Modeling of Nano-Tungsten Sintering Data 

by William S. de Rosset 

ARL-CR-0665 April 2011 

prepared by: 

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Army Research Laboratory 

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ARL-CR-0665 April 2011 

Modeling of Nano-Tungsten Sintering Data 

William S. de Rosset 

Weapons and Materials Research Directorate, ARL 

prepared by: 

Dynamic Science, Inc. 

8433 Black Canyon Hwy. 

Phoenix, AZ 85021 

under contract 

W911QX09-0057 

Approved for public release; distribution is unlimited. 
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Modeling of Nano-Tungsten Sintering Data 

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14. ABSTRACT 

Small metal samples have been made from nano-crystalline tungsten powder by sintering with the goal of producing high 

density parts. The parameters associated with sintering runs have been examined to see if there is an empirical relationship 

between them and the final part density. A function that depends on the time the sample spends above a critical temperature 

has been formulated that represents one specific batch of tungsten powder. The function has been applied to other batches of 

powder with limited success. An attempt was made to use this function to arrive at the sintering parameters that would give a 

final density in excess of 95% that of theoretical density while retaining a nano-crystalline grain structure. 

15. SUBJECT TERMS

sintering theory, nano-tungsten, grain size 

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William S. de Rosset 

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iii

Contents 

List of Figures iv

List of Tables iv

Acknowledgments v

1. Introduction 1

2. Sintering Models 2

3. MS Sintering Model 3

4. Discussion 6

5. Summary 10

6. References 12

Appendix. Grain Size Measurement 13

List of Symbols, Abbreviations, and Acronyms 15

Distribution List 16 
iv

List of Figures 

Figure 1. Comparison of data and functional fit for T = 1400 °C and To = 860 °C. .......................5

Figure 2. Comparison of data and functional fit for T = 1300 °C and To = 860 °C. .......................6

Figure 3. Comparison of data and model for T = 1400 °C, To = 860 °C, Wn-MT-35. ...................7

Figure 4. Comparison of data and model for T = 1400 °C, To = 860 °C, mill run N6. ...................8

Figure-A 1. Micrograph of sample SST-51-P4. ............................................................................14

List of Tables 

Table 1. Sintering run data for Wn-MT-34. ....................................................................................4

Table 2. Sintering parameters for mill run Wn-MT-35. .................................................................7

Table 3. Predicted fraction of fully dense W. .................................................................................9

Table 4. Values of t and T from model predictions. ....................................................................10
v

Acknowledgments 

The author is indebted to Eric Klier and Brady Butler for many helpful discussions on sintering 

theory and data interpretation. They were the ones responsible for generating the data found in 

this report. Thanks also go to James Catalano for providing a final polishing step that brought 

out the grain structure in the micrographs of the nano-crystalline tungsten and made the grain 

size determination more accurate. Finally, the author is grateful for the many micrographs taken 

by Bradley Klotz. 

 
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

Sintering is a process which can be used to produce useful parts from metallic powder. It is 

accomplished by first compacting the powder and then heating the compacted form (greenware) 

until the powder is fused together. This process is currently being used in an attempt to produce 

tungsten penetrators from nano-crystalline tungsten powder. One approach that has been used at 

the U.S. Army Research Laboratory (ARL) is the Plasma Pressure Compaction method (1, 2). 

This report deals with the conventional approach of simply heating the sample in a hydrogen 

atmosphere. The primary goal is to produce large parts that are fully dense but retain a nanocrystalline grain structure. It is surmised that if this can be achieved, the nano-tungsten 

penetrators will exhibit the same high-rate shear behavior during penetration as that of depleted 

uranium-3/4% titanium penetrators. This type of behavior produces superior penetration 

performance (3). 

For the Depleted Uranium Replacement Program (DURP), the source of tungsten powder feed 

stock with a low (~1%) concentration of oxygen was the Chongyi Zhangyuan Tungsten Co., Ltd. 

