Pressure-strain decoupling sensor based on pipeline structure and preparation method
By introducing axisymmetric arcuate pipeline structure and low-dimensional nanomaterials into the sensor, the problem of insufficient signal decoupling ability in high-precision decoupling scenarios is solved, and high-precision detection of human motion composite signals is achieved and Decoupling improves the stability and sensitivity of the sensor.
Patent Information
- Application Number
- CN202410916686.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-07-09
- Publication Date
- 2024-12-17
- Estimated Expiration
- 2044-07-09
AI Technical Summary
Existing pressure-strain sensors show limited decoupling capabilities of strain and pressure signals in scenarios with high decoupling accuracy requirements, which cannot meet the high-precision requirements of complex human motion monitoring.
Adopt a pressure-strain decoupling sensor based on the pipeline structure, and an axisymmetric arcuate hollow structure is set up in the dielectric layer. Through the distribution of the central pipe and the arcuate pipe, the Poisson ratio is improved, the changes in the electrode plate area are reduced, and the signal decoupling is improved. performance and improve device stability through a hybrid conductor of low-dimensional nanomaterials and elastic polymers.
It realizes high-precision decoupling of pressure and strain signals, improves the sensor's detection ability of composite signals, meets the needs of high-precision application scenarios, and enhances the reliability and sensitivity of the device.
Smart Images

Figure CN118654728B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to a flexible wearable sensor, and in particular to a pressure-strain decoupling flexible sensor based on a pipeline structure and a preparation method thereof. Background Art
[0002] With the discovery of new materials such as conductive polymers, carbon nanotubes, and graphene, we have begun to explore how to use the excellent properties of these materials to make bendable and stretchable sensors. These sensors are mainly concentrated in basic types such as resistive, capacitive, and photoelectric, and are used in some occasions with special requirements for flexibility, such as heart rate monitoring of wearable devices, simple pressure sensing, etc. And with the surge in demand in the fields of Internet of Things (IoT), smart wearables, medical health, human-computer interaction, etc., this type of flexible sensor technology has ushered in a period of rapid development, and the types and functions of sensors have become more diversified.
[0003] Among them, the use of wearable flexible sensors to monitor and identify human motion characteristics is a basic function for understanding human activities, and is widely used in emerging application fields such as precision medicine and smart sports. However, human motion is a complex process involving multiple joints. Since flexible sensors can contact the skin on the surface of the human body, these sensors are stimulated by bone compression pressure and skin tensile strain at the same time. Under different stimuli, most sensors produce similar electrical signal outputs, so it is necessary to develop a pressure-strain sensor to monitor and decouple the strain and pressure signals generated simultaneously during joint motion. However, there are currently few studies on such sensors that can achieve decoupling of different mechanical signals. Although the inventor's previous patent can achieve the decoupling function, the accuracy of the decoupling still needs to be improved.
[0004] Although the above-mentioned problem has been addressed in the patent document with application number 202110812835.7 and the invention name is a flexible sensor for simultaneous detection of pressure-strain dual-modal signals, which aims to solve the problem of decoupling strain and pressure signals in different parts of the human body, it was found in actual use that due to the limitations of the original structure, the sensor has limited decoupling ability for strain and pressure signals and is not suitable for scenarios with high decoupling accuracy requirements. Summary of the invention
[0005] The technical problem to be solved by the present invention is to improve the dielectric layer structure of the sensor and the entire process of sensor preparation on the basis of the existing sensor, and propose a pressure-strain decoupled sensor and preparation method based on a pipeline structure, so that the sensor is suitable for scenarios with high decoupling accuracy requirements.
[0006] Technical solution: The present invention proposes a pressure-strain decoupled sensor based on a pipeline structure, including a flexible substrate, a lower flexible electrode plate, a dielectric layer, an upper flexible electrode plate, and a packaging protective layer arranged from bottom to top; a pipeline-shaped hollow structure is arranged inside the dielectric layer.
[0007] The upper flexible electrode plate and the lower flexible electrode plate can be used as strain sensors to sense the tensile strain of the sensor as a whole; they can also be used as the upper and lower plates of the capacitive sensor to sense the pressure applied to the sensor based on the sensing principle of the capacitive sensor; the existence of the pipe-shaped hollow structure can improve the Poisson's ratio of the sensor.
