A silicon-carbon composite material, a preparation method thereof, a pole piece and application, and an electrochemical device

Abstract

The application discloses a silicon-carbon composite material and a preparation method, pole piece and application and electrochemical device thereof. The preparation method of the silicon-carbon composite material comprises the following steps: depositing mixed gas in porous carbon, and then performing carbon coating; the mixed gas comprises magnesium in a gaseous form and a silicon source; the mass ratio of the magnesium and the silicon source in the mixed gas is (0.5-2):1, and the silicon source comprises silicon oxide. The magnesium thermal reaction is performed in a gas phase system, so that the deposition position of the silicon can be effectively controlled. The battery prepared from the silicon-carbon composite material has better cycle performance.

Description

Technical Field

[0001] This invention relates to a silicon-carbon composite material, its preparation method, electrode and application, and electrochemical device. Background Technology

[0002] Silicon-based materials are abundant, relatively inexpensive, and environmentally friendly, possessing a high theoretical specific capacity. This can significantly improve the single-cell capacity of lithium-ion batteries, thereby increasing energy density and making high-energy-density, long-range lithium-ion batteries a possibility. However, silicon undergoes massive volume expansion during charging and discharging (up to 300% or more), leading to pulverization and detachment of electrode materials, resulting in rapid capacity decay and poor cycle performance. This is a key issue limiting the large-scale application of silicon-based anode materials. Furthermore, silicon's relatively poor conductivity also affects the battery's rate performance.

[0003] Porous carbon possesses abundant pore structures and a large specific surface area, providing numerous active sites that facilitate silicon deposition and dispersion, increase the contact area between silicon and the electrolyte, and improve battery charge-discharge efficiency. Simultaneously, the porous structure of porous carbon provides a buffer space for the volume expansion of silicon during charge-discharge processes, effectively mitigating the volume effect of silicon, reducing electrode material pulverization and detachment, and improving battery cycle stability.

[0004] The presence of porous silicon can increase the flexibility of electrode materials. Compared to dense silicon, porous silicon can better adapt to stress changes when its volume changes, and is less prone to cracking and fracture. This improves the mechanical stability of the electrode during charge-discharge cycles and helps reduce volume expansion caused by electrode structure damage.

[0005] Therefore, depositing porous silicon inside porous carbon can significantly reduce the volume expansion of silicon-based anodes. For example, Chinese patent document CN 102569759 A discloses a method for preparing silicon-porous carbon anode materials for lithium-ion batteries. The resulting silicon-porous carbon composite material has a nanoporous structure, which provides a certain buffer space for the volume expansion of silicon and can improve the cycle performance of the material. However, the material does not have good control over the deposition position of silicon, so its improvement in cycle performance still needs to be improved. Summary of the Invention

[0006] The technical problem this invention aims to solve is to overcome the shortcomings of existing silicon-carbon composite materials, which have poor control over the deposition site of silicon, resulting in unsatisfactory cycle performance of batteries made from these materials. This invention provides a silicon-carbon composite material, its preparation method, electrodes, applications, and electrochemical devices. By carrying out a magnesothermic reaction in a gas-phase system, this invention allows for effective control of the silicon deposition site. Batteries made from the silicon-carbon composite material of this invention exhibit superior cycle performance.

[0007] The present invention solves the above-mentioned technical problems by adopting the following technical solution.

[0008] This invention provides a method for preparing a silicon-carbon composite material, comprising the following steps:

[0009] A mixed gas is deposited in porous carbon and then coated with carbon; the mixed gas includes magnesium and silicon sources in gaseous form; the mass ratio of magnesium to SiOx in the mixed gas is (0.5-2):1, and the silicon source includes silicon oxide.

[0010] In this invention, the chemical formula of the silicon oxide is generally SiOx, where 0 < x < 2, preferably 0.8 < x < 1.2, for example, x is 1.

[0011] In this invention, the silicon source preferably further includes silicon.

[0012] In this invention, the mixed gas can be prepared by separately gasifying magnesium powder and silicon source powder and then mixing them.

[0013] The preferred vaporization temperature of the magnesium powder is 300-900℃, more preferably 400-800℃, such as 500, 600 or 700℃.

[0014] The preferred rate of heating from room temperature to the vaporization temperature of the magnesium powder is 3-7°C / min, for example, 4, 5 or 6°C / min.

