Key Insights
The North America Beta-Glucan Market is poised for substantial growth, projected to reach an estimated $488.3 million in 2025. Driven by increasing consumer awareness regarding the health benefits of beta-glucans, particularly their role in immune support and cholesterol management, the market is expected to expand at a robust Compound Annual Growth Rate (CAGR) of 8.8% between 2025 and 2033. Key growth catalysts include the rising demand for functional foods and beverages, where beta-glucans are incorporated to enhance nutritional profiles and appeal to health-conscious consumers. The healthcare and dietary supplements sector also presents a significant avenue for growth, as individuals actively seek out natural ingredients for preventative health and wellness. Emerging applications in other industries, such as cosmetics and animal feed, are also contributing to the market's expansion, reflecting the versatile nature of beta-glucans.
North America Beta-Glucan Market Market Size (In Million)
The market is segmented by type, with both soluble and insoluble beta-glucans finding diverse applications. Soluble beta-glucans are particularly popular in food and beverages for their viscosity-modifying and texturizing properties, alongside their well-documented health advantages. Insoluble beta-glucans, while also offering health benefits, are explored for different functional roles. The primary sources of beta-glucans in North America are cereal grains like oats and barley, followed by yeast and mushrooms, each offering distinct chemical structures and functionalities. The United States currently dominates the market due to its large consumer base and high adoption rate of health-centric products, followed by Canada and Mexico. Despite the positive outlook, potential restraints such as the high cost of extraction and purification processes for certain beta-glucan types and the need for greater consumer education on the specific benefits of different beta-glucan sources could pose challenges.
North America Beta-Glucan Market Company Market Share
This in-depth report delivers a meticulously researched analysis of the North America Beta-Glucan Market. Covering the historical period from 2019 to 2024, a base year of 2025, and a comprehensive forecast period extending to 2033, this study provides critical insights into market dynamics, key trends, and future growth trajectories. With an estimated market size of $XXX million in 2025, the North America Beta-Glucan market is poised for significant expansion.
North America Beta-Glucan Market Market Concentration & Dynamics
The North America Beta-Glucan Market exhibits a moderate to high level of concentration, with key players like Garuda International Inc, ABF Ingredients, Tate & Lyle PLC, Koninklijke DSM N.V., and Cargill Inc. holding substantial market shares. The innovation ecosystem is vibrant, driven by ongoing research into novel applications and enhanced extraction technologies for soluble beta-glucan and insoluble beta-glucan. Regulatory frameworks, particularly concerning health claims in the Food and Beverages and Healthcare and Dietary Supplements sectors, significantly influence market access and product development. Substitute products, such as other soluble fibers and functional ingredients, present a competitive challenge. End-user trends are increasingly favoring naturally derived, health-promoting ingredients, particularly those sourced from Cereal and Yeast. Mergers and acquisitions (M&A) activities are on the rise, with an estimated XX M&A deals in the historical period, signaling consolidation and strategic expansion among leading companies. For instance, recent M&A activities have focused on acquiring specialized beta-glucan producers to enhance product portfolios and expand geographical reach.
North America Beta-Glucan Market Industry Insights & Trends
The North America Beta-Glucan Market is experiencing robust growth, projected to reach $XXX million by 2033, with a Compound Annual Growth Rate (CAGR) of approximately XX% during the forecast period (2025-2033). Several factors are propelling this expansion. Increasing consumer awareness regarding the health benefits of beta-glucans, including their role in cholesterol management, immune system support, and gut health, is a primary driver. This heightened consumer demand translates directly into increased adoption across various applications, most notably within the Food and Beverages industry, where beta-glucans are being incorporated into cereals, baked goods, dairy products, and beverages to enhance nutritional profiles and functional claims. The Healthcare and Dietary Supplements segment also represents a significant market share, with a growing demand for scientifically validated beta-glucan supplements. Technological advancements in extraction and purification processes are enabling higher yields and purer forms of beta-glucans, making them more cost-effective and versatile for manufacturers. For example, enzymatic hydrolysis and advanced filtration techniques are optimizing the production of specific beta-glucan structures with tailored functionalities. Evolving consumer behaviors, such as a preference for clean-label products and natural ingredients, further fuel the demand for beta-glucans derived from Cereal and Yeast. The rising incidence of lifestyle-related diseases and a growing emphasis on preventative healthcare are also contributing to the market's upward trajectory. The market is characterized by continuous product innovation, with companies investing in research and development to explore new sources, such as Mushroom beta-glucans, and to develop novel delivery systems that enhance bioavailability and efficacy. The United States is expected to remain the largest market, driven by its large consumer base and high disposable income, coupled with a proactive approach to health and wellness. Canada and Mexico, along with the rest of North America, are also showing promising growth potential due to increasing health consciousness and a burgeoning food and beverage sector.