[0008] Furthermore, the pipe-shaped hollow structure is an axisymmetric structure; it includes a central pipe, and other pipes distributed on both sides of the central pipe and symmetrically distributed with respect to the central pipe.
[0009] The x-axis direction is defined as the direction along the center of the pipeline, the y-axis direction is perpendicular to the x-axis direction, the plane where the x-axis and y-axis are located is parallel to the plane where the dielectric layer is located, and the z-axis is perpendicular to the plane where the x-axis and y-axis are located. Due to the existence of the pipeline-shaped hollow structure, the contraction in the y direction under the action of tensile strain along the x direction is promoted and the deformation in the z direction is suppressed, thereby improving the decoupling performance of the device for pressure and strain signals.
[0010] Furthermore, the other pipes and the central pipe are located on the same plane, and the plane is parallel to the surface of the dielectric layer.
[0011] If the pipes are not located in the same plane, the Poisson's ratio of the upper and lower surfaces may be inconsistent, resulting in large performance differences when the upper and lower flexible electrode plates are used as resistive strain sensors, and even affecting the stability of the device.
[0012] Furthermore, the other pipes are arcuate, and the opening direction of the arcuate faces away from the central pipe.
[0013] Furthermore, the radius of curvature of the other pipes is:
[0014] ;
[0015] The curvature of the other pipes is:
[0016] ;
[0017] Wherein, L represents the length of the dielectric layer, d represents the width of the dielectric layer, and the central pipe is arranged along the length direction of the dielectric layer.
[0018] Furthermore, the central pipe and other pipes have the same diameter, and the pipe cross-sections of the central pipe and other pipes are circular. If the cross-section is a non-circular pipe, such as a triangle or square, there are sharp corners, which may cause tearing under large strain or pressure, resulting in damage to the device.
[0019] Furthermore, the lower flexible electrode plate and the upper flexible electrode plate are made of the same material, which is a mixed conductor of low-dimensional nanomaterials and elastic polymers; the low-dimensional nanomaterials include one-dimensional carbon nanowires, metal nanowires and two-dimensional graphene nanosheets, MXene nanosheets, etc.;
[0020] The dielectric layer and the flexible substrate are made of the same material, which is an elastic polymer, and the same elastic polymer as that used in the flexible electrode plate. The bottom-up integral casting and curing can be achieved, and the elastic polymer includes but is not limited to elastic materials with high dielectric constant and low conductivity such as PDMS and Ecoflex.
[0021] Furthermore, the present invention provides a pressure-strain decoupled sensor based on a pipeline structure, comprising a central pipeline, a first pipeline, and a second pipeline; the central pipeline is arranged along the length direction of the dielectric layer; the first pipeline and the second pipeline have the same structure and are symmetrically arranged on both sides of the central pipeline; the first pipeline and the second pipeline are both arched, and the opening direction of the arch is away from the central pipeline.
[0022] A pressure-strain decoupling sensor based on a pipeline structure according to the present invention, and a preparation method thereof comprises the following steps:
[0023] Step 1, the elastic polymer is stirred and evenly poured into a template, and after vacuuming and removing bubbles, a flexible substrate is obtained;
[0024] The template includes a rectangular groove structure;
[0025] Step 2, mixing the low-dimensional nanomaterial into the precursor of the elastic polymer, stirring thoroughly, adding a curing agent after uniform dispersion, to obtain a mixture of the low-dimensional nanomaterial and the elastic polymer, and pouring it on the flexible substrate to obtain a lower flexible electrode plate located on the upper surface of the flexible substrate;
[0026] Step 3, placing a sacrificial ice mold on the surface of the lower flexible electrode plate in the template, pouring an elastic polymer to cover the sacrificial ice mold and form a dielectric layer;
[0027] Step 4, pouring a mixture of low-dimensional nanomaterials and elastic polymers on the dielectric layer to obtain an upper flexible electrode plate;
[0028] The mixture of the low-dimensional nanomaterial and the elastic polymer is the same as the mixture of the low-dimensional nanomaterial and the elastic polymer in step 2;
[0029] Step 5, pouring an elastic polymer on the surface of the upper flexible electrode plate as a packaging protective layer;
[0030] In step 6, the template and the material poured in the template are placed in an oven for curing. During the curing process, the sacrificial ice mold melts and evaporates to form a pipeline structure in the dielectric layer.