[0015] The gasification time of the magnesium powder can be 1-3 hours, for example, 2 hours.

[0016] The D50 particle size of the magnesium powder can be 3-7 μm, for example 5 μm.

[0017] The vaporization temperature of the silicon source powder is preferably 1300-190℃, more preferably 1500-1800℃, for example 1600℃.

[0018] The rate at which the temperature rises from room temperature to the vaporization temperature of the silicon source powder can be 3-7°C / min, for example, 4, 5 or 6°C / min.

[0019] The vaporization time of the silicon source powder can be 1-3 hours, for example, 2 hours.

[0020] The gasification of the magnesium powder and silicon source powder can be carried out in a rotary kiln, preferably in two vacuum heating chambers in the rotary kiln.

[0021] In the silicon source powder, the atomic ratio of silicon to oxygen can be 1:(0.8-1.2), for example, 1:1. The D50 particle size of the silicon source powder can be 3-7 μm, for example, 5 μm.

[0022] In this invention, the mass ratio of magnesium to silicon source in the mixed gas is preferably (0.52-1.80):1, for example 0.54:1, 1.09:1 or 1.63:1.

[0023] In this invention, the porous carbon can be conventional in the art. The specific surface area of ​​the porous carbon can be 1800-2000 m². 2 / g. The pore volume of the porous carbon can be 0.7-0.9m. 3 / g. The D50 particle size of the porous carbon can be 7-9 μm.

[0024] In this invention, the deposition can be carried out in a rotary kiln. Preferably, the deposition is carried out under a vacuum of 0.01-0.03 Pa.

[0025] In this invention, the deposition temperature can be 500-700℃, for example 600℃.

[0026] In this invention, the rate of heating from room temperature to the temperature at which the deposition is made can be 3-7°C / min, for example 4, 5 or 6°C / min.

[0027] In this invention, the deposition time can be 8-12 hours, for example, 10 hours.

[0028] In this invention, the mass ratio of the porous carbon to the magnesium in the mixed gas can be 1:(0.5-1.2), preferably 1:(0.8-1.0), for example 1:0.86.

[0029] In this invention, the mass ratio of the porous carbon to the silicon source in the mixed gas can be 1:(0.5-1.2), preferably 1:(0.6-0.9), for example 1:0.79.

[0030] In this invention, after deposition and before gaseous carbon coating, a hydrochloric acid washing step is preferably performed. The molar concentration of the hydrochloric acid solution used for washing can be 5-7 mol / L, for example, 6 mol / L. The washing time can be 1-5 hours, for example, 3 hours.

[0031] In this invention, the carbon coating can be conventional in the art, preferably including liquid phase carbon coating and / or gas phase carbon coating, such as gas phase carbon coating.

[0032] The gas-phase carbon coating can be carried out in a rotary kiln.

[0033] The carbon source gas for the gaseous carbon coating can be conventional in the art, and preferably includes one or more of alkynes, alkanes, alkenes, aromatic compounds and carbohydrates, such as acetylene.

[0034] The temperature of the gas phase carbon coating can be 600-800℃, for example 700℃.

[0035] The time for gaseous carbon coating can be 1-5 hours, for example, 3 hours.

[0036] The rate at which the temperature is increased from room temperature to the temperature of the gaseous carbon coating can be 3-7°C / min, for example 4, 5 or 6°C / min.

[0037] The mass-to-volume ratio of the porous carbon to the carbon source coated with gaseous carbon can be 1g:(0.1-0.2)L, for example, 1g:0.18L.

[0038] The present invention also provides a silicon-carbon composite material, which is prepared by the silicon-carbon composite material preparation method described above.

[0039] In this invention, the magnesium content in the silicon-carbon composite material can be 0.04%-30.1% by mass, for example 6.73%, 0.81%, 0.66%, 0.49%, 0.32%, 0.25%, 0.24%, 0.11% or 0.05%.

[0040] In this invention, the silicon content in the silicon-carbon composite material can be 1.73%-30.7% by mass, for example 30.6%, 30.1%, 29.7%, 28.4%, 25.3%, 21.5%, 15.6% or 12.6%.

[0041] The present invention also provides an electrode comprising the silicon-carbon composite material as described above.

[0042] The present invention also provides an application of the silicon-carbon composite material as described above in an electrochemical device.

