AEML, NTUST

工程材料第五章・擴散Engineering Materials · Ch.05 Diffusion

擴散戰線DIFFUSION RIFT

原子之境Realm of Atoms

擴散戰線DIFFUSION RIFT

原子之境Realm of Atoms

出征準備・選擇英雄Muster · Choose Your Hero

暱稱Nickname

遊戲人數Players

連線方式Mode

戰場Battlefield

難度Difficulty

輸入房號Room code

房號Room------

同學打開同一篇文章 →「開始遊戲」→「加入」輸入房號，或掃 QR code／開下方連結。人齊後按「出征」。Classmates open this post, Start Game, choose Join and type the code, or scan the QR / open the link. Press Deploy when everyone is in.

Cloudflare 雲端中繼Cloudflare relay-

00:000

▲0▲0

C′=0D=1T=300 K

濃度探針 C(x)Concentration probe C(x)

J = −D dC/dx：曲線越陡，通量越大J = −D dC/dx: the steeper the curve, the larger the flux

新手教學Tutorial1/60:00

重生倒數Respawn in0

0/0/0

◉ 0

戰鬥時間Match time

英雄Hero	K/D/A	補兵CS	傷害Damage	答題Quiz	分數Score

戰後實驗報告：擬合濃度剖面Lab Report: fit the concentration profile

你的英雄整場隨身帶著一根「試棒」：一端保持固定表面濃度 C_s（像滲碳爐），D 隨你所在路徑與溫度即時改變。藍點是戰後量到的濃度剖面 C′ = (C_x−C₀)/(C_s−C₀)。拖動滑桿設定 L = 2√(Dt)，讓 1 − erf(x/L) 貼合數據；R² 越高，最多再加 300 分。Your hero carried a test bar all match: one end held at a fixed surface concentration C_s (like a carburizing furnace), with D changing with the path and temperature you stood on. Blue dots are its measured profile C′ = (C_x−C₀)/(C_s−C₀). Drag the slider to set L = 2√(Dt) so 1 − erf(x/L) fits; higher R² earns up to 300 bonus points.

L =R² = 加分Bonus

總分Total score0

排行榜Leaderboard

#	暱稱Name	英雄Hero	結果Result	難度Diff.	分數Score	日期Date

排行榜只顯示暱稱，請勿使用真實姓名或學號。Only nicknames are shown. Please don't use your real name or student ID.

擴散戰線：原子之境・遊戲說明書Diffusion Rift: Realm of Atoms · Game Manual

工程材料 CT3309301 第五章「擴散」的教學 MOBA 遊戲。你和隊友（真人或電腦）組成 5 人的「晶格同盟」，對抗電腦控制的「熔爐軍團」。摧毀對方主堡即獲勝。整張地圖的地面都在即時解菲克第二定律——擴散不只是課本上的公式，而是戰場的規則。A teaching MOBA for Ch.05 Diffusion of CT3309301 Engineering Materials. You and your teammates (human or CPU) form the 5-hero Crystal Concord against the CPU Furnace Legion. Destroy the enemy core to win. The ground of the whole map solves Fick's second law in real time: diffusion isn't just a formula here, it is the rule of the battlefield.

1. 快速開始1. Quick start

按「開始遊戲」，欣賞約 48 秒的片頭動畫（可略過）。Press Start Game and enjoy the ~48 s opening film (skippable).
選擇英雄、人數與難度，按「出征」。Pick a hero, the number of players and a difficulty, then press Deploy.
跟著我方「原子兵」沿路推進，擊敗敵兵賺金與經驗；等級 4 答對研究題解鎖大招。Push with your atomling waves, last-hit enemies for gold and XP, and at level 4 answer a research question to unlock your ultimate.
回到主堡附近打開「實驗室」購買裝備，或隨時答題換取研究補助金。Open the Lab near your base to buy items, or answer questions anytime for research gold.

2. 操作方式2. Controls

動作Action	電腦Keyboard & mouse	手機／平板Phone / tablet	手把Gamepad
移動Move	W A S D / ←↑→↓ 或右鍵點地面or right-click ground	螢幕搖桿（設定開啟），或點地面On-screen stick (enable in Settings) or tap the ground	左類比Left stick
普攻Attack	按住空白鍵；左鍵點敵人鎖定Hold Space; left-click an enemy to focus it	按住「攻擊」；點敵人鎖定Hold ATK; tap an enemy to focus it	A
技能 1／2／大招Skill 1 / 2 / Ult	Q／E／R（朝滑鼠方向施放）Q / E / R (aimed at the mouse)	點一下自動瞄準；拖曳技能鈕可手動瞄準Tap to auto-aim; drag the button to aim	X / Y / RB
回城Recall	B	「回城」鈕（4 秒，受傷會中斷）Recall button (4 s, broken by damage)	B
濃度探針Concentration probe	V（朝滑鼠方向）V (toward the mouse)	「探針」鈕（朝面向）Probe button (facing direction)	–
實驗室／商店Lab / shop	L	「實驗室」鈕Lab button	LB
戰況表Scoreboard	按住 TabHold Tab	☰	Back
暫停／全螢幕Pause / full screen	P, Esc / F	❚❚ / ⛶	Start

按鍵可在「設定 → 自訂按鍵」修改。「鎖英雄」鈕可讓普攻優先打敵方英雄。手機請橫放並按「全螢幕」；全螢幕時畫面固定，不會被拖動或縮放。Rebind keys under Settings → Key bindings. The Heroes button makes attacks prefer enemy heroes. On phones, turn sideways and press Fullscreen; the view is then locked against dragging and zooming.

3. 地圖：三條路徑與濃度場3. The map: three paths and the concentration field

地圖 6000 × 6000，左下是我方晶都，右上是敵方熔爐要塞。三條路的地面圖案直接畫出擴散機制：The 6000 × 6000 map has your citadel at lower left and the Furnace fortress at upper right. Each path's floor shows its mechanism:

晶界捷徑（上路）Grain-boundary path (top)多晶晶粒與發光晶界，D 最大（×4）。Polycrystal grains with glowing boundaries; largest D (×4).
空位之路（中路）Vacancy path (mid)規則原子排列中散布紫色空位，D ×1.6。A regular lattice dotted with violet vacancies; D ×1.6.
間隙之路（下路）Interstitial path (bottom)大原子之間的綠色小原子在間隙中，D ×2.6。Small green atoms in the gaps between large atoms; D ×2.6.
等溫河與野區Isotherm river & jungle河道 D 很小，像擴散屏障；野區 D 較小。The river has a tiny D and acts as a barrier; the jungle has a smaller D.

地面的藍色／紅色就是溶質濃度 C(x,y,t)。每座防禦塔與主堡都是固定表面濃度 C_s 的「源」，原子兵走過會留下溶質，技能也能注入溶質或升溫。細線是等濃度線（每 10%），亮白線是擴散前緣（C = 50%），橘色光暈是升溫區。濃度場由 Fick 第二定律在背景執行緒（Web Worker）即時計算；選「硬體加速」時還會用 WebGPU 運算。The blue/red tint is the solute concentration C(x,y,t). Every tower and core is a source held at a fixed surface concentration C_s; atomlings deposit solute as they march, and skills inject solute or heat. Thin lines are iso-concentration contours (every 10%), the bright line is the diffusion front (C = 50%) and orange glow marks heated ground. Fick’s second law is solved live in a background thread (Web Worker), on WebGPU compute when Hardware is selected.