Key Markets & Segments Leading North America Beta-Glucan Market
The United States stands out as the dominant market within North America, accounting for an estimated XX% of the total market revenue in 2025. This leadership is attributed to several key drivers, including a large and health-conscious consumer base, a well-established food and beverage industry with a strong focus on functional ingredients, and a robust healthcare and dietary supplement sector. The high disposable income and consumer willingness to invest in health-promoting products further bolster demand.
- Category Dominance: Within the beta-glucan categories, Soluble Beta-Glucan is leading the market, driven by its widespread use in functional foods and beverages for its recognized health benefits, particularly cholesterol-lowering properties. Its ability to improve texture and mouthfeel in food products also contributes to its popularity.
- Application Leadership: The Food and Beverages segment is the largest application area, representing approximately XX% of the market. This is due to the increasing trend of fortifying staple foods, dairy products, and beverages with beta-glucans to cater to consumer demand for healthier options.
- Source Preference: Cereal remains the predominant source of beta-glucans, with oats and barley being primary contributors, owing to their established nutritional profiles and widespread availability. However, Yeast-derived beta-glucans are gaining traction due to their potent immune-modulating properties.
- Geographical Breakdown: The United States is the largest market by a significant margin. Canada follows as the second-largest market, with a growing demand for health-focused food products and supplements. Mexico presents a rapidly expanding market, driven by increasing awareness of health and wellness trends and a growing middle class. The Rest of North America also contributes to the overall market size, albeit with smaller individual contributions from various smaller economies.
North America Beta-Glucan Market Product Developments
Product development in the North America Beta-Glucan Market is characterized by innovation focused on enhancing efficacy, expanding applications, and improving consumer appeal. Companies are actively developing beta-glucan ingredients with specific molecular weights and structures to optimize their functional properties for targeted health benefits, such as improved immune support and gut health. Advancements in extraction techniques are yielding higher purity beta-glucans, making them more suitable for sensitive applications like infant nutrition and medical foods. Novel delivery systems are also being explored to improve bioavailability and consumer experience.
Challenges in the North America Beta-Glucan Market Market
Despite its growth, the North America Beta-Glucan Market faces several challenges. Regulatory hurdles related to health claims and labeling requirements can impede market penetration and product innovation. Supply chain complexities, particularly for niche sources like mushroom beta-glucans, can lead to price volatility and availability issues. Intense competition from established players and emerging novel ingredients also exerts pressure on market share and pricing strategies. Furthermore, consumer education regarding the specific benefits and optimal dosage of different beta-glucan types remains a continuous effort to drive widespread adoption.
Forces Driving North America Beta-Glucan Market Growth
The growth of the North America Beta-Glucan Market is propelled by several powerful forces. A significant driver is the growing consumer demand for natural and functional ingredients that offer tangible health benefits, such as improved cardiovascular health and immune function. This trend is amplified by increasing consumer awareness and scientific research validating these benefits. Furthermore, the expanding food and beverage industry's focus on product fortification and the development of healthier alternatives presents a substantial opportunity. Technological advancements in extraction and processing are making beta-glucans more accessible and cost-effective.
Challenges in the North America Beta-Glucan Market Market
Long-term growth catalysts for the North America Beta-Glucan Market lie in sustained innovation and market expansion. Continued investment in research and development to uncover new health benefits and applications of beta-glucans, especially from less conventional sources like Mushroom, will be crucial. Strategic partnerships between ingredient manufacturers and food/supplement companies will foster product development and market penetration. Expansion into emerging applications, such as animal nutrition and cosmetic products, presents further avenues for sustained growth.
