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Q-learning

Q-learning is a model-free reinforcement learning algorithm to learn the value of an action in a particular state. It does not require a model of the environment (hence "model-free"), and it can handle problems with stochastic transitions and rewards without requiring adaptations.

For any finite Markov decision process, Q-learning finds an optimal policy in the sense of maximizing the expected value of the total reward over any and all successive steps, starting from the current state. Q-learning can identify an optimal action-selection policy for any given finite Markov decision process, given infinite exploration time and a partly random policy. "Q" refers to the function that the algorithm computes – the expected rewards for an action taken in a given state.

Reinforcement learning
Reinforcement learning involves an agent, a set of states $$\mathcal{S}$$, and a set $$\mathcal{A}$$ of actions per state. By performing an action $$a \in \mathcal{A}$$, the agent transitions from state to state. Executing an action in a specific state provides the agent with a reward (a numerical score).

The goal of the agent is to maximize its total reward. It does this by adding the maximum reward attainable from future states to the reward for achieving its current state, effectively influencing the current action by the potential future reward. This potential reward is a weighted sum of expected values of the rewards of all future steps starting from the current state.

As an example, consider the process of boarding a train, in which the reward is measured by the negative of the total time spent boarding (alternatively, the cost of boarding the train is equal to the boarding time). One strategy is to enter the train door as soon as they open, minimizing the initial wait time for yourself. If the train is crowded, however, then you will have a slow entry after the initial action of entering the door as people are fighting you to depart the train as you attempt to board. The total boarding time, or cost, is then:

 * 0 seconds wait time + 15 seconds fight time

On the next day, by random chance (exploration), you decide to wait and let other people depart first. This initially results in a longer wait time. However, less time is spent fighting the departing passengers. Overall, this path has a higher reward than that of the previous day, since the total boarding time is now:

 * 5 second wait time + 0 second fight time

Through exploration, despite the initial (patient) action resulting in a larger cost (or negative reward) than in the forceful strategy, the overall cost is lower, thus revealing a more rewarding strategy.

Algorithm
After $$\Delta t$$ steps into the future the agent will decide some next step. The weight for this step is calculated as $$\gamma^{\Delta t}$$, where $$\gamma$$ (the discount factor) is a number between 0 and 1 ($$0 \le \gamma \le 1$$). Assuming $$\gamma < 1$$, it has the effect of valuing rewards received earlier higher than those received later (reflecting the value of a "good start"). $$ \gamma $$ may also be interpreted as the probability to succeed (or survive) at every step $$\Delta t$$.

The algorithm, therefore, has a function that calculates the quality of a state–action combination:

 * $$Q: \mathcal{S} \times \mathcal{A} \to \mathbb{R}$$.

Before learning begins, $Q$ is initialized to a possibly arbitrary fixed value (chosen by the programmer). Then, at each time $$t$$ the agent selects an action $$A_t$$, observes a reward $$R_{t+1}$$, enters a new state $$S_{t+1}$$ (that may depend on both the previous state $$S_t$$ and the selected action), and $$Q$$ is updated. The core of the algorithm is a Bellman equation as a simple value iteration update, using the weighted average of the current value and the new information:

 * $$Q^{new}(S_{t},A_{t}) \leftarrow (1 - \underbrace{\alpha}_{\text{learning rate}}) \cdot \underbrace{Q(S_{t},A_{t})}_{\text{current value}} + \underbrace{\alpha}_{\text{learning rate}} \cdot \bigg( \underbrace{\underbrace{R_{t+1}}_{\text{reward}} + \underbrace{\gamma}_{\text{discount factor}} \cdot \underbrace{\max_{a}Q(S_{t+1}, a)}_{\text{estimate of optimal future value}}}_{\text{new value (temporal difference target)}} \bigg) $$

where $$R_{t+1}$$ is the reward received when moving from the state $$S_{t}$$ to the state $$S_{t+1}$$, and $$\alpha$$ is the learning rate $$(0 < \alpha \le 1)$$.