Currently, this powder is wet-milled to break the agglomerated powder into finely-divided 

particles. The wet-milled slurry is then spray dried to remove the solvent. A compact of the 

powder is formed either by cold iso-static pressing or die pressing. Next, the compact is heated 

in vacuum to burn out contaminants. After heating the sample in a hydrogen atmosphere to 

remove the small amount of remaining oxygen, it is then heated to the sintering temperature at a 

given rate and held there for a given period of time. Clearly, the process of producing metal 

parts from powder requires many steps and processing variables. 

The challenge to producing a fully-dense, nano-crystalline tungsten penetrator is that the 

sintering process leads to a larger grain size as the compacted powder is heated. It would be 

advantageous if the sintering process could be modeled and the parameters governing the process 

could be chosen to optimize the final density while at the same time keeping the grain size as 

small as possible. In section 2, it will be argued that the sintering process is relatively 

complicated, and no acceptable first-principle models of the entire process exist. Some progress 

has been made in establishing semi-empirical models, and the Master Sintering Curve (MSC) 

will be presented and discussed. In section 3, an alternative approach to the MSC is offered. 

This material-specific (MS) model takes a given powder and uses the sintering parameters of 

several different runs to fit a rationalized analytic function. In section 4, the advantages and 

limitations of the model are discussed. An attempt is made to use the MS model to guide the 

choice of sintering parameters to produce the greatest density while limiting grain growth. 

Section 5 summarizes the modeling effort. 
2

2. Sintering Models 

Sintering is such an important commercial process that it has been investigated by the scientific 

community for decades. Most of the fundamental understanding of the process has come 

through the study of two spherical particles that are placed in contact. Three stages of sintering 

have been identified, with the importance of surface diffusion, grain boundary diffusion, and 

volume diffusion being different in each of the stages in chapter 3 of German (4). Some progress 

in modeling the sintering process has been made with computer models that take into account 

more than one form of mass transport. However, most models do not consider the complexity of 

the sintering process. For instance, powders are not made up of single-size, single-component 

spherical particles as assumed by many models. There may be materials added in small 

quantities to assist in the sintering process that are not accounted for. The same can be said of 

small amounts of contaminants. The sintering process is often carried out in a reducing or 

oxidizing atmosphere, generally neglected by most models. Many models assume a rapid 

heating rate that is not possible to impose on the sample without thermal shock and cracking in 

chapter 1 of German (4). 

Alternatively, one can adopt an empirical or semi-empirical approach to modeling the sintering 

process. One such model is the MSC (5). This model relates the shrinkage rate during sintering 

to a function of surface energy γ, absolute temperature T, mean grain diameter G, atomic volume 

Ω, Boltzmann’s constant k, the width of a grain boundary δ, coefficients for grain-boundary and 

volume diffusion (Db and DV, respectively), and two scaling parameters (Гb and ГV). The 

expression can be simplified by assuming only one bulk transport mechanism, either Do = DV or 

Do = δDb. In this case, the scaling parameters, determined experimentally, collapse into one 

value, Г. Then, 

1 ( ) exp( ) 3 ( ( ))

o

n

d Q D

dt kT G RT

  

 

 = - , (1) 

where ρ is the density, n = 3 or 4 depending on the transport mechanism, Q is the apparent 

activation energy, and R is the gas constant. After further manipulation, equation 1 can be 

integrated by separating variables. (This mathematical operation is somewhat questionable, 

since it assumes that G and Г are independent of time. However, both these parameters depend 

on density, which is a function of time as shown in equation 1.) The contention then is that an 

MSC can be generated by conducting a series of experiments that measure the dimensional 

changes at different heating rates. This information then gives the density as a function of the 

sintering parameters. 

Su and Johnson (5) point out the limitations of the MSC. First, it should only be applied to 

powder compacts made from the same powder and same green-body processing. Second, their 

assumption that Г depends only on density has not been generally shown to be true. Third, it 
3

may be that more than one diffusion mechanism is operating during the sintering process. 

However, the authors state that deviations from a single mechanism can be readily observed. 