遊戲規則Game rule	對應觀念Concept
站在我方溶質中（C′ > 0.3）移速 +10% 並回血；在敵方溶質中移速 −10%In your own solute (C′ > 0.3): +10% speed and regen; in enemy solute: −10% speed	濃度場、由高往低擴散Concentration field; flow from high to low
防禦塔若被敵方溶質包圍，受到傷害 +20%Towers surrounded by enemy solute take +20% damage	滲透與累積（非穩態）Penetration and accumulation (non-steady state)
摧毀防禦塔＝移除一個源，領地會逐漸退縮Destroying a tower removes a source, so territory recedes	邊界條件改變後的非穩態擴散Non-steady diffusion after a boundary change
升溫處擴散變快Diffusion speeds up where it's hot	D = D₀ exp(−Q_d/RT)
HUD 顯示所在位置的 C′、D 與 TThe HUD shows C′, D and T where you stand	讀數練習Reading the variables

4. 英雄4. Heroes

英雄Hero	定位Role	技能與觀念Skills & concept
菲克・梯度賢者Fick, the Gradient Sage	法師Mage	梯度衝擊（傷害隨 \|∇C\| 增加）、通量領域、穩態光束——菲克第一定律與穩態擴散Gradient Bolt (damage grows with \|∇C\|), Flux Field, Steady-State Beam: Fick's first law and steady state
布朗・隨機漫步者Brown, the Random Walker	刺客Assassin	隨機漫步（三段隨機跳躍，位移 ≈ √3 步長）、熱振動、布朗風暴——擴散的微觀本質Random Walk (three random hops, displacement ≈ √3 steps), Thermal Jitter, Brownian Storm: the microscopic picture
阿瑞尼斯・熔爐守護者Arrhenius, Keeper of the Furnace	輔助Support	活化火焰、升溫領域、熔爐之陽——溫度與活化能Activation Flame, Heat Zone, Furnace Sun: temperature and activation energy
小碳・間隙遊俠Carbon, the Interstitial Ranger	射手Marksman	間隙穿梭、滲碳箭、表面硬化——間隙擴散與滲碳Interstitial Dash, Carburize Arrow, Case Hardening: interstitial diffusion and carburizing
空位騎士・虛空守衛Vacancy, the Hollow Guardian	坦克Tank	位置交換（與目標互換位置）、空位之盾、晶格崩塌——空位機制Site Swap (trade places with the target), Vacancy Shield, Lattice Collapse: the vacancy mechanism
時之主・時間平方根之主Chronos, Lord of √t	控制Controller	誤差函數波（傷害 ∝ 1−erf）、時間膨脹、√t 擴張（r = k√t）——非穩態擴散與 x ∝ √(Dt)Erf Wave (damage ∝ 1−erf), Time Dilation, √t Expansion (r = k√t): non-steady diffusion and x ∝ √(Dt)

開啟「劇情提示」時，你每次施放技能，畫面下方會顯示該技能對應的擴散觀念。With Story hints on, every skill you cast shows its diffusion concept at the bottom of the screen.

5. 實驗室、研究與裝備5. Lab, research and items

研究補助：在實驗室答題，答對 +130 金；答錯需冷卻 15 秒。題庫 40 題，涵蓋課本 5.1–5.5 節與例題 5.1–5.4。Research grant: answer in the Lab for +130 gold; a wrong answer adds a 15 s cooldown. 40 questions cover Sections 5.1–5.5 and Examples 5.1–5.4.
大招研究：英雄升到 4 級時，答對一題即解鎖大招（答錯 12 秒後可再試）。Ultimate research: at level 4, answer one question correctly to unlock your ultimate (retry 12 s after a miss).
裝備只能在主堡附近或陣亡時購買，最多 6 件；可半價（60%）賣回。Items can be bought only near your base or while dead, up to 6, and sold back for 60%.
單機模式下，開啟實驗室或答題時遊戲會暫停。In solo play the game pauses while the Lab or a question is open.

6. 野區與史詩怪6. Jungle and epic monsters

藍色元素：「通量」增益，冷卻加速 20%（90 秒）。紅色元素：「熾熱」增益，普攻灼燒（90 秒）。Blue elemental: Flux buff, 20% faster cooldowns (90 s). Red elemental: Heat buff, burning attacks (90 s).
活化能巨龍（上河道）：全隊 +12% 傷害並持續散布我方溶質 120 秒。Activation Dragon (upper river): team +12% damage and solute aura for 120 s.
平衡魔像（下河道）：全隊金錢經驗，接下來 150 秒兵線強化。Equilibrium Golem (lower river): team gold and XP, plus stronger waves for 150 s.

7. 勝利、計分與排行榜7. Victory, scoring and the leaderboard

外塔未破時內塔無敵；任一路內塔被破，主堡才會受傷。分數 = 勝負 + 擊殺／助攻 − 陣亡 + 補兵 + 傷害 + 推塔 + 答對題數 × 90 + 勝利時間獎勵，再乘上難度係數（簡單 0.7、普通 1、困難 1.35）。戰後「實驗報告」要你用 1 − erf(x/L) 擬合中路的濃度剖面，R² 越高最多再加 300 分。排行榜只顯示暱稱，請勿使用真實姓名或學號。Inner towers are immune while their outer tower stands; the core takes damage only after an inner tower falls. Score = result + kills/assists − deaths + last hits + damage + towers + correct answers × 90 + a speed bonus on victory, times the difficulty factor (Easy 0.7, Normal 1, Hard 1.35). The post-match Lab Report asks you to fit the mid-path profile with 1 − erf(x/L) for up to 300 bonus points. The leaderboard shows nicknames only; don't use your real name or student ID.

8. 多人連線（1–5 人）8. Online play (1–5 players)

房主：出征準備 → 選擇人數 2–5（自動切到「開房」）→ 出現 6 碼房號、連結與 QR code。Host: in Muster pick 2–5 players (switches to Host) and a 6-character room code, link and QR code appear.
同學：開同一篇文章 → 開始遊戲 → 加入 → 輸入房號（或掃 QR、開連結自動加入），選好英雄與暱稱。Classmates: open the same post → Start Game → Join → enter the code (or scan the QR / open the link), then pick a hero and nickname.
人齊後房主按「出征」。所有人同在藍隊合作對抗電腦。房主的電腦負責運算，請房主使用較穩定的網路與裝置。When everyone is in, the host presses Deploy. Everyone plays together on the blue team against the CPU. The host's device runs the simulation, so the host should use a stable device and network.
有人斷線時由電腦暫代，重新整理同一分頁可回到原座位。If someone drops, the CPU takes over; reloading the same tab returns them to their seat.

9. 設定與存檔9. Settings and saves

語言：中文／中文 English／English。背景音樂（預設關閉）依場景切換：標題、選角、片頭、對線、中期、團戰、實驗室、勝利、失敗、排行榜共 10 首原創樂曲，每首有 9–10 個立體聲左右分佈的樂器音軌加 1 個獨立鼓組音軌，以 General MIDI 音色庫即時演奏（瀏覽器不支援時自動改用合成器版本）。音效預設開啟，三種音量可分別調整。Language: Chinese / bilingual / English. Music (off by default) follows the scene with 10 original pieces (title, muster, opening, lane, mid game, team fight, Lab, victory, defeat, leaderboard); each has 9–10 instrument tracks panned across the stereo field plus 1 independent drum track, played live through a General MIDI sound bank (a synthesized fallback is used if the browser cannot). Sound effects are on by default; three volume sliders.
繪圖：硬體加速（預設，Phaser 3 WebGL 繪圖 + WebGPU 擴散運算）或軟體（Canvas 2D + CPU）；裝置不支援時自動改用軟體。畫質：流暢 720p（預設，地圖貼圖較小、適合手機）／標準 1080p／高 2K（1440p）。Renderer: Hardware (default: Phaser 3 WebGL drawing + WebGPU diffusion compute) or Software (Canvas 2D + CPU); it falls back automatically. Quality: Smooth 720p (default, lighter map textures for phones) / Standard 1080p / High 2K (1440p).
螢幕搖桿（預設關閉）、劇情提示（預設開啟）、濃度場顯示、自訂按鍵、全螢幕（預設關閉）、暫停、重新開始。On-screen stick (off by default), story hints (on), field overlay, key bindings, full screen (off), pause and restart.
存檔：3 個存檔欄 + 每 30 秒自動存檔，可匯出／匯入 .json。連線對戰無法存檔。Saves: 3 slots plus a 30-second auto-save, with .json export/import. Online matches can't be saved.

© 2026 AEML, NTUST（國立臺灣科技大學先進工程材料實驗室）。課本：Callister & Rethwisch, Materials Science and Engineering, 8e, Ch.5。插圖為原創 AI 生成作品。© 2026 AEML, NTUST. Textbook: Callister & Rethwisch, Materials Science and Engineering, 8e, Ch.5. Illustrations are original AI-generated artwork.