[0043] The present invention also provides an electrochemical device comprising the electrode as described above.

[0044] In this invention, the electrochemical device is preferably a lithium-ion battery.

[0045] Based on common knowledge in the field, the above-mentioned preferred conditions can be combined arbitrarily to obtain various preferred embodiments of the present invention.

[0046] The reagents and raw materials used in this invention are all commercially available.

[0047] The positive and progressive effects of this invention are as follows:

[0048] (1) In this invention, the mixed gas includes magnesium and silicon source. The magnesium thermal reaction between magnesium and silicon dioxide in the silicon source will form a structure in which silicon particles are wrapped with magnesium oxide. The magnesium thermal reaction in the gas phase system will make the generated structure more effectively deposited in the pores of porous carbon. The pore structure of porous carbon can effectively alleviate the volume expansion of silicon particles. Preferably, the silicon particles are further transformed into porous silicon by subsequent selective washing. The special structure of porous silicon can further alleviate the stress caused by silicon expansion.

[0049] (2) The battery made of the silicon-carbon composite material of the present invention has better cycle performance. Detailed Implementation

[0050] The present invention is further illustrated below by way of embodiments, but the invention is not limited to the scope of the embodiments described herein. Experimental methods in the following embodiments that do not specify specific conditions were performed according to conventional methods and conditions, or as selected according to the product instructions.

[0051] Example 1

[0052] S1. Gasification process of magnesium powder and SiOx powder: 860g of Mg powder (D50 of 5μm) and 790g of SiOx powder (atomic ratio of silicon to oxygen of 1:1, D50 of 5μm) were placed in two vacuum heating chambers of a rotary kiln respectively. The Mg powder chamber was heated to 600℃ at a heating rate of 5℃ / min and held at the temperature for 2h. Then, nitrogen gas was introduced at a rate of 2L / min for 30min and then stopped for 30min. The SiOx powder chamber was heated to 1500℃ at a heating rate of 5℃ / min and held at the temperature for 2h. Then, nitrogen gas was introduced at a rate of 2L / min for 30min and then stopped for 30min.

[0053] S2. (1) Deposition process: Weigh 1 kg of porous carbon (specific surface area of ​​1800-2000 m²) 2 / g, pore volume can be 0.7-0.9m 3 The vaporized magnesium (Mg, D50 7-9 μm) was placed in a rotary kiln, evacuated to 0.01 Pa, and heated to 600 °C at a rate of 5 °C / min. Then, the vaporized magnesium and vaporized SiOx were introduced into the rotary kiln through the outlet pipes of the two vacuum heating chambers, respectively. After maintaining the temperature for 10 h, the gas source was turned off, and nitrogen was introduced at a rate of 5 L / min. The mass ratio of vaporized magnesium (Mg) to vaporized SiOx was 1.09:1.

[0054] (2) The material was then discharged under nitrogen conditions and washed with 6 mol / L hydrochloric acid for 3 hours.

[0055] (3) The pickled material was then put back into the rotary kiln and heated to 700°C at 5°C / min. Acetylene gas was introduced at 1L / min, and the gas phase carbon was coated for 3 hours before being shut off. The material was then cooled and discharged.

[0056] Example 2

[0057] The only difference from Example 1 is that the amount of Mg powder added in S1 is changed to 430g.

[0058] Example 3

[0059] The only difference from Example 1 is that the amount of Mg powder added in S1 is changed to 1290g.

[0060] Example 4

[0061] The only difference from Example 1 is that the temperature of the Mg powder cavity in S1 is 400°C.

[0062] Example 5

[0063] The only difference from Example 1 is that the temperature of the Mg powder cavity in S1 is 800°C.

[0064] Example 6

[0065] The only difference from Example 1 is that the temperature of the SiOx cavity in S1 is 1800°C.

[0066] Example 7

[0067] The only difference from Example 1 is that the pickling time in S2(2) is 1 hour.

[0068] Example 8

[0069] The only difference from Example 1 is that the pickling time in S2(2) is 5h.

[0070] Example 9

[0071] The only difference from Example 1 is that S2(2) is omitted.

[0072] Comparative Example 1

[0073] The only difference from Example 1 is that no magnesium powder is added.

[0074] Comparative Example 2

[0075] The only difference from Example 1 is that SiOx is not added.