Note that $$Q^{new}(S_t,A_t)$$ is the sum of three factors:

 * $$(1 - \alpha)Q(S_t,A_t)$$: the current value (weighted by one minus the learning rate)
 * $$\alpha \, R_{t+1}$$: the reward $$R_{t+1}$$ to obtain if action $$A_t$$ is taken when in state $$S_t$$ (weighted by learning rate)
 * $$\alpha \gamma \max_{a}Q(S_{t+1},a)$$: the maximum reward that can be obtained from state $$S_{t+1}$$(weighted by learning rate and discount factor)

An episode of the algorithm ends when state $$S_{t+1}$$ is a final or terminal state. However, Q-learning can also learn in non-episodic tasks (as a result of the property of convergent infinite series). If the discount factor is lower than 1, the action values are finite even if the problem can contain infinite loops.

For all final states $$s_f$$, $$Q(s_f, a)$$ is never updated, but is set to the reward value $$r$$ observed for state $$s_f$$. In most cases, $$Q(s_f,a)$$ can be taken to equal zero.

Learning rate
The learning rate or step size determines to what extent newly acquired information overrides old information. A factor of 0 makes the agent learn nothing (exclusively exploiting prior knowledge), while a factor of 1 makes the agent consider only the most recent information (ignoring prior knowledge to explore possibilities). In fully deterministic environments, a learning rate of $$\alpha_t = 1$$ is optimal. When the problem is stochastic, the algorithm converges under some technical conditions on the learning rate that require it to decrease to zero. In practice, often a constant learning rate is used, such as $$\alpha_t = 0.1$$ for all $$t$$.

Discount factor
The discount factor $\gamma$ determines the importance of future rewards. A factor of 0 will make the agent "myopic" (or short-sighted) by only considering current rewards, i.e. $$r_t$$ (in the update rule above), while a factor approaching 1 will make it strive for a long-term high reward. If the discount factor meets or exceeds 1, the action values may diverge. For $\gamma = 1$, without a terminal state, or if the agent never reaches one, all environment histories become infinitely long, and utilities with additive, undiscounted rewards generally become infinite. Even with a discount factor only slightly lower than 1, Q-function learning leads to propagation of errors and instabilities when the value function is approximated with an artificial neural network. In that case, starting with a lower discount factor and increasing it towards its final value accelerates learning.

Initial conditions (Q0)
Since Q-learning is an iterative algorithm, it implicitly assumes an initial condition before the first update occurs. High initial values, also known as "optimistic initial conditions", can encourage exploration: no matter what action is selected, the update rule will cause it to have lower values than the other alternative, thus increasing their choice probability. The first reward $$r$$ can be used to reset the initial conditions. According to this idea, the first time an action is taken the reward is used to set the value of $$Q$$. This allows immediate learning in case of fixed deterministic rewards. A model that incorporates reset of initial conditions (RIC) is expected to predict participants' behavior better than a model that assumes any arbitrary initial condition (AIC). RIC seems to be consistent with human behaviour in repeated binary choice experiments.

Implementation
Q-learning at its simplest stores data in tables. This approach falters with increasing numbers of states/actions since the likelihood of the agent visiting a particular state and performing a particular action is increasingly small.

Function approximation
Q-learning can be combined with function approximation. This makes it possible to apply the algorithm to larger problems, even when the state space is continuous.

One solution is to use an (adapted) artificial neural network as a function approximator. Another possibility is to integrate Fuzzy Rule Interpolation (FRI) and use sparse fuzzy rule-bases instead of discrete Q-tables or ANNs, which has the advantage of being a human-readable knowledge representation form. Function approximation may speed up learning in finite problems, due to the fact that the algorithm can generalize earlier experiences to previously unseen states.