Emerging Opportunities in North America Beta-Glucan Market
Emerging opportunities within the North America Beta-Glucan Market are diverse. The increasing focus on personalized nutrition presents an opportunity for developing beta-glucan formulations tailored to specific individual health needs. The growing demand for plant-based and sustainable ingredients aligns perfectly with the origins of many beta-glucans. Furthermore, the exploration of novel extraction technologies, such as supercritical fluid extraction, could lead to more efficient and environmentally friendly production methods. The untapped potential in markets like pet food and animal feed, where beta-glucans can support immune health, also represents a significant growth avenue.
Leading Players in the North America Beta-Glucan Market Sector
- Garuda International Inc
- ABF Ingredients
- Super Beta-Glucan
- Tate & Lyle PLC
- Koninklijke DSM N V
- Associated British Foods Plc
- Ceapro Inc
- Kerry Group PLC
- Cargill Inc
- Kemin Industries Inc
Key Milestones in North America Beta-Glucan Market Industry
- 2019: Increased research publications highlighting the immune-modulating effects of yeast beta-glucans.
- 2020: Launch of new oat-based beta-glucan fortified breakfast cereals in the US market.
- 2021: FDA approval of new health claims for soluble fiber from oats and barley, boosting demand for beta-glucans.
- 2022: Acquisition of a specialized beta-glucan producer by a major ingredient supplier to expand its portfolio.
- 2023: Growing interest and initial product launches of mushroom-derived beta-glucans in the supplement market.
- 2024: Advancements in enzymatic extraction techniques leading to higher yields of specific beta-glucan fractions.
Strategic Outlook for North America Beta-Glucan Market Market
The strategic outlook for the North America Beta-Glucan Market is exceptionally positive, driven by the continuous convergence of consumer health consciousness, scientific validation, and technological innovation. Key growth accelerators will include further diversification of beta-glucan sources to cater to specific functional benefits and dietary preferences. Strategic focus on product innovation, particularly in the Healthcare and Dietary Supplements and Food and Beverages sectors, will be paramount. Collaboration across the value chain, from raw material suppliers to end-product manufacturers, will be essential for optimizing market penetration and addressing evolving consumer demands for health, wellness, and natural ingredients.
North America Beta-Glucan Market Segmentation
-
1. Category
- 1.1. Soluble
- 1.2. Insoluble
-
2. Application
- 2.1. Food and Beverages
- 2.2. Healthcare and Dietary Supplements
- 2.3. Other Applications
-
3. Source
- 3.1. Cereal
- 3.2. Yeast
- 3.3. Mushroom
- 3.4. Other Sources
-
4. Geography
- 4.1. United States
- 4.2. Canada
- 4.3. Mexixo
- 4.4. Rest of North America
North America Beta-Glucan Market Segmentation By Geography
- 1. United States
- 2. Canada
- 3. Mexixo
- 4. Rest of North America
North America Beta-Glucan Market Regional Market Share
Geographic Coverage of North America Beta-Glucan Market
North America Beta-Glucan Market REPORT HIGHLIGHTS
| Aspects | Details |
|---|---|
| Study Period | 2020-2034 |
| Base Year | 2025 |
| Estimated Year | 2026 |
| Forecast Period | 2026-2034 |
| Historical Period | 2020-2025 |
| Growth Rate | CAGR of 8.8% from 2020-2034 |
| Segmentation |
|
Table of Contents
- 1. Introduction
- 1.1. Research Scope
- 1.2. Market Segmentation
- 1.3. Research Methodology
- 1.4. Definitions and Assumptions
- 2. Executive Summary
- 2.1. Introduction
- 3. Market Dynamics
- 3.1. Introduction
- 3.2. Market Drivers
- 3.2.1. Rising Application in Dietary Supplements
- 3.3. Market Restrains
- 3.3.1. Availability of Low-cost Sweeteners
- 3.4. Market Trends
- 3.4.1. Increased Demand for Soluble Beta-Glucan
- 4. Market Factor Analysis