10. 新功能：教學、課堂快速戰、探針、計算題、戰爭迷霧10. New: tutorial, class quick match, probe, calculation problems, fog of war

新手教學（3 分鐘）Tutorial (3 min)標題畫面按「新手教學」。依序完成：移動 → 普攻訓練史萊姆 → 放技能 → 讀濃度場並使用探針 → 在實驗室答一題 → 回城。Press Tutorial on the title screen and complete: move → attack the training slime → cast a skill → read the field and use the probe → answer one Lab question → recall.
課堂快速戰（6–8 分鐘）Class quick match (6–8 min)只有中路、沒有野區，所有英雄從 3 級、1400 金開始，兵線每 20 秒一波，重生較快。8:00 時若主堡都未被破，由剩餘塔與主堡血量較多的一方獲勝。出征畫面也可切換「完整三路 5v5」或「課堂快速」。Mid lane only, no jungle; every hero starts at level 3 with 1400 gold, waves every 20 s and faster respawns. At 8:00, if neither core has fallen, the side with more structure health wins. You can also switch Battlefield on the muster screen.
濃度探針Concentration probe所有英雄都有：按 V 或「探針」鈕，沿一條 1600 單位的線即時顯示 C(x) 剖面（藍＝我方溶質、紅＝敵方），並標出最陡處（|dC/dx| 最大 ⇒ 通量 J 最大）。持續 9 秒、冷卻 4 秒。Every hero has it: press V or Probe to plot the live C(x) profile along a 1600-unit line (blue = our solute, red = theirs), with the steepest point marked (largest |dC/dx| ⇒ largest flux J). Lasts 9 s, 4 s cooldown.
計算挑戰Calculation challenge實驗室裡的「挑戰計算題」會隨機出例題 5.1（穩態通量）、5.2（滲碳時間，需查 erf 表）、5.3（Dt = 常數）、5.4（查表 5.2 求 D）的變化題。以「數值 × 10 的次方」輸入，誤差 ±3% 內即正確，+200 金，並附詳解。In the Lab, Solve a problem gives randomized versions of Ex. 5.1 (steady-state flux), 5.2 (carburizing time, erf table), 5.3 (Dt = constant) and 5.4 (D from Table 5.2). Enter mantissa × power of ten; within ±3% is correct for +200 gold, with a worked solution.
戰爭迷霧與草叢Fog of war and bushes只看得到我方英雄、小兵、塔與主堡視野內的敵人。站進草叢的英雄會隱身（我方看到半透明），除非敵人走進同一草叢、靠近 280 單位內，或你剛攻擊／施法（1 秒內會現形）。電腦對手與小兵也遵守這個規則。You only see enemies inside the vision of your heroes, minions, towers and core. A hero standing in a bush is hidden (shown translucent to allies) unless an enemy enters the same bush, comes within 280 units, or the hero has just attacked or cast (revealed for 1 s). AI heroes and minions obey the same rule.
美術Art地圖由 OpenArt 生成的材質與道具離線烘焙成 36 張精細貼圖；英雄、小兵、防禦塔、主堡與野怪皆為 OpenArt 角色圖，並加上走路擺動、攻擊衝刺、施法光暈與受擊閃光等動態。The map is baked offline into 36 detailed tiles from OpenArt textures and props; heroes, minions, towers, cores and monsters are OpenArt sprites animated with walk bob, attack lunge, cast glow and hit flash.

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請將手機橫放以獲得最佳體驗Rotate your phone to landscape

Diffusion Rift · AEML, NTUST
此遊戲由 AEML, NTUST 製作。程式碼或版權標示已被修改，遊戲已停用。This game was created by AEML, NTUST. Its code or credits have been modified, so it has been disabled.

Diffusion Rift · AEML, NTUST · CT3309301

無敵Immune

交換位置！Site swap!

晶界捷徑（最快）Grain-Boundary Path (fastest)

空位之路Vacancy Path

間隙之路Interstitial Path

萬物晶格之中，原子從不靜止。Within every crystal, atoms are never still.

晶格同盟守護著湛藍的晶都——他們的溶質，從高濃度流向低濃度。The Crystal Concord guards the sapphire citadel; its solute flows from high concentration to low.

熔爐軍團點燃了深紅要塞，要以烈焰與時間，滲透整個大陸。The Furnace Legion has lit the crimson fortress, to soak the whole continent with heat and time.

三條路徑通往敵陣：晶界捷徑、空位之路、間隙之路——每條的擴散係數 D 都不同。Three paths lead to the enemy: grain boundary, vacancy and interstitial, each with its own diffusion coefficient D.

梯度推動通量，溫度喚醒原子，深度隨時間的平方根成長。Gradients drive flux. Heat wakes the atoms. Depth grows with the square root of time.

召喚師，擴散之戰即將開始。Summoner, the war of diffusion begins.

法師Mage

刺客Assassin

輔助Support

射手Marksman

坦克Tank

控制Controller

菲克Fick

梯度賢者The Gradient Sage

「濃度差，就是我的力量。」"Every gradient is a source of power."

以菲克定律作戰的法師。目標所在處的濃度梯度越陡，她的法術越強。A mage who fights with Fick's laws: the steeper the concentration gradient at the target, the stronger her spells.

梯度衝擊Gradient Bolt

射出光彈；命中處的 |∇C| 越大，傷害最多 +90%。A bolt whose damage rises up to +90% with |∇C| where it hits.

通量領域Flux Field

4 秒領域：持續注入我方溶質，緩速並傷害敵人。A 4 s field that pumps in friendly solute, slowing and damaging enemies.

穩態光束（大招）Steady-State Beam (Ult)

蓄力後射出長光束，沿線留下由高到低的線性濃度剖面。After a short charge, a long beam leaves a linear concentration profile along its path.

菲克第一定律 J = −D dC/dx：梯度越陡，通量越大——傷害也越高。Fick's 1st law, J = −D dC/dx: steeper gradient, larger flux, more damage.

持續注入溶質＝固定的表面濃度 Cs，溶質會沿梯度向外擴散。Constant injection acts like a fixed surface concentration Cs; solute spreads down the gradient.

穩態擴散：通量不隨時間改變，濃度剖面呈直線。Steady-state diffusion: flux is constant in time and the profile is a straight line.

布朗Brown

隨機漫步者The Random Walker

「沒有人指揮我，但我總會抵達。」"No one steers me, yet I always arrive."

以隨機跳躍切入戰場的刺客，象徵擴散的微觀本質——原子的隨機漫步。An assassin who leaps in random hops, the microscopic picture of diffusion: a random walk.

隨機漫步Random Walk

連續三次方向隨機（偏向瞄準方向）的跳躍，每次落地造成傷害。Three hops in random directions (biased toward your aim); each landing deals damage.

熱振動Thermal Jitter

2.6 秒內移速 +45%、受到傷害 −30%。For 2.6 s: +45% move speed and 30% less damage taken.

布朗風暴（大招）Brownian Storm (Ult)

3 秒內在自身周圍隨機落下連續打擊。For 3 s, strikes rain down at random points around you.

隨機漫步 n 步，淨位移約 √n × 步長，而不是 n × 步長。After n random steps the net displacement is about √n steps, not n.

溫度越高，原子振動越劇烈，越容易越過能障跳到鄰位。Hotter atoms vibrate harder and hop over the energy barrier more often.

沒有誰在「推」原子：淨通量來自大量隨機跳躍的統計結果。Nothing pushes the atoms: net flux is the statistics of countless random jumps.

阿瑞尼斯Arrhenius

熔爐守護者Keeper of the Furnace

「溫度升高一點，世界就快上許多。」"Raise the heat a little, and the world moves much faster."

掌控溫度的輔助。她能改變地圖局部溫度，讓擴散係數依阿瑞尼斯方程式指數成長。A support who controls temperature, making D grow exponentially by the Arrhenius equation.

活化火焰Activation Flame

火球爆炸，灼燒敵人並使該處升溫。A fireball that burns enemies and heats the ground.

升溫領域Heat Zone

5 秒領域：區內 D 暴增；隊友攻速、移速提高並持續回血。A 5 s zone: D soars inside; allies gain attack speed, move speed and healing.

熔爐之陽（大招）Furnace Sun (Ult)

治療周圍隊友、灼燒敵人，並使大範圍高溫。Heals nearby allies, burns enemies and heats a wide area.

溫度是擴散的關鍵變數：觀察火球落點，濃度場擴散明顯變快。Temperature is the key variable: watch the field spread faster where the fireball lands.