[0076] Comparative Example 3

[0077] The only difference from Example 1 is that the temperature of the SiOx cavity in S1 is 900°C. In this comparative example 3, because the vaporization temperature of SiOx is low, the silicon suboxide in the cavity cannot be completely vaporized. Therefore, the mass ratio of the vaporized magnesium (Mg) to the vaporized SiOx is greater than 1.09:1.

[0078] Comparative Example 4

[0079] The only difference from Example 1 is that the temperature of the silicon suboxide cavity in S1 is 1200°C. In this comparative example 3, because the vaporization temperature of SiOx is low, the silicon suboxide in the cavity cannot be completely vaporized. Therefore, the mass ratio of the vaporized magnesium (Mg) to the vaporized SiOx is greater than 1.09:1.

[0080] Example 1

[0081] The elemental content of the silicon-carbon composite materials prepared in the above embodiments or comparative examples was tested.

[0082] Among them, the magnesium content was tested by an ICP analyzer; the test method was as follows: (1) Microwave digestion pretreatment: 0.5g of silicon-carbon composite material prepared in the example or comparative example was placed in a digestion vessel, and 3mL of concentrated nitric acid and 9mL of concentrated sulfuric acid were added in sequence with a liquid adder, and shaken; and then kept at 160℃ for 3min, 180℃ for 3min, 200℃ for 10min, and 220℃ for 10min in sequence, and then cooled to room temperature, filtered with quantitative filter paper, and the filtrate was placed in a 50mL volumetric flask as the test solution; (2) The above test solution was tested by a PerkinElmer ICP spectrometer Avio TM The test was conducted in 200 equipment; the test conditions were an air pressure of 2.5 MPa.

[0083] The method for testing silicon content is as follows: 20g of the silicon-carbon composite material prepared in the above examples or comparative examples is placed in a muffle furnace, heated to 1500°C under oxygen conditions, and the mass of the discharged material is weighed to calculate the silicon content.

[0084] The results are shown in Table 1.

[0085] Table 1

[0086] magnesium content Silicon content Example 1 0.32 30.1 Example 2 0.11 25.3 Example 3 0.66 29.7 Example 4 0.04 21.5 Example 5 0.24 30.6 Example 6 0.25 30.7 Example 7 6.73 28.4 Example 8 0.05 30.7 Example 9 30.1 15.6 Comparative Example 1 0 21.7 Comparative Example 2 0.53 0 Comparative Example 3 0.81 1.73 Comparative Example 4 0.49 12.6

[0087] Example 2

[0088] A mixture of silicon-carbon composite material, binder (sodium carboxymethyl cellulose, CMC; styrene-butadiene rubber, SBR), and conductive agent (conductive carbon black, SP; single-walled carbon nanotubes, SWCNT) prepared in the above examples or comparative examples (wherein the mass ratio of silicon-carbon composite material: SP: SWCNT: CMC: SBR = 90: 4.5: 0.5: 2.5: 2.5) was homogenized and coated onto copper foil. The resulting negative electrode was obtained by vacuum drying, rolling, and slitting. A lithium metal sheet was used as the counter electrode. The electrolyte consisted of 1M LiPF6, 89% ethylene carbonate (EC) and diethyl carbonate (DEC) by volume (volume ratio 1:1), 10% fluoroethylene carbonate (FEC) by volume, and 1% vinylene carbonate (VC) by volume. A polypropylene microporous membrane was used as the separator. The mixture was assembled into a coin cell using an argon-filled inert gas glove box system.

[0089] At 25℃, the assembled half-cell was subjected to charge-discharge cycle tests using a Blue Electric testing instrument to obtain data such as capacity, initial coulombic efficiency, and cycle life. The test procedure was: 0.1C discharge to 5mV, 0.05C discharge to 5mV, 0.1C charge to 2V; cycle format: 0.5C discharge to 5mV, 0.05C discharge to 5mV, 0.5C charge to 1V.

[0090] The results are shown in Table 2.