Quantization
Another technique to decrease the state/action space quantizes possible values. Consider the example of learning to balance a stick on a finger. To describe a state at a certain point in time involves the position of the finger in space, its velocity, the angle of the stick and the angular velocity of the stick. This yields a four-element vector that describes one state, i.e. a snapshot of one state encoded into four values. The problem is that infinitely many possible states are present. To shrink the possible space of valid actions multiple values can be assigned to a bucket. The exact distance of the finger from its starting position (-Infinity to Infinity) is not known, but rather whether it is far away or not (Near, Far).

History
Q-learning was introduced by Chris Watkins in 1989. A convergence proof was presented by Watkins and Peter Dayan in 1992.

Watkins was addressing “Learning from delayed rewards”, the title of his PhD thesis. Eight years earlier in 1981 the same problem, under the name of “Delayed reinforcement learning”, was solved by Bozinovski's Crossbar Adaptive Array (CAA). The memory matrix $$W = \|w(a,s)\|$$ was the same as the eight years later Q-table of Q-learning. The architecture introduced the term “state evaluation” in reinforcement learning. The crossbar learning algorithm, written in mathematical pseudocode in the paper, in each iteration performs the following computation:

 * In state $s$ perform action $a$;
 * Receive consequence state $s'$;
 * Compute state evaluation $v(s')$;
 * Update crossbar value $$w'(a,s) = w(a,s) + v(s')$$.

The term “secondary reinforcement” is borrowed from animal learning theory, to model state values via backpropagation: the state value $v(s')$ of the consequence situation is backpropagated to the previously encountered situations. CAA computes state values vertically and actions horizontally (the "crossbar"). Demonstration graphs showing delayed reinforcement learning contained states (desirable, undesirable, and neutral states), which were computed by the state evaluation function. This learning system was a forerunner of the Q-learning algorithm.

In 2014, Google DeepMind patented an application of Q-learning to deep learning, titled "deep reinforcement learning" or "deep Q-learning" that can play Atari 2600 games at expert human levels.

Deep Q-learning
The DeepMind system used a deep convolutional neural network, with layers of tiled convolutional filters to mimic the effects of receptive fields. Reinforcement learning is unstable or divergent when a nonlinear function approximator such as a neural network is used to represent Q. This instability comes from the correlations present in the sequence of observations, the fact that small updates to Q may significantly change the policy of the agent and the data distribution, and the correlations between Q and the target values. The method can be used for stochastic search in various domains and applications.

The technique used experience replay, a biologically inspired mechanism that uses a random sample of prior actions instead of the most recent action to proceed. This removes correlations in the observation sequence and smooths changes in the data distribution. Iterative updates adjust Q towards target values that are only periodically updated, further reducing correlations with the target.

Double Q-learning
Because the future maximum approximated action value in Q-learning is evaluated using the same Q function as in current action selection policy, in noisy environments Q-learning can sometimes overestimate the action values, slowing the learning. A variant called Double Q-learning was proposed to correct this. Double Q-learning is an off-policy reinforcement learning algorithm, where a different policy is used for value evaluation than what is used to select the next action.

In practice, two separate value functions $$Q^A$$ and $$Q^B$$ are trained in a mutually symmetric fashion using separate experiences. The double Q-learning update step is then as follows:
 * $$Q^A_{t+1}(s_{t}, a_{t}) = Q^A_{t}(s_{t}, a_{t}) + \alpha_{t}(s_{t}, a_{t}) \left(r_{t} + \gamma Q^B_{t}\left(s_{t+1}, \mathop\operatorname{arg~max}_{a} Q^A_t(s_{t+1}, a)\right) - Q^A_{t}(s_{t}, a_{t})\right)$$, and
 * $$Q^B_{t+1}(s_{t}, a_{t}) = Q^B_{t}(s_{t}, a_{t}) + \alpha_{t}(s_{t}, a_{t}) \left(r_{t} + \gamma Q^A_{t}\left(s_{t+1}, \mathop\operatorname{arg~max}_{a} Q^B_t(s_{t+1}, a)\right) - Q^B_{t}(s_{t}, a_{t})\right).$$

Now the estimated value of the discounted future is evaluated using a different policy, which solves the overestimation issue.

