matrix will not be governed solely by the external load, it is effected also by the requirement of minimal energy dissipation. Consequently, the shape change of the grains and the inclusions should give rise to as low gradients in the plastic strain rate as possible. Detailed studies prove that the deviations of the Taylor model are usually quite moderate. The mathematics of the Taylor model is simple. When a cylindrical body is subject of homogeneous axial plastic elongation, then co-ordinates of each volume element obey the relations dx=dt ¼ xb;

dy=dt ¼ yb

and

dz=dt ¼ 2zb;

ð9Þ

where the z axis is the axis of the body, and the origin of the co-ordinate system is its mass centre. As a direct consequence of Eq. (9), the volume of each element inside the body remains constant. The parameter b can be expressed with the true strain of wire drawing and swaging e ¼ 2. logðD=dÞ ¼ 2bt;

ð10Þ

where D and d denote the initial and the momentary diameter of the body. Since the solution of Eq. (9) is xðtÞ ¼ xð0Þ

d D

yðtÞ ¼ yð0Þ

d D

and

zðtÞ ¼ zð0Þ

D2 ; d2

ð11Þ

a spherical volume element with surface co-ordinates  2  2  2 xð0Þ yð0Þ zð0Þ þ þ ¼1 ð12Þ A A A will obtain an ellipsoidal shape upon deformation, the surface co-ordinates of which obey the relation  2  2  2 xðtÞ yðtÞ zðtÞ þ þ ¼ 1; ð13Þ a a b where A is the initial radius the sphere, a and b denote the half axes of the ellipsoid, respectively, and a = A Æ d/D and b = A Æ [D/d]2. It is an important point that the same transformation is valid also for ellipsoid shaped inclusions evolving in any step of an axial plastic elongation. In other words: the inclusion will follow the following scaling rules upon swaging and drawing: wðdÞ=wðDÞ ¼ d=D

and

3

f ðdÞ=f ðDÞ ¼ ½D=d  ;

ð14Þ

where d and D denote the diameter of the body at two points of the working schedule and w is width and f is the axial ratio of the ellipsoid. If we would take relation (14) at its face value, then the often quoted scaling rule proposed by Moon and Koo [14], according to which f ðdÞ=f ðDÞ ¼ 1:2½D=d3 ;

ð15Þ

ought to be interpreted as a slight deviation from the Taylor model. Of course, one should not rely of this factor of 1.2 with confidence, because the initial state may have also elongated inclusions and the measurement of the aspect ratio is connected also with serious experimental difficulties [1]. We have to ask now, how large deviation from the Taylor model are compatible with experimental results as

318

I. Gaal et al. / International Journal of Refractory Metals & Hard Materials 24 (2006) 311–320

far as the scaling law of the inclusion width is concerned. Of course, the adequate answer to this question can be given only in a dislocation model. Such a model will be published in an other paper [44]. This model, however, supports a very simple continuum description. When the length of an axially oriented inclusion is few micrometers long (as it is definitely the case in doped tungsten upon drawing), then the length co-ordinate follows the scaling rule of the Taylor model, and the volume change affects merely the diameter of the fluid inclusion. Consequently, the volume change upon a rod diameter reduction from d1 to d2 can be expressed as follows:  2 a22 b2 a22 d 1 V2 ¼ ¼ ; ð16Þ a21 b1 a21 d 2 V1 when instead of the ellipsoid width, the half length of the minor axis is used at the evaluation of the inclusion volume, V. Consequently  1=2  1=2 w2 a2 V2 d2 q d2 ¼ ¼ ¼ 1 ; ð17Þ w1 a1 V1 d1 q2 d1 because the mass of potassium in an inclusion does not change upon volume compression. According to Eq. (17), a density increase of 10% will change the scaling constant of the Taylor model from 1 to 0.95, and one can hardly detect such a slight change in experiments in which the width changes upon plastic deformation are measured [1]. We can, therefore, conclude: a slight compression of a liquid phase is compatible with the scaling laws of the Taylor model, since its effects are too slight to be detected by the methods that are applied at the tests of the Taylor model. 4.4. Coarsening of intragranular spherical inclusions upon annealing Schade [45] studied the coarsening of intragranular dope inclusions in heavily drawn wires (d = 0.2 mm) that were annealed at temperatures above 1573 K. Fig. 3 shows the results obtained below the critical temperature of potassium. The elongated inclusions were transformed into rows of spherical inclusions in the very early period of the applied heat-treatments, consequently the results concerned spherical inclusions. The marked increase of the average inclusion diameter below the critical temperature is in contradiction with the assumption that the inclusions are vapour filled or two-phase inclusions. The details are as follows. The average inclusion diameter was in the range of 30 to 50 nm at the ends of the anneals performed at temperatures ranging from 1573 to 2173 K, respectively. The Laplace pressure at these diameters is higher than 166 and 41.5 MPa, for an interfacial tension of 2 J/m2 and 1 J/m2, respectively, according to point 4.1, and one takes into account that the mean curvature for a sphere is twice as large than for a cylinder, if the diameters are equal. In contrast, the highest vapour pressure below 2120 K is less than 20 MPa according to Fig. 2b. Therefore, the vapour filled bubbles and the two-phase