Butler et al. (6) have successfully applied the MSC approach to the sintering of nano-crystalline 

tungsten. Dilatometer measurements were made at three different heating rates (3 °C/min, 

5 °C/min, and 25 °C/min) for each of three materials with different thermal stabilities. These 

data were then used to generate the MSCs. These curves give the material density as a function 

of the sintering parameters from the initial sintering stage (~40% theoretical density) to the final 

stage (~90% theoretical density). 

3. MS Sintering Model 

An alternative approach to modeling was sought that would try to capture the effect of the 

important sintering parameters in an uncomplicated manner. Equation 2 is an example of such a 

simple model: 

 = - t Q kT exp( / ) , (2) 

where Ψ is the degree of sintering, t is the time, Q is an activation energy, k is Boltzmann’s 

constant, and T is the absolute temperature in chapter 4 of German (4) . It is intuitive that a 

longer sintering time can lead to a greater material density (i.e., degree of sintering). However, 

there is certainly a limit to the process. Thus, some form of time limit would be needed to apply 

equation 2 to a real sintering process. Also, Ψ would have to be defined in terms of measurable 

parameters. 

The first step taken in developing a simple model was to examine a representative sintering data 

set. The intention was to see if some analytical form could be used to fit the data. The form 

would be suggested by the important trends in the data. The data set chosen was all the sintering 

runs taken from mill run Wn-MT-34. (Here, Wn stands for nano-tungsten, MT stands for milling 

trial, and 34 identifies which trial.) This was the first mill run that resulted consistently in 

sintered samples with small (~200 nm) grain sizes. In addition, many sintering runs were made 

from the mill run with a wide variation in sintering parameters so that a better sense of what was 

important could be obtained. 

Results from the sintering runs made with the mill run Wn-MT-34 powder are shown in table 1. 

The first column identifies the sintering run, and the second column shows the hydrogen 

reduction step. The third column shows the sintering temperature and the fourth column the 

heating and cooling rates. Typically, the sample was cooled at twice the heating rate. The 

holding time, HT, at the sintering temperature is shown in the fifth column. The percent 

theoretical density was determined by weighing the sample in air and water and using 19.3 g/cm3

as the theoretical density. The grain size is shown in the last column. The appendix explains 
4

how this value was determined. In the last entry in the table, the heating cycle was not 

completed, so the grain size measurement was not taken. All sintering data were taken from the 

original log entries. Three of the entries in table 1 are marked with an asterisk. These are 

exceptional runs that resulted in a high density (>95% theoretical density) and a small grain size 

(~150 nm or less). 

Table 1. Sintering run data for Wn-MT-34. 

Sintering 

Identification 

Hydrogen 

Reduction 

Sintering 

Temperature 

°C 

Heating/Cooling

Rate 

°C/min 

Sintering 

Time 

Theoretical

Density 

(%) 

Grain

Size 

(nm)