[0031] The sacrificial ice mold described in step 3 includes a left vertical plate, a right vertical plate, a central column connected between the left vertical plate and the right vertical plate, and other columns, the other columns are symmetrical about the central column, and the other columns are arched structures; the left vertical plate, the right vertical plate, the central column and other columns are all formed of ice; after the sacrificial ice mold melts and evaporates in step 6, the position of the central column forms the central pipe in the dielectric layer, and the positions of the other columns form other pipes in the dielectric layer.
[0032] The sacrificial ice mold in step 3 is formed by injecting deionized water into a silicone mold and freezing the mold to prepare a columnar body corresponding to the pipeline structure.
[0033] Beneficial effects: Compared with the prior art, the present invention has the following beneficial effects: On the one hand, the pressure-strain decoupling sensor based on the pipeline structure can detect pressure and strain signals at the same time, and effectively decouple the composite signal of pressure-strain generated by human joint movement. Compared with the inventor's previous porous structure and unstructured dielectric layer design, the existence of the arched pipeline can suppress the overall deformation of the sensor in the z-axis direction, promote the contraction of the sensor in the y direction, and thus reduce the change in the electrode distance and electrode plate area of the capacitive sensor under strain, reduce the sensitivity of its capacitance value to strain, and promote the offset of the capacitance value inflection point of the capacitive sensor; and the pores introduced by the existence of the pipeline structure are also conducive to improving the sensitivity of the capacitive sensor to pressure, so the decoupling ability is greatly improved, which can meet various application scenarios with higher requirements for decoupling accuracy.
[0034] On the other hand, the upper and lower flexible electrode plates are improved, and a mixed conductor precursor is prepared by mixing low-dimensional nanomaterials with elastic polymers. The elastic polymer used is the same as the material of the flexible substrate and the dielectric layer with a pipeline structure, so it can be poured layer by layer and uniformly cured after pouring, which enhances the adhesion between the layers and is conducive to improving the overall reliability of the device. BRIEF DESCRIPTION OF THE DRAWINGS
[0035] Figure 1 Is a schematic diagram of the structure of the flexible sensor of the present invention;
[0036] Figure 2 It is a two-dimensional schematic diagram of the flexible sensor pipeline structure of the present invention;
[0037] Figure 3 This is a schematic diagram of the flexible sensor of the present invention under tensile strain;
[0038] Figure 4 Is a schematic diagram of the flexible sensor of the present invention under pressure;
[0039] Figure 5 It is a flow chart of the preparation of a structural cross-sectional view of step one of the method for preparing a flexible sensor of the present invention;
[0040] Figure 6 Is a flow chart of the structure of the flexible sensor preparation method of the present invention step two cross-sectional view preparation;
[0041] Figure 7 Is a flow chart of the preparation of a flexible sensor structure of step three of the preparation method of the present invention;
[0042] Figure 8 The present invention is a flexible sensor preparation method step four structural cross-sectional view preparation flow chart;
[0043] Fig. 9 Is a flow chart of the preparation of a flexible sensor structure cross-sectional view of step five of the preparation method of the present invention;
[0044] Fig.10 Is a flow chart of the preparation of a flexible sensor structure cross-sectional view of step six of the preparation method of the present invention;
[0045] Fig.11 It is a flow chart of the preparation of the structural cross-sectional view of step seven of the method for preparing a flexible sensor of the present invention.
[0046] Among them, 1. flexible substrate; 2. lower flexible electrode plate; 3. dielectric layer; 4. upper flexible electrode plate; 5. packaging protection layer; 31. first pipeline; 32. central pipeline; 33. second pipeline. DETAILED DESCRIPTION
[0047] The present invention will be further explained below in conjunction with the accompanying drawings.
[0048] like Figure 1 As shown, a pressure-strain decoupled sensor based on a pipeline structure includes a flexible substrate 1 arranged from bottom to top, a lower flexible electrode plate 2, a dielectric layer 3 with a pipeline-shaped hollow structure, an upper flexible electrode plate 4, and a packaging protective layer 5.