Fig. 3. Variation of the average diameter of intergranular potassium bubbles upon annealing in doped tungsten wires having a diameter of 200 lm. The diagram contains those data of [45] that were obtained at subcritical temperatures on AKS doped samples. The squares denote the average diameter, and the asterisks visualise the width of the diameter distribution.

inclusion ought to shrink also in this experiment. The observed increase in the average diameter, therefore, supports the following view: the inclusions should be filled with potassium liquid also in this situation, and their density should be higher than liquid density corresponding to the vapour–liquid equilibrium in the studied temperature range. The discussion of the details of the size evolution is outside the scope of the present paper. 5. Bubble coarsening close to the critical temperature Schade [45] provided detailed data on the coarsening of the dope phase at temperatures ranging from 1973 to 2373 K. This study has given direct pieces of evidences that the size change of an intragranular potassium inclusion population cannot be described in models that neglect the interaction among the individual inclusions. The crucial finding was: the number of inclusions in unit grain boundary area decreases markedly with time, although the coarsening of the grain structure during the coarsening of the dope phase is negligible. In addition, the evolution of the size distribution was typical for an Ostwald type ripening process, i.e. the smallest size fraction vanished, while the occupation frequency of previously unfilled size classes markedly increased. Since it will be very difficult to rationalise any type of inclusion drag through grain boundary or dislocation migration under the condition of these experiments, it is at hand to rationalise the underlying processes through diffusional interaction within the inclusion ensemble. Since the diffusion distance of the bulk diffusion is very limited in this temperature region, one should consider some type of grain boundary mediated diffusional interaction among the intergranular inclusions. Schade rationalised his results through Ostwald ripening that is mediated through the diffusion of solute potassium along the grain boundaries. The assessment of this model

I. Gaal et al. / International Journal of Refractory Metals & Hard Materials 24 (2006) 311–320

should first of all delineate the conditions of the assumed diffusion process. To this end we should quote first of all the study of Gedgvod, Krasovskiy and Novikov [46], who detected the Cs uptake into the grain boundaries of V, Nb, Ta, Mo and W from Cs gas at an oxygen partial pressure of about 106 mbar in a vacuum chamber by means of autoradiographic techniques. The samples were prepared from zone melted single crystals by rolling and annealing, and they had a coarse grain structure. The Cs uptake were followed up by kinetical studies in tungsten at temperatures ranging from 2183 to 1973 K. In this connection, one should mention that the penetration of refractory metals by alkali metals was reviewed by Klueh [47]. Also his review stresses the presence of oxygen at the grain boundaries increases the alkali solubility of V, Nb and Ta. Of course, one may suspect that the potassium mediated Ostwald ripening is simultaneous with an other diffusion process. The reason is as follows. When the fluid pressure and the Laplace pressure are different in a population of single phase inclusion, then also the matrix atoms from the K/W surface are transported through interfacial and grain boundary diffusion from one inclusion into the other. The competition of this process and the potassium mediated ripening requires additional studies. 6. Discussion This work stressed the importance of liquid potassium inclusions in the rationalisation of the annealing processes that take place at subcritical temperatures. It tried to remain within the framework of the continuum aspect at the rationalisation of the phenomena. The effects of the plastic deformation on the shape and volume change of the dope inclusions were solely discussed by means of arguments that tried to use continuum concepts. This framework has definitely its limit, due to the very fine size of the inclusions. In addition, the effect of the internal stresses on the capillary effects has been entirely neglected, although they may play a marked effect in two-phase systems with elongated fibres [48]. Also the dihedral angles at the common lines of K/W interface and a grain boundary segment may be markedly affected by internal stresses, when one takes into account the finite width of the boundaries [49]. The discussion of these effects should be the aim of an other paper. 7. Conclusions (1) There are independent pieces of experimental evidence that the dope inclusion are filled with liquid potassium in heavily drawn lamp grade tungsten wires and rods. (2) A slight volume compression of liquid inclusion upon wire drawing is compatible with the scaling laws of the Taylor model in a sufficient approximation, because the expected deviations are less than the accuracy of the required measurements.