Wn-SST-49-p1* 3 h at 700 °C 1400 5/10 1 s 95.15 143 

Wn-SST-50-p2 3 h at 700 °C 1300 5/10 1 s 90.2 132 

Wn-SST-50-p3 3 h at 700 °C 1300 5/10 1 s 89.12 128 

Wn-SST-51-p4* 3 h at 800 °C 1400 5/10 1 s 95.77 147 

Wn-SST-51-p5* 3 h at 800 °C 1400 5/10 1 s 96.24 128 

Wn-SST-52-p8-1 3 h at 700 °C 1400 25/50 1 s 89.73 124 

Wn-SST-52-p8-2 3 h at 700 °C 1400 25/50 1 s 88.19 108 

Wn-SST-53-p8-3 3 h at 700 °C 1400 25/50 30 min 91.29 838 

Wn-SST-54-p8-4 3 h at 700 °C 1300 25/50 1 s 85.88 111 

Wn-SST-55-p7-1 3 h at 700 °C 1400 25/50 4 h 95.90 189 

Wn-SST-56-p7-2 3 h at 700 °C 1300 25/50 4 h 95.49 300 

Wn-SST-57-p6 2.5 h at 700 °C 1400 

7.3 to 1100 °C 

3.7 to 1400 °C 

7.3 to 500 °C 

30 min 96.21 194 

Wn-SST-58-p7-3 3 h at 700 °C 1200 25/50 30 min 84.80 112 

Wn-SST-59-p7-4 3 h at 700 °C 1500 25/50 30 min 96.68 159 

Wn-SST-60-p7-5 3 h at 700 °C 1500 25/50 1 s 95.17 162 

Wn-SST-61-p8-5 3 h at 700 °C 1300 25/50 30 min 87.83 143 

Wn-SST-64-p9-1 3 h at 700 °C 1500 25/50 4 h 95.16 301 

Wn-SST-65-p9-2 3 h at 700 °C 1600 25/50 1 s 94.74 150 

Wn-SST-66-p10 3 h at 700 °C 1600? N/A 30 min 91.69 — 

These data were examined to extract a functional relationship between the final material 

densities to the sintering parameters, if possible. The two major requirements on the relation 

were that it should incorporate all the major controllable sintering parameters (time, temperature, 

and heating/cooling rates) and that it have a plausible basis. The relationship should also 

conform to physical reality in that the final density should not exceed the theoretical density. A 

desirable feature of the function is that it contains as few fitting parameters as possible. 

All the sintering runs for Wn-MT-34 were conducted at a maximum sintering temperature T

equal to or greater than 1200 °C. It was reasoned that it is important to have the sintering 

temperature above a certain value for a reasonable length of time in order to achieve the highest 

final density. Several functions were considered that fit the data to some extent and met most or 

all of the imposed requirements. The one chosen met all the requirements and was the simplest 

in form. Let the percent theoretical density be ρ/ρo and t the time the sample is above a critical 
5

temperature To. A function that describes the data over the observed range of parameters is 

given by 

( )/ / 1 1/( ) TT T o o o   t = - - .

 (3) 

The percent theoretical density approaches 1.0 as the values of t and (T-To)/To become large. 

Here, t is measured in minutes. Note that the time held at the maximum sintering temperature 

(HT) as well as both the heating and cooling rates are needed to determine t. Realistic heating 

and cooling rates, as well as values of T much larger than To, insure that t is large enough to 

avoid negative values of ρ/ρo. 

Data for a sintering temperature of 1400 °C is shown in figure 1, along with the fitted function 

taken from equation 3. A best fit to the data was obtained by letting To be 860 °C. A similar plot 

is shown in figure 2 for the 1300 °C sintering temperature. 

t, min

0 50 100 150 200 250 300 350



0.86

0.88

0.90

0.92

0.94

0.96

0.98

Figure 1. Comparison of data and functional fit for T = 1400 °C and To = 860 °C. 
6

t, min

0 50 100 150 200 250 300 350

o

0.78

0.80

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

Data

Functional Fit

Figure 2. Comparison of data and functional fit for T = 1300 °C and To = 860 °C. 

One sintering run was conducted at 1200 °C. The model predicts the sintered part to have a 

density 79% that of theoretical density, whereas the actual measured value was 84.8%. This 

disagreement may be due to the steepness of the curve for 0 < t < 50. 

4. Discussion 

The fitting function in equation 3 is restricted to one mill run, so its usefulness in predicting the 

results for sintering runs from other mill runs is open to question. However, the general trend of 

other data sets might be represented by the function. To examine this possibility, another mill 

run was selected that had a large number of sintering trials, Wn-MT-35. Each of the sintering 

runs with this milling batch was done at a fast heating rate, interpreted as 25 °C/min, except for 

Wn-SST-68, which used program 5 (heat at 50 h to 700 °C, hydrogen reduction at 700 °C for 

3 h, heat to 1400 °C at 220 °C/h, hold for 3 h, cool at 440 h to 500 °C, cool to room temperature 

at 50 °C/h). The runs had various hold times at the sintering temperature. Table 2 gives the 

relevant parameters for these sintering runs. None of these runs had grain sizes below 150 nm. 