- 4.1. Porters Five Forces
- 4.2. Supply/Value Chain
- 4.3. PESTEL analysis
- 4.4. Market Entropy
- 4.5. Patent/Trademark Analysis
- 5. North America Beta-Glucan Market Analysis, Insights and Forecast, 2020-2032
- 5.1. Market Analysis, Insights and Forecast - by Category
- 5.1.1. Soluble
- 5.1.2. Insoluble
- 5.2. Market Analysis, Insights and Forecast - by Application
- 5.2.1. Food and Beverages
- 5.2.2. Healthcare and Dietary Supplements
- 5.2.3. Other Applications
- 5.3. Market Analysis, Insights and Forecast - by Source
- 5.3.1. Cereal
- 5.3.2. Yeast
- 5.3.3. Mushroom
- 5.3.4. Other Sources
- 5.4. Market Analysis, Insights and Forecast - by Geography
- 5.4.1. United States
- 5.4.2. Canada
- 5.4.3. Mexixo
- 5.4.4. Rest of North America
- 5.5. Market Analysis, Insights and Forecast - by Region
- 5.5.1. United States
- 5.5.2. Canada
- 5.5.3. Mexixo
- 5.5.4. Rest of North America
- 5.1. Market Analysis, Insights and Forecast - by Category
- 6. United States North America Beta-Glucan Market Analysis, Insights and Forecast, 2020-2032
- 6.1. Market Analysis, Insights and Forecast - by Category
- 6.1.1. Soluble
- 6.1.2. Insoluble
- 6.2. Market Analysis, Insights and Forecast - by Application
- 6.2.1. Food and Beverages
- 6.2.2. Healthcare and Dietary Supplements
- 6.2.3. Other Applications
- 6.3. Market Analysis, Insights and Forecast - by Source
- 6.3.1. Cereal
- 6.3.2. Yeast
- 6.3.3. Mushroom
- 6.3.4. Other Sources
- 6.4. Market Analysis, Insights and Forecast - by Geography
- 6.4.1. United States
- 6.4.2. Canada
- 6.4.3. Mexixo
- 6.4.4. Rest of North America
- 6.1. Market Analysis, Insights and Forecast - by Category
- 7. Canada North America Beta-Glucan Market Analysis, Insights and Forecast, 2020-2032
- 7.1. Market Analysis, Insights and Forecast - by Category
- 7.1.1. Soluble
- 7.1.2. Insoluble
- 7.2. Market Analysis, Insights and Forecast - by Application
- 7.2.1. Food and Beverages
- 7.2.2. Healthcare and Dietary Supplements
- 7.2.3. Other Applications
- 7.3. Market Analysis, Insights and Forecast - by Source
- 7.3.1. Cereal
- 7.3.2. Yeast
- 7.3.3. Mushroom
- 7.3.4. Other Sources
- 7.4. Market Analysis, Insights and Forecast - by Geography
- 7.4.1. United States
- 7.4.2. Canada
- 7.4.3. Mexixo
- 7.4.4. Rest of North America
- 7.1. Market Analysis, Insights and Forecast - by Category
- 8. Mexixo North America Beta-Glucan Market Analysis, Insights and Forecast, 2020-2032
- 8.1. Market Analysis, Insights and Forecast - by Category
- 8.1.1. Soluble
- 8.1.2. Insoluble
- 8.2. Market Analysis, Insights and Forecast - by Application
- 8.2.1. Food and Beverages
- 8.2.2. Healthcare and Dietary Supplements
- 8.2.3. Other Applications
- 8.3. Market Analysis, Insights and Forecast - by Source
- 8.3.1. Cereal
- 8.3.2. Yeast
- 8.3.3. Mushroom
- 8.3.4. Other Sources
- 8.4. Market Analysis, Insights and Forecast - by Geography
- 8.4.1. United States
- 8.4.2. Canada
- 8.4.3. Mexixo
- 8.4.4. Rest of North America
- 8.1. Market Analysis, Insights and Forecast - by Category
- 9. Rest of North America North America Beta-Glucan Market Analysis, Insights and Forecast, 2020-2032
- 9.1. Market Analysis, Insights and Forecast - by Category
- 9.1.1. Soluble
- 9.1.2. Insoluble
- 9.2. Market Analysis, Insights and Forecast - by Application
- 9.2.1. Food and Beverages
- 9.2.2. Healthcare and Dietary Supplements
- 9.2.3. Other Applications
- 9.3. Market Analysis, Insights and Forecast - by Source
- 9.3.1. Cereal
- 9.3.2. Yeast
- 9.3.3. Mushroom
- 9.3.4. Other Sources
- 9.4. Market Analysis, Insights and Forecast - by Geography
- 9.4.1. United States
- 9.4.2. Canada
- 9.4.3. Mexixo
- 9.4.4. Rest of North America
- 9.1. Market Analysis, Insights and Forecast - by Category
- 10. Competitive Analysis
- 10.1. Market Share Analysis 2025
- 10.2. Company Profiles
- 10.2.1 Garuda International Inc
- 10.2.1.1. Overview
- 10.2.1.2. Products
- 10.2.1.3. SWOT Analysis
- 10.2.1.4. Recent Developments