[0049] The flexible substrate 1 is made of elastic polymer material, including but not limited to PDMS, Ecoflex and other elastic polymer materials with high dielectric constant and low conductivity;
[0050] The lower flexible electrode plate 2 and the upper flexible electrode plate 4 have the same structure and use the same materials; the lower flexible electrode plate 2 and the upper flexible electrode plate 4 both use a mixed conductor of low-dimensional nanomaterials and elastic polymers, and the low-dimensional nanomaterials are uniformly dispersed in the elastic polymer substrate; the low-dimensional nanomaterials can be one-dimensional carbon nanowires, metal nanowires, two-dimensional graphene nanosheets or MXene nanosheets, etc.; the elastic polymer material used in the substrate of the lower flexible electrode plate 2 and the upper flexible electrode plate 4 is the same as the elastic polymer material used in the flexible substrate 1; the lower flexible electrode plate 2 and the upper flexible electrode plate 4 can be used as strain sensors to sense the tensile strain exerted on the sensor as a whole, and can also be used as two electrode plates of a capacitive sensor to sense the pressure exerted on the sensor based on the sensing principle of the capacitive sensor.
[0051] The material used for the dielectric layer 3 is the same as that of the flexible substrate 1 , and can be cast and solidified in an integrated manner from bottom to top.
[0052] The dielectric layer 3 having a hollow tube structure can improve the Poisson's ratio of the sensor by the presence of the arched tube. When tensile strain is applied along the x-axis direction, the presence of the arched tube makes the Poisson's ratio V xy Increase and V xz The capacitance sensor has a smaller electrode distance change and a smaller electrode plate area change under the action of strain. Therefore, the sensitivity of the capacitance sensor to strain is weakened, so that the change of the resistance sensor is dominant under the action of strain. The device's ability to decouple strain and pressure is improved, and the Poisson's ratio of the sensor can be further improved by further changing the diameter of the pipe.
[0053] The pipe-shaped hollow structure is an axisymmetric structure; it includes a central pipe 32, and other pipes distributed on both sides of the central pipe 32 and symmetrically distributed about the central pipe 32. The other pipes may be two, four, six, etc.; in this specific embodiment, the other pipes are two, including a first pipe 31 and a second pipe 33; Figure 2 As shown, the dielectric layer 3 includes a central pipe 32 and a first pipe 31 and a second pipe 33 symmetrically having curvatures on both sides. The first pipe 31 and the second pipe 33 are both arc-shaped, also called arc-shaped pipes, and the three pipes have the same pipe diameter. If the sizes are inconsistent, the upper and lower electrode layers will bend in the plane under stretching, and the capacitive sensor will no longer be a parallel plate capacitor. The relationship between the decoupling performance and the change in size is unknown.
[0054] The direction along the center of the pipeline is defined as the x-axis direction, the y-axis direction is perpendicular to the x-axis direction, the plane where the x-axis and y-axis are located is parallel to the plane where the dielectric layer 3 is located, and the z-axis is perpendicular to the plane where the x-axis and y-axis are located; the contraction in the y-direction under the action of tensile strain in the x-direction is promoted and the deformation in the z-direction is suppressed, thereby improving the decoupling performance of the device for pressure and strain signals.
[0055] The curvature radius and curvature of the arched pipe directly affect the decoupling performance of the sensor, such as Figure 3 and 4 As shown in the figure, the principle of improving decoupling performance is as follows:
[0056] When tensile strain is applied to the sensor along the x-axis, the presence of the arched pipe can suppress the overall deformation of the sensor in the z-axis direction and promote the contraction of the sensor in the y-direction, thereby reducing the change in the electrode distance and electrode plate area of the capacitive sensor under the action of strain, reducing the sensitivity of its capacitance value to strain, and reducing the amplitude and curvature radius, arc and inner diameter of the arched structure. Under the same inner diameter, the smaller the radius of curvature and the larger the arc, the lower the sensitivity of the capacitive sensor to strain. The change in the radius of curvature and the arc have little effect on the pressure sensing sensitivity of the capacitive sensor and can be ignored. Under the same radius of curvature and arc, the larger the inner diameter, the more sensitive the capacitive sensor is to pressure, the less sensitive it is to strain, and the stronger the decoupling ability.