However, the theoretical densities were all equal to or above 93% with only one exception. 

The predicted densities are plotted versus the model in figure 3 for T = 1400 °C and To = 860 °C. 

The data appear to fall near the predicted curve, except for 67-FC6. This run produced a 

substantially lower density sample than the identically-processed 67-FC11. One possible 

explanation for this difference is that a large crack was observed in sample 67-FC11. It is 
7

surmised that this crack occurred during the initial hydrogen reduction phase of the process, 

leading to a higher diffusion of hydrogen into the interior of the sample. This was a large sample 

(22 g as opposed to the usual 3–5 g), and the heating rate was high, so differences from this 

crack would magnify the degree of reduction and hence differences in final densities (7). The 

agreement between the predicted and observed values of the density achieved for sintering runs 

using milling run Wn-MT-35 is encouraging, especially considering that the fitting parameter To

was not changed. 

Table 2. Sintering parameters for mill run Wn-MT-35. 

Sintering 

Identification 

T 

(°C)

HT 

(min)

Measured

ρ/ρo

Predicted

ρ/ρo

Grain Size 

(nm) 

61-3p1-1 1300 30 0.9470 0.877 171 

62-3p2 1400 30 0.9707 0.950 182 

63-3p1-2 1400 30 0.9083 0.928 197 

64-3p1-3 1500 240 0.9564 0.986 275 

65-3p3-1 1600 0 0.9624 0.963 212 

67-FC6 1400 240 0.9081 0.972 194 

67-FC11 1400 240 0.9676 0.972 215 

68-FCp9 1400 30 0.9767 0.970 161 

73-3p3-2 1400 120 0.9679 0.959 181 

t, min

0 50 100 150 200 250 300 350

 o

0.90

0.92

0.94

0.96

0.98

1.00

Functional Fit

Data

Figure 3. Comparison of data and model for T = 1400 °C, To = 860 °C, 

Wn-MT-35. 
8

Sintering trials from a third mill run were examined, given the apparent agreement between the 

model and mill runs Wn-MT-34 and -35. The run chosen was labeled N6. Its source was 

Netzsch, Inc., who used the ZX powder as the starting point. However, processing parameters 

used to produce the powder are not known. A plot of the data and model predictions is shown in 

figure 4. Again, 860 °C was used as To. The fit is very poor, and it may be that the parameter t

is inappropriate for this mill run. 

t, min

50 100 150 200 250 300 350

o

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

Functional Fit

Data

Figure 4. Comparison of data and model for T = 1400 °C, To = 860 °C, 

mill run N6. 

Subsequent mill runs were carried out with the intention of scaling the entire process up to make 

larger specimens. The model was used to predict the final densities resulting from sintering runs 

using those mill run outputs. Little success was achieved, however. This could have been due to 

change in mill process parameters or the fact that the specimens were larger in mass. 

The model can be used to suggest the combination of sintering parameters that gives the highest 

density while retaining a small grain size. We expect that as the sintering time increases, grain 

growth will occur. Therefore, we use the model to predict the minimum value of t necessary to 

achieve 95% theoretical density. 

We first start out by assuming that we use a cooling rate is twice the heating rate, HR 

(°C/min). We also assume a 30 min hold at T. Given a value of T and To = 860 °C, we can 

calculate the value of t. This value can then be used in equation 3 to find the per-cent theoretical 

density of the sample. These calculations were carried out with the results shown in table 3. 

(Note that the highest value of HR is outside the sintering parameter range actually used in this 

study.)
9

From an examination of the values of t, the combination of T = 1500 °C and HR = 30 °C/min 

gives the lowest value of t to reach 95% of theoretical density (t = 62 min). 

Table 3. Predicted fraction of fully dense W. 