This algorithm was later modified in 2015 and combined with deep learning, as in the DQN algorithm, resulting in Double DQN, which outperforms the original DQN algorithm.

Others
Delayed Q-learning is an alternative implementation of the online Q-learning algorithm, with probably approximately correct (PAC) learning.

Greedy GQ is a variant of Q-learning to use in combination with (linear) function approximation. The advantage of Greedy GQ is that convergence is guaranteed even when function approximation is used to estimate the action values.

Distributional Q-learning is a variant of Q-learning which seeks to model the distribution of returns rather than the expected return of each action. It has been observed to facilitate estimate by deep neural networks and can enable alternative control methods, such as risk-sensitive control.

Multi-agent learning
Q-learning has been proposed in the multi-agent setting (see Section 4.1.2 in ). One approach consists in pretending the environment is passive. Littman proposes the minimax Q learning algorithm.

Limitations
The standard Q-learning algorithm (using a $$Q$$ table) applies only to discrete action and state spaces. Discretization of these values leads to inefficient learning, largely due to the curse of dimensionality. However, there are adaptations of Q-learning that attempt to solve this problem such as Wire-fitted Neural Network Q-Learning.

title: Q-learning


text: User talk:66.121.127.94

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:66.121.127.94


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Michael Melchior

Michael Melchior (מיכאל מלכיאור; born January 31, 1954) is a Jewish leader, Orthodox rabbi, thinker, and activist. He is a former Minister of Social and Diaspora Affairs, a former Deputy Minister of Foreign Affairs, and a former member of Knesset for Meimad. He is the Rabbi of a community in Talpiyot, Jerusalem (Beit Boyar), while still holding the title of the Chief Rabbi of Norway.

Biography
A descendant of seven generations of rabbis in Denmark, Melchior was born in Copenhagen, Denmark, in 1954. His grandfather, Marcus Melchior, helped orchestrate the escape of Danish Jews during the Nazi occupation of Denmark, and served as the country's chief rabbi until his death in 1969. Michael was ordained as an Orthodox Rabbi at Yeshivat Hakotel of Jerusalem in 1980. Soon afterwards, he returned to Scandinavia to serve as Chief Rabbi of Norway.

In 1986, he immigrated to Israel, and settled down with his family in Jerusalem, while still holding the honorary title of Chief Rabbi of Norway.

Political career
Melchior entered politics with the Meimad party in 1995. When Rabbi Yehuda Amital was appointed minister without portfolio after the assassination of prime minister Yitzhak Rabin in November 1995, Melchior served as Amital's assistant. Melchior was selected chairman of the managing committee of the Meimad party in late 1995.

In the 1999 elections, Meimad ran as part of the One Israel alliance with the Labor Party and Gesher. Melchior won a seat, and was appointed Minister of Social and Diaspora Affairs on 5 August 1999, a post he held until Ariel Sharon became Prime Minister in 2001. Melchior was re-elected to the Knesset as a member of the joint list in 2003 and 2006, as Meimad continued its alliance with the Labor Party. In 2008, Meimad broke away from the alliance, and ran in partnership with the Green Movement in the 2009 elections, but failed to win a seat. On December 14, 2012, on his Facebook page, he said that he will quit Knesset elections.

Extra-parliamentary political and social activities
As part of his official roles in the Knesset and the Israeli government, Melchior was involved with Birthright Israel, an organization that has brought over 200,000 young Jews to explore their heritage in Israel.

In addition, Rabbi Melchior established a number of civil organizations that focus on issues he dealt with in his parliamentary activities, concerning education, inter-Jewish relations, the environment, economics, and Arab-Israeli peace.