D = D₀ exp(−Q_d/RT)：T 一升高，D 呈指數成長。D = D₀ exp(−Q_d/RT): raise T and D grows exponentially.

ln D 對 1/T 作圖是直線，斜率 = −Q_d/R。ln D vs 1/T is a straight line with slope −Q_d/R.

小碳Carbon

間隙遊俠The Interstitial Ranger

「個子小，跑得快！」"Small atom, fast feet!"

嬌小的碳原子射手，在大原子的縫隙間穿梭——間隙擴散比空位擴散快得多。被動：移速提高。A tiny carbon marksman slipping between big atoms: interstitial diffusion is much faster than vacancy diffusion. Passive: extra move speed.

間隙穿梭Interstitial Dash

向前穿越，下一次普攻傷害 +60%。Dash forward; your next basic attack deals +60%.

滲碳箭Carburize Arrow

長距離箭矢：標記目標 5 秒（受傷 +20%），並在命中處注入溶質。A long arrow that marks the target for 5 s (+20% damage taken) and injects solute.

表面硬化（大招）Case Hardening (Ult)

6 秒內攻速 +50%、獲得護盾、普攻濺射。For 6 s: +50% attack speed, a shield and splashing attacks.

間隙擴散：小原子（C、N、H、O）在間隙位置跳躍，鄰近空位多、速度快。Interstitial diffusion: small atoms (C, N, H, O) hop between interstitial sites, which are plentiful, so it is fast.

滲碳：在高溫富碳氣氛中，碳擴散進鋼的表層。Carburizing: at high temperature in a carbon-rich atmosphere, carbon diffuses into the steel surface.

表面硬化：表層含碳量高而變硬，提升耐磨與抗疲勞。Case hardening: the carbon-rich surface layer becomes hard, resisting wear and fatigue.

空位騎士Vacancy

虛空守衛The Hollow Guardian

「總得有人讓位，原子才能前進。」"Someone must step aside for atoms to move."

盾牌中央是一個空位的坦克。空位擴散需要：鄰位有空位，以及足以越過能障的能量。A tank whose shield holds a vacancy. Vacancy diffusion needs an empty neighbouring site and enough energy to cross the barrier.

位置交換Site Swap

衝向目標並與其交換位置，造成傷害與 0.8 秒暈眩。Dash to a target and swap places with it, dealing damage and a 0.8 s stun.

空位之盾Vacancy Shield

4 秒護盾，附近隊友獲得一半。A 4 s shield; nearby allies get half.

晶格崩塌（大招）Lattice Collapse (Ult)

0.4 秒後震盪周圍，暈眩 1.3 秒並造成傷害。After 0.4 s, a shockwave stuns nearby enemies for 1.3 s.

空位機制：原子與相鄰空位交換位置——原子往一邊走，空位就往另一邊走。Vacancy mechanism: an atom swaps with a neighbouring vacancy; atoms and vacancies move in opposite directions.

自擴散與互擴散多以空位機制進行；空位數量隨溫度升高而增加。Self- and interdiffusion usually go by vacancies; the number of vacancies rises with temperature.

空位擴散的活化能包含形成空位與移動原子兩部分，通常比間隙擴散高。Vacancy diffusion's activation energy covers forming and moving a vacancy, so it is usually higher than interstitial.

時之主Chronos

時間平方根之主Lord of the Square Root of Time

「深度與時間的平方根成正比。耐心，也是一種力量。」"Depth grows with the square root of time. Patience is power."

掌握非穩態擴散與誤差函數解的控制型英雄。A controller who wields non-steady-state diffusion and the error-function solution.

誤差函數波Erf Wave

扇形衝擊，傷害依 1 − erf(x/L) 隨距離遞減。A cone blast whose damage falls off with distance as 1 − erf(x/L).

時間膨脹Time Dilation

3 秒強力緩速領域。A 3 s heavy slow zone.

√t 擴張（大招）√t Expansion (Ult)

環狀衝擊波以 r = k√t 向外擴張：越到外圍越慢。A ring expands as r = k√t, slowing as it grows.

(C_x−C₀)/(C_s−C₀) = 1 − erf(x / 2√(Dt))：離表面越遠，濃度越低。(C_x−C₀)/(C_s−C₀) = 1 − erf(x / 2√(Dt)): the farther from the surface, the lower the concentration.

要得到相同的擴散效果，只要 Dt 相同：D 變小，就得等更久。The same diffusion result needs the same Dt: smaller D means waiting longer.

擴散深度 x ∝ √(Dt)：深度加倍，時間要四倍！Depth x ∝ √(Dt): to double the depth, wait four times as long!

提示：地面的藍紅色是即時計算的溶質濃度場 C(x,y,t)。站在我方溶質中會加速並回血。Tip: the blue and red tint is the live solute field C(x,y,t). Standing in your own solute speeds you up and heals you.

∇

⦿

☲

⤨

≋

✺

🔥

♨

☀

➹

➶

⛨

⇄

⛉

✷

⌇

⧗

√

間隙之靴Interstitial Boots

+45 移速+45 move speed

👢

沃斯田鐵之刃Austenite Blade

+24 攻擊+24 attack

🗡

晶界典籍Grain-Boundary Tome

+22% 技能強度+22% skill power

📘

空位鎧甲Vacancy Plate

+260 生命、+14 護甲+260 HP, +14 armor

🛡

熔爐核心Furnace Core

−15% 冷卻、+120 生命−15% cooldown, +120 HP

⚙

滲碳硬化盾Case-Hardened Aegis

+420 生命、+22 護甲、+8% 技能+420 HP, +22 armor, +8% skill

🔰

誤差函數長弓Erf Longbow

+22% 攻速、+12 攻擊+22% attack speed, +12 attack

🏹

活化能藥劑Activation Elixir

+18 攻擊、+12% 技能、10% 吸血+18 attack, +12% skill, 10% lifesteal

⚗

歡迎來到擴散戰線Welcome to the Diffusion Rift

原子兵團出發！Atomlings march!

擊敗敵方英雄！You slew an enemy!

你已陣亡You have been slain

我方英雄陣亡An ally has fallen

敵方英雄陣亡An enemy has fallen

摧毀敵方防禦塔！Enemy tower destroyed!

我方防禦塔被摧毀Our tower has fallen

獲得「通量」增益：冷卻加速Flux buff: faster cooldowns

獲得「熾熱」增益：普攻灼燒Heat buff: attacks burn

我方擊敗活化能巨龍！全隊強化We slew the Activation Dragon! Team empowered

敵方擊敗了活化能巨龍The enemy slew the Activation Dragon

我方擊敗平衡魔像！兵線強化We felled the Equilibrium Golem! Waves empowered

敵方擊敗了平衡魔像The enemy felled the Equilibrium Golem

活化能巨龍已出現The Activation Dragon has appeared

平衡魔像已出現The Equilibrium Golem has appeared

等級 4：答對研究題即可解鎖大招Level 4: answer a research question to unlock your ultimate

我方溶質：加速＋回血Own solute: faster + regen

敵方溶質：減速Enemy solute: slowed

混合區Mixed zone

⦿ 通量Flux

♨ 熾熱Heat

🐉 巨龍之力Dragon might

⛰ 兵線強化Wave surge

簡單Easy

普通Normal

困難Hard

上Up

下Down

左Left

右Right

普攻（按住）Attack (hold)

技能 1Skill 1

技能 2Skill 2

大招Ultimate

回城Recall

實驗室／商店Lab / shop

戰況（按住）Scoreboard (hold)

暫停Pause

全螢幕Fullscreen

存檔檔案無效。That save file isn't valid.

變更Change

已複製連結Link copied

電腦CPU

（空）(empty)

硬體加速運作中Hardware acceleration on

此裝置無法使用硬體加速，已改用軟體繪圖Hardware acceleration unavailable; using software

軟體繪圖Software rendering

房主Host

{p} 加入了房間{p} joined

手機提示：可在設定開啟「螢幕搖桿」；也可直接點地面移動、點敵人攻擊。Tip: turn on the on-screen stick in Settings, or tap the ground to move and tap enemies to attack.