[0091] Table 2

[0092] Reversible specific capacity (mAh / g) First-time coulomb efficiency (%) 500-week cycle retention rate (%) Example 1 1244 85.3 94.8 Example 2 957 81.9 82.3 Example 3 1189 83.4 91.2 Example 4 412 74.7 78.4 Example 5 1267 84.6 93.1 Example 6 1272 84.3 92.7 Example 7 1121 81.2 84.3 Example 8 1058 81.6 89.6 Example 9 794 73.9 82.4 Comparative Example 1 431 74.2 76.5 Comparative Example 2 164 21.9 37.7 Comparative Example 3 186 27.7 46.8 Comparative Example 4 447 68.1 57.2

[0093] As shown in Table 2, compared with Comparative Examples 1-4, Examples 1-9 obtained a first mixed gas by vaporizing magnesium powder and silicon suboxide powder at a specific temperature, and then deposited it in porous carbon for gas-phase carbon coating to prepare silicon-carbon composite materials. The batteries made from the silicon-carbon composite materials obtained in Examples 1-9 exhibit better initial coulombic efficiency and cycle performance.

Claims

1. A method for preparing a silicon-carbon composite material, characterized in that, It includes the following steps: A mixed gas is deposited in porous carbon and then coated with carbon; the mixed gas includes magnesium and silicon source in gaseous form; the mass ratio of magnesium to silicon source in the mixed gas is (1.09-2):1, and the silicon source includes silicon oxide; The carbon coating is a gaseous carbon coating; after deposition and before the gaseous carbon coating, a hydrochloric acid washing step is also performed; The method for preparing the mixed gas includes mixing magnesium powder and silicon source powder after separately vaporizing them; the vaporization temperature of the magnesium powder is 600-800℃; and the vaporization temperature of the silicon source powder is 1300-1800℃.

2. The method for preparing the silicon-carbon composite material as described in claim 1, characterized in that, The preparation method of the silicon-carbon composite material satisfies one or more of the following conditions: A. The chemical formula of the silicon oxide is SiOx, where 0 < x < 2; B. The silicon source also includes silicon; C. In the mixed gas, the mass ratio of magnesium to silicon source is (1.09-1.80):1; D. The specific surface area of ​​the porous carbon is 1800-2000 m². 2 / g; E. The pore volume of the porous carbon is 0.7-0.9 μm. 3 / g; F. The D50 particle size of the porous carbon is 7-9 μm.

3. The method for preparing the silicon-carbon composite material as described in claim 2, characterized in that, The chemical formula of the silicon oxide is SiOx, where 0.8 < x < 1.

2.

4. The method for preparing the silicon-carbon composite material as described in claim 2, characterized in that, The chemical formula of the silicon oxide is SiOx, where x is 1.

5. The method for preparing the silicon-carbon composite material as described in claim 2, characterized in that, In the mixed gas, the mass ratio of magnesium to silicon source is 1.09:1 or 1.63:

1.

6. The method for preparing the silicon-carbon composite material as described in claim 1, characterized in that, The preparation method of the silicon-carbon composite material satisfies one or more of the following conditions: A. The deposition temperature is 500-700℃; B. The rate of heating from room temperature to the temperature at which the deposition is made is 3-7 °C / min; C. The deposition time is 8-12 hours; D. The mass ratio of the porous carbon to the magnesium in the mixed gas is 1:(0.5-1.2). E. The mass ratio of the porous carbon to the silicon source in the mixed gas is 1:(0.5-1.2).

7. The method for preparing the silicon-carbon composite material as described in claim 6, characterized in that, The preparation method of the silicon-carbon composite material satisfies one or more of the following conditions: A. The deposition temperature is 600℃; B. The rate of heating from room temperature to the temperature at which the deposition is made is 4°C / min, 5°C / min, or 6°C / min; C. The deposition time is 10 hours; D. The mass ratio of the porous carbon to the magnesium in the mixed gas is 1:(0.8-1.0). E. The mass ratio of the porous carbon to the silicon source in the mixed gas is 1:(0.6-0.9).

8. The method for preparing the silicon-carbon composite material as described in claim 6, characterized in that, The preparation method of the silicon-carbon composite material satisfies one or more of the following conditions: A. The mass ratio of the porous carbon to the magnesium in the mixed gas is 1:0.86; B. The mass ratio of the porous carbon to the silicon source in the mixed gas is 1:0.

79.