He founded The Yachad Council, which promoted dialogue between religious and secular Jews, and he served as honorary chairman of the Citizens' Accord Forum Between Jews and Arabs in Israel. CAF conducted activities that promoted civic engagement, encouraged active citizenship, and inspired members of the public to play an active role in formulating policies that affect their daily lives and their future. CAF worked to create dialogue and civic action groups of Jews and Arabs across the country, in cooperation with civic and community organizations, including groups of ultra-Orthodox and Arab participants, among them a group of ultra-Orthodox Jewish women and religious Arab women, with the aim of building a shared society in a sustainable democracy in Israel.

While serving in the Knesset, Rabbi Melchior became concerned about the growing polarization between the different Jewish sectors in Israel. One contributing factor was the division of students in different educational streams: religious, secular, ultra-Orthodox, and Arab. These divides in the schools strengthened polarization between the Jewish sectors and reinforced a lack of common narrative, values, and vision for the future of the state. As a response, in 2001, Melchior established Meitarim - a network of pluralistic Jewish schools in Israel where religious and secular students can grow up and learn together in kindergartens, schools, pre- and post-military institutions in a non-coercive approach, enabling multiple interpretations and expressions of Jewish heritage to take responsibility for tolerance and the establishment of a collective narrative. Since 2001, Meitarim has disseminated its experimental infrastructure and curricula to over 15,000 students and instructors in collaboration with 120 educational institutions.

In 2002, Melchior initiated, together with his Palestinian counterpart Sheikh Talal Sider, an inter-religious summit in Alexandria, Egypt, co-sponsored by the Mufti of Egypt, Grand Imam of al-Azhar Mosque, and Grand Sheikh of al-Azhar University, Muhammad Sayyid Tantawi; Archbishop of Canterbury George Carey; and Eliyahu Bakshi-Doron, Israel's Chief Rabbi. The summit, which launched The Alexandria Process, brought together religious leaders from the Middle-East to adopt common principles aimed at preventing the region's religious sensibilities from being exploited during conflicts, and declaring the need to work together towards peaceful solutions of the conflicts. Together with Elie Wiesel, Melchior established the Mosaica Center for Interreligious Cooperation – MERPI (Middle East Religious Peace Initiative) to "lead to the implementation" of the principles of the Alexandria Declaration and continues to serve as chair of both of Mosaica's branches. Melchior is also a member of the Elijah Interfaith Institute Board of World Religious Leaders.

Mosaica's second branch is The Center for Conflict Resolution by Agreement which Rabbi Melchior also serves as Chairman. The center serves as a professional address for establishing understandings and agreements between individuals, organizations, groups, and communities. The center believes in every person's abilities to resolve conflicts and to operate in cooperation with the aid of tools from the realm of mediation. Mosaica supports the activities of 44 mediation and dialogue centers across the country with the aim of making the activities accessible to an array of communities in Israel. Thus, individuals, groups, and communities can contribute to a cohesive and respectful Israeli society.

More controversially, in September 2012, Melchior claimed that extremist Islamic leaders, including the leaders of Hamas, are ready for peaceful co-existence with Israel, and he added that he has "yet to meet with somebody who is not willing to make peace" with the Jewish state of Israel, placing the onus for lack of peace with extremist Islamic movements on Israel.

An additional initiative of Rabbi Melchior was Kulanu. Kulanu sought to strengthen the Jewish character of the State of Israel, enhance democracy, and foster unity through its diverse projects. Kulanu was best known for its Chagim BaKehilla (literally: holidays in the community) program, which provided religious services for secular Jews for many of the Jewish holidays. Kulanu worked with Teva Ivri to launch the Shmita Yisraelit program. Other Kulanu programs included Yom Kippur Shel Kulanu; B'Yachad: For a Shared Jewish Democratic Society; Meitarim: Inclusive Jewish Educational Network; and Shmita Yisraeli (in coordination with Teva Ivri).