[0057] When pressure is applied to the sensor along the z-axis direction, the Young's modulus of the sensor is reduced to a certain extent due to the pores introduced by the pipe-shaped hollow structure, and a larger z-axis deformation can be produced under the same pressure, thereby improving the sensitivity of the capacitive sensor to pressure. Therefore, the decoupling performance is greatly improved under the action of the pipe-shaped hollow structure. The dielectric layer 3 having a pipe-shaped hollow structure, and the pipe-shaped hollow structure with different curvature radii and curvatures is prepared by an ice template sacrificial method. Deionized water is injected into a silicone mold and frozen to form a columnar body corresponding to the pipe-shaped hollow structure. The columnar structure is used as a mold for casting. During the heating and curing process, the ice template dissolves and evaporates due to heat, leaving the required pipe-shaped hollow structure.
[0058] The structural design and preparation method of the pressure-strain decoupling sensor based on the pipeline structure comprises the following steps:
[0059] Step 1: If Figure 5 As shown, the elastic polymer is stirred and evenly poured into the template, and a vacuum drying oven is used to evacuate and remove bubbles to prepare a flexible substrate 1;
[0060] Step 2: Mix the low-dimensional nanomaterials used into the precursor of the elastic polymer and stir them thoroughly with a magnetic stirrer; add a curing agent after the mixture is evenly dispersed and pour it on the flexible substrate 1 to prepare the lower flexible electrode plate 2, such as Figure 6 As shown;
[0061] If the elastic polymer is Ecoflex, the elastic polymer precursor in this step is Ecoflex part A, and the curing agent is Ecoflex part B;
[0062] If the elastic polymer is PDMS, the elastic polymer precursor in this step is a PDMS precursor, and the curing agent is a PDMS curing agent;
[0063] Step 3: Deionized water is injected into the silicone mold with a pipe-shaped hollow structure, fully frozen and demolded to obtain a sacrificial ice mold 6; the sacrificial ice mold is placed in a casting template, such as Figure 7 As shown, it is used for casting the dielectric layer 3 having a pipe-shaped hollow structure;
[0064] Step 4: Figure 8 As shown, an elastic polymer is poured on the sacrificial ice mold 6 to construct a dielectric layer 3 having a pipe-shaped hollow structure on the lower flexible electrode plate 2;
[0065] Step 5: Continue to pour the mixture of low-dimensional nanomaterial and elastic polymer used in step 2 on the dielectric layer 3 with a pipe-shaped hollow structure as the upper flexible electrode plate 4, such as Fig. 9 As shown;
[0066] Step 6: Continue to pour the elastic polymer on the top of the upper flexible electrode plate 4 as the device packaging protection layer 5, such as Fig.10 As shown;
[0067] Step 7: Place the prepared device together with the mold in an oven and cure at 80°C. During the curing process, the sacrificial ice mold melts and evaporates, leaving a pipe-shaped hollow structure in the dielectric layer 3, as shown in FIG. Fig.11 As shown, the device preparation is completed.
[0068] The pressure-strain decoupling sensor of the present invention integrates two sensing structures, resistive and capacitive, to achieve simultaneous output of resistance and capacitance signals. The change in resistance signal is mainly affected by strain, while the change in capacitance signal is mainly affected by pressure, so as to meet the simultaneous detection of strain and pressure signals. Compared with the patent with the published application number 202110812835.7, the former is mainly based on the improvement of the Young's modulus of the sensor pressure direction by the porous structure, and the present invention can not only improve the sensitivity of the capacitive sensor to pressure through the pores introduced by the presence of the pipe-shaped hollow structure, but also suppress the deformation of the sensor as a whole in the z-axis direction, promote the contraction of the sensor in the y direction, and then reduce the change of the electrode distance and the electrode plate area of the capacitive sensor under the action of strain, reduce the sensitivity of its capacitance value to strain, and promote the offset of the capacitance value inflection point of the capacitive sensor, so the decoupling ability is greatly improved, which can meet various application scenarios with higher requirements for decoupling accuracy.
[0069] For example, when the sensor is worn on a finger joint, the pressure of the joint protrusion on the sensor and the tensile strain of the skin on the joint surface can be decoupled when the finger joint is bent. The protrusions of different fingers are different, so when bent at the same angle, the decoupled resistance and capacitance signals are different, which can be used to identify different finger joints.