勝Win

敗Loss

有玩家離線，暫由電腦代打A player disconnected; the CPU takes over

已讀取存檔Game loaded

讀取中…Loading…

大招尚未解鎖：升到 4 級並答對研究題Ultimate locked: reach level 4 and answer a research question

房主已離線The host went offline

目前還沒有紀錄No scores yet

尚未設定全球排行榜網址（設定 → 排行榜網址）No global leaderboard URL yet (Settings → Leaderboard URL)

目前沒有進行中的戰局No match in progress

連線對戰中無法存檔Online matches can't be saved

此欄位沒有存檔No save in that slot

瀏覽器空間不足，請改用匯出Browser storage is full; use Export

線上玩家Online

冷卻中，請稍後再試Cooling down, try again soon

已存檔Saved

已記錄在本機排行榜Saved to this device's leaderboard

賣出Sell

無法連線到全球排行榜（已記錄在本機）Couldn't reach the global leaderboard (kept on this device)

已送出到全球排行榜Sent to the global leaderboard

回到我方主堡附近（或陣亡時）才能購買裝備；答題隨時可進行。Return to your base (or wait while dead) to buy items; questions can be answered anytime.

位於主堡，可以購買裝備。You're at base and can buy items.

等待玩家…Waiting…

你You

晶格同盟Crystal Concord

熔爐軍團Furnace Legion

普攻優先鎖定英雄Attacks prefer heroes

普攻攻擊最近目標Attacks hit the nearest target

★ 大招研究：答對即可解鎖終極技能★ Ultimate research: answer correctly to unlock your ultimate

研究補助考驗：答對 +130 金Research grant: +130 gold if correct

答對了！Correct!

答錯了。Not quite.

大招已解鎖！Ultimate unlocked!

獲得 130 金！+130 gold!

大招解鎖！Ultimate unlocked!

勝利VICTORY

失敗DEFEAT

連線中…Connecting…

房間 {n} 已開啟，等待玩家Room {n} open, waiting for players

已連上中繼，等待房主…Connected; waiting for the host…

連線中斷，重新連線…Connection lost, retrying…

無法連到中繼伺服器Couldn't reach the relay

已加入房間，等待房主出征Joined; waiting for the host to start

房間已滿或已開戰Room full or already started

房號是 6 碼英數字Room codes have 6 letters/digits

「擴散 (diffusion)」最精確的定義是？Which best defines diffusion?

液體因重力流動Liquid flowing under gravity
物質藉由原子運動而進行的質量傳輸Mass transport by atomic motion
電子在導體中的移動Electron flow in a conductor
晶粒在加熱時長大Grain growth on heating

擴散是藉由原子運動造成的質量傳輸，在固體中也會發生。Diffusion is mass transport by atomic motion, and it happens in solids too.

把銅塊與鎳塊緊密接合後長時間加熱，兩種原子互相跨越界面擴散，這稱為？A Cu block and a Ni block are joined and heated for a long time; atoms of each cross the interface. This is called:

自擴散Self-diffusion
再結晶Recrystallization
互擴散（雜質擴散）Interdiffusion (impurity diffusion)
析出硬化Precipitation hardening

兩種金屬原子由高濃度往低濃度互相擴散，稱為互擴散。Atoms of two metals move from high to low concentration into each other: interdiffusion.

純金屬中，同種原子彼此交換位置的擴散稱為？In a pure metal, diffusion of host atoms among themselves is called:

自擴散Self-diffusion
互擴散Interdiffusion
間隙擴散Interstitial diffusion
穩態擴散Steady-state diffusion

純金屬中同種原子的遷移稱為自擴散，因成分不變，常以放射性同位素追蹤。Host atoms migrating in a pure metal is self-diffusion; composition doesn't change, so radioactive tracers are used to follow it.

原子要藉空位機制跳到鄰位，必須具備哪兩個條件？For an atom to move by the vacancy mechanism, which two conditions are needed?

外加電場與磁場An electric and a magnetic field
高壓與液態High pressure and a liquid state
晶界與差排A grain boundary and a dislocation
鄰近有空位，且原子有足夠能量越過能障An adjacent empty site and enough energy to break bonds and squeeze past

兩個條件：(1) 相鄰位置是空的；(2) 原子有足夠的振動能量打斷鍵結、擠過鄰原子。Two conditions: (1) an empty adjacent site; (2) enough vibrational energy to break bonds and distort the lattice while moving.

在多數金屬中，間隙擴散比空位擴散快得多，主要原因是？Interstitial diffusion is usually much faster than vacancy diffusion because:

間隙原子比較重Interstitial atoms are heavier
間隙原子較小、可移動的鄰近間隙位置遠多於空位Interstitial atoms are smaller and empty interstitial sites far outnumber vacancies
間隙擴散不需要能量Interstitial diffusion needs no energy
間隙原子帶電Interstitial atoms are charged

小原子活動性高，而且鄰近的空間隙位置比空位多得多，所以跳躍機率高。Small atoms are more mobile, and there are many more empty interstitial positions than vacancies.

下列哪一組最常以「間隙擴散」在金屬中移動？Which atoms most commonly diffuse interstitially in metals?

H、C、N、OH, C, N, O
Cu、Ni、ZnCu, Ni, Zn
Fe、Cr、MoFe, Cr, Mo
Au、Ag、PtAu, Ag, Pt

原子半徑小的雜質（氫、碳、氮、氧）能塞進主原子之間的間隙。Small impurity atoms (hydrogen, carbon, nitrogen, oxygen) fit into the interstices between host atoms.

「滲碳 (carburizing)」表面硬化的原理是？Case hardening by carburizing works by:

在室溫下以鎚擊表面Hammering the surface at room temperature
讓鋼表面的碳往外擴散Letting carbon diffuse out of the steel
高溫下讓碳從富碳氣氛擴散進鋼的表層Diffusing carbon from a carbon-rich atmosphere into the steel surface at high temperature
把鋼急冷到液態氮溫度Quenching the steel in liquid nitrogen

滲碳在高溫進行，碳原子擴散入表層，提高表面含碳量與硬度、耐磨與抗疲勞性。At high temperature carbon diffuses into the surface layer, raising its carbon content, hardness, wear and fatigue resistance.

擴散通量 J 的 SI 單位是？The SI unit of diffusion flux J is:

m²/sm²/s
kg/m³kg/m³
J/molJ/mol
kg/(m²·s) 或 atoms/(m²·s)kg/(m²·s) or atoms/(m²·s)

通量＝單位時間通過單位截面積的質量（或原子數），J = M/(At)。m²/s 是擴散係數 D 的單位。Flux is mass (or atoms) per unit area per unit time, J = M/(At). m²/s is the unit of D.

菲克第一定律 J = −D dC/dx 中的負號表示？In Fick's first law J = −D dC/dx, the minus sign means:

D 一定是負值D is always negative
擴散方向是沿濃度梯度向下，由高濃度往低濃度Diffusion goes down the gradient, from high to low concentration
通量隨時間遞減Flux decreases with time
只適用於負離子It applies only to anions

負號代表擴散方向與濃度梯度相反：往濃度低的方向走。The sign shows diffusion opposes the gradient: it goes toward lower concentration.

「穩態擴散」的特徵是？Steady-state diffusion is characterized by:

通量不隨時間改變；D 為定值時濃度剖面為直線Flux doesn't change with time; with constant D the profile is linear
濃度處處相同Uniform concentration everywhere
通量隨時間指數遞增Flux growing exponentially with time
只發生在液體中It occurs only in liquids

穩態時 J 與時間無關；若 D 不隨濃度改變，C(x) 是一條直線，斜率即濃度梯度。At steady state J is time-independent; if D is constant, C(x) is a straight line whose slope is the gradient.

例題 5.1：鐵片兩側分別滲碳與脫碳達穩態，5 mm 與 10 mm 處碳濃度為 1.2 與 0.8 kg/m³，D = 3×10⁻¹¹ m²/s。擴散通量 J 為？Ex. 5.1: steady state, C = 1.2 and 0.8 kg/m³ at 5 mm and 10 mm, D = 3×10⁻¹¹ m²/s. The flux J is:

2.4 × 10⁻⁹ kg/(m²·s)
2.4 × 10⁻⁶ kg/(m²·s)
1.2 × 10⁻⁹ kg/(m²·s)
6.0 × 10⁻¹² kg/(m²·s)

J = −D(C₂−C₁)/(x₂−x₁) = −(3×10⁻¹¹)(0.8−1.2)/(0.005) = 2.4×10⁻⁹ kg/(m²·s)。記得 mm 要換成 m！J = −D(C₂−C₁)/(x₂−x₁) = −(3×10⁻¹¹)(0.8−1.2)/(0.005) = 2.4×10⁻⁹ kg/(m²·s). Convert mm to m first!