319

(3) The size increase of the inclusion cannot be explained as the expansion of individual bubbles that do not interact with each other. One has to take into account also Ostwald type ripening processes. References [1] Bewlay BP, Briant CL. Int J Refract Met Hard Mater 1995;14: 137. [2] Yamazaki S, Ogura S, Fukazawa Y, Hatae N. High Temp High Pressures 1978;10:329. [3] Pugh JW, Lasch WA. Metall Trans 1990;21A:2203. [4] Tanaoue K, Matsuda H. Scripta Metall Mater 1995;33:1469. [5] Tanaou K, Okad M, Matsuda HJ. Jpn Inst Metals 1994;58:1103. [6] Fujii K, Ito Y, ITo S, Tanou K. Nippon Tungsten Rev 2002;34:31. [7] Vukcevich MR, Pugh JR, Briant CL. In: Proceedings of the tungsten conference, St. Sebastian Spain. Shewsbury: MPR Publishing; 1992. [8] Briant CL, Walter JL. Acta Metall 1988;36:2503. [9] Zilberstein G. Kneringer, et al, editors. In: Proceedings of the 14th international plansee seminar, vol. 4. Plansee AG, Reutte 1997. p. 100–10. [10] Dawson WC. Metal Trans 1972;3:3103. [11] Horacsek O, To´th CsL, Nagy A. Int J Refract Met Hard Mater 1998;16:51. [12] Schade P. Int J Refract Met Hard Mater 1998;16:77. [13] Schade P. Int J Refract Met Hard Mater 2002;20:301. [14] Moon DM, Koo RC. Metall Trans 1971;2:2115. [15] Briant CL, Horacsek O, Horacsek K. Metall Trans 1993;24A:843. [16] Denissen CJM, Liebe J, Van Rijswick M. Int J Refract Met Hard Mater, this volume, doi:10.1016/j.ijrmhm.2005.10.012. [17] Harmat P, Bartha L, Grosz T. Int J Refract Met Hard Mater 2002; 20:295. [18] Len A, Harmat P, Pepy G, Rosta L, Schade P. J Appl Cryst 2003;36: 621. [19] Len A, Harmat P, Pepy G, Rosta L. Appl Phys A 2002;74:S1418. [20] Hensel F, Stolz M, Hohl G, Go¨tzlaff WR. J de Phys IV 1991;C5/: C1–191. [21] Briant CL. Bildstein H, Eck R, editors. In: 13th International plansee seminar, vol. 1. Plansee Metall AG Reutte, 1993. p. 321–35. [22] Browning PF, Briant CL, Knudsen BA. Bildstein H, Eck R, editors. In: 13th International plansee seminar, vol. 1. Plansee Metall AG Reutte, 1993. p. 336–51. [23] Milman YuV, Zakharowa NP, Ivanshenko RK, Freeze NI. In: Kneringer et al, editors. 14th International plansee seminar, vol. 1. Plansee AG Reutte, 1997. p. 128–47. [24] Briant CL. Metall Trans 1993;24A:1073. [25] Hensel F, Ju¨ngst S, Knuth B, Uchtmann H, Yao M. Physica 1986; 139&140B:90. [26] Shpilrayn EE, Yakumovitch KA, Moegoy AG. Teplofizika Bisokiy Temperatur 1976;14:511. [27] Pfeifer HP, Freyland W, Hensel F. Ber Bunsen-Gesselschaft Phys Chem 1979;83:204. [28] Freyland WF, Hensel F. Berichte der Bunsen-Gesellchaft 1972;76:16. [29] Nagy A. Int J Refract Met Hard Mater 1998;16:45. [30] Wronski A, Fourdeux A. J. Less-Common Metals 1965;8:149. [31] Snow DB. Metall Trans 1974;5:2375. [32] Menyhard M. Surf Interface Anal 1979;1:175. [33] Kele A, Menyhard M. Acta Phys Hung 1980;49:127. [34] Wright PK. Metall Trans 1978;9A:955. [35] Scott LR. In: Henderson D, editor. Physical chemistry. Liquid state, vol. VIIIA. New York/London: Academic Press; 1971. p. 1–261. [36] Ghatee MH, Sanchooli M. Fluid Phase Equilib 2003;214:197. [37] Rah K, Eu BCh. J Phys Chem 2003;B107:4382. [38] Yamazaki Sh. In: Pink E, Bartha L, editors. The metallurgy of doped/ non-sag tungsten. London New-York: Elsevier Applied Science; 1971. p. 47–59.

320

I. Gaal et al. / International Journal of Refractory Metals & Hard Materials 24 (2006) 311–320

[39] Needelman A. Acta Mater 2000;48:105. [40] Boer F, Boom B, Mottens WCM, Miedama AR, Niessen AK. Cohesions in metals. Amsterdam, Oxford, New York, Tokyo: North-Holland; 1988. p. 628. [41] Parteder E, Riedel H, Sun D-Z. Int J Refract Met Hard Mater 2002; 20:287. [42] Parteder E, Riedel H, Kopp R. Mater Sci Eng A 1999;264:17. [43] Segurado J, Parteder E, Planksteiner AF, Bo¨hm HJ. Mater Sci Eng A 2002;333:270.

[44] Gaal I. Scripta Mater, submitted for publication. [45] Schade P. Plansee Berichte fu¨r Pulvermetallurgie 1976;24:243. [46] Gedgovd KN, Krasovskiy AI, Novikon SM. Phys Met Metallog 1980;50:185. [47] Klueh RL. In: Darley JE, Weeks JR, editors. Corrosion by liquid metals. New York: Plenum Press; 1970. p. 177. [48] Sridhar N, Rickman JM, Srolovitz DJ. Acta Mater 1997;45:2715. [49] Genin FY, Mullins WW, Wynblatt P. Acta Metall Mater 1993;41: 3541.