HR T 1300 1400 1500 1600 

5 — 0.929 0.965 0.983 0.992 

10 — 0.906 0.950 0.974 0.986 

15 — 0.893 0.940 0.967 0.982 

20 — 0.883 0.933 0.962 0.979 

25 — 0.877 0.928 0.958 0.976 

30 — 0.871 0.923 0.955 0.974 

Table 1 does not show any run at T = 1500 °C that was exceptional. However, there were two 

runs (WN-SST-59-p7-4 and WN-SST-60-p7-5) that achieved densities over 95% theoretical 

density but had slightly larger grain sizes (159 and 162 nm, respectively). Given the uncertainty 

in the grain size measurement, they also might be considered exceptional. Three runs conducted 

at 1400 °C were exceptional. According to the model, the fastest heating rate that produces a 

sample with 95% theoretical density for a sintering temperature of 1400 °C is 10 °C/min. The 

time t in this case is 112 min, or almost twice that for the 1500 °C sintering temperature run. Of 

course, this result would change if the sample were held at the sintering temperature for times 

different from the 30 min used in this example. 

Most of the sintering runs were conducted at a temperature of 1400 °C. Perhaps if more runs had 

been carried out at 1500 °C there would have been more exceptional runs at this sintering 

temperature. However, the model does not support the supposition that just minimizing the time 

above To will result in a small grain size. The maximum sintering temperature must be 

considered also. 

While no definitive statement can be made concerning the maximum sintering temperature, it 

can be brought into the analysis in the following way. First, we select a desired value of ρ/ρo. 

For our example, 0.95 is chosen. This is an arbitrary choice, and the analysis can be carried out 

for other values. In this case, 

0 95 1 1/( ) o o (T T )/T . t = - - , (4) 

from equation 3. Further manipulation yields 

 (( )/ ) ln ( ) ln 20 TTT t o o - = . (5) 

Noting that 

1 5( )/ o t . T T HR HT =- + . (6) 

where HT is the hold time at the maximum sintering temperature, values of t and T can be 

generated for given values of HT and HR. Here, we have assumed a cooling rate that is twice the 
10

heating rate. We arbitrarily set the value of HT to 30 min and examine the relationship among t, 

T, and HR. These results are shown in table 4. 

Table 4. Values of t and T 

from model predictions. 

HR 

(°C/min)

t 

(min)

T 

(°C)

20 74.8 1457

22.5 70.4 1466

25 66.8 1473

27.5 63.8 1480

30 61.3 1486

32.5 59.1 1491

35 57.3 1496

In addition to minimizing t, it is reasonable to assume that small grain growth will occur for a 

minimum value of T. However, according to the model and our assumptions concerning the 

other parameters, t and T cannot be minimized simultaneously. 

Other considerations may dictate the choice of HR. For instance, large samples may need a 

relatively low value of HR so that the entire sample heats and cools at about the same rate. Low 

values of HR can lead to high values of t. In this case, the holding time can be reduced to 

compensate for this large value. 

5. Summary 

A brief review of sintering theory indicated that no first-principles theory was available to model 

this complicated process. Instead, an empirical approach was used to organize sintering data for 

a specific mill run. One functional form that appeared to fit the data for mill run Wn-MT-34 was 

based on the time t the powder sample was held above a critical temperature, To, used as a singlefitting parameter. This mill run was chosen because there were many sintering runs made from 

the same batch of powder. The same function and fitting parameter were used to fit the data 

from another mill run (Wn-MT-35) with reasonable success. However, further attempts to use 

the same function and fitting parameter on other mill runs did not have as good a result. This 

may have been due to a difference in the starting powder or the fact that many of the sintered 

samples examined were much larger than those in mill runs Wn-MT-34 and -35. Even so, it is 

clear that if some attention is paid to keeping the powder processing parameters the same, as well 

as the sample masses, the possibility exists that the final density can be predicted using values of 

the controllable sintering parameters, based on a model for one mill run. 
11

The functional relationship between the sintering parameters and the final density was used in an 

attempt to predict optimum sintering parameters. It was assumed that a very small grain size 

would be achieved with the minimum time necessary to achieve 95% of theoretical density. The 

model predicted that a minimum time t of 62 min would be needed to obtain 95% of the 

theoretical density for T = 1500 °C and a fast heating rate. While this value of T produced two 

cases (Wn-SST-59-p7-4 and Wn-SST-60-p7-5) where the grain sizes were reasonable (159 and 