9. The method for preparing the silicon-carbon composite material as described in claim 2, characterized in that, The method for preparing the mixed gas satisfies one or more of the following conditions: A. The vaporization temperature of the magnesium powder is 600 or 700℃; B. The rate of heating from room temperature to the vaporization temperature of the magnesium powder is 3-7℃ / min; C. The gasification time of the magnesium powder is 1-3 hours; D. The D50 particle size of the magnesium powder is 3-7 μm; E. The vaporization temperature of the silicon source powder is 1500-1800℃; F. The rate of heating from room temperature to the vaporization temperature of the silicon source powder is 3-7℃ / min; G. The vaporization time of the silicon source powder is 1-3 hours; H. The D50 particle size of the silicon source powder is 3-7 μm; I. The mixing is achieved by simultaneously introducing gaseous magnesium and gaseous silicon sources into the deposition equipment during the deposition process.

10. The method for preparing the silicon-carbon composite material as described in claim 9, characterized in that, The method for preparing the mixed gas satisfies one or more of the following conditions: A. The rate of heating from room temperature to the vaporization temperature of the magnesium powder is 4℃ / min, 5℃ / min or 6℃ / min; B. The gasification time of the magnesium powder is 2 hours; C. The D50 particle size of the magnesium powder is 5 μm; D. The vaporization temperature of the silicon source powder is 1600℃; E. The rate of heating from room temperature to the vaporization temperature of the silicon source powder is 4°C / min, 5°C / min, or 6°C / min. F. The vaporization time of the silicon source powder is 2 hours; G. The D50 particle size of the silicon source powder is 5 μm.

11. The method for preparing the silicon-carbon composite material as described in claim 1, characterized in that, The gas-phase carbon coating satisfies one or more of the following conditions: A. The temperature of the gaseous carbon coating is 600-800℃; B. The time for gaseous carbon coating is 1-5 hours; C. The rate of heating from room temperature to the temperature of the gaseous carbon coating is 3-7℃ / min; D. The carbon source for gas-phase carbon coating includes one or more of alkynes, alkanes, alkenes, aromatic compounds, and carbohydrates; E. The mass-to-volume ratio of the porous carbon to the carbon source coated with gaseous carbon is 1 g: (0.1-0.2) L.

12. The method for preparing the silicon-carbon composite material as described in claim 11, characterized in that, The gas-phase carbon coating satisfies one or more of the following conditions: A. The temperature of the gaseous carbon coating is 700℃; B. The time for gaseous carbon coating is 3 hours; C. The rate of heating from room temperature to the temperature of the gaseous carbon coating is 4℃ / min, 5℃ / min or 6℃ / min; D. The carbon source for the gaseous carbon coating is acetylene; E. The mass-to-volume ratio of the porous carbon and the carbon source coated with gaseous carbon is 1g:0.18L.

13. The method for preparing the silicon-carbon composite material as described in claim 1, characterized in that, The molar concentration of the hydrochloric acid solution used in the hydrochloric acid washing is 5-7 mol / L; And / or, the hydrochloric acid washing time is 1-5 hours.

14. The method for preparing the silicon-carbon composite material as described in claim 13, characterized in that, The molar concentration of the hydrochloric acid solution used in the hydrochloric acid washing is 6 mol / L; And / or, the hydrochloric acid washing time is 3 hours.

15. A silicon-carbon composite material, characterized in that, It is prepared by the method of any one of claims 1-14 for the preparation of silicon-carbon composite materials.

16. The silicon-carbon composite material as described in claim 15, characterized in that, The silicon-carbon composite material satisfies one or more of the following conditions: A. In the silicon-carbon composite material, the magnesium content is 0.04%-30.1% by mass; B. In the silicon-carbon composite material, the mass content of silicon is 1.73%-30.7%.

17. The silicon-carbon composite material as described in claim 16, characterized in that, The silicon-carbon composite material satisfies one or more of the following conditions: A. In the silicon-carbon composite material, the mass content of magnesium is 6.73%, 0.81%, 0.66%, 0.49%, 0.32%, 0.25%, 0.24%, 0.11%, or 0.05%; B. In the silicon-carbon composite material, the mass content of silicon is 30.6%, 30.1%, 29.7%, 28.4%, 25.3%, 21.5%, 15.6% or 12.6%.

18. An electrode sheet, characterized in that, It includes silicon-carbon composite materials as described in any one of claims 15-17.

19. The application of the electrode as described in claim 18 in an electrochemical device.

20. An electrochemical device, characterized in that, It includes the electrode as described in claim 18.