兩處的 D 相同，A 處濃度梯度是 B 處的 3 倍。A 處的通量是 B 處的？D is the same at two places, but the gradient at A is 3× that at B. The flux at A is:

1/3 倍1/3 as large
√3 倍√3 times
3 倍3 times
9 倍9 times

J 與 dC/dx 成正比，梯度 3 倍，通量就 3 倍——菲克英雄的「梯度衝擊」就是這樣算傷害的。J is proportional to dC/dx, so 3× the gradient gives 3× the flux, just like Fick's Gradient Bolt.

菲克第二定律（D 為常數）是？Fick's second law (constant D) is:

J = −D dC/dxJ = −D dC/dx
∂C/∂t = D ∂²C/∂x²∂C/∂t = D ∂²C/∂x²
D = D₀ exp(−Q_d/RT)D = D₀ exp(−Q_d/RT)
x = vtx = vt

第二定律描述非穩態擴散：某點濃度隨時間的變化率與濃度曲線的曲率成正比。遊戲地圖的濃度場正是用它即時計算。The second law describes non-steady diffusion: the rate of change at a point is proportional to the curvature of the profile. The game's field is computed live from it.

「非穩態擴散」的意思是？"Non-steady-state diffusion" means:

完全沒有擴散No diffusion at all
濃度梯度為零Zero gradient
D 隨位置隨機變化D varies randomly with position
某點的通量與濃度都隨時間改變，有累積或耗盡Flux and concentration at a point change with time; there is accumulation or depletion

非穩態時 C = C(x, t)，大多數實際擴散（例如滲碳）都屬於此類。Non-steady: C = C(x, t). Most practical cases, such as carburizing, are like this.

誤差函數解 (C_x−C₀)/(C_s−C₀) = 1 − erf(x/2√(Dt)) 適用的條件是？The error-function solution applies when:

半無限固體、初始濃度 C₀ 均勻、表面濃度 C_s 保持固定The solid is semi-infinite, C₀ is uniform initially and the surface concentration C_s is held constant
已達穩態Steady state has been reached
表面濃度隨時間線性下降Surface concentration falls linearly with time
試片非常薄The specimen is very thin

邊界條件：t = 0 時 C = C₀；t > 0 時 x = 0 處 C = C_s，x = ∞ 處 C = C₀。Boundary conditions: C = C₀ at t = 0; for t > 0, C = C_s at x = 0 and C = C₀ at x = ∞.

例題 5.2：C₀ = 0.25%、C_s = 1.2%、目標 C_x = 0.8%。第一步算出 erf(z) 約為？Ex. 5.2: C₀ = 0.25%, C_s = 1.2%, target C_x = 0.8%. The first step gives erf(z) ≈

0.5789
0.8000
0.4210
0.2500

(0.8−0.25)/(1.2−0.25) = 0.5789 = 1 − erf(z)，所以 erf(z) = 0.4210，查表內插得 z ≈ 0.392。(0.8−0.25)/(1.2−0.25) = 0.5789 = 1 − erf(z), so erf(z) = 0.4210; interpolating the table gives z ≈ 0.392.

承上，x = 0.5 mm、D = 1.6×10⁻¹¹ m²/s、z = 0.392，所需時間 t = x²/(4Dz²) 約為？Continuing: x = 0.5 mm, D = 1.6×10⁻¹¹ m²/s, z = 0.392. The time t = x²/(4Dz²) is about:

25,400 s ≈ 7.1 小時25,400 s ≈ 7.1 h
2,540 s ≈ 0.7 小時2,540 s ≈ 0.7 h
254,000 s ≈ 71 小時254,000 s ≈ 71 h
9,800 s ≈ 2.7 小時9,800 s ≈ 2.7 h

t = (5×10⁻⁴)² / [4(1.6×10⁻¹¹)(0.392)²] ≈ 25,400 s ≈ 7.1 h。t = (5×10⁻⁴)² / [4(1.6×10⁻¹¹)(0.392)²] ≈ 25,400 s ≈ 7.1 h.

在相同位置得到相同濃度（相同擴散效果），需滿足？To reach the same concentration at the same depth (the same diffusion effect) you need:

相同的 DThe same D
相同的 tThe same t
相同的 D/tThe same D/t
相同的 DtThe same Dt

x/2√(Dt) 相同即可，所以 Dt = 常數：溫度低（D 小）就要拉長時間。The same x/2√(Dt) is enough, so Dt is constant: lower temperature (smaller D) needs more time.

例題 5.3：Cu 在 Al 中，600 °C 時 D = 5.3×10⁻¹³、500 °C 時 D = 4.8×10⁻¹⁴ m²/s。600 °C 加熱 10 小時的效果，若改在 500 °C 需時？Ex. 5.3: Cu in Al, D = 5.3×10⁻¹³ at 600 °C and 4.8×10⁻¹⁴ m²/s at 500 °C. The 500 °C time matching 10 h at 600 °C is:

0.9 小時0.9 h
110.4 小時110.4 h
10 小時10 h
33.2 小時33.2 h

t₅₀₀ = D₆₀₀t₆₀₀/D₅₀₀ = (5.3×10⁻¹³)(10)/(4.8×10⁻¹⁴) = 110.4 h。降溫 100 度，要等 11 倍久！t₅₀₀ = D₆₀₀t₆₀₀/D₅₀₀ = (5.3×10⁻¹³)(10)/(4.8×10⁻¹⁴) = 110.4 h. Cool by 100 degrees and you wait 11 times longer!

同一溫度下，想讓滲碳深度變成 2 倍，時間要變成幾倍？At the same temperature, to double the carburized depth the time must be multiplied by:

√2
2
4
8

x ∝ √(Dt)，深度 2 倍需要 t 變 4 倍——這就是時之主大招 r = k√t 的意思。x ∝ √(Dt): doubling depth takes 4× the time, the idea behind Chronos's r = k√t.

關於誤差函數，下列何者正確？Which is true of the error function?

erf(0) = 1erf(0) = 1
erf(0) = 0，z → ∞ 時 erf(z) → 1erf(0) = 0, and erf(z) → 1 as z → ∞
erf(z) 隨 z 線性增加erf(z) grows linearly with z
erf(z) 可以大於 1erf(z) can exceed 1

所以在表面 (x = 0) C = C_s，深處 C → C₀。So at the surface (x = 0) C = C_s, and deep inside C → C₀.

D = 1×10⁻¹¹ m²/s，擴散 1 小時，特徵擴散距離 √(Dt) 約為？With D = 1×10⁻¹¹ m²/s for 1 hour, the characteristic length √(Dt) is about:

19 mm19 mm
1.9 mm1.9 mm
0.019 mm0.019 mm
0.19 mm0.19 mm

√(10⁻¹¹ × 3600) = √(3.6×10⁻⁸) ≈ 1.9×10⁻⁴ m = 0.19 mm。√(10⁻¹¹ × 3600) = √(3.6×10⁻⁸) ≈ 1.9×10⁻⁴ m = 0.19 mm.

擴散係數與溫度的關係（阿瑞尼斯型）是？The temperature dependence of the diffusion coefficient is:

D = D₀ exp(−Q_d/RT)D = D₀ exp(−Q_d/RT)
D = D₀ + Q_dTD = D₀ + Q_dT
D = D₀ exp(+Q_d/RT)D = D₀ exp(+Q_d/RT)
D = RT/Q_dD = RT/Q_d

D 隨溫度呈指數增加；Q_d 為擴散活化能，D₀ 為與溫度無關的前置因子。D rises exponentially with temperature; Q_d is the activation energy and D₀ a temperature-independent pre-exponential.