- 10.2.1.5. Financials (Based on Availability)
- 10.2.2 ABF ingredients
- 10.2.2.1. Overview
- 10.2.2.2. Products
- 10.2.2.3. SWOT Analysis
- 10.2.2.4. Recent Developments
- 10.2.2.5. Financials (Based on Availability)
- 10.2.3 Super Beta-Glucan
- 10.2.3.1. Overview
- 10.2.3.2. Products
- 10.2.3.3. SWOT Analysis
- 10.2.3.4. Recent Developments
- 10.2.3.5. Financials (Based on Availability)
- 10.2.4 Tate & Lyle PLC
- 10.2.4.1. Overview
- 10.2.4.2. Products
- 10.2.4.3. SWOT Analysis
- 10.2.4.4. Recent Developments
- 10.2.4.5. Financials (Based on Availability)
- 10.2.5 Koninklijke DSM N V
- 10.2.5.1. Overview
- 10.2.5.2. Products
- 10.2.5.3. SWOT Analysis
- 10.2.5.4. Recent Developments
- 10.2.5.5. Financials (Based on Availability)
- 10.2.6 Associated British Foods Plc
- 10.2.6.1. Overview
- 10.2.6.2. Products
- 10.2.6.3. SWOT Analysis
- 10.2.6.4. Recent Developments
- 10.2.6.5. Financials (Based on Availability)
- 10.2.7 Ceapro Inc
- 10.2.7.1. Overview
- 10.2.7.2. Products
- 10.2.7.3. SWOT Analysis
- 10.2.7.4. Recent Developments
- 10.2.7.5. Financials (Based on Availability)
- 10.2.8 Kerry Group PLC
- 10.2.8.1. Overview
- 10.2.8.2. Products
- 10.2.8.3. SWOT Analysis
- 10.2.8.4. Recent Developments
- 10.2.8.5. Financials (Based on Availability)
- 10.2.9 Cargill Inc
- 10.2.9.1. Overview
- 10.2.9.2. Products
- 10.2.9.3. SWOT Analysis
- 10.2.9.4. Recent Developments
- 10.2.9.5. Financials (Based on Availability)
- 10.2.10 Kemin Industries Inc
- 10.2.10.1. Overview
- 10.2.10.2. Products
- 10.2.10.3. SWOT Analysis
- 10.2.10.4. Recent Developments
- 10.2.10.5. Financials (Based on Availability)
- 10.2.1 Garuda International Inc
List of Figures
- Figure 1: North America Beta-Glucan Market Revenue Breakdown (million, %) by Product 2025 & 2033
- Figure 2: North America Beta-Glucan Market Share (%) by Company 2025
List of Tables
- Table 1: North America Beta-Glucan Market Revenue million Forecast, by Category 2020 & 2033
- Table 2: North America Beta-Glucan Market Revenue million Forecast, by Application 2020 & 2033
- Table 3: North America Beta-Glucan Market Revenue million Forecast, by Source 2020 & 2033
- Table 4: North America Beta-Glucan Market Revenue million Forecast, by Geography 2020 & 2033
- Table 5: North America Beta-Glucan Market Revenue million Forecast, by Region 2020 & 2033
- Table 6: North America Beta-Glucan Market Revenue million Forecast, by Category 2020 & 2033
- Table 7: North America Beta-Glucan Market Revenue million Forecast, by Application 2020 & 2033
- Table 8: North America Beta-Glucan Market Revenue million Forecast, by Source 2020 & 2033
- Table 9: North America Beta-Glucan Market Revenue million Forecast, by Geography 2020 & 2033
- Table 10: North America Beta-Glucan Market Revenue million Forecast, by Country 2020 & 2033
- Table 11: North America Beta-Glucan Market Revenue million Forecast, by Category 2020 & 2033
- Table 12: North America Beta-Glucan Market Revenue million Forecast, by Application 2020 & 2033
- Table 13: North America Beta-Glucan Market Revenue million Forecast, by Source 2020 & 2033
- Table 14: North America Beta-Glucan Market Revenue million Forecast, by Geography 2020 & 2033
- Table 15: North America Beta-Glucan Market Revenue million Forecast, by Country 2020 & 2033
- Table 16: North America Beta-Glucan Market Revenue million Forecast, by Category 2020 & 2033
- Table 17: North America Beta-Glucan Market Revenue million Forecast, by Application 2020 & 2033
- Table 18: North America Beta-Glucan Market Revenue million Forecast, by Source 2020 & 2033
- Table 19: North America Beta-Glucan Market Revenue million Forecast, by Geography 2020 & 2033
- Table 20: North America Beta-Glucan Market Revenue million Forecast, by Country 2020 & 2033
- Table 21: North America Beta-Glucan Market Revenue million Forecast, by Category 2020 & 2033
- Table 22: North America Beta-Glucan Market Revenue million Forecast, by Application 2020 & 2033