162 μ, respectively), there were other instances where smaller grain sizes were achieved with a 

lower sintering temperature. By restricting the range of parameters and setting the desired final 

density at a fixed percentage of theoretical density, the relationship among t, T, and HR can be 

found. The results indicated that the value of t and T could not be minimized simultaneously. 
12

6. References 

1. Klotz, B. R.; Kellogg, F. R.; Klier, E. M.; Dowding, R. J.; Kyu, C. C. Characterization, 

Processing, and Consolidation of Nanoscale Tungsten Powder; ARL-TR-5045; U.S. Army 

Research Laboratory: Aberdeen Proving Ground, MD, 2009. 

2. Klotz, B. R.; Kellogg, F. R.; Cho, C. C. Characterization and Consolidation of Tungsten 

Nanopowders Produced by Salt-Assisted Combustion Synthesis; ARL-TR-5311; U.S. Army 

Research Laboratory: Aberdeen Proving Ground, MD, 2010. 

3. Magness, L. S. High Strain Rate Deformation Behaviors of Kinetic Energy Penetrator 

Materials during Ballistic Impact. Mechanics of Materials 1994, 17, 147–154. 

4. German, R. M. Sintering Theory and Practice; John Wiley and Sons: New York, 1996. 

5. Su, Hunghai; Johnson, D. L. Master Sintering Curve: A Practical Approach to Sintering. J. 

Am Cer Soc 1996, 79 (12), 3211–3217. 

6. Butler, B.; Klier, E.; Kelley, M.; Gallagher, M.; Maupin, H. Thermal Stability of Milled 

Nanocrystalline Tungsten Powders; ARL technical report to be published. 

7. Butler, B. Private communication, 2011. 

8. ASTM E 112-96. Annu. Book ASTM Stand. Re-approved 2004. 
13

Appendix. Grain Size Measurement 

Samples were prepared for viewing on the scanning electron microscope by polishing them with 

successively finer grit papers. This continues until the surface is smooth enough to be polished 

with 1-micron (or less) powder. Micrographs were then taken of these samples. The grain 

boundary intersection count was used to determine an approximate grain size. This is done by 

overlaying a grid on the micrograph and counting the number of times the grid lines intersect a 

grain boundary. Dividing the length of the grid by the number of intersection points gives the 

result. (Note: this is really the mean lineal intercept length as defined by American Society for 

Testing and Materials (ASTM) (8). Since there is no direct mathematical relationship between 

the mean lineal intercept length and the grain size number G, we simply use it as an average 

grain size.) 

This method works well when the grain boundaries are easily distinguishable. However, the 

polishing resulted in a somewhat blurry picture of the nano-crystalline microstructure. 

Fortunately, a final polishing step was found that brought out the microstructure of the alreadypolished samples. It consisted of using 0.05-micron colloidal silica in conjunction with chromic 

trioxide (an attack etch). An example of a micrograph of a sample after this final step was 

applied is shown in figure A-1. 

The length of this grid is 14 μ, as determined from the scale marking on the photomicrograph. 

The magnification of this particular micrograph was 50,000×. However, for samples with larger 

grain sizes, a smaller magnification was used. This was done to meet the general requirement to 

have at least 50 intersections on the grid. In this case, the grid size changes also. 
14

Figure-A 1. Micrograph of sample SST-51-P4.
15

List of Symbols, Abbreviations, and Acronyms 

ARL U.S. Army Research Laboratory 

ASTM American Society for Testing and Materials 

DURP Depleted Uranium Replacement Program 

MS material-specific 

MSC Master Sintering Curve 

MT milling trial 

SST Self-Sintering Trial 

Wn nano-tungsten 
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