In 2010, Melchior involved himself in economic issues. He co-founded the Israel Civic Action Forum which promotes higher taxation on income from the extraction of natural resources, and the use of the tax income for higher government spending to increase government spending on welfare, education, and health.

In 2014, he, along with Hadassah Froman of Tekoa and Rav Yoel Bin-Nun of Alon Shevut, participated at the Gush Etzion mourning where a kidnapping of three yeshiva students took place on June 12, 2014, who were found dead on June 30, 2014, and are believed to have been killed shortly after being kidnapped.

Prizes and awards
Melchior is a recipient of The Norwegian Award For Tolerance & Bridge Building in the Nobel Institute (1988), The Church Of England's Coventry International Prize For Peace & Reconciliation (2002), and the Liebhaber Prize For The Promotion Of Religious Tolerance And Cultural Pluralism (2007). On January 26, 2016, Rabbi Melchior was awarded The FRRME (Foundation for Relief and Reconciliation in the Middle East) Annual Peace Prize for 2015 at the House of Lords in London. On April 29, 2019 Melchior was awarded the Jean Nussbaum and Eleanor Roosevelt International Award of Excellence at the Global Summit on Religion, Peace, and Security in Geneva which was co-organized by the International Association for the Defense of Religious Liberty and the United Nations Office on Genocide Prevention.

title: Michael Melchior


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Tantalum carbide

Tantalum carbides (TaC) form a family of binary chemical compounds of tantalum and carbon with the empirical formula TaCx, where x usually varies between 0.4 and 1. They are extremely hard, brittle, refractory ceramic materials with metallic electrical conductivity. They appear as brown-gray powders, which are usually processed by sintering.

Being important cermet materials, tantalum carbides are commercially used in tool bits for cutting applications and are sometimes added to tungsten carbide alloys.

The melting points of tantalum carbides was previously estimated to be about 3880 °C depending on the purity and measurement conditions; this value is among the highest for binary compounds. And only tantalum hafnium carbide was estimated to have a higher melting point of 3942 °C. However new tests have conclusively proven that TaC actually has a melting point of 3,768 °C and both tantalum hafnium carbide and hafnium carbide have higher melting points.

Preparation
TaCx powders of desired composition are prepared by heating a mixture of tantalum and graphite powders in vacuum or inert-gas atmosphere (argon). The heating is performed at a temperature of about 2000 °C using a furnace or an arc-melting setup. An alternative technique is reduction of tantalum pentoxide by carbon in vacuum or hydrogen atmosphere at a temperature of 1500–1700 °C. This method was used to obtain tantalum carbide in 1876, but it lacks control over the stoichiometry of the product. Production of TaC directly from the elements has been reported through self-propagating high-temperature synthesis.

Crystal structure
TaCx compounds have a cubic (rock-salt) crystal structure for x = 0.7–1.0; the lattice parameter increases with x. TaC0.5 has two major crystalline forms. The more stable one has an anti-cadmium iodide-type trigonal structure, which transforms upon heating to about 2,000 °C into a hexagonal lattice with no long-range order for the carbon atoms.

Here Z is the number of formula units per unit cell, ρ is the density calculated from lattice parameters.

Properties
The bonding between tantalum and carbon atoms in tantalum carbides is a complex mixture of ionic, metallic and covalent contributions, and because of the strong covalent component, these carbides are very hard and brittle materials. For example, TaC has a microhardness of 1,600–2,000 kg/mm2 (~9 Mohs) and an elastic modulus of 285 GPa, whereas the corresponding values for tantalum are 110 kg/mm2 and 186 GPa.

Tantalum carbides have metallic electrical conductivity, both in terms of its magnitude and temperature dependence. TaC is a superconductor with a relatively high transition temperature of TC = 10.35 K.