以 ln D 對 1/T 作圖，得到的直線斜率是？Plotting ln D against 1/T gives a straight line whose slope is:

D₀D₀
+Q_d/R+Q_d/R
−Q_d/R−Q_d/R
ln D₀ln D₀

ln D = ln D₀ − (Q_d/R)(1/T)：斜率 −Q_d/R，截距 ln D₀。ln D = ln D₀ − (Q_d/R)(1/T): slope −Q_d/R, intercept ln D₀.

例題 5.4：Mg 在 Al 中，D₀ = 1.2×10⁻⁴ m²/s、Q_d = 130 kJ/mol。550 °C 時的 D 約為？Ex. 5.4: Mg in Al, D₀ = 1.2×10⁻⁴ m²/s, Q_d = 130 kJ/mol. D at 550 °C is about:

1.2 × 10⁻⁴ m²/s
6.7 × 10⁻¹³ m²/s
6.7 × 10⁻⁹ m²/s
5.6 × 10⁻¹⁰ m²/s

T = 823 K。D = 1.2×10⁻⁴ exp[−130000/(8.31×823)] = 1.2×10⁻⁴ × e⁻¹⁹ ≈ 6.7×10⁻¹³ m²/s。溫度一定要用 K！T = 823 K. D = 1.2×10⁻⁴ exp[−130000/(8.31×823)] = 1.2×10⁻⁴ × e⁻¹⁹ ≈ 6.7×10⁻¹³ m²/s. Always use kelvin!

若 Q_d 以 eV/atom 表示，阿瑞尼斯式中應搭配哪個常數？If Q_d is given in eV/atom, which constant goes with it?

R = 8.314 J/(mol·K)R = 8.314 J/(mol·K)
亞佛加厥數Avogadro's number
法拉第常數Faraday's constant
k = 8.62×10⁻⁵ eV/(atom·K)k = 8.62×10⁻⁵ eV/(atom·K)

J/mol 配 R = 8.314；eV/atom 配波茲曼常數 k = 8.62×10⁻⁵。單位要一致！J/mol goes with R = 8.314; eV/atom goes with Boltzmann's k = 8.62×10⁻⁵. Keep units consistent!

其他條件相同時，活化能 Q_d 越大，擴散係數 D 會？All else equal, a larger activation energy Q_d makes D:

越小Smaller
越大Larger
不變Unchanged
變成負值Negative

exp(−Q_d/RT) 隨 Q_d 增加而變小：能障越高，越難跳躍。exp(−Q_d/RT) shrinks as Q_d grows: a higher barrier means fewer successful jumps.

在鐵中，下列哪一個在相同溫度下擴散最快？In iron at the same temperature, which diffuses fastest?

Fe 的自擴散Fe self-diffusion
Ni（置換型）Ni (substitutional)
C（間隙型）C (interstitial)
三者一樣快All the same

碳是小的間隙原子，Q_d 較低、D 遠大於置換型原子與自擴散。Carbon is a small interstitial with a lower Q_d, so its D is far larger than substitutional or self-diffusion.

為何空位擴散的活化能通常比間隙擴散高？Why is the activation energy for vacancy diffusion usually higher than for interstitial diffusion?

空位擴散不需要能量Vacancy diffusion needs no energy
除了移動原子，還需要形成空位的能量Energy is needed to form the vacancy as well as to move the atom
空位比間隙原子小Vacancies are smaller than interstitials
空位只在液體中存在Vacancies exist only in liquids

空位擴散的 Q_d 包含空位形成能與原子移動能兩部分。The vacancy Q_d includes both vacancy formation and atom migration energy.

「變軟的齒輪」案件中，爐溫低了 80 °C、時間又從 6 小時縮成 4 小時。最根本的原因是？In the "soft gear" case the furnace ran 80 °C too cold and time was cut from 6 h to 4 h. The root cause is:

鋼材牌號用錯The wrong steel grade
齒輪有裂縫The gears were cracked
碳擴散太快，穿透整個齒輪Carbon diffused too fast and went through the gear
D 與 t 都變小，Dt 大幅下降，碳根本擴散不到應有深度Both D and t fell, so Dt dropped sharply and carbon never reached the required depth

x ∝ √(Dt)：D 約降為 1/3、t 降為 2/3，Dt 約剩 1/4 甚至更少，滲碳深度遠不足。x ∝ √(Dt): D dropped to about a third and t to two-thirds, so Dt fell to a quarter or less and the case was far too thin.

半導體積體電路的「摻雜 (doping)」主要靠什麼製程？Doping of semiconductor integrated circuits relies mainly on:

鑄造Casting
冷軋Cold rolling
讓雜質原子在高溫下擴散進矽晶片Diffusing impurity atoms into the silicon wafer at high temperature
電鍍Electroplating

摻雜的兩步驟（預沉積與驅入）都是擴散控制的熱處理。Both doping steps (predeposition and drive-in) are diffusion-controlled heat treatments.

氣體分子會到處亂跑，但固體中「每個原子都有座位」。固體中原子要移動，關鍵是？Gas molecules wander freely, but in a solid every atom has a seat. For atoms to move in a solid, the key is:

晶格中有缺陷（空位）或間隙可跳，且溫度提供足夠能量Defects (vacancies) or interstitial sites to jump into, and enough thermal energy
固體必須先熔化The solid must melt first
必須施加外力推動An external force must push them
固體中原子完全不能移動Atoms in solids can never move

固體擴散靠空位或間隙，加上熱振動越過能障。「有人讓位，碳才能進去」。Solid-state diffusion uses vacancies or interstitial sites plus thermal energy to cross the barrier: someone has to make room.

隨機漫步中，每一步方向隨機，走了 n 步後的典型淨位移約與什麼成正比？In a random walk of n steps in random directions, the typical net displacement scales with:

n
n²
1/n
√n

均方根位移 ∝ √n，對應巨觀的 x ∝ √(Dt)。The root-mean-square displacement ∝ √n, matching the macroscopic x ∝ √(Dt).

擴散的驅動力（本章的觀點）最主要是？In this chapter, the driving force for diffusion is mainly:

重力Gravity
濃度梯度The concentration gradient
晶粒大小Grain size
磁場A magnetic field

在菲克定律中，濃度梯度是驅動力；沒有梯度就沒有淨通量。In Fick's laws the concentration gradient is the driving force; no gradient means no net flux.

氫氣純化時讓氫氣通過鈀薄片，其他氣體被擋住。這利用的是？Hydrogen is purified by passing it through a thin palladium sheet that blocks other gases. This uses:

穩態擴散：氫原子穿過金屬薄片Steady-state diffusion of hydrogen atoms through the metal sheet
對流Convection
再結晶Recrystallization
凝固Solidification

薄片兩側濃度固定時達到穩態，氫以間隙方式穿透鈀。With fixed concentrations on both faces the flow is steady, and hydrogen diffuses interstitially through the Pd.

遊戲中「升溫領域」讓 T 由 300 K 提高，D 會如何變化？In the game, the Heat Zone raises T above 300 K. How does D change?

線性小幅增加It rises a little, linearly
減少It decreases
依 exp(−Q_d/RT) 呈指數增加It rises exponentially as exp(−Q_d/RT)
不變，溫度只影響 D₀Unchanged; T affects only D₀

指數關係讓小幅升溫也能大幅加快擴散——這就是熱處理要嚴格控制爐溫的原因。The exponential means a modest rise in T speeds diffusion a lot, which is why furnace temperature must be tightly controlled.

在同樣溫度下，碳在 BCC 的 α-Fe 與 FCC 的 γ-Fe 中，何者擴散較快？At the same temperature, is carbon diffusion faster in BCC α-Fe or FCC γ-Fe?

γ-Fe（FCC）γ-Fe (FCC)
α-Fe（BCC）α-Fe (BCC)
兩者相同The same
碳無法在鐵中擴散Carbon can't diffuse in iron

BCC 堆積較鬆（APF 0.68），碳在 α-Fe 的 Q_d 較低（約 87 vs 148 kJ/mol），所以較快。BCC is less densely packed (APF 0.68) and carbon's Q_d in α-Fe is lower (about 87 vs 148 kJ/mol), so it is faster.