[0070] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.
Claims
1. A pressure-strain decoupling sensor based on a pipeline structure, characterized in that: It comprises a flexible substrate (1), a lower flexible electrode plate (2), a dielectric layer (3), an upper flexible electrode plate (4), and a packaging protection layer (5) arranged from bottom to top; A pipe-shaped hollow structure is provided inside the dielectric layer (3); the pipe-shaped hollow structure is an axisymmetric structure; it comprises a central pipe (32), and other pipes distributed on both sides of the central pipe (32) and symmetrically distributed with respect to the central pipe (32); The other pipes and the central pipe (32) are all located on the same plane, and the plane is parallel to the surface of the dielectric layer (3); The other pipes are arcuate, and the opening direction of the arcuate faces away from the central pipe (32); The radius of curvature of the other pipes is: ; The curvature of the other pipes is: ; Wherein, L represents the length of the dielectric layer (3), d represents the width of the dielectric layer (3), and the central pipe (32) is arranged along the length direction of the dielectric layer (3); The central pipe (32) has the same diameter as the other pipes; The lower flexible electrode plate (2) and the upper flexible electrode plate (4) are made of the same material, both of which are mixed conductors of low-dimensional nanomaterials and elastic polymers; The dielectric layer (3) and the flexible substrate (1) are made of the same material, which is an elastic polymer, and are the same as the elastic polymer used in the flexible electrode plate; When tensile strain is applied to the sensor along the x-axis, the presence of the arched tube can suppress the overall deformation of the sensor in the z-axis direction and promote the contraction of the sensor in the y-direction; When pressure is applied to the sensor along the z-axis direction, the pores introduced by the pipe-shaped hollow structure reduce the Young's modulus of the sensor to a certain extent, and a larger z-axis deformation can be generated under the same pressure.
2. According to claim 1, a pressure-strain decoupling sensor based on a pipeline structure is characterized in that: It comprises a central pipeline (32), a first pipeline (31), and a second pipeline (33); The central pipe (32) is arranged along the length direction of the dielectric layer (3); The first pipe (31) and the second pipe (33) have the same structure and are symmetrically arranged on both sides of the central pipe (32); The first pipe (31) and the second pipe (33) are both arcuate, and the opening direction of the arcuate faces away from the central pipe (32).
3. A method for preparing the sensor as claimed in claim 1, characterized in that: The steps include: Step 1, the elastic polymer is stirred and evenly poured into a template, and after vacuuming and removing bubbles, a flexible substrate (1) is obtained; Step 2, mixing the low-dimensional nanomaterial into the precursor of the elastic polymer, stirring thoroughly, adding a curing agent after the mixture is evenly dispersed, to obtain a mixture of the low-dimensional nanomaterial and the elastic polymer, and pouring the mixture on the flexible substrate (1) to obtain a lower flexible electrode plate (2) located on the upper surface of the flexible substrate (1); Step 3, placing a sacrificial ice mold on the surface of the lower flexible electrode plate (2) in the template, pouring an elastic polymer to cover the sacrificial ice mold to form a dielectric layer (3); Step 4, pouring a mixture of low-dimensional nanomaterials and elastic polymers on the dielectric layer (3) to obtain an upper flexible electrode plate (4); The mixture of the low-dimensional nanomaterial and the elastic polymer is the same as the mixture of the low-dimensional nanomaterial and the elastic polymer in step 2; Step 5, pouring an elastic polymer on the surface of the upper flexible electrode plate (4) as a packaging protective layer (5); Step 6, placing the template and the material poured in the template into an oven for curing. During the curing process, the sacrificial ice mold melts and evaporates, forming a pipeline structure in the dielectric layer (3).
4. The preparation method according to claim 3, characterized in that: The sacrificial ice mold in step 3 comprises a left vertical plate, a right vertical plate, a central column connected between the left vertical plate and the right vertical plate, and other columns, the other columns are symmetrical about the central column, and the other columns are arched structures; in step 6, after the sacrificial ice mold is melted and evaporated, a central pipe (32) in the dielectric layer is formed at the position of the central column, and other pipes in the dielectric layer are formed at the positions of the other columns.