The magnetic properties of TaCx change from diamagnetic for x ≤ 0.9 to paramagnetic at larger x. An inverse behavior (para-diamagnetic transition with increasing x) is observed for HfCx, despite that it has the same crystal structure as TaCx.

Application
Tantalum carbide is widely used as sintering additive in ultra-high temperature ceramics (UHTCs) or as a ceramic reinforcement in high-entropy alloys (HEAs) due to its excellent physical properties in melting point, hardness, elastic modulus, thermal conductivity, thermal shock resistance, and chemical stability, which makes it a desirable material for aircraft and rockets in aerospace industries.

Wang et al. have synthesized SiBCN ceramic matrix with TaC addition by mechanical alloying plus reactive hot-pressing sintering methods, in which BN, graphite and TaC powders were mixed with ball-milling and sintered at 1,900 °C to obtain SiBCN-TaC composites. For the synthesis, the ball-milling process refined the TaC powders down to 5 nm without reacting with other components, allowing to form agglomerates that are composed of spherical clusters with a diameter of 100 nm-200 nm. TEM analysis showed that TaC is distributed either randomly in the form of nanoparticles with sizes of 10-20 nm within the matrix or distributed in BN with smaller size of 3-5 nm. As a result, the composite with 10 wt% addition of TaC improved the fracture toughness of the matrix, reaching 399.5 MPa compared to 127.9 MPa of pristine SiBCN ceramics. This is mainly due to the mismatch of thermal expansion coefficients between TaC and SiBCN ceramic matrix. Since TaC has a larger coefficient of thermal expansion than that of SiBCN matrix, TaC particles endures tensile stress while the matrix endures tensile stress in radial direction and compressive stress in tangential direction. This makes the cracks to bypass the particles and absorbs some energy to achieve toughening. In addition, the uniform distribution of TaC particles contributes to the yield stress explained by Hall-Petch relationship due to a decrease in grain size.

Wei et al. have synthesized novel refractory MoNbRe0.5W(TaC)x HEA matrix using vacuum arc melting. XRD patterns showed that the resulting material is mainly composed of a single BCC crystal structure in the base alloy MoNbRe0.5W and a multi-component (MC) type carbide of (Nb, Ta, Mo, W)C to form a lamellar eutectic structure, with the amount of MC phase proportional to TaC addition. TEM analysis showed that the lamellar interface between BCC and MC phase presents a smooth and curvy morphology which exhibits good bonding with no lattice misfit dislocations. As a result, the grain size decreases with increasing TaC addition which improves the yield stress explained by Hall-Petch relationship. The formation of lamellar structure is because at elevated temperature, the decomposition reaction occurs in the MoNbRe0.5W(TaC)x composites: (Mo, Nb, W, Ta)2C → (Mo, Nb, W, Ta) + (Mo, Nb, W, Ta)C in which Re is dissolved in both components to nucleate BCC phase first and MC phase in the following, according to the phase diagrams. In addition, the MC phase also improves the strength of composites, due to its stiffer and more elastic property compared to BCC phase.

Wu et al. have also synthesized Ti(C, N)-based cermets with TaC addition with ball-milling and sintering at 1683 K. TEM analysis showed that TaC helps dissolution of carbonitride phase and converts to TaC-binder phase. The resulting is a formation of “black-core-white rim” structure with decreasing grain size in the region of 3-5 wt% TaC addition and increasing transverse rupture strength (TRS). 0-3 wt% TaC region showed a decrease in the TRS because the TaC addition decreases the wettability between binder and carbonitride phase and creates pores. Further addition of TaC beyond 5 wt% also decreases TRS because TaC agglomerates during sintering and porosity again forms. The best TRS is found at 5wt% addition where fine grains and homogeneous microstructure are achieved for less grain boundary sliding.

Natural occurrence
Tantalcarbide is a natural form of tantalum carbide. It is a cubic, extremely rare mineral.

title: Tantalum carbide