- Table 23: North America Beta-Glucan Market Revenue million Forecast, by Source 2020 & 2033
- Table 24: North America Beta-Glucan Market Revenue million Forecast, by Geography 2020 & 2033
- Table 25: North America Beta-Glucan Market Revenue million Forecast, by Country 2020 & 2033
Frequently Asked Questions
1. What is the projected Compound Annual Growth Rate (CAGR) of the North America Beta-Glucan Market?
The projected CAGR is approximately 8.8%.
2. Which companies are prominent players in the North America Beta-Glucan Market?
Key companies in the market include Garuda International Inc, ABF ingredients, Super Beta-Glucan, Tate & Lyle PLC, Koninklijke DSM N V, Associated British Foods Plc, Ceapro Inc, Kerry Group PLC, Cargill Inc, Kemin Industries Inc.
3. What are the main segments of the North America Beta-Glucan Market?
The market segments include Category, Application, Source, Geography.
4. Can you provide details about the market size?
The market size is estimated to be USD 488.3 million as of 2022.
5. What are some drivers contributing to market growth?
Rising Application in Dietary Supplements.
6. What are the notable trends driving market growth?
Increased Demand for Soluble Beta-Glucan.
7. Are there any restraints impacting market growth?
Availability of Low-cost Sweeteners.
8. Can you provide examples of recent developments in the market?
N/A
9. What pricing options are available for accessing the report?
Pricing options include single-user, multi-user, and enterprise licenses priced at USD 4750, USD 5250, and USD 8750 respectively.
10. Is the market size provided in terms of value or volume?
The market size is provided in terms of value, measured in million.
11. Are there any specific market keywords associated with the report?
Yes, the market keyword associated with the report is "North America Beta-Glucan Market," which aids in identifying and referencing the specific market segment covered.
12. How do I determine which pricing option suits my needs best?
The pricing options vary based on user requirements and access needs. Individual users may opt for single-user licenses, while businesses requiring broader access may choose multi-user or enterprise licenses for cost-effective access to the report.
13. Are there any additional resources or data provided in the North America Beta-Glucan Market report?
While the report offers comprehensive insights, it's advisable to review the specific contents or supplementary materials provided to ascertain if additional resources or data are available.
14. How can I stay updated on further developments or reports in the North America Beta-Glucan Market?
To stay informed about further developments, trends, and reports in the North America Beta-Glucan Market, consider subscribing to industry newsletters, following relevant companies and organizations, or regularly checking reputable industry news sources and publications.
Methodology
Step 1 - Identification of Relevant Samples Size from Population Database
Step 2 - Approaches for Defining Global Market Size (Value, Volume* & Price*)
Note*: In applicable scenarios
Step 3 - Data Sources
Primary Research
- Web Analytics
- Survey Reports
- Research Institute
- Latest Research Reports
- Opinion Leaders
Secondary Research
- Annual Reports
- White Paper
- Latest Press Release
- Industry Association
- Paid Database
- Investor Presentations
Step 4 - Data Triangulation
Involves using different sources of information in order to increase the validity of a study
These sources are likely to be stakeholders in a program - participants, other researchers, program staff, other community members, and so on.
Then we put all data in single framework & apply various statistical tools to find out the dynamic on the market.
During the analysis stage, feedback from the stakeholder groups would be compared to determine areas of agreement as well as areas of divergence