遊戲中三條路的擴散係數「晶界捷徑 > 間隙之路 > 空位之路」，對應的道理是？In the game, D on the paths ranks grain boundary > interstitial > vacancy. The idea is:

結構較開放的路徑（晶界、表面）與小原子的間隙跳躍，阻力較小、擴散較快More open structures (grain boundaries, surfaces) and small interstitial jumps meet less resistance and diffuse faster
路越長擴散越快Longer paths diffuse faster
顏色越亮擴散越快Brighter paths diffuse faster
完全隨機設定It is completely arbitrary

晶界、表面原子排列較鬆散，稱為「短路擴散」；間隙擴散又比需要空位的體擴散快。Atoms at grain boundaries and surfaces are loosely packed ("short-circuit" paths); interstitial diffusion beats vacancy-assisted bulk diffusion.

穩態擴散中，若濃度剖面是直線，則各處的濃度梯度？In steady-state diffusion with a straight-line profile, the gradient along x is:

越往低濃度端越大Larger toward the low end
為零Zero
處處相同The same everywhere
隨時間增加Increasing with time

直線斜率固定，梯度處處相同，所以通量 J 處處相同。A straight line has a single slope, so the gradient, and therefore J, is the same everywhere.

滲碳時若把時間拉長為原來的 9 倍，滲碳深度會變成幾倍？If carburizing time is increased 9-fold, the case depth becomes:

9 倍9×
81 倍81×
4.5 倍4.5×
3 倍3×

x ∝ √t，√9 = 3。x ∝ √t, and √9 = 3.

濃度探針Concentration probe

歡迎來到新手教學！跟著上方提示完成 6 個小任務（約 3 分鐘）。Welcome! Follow the panel above to finish 6 short tasks (about 3 minutes).

① 移動：用 WASD、搖桿，或點地面，走到綠色光圈。Move: use WASD, the stick, or tap the ground to reach the green ring.

② 普攻：靠近訓練史萊姆，按住 {a} 或「攻擊」鈕（也可直接點它）。Attack: walk to the training slime and hold {a} or the ATK button (or tap the slime).

③ 技能：對史萊姆放出技能 1 {q} 或技能 2 {e}。每招都對應一個擴散觀念，提示會跳出來。Skill: cast skill 1 {q} or skill 2 {e} on the slime. Each skill shows the diffusion idea behind it.

④ 讀濃度場：地面的藍／紅色是兩方溶質的濃度 C(x,y)，細線是等濃度線，亮白線是擴散前緣（C = 50%）。按 {v} 或「探針」鈕，看看 C(x) 剖面：曲線最陡處通量最大。Read the field: blue/red ground tint is solute concentration C(x,y), thin lines are iso-concentration contours and the bright line is the diffusion front (C = 50%). Press {v} or Probe to see the C(x) profile: the steepest part carries the largest flux.

⑤ 實驗室：按 {l} 或「實驗室」，回答一題觀念題或計算題（答對有金幣）。Lab: press {l} or Lab and answer one concept or calculation question (gold if correct).

⑥ 回城：按 {b} 或「回城」並站著不動 4 秒。在泉水旁可以買裝備、回血。Recall: press {b} or Recall and stand still for 4 s. At the fountain you heal and can shop.

探針冷卻中Probe cooling down

請輸入數值Please enter a number

正確答案：Answer:

+200 金！+200 gold!

擴散物種 in 母材Diffusing species in host

Fe in α-Fe (BCC)

Fe in γ-Fe (FCC)

C in α-Fe

C in γ-Fe

Cu in Cu

Zn in Cu

Al in Al

Cu in Al

Mg in Al

Cu in Ni

例題 5.1 穩態擴散通量Ex. 5.1 Steady-state flux

例題 5.2 滲碳需時多久Ex. 5.2 Carburizing time

例題 5.3 用時間換溫度Ex. 5.3 Trading temperature for time

例題 5.4 查表求 DEx. 5.4 Compute D from Table 5.2

一鐵片兩側分別置於滲碳與脫碳環境，已達穩態。在 x₁ = {x1} mm 與 x₂ = {x2} mm 處，碳濃度分別為 {c1} 與 {c2} kg/m³，擴散係數 D = {D} m²/s。求擴散通量 J。An iron plate has a carburizing atmosphere on one side and a decarburizing one on the other, at steady state. The carbon concentrations at x₁ = {x1} mm and x₂ = {x2} mm are {c1} and {c2} kg/m³; D = {D} m²/s. Find the diffusion flux J.

某合金含碳量 {C0}%，表面碳濃度提升並保持在 {Cs}%。在 {x} mm 深處，需時多久才達到 {Cx}%？合金視為半無限，D = {D} m²/s。（以小時作答）An alloy initially contains {C0}% C. The surface is raised to and held at {Cs}% C. How long until the concentration at {x} mm depth reaches {Cx}%? Treat it as semi-infinite with D = {D} m²/s. (Answer in hours.)

{sys} 的擴散係數在 {T1} °C 與 {T2} °C 時分別為 {D1} 與 {D2} m²/s。在 {T1} °C 需加熱多久，才能與 {T2} °C 加熱 {t2} 小時的擴散效果相同？（以小時作答）For {sys}, D = {D1} m²/s at {T1} °C and {D2} m²/s at {T2} °C. How long at {T1} °C gives the same diffusion result as {t2} h at {T2} °C? (Answer in hours.)

利用表 5.2，求 {sys} 在 {Tc} °C 時的擴散係數 D。（R = 8.314 J/mol·K）Using Table 5.2, find D for {sys} at {Tc} °C. (R = 8.314 J/mol·K){tbl}

穩態、線性剖面：Steady state, linear profile: J = −D (C₂ − C₁)/(x₂ − x₁) = −({D}) × ({c2} − {c1}) / (({x2} − {x1}) × 10⁻³ m) = {ans} kg/(m²·s). 記得先把 mm 換成 m！Convert mm to m first!

① (C_x − C₀)/(C_s − C₀) = ({Cx} − {C0})/({Cs} − {C0}) = {ratio} = 1 − erf(z)
② erf(z) = {erfz} ⇒ z ≈ {z} （查表 5.1 線性內插）(interpolate Table 5.1)
③ t = x²/(4Dz²) = ({x} × 10⁻³)² / (4 × {D} × {z}²) = {ts} s = {ans} h

「擴散效果相同」= 同深度同濃度 ⇒ Dt 為常數（不需查 erf 表）："Same diffusion result" = same depth at same concentration ⇒ Dt = constant (no erf table needed):
t₁ = D₂t₂/D₁ = {D2} × {t2} / {D1} = {ans} h

D = D₀ exp(−Q_d/RT) = {D0} × exp(−{Q} / (8.314 × {TK})) = {ans} m²/s （溫度要換成 K；Q_d 用 J/mol）(use kelvin, and Q_d in J/mol)

已在新分頁開啟遊戲（手機版全螢幕）Opened the game in a new tab for full screen

語言Language
背景音樂Music	依場景切換・日常／學院／戰鬥changes by scene · daily / academic / battle
音效Sound FX
主音量Master volume
音樂音量Music volume
音效音量FX volume
全螢幕Fullscreen
繪圖Renderer
畫質Quality
螢幕搖桿On-screen stick
劇情提示Story hints
濃度場顯示Field overlay
自訂按鍵Key bindings
中繼伺服器Relay server
排行榜網址Leaderboard URL

擴散戰線DIFFUSION RIFT

原子之境Realm of Atoms

出征準備・選擇英雄Muster · Choose Your Hero

戰後實驗報告：擬合濃度剖面Lab Report: fit the concentration profile

暫停Paused

戰況Scoreboard

擴散實驗室Diffusion Lab◉ 0

教學完成！Tutorial complete!

排行榜Leaderboard

存檔／讀檔Save / Load

設定Settings

自訂按鍵Key Bindings

擴散戰線：原子之境・遊戲說明書Diffusion Rift: Realm of Atoms · Game Manual

1. 快速開始1. Quick start

2. 操作方式2. Controls

3. 地圖：三條路徑與濃度場3. The map: three paths and the concentration field

4. 英雄4. Heroes

5. 實驗室、研究與裝備5. Lab, research and items

6. 野區與史詩怪6. Jungle and epic monsters

7. 勝利、計分與排行榜7. Victory, scoring and the leaderboard

8. 多人連線（1–5 人）8. Online play (1–5 players)

9. 設定與存檔9. Settings and saves

10. 新功能：教學、課堂快速戰、探針、計算題、戰爭迷霧10. New: tutorial, class quick match, probe, calculation problems, fog of war